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Neural self-tuning adaptive control of non-minimum phase system developed for flexible robotic arm

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Title:
Neural self-tuning adaptive control of non-minimum phase system developed for flexible robotic arm
Creator:
Ho, Long T
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Denver, CO
Publisher:
University of Colorado Denver
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English
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56 leaves : illustrations ; 29 cm

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Robots -- Control systems ( lcsh )
Neural networks (Computer science) ( lcsh )
Neural networks (Computer science) ( fast )
Robots -- Control systems ( fast )
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bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

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Bibliography:
Includes bibliographical references (leaves 47-48).
Thesis:
Submitted in partial fulfillment of the requirements for the degree, Master of Science, Department of Electrical Engineering
Statement of Responsibility:
by Long T. Ho.

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|University of Colorado Denver
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Full Text
NEURAL SELF-TUNING ADAPTIVE CONTROL
OF NON-MINIMUM PHASE SYSTEM DEVELOPED
FOR FLEXIBLE ROBOTIC ARM
By
Long T. Ho
B.S.E.E., University of Colorado, 1992
A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in paritial fulfilment
of the requirements for the degree of
Master of Science
Department of Electrical Engineering
1994


This thesis for the Master of Science degree by
Long Thanh Ho
has been approved for the
Department of
Electrical Engineering
by
Miloje Radenkovic
Marvin Anderson


Acknowledgments
The author wishes to express sincere appreciation to Dr. Jan T. Bialasiewicz for
his guidance and joyful support during the course of this thesis. Also, enthusiasms of
Dr. Miloje Radenkovic and Professor Marvin Anderson as the author's thesis
committee are gratefully ackowledged.
Special thanks to the author's family for their never ending support.


4
Ho, Long Thanh (M.S., Electrical Engineering)
Neural Self-Tuning Adaptive Control Strategies of Non-Minimum Phase System
Developed for Flexible Robotic Arm
Thesis directed by Professor Jan T. Bialasiewicz
The motivation of this research came about when a neural network direct adaptive
control schemes were applied to control the tip position of a flexible robotic arm.
Satisfactory control performance was not attainable due to the inherent non-minimum
phase characteristics of the flexible robotic arm. Most of the existing neural network
control algorithms are based on the direct method and exhibit very high sensitivity if
not unstable closed-loop behavior. Therefore a neural self-tuning control (NSTC)
algorithm has been developed and applied to this problem and showed promising
results. Simulation results of the NSTC scheme and the conventional self-tuning
(STR) control scheme are used to examine performance factors such as control
tracking mean square error, estimation mean square error, transient response, and
steady state response.
The form and content of this abstract are approved.
Signed

Jan T. Bialasiewicz


5
NEURAL SELF-TUNING ADAPTIVE CONTROL
OF NON-MINIMUM PHASE SYSTEM DEVELOPED FOR FLEXIBLE
ROBOTIC ARM
Table of Content
1. Introduction.................................................. 9
1.2. Neural Control Survey................................. 12
1.2.1. Stochastic Neural Direct Semi-Adaptive Control. 13
1.2.2. Stochastic Neural Direct Adaptive Control...... 15
1.2.3. Inverse Neural Adaptive Control................ 15
1.2.4. Feedback Error Learning and Control............ 16
1.2.5. Inverse Dynamic Model Reference Control of a
Class of Nonlinear Plants................ 17
1.2.6. Neural Linear State Space Control............ 18
1.2.7. Neural Self-Tuning Control................... 19
2. Stochastic Neural Self-Tuning Adaptive Control Scheme... 21
2.1. Generalized Minimum Variance Control.................. 22
2.2. Neural System Identification....................... 23
3. Flexible Arm Tip Position Dynamics...................... 28
4. Empirical Studies........................................... 35
4.1. Neural Direct Adaptive Control of Arm Hub and Tip. 35
4.2. Neural Self-Tuning Adaptive Control of Arm Tip.... 39
5. Conclusions................................................. 45
5.1 Future Research..................................... 46


6
Bibliography............................................ 47
APPENDIX A Simulation Program........................... 49


7
Figures and Tables
Figures:
1.1 Flexible Arm System........................................ 10
1.2 Specialized Learning Control of Hup Velocity............... 11
1.3 Indirect Neural Adaptive Control Scheme.................... 11
1.4 Adaptive control general block diagram..................... 13
1.5 Direct semi-adaptive control scheme....................... 12
1.6 Neural Network structure................................ 14
1.7 Direct neural adaptive controller........................ 14
1.8 Inverse neural control.................................. 15
1.9 Feedback error learning and control..................... 16
1.10 Inverse dynamic model reference neural control............ 18
1.11 Neural linear state space control........................ 19
1.12 Stochastic neural linear ARM A control.................. 20
2.1 Neural Network Structure................................... 25
3.1 Pole-Zero Diagram of Flexible Arm Tip.......................29
3.2 Servo Motor System Components.............................. 30
3.3 Frequency Magnitude Response of Arm Tip with Five Resonant Modes
........................................................... 32
3.4 Frequency Response of Open-Loop Components................. 33


8
3.5 Frequency Response of the Aggregate filtered Open-Loop..... 33
4.1 Neural Direct Control Scheme of Hub Velocity............... 36
4.2 Hub Velocity Response...................................... 37
4.3 Control Tracking MSE Response.............................. 37
4.4 Unstable Response of Tip Control........................... 38
4.5 Diverging Tracking MSE Of Tip Velocity..................... 38
4.6 NSTC Scheme Block Diagram.................................. 40
4.7 Tip Position Response..................................... 42
4.8 Control Performance Index J(K) of the Adaptive STR andthe NSTC
.......................................................... 43
4.9 Control Signal u(k) of the adapteve STR and the NSTC....... 43
4.10 Identification cost Index V(k) of the Adaptive STR and the NSTC.. 44
4.11 True and Neural Network Estimated Tip Position............ 44
Tables:
3.1 Physical Properties of Arm and Motor...................... 31
3.2 Poles and Zeros
32


9
CHAPTER 1
INTRODUCTION
Most existing neuro control schemes are in the form of the direct method, where
the neural network is trained to approximate the inverse of the plant. In the case where
the plant is non-minimum phase, the inverse approximation introduces instability in the
closed-loop system. Therefore, an indirect neuro control scheme is proposed to deal
with non-minimum phase systems. Specifically, we propose to use a neural network to
identify the plant parameters, then combine this with a minimum variance control law.
The plant in this study is a single degree of freedom flexible robotic arm.
Self-tuning adaptive control used for controlling unknown ARMA plants has
traditionally been based on the minimum variance control law and a recursive
identification algorithm (Astrom and Wittenmark, 1973; Clark and Gawthrop, 1979).
Although the advancement in VLSI has made it more possible to implement real-time
recursive algorithms but it is still computationally intensive and expensive due to the
recursive nature of the algorithm. On the other hand, neural networks VLSI has been
made available commercially with extreme processing capability due to its parallel
architecture. With this in mind the possibility of formulating neural networks to
perform functions of conventional recursive algorithms becomes important. Hence, in
this thesis, the neural self tuning control (NSTC) scheme is used where the implicit
identification is performed by a multilayer neural network (MNN) and the control is
based on the generalized minimum variance (GMV) control law.
Neural networks have undoubtedly demonstrated its effectiveness in controlling
nonlinear systems with knoWn/unknown dynamics and uncertainties (Narendra and
Parthasrathy, 1990; Levin and Narendra, 1993; Werbos et al. 1990; Hunt et al., 1992).
In addition, neural network adaptive control algorithms have also been developed for
specific linear system model such as the state space model (Ho et al., 91a) and the
ARMA model (Ho et al., 1991b). It was shown in the simulation results that neural
network controllers produced comparable results to conventional adaptive controllers.


10
In this thesis, the performance of the NSTC is compared to the conventional adaptive
STR.
The flexible arm to be controlled is shown in Figure 1.1. There are two system
outputs that are of interest, one is the hub angle 0h(t) and the other is the tip angle 0t(t)
of the arm. The goal is to apply a neural network control scheme to control these
outputs to track the command signals. The neural controller will generate a control
voltage signal u(t) that will feed the power amplifier in which will force current through
the motor and cause the arm position to react. The dynamical transfer function of the
hub angle is a linear minimum phase system in which will be shown readily
controllable by a neural network. In fact, the direct adaptive neural control scheme in
Figure 1.2. can be used to control the hub. This control scheme belongs to the type
called specialized learning control (Psaltis et al., 1988; Ho et al., 1991c). However,
the tip of the arm, being at a different location than the actuator point, therefore making
the system to be of the type non-collocated system. The effect of this dynamically is
that there is a zero in the right half of the s-plane. In other words, the transfer function
of the tip angle is of the non-minimum phase type which presents itself to be very
difficult to control when direct adaptive control methodology is applied. This difficulty
may be due to the controller trying to emulate the inverse dynamics of the non-minimun
phase plant and results in an unstable behavior. According to simulation studies, the
specialized learning control algorithm diverges when applied to control the tip angle.
Most other neural control schemes are also based on the inverse dynamics including the
indirect learning method by (Psaltis et al., 1988), the feedback error learning by
(Kawato et al., 1988), and the methods presented by (Narendra and Parthasarathy,
1990).
Figure 1.1. Flexible arm system


11
Figure 1.2. specialized learning control of hub velocity
Figure 1.3 Indirect neural adaptive control scheme
In this thesis, the neural self tuning control scheme which is based on an
indirect control method (Ho et al., 1991c) to control the tip angle. This scheme is


12
shown in Figure 1.3 where the identification is performed by the MNN and the control
is performed by the generalized minimum variance (GMV) controller. The GMV
control algorithm has a dynamic weighting function Q(q_1) applied to the plant control
signal u(k) in the cost function to limit and condition the control energy. Thus, upon
selecting the proper weighting function the controller can be input/output stable and
effective in controlling the non-minimum phase plant. In section 2, the neural self-
tuning control (NSTC) which consists of the minimum variance control algorithm and
the neural identification is presented. Section 3 covers the basic dynamics of the
flexible arm tip position. Section 4 presents a comparative simulation study of the
adaptive STR scheme and the NSTC scheme. Finally, section 5 gives the conclusion
of the results found in this study and addresses the advantages and disadvantages of the
neural control scheme used for treating linear system.
1.2 Neural Control Survey
In the past five years neural network based adaptive control has been a
proliferated and challenging field for researchers in the area of adaptive and nonlinear
control. As technology advances and more and more dynamical systems emerge with
high degree of complexity in coupled and nonlinear characteristics, conventional
modern and adaptive control techniques are showing to be less and less effective in
achieving demanding control performance. This is partly due to the fact that many of
these systems are linearized and decoupled beforehand in order to apply conventional
control techniques, which consequently causes inaccuracies in representing system
dynamics and therefore looses effectiveness in controlling the system. Neural network
based adaptive control (NNBAC) has shown to have some unique and superior
capabilities in controlling stochastic nonlinear time varying systems mainly because
neural networks can model nonlinear complex processes more accurately.
Furthermore, due to the inherent parallel structure of neural networks NNBAC offers
the major computational load advantage because of parallel computations. Hence,
implementation is more possible in cases dealing with large scales and/or high
bandwidth systems where sufficiently fast sampling rate is required. In this section,
we briefly present a survey of existing neural control schemes.
Consider the general block diagram of an adaptive control scheme shown in
Figure 1.4. Now, an adaptive control scheme may assume no a priori knowledge of
the plant, but an effective and prudent adaptive control scheme should utilize and
exercise all the a priori knowledge that is available. Some of the early neural control


13
schemes such as the inverse indirect learning and the specialized learning were
impressive because these schemes required very little a priori information about the
nonlinear plant and treated it like a "black box".
Disturbance
Output
y(k)
Figure 1.4. Adaptive control general block diagram
1.2.1 STOCHASTIC NEURAL DIRECT SEMI-ADAPTIVE CONTROL
Consider the first scheme called the stochastic neural direct semi-adaptive
control shown in Figure 1.5 (Ho et al., 1991c). This scheme is the stochastic weighted
version of the specialized learning (Psaltis et al., 1988) and is formulated with the well
known weighted optimal control cost function
J(k) = \ E{[y(k)-y*(k)]'Q[y(k)-y*(k)] + u(k)'Ru(k)}
(1.1)


14
This control scheme is based on the nonlinear stochastic state space model
x(k+l) = f[x(k)] + B(k)u(k) + w(k)
y(k) = C(k)x(k) + v(k) (1.2)
The a priori information required for this scheme is the input/output dynamic matrices B
and C. This is so that the plant jacobian 9y(k)3u(k) can be computed and used in the
back propagation algorithm. Figure 1.6 shows the typical structure of a multilayered
neural network.
OUTPUT
DELAY
NETWORK
Figure 1.6. Neural Network structure


15
1.2.2. STOCHASTIC NEURAL DIRECT ADAPTIVE CONTROL
This scheme is almost identical to the previous scheme (Ho et al., 199Id)
except that it has no a priori information about the plant input/output dynamics.
Therefore it incorporates an additional neural network so that the plant jacobian can be
estimated. The block diagram of this scheme is shown in Figure 1.7 where the plant
was basically treated to be a "black box" nonlinear system with the general state space
form
x(k+l) = f[x(k),u(k)] + w(k)
y(k) = g[x(k)] + v(k) (1.3)
Figure 1.7. Direct neural adaptive controller
1.2.3. INVERSE NEURAL ADAPTIVE CONTROL
This is one of the first neural adaptive control schemes known as the indirect
learning proposed by (Psaltis et al., 1988) and is shown is Figure 1.8. The plant is
assumed to be a "black box" nonlinear system
y(k) = f[y(k-l), ..., y(k-n); u(k),..., u(k-m)]
(1.4)


16
Figure 1.8. Inverse neural control
Here, the two neural networks at the input and ouput of the plant are identical. The
network is to emulate the inverse of the plant based on optimization of the cost function
J(k)=l[u(k)-u(k)]'[u(k)-u(k)]
which indirectly minimizes the output tracking error [y(k)-y*(k)].
(1.5)
1.2.4. FEEDBACK ERROR LEARNING AND CONTROL (FELC)
This direct adaptive control method proposed by (Kawato et al. 1988) may be
one of the most efficient "black box" neural control scheme as shown in Figure 1.9, the
scheme utilizes a single neural network as an adaptive direct controller performing both
learning and control simultaneously.
Figure 1.9. Feedback error learning and control


17
The neural network directly minimizes the output tracking error cost function
J(k) = i[y*(k)-y(k)],Q[y*(k)-y(k)]
and does not require any a priori infomation such as the Jacobian.
(1.6)
1.2.5. INVERSE DYNAMIC MODEL REFERENCE CONTROL OF A
CLASS OF NONLINEAR PLANTS
This approach was presented by (Narendra and Parathasarathy, 1990)
addressing the issues of identification utilizing neural networks and control of nonlinear
plant using inverse dynamic model reference techniques. The general diagram of this
scheme is shown in Figure 1.10. The four input/output plant models addressed are
Model I:
n-l
y(k+l) = X otiy(k-i) + g[u(k), u(k-l), ..., u(k-m+l)]
i=o (1.7a)
Model II:
m-l
y(k+l) = f[y(k), y(k-l),..., y(k-n+l)] + £ pju(k-i)
i=o (1.7b)
Model HI:
y(k+l) = f[y(k), y(k-l),..., y(k-n+l)] + g[u(k), u(k-l),..., u(k-n+l)]
(1.7c)
Model IV:


y(k+l) = f[y(k), y(k-l), y(k-n+l); u(k), u(k-l), u(k-n+l)]
(1.7d)
Figure 1.10 Inverse dynaiiiics model reference neural control
The a priori information required is which of these specific model fits the plant so that
identification can be performed. However, the inverse dynamic control can be
accomplished provided the representation of the inverse dynamics exists. In other
words, it is recognize that u(k) can be expressed in terms of f(.), g(.), f_1(.) and g_1(.).
1.2.6. NEURAL LINEAR STATE SPACE CONTROL
This scheme shown in Figure 1.11. (Ho and Ho, 1991a) is used for controlling
time varying linear stochastic state space plant
x(k+l) = A(0,k)x(k) + B(0,k)u(k) + w(k)
y(k) = C(0,k) + v(k) (1.8)
where 0 is the parameter vector. The identification is performed by the neural network
and the control can be selected by any modern state space control techniques, in


particular, the tracking per-interval control law. This neural parameter adaptive control
approach is different from the conventional adaptive control approach by the
identification process.
19
Figure 1.11. Neural linear state space control
1.2.7. NEURAL SELF-TUNING CONTROL
This scheme, shown in Figure 1.12. (Ho et al, 1991b) is similar to the state
space control scheme only it is based on the ARMAR plant model
y(k) = q-d^lu(k) +
A(q-)
C(q-1)
A(q-')
m
(1.9)
The identification is performed by the neural network and the control can be selected by
any conventional control techniques, in particular, the minimum variance control. This
neural self-tuning control scheme is different from the conventional self-tuning control
by the identification algorithm.


Figure 1.12. Stochastic neural linear ARMA control


21
CHAPTER 2
STOCHASTIC NEURAL SELF-TUNING
ADAPTIVE CONTROL (NSTC)
The NSTC consists of the minimum variance control law and the neural
identification algorithm. The model assumed for the plant is of ARMA input/output
type having the form
y(k)=q-d5^U(fc) + Sa^lyk) (2.1)
A(q A(q L)
where u(k), y(k), £(k), and d are system input, output, uncertainty, and delay,
respectively. A, B, and C are unknown system dynamics defined as
A(q-1) = 1 + ajq-1 + a^-2 + ... + anaq"na (2.2)
B(q-) = b + bjq"1 + b2q-2 + ... + bbq-nb (2.3)
C(q"*i = 1 + c^-1 + C2q"2 + ... + cncqnc (2.4)
where q is the shift operator. For the above unknown plant, in Figure 1.3, the
objective is to control its output to track a command signal y*(k) based on the
generalized minimum variance control index (Clark and Gawthrop, 1979)
J(k+d) = E{ = E{ [P(q'1)y(k+d)+Q(q"1)u(k)-R(q"1)y*(k)]2}
= E {[y(k+d)+Q(q1 )u(k)-R(q"1 )y *(k)]2} (2.5)
where E is the expectation operator, the weighting dynamics which can be chosen depending on the required response
characteristics.


22
2.1. Generalized minimum variance control
In this section, the generalized minimum variance self-tuning control algorithm
for the above problem statement is summarized (Clark and Gawthrop, 1979). To
obtain the optimal control u(k) which minimizes the performance index (2.5), the
predictive auxiliary output <])y(k+d) in terms of the system dynamics must be
determined. Consider the following identity
Brilgai)=P(q-i)+q-d
A(q-]) A(q_1)
(2.1.1)
where the order of F(q'*) and Gfq*1) are nf=d-l, ng=na-l, respectively. The output
prediction can be shown to have the form
<}>y(k+d) = $y(k+d) + $y(k+d) (2.1.2)
where
$y(k+d) = C(q-l)-1[G(q-l)y(k) + F(q-l)B(q-l)u(k)]
= C(q-l)-1[G(q-l)y(k) + E(q-l)u(k)] (2.1.3)
and
$y(k+d) = F(q-l)^(k+d) (2.1.4)
A ~
())y(k+d) and cj) y(k+d) are the deterministic and uncorrelated random components of
<|)y(k+d). Next, substituting (2.1.2) into (2.5), there results
J(k+d) = E {[ (2.1.5)
Since the second term in (2.1.5) is unpredictable random noise which is
uncompensatable by the control input u(k), and the first term is a linear function of
u(k), J(k+d) can be minimized by setting


(2.1.6)
[$y (k+d)+Q(q-1 )u(k)-R(q-1 )y *(k)] = 0
Solving for the generalized minimum variance control (GMVC) in (2.1.6) gives
R(q-')y*(k)-$v(k+ U------------------------- <21'7)
using (2.1.3), (2.1.7) can also be written as
E(q"1 )+C(q1 )Q(q"1)
Remarks : Recall that E(q_1) is equal to F(q1)B(q'1) where B(q_1) contains the
zeros of the plant. Notice that having the weighting function Q(q_1) additive to
E(q) in (2.1.8) gives the designer the ability to alter the poles of the controller.
Thus with a non-minimum phase plant B(q_1) shall have unstable roots and proper
selection of Q(q_1) in (2.1.8) can assure the control signal u(k) to be bounded.
2.2. Neural system identification
In this section, a stochastic neural identification algorithm is developed for
the self-tuning control scheme in Figure 1.3. Recall the predicted auxiliary output in
(2.1.3) which can also be written as
y(k+d) = C(q-l)-1[G(q-l)y(k) + E(q-l)u(k)] + F(q-l)£(k+d)
= C(q_ 1)1 [G(q1 )y (k) + E(q-)u(k)] + v(k) (2.2.1)
where the uncorrelated noise sequence F(q_1)^(k+d) is replaced by v(k). Also
(2.2.1) can be written as
nc
ng ne
<> (k+d) = ^g-y(k-i) + ^TejU(k-i) / >ci4)y(k-t-d-i) + v(k) (2.2.2)
i=0 i=0 i=l
<>y(k+d) = V(k)0(k)+ v(k) (2.2.3)
where


24
V'(k) = [y(k)...y(k-ng); u(k)...u(k-ne); <>y(k+d-1)...<>y(k+d-nc)] (2.2.4)
0W = [go Si gng; eo ei ... ene; -Cj -c2 ... -cc] (2.2.5)
since the parameter vector 0 is unknown, the estimated form of <|>y(k+d) is given as
$y(k+d) = $'(k)§(k) (2.2.6)
where
V'(k) = [y(k)...y(k-ng); u(k)...u(k-ne); $y(k+d-l)...$(k+d-nc)] (2.2.7)
A A A AAA A AA A
0'(k) = [gog1 gng;eo e 1 -ene*-cl-c2--cncl <2-2-8)
The unknown parameter vector in (2.2.8) (Figure 2.1), is taken from the output of the
neural network
8(k) = [ &l(k) &2(k) ... §j(k) ... §n3(k)]
= [Oi(k) 02(k)... Oj(k)... On3(k)]' (2.2.9)
Where n3 is the number of neurons at the output layer. Consider the system
identification cost function
V(k)=^E{e'(k)A-l(k)e(k)}
= \ E{[ where A(k) is a symmetric positive definite weighting matrix, and V(k) is minimized by
adjusting the weights of the neural identifier.


01
A
02
A
0n3
25
Figure 2.1. Neural network structure
In Figure 2.1, the weights connecting the second layer to the output layer, using the
gradient search (Rumelhart and McClelland, 1987), can be updated as
ij(k+l) = ij(k) + Acoij(k) (2.2.11)
where
Acoij(k) = -Tl{ie(k)A-l(k)e(k)}
acoij(k) z
= -Tl^^ {|[(})y(k)-$y(k)]A-1 (k)[y(k)-$y(k)]
" (2'2'12)
with T| being the search step size. Consider the derivative of with respect to &(k)
in (2.2.12)
d$y'(k)
3§(k)
= \fr(k-d)
(2.2.13)
In (2.2.13) we have assumed that 0(k) ~ 0(k-d), that is, 0 is slowly time varying with
respect to the delay time d. The other partial derivative in (2.2.12) can be determined as
afr(k)
3ij(k)
_______5[f(Neti(k))]1
= i(k)ej a^(k)
(2.2.14)


26
where f(.) is the sigmoidal activation function, Oj(k) is the output of the second layer,
and
Netj(k) = [neti net2 ... netj... netn3]'
(2.2.15)
with
n2
netj(k) = Xcoij(k)i(k)
where n2 is the number of neurons of the second hidden layer as shown in Figure 2.1.
Also ej in (2.2.14) is defined as
ej = [0...0 1 0...0] (2.2.16)
with the j-th element in ej being 1, and other elements are 0. Thus, substituting
(2.2.14) back into (2.2.12) gives
Acoij(k) = Tjej8j(k)Oi(k) (2.2.17)
where
3[f(Netj(k))]'d$v'(k)
SJ= dNetj(k) aS(t)
A'l(k)[(|)y(k)-$y(k)]
(2.2.18)
Next, the weights connecting the first to the second layer, in Figure 2.1, can be updated
by the recursive equation
(Dri(k+l) = (Ori(k) + Acori(k) (2.2.19)
where
A(0ri(k) = -Ti4e'(k)A-l(k)E(k)} (2.2.20)
3o)ri(k) z
Using the similar back propagation approach, (2.2.20) can be shown to result in the
following form
Acori(k) = ii5i(k)Or(k) (2.2.21)
where Or is the output of the first layer and


27
5 ,n r 3f[neti(k)] _,
5i(k)=[coii...a>ij...C0in3] aeti(k) 8j00
(2.2.22)
Lastly, the weights connecting the input to the first layer, in Figure 2.1, can be
updated by the recursive equation
f^srCk+l) = C0sr(k) + Acosr(k)
(2.2.23)
where
Asr(k) = -71^ 4 e(k)A-l(k)e(k)} (2.2.24)
3cosr(k) z
Again, using the back propagation approach, (2.2.24) can be determined as
Acosr(k) = T|8r(k)Is(k) (2.2.25)
where Is(k) is the input from the delay network and
.. 3f(netr(k))3[f(Netj(k))]' 3Netj(k)'
5r(k) [G)ri...a3n...O)m2] anetr(k) 8Neti(k) d[f(Neti(k))]
(2.2.26)
with Netj(k) being defined similarly as Netj(k) in (2.2.14). By adjusting the weights
toij(k), a>ri(k), and ff)sr(k) with the above algorithm, the unknown implicit plant's
parameters can be identified and obtained at the output of the neural identifier, as shown
A
in Figure 2.1. Once the estimate of 0 is available, <))y(k+d) in (2.2.6) can be
computed, and then the control signal can be generated using (2.1.7) as
R(q~1 )y*(k)-$v(k+d)
Q(q_1)
u(k) =
(2.2.27)


28
CHAPTER 3
FLEXIBLE ARM TIP POSITION DYNAMICS
This chapter describes the components and the control model of the flexible arm
tip. A detailed discussion of the dynamics of flexible arm tip and hub can be found in
(Fraser and Daniel, 1991). In order to control the flexible robotic arm shown in Figure
1.1, it is required that the control action produced by the control program running on a
processor board is converted to a voltage by the D/A board and forms the input to the
power amplifier of the motor. The output of the power amplifier is a motor current
directly proportional to the input voltage. The motor then converts this current to a
torque to drive the arm. The resulting motion of the arm is detected by the various
sensors and fed back to the controller.
The adaptive control algorithm design does not require the complete knowledge
of the plant dynamics. However, for the purpose of simulation study, the transfer
function model of the plant needs to be known. This model must incorporate not only
the behavior of the flexible arm itself but also the power amplifier, the motor and the
output sensors. In a servo system, the power amplifier and the sensors usually have a
much higher bandwidth than that of the motor and load therefore they can be
approximated as a constant. The general transfer function of the flexible arm tip is
9t(s) KaKtTT (l-s2/a?j)
u(s) S(s+C0)f={ (l +2^is/COj+S2/(of) (31)
where the physical interpretation of the above equation is as follows:
First, poles and zeros of the system is depicted in Figure 3.1


s-plane
29
Figure 3.1 Pole-zero diagram of flexible arm tip
The above diagram shows the three constituting dynamic components of the plant
which are the motor, the resonant modes of the flexible arm, and the arm non-minimum
phase characteristics. The dynamics of the servo motor system is represented by the
term
KAKt
(s+C0)s (3.2)
where ICj. is the motor torque constant, KA represents the power amplifier and sensor
gain, and CG represents the back emf and viscous damping effects know as the
mechanical time constant. The motor can be seen as a series of subcomponent
connected in series as shown in Figure 3.2.


30
motor motor
velocity position
Figure 3.2 Servo motor system components
Next, the the flexible arm attached to the motor shaft is describe by the term.
ft q-s2/!*?,)
i=! (l+2£jS/ffli+s2/a>?) (3.3)
Here, the denominator of (3.3) represents the set of flexible resonant modes of the arm.
Each flexible mode is associated with the corresponding damping at a frequency ci)j.
Theoretically, there is an infinite number of flexible modes, but in practice only the
sufficiently low frequency modes will be noticeable by the control system. This is
because a real system is always band-limited. Therefore most of the modes are
attenuated by the low-pass frequency behavior. Also, the frequency range of operation
can be limited to be below the major dominant resonant mode so that oscillations will
not be. present in the system response. If higher frequency range of operation is
desired, the dominate resonant modes can be notch filtered out provided their damping
£i's and frequencies cOj's are determinable.
Consider the physical properties of the flexible arm and the servo system given
in Table 3.1. Based on these parameters the transfer function was derived and
measured by experiment (Fraser and Daniel, 1991). Both results agreed as shown in
Table 3.2. The five resonant modes occupy the frequency range from 86 rad/sec to
1445 rad/sec. The frequency response of this system was simulated and is shown in
Figure 3.3. The peaks represent the resonant energy at the specific frequencies. Also
notice that the energy of the modes lessens are the frequency increases.


31
Table 3.1.
Physical properties of arm and motor
effective beam length (m) 0.386
beam thickness (mm) 0.956
beam width 0.03
mass/unit length of arm m (kg/m) 0.222
flexural rigidity of beam(NmA2) 0.426
hub moment of inertia (kg mA2) 0.00009
radius of hub (m) 0.034
Tip mass for loaded arm (kg) 0.065
tip inertia for load arm (kg mA2) 0.000005

continous torgue at rated speed (Nm) 0.177
pulse torque(Nm) 2.913
rated voltage (V) 24
torque constant (Nm/A) 0.048
total inertia (kg mA2) 0.000041

Ka*Kt 3.6
Co (rad/sec) 0.16


32
Table 3.2.
POLES (rad/sect______________________ZEROS (rad/seci
Mode Expmt. Theorv Expmt. Theorv
1 86.1 86.9 48.4 47
2 297.6 285.3 -48.4 -47
3 603.2 601.9
4 1011.6 1065.0
5 1445.1 1658.8
Radian/s
Figure 3.3 Frequency magnitude response of arm tip with five resonant modes


33
For easy controllability it is desirable to filter out these resonance modes. Therefore, a
notch filter is designed to notch out the first resonance mode and a low pass filter is
used to filter out the rest-of the resonance modes. Figure 3.4 shows a block diagram of
the filtering process. The resulting frequency ideal response is shown in Figure 3.5.
notch filter
LP filter
flexible arm
system
Figure 3.4 Frequency response of open-loop components
Figure 3.5 Frequency response of the aggregate filtered open-loop


Since we are primarily interested in learning the controllability and behavior of the non-
minimum phase characteristics of the plant, we can simplify the arm tip transfer
function to have the form
6t(s) KAKT(l-s2/ct2)
u(s) s(s+c0) (3.4)
Lastly, the non-minimum characteristics of the arm tip is describe in (3.1) and (3.4) by
the numerator term.
(l-s2/a2)
This is due to the fact that the control system sensing and actuation do not take place at
the same location and therefore being a non-collocatted system. It should be mentioned
that the non-minimum phase characteristics is very difficult for the neural network to
control (since most neural network adaptive control schemes are based on the direct
method).


35
CHAPTER 4
EMPIRICAL STUDIES
In this chapter we examine some simulation results of the direct and indirect
neural control schemes for controlling the flexible arm hub and tip. We will show that
the hub having a well behaved linear transfer function produced very satisfactory
controlled response; We also attempted to use the direct adaptive control scheme to
control the tip velocity and found unstable response even after numerous controller
parameter changes. Next, the NSTC scheme in section 2 was applied to control the tip
position and produced encouraging results. Lastly, the neural identifier in the NSTC
algorithm is compared with the recursive least square identifier and show faster
convergent rate. The simulation program used in this study is given in the appendix.
4.1. Neural direct adaptive control of arm hub and tip
The neural direct adaptive control scheme was first introduced by (Psaltis et al.,
1988) and was later reformulated for nonlinear/linear state space system by (Ho et al.,
1991c). We will apply this scheme, shown in Figure 4.1. to control the hub velocity of
the arm.
The dynamic transfer function of the hub is a linear minimum phase system.
The numerical transfer function found in (Fraser and Daniel, 1991) is
9h(s) =
U(s)
10.2 (H-£-).
32.72
(s+0.57)(s+2000)
(4.1)


Figure 4.1. Neural direct control scheme of hub velocity
where the resonant modes are assumed to be filtered out. In the simulation process ,
the model in (4.1) was first discretized and then converted to state space form
x(k+l) = Ax(k) + Bu(k) (4.2)
0h(k) = Cx(k)
When using this scheme (Figure 4.1.) there is a priori information that is needed and
39h(k)
that is the jacobian of the plant ^u(k) This term was computed based on the
discretized model and resulted as
d9h(k)_CB
du(k) (4.3)
Information on the neural network algorithm is refered to (Ho et al., 1991c).
Remarks: : The hub position was not suitable for this specialized learning control
scheme because the jacobian turns out to be near zero. Therefore the velocity is the
selected controlled variable and an additional outer control loop may be incorporated to
achieve position control. This outer loop will have a velocity profile generator which
resembles to a proportional controller with saturation (Franklin and Powell, 1981).
Simulation: A smoothed square wave command was presented to the control system,
after 50 iterations (about .3 seconds, sampling period was 6 ms) the hub had tracked
the command signal as shown in Figure 4.2 where the solid line is the desired response
and the dashed line is the actual response.


37
Figure 4.2. Hub velocity response: t(k) & 0t(k)
This trackability is reflected in the mean square tracking error shown in Figure 4.3.
Notice that the convergent time in control application is serveral orders of magnitude
faster than other applications. In this case it took only 50 iterations for the 2-layer
neural network to be maturely trained with initial random weights. This fast convergent
time makes it very practical for real-time control implementation.
Figure 4.3. Control tracking MSE response
Next, the same scheme is applied to control the tip velocity. The numerical tip transfer
function (based on the flexible arm and motor properties in Tables 3.1 and 3.2) is given
in (Fraser and Daniel, 1991) as
3.6 (H-^
6t(s) _ 48.42
)
U(s) s(s+0.16)
(4.4)


38
Here again, we are primarily interested in the non-minimum phase characteristics and
therefore assumed that the resonant modes are filtered out.
Simulation: After numerous attempts to vary the neural network parameters, an
unstable closed-loop response was prevalent as shown in Figures 4.4 and 4.5. This is
due to the fact that the neural network in Figure 4.1. trying to emulate the inverse
dynamics of the plant (4.4.) and in effect produced an unstable pole behavior. Note in
Figure 4.4. that the command signal is small compared to the plant diverging output
response therefore it looks like a straight line.
M 200
<4-1 0
/-N M
-200
'V
V
0 50 100 150 200 250 300 350 400 450 500
Iterations
Figure 4.4. Unstable response of tip control: 0t*(k) and 0t(k)
3000
2000
^ 1000
0
0 50 100 150 200 250 300 350 400 450 500
Iterations
Figure 4.5. Diverging tracking MSE of tip velocity
Neural Network: The 5,10,1 neural network used in this scheme consists of one input
layer, one hidden layer, and one output layer with the number of neurons as 5,10, and
1, respectively. Also at the input of the neural was the desired response vector [y*(k)
y*(k-l) y*(k-2) y*(k-3) y*(k-4)]T. The parameters of the sigmoidal activation function
at the output node were found to be most influential on the tracking error convergentce


39
rate. Predominantly the slope of the activation function was observed to be
proportional to the convergence rate. Also the bipolar sigmoidal saturation levels of the
output neuron needed to be set equal to or greater than the maximum allowable plant
input. The tuning of the sigmoidal functions was done manually by trial and error,
typically for linear system like that of the hub, it takes very few tweaks (around 1 or 2)
before the tracking result was achieved. Auto-tuning of the sigmoidal function
parameters can also be applied to obtain statiscally better results (Yamada and Yabuta,
1992; Proano, 1989).
4.2 Neural self-tuning adaptive control (NSTC) of tip position
In section 4.1. we showed by simulation that the direct neural adaptive control
scheme was unable to control the tip position (Figures 4.4 and 4.5). In fact, this was
why the NSTC algorithm was developed. Recall that this scheme has two distinct
functions, identification and control, which are done by the neural network and the
(GMV) control, respectively. The NSTC scheme is shown again in Figure 4.6.


40
Figure 4.6. NSTC scheme block diagram
In this section we perform the simulations of two schemes which are: The
adaptive STR using recursive least square identification, and the NSTC using the neural
identification. This was done to performis a comparative study in order to assess the
performance of the developed NSTC.
Simulation: The model of the tip position is the discretized model of (4.4). Recall the
control index defined in section 2
J(k+d) = E{2(k+d)}
= E {[P(q"1 )y(k+d)+Q(q1 )u(k)-R(q1 )y *(k)]2} (4.5)
where the weighting functions were chosen as


41
P(q-1)=l; Q(q-1)=.l+.06q-l; R(q-l)=l (4.6)
*
and the desired hub position t(k) was a step command. Beginning with Figure 4.7.
shows the desired step tip response, the controlled tip response based on the adaptive
STR and the tip response from the NSTC. Obviously both controllers manage to track
the command signal. However, the NSTC seems to have a slower settling time.
Figure 4.8. shows the converging tracking control index (2.1.5) where both schemes
seem very comparable to each other. Figure 4.9. displays the comparable control
energy produced by these controllers. Note that the transient control energy was
/N
affected by two factors: one is the initial condition of the estimated parameter vector Go
/N /N
(which was set as Go = [1 1 ... 1]' for both control schemes), the further Go is away
from the optimum 0 in the parameter state space, the longer the convergence of the
tracking control index (2.1.5). The other factor is the selection of the input weighting
function Q(q_1) which has the effect of limiting the control energy with the tradeoff of
slower tracking convergence. Lastly, we compare the recursive least square
identification with the neural network identification. The two identifiers estimate the
parameter vector 0 in (2.2.5) so that the predictive output term ^(k+d) in (2.2.2) can
be computed. Figure 4.10. shows the estimation cost function V(k) in (2.2.10)
response of the RLS and the neural network. V(k) of the RLS has a slightly faster
convergence than the neural network but not by a significant degree. Again, this
indicates that the identification performance of the two algorithms are comparable to
each other. For completeness, the time response of the true output 0t(k) and the
estimated output t(k) produced by the neural network is shown in Figure 4.11.
N3
Neural Network: The three layer neural network 1^2,5,15,po used in this scheme
consists of one input layer, two hidden layers, and one output layer with the number of
neurons as 2, 5, 15, and P0, respectively. P0 is the length of the vector defined in
(2.2.8) which is (ng+l)+(ne+l)+nc, and is 11 for the case of the arm tip plant. The
input of the neural network was a selected as constant vector Is = [1 1]' because it was
desired that the output of the neural network to be correlated to the its input. The
parameters of the sigmoidal activation function at the output node was found to be most
influential on the tracking error convergent rate. Predominantly the slope of the
activation function was observed to be proportional to the estimation convergent rate
V(k). Also the bipolar sigmoidal saturation levels of the output neuron needed to be set
equal to or greater than the maximum component of the parameter vector 0. The tuning


42
of the sigmoidal functions was done manually by trial and error. Autotuning of the
sigmoidal function parameters can also be applied to obtain statiscally better results
(Yamada and Yabuta, 1992; Proano, 1989). However, the optimal dimension of the
neural network in terms of number of layers and nodes was not known and therefore an
N3
initial pick of 2-5.15,po was used throughout the simulation.
#
Figure 4.7. Tip position response: t(k) & 6t(k) 0f the adaptive STR and the NSTC


43
Figure 4.8. Control performance index J(k) of the adaptive SIR and the NSTC
Figure 4.9. Control signal u(k) of the adaptive STR and the NSTC


44
Figure 4.10. Identification cost index V(k) of the adaptive STR and the NSTC
Figure 4.11. True and neural network estimated tip position: t(k) & t(k)


45
CHAPTER 5
CONCLUSION
The neural self-tuning control (NSTC) algorithm was developed and applied to
control the tip of a flexible arm system. The dynamics of the flexible arm tip involves
an unstable zero and therefore making the system non-minimum phase. Most of the
existing neural adaptive control are based on the inverse dynamics and therefore would
not be able to control this type of plant. The NSTC was based on an indirect control
method where the identification is performed by the neural network and the control was
based on the generalized minimum variance (GMV) control law. The performance of
the NSTC was investigated and was compared to the adaptive STR by means of
simulation.
In summary, the NSTC has a very comparable performance to the adaptive STR
shown by simulation results in section 4.2. Unlike other applications of neural
networks where thousands of iterations were required before the network can be
maturely trained, in this application the neural network identification had a convergence
rate comparable to that of the RLS. Another advantage of the NSTC is due to the
availability of neural network VLSI and the massive parallel architecture of the neural
network there will be a computation advantage over conventional recursive algorithms.
This will enable real-time implementation with faster sampling rate for system with
wide bandwidth. Also another advantage of the NSTC is that because the identification
is done by the neural network, it inherits the decentralize property, meaning if there is a
failure in a node or connection the impact on the performance will be minimal.
Whereas with the conventional digital filter a failure in one of the coefficient will have a
major impact on the output. With all the above encouraging characteristics there is one
disadvantage of using the neural network and that is the lack of understanding how the
dimension and activation characteristics of a network is related to its accuracy and
stability. These issues of the recursive algorithms have been addressed and elaborately
analysed in (Kumar, 1990).


5.1 Future research
The NSTC can be modified and extended to control systems that are not only
non-minimum phase but also nonlinear. This is so that the properties of neural
networks can be fully exploited. A system that have the above characteristics is a two
degree of freedom robotic manipulator with the second link being flexible. Most
conventional adaptive control schemes rely heavily on the inverse dynamics and
therefore showed great limitations with this type of system (Centinkunt and Yu, 1990).


47
BIBLIOGRAPHY
Astrom, K., and Wittenmark, B., "On self-tuning regulators", Automatica, 9,
pp.185-199, 1973.
Antsaklis, P., "Neural networks in control systems", IEEE Control Systems
Magazine, April 1990.
Bavarian, B. "Introduction to neural networks for intlligent control", IEEE
Control Systems Magazine, April 1988.
Centinkunt, S., and Book, W., "Performance Limitations of Joint Variable
Feedback Controllers due to Structural Flexibility", IEEE Transaction on Robotics and
Automation, Vol. 6, No. 13, 1990.
Clark, D., and Gawthrop, P., "Self-tuning controller", in proceedings IEE,
126, pp.633-640, 1979.
Franklin, F.G., and Powell, J.D., Digital Control of Dynamic Systems.
Addison Wesley, Reading, Mass. 1981.
Fraser, A., and Daniel, R., Perturbation Techniques for Flexible Manipulators,
Kluwer Academic Publishers, Norwell, Massachusetts, 1991.
Hetch-Nielsen, R. "Theory of the backpropogation neural network" in
proceedings of the International Joint Conference on Neural Networks, Washington
D.C., 1989
Ho, T., Ho, H. "Stochastic state space neural adaptive control", in proceedings
the Third International Conference on Advances in Communication and Control
Systems, Victoria, B.C., Canada, Oct. 16-18, 1991a.
Ho, H., Ho, T., Wall, E., and Bialasiewicz, J., "Stochastic neural self-tuning
adaptive control", in proceedings of the Third International Conference on Advances in
Communication and Control Systems, Victoria, B.C., Canada, Oct. 16-18, 1991b.
Ho, T., Ho, H., Wall, E., and Bialasiewicz, J., "Stochastic Neural Direct
Adaptive Control", in proceedings of the 1991 IEEE International Symposium on
Intelligent Control, Arlington, Virginia, Aug. 13-15,1991c.
Ho, T., Ho, H., and Bialasiewicz, J. "Stochastic neural adaptive control for
nonlinear time varying systems", in proceedings of the 1991 International Conference
on Artificial Neural Networks in Engineering, St. Louis, Missouri, Nov. 10-12,
1991d.
Ho, T., Ho, H., and Bialasiewicz, J. "On stochastic newton adaptive
control", in proceedings of the IASTED International Symposium on Adaptive Control
and Signal Processing, New York, Oct. 10-12, 1990.


48
Hunt K., Sbarbaro, D., Zbikkowski, R., and Gawthrop, P., "Neural
Networks for Control System A survey", Automatica, Vo. 28, No. 6, 1992.
Kawato, M., Setoyama, T. and Suzuki, R. "Feedback-error-learning of
movement by multi-layer neural network", in proceedings of the International Neural
Networks Society First Annual Meeting, 1988.
Kumar, P., "Convergence of Adaptive Control SchemesUsing Least-Squares
Parameter Estimates", IEEE Transactions on Automatic Control, Vol. 35, NO 4, pp.
416-424, April 1990.
Levin. A., and Narendra, K., "Control of Nonlinear SystemsUsing Neural
Networks: Controllability and Stabilization", IEEE Transactions on Neural Networks,
Vol 4. NO. 2, March 1993.
Miller, T., Sutton, R., and Werbos, P., Neural Networks for Control. The
MIT press, Cambridge, Massachusetts, 1990.
Miyamoto, H., Kawato, M., Setoyama, T. and Suzuki, R. "Feedback-error-
learning neural network for trajectory control of a robotic manipulator", Neural
Networks 1:251-265, 1988.
Narendra, K., and Parthasarathy, K., "Identifciation and control of dynamical
systems using neural networks" IEEE Transactions on Neural Networks, Vol 1. NO.
1, March 1990.
Proano, J., Neurodynamic Adaptive Control Systems, Ph.D Dissertation,
University of Colorado, Boulder, 1989.
Psaltis, D., Sideris, A., and Yamamura, A., "A multilayered neural network
controller" IEEE Control Systems Magazine, April 1988.
Rumelhart, D., and McClelland, J., Parallel Distributed Processing: Vol 1,
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Spong, N. M., and Ortega, R. (1988), "Adaptive Motion Control of Rigid
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Method for Nonlinear Functions", IEEE Transactions on Neural Networks, Vo 3, No.
4, July 1992.


49
APPENDIX
Simulation Program
The simulation was perforemed using the software MATLAB. The program shown
below is the NSTC scheme.
clc
clear
%
% ======== BEGIN SIMULATION
N3=l 500; % NUMBER OF ITERATIONS
ndisp=30;
ALGsr = 1; % 1==> Gradient 2==> Newton 3=>MV
ALGri = 1; % 1==> Gradient 2==> Newton 3==>MV
ALGij = 1; % 1=> Gradient 2==> Newton 3=>MV
ID=1; % 1=> RLS 2==> Neural I.D. 0==>Determistic
%
%
% ===== INITIALIZATION =================================
% PLEASE SELECT THE DIMENSION OF THE STATE VECTOR X0,
% INPUT VECTOR U0, OUTPUT VECTOR Y0, AND PARAMETER VECTOR
% P0 BY MODIFYING THE FOLLOWING STATEMENTS:
%
% P0=4;
P00=1; PSI0=1;
%
%
% Plant [al a2 a3...anab0bl b2...bnb];
% A = [.7 .5-.3]'; B=[l .2-.1 .3];
% A = [.7 .5]'; B=[l .2-.1];
% THETAp = [.7 .5 1 .2 -.1]; % minimum phase 2nd order plant
% A = [.7 .5 -3]; B = [1 .2 -.1 3];
% THETAp = [.7 .5 -.3 1 .2 -.1 .3]'; % minimum phase 3rd order
% THETAp = [.7 .5 -.3 1 .2 -.1 3]'; % non-minimum phase 3rd ; ld=4-10
% A=[.7 .5 -.3]'; B=[l .2 -.1 3]';
% THETAp = [-2.58 2.18 -.5965 -429.7 884.8 -430.8]; %missile nmp
% THETAp = [-3.987 5.96 -3.96 .987 -6.94e-5 6.92e-5 6.9e-5 -6.88e-5];
% Mxl
%THETAp = [-3.87 5.63 -3.64 .882 -.0068 .0066 .0065 -.0063]; %missile
%THETAp = [-2.979 2.96 -.979 -.0047 .0094 -.0047]; %Submarine
load plant
num=numd7dend( 1);
den=dend/dend( 1);


B=num;
A=den(2:length(den));
% B=numd A=dend(2,:)
THETAp = [A1 B']';
na=length(A);nb=length(B)-1 ;d= 1;
nf=d-l;
ne=nf+nb;
ng=na-l;
nc=0; %This assumes the noise has no dynamics, i.e. C=l;
PO = (ng+1 )+(ne+1 )+nc; %Dimension of THETA
P0p=na+nb+1; %Dimension of plant's THETA
THETAEST= 1 *ones(P0,1);
%load thetaest
THETAO=THET AEST;
yest=0; w=0;
P=P00*eye(P0,P0);
PSIp=PSIO*ones(POp,l);
PSId = zeros(PO,l);
K=1.5*ones(P0,l);
Yl=zeros(ng+l,l); Ul=zeros(ne+l,l);
Yc=zeros(ng+1,1); Uc=zeros(ne, 1);
Y lp=zeros(na, 1); U1 p=zeros(nb+1,1);
Ud=zeros(d,l); Yd=zeros(d,l); % delayed values of u, y and w
Wd=zeros(d,l);
y=0; u=0; % output y(k), input u(k)
VARV = 0; % Output noise variance
MEANV=0;
n0=2; nl = 5; n2=15; n3=P0; % Dimensions of Neural Network
NETr = zeros(n 1,1);
NETi = zeros(n2,l);
NETj = zeros(n3,l);
Is = zeros(nO,l);
Or = zeros(nl,l);
Oi = zeros(n2,l);
Oj = zeros(n3,l);
ALPHA) = .O3*ones(n3,l);, ALPHAi = .03*ones(n2,l);
ALPHAr = .03*ones(nl,l);
Hj = 0*ones(n3,l);, Hi = 0*ones(n2,l);, Hr = 0*ones(nl,l);
Kj = 3*ones(n3,l);, Ki = 2*ones(n2,l);, Kr = 2*ones(nl,l);
Wsr = rand(nO.nl);
Wri = rand(nl,n2);
Wij = rand(n2,n3);
mu=.8;
lambda0=.99;
lambdak = .995;
LAMBDA = 1;
Pij = 10*ones(n2,n3);
Pri = 5*ones(nl,n2);
Psr = 5.3*ones(n0,nl);


51
Rn = .001;
Re=.9;
%
% ===== end of initialization ============================
rand('seed',10);
%-----------------
% ::::: BEGIN ITERATION :::::
fork=l:N3
% ===== STOCHASTIC ARMA REPRESENTATION OF A LINEAR PLANT
%
% y(k)+a 1 y(k-1)+.. .+anay (k-na)=bOu(k-d)+b 1 u(k-1 -d)+.. .+bnbu(k-nb-d)+v(k)
% y(k)=PSIp'(k)*THETAp(k)+v(k)
% PSIp'=[-y(k-l)...-y(k-na) u(k-d)...u(k-nb-d)]
% THETAp(k)'=[al a2...ana bO bl b2...bnb]
% THETA'(k) = [g0 gl ... gng eO el...ene]
% PSId'(k) = [y(k-d).. y(k-d-ng) u(k-d)..u(k-d-ne)]
% yest(k) = PSId(k)'*THETAEST
%=======Computing THETA ===========
if d==l
E=B*
G=-A; %g(i)=-a(i+l), i=0..ng G=q-l(l-A)
end
THETA = [G'ET;
% PARAMETRIZATION FOR PSI(k).
%
%
%
% u=u(k-l) y=y(k-l)
for i=d-11:1 Ud(i+l)=Ud(i);, end, Ud(l)=u; %[u(k-l)...u(k-d)]
for i=d-l:-l:l Yd(i+l)=Yd(i);, end, Yd(l)=y; %[y(k-l)...y(k-d)]
for i=d-l:-l:l Wd(i+l)=Wd(i);, end, Wd(l)=w; %[w(k-l)...w(k-d)]
%==== PSIp(k) = [-y(k-l).. -y(k-na) u(k-d)..u(k-d-nb)]
for i=na-l:-l:l Ylp(i+l)=Ylp(i);, end; Ylp(l)=-Yd(l);
for i=nb:-l:l Ulp(i+l)=Ulp(i);, end; Ulp(l)=Ud(d);
PSIp = [Yip' UlpT;
%===PSId(k) = [y(k-d).. y(k-d-ng) u(k-d)..u(k-d-ne)]
for i=ng:-l:l Yl(i+l)=Yl(i);, end; Yl(l)=Yd(d); %[y(k-d).. y(k-d-ng)]
for i=ne:-l:l Ul(i+l)=Ul(i);, end; Ul(l)=Ud(d); %[u(k-d)..u(k-d-ne)]
PSId = [YV U1T; % PSI(k-d)


52
%
%----GENERATING NOISE v(k) -
rand('normal')
v=sqrt(VARV)*rand( 1,1 )+MEANV;
%
%----COMPUTING y(k) & w(k)
%
y=PSIp' *THET Ap+v; %y(k)
tau=.5;
w=tau*w + (l-tau)*2; %*sign(sin(0.004*(k))); %command signal w(k)
w=(w/2)+l;
%
%-----END OF PLANT-------------
%
%---ADAPTIVE ESTIMATION---
% ===== the STOCHASTIC LEAST SQUARES ALGORITHM (SLA)
%
%
%----BEGIN ESTIMATION----
% THETAEST = [gO gl ... gng eO el ... ene]'
yest=PSId' *THETAEST; % PREDICTED OUTPUT yest(k)
e=y-yest; % PREDICTION ERROR e(k)
if ID=1
K=P*PSId*inv( l+(PSId'*P*PSId)); % OPTIMAL GAIN
THETAEST=THETAEST+K*e; % PARAMETER ESTIMATION
P=(P-K*PSId'*P);
end
%==========Neural identification ========
if ID=2
Is(l)=l;
Netr = Wsr'*Is-
tempr = (ALPHAr/2).*(Netr+Hr);
Or = Kr.*tanh(tempr);
Or(l)=l;
Neti = Wri'*Or;
tempi = (ALPHAi/2).*(Neti+Hi);
Oi = Ki.*tanh(tempi);
Oi(l)=l;
Netj = Wij'*Oi;


tempj = (ALPHAj/2).*(Netj+Hj);
Oj = Kj.*tanh( tempj);
THETAEST = Oj;
if k==l save thetaest THETAEST, end
PSI=PSId;
tempj2 = cosh(tempj).*cosh(tempj);
tempi2 = cosh(tempi).*cosh(tempi);
tempr2 = cosh(tempr).*cosh(tempr);
Fdotj = (Kj.*ALPHAj/2)./(tempj2);
Fdoti = (Ki. ALPHAi/2)./(tempi2);
Fdotr = (Kr.*ALPHAr/2)./(tempr2);
dj = Fdotj. *PSI;
PSIij = Oi*dj';
di = (Wij*dj).*Fdoti;
PSIri = Of*di';
Q = Fdoti .* (Wij*(Fdotj.*PSI));
dr = Fdotr .* (Wri*Q);
PSIsr = Is*dr';
if ALGij == 1 % Gradient
Lij = mu*PSIij/LAMBDA;
end
if ALGij == 2 % Newton
Sij = (PSIij. *PSIij.*Pij) + (lambdak*LAMBDA*ones(n2,n3));
Lij = (Pij.*PSIij)./Sij;
Pij = (Pij (Lij.*Sij.*Lij))/lambdak;
end
if ALGij = 3 % Minimum Variance
Sij = (PSIij.*PSIij.*Pij) + Re*ones(n2,n3);
Lij = (Pij. PSIij) ./S ij;
Pij = Pij (Lij.*PSIij.*Pij) + Rn*ones(n2,n3);
end
if ALGri == 1 % Gradient
Lri = mu*PSIri/LAMBDA;
end
if ALGri == 2 % Newton
Sri = (PSIri.*PSIri.*Pri) + (lambdak*LAMBDA*ones(nl,n2));
Lri = (Pri.*PSIri)./Sri;
Pri = (Pri (Lri.*Sri.*Lri))/lambdak;
end
if ALGri == 3 % Minimum Variance
Sri = (PSIri.*PSIri.*Pri) + Re*ones(nl,n2);
Lri = (Pri.*PSIri)./Sri;
Pri = Pri (Lri.*PSIri.*Pri) + Rn*ones(nl,n2);
end
if ALGsr == 1 % Gradient
Lsr = mu*PSIsr/LAMBDA;
end
if ALGsr == 2 % Newton


54
Ssr = (PSIsr.*PSIsr.*Psr) + (lambdak*LAMBDA*ones(nO,n 1));
Lsr = (Psr.*PSIsr)./Ssr;
Psr = (Psr (Lsr.*Ssr.*Lsr))/lambdak;
end
if ALGsr = 3 % Minimum Vasrance
Ssr = (PSIsr.*PSIsr.*Psr) + Re*ones(nO,nl);
Lsr = (Psr.*PSIsr)./Ssr;
Psr = Psr (Lsr.*PSIsr.*Psr) + Rn*ones(nO,nl);
end
Wij = Wij + Lij*e;
Wri = Wri + Lri*e;
Wsr = Wsr + Lsr*e;
%LAMBDA = LAMBDA + (e*e'-LAMBDA)/k;
lambdak = lambdaO*lambdak+(l-lambdaO);
Re = Re + (e*e'-Re)/k;
for i=n0:-l:2 Is(i)=Is(i-l);, end
end
%
%----END OF ESTIMATION-----
%
% ===== MINIMUM VARIANCE ADAPTIVE CONTROL
%
%
%
%-----begin adaptive control----------
for i=ng:-l:l Yc(i+l)=Yc(i);, end; Yc(l)=y; %[y(k).. y(k-ng)]
for i=ne-l:-l:l Uc(i+l)=Uc(i);, end; Uc(l)=u; %[u(k-l)..u(k-ne)]
ld2= .06;
ld3=0;
if ID=0 THETAEST=THETA;, end % Q = Id + q-lld2
if k Gest( 1 :ng+l, 1) = THETAc( 1 :ng+1); % G
Eqest(l:ne+l,l) = THETAc(ng+2:ng+2+ne); % E
Eqest( 1) = Eqest( 1 )+ld; % E+Q
Eqest(2) = Eqest(2)+ld2;
%Eqest(3) = Eqest(2)+ld3;
% u(k) = (w(k)-[gOy(k)+...+gncy(k-ng)] -[elu(k-l)+...+eneu(k-ne)]}/eO
SUM1 = Eqest(2:ne+l)'*Uc;
SUM2 = Gest'*Yc;


%roots(Eqest)
%break
u=(w-SUM2-SUM 1 )/Eqest( 1); %u(k)
%u=w;
%----END OF ADAPTIVE CONTROL
%
%
%----SIMULATION ERRORS-------------
%----SAVE THETA(k) & THETAEST(k)
for j=l:P0
THETA1 (k,j)=THETA(j);
THETA1EST (k j)=THET AEST (j);
end
%----SAVE y(k) & yest(k)---
Y(k,l)=y;
YEST(k,l)=yest;
%----SAVE K(k)-----
for j=l:P0
Kl(k,j)=K(j);
end
%----SAVE U(k)-----
U(k)=u;
W(k)=Wd(d);
%
%
%----THE PARAMETER IDENTIFICATION MSE(k)-------
THETAER=THET A1 -THET A1 EST;
for j=l:P0
if k==l, TMSE(k,j)=THETAER(k,j)A2; else
TMSE(k,j)=TMSE(k-1 ,j )+(THET AER(k,j) A2-TMSE(k-1 ,j))/k;
end
end
%----THE OUTPUT PREDICTION MSE(k)-------
YER(k)=y-yest;
if k=l, YMSE(k)=YER(k)A2; else
YMSE(k)=YMSE(k-1)+(YER(k)A2-YMSE(k-1 ))/k;
end
%----THE COST FUNCTION(k) J(k)----
YERc(k)=y-Wd(d);
if k= 1, J(k)=YERc(k)A2; else
J(k)=J(k-1)+(YERc(k)A2-J(k-1 ))/k;
end
% ===== DISPLAY MATRIX ============================
% THIS M-FILE IS USED TO MONITOR SYSTEM PERFORMANCES DURING


56
% SIMULATION.
%----TRANSFER DATA TO MATRIX DISMAT1
DISMATl(l,l)=k;
DISM AT 1 (1,2)=TMSE(k, 1);
DISMAT1 (1,3)=TMSE(k,2);
DISMAT1 (1,4)=TMSE(k,3);
% DISM AT 1(1,5)=TMSE(k,4);
DISM AT 1(1,6)=YMSE(k);
DISM AT 1 (1,7)=U(k);
%----TRANSFER DATA TO MATRIX DISMAT2----------
DISM AT2( 1,1 )=k;
DISMAT2( 1,2)=J(k);
DISMAT2( 1,3)=W d(d);
DISMAT2( 1,4)=y;
DISM AT2( 1,5)=yest;
%----DISPLAY DISMAT1 & DISMAT2--------
if rem(k,ndisp)==0
home
disp(' k TMSE1 TMSE2 TMSE3 TMSE4 YMSE U(k))
disp(DISMATl)
disp(' k J(k) w(k-d) y(k) yest')
disp(DISMAT2)
% [THETA THETAEST]
end
%
% = = = = = END OF DISMAT
%keyboard
end % END OF FOR LOOP(k)
% ::::: END OF ITERATION:::::
%
%----SYSTEMS GRAPHICS------
SYGRAF
%----------------------
%
END
OF
SIMULATION


Table 4.2 Taxa included in study with an unfused symphysis
Taxa with an Unfused Symphysis Sample size
Order- Galago demidoff 1
Primate Nycticebus coucang 1
Order- Canis latrans 12
Carnivora Canis lupis 9
Urocyon cinereoargenteus 8
Vulpus velox 11
Vulpus vulpus 11
Lynx canadensis 3
Lynx rufus 10
Puma concolor 10
Procyon lotor 10
Ursus americanus 9
Ursus arctos 2
Order- Cervus elaphus 2
Artiodactyla Odocoileus hemionus 2
Odocoileus virginianus 3
Mazama americana 2
Total = 106
47


Data Collection Process
The following steps were taken in the data collection process to ensure
that accurate measurements of the specimens were obtained:
Step 1- A Sony digital camera was used to photograph each specimen.
Each specimen was placed on a level work surface. Two photographs were taken
of each individual including both a transverse and occlusal view. When
photographing the occlusal surface, each specimen was held in place with
modeling clay. The modeling clay allowed the specimens to be positioned at a 90
degree angle to the table.
The camera was placed on a tripod and aimed at the specimen. The
camera was positioned as far from the specimen as possible while still filling the
viewfinder with the skull. Filling the viewfinder with as much of the image as
possible maximizes screen resolution which allows measurements to done on a
computer with finer detail, thereby, reducing measurement error (Spencer, 1995).
Step 2- A calibration grid was set up and photographed before pictures of
the specimens were taken. Any time the camera was moved, particularly upon
completion of a species, the calibration grid was repositioned and re-
photographed. This image was used during analysis to determine the size of the
space in which landmarks were located.
48


Step 3- Some of the landmarks that were measured were not clearly visible
(such as the glenoid fossa). These were highlighted with small black dots, which
helped with identification on the computer images.
Step 4- Images were downloaded into a Macintosh computer and
measured within the MacMorph data acquisition package. Each image was
calibrated using the corresponding calibration grid. Measurements were then
taken of the variables for all images.
Step 5- The statistical packages JMP and Statview were used for all
analyses.
Measurements
Three sets of measurements were taken for this study: (1) distances that
represent the observed length of the postcanine dentition, (2) dimensions for
calculating a predicted Region II length, and (3) size adjustment measurements.
Measuring the observed length of the postcanine dentition is necessary for
Hypothesis 1 and 2. The observed length of the postcanine dentition was
measured from the trigon of the maxillary last molar to the trigon of the first
molar and from the trigon of last molar to the trigon of the first premolar. The
trigon of each tooth was chosen because of the biomechanical role it plays; this
feature experiences much of the force that is produced during mastication when it
comes into direct contact with the teeth of the mandible.
49


The main hypothesis of this study is that the molars must lie within
Region II. However, the length of both the premolars and molars were measured
in order to assess the extent to which both types of teeth lie within Region II.
Calculating an estimated Region II length is necessary for Hypothesis 1
and 2. There are five variables that can influence the location of Region II and
must be quantified or estimated in order to calculate a predicted Region II length.
These variables include bicondylar breadth, palatal breadth at Ml, height of the
TMJ relative to the occlusal plane, distance from the TMJ to point of intersection
of the muscle resultant vector and occlusal plane, and the angle of the muscle
resultant vector to the occlusal plane (Fig. 4.1). The first three variables can be
directly quantified. However, it is difficult to quantify muscle resultant position
and orientation due to limited knowledge of the comparative myology and
function of the masticatory muscles among mammals (Throckmorton, 1989;
Spencer, 1999). This study will therefore assume the MRF vector intersects the
occlusal plane directly at the posterior end of the tooth row. This can be directly
quantified as the distance from the TMJ (defined here as the center of the articular
eminence) to the trigon of the last molar. This is consistent with the assumptions
of the constrained model.
A prior study involving muscle resultant force orientation for the
masticatory adductor muscles of anthropoid primates estimated a fixed orientation
of 80 degrees relative to the occlusal plane based on quantified orientations of the
50



rw o1 /
V I
ic
r
w
/ A
\ \
V. ;
."i (
\\
i-.. >
! //
\ //
/




A = Biarticular breadth
B = Palate Breadth at M1
C = Distance from TMJ to point of
intersection of muscle resultant
vector and occlusal plane
D = Height of articular eminence
above occlusal plane
E = Angle of muscle resultant
vector to occlusal plane
(90 equals perpendicular)
Fig. 4.1 Illustration of five variables used in the calculation of Region II length.
51


anterior temporalis, superficial masseter, and medial pterygoid muscles (Spencer,
1999). Muscle resultant orientation cannot be calculated in the present study
since it is likely to be highly dependent on individual muscle force magnitudes.
Collecting these data is beyond the scope of this study.
The length of the molar row is defined as the distance between the trigon
of the right maxillary last molar to the trigon of the first molar. Bicondylar
breadth is the distance between the central articular eminence landmarks. Height
of the TMJ is the perpendicular distance of the right central articular eminence
landmark from the occlusal plane. The occlusal plane is projected onto the
sagittal plane and is defined by the horizontal line connecting the distal end of the
maxillary last molar to the mesial border of the maxillary last molar. Palatal
breadth is the distance between the maxillary tooth rows at the trigons of Ml.
The five variables were placed into an algorithm that calculates expected
Region II length. The equation is shown in Figure 4.2 (Obtained from Spencer,
1995).
Size
Size and shape of the cranium are known to differ drastically among the
mammals included in this study. Only when shape differences are teased apart
from size differences can any meaningful comparison between these groups be
52


/

= arclan
A -
D
/ >
p.^JpJ-p,
o
Ps
= arctan
l^+cj
Pt =
Pi'
sHPl),
Pi *!?<,-Pi
Effective Region II Length = P7 + P4
Fig. 4.2 Equation for estimating the Predicted length of Region II.
53


made (Spencer, 1995). The calculation of the ratios of each dimension divided by
the geometric mean of the cranium allows comparisons of masticatory system
configurations between groups exhibiting different cranial sizes (Darroch and
Mosimann, 1985). These size adjusted shape variables indicate relative
proportions among the groups in this study.
In this study the geometric mean involves four distances within the facial
skeleton in order to get an accurate representation of overall cranial size. This
serves as a size summary by combining multiple size dimensions into a single
value. The equation for the geometric mean is:
GM = (D!*D2...Dn)(1/N)
where D = distance value andN = number of distance values included in the
summary of size.
1 Distances used to assess the geometric mean were chosen as representative
of overall masticatory system size as this is the system of concern to this study.
The distances used are bicondylar breadth, palatal breadth, temporal foramen
length, and molar row length. The landmarks used for measuring bicondylar
breadth, palatal breadth, and molar row length have been defined above.
Temporal foramen length is defined as the distance between the right central
articular eminence landmark and the inferior edge of the zygomatic arch.
An allometric analysis was also done to assess how changes in shape with
size affect the masticatory system. This is necessary because such a wide range
54


of sizes are being sampled. The geometric mean was also used for allometric
analyses.
55


CHAPTER 5
RESULTS
The parameters measured for this study are shown in Tables 5.1 5.3. All
values are represented as sex-specific means and standard deviations (in mm) of
all individuals. Table 5.1 displays mean values for molar row length, the length
of the postcanine dentition, and the length of the molars to the incisors. Table 5.2
lists mean values for the parameters used to calculate predicted Region II length.
These parameters include: bicondylar breadth, palatal breadth, height ofTMJ,
distance from the TMJ to the last molar. Table 5.3 reports the geometric mean, as
well as one additional variable used in this calculation. The three other variables
used in the calculation of the geometric mean, bicondylar breadth, height of the
TMJ, and molar length, were reported in Tables 5.1 and 5.2.
Figure 5.1 shows a box plot comparing postcanine dimensions to predicted
Region II length. These dimensions include the observed length of the molar
dentition, which is expected to be shorter than the predicted length of Region II,
and the observed distance between the most mesial premolar to the most distal
molar. Also included in the plot is the calculated predicted length of Region II.
56


Table 5.1 Mean (x) and standard deviation (sd) of tooth row length. (Molar length =
length of all molars; M-P length = the last molar to the first premolar; M-I length =
last molar to first incisor)
Taxon Sex n Molar Length M-P Length M-I Length
Primates x (mm) sd x (mm) sd x (mm) sd
Cacajao. culvus F 1 5.17 0 9.84 0 27.03 0
Cercoplthecus diana F 1 7.32 . 0 12.82 0 27.85 0
Pongo pygmaeus F 1 23.8 0 39.88 0 75.05 0
Alouaita palliata F 2 32.9 1.27 64.81 5.54 93.14 0
Aotus lemurinus F 1 9.55 0 21.75 0 39.34 0
Gorilla gorilla F 2 33.1 2.68 55.37 0.14 101.2 2.98
Cebus capucbinus F 3 8.64 0.23 19.76 0.41 34.79 1.18
Nycticebus coucang F 1 7.01 0 16.52 0 -
Ateles geoffroyi F 3 10.28 0.48 21.9 0.66 36.73 1.2
Calllthrix argentata F 1 4.09 0 7.36 0 14.66 0
Galago demldoff ? 1 3.76 0 6.93 0 - -
Saguinus geoffroyi M 2 4.61 0.13 8.58 0.12 17.45 0.44
F 2 4.51 0 8.64 0.22 17.39 0.19
Pithecia plthecia F 2 6.88 0.67 14.83 0.81 27.49 1.58
Carnivores
Lynx canadensis M 1 7.81 0 18.47 0 47.39 0
F 1 7.15 0 18.03 0 42.68 0
? 1 7.77 0 17.48 0 46.05 0
57


Table 5.1 (cont.)
Taxon Sex n Molar Length M P Length M 1 Length
Carnivores (Cont.) x (mm) sd x (mm) sd x (mm) sd
Lynx rufus M 4 10.46 1.73 20.5 1.81 47.53 1.55
F 4 9.21 0.83 19.37 1.57 45.03 2.38
? 2 9.25 1.25 19.14 0.33 45.13 0.49
Puma concolor M 2 14.99 0.77 33.37 2.31 62.1 3.53
F 6 17.09 * 1.8 37.56 1.68 66.75 2.14
? 2 15.34 0.77 32.18 2.5 59.23 3.78
Procyon lotor M 9 7.75 0.59 32.8 1.19 52.13 2.04
F 1 7.82 0 32.26 0 51.07 0
Ursus arctos M 1 48.98 0 66.32 0 154.1 0
? 1 31.23 0 58.14 0 125.79 0
Ursus americanus M 4 23.72 1.29 60.85 4.46 100.86 7.56
F 1 22.64 0 62.94 0 107.34 0
7 4 27.37 2.81 65.64 3.10 105.97 3.40
Canis latrans M 6 10 0.47 64.72 3.53 96.23 5.77
F 6 9.54 0.55 62.13 3.9 93 4.87
Canis lupis M 8 13.24 1.42 78.79 5.33 121.9 8.94
F 1 12.26 0 75.72 0 116.7 0
Vulpes vulpes M 7 7.99 2.55 45.42 4.30 69.14 4.94
F 4 * 8.81 2.57 44.74 4.50 68.57 4.97
Vulpes velox M 6 5.81 0.44 40.68 2.13 59.95 3.36
F 5 6.46 0.09 41.67 2.81 60.80 3.73
Urocyon M 5 6.95 0.45 39.59 1.23 58.17 3.77
clnereoargenteus
? 2 6.93 0.07 35.84 2.01 54.51 2.11
58


Table 5.1 (cont.)
Taxon Sex n Molar Length M P Length M 1 Length
Artlodactyls x (mm) sd x (mm) sd x (mm) sd
Cervus elephus M 1 57.38 0 121.53 0 - -
F 1 63.20 0 118.86 0 - -
Odocolleus vlrginlanus M 1 33.52 0 61.59 0 - -
F 2 30.31 . 0.09 67.27 1.65 -
Odocoileus hemionus F 2 35.06 5.66 69.76 3.15
Mazama americana F 2 21.91 2.11 47.25 7.24

Perissodactyls
Equus equus ? 3 45.97 2.54 106.89 5.83 232.17 1.77
Equus greyvi ? 2 46.05 2.00 111.45 6.17 228.14 2.47
Tapims terrestrius F 1 45.59 0 120.82 0 217.59 0
? 1 45.93 0 113.7 0 - -
59


Table 5.2 Mean (x) and standard deviation (sd) of variables used to calculate
Predicted Region II length.
Taxon Sex n Blcondylar Breadth Palatal B readth Height of TMJ
Primates x (mm) sd x (mm) sd x (mm) sd
Cacajao culvus F 1 31.41 0 18.81 0 13.8 0
Cercopithecus dlana F 1 30.14 0 19.69 0 5.5 0
Pongo pygmaeus F 1 84.01 0 51.64 0 27.7 0
Alouatta palliata F 2 128.88 0.91 79.79 3.44 84.8 1.7
Aotus lemurinus F 1 51.83 0 29.56 29.56 7.8 0
Gorilla gorilla F 2 107.86 0.36 63.51 0.25 55.45 9.4
Cebus capuchinus F 3 44.27 3.7 26.29 0.39 6.13 2.15
Nyctlcebus coucang F 1 28.92 0 17.57 0 1.9 0
Ateles geoffroyi F 3 46.65 3.64 26.06 2.03 12.6 2.55
Callithrix argentata F 1 20.29 0 11.48 0 2.76 0
Galago demidoff ? 1 13.74 0 9.02 0 2.65 0
Sagulnus geoffroyi M 2 23.26 0.44 13.74 0.29 3.52 1.65
F 2 23.47 0.44 13.94 0.58 5.31 1.44
Pitheda pitheda F 2 34.91 1.67 17.94 0.33 8.75 3.04
Carnivores
Lynx canadensis M 1 65.76 0 44.83 0 2.2 0
F 1 53.33 0 39.8 0 0.5 0
? 1. 67.24 0 43.61 .0 1 0
Lynx rufus M 4 65.49 3.76 39.32 1.5 1.28 0.92
F 4 64.81 3.13 40.62 4.11 0.8 0.63
? 2 60.83 2.95 38.15 2.01 0.6 0.14
60'


Table 5.2 (cont.)
Taxon Sex n Bicondylar Breadth Palatal 8 readth Height of TMJ
Carnivores (cont.) x (mm) sd x (mm) sd x (mm) sd
Puma concolor M 2 82.78 7.81 54.81 3.97 -0.15 0.21
F 6 87.74 3.94 57.23 2.84 -2.12 0.58
? 2 79.42 4.38 51.86 3.27 -6.55 0.07
Procyon lotor M 9 52.51 1.72 1 33.58 1.16 1.11 1.06
F 1 54.29 0 34.29 0 1.4 0
Ursus arvtos M 1 156.48 0 81.89 0 -18.9 0
? 1 107.31 0 59.36 0 -16.9 0
Ursus americanus M 4 109.43 6.34 52.61 3.54 -8.53 21.75
F 1 115.57 0 52.06 0 -24.2 0
? 4 109.13 5.73 53.82 0.72 5.6 10.87
Canis latrans M 6 64.68 ?-12 44.41 3.02 11.85 3.26
F 6 63.86 4.5 43.53 1.12 11.6 3.08
Canis lupis M 8 89.41 5.93 68.24 4.6 7.39 13.62
F 1 80.74 0 62.17 0 2.2 0
Vulpes vulpes M 7 48.91 3.7 32.02 2.68 0.03 1.71
F 4 46.76 2.63 32.33 2.38 0.58 2.92
Vulpes velox M 6 41.98 1.99 28.98 1.55 1.04 0.5
F 5 43.87 2.21 29.81 1.03 1.75 1.47
Urocyon cinereoargenteus M 5 45.27 1.95 28.18 0.92 6.8 1.61
? 2 41.48 0.64 27.31 1.07 5.3 0.16
Artiodactyls
Cervus elaphus M 1 112.19 0 99.36 0 34 0
F 1 101.69 0 94.92 0 20.4 0
Odocoileus vlrglnlanus M 1 68.74 0 58.8 0 27.5 0
F' 2 71.89 2.6 59.13 5.15 22.45 3.75
61
I


Table 5.2 (cont)
Taxon Sex n Blcondylar Breadth Palatal Breadth Height of TMJ
Artlodactyls (cont.) x (mm) sd x (mm) sd x (mm) sd
Odocoileus hemlonus F 2 77.62 4.75 68.43 4.01 25.5 2.83
Mazama americana F 2 54.15 2.45 9 52.46 3.68 21.1 6.22
Perissodactyls
Equus equus ? 3 105.66 8.33 82.33 3.14 89.65 2.61
Equus greyvl ? 2 103.71 3.56 83.35 2.0 81.8 4.38
Tapirus terrestrlus F 1 123.08 0 89.12 0 51.3 0
? 1 119.88 0 86.54 0 57.1 0
Taxon Sex n Molar to TMJ Est. Reg. II Length
Primates x (mm) sd x (mm) sd
Cacajao culvus F 1 21.15 0 14.13 0
Cercopithecus dlana F 1 20.73 0 14.18 0
Pongo pygmaeus F 1 49.09 0 33.18 0
Alouatta palliate F 2 96.32 2.6 68.87 0.69
Aotus lemurinus F 1 23.64 0 14.27 0
Gorilla gorilla F 2 55.68 8.34 38.55 5.86
Cebus capuchlnus F 3 23.34 3.13 14.47 1.1
Nyctlcebus coucang F 1 17.52 0 10.85 0
Ateles geoffroyl F 3 28.11 3.07 16.95 1.72
Calllthrix argentata F 1 11.43 0 6.74 0
Galago demidoff ? 1 7.55 0 5.26 0
62


Table 5.2 (cont.)
Taxon Sex n Molar to TMJ Est. Reg. II Length
Primates (cont.) x (mm) 9d x (mm) sd
Saguinus geoffroyl M 2 12.76 0.43 7.9 0.41
F 2 12.76 0.15 8.14 0.25
Pitheda pithecla F 2 20.5 0.5 11.33 0.35
Carnivores
Lynx canadensis M 1 38.57 ' 0 26.56 0
F 1 31.7 0 23.72 0
? 1 37.1 0 24.18 0
Lynx rufus M 4 32.32 2.75 21.71 2.48
F 4 34.16 3.06 20.4 1.8
7 2 30.82 0.84 18.32 0.48
Puma concolor M 2 52.25 6.89 37.3 0
F 6 48.47 3.14 31.86 2.18
? 2 47.39 0.87 31.7 0.81
Procyon lotor M 9 31.41 2.76 20.0 0.88
F 1 29.38 0 18,71 0
Ursus arctos M 1 83.94 0 45.67 0
? 1 79.64 0 45.7 0
Ursus americanus M 4 70.2 6 35.41 2.93
F 1 77.83 0 36.98 0
? 4 71.21 5.31 35.99 1.2
Canis latrans M 6 45.2 2.44 33.15 2.58
F 6 46.52 2.55 33.17 1.9
Canls lupis M 8 60.88 7.37 48.13 7.2
F 1 56.57 0 44.09 0
Vulpes vulpes M 7 30.81 1.16 20.3 0.55
F 4 28.95 1.24 20.23 0.94
Vulpes velox M 6 25.95 0.9 18.04 0.62
F 5 25.85 1.22 17.79 0.82
63


Table 5.2 (cont)
Taxon Sex n Molar to TMJ Est. Reg. II Length
Carnivores (cont.) x (mm) sd x (mm) sd
Urocyon clnereoargenteus M 5 30.54 3.28 19.75 2.01
? 2 29.02 0.63 19.74 1.51
Artlodactyls f
Cervus elaphus M 1 93.84 0 88.43 0
F 1 71.75 0 70.34 0
Odocolleus M 1 54.94 0 51.15 0
virglnianus F 2 56.65 3.71 49.91 6.13
Odocoileus hemionus F 2 57.37 10.12 54.23 1.96
Mazama americana F 2 43.31 8.32 45.66 10.25
Perissodactyls
Equus equus ? 3 108.28 4.94 96.85 3.11
Equus greyvi ? 2 105.93 2.8 96.75 0.63
Taplrus terrestrlus F 1 66.88 0 54.99 0
? 1 74.52 0 61.07 0
64


Table 5.3 Mean (x) and standard deviation (sd) of the geometric mean and one
variable used to calculate it The other variables used in this calculation are listed in
Tables 5.1 and 5.2.
Taxon Sex n Geometric Mean Temporal Leno Foramen th
Primates x (mm) sd x (mm) sd
Cacajao culvus F 1 15.60 0 19.38 0
Cercoplthecus diana F 1 16.80 t 0 18.35 0
Pongo pygmaeus F 1 45.02 0 39.84 0
Alouatta palliata F 2 72.11 1.71 79.95 0.49
Aotus lemurinus F 1 20.8 0 12.8 0
Gorilla gorilla F 2 59.51 2.29 55.42 3.63
Cebus capucbinus F 3 22.27 0.94 24.47 1.24
Nycticebus coucang F 1 15.85 0 17.73 0
Ateles geoffroyi F 3 22.94 1.47 22.16 1.39
Callithrix argentata F. 1 9.76 0 9.53 0
Galago demldoff ? 1 7.51 0 6,84 0
Saguinus geoffroyi M 2 11.32 0.08 11.16 0.21
Saguinus geoffroyi F 2 11.11 0.18 10.33 0.04
Pithecia plthecia F 2 16.47 0.76 17.10 0.36
Carnivores
Lynx canadensis M 1 28.62 0 29.16 0
F 1 24.98 0 28.19 0
? 1 27.73 0 23.6 0
Lynx rufus M 4 29.18 3.07 30.89 1.18
F 4 29.30 0.72 26.10 2.08
? 2 26.93 2.24 25.66 0.88
65


Table 5.3 (Cont.)
Taxon Sex n Geometric Mean Temporal Len Foramen qth
Carnivores (cont.) x (mm) sd x (mm) sd
Puma concolor M 2 41.87 2.34 45.39 4.88
F 6 43.61 1.19 42.44 2.08
? 2 40.24 1.68 41.55 0.06
Procyon lotor M 9 25:03 0.63 29.48 2.80
F 1 26.22 0 27.24 0
Ursus arctos M 1 85.20 0 83.95 0
? 1 62.05 0 74.5 0
Ursus americanus M 4 54.20 3.17 63.35 5.93
F 1 55.97 0 72.04 0
? 4 56.92 0.01 65.69 3.78
Canls latrans M 6 33.61 1.14 42.63 3.57
F 6 32.01 1.05 41.53 1.92
Cants lupis M 8 45.97 2.75 55.71 5.64
F 1 42.57 0 53.34 0
Vulpes vulpes M 7 24.62 2.99 29.99 1.75
F 4 24.72 2.86 28.54 1.52
Vulpes velox M 6 20.48 0.82 24.96 0.70
F 5 21.38 0.62 24.78 1.50
U. cinereoargenteus M 5 22.38 0.97 28.35 2.55
? 2 21.37 0.55 26.61 1.82
66


Table 5.3 (cont.)
Taxon Sex n Geometric Mean Temporal Foramen Length
Artiodactyls x (mm) sd x (mm) sd
Cervus elaphus M 1 87.32 0 90.9 0
F 1 85.38 0 87.02 0
Odocoileus M 1 50.77 0 49.03 0
virglnlanus
F 2 51.53 1.05 54.88 2.14
Odocoileus hemionus F 2 55.88 2.05 52.73 0.66
Mazama americana F 2 41.27 3.47 46.68 5.78
Perissodactyls
Equus equus ? 3 67.73 1.90 52.96 5.50
Equus greyvi ? 2 68.26 3.14 54.59 4.49
Taplrus terrestrius F 1 75.89 0 66.32 0
? 1 75.49 0 68.15 0
67


In order to more easily understand these graphs, the data were standardized
against molar row length. This has the effect of reducing differences in size and
shape between the different taxa.
Figure 5.1 shows that the molar length of all species fit well within the
predicted length of Region II. Hypothesis 1, which states the molars should fit
within the predicted length of Region H, is therefore accepted.
Many species also have premolars that fall within the predicted length of
Region II. This is not surprising given the wide range of premolar function that is
represented across mammals. However, what is unusual is that those species with
the typical grinding dentition do not necessarily have premolars that fit into this
region. For example, Cervus elaphus has premolars that fall well outside of
Region II, whereas, Mazama americana's premolars all fall within this region. In
addition, a primate, Saguinus geoffroyi, has premolars that fall well within the
region. It appears that there is no apparent pattern in regard to premolar length
and the predicted length of Region n. This may be due to the wide variety of
functions that premolars serve. Perhaps there is too much variation within the
function of this type of tooth across mammals to be able to predict its location
relative to Region II length.
68


T. terrestrius
E. greyvl
E. equus
0. vlrglnlanus
O. hemlonus
M. amerlcana
C. elaphus
V. vulpes
V. velox
U. cinereoargenteus
U. arctos
U. americanus
P. lotor
P. concolor
L rufus
L. canadensis
C. lupis
C. latrans
S. geoffroyi
P. pygmaeus
P. pithecia
N. coucang
G. gorilla
G. demidoff
C. diana
C. capuchinus
C. calvus
C. argentata
A. palliata
A. lemudnus
A. geoffroyi
O1 234567B9 10
Figure 5.1 Plot comparing postcanine dimensions to predicted Region II length. All distances have
been standardized against molar length. Therefore, the molar dentition fills the distance between 0
and 1 with the first molar being located at 1. The black boxes represent the anterior end of Region
II. Region II, therefore, extends from 0 to this anterior position. The white boxes represent the
anterior end of the premolars, therefore, the total length of the postcanine dentition extends from 0
to this anterior position.
69


Fused vs. Unfused
Mammals with an unfused symphysis are expected to have a theoretically
shorter Region II length due to a laterally located muscle resultant force.
According to Hypothesis 2, it is expected that the ratio between predicted
Region II length and observed molar length will be higher in mammals with an
unfused symphysis because of their theoretically shorter Region II length. A t-
test between predicted Region II length to observed molar length by species with
a fused or unfused symphysis shows that the means for these two groups are
significantly different. The mean for the fused group is 1.75 with a standard
deviation of 0.35 and 2.54 for the unfused group with a standard deviation of 0.81
(t = -5.13; p < 0.01). This is also represented graphically in Fig. 5.2. This plot
shows that mammals with an unfused symphysis tend to have higher predicted
Region II length/observed molar length ratios. Hypothesis 2 is accepted.
Allometric Analysis
An allometric analysis was performed on the variables used to calculate the
predicted length of Region II. It is important to assess how changes in shape with
size affect these variables because of the wide range of cranial sizes represented
in this sample. Figure 5.3 displays bivariate plots of log-transformed data
showing the relationship between the geometric mean, bicondylar breadth, palatal
70


Predicted/Observed Molar Length
3.75
3.5
3.25 1
3
2.75
2.5
2.25
1.75 -
1.5 -
1.25
Fused
Unfused
Figure 5.2 Box plot representing the means of molar length divided by
the geometric mean for mammals with a fuse and unfused symphysis.
These plots show that mammals with an unfused symphysis have a higher
predicted to observed molar length ratio.
71


breadth, height of the TMJ, and molar row length. Table 5.5 lists the slope and Y-
intercept for these variables.
Molar length scales with positive allometry. Its 95% confidence intervals
range from 1.19-1.36. This means that as cranial size increases molar length
increases at an even greater rate. This is surprising considering that increasing the
length of the molars decreases the likelihood that they will fit into Region II.
Other variables must be configured in order to increase the length of this region.
Bicondylar breadth scales with negative allometry. Their 95% confidence
intervals range from 0.81-0.89 and 0.87-0.96, respectively. As cranial size
increases bicondylar breadth does not increase as rapidly. Decreasing the length
between the condyles has the effect of increasing the length of Region II.
TMJ height scales with strong positive allometry. Its 95% confidence
intervals range from 2.35-3.52. However, there is only a weak correlation
between height of the TMJ and cranial size (r2 = 0.62). This is expected
considering that some species with very large cranial sizes, such as horses, have
very tall TMJs, whereas, other species with large crania have very low TMJs,
such as bears. Height of the TMJ is extremely variable within mammals.
The anteroposterior position of the MRF is another variable that affects
the distribution of Region II. The A-P position of the MRF is assumed to be
located at the distal end of the molar tooth row, therefore, the length from the
TMJ to the last molar represents this distance. This variable scales isometrically
72


Figure 5.3 Bivariate plots of log-transformed data showing the relationship between
cranial size (represented by the geometric meap) and other masticatory variables
relevant to the distribution of Region II (+ indicate species without symphyseal
fusion, indicate species with fusion), See Table 5.8 for regression parameters.
73


tn (Height of TMJ) ln (Palatal Breadth)
Figure 5,3 (cont.)
74


Figure 5,3 (cont.)
75


Table 5.4 Variables regressed on the geometric mean to assess allometric
relationships.
Variable (Regressed on GM) Slope Y-intercept 95% Confidence intervals for Slope
Molar Length 0.88 1.28 -0.84 1.19-1.36
Bicondylar Breadth 0.93 0.85 0.51 0.81-0.89
Palatal Breadth 0.94 0.92 0.23 0.88 0.96
Height of TMJ 0.62 2.93 -3.85 2.35-3.52
A-P Position of MRF 0.93 0.99 0.10 0.95-1.04
76


with the geometric mean; its 95% confidence intervals include a slope of 1. The
length from the last molar to the TMJ and cranial size increase at the same rate.
Palatal breadth also scales very close to isometry. Its 95% confidence
intervals range from 0.88 to 0.96. As cranial size increases, palatal breadth
increases at about the same rate.
These allometric analyses have allowed us to look further into the data and
explore how these variables change in relation to cranial size. This helps us to
understand the relationship between cranial size and the masticatory system
variables that are responsible for the distribution of Region II.
77


CHAPTER 6
DISCUSSION
The goal of this study is to test the constrained model on a wide variety of
mammals in order to assess systematic differences in craniofacial configuration
that may be the result of mandibular symphyseal fusion. Of particular concern is
the distribution of Region II. This region should envelope the grinding dentition
because this is where highest magnitude bite forces are produced and these are the
teeth most suitable for these forces. The following is a summary of my
predictions:
> 1. Molar length will be shorter than the predicted length of
Region II for both groups of mammals.
> 2. The ratio between predicted Region II length and observed
molar length is predicted to be higher in mammals with an
unfused symphysis because of the theoretically shorter Region II
length.
The following is a summary of the results obtained from this study:
78


> 1. The molar row of all species fit well within the predicted length
of Region II. Hypothesis 1 is accepted.
> 2. The ratio between predicted Region II length and observed
molar length is higher in mammals with an unfused symphysis.
Hypothesis 2 is accepted.
The results from this study support the general applicability of the
constrained model for both mammals with a fused and unfused symphysis. This
model is able to explain the interactions between some craniofacial variables of a
broad range of mammals including primates, carnivores, perissodactyls, and
artiodactyls. This is in accordance with results from previous studies.
Support for Previous Research
The main assumption of the constrained model is that the TMJ should not
be subjected to joint distraction (Greaves, 1978). This causes limitations on
masticatory system form and function. One such limitation is the distribution of
the Region II. According to the model, Region II should envelope at least all of
the molars. Spencers (1999) morphometric analysis of anthropoid cranial
configuration provides evidence in support of this. Anthropoids appear to exhibit
craniofacial form that is consistent with selection against TMJ distraction by
having the parameters that influence the distribution of the region be configured
79


in such a way so that at least all molars are maintained within this region. This is
consistent with results from the present study. Results from the present study also
indicate that the molars as well as some of the premolars are located within
Region II.
The goal of the present study was to look at the broad-scale applicability
of the constrained model to mammals. Not only were a wider range of taxa used
than in previous studies, this study also focused on a craniofacial feature that
results in fundamental differences in muscle recruitment patterns, the mandibular
symphysis. Previous studies involving galagos, lemurs, and dogs, all of which
have an unfused symphysis, have found that these species exhibit different
working-side to balancing-side muscle force ratios (Leibman and Kussick, 1965;
Hylander, 1979; Dessem, 1989; Hylander et. al., 1998, 2002). Those species with
an unfused symphysis recruit less balancing-side muscle force. Recruiting less
balancing-side muscle force means that the muscle resultant force cannot lie in the
midline. Having a MRF more laterally located consequently reduces the length of
Region II. Because mammals with an unfused symphysis have a theoretically
shorter Region II they are expected to have a higher ratio between predicted
Region II length and observed molar length. This was indeed found to be the
case. Mammals with an unfused symphysis appear to differ in a systematic way
from those with fusion.
80


There has only been a small number of mammalian species whose muscle
recruitment patterns have been studied (Leibman and Kussick, 1965; Hylander,
1979; Dessem, 1989; Hylander et. al., 1998,2002). The finding from this study
provides indirect evidence for the previous research on muscle recruitment
patterns. It can be inferred from this finding that all species with an unfiised
symphysis may recruit less balancing-side muscle force. This would cause them
to have a shorter Region II length which would create a higher ratio between
predicted Region II length and observed molar length as was found in this study.
Allometric Analyses
An allometric analysis has allowed us to understand how changes in size
affect the masticatory system configuration of such a wide range of mammals. It
was found that molar length and height of the TMJ scale with positive allometry,
bicondylar breadth scales with negative allometry, and palatal breadth and the A-
P position of the MRF scale isometrically.
Molar length was found to scale with positive allometry. This means that
as cranial size increases, the molars are increasing in length more rapidly than
expected for isometry. So as cranial size increases the molars are increasing at an
even greater rate.
One would expect that the molars would not increase at this greater rate
given the constraint of Region II length. According to the model, the molars
81


should fall within this region in order to avoid TMJ distraction. However, there
are other factors that act to influence this variable, such as diet.
Diet has been shown to be closely linked to body size, particularly within
primates (Fleagle, 1988). An animals teeth are what allow it to meet its
nutritional requirements. Therefore, molar size is constrained not only by the
need to avoid TMJ distraction but also by diet. This becomes even more complex
when one considers that diet can also be a function of body size.
Within primates, the natural physiological break between insectivores and
folivores occurs at 500 grams and is known as Kays threshold (Kay, 1975;
Fleagle, 1988). In general, folivorous primates have body weights that are no less
than 500 grams and insectivores tend to weigh less than this limit. This is because
as body size increases, metabolic requirements change. A larger animal actually
has relatively lower energy requirements than a smaller one. Although leaves are
generally lower in energy yield than insects or fruit, a large animal can afford this
because they need less energy per kilogram of mass than a small animal (Fleagle,
1988). Results from the allometric analysis of this study show that as cranial size
increases (and therefore body size) molar length is increasing even more rapidly.
This may be because even though a larger animal does not have as high of energy
needs it still requires larger molars in order to process vegetation which is much
tougher than either insects or fruit.
82


This idea also pertains to artiodactyls and perissodactyls. Both of these
groups have very large grinding dentition. This is thought to be an adaptation to
the tough grass material that comprises their diet (L.M. Spencer, 1995). Having
larger molars increases the surface area of the grinding dentition thereby allowing
a more efficient breakdown of the tough vegetation. Although this can explain
why molar length increases more rapidly than body size this does not explain how
the larger molars are able to fit within Region II. Other craniofacial variables
must be working to increase the length of Region H.
TMJ height may be one of those variables. It scales with positive
allometry meaning that it also increases at an even greater rate that cranial size.
Many of the larger animals such as horses, sheep, and deer have very high TMJs.
This has the effect of orienting Region II more anteriorly which increases its
length.
Bicondylar breadth also seems to be working to increase the length of
Region II as cranial size increases. This is because a decrease in bicondylar
breadth increases the length of this region. This parameter scales with negative
allometry; it does not increase as rapidly as cranial size. Therefore, having
bicondylar breadth not increase as rapidly as cranial size may help the molars to
be maintained within Region II even as cranial size increases.
Overall it appears that as cranial size increases across mammals, there are
variables that are configured in such a way to increase the length of Region n.
83


This allows the molars to be maintained within this region even if they are
increasing at a rate greater than cranial size.
Biomechanical Implications for Diet
Spencer (1999) proposes the idea that morphological patterns within the
masticatory system may stem from the selective trade-off between increasing bite
force magnitudes and avoiding joint distraction. Joint distraction is unavoidable
when biting occurs in Region III (Greaves, 1978). Selection should favor a
morphology that does not allow teeth to be located within this region. This idea
has been supported by the present study and the previous research discussed
above. The molars were found to lie only within Region II. However, some
species which require high magnitude bite forces to be produced on either the
incisors or the canines would benefit from having the teeth located more
posteriorly. A more posterior position for the dentition and a relatively anterior
position of the superficial masseter and anterior temporalis muscles enable greater
force production in Region I (the region where the incisors and canines are
expected to be located according to the constrained model) (Spencer and Demes,
1993; Spencer, 1999). This could potentially cause some of the postcanine
dentition to be moved back into Region III.
Some groups that may benefit from greater force production on their
anterior dentition include callitrichids, some pitheciines, Cebus, and some of the
84


carnivores (Spencer, 1999). These groups are specialized for intensive force
production on either the incisors or the canines. Carnivores require especially
high levels of force production on their canines as these are the teeth that are
involved in capturing and killing their prey. The TMJ of this group is also
particularly well-suited to capturing prey. It is locked into the glenoid fossa so
that it can withstand the high magnitude forces that are inevitable as the prey
struggles to free itself from the grip of the predator. This is likely an adaptation to
limit joint distraction.
Selection can limit teeth being located within Region HI through changes
in the configuration of other variables of the masticatory system. Some changes
that would be beneficial include decreasing bicondylar breadth or increasing
palatal breadth. Increasing the height of the TMJ would also be beneficial. All of
these have the effect of increasing the length of Region II which may enable the
molars to be maintained with this region.
The avoidance of TMJ distraction places constraints on masticatory
system morphology. Mandibular symphyseal fusion is another constraint. Fusion
constrains the location of the MRF because it allows more balancing-side muscle
force to be recruited. This has the potential effect of placing the MRF in the
midline. However, the MRF must move laterally when biting in Region II in
order to be maintained within the triangle of support (Spencer and Demes, 1993;
Spencer, 1995,1998,1999). A reduction in balancing-side muscle activity
85


enables this to happen and is necessary to avoid TMJ distraction. Although
having symphyseal fusion allows more balancing-side muscle force to be
recruited, the constrained model limits this in order to maintain the resultant force
within the triangle of support.
Maintaining an equal balancing-side to working-side muscle force ratio
(which would place the MRF in the midline) is acceptable within this model when
biting in Region I. This is because Region I envelopes a midline MRF. What is
interesting, however, is that none of the carnivores, which require high magnitude
force production on their anterior dentition, have a fused symphysis. It is
expected that this group would benefit from having a fused symphysis in order to
recruit more balancing-side muscle activity.
One possible explanation can be approached from a behavioral standpoint.
When a carnivore captures its prey in its jaws it may be biting equally with all of
its anterior dentition. This would make both sides of the jaw the working side and
may allow the predator to maximize force production. They, therefore, do not
need to have symphyseal fusion because they are already maximizing the amount
of force that can be produced. Further behavioral research would serve to clarify
this issue.
Some species which need high magnitude forces produced on their molars,
such as the artiodactyls, also do not have a fused symphysis (Greaves, 1978). It is
possible that selection has favored another aspect of their craniofacial morphology
86


that enables them to efficiently process their food. One such morphology could
be their large selenodont grinding dentition. Having this larger surface area
allows them to more efficiently process the tough vegetation that is characteristic
of their diet.
Colobines, which have a fused mandibular symphysis, also have a diet that
consists of tough vegetable matter (Fleagle, 1988). Perhaps having a fused
symphysis enables them to generate a higher magnitude muscle resultant force on
their molars than the artiodactyls, which allows them to break down their food
material in just as efficient a manner. If this is the case then symphyseal fusion
would be highly advantageous for this group.
This issue is not a simple one due to the numerous confounding variables
that are possible within not only the masticatory system but also the digestive
system. Colobines also have a gut morphology that is specialized to break down
vegetation, including large complex stomachs (Fleagle, 1988). It is not only the
molars that work to break down the food, but their specialized stomachs as well.
The issue of symphyseal fusion and its impact on diet is extremely
complex. There are numerous variables within not only the masticatory system
but the body as a whole that work together to produce an animal that is finely
adapted to its diet and environment. Further research is needed before any
definitive statements can be made regarding symphyseal fusion and diet.
87


Conclusion
Overall it appears that mammals with a fused symphysis do differ in
systematic ways from those with an unfrised symphysis. Both groups have molars
that fit within the estimated length of Region II. However, the fact that mammals
with an unfused symphysis have higher predicted Region II length to observed
molar length ratio shows that these groups differ in craniofacial configuration that
is most likely the result of different working- to balancing-side muscle force ratios
that are in turn the result of differing symphyseal morphologies.
This has important ramifications for the evolution of symphyseal fusion.
Variables within the masticatory system interact with one another to produce a
unique configuration. However, evolution can only be so creative due to the
constraints of these variables having to interact with one anther in order to
maintain the molars within Region II. This is in addition to numerous other
constraints that can be imposed upon this system.
The molars should always be maintained within Region II in order to
maximize masticatory force production. Extinct species both with and without
fusion, should have craniofacial variables, such as bicondylar and palatal breadth,
height of the TMJ, and the A-P position of the MRF that also work in conjunction
to increase the length of Region II. Looking to see if these features are correlated
with early symphyseal fusion would help us to understand how fusion affects the
masticatory system as a whole and why it came to be in the first place.
88


Cranial form and function are extremely important issues to biological
anthropologists. Understanding craniofacial form and the changes in morphology
that have occurred through time is dependent on this biomechanical model. This
model allows us to understand why certain variables are distributed in the way
they are, such as the placement of the molars within the cranium, or the height of
the TMJ above the occlusal plane. This can be used to explain much of the
variation we see in cranial form in both extinct and extant forms, and in both
groups with and without mandibular symphyseal fusion.
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NEURAL SELFTUNING ADAPTIVE CONTROL OF NON-MINIMUM PHASE SYSTEM DEVELOPED FOR FLEXIBLE ROBOTIC ARM By Long T. Ho B.S.E.E., University of Colorado, 1992 A thesis submitted to the Faculty of the Graduate School ofthe University of Colorado in paritial fulfilment of the requirements for the degree of Master of Science Department of Electrical Engineering 1994

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This thesis for the Master of Science degree by Long Thanh Ho has been approved for the Department of Electrical Engineering by Jan T. Bialasiewicz Miloje Radenkovic Marvin Anderson 2

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Acknowledgments The author wishes to express sincere appreciation to Dr. Jan T. Bialasiewicz for his guidance and joyful support during the course of this thesis. Also, enthusiasms of Dr. Miloje Radenk.ovic and Professor Marvin Anderson as the author's thesis committee are gratefully ackowledged. Special thanks to the author's family their never ending support. 3

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Ho, Long Thanh (M.S., Electrical Engineering) Neural Self-Tuning Adaptive Control Strategies of Non-Minimum Phase System Developed for flexible Robotic Arm Thesis directed by Professor JanT. Bialasiewicz The motivation of this research came about when a neural network direct adaptive control schemes were applied to control the tip position of a flexible robotic arm. Satisfactory control performance was not attainable due to the inherent non-minimum phase characteristics of the flexible robotic arm Most of the existing neural network control algorithms are based on the direct method and exhibit very high sensitivity if not unstable closed-loop behavior: Therefore a neural self-tuning control (NSTC) algorithm has been developed and applied to this problem and showed promising results. Simulation results of the NSTC scheme and the conventional self-tuning (STR) control scheme are used to examine performance factors such as control tracking mean square error, estimation mean square error, transient response, and steady state response. The form and content of this abstract are approved. Signed _ v Jan T. Bialasiewicz 4

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NEURAL SELF TUNING ADAPTIVE CONTROL OF NON-MINIMUM PHASE SYSTEM DEVELOPED FOR FLEXIBLE ROBOTIC ARM Table of Content 1. Introduction................................................................. 9 1.2. Neural Control Survey.............................................. 12 1.2.1. Stochastic Neural Direct Semi-Adaptive Control.. .... 13 Stochastic Neural Direct Adaptive Control.. ........... 15 1.2.3. Inverse Neural Adaptive Control........................ 15 1.2.4. Feedback Error Learning and Control.................. 16 1.2.5. Inverse Dynamic Model Reference Control of a Class of Nonlinear Plants ................................ 17 1.2.6. Neural Linear State Space Control.. .................... 18 1.2.7. Neural Self-Tuning Control.. ........................... 19 2. Stochastic Neural Self-Tuning Adaptive Control Scheme........... 21 2.1. Generalized Minimum Variance Control ..... ............. .. 22 2.2. Neural System Identification.................................. 23 3. Flexible Arm Tip Position Dynamics.................................. 28 4. Empirical Studies......................................................... 35 4.1. Neural Direct Adaptive Control of Ann Hub and Tip...... 35 4.2. Neural SelfTuning Adaptive Control of Arm Tip.......... 39 5. Conclusions .............................................................. 45 5.1 Future Research.................................................. 46 5

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6 Bibliography.................................................................. 47 APPENDIX A Simulation Program....................................... 49

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7 Figures and Tables Figures: 1.1 Flexible Arm System...................................................... 10 1.2 Specialized Learning Control of HupVelocity .......................... 11 1.3 Indirect Neural Adaptive Control Scheme.............................. 11 1.4 Adaptive control general block diagram................................. 13 1.5 Direct semi-adaptive control scheme ..................................... 12 1.6 Neural Network structure................................................ 14 1.7 Direct neural adaptive controller ......................................... 14 1.8 Inverse neural control..................................................... 15 1.9 Feedback error learning and control.................................... 16 1.10 Inverse dynamic model reference neural control...................... 18 1.11 Neural linear state space control......................................... 19 1.12 Stochastic neural linear ARMA control................................ 20 2.1 Neural Network Structure................................................ 25 3.1 Pole-Zero Diagram of Flexible Arm Tip ................................. 29 3.2 Servo Motor System Components ....................................... 30 3.3 Frequency Magnitude Response of Ann Tip with Five Resonant Modes 32 3.4 Frequency Response of Open-Loop Components ..................... 33

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3.5 Frequency Response of the Aggregate filtered Open-Loop ........... 33 4.1 Neural Direct Control Scheme of Hub Velocity....................... 36 4.2 Hub Velocity Response.................................................. 37 4.3 Control Tracking MSE Response .................................. ..... 37 4.4 Unstable Response of Tip Control.. ..................................... 38 4.5 Diverging Tracking MSE of Tip Velocity.............................. 38 4.6 NSTC Scheme Block Diagram........................................... 40 4. 7 Tip Position Response ....... .... ........................................ 42 4.8 Control Performance Index J(K) of the Adaptive STR andthe NSTC ........ ...................................................................... 43 4. 9 Control Signal u(k) of the adapteve STR and the NSTC .............. 43 4.10 Identification cost Index V(k) of the Adaptive STRand the NSTC .. 44 4.11 True and Neural Network Estimated Tip Position.................... 44 Tables: 3.1 Physical Properties of Arm and Motor ............................. : .... 31 3.2 Poles and Zeros ............................................................ 32 8

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CHAPTER 1 INTRODUCTION Most existing neuro control schemes are in the form ofthe direct method,. where the neural network is trained to approximate the inverse of the plant. In the case where the plant is non-minimum phase, the inverse approximation introduces instability in the closed-loop system. Therefore, an indirect neuro control scheme is proposed to deal with non-minimum phase systems. we propose to use a neural network to identifythe plant parameters, then combine this with a minimum variance control law. The plant in this study is a single degree of freedom flexible robotic ann. Self-tuning adaptive control used for controlling unknown ARMA plants has. traditionallr been based on the minimum variance control law and a recursive identification algorithm (Astrom and Wittenmark, 1973; Clark and Gawthrop, 1979). Although the advancement in VLSI has made it more possible to implement real-time recursive algorithms but it is still computationally intensive and expensive due to the recursive nature of algorithm. On the other hand;. neural networks VLSI has been made available commercially with extreme processing capability due to its parallel With this in mind the possibility of formulating neural networks to perform functions of conventional recursive algorithms becomes important. Hence, in this thesis, the neural self tuning control (NSTC) scheme is used where the implicit identification is performed by a multilayer neural network (MNN) and the control is based on the generalized minimum variance (GMV) control.law. Neural networks have undoubtedly demonstrated its effectiveness in controlling nonlinear systems with known/unknown dynamics and uncertainties (Narendra and Parthasrathy, 1990; Levin and Narendra, 1993; Werbos.et al. 1990; Hunt et al., 1992). In addition, neural network adaptive control algorithms have also been developed for spec"ific linear system model such as the state space model (Ho et al., 91a) and the ARMA model (Ho et al., 1991b). It was shown in the simulation results that neural network controllers produced comparable results to conventional adaptive controllers. 9

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In this thesis, the performance of the NSTC is compared to the conventional adaptive STR. The flexible arm to be controlled is shown in Figure 1.1. There are two system outputs that are of interest, one is the hub angle 8h(t) and the other is the tip angle 8t(t) of the arm. The goal is to apply a neural network control scheme to control these outputs to the command signals. The neural controller will generate a control voltage signal u(t) that will feed the power amplifier in which will force current through the motor and cause the arm position to react. The dynamical transfer function of the hub angle is a linear minimum phase system in which will be shown readily controllable by a neural network. In fact, the direct adaptive neural control scheme in Figure 1.2. can be used to control the hub. This control scheme belongs to the type called speCialized learning control (Psaltis et al., 1988; Ho et al., 199lc). However, the tip of the arm, being at a different location than the actuator point, therefore making the system to be of the type non-collocated system. The effect of this dynamically is that there is a zero in the right half of the s-'plane. In other words, the transfer function of the tip angle is of the non-minimum phase type which presents itself to be very difficult to control when direct adaptive control methodologY is applied. This difficulty may be due to the controller trying to emulate the inverse dynamics of the non-minimun phase plant and results in an unstable behavior. According to simulation studies, the specialized leani.ing control algorithm diverges when applied to control the tip angle. _Most otherneural control schemes are also based on the inverse dynamics including the indirect learning method by (Psaltis et al., 1988), the feedback error learning by (Kawato et al., 1988), and the methods presented by (Narend'ra and Parthasarathy, 1990): payload tip angle ...... e h hub angle ....... motor ...... ...... ...... ...... ...... ...... ...... ...... ...... Figure 1.1. Flexible arm system 10

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aeh(k) Sh(k) Neural Network u(k) -Controller Flexible Arm -Sh(k) E Figure 1.2. specialized learning control of hub velocity tip position u(k) St(k) = y(k) ....... Tip of Flexible Robotic Arm 8 ; ...... 3-Layer Neural Network Identification ...... -!" "' 81(k) =y(k) 1\ t '" 8 1..., Generalized Minimum Variance Controller -..... )" St (k) = y*(k) desired output Figure 1.3 Indirect neural adaptive control scheme In this thesis, the neural self tuning control scheme which is based on an indirect control method (Ho et al., 1991c) to control the tip angle. This scheme is 11

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shown in Figure 1.3 where the identification is performed by the MNN and the control is performed by the generalized minimum variance (GMV) controller. The GMV control algorithm has a dynamic weighting function Q(q-1) applied to the plant control signal u(k) in the cost function to limit and condition the control energy. Thus, upon selecting the proper weighting function the controller can be input/output stable and effective in controlling the non-minimum phase plant. In section 2, the neural self tuning control (NSTC) which consists of the minimum variance control algorithm and the neural identification is presented. Section 3 covers the basic dynamics of the flexible arm tip position. Section 4 presents a comparative simulation study of the adaptive STR scheme and the NSTC scheme. Finally, section 5 gives the conclusion of the results found in this study and addresses the advantages and disadvantages of the neural control scheme used for treating linear system. 1.2 Neural Control Survey In the past five years neural network based adaptive control has been a proliferated and challenging field for researchers in the area of adaptive and nonlinear control As technology advances and more and more dynamical systems emerge with high degree of complexity in coupled and nonlinear characteristics, conventional modern and adaptive control techniques are showing to be less and less effective in achieving demanding control performance .. This is partly due to the fact that many of these systems are linearized and decoupled beforehand in order to apply conventional control techniques, which consequently causes inaccuracies in representing system dynamics and therefore looses effectiveness in controlling the system. Neural network based adaptive control (NNBAC) has shown to have some unique and superior capabilities in controlling stochastic nonlinear time varying systems mainly because neural networks can model nonlinear complex processes more accurately. Furthermore, due to the inherent parallel structure of neural networks NNBAC offers the major computational load advantage because of parallel computations. Hence, implementation is more possible in cases dealing with large scales and/or high bandwidth systems where sufficiently fast sampling rate is required. In this section, we briefly present a survey of existing neural control schemes. Consider the general block diagram of an adaptive control scheme shown in Figure 1.4. Now, an adaptive control scheme may assume no a priori knowledge of the plant, but an effective and prudent adaptive control scheme should utilize and exercise all the a priori knowledge that is available. Some of the early neural control 12

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schemes such as the inverse indirect learning and the specialized learning were impressive because these schemes required very little a priori information about the nonlinear plant and treated it like a "black box". Input u(k) Disturbance Plant Identifier Output y(k) Figure 1.4. Adaptive control general block diagram 1.2.1 STOCHASTIC NEURAL. DIRECT SEMI-ADAPTIVE CONTROL Consider the first scheme called the stochastic neural direct semi-adaptive control shown in Figure 1.5 (Ho et al., 1991c). This scheme is the stochastic weighted version of the specialized learning (Psaltis et al., 1988) and is formulated with the well known weighted optimal control cost function J(k) = t E{ [y(k)-y*(k)]'Q[y(k)-y*(k)] + u(k)'Ru(k)} ( 1.1) 13

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()y/()u y*(k) controller desired output Figure 1.5. Direct semi-adaptive control scheme This control scheme is based on the nonlinear stochastic state space model x(k+ 1) = f[x(k)] + B(k)u(k) + w(k) y(k) = C(k)x(k) + v(k) (1.2) The a priori infonnation required for this scheme is.the input/output dynamic matrices B and C. This is so that the plant jacobian ay(k)au(k) can be computed and used in the back propagation algorithm. Figure 1.6 shows the typical structure of a multilayered neural network. INPUT DELAY NETWORK OUTPUT Figure 1.6. Neural Network structure 14

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1.2.2. STOCHASTIC NEURAL DIRECT ADAPTIVE CONTROL This scheme is almost identical to the previous scheme (Ho et al., 199ld) except that it has no a priori information about the plant input/output dynamics. Therefore it incorporates an additio.nal neural network so that the plant jacobian can be estimated. The block diagram of this scheme is shown in Figure 1.7 where the plant was basicaily treated to be a "black box" nonlinear system with the general state space form x(k+l) = f[x(k),u(k)] + w(k) y(k) = g[x(k)] + v(k) y*(k) desired -.....--output /' aytau noise estimation error Figure 1.7. Direct neural adaptive controller 1.2.3. INVERSE NEURAL ADAPTIVE CONTROL (1.3) This is one of the first neural adaptive control schemes known as the indirect learning proposed by (Psaltis et al., 1988) and is shown is Figure 1.8. The plant is assumed to be a "black box" nonlinear system y(k) = f[y(k-1), ... y(k-n); u(k), ... u(k-m)] (1.4) 15

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Plant input estimation errore(k) Figure 1.8. Inverse neural control Here, the two neural networks at the input and ouput of the plant are identical. The network is to emulate the inverse of the plant based on optimization of the cost function J(k) = [u(k)-u(k)]'[u(k)-u(k)] (1.5) which indirectly minimizes the output tracking error [y(k)-y*(k)]. 1.2.4. FEEDBACK ERROR LEARNING AND CONTROL (FELC) This direct adaptive control method proposed by (Kawato et al. 1988) may be one of the most efficient "black box" neural control scheme as shown in Figure 1.9, the scheme utilizes a single neural network as an adaptive direct controller performing both learning and control simultaneously. error Figure 1.9. Feedback error learning and control e (k) c 16

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The neural network directly minimizes the output tracking error cost function J(k) = t[y*(k)-y(k)]'Q[y*(k)-y(k)] (1.6) and does not require any a priori infomation such as the Jacobian. 1.2.5. INVERSE DYNAMIC MODEL REFERENCE CONTROL OF A CLASS OF NONLINEAR PLANTS This approach was presented by (Narendra and Parathasarathy, 1990) addressing the issues of identification utilizing neural networks and control of nonlinear plant using inverse dynamic model reference techniques. The general diagram of this scheme is shown in Figure 1.10. The four input/output plant models addressed are Modell: n-1 y(k+l) = L aiy(k-i) + g[u(k), u(k-1), ... u(k-m+1)] i=O (1.7a) Modelll: m-1 y(k+1) = f[y(k), y(k-1), ... y(k-n+1)] + L i=O ( 1.7b) Model ill: y(k+l) = f[y(k), y(k-1), ... y(k-n+1)] + g[u(k), u(k-1), ... u(k-n+1)] (1.7c) ModelN: 17

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y(k+l) = f[y(k), y(k-1), ... y(k-n+1); u(k), u(k-1), ... u(k-n+1)] ( 1.7d) r(k) model input Inverse dynamics controller /\ A f(.), g(.) Reference model Plant Type J,JJ,IIJ,IV Neural networks identifier Qr tracking error e tk) estimation error e jk) Figure 1.10 Inverse dynamics model reference neural control The a priori information required is which of these specific model fits the plant so that identification can be performed. However, the inverse dynamic control can be accomplished provided the representation of the inverse dynamics exists. In other words, it is recognize that u(k) can be expressed in terms off(.), g(.), f-1(.) and g-1(.). 1.2.6. NEURAL LINEAR STATE SPACE CONTROL This scheme shown in Figure 1.11. (Ho and Ho, 1991 a) is used for controlling time varying linear stochastic state space plant x(k+l) = A(S,k)x(k) + B(S,k)u(k) + w(k) y(k) = C(S,k) + v(k) (1.8) where 8 is the parameter vector. The identification is performed by the neural network and the control can be selected by any modern state space control techniques, in 18

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particular, the tracking per-interval control law. This neural parameter adaptive control approach is different from the conventional adaptive control approach by the identification process. y*(k) desired output State space controller A A e, x Plant Neural network identifier estimation error e I Figure 1.11. Neural linear state space control 1.2.7. NEURAL SELF-TUNING CONTROL This scheme, shown in Figure 1.12. (Ho et al, 1991b) is similar to the state space control scheme only it is based on the ARMAR plant model B(q-1) C(q-1) y(k) q-d u(k) + A(q-1) A(q-1) (1.9) The identification is performed by the neural network arid the control can be selected by any conventional control techniques, in particular, the minimum variance control. This neural self-tuning control scheme is different from the conventional self-tuning control by the identification algorithm. 19

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y*{k) desired output Minimum Plant H(s) Neural network identifier error Figure 1.12. Stochastic neural linear ARMA control 20

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CHAPTER 2 STOCHASTIC NEURAL SELF-TUNING ADAPTIVE CONTROL (NSTC) The NSTC consists of the minimum variance control law and the neural identification algorithm. The model assumed for the plant is of ARMA input/output type having the form (2.1) where u(k), y(k), and d are systein input, output, uncertainty, and delay, respectively. A, B, and Care unknown system dynamics defined as A( -1) 1 -1 -2 -na q = + a1 q + + ... + anaq B( -1) b b -1 b -2 b -nb q = 0 + lq + 2q + ... + nbq C( -1) 1 -1 -2 -nc q = + c 1 q + c2q + ... + cncq (2.2) (2.3) (2.4) where q is the shift operator. For the above unknown plant, in Figure 1.3, the objective is to control its output to track a command signal y*(k) based on the generalized minimum variance control index (Clark and Gawthrop, 1979) J(k+d) = E{ c!>2 (k+d)} = E{ [P(q1 )y(k+d)+Q(q1 )u(k)-R(q1 )y*(k)] 2 } = E{ [(j>y(k+d)+Q(q-1 )u(k)-R(q1 )y*(k)]2} (2.5) where E is the expectation operator, c!>yCk+d) is the auxiliary output, and P,Q, and Rare the weighting dynamics which can be chosen depending on the required response characteristics. 21

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2.1. Generalized minimum variance control In this section, the generalized minimum variance self-tuning control algorithm for the above problem statement is summarized (Clark and Gawthrop, 1979). To obtain the optimal control u(k) which minimizes the performance index (2.5), the predictive auxiliary output c!>y(k+d) in terms of the system dynamics must be determined. Consider the following identity (2.1.1) where the order ofF(q-1) and G(q-1) are nf;:d-1, ng=na-1, respectively. The output prediction can be shown to have the form where and = C(q-1f1 [G(q-1)y(k) + F(q-1)B(q-1)u(k)] = C(q-lf1 [G(q-l)y(k) + E(q-l)u(k)] = 1\ (2.1.2) (2.1.3) (2.1.4) c!>yCk+d) and c!>yCk+d) are the deterministic and uncorrelated random components of c!>yCk+d). Next, substituting (2.1.2) into (2.5), there results (2.1.5) Since the second term in (2.1.5) is unpredictable random noise which is uncompensatable by the control input u(k), and the first term is a linear function of u(k), J(k+d) can be minimized by setting 22

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(2.1.6) Solving for the generalized minimum variance control (GMVC) in (2.1.6) gives R(q1 )y*(k)-$y(k+d) u(k) --------.-..J..--Q(q-1) using (2.1.3), (2.1.7) can also be written as u(k) = C(q -1 )R(q -1 )y*(k)-G(q -1 )y(k) E( q -1 )+C( q -1 )Q( q -1) (2.1. 7) (2.1.8) Remarks: Recall that E(q-1) is equal to F(q-1)B(q-1) where B(q-1) contains the zeros of the plant. Notice that having the weighting function Q(q-1) additive to E(ql) in (2.1.8) gives the designer the ability to alter the poles of the controller. Thus with a non-minimum phase plant B(q-1) shall have unstable roots and proper selection of Q( q-1) in (2.1.8) can assure the control signal u(k) to be bounded. 2.2. Neural system identification In this section, a stochastic neural identification algorithm is developed for the self-tuning control scheme in Figure 1.3. Recall the predicted auxiliary output in (2.1.3) which can also be written as cJ>y(k+d) = C(q-If1 [G(q-l)y(k) + E(q-l)u(k)] + = C(q-If1 [G(q-l)y(k) + E(q-l)u(k)] + v(k) (2.2.1) where the uncorrelated noise sequence is by v(k). Also (2.2.1) can be written as (2.2.2) c!>yCk+d) = \jl'(k)9(k)+ v(k) (2.2.3) where 23

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\jl'(k) = [y(k) ... y(k-ng); u(k) ... u(k-ne); (k+d-nc)] (2.2.7) A AA A AA A AA A 9 '(k) = [go g 1 gng; eo e 1 ene; -c 1 -c2 -cncl <2 2 8 ) The unknown parameter vector in (2.2.8) (Figure 2.1), is taken from the output of the neural network A A A 9(k) = [ til (k) t12(k) ... 9j(k) ... en.3(k)l' = [01 (k) 02(k) ... Oj(k) ... On3(k)]' (2.2.9) Where n3 is the number of neurons at the output layer. Consider the system identification cost function V(k) =!E{E'(k)A-1(k)E(k)} =! E{[tj>y(k)-$y(k)]'A-l(k)[$y(k)-$y(k)]} (2.2.10) where A(k) is a symmetric positive definite weighting matrix, and V(k) is minimized by adjusting the weights of the neural identifier. 24

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NEURAL NE1WORK Figure 2.1. Neural network structure In Figure 2.1, the weights connecting the second layer to the output layer, using the gradient search (Rumelhart and McClelland, ), can be updated as ffiij(k+l) = ffi:ij(k) (2.2.11) where () 1 1 AOlij"(k) ::: -11 { 2 E'(k)A(k)E(k)} dffiij(k) = -11 () { dffiij(k) YJ YJ = 11 A-1(k)[cj> dffiijCk) Y YJ (2.2.12) with 11 being the search step size. Consider the derivative of c!ly(k) with respect to in (2.2.12) (2.2.13) In (2.2.13) we have assumed that 9(k) 9(k-d), that is, 9 is slowly time varying with respect to the delay time d. The other partial derivative in (2.2.12) can be determined as = O(k)e d[f(Netj(k))]' dffiij(k) 1 J dNetj(k) (2.2.14) 25

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where f(.) is the sigmoidal activation function, Oi(k) is the output of the second layer, and with Netj(k) = [nell net2 ... netj ... netn3l' n2 netj(k) = LCOij(k)Oi(k) i=1 (2.2.15) where n2 is the number of neurons of the second hidden layer as shown in Figure 2.1. Also ej in (2.2.14) is defined as ej = [0 ... 0 1 0 ... 0] (2.2.16) with the j-th element in ej being 1, and other elements are 0. Thus, substituting (2.2.14) back into (2.2.12) gives (2.2.17) where 1\ (k) = d[f(Netj(k))]' dcl>y'(k) A-l(k)[q, 0J aNetj(k) ae(k) Y (2.2.18) Next, the weights connecting the first to the second layer, in Figure 2.1, can be updated by the recursive equation ronCk+ 1) = ronCk) + L\OJri(k) (2.2.19) where d 1 1 dffiri(k) = -11 { 2 E'(k)A(k)E(k)} dffiri(k) (2.2.20) Using the similar back propagation approach, (2.2.20) can be shown to result in the following form (2.2.21) where Or is the output of the first layer and 26

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df[neti(k)] Oi(k)=[Olil...Olij ... ffiin3] dneti(k) Oj(k) (2.2.22) Lastly, the weights connecting the input to the first layer, in Figure 2.1, can be updated by the recursive equation Olsr(k+ 1) = Olsr(k) + Arosr(k) (2.2.23) where a 1 1 Arosr(k) = -11 { 2 e'(k)A(k)E(k)} dOlsr(k) (2.2.24) Again, using the back propagation approach, (2.2.24) can be detennined as Arosr(k) = 'JlOr{k)Is(k) (2.2.25) where Is(k) is the input from the delay network and df(netr(k)) d[f(Neti(k))]' aNetj(k)' Or(k) = [Olrl...Wri ... rorn2] anetr(k) aNeti(k) a[f(Neti(k))]' Oj(k) (2.2.26) with Neti(k) being defined similarly as Netj(k) in (2.2.14). By adjusting the weights Olij(k), Olri(k), and Olsr(k) with the above algorithm, the unknown implicit plant's parameters can be identified and obtained at the output of the neural identifier, as shown in Figure 2.1. Once the estimate of 9 is available, PyCk+d) in (2.2.6) can be computed, and then the control signal can be generated using (2.1.7) as R(q1 u(k) ------.--L--Q(q-1) (2.2.27) 27

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CHAPTER 3 FLEXIBLE ARM TIP POSITION DYNAMICS This chapter describes the components and the control model of the flexible arm tip. A detailed discussion of the dynamics of flexible arm tip and hub can be found in (Fraser and Daniel, 1991). In order to control the flexible robotic arm shown in Figure -, .. -.-1.1, it is required that the control action produced by the control program running on a processor board is converted to a voltage by the D/A board and forms the input to the power amplifier of the motor. The otltput of the power amplifier is a motor current directly proportional to the input voltage. The motor then converts this current to a torque to drive the arm. The resulting motion of the arm is detected by the various sensors and fed back to the controller. The adaptive control algorithm design does not require the complete knowledge of the plant dynamics. However, for the purpose of simulation study, the transfer function model of the plant needs to be known. This model must incorporate not only the behavior of the flexible arm itself but also the power amplifier, the motor and the output sensors. In a servo system, the power amplifier and the sensors usually have a much higher bandwidth than that of the motor and load therefore they can be approximated as a constant. The general transfer function of the flexible arm tip is 9t(S) = KAKT fl u(s) s(s+co) i=I where the physical interpretation of the above equation is as follows: First, poles and zeros of the system is depicted in Figure 3.1 (3.1) 28

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X X .... X X resonant modes s-plane motor dynamics flexible ami minimum and non-minimun phase zeros Figure 3.1 Pole-zero diagram of flexible arm tip The above diagram shows the three constituting dynamic components of the plant which are the motor, the resonant modes of the flexible arm, and the arm non-minimum phase characteristics. The dynamics of the servo motor system is represented by the term (3.2) where Ky. is the motor torque constant, K A represents the power amplifier and sensor gain, and C0 represents the back emf and viscous damping effects know as the mechanical time constant. The motor can be seen as a series of subcomponent connected in series as shown in Figure 3.2. 29

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motor motor velocity position voltag:-1 I torq:l ' _I_ I I ._l I KA KT (s+co) s Figure 3.2 Servo motor system components Next, the the flexible arm attached to the motor shaft is describe by the term. (3.3) Here, the of (3.3) represents the set of flexible resonant modes of the aim. Each flexible mode is associated with the corresponding damping at a frequency Olj. Theoretically, there is an infinite number of flexible modes, but in practice only the sufficiently low frequency modes will be noticeable by the control system. This is because a real system is always band-limited. Therefore most of the modes are attenuated by the low-pass frequency behavior. Also, the frequency range of operation can be limited to be below-the inajor do.minant resonant mode so that oscillations will not be. present in. the system response. If higher frequency range of operation is desired, the dominate resonant modes can be notch filtered out provided their damping and frequencies Olj'S are determinable, Consider the physical properties of the flexible arm and the servo system given in Table 3.1. Based. on these parameters the transfer function was derived and measured by experiment (Fraser and l)aniel, 1991) .. Both results agreed as shown in Table 3.2. The five resonant modes occupy the frequency range from 86 rad/sec to 1445 rad/sec. The frequency response of this system was simulated and is shown in Figure 3.3. The peaks represent the resonant energy at the specific freqQencies. Also notice that the energy of the modes lessens are the frequency increases. 30

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31 Table 3.1. Ph I IYSICa r proper 1es o f arm an d motor effective beam lenath Jm) 0.386 beam thickness (mm) 0.956 .... beam width .. 0.03 mass/unit length of arm m (kg/m) 0.222 flexural rigldity of beam(NmA2) 0.426 hub moment of inertia (kg mA2) 0.00009 radius of hub (m) 0.034 Tip mass for loaded arm (ka) 0.065 tip inertia for. load arm (kg mA2) 0.000.005 continous torQue at rated speed (Nm) 0.177 pulse torque(Nm) 2.913 rated voltaae (V) 24 torQue constant (Nm/A) 0.048 total inertia (ka mA2) 0.000041 Ka*Kt 3.6 Co (rad/secl 0.16

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Table 3.2. POLES (rad/sec) ZEROS (rad/sec) Mode Expmt. Theory Expmt. Theory 1 86.1 86.9 48.4 47 2 297.6 285.3 -48.4 -47 3 603.2 60L9 4 1011.6 1065.0 5 1445.1 1658.8 20 r---:--c.----:-. -:-. -:-. -:-.:-:-. -:-. --;-----,,;--;-, -:-. -:-. -;-, -:-;-, -;-, ----:.-.,,.......,..--, .---:.--;-. --,..., ,,....,.. ----..,...--,----,--.,........,....,-,--,-, .... .. . ...... . .. '. . . . . . . . . . . o : ne<: -20 ....... ... :. .. : .. : : : ...... : ... : .. :.-: .. : ...... ... : ... : : :::::::: ::::::::: ::::::::h .. :: : : j : IT][ iii 1 ][iiiiL L[ il:tlll \ I I I ....... _! .... .. 1"[ .. : :t ...... : .... : .. : .. f r i j-:-.... : .... : .. : : i ..... \ t ... : : -100 ....... ... : .. .. : .. : ...... .... ; .. > >: ...... : .... .. : ::-:...... .. \ .... : . : 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -120 ....... ; ... .; .. : .. ; .. ;. ...... :-...; ... ; .. .;.:: .; ....... ; .... .. ; :-: .;. :.;.; ....... ... : .... ; ... ;. 100 lOt 102 Radianls 103 Figure 3.3 Frequency magnitude response of arm tip with five resonant modes 104 32

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For easy controllability it is desirable to filter out these resonance modes. Therefore, a notch filter is designed to notch out the first resonance mode and a low pass filter is used to filter out the rest of the resonance modes. Figure 3.4 shows a block diagram of the filtering process. The resulting frequency ideal response is shown in Figure 3.5. notch filter LP filter flexible arm system voltage .__ _____ __. Figure 3.4. Frequency response of open-loop components Tip position 0 Q ......... ; ..... .... : .... : ... ;.; .. ......... : ... . . o 0 o o o I 0 .g .a : : : : : : : :: : : .. : : : : : :: : : : : : : : :. :: :: : : : : :::: : : : -100 ........ ; .. ".: .... : ... : .. :. .:. : ....... .-:.... : ... : .. :.:. ;, ; .: ;_"" .... : .... ,; ".: .. . 0 0 -150 ......... ; ..... : .... : ... : .. > -:........ -:... ... : .. -:-:-" "." -:... ... 0 0 0 0 0 0 0 . . . . . . . . . . . . . . . ::\ 101 102 103 104 Radian/s Figure 3.5 Frequency response of the aggregate filtered open-loop 33

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Since we are primarily interested in learning the controllability and behavior of the non minimum phase characteristics of the plant, we can simplify the arm tip transfer function to have the form (3.4) Lastly, the non-minimum characteristics of the ann tip is describe in (3.1) and (3.4) by the numerator term. This is due to the fact that the control system sensing and actuation do not take place at the same location and therefore being a non-collocatted system. It should be mentioned that the non-minimum phase characteristics is very difficult for the neural network to control (since most neural network adaptive control schemes are based on the direct method). 34

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CHAPTER 4 EMPIRICAL STUDIES In this chapter we examine some simulation results of the direct and indirect neural control schemes for controlling the flexible arm hub and tip. We will show that the hub having a well behaved linear transfer function produced very satisfactory controlled response; We also attempted to use the direct adaptive control scheme to control the tip velocity and found unstable response even after numerous controller parameter changes. Next, the NSTC scheme in section 2 was applied to control the tip position and produced encouraging results. Lastly; the neural identifier in the NSTC algorithm is compared with the recursive least square identifier and show faster . convergent rate. The simulation program used in this study is given: in the appendix. 4.1. Neural direct adaptive control of arm bub and tip The neural direct adaptive control scheme was frrst introduced by (Psaltis et al., 1988) and was Jater reformulated for nonlinearninear state space system by (Ho et al., 1991c). We will apply this scheme, shown in Figure 4.1. to control the hub velocity of the arm. The dynamic function of the hub is a linear minimum phase system. The numerical transfer function found in (Fraser and Daniel, 1991) is s2 9 ( ) 10.2 ( 1 I ) h s 32.72 U(s) (s+0.57)(s+2000) (4.1) 35

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aeh(k) 8h(k) Neural Network u(k) -Controller Flexible Ann -8h(k) '\ E Figure 4.1. Neural direct control scheme of hub velocity where the resonant modes are assumed to be filtered out. In the simulation process, the model in ( 4.1) was first discretized and then converted to state space form x(k+1) = Ax(k) + Bu(k) (4.2) 8h(k) = Cx(k) When using this scheme (Figure 4.1.) there is a priori information that is needed and aeh(k) that is the jacobian of the plant au(k) This term was computed based on the discretized model and resulted as aeh(k) = cs du(k) (4.3) Information on the neural network algorithm is refered to (Ho et al., 1991c). Remarks: : The hub position was not suitable for this specialized learning control scheme because the jacobian turns out to be near zero. Therefore the velocity is the selected controlled variable and an additional outer control loop may be incorporated to achieve position control. This outer loop will have a velocity profile generator which resembles to a proportional controller with saturation (Franklin and Powell, 1981 ). Simulation: A smoothed square wave command was presented to the control system, after 50 iterations (about .3 seconds, sampling period was 6 ms) the hub had tracked the command signal as shown in Figure 4.2 where the solid line is the desired response and the dashed line is the actual response. 36

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./: . . 0 I ...... ....... ....... ... --. -.. -.. .. --.. -. ... --.... ---:........ '1:' .. '> . . . . . . . . . . . . . . . . 0 50 100 150 200 250 300 350 400 450 500 Iterations "' Figure 4.2. Hub velocity response: St (k) & St(k) This trackability is reflected in the mean square tracking error_ shown in Figure 4.3. Notice that the convergent time in control application is serveral orders of magnitude faster than other applications. In this case it took only 50 iterations for the 2-layer neural network to be maturely trained with initial random weights. This fast convergent time makes it very practical for real-time control irriplementation; 0.2 ;:;;;;-0.1 '-' ..... 0 0 50 100 150 200 250 300 350 400 450 500 Iterations Figure 4.3. Control tracking MSE response Next, the same scheme is applied to control the tip velocity. The numerical tip transfer function (based on the flexible arm and motor properties in Tables 3.1 and 3.2) is given in (Fraser and Daniel, 1991) as 3.6 (11 s2 ) St(S) 48.42 U(s)-s(s+0.16) (4.4). 37

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Here again, we are primarily interested in the non-minimum phase characteristics and . therefore assumed that the resonant modes are filtered out. Simulation: After numerous attempts to vary the neural network parameters, an unstable closed-loop response was prevalent as shown in Figures 4.4 and 4.5._ This is due to the fact that the neural network in Figure 4.1. trying to emulate the inverse dynamics of the plant (4.4.) and in effect produced an unstable pole behavior. Note in Figure 4.4. that the command signal is small compared to the plant diverging output response therefore it looks like a straight line. :,,._ e . . ...... . ;;.... 0 ......... : ......... : ......... ; .,, ..... : ......... .; ___ ........ ...... : ......... : .. :.... . -: 0 0 0 : : I ....._ :_\ r / _______________ : 0 50 100 150 200 250 300 350 400 450 500 Iterations . Figure 4.4. Unstable response of tip control: 8t*(k) and 8t(k) 3ooo L ! rI:I: : :: :. v : : . . . . . . . 0 : : : 0 0 : : 0 50 100 150 200 250 300 350 400 450 SOD Iterations Figure 4.5. Diverging tracking MSE of tip velocity N2 Neural Network: The 5,10,1 neural network used in this scheme consists of one input layer, one hidden layer, i;ind one output layer with the number of neurons as 5, 10, and 1, respectively. Also at the input of the neural was the desired response vector [y*(k) y*(k-1) y*(k-2) y*(k-3) y*(k-4)]T. The parameters of the sigmoidal activation function at the output node were found to be most influential on the tracking error convergentce 38

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rate. Predominantly the slope of the activation function was observed to be proportional to the convergence rate. Also the bipolar sigmoidal saturation levels of the output neuron needed to be set equal to or greater than the maximum allowable plant input. The tuning of the sigmoidal functions was done manually by trial and error, typically for linear system like that of the hub, it takes very few tweaks (around 1 or 2) before the tracking result was achieved. Auto-tuning of the sigmoidal function parameters can also be applied to obtain statiscally better results (Yamada and Yabuta, 1992; Proano, 1989). 4.2 Neural self-tuning adaptive control (NSTC) of tip position In section 4.1. we showed by simulation that the direct neural adaptive control scheme was unable to control the tip position (Figures 4.4 and 4.5). In fact, this was why the NSTC algorithm was Recall that this scheme has two distinct functions, identification and control, which are done by the neural network and the (GMV) control, respectively. The NSTC scheme is shown again in Figure 4.6. 39

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tip position u(k) 6t(k) = y(k) "Tip of Flexible Robotic Arm ,.. e +' f'e(k) ; .. _J, "'3-Layer Neural Network Identification ""' -,. ..,.... 6t(k) =y(k) t l.....r Generalized Minimum Variance Controller "' et (k) = y*(k) desired output Figure 4.6. NSTC scheme block diagram In this section we perform the simulations of two schemes which are: The adaptive STR using recursive least square identification, and the NSTC using the neural identification. This was done to performis a comparative study in order to assess the performance of the developed NSTC. Simulation: The model of the tip position is the discretized model of ( 4.4). Recall the control index defined in section 2 J(k+d) = E{ cp2 (k+d)} = E{ [P(q1 )y(k+d)+Q(q1 )u(k)-R(q1 )y*(k)] 2 } (4.5) where the weighting functions were chosen as 40

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P(q-1 )=1; Q(q-1 )=.1 +.06q-1; R(q-1)=1 (4.6) and the desired hub position 8t (k) was a step command. Beginning with Figure 4.7. shows the desired step tip response, the controlled tip response based on the adaptive STR and the tip response from the NSTC. Obviously both controllers manage to track the command signal. However, the NSTC seems to have a slower settling time. Figure 4.8. shows the converging tracking control index (2.1.5) where both schemes seem very comparable to each other. Figure 4.9. displays the comparable control energy produced by-tliese.controlieri. Note that tlie transient control energy was ...... affected by two factors: one is the initial condition of the estimated parameter vector 8o ...... ...... (which was set as 8o = [1 1 ... 1]' for both control schemes), the further 8o is away from the optimum 8 in the parameter state space, the longer the convergence of the tracking control index (2.1.5). The other factor is the selection of the input weighting function Q( q-1) which has the effect of limiting the control energy with the tradeoff of slower tracking convergence. Lastly, we compare the recursive least square identification with the neural network identification. The two identifiers estimate the parameter vector e in (2.2.5) so that the predictive output term cj}y(k+d) in (2.2.2) can be computed. Figure 4.1 0. shows the estimation cost function V (k) in (2.2.1 0) response of the RLS and the neural network. V(k) of the RLS has a slightly faster convergence than the neural network but not by a significant degree. Again, this indicates that the identification performance of the two algorithms are comparable to each other. For completeness, the time response of the true output 81(k) and the estimated output 8t(k) produced by the neural network is shown in Figure 4.11. Neural Network: The three layer neural network Nts.I5,PO used in this scheme consists of one input layer, two hidden layers, and one output layer with the number of neurons as 2, 5, 15, and PO, respectively. PO is the length of the vector defined in (2.2.8) which is (ng+ 1 )+(ne+ 1 )+nc, and is 11 for the case of the arm tip plant. The input of the neural network was a selected as constant vector Is= [1 1]' because it was desired that the output of the neural network to be correlated to the its input. The parameters of the sigmoidal activation function at the output node was found to be most influential on the tracking error convergent rate. Predominantly the slope of the activation function was observed to be proportional to the estimation convergent rate V(k). Also the bipolar sigmoidal saturation levels of the output neuron needed to be set equal to or greater than the maximum component of the parameter vector 8. The tuning 41

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of the sigmoidal functions was done manually by trial and error. Autotuning of the sigmoidal function parameters can also be applied to obtain statiscally better results (Yamada and Yabuta, 1992; Proano, 1989). However, the optimal dimension of the neural network in terms of number of layers and nodes was not known and therefore an N3 initial pick of 2,5,15,PO was used throughout the simulation. 1 0 I r I II -1111 ,u ,w Adaptive STR --NSTC 0 0.5 1 1.5 2 2.5 3 Time (sec) Figure 4.7. Tip position response: 6t (k)" & 6t(k) of the adaptive STRand the NSTC 42

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8 6 4 2 0.5 1 Adaptive STR --NSTC 1.5 Time (sec) 2 2.5 Figure 4.8. Control performance index J(k) of the adaptive STR and the NSTC 50 40 II ,, l1 30 /I I\ 20 /I I I I 10 I 0 1 -10 0 /: N I \I ,, 0.1 ' 0.2 Adaptive STR --NSTC 0.3 Time (sec) 0.4 0.5 Figure 4.9. Control signal u(k) of the adaptive STRand the NSTC 3 0.6 43

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2.5 I II\ 2 \ I I I I I 1.5 : \ I I I l r n. \ I I I 0.5 7 Adaptive STR --NSTC 0.5 1 1.5 2 2.5 Time (sec) Figure 4.10. Identification cost index V(k) of the adaptive STR and the NSTC -3 I ll -------... ---1 -1 --estimated hub position -2 actual hub position -3 -4 0 0.1 0.2 0.3 0.4 0.5 0.6 Time (sec) Figure 4.11. True and neural network estimated tip position: 9t(k) & 9t(k) 44

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CHAPTER 5 CONCLUSION The neural self-tuning control (NSTC) algorithm was developed and applied to control the tip of a flexible ann system. The dynamics of the flexible arm tip involves an unstable zero and therefore making the system non-minimum phase. Most of the existing neural adaptive control are based on the inverse dynamics and therefore would not be able to control this type of plant. The NSTC was based on an indirect control method where the identification is performed by the neural network and the control was based on the generalized minimum variance (GMV) control law. The performance of the NSTC was investigated and was compared to the adaptive STR by means of simulation. In summary, the NSTC has a very comparable performance to the adaptive STR shown by simulation results in section 4.2. Unlike other applications of neural networks where thousands of iterations were required before the network can be maturely trained, in thisapplication the neural network identification had a convergence rate comparable to that of the RLS. Another advantage of the NSTC is due to the availability of neural network VLSI and the massive parallel architecture of the neural network there will be a computation advantage over conventional recursive algorithms. This will enable .real-time implementation with faster sampling rate for system with wide bandwidth. Also another advantage of the NSTC is that because the identification is done by the neural network, it inherits the decentralize property, meaning if there is a failure in a node or connection the impact on the performance will be minimal. Whereas with the conventional digital filter a failure in one of the coefficient will have a major impact on the output. With all the above encouraging characteristics there is one disadvantage of using the neural network and that is the lack of understanding how the dimension and activation characteristics of a network is related to its accuracy and stability. These issues of the recursive algorithms have been addressed and elaborately analysed in (Kumar, 1990). 45

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. 5.1 Future research The NSTC can be modified and extended to control systems that are not only non-minimum phase but also nonlinear. This is so that the properties of neural networks can be fully exploited. A system that have the above characteristics is a two degree of freedom robotic manipulator with the second link being flexible. Most conventional adaptive control schemes rely heavily on the inverse dynamics and therefore showed great limitations with this type of system (Centinkunt and Yu, 1990). 46

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BIBLIOGRAPHY Astrom, K. and Wittenmark, B., "On self-tuning regulators", Automatica, 9, pp.185-199, 1973. Antsaklis, P., "Neural networks in control systems", IEEE Control Systems Magazine, April 1990. Bavarian, B. "Introduction to neural networks for intlligent control", IEEE Control Systems Magazine, April 1988. Centinkunt, S., and Book, W., "Performance Limitations of Joint Variable Feedback Controllers due to Structural Flexibility", IEEE Transaction on Robotics and Automation, Vol. 6, No. 13, 1990. Clark, D., and Gawthrop, P., "Self-tuning controller", in proceedings lEE, 126, pp.633-640, 1979. Franklin, F.G., and Powell, J.D., Digital Control of Dynamic Systems, Addison Wesley, Reading, Mass. 1981. Fraser, A., and Daniel, R., Perturbation Techniques for Flexible Manipulators, Kluwer Academic Publishers, Norwell, Massachusetts, 1991. Hetch:.Nielsen, R. "Theory of the backpropogation neural network" in proceedings of the International Joint Conference on Neural Networks, Washington D.C., 1989 Ho, T., Ho, H. "Stochastic state space neural adaptive control", in proceedings the Third International Conference on Advances in Communication and Control Systems, Victoria, B.C., Canada, Oct. 16-18, 1991a. Ho, H., Ho, T., Wall, E., and Bialasiewicz, J., "Stochastic neural self-tuning adaptive control", in proceedings of the Third International Conference on Advances in Communication and Control Systems, Victoria, B.C., Canada, Oct. 16-18, 1991b. Ho, T., Ho, H., Wall, E., and Bialasiewicz, J., "Stochastic Neural Direct Adaptive Control", in proceedings of the 1991 IEEE International Symposium on Intelligent Control, Arlington, Virginia, Aug. 13-15, 1991c. Ho, T., Ho, H., and Bialasiewicz, J. "Stochastic neural adaptive control for nonlinear time varying systems", in proceedings of the 1991 International Conference on Artificial Neural Networks in Engineering, St. Louis, Missouri, Nov. 10-12, 1991d. Ho, T., Ho, H., and Bialasiewicz, J. "On stochastic newton adaptive control", in proceedings of the lASTED International Symposium on Adaptive Control and Signal Processing, New York, Oct. 10-12, 1990. 47

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Hunt K., Sbarbaro, D., Zbikkowski, R., and Gawthrop, P., "Neural Networks for Control SystemA survey", Automatica, Vo. 28, No.6, 1992. Kawato, M., Setoyama, T. and Suzuki, R. "Feedback-error-learning of movement by multi-layer neural network", in proceedings of the International Neural Networks Society First Annual Meeting; 1988. Kumar, P., "Convergence of Adaptive Control SchemesUsing Least-Squares Parameter Estimates", IEEE Transactions on Automatic Control, Vol. 35, NO 4, pp. 416-424, April1990. Levin. A., and Narendra, K.; "Control of Nonlinear SystemsUsing Neural Networks: Controllability andStablilization", IEEE Transactions on Neural Networks, Vol4. NO.2, March 1993. Miller, T., Sutton, R., and Werbos, P., Neural Networks for Control, The MIT press, Cambridge, Massachusetts, 1990. Miyamoto, H., Kawato, M., Setoyama, T. and Suzuki, R. "Feedback-error learning neural network for trajectory control of. a robotic manipulator", Neural Networks 1:251.,.265, 1988. Narendra, K., and Parthasarathy, K., "Identifciation and control of dynamical systems using neural networks" IEEE Transactions on Neural Networks, Vol 1. NO. 1, March 1990. Proano, J., Neurodynamic Adaptive Control .Systems, Ph.D Dissertation, University of Colorado, Boulder, 1989. Psaltis, D., Sideris, A., and Yamamura, A., "A multilayered neural network controller" IEEE Control Systems Magazine, April 1988. Rumelhart, D., and McClelland, J., Parallel Distributed Processing: Vol 1, Foundations, The MIT Press, 1987. Spong, N. M., and Ortega, R. (1988), "Adaptive Motion Control of Rigid Robots: A tutorial" in Proceedings of the 27th IEEE conf. on Decision and Control, pp. 1575-1584, Dec. 1988. Yamada, T., and Yabuta, T., "Neural Network Controller Using Autotuning Method for Nonlinear Functions", IEEE Transactions on Neural Networks, Vo 3, No. 4, July 1992. 48

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APPENDIX Simulation Program The simulation was perforemed using the software MATLAB. The program shown below is the NSTC scheme. clc clear. % % BEGIN ---------------------------------------------------------------------------------------------%NUMBER OF iTERATIONS SIMULATION N3=1500; ndisp=30; ALGsr= 1; ALGri = 1; ALGij = 1; ID=1; % 1==> Gradient 2==> Newton .3=>MV % 1==> Gradient 2==> Newton 3==>MV % 1=> Gradient 2==> Newton 3=>MV % 1=> RLS 2==> Neural J.D. O==>Determistic % ----------------------------------------------------------------------------------------------------------------------% % ===== INITIALIZATION ==================================== %PLEASE SELECT THE DIMENSION OF THE STATE VECTOR XO, %INPUT VECTOR UO, OUTPUT VECTOR YO, AND PARAMETER VECTOR %PO BY MODIFYING THE FOLLOWING STATEMENTS: % % P0=4; POO= 1; PSIO= 1; % ----------------------------------------------------------------------------------------------------------------------% %Plant [a1 a2 a3.:.ana bO b1 b2 ... bnb]; % A= [.7 .5 -.3]'; B=[1 .2 -.1 .3]'; % A = [.7 .5]'; B=[1 .2 -.1]'; % THETAp = [.7 .5 1 .2 -.1]'; %minimum phase 2nd order plant % A= [.7 .5-3]'; B = [1 .2 -.1 3]'; % THETAp = [.7 .5 -.3 1 .2 -.1 .3]'; %minimum phase 3rd order % THETAp = [.7 .5 -.3 1 .2 -.1 3]';% non-minimum phase 3rd; ld=4-10 % A=[.7 .5 -.3]'; B=[l .2 -.1 3]'; % THETAp = [-2.58 2.18 -.5965 -429.7 884.8 -430.8]'; %missile nmp % THETAp = [-3,987 5.96 -3.96 .987 -6.94e-5 6.92e-5 6.9e-5 -6.88e-5]'; % Mxl %THETAp = [-3.87 5.63 -3.64 .882 -.0068 .0066 .0065 -.0063]'; %missile %THETAp = [-2.979 2.96-.979-.0047 .0094 -.0047]'; %Submarine load plant num=numd'/dend(l ); den=dend'/dend( 1); 49

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B=num; A=den(2:length( den)); % B=numd A=dend(2,:) THET Ap = [A' B']'; na=length(A);nb=length(B)-I ;d=l; nf=d-I; ne=nf+nb; ng=na-I; nc=O; %This assumes the noise has no dynamics, Le. C= I; PO= (ng+1)+(ne+1)+nc; %Dimension of THETA POp=na+nb+ I; %Dimension of plant's THETA THETAEST=I *ones(P0,1); %load thetaest THETAO=THET AEST; yest=O; w=O; P=POO*eye(PO,PO); PSip=PSIO*ones(POp, 1 ); PSid = zeros(PO,I); K= I.S *ones(PO, 1); Y1=zeros(ng+ I,I); UI=zeros(ne+ 1,1); Yc=zeros(ng+l,l); Uc=zeros(ile,l); Yl p=zeros(na, 1 ); U lp=zeros(nb+ 1, I); Ud=zeros(d,l); Yd=zeros(d,I); %delayed values ofu, y and w W d=zeros( d, 1); y=O; u=O; VARV=O; :MEANV=O; % output y(k), input u(k) % Output noise variance n0=2; ni = 5; n2=15; n3::::Po; %Dimensions of Neural Network NETr = zeros(n1,1); NETi = zeros(n2,1); NETj = zeros(n3,I); Is= zeros(nO,l);. Or= zeros(n1,1); Oi = zeros(n2,1); Oj = zeros(n3,1); ALPHAj = .03*ones(n3,1);, ALPHAi = .03*ones(n2,1); ALPHAr = .03*ones(nl,l); Hj = O*ones(n3,1);, Hi= O*ones(n2,I);, Hr = O*ones(n1,1); Kj = 3*ones(n3,1);, Ki = 2*ones(n2,1);, Kr = 2*ones(nl,I); Wsr = rand(nO,nl); wri = raitd(nl,n2); Wij = rand(n2,n3); mu::.8; lambda0=.99; lambdak = .995; LAMBDA =I; Pij = 10*ones(n2,n3); Pfi = 5*ones(nl,n2); Psr = 5.3*ones(n0,nl); 50

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Rn = .001; Re=.9; % % ===== END OF INITIALIZATION ============================ rand(' seed', 10); % --------------------------%:::::BEGIN ITERATION::::: fork=l:N3 % ===== STOCHASTIC ARMA REPRESENTATION OF A LINEAR PLANT ----------------% % y(k)+a1y(k-1)+ ... +anay(k-na)=bOu(k-d)+b I u(k-1-d)+ ... +bnbu(k-nb-d)+v(k) % y(k)=PSip'(k)*THET Ap(k)+v(k) % PSip'=[ -y(k-1 ) ... -y(k-na) u(k-d) ... u(k-nb-d)] % THETAp(k)'=[a1 a2 ... ana bO b1 b2 ... bnb] % THETA'(k) =[gO g1 ... gng eO el...ene] % PSid'(k) = [y(k-d) .. y(k-d-ng) u(k-d) .. % yest(k) = PSid(k)'*THET AEST %=======Computmg THETA ======::;:===== if d==1 E=B; G=-A; %g(i)=-a(i+1), i=O .. ng G=q-1(1-A) end THETA = [G'E']'; %PARAMETRIZATION FOR PSI(k). % % ---------------------------------------------------------------------------------------------------------------------% % u=u(k-1) y=y(k-1). for i=d-1:-1:1 Ud(i+1)=Ud(i);, end, Ud(1)=u; %[u(k-1) ... u(k-d)] for i=d-1:-1:1 Yd(i+l)=Yd(i);, end, Yd(l)=y; %[y(k-l) ... y(k-d)] for i=d-1:-1:1 Wd(i+l)=Wd(i);, end, Wd(1)=w; %[w(k-1) ... w(k-d)] %==== PSip(k) = [-y(k-1) .. -y(k-na) u(k-d) .. li(k-d-nb)]' for i=na-1:-1:1 Ylp(i+1)=Y1p(i);, end; Ylp(l)=-Yd(l); for i=nb:-1:1 Ulp(i+1)=U1p(i);, end; Ulp(l)=Ud(d); PSip = [Ylp' U1p']'; %====PSid(k) = [y(k-d) .. y(k-d-ng) u(k-d) .. u(k-d-ne)] for i=ng:-1:1 Y1(i+l)=Yl(i);, end; Yl(l)=Yd(d); %[y(k-d) .. y(k-d-ng)] for i=ne:-1:1 U1(i+l)=Ul(i);, end; U1(1)=Ud(d); %[u(k-d) .. u(k-d-ne)] PSid = [Yl I U1 ']'; % PSI(k-d) 51

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% =========================================================== % ----GENERATING NOISE v(k) ---rand(' normal') v=sqrt(VARV)*rand(l,l)+!\ffiANV; % % ----COMPUTING y(k) & w(k) ----% y=PSip'*THET Ap+v; %y(k) tau=.5; w=tau*w + (1-tau)*2; %*sign(sin(0.004*(k))); %command signal w(k) w=(w/2)+1; % % ----END OF PLANT -------------% ----------------------------------------------------------------------------------------------------------------------%-----ADAPTIVE ESTIMATION----% ===== THE STOCHASTIC LEAST SQUARES ALGORITHM (SLA) -------------------% % =========================================================== % ----BEGIN ESTIMATION ----% THETAEST =[gO gl ... gng eO el ... ene]' yest=PSid'*THET AEST; % PREDICTED OUTPUT yest(k) e=y-yest; % PREDICTION ERROR e(k) ifiD=l K=P*PSid*inv(l+(PSid'*P*PSid)); %OPTIMAL GAIN THETAEST=THETAEST+K*e; %PARAMETER ESTIMATION P=(P-K*PSid'*P); end %==========Neural identification ======== ifiD=2 ls(l)=l; Netr = Wsr'*ls; tempr = (ALPHAr/2). *(Netr+Hr); Or= Kr. *tanh(tempr); Or(l)=l; Neti = Wri'*Or; tempi = (ALPHAi/2). *(Neti+Hi); Oi = Ki.*tanh(tempi); Oi(l)=l; Netj = Wij'*Oi; 52

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tempj = (ALPHAj/2). *(Netj+Hj); Oj = Kj. *tanh(tempj); THET AEST = Oj; if k== 1 save thetaest THET AEST, end PSI=PSid; tempj2 = cosh(tempj). *cosh(tempj); tempi2 =cosh( tempi). *cosh( tempi); tempr2 = cosh(tempr). *cosh(tempr); Fdotj = (Kj. ALPHAj/2)./(tempj2); Fdoti = (Ki. ALPHAi/2)./(tempi2); Fdotr = (Kr. ALPHAr/2)./(tempr2); dj = Fdotj. *PSI; PSiij = Oi*dj'; di = (Wij*dj). *Fdoti; PSiri = Ot*di'; Q = Fdoti .* (Wij*(Fdotj.*PSI)); dr = Fdotr .* (Wri*Q); PSisr = Is*dr'; if ALGij == 1 % Gradient Lij = mu*PSiij/LAMBDA; end if ALGij == 2 %Newton Sij = (PSiij. *PSiij;*Pij) + (lambdak*LAMBDA *ones(n2,n3)); Lij = (Pij.*PSiij)./Sij; . . Pij = (Pij(Lij.*Sij.*Lij))/hunbdak; end if ALGij = 3 % Minimum Sij = (PSiij.*PSiij.*Pij) + Re*ones(n2,n3); Lij = (Pij.*PSiij)./Sij; Pij = Pij (Lij. *PSiij. *Pij) + Rn*ones(n2,n3); end if ALGri == 1 % Gradient Lri = mu*PSiri/LAMBDA; end if ALGri == 2 % Newton Sri= (PSiri.*PSiri.*Pri) + (lambdak*LAMBDA*ones(nl,n2)); Lri = (Pri. *PSiri)./Sri; Pri = (Pri (Lri. *Sri. *Lri) )llambdak; end if ALGri == 3 % Minimum Variance Sri= (PSiri.*PSiri.*Pri) + Re*ones(ri1,n2); Lri = (Pri. *PSiri)./Sri; Pri = Pri-(Lri.*PSiri.*Pri) + Rn*ones(n1,n2); end if ALGsr == 1 % Gradient Lsr = mu*PSisr/LAMBDA; end if ALGsr == 2 % Newton 53

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Ssr = (PSisr.*PSisr.*Psr) + (lambdak*LAMBDA*ones(n0,n1)); Lsr = (Psr. *PSisr)./Ssr; Psr = (Psr(Lsr.*Ssr.*Lsr))/lambdak; end if ALGsr = 3 % Minimum Vasrance Ssr = (PSisr. *PSisr. *Psr) + Re*ones(n0,n1); Lsr = (Psr. *PSisr)./Ssr; Psr = Psr-(Lsr.*PSisr.*Psr) + Rn*ones(nO,n1); end Wij = Wij + Lij*e; Wri = Wri + Lri*e; Wsr = Wsr + Lsr*e; %LAMBDA= LAMBDA +(e*e'-LAMBDA)Ik:; lambdak = lambdaO*lambdak+( 1-lambdaO); Re = Re + (e*e'-Re)lk:; for i=n0:-1:2 Is(i)=Is(i-1);, end end % % ----END OF ESTIMATION ----% % ==== MINIMUM VARIANCE ADAPTIVE CONTROL ----------------------------------------------------% % % ------------------------------------------------------------------------------------------------------------------------------------% ----BEGIN ADAPTIVE CONTROL----for i=ng:-1: 1 Yc(i+1)=Yc(i);, end; Yc(l)=y; %[y(k) .. y(k-ng)] for i=ne-1:-1:1 Uc(i+l)=Uc(i);, end; Uc(1)=u; %[u(k-l) .. u(k-ne)] ld = .1; ld2= .06; ld3=0; if ID=O THET AEST=THETA;, end % Q = ld + q-lld2 if k<1 THETAc=THETA;, else THETAc=THET AEST;, end Gest(l:ng+l,l) = THETAc(1:ng+1); % G Eqest(l:ne+1,1) = THETAc(ng+2:ng+2+ne); % E Eqest(1) = Eqest(1)+ld; % E+Q Eqest(2) = Eqest(2)+ld2; %Eqest(3) = Eqest(2)+ld3; % u(k) = {w(k)-[gOy(k)+ ... +gncy(k-ng)] -[elu(k-l)+ ... +eneu(k-ne)] }leO SUMl = Eqest(2:ne+l)'*Uc; SUM2 = Gest'*Y c; 54

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%roots(Eqest) %break u=(w-SUM2-SUM1)/Eqest(1); %u(k) %u=w; % -----END OF ADAPTIVE CONTROL---% =========================================================== -------------% % ----SIMULATION ERRORS ----------% ----SAVE THET A(k) & THET AEST(k) ---for j=1:PO THETA1 (k,j)=THETA(j); THETA1EST(kj)=THET AEST(j); end % ----SAVE y(k) & yest(k) ---Y(k,1)=y; YEST(k, 1 )=yest; % ----SAVE K(k) ---for j=1:PO K1(k,j)=K(j); end %-----SAVE U(k) ---U(k)=u; W(k)=Wd(d); % ---------------------------------------------------------------------------------------------------------------------% % -----THE PARAMETER IDENTIFICATION MSE(k) ---THETAER=TIIETAlTHETA lEST; forj=l:PO if k== 1, TMSE(k,j)=THET AER(k,j)"2; else TMSE(k,j)=TMSE(k-1,j)+(THETAER(k,j)"2-TMSE(k-1,j))/k; end end % -----THE OUTPUT PREDICTION MSE(k) ---YER(k)=y-yest; ifk=1, YMSE(k)=YER(k)"2; else YMSE(k)= YMSE(k-1 )+(YER(k)"2YMSE(k-1 ))/k; end %-----THE COST FUNCTION(k) J(k)---YERc(k)=y-Wd(d); if k=1, J(k)=YERc(k)"2; else J (k)=J (k -1 )+(YERc(k)"2-J (k -1) )/k; end % ===== DISPLAY MATRIX ===.:...-======================== % TIDS M-FILE IS USED TO MONITOR SYSTEM PERFORMANCES DURING 55

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%SIMULATION. % ========================================================= %-----TRANSFER DATA TO MATRIX DISMAT1---. DISMAT1{1,1)=k; DIS MAT 1 ( 1 ,2)=TMSE(k, 1 ); DISMAT1(1,3)=TMSE(k,2); DISMATl ( 1,4 )=TMSE(k,3); % DISMAT1(1,5)=TMSE(k,4); DISMAT1(1,6)=YMSE(k); DISMAT1 (1 ,7)=U(k); % -----TRANSFER DATA TO MATRIX DISMAT2 ---DISMAT2(1,l)=k; DISMAT2( 1 ,2)=J(k); DISMAT2(1,3)=Wd(d); DISMAT2(1,4)=y; DISMAT2( 1 ,5)=yest; % -----DISPLAY DISMATl & DISMAT2 ----if rem(k,ndisp )==0 home disp(' k TMSE1 TMSE2 TMSE3 TMSE4 YMSE U(k)') disp(DISMAT1) disp(' k J(k) w(k-d) y(k) yest'). disp(DISMA T2) %[fHETA THET AEST] end % % ----------END OF -------------------------------------------------------------------------------------------------%keyboard end %END OF FOR LOOP(k) %:::::END OF ITERATION::::: .. % % ----SYSTEMS GRAPHICS ----------SYGRAF % ----------------------------------DIS MAT % END OF SIMULATION --------------------------------------------------------------------------------------56



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FUSION OF THE MANDffiULAR SYMPHYSIS AND CRANIAL EVOLUTION IN MAMMALS by Rachel A. Hogard B.A., Washington University in St. Louis, 1999 A thesis submitted to the University of Colorado at Denver in partial fulfillment of the requirements for the degree of Master of Arts Anthropology 2003

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This thesis for the Master of Arts degree by Rachel A. Hogard has been approved by A. Spencer David Tracer -=Date

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Hogard, Rachel A. (M.A., Anthropology) Fusion of the Mandibular Symphysis and Cranial Evolution in Mammals Thesis directed by Assistant Professor Mark A. Spencer ABSTRACT Hypotheses regarding masticatory force production and the distribution of craniofacial variables are based on biomechanical models. One such model, the constrained model, has been particularly influential within the field of jaw biomechanics. This model is tested on a wide range of mammals, both with and without mandibular symphyseal fusion, to assess systematic differences in craniofacial configuration that may exist between these two groups. These groups are expected to differ in systematic ways due to differing muscle recruitment patterns. Mammals with an unfused symphysis recruit less balancing-side muscle force. This has important implications for the distribution of craniofacial variables within this model. Of particular interest is the distribution of the molars. The molars are expected to fit into the area of the cranium where highest magnitude bite forces can be produced. Results indicate that for both groups of mammals, the molars are indeed maintained within that region However, there are other configurations of the masticatory system that appear to differ between. mammals with an unfused symphysis and those with symphyseal fusion. Therefore, these two groups do appear to differ in systematic ways. This has important implications for the evolution of symphseal fusion and the role fusion may play in dietary adaptation. This abstract accurately represents the content of the candidate's thesis. I recommend its publication. iii signed MarklrSpencer

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DEDICATION I dedicate this thesis to my mother and father for their unfaltering support while I attended graduate school. I also dedicate this thesis to Karl Kesti for providing me with generous understanding and support throughout this process.

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ACKNOWLEDGEMENT My thanks and gratitude to my advisor, Mark Spencer, for his help and dedication to me during this process. I would also like to thank Cheri Jones of the Denver Museum of Nature and Science and Rosanne Humphrey of the University of Colorado at Boulder Natural History Museum.

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CONTENTS Figures ................................................................................................. v1n Tables ..................................................................................................... x CHAPTER 1. IN"TRODUCTION ............................................................................... 1 2. BIOMECHANICAL MODELS OF THE MASTICATORY SYSTEM ................................................................ 7 The Mandible as a Lever ............................................................. 7 The Constrained Model ............................................................. 1 0 Modifications to the Constrained Model ................................... 19 Muscle Force Recruitment ......................................................... 25 Why Fuse the Mandibular Symphysis? ..................................... 29 Stiffness Hypothesis .......................................................... 30 Strength Hypothesis ........................................................... 33 3. HYPOTHESES AND STUDY DESIGN .................................................. 37 The Effects of an Unfused Symphysis on the Model ........................ 39 Specific Predictions ........................................................................... 42 Study Design ..................................................................................... 42 Tests ........................................................................................... 43 4. MATERIALS AND METHODS .............................................................. 45 Vl

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Sample ........................................................................................... 45 Data Collection Process ..................................................................... 48 Measurements .................................................................................... 49 Size .................................. ........................................................ 52 5. RESULTS .................................................................................. ........ 56 Fused vs. Unfused .............................................................................. 70 Allometric Analysis ........................................................................... 70 6. DISCUSSION ........................................................................................... 78 Support for Previous Research .......................................................... 79 Allometric Analysis ........................................................................... 81 Biomechanical Implications for Diet.. ............................................... 84 Conclusion ......................................................................................... 88 LITERATURE CITED ...................................................................................... 90 Vll

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FIGURES Figure 1.1 Phylogenetic hypothesis concerning the relationships between the orders within eutherian mammals ................................................................ 4 2.1 Occlusal view of the mandible with a midline muscle resultant force ...... 13 2.2 For more posterior bite points, the triangle of support may not envelope a midline muscle resultant force producing tensile forces in the working-side joint .............................. : ............................................. l4 2.3 Occlusal view of the mandible showing the distributions of Regions I, II, and III for a mammal with a fused mandibular symphysis ............... 16 2.4 Predicted bite force and joint reaction force values using the assumptions ofthe constrained model. ...................................................... l8 2.5 Occlusal view ofthe mandible demonstrating how bicondylar breadth affects the length ofRegion II ...................................................... 23 2.6 An additional variable that alters Region II length is the anteroposterior position of the midline muscle resultant force ........................... 24 2. 7 Occlusal view of mandible showing how palatal breadth affects the length of Region II ............................................................................... 26 3.1 Occlusal views of mandible showing the effects of changes in the position of the muscle resultant force on the distribution of Region II .... .40 4.1 Illustration of five variables used in the calculation ofRegion II length .. 51 4.2 Equation for estimating the predicted length ofRegion II ........................ 53 5.1 Plot comparing postcanine dimensions to predicted Region II length ...... 69 viii

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5.2 Box plot representing the means of molar length divided by the geometric mean for mammals with a fused and unfused symphysis ......... 71 5.3 Bivariate plots oflog-transformed data showing the relationship between cranial size and other masticatory variables relevant to the distribution of Region II ...................................................................... 73 lX

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TABLES Table 4.1 Taxa indicated in study with a fused symphysis ...................................... .46 4.2 Taxa indicated in study with an unfused symphysis ................................ .47 5.1 Mean and standard deviation of tooth row length ..................................... 57 5.2 Mean and standard deviation of variables used to calculate predicted Region II length ......................................................................... 60 5.3 Mean and standard deviation of the geometric mean and one variable used to calculate it.. ...................................................................... 65 5.4 Variables regressed on the geometric mean to assess allometic relationships ........................................................................................... 76 X

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CHAPTER 1 INTRODUCTION Diet plays an influential role in almost all aspects of an organism's life. The selection pressures associated with the acquisition and processing of food have an influence on many aspects of a species' anatomy and behavior (Cartmill, 1975; Clutton-Brock, 1974; Rylander, 1979a; Hiiemae, 1984; Fleagle, 1988; Smith, 1993). Selection pressures range from the processes involved in finding food to being able to efficiently process this food once ingested. These pressures can change as a result of a changing environment. The ability of a species to adapt to these changing selection pressures is essential to its survival. The study of the masticatory system is therefore crucial to our understanding of the evolutionary relationship between diet and morphology as it is this structure that enables food acquisition and processing. This is because the masticatory system, including the jaws, teeth, and craniofacial muscles, is a direct link between an animal's external environment and its internal requirements. Previous studies have established that features of masticatory morphology are functionally correlated with dietary pattern (Kay, 1975; Kay and Covert, 1984; Lucas et. a!., 1986; Fleagle, 1988). This association between morphology 1

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and diet in extant species is often used to guide our explanations of the fossil record (Hiiemae, 1984). Therefore, studying this system can lead to a greater understanding ofthe functional role morphology plays in the life of the animal and the evolution of a species. Forces are generated within the masticatory system to break down food objects. Cranial configuration (i.e., the interaction between morphological variables of the cranium) is known to influence force production (Greaves, 1978; Smith, 1978; Spencer, 1998, 1999). Hypotheses regarding force production and the factors that relate to this are based on biomechanical models (Roberts and Tattersall, 1974; Greaves, 1978; Smith 1978; Spencer, 1999). From these biomechanical models functional predictions are generated about ways in which the masticatory system should be configured. Testing these predictions helps to increase our understanding of morphological design and change over time . While biomechanical models are needed to understand masticatory system function and evolution, we frequently do not know how widely applicable they are. It is important, therefore, to look at a wide range of mammalian taxa in order to understand the broad applicability of these models to the masticatory system. Mammals, in general, and primates in particular, differ drastically in the morphological structure and configuration of their masticatory system. One such structure that has marked effects on this system is the mandibular symphysis. The 2

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mandibular symphysis is the structure that separates the two dentaries in the sagittal plane. An unfused mandibular symphysis, in which the two dentaries are joined by non-ossified connective tissue, is the ancestral condition among mammals. However, symphyseal fusion has evolved independently in several mammalian lineages (Greaves, 1988). Within primates, extant prosimians retain the primitive condition of having an unfused mandibular symphysis, though they vary in symphyseal morphology, while anthropoid primates have evolved symphyseal fusion (Beecher, 1977; Rylander, 1979b; Ravosa, 1991). Other groups of mammals that have evolved the derived condition of fusion include perissodactyls, hyracoids, families of chiropterans and several artiodactyl taxa (Suidae, Tayassuidae, Hippoptomidae, and Camelidae) (Beecher, 1977). Figure 1.1 outlines the relationships between the orders within eutherian mammals. This phylogenetic hypothesis shows that fusion has evolved multiple times within the order Mammalia in several different lineages (Allard et. a/., 1996). Some orders that contain families with symphyseal fusion also include families without fusion. The repeated evolution of symphyseal fusion within mammals has generated substantial interest concerning the adaptive significance (and biomechanical consequences) of this feature (Beecher, 1979; Rylander, 1979a, 1979b, 1984; Scapino, 1981; Greaves, 1988, 1993; Ravosa, 1991, 1993; Rylander and Johnson, 1994; Ravosa and Rylander, 1994; Rylander et. al., 1998; Ravosa et. a/., 2000). The biomechanical consequences of symphyseal fusion within the 3

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Lagomorpha Rodentia Macros eel idea r------Artiodactyla Cetacea Perissodactyla Hyraooidea Sirenia Proboscidea Dermoptera Chiroptera Primates Sca:nden t i a E!3entata Pholidota Carnivora Tublidentata Insectivora Out group Fig. 1.1 Phylogenetic hypothesis concerning the relationship between the orders within mammals . This strict consensus tree was obtained from 88 morphological characters from fossils and recent evidence (Allard et. al., 1996). indicates orders containing some families with symphyseal fusion. 4

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various mammalian taxa can provide a greater understanding of the adaptive significance of this feature. Experimental data indicate that there are differences in masticatory muscle recruitment patterns between animals with a fused and unfused symphysis. Regardless of how symphyseal fusion leads to these differences, they have important theoretical implications because it is unknown how these differences may affect other aspects of masticatory function or whether they lead to systematic differences in overall cranial morphology. The goal of this study is to test predictions regarding differences in masticatory system configuration and how this differs between mammals with a fused and unfused symphysis. These groups are known to differ in systematic ways in masticatory system morphology (Beecher, 1977; Hiiemae, 1984; Greaves, 1988; Spencer, 1999). An analysis ofthe functional significance ofthese differences should improve our understanding ofthe adaptive basis of cranial form in primates, including humans. Adaptive hypotheses must be tested using the comparative method (Clutton-Brock and Harvey, 1979). In this method, features are inferred to be associated if they evolved together repeatedly in several independent lineages. An understanding of the relationship between symphyseal fusion and masticatory system configuration must be based, therefore, on identifying commonalities among multiple lineages that have evolved symphyseal fusion. This study 5

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quantifies cranial form in a wide range of mammals with fused and unfused mandibular symphyses in order to assess the biomechanical consequences of symphyseal fusion. 6

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CHAPTER2 BIOMECHANICAL MODELS OF THE MASTICATORY SYSTEM The functional significance of masticatory system morphology has long been of great interest within the field of biological anthropology. Descriptions of this system often rely on simplified models. The earliest studies (Gysi, 1921, Maynard-Smith and Savage, 1959) argued that the mandible acts as a lever. This model was challenged in the early 1970's (e.g., Roberts and Tattersall, 1974) but is now the most generally accepted biomechanical analogy for the mandible (Rylander, 1975; DuBrul, 1977; Greaves, 1978; Smith, 1978). Other models, such as the constrained model, have expanded upon the principles of the lever model to include more complex factors. The Mandible as a Lever The mammalian masticatory system has historically been modeled as a lever (Gysi, 1921; Maynard-Smith and Savage, 1959; DuBrul, 1977; Greaves, 1978, 1982, 1988; Rylander, 1975, 1977; Smith, 1978, Demes and Creel, 1988). In this model, the skull is analyzed in the sagittal plane. The condyle acts as the fulcrum, the masticatory muscles are the applied force, and the bite point is the 7

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resistance. The masticatory muscles apply an adducting force to the mandible, which serves to close the jaw. This is resisted by reaction forces at the temporomandibular joints (TMJ) and the bite point. The muscles responsible for closing the jaw are the masseter, medial pterygoid, and temporalis. All of the muscle, TMJ, and bite forces are typically simplified into individual force vectors (Spencer, 1995). The vector representing the sum of the individual muscle force vectors from both sides of the head is termed the muscle resultant force (MRF). During the 1950's and 1960's this model was generally well accepted (Maynard-Smith and Savage, 1959) and was used in evolutionary interpretations of cranial form in various taxa. However, some argued that this is a mechanically inefficient system and proposed alternative models (Gingerich, 1971; Roberts and Tattersall, 1974). Those who opposed the lever model assumed that the process of adaptation should result in greater efficiency of the masticatory system than they believed the lever model allowed. Smith (1978) points out, however, that the process of evolution is not necessarily a process of optimal design, rather, it probably results in species that are just a little better than their competitors. The alternative models centered aroWld the idea that during biting, the jaw fimctions as a link between the adductor muscle force and the bite force (Gingerich, 1971). The argument that the mandible does not fimction as a lever and instead acts as a 'link' was based on two assertions: (1) the resultant ofthe forces produced by the masticatory muscles (i.e., the muscle resultant force) 8

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always passes through the bite point; (2) the condylar neck and/or TMJ is poorly suited to withstand reaction forces. These two assertions have since been disproved. The muscle resultant force (MRF) does not always pass through the bite point (Greaves, 1978). It has been proposed that the MRF actually lies in the midline between the last molars during equal working side and balancing side muscle activity and that differential activity between the muscles produces mediolateral movement ofthe MRF (Rylander, 1985; Spencer, 1998). Also in opposition to the 'link' models, electromyographic data have demonstrated that large condylar reaction forces do exist and that the condyle is strong enough to withstand reaction forces during lever action (Rylander, 1975). Early work within the field of biomechanics suggested that the strongest force is exerted at the balancing condyle (i.e., the TMJ on the side opposite from the bite point) (Gysi, 1921). It was theorized that the balancing-side muscle force, which gets transmitted through the mandibular symphysis to the working side, reduces the force at the working-side condyle. Modelling data showed a reduction in working-side joint reaction force as the bite point moved distally along the tooth row (Gysi, 1921). When food is crushed at the premolars, the force at the working-side condyle is slightly reduced compared to more anterior bite points. The working-side joint reaction force gets neutralized as the bite point moves distally to the second molar. It was also hypothesized that hard 9

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foods cannot be crushed on the third molars because the downward pull of the working-side TMJ could lead to joint distraction. Later studies have indeed confirmed that condylar forces are greater on the balancing side than the working side (Rylander, 1979a). This work is particularly important to later research for two reasons. First, it supports the hypothesis that the mandible can be modeled as a lever. Although alternative models have been offered to explain the biomechanical actions of the mandible, the lever is still upheld as the dominant model today. Second, it shows that balancing-side muscles contribute to the muscle resultant force, which gets transmitted through the symphysis to the working side. These two points have had important implications in the development ofbiomechanical models of the masticatory system and explaining the functional significance of the mandibular symphysis. The Constrained Model Walter Greaves' constrained model of the jaw lever system of ungulates has been particularly influential in the field of jaw biomechanics. This model was developed based on the view that in Homo muscle forces on both sides of the head are transmitted to the working tooth row and that there are reaction forces at both TMJs (Greaves 1978). This model is applied, however, to selenodont artiodactyls, which differ drastically in their mandibular morphologies. Homo, as 10

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an anthropoid primate, has a fused mandibular symphysis. Selenodont artiodactyls, with the exception of the camelids, have an unfused mandibular symphysis (Greaves, 1978). Although Greaves recognized this, the model is based on an assumption that applies only to those mammals with a fused symphysis. It was not known at the time that differences in muscle recruitment patterns between fused and unfused species could have an effect on the model, which will be discussed later. In Greaves' model, masticatory forces are examined in the occlusal view (Fig. 2.1 ). During mastication the mandible is pulled toward the skull by the adductor muscles, the masseter, temporalis, and medial pterygoid. These muscle forces are resisted by reaction forces at three regions of the craniwn: the bite point, the working side TMJ, and the balancing side TMJ (Greaves, 1978). It is these three points that form what has been termed the triangle of support. Because there are multiple muscles on both sides of the head working to adduct the mandible, these muscle forces can be combined into a single vector termed the muscle resultant force (.MRF) through simple vector addition as is the case with the lever model. Greaves argued that distraction ofthe temporomandibular joint (i.e., forces that can separate the mandibular condyle from the articular eminence) could lead to potentially serious injury. He therefore proposed that natural selection should favor a morphology that limits TMJ distraction. This is the fundamental 11

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assumption of the constrained model and is predicted to cause limitations on the evolution of masticatory morphology in mammals. There should be limitations on masticatory morphology because jaw distraction is avoided only if the MRF lies within the triangle of support (Fig. 2.1) (Adapted from Spencer, 1995) (Greaves, 1978). The mediolaterallocation of the .MRF is determined by the positions and relative force contributions of the adductor muscles from both the working and balancing sides (Spencer, 1999). The muscle resultant will lie in the midline when the balancing and working side muscles are equally active. However, this is not always the case. Differential activity between these muscles produces mediolateral movement of the .MRF (Rylander, 1985; Spencer, 1998). The MRF will also lie at different positions relative to the triangle of support depending on the bite point. Biting on more anterior teeth creates a relatively large triangle of support that will enclose a midline MRF (see Fig. 2.1 ). However, during biting on more posterior teeth, the triangle of support is smaller and gets shifted laterally toward the working side (Spencer, 1999). This smaller triangle may not encompass a midline .MRF. If the midline .MRF falls outside of the triangle of support, the mandible could potentially rotate around the bite point and the balancing-side joint causing distraction of the working-side mandibular condyle (Fig. 2.2). 12

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Balancing Side Joint Reaction Force Side Joint Reaction Force Fig. 2.1 Occlusal view of the mandible with a midline muscle resultant force (B). During biting on more anterior teeth, this midline muscle resultant force passes through the triangle of support (shaded zone). The comers of this triangle are positioned at the bite force (F), the balancing-side joint reaction force (J), and the working-side joint reaction force (0). 13

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Balancing Side Joint Reaction Force Working Side Joint Reaction Force Fig. 2.2 For more posterior bite points, the triangle of support may not envelope a midline muscle resultant force producing tensile forces in the working-side joint. .Greaves (1978) suggested that the muscle resultant could be moved back into the triangle of support (arrow) through a reduction in balancing-side muscle activity. 14

PAGE 25

As stated earlier, the main assumption ofthe constrained model is that the TMJ should not be subjected to distraction. Therefore, the muscle resultant force must move so that it will always lie within the triangle of support. Differential muscle activity between the working side and balancing side (i.e. less balancing side muscle activity) will enable the MRF to shift laterally toward the working side. This means that a smaller triangle of support characteristic of the more posterior dentition will still encompass the MRF. Changes in muscle activity and joint loading have lead to the division of three zones of potential bite points termed Regions I, II, and III (Spencer and Demes, 1993; Spencer, 1995, 1998, 1999) (Fig. 2.3). Region I encompasses the anterior dentition. It is separated from Region II by an oblique line, which passes through the balancing-side joint reaction force and a midline :MRF. The triangle of support for this region is able to enclose a midline MRF; it is unnecessary for the l\1RF to shift toward the working side. Regions II and ill are separated by a transverse line passing through the muscle resultant force. Region II is characterized by a relatively small triangle of support through which a midline l\1RF will not pass ifthe adductor muscles on both sides of the head are equally active. The l\1RF must shift toward the working side in order to avoid distraction of the working-side TMJ. This is done through a reduction in balancing-side muscle activity. 15

PAGE 26

Balancing Side Joint Reaction Force REGION I ........... ... :/ REGION II REGION Ill Working Side Joint Reaotlon Force Fig. 2.3 Occlusal view o( the mandible showing the distributions of Regions I, II, and III for a mammal with a fused mandibular symphysis. Any bite point located in the anterior region (Region I) will produce a triangle of support that envelopes a midline muscle resultant force; no reduction in balancing side activity must therefore occur. In Region II, the muscle resultant must shift toward the working side through a reduction in balancing side activity or tension will be produced in the working In the most P.osterior region III) the muscle resultant cannot be repositioned so that tensile forces are avmded. 16

PAGE 27

A triangle of support in Region III will not be able to encompass a MRF even if it were to move laterally. Biting in this region will be unavoidably associated with TMJ distraction because the MRF cannot fall within the triangle of support (Spencer, 1999). Therefore, no teeth should lie here. Bite force and joint reaction force values differ between Regions I, II, and III. These provide evidence for demarcating the boundaries of these regions. This has been demonstrated in a theoretical model of reaction force values based on the predictions of the constrained model (Spencer, 1998) (Fig. 2.4). Region I is characterized by bite force values that are the lowest in magnitude yet increase as the bite point moves distally. Region II maintains the highest magnitude bite forces, however, all values are equal within this region. Region Til does not have any bite force values as biting in this region should be avoided due to the possibility of joint distraction. The joint reaction forces show a different pattern of magnitude from the bite forces. The working-side joint reaction force decreases as the bite point moves distally through Region I. In Region II, the working-side joint reaction force maintains a value of zero. The balancing-side joint reaction force values decrease slightly within Region I and then drastically in Region II as the bite point moves distally (Spencer, 1998). The constrained model has been very influential within the field of functional morphology (Greaves, 1982; Dessem and Druzinsky, 1992; Spencer 17

PAGE 28

Region II Re lon I 41126 )100 i 75 :i 60 J -50 ........... 1 o 20 30 40 50 eo 10 eo Distance of Bite Point Anterior to TMJ Fig 2.4 Predicted bite force and joint reaction force values using the assumptions of the constrained model. The magnitude of the bite force (F), the working-side.joint reaction force (0), and the balancing-side joint reaction force (J) is predicted under the conditions of maximum bite force production and the avoidance of tensile working-side joint reaction forces. Bite force values in Region I are lowest in magnitude but increase as the bite point moves posteriorly. Region II bite force values are highest in magnitude but are all equal. Joint reaction force values show a different pattern. Within Region I, the working-side joint reaction force values decrease as the bite point moves posteriorly .. In Region II, the working-side joint reaction force values are all zero. The balancing-side joint reaction force values decrease slightly within Region I and then drastically in Region II as the bite point moves distally. 18

PAGE 29

and Demes, 1993; Spencer, 1995, 1998, 1999; Dumont and Herrel, 2003; Thompson, et. a/., 2003). Predictions concerning masticatory configuration have been generated using this theoretical model. Subsequent testing of this model has led to the modification of the constrained model to include variables and ideas that are discussed in the next section. Modifications to the Constrained Model Subsequent studies have provided support for Greaves' fundamental assumption that the TMJ should not be loaded by distractive forces. A morphometric analysis of anthropoid masticatory system configurations suggests that the phenotypic diversity in cranial morphology in this group is limited by the need to avoid TMJ distraction (Spencer, 1999). However, this study also highlights some discrepancies between the constrained model and observed anthropoid cranial morphology. One such discrepancy involves the observation that the masticatory adductor muscles are positioned more posteriorly than proposed in the constrained model. This means that the MRF may not be produced directly at the posterior end of the tooth row during forceful isometric biting as Greaves had assumed. Spencer (1999) proposes a more posterior position of the MRF than originally conceived ofby Greaves (1978). A more posteriorly oriented MRF means that even ifthe MRF were to migrate forward at larger gapes, it would still 19

PAGE 30

be maintained within the triangle of support. This creates a more conservative masticatory system configuration in which combined muscle force is positioned more posteriorly than predicted by Greaves. The constrained model has been tested through studies of force production in the primate, canid, and opossum masticatory system. If this model is correct, muscle activity should change with bite point position. This must occur in order to maintain the basic assumption of this model, which is that the TMJ should not be loaded with distractive forces. The TMJ will not be subjected to distractive forces as long as the MRF is maintained within the triangle of support. Biting on more posterior teeth will, therefore, cause the MRF to shift laterally in order to be maintained within the relatively smaller triangle of support. This is done through a reduction ofbalancing-side muscle activity. Electromyographic data have shown that in humans the activity of the largest masticatory adductor muscles, the superficial masseter and anterior temporalis, changes with bite point (Spencer, 1998). Maximum muscle force magnitudes were found to be greatest for the first molar, decreasing both anterior and posterior to this bite point. Balancing-side to working-side muscle force ratios were also found to differ by bite point. Balancing-side muscle activity was found to be lowest during biting on the third molars. This decreased balancing side muscle activity may serve as a mechanism for avoiding TMJ distraction by enabling the MRF to be maintained within the triangle of support (Spencer, 1998). 20

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An electromyographic study of Canis fami/iaris also supports the constrained model (Dessem, 1989). This study found greater working-side than balancing-side muscle activity which supports the notion that the MRF should always fall within the triangle of support in order to maintain jaw-joint stability. The constrained model also enables specific predictions to be made about bite forces in Regions I and IT. Those predictions are that bite force increases as the bite point moves posteriorly within Region I and then reaches its highest magnitude within Region II. All bite points within Region II are of equal value. A study involving Monodelphis domestica, an opossum, supports these predictions (Thompson et. al., 2003). They found that within this species, both juveniles and adults, maintain at least three molariform teeth within Region II and that it is within this region where highest magnitude bite forces are produced. Also, within Region I, the bite force generated at the premolars was stronger than that generated at the incisors or canines and forces were equivalently strong within Region II. Other studies (Spencer, 1999) have also tested the constrained model and found that there are many variables related to the configuration of the masticatory system. that interact to determine the distribution of Regions I, II, and ill. These variables include bicondylar breadth, the anteroposterior position of the :MRF, palatal breadth, and height of the TMJ. 21

PAGE 32

Changes in bicondylar breadth can alter the anterior border of Region II (Fig. 2.5). The border this Region II moves posteriorly as the distance between the two condyles increases, causing less of the molar tooth row to fall within Region II (Spencer, 1999). Therefore, increasing bicondylar breadth decreases the length of Region II. Another variable relating to the distribution of Region II is the anteroposterior (A-P) position of the MRF. In the constrained model the MRF is located at the midline between the last molars. Bite points posterior to the MRF will not allow it to pass through the triangle of support even if it were to shift laterally toward the working side (Spencer, 1999). However, a MRF that is positioned posterior to the last molars will create a more posterior, and shorter, Region II (Fig. 2.6). The height of the TMJ above the occlusal plane is another variable that affects the distribution of Region II. TMJs that are positioned above the occlusal plane will cause the MRF vector to pass more anteriorly through the triangle of support (Spencer, 1999). This is because the triangle of support will become inclined as the TMJ gets raised above the occlusal plane. The taller the TMJ, the more inclined the triangle of support will be. This inclination may cause the MRF to pass anterior to the triangle of support. Therefore, mammals with taller TMJs may have a more anteriorly positioned Region II. This has the effect of increasing the length of Region II. 22

PAGE 33

c B A Working Sld.e Joint Reaction Force Fig. 2.5 Occlusal view of the mandible demonStrating how bicondylar breadth affects the length of Region IT. A decrease in bicondylar breadth, from positions A through C, causes an inc!ease in the length of Region II. This enables more of the molar row to fall within this region. 23

PAGE 34

' ...... .. / \ I "' .... \ Midline '\""< J 1 Muscle \ '\ i 1 Resultant \ \-. 1 (.. --( 'l !. I \ \ i \ \ \ jt I \.\\\\ . --: "I i i -.. "'::l.. ,..---/ \ \. ---o -' .. ,. ___ _.// ____ Balancing Side Joint Reaction Force Working Side Joint Reaction Force Fig. 2.6 An additional varianle that alters Region II length is the anteroposterior position of the midline muscle resultant force (B). As the muscle resultant force is moved from positions A through C, the boundary line between Region II and lli (bold transverse lines passing through the muscle resultant force) is moved posteriorly. As the muscle resultant force is reoriented posteriorly, the length of Region II decreases (shaded boxes). 24

PAGE 35

Palatal width also affects the distribution of Region II (Fig. 2.7). The mediolateral position of the tooth row determines the length of the postcanine dentition that will fall within Region II. Thus, movement of the palate laterally increases the length of Region II. These variables all interact with one another to produce a masticatory system configuration unique to each species. Studying the configuration of this system and how these variables interact to determine the distribution of Region II is crucial to testing the broad scale applicability of the constrained model. Muscle Force Recruitment The purpose of this study is to assess the differences in masticatory system configuration between mammals with an unfused mandibular symphysis and those with a fused one. The justification for studying these differences lies in the finding that animals with a fused symphysis differ in muscle recruitment patterns from those with an unfused symphysis (Rylander, 1979b; Dessem, 1989; Rylander et. a/., 1998). It has been shown that primates with a fused symphysis recruit more balancing-side muscle force during powerful mastication than those with an unfused symphysis. In 1979, Rylander reported results from an experiment that showed that only a small percentage of the bite force of Galago crassicaudatus, which has an unfused symphysis, is due to balancing-side muscle force during 25

PAGE 36

. / ( "-..(_ 1 ,,, / J /_ r ', l \,, ....... -, / Balancing Side Joint Reaction Force \ \, ,,, .. .. l; '' I 1 Midline "''"t.>,_ I I Muscle \ r; 1 Resultant \ \ \ \. \ '\. \ l \. '\ \. \ \ '\ 1 -...,.r-/ . .r,(_rO,., _,/ ...... :,..---.... __ Working Side Joint Reaotlon Force Fig. 2.7 Occlusal view of mandible showing how palatal breadth affects the length of Region II. The boundary line between Regions I and II is represented as a thin diagonal line and Regions II and III are separated by a bolder horizontal line. The. rnediolateral position of the tooth row relative to these lines will determine the length of postcanine dentition that will fall within Region II (boxes). Thus, movement of the tooth row laterally (i.e., increasing palatal breadth) from positions A through C results in an increase in Region II length. 26

PAGE 37

isometric unilateral molar biting. This suggests that the working-side jaw musculature of galagos is much more active than the balancing side. Bone strain data and moment arm calculations demonstrated that the working-side muscles generate at least four to five times more force than the balancing side (Rylander, 1979b). This decreased emphasis on balancing-side musculature for galagos contrasts with electromyographic results for humans. Results show that the balancing-side muscles of humans are only slightly less active than the working side masticatory muscles during powerful unilateral biting (Meller, 1966 as discussed in Rylander, 1979b). One reason this may be the case can be seen in the mandibular morphology of these two groups. Galagos have an unfused mandibular symphysis whereas humans, including all anthropoids, have fusion. The fused mandibular symphysis of this group of primates has been suggested to be an adaptation to counter increased symphyseal stress due to increases in balancing-side muscle force during powerful unilateral biting (Rylander, 1979b ). This is because symphyseal fusion functions to prevent structural failure ofthe symphysis by strengthening it. This becomes important as more balancing-side muscle force is recruited during mastication. The bone strain data of galagos discussed above support this suggestion. Data comparing long-tailed macaques (Macacafascicularis) and thick tailed galagos (Otolemur crassicaudatus) also support the symphyseal fusion-27

PAGE 38

muscle recruitment hypothesis. Advocates for this hypothesis argue that symphyseal fusion and balancing-side jaw adductor muscle force are functionally linked (Rylander, 1979a). Long-tailed macaques, which have a fused mandibular symphysis, have about 1.5 to 2 times more bone strain along the working-side corpus than along the balancing-side corpus. Thick-tailed galagos, whose symphysis is unfused, have about 7 times more bone strain along the working side corpus. This shows that the macaques recruit more balancing-side adductor muscles during isometric molar biting, whereas the galagos rely more exclusively on the working-side jaw muscles. Lemurs, which have an unfused symphysis, have also been found to recruit relatively little force from their balancing-side deep masseter. The recruitment and firing pattern of this muscle is much more similar to that of the galagos than anthropoids (Rylander et. a/., 2002). Owl monkey data show a pattern of strain that is similar to that recorded for the macaques (Rylander et. a/., 1998). This pattern differs markedly from the galagos' strain pattern. This is expected considering owl monkeys and macaques both have a fused symphysis whereas galagos do not. Owl which are smaller than galagos, show that differences in muscle recruitment patterns are related to symphyseal fusion rather than allometric constraints. A study of electromyographic activity from jaw-adductor muscles of Canis familiaris (domestic dog) also supports the idea that animals with an 28

PAGE 39

unfused symphysis recruit less balancing-side muscle force. During mastication and bone crushing working-side muscle activity was reported to be greater than balancing-side muscle activity in this species, which has an unfused symphysis (Leibman and Kussick, 1965; Dessem, 1989). It was found that even during the production ofthe largest bite forces, balancing-side muscle activity was never maximally recruited. This finding supports the previously discussed studies involving primates. Much evidence demonstrates that there are marked differences in muscle recruitment patterns between mammals with a fused and unfused symphysis. These muscle recruitment patterns may cause systematic differences in the configuration of the masticatory system between mammals with a fused and unfused symphysis. Why Fuse the Mandibular Symphysis? Symphyseal fusion enables those animals with this morphological trait to recruit more balancing side muscle force as discussed in the previous section. However, this does not explain the convergent acquisition of this feature in a number of mammalian taxa. During the past thirty years, morphological and experimental analyses of primates have been done in an attempt to answer this question. Although it has been confirmed through histological studies (Beecher, 1977) that a fused symphysis is stronger than an unfused one (because bone is 29

PAGE 40

stronger than the fibrocartilage and ligaments of an unfused joint) it is unclear exactly why fusion has occurred. Two groups of hypotheses have been offered to explain the functional and evolutionary significance of symphyseal fusion. These hypotheses involve stiffness and strength in the mandibular symphysis. Stiffness Hypotheses One group of hypotheses involves the idea that fusion of the symphysis occurred within anthropoid primates as an adaptation to stiffen the symphysis. Stiffness is defined as the ability to resist deformation in response to applied forces and is the primary mechanical property of bone that enables force transfer (Lieberman and Crompton, 2000). One idea that has been suggested is that syniphyseal fusion occurred within anthropoids to stiffen the symphysis in order to better resist the forces associated with the crushing of small, hard objects, such as seeds, along the incisors (Kay and Hiiemae, 1974; Greaves, 1988). This would prevent an inefficient dissipation of dorsally-directed force across the symphysis. This is because an unfused symphysis allows some independent movement between the two dentaries during incision. Theoretically, this independent movement enables the balancing side incisors to contact one another during the crushing of small, hard objects. Therefore, the balancing-side muscle force gets dissipated along the incisors on the balancing side rather than being transmitted across the symphysis 30

PAGE 41

(Ravosa and Rylander, 1994). The muscle force that was generated on the balancing side gets wasted on the balancing-side incisors rather than contributing to the force production on the working side. This scenario differs as the size ofthe food object gets larger. As the diameter of a food object increases the balancing-side upper and lower incisors will not come into contact. The balancing-side muscle force is then able to get transmitted across the symphysis and contribute to the crushing of the food object. This argument is, therefore, relevant only for the crushing of small, hard food objects, such as seeds. In conclusion, according to this hypothesis, the function of a fused symphysis is to stiffen the symphyseal joint in order to reduce independent movement of the two dentaries during the crushing of small, hard objects. This hypothesis is faulty for a number of reasons. The majority of anthropoid primates do not include small, hard objects, such as seeds, as a major component of their diet. Also, according to Ravosa and Rylander (1993), the literature provides no evidence for the processing of small, hard objects on the incisors when objects such as seeds are ingested. The apparent discrepancy between observed behaviors in anthropoid primates and this argument has led others to modify the stiffuess hypothesis. Another problem with this argument is that it does not address the extent to. which independent movement between the two dentaries occurs. Greaves' 31

PAGE 42

(1988) schematic drawings make it look as though each dentary moves completely independent of one another. This is not the case; the ligaments of an unfused symphysis do not allow such an amount of independent movement between the two dentaries. In fact, Beecher (1977) has shown that the pattern of fibrocartilage and ligaments of pro simians are arranged to resist movement during mastication. The functional association between symphyseal fusion and increased stiffness has been hypothesized to be important for molar biting rather than for the incisors (Ravosa and Rylander, 1994). Symphyseal fusion could be an adaptation to reduce the amount of balancing-side tooth contacts during unilateral mastication on the molars. This is similar to the proposition that balancing side tooth contact should be avoided so that balancing-side muscle force can contribute to the crushing of the food object on the working side. This idea is also problematic because anisognathic jaws and an unfused symphysis probably represent the primitive condition for primates (Ravosa and Rylander, 1994). Anisognathy refers to the condition of having different widths of the upper and lower dental arches. Therefore, these teeth are not likely to come into contact anyway even with an unfused symphysis. Fusion of the mandibular symphysis does not appear to be necessary to prevent balancing-side tooth contacts during either incision or mastication. 32

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An alternative hypothesis focuses on the orientation of the occlusal wear facets. The main idea is that fusion functions to stiffen the mandible in taxa with more horizontally oriented occlusal wear facets. Both fused and unfused symphyses are effective at transferring dorsally-oriented force. However, a fused symphysis, because of its stiffness in all planes, is likely to be more effective at transferring force in the transverse plane. Taxa with a strong transverse component to chewing will benefit more from a fused symphysis. This idea has been supported in studies of mammals; symphyseal fusion does in fact correlate with transversely oriented occlusal planes (Lieberman and Crompton, 2000). It can be concluded, therefore, that fusion is most likely an adaptation for increasing the efficiency of transversely-oriented occlusal forces. Strength Hypotheses Another set of hypotheses regarding the functional significance of symphyseal fusion is that fusion strengthens the symphysis to help counter symphyseal stress during incisal biting and unilateral mastication (Ravosa and Rylander, 1994). Hypotheses that have been offered in support of the strengthening hypothesis involve the idea that symphyseal fusion occurred to counter wishboning stress and/or dorsoventral shear. It has been found that during incision and mastication, the mandibles of macaques are twisted about their long axes (Rylander, 1979a). This twisting 33

PAGE 44

everts the lower borders of the mandible so that the symphysis gets bent vertically. Compression occurs along the upper or alveolar surface of the symphysis and tension is present along the inferior surface ofthe symphysis. This stress is most effectively countered by complete symphyseal fusion. Two loading patterns are also present among anthropoid primates during unilateral mastication: wishboning and dorsoventral shear of the symphysis. Wishboning stress results from the laterally directed component of muscle force on the balancing side and a laterally directed component of bite force, and perhaps working-side muscle force. The symphysis therefore experiences high stress concentrations and high strain magnitudes, especially along the lingual surface. This is best countered by fusion because bone is much stronger than ligaments (Ravosa and Rylander, 1994). Dorsoventral shear is another explanation concerning symphyseal fusion. In an experimental study with macaques it was found that the symphysis is sheared dorsoventrally during mastication and incision (Rylander, 1984). The amount of dorsoventral shear stress is directly related to the amount of vertically oriented muscle force transmitted across the symphysis from the balancing to the working side. As balancing-side muscle recruitment increases so does the amount of dorsoventral shear. This is also best countered by complete fusion of this joint (Ravosa and Rylander, 1994). 34

PAGE 45

Wishboning and dorsoventral shear are both present during mastication. However, according to Ravosa and Rylander (1994), it appears that wishboning is the most important determinant of symphyseal fusion in primates because the stresses associated with this loading regime are considerably higher than those associated with dorsoventral shear. It has been suggested, however, that fusion in early anthropoids may be due to increased dorsoventral shear resulting from increased recruitment of vertically directed balancing-side muscle force. Evidence for this can be seen in prosimians that seem to have evolved morphologies to resist dorsoventral shear such as having calcified or ossified ligaments. Also, wishboning stress appears to occur only in taxa who have already evolved complete fusion. Therefore, fusion in early anthropoids may have followed from increased dorsoventral shear and after fusion was attained, wishboning stress increased as greater amounts of horizontally-directed forces could be recruited. Both stiffness and strength hypotheses have been offered to explain the functional and evolutionary significance of symphyseal fusion. Histological and experimental evidence provide greater support for the strength hypothesis. Whether fusion occurred due to wishboning or dorsoventral shear stresses associated with the strength hypothesis is not entirely clear. It seems as though these loading patterns may both be involved in the evolution of symphyseal 35

PAGE 46

fusion. Experimental studies including other mammalian taxa need to be performed in order to clarify this issue. 36

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CHAPTER3 HYPOTHESES AND STUDY DESIGN The main goal of this study is to determine how applicable the constrained model, discussed in the previous chapter, is for understanding masticatory form and evolution in a wide range of mammals. The particular topic of concern here is whether the configuration of the masticatory system differs in a patterned way between mammals with symphyseal fusion and those without it and if this can be explained using the biomechanical principles underlying the constrained model. Of substantial interest is the distribution of Region II, the region where highest magnitude bite forces occur. Because of this the grinding dentition, particularly the molars, should be located here. This hypothesis has been tested and supported for anthropoid primates (Spencer, 1999), but it is not clear whether it holds true for other groups of mammals, particularly those without symphyseal fusion. If there is an association between the length of Region II and molar length within the mammals studied here, it will imply that masticatory system configuration, although highly can be explained through general principles ofthe constrained model. If such an association does not exist, it will suggest that the model is incorrect. 37

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The main prediction of this study is that observed molar length will be shorter than the predicted Region II length in all taxa. This is because according to the constrained model, the molars must lie within Region II. Maintaining the molars within this region can be accomplished by either decreasing overall molar length or increasing the length of Region II through the interaction of other craniofacial variables. This is because the distribution of Region II is determined by a number of variables, including bicondylar breadth, height of the TMJ, palatal breadth, and the anteroposterior position of the muscle resultant force. Determining the length ofRegion II is essential to understanding how these craniofacial variables interact within this model. Predicting the position of the premolars within Region II is problematic. The constrained model maintains that the grinding dentition should lie within this region. When applied to selenodont artiodactyls this may not be an issue because the molars and premolars are functionally similar and consequently labeled together as grinding dentition. However, other mammals, such as primates, have premolars that are distinct from the molars and may be functionally different. The diversity of premolar form within primates suggests functions including: puncturing, slicing, and perhaps molar-like crushing (Spencer, 1995). Therefore, many of these teeth cannot be defined as grinding dentition. Because ofthis diversity in premolar function, this study will focus on the molars. The model is accepted if at least all of the molars are located within Region II. 38

PAGE 49

The Effects of an Unfused Symphysis on the Model The constrained model was developed based on a mammal with a fused mandibular symphysis. Although this model was applied to selenodont artiodactyls, a mammal with an unfused symphysis, Greaves (1978) did not address how lack of fusion may affect the configuration of this model. Mammals with an unfused symphysis have a MRF that is located closer to the working side. This has the effect of shortening the length of Region II (Fig. 3.1). However, the constrained model assumes equal balancing side (B/S) to working side (W/S) muscle force ratio, which places the MRF in the midline. The length of Region II will, therefore, be calculated under the assumption of a midline MRF. This assumption is problematic, however, for mammals with an unfused mandibular symphysis as .it is essential to know the location ofthe MRF in order to calculate an exact length for Region II. This is because we know that the MRF in mammals with an unfused symphysis does not lie in the midline during mastication. These animals recruit relatively less balancing side muscle force than those with a fused symphysis, therefore, the MRF will be located more laterally. A laterally located MRF would cause the length of Region II to be shorter than if the MRF passed through the midline (Fig. 3.1 ). Although I assume that it is more laterally located, the exact location cannot be determined. This is because we do not know the working side/balancing side 39

PAGE 50

REGION I a .. ,' .. ,' ' ' ' ReGION I / ReatoNII 8atan<:ing Side Joint Reaaion Foree Working Side Joint Aeaalon Foroe B Midline muscle resultant force for mammids with a fused symphysis t:=::==J Laterally positioned muscle resultant force for mammals with an unfused symphysis Fig.1 Occlusal view of the mandible showing the effects of changes in the position of the muscle resultant force on the distribution of Region II. (a) The distribution of Region II with a midline muscle resultant force. (b) A laterally positioned muscle resultant force will create a Region II length, therefore, less of the molar tooth row will fall within this region. 40

PAGE 51

muscle force ratios for all species of mammals with an unfused symphysis. Therefore, an actual Region II length cannot be calculated for this group of mammals. This dilemma can be overcome by estimating the length of Region II for both groups of mammals assuming a midline MRF. This predicted length can then be compared to actual molar length in both groups. It is necessary to use this calculated length for mammals with an unfused symphysis because it is the only estimate of Region IT length that can be attained. It is predicted that the molar row will not fall outside of Region II for both groups. Mammals without fusion can compensate for having a theoretically shorter Region TI through cranial variables. The variables that impact the distribution of this region are bicondylar breadth, palatal breadth, height of the TMJ, and the A-Pposition ofthe MRF. All of these variables can increase the length of Region II regardless ofwhether an animal has a fused or unfused symphysis. The model demonstrates, however, that a lack of fusion creates a theoretically shorter Region II length in this group. These variables provide mammals with an Wlfused symphysis multiple ways to overcome this constraint. The variables discussed above provide mammals with a way to increase the length of Region II. These variables, however, are not independent of one another. Change in one variable is likely to covary with other variables (for example, an animal with a wide palatal length could have a decreased bicondylar 41

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breadth) perhaps still resulting in an increased Region II length. The main prediction will be upheld as long as the molars are maintained within the estimated Region II length, whether this is accomplished through a decrease in molar size/number or by increasing Region II length through these other craniofacial variables. The second prediction of this study is that the ratio between predicted Region II length and observed molar length will be higher in mammals with an unfused symphysis. This is because mammals with an unfused symphysis have a theoretically shorter Region II length due to the MRF being located laterally. Specific Predictions 1. Molar length will be shorter than the predicted length ofRegion II for both groups of mammals. 2. The ratio between predicted Region II length and observed molar length is predicted to be higher in mammals with an unfused symphysis because of the theoretically shorter Region II length. Study Design The variables related to the distributions of Region II were quantified so that comparisons between the two groups could be performed. A broad-scale comparative test was carried out on these hypotheses to determine the degree to 42

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which overall masticatory form in mammals is consistent with the principles of the constrained model. It is expected that the development of fusion and the changes it brought about in muscle force ratios, also brought about changes in cranial configuration. This was assessed through the principle of comparison (Fleagle, 1988). The comparative approach enables us to identify morphological trends of the masticatory system between taxa with symphyseal fusion and those without it. Identifying these trends is essential to uncovering the functional and evolutionary explanation for the change in morphological structure from an unfused mandibular symphysis to a fused one. Tests The main prediction of this study (Hypothesis 1) is that the molar tooth row will lie within Region II for all taxa. This hypothesis will be rejected if the observed length of the molars is greater than the predicted Region II length in any taxa. This test requires that the length of Region IT be predicted as it cannot be directly measured. This will be done under the assumption of a midline MRF for all mammals as previously discussed. The predicted length of Region II can then be compared to actual molar length in all taxa. Hypothesis 2 states that the ratio between predicted Region II length and molar row length will be higher in mammals with an unfused symphysis because 43

PAGE 54

their actual Region II length (although not able to be calculated) should be shorter. This hypothesis will be rejected if mammals with an unfused symphysis do not have a higher ratio of predicted Region II length to molar length. An allometric analysis is also important to explore patterns of variation in individual dimensions. An allometric analysis enables us to understand how changes in shape and size allow different taxa to accomplish the same behavior, which in this case is the maximization of force production in the masticatory system (Clutton-Brock and Harvey, 1979; Fleagle, 1988). 44

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CHAPTER4 MATERIALS AND METHODS Sample Measurements were taken on a total of 135 individuals representing 15 species with symphyseal fusion and 16 species without fusion (Table 4.1 and 4.2). Sample size ranges from 1 to 12 for the different species. Representative species from the orders Primates, Perissodactyla, Carnivora, and Artiodactyla were included in order to obtain a large sampling oftaxa in which to compare those with symphyseal fusion to those without fusion. It is important to include a variety of mammalian taxa in order to assess the generalizability of this model within the class Mammalia. Four species of artiodactyls, 12 carnivore species, 3 perissodactyl species, and 15 primate species are represented in this sample. Only adult crania (all teeth fully erupted) and their associated mandibles were used. Specimens were obtained from the Denver Museum of Nature and Science and the University of Colorado at Boulder Museum of Natural History. Although some species are represented by only a minimal number of specimens they were included in the study as my purpose is to sample a broad range of species within the mammalian order. 45

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Table 4.1 Taxa included in study with a fused symphysis Taxa with a Fused Symphysis Sample size Order-Cebus capuchinus 3 Primate Pithecia pithecia 2 Ateles geoffroyi 3 Aotus lemurinus 1 Alouatta palliata 4 Callithrix argentata 1 Saguinus geoffroyi 3 Cercopithecus diana 1 Cacajao calvus 1 Gorilla gorilla 2 Pongo pygmaeus 1 Order-Equus grevyi 2 Perissodactyla Equus equus 3 Tapirus terrestrius 2 Total =29 46

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Table 4.2 Taxa included in study with an unfused symphysis Taxa with an Unfused Symphysis Sample size OrderGalago demidoff 1 Primate Nycticebus coucang 1 OrderCanis latrans 12 Carnivora Canis lupis 9 Urocyon cinereoargenteus 8 Vulpus velox 11 Vulpus vulpus 11 Lynx canadensis 3 Lynx rufus 10 Puma concolor 10 Procyon lotor 10 Ursus americanus 9 Ursus arctos 2 OrderCervus elaphus 2 Artiodactyla Odocoileus hemionus 2 Odocoileus virginianus 3 Mazama americana 2 Total= 106 47

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Data Collection Process The following steps were taken in the data collection process to ensure that accurate measurements of the specimens were obtained: Step 1A Sony digital camera was used to photograph each specimen. Each specimen was placed on a level work surface. Two photographs were taken of each individual including both a transverse and occlusal view. When photographing the occlusal surface, each specimen was held in place with modeling clay. The modeling clay allowed the specimens to be positioned at a 90 degree angle to the table. The camera was placed on a tripod and aimed at the specimen. The camera was positioned as far from the specimen as possible while still filling the viewfinder with the skull. Filling the viewfinder with as much of the image as possible maximizes screen resolution which allows measurements to done on a computer with finer detail, thereby, reducing measurement error (Spencer, 1995). Step 2A calibration grid was set up and photographed before pictures of the specimens were taken. Any time the camera was moved, particularly upon completion ofa species, the calibration grid was repositioned andrephotographed. This image was used during analysis to determine the size of the space in which landmarks were located. 48

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Step 3Some of the landmarks that were measured were not clearly visible (such as the glenoid fossa). These were highlighted with small black dots, which helped with identification on the computer images. Step 4Images were downloaded into a Macintosh computer and measured within the MacMorph data acquisition package. Each image was calibrated using the corresponding calibration grid. Measurements were then taken of the variables for all images. Step 5The statistical packages J1v1P and Statview were used for all analyses. Measurements Three sets ofmeasurements were taken for this study: (1) distances that represent the observed length of the postcanine dentition, (2) dimensions for calculating a predicted Region II length, and (3) size adjustment measurements. Measuring the observed length of the postcanine dentition is necessary for Hypothesis 1 and 2. The observed length of the postcanine dentition was measured from the trigon of the maxillary last molar to the trigon of the first molar and from the trigon oflast molar to the trigon of the first premolar. The trigon of each tooth was chosen because of the biomechanical role it plays; this feature experiences much of the force that is produced during mastication when it comes into direct contact with the teeth of the mandible. 49

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The main hypothesis of this study is that the molars must lie within Region II. However, the length of both the premolars and molars were measured in order to assess the extent to which both types of teeth lie within Region II. Calculating an estimated Region II length is necessary for Hypothesis 1 and 2. There are five variables that can influence the location of Region II and must be quantified or estimated in order to calculate a predicted Region II length. These variables include bicondylar breadth, palatal breadth at M1, height of the TMJ relative to the occlusal plane, distance from the TMJ to point of intersection of the muscle resultant vector and occlusal plane, and the angle of the muscle resultant vector to the occlusal plane (Fig. 4.1). The first three variables can be directly quantified. However, it is difficult to quantify muscle resultant position and orientation due to limited knowledge of the comparative myology and function ofthe masticatory muscles among mammals (Throckmorton, 1989; Spencer, 1999). This study will therefore assume the MRF vector intersects the occlusal plane directly at the posterior end of the tooth row. This can be directly quantified as the distance from the TMJ (defined here as the center of the articular eminence) to the trigon of the last molar. This is consistent with the assumptions of the constrained model. A prior study involving muscle resultant force orientation for the masticatory adductor muscles of anthropoid primates estimated a fixed orientation of 80 degrees relative to the occlusal plane based on quantified orientations of the 50

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Fig. 4.1 I I I I I'' : : : 1., ; ..J'I: ........ ''t. : I /0 M .. ... I I I I; I I l I '' ". :. ......... I J A A = Biarticular breadth B = Palate Breadth at M 1 C = Distance from TMJ to point of intersection of muscle resultant vector and occlusal plane D =Height of articular eminence above occlusal plane E =Angle of muscle resultant vector to occlusal plane (90 equals perpendicular) Illustration of five variables used in the calculation of Region II length. 51

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anterior temporalis, superficial masseter, and medial pterygoid muscles (Spencer, 1999). Muscle resultant orientation cannot be calculated in the present study since it is likely to be highly dependent on individual muscle force magnitudes. Collecting these data is beyond the scope of this study. The length of the molar row is defined as the distance between the trigon of the right maxillary last molar to the trigon of the first molar. Bicondylar breadth is the distance between the central articular eminence landmarks. Height of the TMJ is the perpendicular distance of the right central articular eminence landmark from the occlusal plane. The occlusal plane is projected onto the sagittal plane and is defmed by the horizontal line connecting the distal end of the maxillary last molar to the mesial border of the maxillary last molar. Palatal breadth is the distance between the maxillary tooth rows at the trigons of M 1. The five variables were placed into an algorithm that calculates expected Region II length. The equation is shown in Figure 4.2 (Obtained from Spencer, 1995). Size Size and shape of the cranium are known to differ drastically among the mammals included in this study. Only when shape differences are teased apart from size differences can any meaningful comparison between these groups be 52

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o (DP2) Ps = arctan --C Effective Re.gion ll Length::;: P7 + P4 Fig. 4.2 Equation for estimating the Predicted length of Region II. 53

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made (Spencer, 1995). The calculation of the ratios of each dimension divided by the geometric mean of the cranium allows comparisons of masticatory system configurations between groups exhibiting different cranial sizes (Darroch and Mosimann, 1985). These size adjusted "shape variables" indicate relative proportions among the groups in this study. In this study the geometric mean involves four distances within the facial skeleton in order to get an accurate representation of overall cranial size. This serves as a size summary by combining multiple size dimensions into a single value. The equation for the geometric mean is: GM = (Dl D2 ... DN)(l/N) where D = distance value and N = number of distance values included in the summary of size. Distances used to assess the geometric mean were chosen as representative of overall masticatory system size as this is the system of concern to this study. The distances used are bicondylar breadth, palatal breadth, temporal foramen length, and molar row length. The landmarks used for measuring bicondylar breadth, palatal breadth, and molar row length have been defined above. Temporal foramen length is defmed as the distance between the right central articular eminence landmark and the inferior edge of the zygomatic arch. An allometric analysis was also done to assess how changes in shape with size affect the masticatory system. This is necessary because such a wide range 54

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of sizes are being sampled. The geometric mean was also used for allometric analyses. 55

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CHAPTERS RESULTS The parameters measured for this study are shown in Tables 5.1-5.3. All values are represented as sex-specific means and standard deviations (in mm) of all individuals. Table 5.1 displays mean values for molar row length, the length of the postcanine dentition, and the length of the molars to the incisors. Table. 5.2 lists mean values for the parameters used to calculate predicted Region II length. These parameters include: bicondylar breadth, palatal breadth, height ofTMJ, distance from the TMJ to the last molar. Table 5.3 reports the geometric mean, as well as one additional variable used in this calculation. The three other variables used in the calculation of the geometric mean, bicondylar breadth, height of the TMJ, and molar length, were reported in Tables 5.1 and 5.2. Figure 5.1 shows a box plot comparing postcanine dimensions to predicted Region II length. These dimensions include the observed length of the molar dentition, which is expected to be shorter than the predicted length of Region IT, and the observed distance between the most mesial premolar to the most distal molar. Also included in the plot is the calculated predicted length of Region II. 56

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Table 5.1 Mean (x) and standard deviation (sd) of tooth row length. (Molar length= length of all molars; M-P length= the last molar to the first premolar; M-1 length= last molar to first incisor) Taxon Sex n Molar Length M -P Length M-I Length PrlmatE!S x (mm) sd x (mm) sd x (mm) sd Cacajao. cufvus F 1 5.17 0 9.84 0 27.03 0 Cercopithecus diana F 1 7.32 0 1.82 0 27.85 0 Pongo pygmaeus F 1 23.8 0 39.88 0 75.05 0 Alouatta palliata F 2 32.9 1.27 64.81 5.54 93.14 0 Aotus lemurinus F 1 9.55 0 21.75 0 39.34 0 Gorilla _gorilla F 2 33.1 2:68 55.37 0.14 101.2 2.98 Cebus capuchinus F 3 8.64 0.23 19.76 0.41 34.79 1.18 Nycticebus coucang F 1 7.01 0 16.52 0 -A te/es F 3 10.28 0.48 21.9 0.66 36:73 1.2 Callithrfx argentata F 1 4.09 0 7.36 0 14.66 0 Galago demldoff ? 1 3.76 0 6.93 0 Saguinus geoffroyi M 2 4.61 0.13 8.58 0.12 17.45 0.44 F 2 4.51 0 8.64 0.22 17.39 0.19 Pithecia plthec/a F 2 6.88 0.67 14.83 0.81 27.49 1.58 Carnivores Lynx canadensis M 1 7.81 0 18.47 0 47.39 0 F 1 7.15 0 18.03 0 42.68 0 7 1 7.77 0 17.48 0 46.05 0 57

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Table 5.1 (cont.) Taxon Sex n Molar Length M P Length M-I Length Carnivores (Cont.) x (mm) sd x (mm) sd x (mm) sd Lynx rufus M 4 10.46 1.73 20."5 1.81 47.53 1.55 F 4 9.21 0.83 19.37 1.57 45.03 2.38 7 2 9.25 1.25 19.14 0.33 45.13 0.49 Puma concolor M 2 14.99 0.77 33.37 2.31 62.1 3.53 F 6 17.09 ( 1.8 37.56 1.68 66.75 2.14 1 2 15.34 0.77 32.18 2.5 59.23 3.78 Procyon lotor M 9 7.75 0.59 32.8 1.19 52.13 2.04 F 1 7.82 0 32.26 0 51.07 0 Ursus arctos M 1 48.98 0 66.32 0 154.1 0 7 1 31.23 0 58.14 0 125.79 0 Ursus americanus M 4 23.72 1.29 60.85 4.46 100.86 7.56 F 1 22.64 0 62.94 0 107.34 0 ? 4 27.37 2.81 65.64 3.10 105.97 3.40 Canis latrans M 6 10 0.47 64.72 3.53 96.23 .77 F 6 9.54 0.55 62.13 3.9 93 4.87 Canis lupls M 8 13.24 1.42 78.79 5.33 121.9 8.94 F 1 12.26 0 75.72 0 116.7 0 Vu/pes vulpes M 7 7.99 2.55 45.42 4.30 69.14 4.94 F 4 '8.81 2.57 44.74 4.50 68.57 4.97 Vulpes velox M 6 5.81 0.44 40.68 2.13 59.95 3.36 F 5 6.46 0.09 41.6.7 2.81 60.80 3.73 Urocyon M 5 6.95 0.45 39.59 1.23 58.17 3.77 clnereoargenteqs 7 2 6.93 0.07 35.84 2.01 54.51 2.11 58

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Table 5.1 (cont.) Taxon Sex n Molar Length MP Length M-I Length Artlodactyls x (mm) sd x (mm) sd x (mm) sd Cervus elephus M 1 57.38 0 121.53 0 --F 1 63.28 0 118.86 0 --Odoco/leus virgin/anus M 1 33.52 0 61.59 0 --F 2 30.31 0.09 67.27 1.65 -Odocoileus hemionus F 2 35.06 5.66 69.76 3.15 --Mazama americana F 2 21.91 2.11 47.25 7.24 -Perissodactyls Equus equus ? 3 45.97 2.54 106.89 5.83 232.17 1.77 Equus greyvi ? 2 46.05 2.00 1 1 1.45 6.17 228.14 2.47 Tapirus terrestrius F 1 45.59 0 120.82 0 217.59 0 ? 1 45.93 0 113.7 0 --59

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Table 5.2 Mean (x) and standard deviation (sd) of variables used to calculate Predicted Region II length. Taxon Sex n Blcondylar Breadth Palatal Breadth Height of TMJ Primates x (mm) sd x (mm) sd x (mm) sd Cacajao cu/vus F 1 31.41 0 18.81 0 13.8 0 Cercopithecus diana F 1 30.14 0 19.69 0 5.5 0 Pongo pygmaeus F 1 84.01 0 51.64 0 27.7 0 Alouatta pa/liata F 2 128.88 0.91 79.79 3.44 84.8 1.7 Aotus /emurlnus F 1 51.83 0 29.56 29.56 7.8 0 Gorilla gorilla F 2 107.86 0.36 63.51 0.25 55.45 9.4 Cebus capuchinus F 3 44.27 3.7 26.29 0.39 6.13 2.15 Nyctlcebus coucang F 1 28.92 0 17.57 0 1.9 0 Ate/es geoffroyi F 3 46.65 3.64 26.06 2.03 12.6 2.55 Callfthrix argentata F 1 20.29 0 11.48 0 2.76 0 Ga/ago demidoff ? 1 13.74 0 9.02 0 2.65 0 Sagulni.Js geoffroyi M 2 23.26 0.44 13.74 0.29 3.52 1.65 F 2 23.47 0.44 13.94 0.58 5.31 1.44 Pithec/a pithecia F 2 34.91 1.67 17.94 0.33 8.75 3.04 Carnivores Lynx canadensis M 1 65.76 0 44.83 0 2.2 0 F 1 53.33 0 39.8 0 o.s 0 7 1 67.24 0 43.61 0 1 0 Lynx rufus M 4 65.49 3.76 39.32 1.5 1.28 0.92 F 4 64.81 3.13 40.62. 4.11 0.8 0.63 7 2 60.83 2.95 38.15 2.01 0.6 0.14 60'

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Table S.l (cont.) Taxon Sex n Blcondylar Breadth Palatal Breadth Height of TMJ Carnivores x (mm) sd x (mm) sd x (mm) sd !(cont.) Puma concolor M 2 82.78 7.81 54.81 3.97 -0.15 0.21 F 6 87.74 3.94 57.23 2.84 -2.12 0.58 7 2 79.42 4.38 51.86 3.27 -6.55 0.07 Procyon lotor M 9 52.51 1.72 .. 33.58 1.16 1.1 1 1.06 F 1 54.29 0 34.29 0 1.4 0 Ursus arctos M 1 156.48 0 81.89 0 -18.9 0 7 1 107.31 0 59.36 0 -16.9 0 Ursus americanus M 4 109.43 6.34 52.61 3.54 -8.53 21.75 F 1 115.57 0 52.06 0 -24.2 0 ? 4 109.13 5.73 53.82 0.72 5.6 10.87 Canis latrans M 6 64.68 12 44.41 3.02 11.85 3.26 F 6 63.86 4.5 43.53 1.12 11.6 3.08 Canis lupis M 8 89.41 5:93 68.24 4.6 7.39 13.62 F 1 80.74 0 62.17 o 2.2 0 Vulpes vulpes M 7 48.91 3.7 32.02 2.68 0.03 1.71 F 4 46.76 2.63 32.33 2.38 0.58 2.92 Vulpes velox M 6 41.98 1.99 28.98 1.55 1.04 0.5 F 5 43.87 2.21 29.81 1.03 1.75 1.47 Urocyon M 5 45.27 1.95 28.18 0.92 6.8 1.61 clnereoargenteus ? 2 41.48 0.64 27.31 1.07 5.3 0.16 Artlodactyls Cervus elaphus M 1 112.19 0 99.36 0 34 0 F 1 101.69 0 94.92 0 20.4 0 Odocoi/eus M 1 68.74 0 58.8 0 27.5 0 virgin/anus F 2 71.89 2.6 59.13 5.15 22.45 3.75 61

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Table S.2 (cant) Taxon Sex n Bicondylar Breadth Palatal Breadth Height of TMJ Artlodactyls x (mm) sd x (mm) sd x (mm) sd l(cont.) Odocoileus F 2 77.62 4.75 68.43 4.01 25.5 2.83 hem/onus Mazama americana F 2 54.15 2.45 52.46 3.68 21.1 6.22 Perlssodactyls Equus equus 7 3 105.66 8.33 82.33 3.14 89.65 2.61 Equus greyv/ 7 2 103.71 3.56 83.35 2.0 81.8 4.38 Tapirus terrestr/us F 1 123.08 0 89.12 0 51.3 0 7 1 119.88 0 86.54 0 57.1 0 Taxon Sex n Molar to TMJ Est. Reg. II Length Primates x (mm) sd x (mm) sd Caca}ao culvus F 1 21.15 0 14.13 0 Cercopithecus diana F 1 20.73 0 14.18 0 Pongo pygmaeus F 1 49.09 0 33.18 0 Alouatta palliats F 2 96.32 2.6 68.87 0.69 Aotus lemurinus F 1 23.64 0 14.27 0 Gorilla gorilla F 2 55.68 8.34 38.55 5.86 Cebus capuchlnus F 3 23.34 3.13 14.47 1.1 Nyctlcebus coucang F 1 17.52 0 10.85 0 Ate/es geoffroyl F 3 28.11 3.07 16.95 .72 Calllthrix argentata F 1 11.43 0 6.74 0 Gslago demidoff 7 7.55 0 5.26 0 62

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Table S.2 (cont.) Taxon Sex n Molar to TMJ Est. Reg. II Length Primates (cont.) x (mm) sd x (mm) sd Sagu/nus geoffroy/ M 2 12.76 0.43 7.9 0.41 F 2 12.76 0.15 8.14 0.25 P/thec/a pfthecfa F 2 20.5 0.5 11.33 0.35 Carnivores Lynx canadensis M 1 38.57 0 26.56 0 F 1 31.7 0 23.72 0 7 1 37.1 0 24.18 0 Lynx rufus M 4 32.32 2.75 21.71 2.48 F 4 34.16 3.06 20.4 1.8 7 2 30.82 0.84 18.32 0.48 Puma concolor M 2 52.25 6.89 37.3 0 F 6 48.47 3.14 31.86 2.18 ? 2 47.39 0.87 31.7 0.81 Procyon Jotor M 9 31.41 2.76 20.0 0.88 F 1 29.38 0 18 .. 71 0 Ursus arctos M 1 83.94 0 45.67 0 7 1 79.64 0 45.7 0 Ursus americanus M 4 70.2 6 35.41 2.93 F 1 77.83 0 36.98 0 7 4 71.21 5.31 35.99 1.2 Canis Jatrans M 6 45.2 2.44 33.15 2.58 F 6 46.52 2.55 33.17 1.9 Canis lupis M 8 60.88 7.37 48.13 7.2 F 1 56.57 0 44.09 0 Vulpes vulpes M 7 30.81 1.16 20.3 0.55 F 4 28.95 1.24 20.23 0.94 Vu/pes velox M 6 25.95 0.9 18.04 0.62 F 5 25.85 1.22 17.79 0.82 63

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Table 5.2 (cont) Taxon Sex n Molar to TMJ Est. Reg. II Length Carnivores x (mm) sd x (mm) sd (cont.) Urocyon M 5 30.54 3.28 19.75 2.01 clnereoargenteus 7 2 29.02 0.63 19.74 1.51 Artlodactyls Cervus e/aphus M 1 93.84 0 88.43 0 F 1 71.75 0 70.34 0 Odocolfeus M 1 54.94 0 51.15 0 virg/nianus F 2 56.65 3.71 49.91 6.13 Odocoileus F 2 57.37 10.12 54.23 1.96 hemionus Mazama americana F 2 43.31 8.32 45.66 10.25 Perissodactyls Equus equus 7 3 108.28 4.94 96.85 3.11 Equus greyvi 1 2 105.93 2.8 96.75 0.63 Taplrus terrestrlus F 1 66.88 0 54.99 0 7 1 74.52 0 61.07 0 64

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Table 5.3 Mean (x) and standard deviation (sd) of the geometric mean and one variabfe used to calculate it. The other variables used in this calculation are listed in Tables 5.1 and 5.2. Taxon Sex n Geometric Mean Temporal Foramen Length Primates x (mm) sd x (mm) sd Cacajao cu/vus F 1 15.60 0 19.38 0 Cercopithecus diana F 1 16.80 0 18.35 0 Pongo pygmaeus F 1 45.02 0 39.84 0 A/ouatta palliats F 2 72.11 1.71 79.95 0.49 Aotus lemur/nus F 1 20.8 0 12.8 0 Gorilla gorilla F 2 59.51 2.29 55.42 3.63 Cebus capuchinus F 3 22.27 0.94 24.47 1.24 Nycticebus coucang F 1 15.85 0 17.73 0 Ateles geoffroyi F 3 22.94 1.47 22.16 1.39 Calllthrix argentata F. 1 9.76 0 9.53 0 Galago demldoff 7 1 7.51 0 6 .. 84 0 Saguinus geoffroyi M 2 11.32 0.08 11.16 0.21 Sagu/nus geoffroyi F 2 11.11 0.18 10.33 0.04 Pithecia plthecia F 2 16.47 0.76 17.10 0.36 Carnivores Lynx canadensis M 1 28.62 0 29.16 0 F 1 24.98 0 28.19 0 7 27.73 0 23.6 0 Lynx rufus M 4 29.18 3.07 30.89 1.18 F 4 29.30 0.72 26.10 2.08 7 2 26.93 2.24 25.66 0.88 65

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Table S.3 (Cont.) Taxon Sex n Geometric Mean Temporal Foramen Len :Jth Carnivores (cont.) x (mm) sd x (mm) sd Puma conco/or M 2 41.87 2.34 45.39 4.88 F 6 43.61 1.19 42.44 2.08 ? 2 40.24 1.68 41.55 0.06 Procyon lotor M 9 0.63 29.48 2.80 F 1 26.22 0 27.24 0 Ursus arctos M 1 85.20 0 83.95 0 ? 1 62.05 0 74.5 0 Ursus americanus M 4 54.20 3.17 63.35 5.93 F 1 55.97 0 72.04 0 7 4 56.92 0.01 65.69 3.78 Canis latrans M 6 33.61 1.14 42.63 3.57 F 6 32.01 1.05 41.53 1.92 Canis lupis M 8 45.97 2.75 55.71 5.64 F 1 42.57 0 53.34 0 Vu/pes vulpes M 7 24.62 2.99 29.99 1.75 F 4 24.72 2.86 28.54 1.52 Vu/pes velox M 6 20.48 0.82 24.96 0.70 F 5 21.38 0.62 24.78 1.50 U. cinereoargenteus M 5 22.38 0.97 28.35 2.55 7 2 21.37 0.55 26.61 1.82 .66

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Table S.3 (cont.) Taxon Sex n Geometric Mean Temporal Foramen Length Artiodactyls x (mm) sd x (mm) sd Cervus e/aphus M 1 87.32 0 90.9 0 F 1 85.38 0 87.02 0 Odocoileus M 1 50.77 0 49.03 0 virgin/anus F 2 51.53 1.05 54.88 2.14 Odocoi/eus hemionus F 2 55.88 2.05 52.73 0.66 Mazama americana F 2 41.27 3.47 46.68 5.78 Perlssodactyls Equus equus ? 3 67.73 1.90 52.96 5.50 Equus greyvi ? 2 68.26 3.14 54.59 4.49 Taplrus terrestrlus F 1 75.89 0 66.32 0 ? 1 75.49 0 68.15 0

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In order to more easily understand these graphs, the data were standardized against molar row length. This has the effect of reducing differences in size and shape between the different taxa. Figure 5.1 shows that the molar length of all species fit well within the predicted length ofRegion II. Hypothesis 1, which states the molars should fit within the predicted length of Region II, is therefore accepted. Many species also have premolars that fall within the predicted length of Region II. This is not surprising given the wide range of premolar function that is represented across mammals. However, what is unusual is that those species with the typical "grinding dentition" do not necessarily have premolars that fit into this region. For example, Cervus elaphus has premolars that fall well outside of Region II, whereas, Mazama americana's premolars all fall within this region. In addition, a primate, Saguinus geoffroyi, has premolars that fall well within the region. It appears that there is no apparent pattern in regard to premolar length and the predicted length of Region II. This may be due to the wide variety of functions that premolars serve. Perhaps there is too much variation within the function of this type of tooth across mammals to be able to predict its location relative to Region II length. 68

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T. terrestrius I OJ E. greyvl I i E. equus 1 m 0. virgin/anus ID 0. hem/onus rn M. americana C. elaphus rn V. vulpes I f, V. velox ,._, t-Q]----; U. cinereoargenteus ...--. r-{D U. arctos U. americanus 4 {]}. P. lotor t-[]}-; P. concolor L. rufus ..... L. canadensis C. /upis C.latrans t{[}; 5. geoffroyi I P. pygmaeus I I P. pithec/a m N. coucang I I .G. gorilla [[] G. demidoff I I C. diana II C. capuchlnus I u C. calvus IiI C. argentata A. pal/lata [) A. lemur/nus I I A. geoffroy/ I m I I I I 0 2. 3 4 5 6 7 B 9 10 Figure 5.1 Plot comparing postcanine dimensions to predicted Region II length. All distances have been standardized against molar length. Therefore, the molar dentition fills the distance between 0 and I with the first molar being located at 1. The black boxes represent the anterior end of Region II. Region II, therefore, extends from 0 to this anterior position. The white boxes represent the anterior end of the premolars, therefore, the total length of the postcanine dentition extends from 0 to this anterior position. 69

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Fused vs. Unfused Mammals with an unfused symphysis are expected to have a theoretically shorter Region II length due to a laterally located muscle resultant force. According to Hypothesis 2, it is expected that the ratio between predicted Region II length and observed molar length will be higher in mammals with an unfused symphysis because of their theoretically shorter Region II length. A t test between predicted Region II length to observed molar length by species with a fused or unfused symphysis shows that the means for these two groups are significantly different. The mean for the fused group is 1. 75 with a standard deviation of0.35 and 2.54 for the unfused group with a standard deviation of0.81 (t = -5.13; p < 0.01). This is also represented graphically in Fig. 5.2. This plot shows that mammals with an unfused symphysis tend to have higher predicted Region II length/observed molar length ratios. Hypothesis 2 is accepted. Allometric Analysis An allometric analysis was performed on the variables used to calculate the predicted length of Region II. It is important to assess how changes in shape with size affect these variables because of the wide range of cranial sizes represented in this sample. Figure 5.3 displays bivariate plots of log-transformed data showing the relationship between the geometric mean, bicondylar breadth, palatal 70

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3.75 3.5 -r---' 3.25 3 c ol1l 2.75 _. 1.. .!2 0 I: 2.5 ., ol1l > 1.. ol1l 1:l 2.25 0 ........ ., ol1l .... 2 ., T f: a.. 1.75 1.5 j_ -'----1.25 Fused Unfused Figure 5.2 Box plot representing the means of molar length divided by the geometric mean for mammals with a fuse and unfused symphysis. These plots show that mammals with an unfused symphysis have a higher predicted to observed molar length ratio.

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breadth, height of the TMJ, and molar row length. Table 5.51ists the slope andYintercept for these variables. Molar length scales with positive allometry. Its 95% confidence intervals range from 1.19-1.36. This means that as cranial size increases molar length increases at an even greater rate. This is surprising considering that increasing the length ofthe molars decreases the likelihood that they will fit into Region II. Other variables must be configured in order to increase the length of this region. Bicondylar breadth scales with negative allometry. Their 95% confidence intervals range from 0.81-0.89 and 0.87-0.96, respectively. As cranial size increases bicondylar breadth does not increase as rapidly. Decreasing the length between the condyles has the effect of increasing the length ofRegion II. TMJ height scales with strong positive allometry. Its 95% confidence intervals range from 2.35-3.52. However, there is only a weak correlation between height of the TMJ and cranial size(?= 0.62). This is expected considering that some species with very large cranial sizes, such as horses, have very tall TMJ's, whereas, other species with large crania have very low TMJ's, such as bears. Height of the TMJ is extremely variable within mammals. The anteroposterior position of the MRF is another variable that affects the distribution ofRegion II. The A-Pposition of the MRF is assumed to be located at the distal end of the molar tooth row, therefore, the length from the TMJ to the last molar represents this distance. This variable scales isometrically 72

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Figure 5.3 Bivariate plots of log-transfonned data showing the relationship between cranial size (represented by the geometric mean) and other masticatory variables relevant to the distribution of Region II (+indicate species without symphyseal fusion, indicate species with fusion). See Table 5.8 for regression parameters. 1.9 1.8 1.7 .6 1.5 ,...... 1.4 :15 Q'l 1.3 c Q/ ...J 1 .2 II. 1.1 0 6 1.0 . 0.9 0.8 0.7 0.6 0.5 2.2 2.1 2".0 1.9 t 1.8 1.7 .!!! f 1.6 1.5 e 1.4 .s 1.3 1.2 .8 .9 1.0 1.1 1.3 1.5 1.7 1.9 In (Geometric Mean) y = 0.85x + 0.51 1.1 .8 .9 1.0 1.1 1.3 1.5 1.7 1.9 In (Geometric Mean)

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Figure 5.3 (cont.) 2.0 1.9 1.8 .......... 1.7 1.6 Cll 1.5 L. m 1.4 .!!1. "' 1 3 0.. '-" 1.2 1.1 1.0 + y = 0 .92x + 0.23 0.9 .8 .9 1.0 1.1 1.3 1.5 1.7 1.9 In (Geometric Mean) 2.0-r---------------,...,---. 1.5 1.0 ,._ 0 -:. 0.5 .5 0.0 -0.5 a II B y= 2.93x-3.85 .8 .9 1.0 1.1 1.3 1.5 1.7 1.9 In (Geometric Mean) 74

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Figure 5.3 (cont.) 2.1 ,...., 1.9 1.8 1.7 1.6 1.5 1.4 .k 1.3 '-" .!: 1.2 1.1 1.0 0.9 + 0.8 y = 0.99x + 0.10 .. 8 .9 1.0 1.1 1.3 1.5 1.7 1.9 In (Geometric Mean) 75

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Table 5.4 Variables regressed on the geometric mean to assess allometric relationships. Variable r:z. Slope -intercept 95% Confidence (Re2ressed on GM) intervals for Slope Molar Length 0.88 1.28 -0.84 1.19-1.36 Bicondylar Breadth 0.93 0.85 0.51 0.81-0.89 Palatal Breadth 0.94 0.92 0.23 0.88-0.96 Height of TMJ 0.62 2.93 -3.85 2.35-3.52 A-P Position of 0.93 0.99 0.10 0.95-1.04 MRF 76

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with the geometric mean; its 95% confidence intervals include a slope of I. The length from the last molar to the TMJ and cranial size increase at the same rate. Palatal breadth also scales very close to isometry. Its 95% confidence intervals range from 0.88 to 0.96. As cranial size increases, palatal breadth increases at about the same rate. These allometric analyses have allowed us to look further into the data and explore how these variables change in relation to cranial size. This helps us to understand the relationship between cranial size and the masticatory system variables that are responsible for the distribution of Region II. 77

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CHAPTER6 DISCUSSION The goal of this study is to test the constrained model on a wide variety of mammals in order to assess systematic differences in craniofacial configuration that may be the result of mandibular symphyseal fusion. Of particular concern is the distribution of Region IT. This region should envelope the grinding dentition because this is where highest magnitude bite forces are produced and these are the teeth most suitable for these forces. The following is a summary of my predictions: 1. Molar length will be shorter than the predicted length of Region II for both groups of mammals. 2. The ratio between predicted Region II length and observed molar length is predicted to be higher in mammals with an unfused symphysis because of the theoretically shorter Region II length. The following is a summary of the results obtained from this study: 78

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1. The molar row of all species fit well within the predicted length ofRegion II. Hypothesis 1 is accepted. 2. The ratio between predicted Region II length and observed molar length is higher in mammals with an unfused symphysis. Hypothesis 2 is accepted. The results from this study support the general applicability of the constrained model for both mammals with a fused and unfused symphysis. This model is able to explain the interactions between some craniofacial variables of a broad range of mammals including primates, carnivores, perissodactyls, and artiodactyls. This is in accordance with results from previous studies. Support for Previous Research The main assumption of the constrained model is that the TMJ should not be subjected to joint distraction (Greaves, 1978). This causes limitations on masticatory system form and function. One such limitation is the distribution of the Region II. According to the model, Region II should envelope at least all of the molars. Spencer's (1999) morphometric analysis of anthropoid cranial configuration provides evidence in support of this. Anthropoids appear to exhibit craniofacial form that is consistent with selection against TMJ distraction by having the parameters that influence the distribution of the region be configured 79

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in such a way so that at least all molars are maintained within this region. This is consistent with results from the present study. Results from the present study also indicate that the molars as well as some of the premolars are located within Region II. The goal ofthe present study was to look at the broad-scale applicability ofthe constrained model to mammals. Not only were a wider range oftaxa used than in previous studies, this study also focused on a craniofacial feature that results in fundamental differences in muscle recruitment patterns, the mandibular symphysis. Previous studies involving galagos, lemurs, and dogs, all ofwhich have an unfused symphysis, have found that these species exhibit different working-side to balancing-side muscle force ratios (Leibman and Kussick, 1965; Rylander, 1979; Dessem, 1989; Rylander et. a!., 1998, 2002). Those species with an unfused symphysis recruit less balancing-side muscle force. Recruiting less balancing-side muscle force means that the muscle resultant force cannot lie in the midline. Having a l\t1RF more laterally located consequently reduces the length of Region II. Because mammals with an unfused symphysis have a theoretically shorter Region II they are expected to have a higher ratio between predicted Region II length and observed molar length. This was indeed found to be the case. Mammals with an unfused symphysis appear to differ in a systematic way from those with fusion. 80

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There has only been a small number of mammalian species whose muscle recruitment patterns have been studied (Leibman and Kussick, 1965; Rylander, 1979; Dessem, 1989; Rylander et. a/., 1998, 2002). The finding from this study provides indirect evidence for the previous research on muscle recruitment patterns. It can be inferred from this fmding that all species with an unfused symphysis may recruit less balancing-side muscle force. This would cause them to have a shorter Region II length which would .create a higher ratio between predicted Region II length and observed molar length as was found in this study. Allometric Analyses An allometric analysis has allowed us to understand how changes in size affect the masticatory system configuration of such a wide range of mammals. It was found that molar length and height of the TMJ scale with positive allometry, bicondylar breadth scales with negative allometry, and palatal breadth and the Apposition of the MRF scale isometrically. Molar length was found to scale with positive allometry. This means that as cranial size increases, the molars are increasing in length more rapidly than expected for isometry. So as cranial size increases the molars are increasing at an even greater rate. One would expect that the molars would not increase at this greater rate given the constraint of Region II length. According to the model, the molars 81

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should fall within this region in order to avoid TMJ distraction. However, there are other factors that act to influence this variable, such as diet. Diet has been shown to be closely linked to body size, particularly within primates (Fleagle, 1988). An animal's teeth are what allow it to meet its nutritional requirements. Therefore, molar size is constrained not only by the need to avoid TMJ distraction but also by diet. This becomes even more complex when one considers that diet can also be a function of body size. Within primates, the natural physiological break between insectivores and folivores occurs at 500 grams and is known as Kay's threshold (Kay, 1975; Fleagle, 1988). In general, folivorous primates have body weights that are no less than 500 grams and insectivores tend to weigh less than this limit. This is because as body size increases, metabolic requirements change. A larger animal actually has relatively lower energy requirements than a smaller one. Although leaves are generally lower in energy yield than insects or fruit, a large animal can afford this because they need less energy per kilogram of mass than a small animal (Fleagle, 1988). Results from the allometric analysis of this study show that as cranial size increases (and therefore body size) molar length is increasing even more rapidly. This may be because even though a larger animal does not have as high of energy needs it still requires larger molars in order to process vegetation which is much tougher than either insects or fruit. 82

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This idea also pertains to artiodactyls and perissodactyls. Both of these groups have very large grinding dentition. This is thought to be an adaptation to the tough grass material that comprises their diet (L.M. Spencer, 1995). Having larger molars increases the surface area of the grinding dentition thereby allowing a more efficient breakdown of the tough vegetation. Although this can explain why molar length increases more rapidly than body size this does not explain how the larger molars are able to fit within Region II. Other craniofacial variables must be working to increase the length of Region II. TMJ height may be one of those variables. It scales with positive allometry meaning that it also increases at an even greater rate that cranial size. Many of the larger animals such as horses, sheep, and deer have very high TMJ's. This has the effect of orienting Region II more anteriorly which increases its length. Bicondylar breadth also seems to be working to increase the length of Region II as cranial size increases. This is becausea decrease in bicondylar breadth increases the length of this region. This parameter scales with negative allometry; it does not increase as rapidly as cranial size. Therefore, having bicondylar breadth not increase as rapidly as cranial size may help the molars to be maintained within Region II even as cranial size increases. Overall it appears that as cranial size increases across mammals, there are variables that are configured in such a way to increase the length ofRegion II. 83

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This allows the molars to be maintained within this region even if they are increasing at a rate greater than cranial size. Biomechanical hnplications for Diet Spencer (1999) proposes the idea that morphological patterns within the masticatory system may stem from the selective trade-off between increasing bite force magnitudes and avoiding joint distraction. Joint distraction is unavoidable when biting occurs in Region III (Greaves, 1978). Selection should favor a morphology that does not allow teeth to be located within this region This idea has been supported by the present study and the previous research discussed above. The molars were found to lie only within Region II. However, some species which require high magnitude bite forces to be produced on either the incisors or the canines would benefit from having the teeth located more posteriorly. A more posterior position for the dentition and a relatively anterior position ofthe superficial masseter and anterior temporalis muscles enable greater force production in Region I (the region where the incisors and canines are expected to be located according to the constrained model) (Spencer and Demes, 1993; Spencer, 1999). This could potentially cause some ofthe postcanine dentition to be moved back into Region ill. Some groups that may benefit from greater force production on their anterior dentition include callitrichids, some pitheciines, Cebus, and some of the 84

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carnivores (Spencer, 1999). These groups are specialized for intensive force production on either the incisors or the canines. Carnivores require especially high levels of force production on their canines as these are the teeth that are involved in capturing and killing their prey. The TMJ of this group is also particularly well-suited to capturing prey. It is "locked" into the glenoid fossa so that it can withstand the high magnitude forces that are inevitable as the prey struggles to free itself from the grip ofthe predator. This is likely an adaptation to limit joint distraction. Selection can limit teeth being located within Region ill through changes in the configuration of other variables of the masticatory system. Some changes that would be beneficial include decreasing bicondylar breadth or increasing palatal breadth. Increasing the height of the TMJ would also be beneficial. All of these have the effect of increasing the length ofRegion II which may enable the molars to be maintained with this region. The avoidance of TMJ distraction places constraints on masticatory system morphology. Mandibular symphyseal fusion is another constraint. Fusion constrains the location of the MRF because it allows more balancing-side muscle force to be recruited. This has the potential effect of placing the MRF in the midline. However, the MRF must move laterally when biting in Region IT in order to be maintained within the triangle of support (Spencer and Demes, 1993; Spencer, 1995, 1998, 1999). A reduction in balancing-side muscle activity 85

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enables this to happen and is necessary to avoid TMJ distraction. Although having symphyseal fusion allows more balancing-side muscle force to be recruited, the constrained model limits this in order to maintain the resultant force within the triangle of support. Maintaining an equal balancing-side to working-side muscle force ratio (which would place the MRF in the midline) is acceptable within this model when biting in Region I. This is because Region I envelopes a midline MRF. What is interesting, however, is that none of the carnivores, which require high magnitude force production on their anterior dentition, have a fused symphysis. It is expected that this group would benefit from having a fused symphysis in order to recruit more balancing-side muscle activity. One possible explanation can be approached from a behavioral standpoint. When a carnivore captures its prey in its jaws it may be biting equally with all of its anterior dentition. This would make both sides of the jaw the working side and may allow the predator to maximize force production. They, therefore, do not need to have symphyseal fusion because they are already maximizing the amount of force that can be produced. Further behavioral research would serve to clarify this issue. Some species which need high magnitude forces produced on their molars, such as the artiodactyls, also do not have a fused symphysis (Greaves, 1978). It is possible that selection has favored another aspect of their craniofacial morphology 86

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that enables them to efficiently process their food. One such morphology could be their large selenodont grinding dentition Having this larger surface area allows them to more efficiently process the tough vegetation that is characteristic of their diet. Colobines, which have a fused mandibular symphysis, also have a diet that consists of tough vegetable matter (Fleagle, 1988). Perhaps having a fused symphysis enables them to generate a higher magnitude muscle resultant force on their molars than the artiodactyls, which allows them to break down their food material in just as efficient a manner. lfthis is the case then symphyseal fusion would be highly advantageous for this group. This issue is not a simple one due to the numerous confounding variables that are possible within not only the masticatory system but also the digestive system. Colobines also have a gut morphology that is specialized to break down vegetation, including large complex stomachs (Fleagle, 1988). It is not only the molars that work to break down the food, but their specialized stomachs as well. The issue of symphyseal fusion and its impact on diet is extremely complex. There are numerous variables within not only the masticatory system but the body as a whole that work together to produce an animal that is finely adapted to its diet and environment. Further research is needed before any definitive statements can be made regarding symphyseal fusion and diet. 87

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Conclusion Overall it appears that mammals with a fused symphysis do differ in systematic ways from those with an unfused symphysis. Both groups have molars that fit within the estimated length of Region II. However, the fact that mammals with an unfused symphysis have higher predicted Region II length to observed molar length ratio shows that these groups differ in craniofacial configuration that is most likely the result of different workingto balancing-side muscle force ratios that are in turn the result of differing symphyseal morphologies. This has important ramifications for the evolution of symphyseal fusion. Variables within the masticatory system interact with one another to produce a utiique configuration. However, "evolution" can only be so creative due to the constraints of these variables having to interact with one anther in order to maintain the molars within Region II. This is in addition to numerous other constraints that can be imposed upon this system. The molars should always be maintained within Region II in order to maximize masticatory force production. Extinct species both with and without fusion, should have craniofacial variables, such as bicondylar and palatal breadth, height ofthe TMJ, and the A-Pposition of the MRF that also work in conjunction to increase the length of Region II. Looking to see if these features are correlated with early symphyseal fusion would help us to understand how fusion affects the masticatory system as a whole and why it came to be in the first place. 88

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Cranial form and function are extremely important issues to biological anthropologists. Understanding craniofacial form and the changes in morphology that have occurred through time is dependent on this biomechanical model. This model allows us to understand why certain variables are distributed in the way they are, such as the placement ofthe molars within the cranium, or the height of the TMJ above the occlusal plane. This can be used to explain much of the variation we see in cranial form in both extinct and extant forms, and in both groups with and without mandibular symphyseal fusion. 89

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DuBrul, E. L. (1977). Early hominid feeding mechanisms. American Journal of Physical Anthropolology 47:305-320. Dumont, E.R. and A. Rerrel (2003). The effects of gape angle and bite point on bite force in bats. Journal of Experimental Biology 206:2117-2123. Fleagle, J.G. (1988). Primate Adaptation and Evolution. New York: Academic Press. Gingerich, P.D. (1971). Functional significance of mandibular translation in vertebrate jaws. Postilla 152:1-10. Greaves, W.S. (1978). The jaw lever system in ungulates: a new model. Journal ofZoology 184, 271-285. Greaves, W.S. (1982). A mechanical limitation on the position of the jaw muscles of mammals: the one-third rule. Journal ofMammalogy 63: 261-266. Greaves, W.S. (1988). A functional consequence of an ossified mandibular symphysis. American Journal of Physical Anthropology 77:53-56. Gysi, A. (1921). Studies on the leverage problems ofthe mandible. Dental Digest 27, 74-84, 144-150, 203-208. Riiemae, K. (1984). Functional aspects of primate jaw morphology. In D.J. Chivers, B.A. Wood, and A. Bilsborough (eds.): Food Acquisition and Processing in Primates. New York: Plenum Press, pp. 257-281. Rylander, W.L.(1975). The human mandible: lever or link? American Journal of Physical Anthropology 43, 227-242. Rylander, W.L. (1977). In vivo bone strain in the mandible of Galago crassicaudatus. American Journal of Physical Anthropology 46:309-326. Rylander, W.L.(1979a). An experimental analysis oftemporomandibular joint reaction force in macaques. American Journal of Physical Anthropology 51, 433-456. Rylander, W.L. (1979b). The functional significance of primate mandibular form. Journal of Morphology 160, 223-240. 91

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Rylander, W.L. (1984). Stress and strain in the mandibular symphysis of primates: a test of competing hypotheses. American Journal of Physical Anthropology 64:1-46. Rylander, W.L. (1985). Mandibular function and temporomandibular joint loading. In D.S. Carlson, J.A. McNamara, and K.A. Ribbens (eds.): Developmental Aspects of Temporomandibular Joint Disorders. Ann Arbor, MI: University of Michigan Rylander, W.L. and K.R. Johnson (1994). Jaw muscle function and wishboning of the mandible during mastication in macaques and baboons. American Journal of Physical Anthropology 94:523-547. Rylander, W.L., Ravosa, M.J., Ross, C.F., and K.R. Johnson (1998). Mandibular corpus strain in primates: further evidence for a functional link between symphyseal fusion and jaw-adductor muscle force. American Journal of Physical Anthropology 107:257-271. Rylander, W.L., Vinyard, C.J., Wall, C.E., Williams, S.R., and K.R. Johnson (2002). Recruitment and firing patterns of jaw muscles during mastication in ring-tailed lemurs. American Journal of Physical Anthropology Supplement 117:88. Kay, R.F. (1975). The functional adaptations ofprimate molar teeth. American Journal of Physical Anthropology 43: 195-216. Kay, R.F. and R.R. Covert (1984). Anatomy and behaviour of extinct primates. In D.J. Chivers, B.A. Wood, and A. Bilsborough (eds.): Food Acquisition and Processing in Primates. New York: Plenum Press Kay, R.F. and K.M. Riiemae (1974). Jaw movement and tooth use in recent and fossil primates. American Journal of Physical Anthropology 40:227-256. Leibman, F.M. and L. Kussick (1965). An electromyographic analysis of masticatory muscle imbalance with relation to skeletal growth in dogs. Journal of Dental Research 44:768-774. Lieberman, D.E. and A.W. Crompton (2000). Why fuse the mandibular symphysis? A comparative analysis. American Journal of Physical Anthropology 112:517-540. 92

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Lucas, P.W., R. T. Corlett, and D.A. Luke (1986). Postcanine tooth size and diet in anthropoid primates. Z. Morph. Anthrop. 76:253-276. Maynard-Smith, J. and J.G. Savage (1959). The mechanics of mammalian jaws. School Science Review 40:289-301. Ravosa, M.J. (1991) Structural allometry of the prosimian mandibular corpus and symphysis. Journal of Human Evolution 20, 3-20. Ravosa, M.J. and W.L. Rylander (1993). Functional significance of an ossified mandibular symphysis: a reply. American Journal of Physical Anthropology 90:509:512. Ravosa, M.J. and W.L. Rylander (1994). Function and fusion of the mandibular symphysis in primates: Stiffness or strength? In J.G. Fleagle and R.F. Kay (eds): Anthropoid Origins. New York: Plenum Press Roberts, D. and I. Tattersall (1974). Skull form and the mechanics of mandibular elevation in mammals. American Museum Novitates No. 2536. Scapino, R. (1981 ). Morphological investigation into functions of the jaw symphysis in carnivorans. Journal of Morphology 167:339-375. Smith, R.J. (1978). Mandibular biomechanics and temporomandibular joint function in primates. American Journal of Physical Anthropology 49:341-350. Smith, R.J. (1993). Categories of allometry: body size versus biomechanics. Journal of Human Evolution 24:173-182. Spencer, L.M. (1995). Morphological correlates of dietary resource partitioning in the African Bovidae. Journal ofMammalogy 76:448-471. Spencer, M.A. (1995). Masticatory system configuration and diet in anthropoid primates. Ph.D. Dissertation, State University of New York, Stony Brook. Spencer, M.A. (1998). Force production in the primate masticatory system: electromyographic tests ofbiomechanical hypotheses. Journal of Human Evolution 34, 25-54. Spencer, M.A. (1999). Constraints on masticatory system evolution in anthropoid primates. American Journal of Physical Anthropology 108, 483-506. 93

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M.A. and B. Demes (1993). Biomechanical analysis of masticatory system configuration in Neandertals and Inuits. American Journal of Physical Anthropology 91, 1-20. Spencer, M.A. and G.S. Spencer (1993). MacMorph Data Acquisition Package. Stony Brook, NY: State University ofNew York Department of Anthropology. Thompson, E. N., A. R. Biknevicius, and R.Z. German (2003). Ontogeny of feeding function in the gray short-tailed opossum Monodelphis domestica: empirical support for the constrained model of jaw biomechanics. Journal of Experimental Biology 206:923-932. Throckmorton, G.S. (1989). Sensitivity of temporomandibular joint force calculations to errors in muscle force measurements. Journal of Biomechanics 22:455-468. Turnbull, W.D. (1970). Mammalian masticatory apparatus. Fie/diana-Geology 18. 94