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Combining passive and active impedance parameters for optimized manipulator control

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Title:
Combining passive and active impedance parameters for optimized manipulator control
Creator:
Scholz, Stephen Matthew
Place of Publication:
Denver, CO
Publisher:
University of Colorado Denver
Publication Date:
Language:
English
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viii, 76 leaves : illustrations ; 29 cm

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Subjects / Keywords:
Manipulators (Mechanism) ( lcsh )
Automatic control ( lcsh )
Automatic control ( fast )
Manipulators (Mechanism) ( fast )
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bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

Notes

Bibliography:
Includes bibliographical references (leaves 73-76).
Thesis:
Submitted in partial fulfillment of the requirements for the degree, Master of Science, Electrical Engineering
General Note:
Department of Electrical Engineering
Statement of Responsibility:
by Stephen Matthew Scholz.

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|University of Colorado Denver
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|Auraria Library
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All applicable rights reserved by the source institution and holding location.
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34585923 ( OCLC )
ocm34585923
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LD1190.E54 1995m .S36 ( lcc )

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Full Text
COMBINING PASSIVE AND ACTIVE IMPEDANCE PARAMETERS
FOR OPTIMIZED MANIPULATOR CONTROL
B .A., University of Colorado at Boulder, 1991
A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado at Denver
in partial fulfillment
of the requirements for the degree of
Master of Science
Electrical Engineering
by
Stephen Matthew Scholz
1995


This thesis for the Master of Science
degree by
Stephen Matthew Scholz
has been approved for the
Department of
Electrical Engineering
by
Mijole Radenkovic
Thomas Depkovich


Scholz, Stephen Matthew (M S. Electrical Engineering)
Combining Passive and Active Impedance Parameters for Optimized Manipulator
Control
Thesis directed by Professor Gary G. Leininger
ABSTRACT
Most error correction in manipulators is in the form of active control. This
is usually expensive and cumbersome. If a certain amount of this active control
could be replaced by a passive element much of the fixturing and bandwidth
requirements for active control could be greatly reduced, simplifying manipulator
operation. This thesis addresses the issue of combining passive and active
parameters to reduce the measured environmental forces from a hole on a peg
during a simple peg-in-hole insertion experiment. Relevant background work is
cited, preliminary control experiments are performed and passive + active
experiments are analyzed. The conclusion states that indeed an inexpensive
passive element can replace a significant amount of active control reducing costs to
the user and simplifying operation.
This abstract accurately represents the content of the candidates thesis. I
recommend its publication.
Sign
ui


This thesis is dedicated to my parents. Without them none of this would be
possible.


CONTENTS
Chapter
1. Review of Passive vs. Active Control.................................1
1.1 Problem Statement....................................................3
1.2 Overall Approach to the Thesis.......................................3
2. Technical Background.................................................6
2.1 Historical Overview..................................................6
2.2 Manipulator Kinematics and Dynamics.................................7
2.2.1 Kinematics.........................................................8
2.2.2 Dynamics..........................................................11
2.2.3 Actuator Dynamics.................................................13
2.3 Control Overview..................................................14
2.3.1 Independant Joint Control.........................................14
2.3.2 Compliant Control.................................................17
2.3.3 Impedance Control.................................................18
3. Hardware Setup and Description......................................19
3.1 The Arm.............................................................20
3.2 Data Manipulation..................................................21
v


4. DOF Model for Impedance Control.....................................22
4.1 Rational............................................................22
4.2 Study Overview.....................................................24
5. Impedance Control Models............................................26
5.1 Active Only Approach................................................26
5.2 Passive + Active Control............................................31
6. Analysis of Results.................................................40
6.1 The Scenario........................................................40
6.2 Passive Element....................................................41
6.3 Preliminary Results................................................47
6.4 Experimental Results...............................................55
7. Conclusions.........................................................63
7.1 Model Validity......................................................63
7.2 Potential Applications.............................................65
7.3 Recommended Future Research........................................66
Appendix
A. Control Files.......................................................68
B. Data Matrix.........................................................71
References..............................................................73
vi


Figures
2.1 Robot Coordinate System 7
2.2 Coordinate Reference System 9
2.3 Transformation of Coordinates 9
2.4 Actuator Block Diagram 13
2.5 Simplified System Block Diagram 18
3.1 Picture of Manipulator 19
3.2 Lord Force/Torque Sensor 20
5.1 General Model for Motion 26
5.2 Manipulator Impedance Block Diagram 27
5.3 System with Impedance Specification Filter 28
5.4 Coupled Mass Model for End Effector and Manipulator 31
5.5 Damper Only System 35
6.1 Typical Run 40
6.2 Passive Element 41
6.3 Determining Parameter K for Passive Element 42
6.4 Y Axis Force vs. Displacement 44
6.5 X Axis Force vs. Displacement 45
6.6 Both Axes Force vs. Displacement 46
6.7 Preliminary Run 47
6.8 X Force vs. Time for Preliminary Run 49
6.9 Closeupof6.8 51
6.10 Y Force vs. Time for Preliminary Run 52
6.11 Closeup of 6.10 53
6.12 Accuracy of Impedance Specification Filter 54
6.13 Qualitative Graph of Run Phases 56
6.14 X Force Comparison at G = 50, 150 57
6.15 Y Force Comparison at G = 0.5 59
6.16 Y Force Comparison at G = 1 60
6.17 Y Force Comparison at G = 5 61
6.18 Y Force Comparison at G = 10, 50, 500 62
vii


ACKNOWLEDGMENTS
I would like to thank the people at Martin Marietta and especially Thomas
Depkovich Ph. D. for providing the manipulator and crucial guidance necessary in
producing this thesis. I would also like to thank Professor Gary Leinniger for his
valuable advice as my Thesis Advisor.
vm


1. Review of Passive vs. Active Control
Robot manipulators and their controllers like all real physical control
systems experience a certain degree of error when performing a task. For a robot
arm this task may be assembly, welding or mating type tasks. These errors may
result in torques and forces damaging to the manipulator and the task. There are
two ways this can be fixed. The first is to gain a better knowledge of the
manipulator and the task and their relation to each other through structural
additions and constraints like sensors and fixturing Fixturing is modifying the
work area or the environment such that the errors are reduced. This method can
become cumbersome, expensive and inflexible. The second method is to design
systems and controllers that accommodate the forces and torques. The system or
the controller may incorporate the ability to change the path of the manipulator as
forces and torques evolve such that peak forces and torques stay below a pre-
determined level.
A certain amount of fixturing is inherent in most industrial applications of
robotic arms due to the highly structured nature of factories. This thesis,
therefore, will not explore the varying degrees of fixturing. Instead, it will explore
manipulator system and controller designs that meet manipulator task interaction
requirements such as speed, peak force and torque while simultaneously satisfying
all other system requirements that arise as a result of non-interactive tasks like free
space positioning and slewing.
1


There are two basic approaches to designing a robotic manipulator system
capable of reacting in a predictable manner to forces and torques that arise as a
result of interaction with the environment. The first of these is generally referred
to as the passive approach while the second is referred to as the active approach.
In the passive approach it is assumed that either a material or mechanism inserted
between the task element grasped by the manipulator and the end effector will, as a
function of structural properties, relieve forces and torques as they occur. The
simplest example of a passive device is some type of rubber material inserted
between the task and the end effector. Another approach is to have a spring-like
mechanism between the manipulator and the grasped task. In the active approach
it is assumed that forces and torques generated as a result of interaction can be
sensed, fed back and utilized by the manipulator controller in such a manner that
evolving forces and torques are adequately controlled or minimized.
The passive approach brings simplicity to the system. It does not require
extra sensors, electronics, programming, computation etc. However, once it is in
place it stays in place. While the passive approach is desirable for certain jobs such
as mating it could fall short when it comes to other jobs requiring heavy lifting or
torque exertion such as assembly tasks. The active approach may be turned on or
off at will or as a function of the task being performed. The price for this flexibility
is more sensors, electronics, programming and more complicated computation.
2


1.1 Problem Statement
At first it may seem that the purpose of this thesis is to strike some sort of
happy medium with a trade off between the active and passive approaches. One
could look beyond this by asking: can a manipulator system be designed with both
active and passive attributes such that the strengths of both approaches can be
maximized while minimizing their weaknesses? This is a more subtle question than
the question of attaining a happy medium because it asks if using both approaches
could create a system that surpasses in a practical and economic sense another
system using only one approach.
1.2 Overall Approach to the Thesis
The next chapter will set the stage for the investigation by reviewing some
basic concepts of robotic manipulators and formalizing the conventions and
notation that will be used. Of particular importance, dynamic models will be
developed that in addition to standardized manipulator link/actuator models will
address the modeling of passive devices as well as models for the task interaction.
The chapter will conclude with an overview of manipulator control strategies that
emphasizes currently used approaches for active control.
Chapter three describes the hardware used in the thesis. This includes the
arm structure, the joint motors and the computer hardware. Details of data
storage can be found in the appendices.
The fourth chapter will be devoted to the specific methodology that will be
employed in this investigation. The discussion in this chapter will include the
selection of specific task and manipulator parameters, the range of experimental
3


actions to be investigated and, most importantly, a set of performance metrics to
be used for analyzing the results of the investigation. These metrics will address
system performance, system cost and, if possible, issues associated with reliability
and maintenance. This is critically important since a tradeoff between
combinations of competing approaches is of little use if the investigation does not
also begin to explore the means by which cost sensitive users can utilize the results
to establish their own system requirements.
The fifth chapter discusses the models used for impedance control. The
three models used are active only, passive only, and active + passive control. The
chapter develops the models through standard impedance control methods. At
first the system is developed without an impedance specification filter for the
active only model. Then the impedance specification filter is incorporated into the
system through a feedback loop for the active + passive control model. Finally,
the passive only controller is developed by incorporating the passive element into
the system while setting very stiff impedance parameters for the impedance
specification filter. This effectively turns off the filter feedback loop.
The sixth chapter presents the analytical results of the study. This will
include the specific dynamic system models selected, the system simulations
performed, and the analysis of the simulation studies. A broad set of simulation
scenarios will be analyzed and used to select a small, specific set for hardware
implementation.
The seventh and final chapter will address results and conclusions. Model
validity will be examined and a discussion on potential applications in industry will
4


follow. Finally, future research possibilities will be examined which would use the
results found in this thesis.
5


2. Technical Background
In this chapter the technical background for the thesis will be developed. A
history of research on active and passive compliance in robotic arms details the
work done by others in this field and establishes the groundwork on which the
thesis is based. Manipulator dynamics follow and give a detailed mathematical and
graphic representation of the 3 degree of freedom (DOF) arm used. Finally, the
control schemes used will be detailed at the end of the chapter.
2.1 Historical Overview
This section will present an historic overview of passive and active
compliance in robotic arms. Many articles have been written on implementing
passive devices to achieve task completion [1] and contact control [2], Active
compliance has made its way into many articles as well [3], These articles deal
with highly suitable applications, however, none deal with combining the two
approaches to determine the optimal combination. Furthermore, the articles have
been dealing with a very specific part of passive/active compliant application.
Force and impact control [4], contact control [5], constrained motion tasks [6] and
stability in contact tasks [7] are thoroughly covered in these and other articles
This thesis will deal with manipulator/task interaction after contact has been made.
It will approach the problem with the hypothesis that it is possible to incorporate
6


both active and passive compliance to optimize manipulator control. This has not
been done before.
2.2 Manipulator Kinematics and Dynamics
In this section the kinematics and dynamics of the manipulator system will
be defined Once the manipulator comes in contact with the environment it
becomes part of the manipulator system and therefore changes the system and
influences the job the manipulator has to do.
Fig 2.1 is a representation of the system used. It can be assumed that there
is some type of passive coupling between the manipulator end-effector and the
task. For the sake of simplicity, a planar arm was used.
7


This coupling can be represented in terms of stiffness and damping parameters.
2.2.1 Kinematics
For the main purpose of establishing notation, this section will give a brief
discussion of the kinematics. Specific solutions to inverse kinematics and specific
convention for assigning link reference frames will not be covered. Please see [8]
for more details.
We can easily determine the relative orientation of two reference frames in
space by using the rotation matrix. The simplest case is where the origins of the
two reference frames are coincident. In this case the rotation matrix can be used
to transform a vector in one reference frame into the other frame. In this case we
will use the following notation to define a rotation matrix R that transforms a
vector P from frame B into frame A:
where
ap=abr BP (2.1)
cos -sin o'
bR = sin cos 0 (2.2)
0 0 1
8


Fig 2.2 is a graphical representation relating frame A to frame B.
Figure 2.2 Coordinate Reference System
In the more general case when the origins are not coincident a 4x4
Homogeneous transform matrix must be used. The vector requires a rotation
matrix as before with the addition of a vector relating the separation of the origins
of the two different reference frames. This equation is shown below.
Figure 2.3 Transformation of Coordinates
AP=ABRBP+APBor=ABTBP (2.3)
The transform matrix simply combines the two operations into a single step. This
matrix has the form shown below and will be designated T.
(2.4)
9


In the case of robotic arms which will link multiple reference frames this can be
further generalized by assigning coordinate frames to each link. The product of
the resulting transformation matrices will yield the end-effector position and
orientation.
The forward kinematic relationship that relates manipulator joint angles to
end effector position and orientation in a designated coordinate frame can be
defined as L(0). This function yields both a position vector and a rotation matrix.
The inverse of this function yields manipulator joint angles as a function of
manipulator position and orientation and is called L_1(X). Therefore:
The forward kinematic function relates manipulator position and attitude to
manipulator joint angles. By calculating the differential of this function, an
additional important function called the Jacobean is obtained. This operation
results in a linear differential relationship between Cartesian position and attitude
and manipulator joint angles as well as a relationship between manipulator linear
and angular velocities and joint velocities.
L(0) = X
L'^) = X
(2.5)
(2.6)
(2.7)
dX = J(0)<5
(2.8)
10


2.7 above is also useful in relating end effector forces and torques to joint
torques. F represents the forces and torques at the manipulator end effector, and x
represents joint torques. Then from the principle of virtual work:
Fr dX rTS& (2.9)
and from the relationship between dd and dX:
t = JTF
(2.10)
2.2.2 Dynamics
The general equation of motion for a robotic manipulator is given below.
t = A/(0)0+F(0,0) (2.11)
M(0) is a 3x3 mass matrix and C(0,0 Represent Coriolis and centripetal forces.
In Cartesian variables the equations would take the form:
Mx =
Vx (0,0) = J~T (0)(F(0,0) A/(0)7 (0) 7(0) 0)
(2.12)
(2.13)
11


This is called operational space representation and results in a system of equations
expressed in terms of Cartesian displacement and orientation. The equations
below relate the terms in this system to those in the configuration space equation.
First multiply both sides by the Jacobian transpose.
(2.14)
J~tt = J~TM{)+ J~TV(,)
or (2.15)
F = J~TM{)+ rTV{,)
Where F is a force-torque vector acting on the end-effector of the robot. F can
also take the form
F = Mx()X+Vx(Q,) (2.16)
Here, X is an appropriate Cartesian vector representing position and orientation of
the end-effector. Mx(0) is the Cartesian mass matrix and Vx(0,0) is a vector of
velocity terms in Cartesian space. A relationship can then be developed between
joint space and Cartesian acceleration starting with the Jacobian.
X = J0
then differentiating to get
(2.17)
X = J+ J
(2.18)
12


Then solve for the joint space acceleration to get
0 = J~'X-J-'J0 (2.19)
Substituting this into the force-torque vector equation to get
F = J-tM(Q)J-' X- J-TM(G)J-' J0+ J-rV(0,0) (2.20)
It is from this equation that the terms for the Cartesian dynamics are obtained.
2.2.3 Actuator Dynamics
The equations above describe a perfect robotic arm. In reality, pure torque
generators do not exist and actuator dynamics must be accounted for. Fig. 2.4 is a
block diagram of the actuator system used.
386 proc.
Digit. Cont. f
7K------*
* D/A
A/D
U1
Analog Controller
& Sensor Electmc'
A/D *
PWM
Amps
3 DOF
* Manipulator
Accelerometers, Tach
& F/T sensor
Resolvers
Force/Torque
Processor
Force/Torque
Transducer
Figure 2.4 Actuator Block Diagram
13


This is still a very complicated control system design even when non-linearities due
to the compliance present in almost all power transmission mechanisms are
neglected. If this compliance is not compensated for in some way the addition of
dynamic effects due to electronics will produce a grossly inaccurate model.
2.3 Control Overview
When robotics gained national attention in the 1970s it was eagerly
embraced by the control system community as a whole. It was recognized, almost
immediately that the nonlinear, coupled robotic system represented a prime
application area for many of the key disciplines within the control system area.
The techniques that appeared ideal for application included nonlinear, optimal,
multivariable, and adaptive control, One problem that arose, however, was the
lack of available testbeds for hardware experimentation. Without hardware
testbeds to focus research activities, much of the work remained very theoretical,
utilizing simplistic models to represent manipulator dynamics. The result of this
was an overemphasis of the nonlinear control issues and a deemphasis of the
practical issues associated with manipulator control.
2.3.1 Independent Joint Control
With very rare exceptions the vast majority of robotic manipulators in use
today are controlled using independent joint servo controllers that are designed
using techniques that have been more or less standard for forty years. Even in the
cases where more refined linearizing, decoupling control is implemented, joint
14


torque controllers are required to generate the desired output torque profiles on
which these methods depend. Therefore, the issues associated with the design of
independent joint controllers remain important. Their importance will probably
grow with time, since successful implementation of the more exotic control
strategies will have to rely on the incorporation of the lessons learned in successful
independent joint control studies.
There are two primary problems associated with independent joint control.
The first problem arises from the structure of the typical actuator mechanism used
in robotic systems that was discussed at the end of the previous section. With the
exception of manipulators that use direct drive actuators, almost all systems use
some means of torque amplification. In most cases this is achieved through the use
of spur gears, but examples of harmonic drives, belts, cables and ball screws also
exist. These power transmission elements generally have two undesirable side
effects: compliance and nonlinear friction. Both complicate the control design
process and require special techniques.
The second problem arises from the nonlinear, coupled nature of the
manipulator dynamic system. From the standpoint of the independent joint
controller, the variation in manipulator dynamic properties represents both a
change in the inertia of the plant that is being controlled as well as a source of
torque disturbances. These disturbances are a function of the position, velocity,
and acceleration of the other manipulator joints. Therefore, controller design must
be robust in the sense that adequate performance can be maintained as both the
inertia of the plant varies and disturbances are encountered.
15


In general, there are two different structures through which independent
joint control can be implemented. These are shown below and are referred to as
joint based and Cartesian based techniques.
Of these, the joint based approach is the most straightforward to implement
since it requires only an inverse kinematic transform. In the Cartesian approach,
two stages of transformation are required: a forward kinematic transform of the
vector of joint position sensor signals and an inverse Jacobean on the Cartesian
error signal. In reality, with modem microprocessors both strategies can be
implemented at real time rates. The joint based approach is probably the most
popular of the two since it simplifies the servo design process and makes it much
easier to implement direct joint control modes where the operator can control one
joint at a time.
There is no general technique for the design of the individual joint control
loops. Like most control problems, each must be treated individually. There are,
however, two basic trains of thought that appear in a variety of sources. One train
is that conventional proportional-derivative control techniques are presented for
controlling a single actuator. These discussions also address the issues of
disturbance rejection and typical nonlinearities such as static friction. Most
references also contain some discussion of the use of PID controllers. These have
been very popular in industrial implementations. The general rationale normally
cited for the use of a PID controller is the role of the forward loop integrator in
achieving zero steadystate error to a step disturbance, which is a reasonable model
for gravity loading. While both of these approaches represent reasonable starting
16


points for a design effort, the problems encountered in producing a final design
rarely succumb to such simple solutions.
2.3.2 Compliant Control
This section starts to deal with when there is environmental interaction
with the robot For the purpose of this discussion, a controller that must contend
with reaction forces and torques from the environment will be referred to as a
compliant or force controller.
Many promising demonstrations have been performed in a number of
laboratories, but there is currently no mature, standardized set of algorithms with a
sufficiently good track record for general implementation. As in the case of the
development of single arm control laws, part of the problem has been the
availability of hardware testbeds for experimentation. This problem is now
diminishing with a larger number of available testbeds and the availability of
commercial force and torque sensing devices.
In compliant control tasks, the manipulator control system must deal with
kinematic constraints that are associated with the task and, in some cases, be
capable of applying forces and torques at the end effector of the manipulator as
dictated by the task.
There must be a relatively accurate estimation of the end effector forces
and torques. A good way to achieve this is through the use of a force/torque
sensor mounted on the proximity of the end effector. Force/torque sensors employ
17


strain gauge sensing elements. One of the more popular configurations for this
type of device is shown and discussed in chapter 3.
2.3.3 Impedance Control
Position based impedance control is the method of control in the thesis. It
makes the robot arm behave like a second order spring/damper system.
While the controller normally acts on nominal position and orientation
commands, it also has the capability for commanding applied forces and torques
through the inputs shown in Fig 2.5 below.

Manipulator

B
0-
End Effector
Position Control
Impedance
Specification
_ Filter-
B,s + Kj
Figure 2.5 Simplified System Block Diagram
The desired impedance relationships are achieved by filtering measured forces and
torques through impedance specification filters as shown. The outputs of these
filters are position and orientation perturbations which are used to modify the
nominal, commanded reference signals. Currently, translational commands are
modified through the use of transition matrices. The fine details of impedance
control will be dealt with in chapter 5.
18


3. Hardware Setup and Description
Martin Marietta has provided the manipulator, testbed and all sensing and
data manipulation hardware. This chapter will describe the hardware used in the
thesis. Fig 3.1 is a picture of the arm on a nearly frictionless surface.
Figure 3.1 Picture of Manipulator Used in Thesis
19


3.1 The Arm
The manipulator used is a 3-DOF arm supported by air bearings on a flat
surface. This enables the arm to achieve 2 dimensional space-like motion. The
arm itself has about a 4.5 ft. reach. The links are thin and plate-like to provide
compliance in the vertical plane. This will compensate somewhat for imperfections
in the flatness of the test bed and will prevent the air bearings from grounding.
A Lord force/torque sensor is at the end of the wrist and can measure force
in the x and y directions as well as the torque exerted on it. It is a six DOF sensor
able to measure forces and torques along and around all axes. The basic structure
is shown below in Fig 3.2.
Figure 3.2 Lord Force/Torque Sensor
20


Each joint consists of a harmonic drive with a gear ratio of 80:1 driven by a DC
motor. Above the motor housing are the sensors consisting of a motor side
tachometer, an output torque transducer and a resolver to measure the joint angle.
Cartesian accelerations are measured through accelerometers mounted on the
wrist.
3.2 Data Manipulation
The controller package consists of a modified Intel 310 system using single
or double 386 processors. The program is written in C and enables the user to
select sampling rates (50 Hz was used for this thesis), open or closed loop control
modes, joint space or Cartesian space reference commands, and sensor/actuator
sets as needed to control from one to three degrees of freedom. The user can
record item-by-item a selection of data such as sensor measurements and/or
control law variables and the rate of data selected. These data files can then be
transferred via ETHERNET to a work station for further analysis with commercial
software (Matlab was used). See Appendix A for more on data setup and transfer.
Appendix B has more on how Matlab was used to analyze data.
21


4. DOF Model for Impedance Control
This chapter will take a look at the way the thesis approaches the issue of
fitting the robotic arm with a passive element for the purpose of enhanced
performance with a decreased amount of active control.
4.1 Rational
When robotic manipulators are interacting with the environment as a part
of task execution small errors in task alignment or manipulator positioning can lead
to extremely large forces and torques being generated. These have the potential
for damaging the manipulator and/or the task. This problem was recognized very
early in the development of robotic technology [9], [10], [11], A number of
approaches have evolved to help solve this problem. In general, there are two
basic strategies that are used. The most obvious approach is to have some type of
element between the manipulator end effector grasping part of the task and the
primary manipulator system. This is generally known as the Passive approach.
The passive element can be something as simple as a compliant material such as a
rubber pad. Other approaches to passively relieving forces and torques have
focused on more complex arrangements of springs and dampers. The second, and
22


more complex approach, is to use more sophisticated active control strategies that
directly modify the closed loop control of the manipulator to achieve the desired
characteristics. In general, these active approaches require the use of a wrist
mounted force/torque sensor to provide inputs to the controller.
Passive approaches have the advantage of simplicity but they lack
flexibility. Once in place, it stays in place for all tasks. This is not always
desirable. Active approaches have an inherent flexibility but they also come with a
number of problems due to increased sensing and computational requirements In
most application examples the final implementation ends up being a combination of
both active and passive control. The passive element of implemented controllers
arises from the fact that most end effector/gripper designs have some type of
passive compliance already built in. Inevitably, this ends up making the job of
active controller design much easier.
Therefore, although passive and active approaches have been used in
conjunction many times in the past, combined active and passive control has never
been studied from the standpoint of a shared control perspective. It seems
reasonable that this topic can be investigated with the objective of determining,
based on known characteristics:
23


Asa function of the tasks, what overall impedance properties are required?
(The tasks include all tasks, interactive or otherwise.)
What options are realistic for passive and active elements?
How control functionality should be distributed between passive and active
elements, (i.e. design guidelines for selecting passive and active parameters.)
A key element of this effort will be the examination of ways in which the
control can be partitioned with respect to some global measure of optimality.
4.2 Study Overview
At a summary level, this study will use analysis and simulation techniques
to study the basic problem. Results from the analysis and simulation will be
verified experimentally in hardware. Analysis, simulation, and hardware
experimentation will be limited to a planar 3-DOF case. At the conclusion
extension of results to the more general 6-DOF case will be discussed.
Because a study of this type has a number of subtle complexities, it is
useful to break the overall study down into the smaller tasks that are listed below:
24


Manipulator/task/controller modeling
Cost function synthesis
Task performance analysis
Design guideline formulation
Hardware validation
25


5. Impedance Control Models
This chapter deals with a description of the active and active + passive
control of the arm. First, an active only scenario is developed to demonstrate the
inherent problems with increased active control. Then a passive element is
introduced to the arm and the effect is examined.
5.1 Active Only Approach
The general model for motion is shown below in Fig 5.1.
(5.1)
(5.2)
Figure 5.1 General Model for Motion
The equation for this is
mx = + F.
where
c e
F = -Kx Bx+ xR (K)
26


The K term is the position feedback, the B term is the velocity feedback and the
x(K) term is the reference command scaled by K.
Then
(5.3)
(5.4)
(5.5)
This is shown graphically below in Fig 5.2.
mx+ Bx+ Kx = Kxr + Ft
(ms2 + Bs + K)x + Kxr + Ft
K Ft
X = -;--------+---;--£----
ms+Bs + K ms'+Bs + K
Initially, the FJK term is ignored. This is because an environmental force that
could damage the task say about 10 lbs could be insignificant with respect to
the manipulator (about 3000 lbs ).
27


Now it is assumed that \r is generated as a result of active impedance
control, with a commanded position of zero as shown below in Fig 5.3.
Figure 5.3 System with Impedance Specification Filter
Then
x =
K
ms' + Bs + K
____C_
<(B,s + Kt) sj
F
or
x_______________KC____________
Ft ~ (ms2 +Bs + K)(s(B,s + K,))
(5.6)
(5.7)
So to achieve a desired closed loop impedance it is necessary to deal with
manipulator position control dynamics. That is K/(ms2+Bs+K).
28


When the direct action of Fe on the manipulator is not neglected this last
equation becomes
________(AC + 1)______
Fe ~ s(ms2 + Bs + K)(Bjs + Ki)
In either case achieving a desired closed loop impedance
CD
Bds + Kp
requires adjusting impedance control and position control
CD__________________KC____________
s(Bds + Kd) s(ms2 +Bs + K)(Brs + Ki)
Now this can be written as
{
KC/
'mB.
2 B K
s + s +
m m
s +
KA
B,J
or
b __________________d____________
s(s + a) s(s2 + 2£a> ns + a>2n)(s + e)
Now take the Laplace Transform to find
(5.8)
(5.9)
(5.10)
(5.11)
(5.12)
29


r-1
die
s(s2 +2£ = 1-
f'a>\
1-2fCC0n -r f 2 CO
A'"")
+
(5.13)
e~w sin(conyl&co+f2co2n)
j\-<;2{\-2tfcon+f2co2n)
This last term is of no concern but the one and the second term yield important
results for the complete range of co.
Notice

f-
1-2 fQan+f2co\ l/,_2 ft/ +
/(O'. J /o)n J
= r(o)J
and
lim r(iyn) = 0
ajn*0
and
lim r( (5.14)
(5.15),
(5.16)
So a higher bandwidth is required for active only control.
The next section will look at active control with a passive element to
decrease bandwidth and maintain acceptable accuracy.
30


5.2 Passive + Active Control
The passive element between the wrist and the peg is a compliant substance
in this case foam. Fig 5.4 is a model of the arm/end effector coupled system.
Figure 5.4 Coupled Mass Model for End-Effector and Manipulator
where m is the mass of the arm, n^ is the mass of the manipulator, xi is the
distance the arm moves, x2 is the distance the end effector moves, Fc is the force
from the arm, Fe is the force the environment exerts on the end effector, and Bp
and Kp are the damping and spring stiffness constants. Below are the equations of
motion for this coupled system:
31


(5.17)
(5.18)
mx\ + Bp(xx-X2) + Kp(xx-x,) = Fe
mtX2 + Bp(xi-x\) + Kp{xz -xx) = Ft
after a Laplace Transform these become
(ms2 + Bps + Kp)xt=Fc + (Bps + Kp)x2 V '
(n,'S:+Brs + Kt)x.=F'+(Bps+Kr)x, (5-20)
with some algebra these equations become
Fe+(Bps + Kp)x2
(ms2 +Bps+Kp) (5.21)
The commanded inputs are
Fc = (Bs + K)xi +xrG
xr ------------F&
(B'S + KJs
(5.22)
(5.23)
where the first term in equation (5.22) is the PD control loop, the coefficient of G
in equation (5.23), and G is a general gain term.
Substitution in Xi yields


(5.24)
Gc
xi =
_ (B,s + K,)s
+ (BPS + Kp)x2
ms' + (Bp + B)s + (K p + /l)
now this can be substituted back into equation (5.20) to get
(m.s2+B,s + Kf)x2=F.+{B,s + Kp)
Gc
(B,s + K,)s
F, + (B s + K )x2
ms2 +B's + K'
(5.25)
where B=BP+B and K=KP+K
Now for simplicity let
me(s) = my- +Bps+Kp
m(s) = ms2 + B's + K'
(5.26)
(5.27)
then equation (5.25) becomes
Gc(Bps+Kp)Ft (B s + K )2xz
ffl, (s)x, = F.+------------+ ------------
s(Bts + Kj)m(s) m(s)
(5.28)
or
[. (M) (BPS + KP)2 ]x2
m(s) +
GcjBpS + Kp)
s(B'S + Kt)
F.
(5.29)
33


notice that in matrix form the determinant is
me(s)m(s)-(Bps + Kpy- = A (s)
This results in an impedance of
x: mjsMB.s + KM + GcjB'S + K,)
F. A (s)[s(B,s + K,)}
If the arm undergoes completely active control Kp and Bp can be ignored. That
means that equation (5.17) becomes
(5.30)
(5.31)
mil Fc- -Bxi- Ajc, +xrG (5.32)
GcF (ms2 +Bs + K)x.= s(B,s + K,) (5.33)
or
GcF, X' siB'S + KJims2 +Bs + K) (5.34)
substituting the last equation into (5.20)
Cm.s' + Bps + Kp)xz
F, + (Bps + Kp )GcFt
s(Bts + K( )(ms2 + Bs + K)
(5.35)
for an impedance of
34


x2 s(B'S+ Kt )(ms2 +Bs + K) + Gc(Bps + Kp)
Fe ~ s(B! s + Kt ){mes2 +Bps + Kp )(ms2 +Bs + K)
This last equation can be simplified to show the two components that contribute to
the expression for the overall impedance.
X2
F.
Mm.

m, m.
GcBn Kb
-----(* + )
Bp
s(s + ^-)(s2 +-S + -)
B. m m
(5.37)
Notice what the equation becomes if Kp = 0 as in the Fig 5.5
Figure 5.5 Damper Only System
35


£2
F.
1
r
m_
s(s + ^)
m.
GcBn
1+ '
B.m
5 +
K,
\
s1 +
B K
s+ -
m mJ j
(5.38)
This equation can now be used to find the value of Bp. To do this, the
inverse Laplace Transform must be taken.
LT'\
F.
f
1 e
V

y
(5.39)
(5.40)
and through partial fraction expansion the second term becomes
36


(5.41)
r -I
GcB,'
C
s
where
A = -
B = -
'K (K B\ jvi p \B, mj Sme' ] ! BPB k) + mtm my
N UJ (B. p i V, B,J 2 K,B K + mB, m ^
C =
mtBm
bpk,k
(5.42)
(5.43)
(5.44)
and poly is the polynomial representing the fourth and final factor in the
denominator. This then becomes
GcBp
Bmmt
For simplicity assume robot closed loop control.
(B,
C Ae
-Be

+ r' {poly}
Then
(5.45)
37


(5.46)
2 B K . v
S +S + =(5 + a)(5 + £>)
m m
and
Z'1 {poly} = Z-'
where
D_____£_1
s+a s+bJ
f Bp) f K,' (- a + b)
a -a + -a +
V rrij V B, >
1
r BA f K \
b -b + -^~ -b + ^- (a ~b)
y mj BJ
These expressions for D and E come from the following
s2+ (a+b)s + ab = s2 + (B/m)s + (K/m)
(5.47)
(5.48)
(5.49)
(5.50)
so
ab = K/m (5.51)
and
a + b = B/m (5.52)
now set m = 100 to get
lOOab = K (5.53)
and
100(a + b) = B (5.54)
These last two equations are expressions for finding K and B to achieve a desired a
and b.
38


So the final equation for finding the poles of the system is
GcB, )(
-I' -I-'
C-Ae -Be De
[
-Ee'1
(5.55)
\BmmJ ^
pole
associated
with passive
element
me and Bp
associated
with imped
spec, filter
Bj and Kj
pole
poles from
closed loop
position
control
s2+B/ms+K/m
It should be noted here that the system can be made passive only simply by
choosing very stiff impedance parameters (on the order ofK, and Bj = 100). This
effectively makes the spring and damper system so stiff it is as if the impedance
controller was turned off and the arm would be operating on position control only.
This option is explored in the experimental analysis.
39


6. Analysis of Results
This chapter will discuss the scenario used in gathering data, a detailed
description and analysis of the passive element used in the thesis, and typical
results of runs under active and active + passive control.
6.1 The Scenario
The typical experiment for this thesis was a peg-in-hole insertion run. Fig
6.1 is a figure showing the set up and the path the peg took.
40


The commanded path is slightly offset from the actual path. This error
creates forces in the force torque loops due to the position based impedance
controller. These forces can then be recorded through the force/torque sensor and
stored via the Intel 386 processor. It is these forces and torques that the thesis
addresses. The passive element introduced into the wrist deforms and compresses
to partially compensate for these forces. This allows less stiff impedance
characteristics of the controller hence reducing the bandwidth and fixturing of the
system.
6.2 Passive Element
Below is a picture of the passive element used in the thesis.
Figure 6.2 Passive Element
It consists of foam epoxied between two threaded metal plates. It screws in place
between the force/torque sensor and the peg and is held in place with two collars.
The stiffness characteristics of the passive element can be accurately estimated by
41


compressing the element a certain amount and measuring the force of resistance.
According to Hookes Law this yields the spring constant through the equation
F = -kx (6.1)
The quantitative stiffness was obtained by compressing the element while attached
to the force/torque sensor. The sensor detected the force necessary to deform the
element a certain distance and transferred that information to the Intel 310. This
information along with the distance the element was compressed was used to
generate a Matlab plot to determine the linearity of the elements stiffness. The
picture below shows the process in which this was done.
Figure 6.3 Determining Parameter K for the Passive Element
42


Fig 6.4 shows the Force vs. Displacement graph for the passive element along the
Y axis. As with any real physical system the graph is not completely linear. The
non-linearity arises out of the natural properties of the element. For the most part
the thesis dealt with forces in the range from 0 to 5 lbs. Fig 6.5 shows the same
graph for the X axis. As one would expect this graph is similar to the one for the
Y axis. Fig 6.6 shows the two graphs together with the bias subtracted. This
shows the behavior of the element under small forces all the way to the origin.
According to these graphs the element has a stiffness k of about .25 lbs./mm or
6.25 lbs/in.
In the graphs for the experimental results the x forces change sign while the
y forces dip only slightly below zero. This is opposite of the steel wall tracking
runs. This is because when the peg tracks the chamfer the forces generated are
similar to those when the peg is tracking the wall. However, when the peg enters
the hole it comes in contact with two other walls (the hole) that are rotated 90
degrees from the frame of reference outside the hole. So in this instance one
would expect negative x forces and nearly zero y forces.
43


IUIU-|U3IU73<|dS|C]
PP
^7 9 aanBxj
Force-lbs
t
\
\
\
KJ
o
Displacement vj Foice fi* Passive Element Y axis


Displacement vs Force for Passive l lcmcnl X
r t--------7
i _______________i__________________i-
8 10 12
Displacement mm.


7
46


6.3 Preliminary Results
Before peg-in-hole insertion was attempted the impedance controller
needed to be verified as working properly and a proper set of impedances
determined. These runs were done with no passive element present and with a
variety of impedances. Instead of peg-in-hole insertion the peg was pushed along
a steel wall (the peg is made of aluminum) to determine if the forces generated are
what would be expected. Fig 6.7 below shows the path taken for these runs.
Figure 6.7 Preliminary Run
47


Before 5.5 seconds the peg is not in contact with the steel wall so no force is
generated. At t = 5.5 the peg is in contact with the wall and begins tracking along
the wall until about 8.2 seconds at which the peg comes to rest for two seconds
then returns back to the starting point to repeat the run. Since the final position is
not the same as the commanded position and since the controller returns the peg
by simply reversing the commanded forward path one would expect the force in
the x direction (Fx) to always be positive. A negative Fx would mean some force is
pulling the peg away from the force/torque sensor. This is impossible under the
given configuration However, since the peg will be back tracking along the wall
on the return run one would expect the force in the y direction (Fy) to quickly
reverse and become negative after the return path began.
Fig. 6.8 shows the force vs. time for Fx. It is important to notice here the
different levels the impedance specification filter produces between 8 and 10
seconds. As the gain on the filter is increased from one (on the bottom graph) to
100 (the top graph) the force measured during this period when the peg is at its
final point is increased. The non-linearity of the arm is apparent especially when
the graphs of G = 50 and G = 100 are compared. The forces shown in the graphs
of the lower gains are closer to being directly proportional than those in the higher
gains. This is due to the fact that at higher gains (more power supplied to the
motors) the rigidity of the links is compromised. This effect is more easily seen in
48


X Forces G-1.5.10.50.100, 4/12/95
Time-sec M7I-475 data
CO


Fig. 6.9 in which the level off times are blown up for clarity. The next graph, Fig.
6.10, shows the same force vs. time graphs but for Fy. Again the stiffest
parameters are the least linear with respect to the closest gains. Fig. 6.11 again
shows the max deflection Y force for gains 1 through 100 blown up for clarity.
Fig. 6.12 shows the accuracy of the impedance specification filter. The
commanded K for the filter in this graph is 5. After the initial transients one would
hope that the measured K would be somewhat close to this value. As it turns out
the measured K was very close to 5 for both the x direction (top graph) and the y
direction (bottom graph).
All the graphs in this section have shown that the impedance characteristics
of the arm and the controller are quite sufficient to proceed with the experiments
of incorporating the passive element into the arm to reduce the forces generated
at the task. The next section will show conclusively that the passive element will
indeed enable an arm with a lesser degree of active control to perform as well as an
arm with a greater degree of active control.
50


Force-ibs
X Force a! max deflection; G* I -100. 4/12/95
Timc-20msec inc r471-r475


Timc-sec 1471 -475 data

019
Force-lbs
Y Forets 0=1,5.10,50.100, 4/12/95


7
Y Foace al mat deflection, G= 1-10(1. 4/12/95
2
I HO
190
200 210 220 230 240 250 200 270 780
0
Time-20niicc ir>c r471 -475


Timc-jcc, 4/16/95 ficmi Icst.dal; imp_no_55

Zl9
Measured force (lbs)/ measured pos (in)
Expeiimcnlal Impedance (Foice/Displacemenl) G=5; x & y


6.4 Experimental Results
This section will discuss the results of the peg-in-hole insertion
experiments. Fig. 6.1 shows the execution of this experiment.
Fig. 6.13 is just for qualitative purposes. It shows graphically the different
stages of the experimental runs. During the first two seconds the peg tracks the
chamfer and enters the hole. After the peg has been inserted into the hole there is
a significant drop off in forces. This decrease in force is expected. After the peg
has been inserted it spends two seconds resting while the force/torque sensor
measures the environmental forces exerted on the peg. After this two second
period is done the peg is removed over another two second time period. The
graph shows the expected drop off in forces during this time. When the peg is
finally removed there is another rest period before the manipulator starts another
run. These sections occur throughout the experimental graphs.
Fig. 6.14 is very revealing. The lowest graph line corresponds to a purely
active run with gain equal to 150. The graph line just on top of that corresponds
to a gain of 50. The top solid graph line corresponds to a run with a gain of 50
and the second solid graph is a run with gain 150. The only difference between the
solid graphs and the dashed graphs is that the solid graphs represent runs made
with the passive element in place. It can easily be seen that incorporating the
passive element significantly reduces the environmental forces on the peg. The
55


Time-icc inc no_5,51,55,6,7


Sq|-3DJ0£
Figure 6.14
X Forces Pass+Act vs. Acl; G=50,150
Time-sec 4/21/95


anomaly between 5 and 6 seconds is from the peg momentarily getting caught on
the edge of the chamfer. The same anomaly can be seen in the graphs for y forces.
The next few graphs will compare different gains with and without the
passive element. The gain of the active + passive runs will always be 50.
Fig. 6.15 shows an active + passive run with a gain of 50 compared to a
purely active run with a gain of 0.5. In this case the active + passive run has
generally higher forces than the purely active run. The top graph is the active +
passive run. It is clear from this graph that the active + passive approach has its
limitations.
However, Fig. 6.16 shows the purely active run with a gain of 1 compared
to the active + passive run with gain 50. The purely active run is the one with the
anomaly between 5 and 6 seconds. One can see that these two runs produce very
similar forces after the transients have died down.
Fig 6.17 shows a very significant difference between the two graphs. The
purely active run has a gain of 5 and produces much higher forces than the active +
passive run.
Fig 6.18 compares the active + passive approach (the bottom most graph)
to a series of high gain runs of 10, 50, and 500. In each case the active + passive
runs reduced the environmental forces much better than the stiff purely active runs.
58


Force-lbs
Time-sec


Force-lbs
Activc+passive vs. active act. G=1
Time-scc


Aclivc+passive vs. active act. G=5
Time-sec


Passive vs Active G=10,50,500
Time-sec


7. Conclusions
This final chapter will discuss the conclusions to be made from the results
of the last chapter and the rest of the thesis. The first section will cover model
validity and conclusions to be made from the models. The next section will engage
a discussion of potential applications of this thesis. The final section will discuss
recommended future research that would use this thesis as a starting point or at
least as a reference.
7.1 Model Validity
The graphs in chapter 6 yield some very important information:
The active + passive runs with a gain of 50...
Were not quite as good as purely active with a gain of 0.5 (very soft
impedance)
About equal to the purely active run with a gain of 1 (still pretty soft)
Were better than the purely active run with a gain of 5 (only slightly stiffer)
Were much better than runs in the stiff range from 10 to 500
One of the key conclusions that one must come to is that an active + passive arm
with a gain of 50 (pretty stiff/less active) is as good as a purely active arm with a
63


gain of 1. Indeed the passive element can be incorporated into the arm, increase
the stiffness of the arm thereby reducing the bandwidth and the need for excessive
fixturing while still maintaining or exceeding the performance of lower gain, softer
impedance control schemes.
From the characterization data for the passive element the active + passive
runs agree reasonably well with the measured stiffness of the passive element.
From this the following conclusions can be made:
The thesis shows successful performance of active impedance control (could
not go below G = 0.5 due to stability problems)
The passive element has been successfully characterized
The active + passive vs. purely active runs show expected performance gain
Passive element on arm agrees with characterization
As far as cost analysis goes, clearly the softer the impedance the greater the
bandwidth and the greater the need for fixturing such as more or better sensors
better resolvers and more accurate electronics. This thesis shows that a relatively
stiff impedance control can be supplemented with a passive element to take the
compliance out of the software and put it into the arm itself thereby increasing job
performance. In a sense a person could trade expensive fixturing and bandwidth
constraints for a quite inexpensive passive element in their robot and still maintain
64


a desired level of performance. Of course, this approach has its limitations as was
detailed in Fig. 6.15 in which the purely active run performed better than the active
+ passive run. One may think that this issue could be easily addressed by making
the element arbitrarily compliant. The problem with that is in industry a passive
element that is too compliant will tend to droop due to gravity. Also, a passive
element that is too soft may wobble too much at the end of its path to be of much
use. It may have superior force reduction characteristics but it may take too long
to stop oscillating.
7.2 Potential Applications
There are a number of potential applications for the information in this
thesis. Since weight is of prime importance to NASA launch vehicles a light
passive element replacing a certain amount of sensors and other fixturing could be
very beneficial to a launch mission.
In addition to this one should keep in mind that manipulator manufacturers
do not want their customers to tamper with the position controller trying to make
it a better controller. The higher forces produced using stiffer impedances in the
impedance specification filter may be reduced through active means but this
ultimately means altering the position controller and is therefore unfeasible.
65


Incorporating the passive element into an industrial robot manipulator would
reduce those forces without all the fixturing and necessary controller tampering
and is therefore much more practical.
7.3 Recommended Future Research
A good idea for a future topic of research is to invent a variable passive
element. The passive element used in this thesis served its purpose well but it only
had one value for K and B (in the range of forces the arm was operating in).
NASA as well as other industries may want to incorporate a passive element into a
robotic arm for assembly and/or mating tasks. Once the controller has achieved
peg-in-hole insertion the task may require a torque to be exerted on the task to
move the object to a different location or orientation. This may require a stiffer
passive element. If the stiffness and damping parameters could be varied and made
stiffer after insertion the task of assembly would be made easier.
Another topic may be to alter the position controller to respond faster to
environmental forces and compare the resulting data to the active + passive
approach.
Finally, one could accurately determine the value of B for a passive
element. This is a much more formidable task than finding K the spring stiffness
66


parameter. Ultimately, this thesis opens the door for a wide range of research both
practical and theoretical.
67


APPENDIX A
Control Files
An important tool in data generation and manipulation in this thesis is
known as control files (ctl files). The next few pages contain print outs of ctl files.
These files set up the different internal impedances for different runs and they
basically controlled the robot arm impedance characteristics.
The impedance control flag indicates whether the user wants the impedance
controller on or off. The user can also adjust the proportional gains for the
shoulder, elbow and wrist joints as well as the derivative gains. K and B, the
impedance parameters, could be determined for each DOF and the user can also
input whether or not this particular set up would be using a peg. Finally, the user
may choose a set of inertia matrix parameters depending on the size of the load.
The ctl files proved to be a powerful tool in experiment runs.
68


7-pec- 94 09:35________-scivni t z/I sm/Cer/Rt Code/hdn hbw.ctl
' This is a .C7L file for tne dua 1 -proc executable: HDMj.EXE
' Assumed sample time: 2 0rr.s
: NC IMPEDANCE CONTROL 1.2 ho on ShiEl :. C no or. W: Joint
der discrete filters v(o) (Num.Den)u(o) are computed as :
urn ; 0 ] *u £ k) Sum [ 1 ] *u pc 1 ) Nun[2!* u(x-2) . * Nun [ ; n ) u ( x n)
- Den. (0; y (< -1 ) Den ; 11*y(k-2) . . Den! : r. -1 j V ( X n
Impedance control flag: O-OFF l*ON
0
Vioranon Av oidance (VA) pres nap mg parameters (NOTE: KAXNF 2):
0 VAfl ag: 0-05 f 1-Cr. 2Change Parameters ; Otherwise
' No. of filte rs to i nolemer.t ( < KAXNF )
3 3 ' No . of impui ses for MAXNF filoers ( 2 <- NlT.p <- 4
1 Prin ;.s f ilce r coef:
real
Proportional gains f or [ Sh El wr ]
! 55.35 56.35 55.35 56. 35 56 .85 266.7 1 2.5 He bw or. Wr
56.35 56.35 157.9 he pw :r. Wr
Derivative g a ins for [ 3h El Wr 1 :
' 10.55 10.55 10.55 10.55 10.55 21.99 ! 2.5 He pw on Wr
10.55 10.55 17.59 : 2.0 He bw on Wr
Impedance Stiffness K fo
5.0 5.C 100.0
Fy To 1 units { iP/ir. Id/in in-lb/rad):
! Impedance Damping 5 for [ Fx Fy To ) units
2.5 2.5 50.0
O-sec'rail:
Distance (m inches) from. F/T sensor
f rane to tip or pe
6.3 33 ' using the Long t-Smi pep
1.0 ' r. o pe g
Inertia matrix parameters for Theta [ 000 1:
! Oil J21 J31 032 022 023
:: 53.16 27.16 6.44 3.7; 5.:: 1.033 -> 6C lb load
25.76 11.62 1.33 1.03 5.36 0.265 ' - > 20 lb load
1 2r.c-order Inner FeedF o rward filter fc: S.-.l = r:
1 0.337 0.0 0.0 1 Nun I N-m;i! Nun;2 >
-0.996 0.333 * Den 0. Den 1!
1.0 0.0 C 0 2r.c-crder Inner FeedF c.o 0 0 or-3.'5 ol-
1.0 0.0 f\ *' * --.a a " 0.0
Page 1
69


70


APPENDIX B
Data Matrix
Matlab has been very instrumental in this thesis. It was used primarily as a
plot generator. It would accept data from the Intel 310 (via ETHERNET) in
matrix form and produce a plot with a predetermined column acting as the x axis
and one or more columns yielding information along the y axis. Below is a
representation of the data matrix the Intel 310 could generate with all 16 columns.
Time| Joint Reference! Joint Measured| Cart Reference! Measured Forces! Joint Ref
Radians Position Rad x, y, Fx, Fy, Torque Rad
1 2,3,4 5,6,7 8,9,10 11,12,13 14,15,16
The numbers correspond to the columns in the generated matrix. The rows would
be successive samples. Since 250 samples were taken per run the matrix generated
would be 250 x 16.
Once this matrix was generated Matlab could then assign column 1 to the x
axis and column(s) for what ever information required for this particular section of
the thesis to the y axis. Columns from different runs (different matrices) could also
71


be compared and graphed together. This set up greatly simplified data
manipulation and plot generation.
72


References
[1] Oh Y. H., Chung W. K., Jeong K. W., Youm Y., Implementation of Passive
Hardware Damper for Force and Impact Control, 1994
[2] Vukobratovic M., Contact Control Concepts in Manipulation Robotics An
Overview, IEEE Transac. on Indust. Electronics., Vol. 41, February 1994
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Full Text

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COMBINING PASSIVE AND ACTIVE IMPEDANCE PARAMETERS FOR OPTIMIZED MANIPULATOR CONTROL by Stephen Matthew Scholz B.A. University of Colorado at Boulder 1991 A thesis submitted to the Faculty of the Graduate School of the University of Colorado at Denver in partial fulfillment of the requirements for the degree of Master of Science Electrical Engineering 1995

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This thesis for the Master of Science degree by Stephen Matthew Scholz has been approved for the Department of Electrical Engineering by Gary G. Leininger Mijole Radenkovic Thomas Depkovich

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Scholz Stephen Matthew (M S Electrical Engineering) Combining Passive and Active Impedance Parameters for Optimized Manipulator Control Thesis directed by Professor Gary G Leininger ABSTRACT Most error correction in manipulators is in the form of active control. This is usually expensive and cumbersome If a certain amount of this active control could be replaced by a passive element much of the fixturing and bandwidth requirements for active control could be greatly reduced simplifying manipulator operation This thesis addresses the issue of combining passive and active parameters to reduce the measured environmental forces from a hole on a peg during a simple peg-in-hole insertion experiment. Relevant background work is cited preliminary control experiments are performed and passive + active experiments are analyzed The conclusion states that indeed an inexpensive passive element can replace a significant amount of active control reducing costs to the user and simplifying operation This abstract accurately represents the content of the candidate's thesis. I recommend its publication 111

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This thesis is dedicated to my parents Without them none of this would be possible

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CONTENTS Chapter 1. Review ofPassive vs Active Control... ....... ..... ....... ...... ..... .............. ..... ..... 1 1 1 Problem Statement. .. ..... ........... ............ ............. ..... ................ .... ............ ..... 3 1 2 Overall Approach to the Thesis ..................................... . ............................... 3 2 Technical Background .................... ......................... ....... ..... ................ ......... 6 2 1 Historical Overview ............................... ................. ..... .... ............................ 6 2 2 Manipulator Kinematics and Dynamics ....... .......... .... .......... ........................... 7 2 .2.1 Kinematics .......... ..................... ........ ...................................... .... ....... ........ 8 2 2 2 Dynamics .......... ........... ............. ..... ............................. ......................... .... 11 2 2 3 Actuator Dynamics ......... ..................... . .... .... . ............................. ........ 13 2.3 Control Overview ........................... .... ................ . ........................... ...... .. 2 3 1 Independant Joint Control.. ....................................................................... 14 2.3 2 Compliant Control ........ ........... ............................ ................ ......... .......... 17 2 3 3 Impedance Control ................ ......................... ..... . .... ....... .... ...... .......... 18 3 Hardware Setup and Description ................................................................... 19 3 1 The Arm .......... ............ ........................ .............. .................................... .... 20 3 2 Data Manipulation ......................................................... ...... ..................... . 21 v

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4 DOF Model for Impedance Control... ......................... ... ....... ...... . ............... 22 4 1 Rational. .... ...... ......... ... .... . ....... ....... .................. ................. .... ....... ... ....... 22 4 2 Study Overview .......... ................ ............... ............. .... ......... ............ ....... 24 5 Impedance Control Models ...... .... . ............... ... ... ......... ...... ...... . ........ .... .... 26 5 1 Active Only Approach . ..... .... ..... ... ... ........ ......... ... ..... ... ..... . .... .................. 26 5 2 Passive + Active Control. .... ................ ........... ................... .... ..... ..... ........... 3 1 6 Analysis ofResults ................... . ... .... ........................................... .... ........... 40 6 1 The Scenario ......... .... ...... ................. ..... ........ .......... . .... ....... ......... ......... .40 6 2 Passive Element. ............... .... ......... .................. .... ... ........... ............ ... ...... .... 41 6 3 Preliminary Results ........ ......... .................................. . .................. .... . .... .... .47 6.4 Experimental Results .... ........ ........ ............. ........ ... .... ........... .................... 55 7 Conclusions ......... . ....... . ... ... ..... ............. ....................... ........... .... ........ .... . 63 7.1 Model Validity .... .... ....... ... ........................ ........... ...... ... .... ... .. ... ......... ....... 63 7 2 Potential Applications ..... ............ ........................... .......... ... . .... .......... ...... 65 7.3 Recommended Future Research .... ........ . ................ ... .... ....... ....... .... ....... ... 66 Appendix A Control Files ... .......... ............................. ...... .... .......... ............. ................... 68 B Data Matrix ............................................ ............................ .... ....... .............. 71 References ..... ... ... ........... ...................... ...... .... ... .......... ................................... 73 VI

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Figures 2 1 Robot Coordinate System 7 2 2 Coordinate Reference System 9 2 3 Transfonnation of Coordinates 9 2.4 Actuator Block Diagram 13 2 5 Simplified System Block Diagram 18 3.1 Picture of Manipulator 19 3.2 Lord Force/Torque Sensor 20 5 1 General Model for Motion 26 5 2 Manipulator Impedance Block Diagram 27 5 3 System with Impedance Specification Filter 28 5.4 Coupled Mass Model for End Effector and Manipulator 31 5 5 Damper Only System 35 6.1 Typical Run 40 6 2 Passive Element 41 6 3 Determining Parameter K for Passive Element 42 6.4 Y Axis Force vs. Displacement 44 6.5 X Axis Force vs Displacement 45 6 6 Both Axes Force vs Displacement 46 6 7 Preliminary Run 47 6 8 X Force vs. Time for Preliminary Run 49 6 9 Closeup of6. 8 51 6 .10 Y Force vs Time for Preliminary Run 52 6 .11 Closeup of 6 .10 53 6 .12 Accuracy oflmpedance Specification Filter 54 6 .13 Qualitative Graph of Run Phases 56 6 .14 X Force Comparison at G =50, 150 57 6 .15 Y Force Comparison at G = 0 5 59 6 .16 Y Force Comparison at G = 1 60 6 .17 Y Force Comparison at G = 5 61 6 .18 Y Force Comparison at G = 10, 50 500 62 Vll

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ACKNOWLEDGMENTS I would like to thank the people at Martin Marietta and especially Thomas Depkovich Ph D for providing the manipulator and crucial guidance necessary in producing this thesis I would also like to thank Professor Gary Leinniger for his valuable advice as my Thesis Advisor. Vlll

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1. Review of Passive vs. Active Control Robot manipulators and their controllers like all real physical control systems experience a certain degree of error when performing a task For a robot arm this task may be assembly welding or mating type tasks These errors may result in torques and forces damaging to the manipulator and the task There are two ways this can be fixed The first is to gain a better knowledge ofthe manipulator and the task and their relation to each other through structural additions and constraints like sensors and fixturing Fixturing is modifying the work area or the environment such that the errors are reduced This method can become cumbersome expensive and inflexible The second method is to design systems and controllers that accommodate the forces and torques. The system or the controller may incorporate the ability to change the path of the manipulator as forces and torques evolve such that peak forces and torques stay below a pre determined level. A certain amount of fixturing is inherent in most industrial applications of robotic arms due to the highly structured nature of factories This thesis therefore will not explore the varying degrees of fixturing Instead it will explore manipulator system and controller designs that meet manipulator task interaction requirements such as speed peak force and torque while simultaneously satisfying all other system requirements that arise as a result of non-interactive tasks like free space positioning and slewing

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There are two basic approaches to designing a robotic manipulator system capable of reacting in a predictable manner to forces and torques that arise as a result of interaction with the environment. The first of these is generally referred to as the passive approach while the second is referred to as the active approach In the passive approach it is assumed that either a material or mechanism inserted between the task element grasped by the manipulator and the end effector will as a function of structural properties relieve forces and torques as they occur The simplest example of a passive device is some type of rubber material inserted between the task and the end effector Another approach is to have a spring-like mechanism between the manipulator and the grasped task. In the active approach it is assumed that forces and torques generated as a result of interaction can be sensed fed back and utilized by the manipulator controller in such a manner that evolving forces and torques are adequately controlled or minimized The pa ssive approach brings simplicity to the system It does not require extra sensors electronics programming computation etc However once it is in place it stays in place While the passive approach is desirable for certain jobs such as mating it could fall short when it comes to other jobs requiring heavy lifting or torque exertion such as assembly tasks The active approach may be turned on or off at will or as a function of the task being performed. The price for this flexibility is more sensors electronics programming and more complicated computation 2

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1.1 Problem Statement At first it ma y seem that the purpose of this thesis is to strike some sort o f happ y medium with a trade offbetween the active and passive approaches One could look beyond this b y asking : can a manipulator system be designed with both active and passive attributes such that the strengths ofboth approaches can be maximized while minimizing their weaknesses ? This is a more subtle question than the question of attaining a happy medium because it asks if using both approaches could create a system that surpasses in a practical and economic sense another system using only one approach 1.2 Overall Approach to the Thesis The next chapter will set the stage for the investigation by reviewing some basic concepts of robotic manipulators and formalizing the conventions and notation that will be used Of particular importance dynamic models will be developed that in addition to standardized manipulator link/actuator models will address the modeling of passive devices as well as models for the task interac t ion The chapter will conclude with an overview of manipulator control strategies that emphasizes currently used approaches for active control. Chapter three describes the hardware used in the thesis This includes the arm structure the joint motors and the computer hardware Details of data storage can be found in the appendices The fourth chapter will be devoted to the specific methodology that will be employed in this investigation The discussion in this chapter will include the selection o f specific task and manipulator parameters the range of experimental 3

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actions to be investigated and most importantly a set of performance metrics to be used for analyzing the results of the investigation. These metrics will address system performance system cost and if possible issues associated with reliability and maintenance This is critically important since a tradeoffbetween combinations of competing approaches is of little use if the investigation does not also begin to explore the means by which cost sensitive users can utilize the results to establish their own system requirements The fifth chapter discusses the models used for impedance control. The three models used are active only passive only and active+ passive control. The chapter develops the models through standard impedance control methods At first the system is developed without an impedance specification filter for the active only model. Then the impedance specification filter is incorporated into the system through a feedback loop for the active + passive control model. Finally the passive only controller is developed by incorporating the passive element into the system while setting very stiff impedance parameters for the impedance specification filter. This effectively turns off the filter feedback loop The sixth chapter presents the analytical results of the study This will include the specific dynamic system models selected the system simulations performed and the analysis of the simulation studies A broad set of simulation scenarios will be analyzed and used to select a small specific set for hardware implementation. The seventh and final chapter will address results and conclusions Model validity will be examined and a discussion on potential applications in industry will 4

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follow Finally future research possibilities will be examined which would use the results found in this thesis 5

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2. Technical Background In this chapter the technical background for the thesis will be developed A history of research on active and passive compliance in robotic arms details the work done by others in this field and establishes the groundwork on which the thesis is based Manipulator dynamics follow and give a detailed mathematical and graphic representation ofthe 3 degree of freedom (DOF) arm used Finally, the control schemes used will be detailed at the end ofthe chapter 2.1 Historical Overview This section will present an historic overview of passive and active compliance in robotic arms Many articles have been written on implementing passive devices to achieve task completion [1] and contact control [2]. Active compliance has made its way into many articles as well [3]. These articles deal with highly suitable applications however none deal with combining the two approaches to determine the optimal combination. Furthermore the articles have been dealing with a very specific part of passive/active compliant application Force and impact control [4], contact control [5] constrained motion tasks [6] and stability in contact tasks [7] are thoroughly covered in these and other articles This thesis will deal with manipulator / task interaction after contact has been made It will approach the problem with the hypothesis that it is possible to incorporate 6

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both active and passive compliance to optimize manipulator control. This has not been done before 2.2 Manipulator Kinematics and Dynamics In this section the kinematics and dynamics of the manipulator system will be defined Once the manipulator comes in contact with the environment it becomes part of the manipulator system and therefore changes the system and influences the job the manipulator has to do. Fig 2 1 is a representation ofthe system used It can be assumed that there is some type of passive coupling between the manipulator end-effector and the task For the sake of simplicity a planar arm was used. Figure 2 1 Robot Coordinate System 7

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This coupling can be represented in terms of stiffness and damping parameters 2.2.1 Kinematics For the main purpose of establishing notation this section will give a brief discussion of the kinematics. Specific solutions to inverse kinematics and specific convention for assignin g link reference frames will not be covered Please see [8] for more details We can easily determine the relative orientation of two reference frames in space by using the rotation matrix The simplest case is where the origins of the two reference frames are coincident. In this case the rotation matrix can be used to transform a vector in one reference frame into the other frame In this case we will use the following notation to define a rotation matrix R that transforms a vector P from frame B into frame A: where cos0 -si0ne 8 (2 .1) (2 2)

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Fig 2 2 is a graphical representation relating frame A to frame B Ys YA Figure 2.2 Coordinate Reference System In the more general case when the origins are not coincident a 4x4 Homogeneous transform matrix must be used The vector requires a rotation matrix as before with the addition of a vector relating the separation of the origins of the two different reference frames This equation is shown below Figure 2.3 Transformation of Coordinates A p= A R BP+A P =ATBP B Borg B (2.3) The transform matri x simply combines the two operations into a single step This matrix has the form shown below and will be designated T. (2.4) 9

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In the case of robotic arms which will link multiple reference frames this can be further generalized by assigning coordinate frames to each link The product of the resulting transformation matrices will yield the end-effector position and orientation. The forward kinematic relationship that relates manipulator joint angles to end effector position and orientation in a designated coordinate frame can be defined as L(0). This function yields both a position vector and a rotation matrix The inverse ofthis function yields manipulator joint angles as a function of manipulator position and orientation and is called L-1(X) Therefore : L(0) =X L-1(0) =X The forward kinematic function relates manipulator position and attitude to manipulator joint angles By calculating the differential of this function an additional important function called the Jacobean is obtained This operation (2 5) (2 6) results in a linear differential relationship between Cartesian position and attitude and manipulator joint angles as well as a relationship between manipulator linear and angular velocities and joint velocities ax= dX = J(0)d0 dt dt 10 (2 7) (2 8)

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2 7 above is also useful in relating end effector forces and torques to joint torques. F represents the forces and torques at the manipulator end effector, and 1 represents joint torques Then from the principle of virtual work : (2.9) and from the relationship between C0 and oX : (2 1 0) 2.2.2 Dynamics The general equation of motion for a robotic manipulator is given below. r = M(E>)E>+V(E>,E>) (2 11) M(E>) is a 3x3 mass matrix and C( 0 0 )represent Coriolis and centripetal forces In Cartesian variables the equations would take the form : Mx = J -r (E>)M(E>)J -1 (0) Vx(E>,E>) = J -r (0)(V(0,0)-M(E>)J -1 (0)J(0)0) 11 (2 12) (2 13)

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This is called operational space representation and results in a system of equations expressed in terms of Cartesian displacement and orientation The equations below relate the terms in this system to those in the configuration space equation. First multiply both sides by the Jacobian transpose 1-T T = 1-T M(0)0+ 1 TV(0,0) or (2. 14) (2. 15) Where F is a force-torque vector acting on the end-effector of the robot. F can also take the form (2. 16) Here X is an appropriate Cartesian vector representing position and orientation of the end-effector. Mx(0) is the Cartesian mass matrix and Yx(0 ,0) is a vector of velocity terms in Cartesian space. A relationship can then be developed between joint space and Cartesian acceleration starting with the Jacobian X=10 (2.17) then differentiating to get X= 10+10 (2. 18) 12

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Then solve for the joint space acceleration to get (2 .19) Substituting this into the force-torque vector equation to get (2.20) It is from this equation that the terms for the Cartesian dynamics are obtained 2.2.3 Actuator Dynamics The equations above describe a perfect robotic arm In reality pure torque generators do not exist and actuator dynamics must be accounted for. Fig. 2.4 IS a block diagram of the actuator system used 386 proc J D /A I Analog Controller PWM 3 -DOF Digit. Cont. & Sensor Electrnc Amps Manipulator jl' AID Accelerometers, Tach &FIT sensor lAID Resolvers Force/Torque Force/Torque Processor Transducer ifFigure 2.4 Actuator Block Diagram 13

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This is still a very complicated control system design even when non-linearities due to the compliance presen t in almost all power transmission mechanisms are neglected If this compliance is not comp ensated for in some way the addition of dynamic effects due to electronics will produce a grossly inaccurate model. 2.3 Control Overview When robotics gained national attention in the 1970s it was eagerly embraced by the control system community as a whole It was recognized almost immediately that the nonlinear coupled robotic system represented a prime application area for many of the key disciplines within the control system area The techniques that appeared ideal for application included nonlinear optimal multivariable and adaptive control, One problem that arose however was the lack of available testbeds for hardware experimentation Without hardware test beds to focus research activities much of the work remained very theoretical utilizing simplistic models to represent manipulator dynamics The result of this was an overemphasis of the nonlinear control issues and a deemphasis of the practical issues associated with manipulator control. 2.3.1 Independent Joint Control With very rare exceptions the vast majority of robotic manipulators in use today are controlled using independent joint servo controllers that are designed using techniques that have been more or less standard for forty years Even in the cases where more refined linearizing decoupling control is implemented joint 14

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torque controllers are required to generate the desired output torque profiles on which these methods depend Therefore the issues associated with the design of independent joint controllers remain important. Their importance will probably grow with time since successful implementation of the more exotic control strategies will have to rely on the incorporation of the lessons learned in successful independent joint control studies There are two primary problems associated with independent joint control. The first problem arises from the structure of the typical actuator mechanism used in robotic systems that was discussed at the end ofthe previous section With the exception of manipulators that use direct drive actuators almost all systems use some means of torque amplification . In most cases this is achieved through the use of spur gears but examples of harmonic drives belts cables and ball screws also exist. These power transmission elements generally have two undesirable side effects : compliance and nonlinear friction Both complicate the control design process and require special techniques The second problem arises from the nonlinear coupled nature of the manipulator dynamic system From the standpoint of the independent joint controller the variation in manipulator dynamic properties represents both a change in the inertia of the plant that is being controlled as well as a source of torque disturbances These disturbances are a function ofthe position velocity and acceleration of the other manipulator joints. Therefore controller design must be robust in the sense that adequate performance can be maintained as both the inertia of the plant varies and disturbances are encountered 15

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In general there are two different structures through which independent joint control can be implemented These are shown below and are referred to as joint based and "Cartesian based techniques Of these the joint based approach is the most straightforward to implement since it requires only an inverse kinematic transform In the Cartesian approach, two stages of transformation are required : a forward kinematic transform of the vector of joint position sensor signals and an inverse Jacobean on the Cartesian error signal. In reality with modem microprocessors both strategies can be implemented at real time rates The joint based approach is probably the most popular of the two since it simplifies the servo design process and makes it much easier to implement direct joint control modes where the operator can control one joint at a time There is no general technique for the design of the individual joint control loops Like most control problems each must be treated individually There are however two basic trains of thought that appear in a variety of sources One train is that conventional proportional-derivative control techniques are presented for controlling a single actuator. These discussions also address the issues of disturbance rejection and typical nonlinearities such as static friction Most references also contain some discussion of the use of PID controllers. These have been very popular in industrial implementations The general rationale normally cited for the use of a PID controller is the role of the forward loop integrator in achieving zero steadystate error to a step disturbance which is a reasonable model for gravity loading While both of these approaches represent reasonable starting 16

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points for a design effort the problems encountered in producing a final design rarel y succumb to such simple solutions 2.3.2 Compliant Control This section starts to deal with when there is environmental interaction with the robot. For the purpose of this discussion a controller that must contend with reaction forces and torques from the environment will be referred to as a compliant or force controller Many promising demonstrations have been performed in a number of laboratories but there is currently no mature standardized set of algorithms with a sufficiently good track record for general implementation As in the case of the development of single arm control laws part of the problem has been the availability of hardware testbeds for experimentation This problem is now diminishing with a larger number of available testbeds and the availability of commercial force and torque sensing devices In compliant control tasks the manipulator control system must deal with kinematic constraints that are associated with the task and in some cases be capable of applying forces and torques at the end effector of the manipulator as dictated by the task. There must be a relatively accurate estimation of the end effector forces and torques A good way to achieve this is through the use of a force/torque sensor mounted on the proximity ofthe end effector. Force/torque sensors employ 17

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strain gauge sensing elements One of the more popular configurations for this type of device is shown and discussed in chapter 3 2.3.3 Impedance Control Position based impedance control is the method of control in the thesis It makes the robot arm behave like a second order spring/damper system While the controller normally acts on nominal position and orientation commands it also has the capability for commanding applied forces and torques t hr h h ougJ t e mputs s h F 25b I ownm lg eow. Manipulator End Effector E-B l -Position Control Impedance Specification EFilter 1 B ; s + K ; Figure 2 5 Simplified System Block Diagram The desired impedance relationships are achieved by filtering measured forces and torques through impedance specification filters as shown The outputs of these filters are position and orientation perturbations which are used to modify the nominal commanded reference signals Currently translational commands are modified through the use of transition matrices The fine details of impedance control will be dealt with in chapter 5 18

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3. Hardware Setup and Description Martin Marietta has provided the manipulator testbed and all sensing and data manipulation hardware This chapter will describe the hardware used in the thesis Fig 3 1 is a picture of the arm on a nearl y frictionless surface Figure 3 1 Picture of Manipulator Used in Thesis 19

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3.1 The Arm The manipulator used is a 3-DOF arm supported b y a i r bearings on a flat surface This enables the arm to achieve 2 dimensional space-like motion The arm itself has about a 4 5 ft reach The links are thin and plate-like to prov i de compliance in the vertical plane This will compensate somewhat for imperfections in the flatness of the test bed and will prevent the air bearings from grounding A Lord force / torque sensor is at the end of the wrist and can measure force in the x and y directions as well as the torque exerted on it. It is a six DOF sensor able to measure forces and torques along and around all axes The basic structure is shown below in Fig 3 2 Forcefforque sensor with Z axis out of the page with strain gauges Figure 3 2 Lord Forcefforque Sensor 20

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Each joint consists of a harmonic drive with a gear ratio of 80 : 1 driven by a DC motor. Above the motor housing are the sensors consisting of a motor side tachometer an output torque transducer and a resolver to measure the joint angle Cartesian accelerations are measured through accelerometers mounted on the wrist. 3.2 Data Manipulation The controller package consists of a modified Intel 310 system using single or double 3 86 processors The program is written in C and enables the user to select sampling rates (50 Hz was used for this thesis) open or closed loop control modes joint space or Cartesian space reference commands and sensor/actuator sets as needed to control from one to three degrees of freedom The user can record item-by-item a selection of data such as sensor measurements and/or control law variables and the rate of data selected These data files can then be transferred via ETHERNET to a work station for further analysis with commercial software (Matlab was used) See Appendix A for more on data setup and transfer Appendix B has more on how Matlab was used to analyze data 21

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4. DOF Model for Impedance Control This chapter will take a look at the way the thesis approaches the issue of fitting the robotic arm with a passive element for the purpose of enhanced performance with a decreased amount of active control. 4.1 Rational When robotic manipulators are interacting with the environment as a part of task execution small errors in task alignment or manipulator positioning can lead to extremely large forces and torques being generated These have the potential for damaging the manipulator and / or the task. This problem was recognized very early in the development of robotic technology [9] [10] [11]. A number of approaches have evolved to help solve this problem. In general there are two basic strategies that are used The most obvious approach is to have some type of element between the manipulator end effector grasping part of the task and the primary manipulator system This is generally known as the Passive approach The passive element can be something as simple as a compliant material such as a rubber pad Other approaches to passively relieving forces and torques have focused on more complex arrangements of springs and dampers The second and 22

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more complex approach is to use more sophisticated active control strategies that directl y modify the closed loop control of the manipulator to achieve the desired characteristics In general these active approaches require the use of a wrist mounted force / torque sensor to provide inputs to the controller Passive approaches have the advantage of simplicity but they lack flexibility Once in place it stays in place for all tasks This is not always desirable Active approaches have an inherent flexibility but they also come with a number of problems due to increased sensing and computational requirements In most application examples the final implementation ends up being a combination of both active and passive control. The passive element of implemented controllers arises from the fact that most end effector/gripper designs have some type of passive already built in Inevitably this ends up making the job of active controller design much easier Therefore although passiv e and active approaches have been used in conjunction many times in the past combined active and passive control has never been studied from the standpoint of a shared control perspective. It seems reasonable that this topic can be investigated with the objective of determining based on known characteristics : 23

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As a function of the tasks what overall impedance properties are required ? (The tasks include all tasks interactive or otherwise ) What options are realistic for passive and active elements ? How control functionality should be distributed between passive and active elements (i. e design guidelines for selecting passive and active parameters ) A key element of this effort will be the examination of ways in which the control can be partitioned with respect to some global measure of optimality 4.2 Study Overview At a summary level this study will use analysis and simulation techniques to study the basic problem Results from the analysis and simulation will be verified experimentally in hardware Analysis simulation and hardware experimentation will be limited to a planar 3-DOF case. At the conclusion extension of results to the more general 6-DOF case will be discussed Because a study of this type has a number of subtle complexities it is useful to break the overall study down into the smaller tasks that are listed below : 24

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Manipulator / task/controller modeling Cost function synthesis Task performance analysis Design guideline formulation Hardware validation 25

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5. Impedance Control Models This chapter deals with a description of the active and active + passive control of the arm First an active only scenario is developed to demonstrate the inherent problems with increased active control. Then a passive element is introduced to the arm and the effect is examined 5.1 Active Only Approach The general model for motion is shown below in Fig 5 1 --+X Figure 5 1 General Model for Motion The equation for this is .. mx=Fc+Fe (5.1) where (5. 2) 26

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The K term is the position feedback the B term is the velocity feedback and the x(K) term is the reference command scaled by K. Then mx+Bx+Kx = KxR +F, +Bs+K)x+KxR +F, K F, X = + --, ----"--mS" + Bs + K mr + Bs + K This is shown graphically below in Fig 5.2 K +Bs+K 1----+ X FJK Figure 5 2 Manipulator Impedance Block Diagram Initially the F JK term is ignored This is because an environmental force that (5. 3) (5. 4) (5.5) could damage the task -say about I 0 lbs could be insignificant with respect to the manipulator (about 3000 lbs. ) 27

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Now it is assumed that XR is generated as a result of active impedance control with a commanded position of zero as shown below in Fig 5 .3. :xc=O XR The rest of the system __.. Fe impedance specification filter c (B,.s+ K,).s Figure 5.3 System with Impedance Specification Filter Then x = K ( C ) F +Bs+K (B,s+K,)s (5. 6) or X KC =--------------------F. +Bs+K)(s(B;s+K;)) (5.7) So to achieve a desired closed loop impedance it is necessary to deal with manipulator position control dynamics That is K/(ms2+ Bs+K). 28

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When the direct action ofFe on the manipulator is not neglected this last equation becomes X (KC + l) F e s(ms2 + Bs + K)(B; s + K ; ) (5.8) In either case achieving a desired closed loop impedance (5. 9) requires adjusting impedance control and position control (5 1 0) Now this can be written as (5.11) or b d -----= ------------------s(s+ a) s(s2 + 2(m ns + m!)(s +e) (5.12) Now take the Laplace Transform to find 29

PAGE 38

L 1 { d / e } = 1 f: OJ; ( etlf) + +2(0JnS+OJ;)(Js+!) 1-2/(0J" e-'tlJI + j:OJ;) (5.13) This last tenn is of no concern but the one and the second tenn yield important results for the complete range of w Notice (5.14) and lim r(OJ J = O m,. -.o (5. 15) and lim r(OJJ = /: = 1 cu,.-.co j-(5 16) So a higher bandwidth is required for active only control. The next section will look at active control with a passive element to decrease bandwidth and maintain acceptable accuracy 30

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5.2 Passive + Active Control The passive element between the wrist and the peg is a compliant substance in this case foam Fig 5.4 is a model ofthe arm/end effector coupled system Figure 5 4 Coupled Mass Model for End-Effector and Manipulator where m is the mass of the arm m e is the mass of the manipulator x1 is the distance the arm moves x2 is the distance the end effector moves F e is the force from the arm Fe is the force the environment exerts on the end effector and B p and Kp are the damping and spring stiffness constants Below are the equations of motion for this coupled system : 31

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after a Laplace Transform these become (ms2 +BPs+KP)x1 =Fc+(BPs+KP)x2 (m,s2 + BPs+KP)x2 = F, +(BPs+KP)x1 with some algebra these equations become The commanded inputs are F e = (Bs + K)x1 + xRG c FG XR=---(B;S+ K;)s (5. 17) (5.18) (5.19) (5. 20) (5.21) (5. 22) (5.23) where the first term in equation ( 5 22) is the PD control loop the coefficient of G in equation (5.23), and G is a general gain term Substitution in x1 yields 32

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Gc ----F +(B s+K )x (B ,s+K.)s P P 2 X I ms +(B p +B)s+(Kp +K) (5 24) now this can be substituted back into equation (5 .20) to get Gc ----F +(B s+K )x, ( B s + K ) s p p -(m,s2 +BPs+ KP )x2 = F, +(BPs+ KP) ....:....__:..___..:...:...__, -----ms+B's+K' (5 .25) where B'=Bp+B and K'=Kp+K Now for simplicity let m, ( s) = m,s2 +BPs+ KP m(s) = ms2 + B s + K' then equation (5.25) becomes or 33 (5 26) (5. 27) (5.28) (5 29)

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notice that in matrix fonn the detenninant is (5. 30) This results in an impedance of (5. 31) Ifthe ann undergoes completely active control Kp and Bp can be ignored That means that equation (5. 17) becomes mx, =Fe =-Bx,-Kx1 +xRG GcF (ms+ Bs + K)x1 = s(B,s+ K.) or GcF, x = s(B; s + K, )(ms+ Bs + K) substitutin g the last equation into (5.20) for an impedance of 34 (5. 32) (5 33) (5. 34) (5. 35)

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(5. 36) This last equation can be simplified to show the two components that contribute to the expression for the overall impedance (5. 37) l+ K 2 B K s(s+-' )(s + -s+) B ; m m Notice what the equation becomes ifKp = 0 as in the Fig 5 5 B m IDe Figure 5 5 Damper Only System 35

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1 x 2 m GcBP 1 (5 38) = 1 +-F. B B ; m (s+K/BJ(s2 +! s+ s(s+-P) m This equation can now be used to find the value ofBp. To do this the inverse Laplace Transform must be taken 1 L -' x 2 m GcB P 1 (5 39) = (s+K/BJ(s2 +! s+ F. B B ; m s(s+ _P) m 1 /Bp /Bp r' m = L -' ---B s s(s+-P) m m (5 .40) and through partial fraction expansion the second term becomes 36

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(5.4 1 ) wher e A = ( BPJ(K ; BPJ((1 B,J2 B P B + K J m, B m, m, m,m m (5.42) B = ( J ( B J ((l J : J K p K K K ; B K B, m, B, B, -mB, + m ( 5.43) (5.44) and "poly i s the polynomial representing the fourth and final factor in the denominator Jhis then becomes (5.4 5 ) For simplicity assume robot closed loop control. Then 3 7

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, B K s+-s +-= ( s + a)( s + b) m m and L-' {poly}= r'{ _!}_ _} s+a s+b where D= 1 a( -a+ !:)(-a+ E= 1 b ( -b + -b+ -b) These expressions for D and E come from the following s2+ (a+b)s + ab = s2 + (B/m)s + (K/m) so ab = K/m and a+ b = B/m now set m = 100 to get 100ab = K and lOO(a +b)= B (5.46) (5.47) (5.48) (5.49) (5.50) (5. 51) (5. 52) (5.53) (5.54) These last two equations are expressions for finding K and B to achieve a desired a and b 38

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So the final equation for finding the poles of the system is {x} (GeE)[ _rs,) t (!i) t J r ;, = C-Ae l . -Be r (5. 55) pole \_ poles from associated pole closed loop with passive element rr1e and Bp associated position with imped control spec filter s2+ B/ms +K/m B; and K; It should be noted here that the system can be made passive only simply by choosing very stiffimpedance parameters (on the order ofK; and B ; = 100) This effectively makes the spring and damper system so stiff it is as if the i mpedance controller was turned off and the ann would be operating on position control only. This option is explored in the experimental analysis 39

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6. Analysis of Results This chapter will discuss the scenario used in gathering data a detailed description and analysis of the passive element used in the thesis and typical results of runs under active and active + passive control. 6.1 The Scenario The typical experiment for this thesis was a peg-in-hole insertion run Fig 6 1 is a figure showing the set up and the path the peg took. PEG actual path (due to chamfer) Figure 6 1 Typical Run chamfer 40 command path

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The commanded path is slightly offset from the actual path. This error creates forces in the force torque loops due to the position based impedance controller These forces can then be recorded through the force/torque sensor and stored via the Intel 3 86 processor It is these forces and torques that the thesis addresses The passive element introduced into the wrist deforms and compresses to partially compensate for these forces This allows less stiff impedance characteristics of the controller hence reducing the bandwidth and fixturing of the system 6.2 Figure 6.2 Passive Element It consists of foam epoxied between two threaded metal plates It screws in place between the force / torque sensor and the peg and is held in place with two collars . The stiffness characteristics of the passive element can be accurately estimated by 41

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compressing the element a certain amount and measuring the force of resistance According to Hooke s Law this yields the spring constant through the equation F = -kx (6 .1) The quantitative stiffness was obtained by compressing the element while attached to the force / torque sensor The sensor detected the force necessary to deform the element a certain distance and transferred that information to the Intel 310 This information along with the distance the element was compressed was used to generate a Mat lab plot to determine the linearity of the element's stiffness The picture below shows the process in which this was done. ------------------------------------------------------------------------------------------------------------------------Force/Torque Sensor Passive Element Matlab Plot F x = F = y Torque= Intel 310 1 jForce ./ en Figure 6.3 Determining Parameter K for the Passive Element 42

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Fig 6.4 shows the Force vs Displacement graph for the passive element along the Y axis As with any real physical system the graph is not completely linear. The non-linearity arises out of the natural properties of the element. For the most part the thesis dealt with forces in the range from 0 to 5 lbs Fig 6 5 shows the same graph for the X axis. As one would expect this graph is similar to the one for the Y axis Fig 6 6 shows the two graphs together with the bias subtracted This shows the behavior of the element under small forces all the way to the origin According to these graphs the element has a stiffness k of about 25 lbs./mrn or 6 .25 lbs/in In the graphs for the experimental results the x forces change sign while the y forces dip only slightly below zero This is opposite of the steel wall tracking runs This is because when the peg tracks the chamfer the forces generated are similar to those when the peg is tracking the wall However when the peg enters the hole it comes in contact with two other walls (the hole) that are rotated 90 degrees from the frame of reference outside the hole So in this instance one would expect negative x forces and nearly zero y forces 43

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0 :"' I I \ I I I I \ I \ I .J 00 \ [ I I i \ I I I \ I L !: M .. .. > c: :::::: "' c "' Ui "' .. c. c c "' "-I 0 [ c .. > c ]. j 5 5 sql:l:>JO.:! Figure 6.4 44

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I ..... \ \ I \ oe : I I \ I ', I I I I \ \ I \ I \ I i \ I I ..: .,. I i I >< I 'E !::! g I u i:i i > I ::.. 5 5 -= r " .. v 0 "'-0 .. > c g u .... 00 v .. g l-D I I I I I I I I I I "" " I ' :::> 1'"1 ""' ""' = ,... 0 sqj:IJJO,j Figure 6.5 45

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-i i I I .... \ \ I I \ .. I \ \ \ I I \ \ \ \ \\ sq{ -;l:lJO;i Figure 6.6 46 i I I I I -... 5 "' c ;; u .. ].

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6.3 Preliminary Results Before peg-in-hole insertion was attempted the impedance controller needed to be verified a s working properly and a proper set of impedances determined These runs were done with no passive element present and with a variety of impedances Instead of peg-in-hole insertion the peg was pushed along a steel wall (the peg is made of aluminum) to determine if the forces generated are what would be expected Fig 6 7 below shows the path taken for these runs Figure 6 7 Preliminary Run Force / Torque Sensor Peg (initial) Steel Wall Commanded Path Wrist I I I I I I I I I I I I I I I I I I ----__ .J_--------__ I_------I I I I : Force/Torque : : Sensor : I I I I 47 Peg (final)

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Before 5 5 seconds the peg is not in contact with the steel wall so no force is generated At t = 5 5 the peg is in contact with the wall and begins tracking along the wall until about 8 2 seconds at which the peg comes to rest for two seconds then returns back to the starting point to repeat the run Since the final position is not the same as the commanded position and since the controller returns the peg by simply reversing the commanded forward path one would expect the force in the x direction (F x ) to always be positive A negative F x would mean some force is pulling the peg away from the force / torque sensor This is impossible under the given configuration However, since the peg will be back tracking along the wall on the return run one would expect the force in they direction (Fy ) to quickly reverse and become negative after the return path began Fig 6. 8 shows the force vs time for Fx. It is important to notice here the different levels the impedance specification filter produces between 8 and 10 seconds As the gain on the filter is increased from one (on the bottom graph) to 100 (the top graph) the force measured during this period when the peg is at its final point is increased The non-linearity of the arm is apparent especially when the graphs of G = 50 and G = 100 are compared The forces shown in the graphs of the lower gains are closer to being directly proportional than those in the higher gains This is due to the fact that at higher gains (more power supplied to the motors) the rigidity ofthe links is compromised This effect is more easily seen in 48

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I 0 .. u u >< ..,. ___ --7-,--__ ...:.._,"'::.._. ... -' ':,. -\--,,-=..... -=.-,. .. ...:-; .. __ -... :...::. .. --"' 1 l ', c sqf :l:uo 3 Figure 6.8 49 0 00 I ..,. 0 N .. ... ..., ..... ,. ..... "! u u = i=

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Fig. 6 9 in which the level offtimes are blown up for clarity The next graph Fig. 6 1 0 shows the same force vs time graphs but for F y Again the stiffest parameters are the least linear with respect to the closest gains Fig 6 .11 again shows the max deflection Y force for gains 1 through 100 blown up for clarity Fig 6 .12 shows the accuracy of the impedance specification filter. The commanded K for the filter in this graph is 5 After the initial transients one would hope that the measured K would be somewhat close to this value As it turns out the measured K was very close to 5 for both the x direction (top graph) and they direction (bottom graph) All the graphs in this section have shown that the impedance characteristics of the arm and the controller are quite sufficient to proceed with the experiments of incorporating the passive element into the arm to reduce the forces generated at the task. The next section will show conclusively that the passive element will indeed enable an arm with a Jesser degree of active control to perform as well as an arm with a greater degree of active control. 50

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c ' oc N I I I 8 "') I r-' I _;;2 \ N I I ' I ' 0 N I I I I 0 .., I N I I \ I I I I I .., I N 0 -.... ..... N g .., ..... -;-I -:;: I II I ..... 0 I -:: c I v 2 I :> = E [ I '"' N j c::: I u I "" I "' I .. E I ;;; I i= I I I I 0 0 I I N l N 1..:.. I I X I I I I \ I I \ :> ' I I I I \ I \ I I ,, I I ...! \ ( N I I I I I I I I I I \ I i I ) J ( I i :;:; ) I I I I I I I I I ,I x N ::: X -.!) :::Figure 6.9 51

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..,., II c j > l c X -___ .... --:::: .,---.=. ... -... ::: .. -; I :l I J I I sqj-;JJJO.::! Figure 6.10 52 --. !! .. ., ..... ..... ,. 0 ..... "! E i= I -.:

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-.... II 0 = = E c:::: ... .. E ; u >[ I / ---' I / I I I \ I I J / I I I I I \ I I I I I \ I I r I I / I I \ I I I I I / I I I I I I -,----) / \ ____ Q ,----------1 ,-.:.---(--, ________ .:' p ------I I I I I I ........ J I / I \ I I I I I I I I I I I I I I I I I I I I I / I f I I \ / I Figure 6.11 53 ( ( i I \ \ ) = I I l c ..... ..... c 1 "' ..... J: .... 0 t ,.., .... -,... .... r I 1 I l o j :'\ -0 X ";'-.... ..... ,. ..... "! c .... t=

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>. o(j ...., 0 5 ;: tl "' f i5 :::.. u v c: .., u e--= -;; c: "' w I L i I I I I I I I ...., ..,. ,..., 0 ,..., ...., c ,.... ,.... (u1) sod t(sql) Figure 6.12 54 c oc .., .., I 0 c: I ;; "C ..... 5 .:: "' -..,. u > f.: _;,s --:;::::::. "?! J...., f! f 1 I I ' \ ... ::::> ":'

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6.4 Experimental Results This section will discuss the results ofthe peg-in-hole insertion experiments Fig 6 1 shows the execution of this experiment. Fig 6 .13 is just for qualitative purposes It shows graphically the different stages of the experimental runs During the first two seconds the peg tracks the chamfer and enters the hole After the peg has been inserted into the hole there is a significant drop off in forces This decrease in force is expected After the peg has been inserted it spends two seconds resting while the force/torque sensor measures the environmental forces exerted on the peg After this two second period is done the peg is removed over another two second time period The graph shows the expected drop off in forces during this time When the peg is finally removed there is another rest period before the manipulator starts another run These sections occur throughout the experimental graphs Fig 6 .14 is very revealing The lowest graph line corresponds to a purely active run with gain equal to 150 The graph line just on top of that corresponds to a gain of 50 The top solid graph line corresponds to a run with a gain of 50 and the second solid graph is a run with gain 150 The only difference between the solid graphs and the dashed graphs is that the solid graphs represent runs made with the passive element in place It can easily be seen that incorporating the passive element significantly reduces the environmental forces on the peg The 55

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--"' ,-. ... ; -------::-: _-;::::::=-=-:=: __ ----__ ..,'!.,.' _________ ,-( .. .... -:: ;. _:: r;: -: = -. ':.': ............... -.. Fl.gure 6.13. 56 ' I ' ,,-'-.-I ,---' .. -:ao .... ..... -c' ..... ..... ..... ..... I 0 c u .: u i=

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Vl -....J '"':1 f-' t.Q c Vl ti..D ro,-v O'U L. 1--'0 >1>-U... X Forces Pass+Act vs. Act; G=50,150 -,--------0 5 () --0 5 -I --I. 5 . -2 4 1''1 I I I I I I 1111 1 tJ lh I I 1 I I I II 1 I II 11 II 11 "' II '11 II :,, 11 '1 .I} II 1 \ 11 u 1, 1 ) l 111 1 "' ' '' r< '" "' '' _,' lulh/\JAil111' ' \ -r'-.fV ;,! :: 11 ,I 11 I I J \ J11';./l\f 11 1 IJ } 1 1 r 1._ II 1111 1 1 J11111-,,1 1 1 J I I 1,/lllllr.l 1 '-.._\1\l\ J l 1 f l" I 1 I I I \ I IJ I )o. I II 1 I 1 v' 11 Ill 111 \ 1111!11 11 1111 I 1111 1111 1111 1111 ----'------------......L.. ___ _t_ ___ ___ ___L ___ ___.i. _____ ___.JL__ __ __j 5 6 7 8 9 10 II 12 Time sec 4/21/95 13

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anomal y be t ween 5 and 6 seconds i s from the pe g momentaril y gettin g caught on the edge o f the chamfer The same anomaly can be seen in the graphs for y forces The next few graphs will compare different gains with and without the passi v e element. The gain ofthe active + passive runs will alwa y s be 5 0 Fig 6 .15 shows an active + passive run with a gain of 50 compared to a purel y act i ve run with a gain of0. 5 In this case the active + passive run has generally higher forces than the purely active run The top graph is the active + passive run It is clear from this graph that the ac t ive + passive approach has its limitations However Fig. 6 .16 shows the purely active run with a gain of 1 compared to the active + passive run with gain 50 The purely active run is the one with the anomal y betv.-:een 5 and 6 seconds One can see t hat these two runs produce very similar forces after the transients have died down Fig 6 .17 shows a very significant difference between the two graphs The purely active run has a gain of 5 and produces much higher forces than the active + passive run Fig 6 .18 compares the active + passive approach (the bottom most graph) to a series of high gain runs of 10, 50 and 500 In each case the active + passive runs reduced the environmental forces much better than the stiff purely active runs 58

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0 Vi II 0 u ell 0 C"' :> u u C':l 0 "' I 0 E 0 :> "' "' 00 C':l 0. + 0 :> u sqj -;:JJJO .:I Figure 6.15 59

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II 0 u "" II) > u c:: vi > u > Vl Vl c:: 0. + u > u <( V) N Figure 6 .16 60 0 0 C\ 00 I"' u u Vl I II) f.=

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0 Vi II 0 -u C';l u ;> C\ u u C';l Q.) V) vi I ;> Q.) Q.) E ;> f= V) Vl 00 C';l c. + u ;> -u -< Figure 6 17 61

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0 0 Vi 0 Vi 0 .. .... ,_ ... "' -_ _, ) ..,_ --, ""-, --.:....!_,---' -._r \ -::""' -, I _--:,.. ... "" -=--... (-r.,_ -:., ,_, _, r I ::..., ...-.., '"-_....--''' ,.-__ .. I ) ,-""-"'-, -.... _-... \ -:-.. --:z:-.-_ Figure 6.18 62 0 00 (.) 0 VI I 0 E f=

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7. Conclusions This final chapter will discuss the conclusions to be made from the results of the last chapter and the rest of the thesis The first section will cover model validity and conclusions to be made from the models The next section will engage a discussion of potential applications of this thesis The final section will discuss recommended future research that would use this thesis as a starting point or at least as a reference 7.1 Model Validity The graphs in chapter 6 yield some very important information : The active+ passive runs with a gain of 50 ... Were not quite as good as purely active with a gain of 0 5 (very soft impedance) About equal to the purely active run with a gain of 1 (still pretty soft) Were better than the purely active run with a gain of 5 (only slightly stiffer) Were much better than runs in the stiff range from 1 0 to 5 00 One of the key conclusions that one must come to is that an active + passive arm with a gain of 50 (pretty stiff/less active) is as good as a purely active arm with a 63

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gain of 1 Indeed the passive element can be incorporated into the arm increase the stiffness of the arm thereby reducing the bandwidth and the need for excessive fixturing while still maintaining or exceeding the performance of lower gain softer impedance control schemes From the characterization data for the passive element the active+ passive runs agree reasonably well with the measured stiffness of the passive element. From this the following conclusions can be made : The thesis shows successful performance of active impedance control (coul d not go below G = 0 5 due to stability problems) The passive element has been successfully characterized The active + passive vs. purely active runs show expected performance gain Passive element on arm agrees with characterization As far as cost analysis goes, clearly the softer the impedance the greater the bandwidth and the greater the need for fixturing such as more or better sensors better resolvers and more accurate electronics. This thesis shows that a relatively stiff impedance control can be supplemented with a passive element to take the compliance out of the software and put it into the arm itself thereby increasing job performance. In a sense a person could trade expensive fixturing and bandwidth constraints for a quite inexpensive passive element in their robot and still maintain 64

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a desired level of performance Of course this approach has its limitations as was detailed in Fig 6 .15 in which the purely active run performed better than the active + passive run One may think that this issue could be easily addressed by making the element arbitrarily compliant. The problem with that is in industry a passive element that is too compliant will tend to droop due to gravity Also a passive element that is too soft may wobble too much at the end of its path to be of much use It may have superior force reduction characteristics but it may take too long to stop oscillating 7.2 Potential Applications There are a number of potential applications for the information in this thesis Since weight is of prime importance to NASA launch vehicles a light passive element replacing a certain amount of sensors and other fixturing could be very beneficial to a launch mission In addition to this one should keep in mind that manipulator manufacturers do not want their customers to tamper with the position controller trying to make it a better controller The higher forces produced using stiffer impedances in the impedance specification filter may be reduced through active means but this ultimately means altering the position controller and is therefore unfeasible 65

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Incorporating the passive element into an industrial robot manipulator would reduce those forces without all the fixturing and necessary controller tampering and is therefore much more practical. 7.3 Recommended Future Research A good idea for a future topic of research is to invent a variable passive element. The passive element used in this thesis served its purpose well but it only had one value for K and B (in the range of forces the arm was operating in) NASA as well as other industries may want to incorporate a passive element into a robotic arm for assembly and/or mating tasks Once the controller has achieved peg-in-hole insertion the task may require a torque to be exerted on the task to move the object to a different location or orientation This may require a stiffer passive element If the stiffuess and damping parameters could be varied and made stiffer after insertion the task of assembly would be made easier Another topic may be to alter the position controller to respond faster to environmental forces and compare the resulting data to the active + passive approach Finally, one could accurately determine the value ofB for a passive element. This is a much more formidable task than finding K the spring stiffuess 66

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parameter Ultimatel y this thesis opens the door for a wide range of research both practical and theoretical. 67

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APPENDIX A Control Files An important tool in data generation and manipulation in this thesis is known as control files ( ctl files) The next few pages contain print outs of ctl files These files set up the different internal impedances for different runs and they basicall y controlled the robot arm impedance characteristics The impedance control flag indicates whether the user wants the impedance controller on or off. The user can also adjust the proportional gains for the shoulder elbow and wrist joints as well as the derivative gains K and B the impedance parameters could be determined for each DOF and the user can also input whether or not this particular set up would be using a peg Finally the user may choose a set of inertia matrix parameters depending on the size ofthe load The ctl files proved to be a powerful tool in experiment runs 68

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7-Dec-94 09 : 35 -schmitz/lsm /Cer/Rt Code/ hdrn hbw.ctl .c:: o : :ne HDMJ. EXE ylkl Nc;,-,[ 1 1 ut"< l y l 2 J u ( k.-2 ) :-ec.::l y !k-c> u I y lk-n> .ir.: 0 VA!!a:;: 0 0!! lC!: :-::;a!"lge Parameter-s : Ot!'lerwi=se, i:1qw1.re. No. o: f>lters to < > No. o: .:.mpulses !'o : ( < < 4 ) ?:ints : :ea!. !'ropor:ional :or ( Sh El Wr I : 55 9 5 55.85 56 85 ! 56.B5 56 35 246. 7 c. :I:'"'. Wr 55.SS 50. 35 1 57 9 ' U ::; ... w -... De:i,ative qa i:-. s !o: ( Sh .1 W: I : 10. 55 '0. 55 10 55 10. s s 10.55 :1. 99 2 5 :i: ow CCI Wr : 0. 5 s 10. 55 '7. 59 .0 u-:.:'1. Wr !r;,pecar.ce K o r fx !'y T: I I lb/ 'c. 5 0 5.C 100.0 Fx F y T : I I :.:-: 2 5 so.o 11c ,c. ::-oes l F / 7 ser.so : !:ame 5. 5)) .C u s1.nc; :he !-5:.:"":1 r.o peg Iner:1a ma:r1x for 0 0 0 5). 1-i 25.76 : l c ' "' 11.. 62 0 -0. c v :l.O c : 0 995 JJ: J):: "" 6 4 4 ) : 5 : 9) : 0 3 ? J.; 0 0 1';.;:-..::-L : : 0 JaJ : c . 0 :::1-?:-o.: C .'J 69 .;") i J 3 --> ,.. "'I .... ':) .. --> in-1b/rad) 6 C lb 2 0 ; P aqe 1

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09 .35 J J -schmitz/lsm/Cer/Rt 0. .0 Nu:n Den parame:e:s (No:e: 2 l u:-.:::a::-.pe= ::e::; (:.:-: !-{;::) o : :r.odes : o ""\,::0 avo.:.::::ec 70 Paqe 2

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APPENDIX 8 Data Matrix Matlab has been very instrumental in this thesis It was used primarily as a plot generator It would accept data from the Intel 310 ( via ETHERNET) in matrix form and produce a plot with a predetermined column acting as the x axis and one or more columns yielding information along the y axis Below is a representation of the data matrix the Intel 31 0 could generate with all 16 columns Time Joint Reference Joint Measured! Cart Reference! Measured Forces Joint Ref Radians Position Rad x y, Fx, Fy, Torque Rad 1 2 3 4 5 6 7 8 9 10 11' 12, 13 14 15, 16 The numbers correspond to the columns in the generated matrix. The rows would be successive samples Since 250 samples were taken per run the matrix generated would be 250 x 16. Once this matrix was generated Matlab could then assign column 1 to the x axis and column(s) for what ever information required for this particular section of the thesis to the y axis Columns from different runs (different matrices) could also 71

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be compared and graphed together This set up greatly simplified data manipulation and plot generation 72

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References [1] Oh Y. H. Chung W K., Jeong K. W., Youm Y., Implementation of Passive Hardware Damper for Force and Impact Control ," 1994 [2] Vukobratovic M., "Contact Control Concepts in Manipulation Robotics-An Overview ," IEEE Transac. on Indust. Electronics ., Vol. 41, February 1994 [3] Salisbury J. K., Active Stiffness Control of a Manipulator in Cartesian Coordinates Proc. of the 19th Conf on Dec. and Cont. 1980 [4] Volpe R., Khosla P., Experimental Verification of a Strategy for Impact Control IEEE International Conference on Robotics and Automation, I99I. [5] Asada H and Ogawa K., On the Dynamic Analysis of a Manipulator and Its End Effector Interacting with the Environment IEEE International Conference on Robotics and Automation, I986. [6] Mills J. K., Goldenberg A. A., Force and Position Control ofManipulators During Constrained Motion Tasks ," IEEE Trans. Robot. Automat., Vol. 5 February I989 [7] Hogan N., On the Stability of Manipulators Performing Contact Tasks" IEEE Journal of Robotics and Automation. Vol. 4 No.6. December I988 [8] Craig J. J., Introduction to Robotics ; Mechanics and Control. Addison Wesley 1986 [9] Sturges R H. ''A Quantification of Machine Dexterity Applied to an Assembly Task International Journal of Robotics Research. Vol. 9 No. 3 June 1990. [10] Zheng Y.-F., Hemami H. Mathematical Modeling of a Robot Collision with its Environment" Journal of Robotic Systems 2(3) John Wiley and Sons 1985 [II] Rajan V. T., et al. Dynamics of a Rigid Body in Frictional Contact with Rigid Walls," IEEE International Conference on Robotics and Automation, I987. [I2] Beer F P and Johnston E R. Jr., Vector Mechanics for Engineers: Statics and Dynamics. New York: McGraw-Hill I962. 73

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[13] Mason M Compliance and Force Control for Computer Controlled Manipulators," IEEE Trans. on Systems Man and Cybe rn etics Vol. SMC-11 No. 6 June 1981. [14] Sturges R H Task/Effector System Models of Dexterity for Machine Assembly," IEEE International Symposium on Intelligent C ontrol. September 1989 [15] De Schutter J., Van Brussel H., Compliant Control Motion I. A Formalism for Specifying Compliant Motion Tasks International Journal of Robotics R ese arch. Vol. 7 No 4 August 1988 [16] Huang H-P., The Unified Formulation of Constrained Robot Systems IEEE International Conference on Robotics and Automation 1988 [ 17] F aessler H., ' Manipulators Constrained by Stiff Contact : Dynamics Control and The International Journal of Robotics Research. Vol. 9, No 4 August 1990 [18] Wang Y., Mason M Modeling Impact Dynamics for Robotic Operations ,'' I EEE International C onference on Robotics and Automation 1987 [19] Zheng Y.-F., Fan Y., Robot Force Sensor Interacting with Environments,'' IEEE Transactions on Robotics and Automation Vol. 7 No 1. February 1991. [20] Whitney D E Quasi-Static Assembly of Compliantly Supported Rigid Parts Journal of Dynamic Systems Measurement, and Control. Vol. 104 March 1982 . [21] Peshkin M A., Sanderson A. C., Manipulation of a Sliding Object IEEE International Conference on Robotics and Automation. 1986 [22] Strip D. R., "Insertions Using Geometric Analysis and Hybrid Force-Position Control ; Method and Analysis IEEE International Conference on Robotics and Automation 1988. [23] Strip D R., A Passive Mechanism for Insertion of Convex Pegs ,'' IEEE International Conference on Robotics and Automation, 1989. [24] Shahinpoor M., Zoboor H., Analysis of Dynamic Insertion Type Assembly For Manufacturing Automation," IEEE International Conference on Robotics and Automation. April 1991. [25] PaiD. K. Leu M C., Feasible Tasks for Manipulators with Uncertainty and Compliance ,' IEEE International C onferenc e on Robotics and Automation 1991. 74

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[26] Hollerbach J. M., A Recursive Lagrangian Formulation of Manipulator Dynamics and a Comparative Study ofDynamics Formulation Complexity" IEEE Trans. Syst., Man andCyber., Vol. SMC-10 No 11, Nov. 1980 [27] Lee C. S G., Lee B. H. and Nigam R., Development of the Generalized d'Alembert Equations of Motion for Mechanical Manipulators" Proc of the 22d Conf on Decision and Con trol December 1983 [28] Khatib 0., A Unified Approach for Motion and Force Control of Robot Manipulators : The Operational Space Formulation" IEEE Jour of Rob. and Auto ., Vol. RA-3 No 1 February 1987 [29] Sweet L. M and Good M C., Re-Definition of the Robot Motion Control Problem : Effects of Plant Dynamics Drive System Constraints and User Requirements" 1984 Conf on Decision and Control, Las Vegas NV, December 1984 [30] Asada H. Kanade T., and Takeyama I. Control of a Direct-Drive Arm", Trans. of the ASME, Jour. of Dyn. Syst., Meas., and C ontrol Vol. 105, September 1983 [31] Fu K. S., Gonzalez R. C., Lee C. S G., Robotics; Control, Sensing Vision, and Intelligence McGraw-Hill 1987 [32] Shinners. S M., Modern Control System Theory and Application Addison Wesley 1978 [33] Lee C. S. G., and Chung M J., "An Adaptive Control Strategy for Mechanical Manipulators", IEEE Trans. on Auto Cant Vol. AC-29 No. 9, 1984 [34] Luh J. Y. S., Walker M W., Paul R. P., On-Line Computational Scheme for Mechanical Manipulators," Trans. of the ASME Jour. of Dyn. Syst. Meas., and Cont., Vol. 102 June 1980. [35] Luh J. Y. S., Fisher W D Paul R. P Joint Torque Control by a Direct Feedback for Industrial Manipulators IEEE Trans. on Auto. Cont., Vol. AC-28 No. 2 February 1983 [3 6] Wu C., "Compliance Control of a Robot Manipulator Based on Joint Torque Servo," Vol. 4 No. 3 Fall 1985 [37] Freund E., Direct Design Methods for the Control of Industrial Robots Computers in Mechanical Engineering, April 1983 75

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[38] Fournier S J., Schilling R. J., "Decoupling of a Two-Axis Robotic Manipulator Using Nonlinear State Feedback; A Case Study Int. Jour of Rob. Research Vol. 3 No. 3 Fall 1983 [39] Slotine J.-J. E., The Robust Control ofRobot Manipulators Inter Jour. of Rob. Research Vol. 4 No. 2 Summer 1985 [40] Mason M T., Compliance and Force Control for Computer Controlled Manipulators IEEE Trans. on Syst. Man, and Cyber. Vol. SMC-11 No. 6 June 1981. [ 41] Raibert M H., Craig J. J., Hybrid Position!F orce Control of Manipulators Trans. of the ASME Jour. of Dyn. Syst., Meas., and Control, Vol. 102, June 1981 [42] Zhang H. Paul R. P "Hybrid Control ofRobot Manipulators IEEE Conf o42Dec. and Cont., 1985 [43] Hogan N., "Impedance Control : An Approach to Manipulation Jour. of Dyn. Syst., Meas., and Cont ., Vol. 107, March, 1985 [ 44] Lawrence D. A., "Impedance Control Stability Properties in Common Implementations 1987 [45] Whitney D E., Force Feedback Control of Manipulator Fine Motions ASME Jour. of Dyn. Syst., Me as., and Cont., June 1971 [46] Simunovic, S Force Information in Assembly Processes," 5th International Symposium on Industrial Robotics, September, 1975. 76