THE REDESIGN OF A RECUMBENT TRICYCLE USING A CRANK ROCKER

MECHANISM TO INCREASE POWER THROUGHPUT IN FES CYCLING

Thesis

Submitted to

The School of Engineering of the

UNIVERSITY OF DAYTON

In Partial Fulfillment of the Requirements for

The Degree of

Master of Science in Mechanical Engineering

By

Anthony Lee Bazler II

Dayton, Ohio

December 2020

THE REDESIGN OF A RECUMBENT TRICYCLE USING A CRANK ROCKER

MECHANISM TO INCREASE POWER THROUGHPUT IN FES CYCLING

Name: Bazler, Anthony Lee, II APPROVED BY:

Andrew Murray, Ph.D David Myszka, Ph.D, M.B.A, P.E Advisory Committee Chairman Committee Member Professor Professor Mechanical and Aerospace Engineering Mechanical and Aerospace Engineering

Reissman, Timothy, Ph.D Committee Member Associate Professor Mechanical and Aerospace Engineering

Robert J. Wilkens, Ph.D., P.E. Eddy M. Rojas, Ph.D., M.A., P.E. Associate Dean for Research and Innovation Dean, School of Engineering Professor School of Engineering

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ABSTRACT THE REDESIGN OF A RECUMBENT TRICYCLE USING A CRANK ROCKER

MECHANISM TO INCREASE POWER THROUGHPUT IN FES CYCLING

Name: Bazler, Anthony Lee, II University of Dayton

Advisors: Dr. Andrew Murray, Dr. David Myszka

This thesis presents an investigation of alternative mechanisms to improve the power throughput of persons with tetra- or paraplegia pedaling via functional electrical stimulation (FES). FES stimulates muscle contraction with small electrical currents and has proven useful in building muscle in patients while relieving soreness and promoting cardiovascular health. An FES-stimulated cyclist produces power that is an order of magnitude less than an able-bodied cyclist. At these reduced power levels, many difficulties associated with FES cycling become apparent, namely inactive zones. Inactive zones are defined by the leg being in a position where muscle stimulation is unable to produce power to propel the tricycle forward. A possibility for reducing inactive zones and increasing the power throughput of the cyclist is to alter the motion of a cyclist’s legs.

Bicycles have recently been marketed that feature pedaling mechanisms that employ alternate leg motions. This work considers using four-bar and ratchet-and-pawl linkages in the redesign of a performance tricycle piloted by an FES-stimulated rider. Quasi-static and power models have been optimized for this cycling architecture yielding design that suggest a 79% increase in throughput power for some FES cyclist. Multiple designs were compared against design criteria to identify an ideal design.

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ACKNOWLEDGEMENTS

I must begin by thanking my advisors and mentors, Dr. Andrew Murray, Dr. David

Myszka, and Dr. Timothy Reissman. Nearly three and a half years ago, Dr. Murray and Dr.

Myszka introduced me to field of machine design and I have been captivated ever since.

Their enthusiasm, patience, and desire to teach both in the classroom and in the lab setting are unparalleled. The opportunities that they offered and challenges they placed before me that have led to the success of this thesis and has greatly influenced my academic career.

This project would not have been available to me nor would have been successful if it were not for our friends at the University of Montpellier. Dr. Christen Azevedo Coste and her team were incredibly supportive throughout this research effort. Their expertise in biomechanics and FES were influential in how the project was approached and carried out.

I am further grateful for the DIMLab and my fellow lab members. Nick Lanese, Bennett

Synder, and many others have been supportive of my efforts toward achieving this goal.

The lab provides an open environment to put your skills to the test on a variety of projects from tensegrity structures to advanced modeling techniques within Solidworks. I cannot express how important the exposure to such a wide variety of projects has been to my young career. The friendships I have developed while a member of the lab are bound to last for decades to come.

I am extremely grateful to my parents for their love, prayers, and caring sacrifices that have made this opportunity of a graduate degree possible to me. I owe it to my brother,

Aaron, for grinding out long nights of studying along my side in Kettering Lab. Lastly, thank you to all my friends for maintaining an open ear as I rambled on about my projects.

In some small way, the work we have completed may very well improve other’s lives.

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TABLE OF CONTENTS

ABSTRACT ...... iii

ACKNOWLEDGMENTS ...... iv

LIST OF FIGURES ...... ix

LIST OF TABLES ...... x

LIST OF ABBREVIATIONS AND NOTATIONS ...... xi

CHAPTER I INTRODUCTION ...... 1

1.1. Motivation ...... 1

1.2. Neuromuscular Electrical Stimulation Therapy ...... 2

1.3. Functional Electrical Stimulated Cycling ...... 4

1.4. Innovative Cycling Claims ...... 8

1.5. Contribution ...... 11

CHAPTER II TRADITIONAL RECUMBENT TRIKE MODEL ...... 12

2.1. Modeling Overview ...... 12

2.2. TRT Model Overview ...... 13

2.3. TRT Model Inputs ...... 15

2.4. TRT Kinematic Analysis ...... 17

2.5. TRT Quasi Static Analysis ...... 22

2.6. TRT Power Analysis ...... 25

2.7. Tuning Cyclist Dimensions ...... 28

CHAPTER III JOINT RELATIONSHIPS ...... 30

3.1. Joint Relationship Introduction ...... 30

3.2. Torque as a Function of Joint Angle and Joint Motion ...... 31

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3.3. Joint Constraints ...... 34

3.4. Torque Discontinuities ...... 35

CHAPTER IV CRANK ROCKER TRIKE MODEL ...... 36

4.1. CRT Model Overview ...... 36

4.2. CRT Model Inputs ...... 37

4.3. CRT Dimensional Constraints ...... 39

4.4. CRT Kinematic Analysis ...... 41

4.5. CRT Quasi Statics Analysis...... 45

4.6. CRT Power Analysis ...... 48

CHAPTER V CRT OPTIMIZATION AND RESULTS ...... 49

5.1. Optimization Overview ...... 49

5.2. Optimization Results ...... 51

5.3. Design Evaluation ...... 54

CHAPTER VI DISCUSSION AND CONCLUSION ...... 58

6.1. TRT Model Review ...... 58

6.2. Final Design Selection ...... 59

6.3. Explanation of Results ...... 60

6.4. Future Work ...... 61

6.5. Conclusion ...... 61

BIBLIOGRAPHY ...... 63

APPENDICES

A. Optimization Results ...... 66

B. Optimized Design Torque Curves ...... 68

vi

C. Optimized Design Power Curves ...... 75

D. Optimized Design Joint Motion ...... 80

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LIST OF FIGURES

1.1. Traditional Recumbent Trike ...... 1

1.2. FES Cycling Stimulation Patterns ...... 6

1.3. Power for the P1P2 and P1P3 as a Function of Crank Angle ...... 7

1.4. Conventional Recumbent Trike Drivetrain ...... 9

1.5. Alternative Trike Designs ...... 10

2.1. Model Information Flow ...... 13

2.2. TRT Kinematic Sketch ...... 14

2.3. Joint Torque as a Function of Crank Angle ...... 16

2.4. TRT Configuration...... 17

2.5. TRT Vector Diagram ...... 18

2.6. TRT Kinematic Results...... 21

2.7. Free Body Diagrams for the TRT Model ...... 23

2.8. Resultant Torque for the P1P2 Group per Cycle of Crank ...... 24

2.9. Resultant Torque for the P1P3 Group per Cycle of Crank ...... 25

2.10. Power Matching for the P1P2 Group ...... 27

2.11. Power Matching for the P1P3 Group ...... 27

3.1. Example Joint Motion Trace Comparison ...... 31

3.2. P1P2 – Joint Torque as a Function of Joint Angle...... 32

3.3. P1P3 – Joint Torque as a Function of Joint Angle...... 33

3.4. Torque Discontinuity ...... 35

4.1. Leg Press Motion Isolates Quadriceps ...... 37

4.2. CRT Kinematic Sketch ...... 38

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4.3. CRT Vector Diagram ...... 39

4.4. Design Space Constraints ...... 40

4.5. Free Body Diagrams for the CRT Model ...... 46

5.1. Optimization Flow Diagram ...... 50

5.2. Scatter Plot of Optimization Results ...... 52

5.3. Histogram of Optimization Results ...... 52

5.4. Joint Torque Curve Continuity Comparison ...... 54

6.1. Final CRT Design ...... 59

6.2. Final CRT Design Joint Range ...... 60

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LIST OF TABLES

2.1. Tuned Cyclist Dimensions ...... 29

3.1. Kinematic Joint Constraints ...... 34

4.1. Design Space Constraints ...... 40

5.1. Optimization Results ...... 53

5.2. Cross Comparison of Power ...... 57

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LIST OF ABBREVIATION AND NOTATIONS

AB Able-bodied

BDC Bottom Dead Center

CRT Crank Rocker Trike

ESTIM Electrical Stimulation

FES Functional Electrical Stimulation

MN Motor Neurons

NMES Neuromuscular Electrical Stimulation

SCI Spinal Cord Injured

TDC Top Dead Center

TRT Traditional Recumbent Trike

푙푖 Length of a planar position vector

휃푖 Angle of position vector with respect to the positive horizontal x-axis

∅푗표푖푛푡 Relative angle of a joint

푥푖 Horizontal component of position vector

푦푖 Vertical component of position vector

푅⃗ 푖 Planar position vector composed of (푙푖, 휃푖) and/or (푥푖, 푦푖)

푉⃗ 푖 Derivative of the position vector

퐴 푖 Second derivative of the position vector

Ξ Gear Reduction

푀푖 Moment applied to a linkage

푃푖 Instantaneous power at a particular joint

xi

휇푐푟푎푛푘 Average power calculated at the crank center

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CHAPTER 1

INTRODUCTION

1.1 Motivation

Functional electrical stimulated (FES) cycling has proven to be a successful means of exercise and rehabilitation for patients with Spinal Cord Injuries (SCI) and other neurological impairments of the lower body. FES cycling, however, is limited to ideal track-like conditions or stationary trikes due to the extremely low power generation of the trike-rider system. Current FES trikes use a traditional bicycle/tricycle (trike) drivetrain that was designed around power delivered from an able-bodied (AB) cyclist using voluntary movement of the lower limbs. This thesis investigates an alternative drivetrain design in an attempt to increase the power throughput of the traditional recumbent trike

(TRT) and rider system seen in Fig 1.1. This introduction reviews the motivation and terminology associated with FES Cycling and establishes the need for investigating an alternative trike design.

Figure 1.1: Traditional Recumbent Trike. Trike used by the 2016 FreeWheels Team at the Cybathlon competition [16].

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1.2 Neuromuscular Electrical Stimulation Therapy

Neuromuscular electrical stimulation (NMES) is the application of electrical current through muscle fiber to induce muscle contraction [5]. NMES, more commonly referred to as electrical stimulation (ESTIM), has provided a method of chronic pain relief, exercise, and rehabilitation for centuries. Egyptians, as early as 2750 BCE, were believed to be the first to use ESTIM as a form of pain relief [2]. Early uses note Benjamin Franklin and other noteworthy scientists began experimenting more prolifically with the effects of electrical current on the body in the 18th century. These studies even inspired authors to write on the subject, notably Mary Shelly who wrote Frankenstein in 1816 [1,2]. A new wave of experiments and studies began in the mid to late-20th century when scientists and doctors began investigating the restorative nature an electrical shock can have on ineffective muscle groups.

That new application of ESTIM was labeled FES and is now used to generate muscle contractions that perform a task instead of providing pain relief or other strictly therapeutic actions [3,7]. FES technologies have given people suffering from paralysis (due to SCI), strokes, and other physically disabling conditions the ability to stand, grasp objects, and exercise using their own muscle groups. FES technologies is a commonplace technology used at many hospitals and therapy office. FES technologies have been the spotlight of many biomechanical events, namely the Cybathlon held in Switzerland. The Cybathlon competition, held every four years, is an event very similar to the special Olympics with an emphasis on assistive technologies.

Voluntary muscle contractions of AB cyclists contrast greatly with the involuntary contractions induced by ESTIM. Therefore, understanding the biomechanics behind this

2 phenomenon provides key insights into redesigning the pedaling mechanism on a TRT.

Muscular contraction occurs when electrical current passes through motor neurons (MN) in muscular fiber. The current depolarizes the MN thus inducing muscular contraction. A significant difference in contraction force and response time has been observed between abled body (AB) and SCI ESTIM. This has been credited to the difference in the MN between AB and SCI participants. The cross-sectional diameter of the MN in healthy, innervated muscle tissue is typically much greater than that of non-innervated muscle tissue. The reduced size of MN in non-innervated muscle tissue inherently requires an increased “charge-up” time to induce contraction in the surrounding muscular fiber. This delayed response in contraction is not seen in AB cyclists. Thus, a TRT may not provide the most opportune cycling motion for SCI riders.

Electrical stimulation can be applied to a patient in two ways: 1) electrodes implanted into the patient’s muscle tissue or 2) electrodes secured to the skin above the desired muscle of actuation known as transcutaneous electrodes [12]. Implanted electrodes come in a variety of forms. Some electrical leads ran through limbs to a central control in the trunk, others are leadless and are controlled by devices held to the skin just above the implant.

Implants are able to stimulate both surface and deep muscle tissue. Transcutaneous electrodes are electrodes attached to the skin and pass current through the skin to another electrode on the opposite end to the muscle fiber. All aspects of this device are external to the patient’s skin and must be applied each time the device is to be used. Transcutaneous electrodes do not effectively stimulate deep muscle tissue. This study used joint moment data induced exclusively by transcutaneous electrodes. One observed problem is that

3 stimulation patterns and electrode placement may also affect the magnitude of muscular contraction force [14].

There are many health benefits associated with FES, especially for those with complete

SCI, but also for those with incomplete SCI. These benefits tend to be divided into three primary categories: functional gains, physical health improvement, and psychological health improvements [7,8]. Many functional gains have been achieved with FES technologies including regaining the ability to stand, grasp items, perform assisted walking.

There are many health benefits including improved circulation, reduced risk of osteoporosis, increased bone density, less postural problems, stronger muscles, and prevention of muscle atrophy [7]. Due to functional gains and improved health, patient psyche can vastly improve. The ability to stand and address others at eye level, put on clothes, and cleanse themselves drastically improves the quality of life for SCI individuals

[8].

1.3 Functional Electrical Stimulated Cycling

Stationary FES cycling became prominent in the late 1980s and was used as a means of exercise [11]. Since the early 2000s, scientist began experimenting with FES cycling using recumbent trikes (Fig. 1.1).

FES cycling builds upon the health benefits seen in traditional functional electrical stimulation. Stimulation of the legs coupled with the resistive torque at the crank center has been found to restore bone density to a subject’s lower extremities [10]. Increased mobility, lower leg muscular strength, cardiovascular health, and reduction of pressure ulcers have all been observed after training with FES cycling routines over the course of a

4 year [11]. Just like with traditional FES therapy, FES cycling has drastically improved subject mental health and quality of life.

The involuntary muscle contractions associated with FES cycling present biomechanical challenges to engineers and scientists. First, FES cannot reliably stimulate the lower leg to acuate the ankle, therefore SCI participants must constrain their ankle in a boot. Second, transcutaneous electrodes are not capable of precisely stimulating one specific muscle as AB rider are through voluntary muscle contraction. Instead the electrodes stimulate an entire muscle group. The quadriceps and hamstrings are bi-joint muscle groups, meaning that when contracted a moment is produced at the hip and knee simultaneously. AB riders have precise control of each joint by voluntarily contracting specific muscles within these groups to achieve a desired motion. SCI FES subjects do not have this precise control. When the quadriceps are stimulated the moments at the hip and knee will move the hip into flexion and the knee into extension. Inversely, stimulation of the hamstring will result in hip extension and knee flexion. This limited control becomes conflicting when pedaling on a traditional crank because portions of the cycle require the leg to be in both hip extension and knee extension or hip flexion and knee flexion simultaneously. Therefore, power generated at the agonist joint must divert some power to overcome the antagonist joint power and send the remaining to the driving wheel to move the trike forward. Many studies have been conducted to finely tune muscle stimulation as a function of crank angle to produce the greatest power output at the driving wheel despite conflicting muscle stimulation. This has led to portions of the cycle where solely quadriceps or hamstrings are stimulated, where both quadricep and hamstrings are stimulated, or where neither muscle group is stimulated. This portion of the cycle where

5 neither muscle group is stimulated is referred to as inactive zones and may consume 1/3 of the crank cycle as seen in Fig. 1.2.

Muscle atrophy cause significant variations in FES SCI cycling. The stimulation patterns and power cycles of SCI FES cycling have been the subject of several reports.

Experimental findings claim that there are primarily two types of SCI FES cyclist power cycles that are relatable to the four power peaks seen through the duration of AB cycling.

The first group, which make up 75% in the Szecsi study, primarily produced power to the driving wheel through the first two peaks found in AB cycling and is referred to as the

P1P2 group [14]. The second group produced the most power in the first and third peaks found in AB cycling and is referred to as the P1P3 group. Each group was evaluated separately during this study. Figures 1.3 show the power cycle as a function of crank angle for groups P1P2 and P1P3.

Figure 1.2: FES cycling stimulation patterns for the quadriceps and hamstring used by the 2016 FreeWheels team at the Cybathlon competition [16].

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(a)

(b) Figure 1.3: Power for the P1P2 and P1P3 as a function of crank angle. Plot (a) displays the average power produced by the P1P2 group. Whereas plot (b) displays the power produce by the P1P3 group. KEY: The bold continuous line represent the power measured at the crank center. The bold dashed line represents the power produced by the FES cyclist at the knee. The bold dash-dotted line represents the power produced by the FES cyclist at the hip.

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The first major FES cycling competition occurred at the Cybathlon summit in 2016 in

Switzerland. Eleven teams representing ten countries brought customized recumbent trikes and FES technologies to compete and share their progress in advancing FES cycling. There are two distinct mechanical challenges, observed at the summit, that face FES cycling: 1) extremely low power throughput from rider to driving wheel, and 2) discontinuous and choppy pedaling motion.

FES Cycling produces an average of 30 watts per cycle of the crank. This is an order of magnitude less than that of AB cyclist. This significant power reduction by the FES cyclist is likely a result of 1) atrophied muscles, 2) agonist and antagonist muscle contractions, and 3) inactive zones. The combination of all three lead to this drastic reduction of power output limiting SCI FES cyclists to cycle solely on tracks or other experimental setups. Muscular contraction force is so low that SCI FES cyclists must begin on a ramp to produce linear momentum of the trike necessary to put the legs in motion to pedal properly. To reduce the choppiness of the on/off effect of FES contractions, a fixed drive is required to convert the linear momentum of the trike to angular momentum of the crank to keep the pedals moving through TDC, BDC, and inactive zones[16].

1.4 Innovative Cycling Claims

The drive train for the bicycle remains relatively unchanged since being invented by

John Kemp Starley in 1885 [17]. The pedals, driven solely by the cyclist’s legs, rotate on crank ends which travel 360 degrees about their center, driving a front sprocket rigidly connected to the cranks. This front sprocket drives the rear sprocket, known as the cassette, and wheel via a chain and in turns moves the bike forward. A traditional recumbent trike

(TRC) maintains these characteristics and is shown in Fig. 1.4.

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Figure 1.4: Conventional Recumbent Trike Drivetrain (adapted from [21])

Several variations of drivetrains have been proposed but all remain relatively obscure as compared to the traditional bicycle drivetrain despite boasting advantages like improved power throughput for AB cyclists. One such alternative is Nubike (Fig. 1.5a) which claimed an increased power output via improved mechanical advantage and an altered leg motion [18]. A coupler drive has also been proposed by the Gföhler (Fig. 1.5b), that changes the circular motion of the crank into on that is more ovate [19].

The Berkle bike (Fig. 1.5c) is yet another alternative and has been considered for SCI

FES use. Instead of powering the trike solely with the involuntary FES stimulation of legs, the drive train also incorporates cranks that are driven by voluntary moments of the arms.

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(a)

(b)

(c) Figure 1.5: Alternative Trike Designs. Photo (a) is adapted from the Nubike’s marketing campaign and highlights a linear motion of the pedal. Sketch (b) is adapted from [19] and shows a ovate pedaling motion. Photo (c) is adapted from the Berkle Bike’s marketing campaign and shows a combination of pedaling from the upper and

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1.5 Contribution

There are many challenges that FES cycling must overcome in order to provide SCI cyclists a reliable method of exercises and a means of increasing self-worth. This thesis pursues an alternative TRT designs in an effort to increase the power throughput of SCI

FES cyclists to overcome the three leading challenges of FES: 1) atrophied muscles, 2) agonist and antagonist muscle contractions, and 3) inactive zones.

This thesis accomplishes this goal in three steps. First, a model average power per cycle of the crank of the TRT was necessary to establish a baseline. Second, the TRT model was used to derive necessary inputs to support alternative designs. Third, a model of the alternative design was constructed and optimized for the greatest average power throughput for one cycle of the crank.

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CHAPTER 2

TRADITIONAL RECUMBENT TRIKE MODEL

2.1 Modeling Overview

The goal of this work is to consider alternative architectures to the traditional recumbent trike drivetrain to enable persons with SCI the opportunity to use FES cycling as a means of exercise and rehabilitation. To facilitate this design work, a power model was developed for both the traditional recumbent trike (TRT) and an alternative design called the Crank Rocker Trike (CRT). A model of the TRT served as the baseline design when comparing the effectiveness of alternative trike designs. The effectiveness of all designs was assessed by the power production at the crank center averaged over one cycle of the crank provided each trike was modeled in similar environments (i.e. similar velocity, wheel diameter, etc.).

As the TRT model is the source of most experimentation and data, the TRT was used to derive data useful in modeling the CRT model. The derived data includes joint constraints and the joint torque relationship. An information flow diagram of the two models is shown in Figure 2.1.

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Experimental Experimental Trike/Cyclist Joint Torque Initial Guess: Crank Rocker Dimensions Data Trike Parameters

TRT Model Derived Joint CRT Model Optimize Kinematics Kinematic Kinematics Constraints Parameters

Quasi-Statics Quasi-Statics Derived Joint Power Calculations Angle to Joint Power Calculations Torque Relationship

Average Power per Optimum Average Power Crank Cycle Baseline per Crank Cycle

Figure 2.1: TRT and CRT Model Information Flow

2.2 TRT Model Overview

The interaction of a single leg with the TRT drive-chain is modeled by a crank-rocker mechanism. The upper leg serves as the rocking input link, lower leg, and foot (constrained by a boot) serve as the coupler, and the cycle crank is the crank as labeled in Fig. 2.2.

Driving torques are applied at the hip and knee joints. A torque applied at the crank center creates static equilibrium in the mechanism. By combining this applied torque with the angular velocity of the crank, the power at the crank center is found.

The literature provides two data sets for the TRT: joint torque as a function of crank angle, and the physical parameters of the recumbent trike and cyclist (discussed in detail in Section 2.3). Note that these data sets are not conducive to modeling alternative designs.

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The primary objective in those studies was to determine the muscle groups in the leg that contribute the greatest amount of power throughout one cycle of the crank. Similarly, the

TRT model primary goal was to calculate a baseline power output at the crank center.

Secondary objectives include developing the inputs for the CRT model. These included relating joint angle to joint torque and establishing kinematic constraints for the CRT optimization. The information flow in Fig. 2.1 depicts these inputs and outputs.

Although biomechanics has its own standards, to align the diagram more closely to traditional machine design theory, all angles were measured from the positive x axis (Fig.

2.2). Therefore, the crank angle is assigned to be zero when pointed in the direction of the positive x-axis. This contradicts angle measurements which placed the crank angle at zero when the leg reached TDC [14,15]. In addition to reassigning the crank angle, the background image in Figure 2.2 has been horizontally mirrored to simplify the alignment with machine design theory. Lastly it should be acknowledged that power cannot be extracted from the ankle joint, therefore the ankle is constrained in a boot holding the foot perpendicular to the tibia. This image was taken directly from Szecsi’s report noting it as the reflected image and that labels are not utilized here in.

Rocking Coupler Input Link

Crank y- axis x- axis Figure 2.2: TRT Kinematic Sketch

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2.3 TRT Model Inputs

The traditional recumbent trike model requires two inputs (Fig. 2.1). Few studies have reported joint moments for SCI cyclists while cycling. Therefore, this design work uses a single source for obtaining the data [14,15].

Experimental Joint Torque Data

The joint torque data obtained for this study was derived from joint moments published by Szecsi [14]. Using force plates at the pedals and recording the leg’s kinematics, inverse dynamic calculations produced joint moments for FES cyclists. After analyzing the power peaks of 16 SCI subjects, two groups of FES cyclist emerged, labeled P1P2 and P1P3, corresponding to the portion of the cycle when the cyclist was most powerful.

Not having access to those raw data sets, curve matching techniques are used in this work to match the data and to achieve a joint torque to crank angle relationship necessary for the model. Between 20 and 30 points were selected from Szecsi’s curves. Using the spline interpolation tool in MATLAB 2020b, the blue traces are produced, providing joint torque as a function of crank angle. The original curves from the Szecsi study are overlayed with the curves used in this study in Figure 2.3. These plots also highlight the points selected to produce the splines. The maximum observed error between the reported data

푁 and the approximation is roughly 3[ ⁄푚] and is circled in red. The effect of renaming the crank angle to align with the positive x-axis is seen here as well.

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Figure 2.3: Curve Matching Joint Torque as a Function of Crank Angle. Blue circles represent data points selected to use as spline interpolation points to produce the blue curve. The blue curve overlays black line representing the data Szecsi reported [14]. The red circle show the greatest discrepancy in the interpolated curve match and the reported data.

Experimental Recumbent Trike Dimensions

To maintain continuity with the joint moment data, the dimensions for the traditional recumbent trike model were taken from the trike configuration used in Szecsi’s study.

Szecsi performed two concurrent studies of joint moments, one for AB cyclist and one for SCI cyclists [15]. Figure 2.4 depicts the trike used for the AB. Although not explicitly stated, the fact that this trike was also used for the SCI cyclists is assumed. This photo was used to derive dimensions using the length specified in the report of 15 [푐푚] long

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[14,15] for a pedal length. Placing the origin of the model at the hip center, the crank center was found to be (-0.7357, -0.3036) [m]. The dimensions for the cyclist’s upper leg, lower leg and foot were initially scaled from this figure. In an effort to match the power curves (Fig. 1.3), the cyclist dimensions were altered. This is further examined in section

2.7. The cyclist’s torso was measured to be inclined at an angle of 60° from the positive x-axis.

Figure 2.4: The TRT configuration used by Szecsi to collect data for calculating the joint moments of AB cyclists [15]

2.4 TRT Kinematic Analysis

The vector diagram for modeling the rider/trike interaction is seen in Figure 5. The crank-rocker mechanism has a single degree of freedom. As the output power is desired at the crank, the crank angle is selected as the single input to this model. The position, velocity, and acceleration analysis are solved at each degree angle of the crank provided that the cyclist pedals at a constant angular velocity (cadence) of 60 RPM. A cadence of

60 RPM is consistent with Szecsi’s experimental work [14] despite current work suggesting that a cadence of 50 RPM better supports SCI FES cyclists [16]. Assuming a

17 gear reduction of 1.31 between the driven sprocket and the driving wheel, a linear velocity of the TRT was calculated (using a wheel diameter of 26 inches, 0.66 meters) to be 5.71 km/h, roughly the average speed of cyclist at the Cybathlon in 2016. Identifying the knee and hip angle is necessary as they serve as constraints for the alternate design.

휃3 ∅ℎ푖푝

푅⃗ y 3 푅⃗ 4 ∅푘푛푒푒

휃4 푅⃗ 5 x

푅⃗ 6 푅⃗ ⃗ 1 푅2 휃2

Figure 2.5: TRT Vector Diagram

Position Analysis

Vectors are denoted in either their polar or cartesian form as seen in Eq 2.1.

푥푖 cos 휃푖 푅⃗ 푖 = [ ] = 푙푖 [ ] (2.1) 푦푖 sin 휃푖

The following loop closure equations are observed from the vector diagram:

푅⃗ 1 + 푅⃗ 2 − 푅⃗ 3 − 푅⃗ 4 = ⃗0 , (2.2a)

푅⃗ 3 − 푅⃗ 5 − 푅⃗ 6 = ⃗0 . (2.2b)

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Where 푙3 is

2 2 푙3 = √(푙5) + (푙6) (2.3)

and 휃3 and 휃4 are

2 2 2 2 2 퐴 = 푥1 + 푦1 + 푙2 + 푙3 + 2 ∙ 푥1 ∙ 푙2 cos 휃2 + 2 ∙ 푦1 ∙ 푙2 sin 휃2 − 푙4, (2.4a)

퐵 = −2 ∙ 푥1 ∙ 푙3 − 2 ∙ 푙2 ∙ 푙3 ∙ cos 휃2 , (2.4b)

퐶 = −2 ∙ 푦1 ∙ 푙3 − 2 ∙ 푙2 ∙ 푙3 ∙ sin 휃2 , (2.4c)

−1 퐶 −1 −퐴 휃3 = tan ( ) − cos ( ) , (2.4d) 퐵 √퐵2 + 퐶2

푥4 = 푥1 + 푙2 cos 휃2 + 푙3 cos 휃3 , (2.4e)

푦4 = 푦1 + 푙2 sin 휃2 + 푙3 sin 휃3 , (2.4f)

−1 푦4 휃4 = tan ( ) . (2.4g) 푥4

The joint angles are

⃗ −1 |푅6| ∅퐾푛푒푒 = 휃4 − 휃3 − tan ( ) − 180, (2.5) |푅⃗ 5|

∅퐻푖푝 = 휃4 − 훿푡표푟푠표. (2.6)

As seen in Figure 2.5, all angles are measured in reference to the positive x-axis.

Equations 2.1-2.6 may be solved for any angle of 휃2. In this work these equations are solved at 1° increments from 1° to 360°.

Velocity Analysis

A complete position analysis provides data necessary calculate a velocity analysis provided that the angular velocity is known at any instant. The derivative of the position analysis, Eq. 2.2, yields

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푉⃗ 1 + 푉⃗ 2 − 푉⃗ 3 − 푉⃗ 4 = ⃗0 , (2.7a)

푉⃗ 3 − 푉⃗ 5 − 푉⃗ 6 = ⃗0 . (2.7b)

When expanded, Eq. 2.7 becomes

0 − sin 휃2 − sin 휃3 − sin 휃4 0 { } + 푙2 ∙ 휃̇2 ∙ [ ] − 푙3 ∙ 휃̇3 ∙ [ ] − 푙4 ∙ 휃4̇ ∙ [ ] = { } , (2.8a) 0 cos 휃2 cos 휃3 cos 휃4 0

휃̇3 = 휃̇5 = 휃̇6. (2.8b) where 휃3̇ and 휃4̇ must be solved using the set of linear equation. The angular velocity of the knee and hip joints are provided as

̇ ∅푘푛푒푒 = 휃4̇ − 휃̇3, (2.9)

̇ ∅ℎ푖푝 = 휃̇4. (2.10)

Acceleration Analysis

With the complete position and velocity analysis, an acceleration analysis follows given that the angular acceleration for the crank is known. Under the assumption that the cyclist pedals at a constant rate, the angular acceleration of the crank is 0 [rads/second].

The derivative of Eq. 2.7 yields

퐴 1 + 퐴 2 − 퐴 3 − 퐴 4 = ⃗0 , (2.11a)

퐴 3 − 퐴 5 − 퐴 6 = ⃗0 . (2.11b)

When expanded, Eq. 2.11 becomes

cos 휃 − sin 휃 cos 휃 − sin 휃 푙 휃̇ 2 [ 2] − 푙 휃̈ [ 3] + 푙 휃̇ 2 [ 3] − 푙 휃̈ [ 4] 2 2 sin 휃 3 3 cos 휃 3 3 sin 휃 4 4 cos 휃 2 3 3 4 2 cos 휃4 +푙4휃4̇ [ ] = ⃗0 , (2.12a) sin 휃4

휃̈3 = 휃̈5 = 휃̈6. (2.12b) where 휃̈3 and 휃4̈ must be solved using the set of linear equation. The angular acceleration of the knee and hip joints are provided as

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̈ ∅푘푛푒푒 = 휃4̈ − 휃̈3, (2.13)

̈ ∅ℎ푖푝 = 휃̈4. (2.14)

The position, velocity, and acceleration analysis results for the hip and knee joint angles are shown in Figure 2.6. These curves identify joint movements and joint constraints discussed in more detail Chapter 3.

(a) (d)

(b) (e)

(c) (f) Figure 2.6: TRT Kinematic Results. Plots a, b, and c describe the motion of the knee joint. Plots d, e, and f describe the motion of the hip joint.

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2.5 TRT Quasi-Static Analysis

A quasi-static model was chosen to calculate the torque production at the crank by observing the low angular velocities and accelerations, which in turn, justify neglecting the dynamic effects of the leg’s mass. Thus, given the complete kinematics, a quasi-static analysis generates the statically equivalent torque given the torque produced by the cyclist at the knee and hip at any given angle of the crank.

The masses of all links was ignored when examining the statics throughout the motion.

This is deemed acceptable because the comparison parameter between the TRT and the alternative design focuses on the average throughput power per one cycle of the crank, which is directly related to energy throughput over one cycle of the crank. Therefore, over on cycle of the crank, the potential and kinetic energy, developed by the masses, would sum to zero and would have no effect on the average power throughput.

The resultant torque may be calculated at any angle of the crank provided the torque at the knee and hip. In this work the resultant crank torque is calculated for the P1P2 and

P1P3 groups separately by incriminating from 1° to 360° by one degree. The input torque for each increment was interpolated from the data shown in Fig. 2.3. The free body diagrams (FBD) are shown in Fig. 2.7.

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퐹퐵푦 퐹퐶푦

퐹퐵푥 퐹퐶푥 ⃗ 푅2 푀 퐶푟푎푛푘 푅⃗ 3 푀 퐹 퐾푛푒푒 퐹퐴푥 퐵푥

퐹퐴푦 퐹퐵 FBD 1 푦 FBD 2

퐶 퐹퐶 ⃗ 푥 푅3 푅⃗ 4 푀퐾푛푒푒 퐷 푀퐻푖푝 퐵 퐹퐶 푦 푅⃗ 4 퐹퐷푥

푅⃗ 1 푅⃗ 2 퐹퐷 퐴 FBD 3 푦

Figure 2.7: Free Body Diagrams for the TRT Model

The quasi-static model is solved by summing all forces and moments for and setting them equal to zero. The summation of forces for FBD 1 yields:

퐹퐴푥 퐹퐵푥 0 ∑퐹 = [ ] − [ ] = [ ] , (2.15a) 퐹퐴푦 퐹퐵푦 0

∑푀퐴 = 푙2 ∙ (FBxsin 휃2 − 퐹퐵푦 cos 휃2) − 푀퐶푟푎푛푘 = 0. (2.15b)

The summation of forces from FBD 2 yields:

퐹퐵푥 퐹퐶푥 0 ∑퐹 = [ ] − [ ] = [ ] , (2.16a) 퐹퐵푦 퐹퐶푦 0

∑푀퐶 = 푙3 ∙ (−FBxsin 휃3 + 퐹퐵푦 cos 휃3) − 푀퐾푛푒푒 = 0. (2.16b)

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The summation of forces from FBD 3 yields:

퐹퐷푥 퐹퐶푥 0 ∑퐹 = [ ] + [ ] = [ ] , (2.17a) 퐹퐷푦 퐹퐶푦 0

∑푀퐶 = 푙4 ∙ (−FCxsin 휃3 + 퐹퐶푦 cos 휃3) − 푀퐾푛푒푒 − 푀퐻푖푝 = 0. (2.17b)

The resultant torque at the crank is calculated by solving this set of linear equations.

Figures 2.8 and 2.9 shows the resulting torque at the crank over one cycle for both the P1P2 and P1P3 FES cyclist.

Figure 2.8: Resultant Torque for the P1P2 Group per Cycle of Crank

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Figure 2.9: Resultant Torque for the P1P3 Group per Cycle of Crank

2.6 TRT Power Analysis

The power analysis was achieved by calculating the instantaneous power at each increment of the crank angle. Instantaneous power is determined by multiplying the angular velocity of a joint by the resultant joint torque as seen in Eq. 2.18.

푃퐶푟푎푛푘 = 휃̇2 ∙ 푀퐶푟푎푛푘, (2.18a)

푃퐾푛푒푒 = 휃̇퐾푛푒푒 ∙ 푀퐾푛푒푒, (2.18b)

푃퐻푖푝 = 휃̇퐻푖푝 ∙ 푀퐻푖푝. (2.18c)

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The area under the instantaneous power curve for the crank is equivalent to the sum of the instantaneous power curve of the knee and hip, Eq. 2.19. This equivalency verifies that power in is equivalent to power out and provided validation of the calculations. The average power, 휇퐶푟푎푛푘, was calculated using MATLAB’s mean function which follows

Equation 2.20.

∑퐻퐶푟푎푛푘 − (∑퐻퐾푛푒푒 + 퐻퐻푖푝) = 0 (2.19)

∑360(퐻 ) 휇 = 푖=1 퐶푟푎푛푘 푖 (2.20) 퐶푟푎푛푘 360

Szecsi reports that each SCI FES cyclist produced 30 Watts of power per cycle of the crank. Assuming that each leg equally contributes to the power generation, the expectation is that the TRT model should report an average power of 15 Watts (given that the analysis is of a single leg). The power curves that Szecsi reports for both P1P2 and P1P3 are overlayed with the TRT model’s power curves in Figures 2.10 and 2.11 where the maximum observed error is roughly 8.5 Watts (circled in red). The TRT model was not expected to match Szecsi curves exactly, as they are an average of several cyclist each with their physical dimensions. To account for the variation in rider dimensions, the TRT was tuned to better match the power curves. This tuning is discussed in section 2.7.

Observing that the TRT model curves very nearly match the P1P2 and P1P3 curves published by Szecsi, it was concluded that the TRT model served as a valid baseline model to compare alternative designs to.

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Figure 2.10: Power Matching for the P1P2 Group. The power calculated by the TRT at the crank center is represented with the blue line. This plot overlays the power reported at the crank center by Szecsi represented by the continuous black line.

Figure 2.11: Power Matching for the P1P3 Group. Power Matching for the P1P2 Group. The power calculated by the TRT at the crank center is represented with the blue line. This plot overlays the power reported at the crank center by Szecsi represented by the continuous black line. The red circle highlights the greatest discrepancy in the data.

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2.7 Tuning Cyclist Dimensions

The length of the upper and lower leg and distance from the ankle to the pedal center has a significant effect on the average power production through one cycle of the crank.

The TRT model accepts the dimensions of a single rider, whereas the experimental joint moment data presents the average joint moments of several SCI FES cyclist, each of which had have different leg dimensions and slightly different seat settings. At the outset of this work, the dimension for the modeled cyclist were unknown and an initial guess was established by deriving dimensions from Fig. 2.4. These scaled dimensions produced power in excess of 18W per cycle, slightly greater than the targeted 15W. To obtain the ideal set of leg dimensions for the TRT modeled rider, the leg dimensions were iteratively changed to a bring the TRT modeled power curves into alignment with Szecsi’s power curves.

The distance between the ankle and pedal center would have remained constant for all riders due to the boot used to stabilize the ankle and fasten the leg to the crank. To determine the ideal dimensions for the upper and lower leg it was assumed that these lengths were equivalent [22]. Ideal dimensions were found for the P1P2 and P1P3 Groups.

A length that best represented both groups was used in the TRT model and the CRT model.

These dimensions reference Figure 2.5 and are listed in Table 2.1.

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Table 2.1: Tuned Cyclist and Trike Dimensions Pedal Ankle to Pedal Upper Leg Lower Leg Pedal Center Length Center (푙6) (푙4) (푙5) (푅⃗ 1) (푙2) Ideal Dimensions −0.7357 0.06875m 0.5148m 0.5148m { } m 0.15m for P1P2 −0.3036 Group Ideal

Dimensions −0.7357 0.06875m 0.5417m 0.5417m { } m 0.15m for P1P3 −0.3036 Group

Modeling −0.7357 0.06875m 0.52m 0.52m { } m 0.15m Dimensions −0.3036

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CHAPTER 3

JOINT RELATIONSHIPS

3.1 Joint Relationship Introduction

As seen in Figure 2.1, joint relationships serve as primary inputs to the alternative design: the CRT model. These joint relationships and constraints are necessary due to the modified kinematics of the leg in the CRT model. In the TRT model the legs followed the circular motion of the pedal crank. The fundamental hypothesis of this design is that the kinematics of a crank rocker pedaling mechanism can increase the throughput of power by positioning the leg in regions of high joint torque and using the rocker arm for increased leverage. The leg kinematics associated with a crank rocker mechanism are significantly different than that of the TRT model (Fig.3.1). Therefore, the application of joint torque as a function of crank angle is no longer suitable for the CRT model.

Joint torque is known to be a function of three inputs: muscular contraction speed, joint angle, and joint motion [23]. Ideally, a field of this empirical data would be available which is not the case for electrically stimulated muscle groups. This research separated the three inputs into two separate groups. First, the joint torque was directly related to joint angle and joint motion (joint extension and flexion). Second, the kinematics of the hip and knee joint of the crank rocker trike are constrained not to exceed that seen in the TRT model.

These constraints on the joint are assumed to account for the linear contraction of the bi- joint muscle groups in the upper leg.

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TRT Model CRT Model Knee Rocker Arm Knee Pedal Hip Center Pivot

Crank Center Hip Center Crank Center Pedal Figure 3.1: Comparison of joint motion between the TRT and CRT model

3.2 Torque as a Function of Joint Angle and Joint Motion

Szecsi’s set of published curves displayed joint moments as a function of crank angle as seen in Figure 2.3. The results from the kinematic analysis also relate the joint angle as a function of crank angle, Figure 2.6. The combination of this data yields a continuous curve of joint torque as a function of joint angle as seen in Figure 3.2 and 3.3. However, this plot cannot serve as the single input to CRT model because there is no distinction between joint extension and flexion, which corresponds to stimulation of specific muscle groups and therefore effects the joint torque.

To determine joint motion, the angular velocity of each joint was considered. If the joint angular velocity was positive (or zero), the joint motion was recorded as extension.

Inversely, if the angular was negative, the joint was considered to be in flexion. The joint motion is denoted in Figures 3.2 and 3.3 for the P1P2 and P1P3 Groups. Note that alterations to the kinematics of the TRT model will manipulate these curves.

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Figure 3.2: P1P2 - Joint Torque as a Function of Joint Angle

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Figure 3.3: P1P3 - Joint Torque as a Function of Joint Angle

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3.3 Joint Constraints

Kinematic constraints were placed on the alternative design to ensure that the model remained in the known angular joint range and therefore know data range. These constraints were applied in two ways. First the magnitude of joint angle calculated on the

TRT model was considered. When examining the knee torque as a function of knee joint angle, it can be observed that there is no data plotted when the knee joint is less than 69° or greater than 122°. Therefore, the knee kinematics of the alternative design must operate within these limits.

The angular velocity and acceleration were also considered when bounding the alternative design kinematics. Joint torque is partially a function of muscular contraction speed, therefore this study had to maintain realistic limits on angular velocities and accelerations. Realistic bounds were set by observing the minimum and maximum angular velocities and accelerations calculated in the TRT model. The constraints used to bound alternative designs’ kinematics is listed in Table 3.1. The application of these constraints are used when determining the dimensions for the CRT.

Table 3.1: Kinematic Joint Constraints Angular Angular Velocity Joint Position [Rad/s] max 122° 2.82 Knee min 69° -2.82 max 120° 1.99 Hip min 85° -1.84

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3.4 Torque Discontinuities

By establishing angular constraints, it is assumed that any value up to the limit is plausible. For example, it is possible that the maximum knee angle for the CRT model will not exceed 120°. This will cause discontinuities in the applied joint toque in the CRT model as seen in the knee joint example in Figure 3.4b. When the entire joint angle range (Fig

3.4a) is not used the model will jump from the extension portion of the joint torque curve to the flexion portion of the curve (or inversely from flexion to extension) corresponding at that particular joint angle (Fig 3.4b). This leap from a potential negative joint torque to a positive joint torque instantaneously is not expected to occur among SCI FES cyclist, however this discontinuity is an acceptable assumption for the models as there is insufficient data to develop an alternative model.

(a) (b) Figure 3.4: Torque Discontinuity. Plot (a) shows the continuous torque curve from the TRT model that uses the full range of the knee joint range. Plot (b) shows a disjointed torque curve as it does not use the full angular knee joint range.

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CHAPTER 4

Alternative Design: Crank Rocker Trike Model

4.1 CRT Model Overview

The alternative pedaling mechanism selected to improve power throughput of the FES cyclist is crank rocker depicted in Figure 3.1. This drastically simplifies the motion of the leg. The TRT requires cyclists to follow the motion of the crank and essentially trace a circle with their foot. This motion is not as trivial as it may seem. AB cyclists perform this motion with ease as they stimulate several different muscles through the course of one cycle of the crank. FES cycling is more limited. The push/pull mechanic that the CRT offers may be more suitable for FES cyclist (and potentially even the AB cyclist).

The pedal motion of the CRT is similar to the foot path of a leg press machine (Fig.

4.1). The leg press machine is used to strengthen the quadriceps, a key muscle that is readily accessible by FES. Although the CRT does not produce a linear motion of the foot, as in the leg press, it does offer a motion far closet to linear as by the semicircular trace in Figure

3.1. This mechanism, however, may not be ideal when considering antagonist joint torques

(as is the case for the majority of the pedaling motion in the TRT). As the leg moves the pedal away from the body, the knee and hip undergo joint extension as the quadricep is stimulated. The positive torque developed by contraction of the quadricep combined with the positive angular velocity associated with joint extension produces a positive power production at the knee. Some of this power is lost at the hip as the quadriceps contraction invokes a negative torque (generally low in magnitude)at the hip. The negative toque coupled with the positive angular velocity of hip extension lends to negative power

36 production at the hip joint. Despite the low adverse effects of the joint motions, it is hypothesized that the alternative motion of the leg resulting from the crank rocker mechanism will result in a greater average of throughput power calculated at the crank center.

Figure 4.1: Leg Press Motion Isolates Quadriceps (adapted from [24])

4.2 CRT Model Inputs

The CRT model requires several inputs: the physical dimensions of the rider, the dimension of the CRT mechanism, joint torque as a function of joint angle(Chapter 3 Fig), and joint angular position, velocity, and acceleration constraints.

Cyclist’s Dimensions

Three dimensions of the cyclist were considered in the TRT model: the length of the upper leg, length of the lower leg, and the distance from the pedal center to the ankle (Table

2.1). The length for the upper and lower leg (which were assumed to be equivalent [23]) used in the TRT model were also used in the CRT model. The distance from the pedal center to the ankle however was left to vary as an optimization parameter. The dimension

37 from the pedal center to ankle was not assigned to CRT model because of the nature of the boot that rigidly connects the foot to the pedal. The distance from ankle to pedal center is not a function of the rider’s foot length, rather is dependent on where the clips are located on the boot. Therefore this length was left as an optimization parameter. A sensitivity has not been completed at this time to identify the importance of the cyclist’s dimensions. It is possible that a shorter or taller cyclist would yield different optimization results. An adjustable seat may account for a variety of cyclist heights.

CRT Dimensions

A kinematic sketch and vector diagram for the crank rocker mechanism is seen in Figs.

4.2 and 4.3. Similar to the TRT, the origin of the vector diagram is placed at the hip center.

The parameters, referred to as the CRT mechanical parameters (CRT MP), required to fully define the mechanical aspect of the CRT model (excluding the cyclist’s dimensions) are 푙3, 푙4, 푙5, 푙6, 푅⃗ 7, 푅⃗ 8, 푙10, and Ξ. Ξ, discussed in detail in the CRT kinematics, represents the gear reduction from the crank to the driven wheel. These parameters leave the mechanism a single degree of freedom. The CRT model depicted in the kinematic sketch displays a random set of dimensions that satisfy a series of constraints. Note that 푅⃗ 3 and

푅⃗ 4 are colinear. Knee Rocker Arm Pivot Hip Center

Crank Center

Pedal Center Figure 4.2: CRT Kinematic Sketch

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⃗⃗ 푹⃗⃗ 푹ퟖ ퟕ ∅풉풊풑

푹⃗⃗ ퟔ 푹⃗⃗ ퟒ 푹⃗⃗ ퟓ

푹⃗⃗ ∅풌풏풆풆 Vector Key ퟐ 푹⃗⃗ ퟏ 푹⃗⃗ ퟑ 푹⃗⃗ ퟗ 풍풊 푅⃗⃗ 푖

휽 풊 푹⃗⃗ ퟏퟎ Figure 4.3: CRT Vector Diagram

4.3 CRT Dimensional Constraints

Constraints applied to the CRT model include: Grashof’s law, bounds on the design space, model construction requirement, joint angular position bounds, and joint angular velocity bounds. If the CRT model did not satisfy these constraints, the CRT MP is rejected, and the model fails to produce an average power at the crank center. Therefore, careful consideration of the ten CRT MP is required to produce a successful model.

Grashof’s Law

Grashof’s Law was considered to ensure that the crank would make a full rotation about the crank center. The variation of Grashof’s law listed as Equation 4.1 defines a crank rocker four bar mechanism for vectors 푅⃗ 4, 푅⃗ 5, 푅⃗ 6, and 푅⃗ 7.

푙6 + 푙5 − 푙7 − 푙4 ≤ 0 (4.1)

Design Space Bounds

To ensure that the trike maintains practical and manufacturable dimensions, several dimensions are bounded by constraints. First, the position of the crank and rocker arm pivots must be positioned so that they do not inhibit the cyclist’s view. To satisfy this

39

criteria, 푅⃗ 8 and 푅⃗ 7 + 푅⃗ 8 could not exceed the vertical bounds listed in the table. Second,

to prevent the trike from becoming to long and to maintain manufacturability, horizontal

constraints were placed on the afore mentioned pivots. Third, to ensure functionality of the

crank, the crank and rocker crank had to possess a length larger than 1 inch (commonly

found crank length in engines). Lastly, the length from ankle to pedal center was limited to

maximum length of 14 inches based on notions of design. Constraints placed on this design

space is defined relative to the origin at the hip center and are list in Table 4.1. These

bounds were selected by the designer and are depicted by the green square the example

depicted in Figure 4.1.

Table 4.1: Design Space Constraints

Minimum [푚] Maximum[푚] Horizontal Bounds -1.5 -0.25 Vertical Bounds -0.6 0.5

Crank Length (푙6) 0.0254 -

Rocker Crank Length (푙4) 0.0254 -

Pedal Center to Ankle (푙10) 0 0.3556

Figure 4.4: Design Space Constraints. The green box bounds the design space for the

position of the crank center and rocker arm pivot. Note the grid is in units of meters.

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Model Construction

The model construction ensured that the model would be successful, i.e. that the kinematics could perform a full rotation of the crank. Imaginary dimensions may appear if one of the links is required to change in length to maintain a closed kinematic chain.

Imaginary dimensions are not applicable in this case of mechanical design and therefore

CRT MP are rejected if this constraint is not passed. This was accomplished using the try/catch command in MATLAB.

Joint Constraints

The constraints on angular position and velocity are discussed in detail in the respective section of the CRT model.

4.4 CRT Kinematic Analysis

The vector diagram for the CRT kinematic model is seen in Figure 5. The CRT model has a single degree of freedom, and because the power output at the crank center is desired, the crank was chosen as the single input to this model. The position, velocity, and acceleration analysis are solved as a function of this crank angle. The angular velocity of the crank, assumed to be constant, is a function of the gear reduction, Ξ, between the wheel and driven front sprocket. The trike is assumed to have a constant linear velocity of 3.55 mph (5.71 km/h), as seen in the Cybathlon, and a wheel diameter, 휙푤ℎ푒푒푙, of 26 inches

(0.66 m). The angular velocity of the crank (푅⃗ 6) is:

푣푡푟푖푘푒 휃̇6 = 2 ∙ Ξ ∙ , (4.2) 휙푤ℎ푒푒푙

where 푣푡푟푖푘푒 is the linear velocity of the trike and ∅푤ℎ푒푒푙 is the diameter of the wheel.

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Position Analysis Vectors are denoted by equation ## (Chapter 2). The following vector loops are observed from Fig. 3.

푅⃗ 9 + 푅⃗ 10 − 푅⃗ 2 = ⃗0 , (4.3a)

푅⃗ 7 + 푅⃗ 6 − 푅⃗ 5 − 푅⃗ 4 = ⃗0 , (4.3b)

푅⃗ 1 + 푅⃗ 2 + 푅⃗ 3 − 푅⃗ 8 = ⃗0 . (4.3c)

The length of 푅⃗ 2 is

2 2 푙2 = √(푙9) + (푙10) . (4.4)

The angles of 푅⃗ 4 and 푅⃗ 5 are

cos 휃6 푅⃗ 퐷 = 푅⃗ 7 + 푙6 ∙ [ ] , (4.5a) sin 휃6

( 2 2 2 2) −1 −2 ∙ 푙5 ∙ 푦퐷 −1 − 푥퐷 + 푦퐷 + 푙5 − 푙4 휃5 = tan ( ) + cos ( ) , (4.5b) −2 ∙ 푙 ∙ 푥 2 2 5 퐷 √(2 ∙ 푙5 ∙ 푥퐷) + (2 ∙ 푙5 ∙ 푦퐷)

푥퐷 − 푙5 cos 휃5 푥4 = , (4.5c) 푙4

푦퐷 − 푙5 sin 휃5 푦4 = , (4.5d) 푙4

−1 푦4 휃4 = tan ( ) . (4.5e) 푥4

Vectors 푅⃗ 3 and 푅⃗ 4 are antiparallel,

휃3 = 휃4 − 180. (4.6)

The angle of the cyclist’s legs are the only parameters left to solve. This final four bar mechanism is solved with:

cos 휃3 푅⃗ 퐸 = 푅⃗ 8 − 푙3 ∙ [ ] , (4.7a) sin 휃3

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( 2 2 2 2) −1 −2 ∙ 푙2 ∙ 푦퐸 −1 − 푥퐸 + 푦퐸 + 푙2 − 푙1 휃5 = tan ( ) + cos ( ) , (4.7b) −2 ∙ 푙 ∙ 푥 2 2 2 퐸 √(2 ∙ 푙2 ∙ 푥퐸) + (2 ∙ 푙2 ∙ 푦퐸)

푥퐻 − 푙2 cos 휃2 푥1 = , (4.7c) 푙1

푦퐻 − 푙2 sin 휃2 푦1 = , (4.7d) 푙1

−1 푦1 휃1 = tan ( ) . (4.7e) 푥1

The joint angles are

⃗ −1 |푅10| ∅퐾푛푒푒 = 휃1 − 휃2 − tan ( ) + 180, (4.8) |푅⃗ 9|

∅퐻푖푝 = 휃1 − 훿푡표푟푠표. (4.9)

As seen in Fig 3, all angles are measured in reference to the positive x-axis. Equations

4.3-4.9 may be solved for any angle of 휃6.

Velocity Analysis A complete position analysis provides data necessary to perform a velocity analysis provided that the angular velocity of one link is known at any instant. The derivative of the vector loops in, Eq.4.3, yields

푉⃗ 9 + 푉⃗ 10 − 푉⃗ 2 = ⃗0 , (4.10a)

푉⃗ 7 + 푉⃗ 6 − 푉⃗ 5 − 푉⃗ 4 = ⃗0 , (4.10b)

푉⃗ 1 + 푉⃗ 2 + 푉⃗ 3 − 푉⃗ 8 = ⃗0 . (4.10c)

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Observing that 푅⃗ 2, 푅⃗ 9, and 푅⃗ 10 form a rigid body as well as 푅⃗ 3 and 푅⃗ 4, the angular velocity is equivalent among all links in the loop.

휃̇2 = 휃̇9 = 휃10̇ , (4.11a)

휃̇3 = 휃4̇ . (4.11b)

Completing the velocity analysis by expanding Eq. 4.10b and 4.10c yields

0 cos 휃6 cos 휃4 cos 휃5 0 { } + 푙6 ∙ 휃̇6 [ ] − 푙4 ∙ 휃̇4 [ ] − 푙5 ∙ 휃̇5 [ ] = { } , (4.11c) 0 sin 휃6 sin 휃4 sin 휃5 0

cos 휃1 cos 휃2 cos 휃3 0 0 푙1 ∙ 휃1̇ [ ] − 푙2 ∙ 휃̇2 [ ] + 푙3 ∙ 휃̇3 [ ] − { } = { } . (4.11d) sin 휃1 sin 휃2 sin 휃3 0 0

These equations are linear in the unknown velocity terms. The angular velocity of the knee and hip joints are

휃̇푘푛푒푒 = 휃1̇ − 휃̇2, (4.12)

휃̇ℎ푖푝 = 휃̇1. (4.13)

Acceleration Analysis

With the complete position and velocity analysis, an acceleration analysis follows given that the angular acceleration for the crank is known. Under the assumption that the cyclist pedals at a constant rate, the angular acceleration of the crank is 0. The derivative of Eq 4.10 yields:

퐴 9 + 퐴 10 − 퐴 2 = ⃗0 , (4.14a)

퐴 7 + 퐴 6 − 퐴 5 − 퐴 4 = ⃗0 , (4.14b)

퐴 1 + 퐴 2 + 퐴 3 − 퐴 8 = ⃗0 . (4.14c)

Expanding the derivative of 4.11,

휃̈2 = 휃̈9 = 휃10̈ , (4.15a)

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휃̈3 = 휃4̈ . (4.15b)

Completing the velocity analysis by expanding Eq. 4.14b and 4.14c yields

̇ 2 cos 휃6 ̈ − sin 휃4 ̇ 2 cos 휃4 ̈ − sin 휃5 ̇ 2 cos 휃5 푙6휃6 [ ] + 푙4휃4 [ ] − 푙4휃4 [ ] + 푙5휃5 [ ] − 푙5휃5 [ ] = ⃗0 , (4.15c) sin 휃6 cos 휃4 sin 휃4 cos 휃5 sin 휃5

− sin 휃 cos 휃 − sin 휃 cos 휃 − sin 휃 푙 휃̈ [ 4] − 푙 휃̇ 2 [ 1] + 푙 휃̈ [ 2] − 푙 휃̇ 2 [ 2] + 푙 휃̈ [ 3] 1 1 cos 휃 1 1 sin 휃 2 2 cos 휃 2 2 sin 휃 3 3 cos 휃 4 1 2 2 3 ̇ 2 cos 휃3 0 −푙3휃3 [ ] = { } . (4.15d) sin 휃3 0

This set of acceleration equations is linear in the unknown acceleration terms. The angular acceleration of the knee and hip joints are

휃̈푘푛푒푒 = 휃1̈ − 휃̈2, (4.16)

휃̈ℎ푖푝 = 휃̈1. (4.17)

4.5 CRT Quasi-Static Analysis

Similar to the TRT model, a quasi-static model was chosen to calculate the torque production at the crank. The dynamic effects of the leg and the linkage masses are neglected by acknowledging the low angular velocities and accelerations experienced through one cycle of the crank. Thus, given the complete kinematics, a quasi-static analysis generates the statically equivalent torque at the crank given the torque produced by the cyclist at the knee and hip joints for any given angle of the crank.

Over the course of one cycle of the crank all potential and kinetic energies sum to zero.

As in the TRT, the mass of each member was ignored as the concern of this study is the average power produced at the crank.

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The resultant torque at the crank center can be calculated at any angle of the crank. In this work the resultant crank torque is calculated for the P1P2 and P1P3 groups separately by incrementing from 1° to 360°. The input torque for each increment was interpolated from the data shown in Figures 3.2 and 3.3 depending on the group. The free body diagrams

(FBD) are shown in Fig. 4.5.

푅⃗⃗ 8 푅⃗⃗ 7 G F E ⃗⃗ 푅6 푅⃗⃗ 4 B 푅⃗⃗ 5 D 푅⃗⃗ 2 푅⃗⃗ ퟏ 푅⃗⃗ 3 A 퐹 C 퐵푦 푀 푀 퐹 퐾푛푒푒 퐾푛푒푒 퐵푥 퐹퐵푥

퐹퐵푦 푀퐻푖푝 푅⃗⃗ ퟏ 푅⃗⃗ ퟐ FBD 1 퐹퐴푥

퐹퐴 푦 퐹퐶푥 FBD 2

퐹퐶푦 퐹퐸푦 퐹퐸푥

퐹퐹 푙 푥 4 퐹퐷푦 퐹퐷푥 퐹퐹푦 FBD 4 퐹퐷 퐹퐷푦 푥 퐹 FBD 3 퐶푥 퐹 퐺푦 퐹퐹푦 퐹퐶푦 퐹퐺푥

퐹퐹푥 FBD 5 푀퐶푟푎푛푘 Figure 4.5: Free Body Diagrams for the CRT Model

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The quasi-static model is solved by summing all forces and moments and setting them equal to zero. The summation of forces for FBD 1 yields

퐹퐴푥 퐹퐵푥 0 ∑퐹 = [ ] + [ ] = [ ] , (4.18a) 퐹퐴푦 퐹퐵푦 0

∑푀퐴 = 푙1 ∙ (F퐴xsin 휃1 − 퐹퐴푦 cos 휃2) + 푀퐾푛푒푒 + 푀퐻푖푝 = 0. (4.18b)

The summation of forces from FBD 2 yields

퐹퐶푥 퐹퐵푥 0 ∑퐹 = [ ] − [ ] = [ ] , (4.19a) 퐹퐶푦 퐹퐵푦 0

∑푀퐵 = 푙2 ∙ (−FCxsin 휃2 + 퐹퐶푦 cos 휃2) − 푀퐾푛푒푒 = 0. (4.19b)

The summation of forces from FBD 3 yields

퐹퐶푥 퐹퐷푥 퐹퐸푥 0 ∑퐹 = − [ ] + [ ] + [ ] = [ ] , (4.20a) 퐹퐶푦 퐹퐷푦 퐹퐸푦 0

∑푀퐶 = (푙3 − 푙4) ∙ (−퐹퐷푥 sin 휃3 + 퐹퐷푦 cos 휃3) − 푙3 ∙ (−퐹퐸푥 sin 휃3 + 퐹퐸푦 cos 휃3) = 0. (4.20b)

The summation of two force member in FBD 4 yields

퐹퐹푥 퐹퐷푥 0 ∑퐹 = [ ] − [ ] = [ ] , (4.21a) 퐹퐹푦 퐹퐷푦 0

∑푀퐵 = 푙5 ∙ (−퐹퐹푥 sin 휃5 + 퐹퐹푦 cos 휃5) = 0. (4.21b)

The summation of forces from FBD 5 yields

퐹퐺푥 퐹퐹푥 0 ∑퐹 = [ ] − [ ] = [ ] , (4.22a) 퐹퐺푦 퐹퐹푦 0

∑푀퐵 = 푙6 ∙ (퐹퐹푥 sin 휃6 − 퐹퐹푦 cos 휃6) − 푀퐶푟푎푛푘 = 0. (4.22b)

The resultant torque at the crank is calculated by solving this set of linear equations.

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4.6 CRT Power Analysis

The power analysis was achieved by calculating the instantaneous power any value of the crank angle. Instantaneous power is determined by multiplying the angular velocity of a joint by the resultant joint torque as seen in Equation 4.23,

푃퐶푟푎푛푘 = 휃̇6 ∙ 푀퐶푟푎푛푘, (4.23a)

푃퐾푛푒푒 = 휃̇퐾푛푒푒 ∙ 푀퐾푛푒푒, (4.23b)

푃퐻푖푝 = 휃̇퐻푖푝 ∙ 푀퐻푖푝. (4.23c)

As mentioned in the TRT model, the area under the instantaneous power curve for the crank is equivalent to the sum of the instantaneous power curve of the knee and hip, Eq.

2.19. This equivalency verifies that the power into the system is equivalent to the power output of the system. This provides validation of the calculations. The average power is calculated according to Eq. 2.20.

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CHAPTER 5

CRT O PTIMIZATION AND RESULTS

5.1 Optimization Overview The 10 crank rocker trike mechanical parameters (CRT MP) for each group were optimized to produce the greatest average power per cycle of the crank. The MATLAB optimization function, fmincon, was initially used in an attempt to optimize the design. No successful design was generated in this way. This is believed to be due to a combination of nonlinearities and a significant number of inequality constraints. Therefore a “poor man’s genetic algorithm” was generated to optimize the design. The resultant torque at the crank is then calculated using this initial guess and the joint torque curves. The average power calculated at the crank center for one cycle of the crank is then solved and is considered herein as the objective function.

This optimization begins with an initial guess of the 10 CRT MP that satisfies all constraints. The power is then calculated by the objective function and the initial CRT MP parameters are considered held. A new guess is then developed by making small, random changes to the held CRT MP. These new values must satisfy the constraint criteria and, if so, are passed to the objective function. The newly computed power is compared against the power developed from the held CRT MP. If the newly computed power is found to be greater, the new CRT MP values become the held parameters. If the newly computed power is not greater, the held parameters remain unchanged. Then a new set of CRT MP parameters is generated from the held CRT MP. This pattern of randomly incrementing the

CRT MP on the held dimensions continues until the convergence criteria is met. This optimization is outlined in Figure 5.1.

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CRT MP Initial Guess Alter CRT MP Φ푖 = Φ퐻푒푙푑 + 훿 Φ퐻푒푙푑 = [푙3 푙4 푙5 푙6 푅⃗⃗ 7 푅⃗⃗ 8 푙10 Ξ]

Check Constraint Criteria Check Constraint Criteria Pass Fail Pass Fail

푖 = 1 Objective Function Objective Function 푘 = 0 휇푖 = 푓(Φ푖) 휇 = 휇 = 푓(Φ퐻푒푙푑) 푐푟푎푛푘 퐻푒푙푑

휇푖 > 휇퐻푒푙푑 Pass Fail

휇 = 휇 푖 = 푖 + 1 퐻푒푙푑 푖

Φ퐻푒푙푑 = Φ푖 푘 = 푘 + 1

푘 = 0

Check Convergence Criteria 푘 > 10000 Pass Fail

Optimized Design Figure 5.1: Optimization Flow Diagram

Altering CRT MP

The initial guess and subsequent improved CRT MP were incrementally changed to supply the objective function a new set of parameters. This magnitude of each change was randomly generated using MATLAB’s pseudo rand number generator as seen in Eq. 5.1.

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푙3 + 훿1 푙 + 훿 4 2 푙5 + 훿3 푙 + 훿 6 4 푥7 + 훿5 cj Φ푖 = , where 훿푗 = , and − 1 ≤ c푗 ≤ 1. (5.1) 푦7 + 훿6 100 푥8 + 훿7 푦 + 훿 7 8 푙10 + 훿9 { Ξ + 훿10

Convergence Criteria

Convergence was determined by counting the number of iterations since the last improvement. This optimization has 10 degrees of freedom; therefore, it would take 210

1024 iterations to explore the entire design space surrounding a particular guess. Due to the randomness associated with altering parameters and to be confident that an optimized design has been obtained, the convergence criteria was established to be 10,000 consecutive iterations of parameters that satisfy the constraints without improving the average power calculated at the crank center.

5.2 Optimization results The optimization was deemed to produce be successful in finding local maximums as results showed convergence at specific average power throughput at the crank center. This can be observed in the histograms in Figure 5.2 and 5.3 that display the number of initial guess for a grouping of throughput power. The severe skewness to the left suggests that there are no designs that can satisfy the constraints and produce a significantly greater throughput power.

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Figure 5.2: Histogram of the P1P2 group optimization. The severe skewness to the left suggests that a maximum power throughput for the CRT has been identified.

Figure 5.3: Histogram of the P1P3 group optimization. The severe skewness to the left suggests that a maximum power throughput for the CRT has been identified.

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A total of eight sets of CRT MP optimization results were selected (four from each group) for further analysis to identify the optimal CRT design for SCI FES cycling. Of the four designs per group, two designs were based objectively on the greatest power throughput and two designs were selected subjectively by criteria of manufacturability and the greatest power throughput. Manufacturability considers several design criteria such as distance from the hip center to crank center, length of rocker arm, height of rocker arm pivot relative to rider’s line of sight, estimated weight of design, etc. The CRT MP associated with each of the eight designs are listed in Table 5.1. The kinematic sketches associated with these dimensions are found in Appendix A. The torque and power curve for each design is found in Appendix B.

Table 5.1: Best Eight Optimization Results. Designs A-H were selected for further analysis to identify the optimal CRT design for SCI FES cycling. Power Design Throughput 푙3 푙4 푙5 푙6 푙10 푅⃗ 7 푅⃗ 8 Ξ [W] −0.50 −0.70 A 23.98 0.22 0.05 0.51 0.03 0.07 [ ] [ ] 2.03 0.14 0.11 −0.56 −0.70

B 23.79 0.22 0.05 0.60 0.03 0.09 [ ] [ ] 2.01 0.21 0.10

P1P2 −0.18 −0.63 C 23.57 0.24 0.05 0.17 0.03 0.00 [ ] [ ] 2.00 0.01 0.07 −0.53 −0.74 D 23.54 0.20 0.14 0.50 0.07 0.10 [ ] [ ] 1.82 0.16 0.10 −0.49 −0.62 E 29.00 0.24 0.09 0.44 0.04 0.08 [ ] [ ] 1.81 0.01 0.32 −0.52 −0.64

F 28.78 0.18 0.04 0.55 0.03 −0.01 [ ] [ ] 2.06 0.33 0.21

P1P3 −0.45 −0.61 G 27.74 0.22 0.06 0.45 0.04 0.00 [ ] [ ] 1.64 0.14 0.27 −0.28 −0.68 H 27.71 0.17 0.12 0.25 0.07 0.02 [ ] [ ] 2.01 0.13 0.22

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5.3 Design Evaluation

Designs A-H were further evaluated to identify the optimal CRT design based upon the following set of metrics: joint torque curve continuity, equal power distribution over the entire crank cycle, and percent increase in average power for each group.

Joint Torque Curve Continuity

As seen in Figure 3.4, the joint torque curves may become truncated in a way that is not biomechanically correct when the entire joint range is not utilized. Although the expectation is that FES cyclists would produce a continuous curve of joint torques as a function of joint angles, there is not enough experimental data to generate these continuous curves as inputs to the CRT. Therefore, discontinuous joint curves were accepted as reasonable inputs to the model. Thus, the least disjointed curves are assumed to accurately model the true dynamics of a cyclist riding that specific CRT. As such, designs with the least observed distance between the truncated ends of the curve partitions became more favorable designs. For example, the continuity of plot (a) is favorable over plot (b) in Figure

5.4.

(a) (b) Figure 5.4: Joint Torque Curve Continuity Comparison. Plot (a) displays a curve where the distance been the ends of the truncated data is minimal. Plot (b) displays a curve where there is a significant difference in torque at the ends of the truncated data.

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The joint torque curves are plotted in Appendix B, Figure B.1 through B.4. All eight designs are found to limit the joint range of the knee for the P1P2 and P1P3 group, thus causing discontinuous torque values when the knee transitions from extension to flexion and flexion to extension. All designs require an average instantaneous shift in knee joint torque of roughly 20 Newton-meters. The optimization significantly reduced the joint range of the hip from 35° to an average between these eight designs of roughly 10°. Despite removing large portions of the hip torque curve, Design A-G require an average shift of roughly 10 Newton-meters when the hip transitions from extension to flexion or flexion to extension.

By examinations Designs A-D would best serve the P1P2 group when considering the metric of torque curve continuity. Likewise, Designs E-G would best serve the P1P3 group.

Design E effectively accounts for the continuity of the torque for both groups.

Power Distribution

A design that would be well suited for FES Cycling would ensure that the cyclist would produce a constant level of power at the crank. If this were achievable, FES cyclists would easily overcome inactive zones. However, this is not a realistic expectation for a mechanical device because FES cyclists are unable to produce a constant power output as reported by Szesci [14]. Instead, a more realistic expectation is for an alternative design to prevent the power calculated at the crank from becoming a negative value. When the power at the crank becomes negative, it must extract power from the linear forward momentum of the trike to move the legs. This extraction of power from the forward momentum of the trike to move legs is referred to in Chapter 1 as inactive zones where the leg is not in a configuration that can be stimulated to produce positive work. This creates a choppy

55 pedaling motion and can lead to the cyclist becoming stuck at TDC and BDC. ANTS Asso. has posted footage from the cybathlon that demonstrates this phenomena [25].

The power curves for each optimized is shown in Appendix C, Figures C.1 - C.8. Each of the power curves in the P1P2 group are very similar and trace as expected. Most of the power is generated in one phase of the TRT model, therefore it is expected that there would be one predominately large peak for the P1P2 CRT optimization power curves. Despite the similarities, Design H stands out among the rest of the P1P2 group as the power calculated at the crank remains positive for the greatest portion of the crank cycle; and in doing so reduces the inactive zone. In a similar fashion, the P1P3 group was expected to produce power curves with two distinct power peaks. Among these designs, Design H stands out as the power calculated at the crank is only approaches zero at one point throughout the entire cycle of the crank. Each of these designs eliminate all inactive zones for P1P3 cyclist and would be a significant improvement over the TRT.

Design Group Cross Comparison

Designs A-D were optimized for the P1P2 group while Designs E-H were optimized for the P1P3 group. Therefore, an analysis of the average power increase at the crank center for each design for each group of FES cyclists is desirable to ensure an optimal CRT design for all potential riders. Table 5.2 shows the percent increase for each design and group.

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Table 5.2: Cross Comparison of Power P1P2 P1P3 Average Percent Average Percent

Power [W] Increase Power [W] Increase TRT Model 14.6 - 16.2 - Design A 23.98 64% 22.79 41% Design B 23.79 63% 23.27 44% Design C 23.57 61% 23.62 46% Design D 23.54 61% 22.09 36% Design E 21.36 46% 29.00 79% Design F 20.66 42% 28.78 78% Design G 19.47 33% 27.74 71% Design H 19.43 33% 27.71 71%

Designs A-D would best serve the P1P2 group when considering the metric of percent power increase. Likewise, Designs E and F would best serve the P1P3 group. Design E and

G effectively accounts for the percent power increase for both groups.

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CHAPTER 6

DISCUSSION AND CONCLUSION

6.1 TRT Model Review

The TRT trike power curves were modeled using a single source of torque data developed by Szecsi [14] and a single set of tuned kinematic inputs. Szecsi acknowledges that 75% of SCI FES cyclists are most powerful in the early stages of the cycling process, referred to as the P1P2 group. The remaining 25% of cyclists are found to produce power more evenly throughout the pedaling cycle and form the P1P3 group. The TRT model was provided torque as a function of crank angle for each group and computed the average power over one cycle of the crank. In the study conducted by Szecsi, all cyclists were constrained to produce 30 watts of power per cycle of the crank. Each leg is assumed to produce 15 watts of power per cycle of the crank. Therefore, the TRT model was expected to reproduce these results provided the joint torque data measured by Szecsi. Upon tuning the models leg dimensions, which are unknown, the TRT model produced 14.7 Watts for the P1P2 group and 16.1 Watts for the P1P3 group. Acknowledging that the joint torque curves reported by Szecsi were an average of several cyclist, it is reasonable to assume that the TRT model would not match 15 Watts exactly given a single set of kinematic dimensions.

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6.2 Final Design Selection

A significant amount of work is required to determine whether an FES cyclist fits into the P1P2 or P1P3 group. Therefore, it is not practical to have two separate designs designated as the optimal design for a particular group. Instead one CRT configuration that is beneficial to both groups is most desirable.

The CRT optimized produced serval optimized designs of which eight were selected based upon average power throughput at the crank and manufacturability. These eight designs were further analyzed against the metrics of joint torque curve continuity, equal power distribution over the entire crank cycle, and percent increase in average power for each group. By observation Design E shows the best performance by all three metrics. The kinematic sketch of Design E is shown in Figure 6.1. The kinematic joint range is shown in Figure 6.2. The power curves for the P1P2 and P1P3 group is shown in Figure C.5.

Rocker Pivot Knee Center Crank Center

Pedal Center

Hip Center

Figure 6.1: Final CRT Design

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Figure 6.2: Final CRT Design Joint Range

6.3 Explanation of Results

Design E increases the power throughput for the P1P2 group by 46% and for the P1P3 group by 79% without requiring the cyclist to produce more torque than what is used to pedal a TRT. Design E achieved this by changing the motion of the leg and holding the joint angle in regions of high toque to ensure that positive power is produced for the entire duration of the crank cycle. The optimization recognized that a significant portion of the hip range did not produce positive power at the crank, therefore the kinematics of the CRT

Design E prevents the hip from entering this region. Design E of the CRT further limits any antagonistic joint moments created at the hip through limiting the joint range of the hip. Furthermore, the rocker arm provides significant mechanical advantage to the crank arm, multiplying any force generated by the leg.

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6.4 Future Work

Future work should be directed into two areas: model refinement and alternative mechanisms. Current models rely on a single set of torque data that is limited to a particular set of cyclist dimensions, and therefore limited joint ranges. A comprehensive set of joint torque as a function of joint angle and joint motion that covers the full joint range is desirable as this would remove kinematic constraints on the optimization. Further analysis of the intricacies of joint torque as a function of joint angular position, joint angular velocity, and extension/flexion would refine the models presented herein.

Other mechanisms, such as a coupler driver and pulley systems, also may alter the motion in a beneficial manner and should be pursued.

6.5 Conclusion

This thesis presents alternative tricycle designs and modeling techniques for SCI FES cycling. Traditional cycling for AB subjects is vastly different from FES cycling. Average power production for FES cyclist is an order of magnitude less than AB cyclist.

Additionally, the four power peaks observed in AB cycling are not seen in FES cycling.

Therefore, alternative drivetrains may have better force/power transmission to the driving wheel. The crank rocker trike (CRT) employs a four-bar architecture that allows the pedal, and therefore leg, to travel in a back and forth semi-circular motion instead of the traditional circular motion of the pedal (Fig 3.1). Optimization of this architecture lead to designs that improved the throughput power of P1P2 cyclist by as much as 64% and P1P3 cyclist by as much as79%. These designs take advantage of joint angular ranges where cyclist can produce large amounts of torque. Furthermore, the back-and-forth motion more evenly distributes power throughout the cycle of the crank, and thus reduce inactive zones.

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Increased power throughput to the driving wheel and a smoother pedaling cycle are believed to increase SCI FES cycling.

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[24] HOME PAGE. (n.d.). Retrieved November 2020, from https://www.pumphousefitness.com/LEG%20PRESS%20AND%20SEATED%20C ALF%20RAISE%20BY%20INFLIGHT%20FITNESS.htm

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[25] ANTS Asso. (June 2018). Vance B. - FES cycling race Cybathlon 2016 [Video]. https://www.youtube.com/watch?v=RWiN5ihf3og&ab_channel=ANTSAsso

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APPENDIX A

Optimization Results

Design A Design B

Design C Design D Figure A.1: P1P2 Optimization Results

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Design E Design F

Design G Design H Figure A.2: P1P3 Optimization Results

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APPENDIX B

Optimized Design Torque Curves

(a) (b)

Figure B.1: Joint Torque Curves for Design A. Plot (a) show the joint torque curves for the P1P2 group. Plot (b) show the joint torque curves for the P1P3 group.

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(a) (b)

Figure B.2: Joint Torque Curves for Design B. Plot (a) show the joint torque curves for the P1P2 group. Plot (b) show the joint torque curves for the P1P3 group.

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(a) (b)

Figure B.3: Joint Torque Curves for Design C. Plot (a) show the joint torque curves for the P1P2 group. Plot (b) show the joint torque curves for the P1P3 group.

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(a) (b)

Figure B.4: Joint Torque Curves for Design D. Plot (a) show the joint torque curves for the P1P2 group. Plot (b) show the joint torque curves for the P1P3 group.

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(a) (b)

Figure B.5: Joint Torque Curves for Design E. Plot (a) show the joint torque curves for the P1P2 group. Plot (b) show the joint torque curves for the P1P3 group.

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(a) (b) Figure B.6: Joint Torque Curves for Design F. Plot (a) show the joint torque curves for the P1P2 group. Plot (b) show the joint torque curves for the P1P3 group.

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(a) (b)

Figure B.7: Joint Torque Curves for Design G. Plot (a) show the joint torque curves for the P1P2 group. Plot (b) show the joint torque curves for the P1P3 group.

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(a) (b) Figure B.8: Joint Torque Curves for Design H. Plot (a) show the joint torque curves for the P1P2 group. Plot (b) show the joint torque curves for the P1P3 group.

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APPENDIX C

Optimized Design Power Curves

(a) (b)

Figure C.1: Instantaneous Power Curve at Crank Center for Design A. Plot (a) shows the instantaneous power curve for the P1P2 group. Plot (b) shows the instantaneous power curve for the P1P3 group.

(a) (b) Figure C.2: Instantaneous Power Curve at Crank Center for Design B. Plot (a) shows the instantaneous power curve for the P1P2 group. Plot (b) shows the instantaneous power curve for the P1P3 group.

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(a) (b) Figure C.3: Instantaneous Power Curve at Crank Center for Design C. Plot (a) shows the instantaneous power curve for the P1P2 group. Plot (b) shows the instantaneous power curve for the P1P3 group.

(a) (b) Figure C.4: Instantaneous Power Curve at Crank Center for Design D. Plot (a) shows the instantaneous power curve for the P1P2 group. Plot (b) shows the instantaneous power curve for the P1P3 group.

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(a) (b) Figure C.5: Instantaneous Power Curve at Crank Center for Design E. Plot (a) shows the instantaneous power curve for the P1P2 group. Plot (b) shows the instantaneous power curve for the P1P3 group.

(a) (b) Figure C.6: Instantaneous Power Curve at Crank Center for Design F. Plot (a) shows the instantaneous power curve for the P1P2 group. Plot (b) shows the instantaneous power curve for the P1P3 group.

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(a) (b)

Figure C.7: Instantaneous Power Curve at Crank Center for Design G. Plot (a) shows the instantaneous power curve for the P1P2 group. Plot (b) shows the instantaneous power curve for the P1P3 group.

(a) (b) Figure C.8: Instantaneous Power Curve at Crank Center for Design H. Plot (a) shows the instantaneous power curve for the P1P2 group. Plot (b) shows the instantaneous power curve for the P1P3 group.

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APPENDIX D

Optimized Design Joint Motion

(a) (b)

(c) (d)

Figure D.1: Joint Motion for alternative designs A-D. (a) Design A. (b) Design B. (c) Design C. (d) Design D.

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(a) (b)

(c) (d) Figure D.2: Joint Motion for alternative designs E-G. (a) Design E. (b) Design F. (c) Design G. (d) Design H.

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