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Comparison mechanical parameters between a stewart platform theoretical model and n experimental static test specific to the stance phase of gait in foot-ankle complex

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Comparison mechanical parameters between a stewart platform theoretical model and n experimental static test specific to the stance phase of gait in foot-ankle complex
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Hensley, Sarah ( author )
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Denver, Colo.
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Master's ( Master of science)
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University of Colorado Denver
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Department of Bioengineering, CU Denver
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Bioengineering

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Ankle ( lcsh )
Foot ( lcsh )
Ankle ( fast )
Foot ( fast )
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bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

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Understanding true kinematics and kinetics of the foot and ankle during stance phase of gait will lead to better methods of injury prevention, diagnosis, and rehabilitation. More specifically, these parameters can provide clinically useful information in studying bone stress fractures, acute ligamentous injuries, external devices like shoe orthotics, and internal devices such as ankle implants. The use of a robotic gait simulator (RGS) has proven to be an accurate method of achieving this goal. This study was the first step in creating a RGS that can produce repetitive cycles of stance phase of gait to measure kinematics and kinetics. A computer-simulated model based on a Stewart platform was created and modified specific to gait in the foot and ankle. The model replicated the repeated walking motion during stance phase using average body weight (1.5 times body weight). This model was then validated by building a static experimental test setup that replicated the same geometry and forces. The experimental test compared its distributed loading during stance phase vs the distributed loading predicted with the model. Results showed that the experimental data trended well with the model and resulted in less than 20% error at the midstance, or 0 degree position. The next step in this project will be to build a dynamic RGS that can produce repetitive cycles of gait during stance phase.
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Includes bibliographical references.
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by sarah Hensley.

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Full Text
COMPARISON OF MECHANICAL PARAMETERS BETWEEN A STEWART PLATFORM
THEORETICAL MODEL AND AN EXPERIMENTAL STATIC TEST SPECIFIC TO THE STANCE PHASE OF GAIT IN FOOT-ANKLE COMPLEX
by
SARAH HENSLEY
BS, Rose-Hulman Institute of Technology, 2015
A thesis submitted to the Faculty of the Graduate School of the University of Colorado in partial fulfillment of the requirements for the degree of Master of Science Bioengineering Program 2017
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This thesis for the Master of Science degree by Sarah Hensley has been approved for the Bioengineering Program by
Cathy Bodine, Chair Kenneth Hunt Steven Lammers
Levin Sliker


Hensley, Sarah (M.S., Bioengineering Program)
Comparison of Mechanical Parameters Between a Stewart Platform Theoretical Model and an Experimental Static Test Specific to the Stance Phase of Gait in Foot-Ankle Complex Thesis directed by Associate Professor Cathy Bodine.
ABSTRACT
Understanding true kinematics and kinetics of the foot and ankle during stance phase of gait will lead to better methods of injury prevention, diagnosis, and rehabilitation. More specifically, these parameters can provide clinically useful information in studying bone stress fractures, acute ligamentous injuries, external devices like shoe orthotics, and internal devices such as ankle implants. The use of a robotic gait simulator (RGS) has proven to be an accurate method of achieving this goal. This study was the first step in creating a RGS that can produce repetitive cycles of stance phase of gait to measure kinematics and kinetics. A computer-simulated model based on a Stewart platform was created and modified specific to gait in the foot and ankle. The model replicated the repeated walking motion during stance phase using average body weight (1.5 times body weight). This model was then validated by building a static experimental test setup that replicated the same geometry and forces. The experimental test compared its distributed loading during stance phase vs the distributed loading predicted with the model. Results showed that the experimental data trended well with the model and resulted in less than 20% error at the midstance, or 0 degree position. The next step in this project will be to build a dynamic RGS that can produce repetitive cycles of gait during stance phase.
The form and content of this abstract are approved. I recommend its publication.
Approved: Cathy Bodine
m


ACKNOWLEDGMENTS
I would like to thank Todd Baldini and my committee members for advising me throughout my project. Thank you to my many professors that have inspired me to pursue my academic goals. I would also like to thank my family and friends for their continued and unwavering support.
IV


TABLE OF CONTENTS
CHAPTERS
I. INTRODUCTION........................................................1
Foot Injury Gait Analysis......................................1
In-Vitro Robotic Gait Simulators...............................1
Project Goals..................................................2
II. REVIEW 01 THE LITERATURE............................................3
Overview of Gait...............................................3
Phases of Gait.................................................3
Parameters of Gait.............................................4
Early methods of gait analysis.................................6
Motion Capture Gait Analysis...................................7
In-vitro Robotic Gait Simulators...............................9
Limitations to Current Robotic Gait Simulators................10
Significance..................................................10
III. HYPOTHESIS AND SPECIFIC AIMS.......................................12
IV. METHODS............................................................14
Overview......................................................14
Matlab Simulation.............................................14
Matlab Stewart Platform Template..............................14
Modifications Specific to the Foot and Ankle..................17
Experimental Test for Validation..............................19
Materials and CAD Modeling....................................19
Device Construction...........................................20
Load Cell Implementation......................................21
Load Cell Configuration in LabView............................23
v


Test Protocol and Data Collection...........................25
Test Summary................................................26
Data Analysis...............................................26
V. RESULTS...........................................................29
Experimental vs Computer Model Comparison...................29
Statics Analysis............................................33
VI. DISCUSSION AND LIMITATIONS........................................34
Model Comparisons...........................................34
Experimental Limitations....................................34
Matlab Model Limitations....................................36
VII. I l.Tl. RI] WORK..................................................37
Next Steps..................................................37
Luture Modifications to the RGS.............................37
VIII. CONCLUSION........................................................38
IX. ADDENDUM..........................................................39
REFERENCES.............................................................43
APPENDIX...............................................................45
A. Matlab Script...........................................45
B. Solid Works Drawings of Components......................48
C. Data Collection VI Block Diagram........................55
D. Modified Data Collection VI.............................56
E. Matlab Script for Statics Analysis......................57
vi


CHAPTER I
INTRODUCTION Foot Injury Gait Analysis
Ease of gait in human locomotion is a critical component in individuals daily lives. Gait is a multi-segmental process, requiring movement in most of the body but largely from the lower half, including the hip, knee, foot, and ankle. Injuries to the foot and ankle negatively alter normal gait, and it does not remain exclusive to the foot and ankle. Other parts of the body are also abnormally affected when an injury occurs (1). For example, if someone fractures their foot, it would alter gait and affect not only the foot but other joints such as the hip and knee joints. This one, multi-segmental, symbiotic system, the human body, has been the reason for sparked interest in gait analysis, which is the study of human motion (1, 2, 3). Compared to numerous orthopedic surgeries being performed in the hip and knee joints, there are less currently being done in the foot-ankle complex (4). One reason for this is due to more difficulty in measuring kinematics (motion without reference to the forces that causes the motion) and kinetics (study of forces that cause motion) of the foot-ankle complex.
Two main methods of gait analysis currently used are clinical observational analysis and motion capture analysis. However, both methods produce variability and non-reproducibility (5, 6, 7, 8). Clinical observational analysis is based on temporal-spatial parameters. This includes gait speed and stance duration, both of which change according to global parameters such as a persons age or height. Motion capture analysis falters due to size, complexity, and cost. Measuring motion in the arm or leg might be feasible but can be very difficult in the much smaller foot-ankle complex. Additionally, a complete motion capture setup can exceed $100,000 in cost. Because of these inconsistencies, this limits the validity of these methods in measuring true kinematics and kinetics of the foot and ankle during gait (9, 10).
In-vitro Robotic Gait Simulators
Most recently, in-vitro robotic gait simulators that simulate the stance phase of gait have been used [9, 10, 11, 12, 13], These in-vitro simulators show potential in eliminating error produced in
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current methods of gait analysis, but there is still a great need to improve these devices in accuracy and validation. There is a critical need to more accurately measure kinematics and kinetics in the foot-ankle complex. Taking advantage of a controlled gait simulator with repetitive gait cycles could more accurately measure these parameters. This tool could then be used to compare simulated normal gait to current methods of correcting gait. By doing so, methods of injury prevention, diagnosis, and rehabilitation of foot-ankle injuries could be improved. Ultimately, this method of gait analysis could help individuals return to their normal, active lives.
Project Goals
The long-term goal of this project is to design a 6 degree of freedom robotic gait simulator that produces the repetitive cycles of gait in an effort to improve understanding of physiological loading (vertical ground reaction force and loading distribution) and kinematics during the stance phase of gait. The first step in this multi-stage process is to develop a computer model to simulate loading and repetitive cycles of gait during stance phase. The second step is to validate the models mechanical feasibility by replicating the model using a fixed structure that uses adjustable rigid rods as opposed to linear actuators.
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CHAPTER II
REVIEW OF THE LITERATURE
Overview of Gait
Gait is defined as an individuals ambulation or locomotion of the entire body (14). Its necessity on a daily basis makes it one of the most studied components of human movement. Because it is an easily observable activity that involves multi-segment motion of the body, the systematic study of human walking, or gait analysis, is prevalent in literature in continued efforts to fully understand human biomechanics (1, 2, 3, 5, 9).
Observations of gait can be dated back to Aristotle (384-322 BCE) when he walked adjacent to a wall with an ink utensil attached to the top of head. He observed that the ink did not draw a straight line but rather a zig-zag, indicating a vertical movement (bending and straightening) as he walked (15). As technology has advanced, gait analysis has progressed past purely observational research to computer-driven measurement techniques. In a clinical setting commonly with orthopedic foot and ankle surgeons, visual gait evaluation is used to assess gait patterns or abnormalities. Visual gait evaluation differs from formal gait analysis, which takes place in a scientific research-driven setting. Formal gait analysis records and analyzes human locomotion for reasons beyond the clinical scope including: understanding biomechanical characteristics of gait, analyzing separate components of gait (e.g., foot, ankle, and rest of body), and performing further analysis to improve clinical treatments.
Phases of Gait
Gait is classified into two main phases, stance and swing, with both featuring various periods and events that occur in each phase (Figure 1). Gait begins with the stance phase and comprises 60% of one complete gait cycle. The stance phase is comprised of three rockers with the first rocker being the initial heel strike, marking the beginning of a gait cycle. As an individual takes a step forward, their heel strikes the ground, causing eccentric contraction of ankle dorsiflexors and allowing the ankle to begin plantar flexing. As forward movement progresses, the foot becomes flat
3


with the ground, indicating the second rocker. During this period, the tibia rolls over the ankle as the foot remains planted to the ground. Forward movement continues into the third rocker as the foot begins to dorsiflex, ultimately leading to toe-off Throughout the stance phase, muscle groups are equally important in the function of gait; their main function is to make this phase of gait an energetically efficient process (15). In summary, the stance phase of gait consists of the body bearing weight with initial contact at the heel to translate the lower extremity over the fixed foot and continue the forward movement of the individual.
The other 40% of gait consists of the swing phase, which primarily functions in foot clearance as forward movement continues. The plantar flexed foot at toe-off rotates into a more neutral position due to concentric contraction of anterior leg musculature. As this rotation occurs, the foot clears the floor, preventing foot drag on the ground, and the leg continues forward as a result of stored energy in the pelvis and proximal thigh musculature. A major benefit to the swing phase of gait is its process of allowing the high loads that occurred at heel strike to dissipate over time before a new step is taken (1, 3)
Figure 1. One complete gait cycle consisting of stance phase and swing phase (16). Parameters of Gait
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In addition to understanding gait mechanics, it is also important to understand various parameters of gait analysis. Knowing these parameters and how they affect gait are vital in developing any type of device for simulating gait in addition to providing clinical relevance. Three key gait parameters include temporal-spatial, kinematic, and kinetic.
Two major gait parameters are kinematics and kinetics, both used heavily in research setting for quantitatively analyzing gait. Kinematics is the study of motion, more specifically defined as the movement of a body in isolation of the forces. Kinematic parameters often include displacement, velocity, and acceleration, and are often recorded using motion tracking devices. Alternatively, kinetics is the study of the forces and moments that cause the movement. Parameters often include ground reaction forces (GRFs), joint moments, and joint power. Both kinematic and kinetic parameters are key in understanding and simulating human gait (10, 17, 18).
Another classification of gait is based on temporal-spatial parameters, having been referred to as the, vital signs of gait (19, pg21). These observable parameters are often used in clinical settings and include gait speed and stance duration, among others (Figure 2). Gait speed is defined by the equation:
m
Gait speed () = s
Cadence ( si:eVs \ ^ g^rige length(i^n^gr ) Vminute/ a v2 stepsJ
120 (seconds)
Cadence represents the number of steps per minute, and stride length represents the total length of an individual taking two steps (i.e. left heel strike to the next left heel strike). The stance duration parameter is defined as the amount of time the foot is in contact with the ground over the period of one step.
These two parameters, along with several others, are very intuitive to clinicians and can be used to quantify pathological gait patterns (21). Though quite useful for clinicians, temporal-spatial parameters are a global function of gait and can be influenced by many factors, some of which include age, sex, human error in data collection, and auditory cueing when taking measurements.
5


Temporal-spatial parameters can be very useful in diagnosing gait abnormalities, but they cannot be
relied on independently (21).
Figure 2. Example of clinical gait analysis based on temporal-spatial parameters (20).
Of all the gait parameters, temporal-spatial parameters are considered the most clinically applicable (14). Gait speed has been correlated with orthopedic research, often with slower speeds indicating foot pain and abnormal gait (14). Additionally, it has also been correlated with higher prevalence in elderly individuals (14). Stance duration has also been clinically observed with decreased stance duration due to avoidance of weight bearing or conversely, increased stance duration due to weakened muscles or decreased foot mechanics preventing a normal rate of stance phase (14). These observable parameters, though clinically effective, are still global parameters easily influenced by uncontrollable factors such as age, sex, and others (14). Because of this, it is vital to better understand true kinematics and kinetics of gait. To achieve this, these parameters must be quantified by more advanced techniques than pure observation.
Early Methods of Gait Analysis
Before motion capture and other current methods of gait analysis, primitive methods dating back to the 1950s were used (5). After WWII, there was increased demand for understanding locomotion to improve treatment of war veterans (5, 6) Using a strobe light, reflective strips, and a manual goniometer, gait analysis was conducted by having an individual walk under the illumination of the strobe light. With the reflective strips attached to anatomical landmarks, photographs were
6


taken to manually make joint measurements. Though this early method was not exceptionally accurate and had problems in measuring rotations in the transverse plane, it did measure sagittal plane joint angles that are similar to results still found today (5).
As technology advanced, the Karpovich brothers among others began to take advantage of this technology and use electrogoniometers for gait analysis (5). Compared to manual goniometers, electrogoniometers significantly reduced data processing via computers. Additionally, this method was simple, reliable, and inexpensive, which allowed more researchers to perform gait analysis. Most importantly, a triaxial goniometer was the first attempt to capture motion in 3 directions, something which is now known to be critical in making accurate, physiologically-mimicking parameters (5). Limitations to electrogoniometers include their inability to obtain simultaneous measurements of all moving segments and their reduced accuracy because the device was positioned offset to the side of limb segments. Additionally, this method was not uniform across different demographics such as height and weight, which made data normalization difficult (5).
Motion Capture Gait Analysis
The next iteration of gait analysis developed rapidly with the invention of passive marker systems based on those used in cine film. Photographic techniques by Marey and Muybridge were able to capture the entire human body while simultaneously viewing relative motions between body segments (5). With this large field of view, individuals of all sizes could be measured, greatly increasing robustness of this method. The major limitation of this gait analysis was the inadequacy of computers during this time period. It took significant time to process the data, and there were major computer storage limitations.
By the late 1960s, television-motion analysis systems were being developed with automated recording of reflective markers. One of the early motion capture tests was measuring multiple joint angles of a cat running on a treadmill (5). The company, VICON, was established in 1984 in Oxford, UK, and developed the first commercial motion capture system. Once the film industry realized the potential of motion capture, advances in this field increased. Currently, VICON alone has won an
7


Academy-Award, an Emmy, and other awards for their expertise in motion capture (10, 11). Though motion capture is heavily used for entertainment purposes, the film industry has driven the market, which resulted in high quality motion capture systems.
Motion capture systems are one of the most used methods of gait analysis. Compared to clinical gait analysis, motion capture systems provide more quantitative than qualitative results (7,
22). The basics of motion capture gait analysis consist of placing reflective markers on the skin's surface. To capture 3-D motion of a particular segment, 3 markers are placed on each point of interest. A model limb segment is represented as a rigid body and then an algorithm is applied for rigid body motion. Cameras are positioned around the individual, and as he or she moves, the cameras capture the movement of the reflective markers, which are representing body segmental movements (Figure 3). Motion capture systems have been used in analyzing gait of larger body movements, such as the femur-tibia, humerus-ulna/radius, or posture. However, measuring kinematics of more complex, smaller structures, such as the foot-ankle, is much more difficult (9, 10).
Figure 3. General setup of motion capture gait analysis (23).
Two major factors in motion capture are size and complexity. For example, the knee joint connects the femur to the tibia, both large bones of the body. Furthermore, the knee joint and corresponding bones produce large amounts of motion during gait, resulting in accurate
8


measurements of the knee joint complex when using motion capture systems. However, the foot-ankle complex is significantly smaller. Not only is the motion of the foot and ankle during gait much smaller in magnitude, but this motion involves many more bones. There are 26 bones and 33 joints in the foot alone that provide structural support compared to 3 bones (femur, tibia, fibula) surrounding the knee joint.
This limitation is one major reason why motion capture is not ideal for gait analysis in the foot and ankle. Secondly, motion capture systems have one major limitation across any measurement of the body. There is error and variability in measuring kinematics due to skin movement relative to the underlying bone. Reflective markers are placed on the skins surface. As an individual walks, motion capture systems cannot differentiate between what motion is due to bone movement or skin or muscle movement (7, 8).
In-vitro Robotic Gait Simulators
Due to motion captures limitations in determining accurate bony motion without skin or other artifact error, research led to the development of in vitro robotic gait simulators (RGS) (9, 10, 11). The basic components of a RGS are a fixed base plate attached to six linear actuators with a moveable platform attached on top. The six linear actuators control the position of the RGS and allows for 6 degrees of freedom. Six degrees of freedom is critical to the RGSs design because it allows for not only translation in the x, y, z-directions but also rotation in the form of pitch, roll, and yaw.
It is important to know that gait in the foot-ankle complex is not simply dorsi-plantarflexion; with each step in the walking motion, there is also rotation and distribution of the loading also in the medial-lateral directions. The RGS simulates the stance phase of gait by setting initial position of the moveable plate to correspond to initial heel strike, also known as the beginning point of stance phase of gait. The moveable plate, controlled by the six linear actuators, then moves through the stance phase of gait, ending at toe off. The linear actuators are working in tight coordination with each other
9


to accurately simulate gait. Additionally, the moveable plate is attached to a force plate allowing for the recording of vertical ground reaction forces and loading distribution along the foot (11, 12).
When testing cadaveric specimens in the RGS, they are potted in poly(methyl methacrylate) and connected to a biaxial testing machine via a custom mounting jig. Similar to previously discussed motion capture methods, the RGS also works in tandem with motion capture cameras to determine kinematics. Studies suggest that using a RGS as the method of gait analysis reduces significant error and test variability found in clinical and motion capture gait analyses methods (9, 10, 11, 12, 13, 18). One major design feature in RGSs is that the motion simulating gait is coming from the moving platform inferior to the foot-ankle as opposed to the foot and ankle having to perform the loading and position change. Cadaver specimens are limited in their overall strength; using the RGS to simulate the motion reduces the wear and tear the cadaveric specimens must withstand for each test.
Limitation to Current Robotic Gait Simulators
Studies indicate that an in vitro approach to gait analysis through the use of RGSs shows significant promise in translating to clinical use (12, 13). However, the major limitation to current RGSs is that they produce just one cycle of gait per test. Many injuries to the foot and ankle are caused by overuse, such as stress fractures. To more accurately measure gait in its true repetitive form, RGSs need to be able to run multiple cycles of gait for each iteration. This repetitive cyclic loading is a more accurate simulation of normal gait, especially for future applications in clinical settings.
Significance
Human locomotion is a daily activity for almost all individuals. This includes walking, running, skipping, and any other motions that people need to go about their normal daily activities. A persons gait can be altered due to acute injuries but more commonly, gait is altered due to overuse. Overuse can be caused by excessive repetitions in running, because of a sudden change in gait (i.e. deciding to run 5 miles after having not exercised in a month), being overweight or obese, and other factors (24, 25, 26). Prolonged overuse leads to injury, most commonly stress fractures. Once an
10


injury has occurred, the likelihood of sustaining another injury at the same site is increased. Current treatments of stress fractures in the foot are most commonly rest, ice, compression, and elevation (RICE). This can be frustrating for individuals with these injuries because the main course of treatment is simply to wait for it to heal. In some cases, shoe orthotics are recommended as treatment or prevention due to high or low foot arches. However, these recommendations are based on static observations of the foot and not on dynamic loading of the foot. Shoe orthotics recommended based on foot arch abnormalities are not sufficient, especially because foot injuries usually occur during dynamic situations, such as running or jumping (25, 26).
Current forms of gait analysis include observational gait analysis and the more recent clinical gait analysis using motion-tracking systems. Though both have shown certain qualities in determining gait patterns, both have variability in their methods that reduces efficacy and reliability. Studies have shown poor reproducibility in motion-tracking gait analysis, which is a key factor in sustainable research (7, 8). The new method of studying gait using an in vitro robotic gait simulator has shown hopeful preliminary results. However, accurate kinematic and kinetic measurements must be improved in these simulators to translate into a clinically useful tool. Based on previous research and the evident need for better understanding of kinematics and kinetics during gait, a robotic gait simulator, designed specifically for analyzing the stance phase of gait in the foot and ankle is needed to advance this field of research with the future goal of applying its knowledge in a clinical setting.
11


CHAPTER III
HYPOTHESIS AND SPECIFIC AIMS
Hypothesis.
A theoretical model that simulates 1.5 times body weight and repetitive cycles of gait during stance phase in the foot and ankle can be validated using a fixed mechanical Stewart platform within 20 percent error.
Specific Aim One: To develop a theoretical, computer model for a robotic gait simulator (RGS) to simulate the repetitive cycles of stance phase of gait in the foot and ankle.
Based on a Stewart platform, a Matlab model was created to simulate the repetitive cycles of stance phase of gait in the foot-ankle complex. A Matlab-created Stewart platform template was modified using Simulink and one of its add-ons, SimMechanics (27). The program consisted of an inverse kinematics module that allowed a user to input translational and rotational coordinates to control the positioning of the top, mobile plate in reference to time. Other modifications included adding appropriate loading to simulate heel strike loading (1.5 times body weight) and designing the model according to the size of an average human foot. As the simulation runs, scope outputs includes the following parameters in real time: length of each actuator, force of each actuator, and acceleration of each actuator.
Specific Aim Two: To build a static experimental Stewart platform using fixed cylinders in substitute of linear actuators at 1.5 times body weight.
The experimental test included all necessary components to build a fully functioning RGS, apart from substituting the actuators for fixed rods and cylinders. In reference from the ground up, the simulator begins with a fixed platform (add dimensions). Connected via ball bearings are six hollow cylinders (as linear actuator substitutes) in a Stewart Platform configuration. Prior to assembly, each cylinder was cut to size, load cells (Phidgets) were placed in between the two halves, and the cylinders were set to their original orientation. Six more ball bearings were connected to the opposite
12


ends of the cylinders, which were then connected to a separate mobile aluminum plate with the top plate offset 60 degrees from the fixed platform.
Specific Aim Three: To use the experimental Stewart platform to validate the theoretical model
With the experimental Stewart platform fully constructed, a 1050 Newton force was applied using an Instron to simulate 1.5 times body weight at various positions during stance phase while load cells measured the distributed loading using Labview. Force data was captured over a period of three, 10 second trials for each position at a 1050 Newton force loading. Theoretical forces from the computer model in each leg actuator were then compared with the experimental force data collected from the static test. With the top plate at 0 degrees, the experimental test performed with less than 20 percent error.
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CHAPTER IV
METHODS
Overview
Based on the research hypothesis, a Matlab simulation of a Stewart platform was developed to produce repetitive cycles of stance phase of gait. The simulation was programmed to simulate stance phase of gait based on normal vertical ground reaction forces produced by an average persons body weight and kinematics (1,2, 12). Once completed, a rigid version of a RGS was constructed to validate the Matlab simulation.
Matlab Simulation
The first step in designing and building a robotic gait simulator for the foot and ankle during stance phase of gait was to develop a computer model that could simulate the repetitive cycles of gait. This was accomplished in two main steps:
1. Identifying a generic Stewart Platform model with specific controlling capabilities
2. Modifying the Stewart Platform specific to stance phase of gait in the foot and ankle Matlab Stewart Platform Template
A Matlab-created Stewart platform simulation template was controlled using Simulink and one of its add-ons, SimMechanics. Matlab separated the simulation template into three subsystems: the inverse kinematics module (also called leg trajectory), proportional-integral-derivative (PID) controller, and plant (Figure 4).
14


len
pos
vel
os
Force

Lee
Leg Trajectory
*
Controller
(PID)
&
Forces
ces
Force Pos
Vel
L( g length.
du/dt
9
s
Body
Position
Sensor
Plant
Derivative
MyScopel
Body Position
Stewart Platform

errors w L_


OnginalScope
Figure 4. Graphic of Matlab Stewart Platform template using Simulink.
The leg trajectory subsystem consisted of an inverse kinematics module that allowed a user to input translational and rotational coordinates (x, y, z, pitch, roll, and yaw) to control the positioning of the top, mobile plate in reference to time and the fixed bottom plate (Figure 5).
Figure 5. Desired leg trajectory using translational and rotational coordinates.
Once coordinates were defined, the Matlab template transformed those input values into leg
lengths for each of the six actuators that created the Stewart platform's geometry.
These leg lengths then became the input for the PID controller subsystem, see Figure 6. The
PID controller acted as a control loop feedback mechanism for calculating and adjusting for error in
15


the desired leg lengths compared to actual leg lengths. The controller had three inputs: desired position, actual position, and actual leg velocity. The PID then adjusted according to error and outputted the required force of each leg (actuator) to achieve the desired position.
These forces then became the input for the plant subsystem, which was a Matlab representation of the mechanical components of the Stewart Platform (Figure 7). The mechanical components consisted of atop plate, bottom plate, six legs connecting the two plates, and universal joints connecting each end of a leg to its corresponding plate. Using SimMechanics, the legs were given the freedom to translate, as linear actuators do, using joint actuator blocks from the library in SimMechanics. Geometry of the plant subsystem was defined using an M-file script template, shown in Appendix A. With defined geometry and force input, the plant subsystem was used to output leg lengths and leg velocities.
16


1 ^ Force
Figure 7. Plant subsystem.
As a final step, a simple scope block was used to graphically output in real time leg lengths, velocities (and acceleration using a derivative function), and forces of each actuator. These three parameters were compared and evaluated to verify the simulation while also providing necessary value to build the experimental validation test.
Modifications Specific to the Foot and Ankle
Using the Matlab-created template, three major modifications were made to create a simulation specific to the stance phase of gait in the foot-ankle complex:
Modification 1:
17


In the leg trajectory subsystem, six desired coordinate trajectories (x, y, z, pitch, roll, and yaw) were defined to simulate repetitive cycles of stance phase of gait. The kinematics of stance phase of gait include initial foot strike, mid-stance, and toe-off To simulate repetitive cycles of these this phase, the x, y, z, roll, and yaw coordinates were set to zero. Pitch was defined using a sine wave function with an amplitude of 0.436 radians and frequency of 8 radians/second. The sine wave guaranteed continuous and repetitive cycles simulating walking motion. An amplitude of 0.436 radians, or 25 degrees, simulated the angle at which the foot strikes the ground during heel-strike, or initial step of stance phase. A frequency of 8 rad/s simulated a walking speed that correlates to approximately 3 m/s, which is the average walking speed of a human.
Modification 2:
In the plant subsystem, the Stewart platform template automatically determined the forces required of each of the six actuator legs to maintain equilibrium of the system. However, to modify the system to account for the force of a human of average weight, an adjusted density method was used for the top plate. The constant value of 1050 Newtons, or 1.5 times body weight, was set according to the average force created by normal walking motion. To ensure a distributed load on the top plate, the weight of the top plate, polyethylene block, six ball joints, six rods (rod 1), two clamps, and 1050 Newton force were combined as one mass to determine an adjusted density of 83,328.195 kg/m3 for each different position. After adding this external force, the six leg actuators took this additional force into account when measuring the forces required to keep the system in equilibrium. Modification 3:
In the Matlab script, the following parameters were changed: height of the overall device, radii of top and bottom plate, thickness of top and bottom plate, inner and outer radii of each leg, density of materials used, and PID controller gain values. In the Matlab script template, the default geometry of the device was significantly larger than the projects scaled version suited for a foot-ankle complex. The default and modified values are shown in Table 1 below.
Table 1. Comparison of default and modified values in Matlab script.
18


Part Default Value Modified Value (scaled specific for a foot-ankle system)
Height of overall device 3.0 meters 0.75 meters
Top plate radius 1.0 meter 0.1905 meters (7.5)
Top plate thickness 0.05 meters 0.0127 meters ('A)
Bottom plate radius 3.0 meters 0.1905 meters
Bottom plate thickness 0.05 meters 0.0127 meters
Leg actuator inner radius 0.03 meters 0.020 meters
Leg actuator outer radius 0.05 meters 0.037 meters
Density 1 76e3/9.81kg/m3 2800 kg/m3
Adjusted density NA 83,328.2 kg/m3
Alphab 2.5 degrees 10 degrees
Alphat 10 degrees 10 degrees
Experimental Test for Validation
The experimental test included all necessary components to build a fully functioning RGS, based on the Matlab model, apart from substituting the actuators for fixed rods and cylinders. The goal was to apply static loads to the RGS at different positions during the stance phase of gait and compare the experimental load distribution to the theoretical load distribution based on the Matlab model.
Materials and CAD Modeling
A SolidWorks assembly of the RGS was created to accurately size all components and component connections. The following is a comprehensive list of each SolidWorks part created (Table 2). See Appendix B for drawings.
Table 2. Device component list.
Part Name Details # Units
Top plate 18x18, A thick 6061 A1 1
Plate screw %-28 screw 2
Ball-socket joint %-28 threaded hole (Midwest Controls) 12
Rod 1 3/4 diam., 7 long solid rod (McMaster) 6
Load cell Max 500kg compressive/tensile S type Phidget) 6
Load cell threaded rod 1 length, Ml2x1.75 12
Rod 2 % diam., 9 long solid rod 6
Set screws A-28, '//'length 18
Cylinder 1 OD, 3/4 ID, 17 length 6061 A1 6
Rod-cylinder fixation screws %-28 set screw 18
Cylinder-joint coupler 3/4D, 4 length 6061 A1 rod 6
19


Bottom plate
16x24 A thick A1 plate
Device Construction
In reference to the ground up, the simulator began with a fixed platform (bottom plate). Six A-28 thru holes were drilled into the platform in a specific orientation. These holes served as connection points for ball and socket joints, which had threaded female ends. Screws were inserted into the platform on the inferior side and connected to the ball and socket joints. The screws were countersunk into the platform to ensure a flat surface on the platform.
Next, the cylinder-joint coupler was machined to connect the ball and socket joint to the cylinder. A %-28 threaded hole was drilled into the inferior side of the cylinder-joint coupler. This allowed the ball and socket joint to be connected to the coupler. To connect the coupler to the cylinder, a %-20 thru hole was drilled in both the coupler and the cylinder. A 'A length %-20 screw was placed through both components and held together using a washer and bolt.
Next, rod 2 was place inside the cylinder to allows for fine adjustability (0 to 6 inches) in length. To do this, three equally-spaced %-20 holes were drilled through the cylinders thickness and placed A distal to the superior side of the cylinder. Set screws were inserted to fix any desired position within a range of 0-6 inches of extension.
The S type load cell was placed in between rod 1 and rod 2. Each end of the load cell had M12xl.75 sized threaded holes serving as connection points. A % long threaded hole was drilled into both the inferior end of rod 1 and superior end of rod 2. A threaded rod was place into each of these holes to connect both ends to the load cell.
To complete the assembly a %-28 threaded hole was drilled into the superior side of rod 1 to attach the ball and socket joint. Then the other end of the joint was connected to the top plate. This was done using the same orientation described for the bottom plate but with a 60 degree offset. See Figure 8 below for the complete assembly.
20


Top plate
Ball-socket joint
Rod 1
S type load cell
Rod 2
Cylinder
Bottom plate
Figure 8. Fully assembled Stewart platform device.
Load Cell Implementation
Prior to attaching the load cells to the device, they were first tared. Each load cell came precalibrated by the manufacturing company but additional calibration was performed. Each load cell was connected to a 1046 PhidgetBridge 4-input board via a four-wire configuration shown in Figure
9. The red wire attached to the 5 V input, green wire to the positive input, white wire to the negative input, and black and yellow wires to the ground input.
21


Figure 9. Wire configuration for load cell-Phidget bridge connection (28).
The board acted as an interface with the load cells to measure each of their outputs. These load cells were configured using a built-in wheatstone bridge connection. Finally, the micro-USB output from the amplifier was connected to a LabView-enabled laptop via USB port. This setup is shown below in Figure 10. With no external loads being applied, data was collected for a period of 30 seconds and repeated five times. These load measurements were averaged among the five trials and then subtracted from each load cells. These subtracted values were integrated in the LabView virtual instrument (VI), which is discussed in more detail below. With the load cells calibrated, they were then attached to the device.
22


Figure 10. Hardware setup: (2) 1046PhidgetBridges connected via USB port and Labview-enabled
laptop.
Load cell configuration in Labview
Phidget template VI
Using the company, Phidgets, allowed for both the use of their online code examples in Labview specific to their load cells and their own function palette. Using the six S-type load cells in combination with 2 PhidgetBridge boards, a template VI named "Bridge Example.vi" was modified for specific use with the Stewart platform. Prior to modification, the VI functioned by reading the serial number for a given PhidgetBridge board. With the board identified, a user could define the number of bridges (in this case, two), pick a specific channel (load cell), and set the gain and data rate. Running the VI resulted in a bridge value output reading in millivolts/volt of the channel chosen, see Figure 11. Refer to Appendix C for the block diagram.
23


Serial Number
0
Enable
Channel
* Bridges:
0
Gain
error out status code
0 V 1 V
DataRate
8
£o
source
8 1000 The increment should be 8 and coerced to the nearest digit. Bridge Value (mV/V)
0
Converted Value
0
STOP
Figure 11. Front Panel of "Bridge Example.vi.
Modifications to VI
The following modifications to the "Bridge Example.vi" were performed:
1. Creating a "copy of the VI and inserting into the same VI. Connecting this copy to the original components within the same loop allowed the program to read two PhidgetBridge boards so six load cells could be used. One board could only read four load cells at a time.
2. Each load cell came with a data sheet, including the output rate. More specifically, it gave an output rate in millivolts per volt at the maximum load of 500 kilograms. A numeric multiply constant of 500kg was inserted into the VI to convert the units from millivolts Volt to kilograms
3. As mentioned earlier, the load cells were zeroed prior to any loading. This averaged constant for each load cell was subtracted using a numeric subtract constant.
4. For visual purposes during testing, a Waveform Chart was inserted into each load cell's output. This resulted in a real-time vs loading graph displayed in LabView's panel.
5. Load cell readings during testing needed to be recorded and saved for future data analysis. To achieve this, a "Write to Measurement File block was used to take the collected data and save it as an Microsoft Excel file.
Refer to Appendix D for the modified VI titled "Record.vi.
24


Test Protocol and Data Collection
With the device fully assembled and connected to Labview, the test platform was position in an Instron materials and testing machine. A custom point load attachment, shown in Figure 12, was installed into the Instron. This acted as the loading contact point on the top plate.
The Instron's load cell was calibrated prior to testing. Additionally, a polypropylene block was placed between the top plate and the point load attachment. To prevent harsh metal-on-metal, this block allowed for a smoother initial loading.
For each testing position, shown in Table 3, a 50 Newton preload was applied. Next, the device was loaded at a rate of lOONewton/second until the load reached 105 ON, where it was held for 10 seconds. This 105ON load represented a heel strike load, which is the maximum point of loading during stance phase of gait. After this 10 second hold, the device was unloaded at the same rate of 100 Newtons/second, see Figure 13. Simultaneously, Labview collected load cell data via interface board and LabView software. To run the LabView VI, the VI was started by clicking, "Operate-Run." This prompted the user to select a file to save the data file. Once selected, data collection began and ran until the user clicked, Stop/ on the user interface.
Instron end effector
Custom point load
Top plate
Figure 12. Custom point load attachment to Instron.
25


Table 3. Testing positions based on angle of top plate. Positive angles indicate dorsiflexion. Negative
angles indicate plantar flexion.
Test # Top Plate Angle (degrees)
1 25
2 15
3 10
4 0
5 -10
6 -15
7 -25
Load (Newtons)
Figure 13. Instron loading pattern for each test.
Test Summary
In summary, the complete setup included the following. A fully assembled fixed Stewart platform including six load cells was placed in an Instron that provided a point load. The load cells were connected to two PhidgetBridge interface boards that communicated with a Labview-enabled laptop. Labview recorded readings from the load cells and translated these measurements from millivolts per volt to kilograms. Using this setup, the test protocol was completed, including seven
different top plate positions. Data Analysis
26


Experimental data
Following testing, the experimental data was analyzed to compare the loading distribution in the experimental test versus the Matlab simulation. The raw data files contained a time stamp followed by six columns of force data for each of the six legs. First, the three cycles of the constant 105 ON force data was extracted from the raw data from each of the seven plate positions. This resulted in three clusters of 10 seconds of data for each position. All data were divided by 5 and multiplied by 9.8 to account for a unit change from mV/V to N. This produced a final collection of data in the unit of Newtons, which was the same units defined in the Matlab simulation. To summarize, this post-processing resulted in 30 seconds (3 trials at 10 seconds each) of experimental data for 7 different top plate positions for legs 1 through 6. The last step in data processing was to average each cycle of data in each position. This gave one experimental data point in each position to compare with the Matlab model. Standard deviation was also calculated in each leg.
Matlab simulation data
The next step was to extract forces in the computer model for each angle so it could be compared to the experimental data. To do this, model forces were found using the cursor measurement tool in the models scope output (Figure 14). To remain consistent, forces for each position were recorded at a simulation time of 1.85 seconds for positive angle positions and 2.38 seconds for negative angles.
27


v Trace Selection
x
Leg length !
t T Cursor Measurements
Settings 9 Screen cursors
Horizontal Vertical
O Waveform cursors
1 I Leg Forces: 1 v
2 | Leg Forces:3 s/
Lock cursor spacing
Snap to data
Measurements
Time Value
1 1.85 2.015e+02
1 2.38 2.016e+02
AT 530.000 ms AY 1 035e-01
1 / AT 1.887 Hz
AY/AT 195 223 (/ks)
Figure 14. The cursor measurement tool measuring forces (N) in Legs
plate position.
and 3 with a 0 degree top
Total distributed loading in experimental test and Matlab model
As another mode of data comparison, the sum of legs 1 through 6 were calculated in both the
experimental test and the Matlab model. This was an additional comparison made to validate the computer model.
28


CHAPTER V
RESULTS
Experimental vs Computer Model Comparison
Following post-processing of data from both the experimental test and Matlab model, the two sets of data were compared in each leg, see Figures 15 through 20. The Matlab model was created by defining recorded data points at each angle (-25, -15, -10, 0, 10, 15, 25 degrees) using a scatter plot with black smooth lines. Then, using a scatter pattern, the experimental data points were overlaid onto the same plot using red dots. Standard deviation bars were also created for each experimental data point.
Figure 15. Experimental (red dots) vs Model (blue line) data comparison in Leg 1.
29


Figure 17 Experimental vs Model data comparison in Leg 3.
30


31


Leg 6
I I 800

200;
I 1 ^
30 -2 5 -2 0 -1 5 -1 0 5 ) 5 1 0 1 5 20 2 5 3
-400
-600 <
Top Plate Angle (degrees)
Figure 20. Experimental vs Model data comparison in Leg 6.
With the top plate in neutral (0 degree) position, all leg lengths were equivalent. The Matlab model was compared to the experimental test and a percent error was calculated (Table 4).
Table 4. Loading distribution comparison at flat plate position.
Leg Model Experimental % Error
1 201.50 217.37 7.87
2 201.40 220.10 9.29
3 201.40 194.90 3.23
4 201.50 195.86 2.80
5 201.40 222.73 10.59
6 201.40 172.11 -14.54
To compare the sum of the distributed loading, forces in each leg were added together in both the experimental and Matlab model (Table 6).
Table 5. Comparison of total loading in experimental test and model.
32


Experimental Sum of Forces Theoretical Model Sum of Forces Difference between Experimental and Theoretical Sum of Forces
1202.10 1361.40 159.30
1211.79 1260.80 49.01
1209.38 1230.00 20.62
1223.06 1208.60 -14.46
1220.62 1229.51 8.89
1218.24 1256.80 38.56
1198.14 1321.10 122.96
Statics Analysis
Following these results, further analytics were performed with a statics analysis of the device. The analysis solved for each of the six leg forces required to achieve the same angled top plate positions acquired in the experimental testing. This analysis provided further insight into both the experimental setup as well as the computer model. The addendum provides full text of the statics analysis.
33


CHAPTER VI
DISCUSSION AND LIMITATIONS
Model Comparisons
Based on Figures 15 through 20, the experimental test showed trends in the direction of the theoretical model. When comparing model and experimental results at 0 degrees (Table 5), the experimental test was within 14.54% accuracy. At 0 degrees, the six legs were equivalent in length and not extended in the experimental model. This position provided a solid foundation on which to compare the two models. As seen in Table 5, each leg supported approximately 200N of load, which correlated to equal leg lengths. However, as the top plate angle increased, the difference in forces between the experimental and Matlab model also increased. As the angle increased, leg lengths varied in all six legs, which resulted in a non-equivalent loading distribution.
When comparing only the experimental data at different top plate angle positions, clear trends were seen in each of the legs, illustrated with red dots. Though this data varied significantly with the Matlab model, it showed consistency in the experimental testing. Additionally, when comparing only the Matlab model data at different top plate angle positions, there were also clear trends in each of the legs, illustrated with black lines. Also, the Matlab data showed clear similarities in leg couplets (Legs 1 and 2, Legs 3 and 6, and Legs 4 and 5). This was to be expected due to the Stewart platform orientation of the legs.
Experimental Limitations
After construction and testing of the experimental test, considerations were made regarding its overall ability to precisely and accurately record force readings in each leg. The overall devices height was based on a Stewart platform in neutral, or zero tilt, position. Additionally, its leg lengths were determined by approximating equivalent linear actuators from the company, LinMot. By doing so, the overall height of the device limited full extension in the y-direction. The limiting factor was the amount of working space available once the device was placed in an Instron. The device was able to achieve the maximum angle desired, plus and minus 25 degrees. However, ensuring there was zero
34


yaw or roll (tilting) of the plate proved more difficult with these larger angles. Even small amounts of tilt in the plate could result in slightly incorrect distributed loading in the six legs.
Another limitation involved the many parts necessary to build the static test. Unlike linear actuators, which would consist of two main parts (slider and stator), the experimental test used five main components. Rod 1, which acted as the slider, was cut into two pieces for the attachment of the load cell. The cylinder, which acted as the stator, required the joint-cylinder connector as an attachment for the ball joint. Each of these components were manually measured and machined separately for all six legs, leaving room for potential human and equipment error. For example, if one leg was machined slightly smaller in length than the other five legs, this would result in an incorrect loading distribution. If, for example, testing was taking place with the top plate at 25 degrees, and the shorter leg was positioned at the toe-off region of stance phase, the loading in this leg could be much lower than its other corresponding leg and therefore higher in the other legs.
During construction of the experimental test, the top plate was defined as a square (18x18 inch) plate, as opposed to a circular (15 inch diameter) plate defined in the Matlab model. This added significant additional weight to the experimental test that the Matlab model did not take into account. To compare, the difference in weight would be 3.38 kilograms, or 33.124 Newtons. This could account for a portion of the loading differences between the experimental and Matlab model.
Lastly, there were slight differences in the location of the point load in the experimental test. In the Matlab model, the external load to the top plate was consistently placed in the direct middle of the circular plate. However, the point load in the experimental test varied some, especially at the larger angles (15 and 25 degrees). The bottom plate was firmly locked into the base of the Instron. To remain consistent in testing and positioning, the bottom plates position remained fixed. This led to variances in the positioning of the point load, with the exception of testing at the neutral (0 degree) position. The Matlab simulation was run using a sine wave with the amplitude representing the desired angle of the top plate. The sine wave acted as a teeter-totter, always making sure the external load point of contact was in the center of the top plate. With small changes in the position of the point
35


load in the experimental test, there would be an altered distributed loading. For example, if the point load was slightly more towards the heel strike area of gait, it would be reasonable that the two legs closer to the heel strike area would be sustaining a higher load compared to a correctly positioned point load.
Matlab Model Limitations
Using a pre-defined Matlab template for a Stewart Platform acted as a good starting point for creating repetitive cycles during the stance phase of gait. However, the template was originally defined as a significantly larger device, meant to withstand loads a factor of ten higher than loads necessary for
1.5 times body weight in simulating the walking motion in the foot and ankle. Adjustments were made to scale down the Stewart Platform; this included the devices height, plate geometry, and material densities. Additionally, the PID controllers values were modified to account for this large scaling process. It is possible that the original values were proportional in a specific manner. Modifications to the Matlab template were made to replicate the dimensions in the linear actuators from LinMot. There is a possibility that by doing so, the modified values altered the specific proportional requirements of the Stewart Platform template. This, too, could affect the loading distribution in the six legs.
Lastly, specific assumptions were made with the theoretical model. As discussed earlier, the
1.5 times body weight was built within the top plates density. The experimental test included extra weight, including the polyethylene block, six rod 1 components, six joints, and the external 1050 N force. This added weight was represented in the model by including this weight in the top plate, resulting in an increased top plate density. This distributed loading in the model is not a perfect comparison to the loading in the experimental setup; however, this modification resulted in a model that more closely resembled the experimental setup.
36


CHAPTER VII
FUTURE WORK
Next Steps
Despite limitations regarding the experimental test, the data showed clear trends showing similarity in the experimental test and the theoretical model. More importantly, the experimental data showed that the loads required of each leg was more evenly distributed than depicted in the Matlab model. This is most clearly seen in a top plate angle range of -15 to +15 degrees. Based on these results, the next step in the project is to size appropriate linear actuators and construct a dynamic Stewart platform robotic gait simulator.
Original estimates for linear actuators were sized to withstand 1,050 Newtons per actuator. Following device construction, a correction was made to the Matlab model that produced maximum forces of approximately 700 Newtons. This correction was made in the Matlab script, which created an external load that distributed correctly. Additionally, the experimental data showed that actual loads were less than the 661 Newton force from the Matlab model. Next steps should involve additional experimental testing to verify this initial data collection. Once verified, accurate sizing of actuators can be performed with the goal of building the dynamic gait simulator.
Future Modifications to the RGS
Once linear actuators are selected, the next step would be to translate from using a point load to an accurate distributed loading using a prosthetic foot or cadaveric specimen. This would be a significant step towards achieving more physiological loading. Additionally, the use of motion capture techniques could be used simultaneously with the RGS. Taking advantage of this method with the use of a cadaveric specimen would allow for kinematics in specific joints in the foot to be measured. Using the combination of RGS and motion capture technology would begin to achieve more accurate kinematic and kinetic measurements during repetitive cycles of stance phase of gait. At this point, clinical studies could begin to compare RGS and motion capture data with both healthy and injured patients with foot injuries. This could mark the transition to translational research.
37


CHAPTER VIII
CONCLUSION
A Matlab model of a Stewart Platform that simulated average body weight repetitive loading cycles of gait during the stance phase was validated using an experimental static Stewart platform device. A Matlab Stewart platform template was modified and scaled down to match dimensions needed for an average size human foot. Using Simulink in Matlab, a dynamic simulation of repetitive cycles of stance phase of gait was achieved. Based on this modified computer model, a static experimental Stewart platform was constructed. Average walking loads (1.5 times body weight) were applied at various angled positions, replicating heel strike, midstance, and toe off regions of gait. Comparing experimental and Matlab model values resulted in similar trends across each of the legs. Experimental loads showed more equal loading distribution in the six legs. The next step in this project will be to substitute the rigid rods for linear actuators and build a dynamic robotic gait simulator based on the geometry and loading distribution of the experimental and Matlab model.
38


CHAPTER IX
ADDENDUM
Statics Analysis
Following the results obtained from the theoretical model and experimental data, a 2-D statics analysis was performed on the device. The goal was to utilize an additional method of calculating forces in each leg of the device. Prior to analysis, certain assumptions were made. First, the external force assumes zero friction, which is untrue when compared to the experimental test setup. Additionally, the moment calculation defined an origin in the center position of the top plate. This also assumed that legs 3 and 6 were located at this center location. However, in the experimental setup, legs 3 and 6 are slightly offset from the center, closer to legs 4 and 5. Lastly, this statics analysis was performed in 2-D orientation while of course, the experimental setup is in three dimensions. By using a 2-D orientation, symmetry was assumed so that when calculating for each of the legs, legs 1 and 2 would show equivalent loads (also for legs 3/6 and 4/5). Even with these assumptions, the statics analysis could provide useful information when comparing its results to the experimental data.
Using the following equations, leg forces were calculated. Theta values represented each of the six leg angles in relation to the top plate and were found by creating a 2-D simplified view of the device in SolidWorks.
(1) £Fy = sin(thetal) + F2 sm(theta2) + F3 sm(theta3) = Fo + Fey
(2) £Fx = FI cos(thetal) + F2 cos(theta2) + F3 cos(theta3) = Fx
(3) = -Fyla + Fy3b = 0
Shown below in Figures 1-6 is the results from the statics analysis compared with the experimental data for each leg.
39


Figure 1. Experimental data vs Statics Calculation comparison in Leg 1.
Figure 2. Experimental data vs Statics Calculation comparison in Leg 2.
40


Leg 3
1500
-2000
Top Plate Angle (degrees)
-Statics Calculation -Experimental
Figure 3. Experimental data vs Statics Calculation comparison in Leg 3.
Figure 4. Experimental data vs Statics Calculation comparison in Leg 4.
41


Figure 5. Experimental data vs Statics Calculation comparison in Leg 5.
Figure 6. Experimental data vs Statics Calculation comparison in Leg 6.
For the most part, the statics analysis stays within an order of magnitude of the experimental data. However, there are significant differences in the general trends. Recommendations for future work
include taking a closer look at the experimental design and perhaps making modifications to better match the model and the statics analysis. Eliminating assumptions made in the model and statics
analysis would more accurately compare their results to the experimental data.
42


REFERENCES
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4. Dunn, Lindsey. "11 Statistics and Facts About Orthopedics and Orthopedic Practices."Becker's Spine Review. N.p., 9 Dec. 2009. Web.
5. Sutherland, David H. "The evolution of clinical gait analysis: Part II Kinematics." Gait & posture 16.2 (2002): 159-179.
6. Whittle, Michael W. "Clinical gait analysis: A review." Human Movement Science 15.3 (1996): 369-387.
7. Corazza, Stefano, et al. "A markerless motion capture system to study musculoskeletal biomechanics: visual hull and simulated annealing approach." Annals of biomedical engineering 34.6 (2006): 1019-1029.
8. Miindermann, Lars, Stefano Corazza, and Thomas P. Andriacchi. "The evolution of methods for the capture of human movement leading to markerless motion capture for biomechanical applications." Journal of NeuroEngineering and Rehabilitation 3.1 (2006).
9. Whittaker, Eric C., Patrick M. Aubin, and William R. Ledoux. "Foot bone kinematics as measured in a cadaveric robotic gait simulator." Gait & posture 33.4 (2011): 645-650.
10. Sharkey, Neil A., and Andrew J. Hamel. "A dynamic cadaver model of the stance phase of gait: performance characteristics and kinetic validation."Clinical Biomechanics 13.6 (1998): 420-433.
11. Aubin, Patrick M., Matthew S. Cowley, and William R. Ledoux. "Gait simulation via a 6-DOF parallel robot with iterative learning control."Biomedical Engineering, IEEE Transactions
on 55.3 (2008): 1237-1240.
12. Aubin, Patrick M., Eric Whittaker, and William R. Ledoux. "A robotic cadaveric gait simulator with fuzzy logic vertical ground reaction force control." Robotics, IEEE Transactions on 28.1 (2012): 246-255.
13. Hurschler, Christof, Judith Emmerich, and Nikolaus Wiilker. "In vitro simulation of stance phase gait part I: Model verification." Foot & ankle international 24.8 (2003): 614-622.
14. Washington. "Gait I: Overview, Overall Measures, and Phases of Gait." Kinesiology> (n.d.): n. pag. Web.
15. Baker, Richard. "The history of gait analysis before the advent of modem computers." Gait & /wwft/n? 26.3 (2007): 331-342.
16. "Gait Phases." Microgate OptoGait Gait Phases. N.p., n.d. Web.
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17. Noble, Lawrence D., et al. "Design and validation of a general purpose robotic testing system for musculoskeletal applications." Journal of biomechanical engineering 132.2 (2010): 025001.
18. Baxter, Josh R., et al. "Cadaveric gait simulation reproduces foot and ankle kinematics from population-specific inputs." Journal of Orthopaedic Research 34.9 (2016): 1663-1668.
19. Kirtley, Christopher. Clinical gait analysis: theory and practice. Elsevier Health Sciences, 2006.
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21. Cutlip, Robert G., et al. "Evaluation of an instrumented walkway for measurement of the kinematic parameters of gait." Gait & posture 12.2 (2000): 134-138.
22. "The History and Current State of Motion Capture." Motion Capture Society. N.p., n.d. Web.
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44


APPENDIX A
Matlab Script
% This M-file creates the geometry and dynamic information for the Stewart
% platform in the home configuration. This Stewart platform consists of a
% top plate, a bottom plate, and six legs connecting the top plate to the
% bottom plate. The overall system has six degrees of freedom. Each leg has
% six degrees of freedom and is composed of two bodies, two u-joints, and
% one cylindrical joint. Controller data is also loaded into the
workspace.
0/
7o
% Copyright The MathWorks, Inc.
deg2rad = pi/180;
x_axis = [100];
y_axi s = [0 10];
z_axi s = [0 0 1];
% Connection points on base and top plate w.r.t. World frame at the center
% of the base plate pos_base = []; pos_top = [];
alpha_b = 10*deg2rad; % +- offset angle from 120 degree spacing on base
alpha_t = 10*deg2rad; % +- offset angle from 120 degree spacing on top
height = 0.75; % height in home configuration radius_b = 0.1905; % base radius in meters radius_t = 0.1900; % top radius in meters for i = 1:5,
% base points
angle_m_b = (2*pi/3)* (i-1) alpha_b; angle_p_b = (2*pi/3)* (i-1) + alpha_b;
pos_base(2*i-1, :) = radius_b* [cos(angle_m_b) sin(angle_m_b),
45


0.0];
pos_base(2*i,:) = radius_b* [cos(angle_p_b), sin(angle_p_b),
0.0];
% top points
% Top points are 60 degrees offset ang]e_m_t = (2*pi/3)* (i-1) a]pha_t + 2*pi/6; ang]e_p_t = (2*pi/3)* (i-1) + a]pha_t + 2*pi/6; pos_top(2*i-1,:) = radius_t* [cos(ang]e_m_t), sin(ang]e_m_t), height];
pos_top(2*i,:) = radius_t* [cos(ang]e_p_t), sin(ang]e_p_t), height]; end
% permute pos_top points so that legs are end points of base and top points
pos_top = [pos_top(6,:); pos_top(l:5,:)]; %6th point on top connects to 1st on bottom
% Compute points w.r.t. to the body frame in a 3x6 matrix body_pts = pos_top' height*[zeros(2,6);ones(1,6)];
% leg vectors
legs = pos_top pos_base;
]eg_]ength = [ ];
]eg_vectors = [ ]; for i = 1:6,
]eg_]ength(i) = norm(]egs(i :)) ;
] eg_vectors (i :) = ]egs(i,:) / ] eg_] ength (i) ;
end
% Calculate revolute and cylindrical axes for i = 1:6,
revl(i,:) = cross(1eg_vectors(i,:), z_axis); revl(i,:) = revl(i,:) / norm(revl(i :)) ; rev2(i,:) = cross(revl(i :) 1 eg_vectors (i :)) ; rev2(i,:) = rev2(i,:) / norm(rev2 (i :)) ; cyll(i,:) = leg_vectors(i :) ; rev3(i,:) = revl(i,:); rev4(i,:) = rev2(i,:); end
% Coordinate systems
46


lower_leg = struct('origin', [000], 'rotation', eye(B), 'encLpoint', [0 0 0]);
upper_leg = struct('origin', [000], 'rotation', eye(B), 'encLpoint', [0 0 0]);
for i = 1:6,
]ower_]eg(i) .origin = pos_base(i , + (3/8)*]egs(i :) ;
]ower_]eg(i) .encLpoint = pos_base(i :) + (3/4)*]egs(i :) ;
] owe r_] eg (i). rotation = [revl(i,:)', rev2(i,:)', cy]l(i,:)']; upper_]eg(i) .origin = pos_base(i :) + (l-3/8)*legs(i :) ; upper_]eg(i) .encLpoint = pos_base(i :) + (1/4)*]egs(i :) ;
u pper_] eg (i). rotation = [revl(i,:)', rev2(i,:)', cy]l(i,:)']; end
% Inertia and mass calculation top_thickness = 0.0127; % 1/2 inch base_thickness = 0.0127;
inner_radius = 0.020; % based on LinMot actuator (estimate) -in meters
outer_radius = 0.037;
density = 2800; % aluminum density in kg/mAB density2 = 0.01;
adjusteddensity = 83328.20; %at0degrees %leg inertia and mass
[lower_leg_mass, 1ower_leg_inertia] = inertiaCylinder(density2, ...
0.75*leg_length(l),outer_radius, inner_radius); [upper_leg_mass, upper_leg_inertia] = inertiaCylinder(density, ... 0.75*leg_length(l),inner_radius, 0);
% top and base plate mass and inertia
[top_mass, top_inertia] = inertiaCylinder(adjusteddensity, ... top_thickness, radius_t, 0);
[base_mass, base_inertia] = inertiaCylinder(density, ... base_thickness,radius_b, 0);
% PID controller gains
% Kp = 2e5; Ki = le3; Kd = 4.5e3; % default values Kp = 100000; Ki = 500; Kd = 2250;
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Appendix B
Solid Works Drawings of Components
Top plate:
48


Bottom plate:
49


7.00
Rod 1:
50


Rod 2:
51


Cylinder:
O
O
I--!
Cylinder-joint coupler:


2.00
53


Ball-socket Joint:
54


Appendix C
Data Collection VI Block Diagram
55


Appendix D
Modified Data Collection VI
56


APPENDIX E
Matlab Script for Statics Analysis
clear all clc
fo = 28.5; % force plate+block
fe = 525; % external force applied
phi = [-25, -15, -10, 0, 10, 15, 25]; %top plate angle
al = [3.12 3.64 3.8 3.94 4.07 4.4 5.49]; % angle 1 from Solidworks drawing
a2 = [-7.2 -6.5 -6.3 -6.2 -5.63 -5.15 -3.87]; % angle 2 from SWs
a3 = [1.01 2.68 3.28 3.94 4.01 4.1 4.39]; % angle 3 from Sws
a = 2; % Distance between middle and left pin on top plate b = 2; % Distance between middle and right pin on top plate
% theta calculations based on unit circle and al thru a3 thetal = pi/2-al *pi/l 80; theta2 = pi/2-a2*pi/180; theta3 = pi/2-a3*pi/180;
% force in y-direction; fly=f2y=f3y fori = l:length(phi)
fy(i) = -fo-fe*sin((90-phi(i))*pi/180); % Force is dirested fx(i) = -fe*cos((90-phi(i))*pi/180);
tl = thetal(i); t2 = theta2(i); t3 =theta3(i);
alpha = sin(tl)*(l+a/b);
beta = a*cos(t3)*sin(tl)/(b*sin(t3));
f2(i) = (fx(i) fy(i)*(cos(tl)/alpha + beta/alpha)) / (cos(t2) sin(t2)*cos(tl)/alpha -
57


beta/alpha* sin(t2));
fl(i) = (fy(i) f2(i)*sin(t2)) / (sin(tl)*(l+a/b)); f3(i) = (fl(i)*sin(tl)*a)/(sin(t3)*b); end
fxl =fl .* cos(thetal); fx2 = £2 .* cos(theta2); fx3 = f3 .* cos(tlieta3);
test_fx = fxl + fx2 + fx3;
fyl =fl .* sin(tlietal); fy2 = f2 .* sin(theta2); fy3 = f3 .* sin(theta3);
test_fy = fyl + fy2 + fy3;
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Full Text

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i COMPARISON OF MECHANICAL PARAMETERS BETWEEN A STEWART PLATFORM THEORETICAL MODEL AND AN EXPERIMENTAL STATIC TEST SPECIFIC TO THE STANCE PHASE OF GAIT IN FOOT ANKLE COMPLEX by SARAH HENSLEY BS Rose Hulman Institute of Technology, 2015 A thesis su bmit ted to the Faculty of the Graduate School of the University of Colorado in partial fulfillment of the requirements for the degree of Master of Science Bioengineering Program 2017

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ii This thesis for the Master of Science degree by Sarah Hensley h as bee n approved for the Bioengineering Program b y Cathy Bodine Chair Kenneth Hunt Steven Lammers Levin Sliker Date: July 29 2017

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iii Hensley, Sarah (M .S., Bioengineering Program ) Co mparison of Mechanical Parameters Between a Stewart Platform Theoretical Model and an Experimental Static Test Specific to the Stance Phase of Gait in Foot Ankle Complex Thesis directed by Associate Professor Cathy Bodine. ABSTRACT Understanding true kinematics and kinetics of the foot and ankle during stance phase of gait will lea d to better methods of injury prevention, diagnosis, and rehabilitation. More specifically, these parameters can provide clinically useful information in studying bone stress fractures, acute ligamentous injuries, external devices like shoe orthotics, and internal devices such as ankle implants. The use of a robotic gait simulator (RGS) has proven to be an accurate method of achieving this goal. This study was the first step in creating a RGS that can produce repetitive cycles of stance phase of gait to mea sure kinematics and kinetics. A computer simulated model based on a Stewart platform was created and modified specific to gait in the foot and ankle. The model replicated the repeated walking motion during stance phase using average body weight (1.5 times body weight). This model was then validated by building a static experimental test setup that replicated the same geometry and forces. The experimental test compared its distributed loading during stance phase vs the distributed loading predicted with the model. Results show ed that the experimental data trended well with the model and resulted in less than 20% error at the midstance, or 0 degree position. The next step in this project will be to build a dynamic RGS that can produce repetitive cycles of gait during stance phase. The form and content of this abstract are approved. I recommend its publication. Approved: Cathy Bodine

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iv ACKNOWLEDGMENTS I would like to thank Todd Baldini and my committee members for advising me through out my project. Thank you to my many professors that have inspired me to pursue my academic goals. I would also like to thank my family and friends for their continued and unwavering support.

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v TABLE OF CONTENTS CHAPTERS I. ... ......1 Foot Injury G 1 In 1 ... 2 II. REVIEW OF THE .... 3 ...... 3 Phases of Ga .... .......... 3 .... 4 6 7 In 9 Limitations t 10 ....... ........ 10 III. HYPOTHESIS AND SPECIFIC IV. ME ...... .......1 4 .... 14 14 14 ... 17 Experimental Te 19 19 Devi 20 Load Cel 21 23

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vi Test Protocol and Data Collection ........ 25 ..... ... ..... 26 ......... .... 26 V. RESULTS 29 S t a t i c s A n a l y s i s 3 3 VI. DISCUSSION AND L Model Compa .. 34 Expe rimental ... .36 VII. FUTURE WORK 37 37 VIII. CONCLUSION IX. A D D E N D U M 3 9 4 3 4 5 4 5 B. Solid Works Drawings of Components .4 8 C. Data Collection VI Block Diagram 5 D. Modified Data Collection VI 6 E M a t l a b S c r i p t f o r S t a t i c s A n a l y s i s 5 7

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1 CHAPTER I INTRODUCTION Foot Injury Gait Analysis Gait is a multi segmental process, requiring movement in mos t of the body but largely from the lower half, including the hip, knee, foot, and ankl e. Injuries to the foot and ankle negatively alter normal gait and it does not remain exclusive to the foot and ankle. Other parts of the body are also abnormally affect ed when an injury occurs (1 ). For example, if someone fractures their foot, it would alter gait and a ffect not only the foot but other joints such as the hip and knee joints. This one, multi segmental, symbiotic system, the human body, has been the reason for sparked interest in gait analysis, which is the study of human motion (1, 2, 3 ). Compared to numerous orthopedic surgeries being performed in the hi p and knee joints, there are less currently being done in the foot ankle complex (4). One reason for thi s is due to more difficulty in measuring kinematics ( motion without reference to the forces that causes the motion ) and kinetics ( study of forces that cause motion ) of the foot ankle complex. Two main methods of gait analysis currently used are clinical o bservational analysis and motion capture analysis. However, both methods produce variability and non reproducibility ( 5, 6, 7, 8 ). Clinical observational analysis is based on temporal spatial parameters. This includes gait speed and stance duration, both o height. Motion capt ure analysis falters due to size complexity and cost. Measuring motion in the arm or leg might be feasible but can be very difficult in the much smaller foot ankle complex. Additionally, a complete motion capture setup can exceed $100,000 in cost. Because of these inconsistencies, this limits the validity of these methods in measuring true kinematics and kinetics of the foot and ankle during gait ( 9, 10 ). In vitro R obotic Gait Simulators Most recently, in vitro robotic gait simulators that simulate the stance phase of gait have been used [ 9, 10, 11, 12, 13 ]. These in vitro simulators show potential in eliminating error produced in

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2 current methods of gait analysis, bu t there is still a great need to improve these devices in accuracy and validation There is a critical need to more accurately measure kinematics and kinetics in the foot ankle complex Taking advantage of a controlled gait simulator with repetitive gait c ycles could more accurately measure these parameters This tool could then be used to compare simulat ed normal gait to current methods of correcting gait. By doing so, methods of injury prevention, diagnosis, and rehabilitation of foot ankle injuries could be improved. Ultimately, this method of ga it analysis could help individuals return to their normal, active lives. Project Goals The long term goal of this project is to design a 6 degree of freedom robotic gait simulator that produces the repetitive cycl es of gait in an effort to improve understanding of physiological loading (vertical ground reaction force and loading distribution) and kinematics during the stance phase of gait T he first step in this multi stage process is to develop a computer model t o simulate loading and repetitive cycles of gait during stance phase. The second step is to validate the mod mechanical feasibility by replicating the model using a fixed structure that uses adjustable rigid rods as opposed to linear actuators.

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3 CHAPT ER II REVIEW OF THE LITERATURE Overview of Gait Gait is locomotion of the entire body (1 4 ) Its necessity on a daily basis makes it one of the most studied components of human movement. Because it is an easily obser vable activity that involves multi segment motion of the body, the systematic study of human walking, or gait analysis, is prevalent in literature in continued efforts to fully understand human biomechanics ( 1, 2 3, 5, 9 ) Observations of gait can be dat ed back to Aristotle (384 322 BCE) when he walked adjacent to a wall with an ink utensil attached t o the top of head He observed that the ink did not draw a straight line but rather a zig zag, indicating a vertical movement (bending and straightening) as he walked ( 15 ) As technology has advanced, gait analysis has progressed past purely observational research to computer driven measurement techniques. In a clinical setting commonly with orthopedic foot and ankle surgeons, visual gait evaluation is used to assess gait patterns or abnormalities. Visual gait evaluation differs from formal gait analysis, which takes place in a scientific research driven setting. Formal gait analysis records and analyzes human locomotion for reasons beyond the clinical scope in cluding: understanding biomechanical characteristics of gait, analyzing separate components of gait ( e.g., foot, ankle, and rest of body), and performing further analysis to improve clinical treatments Phases of Gait Gait is classified into two main phas es, stance and swing, w ith both featuring various p eriods and events that occur in each phase (Figure 1) Gait begins with the stance phase and comprises 60% of one complete gait cycle. The stance phase is comprised of three being the initial heel strike, marking the beginning of a gait cycle. As an individual takes a step forward, their heel strikes the ground, causing eccentric contraction of ankle dorsiflexors and allowing the ankle to begin plantar flexing. As forward mov ement progresses, the foot becomes flat

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4 with the ground, indicating the second rocker. During this period, the tibia rolls over the ankle as the foot remains planted to the ground. Forward movement continues into the third rocker as the foot begins to dors iflex, ultimately leading to toe off. Throughout the stance phase, muscle groups are equally important in the function of gait; their main function is to make this phase of gait an e n ergetically efficient process (15 ) In summary, the stance phase of gait consists of the body bearing weight with initial contact at the heel to translate the lower extremity over the fixed foot and continue the forward movement of the individual. The other 40% of gait consists of the swing phase, which primarily functions in foot clearance as forward movement continues. The plantar flexed foot at toe off rotates into a more neutral position due to concentric contraction of anterior leg musculature. As this rotation occurs, the foot clears the floor, preventing foot drag on the ground, and the leg continues forward as a result of stored energy in the pelvis and proximal thigh musculature. A major benefit to the swing phase of gait is its process of allowing the high loads that occurred at heel strike to dissipate over time befor e a new step is taken ( 1, 3 ) Figure 1 One complete gait cycle consisting of stance phase and swing phase (16) Parameters of Gait

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5 In addition to understanding gait mechanics, it is also important to understand various parameters of gait analysis. Knowi ng these parameters and how they affect gait are vital in developing any type of device for simulating gait in addition to providing clinical relevance. Three key gait parameters include temporal spatial, kinematic, and kinetic. Two major gait parameters a re kinematics and kinetics, both used heavily in research setting for quantitatively analyzing gait. Kinematics is the study of motion, more specifically defined as the movement of a body in isolation of the forces. Kinematic parameters often include displ acement, velocity, and acceleration, and are often recorded using motion tracking devices. Alternatively, kinetics is the study of the forces and moments that cause the movement. Parameters often include ground reaction forces (GRFs), joint moments, and jo int power. Both kinematic and kinetic parameters are key in understanding and simulating human gait ( 10, 17, 18 ) Another classification of gait is based on temporal spatial parameters, having been referred to as 19, pg21). The se observabl e parameters are often used in clinical setting s and include gait speed and stance duration, among others (Figure 2). Gait speed is defined by the equation: Cadence represents the number of steps per minute, and stride length represents the total length of an individual t aking two steps (i.e. left heel strike to the next left heel strike). The stance duration parameter is defined as the amount of time the foot is in contact with the grou nd over the period of one step. These two parameters, along with several others, are ve ry intuitive to clinicians and can be used to quantify pathological gait patterns (21 ) Though quite useful for clinicians, temporal spatial parameters are a global function of gait and can be influenced by many factors, some of which include age, sex, hum an error in data collection, and auditory cueing when taking measurements.

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6 Temporal spatial parameters can be very useful in diagnosing gait abnormalities, but they cannot be relied on independently (21 ) Figure 2 E xample of clinical gait analysis based on temporal spatial parameters (20) Of all the gait parameters, temporal spatial parameters are considered the most clinically applicable ( 1 4) Gait speed has been correlated with orthopedic research, often with slower speeds indicating foot pain and ab normal gait ( 14 ) Additionally, it has also been correlated with higher prevalence in elderly individuals ( 14 ) Stance duration has also been clinically observed with decreased stance duration due to avoidance of weight bearing or conversely, increased sta nce duration due to weakened muscles or decreased foot mechanics preventing a normal rate of stance phase (14 ) These observable parameters, though clinically effective, are still global parameters easily influenced by uncontrollable factors such as age, s ex, and others (14 ) Because of this, it is vital to better understand true kinematics and kinetics of gait. To achieve this, these parameters must be quantified by more advanced techniques than pure observation. Early Methods of G ait A nalysis Before moti on capture and other current methods of gait analysis, primiti ve methods dating back to the 1950s were used ( 5 ) After WWII, there was increased demand for understanding locomotion to improve treatment of war veterans ( 5, 6 ) Using a strobe light, reflectiv e strips, and a manual goniometer, gait analysis was conducted by having an individual walk under the illumination of the strobe light. With the reflective strips attached to anatomical landmarks, photographs were

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7 taken to manually make joint measurements. Though this early method was not exceptionally accurate and had problems in measuring rotations in the transverse plane, it did measure sagittal plane joint angles that are similar to results still found today (5 ) As technology advanced, the Karpovich b rothers among others began to take advantage of this technology and use electrogoniometers for gait analysis ( 5 ). Compared to manual goniometers, electrogoniometers significantly reduced data processing via computers. Additionally, this method was simple, reliable, and inexpensive, which allowed more researchers to perform gait analysis. Most importantly, a triaxial goniometer was the first attempt to capture motion in 3 directions, something which is now known to be critical in making accurate, physiologic ally mimicking parameters (5 ) Limitations to electrogoniometers include their inability to obtain simultaneous measurements of all moving segments and their reduced accuracy because the device was positioned offset to the side of limb segments. Additional ly, this method was not uniform across different demographics such as height and weight, wh ich made data normalization difficult (5 ) Motion C apture G ait A nalysis The next iteration of gait analysis develop ed rapidly with the invention of passive marker s ystems based on those used in cine film. Photographic techniques by Marey and Muybridge were able to capture the entire human body while simultaneous ly viewing relative motions between body segments ( 5 ) With this large field of view, individuals of all si zes could be measured, greatly increasing robustness of this method. The major limitation of this gait analysis was the inadequacy of computers during this time period. It took significant time to process the data, and there were major c omputer storage lim itation s. By the late 1960s, television motion analysis systems were being developed with automated recording of reflective markers. One of the early motion capture tests was measuring multiple joint angles of a cat running on a treadmill ( 5 ) The company VICON, was established in 1984 in Oxford, UK, and devel oped the first commercial motion capture system. Once the film industry realized the potential of motion capture, advances in this field increased. Currently, VICON alone has won an

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8 Academy Award, an Emmy, and other awards for t heir expertise in motion capture (10, 11 ). Though motion capture is heavily used for entertainment purposes, the film industry has driven the market, which resulted in high quality motion capture systems. Motion capture system s are one of the most used methods of gait analysis. Compared to clinical gait analysis, motion capture systems provide more quantitative than qualitative results ( 7, 22 ) The basics of motion capture gait analysis consist of placing reflective markers on surface. To capture 3 D motion of a particular segment, 3 markers are placed on each point of interest. A model limb segment is represented as a rigid body and then an algorithm is applied for rigid body motion. C ameras are positioned around the individual and as he or she moves, the cameras capture the movement of the reflective markers, which are representing body segmental movements (Figure 3) Motion capture systems have been used in analyzing gait of larger body movements, such as the femur tibia, humerus ulna/radius, or posture. However, measuring kinematics of more complex, smaller structures, such as the foot ankle, is much more difficult ( 9, 10 ) Figure 3 General setup of motion capture gait analysis (23) Two major factors in m otion capture are size and complexity. For example, the knee joint connects the femur to the tibia, both large bones of the body. Further more the knee joint and corresponding bones produce large amounts of motion during gait r esulting in accurate

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9 measurements of the knee joint complex when using motion capture systems. However, the foot ankle complex is significantly smaller. Not onl y is the motion of the foot and ankle during gait much smaller in magnitude, but this motion involves many more bones. There are 26 bones and 33 joints in the foot alone that provide structural support compared to 3 bones (femur, tibia, fibula) surrounding the knee joint. This limi tation is one major reason why motion capture is not ideal for gait analysis in the foot and ankle. Se condly, motion capture systems have one major limitation across any measurement of the body. There is error and variability in measuring kinematics due to skin movement relative to s an individual walks, motion capture systems cannot differentiate between what motion is due to bone movement or skin or muscle movement ( 7, 8 ) In vitro Robotic Gait S imulators s in determining accurate bony motion with out skin or other artifact error, research led to the development of in vitro robotic gait simulators (RGS) ( 9, 10, 11 ) The basic components of a RGS are a fixed base plate attached to six linear actuators with a moveable platform attached on top. The six linear actuators control the position of the RGS and allows for not only translation in the x, y, z directions but also rotation in the form of pitch, roll, and yaw. It is important to know that gait in the foot ankle complex is not simply dorsi plantarflexion; with each step in the walking motion there is also rotation and distribution of the loading also in the medial lateral directions. The RGS simu lates the stance phase of gait b y setting initial position of the moveable plate to correspond to initial heel strike, also known as the beginning point of stance phase of gait. The moveable plate, controlled by the six linear actuators, then moves through the stance phase of gait, ending at toe off. The linear actuators are working in tight coordination with each other

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10 to accurately simulate gait. Additionally, the moveable plate is attached to a force plate allowing for the recording of vertical ground reactio n forces and loading distribution along the foot ( 11, 12 ) When testing cadaveric specimens in the RGS, they are potted in poly(methyl methacrylate) and connected to a biaxial testing machine via a custom mounting jig. Similar to previously discussed motio n capture methods, the RGS also works in tandem with motion capture cameras to determine kinematics. Studies suggest that using a RGS as the method of gait analysis reduces significant error and test variability found in clinical and motion capture gait an alyses methods ( 9, 10, 11, 12, 13, 18 ). platform inferior to the foot ankle as opposed to the foot and ankle having to perform the loading and position change. C adaver specimens are limited in their overall strength; using the RGS to s imulate the motion reduces the wear and tear the cadaveric specimens must withstand for each test. Limitation to Current Robotic Gait Simulators Studies indicate that an in vitro ap significant promise in translating to clinical use ( 12, 13 ) However, the major limitation to current caused by overuse, such as stress fractures. To more accurately measure gait in its true repetitive form, RGSs need to be able to run multiple cycles of gait for each iteration. This repetitive cyclic loading is a more accurate simulation of normal gait, especia lly for future applications in clinical setting s Significance Human locomotion is a daily activity for almost all individuals. This includes walking, running, skipping, and any other motions that people need to go about their normal daily activiti es. A gait is altered due to overuse. Overuse can be caused by excessive repetitions in running, because of a sudden change in gait (i.e. deciding to run 5 miles after having not exercis ed in a month), being overweight or obese, and other factors (24, 25, 26 ) Prolonged o veruse leads to injury, m ost commonly stress fractures. Once an

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11 injury h as occu rred, the likelihood of sustaining another injury at the same site is increased Current tr eatments of stress fractures in the foot are most commonly rest, ice, compressio n, and elevation (RICE). This can be frustrating for individuals with these injuries because the main course of treatment i s simply to wait for it to heal. In some cases, shoe orthotics are recommended as treatment or prevention due to high or low foot arches. However, these recommendations are based on static observations of the foot and not on dynamic loading of the foot. Shoe orthotics recommended based on foot arch abnormali ties are not sufficient, especially because foot injuries usually occur during dynamic situations, such as running or jumping (25, 26 ) Current forms of gait analysis include observational gait analysis and the more recent clinical gait analysis using mot ion tracking systems. Though both have shown certain qualities in determining gait patterns, both have variability in their methods that reduces efficacy and reliability. Studies have shown poor reproducibility in motion tracking gait analysis, which is a key factor in sustainable research (7, 8 ). The new method of studying gait using an in vitro robotic gait simulator has shown hopeful pr eliminary results. However, accurate kinematic and kinetic measurements must be improved in these simulators to translat e into a clinically useful tool Based on previous research and the evident need for better understanding of kinematics and kinetics during gait, a robotic gait simulator, designed specifically for analyzing the stance phase of gait in the foot and ankle i s needed to advance this field of research with the future goal of applying its knowledge in a clinical setting.

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12 CHAPTER III HYPOTHESIS AND SPECIFIC AIMS Hypothesis A theoretical model that simulates 1.5 times body weight and repetitive cycles of gait during stance phase in the foot and ankle can be validated using a fixed mechanical Stewart platform within 20 percent error Specific Aim One: To develop a theoretical, computer model for a robotic gait simulator (RGS) to simulate the repetitive cycles of stance phase of gait in the foot and ankle. Based on a Stewart platform, a Matlab model was created to simulate the repetitive cycles of stance phase of gait in the foot ankle complex. A Matlab created Stewart platform template was modified using Simulin k and one of its add ons, SimMechanics (27 ) The program consisted of an inverse kinematics module that allowed a user to input translational and rotational coordinates to control the positioning of the top, mobile plate in reference to time. Other modific ations included adding appropriate loading to simulate heel strike loading (1.5 times body weight) and designing the model according to the size of an average human foot. As the simulation runs, scope outputs includes the following parameters in real time: length of each actuator, force of each actuator, and acceleration of each actuator. Specific Aim Two: To build a static experimental Stewart platform using fixed cylinders in substitute of linear actuators at 1.5 times body weight. The experimental test i ncluded all necessary components to build a fully functioning RGS, apart from substituting the actuators for fixed rods and cylinders. In reference from the ground up, the simulator begins with a fixed platform (add dimensions). Connected via ball bearings are six hollow cylinders (as linear actuator substitutes) in a Stewart Platform configuration. Prior to assembly, each cylinder was cut to size, load cells (Phidgets) were placed in between the two halves, and the cylinders were set to their original orie ntation. Six more ball bearings were connected to the opposite

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13 ends of the cylinders, which were then connected to a separate mobile aluminum plate with the top plate offset 60 degrees from the fixed platform. Specific Aim Three: To use the experimental S tewart platform to validate the theoretical model With the experimental Stewart platform fully constructed, a 1050 Newton force was applied using an Instron to simulate 1.5 times body weight at various positions during stance phase while load cells measure d the distributed loading using Labview. Force data was captured over a period of three, 10 second trials for each position at a 1050 Newton force loading. Theoretical forces from the computer model in each leg actuator were then compared with the experime ntal force data collected from the static test. With the top plate at 0 degrees, the experimental test performed with less than 20 percent error.

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14 CHAPTER IV METHODS Overview Based on the research hypothesis, a Matlab simulation of a Stewart platform was developed to produce repetitive cycles of stance phase of gait. The simulation was programmed to simulate body weight and kinematics (1, 2, 12 ) Once c omp leted, a rigid version of a RGS was constructed to validate the Matlab simulation. Matlab Simulation The first step in designing and building a robotic gait simulator for the foot and ankle during stance phase of gait was to develop a computer model that could simulate the repetitive cycles of gait. This was accomplished in two main steps: 1. Identifying a generic Stewart Platform model with specific controlling capabilities 2. Modifying the Stewart Platform specific to stance phase of gait in the foot and ankle Matlab Stewart Platform Template A Matlab created Stewart platform simulation template was controlled using Simulink and one of its add ons, SimMechanics. Matlab separated the simulation template into three subsystems: the inverse kinematics module proportional integral derivative ( PID ) controller, and plant ( Figure 4 ).

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15 Figure 4 Graphic of Matlab Stewart Platform template using Simulink. The leg trajectory subsystem consisted of an inverse kinematics module that a llowed a user to input translational and rotational coordinates (x, y, z, pitch, roll, and yaw) to control the positioning of the top, mobile plate in reference to time and the fixed bottom plate (Figure 5 ). Figure 5 Desired leg trajectory using transl ational and rotational coordinates. Once coordinates were defined, the Matlab template transformed those input values into leg These leg lengths then became the input for t he PID controller subsystem see Figure 6 The PID controller acted as a control loop feedback mechanism for calculating and adjusting for error in

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16 the desired leg lengths compared to actual leg lengths. The controller had three inputs: desired position, a ctual position, and actual leg velocity. The PID then adjusted according to error and output ted the required force of each leg (actuator) to achieve the desired position. Figure 6 PID controller subsystem. These forces then became the input for the pla nt subsystem, which was a Matlab representation of the mechanical components of the Stewart Platform (Figure 7) The mechanical components consisted of a top plate, bottom plate, six legs connecting the two plates, and universal joints connecting each end of a leg to its corresponding plate. Using SimMechanics, the legs were given the freedom to translate, as linear actuators do, using joint actuator blocks from the library in SimMechanics. Geometry of the plant subsystem was defined using an M file script template, shown in Appendix A With defined geometry and force input, the plant subsystem was used to output leg lengths and leg velocities.

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17 Figure 7. Plant subsystem. As a final step, a simple scope block was used to graphically output in real time leg lengths, velocities (and acceleration using a derivative function), and forces of each actuator. These three parameters were compared and evaluated to verify the simulation while also providing necessary value to build the experimental validation test. Mo difications Specific to the Foot and Ankle Using the Matlab created template, three major modifications were made to create a simulation specific to the stance phase of gait in the foot ankle complex: Modification 1:

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18 In the leg trajectory subsystem six d esired coordinate trajectories (x, y, z, pitch, roll, and yaw) were defined to simulate repetitive cycles of stance phase of gait. The kinematics of stance phase of gait include initial foot strike, mid stance, and toe off. To simulate repetitive cycles of these this phase, the x, y, z, roll, and yaw coordinates were set to zero. Pitch was defined using a sine wave function with an amplitude of 0.436 radians and frequency of 8 radians/second. The sine wave guaranteed continuous and repetitive cycles simulat ing walking motion. An amplitude of 0.436 radians, or 25 degrees, simulated the angle at which the foot strikes the ground during heel strike, or initial step of stance phase. A frequency of 8 rad/s simulated a walking speed that correlates to approximatel y 3 m/s, which is the average walking speed of a human. Modification 2: In the plant subsystem, t he Stewart platform template automatically determined the forces required of each of the six actuator legs to maintain equilibrium of the system. However, to modify the system to account for the force of a human of average weight, an adjusted density method was used for the top plate. The constant value of 1050 Newtons, or 1.5 times body weight, was set according to the average force created by normal walking m otion. To ensure a distributed load on the top plate, the weight of the top plate, polyethylene block, six ball joints, six rods (rod 1), two clamps, and 1050 Newton force were combined as one mass to determine an adjusted density of 83,328.195 kg/m 3 for e ach different position. After adding this external force, the six leg actuators took this additional force into account when measuring the forces required to keep the system in equilibrium. Modification 3: In the Matlab script, the following parameters we re changed: height of the overall device, radii of top and bottom plate, thickness of top and bottom plate, inner and outer radii of each leg, density of materials used, and PID controller gain values. In the Matlab script template, the default geometry of ankle complex. The default and modified values are shown in Table 1 below. Table 1 Comparison of default and modified values in Matlab script.

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19 Part Default Value Mod ified Value (scaled specific for a foot ankle system) Height of overall device 3.0 meters 0.7 5 meters Top plate radius 1.0 meter Top plate thickness 0.05 meters Bottom plate radius 3.0 meters 0.1905 meters Bott om plate thickness 0.05 meters 0.0127 meters Leg actuator inner radius 0.03 meters 0.020 meters Leg actuator outer radius 0.05 meters 0.037 meters Density 1 76e3/9.81kg/m 3 2800 kg/m 3 Adjusted density NA 83,328.2 kg/m 3 Alpha_b 2.5 degrees 10 degrees A lpha_t 10 degrees 10 degrees Experimental Test for Validation The experimental test included all necessary components to build a fully functioning RGS, based on the Matlab model, apart from substituting the actuators for fixed rods and cylinders. The goa l was to apply static loads to the RGS at different positions during the stance phase of gait and compare the experimental load distribution to the theoretical load distribution based on the Matlab model. Materials and CAD Modeling A SolidWorks assembly o f the RGS was created to accurately size all components and component connections. The following is a comprehensive list of each SolidWorks part created (Table 2) See Appendix B for drawings. Table 2 Device component list. Part Name De tails # Units Top plate 6061 Al 1 Plate screw 28 screw 2 Ball socket joint 28 threaded hole (Midwest Controls) 12 Rod 1 (McMaster) 6 Load cell Max 5 00kg compressive /tensile S type Phidget ) 6 Load cell threaded rod length, M12x1.75 12 Rod 2 6 Set screws length 18 Cylinder 6 Rod cylinder fixation screws 28 set screw 18 Cylinder joint coupler 6

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20 Bottom plate thick Al plate 1 Device Construction In reference to the ground up the simulator be gan with a fixed platform (bottom plate ). Six 28 thru holes were drilled into the platform in a specific orientation. These holes serve d as connection points for ball and socket joints, which had threaded female ends. Screws were inserted into the platform on the inferior side and connected to the ball and socket joints. The screws were countersunk into the platform to ensure a flat surface on the platform. Next, the c ylinder joint coupler was machined to connect the ball and socket joint to the cylinder A 28 threaded hole was drilled into the inferior side of the cylinder joint coupler. This allowed the ball and socket joint to be connected to the coupler. To connec t the coupler to the cylinder, a 20 screw was placed through both components and held toget her using a washer and bolt Next, rod 2 was place inside the cylinder to allows for f ine adjustability (0 to 6 inches) in length. To do this, three equally spaced e superior side of the cylinder Set screws were inserted to fix any desired position within a range of 0 6 inches of extension. The S type load cell was placed i n between rod 1 and rod 2 E ach end of the load cell had M12x1.75 size d threaded holes s long threaded hole was drilled into both the inferior end of rod 1 and superior end of rod 2. A threaded rod was place into each of these holes to con nect both ends to the load cell To complete the assembly a 28 threaded hole was drilled into the superior side of rod 1 to attach the ball and socket joint. Then the ot her end of the joint was connected to the top plate. This was done using the same orientation described for the bottom plate but wit h a 60 degree offset. See Figure 8 below for the complete assembly.

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21 Figure 8 Fully assemble d Stewart platform device. Load Cell Implementation Prior to attaching the load cells to th e device, they were first tared Each load cell came pre calibrated by the manufacturing company but additional calibration was performed. E ach load cell was conne cted to a 1046 PhidgetBridge 4 input board via a four wire configuration shown in Figure 9 The red wire attached to the 5V input green wire to the positive input, white wire to the neg ative input, and black and yellow wires to the ground input. Top pla te Rod 1 Cylinder joint coupler S type load cell Rod 2 Bottom plate Cylinder Ball socket joint

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22 Figure 9 Wire configuration for load cell Phidget bridge connection ( 28 ) The board acted as an interface with the load cells to measure each of their outputs These load cells were configured using a built in wheatstone bridge connection. Finally, the m icro US B output from the amplifier was connected to a LabView enabled laptop via USB port. This setup is shown below in Figure 10 With no external loads being applied, data w as collected for a period of 30 seconds and repeated five times. These load measurements were averaged among the five trials and then subtracted from each load cells. These subtracted values were integrated in the LabView virtual instrument ( VI ) which is discussed in more detail below. With the load cells calibrated, they were then attached to the device.

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23 Figure 10 Hardware setup: (2) 1046PhidgetBridge s connected via USB port and Labview enabled laptop. Load cell configuration in Labview Phidget template VI Using the company, Phidgets, allowed for both the use of their online code exampl es in Labview specific to their load cells and their own function palette Using the six S type load cells in for specific use with the Stewart platform. Prior to modification, the VI functioned by reading the serial number for a given PhidgetBridge board. With the board identified, a user could define the number of bridges (in this case, two), pick a specific channel (load cell), and set the gain and data rate. Ru nning the VI resulted in a bridge value output reading in millivolts/volt of the channel chosen see Figure 11 Refer to Appendix C for the block diagram.

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24 Figure 11 Modifications to VI The following modifications to 1. components within the same loop allowed the program to read two PhidgetBridge boards so six load cells could be used. One board could only read four load cells at a time. 2. Each load cell came with a data sheet, including the output rate. More specifically, it gave an output rate in millivolts per volt at the maximum load of 500 kilograms. A numeric multiply constant of 500 kg was inserted into the VI to convert the units from millivolts/volt to kilograms 3. As mentioned earlier, the load cells were zeroed prior to any loading. This averaged constant for each load cell was subtracted using a numeric subtract constant. 4. For visual This resulted in a real 5. Load cell readings during testing needed to be recorded and saved for future data analysis. To as an Microsoft Excel file. Refer to Appendix D

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25 Test Protocol and Data Collection With the device fully assembled and connected to Labview, the test platform was position in an Instron materials and testing machine. A custom point load attachment, shown in Figure 12 was installed into the Instron. This acted as the loading contact point on the top plate. Figure 12 Custom point load attachment to Instron Additionally, a polypropylene block was placed between the top plate and the point load attachment. To prevent harsh metal on metal, this block allowed for a smoother initial loading. For each testing position, shown in Table 3 a 50 Newton pre load was applied. Next, the device was loaded at a rate of 100Newton/second until the load reached 1050N, where it was held for 10 seconds. This 1050N load represented a heel strike load, which is the maximum point of loading during stance phase of gait. After this 10 second hold, the device was unloaded at the same rate of 100 Newtons/second, see Figure 1 3 Simultaneously, Labview collected load c ell data via interface board and LabView software. To run the LabVie w VI, the VI was started by clic This prompted the user to select a file to save the data file. Once selected, data collection began and Instron end effector Custom point load Top plate

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26 Table 3 Testing positions based on angle of top plate. Positive angles indicate dorsiflexion. Negative angles indicate plantar flexion. Test # Top Plate Angle (degrees) 1 25 2 15 3 10 4 0 5 10 6 15 7 25 Figure 1 3 Instron loading pattern for each test. Test Summary In summary, the complete setup included the following. A fully assembled fixed Stewart platform including six load cells was placed in an Instron that provided a point load. The load cells were connect ed to two PhidgetBridge interface boards that communicated with a Labview enabled laptop. Labview recorded readings from the load cells and translated these measurements from millivolts per volt to kilograms. Using this setup, the test protocol was complet ed, including seven different top plate positions. Data Analysis

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27 Experimental data Following testing, the experimental data was analyzed to compare the loading distribution in the experimental test versus the Matlab sim ulation. The raw data files contained a time stamp followed by six columns of force data for each of the six legs. First, the three cycles of the constant 1050N force data was extracted from the raw data from each of the seven plate positions. This resulted in three clusters of 10 seconds of data for each position. All data were divided by 5 and multiplied by 9.8 to account for a unit change from mV/V to N This produced a final collection of data in the unit of Newtons, which was the same units defined in the Matlab simulation. To summarize, this post processing resulted in 30 seconds (3 trials at 10 seconds each) of experimental data for 7 different top plate position s for legs 1 through 6. The last step in data processing was to average each cycle of data in each position. This gave one expe rimental data point in each position to compare with the Matlab model. Standard deviation was also calculated in each leg. Matlab simulation data The next step was to extract forces in the computer model for each angle so it could be compared to the experi mental data. To do this, model forces were found using the cursor measurement tool i To remain consistent, forces for each position were recorded at a simulation time of 1.85 seconds for positive angle positions and 2 38 seconds for negative angles.

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28 Figure 14 The cursor measurement tool measuring forces (N) in Legs 1 and 3 with a 0 degree top plate position. Total distributed loading in experimental test and Matlab model As another mode of data comparison, the sum of legs 1 through 6 were calculated in both the experimental test and the Matlab model. This was an additional comparison made to validate the computer model.

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29 CHAPTER V RESULTS Experimental vs Computer Model Comparison Following post processing of data from both the experimental test and Matlab model, the two sets of data were compared in each leg, see Figures 1 5 through 20 The Matlab model was created by defining recorded da ta points at each angle ( 25, 1 5, 10, 0, 10, 15, 25 degrees) using a scatter plot with black smooth lines. Then, using a scatter pattern, the experimental data points were overlaid onto the same plot using red dots. Standard deviation bars were also created for each experimental data point. Figure 15 Experimental ( red dots) vs Model (blue line) data comparison in Leg 1. -1000 -800 -600 -400 -200 0 200 400 600 800 1000 1200 1400 -30 -25 -20 -15 -10 -5 0 5 10 15 20 25 30 Force (N) Top Plate Angle (degrees) Leg 1

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30 Figure 16 Experimental vs Model data comparison in Leg 2. Figure 17 Experimental vs Model data comparison in Leg 3. -1000 -800 -600 -400 -200 0 200 400 600 800 1000 1200 1400 -30 -25 -20 -15 -10 -5 0 5 10 15 20 25 30 Force (N) Top Plate Angle (degrees) Leg 2 -600 -400 -200 0 200 400 600 800 -30 -25 -20 -15 -10 -5 0 5 10 15 20 25 30 Force (N) To Plate Angle (degrees) Leg 3

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31 Fig ure 18 Experimental vs Model data comparison in Leg 4. Figure 19 Experimental v s Model data comparison in Leg 5. -800 -600 -400 -200 0 200 400 600 800 1000 -30 -25 -20 -15 -10 -5 0 5 10 15 20 25 30 Force (N) Top Plate Angle (degrees) Leg 4 -800 -600 -400 -200 0 200 400 600 800 1000 -30 -25 -20 -15 -10 -5 0 5 10 15 20 25 30 Force (N) Top Plate Angle (degrees) Leg 5

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32 Figure 20 Experimental vs Model data comparison in Leg 6. With the top plate in neutral (0 degree) position, all leg lengths were equivalent. The Matlab model was compared to the experimental test and a pe rcent error was calculated ( Tabl e 4 ) Table 4. Loading distribution comparison at flat plate position. Leg Model Experimental % Error 1 201.50 217.37 7.87 2 201.40 220.10 9.29 3 201.40 194.90 3.23 4 201.50 195.86 2.80 5 201.40 222.73 10.59 6 201.40 172.11 14.5 4 To compare the sum of the distributed loading, forces in each leg were added together in both the expe rimental and Matlab model (Table 6) Table 5 Comparison of total loading in experimental test and model. -600 -400 -200 0 200 400 600 800 -30 -25 -20 -15 -10 -5 0 5 10 15 20 25 30 Force (N) Top Plate Angle (degrees) Leg 6

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33 Experimental Sum of Forces Theoretical Mode l Sum of Forces Difference between Experimental and Theoretical Sum of Forces 1202.10 1361.40 159.30 1211.79 1260.80 49.01 1209.38 1230.00 20.62 1223.06 1208.60 14.46 1220.62 1229.51 8.89 1218.24 1256.80 38.56 1198.14 1321.10 122.96 Statics Analy sis Following these results, further analytics were performed with a statics analysis of the device. The analysis solved for each of the six leg forces required to achieve the same angled top plate positions acquired in the experimental testing. This analy sis provided further insight into both the experimental setup as well as the computer model. The addendum provides full text of the statics analysis.

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34 CHAPTER V I DISCUSSION AND LIMITATIONS Model Comparisons Based on Figures 1 5 through 20 the experimental test showed trends in the direction of the theoretical model. When comparing model and experimental results at 0 degrees (Table 5 ), the experimental test was within 14 .54 % accuracy At 0 degrees, the six legs were equivalent in length and not extended in the experimental model. This position provided a solid foundation on which to compare the two models. As seen in Table 5 each leg supported approximately 200N of load, which correlated to equal leg lengths. However, as the top plate angle increased, the d ifference in forces between the experimental and Matlab model also increased. As the angle increased, leg lengths varied in all six legs, which resulted in a non equivalent loading distribution. When comparing only the experimental data at different top p late angle positions, clear trends were seen in each of the legs illustrated with red dots Though this data varied significantly with the Matlab model, it showed consistency in the experimental testing. Additionally, when comparing only the Matlab model data at different top plate angle positions, there were also clear trends in each of the legs, illustrated with black lines Also, the Matlab data showed clear similarities in leg couplets (Legs 1 and 2, Legs 3 and 6, and Legs 4 and 5). This was to be expe cted due to the Stewart platform orientation of the legs. Experimental Limitations After construction and te sting of the experimental test, considerations were made regarding its overall ability to precisely and accurately record force readings in each le height was based on a Stewart platform in neutral, or zero tilt, position. Additionally, its leg lengths were determined by approximating equivalent linear actuators from the company, LinMot. By doing so, the overall height of the d evice limited full extension in the y direction. The limiting factor was the amount of working space available once the device was placed in an Instron. The device was able to achieve the maximum angle desired, plus and minus 25 degrees. However, ensuring there was zero

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35 yaw or roll (tilting) of the plate proved more difficult with these larger angles. Even small amounts of tilt in the plate could result in slightly incorrect distributed loading in the six legs. Another limitation involved the many parts ne cessary to build the static test. Unlike linear actuators, which would consist of two main parts (slider and stator), the experimental test used five main components. Rod 1, which acted as the slider, was cut into two pieces for the attachment of the load cell. The cylinder, which acted as the stator, required the joint cylinder connector as an attachment for the ball joint. Each of these components were manually measured and machined separately for all six legs, leaving room for potential human and equipme nt error. For example, if one leg was machined slightly smaller in length than the other five legs, this would result in an incorrect load ing distribution. If, for example, testing was taking place with the top plate at 25 degrees and t he shorter leg was positioned at the toe off region of stance phase, the loading in this leg could be much lower than its other corresponding leg and therefore higher in the other leg s During construction of the experimental test, the top plate was defined as a square (18x 18 inch) plate, as opposed to a circular (15 inch diameter) plate defined in the Matlab model. This added significant additional weight to the experimental test that the Matlab model did not take into account. To compare, the difference in weight would be 3.38 kilograms, or 33.124 Newtons. This could account for a portion of the loading differences between the experimental and Matlab model. Lastly, there were slight differences in the location of the point load in the experimental test. In the Matlab model the ex ternal load to the top plate was consistently placed in the direct middle of the circular plate. However, the point load in the experimental test varied some, especially at the larger angles (15 and 25 degrees). The bottom plate was firmly locked i nto the base of the Instron. To remain consistent in testing a This led to variances in the positioning of the point load, with the exception of testing at the neutral (0 degree) position. The Matl ab simulation was run using a sine wave with the amplitude representing the desired angle of the top plate. The sine wave acted as a teeter totter, always making sure the external load point of contact was in the center of the top plate. With small changes in the position of the point

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36 load in the experimental test, there would be an altered distributed loading. For example, if the point load was slight ly more towards the heel strike area of gait, it would be reasonable that the two legs closer to the heel s trike area would be sustaining a higher load compared to a correctly positioned point load. Matlab Model Limitations Using a pre defined Matlab template for a Stewart Platform acted as a good starting point for creating repetitive cycles during the stance phase of gait. However, the template was originally defined as a significantly larger device, meant to withstand loads a factor of ten higher than loads necessary for 1.5 times body weight in simulating the walking motion in the foot and ankle. Adjustments were made to scale down the Stewart Platform; this include height, plate geometry, and scaling process. It is possible that the original va lues were proportional in a specific manner. Modifications to the Matlab template were made to replicate the dimensions in the linear actuators from LinMot. There is a possibility that by doing so, the modified values altered the specific proportional requ irements of the Stewart Platform template. This, too, could affect the loading distribution in the six legs. Lastly, specific assumptions were made with the theoretical model. As discussed earlier, the s density. The experimental test included extra weight, including the polyethylene block, six rod 1 components, six joints, and the external 1050 N force. This added weight was represented in the model by including this weight in the top plate, resulting i n an increased top plate density. This distributed loading in the model is not a perfect comparison to the loading in the experimental setup; however, this modification resulted in a model that more closely resembled the experimental setup.

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37 CHAPTER VI I F UTURE WORK Next Steps Despite limitations regarding the experimental test, the data showed clear trends showing similarity in the experimental test and the theoretical model. More importantly, the experimental data showed that t he loads required of each l eg was more evenly distributed than depicted in the Matlab model. This is most clearly seen in a top plate angle range of 15 to +15 degrees. Based on these results, the next step in the project is to size appropriate linear actuators and construct a dynam ic Stewart platform robotic gait simulator. Original estimates for linear actuators were sized to withstand 1,050 Newtons per actuator. Following device construction, a correction was made to the Matlab model that produced maximum forces of approximately 700 Newtons. This correction was made in the Matlab script, which created an external load that distributed correctly. Additionally, the experimental data showed that actual loads were less than the 661 Newton force from the Matlab model. Next steps shoul d involve additional experimental testing to verify this initial data collection. Once verified, accurate sizing of actuators can be performed with the goal of building the dynamic gait simulator. Future Modifications to the RGS Once linear actuators are selected, the next step would be to translate from using a point load to an accurate distributed loading using a prosthetic foot or cadaveric specimen. This would be a significant step towards achieving more physiological loading. Additionally, the use of motion capture techniques could be used simultaneously with the RGS. Taking advantage of this method with the use of a cadaveric specimen would allow for kinematics in specific joints in the foot to be measured. Using the combination of RGS and motion capt ure technology would begin to achieve more accurate kinematic and kinetic measurements during repetitive cycles of stance phase of gait. At this point, clinical studies could begin to compare RGS and motion capture data with both healthy and injured patien ts with foot injuries. This could mark the transition to translational research.

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38 CHAPTER VII I CONCLUSION A Matlab model of a Stewart Platform that simulated average body weight repetitive loading cycles of gait during the stance phase was validated using an experimental static Stewart platform device. A Matlab Stewart platform template was modified and scaled down to match dimensions needed for an average size human foot. Using Simulink in Matlab, a dynamic simulation of repetitive cycles of stance phase o f gait was achieved. Based on this modified computer model, a static experimental Stewart platform was constructed. Average walking loads (1.5 times body weight) were applied at various angled positions, replicating heel strike, midstance, and toe off regi ons of gait. Comparing experimental and Matlab model values resulted in similar trends across each of the legs. Experimental loads showed more equal loading distribution in the six legs. The next step in this project will be to substitute the rigid rods fo r linear actuators and build a dynamic robotic gait simulator based on the geometry and loading distribution of the experimental and Matlab model.

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39 CHAPTER IX ADDENDUM Statics Analysis Following the results obtained from the theoretical model and experim ental data, a 2 D statics analysis was performed on the device. The goal was to utilize an additional method of calculating forces in each leg of the device. Prior to analysis, certain assumptions were made. First, the external force assumes zero friction, which is untrue when compared to the experimental test setup. Additionally, the moment calculation defined an origin in the center position of the top plate. This also assumed that legs 3 and 6 were located at this center location. However, in the experim ental setup, legs 3 and 6 are slightly offset from the center, closer to legs 4 and 5. Lastly, this statics analysis was performed in 2 D orientation while of course, the experimental setup is in three dimensions. By using a 2 D orientation, symmetry was a ssumed so that when calculating for each of the legs, legs 1 and 2 would show equivalent loads (also for legs 3/6 and 4/5). Even with these assumptions, the statics analysis could provide useful information when comparing its results to the experimental da ta. Using the following equations, leg forces were calculated. Theta values represented each of the six leg angles in relation to the top plate and were found by creating a 2 D simplified view of the device in SolidWorks. (1) (2) (3) Shown below in Figures 1 6 is the results from the statics analysis compared with the experimental data for each leg.

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40 Figure 1. Experimental data vs Statics Calculation comparison in Leg 1. Figure 2. Experimental data vs Statics Calculation comparison in Leg 2. -1000 -800 -600 -400 -200 0 200 400 600 -30 -20 -10 0 10 20 30 Force (N) Top Plate Angle (degrees) Leg 1 Statics Calculation Experimental -1000 -800 -600 -400 -200 0 200 400 600 -30 -20 -10 0 10 20 30 Force (N) Top Plate Angle (degrees) Leg 2 Statics Calculation Experimental

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41 Figure 3. Experimental data vs Statics Calculation comparison in Leg 3. Figure 4. Experimental data vs Statics Calculation comparison in Leg 4. -2000 -1500 -1000 -500 0 500 1000 1500 -30 -20 -10 0 10 20 30 Force (N) Top Plate Angle (degrees) Leg 3 Statics Calculation Experimental -1000 -800 -600 -400 -200 0 200 400 600 800 -30 -20 -10 0 10 20 30 Force (N) Top Plate Angle (degrees) Leg 4 Statics Calculation Experimental

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42 Figure 5. Experimental data vs Statics Calculation comparison in Leg 5. Figure 6. Experimental data vs Statics Calculation comparison in Leg 6. For the most part, the stati cs analysis stays within an order of magnitude of the experimental data. However, there are significant differences in the general trends. Recommendations for future work include taking a closer look at the experimental design and perhaps making modificati ons to better match the model and the statics analysis. Eliminating assumptions made in the model and statics analysis would more accurately compare their results to the experimental data. -1000 -800 -600 -400 -200 0 200 400 600 800 -30 -20 -10 0 10 20 30 Force (N) Top Plate Angle (degrees) Leg 5 Statics Calculation Experimental -2000 -1500 -1000 -500 0 500 1000 1500 -30 -20 -10 0 10 20 30 Force (N) Top Plate Angle (degrees) Leg 6 Statics Calculation Experimental

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43 REFERENCES 1. Karsten, De Koster. "Gait." Physiopedia N.p., n.d. W eb. 2. Whittle, Michael W. Gait analysis: an introduction Butterworth Heinemann, 2014. 3. Wheeless, Clifford R. "Gait." Duke Orthopaedics: Wheeless' Textbook of Orthopaedics N.p., 24 May 2012. Web. 4. Dunn, Lindsey. "11 Statistics and Facts About Orthopedics a nd Orthopedic Practices." Becker's Spine Review N.p., 9 Dec. 2009. Web. 5. Sutherland, David H. "The evolution of clinical gait analysis: Part II Kinematics." Gait & posture 16.2 (2002): 159 179. 6. Whittle, Michael W. "Clinical gait analysis: A review." Human Movement Science 15.3 (1996): 369 387. 7. Corazza, Stefano, et al. "A markerless motion capture system to study musculoskeletal biomechanics: visual hull and simulated annealing approach." Annals of biomedical engineering 34.6 (2006): 1019 1029. 8. Mndermann Lars, Stefano Corazza, and Thomas P. Andriacchi. "The evolution of methods for the capture of human movement leading to markerless motion capture for biomechanical applications." Journal of NeuroEngineering and Rehabilitation 3.1 (2006). 9. Whittaker, Eric C., Patrick M. Aubin, and William R. Ledoux. "Foot bone kinematics as measured in a cadaveric robotic gait simulator." Gait & posture 33.4 (2011): 645 650. 10. Sharkey, Neil A., and Andrew J. Hamel. "A dynamic cadaver model of the stance phase of gait: perfo rmance characteristics and kinetic validation." Clinical Biomechanics 13.6 (1998): 420 433. 11. Aubin, Patrick M., Matthew S. Cowley, and William R. Ledoux. "Gait simulation via a 6 DOF parallel robot with iterative learning control." Biomedical Engineering, IE EE Transactions on 55.3 (2008): 1237 1240. 12. Aubin, Patrick M., Eric Whittaker, and William R. Ledoux. "A robotic cadaveric gait simulator with fuzzy logic vertical ground reaction force control." Robotics, IEEE Transactions on 28.1 (2012): 246 255. 13. Hursch ler, Christof, Judith Emmerich, and Nikolaus Wlker. "In vitro simulation of stance phase gait part I: Model verification." Foot & ankle international 24.8 (2003): 614 622. 14. Washington. "Gait I: Overview, Overall Measures, and Phases of Gait." Kinesiology (n.d.): n. pag. Web. 15. Baker, Richard. "The history of gait analysis before the advent of modern computers." Gait & posture 26.3 (2007): 331 342. 16. "Gait Phases." Microgate OptoGait Gait Phases N.p., n.d. Web.

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44 17. Noble, Lawrence D., et al. "Design and valid ation of a general purpose robotic testing system for musculoskeletal applications." Journal of biomechanical engineering 132.2 (2010): 025001. 18. Baxter, Josh R., et al. "Cadaveric gait simulation reproduces foot and ankle kinematics from Journal of Orthopaedic Research 34.9 (2016): 1663 1668. 19. Kirtley, Christopher. Clinical gait analysis: theory and practice Elsevier H ealth Sciences, 2006. 20. Portal, McMaster Optimal Aging. "Healthy Aging with McMaster Optimal Aging Portal." Healthy Aging Research McMaster Optimal Aging Portal, 21 July 2014. Web 21. Cutlip, Robert G., et al. "Evaluation of an instrumented walkway for measur ement of the kinematic parameters of gait." Gait & posture 12.2 (2000): 134 138. 22. "The History and Current State of Motion Capture." Motion Capture Society N.p., n.d. Web. 23. "Gait Research." Qualisys Motion Capture Systems N.p., n.d. Web. 24. Kaufman, Kenton R., et al. "The effect of foot structure and range of motion on musculoskeletal overuse injuries." The American Journal of Sports Medicine 27.5 (1999): 585 593. 25. Matheson, G. O., et al. "Stress fractures in athletes: a study of 320 cases." The American J ournal of Sports Medicine 15.1 (1987): 46 58. 26. Milgrom, C., et al. "Youth is a risk factor for stress fracture. A study of 783 infantry recruits." Bone & Joint Journal 76.1 (1994): 20 22. 27. MathWorks. "Creating a Stewart Platform Model Using SimMechanics." Technical Articles and Newsletters N.p., n.d. Web 28. Phidgets. "Phidgets Inc. 3140_0 S Type Load Cell (0 500kg) CZL301." Phidgets N.p., n.d. Web.

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45 APPENDIX A Matlab Script % This M file creates the geometry and dynamic information for the Stewart % platform in the home configuration. This Stewart platform consists of a % top plate, a bottom plate, and six legs connecting the top plate to the % bottom plate. The overall system has six degrees of freedom. Each leg has % six degrees of freedom and is c omposed of two bodies, two u joints, and % one cylindrical joint. Controller data is also loaded into the workspace. % % Copyright The MathWorks, Inc. deg2rad = pi/180; x_axis = [1 0 0]; y_axis = [0 1 0]; z_axis = [0 0 1]; % Connection points on base and top plate w.r.t. World frame at the center % of the base plate pos_base = []; pos_top = []; alpha_b = 10*deg2rad; % + offset angle from 120 degree spacing on base alpha_t = 10*deg2rad; % + offset angle from 120 degree spacing on top he ight = 0.75; % height in home configuration radius_b = 0.1905; % base radius in meters radius_t = 0.1900; % top radius in meters for i = 1:3, % base points angle_m_b = (2*pi/3)* (i 1) alpha_b; angle_p_b = (2*pi/3)* (i 1) + alpha_b; pos_base(2*i 1 ,:) = radius_b* [cos(angle_m_b), sin(angle_m_b),

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46 0.0]; pos_base(2*i,:) = radius_b* [cos(angle_p_b), sin(angle_p_b), 0.0]; % top points % Top points are 60 degrees offset angle_m_t = (2*pi/3)* (i 1) alpha_t + 2*pi/6; angle_p_t = (2*pi/3)* (i 1) + alpha_t + 2*pi/6; pos_top(2*i 1,:) = radius_t* [cos(angle_m_t), sin(angle_m_t), height]; pos_top(2*i,:) = radius_t* [cos(angle_p_t), sin(angle_p_t), height]; end % permute pos_top points so that legs are end points of base and top points pos_top = [ pos_top(6,:); pos_top(1:5,:)]; %6th point on top connects to 1st on bottom % Compute points w.r.t. to the body frame in a 3x6 matrix body_pts = pos_top' height*[zeros(2,6);ones(1,6)]; % leg vectors legs = pos_top pos_base; leg_length = [ ]; leg_vectors = [ ]; for i = 1:6, leg_length(i) = norm(legs(i,:)); leg_vectors(i,:) = legs(i,:) / leg_length(i); end % Calculate revolute and cylindrical axes for i = 1:6, rev1(i,:) = cross(leg_vectors(i,:), z_axis); rev1(i,:) = rev1(i,:) / norm(rev1(i,:)); rev2(i,:) = cross(rev1(i,:), leg_vectors(i,:)); rev2(i,:) = rev2(i,:) / norm(rev2(i,:)); cyl1(i,:) = leg_vectors(i,:); rev3(i,:) = rev1(i,:); rev4(i,:) = rev2(i,:); end % Coordinate systems

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47 lower_leg = struct( 'origin' [0 0 0], 'rotation' eye(3 ), 'end_point' [0 0 0]); upper_leg = struct( 'origin' [0 0 0], 'rotation' eye(3), 'end_point' [0 0 0]); for i = 1:6, lower_leg(i).origin = pos_base(i,:) + (3/8)*legs(i,:); lower_leg(i).end_point = pos_base(i,:) + (3/4)*legs(i,:); lower_leg(i).ro tation = [rev1(i,:)', rev2(i,:)', cyl1(i,:)']; upper_leg(i).origin = pos_base(i,:) + (1 3/8)*legs(i,:); upper_leg(i).end_point = pos_base(i,:) + (1/4)*legs(i,:); upper_leg(i).rotation = [rev1(i,:)', rev2(i,:)', cyl1(i,:)']; end % Inertia and mass c alculation top_thickness = 0.0127; % 1/2 inch base_thickness = 0.0127; inner_radius = 0.020; % based on LinMot actuator (estimate) in meters outer_radius = 0.037; density = 2800; % aluminum density in kg/m^3 density2 = 0.01; adjusteddensity = 83328.20; %a t0degrees %leg inertia and mass [lower_leg_mass, lower_leg_inertia] = inertiaCylinder(density2, ... 0.75*leg_length(1),outer_radius, inner_radius); [upper_leg_mass, upper_leg_inertia] = inertiaCylinder(density, ... 0.75*leg_le ngth(1),inner_radius, 0); % top and base plate mass and inertia [top_mass, top_inertia] = inertiaCylinder(adjusteddensity, ... top_thickness, radius_t, 0); [base_mass, base_inertia] = inertiaCylinder(density, ... base_thickness ,radius_b, 0); % PID controller gains % Kp = 2e5; Ki = 1e3; Kd = 4.5e3; % default values Kp = 100000; Ki = 500; Kd = 2250;

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48 Appendix B Solid Works Drawings of Components Top plate:

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49 Bottom plate:

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50 Rod 1:

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51 Rod 2:

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52 Cylinder: Cylinder joint coupler :

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53

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54 Ball socket Joint:

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55 Appendix C Data Collection VI Block Diagram

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56 Appendix D Modified Data Collection VI

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57 A P P E N D I X E M a t l a b S c r i p t f o r S t a t i c s A n a l y s i s clear all clc fo = 28.5; % force plate+block fe = 525; % external force applied phi = [ 25, 15, 10, 0, 10, 15, 25]; % top plate angle a1 = [3.12 3.64 3.8 3.94 4.07 4.4 5.49]; % angle 1 from Solidworks drawing a2 = [ 7.2 6.5 6.3 6.2 5.63 5.15 3.87]; % angle 2 from SWs a3 = [1.01 2.68 3.28 3.94 4.01 4.1 4.39]; % angle 3 from Sws a = 2; % Distance between middle and left pin on top plate b = 2; % Distance between middle and right pin on top plate % theta c alculations based on unit circle and a1 thru a3 theta1 = pi/2 a1*pi/180; theta2 = pi/2 a2*pi/180; theta3 = pi/2 a3*pi/180; % force in y direction; f1y=f2y=f3y for i = 1:length(phi) fy(i) = fo fe*sin((90 phi(i))*pi/180); % Force is dirested fx(i) = fe*cos((90 phi(i))*pi/180); t1 = theta1(i); t2 = theta2(i); t3 = theta3(i); alpha = sin(t1)*(1+a/b); beta = a*cos(t3)*sin(t1)/(b*sin(t3)); f2(i) = (fx(i) fy(i)*(cos(t1)/alpha + beta/alpha)) / (cos(t2) sin(t2)*cos(t1)/alpha

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58 beta/alpha*sin(t2)); f1(i) = (fy(i) f2(i)*sin(t2)) / (sin(t1)*(1+a/b)); f3(i) = (f1(i)*sin(t1)*a)/(sin(t3)*b); end fx1 = f1 .* cos(theta1); fx2 = f2 .* cos(theta2); fx3 = f3 .* cos(theta3); test_fx = fx1 + fx2 + fx3; fy1 = f1 .* sin(thet a1); fy2 = f2 .* sin(theta2); fy3 = f3 .* sin(theta3); test_fy = fy1 + fy2 + fy3;