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Roetter, Adam Daniel.
Compliant prosthetic knee extension aid :
b a finite elements analysis investigation of proprioceptive feedback during the swing phase of ambulation
h [electronic resource] /
by Adam Daniel Roetter.
[Tampa, Fla] :
University of South Florida,
Title from PDF of title page.
Document formatted into pages; contains 149 pages.
Thesis (M.S.M.E.)--University of South Florida, 2008.
Includes bibliographical references.
Text (Electronic thesis) in PDF format.
ABSTRACT: Compliant mechanisms offer several design advantages which may be exploited in prosthetic joint research and development: they are light-weight, have low cost, are easy to manufacture, have high-reliability, and have the ability to be designed for displacement loads. Designing a mechanism to perform optimally under displacement rather than force loading allows underlying characteristics of the swing phase of gait, such as the maximum heel rise and terminal swing to be developed into a prosthetic knee joint. The objective of this thesis was to develop a mechanical add-on compliant link to an existing prosthetic knee which would perform to optimal standards of prosthetic gait, specifically during the swing phase, and to introduce a feasible method for increasing proprioceptive feedback to the amputee via transferred moments and varying surface tractions on the inner part of a prosthetic socket.A finite elements model was created with ANSYS to design the prosthetic knee compliant add-on and used to select the geometry to meet prosthetic-swing criteria. Data collected from the knee FEA model was used to apply correct loading at the knee in a SolidWorks model of an above-knee prosthesis and residual limb. Another finite element model was creating using COSMOSWorks to determine the induced stresses within a prosthetic socket brought on by the compliant link, and then used to determine stress patterns over 60 degrees of knee flexion (standard swing). The compliant knee add-on performed to the optimal resistance during swing allowing for a moment maxima of 20.2 Newton-meters (N-m) at a knee flexion of 62 degrees.The moments applied to the prosthetic socket via the compliant link during knee flexion and extension ranged from 5.2 N-m (0 degrees) in flexion, to 20.2 N-m (62 degrees) in extension and induced a varying surface tractions on the inner surface of the socket over the duration, thus posing a possible method of providing proprioceptive feedback via surface tractions. Developing a method for determining the level of proprioceptive feedback would allow for less expensive and more efficient methods of bringing greater control of a prosthesis to its user.
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Advisor: Craig Lusk, Ph.D.
x Mechanical Engineering
t USF Electronic Theses and Dissertations.
Compliant Prosthetic Knee Extension Aid: A Finite Elements Analysis Investigation of Proprioceptive Feedback Duri ng the Swing Phase of Ambulation by Adam Daniel Roetter A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Mechanical Engineering Department of Mechanical Engineering College of Engineering University of South Florida Major Professor: Craig Lusk, Ph.D. Rajiv Dubey, Ph.D. Nathan Crane, Ph.D. Date of Approval: October 28, 2008 Keywords: compliant mechanisms, proprioce ption, knee disarticulation, polycentric 4bar, prosthetic, interface mechanics, design by specialization Copyright 2008, Adam Daniel Roetter
Acknowledgments There are many people who have contribute d to my life during the course of this thesis and the road to it, wh ich I would like to recognize. First and foremost, I am truly grateful to my father, who has sacrificed so much my entire life a nd has been my shining mentor; who has taught me to learn from my mistakes and to love and appreciate everything along the way, who has been my best friend and an irrepl aceable part of my world. To my loving step-mother, Bev, for al l the support and love she has given to me and for being the piece of my family I had been missing for so long. To my beautiful wife-to-be, Laura, for her c onstant love, support, admira tion and understanding; I love and appreciate all she has done for me and si ncerely hope her sacrifices will be well rewarded. To LauraÂ’s family, Patty, Glenn, Audr ey and Calvin, for their constant interest in my work and my life, and for showing me support whenever they knew I needed it. I greatly appreciate my fello w peers, whose friendships ha ve kept my life true and meaningful. To Pete for all the times we st udied and worked on finite elements and for the great future to-come w ith our families. To my friends Anthony, Jason, Jeff and Aaron, your friendshipsÂ’ have truly made my life richer. To my professors who have nourished my desire for learning ove r these past years I cannot thank you enough. To my committee who has taken so much time to review my work and offer their thoughts and guidance I am forever grateful, I am particularly grateful to my advisor Dr. Lusk, for all his sup port and encouragement. Finally, I am eternally grat eful for the abundant blessings God has offered me and the strength he has given me to push forward.
i Table of Contents List of Tables................................................................................................................. ....iii List of Figures................................................................................................................ ....iv Abstract....................................................................................................................... .......vi Chapter 1 Overview....................................................................................................1 1.1 Background.........................................................................................................3 1.1.1 Background Â– History of Prosthet ics and the Prosthetic Knee.......................3 1.1.2 Background Â– Compliant Mechanis ms and Current Research.......................5 1.1.3 Background Â– Compliant Mechanism Prosthetic Joint Research...................8 1.2 Phases of Gait...................................................................................................16 1.3 Knee Disarticulation.........................................................................................18 1.3.1 Advantages and Disadvantages of Knee Disarticulation..............................19 1.4 Prosthetic Knee Inherent Stability....................................................................23 Chapter 2 Prosthetic Knee Classifications................................................................25 2.1 Classification Â– Functional...............................................................................25 2.2 Classification Â– Mechanical..............................................................................27 2.3 User Aspects of Swing and Stance...................................................................30 2.4 Medicare Functional Modifier System.............................................................32 2.4.1 K-Scores........................................................................................................32 Chapter 3 Interface Mechan ics Literature Review...................................................35 3.1 Finite Element Analysis Design........................................................................37 3.2 Finite Element Analysis Techniques................................................................40 3.2.1 Geometry.......................................................................................................40 184.108.40.206 Totally-Glued Interface.............................................................................41 220.127.116.11 Partially-Glued Interface...........................................................................42 18.104.22.168 Slip Permitted at Interface........................................................................43 3.2.2 Element Properties........................................................................................45 3.2.3 Boundary Conditions....................................................................................47 3.3 Modeling the Residual Limb............................................................................48 3.4 Experimental Analysis......................................................................................50 3.5 Numerical Analysis...........................................................................................52 3.6 Validation of the FE Analysis...........................................................................52 3.7 Parametric Analysis..........................................................................................54 3.8 Conclusions on Interface Mechanics Review...................................................58
ii Chapter 4 Bistable Compliant Extension Aid...........................................................60 4.1 Design by Specialization...................................................................................60 4.2 Background.......................................................................................................62 4.3 Functional Criteria............................................................................................63 4.4 Concept of Bistability.......................................................................................67 4.5 Bistable Compliant Extension Aid (BCEA) Design.........................................68 4.6 Analysis and Results.........................................................................................72 4.7 Knee and BCEA Unloading After Snap...........................................................77 4.8 BCEA Stress Analysis and Factor of Safety.....................................................79 4.9 BCEA Design Conclusion................................................................................81 Chapter 5 Proprioception via Variable Internal Socket Stress Patterns...................82 5.1 Interface Mechanics and Proprioception..........................................................82 5.2 Finite Element Design Characteristics..............................................................84 5.3 Modeling...........................................................................................................86 5.4 Applied Loads...................................................................................................88 5.5 Analysis and Results.........................................................................................90 5.6 Proprioception and Variable Stress Conclusions and Future Work..................96 Chapter 6 Conclusions..............................................................................................97 6.1 Contributions.....................................................................................................97 6.2 Suggestions for Future Work............................................................................99 List of References...........................................................................................................10 0 Appendices..................................................................................................................... .103 Appendix I: ANSYS Knee Code...........................................................................104 Appendix II: ANSYS Results File ( =/2).............................................................115 Appendix III: Matlab Code for Plotti ng Flexion and Extension Moments.............131 Appendix IV: Matlab Code for Plotting Reaction Forces.......................................134 Appendix V: Reaction Force Plots.........................................................................138 Appendix VI: COSMOSWorks Repor t File Â– Socket and Knee............................142
iii List of Tables Table 2-1. Functional Cl assification Examples..........................................................26 Table 2-2. Mechanical Cl assification Breakdown.....................................................29 Table 2-3. MFMS K-Scores.......................................................................................34 Table 3-1. Parametric Analysis..................................................................................55 Table 4-1. Summary of Sw ing Phase Requirements..................................................64 Table 4-2. Extension Moment Data for Optimized LBCEA.........................................76 Table 4-3. BCEA Stress Summary.............................................................................80 Table 5-1. Summary of BCEA Ap plied Extension Moments....................................88 Table 5-2. Summary of BCEA Applied Reaction Forces..........................................89 Table 5-3. Surface Stress Summary at 62 Degrees of Flexion...................................93 Table 5-4. Surface Strain Summary at 62 Degrees of Flexion...................................94
iv List of Figures Figure 1-1. Photograph of Otto Bock 3R 21 Modular 4-Bar Linkage Knee Joint.........2 Figure 1-2. CAD Model of Otto Bock 3R 21 Modular 4-Bar Linkage Knee Joint........2 Figure 1-3. Otto Bock 3R21 with Bistab le Compliant Extension Aid Concept............2 Figure 1-4. Prosthetic Toe in Cairo Museum................................................................3 Figure 1-5. Ambroise Pare: Founder of Prosthetics......................................................4 Figure 1-6. Common Compliant Mechanisms...............................................................6 Figure 1-7. Crimping Mechanism, Co mpliant & Rigid-Body Counterpart...................7 Figure 1-8. Overrunning Clutch, Co mpliant & Rigid-Body Counterpart.....................7 Figure 1-9. GurinotÂ’s Inversion HCCM Concept......................................................10 Figure 1-10. GurinotÂ’s Isolation HCCM Concept.......................................................11 Figure 1-11. GurinotÂ’s Tested Inverted Cross-Axis Flexur al Pivot Knee Prototype...12 Figure 1-12. MahlerÂ’s Pediatric Prosthetic Knee Prototype..........................................13 Figure 1-13. MahlerÂ’s Knee Instantaneous Center........................................................14 Figure 1-14. WiersdorfÂ’s Modular Expe rimental Research Ankle (MERA)................15 Figure 1-15. Sub-Phases of Stance................................................................................17 Figure 1-16. Swing Phase of Gait..................................................................................18 Figure 1-17. Distances from Distal End to Prosthetic Knee Center..............................21 Figure 1-18. Stability vs. Control..................................................................................24 Figure 2-1. Constant Friction Single Axis Knee by Ossur..........................................28 Figure 2-2. Variable Fric tion Single Axis Knee..........................................................28 Figure 2-3. Multiple Axial Knee Mechanisms............................................................29 Figure 3-1. Mesh of Above-Knee St ump and Socket (Zhang and Mak).....................44 Figure 3-2. Distal-End Boundary Conditions..............................................................47 Figure 3-3. FE Modeling.............................................................................................48 Figure 4-1. 3R32 with Manual Lock (a) and 3R55 with Pneumatic Cylinder (b).......61 Figure 4-2. Knee Angle vs. Gait Â– Shown w ith and without Excessive Heel Rise.....66 Figure 4-3. Optimal Influence of Pr osthetic Knee Extension Assist...........................67 Figure 4-4. Bistability Anal ogy with a Ball and Hill...................................................68 Figure 4-5. Knee Mechanism Simplification Model...................................................69 Figure 4-6. Otto Bock Knee Mechanism with BCEA Assembly................................69 Figure 4-7. Design Approximati on of the BCEA Geometry.......................................71 Figure 4-8. Free-Body Diagram of Knee and BCEA..................................................72 Figure 4-9. BCEA Extension Moment vs. Knee Flexion............................................73 Figure 4-10. BCEA Snap Phenomena...........................................................................74 Figure 4-11. BCEA Extension Moment Graph with Labeled Key-points.....................75 Figure 4-12. BCEA Extension Moment vs. Kn ee Flexion Â– Optimal Geometry Sets..76 Figure 4-13. BCEA Unloading Curve...........................................................................78 Figure 4-14. Complete BCEA Cycle: 90 De grees of Flexion and Extension...............79
v Figure 4-15. BCEA Stress Magnitude and Di stribution at Maximum Stress State.......80 Figure 5-1. Complete Model of Lower-Limb Prosthesis.............................................86 Figure 5-2. Applied BCEA Moments..........................................................................88 Figure 5-3. BCEA Reaction Forces vs. Knee Flexion.................................................89 Figure 5-4. Free Body Diagram of the Pros thetic KneeÂ’s Top Bracket and Socket....90 Figure 5-5. Stress Patterns on Inner Part of Prosthetic Socket by Knee Flexion........92 Figure 5-6. Stress Pattern Summ ary Over Key Knee Flexions...................................92 Figure 5-7. Strain at Ma ximum Knee Flexion.............................................................93 Figure 5-8. Stress Anomaly Due to Knee Fixation......................................................95 Figure 5-9. Socket and Kn ee Fixation/Contact Area...................................................95 Figure A-1. Anterior Force in x-Direction vs. Knee Angle........................................138 Figure A-2. Anterior Force in y-Direction vs. Knee Angle........................................138 Figure A-3. Magnitude of Anteri or Force vs. Knee Angle........................................139 Figure A-4. Magnitude of Ante rior Force vs. Direction.............................................139 Figure A-5. Posterior Force in x-Direction vs. Knee Angle.......................................140 Figure A-6. Posterior Force in y-Direction vs. Knee Angle.......................................140 Figure A-7. Magnitude of Posterior Force vs. Knee Angle........................................141 Figure A-8. Magnitude of Posterior Force vs. Direction............................................141
vi Compliant Prosthetic Knee Extension Aid: A Finite Elements Analysis Investigation of Proprioceptive Feedback Duri ng the Swing Phase of Ambulation Adam Daniel Roetter ABSTRACT Compliant mechanisms offer several de sign advantages whic h may be exploited in prosthetic joint research and development: they are li ght-weight, have low cost, are easy to manufacture, have high-reliability, and have the ability to be designed for displacement loads. Designing a mechanism to perform optimally under displacement rather than force loading allows underlying characteristics of the swing phase of gait, such as the maximum heel rise and terminal swing to be developed into a prosthetic knee joint. The objective of this thesis was to develop a mechanical add-on compliant link to an existing prosthetic knee whic h would perform to optimal st andards of prosthetic gait, specifically during the swing phase, and to introduce a feasible method for increasing proprioceptive feedback to the amputee via transferred moments and varying surface tractions on the inner part of a prosthetic socket. A finite elements model was created with ANSYS to design the prosthetic knee compliant add-on and used to select the geometry to meet prosthetic-swing criteria. Data collected from the knee FEA model was used to apply correct loading at the knee in a SolidWorks model of an above-knee prosthesis and residual limb. Another finite element model was creating using COSMOSWorks to determine the induced stress es within a prosthe tic socket brought on
vii by the compliant link, and then used to dete rmine stress patterns ove r 60 degrees of knee flexion (standard swing). The compliant kne e add-on performed to the optimal resistance during swing allowing for a moment maxima of 20.2 Newton-meters (N-m) at a knee flexion of 62 degrees. The moments applied to the prosthetic socket via the compliant link during knee flexion and extension ranged fr om 5.2 N-m (0 degrees) in flexion, to 20.2 N-m (62 degrees) in extension and induced a varying surface tractions on the inner surface of the socket over the duration, thus posing a possibl e method of providing proprioceptive feedback via surface tractions Developing a method for determining the level of proprioceptive feedback would allo w for less expensive and more efficient methods of bringing greater contro l of a prosthesis to its user.
1 Chapter 1 Overview The objective of this thesis was to deve lop a compliant linkage add-on as a design specialization to the Otto Bock 3R21 frame (Figures 1-1 and 1-2) and to test the hypothesis that the extension moments brought about by the compliant extension aid offer a method of providing proprioceptive fee dback to the amputee via variable stress patterns on the inner part of the prosthetic so cket over the swing phase of the gait cycle. This hypothesis was tested by developing a Computer Assisted Drawing (CAD) and Finite Element (FE) model of the knee with the bistable compliant extension aid (Figure 1-3), a prosthetic socket and residual limb with simplified geometry. Knee flexion (0-90 degrees) and the resulting forces and mome nts were analyzed with ANSYS, and the resulting tractions on the socket analy zed using SolidWorks (COSMOSWorks). The criterion we adopted for analyzi ng proprioception was that the tractions applied to the inner part of the socket show ed distinct variation over the swing phase, remained tolerable by the user and did not cau se failure of the polypropylene socket. This criterion provided the basis for analytical work but should be refined through clinical testing.
2 Figure 1-1. Photograph of Otto Bock 3R21 Modular 4-Bar Linkage Knee Joint Figure 1-2. CAD Model of Otto Bock 3R21 Modular 4-Bar Linkage Knee Joint Figure 1-3. Otto Bock 3R21 with Bistable Compliant Extension Aid Concept
3 1.1 Background The introduction of compliant mechanism technology offers several advantages in prosthetic joint design: low friction and wear, low part count, lighter weight, high reliability and efficient manufacturing and a ssembly. These advantages, as well as the ability to design for displacement loading, fit compliant mechanisms well into the design of an efficient prosthetic knee during swing. 1.1.1 Background Â– History of Prosthet ics and the Prosthetic Knee Prosthetics are said to have existed fr om the times of the ancient Egyptians. Prosthetics were used in many applicati ons: function, cosmetic appearance and most important to the ancient Egyptians, psycho-spiritu al sense of being whol e. It was feared by many that when an amputation was perfor med the individual would be left un-whole in the afterlife. Once pe rformed, the amputated limb was buried until the individual passed when it would be placed with the body so as to make them whole for the afterlife. One of the earliest known examples of a co smetic prosthesis da te back to the 18th dynasty of ancient Egypt where a mummy was found with a prosthetic toe made of leather and wood (Figure 1-4). Greek and Roman civilizat ions are sometimes credited for creating prostheses for rehabilitation aids.  Figure 1-4. Prosthetic Toe in Cairo Museum 
4 Modern prostheses are said to have or iginated from a man known as Ambroise Par (Figure 1-5). The French surgeon contributed to the origination and perfection of the amputation procedure itself and among the fi rst to show interest in the design of a functional prosthesis. Par instructed a Pari sian armor maker (Le petit Lorrain) to construct a metal above-knee prosthesis which consisted of a locking knee joint as well as an ankle joint. His prosthesis weighed 7 kg and was only suitabl e for equestrians. Functional prostheses were not used at that time mainly because the distal end of the residual limb could not be loaded without da mage; this limited people to using crutches, peg legs or even crawling as means of locomotion.  Figure 1-5. Ambroise Pare: Founder of Prosthetics  After 1816, functional wooden prostheses were built which consisted of a mechanism which synchronized the motion of th e knee and ankle joints. This ingenious mechanism was invented by James Potts, who also is credited with the use of the trumpet socket. This Total-Surface-Bearing-type socket along with the joint mechanism was made famous by the Count of Uxbridge, also known as the Marquees of Anglesey who lost his leg in the Battle of Mont St. Jean in 1815. 
5 Over the course of history, large scale wars have directed government interests towards research and development of more efficient and functional prostheses. Following World War I, materials such as al uminum and rubber were being tested as alternative materials which led to the cu rrent research on space-age materials and mechanism designed to improve user comfor t, mechanical efficiency, and cosmetic symmetry. 1.1.2 Background Â– Compliant Mechanisms and Current Research Compliant mechanisms are mechanisms that gain some or all of their motion from the deflection of flexible segments . Compliant mechanisms store and release strain energy as they move. Input forces are requir ed to store strain energy and output forces can be provided when strain energy is rel eased. Most compliant mechanisms have an unstressed (or minimum energy) state which they naturally assume. In a bistable mechanism (a mechanism which contains two stable equilibrium positions), the mechanism has two distinct locally minimu m energy states. A bistable mechanism will oppose forces that drive the mechanism from eith er one of the stable positions. Bistable mechanisms make sense for prosthetic knees because they offer two home positions (straight and bent) for the leg. The straight -leg position is the preferred home position while walking or standing, and the bent-leg position is the preferred home position when sitting. Furthermore, the change in stored strain energy between stable points offers a characteristic moment-rotation profile or Â‘f eelÂ’ to leg motion, thus increasing user proprioception over the position of their knee and lower leg. Other potential advantages
6 of compliant mechanisms are the lower cost s of manufacturing and assembly through lower part count as well as the reduction of weight when compared with rigid-body counterparts. Compliant mechanisms are used by th e public everyday, and a few are so commonplace that their compliance is consid ered unremarkable. The paperclip and shampoo bottle cap are examples of such Â‘ unremarkableÂ’ compliant mechanisms. The paperclip utilizes stored strain energy to hol d paper together by attempting to return to its original shape. The shampoo bottle incorpor ates small plastic flexures known as living hinges in the cap. These are some of the si mplest forms of compliant mechanisms. Figure 1-6. Common Compliant Mechanisms Other more advanced mechanisms can be designed compliant or can be transformed via compliant mechanism synthe sis. The crimping device shown in Figure 1-7 (a) is very similar to its rigid-body count erpart (Figure 1-7 b). The locking jaws serve similar functions, while the weaker materi al used to construct the compliant version limits its applicability. The plastic construction of the compliant crimping device limits its maximum output force, and its compliant members store some of the work provided by the input force in the form of strain en ergy making less work available for output at the jaws. As shown, the design of th e crimping device was based upon a rigid body mechanism which has four separate parts, but is realized using a single monolithic part.
7 (a) (b) Figure 1-7. Crimping Mechanism, Compliant & Rigid-Body Counterpart Courtesy of the Compliant Mechanisms Resea rch Group (CMR) at Brigham Young University A reduction in part count is one of th e most noticeable differences between compliant mechanisms and their rigid-body co unterparts. The Compliant Mechanisms Research group at Brigham Young Univers ity designed and prototyped an overrunning clutch using only two links and a pin. Figure 18 (a) shows the latter, while (b) depicts its rigid-body counterpart which has a si gnificant increase in part count. (a) (b) Figure 1-8. Overrunning Clutch, Compliant & Rigid-Body Counterpart Courtesy of the Compliant Mechanisms Resea rch Group (CMR) at Brigham Young University
8 Compliant mechanisms have advantages and disadvantages when compared with their rigid-body counterparts, whose impor tance depends upon the requirements of a given application. For example, some appli cations have requirements for high precision, some for high strength, and some require both. Both of these requirements have been demonstrated in compliant mechanism design. For example, the concept of high strength has been demonstrated in High Compre ssion Compliant Mechanisms (HCCMs) by Alexandre Gurinot in his desi gn of a compliant prosthetic kn ee  (discussed in the next section). High precision mechanisms have been applied to Micro-Electromechanical systems (MEMS). The prosthesis industry is a recent target for compliant mechanism designs, which are discusse d in the next section. 1.1.3 Background Â– Compliant Mechanism Prosthetic Joint Research Compliant mechanisms have made the transition to prosthetics joint research. For example, compliant prosthetic knees have b een researched at The University of South Florida under the direction of Dr. Craig Lu sk  and at Brigham Young UniversityÂ’s CMR under Dr. Larry Howell . A complia nt prosthetic ankle was designed and analyzed at BYU by Jason Wiersdorf  under Dr. Howell and Dr. Magleby. The introduction of compliant mechanis ms under loading more appropriate to rigid-body mechanisms is a challenging task and must be done under heavy scrutiny. Prosthetic knees and ankles see very large compressive loads which are not suited for compliant mechanisms. Theories have been developed to alleviate these major design issues and are discussed.
9 Prosthesis design and engineering has ma de transitions from new materials to exotic mechanism design (including CPU contro l), and has traditionally been constructed to withstand any and all buckling of memb ers comprising the mechanisms. Compliant mechanism design is counter to the concept of th e rigid structure as they gain all of their motion from the bending/buckling of the comp liant members. Nature employs compliant structures to provide both movement and strength. Ligaments are made of flexible, fibrous tissue which binds bones together, an d helps form the joints necessary for locomotion and movement. A common misp erception is that strength and safety necessarily go hand-in-hand with stiffness. This is one reason why the prosthesis industry is dominated by rigi d-body mechanisms which use pins and friction rather than compliant parts. The concept that stiffness equals strength is, in fact, incorrect as a healthy biological knee shows. It is quite contrary to the Â‘stiffness equals safetyÂ’ argument since as a knee gets stiffer, a decreas e in function is noticed (i.e. arthritis). Â“This design preference can largely be attribut ed to the long legacy of design for force loads rather than design for displacement loads that has influenced the engineering communityÂ” . Prosthetic knees are designed to meet stri ct safety criteria and must be able to withstand high compressive loading. On the other hand, compliant mechanisms are more typically designed under tensile loads rather th an the compressive ones that the knee joint sees. Work done by Alexandre Gurinot [5,6] with High Compression Compliant Mechanisms (HCCMs) have opened new door s to the applicability of compliant mechanisms to high compression situations sim ilar to those faced in the prosthetic knee
10 joint. He laid the groundwork for design of compliant mechanisms which can carry high compressive loading by using two design pr inciples: inversion and isolation. Inversion is the ability of the compliant mechanism to Â‘invertÂ’ a compressive load into a tensile load by the desi gn of the mechanismÂ’s geometry. Â“The concept of inversion builds on the proposition of tensurial pivots, which are fle xures loaded in tensionÂ” . The geometry of the rigid links invert the top and bottom of the mechanism thus transforming the load more appropriately for a compliant mechanism. Figure 1-9 depicts one of GurnotÂ’s inversion concepts of a knee prototype. Notice the top and bottom brackets invert the loading and thus allow the compliant segm ents to see a tensile load rather than compressive. (a) (b) Figure 1-9. GurinotÂ’s Inversion HCCM Concept (a) Compressive Configuration, (b) Inverted Tensile Configuration  The second principle Gurinot discusses is the concept of isol ation. Isolation is the ability to remove th e load from the flexible segments and redirect it through the rigidbody segments. Isolation can be applied when compressive loads are in alignment. In prosthetic knees, isolation will allow the co mpliant knee to withst and the stance loading
11 while Â‘feelingÂ’ rigid and Â‘strongÂ’ to the user while at the same time during motion the compliance is unchanged and fully effective as a compliant mechanism. The true advantage of isolation is to harness the stiffness of the rigid body mechanism while still utilizing the flexibility of the compliant mechanism, thus increasing the overall compressive load capability of the compliant mechanism. Figure 1-10. GurinotÂ’s Isolation HCCM Concept  GurinotÂ’s design of a compliant knee jo int included these con cepts of inversion and isolation and was successful in supporting heavy compressive loads. Under testing, the knee, shown in Figure 1-11, was able to withstand close to 700 lbf in compression with roughly a mere 0.14-0.15 inches of disp lacement . The success of a compliant mechanism being able to hold such high levels of compressive loads has been tested against the inversion and isol ation theories and proved to be highly successful. These HCCM concepts are crucial for a fully compliant knee joint to be able to withstand the loading during the stance phase of gait (discussed later).
12 Figure 1-11. GurinotÂ’s Tested Inverted Cross-Axis Flexural Pivot Knee Prototype  Further compliant knee joint research wa s conducted at the University of South Florida by Sebastian Mahler, under the direction of Dr. Craig Lusk . Mahler designed and prototyped a pediatric prosthet ic knee that introduced compliance into the mechanism shown in Figure 1-12. The major in fluential factor driving the design of a compliant pediatric prosthetic knee was the ove rall reduction in weight allowing the child to wear their prosthesis for longer pe riods of time. Children with above-knee amputations are typically given a peg leg to learn to walk on. The prosthetic leg must be shorter than the sound limb in order to cl ear the ground during swi ng, but this creates a gait pattern similar to walking with Â“one foot constantly in a holeÂ” [ 10]. These major gait deviations are exacerbated later in life when learned at an early stage. The lighter knee, and thus a lighter prosthesis, allows the child to wear thei r prosthesis for longer periods of time without the discomfort of heavier pr ostheses. With longer wear, the child can learn to walk with a standard polycentric knee similar to that of an adult prosthesis, thus lowering or eliminating the ga it deviations early.
13 Figure 1-12. MahlerÂ’s Pediatric Prosthetic Knee Prototype  Mahler was able to analyze the motion of the knee prototype by using nonlinear finite elements analysis and the calculation of the mechanisms instant center of rotation. The reaction forces and resultant mechan ismÂ’s stresses were also analyzed under deflections from 0 to 120. MahlerÂ’s wo rk focussed heavily on the concept of the instantaneous center of rotation. The instantaneous center (IC) of rotation is defined as a Â‘key pointÂ’ where the body rotates about at a particular instant in time. This IC is at rest and is the only point at rest in the body at this particular instant. Mahler explains how the instant center of rotation and the stability of a pros thetic knee go hand in hand. A Â‘well placedÂ’ IC can give the prosthesis adequate to e clearance as well as provide the necessary trade-off from stability to control (discusse d in detail later in the chapter). The instantaneous center of rotation is crucial po int of design when considering polycentric prosthetic knee mechanisms, a mechanism with a varying IC through rotation. For a simpler single axis knee mechanism, the IC is constant and does not lend advantages such as those listed above.
14 Figure 1-13. MahlerÂ’s Knee Instantaneous Center  Mahler explains the four most importan t design characteristics for a pediatric prosthetic knee: toe clearance, stability, lightweight and adju stability. The toe clearance, and stability were analyzed under the nonlinea r FEA, while the lightweight requirement was met with the compliant mechanism design. Adjustability was one of the foremost design challenges met with MahlerÂ’s pediatric compliant knee prototype. Adjustability of a prosthesis holds a high le vel of importance based upon the fact that no two people are exactly alike. Size and shape differences vary the gait pattern slightly from one individual to another, thus requiring the need for prosthesis adjustab ility. Mahler posed a design which could adjust the required torque necessary to initiate motion of the knee, thus allowing for differences in the childÂ’s act ivity level. The latter goes so far as to allow Â‘on-siteÂ’ adjustability allo wing the prosthesis to be set for standard walking and to be adjusted immediately for a higher level of activity.
15 The knee was evaluated at different complia nt segment angles, i.e. at different levels of adjustments. The stresses and fo rce data was evaluated for the mechanism at these different points. Stresses appeared to be higher than the mate rials yield strength and thus a method for removing or redirecting th ese stresses is needed in future work. These stresses brought about by prescribed compressive loading could be alleviated utilizing one or both of GurinotÂ’s theories inversion and isolat ion, thus improving and perhaps perfecting a pediatric compliant prosthetic knee. Compliant joint research also evaluated a prosthetic ankle joint with three degrees of freedom (the knee consists of just one degree of freedom). Jason Wiersdorf researched this project under the direction of Dr. Magle by at BYUÂ’s CMR . While this projectÂ’s emphasis is different than this thesisÂ’s, it is important to note that prosthesis research has been developed for other appli cations than the knee joint. Figure 1-14. WiersdorfÂ’s Modular Experimental Research Ankle (MERA) 
16 1.2 Phases of Gait Gait, or the means of forward locomoti on, has been standardized and broken into two distinct phases, stance and swing. Popul ar conventions have denoted particular points in the gait cycle by pe rcentages. These percentages follow symmetry with one heel strike of a limb denoted 0% and the h eel strike of the same limb as 100%. Each phase of gait can thus be characterized by a percentage of the cycle; stance accounts for the majority of the gait cycle with 60%, and swing owning the re maining 40%. Each phase of gait holds characteristics unique and easily de finable.  Stance includes four Â‘sub-phasesÂ’: loadi ng, midstance, terminal stance and preswing or toe-off. Loading refers to the porti on of stance just at and following heel strike when the alignment of the hip, knee and a nkle allow loading of the foot. Loading accounts for the first 10% of gait and is also defined as the period from heel strike to contralateral toe-off, depicted in Figure 115 (a & b). Some include a separate subsection just before loading and label it the in itial heel strike. Midstance refers to the loading of the full body weight on one leg, the knee is slightly bent and the ankle is in the neutral position, Figure 115 (c). Terminal stance is the progression of the weight line through the ball of the foot, ante rior to the knee and posterior to the hip. Terminal stance also includes what some have labeled heel-off from observational analysis and is depicted in Figure 1-15 (d). Midstance and terminal stance account for the next 40% of the gait cycle (10%-50% respectively), and overall is characterized by an external rotation of the entire lower limb with respect to the line of progress. Pre-swing, commonly known as toe-off is the portion of stance when the weight line passes from the ball of the foot to the toes, causing the knee to bend and the weight line running closer through the knee and
17 hip together, Figure 1-15 (e). Toe-off ends at toe-lift and thus begins the next portion of the gait cycle, the swing phase. [19,30] (a) (b) (c) (d) (e) Figure 1-15. Sub-Phases of Stance Mahler  Red line is the weight line, and the black lines represent upper and lower leg and foot Just as the stance phase is broken into sub-phases, so is the swing phase. There are three distinct sub-phases during swing: init ial swing, mid-swing and terminal swing, shown in Figure 1-16. The swing phase is 40% of the entire cycle and is critical when analyzing the dynamics of gait. The initial swing begins fo llowing toe-off of the stance phase and continues until the knee reaches its maximum flexion of 60 degrees. The primary purpose for the initial swing is to clear the foot, meani ng that tripping or stubbing of the toe is avoided, and prepare fo r swing. Clearance is achieved through flexion of the hip, knee and ankle. Follo wing maximum knee flexion and the initial swing phase, mid-swing begins from ma ximum knee flexion until the tibia is perpendicular to the ground. Finally, terminal swing fini shes the swing phase from perpendicular tibia location to initial heel contact with the ground, thus starting the stance phase again.
18 Figure 1-16. Swing Phase of Gait Normal gait holds key features which must be mimicked in prosthetic design. To prevent excessive heel rise and to initiate the forward swing of the leg, the quadriceps contract before toe-off. To dampen forw ard motion of the leg at terminal swing and control where the foot is just prior to heel strike, the hamstring muscles become active. In order to achieve the latter, prostheses have introduced several design features including constant friction, hydraulic and pne umatic dampers as well as other high technological options such as CPU control. Toe cleara nce during swing is also a challenge; during normal gait, ankle dorsiflex ion gives clearance but in the case of an amputee, the muscles are not present and the knee prosthesis or combination of knee and ankle prostheses must provide the necessary clearance to preven t stubbing the toe and tripping. These characteristics of normal gait must be include d in the engineering of a prosthesis that is fully suitable to sust ain as close to normal gait as possible. 1.3 Knee Disarticulation A disarticulation is the amputation of a limb through the joint without cutting of the bone. The disarticulation of the knee is a surgery that is done between bone surfaces removing the tibia and fibula while either keep ing or removing the knee cap (which is the
19 judgment of the surgeon). Knee disarticul ations are considered somewhat rare and account for only about two percent of major lim b loss within North America. The first knee disarticulation in the United States wa s performed in 1824 and since has received strong support as well as strict skepticism.  1.3.1 Advantages and Disadvantage s of Knee Disarticulation Disadvantages of the knee disarticulat ion lie within function and cosmetic rationale. Earlier in the development of th e knee and ankle disart iculations (1800Â’s) a drop in mortality rates were of utmost importance as the disarticulation decreased infection, bleeding and surgical shock. Modern day healthcare and surgical procedures have decreased the aforementioned mortality ra tes for all amputations and therefor can no longer be considered the deciding factor in the surgeonÂ’s decision. Why then, if the knee disarticulation was so popular when first introduced is ther e skepticism now? Primarily, complaints have been made based upon the prosthesis fit and the bulbous distal end of the residuum. A particular paper written in 1940 by Dr. S. Perry R ogers, an orthopedic surgeon with a knee disarticulation (from a wa r injury), highlighted the differing opinions on the amputation. He based the divided opinion on Â“erroneous conclusions by some physicians and prosthetistsÂ” , noti ng the Association of Artificial Limb Manufacturers of America claiming that kn ee disarticulations were Â“impeding to successful prosthesisÂ” . Objecting to this statement, Dr. Rogers claimed that it was Â“no longer grounded in factÂ” . The claim th at the bulbous shape of the distal end of the residual limb was a problem to the patien t was also addressed by Dr. Rogers whose
20 photographic evidence proved that the femoral lower extremity proved to assist in the lifting of the prosthesis as well as increas e control over the rotation. Still, many people object to the disarticul ation based upon cosmetic reasoni ng that the bulbous end of the residual limb was unappealing. The bulbous end of the residuum caused issues relating to function as well; creating a socket with the co rrect fit was challenging, even to the point that some prosthetists were reluctant to make one fearing an unsuccessful fitting. Dr. Rogers commented on this as well stating that the bulbous end essentially makes the socket Â“self-suspendingÂ” . Amongs t the cosmetic downside of the knee disarticulation, many people with the amput ation note the longer thigh length of the residuum with prosthesis ove r the sound leg. The residu um, distal padding, socket, connector and knee unit add a few inches to the overall length thus creating a nonsymmetric appearance while sitting. Four-b ar prosthetic knees (polycentric) reduce the overall length of the amputated limb, but not completely. Figure 1-17 depicts the notable differences in distance from the distal end to the prosthetic knee center (note the right picture is of a polycentric knee). Sitting is cosmetically asymmetric, but standing also has its cosmetic symmetry issues that some dislike. When standing the knee center of the residual limb is a few inches closer to the gr ound which some say is a problem. As noted by Dr. Smith, Â“as long as the prosthesis is design ed so that the total le ngth of both legs is equal and the hips remain level, the back can be straight, and for many there is no discomfortÂ” . Noting that over sixty year s has past since the release of Dr. RogersÂ’ paper, controversy over the drawbacks of th e knee disarticulation still remain and are discussed today.
21 (a) (b) Figure 1-17. Distances from Distal End to Prosthetic Knee Center (a) Higher transfemoral (TF) amputation (b) lower TF amputation with polycentric knee Image by USF College of Medicine School of Physical Therapy and Rehabilitation Sciences  Advantages of the knee disarticulation over the transfemoral counterpart lie within both functional and surgical rationale Many individuals unfortunate enough to require lower limb amputation near the knee joint are fortunate enough to hold the option of a transtibial (below-knee) amputation thus leaving the knee intact. For some, there is no choice but to amputate higher up the thi gh and through the femur. Though rare in comparison and controversial, the knee disartic ulation may be the best option for several groups of individuals: Children Cancer/Trauma Patients Spasticity Patients Children benefit from the knee disartic ulation over transfemoral simply by preserving the growth plates located at the e nds of the femur. The bottom growth plate accounts for the majority of femurÂ’s growth a nd with the leg being amputated through the joint the plate is preserved and the femur able to grow through the childÂ’s life. If the child undergoes a transfemoral amputa tion, the residual limb though long when
22 amputated will result in a shorter residuum as an adult. The growth of the femur without the growth plate would not be able to keep pace with the sound leg and thus result in a short residuum during adulthood. The knee disa rticulation also eliminates the childhood condition of painful bone overgrowth, which is a result of new bone growth that forms a spike or bone spur at the amputated end afte r the bone is transected . Cancer or trauma patients undergo a knee disarticulation if the tibia cannot be saved and the soft tissue that would be located at the distal end is good for Â“paddingÂ” . Patients suffering from problems with sp asticity or contractures, wh ich typically are results of spinal cord or brain injuries, can leave their legs in a bent position and are susceptible to being fixed in that position. In these particul ar cases, Â“the knee disarticulation can offer some unique advantages over either a tr anstibial or transf emoral (above-knee) amputationÂ” . One of the most notable advantages of the knee disarticulati on over transfemoral is the remaining muscle that is left intact. A full-length femur is left and the thigh muscles tend to be stronger because they are not transected in the middle of the muscle but rather at the end where th ere is fascia (connecting tissue) Muscles that are dissected mid-length tend to become swollen, need more time to heal, retract and never quite regain the strength. The knee disarticulation is typi cally an end loading (weightbearing at the distal end) amputation and provides a long mechanical lever-arm with the maximum amount of muscle present to provide neces sary moments to control the prosthesis adequately (this is discusse d further in section 1-4).
23 1.4 Prosthetic Knee Inherent Stability To better understand the required stability needed for a particular patient (over another patient with a different level of tran sfemoral amputation), the concept of torque must be mastered. In physics, torque, also known as a moment, is the measure of Â“the tendency of a force to rotate an object abou t some axis (center)Â” . Torque can be quantified by the product of a force and the leng th of the lever arm to which it is applied to the body. In simpler terms, torque is equal to force time s distance. The force applied on a residual limb is directed and applied by the remaining muscles of the residuum. The length of the Â‘lever armÂ’ is the length of th e femur (with an above knee amputation). It is interesting to note that the length of the re sidual femur affects both the force and lever arm because the longer the residuum, the more residual musculature; therefore the length of the femur determines the amount of torque a patient can apply and the more control they will have. USF O&P  describes an example which illustrates this idea; a short transfemoral limb will require a larger prosthesis, thus having a higher mass, and is placed at a shorter lever length. The concept of Â“inherent stabilityÂ”  is based upon the type of prosthetic knee used and the Â“alignment or position of the knees COR (Center of Rotation) relative to the TKA (trochanter-knee-ankle) weig ht line. The type of prosthetic knee determines the ability of the prosthes is to allow or withstand buckling, either during swing or stance. This is a crucial part of the knee classification, but the concept of control versus stability focuses around residuumÂ’s torque capabilities and this idea of alignment. With a long transfemoral amputation (e.g. knee disarticulatio n), the TKA weight lin e falls posterior to the knees COR and thus is in an unstable position. With this unstable position, the
24 patient must have the ability to have more control over the prosthesis. With the greater amount of residual musculature, this control is easier than with a shorter transfemoral amputation. Those with the knee disarticulation seem to prefer to have more control over their prosthesis rather than have it heavily st able . A shorter transfemoral amputation requires more stability then a knee disarticula tion as the residuum would have less ability to control the prosthesis (less muscle presen t). The TKA weight line would need to lie anterior to the kneeÂ’s COR to withstand ro tation during loading thus increasing the stability during stance. Fi gure 1-18, from a presentation put together by Dr. Jason Highsmith and Dr. Jason Kahle  depicts the co ncept of inherent stab ility versus control and how they relate to residual limb lengths. Figure 1-18. Stability vs. Control Image by USF College of Medicine School of Physical Therapy and Rehabilitation Sciences 
25 Chapter 2 Prosthetic Knee Classifications The prosthetic knee market is saturated with over 200 different knee joints from dozens of manufacturers and each year that number grows. With the abundance of knee mechanisms it makes it very difficult for the prosthetist to choose the Â‘correctÂ’ knee for the user as there is typically more than one knee which is appropria te for a particular application. The reason behind such large num bers of knee designs can be attributed to two different explanations: designerÂ’s choice and contradictory dema nds made by users. A newly designed prosthetic knee is difficu lt and expensive to evaluate, typically requiring time-consuming experimentation and clinical trials. Classification of a prosthetic knee is a technical process and is done in several different ways. In this chapter, the following classification scheme s are described: func tion-based schemes, mechanical-design-based schemes, and scheme s based on the level of amputation of the user. The tradeoffs between stability and control are also described. 2.1 Classification Â– Functional Dr. ir. P.G. van de Veen  descri bes two subcategorie s under the functional classification of knee prostheses: locking an d braking mechanisms. Each of these types has a unique characteristic that makes them mo re suitable for different environments as
26 well as different levels of user activ ity. Table 2-1 summarizes the functional classification of knee mechanisms and gives a few examples of each. Table 2-1. Functional Classification Examples Locking: Continuously Locking Automatically Locking Geometrically Locking Brake: Load-Dependent Load-Independent Locking mechanisms mechanically rest rict all motion (while in the locked position), regardless of the forces applied (neglecting those which cause mechanical failure). As mentioned, there are three di fferent locking mechanisms which restrict flexion. The first is the continuously locki ng mechanism which is the simplest form of the locking prosthetic knee. The continuous lock is a manual lock which is enabled or disengaged by a user command alone, such as pushing a button. The second, the automatically-locking mechanism, applies restriction through th e knee joint when triggered by either position, load or during a particular input response (flexion of the foot/ankle or other means). The automatica lly locking knee also includes a point at which the mechanism Â‘unlocksÂ’ and is able to flex. Finally, the geometrically locking knee utilizes the knees center of rotation (COR) to lock the mechanism. The knee is able to lock if the kneeÂ’s COR lies posterior to the weight line (or load line) during all instances and circumstances. Only when the loading is removed from the knee is it able to flex. Locking knees are worn by those w ho require the highest le vel of stability, but many who ambulate with such knees develop gait abnormalities similar to the hiking of
27 the hip to compensate for the lack of knee flexion (and thus the inability of the leg to shorten through initial swing). Braking mechanisms provide a Â“flexioncounteracting momentÂ”  to prevent rapid flexion. While this applied moment can be large, it will ne ver be infinite and therefore cannot prevent motion completely (lik e that of the locking mechanisms above). As listed, two functional braking mechanisms are prominent on the market: loaddependent and independent brake mechanisms The load-dependent braking mechanism is a friction brake that exerts a counteracti ng moment that is proportional to the loading on it. Usually, motion is prevented, but is done so by the equilibrium of forces and not a locking mechanism. The load-independent braking mechanism provides counteracting forces that are independent of the applied loading but rather to the speed of rotation (flexion). Load-independent braking knees offe r more controlled flexion rather than the strict stability offered by the locking mechanisms.  2.2 Classification Â– Mechanical The mechanical classification system fo cuses primarily on the type of linkagebased mechanism the knee employs. Prosth etic knees can be broken into three mechanical categories: single-axis knee mechanisms, multiple-axis (polycentric) knee mechanisms and Â‘exoticÂ’ knee mechanisms. Single-axis knee mechanisms tend to be the simplest models, and have a wide range of applicability. There are several t ypes of single-axis knees which incorporate additional features like manual locks or hydra ulic cylinders. Singleaxis knees tend to
28 work well with friction, either constant or va riable, introduced into the mechanism, which allows for user comfort and safety . The single-axis constant-friction knee is rare in comparison to most prostheses on the market It is typically designed and limited for pediatric users as it is very dur able and light in weight. The design is simple and is ideal for children. Figure 2-1 shows an example of a single-axis cons tant-friction knee manufactured by Ossur. While constant-friction single-axis knees are limited in number, there are several single-axis knees constructe d with variable fric tion. Microprocessor knees, SNS, pneumatic and other forms of kne e designs incorporate the idea of variable friction into the knee mechanism (shown in Figure 2-2). Figure 2-1. Constant Friction Single Axis Knee by Ossur  OSSUR OTTO BOCK Figure 2-2. Variable Friction Single Axis Knee [13,14] Multiple-axis knee mechanisms are characterized by the number of links present in the system. Utilizing multiple links, the engineer can alter the location of the instant
29 center of rotation and thus the motion of th e shank in comparison to the residuum. Manual locks and condylar mechanisms are also incorporated into these types of knees. This thesis focuses on a polycentric f our-bar knee manufactur ed by Otto Bock. Polycentric is a term which refers to the instant center of rotation of the mechanism and is used primarily to allow for the toe to clea r the ground during the sw ing phase (discussed later). Figure 2-3. Multiple Axial Knee Mechanisms  Exotic knees are a classification which is given to those knees which do not Â‘neatlyÂ’ fit into one of the ot her two mechanical cl assification systems. These knees can be either single axis, multiple axial or some other type not yet discussed. The exotic approach is new and upcoming a nd is not widely applied as mo st have yet to be tested rigorously enough to be app lied widely as of yet. Table 2-2. Mechanical Classification Breakdown Single Axis Knee: Manual Lock Backward Center of Rotation Friction Brake Hydraulic Cylinder Multiple Axial Knee Manual Lock Condylar Mechanisms 3 Bar, 4,5,6 and 7 Bar Mechanisms Exotic Knee Single or Multiple Axial Knees
30 2.3 User Aspects of Swing and Stance User aspects of prostheses define the necessary attributes of prosthetic knees especially, and sifts knees into finely differe ntiated categories. These categories enable the prosthetist to confidently fit a patient knowing that the knee will meet safety and appropriateness criteria during both stance phase and swing phase. The criteria for determining safety and appropriateness for each of the sub-gait categories used by the prosthetist depend on the activity level and abilities of the patient. The safety of a knee during the stance phase is determined by its stability. Stability refers to the ability of the prosthes is to support its user without buck ling, and is one of the first attributes of the prosthesis noticed by its user. If the prosthesis does not feel stable to the user duri ng stance, rejection is common. Typically, mo re active patients can tolerate lower levels of stability because th ey are better able to control their residual limb. Also, as mentioned, those with a larger residuum musculature have the ability to apply larger torques and are better suited for le ss stable knees. The prostheses of more active patients see more use and long-term wear, and the reliability or long term performance becomes a greater concern. Stability is a necessary part of safe knee performance, but adjustment of the kneeÂ’s stability is also importa nt. The knee must be able to initiate swing phase without much difficulty. There must also be so me flexion under loading, which itself seems counterintuitive to the stability argument. Normal gait includes small knee flexion at heel strike. This flexion serves several purpos es: reduce the initial shock brought on by heel strike and reduces the vertical body center oscillation thus reduci ng energy expended.
31 The behavior of the knee during swing pha se is also very important for user success with the prosthesis. It is important to note that the vast majority of prosthetic knees are passive joints they do not add any energy to the amputees walking cycle. The swing phase is initiated upon motion of the sh ank to the posterior. The knee joint must prevent excessive heel rise as this causes delays during the exte nsion phase which can result in the loss of user comf ort and confidence as well as increase falling rate when heel strike is not synchronized with shank position. In what is known as mid-swing phase, the shank moves anteriorly under the influence of gravity, inertia and an extension assist device. The motion of the prosthetic shank moves more slowly than the sound limb during extension, thus requiring an extensi on aid. The introduction of this extension device poses other issues which must be reso lved; the extension aid increases terminal impact of the kneeÂ’s linkage system on the hyperextension stop. In short the knee joint must meet the following criteria relating to the swing phase: Dampen flexion to prevent excessive heel rise. Assist extension. Dampen terminal impact at end of extension phase. There are also generalized needs of the us ers which the knee must also satisfy; the prosthesis is used not only for ambulati on but also for everyday activities such as kneeling, sitting and others like driving a car. All of thes e activities require the knee to bend in a manner that does not impose disc omfort or restriction on the user. Cosmetically, during sitting the knee must not protrude far beyond the sound limb. As discussed previously, polycentric 4-bar knees are designed to meet this need. It is
32 important to note these characteristics of the prosthetic gait in terms of the users needs as this typically determines the success of the amputee with his/her pros thesis (rather than the prosthetic limbÂ’s success). 2.4 Medicare Functional Modifier System The medicare functional modifier system (MFMS) of prosthetic knees (and feet) is unique over the other classi fication methods/systems discu ssed in that it evaluates the usersÂ’ abilities and needs to fit them with the Â‘most appropriateÂ’ prosthesis. Up to now the prosthesis itself and the mechanism have be en evaluated in order to classify them for need, but as mentioned, the MFMS evaluates the amputee for their abilities and activity levels, thus creating a prosthesis that would best fit their everyday activities. The MFMS is broken into K-scores ranging from K0 to K4 each having its own designations for activity and ability levels associat ed with everyday activities. 2.4.1 K-Scores The K-score is assigned by a prosthetist, and as menti oned, determines the level of activity and the appropriateness of a prosth esis for an amputee. The lowest K-score is the K0 level; the K0 score is indicative of an amputee who does not have the ability or the potential to ambulate safely either with or without assi stance, and a prosthesis would not enhance the quality of life. The K0 le vel patient is not a candidate for either a prosthetic knee or foot and would therefor be limited to mobility via wheelchair.  
33 The K1 level patient shows the ability to ambulate or transfer safely with a prosthesis and has limited (and sometimes unlimited) household use. The amputee can ambulate on level surfaces with a fixed gait speed (cadence). This level is indicative of an amputee who uses their prosthesis for ther apeutic purposes and is a candidate for the basic prosthetic knees and feet. [7,19] An amputee showing the ability to be a community ambulator and is able to negotiate low-level environmenta l barriers such as curbs, ra mps, stairs and small uneven surfaces is designated the K2 score of the MFMS Those able to perform to this level of activity are candidates for higher levels of pr osthetic feet (i.e. multi-axial) and basic prosthetic knees. [7,19] K3 level individuals show the ability to traverse most environmental barriers and are considered a community ambulator. Th ey are also able to uphold or have the potential to ambulate at a variable cadence, and may have the therapeutic, recreational or exercise activity that demands prosthesis us e beyond that of the simple locomotion. In order to perform up to this patientÂ’s level of activity, higher end prostheses are used such as dynamic response feet and fluid/pneumatic knees. [7,19] Finally, the highest level of activity is in dicated by the K4 score and is typically assigned to children, bilateral cases, active ad ults and athletes. Th ese individuals have the ability (or potential) for hi gher levels of ambulation that possess high impact, stress or energy levels. These amputees are candidates for all the prostheses on the market and are considered to have high levels of control and ability [7,19]. Table 2-3 summarizes the MFMS K-score and the requirements of each.
34 Table 2-3. MFMS K-Scores [7,19] K Score Amputee Activity Level Pr osthetic Knee Prosthetic Feet K0 Non-ambulator NONE NONE K1 Limited household use, level surfaces and fixed cadence Basic Basic K2 Community ambulator, able to traverse low-level boundaries Basic Multi-axial & alike K3 Environmental barriers at variable cadence Fluid/pneumatic Dynamic response K4 Children, Bilateral Cases, Active Adults and Athletes ALL ALL
35 Chapter 3 Interface Mechanics Literature Review The technological advance of lower-limb prostheses has been rapid over the past several years. Recent advances in prostheses have occurred in the materials used to construct the prosthetic limbs, the comple x systems of knees with CPU controlled motion, and the interaction between prosthetic foot and ground. Current research that is being applied for the advancement of pros theses, both in manufacturing and patient adaptability, has been primarily done with in the Â“commercial sector: new suspension options, innovative socket conf igurations, advances in kne e mechanisms, and guidelines for prescription and reimbursement of prosthesesÂ” . Zahedi  reports an Â“overall amputee satisfactionÂ” varying 70-75% among po lled patients, while a 20% reduction in patient care budget was reported. Computer-aided technology has advanced the manufacturing of the prosthesis tremendously; what took days is now conceived in hours. The prosthetic socket is most affected by the introduc tion of computer-aided manufactur ing. In practice, prosthetists form the residuum geometry via plaster molds (typical), and then cr eate the prosthetic socket around the limb geometry. This practice requires much skill and experience as it is typically a trial and error method. The patient makes a coupl e of visits for this method of manufacturing, and sometimes even more if the prosthetistÂ’s desi red fit does not match at first. Engineers have propos ed an interactive lab for the pr osthetist in which he/she can form the geometry in CAD-Space and from there, a lathe receives geometric inputs from
36 the CAD-file and carves Â“a positiv e of the socket from a plaste r composite materialÂ” . Finally, the socket is created by vacuum forming a piece of polypropylene over the positive socket cut. While fit adjustments and design alterati on considerations are always present, correct fitting between the prosthetic sock et and the patientÂ’s residual limb has the following consequences: It prevents further injury to the residuum via an inflammatory response (followed by necrosis). It allows the patient sufficient control of the prosthetic limb. It enhances the patientÂ’s comfort. These are generalized concepts which can le ad to a successful prosthetic limb. The socket is the starting point for any pros thesis design phase, primarily because if the patient-prosthesis interface is not created to perfection, problems are inevitable. This chapter deals with the underlying prin ciples of the interaction between the patientÂ’s residual limb and the prosthetic socket (and liner) also referred to as the interface mechanics. Interface mechanics in these terms, re ference the interface stresses induced upon the residuum via the prosthes is and loading during ambulation. Shear stresses are felt as friction by the patient, and normal stresses correlat e with the pressure caused by stance and ambulation. Stress concentrated around the interface between a residual limb and a prosthetic socket is a crucial piece of information when designing the socket to an individual with an amputation. As mentioned, the prosthes is must be safe to the surrounding tissue, provide some sense of comfort to the individu al, and not fall off.
37 Finite element techniques have posed a po ssible route to unc overing the stresses on a modeled residual limb. These techniques can facilitate designing a socket which alleviates stresses which cause tissue trauma and/or discomfort to the patient, or designing a prosthesis which can optimize these stresses to better serve user control. Finite element techniques, in a nut-shell, allow for the small Â‘finiteÂ’ division of a complex geometry. This allows for geometries and loads which are very difficult to analyze via analytical methods to be brok en into smaller Â‘elementsÂ’ which can be analyzed. These techniques have been identifi ed as a tool to enab le the in-house lab to create an optimal prosthetic socket one which ensures the most control over the prosthesis as well as safety to the patient. This review encapsulates the ideas of interface mechanics, how they relate towards control and their importance within external prosthetics as well as the idealizations of finite element analysis, the assumptions and complications therein, which permit the creation of the Â‘optimalÂ’ prosthesis for each patient. 3.1 Finite Element Analysis Design The objective of the socket shaping process essentially is to Â“optimally distribute the interface stresses between the residual limb and socket while providing adequate stability and efficient control of the prosthes isÂ” . There are ot her design criteria besides the geometry of the socket which aff ect the overall stress distribution; material properties of the inner liner and socket wa ll also have significant influence.
38 Finite element analysis (FEA) is an e ngineering tool which has earned great respect within industry and research institu tions and is being incorporated within prosthetics in order to unders tand the Â“relevant biomechani cal rationale, especially the biomechanical interaction between the stum p and the socketÂ” . FEA is widely applied in engineering practice in order to obtain approxim ate analytical solutions to problems for which no simple closed-form solution exists. To initialize the model, the geometry wh ich represents the residuum and socket alike, is generated and divided into finite se gments (elements) which when put together is referred to as the element mesh. The nodes of the mesh are the points at which there are interface Â“verticesÂ” . These nodes are cruc ial in the design phase of modeling as they determine the slip parameters of the interface, which tells the program that the socket and residual limb are not one material and must allow slip as well as no tensile stresses to be induced. The method in which slip is impl emented differentiates between research approaches and is described later. FEA requires distinct knowledge of several overall features of the model itself. Several design characteristics are of critical importance, because of their affect on the accuracy of the model: The material properties of the soft tissues which Â“exhibit nonlinear and non-uniform behaviorÂ”.  The way that interface nodes between the socket wall and the residual limb are modeled. The accuracy of the residuum geometry: soft tissue, bone, and location.
39 The inclusion of pre-st resses within the soft tissue (as a result of wearing the prosthetic socket, Â‘snug fitÂ’/donni ng of the socket on the limb). Each of the above items has been simp lified in different ways by different researchers, which allows fo r variations in results lead ing to skepticism about the accuracy of FEA of external pr osthetic sockets (and interface mechanics). The variations in the research results are discussed later in this chapter. Finite element analysis, as it applies to wards interface mechanics, has progressed tremendously from only accounting for 2-dimens ional geometries with linear properties to now integrating 3-dimensional residual limb geometry and incorporating nonlinear tissue properties (bone, epidermis and other soft tissue) as well as pre-stressing of the epidermis due to the donning of the prosthesis. Other newly integrated approaches attempt to find better models by incorporat ing different distal-end boundary conditions . To summarize the key aspects of the Fin ite Element techniques, in order to have a working analysis, the inputs into the program are as follows: Geometries Element Properties Boundary Conditions Each of these inputs allows for the a pplication of different approaches and variations in the design and analysis of th e interface stresses, thus creating a need for model validation.
40 3.2 Finite Element Analysis Techniques Variations in the three major components of the FE model result in different model predictions. In the next few secti ons, the different approaches to interface mechanics are reviewed based on their de cisions in creating the FE model. 3.2.1 Geometry The model geometry is one of the more co mplex areas of focus within any FEA. Within interface mechanics the model geometry varies from researcher to researcher through many facets: interface methods, residuum modeling and interaction with fibula/tibia location within the residual lim b. The first two are debated within many papers of the field and are discussed here. The Â‘interface methodsÂ’ describe the type of methodology called upon to describe the interaction of the residuum epidermis a nd the socket liner and socket itself. Zachariah and Sanders  describe three diffe rent types of interact ion analysis, each of which is analyzed within this section: Totally Â“gluedÂ” interf ace , ,  Partially Â“gluedÂ” interface  Slip permitted at the interface 
41 22.214.171.124 Totally-Glued Interface The totally-glued interface is an assumpti on that the residual limb and the socket or socket liner (and socket) are modeled as sharing nodes. Sharing these nodes implies that no slip or separation is allowable and thus acts just as a glued interface would. Â“The interface stress estimated by the FE solution is the nodal stress at the set of common nodesÂ” . Zachariah and Sande rs  describe the main a dvantage of the totally-glued interface as its simplicity, both in setup and in computation as well as the low cost of the tools required to perform the computation. Brennan  used a model which employed the method of totally-glued interface between the skin and the socket in an above knee prosthetic socket. The socket was modeled as a rigid structure and no socket liner was employed in the model. The residuum epidermis was not modeled separate ly and thus shared the common nodes with the rigid socket wall. The PoissonÂ’s Ratio and YoungÂ’s Modulus were standardized (in reference literature), while the material be havior of the soft ti ssue was based upon other research noted in the paper. Brennan compared the data collected from the FE model to experimental data which was set up to meas ure the pressure at key points within a modified socket which held piezoresistive pr essure transducers in key locations within the socket wall. Sanders  also utilized the totally-glued in terface to make early assumptions to simplify analysis. While the method seem s common, Sanders did utilize a unique approach in material modeling; both fat, so ft tissue and muscle were included in the geometry of the residuum, trying to crea te a more accurate model of the residuum.
42 Reynolds [17,18] also employed the assump tion of a totally-glued interface in a patellar-tendon-bearing (PTB) below-knee sock et Â“to assess interface pressure sensitivity to socket rectification (socke t shape), tissue material proper ties (modulus), and alignment (force direction at the model boundary)Â” [ 33]. The idea behind the assessment of pressure sensitivity to changes in the socket shape is one of the driving forces behind the application of the finite elemen t approach in optimal socket rectification (as mentioned in the introduction). The application of this approach is one way to apply the in-house lab, which could revolutionize the prosthetic indus try and become a priceless tool for the prosthetist (beyond what it is currently doing). 126.96.36.199 Partially-Glued Interface The partially-glued interface is one which was first modeled as totally glued (the socket wall or wall and liner shared a co mmon node with the residuum thus creating the single geometry) but during th e post-processing of the FE A a noticeable tensile stress was identified and a modified model was created to eliminate the tension that was present. Different approaches to eliminati ng the tension are reported; creating separation between pairs (i.e. introducing discontinuities), or defining an extremely low socket modulus at the point of tens ion are both strategies for the partially-glued interface correction. Steege  reported the existence of the tension and thus utilized the partiallyglued interface assumption. Socket information was gathered using CT scans of patients wearing PTB socket with liner (methods of geometric formulation of the residuum is
43 discussed later in this section). Intere stingly, Steege modeled the cartilage as a completely different material than the rest introducing some of the nonlinearities. This method is generally ignored, as ma ny feel that the simplification of the totally glued interface may work for their application while other researchers tend to model the interface with more of a slip nature Â– allowing slip between surfaces and eliminating any tensile stresses induced (i.e. allow separation). The latter refers to the final interface method mentioned prev iously, slip permitted at interface. 188.8.131.52 Slip Permitted at Interface The method of allowing slip at the in terface between the socket and residuum is one which incorporates more complexities in the FE mesh and model. Different tools to incorporate the slip permitted interface are difficult and include slipÂ–elements which are introduced as either springs , coulomb frictional elements or by using FE add-ons which allow for the use of slip elements (ABAQUS v6.3, .). The concept of allowing slip is one wh ich is being approached more when modeling interface mechanics; in fact Zhang and Mak  were attempting to design a model to test whether the distal-end loadin g had much of an influence on the overall accuracy of the model. In doing so, they applied the slip-permitted interface method using ABAQUS. They modeled the nodes betw een the socket and residual limb to be separate which allowed slip and separation. Notice in Figure 3-1 (taken from Zhang and Mak ), that the residuum and socket ar e modeled separately a nd using ABAQUS are
44 allowed to separate and slide tangentially pa st one another while still contacting, creating frictional and normal stresses (interface stress es) that are crucial to the science. Figure 3-1. Mesh of Above-Knee Stump and Socket (Zhang and Mak) Zhang and MakÂ’s  rendering of (a) Â“Mesh based on a sagittal plane geometry of an above-knee stump and its socket.Â” And (b) Â“Int erface element consisting of nodes 1 to 4, nodes 1 and 2 on the skin surface and nodes 3 a nd 4 on the internal surface of the socketÂ”. Figure 3-1 illustrates the en tire residuum geometry in cluding: bony tissue, soft tissue, socket as well as the dist al-end boundary condition (discussed under Boundary Conditions ). Silver-Thorn  was mainly interested in determining the importance of the complexity of the residuum geometry to th e accuracy of the model. Three different models of varying the complexity (successi vely increasing the accuracy) were created and tested to determine the point at which simplification to the model is allowable without much tradeoff to the accuracy of the solution. PTB below-knee models were created for this analysis and were also modifi ed to include the joint spacing and cartilage (most considered rigid in the simpler model).
45 As the demand for the understanding of interface mechanics grows, this method of slip permission at the interface is beco ming more appreciated. Determining the route by which one applies this idea is what vari es researcher to researcher. Overall the understanding of slip is vital to the success and accuracy of the FE model in general. 3.2.2 Element Properties Employing the totally-glued interface assumption raises several questions about the material behavior. Fundamentally, knowin g that the socket and residuum epidermis have very different material properties, how does the totally-glued interface take this into account? This question led resear chers to try to understand th e material properties of the different tissues as well as the mate rial properties of the socket. Noticeably, in most models the material behavior of all the elements Â– bone, cartilage, soft tissue, liner and socket Â– we re assumed to be homogeneous, isotropic and linearly elastic. It has been shown thr ough extensive modeling that the material properties have an extensive impact on the ove rall stresses within the prosthetic socket. The material properties of the socket wall have effects on the overall stress distribution within the socket. Quesada  showed that decreasing the overall socket modulus and making the socket Â‘less stiffÂ’ decreased the normal stresses within the socket greatly. Decreasing the stiffness can also be achieved by cha nging the thickness. Quesada also showed that decreasing the thic kness of the socket did affect of stress within the socket greatly and therefore could be applied to situations where the normal stresses were too high. Silv er-Thorn  reporte d that the normal and shear stresses
46 within the socket wall were much more sens itive to the changes in the socket liner stiffness than to the stiffness of the socket its elf. These findings seem to be indicative of displacement loading. The socket liner stiffness al so has an effect on the stre sses within the prosthesis, but care must be taken not to be too liberal with the softening of the liner as there are tradeoffs. While decreasing the stiffness of th e liner eases the stresse s within the socket, too the same degree does the patient loose stabil ity in the prosthesis. A certain degree of stress is therefore required to maintain c ontrol of the prosthesis, while too much stress causes discomfort and even trauma to the surrounding tissues. The soft tissue of the residual limb al so shows impact on the predicted stresses within the socket. As the tissue grows t ougher it exaggerates the stresses within the socket. The shear stresses were shown to be affected more by the increased tissue stiffness than the normal stresses were. Skin (not to be confused with the soft tissue) was only mode led separately by Sanders . It was shown that the result of the increased stiffness of the skin was opposite to that of the soft tissue with rega rds to both shear and normal stresses. This may suggest that Â“membrane elements capable of transferring only tension may play an important role in th e FE modelÂ” . Bones need to be studied further to dete rmine whether the material properties vary with stress distribution. Curre ntly bones are usually modeled as rigid bodies, but due to some bending of the bones under loading led Steege  to use the properties of cortical bone to try and model the phenomena.
47 3.2.3 Boundary Conditions The application of boundary conditions has dramatic affects on the overall analysis of the model. One of the most prominent loading assumptions is made based upon the body weight of the person at the hip jo int. Typically an assumption (for stance) of one-half of the body weight is loaded dire ctly onto the femur (or is transmitted based upon gait location for belo w-knee prostheses). One very interesting analysis was c onducted by Zhang and Mak  testing whether the distal-end boundary condition had an effect on the interface stresses. Three models were used, one with no distal-end lo ading (modeled with an air gap between the distal end of the residual limb and the bottom of the gap, Model A), one with full contact between the distal-end and the socket (Model B) and a final model with an air gap simulating a partial loading of the distal end (suction socket with sealed air, Model C). Included here is a Figure which Zhang and Mak  used to describe the loading and is depicted here as Figure 3-2 not only for its explanation of th e distal-end loading condition. Figure 3-2. Distal-End Boundary Conditions Image by Zhang and Mak 
48 3.3 Modeling the Residual Limb Modeling the residual limb in order to input it into the FE model can be costly if done in detail or simple if appropriate a ssumptions are made. Computed Tomography (CT) is one of the medical approaches atte mpted to model the internal tissues of the residuum accurately. This is still somewhat difficult and others have approached the more interactive Magnetic Resonance Image (MR I). The main disadvantage of having to use these approaches is their soaring cost. Preliminary research allows such an expense but some assumptions must be made in order to allow simplifications and/or addition of data to model the current patientÂ’s residuum accurately, quickly, and inexpensively. Brennan determined the Â“shape of the un-deformed residual limbÂ… by digitizing a loose plaster wrap-cast of the subjectÂ’s re sidual limbÂ” . The shape and location of the bone structure of the residua l limb was constructed using CT scans of a person with similar stature. This applies some of the concepts of complexity management in an inexpensive route via the use of CT scans from another patient with similar stature to that of the current patient. (a) (b) (c) Figure 3-3. FE Modeling (a) Bone, (b) Soft Tissue and (c) Socket Liner Image by Faustini et al. 
49 Faustini et al.  depicts gr aphically the FE model used in their estimation of the stresses. The layered geometry allows the bo ne, soft tissue and so cket/liner to all be incorporated into the model in order to crea te a more realistic model for analysis. Moreno et al.  used MRI as a basis for the reconstructi on of the residual geometry. As, mentioned the high costs limit the use of MRI, but not many other methods can match the accuracy of the model generated from such methods. Â“Magnetic resonance imaging (MRI) was selected as an ideal diagnostic and rese arch tool to study the behavior of hydrogen atoms in the body tis suesÂ” . As the hydrogen atoms reflect the frequency emitted by the MRI, a local c oncentration of the hydrogen atoms allows the differentiation of the tissues within the resi duum. This differen tiation of the tissues within the limb allows the scientist to model the residuum within 3-dimensional space accurately and efficiently. In order to perform the MRI without inducing a defo rmed geometry (from the patient lying down), Moreno et al.  fitted the residual limb with a plaster cast which was fit onto the limb slowly a nd diligently. Care was taken not to alter the anatomically unloaded Â“topographyÂ” of the limb. It is ideal to use such tools as the MRI a nd CT that are available to us, but the cost of each of the tests limits the amount of uses that can be applied in a research setting. With a database of such measurements from research, it may be possible to use data from a prior patient to estimate th e geometry of a new patientÂ’s residuum. These tools, which take cross sectional Â‘picturesÂ’ of the tissues, allow researchers to model the tissues of the residual limb, which differ greatly from the normal limb. It is only through these
50 techniques that an understanding of the intern al tissue orientation can be discovered and modeled for use within finite elements (or other medical purposes). Future applications of MRI as a research tool include the response of the tissues, bones and epidermis to the mechanical loadi ng applied through the prosthesis. The use of these techniques is limited by costs but th ere are endless research possibilities. There are limitations to the use of differe nt imaging techniques: X-ray and CT scans expose the patient to ionizing radiation and X-rays produce a 2-dimensional model (a planar projection of a 3-dimensional imag e) requiring at least tw o views in order to extrapolate a 3-dimensional image (resulting in substantial error). As mentioned care must be taken not to influence tissue loca tion based upon the gravita tional field; if the patient is lying down, the limb must not dist ort from that of the limb while in stance. 3.4 Experimental Analysis Tests have been conducted to define the stresses within the prosthetic socket within laboratory as well as clinical set tings by several groups. Both above-knee and below-knee amputations were analyzed and re searched for the studies. Pressures were recorded within the socket in order to ex plore the effects of Â“p rosthetic alignment, relative weight-bearing, muscle contraction, socket liners, an d suspension mechanisms on the interface pressure distri butionÂ” . Pressure meas urements were recorded at discrete points within the socket, which was limited due to the discrete number of locations a transducer could be placed. They were put in locations deemed of Â‘high interestÂ’ within stress analysis (Â‘high interestÂ’ termed to describe a point of high stress).
51 It is difficult to equate the stress measurem ents from researcher to researcher as the pressures varied as a result of the type of transducer used as well as the method for calibration standard to a particular laboratory. In most of the completed experimental te sts, a special socket was fabricated to house the transducers for measurement. Th is method is preferred over use of the subjectÂ’s own prosthesis, as tapped/drilled holes permanently alter the prosthesis. One disadvantage to the experiment al techniques is the high cost for transducers and the relatively low area covered in the measurement of stress per transducer. In addition, some transducers have difficulty with quick response and are therefore not suitable for dynamic testing. In laboratory and clinical te sting, the finite thickness of the transducer can also play into the role of a stress con centration within the sock et and measure stresses higher than what would normally be e xperienced by the residual limb. There have been commercial developments within the field of these transducers and are currently being employed as an alternative to the slower less evolved ones in use for earlier testing. Â“Teksan, Inc. (Boston, MA) markets several biomedical pressure measurement systemsÂ… utilizing a grid-based sensor in which the rows and columns are separated by a polymer whose electrical resistance varies with forceÂ” . The limiting factors in experimental data collection leave room for the introduction of error, thus preventing of a direct comparison between computational stress methods like FEA, and experimental stress measurements. Further improvements need to be made in experimental approaches as well as to the finite-elements method to get validated stress measurements using interface mechanics.
52 3.5 Numerical Analysis One of the primary advantages of using numerical analyses (finite elements) over experimental analyses is the potential to estimate the interface pressures over the entire residuum. In some research, the data collecte d is not limited to the interface stresses but can also include Â‘subcutaneous stressesÂ’. Th e latter can be used to evaluate the overall longevity and success of the pros thesis per the individual, as well as other influential factors within the residuum affecting the pr osthesisÂ’s success with the patient Â– thus defining the problem at hand. For the past twenty years, finite elements has been the leading choice when using a numeric methodology; finite elements is chosen primarily based upon the endless boundaries within the software, the analys is is only bounded by the hardware in use (which can be upgraded when needed). 3.6 Validation of the FE Analysis Currently, validation of the models can only be achieved through experimental means (which possess errors with in the test setup as discussed previously). Only discrete points within the socket have been measured leaving holes within the validation of the model. These holes are only filled through theory and/or extrapolation of data (which in itself is theory). Some researchers have qua ntified data leading to verification of FEA within the range of experi mentally recorded data. Qualitative analytic and experimental stress waveforms were created by Zhang  and showed similarities within a doubl e peak. Â“The predic ted resultant shear stresses were less than the experimental va lues at all measured sites.Â” Zhang 
53 reported a rough 30% difference (l ower) in analytic results ove r experimental. One of the sources of this error is however due to an assumption made within the FE model; the FE model was analyzed under st ance where half the body weight is applied through the Femur, while experimentally dynamic anal ysis was conducted during various stages within the gait cycle . Sanders et al.  reported interface shear stresses high enough to cause blisters on the epidermis which fall in the range 4 kPa to 23 kPa (running between 22 and 118 cycles and average coefficient of friction 0.5) The magnitude of the experimental stress varied slightly from the analytical due to the type of socket used in experimental analysis (Berkeley jigs, which are substantially heav ier than the typical thermoplastic socket) along with the patients not wearing socks whic h exaggerates the coefficient of friction (and intuitively causing blisters).  Sanders and Daly [20,21] also reported double-peaked interface stress curves which matched Â“the general trend in clinical dataÂ”. They re ported a Â‘best matchÂ’ between the analytic and experimental data at the pos tero-distal and antero-proximal sites, while Â“consistent mismatches were seen in antero-l ateral distal normal stress waveformsÂ… and postero-proximal normal stress waveformsÂ”  Much effort was invested into the discussion of the analytical matches with char acteristics of waveform shapes  and is broken into: loading delays, high frequency events, central stance and toe-off. With collected data, the recurring similarities betw een the 3-dimensional model (created with MRI technology) and the experimental analysis were proven to be substantial leading to the effective prediction of interf ace stress with the FE model.
54 When attempting to quantitatively compare the data it is important to recognize that the technique and methods used to evaluate the stresses varied from study to study. Â“The type of activity, type of transducer, and location of the tran sducers on the residual limb surface differed between laboratoriesÂ” , and thus accounts for some of the discrepancies within the data. Clinical data was not always measured within the lab, but was taken from other sources wh ich may have evaluated the st resses at a diffe rent period in the gait cycle. In a general view, the differences in the models are results of the techniques and methods used in the interface model. 3.7 Parametric Analysis Zachariah et al. explored in detail the idea of parametric analysis . Parametric analysis is performed via altering one variable in the system and relating it to a change in a particular quantity output; wh en Â“the magnitude of one variable in the model (or one feature of the model) is pert urbed about its chosen value and the relative change in the estimated quantity evaluatedÂ” . This type of analysis is particularly important now with the technological advanc e of the finite elements method within interface mechanics in part due to its ability to point out wi th some level of assuredness that parameter Â‘XÂ’ must be specified to accur acy Â‘%Â’ in order to create a model with as little of error as possible w ithout creating complexities far greater than the level of technology available. In simpler terms, it a llows one to conclude just how precise a parameter within the model (material prope rty, geometric measurement/differentiation
55 etc) must be with the application of simplifications and/or assumptions to the model for ease of calculation or analysis. Parametric analysis was conducted on ge ometric properties, element properties and boundary conditions (the three main aspect s behind the finite element modeling) in order to determine the Â‘optimalÂ’ model for accuracy and simplicity. All parametric analysis was based upon Silver-T hornÂ’s definition of sensitivity Â– the ratio of the relative change in the finite element estimate to the parameter disturbed. Table 3-1. Parametric Analysis Zachariah et al.Â’s tabular review of Â“E xperimental Comparisons and Parametric AnalysesÂ”  Parametric analyses Type of Interface Investigator (year) Loading Condition Experimental, data comparison Geometry Element Properties Boundary Conditions Brennan (1991) Standing Std. prosthesis Modified socket Socket shape Reynolds (1992) Standing Socket rectification Soft tissue stiffness Alignment Glued Sanders (1993) Stance phase Std. prosthesis Modified alignment Skin stiffness Force, moment directions Steege (1988) Standing Std. prosthesis Modified alignment Steege (1995) Stance phase Bone stiffness Tension Released Silver-Thorn (1991) Standing Std. prosthesis Modified socket Socket rectification Absence of fibula Stump length Bone Shape Socket shape Socket stiffness Liner stiffness Soft tissue stiffness Soft tissue PoissonÂ’s Ratio Quesada (1991) Heel strike stump length Socket stiffness Soft tissue stiffness No release of tension Slip Permitted Zhang (1995) standing Coefficient of friction
56 Table 3-1 Â– from Zachariah et al. Â– summar izes the work of several researchers in the area and the area on which they focused thei r work within interface mechanics. It is included here as a well defined summary of th e work in the area up to the year 1996. The parametric analyses conducted by each researcher tend to form the overall picture of the importance of elements within the main sections of finite element modeling. The geometric parametric analysis ranged from the socket (shaping and rectification therein) to th e very distinct realities of the residuum biological tissue differentiation. Zachariah et al. report that Silver-ThornÂ’ s analysis of a short socket with PTB rectification experienced small deviations in normal stresses but noticeable variations within shear with the absence of the fibula in the model. The residuum length had an affect on the normal stresses within the socket; the shorter the residuum the higher the normal stre sses recorded (Quesa daÂ’s model of heel strike). In theory, the variations in normal st resses are a result of the change in the lever arm acting with the bending moment of the limb as well as the area which is exposed to the loading of the person (dynamic or static). As noticed in Zachariah et al.Â’s table su mmarizing the parametric analyses (Table 3-1), the element properties were believed to ha ve just as much impact on the accuracy of the models as that of the geometry. First a nd foremost, the modeling of the socket itself requires insight into the material behavior a nd its properties to model the stresses stored within the thermoplastic. Studies varying th e thickness of the socket (stiffness) achieved by Silver-Thorn reflects the latter theory of element property importance, in that as the thickness of the socket wall decreased so did th at of the normal stresses within. Even
57 with these stress alleviations, S ilver-Thorn reported that the sens itivity of stress relief was not as great for the introduction of compliance into the socket as it was when the socket liner was analyzed. Silver-ThornÂ’s model also va ried the socket liner stiffness and recorded the changes in stresses within the socket. As the socket liner stiffness increased the normal stress increased greatly. There is a trade-o ff involved in liner stiffness decrease though, as a relief in stress is seen with a less-stiff socket a reducti on of socket stability is also noticeable.  When modeling the geometry of the system the location of the tissues were of great importance (which is why more elabor ate methods of tissue differentiation are being utilized more often), and we re thus exposed to the parametric analysis as well. The stiffness of the soft tissues were increased and reports by Quesada, Reynolds and SilverThorn all showed an increase in the stress. This is parall el to the medical knowledge of tissues which have been injured and have heal ed to a permanently hardened state (which reflects that of scar tissue). These tissues have less ability to flex under loading and as such experience much higher stresses than t hose which can yield to the applied loading. SteegeÂ’s  test of transtibial prosth etic gait showed significant Â“bone bendingÂ” which led to the use of the material propertie s of cortical bone as opposed to cancellous (1.5 GPa as opposed to 10 MPa). Zachariah et al. report that the parametric analyses of these preliminary results are essential to the full understand ing of bone properties within the finite element model. The final aspect of the FE modeling Â– Boundary Conditions Â– is also reviewed parametrically and touches upon the interface, external loading and alignment issues.
58 Modeling of the interface is the key with in the finite element model and the conditions for slip/stick are of utmost impor tance. The three methodologies used were analyzed including the totally-glued interf ace, which revealed th at tensile interface stresses reduce the peak compressi ve stresses (60-85%) . External loading is fairly intuitive as the model is most susceptible to variations in stress magnitudes through axia l and bending moment s in the sagittal plane. Sanders reported parametric analyses of these conditions and noted that the normal stresses and shear stresses were most susceptible to the axial force and sagittal bending moment while the normal stresses were also sensitive to alte rations in the sagittal shear force while the shear stresses were more sensitiv e to the torsional moments applied. As a general research investment, para metric analyses are highly informative towards the future direction of the computa tional analysis of the interface stresses (or estimation thereof). It allows the scientis t to model complex residuum geometry with appropriate assumptions which are not detrimen tal to the success of the model itself and are able to provide the necessary information to differentiate the simplifications made in the model. 3.8 Conclusions on Interface Mechanics Review Through extensive research it has been s hown that the finite elements method has the possibility to be an ex tremely powerful computational tool for the estimation of interface stresses within extern al prosthetics studies. As technology advances and computers become more powerful, the bounds upon which finite elements can be applied
59 are approaching limitless. Ther e is a strong possibility that finite elements can contribute greatly towards efficiency in prosthesis care as a tool for the estima tion of stress as well as that of parametric analysis. While many of the experimental technique s are not suitable for routine clinical settings, it is clear that with the incorpor ation of the thin pressure membranes into a smaller transducer-like function, it is possible to enhanc e the clinical measurement of each patient as prostheses are manufactured. It is one of the main goals of the further understanding of interface mechanics to enhance patient care and prosth etic efficiency. It is clear that the interface is of great importance within prostheses, and as the research has shown, the experimental grounds behind measurement yield limitations in the discrete number of locations force m easurements can be taken without inducing higher errors in the form of stress concentrators. Numerical finite element analysis is becoming useful in its ability to evaluate over the entire su rface and even subcutaneously.
60 Chapter 4 Bistable Compliant Extension Aid A Bistable Compliant Extension Aid (BCEA) designed to be added to an existing polycentric prosthetic knee, was developed and analyzed using a finite elements software package (ANSYS). Design criteria for th e BCEA were based on swing control requirements that are not inherently satisfie d by the geometry of the polycentric kneeÂ’s four-bar frame. The requirements of the prev ention of excessive heel rise and a stable sitting position, were achieved by optimiz ing the BCEAÂ’s geometry. The optimization procedure was based on knee flexions ranging between 0 and 90 degrees and the resulting reaction moments experienced by the compliant segment. 4.1 Design by Specialization The majority of commercially available prosthetic knee joints are designed to meet the userÂ’s level of performance, whet her it is being fitted to a limited household ambulator or an Olympic athlete, the prosth etic knee must perform optimally. There is no one-size-fits-all prosthetic knee; therefore, performan ce is Â‘designedÂ’ on a case-bycase user-defined basis, meaning that there is a spectrum of knee prostheses which meet high-stability needs while others meet the ma ximum user-control pref erence and all those between.
61 The tradeoff between control and stability is among the first attributes the user notices when getting acquainted with a new pros thesis. Typically the higher the level of amputation, the more stability is required of the knee mechanism, as there is less residual musculature the amputee has at their dis posal; conversely the lower the level of amputation, leading to the knee disarticulation, leaves much more thigh muscle intact along with a longer lever-arm, thus leaving a higher ability to a pply control over the prosthetic limb . Achieving necessary tradeoffs, while m eeting basic functional requirements is accomplished by design by specialization In a prosthetic knee, many basic functional requirements are achieved by the polycentric (four-bar) knee design, while important functional tradeoffs can be accomplished by de sign specializations. As examples, Figure 4-1 depicts two different speci alizations of the same poly centric knee mechanism: the Otto Bock (a) 3R32 and (b) 3R55. The Otto Bock 3R32 specializat ion consists of a manual lock and facilitates a K1 level amput ee (Â“poor voluntary controlÂ” and Â“transfer onlyÂ” ), while the 3R55 specialization is designed to meet the K3-K4 level amputee (Â“good voluntary controlÂ” and Â“community am bulators who can walk with variable cadence and for patients who participate in hi gh impact activities such as runningÂ” ). (a) (b) Figure 4-1. 3R32 with Manual Lock (a) and 3R55 with Pneumatic Cylinder (b) 
62 The purpose of this chapter is to introduce a compliant link add-on as a specialization of the Otto Bo ck 3R32 and 3R55 frame, acting as an extension aid which prevents excessive heel rise and provides a stable sitting position. 4.2 Background Traditionally, prosthetic knees are desi gned as rigid frames with pin joints accommodating motion (if appropriate). They are typically analyzed using force loading and failure is determined by stance criteria and buckling. Compliant mechanisms, on the other hand, gain some or all of their motion fr om the deflection of flexible segments, thus producing a form of directed buckling of the linkage. Because of the buckling effects, compliant mechanisms are more effectively analyzed under displacem ent loading rather than applied force. Why then would compliant mechanisms be a good fit for prostheses? The general advantages of compliant mechanisms within the prosthetics area include relative lighterweights, lower costs both in a reduction of part count as well as manufacturing (most are polymers), they hold high reliability and can be designed for high-precision applications . This chapter will also introduce a specialized design advantage the compliant mechanism add-on can offer Â– the introduction a nd transfer of moment s that vary over a given displacement, needed for proper swing control. Compliant mechanisms have been studied as a feasible alternative to rigid-body mechanisms within prosthetic joint design. Prominently, work done by Gurinot et al.  introduced methods of using compliant mechan isms Â– which are predominantly used with
63 tensile loading Â– under high compression situati ons similar to those seen by a prosthetic knee joint, known as High Compression Co mpliant Mechanisms (HCCM). These methods have proven to be the foundation fo r the introduction of compliant mechanisms to prosthetic knee and ankle design. The two methods, inversion and isolation, either transform a compressive load to tensile via ge ometric alterations (inve rsion) or transfer the compressive loads through rigid links and away from the compliant links, similar to traditional prosthetic knee mechanisms (isolation). More recently a project undertaken by Ma hler  also combined compliant mechanisms and prostheses by designing an adjustable pediatric compliant prosthetic knee mechanism to better suit the needs of a growing child who w ould be subjected to harsher, more active environments. His research focused on the kinematic instant center of rotation of the mechanism in order to understand motion (extension and flexion) relative to Â‘knee adjustmentÂ’. Both projects along with several others including a prosthetic ankle  have proven the validity of compliant mechanisms t echnology within prosthetic joint research. This chapter further introduces compliant mechanism technology and its inherent advantages to the field of prosthetics via a design specialization of the Otto Bock 3R55 and 3R32 knee frame. 4.3 Functional Criteria The compliant extension aid was designed to meet functional criteria for efficient prosthetic swing control, which have been standardized by the prosthetics industry over
64 years of practice. The standards of stance co ntrol are also of importance in prosthetics, however for this project, stance was not evaluated since a prosthetic knee mechanism Â– the Otto Bock 3R55 and 3R32 Â– which has been tested and validated, was used as a base and therefore does not require further scrutiny (only to the extent that the BCEA does not interfere with its function). The functional criteria which were pre-defined are those of: sufficient ground clearance, prevention of excess ive heel rise at the end of knee flexion, a fast extension phase and in some cases a term inal impact stop just before heel strike. Table 4-1 summarizes these functional criteria. Table 4-1. Summary of Swing Phase Requirements Swing Phase Requirements Purpose Ground Clearance Prevents stubbing of the toe. Prevent Excessive Heel Rise Allows the shank to be in position for stance phase. Fast Extension Phase Ensures the shank moves into position. Terminal Impact Hyperextension Stop Provides a signal that shank is in position for load bearing (although not a strict requirement). With the exception of recent advanced prosthetic knees on the market (i.e. bionic technology by Ossur Â– Power Knee), prosthetic kn ees are passive knee joints; they do not add energy to the amputeeÂ’s gait. Since thes e prostheses do not a dd energy to the swing phase they must conform to certain principles in order to function properly to maintain a proper gait pattern. Normal prosthetic swin g is initiated by movement of the shank posteriorly under the influence of inertia. It is imperative that exce ssive heel rise is
65 prevented since it slows the extension phase (thus causing falling). During mid-swing, the shank moves anteriorly under the influences of inertia (providing th ere is an extension aid present) and gravitational forces. The rotational speed of the prosthetic limb (knee) is slower than that of the sound limb not only due to the lack of input energy via musculature, but also due to differing mass distribution. The result is a slower -abnormal gait. This gait abnormality is addressed and alleviated by an extension assist device built wi thin the prosthetic knee. With most extension aids, a terminal impact results at the end of the swing phase (caused by the contact of the kneeÂ’s mechanism a nd the hyperextension stop), however flawed this may seem to the designer concerned a bout impact loads, many amputees prefer a noticeable signal that the limb is in position for loading. Ground clearance is the fundamental desi gn goal of prosth etic knees when considering the swing phase of pr osthetic gait. Polycentric four-bar knees, like that of the Otto Bock 3R55, were designed to Â‘shortenÂ’ the limb in order to achieve clearance between the toe and ground during mid-swing, preventing the stubbing of the toe (leading to falls). Beyond ground cleara nce, prevention of excessive heel rise is paramount; during flexion, if the heel ri ses too far (knee angle exceed ing 60 degrees), the rotational speed of the prosthetic shank is too slow under the action of gravity to ready the prosthesis for heel strike, thus resulting in Â‘excessiveÂ’ knee flexion leading to buckling under stance loading. The prosthesis thus requires a fast extension phase, and the extension assist device must provide the nece ssary moments to perform to these optimal swing characteristics.
66 Figure 4-2 illustrates the dyna mics of the knee angle over the gait cycle and the importance of the knee angle between the flex ion and extension phase As shown, if the knee does not resist motion beyond 60 degrees during mid-swing, the swing phase ends at a knee flexion exceeding what it should during the beginning of stance, which will inevitably lead to buckling and falling. Figure 4-2. Knee Angle vs. Gait Â– Shown with and without Excessive Heel Rise With excessive heel rise is shown as gray, without, in red. Under these conditions it seems as though a simple elastic strap would suffice to meet these extension characte ristics, however, though the mo ments exerted by the strap do meet the necessary criteria of e liminating excessive heel rise, it does not meet the behavior necessary for the amputee to sit (a common position of everyday life). When seated, the prosthetic knee and extension assi st device must not exert extension moments causing the prosthetic limb to Â‘kick-outÂ’ to fu ll extension; they must be designed in such a way that their influences (applied extens ion moments) are at a maximum near 60 degrees of knee flexion (to account for prop er swing) and then begin to decrease afterwards to zero near 90 degrees to account for the seated position.
67 Figure 4-3 depicts the optimal influence of an extension aid on a prosthetic knee. Notice the gradual increase of the applied extension moment to a maximum at a knee flexion nearest 60 degrees to account for norma l gait, and a sharp decrease following to minimum near 80-90 degrees in order to prepare the knee for the seated position.  Figure 4-3. Optimal Influence of Prosthetic Knee Extension Assist 4.4 Concept of Bistability Bistability is easily associated with the well-known ball and hill analogy shown in Figure 4-4. Bistability occurs when an object has two point s where itsÂ’ potential energy is at a minimum. These points are known as stable equilibrium points, labeled (A) and (B) in Figure 4-4. In order for the particle to deviate from either of these positions, an external energy must act in a way to force the pa rticle from its resting state. If the ball is resting in position (A) and is pushed to the ri ght to point (C), the ball has the ability to balance itself at this point and be in equ ilibrium also; point (C) is known as the unstable equilibrium. If any external energy is added to the ball at the unstable equilibrium point,
68 it will always assume one of its stable equilibr ium points (A) or (B). Stability refers to the ability to resist or rec over from small displacements. Figure 4-4. Bistability Analogy with a Ball and Hill The bistable compliant extension aid addre ssed in this paper must perform as the ball would on the hill. Point (A), when the ba ll is at its first equi librium point coincides to the prosthetic knee during st ance. When energy is added to the knee, via inertia during swing, it will tend to return back to its or iginal position accomm odating stance at heel strike. When enough energy is added to the knee, like when crouching to sit down, it transitions to a second equilibrium point, just as the ball does at point (B). 4.5 Bistable Compliant Extens ion Aid (BCEA) Design The BCEA was designed on the existing polycentric frame of the Otto Bock 3R55. A simplified four-bar schematic wa s used by converting the top link of the mechanism to ground, shown in Figure 4-5. Th e BCEA was pre-assembled as a straight, unstressed polypropylene copolymer beam measuring 1mm x 5mm x LBCEA, where LBCEA is the length parameter whose optimal length was determined and added to the four-bar frame via pinning it to the existing anterior-b ottom pin (pin 2 in Figure 4-6(a)).
69 Figure 4-5. Knee Mechanism Simplification Model After the BCEA was inserted into the model as a straight unstressed beam, displacement loading was applied to the t op of the BCEA link and it was moved in the manner shown in Figure 4-6(b). Once the to p of the BCEA was aligned with Pin 1, it was fixed to the top link of the mechanism (shown as ground) as seen in Figure 4-6(c). (a) (b) (c) Figure 4-6. Otto Bock Knee Mechanism with BCEA Assembly (a) pre-assembly, (b) mid-assembly showing pre-stress stepping motion of top pin, (c) final assembly.
70 Pre-stressing the BCEA into position (as opposed to creating an un-stressed curved beam) was a critical step ensuring that bistability was achieved. Bistability allows the knee mechanism to reach a stable equilibrium point, like those needed when standing and sitting (0 and 90 degrees of flexion). By pre-stressing the BCEA, we were able to achieve a Â‘snap-phenomenaÂ’ resu lting in the desired bistabil ity and extension moments at the appropriate degree of knee flexion (discu ssed in more detail in results section). The length of the BCEA was optimized by evaluating the lengths which produced an arc-angle, ranging from 0 to /2, shown in Figure 4-6. No te that the arc-angle is defined after pre-stressing the assembly in to the position shown in Figure 4-6(c). For simplified design purposes, the final shape of the BCEA (shown in Figure 4-6(c)) was assumed to be circular. Arc-angles which produced BCEA curvatures exceeding a quarter-circle were also evalua ted but produced results outside of the set criteria and thus were not included here. Optimization of the geometry was conduc ted by looping an FEA model to run a knee flexion simulation from 0 to 90 degr ees, over a series of BCEA lengths (LBCEA), and the resulting extension moment characteristics were compared with the optimal influence shown in Figure 4-3. The overall arc-length, LBCEA, was evaluated using geometry parameters shown in Figure 4-7. The max arc-angle, max, was defined and used to alter LBCEA incrementally in order to better understand how the length of the BCEA affected its function.
71 Figure 4-7. Design Approximation of the BCEA Geometry The geometric input max, was broken into a prescribed number, of segments, yielding the step arc-length (3) of the BCEA which was then used to create the overall length, LBCEA. 2 0 (1) angle max 2max (2) maxincrement (3) incrementj j=1,2,3Â…(1) (4) From Figure 4-6, LBCEA can be equated to the arc-angle by equations (5) and (6) as a function of the length of the anterior link, LANT, of the 3R55 knee mechanism. 2 sin 2ANTL R (5) LBCEA= R 0 < < /2 (6)
72 For the extreme conditions, =0 and /2, LBCEA was calculated using equations (7) and (8). LBCEA = LANT =0 (7) LBCEA = 4 sin 4 ANTL = /2 (8) Once the desired LBCEA was calculated, it was inserted into the model as described previously, and was then pre-stress ed into the anal ysis-ready position. 4.6 Analysis and Results Figure 4-8 depicts the freebody-diagram of the knee model used for analysis. Reaction forces at pin joints 1 and 2 were calculated ove r a knee flexion from 0 to 90 degrees, as well as the reaction moment app lied at pin 1 as a result of the BCEA. Figure 4-8. Free-Body Diagram of Knee and BCEA
73 Successful geometry optimization was de termined by the criteria of maximum reaction moment closest to 60 degrees of flexion followed by a sharp decrease in the reaction moment closest to 0 N-m on, or befo re 90 degrees of flexi on (as shown in Figure 4-3). Figure 4-9. BCEA Extension Moment vs. Knee Flexion The extension moment and knee flexion da ta, when graphed over 0-90 degrees of knee flexion yield results that model closely to that of the pre-desi gn criteria depicted in Figure 4-3. Figure 4-9 graphs the entirety of the extens ion moment resu lts defined by equations (1)-(3), with = 30, and also shows the varia tion of the extension moment magnitude with respect to LBCEA. Each curve represents a different value for LBCEA, with LBCEA( =0) furthest to the left, and LBCEA( =/2) furthest to the right. Using Figure 4-9, it can be said that LBCEA would be the most functional, with regards to prevention of excessive heel rise and correct extension characteristics during 90 degrees flexion, at an arc-angle closest to /2 ( =/2). As shown in Figure 4-9, the moment increases over flexion, th en Â‘snapsÂ’ to zero (or near ly zero); this Â‘zeroÂ’-moment
74 point is a stable-equilibrium point, and define s when the knee is in the sitting position and will not return back to stance unless acted upon. It seems as though the data smoothly returns to zero, but in fact the data points ju mp from a high magnitude to relative zero in one step, which is a result of the Â‘snap phenomenaÂ’. The snap phenomena is brought upon the BCEA when the flexion of the knee has reached a point of instability, and is then pushed passed that point until the BCEA snaps into its second equilibrium pos ition. The end-conditions of the BCEA allow it to rotate freely at its bottom while the top remains fi xed to the kneeÂ’s top link, which causes the segment to rotate uniquely. Figure 4-10 illustrates the snap phenomena by showing the BCEA in its initial point (a), its maximu m-extension-moment point, (b), pre-snap position (c) and the seated pos ition (d). Between 60-85 degrees of flexion, the BCEA snaps through as a result of the extension mo ments being relieved by a rotation in pin 2. Figure 4-11 highlights the ex tension moment key-points: maximum extension, snap phenomena and stable equilibrium. Figure 4-10. BCEA Snap Phenomena Knee flexions corresponding to : (a) initial position, (b) maximum extension moment, (c) pre-snap, (d) seated position
75 Figure 4-11. BCEA Extension Moment Graph with Labeled Key-points The data for the curves nearest an arc-angle of /2 ( = /2) is tabulated in Table 4-2. As LBCEA increases in length, the maximum a pplied extension moment increases as well. With that, the angle of knee flex ion corresponding to the maximum extension moment increases. The increase in maximu m extension moment with respect to the length of the BCEA corresponds to the complia nt member storing more of the strainenergy during rotation. The longer the comp liant link, the more strain-energy can be stored, which will then be released at the point of snap.
76 Figure 4-12. BCEA Extension Moment vs. Knee Flexion Â– Optimal Geometry Sets Table 4-2. Extension Moment Data for Optimized LBCEA Optimally, the maximum extension moment should correspond to a knee flexion of 60 degrees, along with a snap angle between 80 and 90 degrees. The particular data sets listed in Table 4-2 and Figure 4-12 depict the LBCEA values necessary to optimize the geometry to meet the design requirements. LBCEA Extension Moment (mm) MaximumKnee Angle at maximum Knee Angle at snap (N-m) (Degrees) (Degrees) 93.584 17.942 54 72 94.172 18.525 56 75 94.787 19.096 58 78 95.430 19.654 60 80 96.103 20.201 62 83
77 4.7 Knee and BCEA Unloading After Snap The data depicted in Figures 4-9, 4-11 and 4-12 all have not definitively shown the second stable equilibrium position (which would naturally follow BCEA snap, and is defined by the resulting BCEA moments to be 0 N-m). The resulting loads which were continually placed on the mechanism during anal ysis in ANSYS, in order to force the mechanism through 90 degrees of flexion, prohibit the BCEA from being unloaded completely. In order to define this second point of stable equilibrium, unloading of the knee and the consequent resu lts were analyzed. Figure 4-13 depicts the unloading characteristics of the mechanism from 90 degrees of flexion (post-snap) to zero degrees of flexion (stance). It can be shown that the unloading curve has many of the same char acteristics of the lo ading curve (shown in Figure 4-11), with a resultant maximum flexion moment as well as a snap-phenomena resulting in the release of stra in-energy via a rotation in Pin 2.
78 Figure 4-13. BCEA Unloading Curve The characteristics of the unloading curv e distinctly define the second stable equilibrium point (seated position). This point was unable to be correctly shown in previous Â‘loadingÂ’ curves due to the fact that displacement loading was continuously being applied following snap. Physically speaking, if the mechanism were loaded beyond the snap point of knee flexion and th e load released, the knee would assume a stable equilibrium point at a slight decrease in knee flexion all due to the presence of a small knee extension moment following snap. Figure 4-14 overlays the unl oading curve on the loading curve and labels the inherent key-points during both situations. The sn ap-phenomena of flexion and the snap phenomena of extension result in each of the positions Â‘secondÂ’-equilibrium points; the flexion-snap phenomenaÂ’s second equilibrium point being the seated position, while the
79 extension-snap phenomena results in the kn eeÂ’s stance position. All stable equilibrium points are also defined and labeled on the graph. Figure 4-14. Complete BCEA Cycle: 90 Degrees of Flexion and Extension 4.8 BCEA Stress Analysis and Factor of Safety Stresses which were induced within th e BCEA during flexion were calculated using ANSYS. A static stress analysis was conducted at each degree of knee flexion and the maximum stress state analyzed (i.e. the position of the knee and BCEA for which the maximum stress was discovered). Geometrica lly and analytically, the maximum stress state was found to be the position just before snap, 82 degrees of flexion. The stresses ranged from 6.1252 MPa to 28.047 MPa, and the stress magnitudes over the length (LBCEA) are shown in Figure 4-15.
80 Fully reversed stress cycles occur ove r the knee flexions/extensions shown in Figure 4-15 (90 degrees of fl exion and subsequent extensi on) as a result of similar force/moment reactions over flexion and ex tension. The BCEA stress analysis and the corresponding factor of safety fo r the optimized BCEA geometry ( = /2) are summarized in Table 4-3. These values are ba sed off the yield stre ngth of polypropylene Sy = 34 MPa . Figure 4-15. BCEA Stress Magnitude and Distribution at Maximum Stress State Table 4-3. BCEA Stress Summary Max. Stress (MPa) Min. Stress (MPa) Factor of Safety 28.047 6.1252 1.2123
81 4.9 BCEA Design Conclusion The bistable compliant extension aid deve loped within this CAD structure shows promise in its ability to conform to the prin ciples of prosthetic swing needed for normal gait. While the data shows the selection of geometric variations to be of a wide-range, the corresponding data shown in Figure 4-12 develops itself well into those criteria outlined in its design and allows for further geometric refinement based on amputee needs. The data collected from the design of the BCEA contributes to the validation of compliant mechanisms even further into th e prosthetics industry. This chapter has introduced a compliant prosthetic knee extension aid design that has the ability to apply the necessary extension moments in order fo r a prosthetic knee to function properly during the swing phase and while seated.
82 Chapter 5 Proprioception via Variab le Internal Socket Stress Patterns A finite element model was constructed to simulate a below-knee prosthesis during the swing phase of gait in order to show the stress va riations on the inner surface of a prosthetic socket and to pose the hypot hesis of increased proprioception based on these variable stress patterns The hypothesis is that the changing loads caused by the bending of the BCEA will be felt by the amputee and will give him/her a sense (proprioception) of the amount of flexion in the prosthetic knee. The external forces applied to the system were based on the Bistable Compliant Extension Aid design and were applied via direct loading of the finite element model. The criterion adopted for the proprioception hypothesis was variable stress patterns on the prosthetic socket over different degrees of knee fl exion without failure of th e polypropylene socket. The interface stresses varied in magnitude and lo cation over knee flexion angles and can be used to develop the hypothesis of proprioceptiv e feedback via variable stress patterns on the inner surface of the prosthetic socket. 5.1 Interface Mechanics and Proprioception The biomechanical interaction between the residuum and socket, also known as interface mechanics, has evolved into discrete numerical stress analyses with the use of modern finite elements software packages. Finite element analysis is able to produce
83 approximate analytical solutions in the prosthetics field (as well as many other engineering fields) to problems in which no simple closed-form solution exists. The advantages that finite elements and interface mechanics bring to the prosthetics industry are inherent in the functional design of each prosthesis: the ability to discretely analyze the stresses and their s ubsequent capacity to affect the functional design of the prosthesis. Research centered on interface shear and compressive stresses have consistently been focussed on their abil ity to apply on-hand data to form-fit a better prosthetic socket allowing fo r an Â‘optimal prosthesisÂ’ both in comfort and control. Control over lower-limb pros theses during swing has b een an issue addressed heavily as of late through m eans of developing prosthetic knees which are able to apply active moments at key points during swing. The Power Knee by Ossur is an example of such a knee which can adapt to swing phase characteristics in order for the amputee to hold better control over their gait cycle. These knees hold state-of-the-art technology and also own a price-tag to match, thus restri cting its commonality within the lower-limb amputee population. Proprioception, the sense of th e orientation of oneÂ’s limbs, is the Â‘naturalÂ’ method of offering complete control over biological gait. Prosthesis control via proprioceptive feedback could offer potential advantages ev en over the active prosthetic knee(s) on the market by providing a more na tural/biological control as opposed to motor driven control. Even better, by enhancing proprioce ption via inexpensive mechanical means, the greater majority of the lower-limb amputation population would be offered the ability to share these control advantages.
84 The BCEA, developed in the last chapter, offers Â‘optimizedÂ’ extension moments applied to the knee mechanism and thus the pros thetic socket over a range of knee flexion angles. This chapter proposes a method of providing and anal yzing proprioceptive feedback via variable stress patterns imposed on the inner part of a pr osthetic socket as a result of forces and moments induced by the BCEA. 5.2 Finite Element Desi gn Characteristics Geometry, element properties and bounda ry conditions develop the accuracy, complexity and computational intensity of the finite element model, each being developed and enhanced by continuing researc h. While simplifications of each design characteristic can offer the foundation for st ate-of-the-art research, increasing the complexity allows for the accuracy of the an alytical solution to mirror itself closer to physical results (that may be experimentally determined). The input geometry emphasizes the importa nce of stress concentrations and the correct loading of differing materials over their boundaries (i.e. over solid-contact points). Variations of geometries are co mmon when modeling the residual-limb; these differentiations from one researcher to another associate themselves with those complexities of biology: bone, soft-tissue, epider mis, cartilage etc. In order to increase the complexity of the residuum geometry, mo re complex methods of modeling must be used: X-ray, computed-tomography (CT) scan and MRI as examples . The increasing complexities of the model ge ometries bring with it the need for the introduction of new materials and their properties; when th e bones, muscles and skin are
85 introduced into the model, these properties mu st also accompany them in order to justify their interactions. The material properties of biological tissues bring complexities that can only be solved for by clin ical testing. Many of these pr operties are approximations to better serve the models to which they are applied, but the clinical-data is becoming readily available and can be used to r un statistical measurement and approximation determination. Finally, the boundary conditions within the finite element model tell the program how to treat two nodes in contac t from two different bodies, i.e. the residual limb and the prosthetic socket. Heavy approximations have been made relating to the boundary conditions in interface mechanics due to: (1) the complexities of the materials undergoing loads, and (2) the computational intensity increases and in some cases a solution is indeterminate. Three cases of socket-residuum contact have been employed by researchers: the fixed interface, partially fi xed interface and free . The completely fixed interface is the least computationally intensive, however th is boundary condition allows for the residual tissu e to undergo tensile loading (which is not the case with prosthetic limbs, unless suction is present). Partially fixed allows the researcher to remove these tensile loads through a post-processing command, and the free interaction removes these loads completely before the analysis is run. The approximation of the boundary condition must be analyzed in order to determine the validity of the analysis and can in turn lead to fals e results if not done carefully.
86 5.3 Modeling The CAD model of the lower-limb prosthesis used in this analysis was constructed in a SolidWorks environmen t and included a simplified residual limb, prosthetic socket, the Otto Bock 3R21 knee fr ame, and simplified sh ank and foot (shown in Figure 5-1). In order to introduce the BCEA to the model shown, the reaction forces and moments that were calculated using ANSYS were directly applied to the knee frame; this was done due to the inefficiencies a nd nonlinearities in the model and the errors induced by solving the FEA with large defl ection as well as rigid body motion (quite simply, large errors were experienced wh en the BCEA was directly modeled in the system). Figure 5-1. Complete Model of Lower-Limb Prosthesis
87 The prosthetic socket and residuum were modeled with a bonded global contact; the program bonds the source and target entitie s, which may be touching or within small distances from each other. As mentioned, th is is the simplest approximation in the boundary conditions listed previously, and will a ffect tensile stresses on the residual limb (which were not being analyzed here). Th e socket was constructed out of polypropylene copolymer (Elastic Modulus = 8.96e08 N/m^2, PoissonÂ’s Ratio = 0.4103, Shear Modulus = 3.16e08 N/m^2 and Density = 2.77e-05 kg/m^3), and the residuum was simplified and modeled as rubber (Elastic Modulus = 6.099e06 N/m^2, PoissonÂ’s Ratio = 0.49, Shear Modulus = 2.899e06 N/m^2). Each of these elementsÂ’ geometries were simplified and approximated as cylinders in order to ea se the computational intensity (high-end modeling techniques as described in Chapter 3 could propose further advancements to this simplification). The purpose of this chapter is to develop a finite element analysis simulation of the introduction of the BCEA to the model and the resulting stresses between these two surfaces (residuum and socket) analyzed. The knee mechanism was constructed from titanium and steel (as built) with very few design simplifications only associated with the pinning of the mechanism together. The shank and foot were also constructed of metal and were assigned the properties of steel. The simplifications imposed on this model (i.e. residuum geometry) are such that the work here should only be used for anal ytical approximations a nd further refinement of the model should be considered in orde r to closely mirror real-world situations.
88 5.4 Applied Loads In order to introduce the BCEA to the model shown in Figure 5-1, the reaction loads as calculated from the BCEA were applied to the top bracket of the knee mechanism (as determined by the ANSYS FEA) and were introduced to the model using direct transfer loading in the finite elemen t programming. Figure 52 depicts the resultant extension moments induced by the BCEA which was applied to the model, summarized in Table 5-1. Figure 5-2. Applied BCEA Moments Table 5-1. Summary of BCEA Applied Extension Moments Extension Moment MaximumKnee Angle at maximum Knee Angle at snap (N-m) (Degrees) (Degrees) 20.201 62 83
89 The reaction forces at the top anterior and posterior pins of the prosthetic knee were also developed in the BCEA design an d are shown graphically in Figure 5-3. Figure 5-3 depicts the magnitude of the anteri or and posterior pin reaction forces versus the knee flexion angle. The reaction forces of the posterior pin were very small and are shown near zero, while the anterior pin forces were more prominent. These reaction forces were separated into the x and y dir ections and applied to their respective pin locations labeled in Figure 5-4 along w ith the extension moments discussed. Figure 5-3. BCEA Reaction Forces vs. Knee Flexion Anterior (black) and Posterior (Blue) Table 5-2. Summary of BCEA Applied Reaction Forces Reaction Force Pin Maximum Min |FX| |FY| |FX| |FY| (N) (N) Top-Anterior 2.287 1.199 0.00116 0.135 Top-Posterior4.74e-51.929e-52.11e-102.12e-10
90 Figure 5-4. Free Body Diagram of the Pr osthetic KneeÂ’s Top Bracket and Socket These variable external loads were the driving forces behind stress variations within the prosthesis and were analyti cally calculated usi ng ANSYS. Based on magnitudes of the forces and moments calculate d, the moments applied as a result of the BCEA will have more impact on the stresses in duced within the prosthetic socket than those of the reactions forces (whi ch were induced as good measure). 5.5 Analysis and Results Stresses induced on the inner part of the pr osthetic socket were evaluated for each degree of knee flexion and stress magnitude photos were created using SolidWorks (COSMOSWorks). The external loads (as di scussed previously) varied from small knee flexions to large and induced varying stress magnitudes. The criterion we adopted for analyzing proprioception was that the stresses ap plied to the inner part of the prosthetic
91 socket showed distinct variation over knee flexions and did not cause failure of the polypropylene socket. When conducting interface mechanics research it is crucial to determine whether the stresses exposed to the residuum cause ti ssue damage, which could lead to necrosis and further injury to the re sidual limb and surrounding tissues. This chapter evaluates stresses at the inner socket interface and were analyzed as opposed to the interface stresses on the residual limb due to higher am ounts of error present as a result/lack of residual sock (which was unaccounted for in the present model), soft-tissue, epidermis and bone. The simplifications imposed on th e model to make it less computationally intensive also lend themselves to higher error and less justification when crossing material interfaces, therefore only the stresse s on the inner socket wall were analyzed. For this reason, ensuring that the stresses induced on the residual tissue would not cause further tissue damage, was not adopted for this thesis, but should be analyzed with further expansion of the geometry and before any clinical testing. Stress Â‘picturesÂ’ of the inner cone of the prosthetic socket were developed for each degree of knee flexion ranging 0-90 degrees as a result of static failure analysis by the von Mises principle. These results are summarized in Figure 5-5, which shows the stresses from 15-90 degrees for every 15 degr ee step. As shown, the stresses increase over flexions from 0 degrees to 60 degrees and then begi n to decrease. This is characteristic of the moments shown in Fi gure 5-2; Figure 5-5 (F) occurs post-snap, which is a result of the knee and BCEA ready for the seated position. These stress patterns offer initial validation that the BCEA will offer variable stress patterns on the inner part of the prosthetic socket.
92 Figure 5-6 emphasizes the stress variations over the results. Figure 5-6 (A) and (D) depicts the lowest stress states, resulti ng near each of the kneeÂ’s stable equilibrium points (stance and sitting). Figure 5-6 (C) is a result of the maximum extension moments which occur at 62 degrees of flexion, while (D ) is an intermediate point between (A) and (C). These stress variations are a direct result of the BCEA and pose the hypothesis of increased proprioception via stress variat ions over the swing phase. Table 5-3 summarizes the stress results at maximum knee flexion (62 de grees), shown in Figure 5-6 (C). Figure 5-5. Stress Patterns on Inner Part of Prosthetic Socket by Knee Flexion (A) 15 degrees, (B) 30 degrees, (C) 45 degrees, (D) 60 degrees, (E) 75 degrees, (F) 90 degrees Figure 5-6. Stress Pattern Summary Over Key Knee Flexions (A) 8 degrees, (B) 20 degrees, (C) Max Extension Moment (62 degrees), (D) 83 degrees
93 Table 5-3. Surface Stress Summary at 62 Degrees of Flexion Surface Stress Summary at 62 Degrees Value Sum 10.613 MPa Average 14.719 kPa Maximum 51.705 kPa Minimum 0.98013 kPa RMS 18.103 kPa Safety Factor 657.64 These results correspond to the BCEA desi gned previously; the magnitude of the forces applied by the compliant segment (BCEA) can be increased or decreased with the geometry (its width). The safety factor above is large and can be decreased with increasing the applied forces. The resul ting strains caused at maximum knee flexion were also analyzed and are shown in Figure 5-7, and summarized in Table 5-4. Figure 5-7. Strain at Maximum Knee Flexion
94 Table 5-4. Surface Strain Summary at 62 Degrees of Flexion Surface Strain Summary at Max Knee Flexion Value Sum 0.059192 Average 8.2097e-005 Maximum 0.00028839 Minimum 5.4667e-006 RMS 0.00010097 In each of the stress photos, a noticeable stress vari ation is located near the top of the illustration, as shown in Figure 5-8. This stress Â‘anomalyÂ’ is not an anomaly, but a result of the fixation applied to the prosth etic kneeÂ’s top bracket and the socket. The model is constructed in a way that the pol ypropylene socket at the surface of contact between the kneeÂ’s top bracket is fixed at each node (as mentioned previously in boundary conditions), and restricts stresses, stra ins and deflections at the contact pairs. Figure 5-9 shows the area of c ontact and the pairs that are bonded together which form this stress diagram anomaly. In reality, the fi xation methods of the prosthetic socket to the knee will cause stress concentrators via sc rews, pins or bolts, and will result in higher stresses in the area highlighted as oppos ed to the lower stresses calculated here.
95 Figure 5-8. Stress Anomaly Due to Knee Fixation Figure 5-9. Socket and Knee Fixation/Contact Area
96 5.6 Proprioception and Variable Stress Conclusions and Future Work The stresses and strains induced over 90 degr ees of flexion varied over the results as shown, and can be used to develop the hypothesis of increased proprioception based off variable stress patterns in a prosthetic socket as a result of a bistable compliant extension aid. The strains calculated at th e maximum flexion moment knee angle were also analyzed in order to develop the con cept of socket deflecti on under flexion, and showed that the moments caused by the BCEA were sufficient to cause small strains which can be Â‘optimizedÂ’ further by alteri ng the reaction forces brought on by the BCEA (via altering its geometry). These analytical results form the foundation for the measurement/increase of proprioceptive feedback in lower-limb prosth eses by analyzing stress variations within the prosthetic socket. These surface stresses and strains can be used to justify further complexity to the FEA model and calcula tion of the surface stresses induced on a modeled residual limb. Stresses on the modeled residual limb, when calculated efficiently, can produce pre-clinical testing re sults and the basis for experimentation of stress patterns versus proprioception.
97 Chapter 6 Conclusions This chapter focuses on the conclusions of this thesis, the cont ributions made to the mechanical engineering field and recommendations for future work are provided. A bistable compliant extension aid (BCEA) which has the ability to conform to the necessary functional requirements of pros thetic swing of an above-knee prosthesis has been developed and optimized. The resulting BCEA extension moments were analytically calculated using ANSYS and were shown to provide th e necessary moment characteristics of a prosthetic knee extens ion aid. A method for evaluating prosthetic proprioception over the swing pha se by interface stresses between the prosthetic socket and residual limb, as a result of the bistable compliant extension aid, has been introduced. These stresses on the inner cone of the prosthetic socket were calculated using COSMOSWorks. The results were plotted as stress magnitude photos and were visually analyzed over knee flexions from 0 to 90 degrees, and showed the necessary magnitude variations for validation of the propr ioception-via-stress-variation hypothesis. 6.1 Contributions As discussed in Chapter 3, interface mechanics have been researched in depth and have shown promise in future research and a pplication. Interface mechanics, as shown in this thesis as well as other research, have the ability to provide ground-breaking
98 advancements in the field of prosthetics in the form of proprioceptive feedback. Compliant mechanism technology has also shown promise in thei r applications in prosthetic joint research. The primary contribut ions made by this research are as follows: A compliant link (BCEA) has b een developed that can act efficiently as a prosthetic exte nsion aid by providing the necessary extension moments during key knee fl exions. The geometry of the BCEA was optimized in order to meet these functional swingmoment requirements, and the resulting forces on the knee mechanism were analytically calc ulated in order to analyze the stresses induced on the prosthetic socket by the compliant add-on. The BCEA design specialization also offers a way of optimizing a prosthetic knee during swing (app lied extension moments) to any particular patient by altering the geometry of the compliant segment. The external forces and moments induced by the bistable compliant extension aid were applied to a finite element model of an above-knee prosthesis. The interface stresse s on the innersurface of the prosthetic socket we re analyzed in order to lay the foundation for the measurement of proprioceptive feedback by means of induced variable stress patterns. This hypothesis was analytically validated by a simplified finite element model and laid the groundwork for further model refinement.
99 6.2 Suggestions for Future Work Future work should be directed at the re finement and advancement of the finite element model geometries and material properti es used in order to better estimate the interface mechanics developed in this thesis. Residuum tissu es such as the epidermis, bone and cartilage should be added to the re sidual geometry. A residual sock should be included, and the material properties should be verified. Once the model closely reflects actual geometries, the analysis should focus on the stresses on the residual limb rather than the prosthetic socketÂ’s inner surface. Analytical re sults should verify that the induced stresses do not cause tissue trauma to the residual limb. The use of compliant mechanism technology offers several design advantages: if the stresses induced on the residual limb are too high, the BCEA extension moments could be optimized by optimizing the width of the BCEA geometry; while if the stresses on the BCEA are too high, the thickness of the link could be optimized thinner to reduce these stresses. These design advantages allo w for further modificat ion and research of the BCEA and how it affects the inte rface-stress proprioception theory.
100 List of References  Brennan, J.M. and Childress, D.S., Â“ Finite element and exper imental investigation of above-knee amputee limb/prosthesi s systems: A comparative study Â”, Advances in Bioengineering, ASME, Vol. 20 pp. 547-550, 1991.  Britt, M. and Ellen, L. and Thomas, H., Â“ Prosthetic Devices Â”, Retrieved June 15, 2008 from http://www.unc.edu/~mbritt/  Childress, D.S. and Steeg e, J.W. and Wu, Y. et al., Â“ Finite element methods for below-knee socket designÂ”, Rehabilitation Res. Devel opment Prog. Reports, Vol. 29, pp. 22-23, 1992.  Faustini, M. and Neptun e, R. and Crawford, R., Â“ The quasi-static response of compliant prosthetic sockets for transtib ial amputees using fi nite element methods Â”, Medical Engineering & Physics, Vol. 18, pp. 114-121, 2006.  Gurinot, A.E., Magleby, S.P., and Howell, L. L., Â“ Preliminary Design Concepts for Compliant Mechanism Prosthetic Knee JointsÂ” ASME Design Engineering Technical Conference, Salt Lake City, Utah, 2004.  Gurinot, A. E., Magleby, S. P ., Howell, L. L., and Todd, R, H, Â“ Compliant Joint Design Principles for High Compressive Load Situations Â”, Journal of Mechanical Design, in press, 2004.  Highsmith, J.M., Kahle, J.T., Â“ Prosthetic Knees: Classification & Overview Â”, Retrieved May 10, 2008 from http://oandp.health.usf.edu/prosthetics_main.html  Howell, L. L., Â“ Compliant MechanismsÂ” Wiley, New York, 2001.  Jia, X. and Zhang, M. and Li, X. and Lee, W., Â“ A quasi-dynamic nonlinear finite element model to investigate prosthetic interface stresses during walking for transtibial amputees Â”, Clinical Biomechanics Vol. 20 Num. 6, pp. 630-635.  Mahler, S., Â“ Compliant Pediatric Prosthetic KneeÂ”, M.S. Thesis, University of South Florida, Tampa, Florida, 2007.  Advameg, Inc., 2007, Â“ Medical Discoveries Â”, Retrieved May 17, 2008 from http://www.discoveriesinmedicine.com/index.html
101  Moreno, R. and Jones, D. a nd Solomonidis, S. and Mackie, H., Â“ Magnetic Resonance Imaging of Residual Soft Tissu es for Computer-Aided Technology Applications in Prosthetics Â– A Case Study Â”, JPO, Vol. 11 Num. 1 p. 6, 1999.  Ossur, (n.d.), Re trieved December 2007 from http://www.ossur.com/  Otto Bock, (n.d.), Retrieved December 2007 from http://www.ottobockus.com/  Pitkin, M., Â“ Effects of Design Variants in Lower-Limb Prostheses on Gait Synergy Â”, Journal of Prosthetics and Ort hotics, Vol. 9 Num. 3, pp. 113-122, 1997.  Quesada, P. and Skinner, H.B., Â“ Analysis of a below-knee patellar tendon-bearing prosthesis: A Finite element study Â”, J. Rahab. Res. Develop., Vol. 28, Num. 3, pp. 1-12, 1991.  Reynolds, D.P., Â“ Shape design and interface load anal ysis of belowknee prosthetic sockets Â”, Ph.D. dissertation, University of London, 1988.  Reynolds, D.P. and Lord, M., Â“ Interface load analysis for computer-aided design of below-knee prosthetic sockets Â”, Med. Biol. Eng. Com put., Vol. 30, pp. 419-426, 1992.  Rosenberger, B., Â“ Medicare O & P Reimbursement: Part 3 of 3: Can I Have a Cadillac? Â”, inMotion, Vol. 10 Issue 5, 2000.  Sanders, J. and Daly, C., Â“ Normal and shear stresses on a residual limb in a prosthetic socket during ambulation: Com parison of finite element results with experimental measurements Â”, Journal of Rehabilitation Research and Development Vol. 30 Num. 2, pp. 191-204, 1993.  Sanders, J. and Daly, C. and Burgess, E., Â“ Interface shear stresses during ambulation with a below-knee prosthetic limb Â”, Journal of Reha bilitation Research and Development Vol. 29 Num. 4, pp. 1-8, 1992.  Serway, R. A. and Jewett, Jr. J. W., Â“ Physics for Scientists and Engineers Â”, 6th Ed., Brooks Cole, 2003.  Silver-Thorn, M.B., Â“ Estimation and experimental verification of residual limb/prosthetic socket interface pressures for below-knee amputees Â”, Ph.D. dissertation, Northwestern University, 1991.  Silver-Thorn, M.B. and Childress, D.S., Â“ Parametric analysis using the finite element method to investigate prosthetic interface stresse s for persons with transtibial amputation Â”, Journal of Rehabilitation Res earch and Development, Vol. 33 Issue 3, pp. 227-238, 1996.
102  Silver-Thorne, B. and Steege, J., Â“ A review of prosthetic interface stress investigations Â”, Journal of Rehabilitation Resear ch & Development, Vol. 33 Issue 3, pp. 253-266, 1996.  Smith, D., Â“ The Knee Disarticulation: ItÂ’s Better When ItÂ’s Better and ItÂ’s Not When ItÂ’s Not Â”, inMotion, Vol. 14 Issue 1, 2004.  Stark, G., Â“ Overview of Knee Disarticulation, Â” Journal of Prosthe tics and Orthotics, Vol. 16 Issue 4, pp.130-137, 2004.  Steege, J.W. and Childress, D.S., Â“ Analysis of trans-tibial prosthetic gait using the finite element technique Â”, American Academy of Orthotists and Prosthetists, New Orleans, LA, pp. 13-14, 1995.  van de Veen, P.G., Â“ Above-knee Prosthesis Technology Â”, P.G. van de Veen Consultancy, The Netherlands, 2001.  Wheeless, C., Â“ WheelessÂ’ Texbook of Orthopaedics ,Â” Duke University, 2000, retrieved June 29, 2008 from http://www.wheelessonline.com/ortho/gait  Winter, D.A., Biomechanics and Motor Cont rol of Human Movement Wiley, New Jersey, 2005.  Wiersdorf, J., Â“ Preliminary Design Apporoach for Prosthetic Ankle Joints Using Compliant Mechanisms Â”, M.S. Thesis, Brigham Young University, Provo, Utah, 2005.  Zachariah, S. and Sanders, J., Â“ Interface Mechanics in Lower-Limb External Prosthetics: A Review of Finite Element Models Â”, IEEE Transactions on Rehabilitation Engineering, Vol. 4 Num. 4, December 1996.  Zahedi, S., Â“ Lower Limb Prosthetic Research in the 21st Century Â”, ATLAS of Prosthetics, (n.d.).  Zhang, M. and Mak, A., Â“ A Finite Element Analysis of the Load Transfer Between an Above-Knee Residual Limb and Its Pros thetic Socket Â– Roles of Interface Friction and Distal-End Boundary Conditions Â”, IEEE Transactions of Rehabilitation Engineering, Vol. 4, Num. 4, pp. 337-346, December 1996.  Zhang, M. and Mak, A. and Roberts, V.V., Â“ Finite element modeling of a residual lower-limb in a prosthetic socket: a survey of the development in the first decade Â”, Medical Engineering & Physics, Vol. 20, pp. 360-373, 1998.
104 Appendix I: ANSYS Knee Code !*********************************************************************** !/INPUT,C:\DOCUME~1\aroette r\Desktop\KneeCode2,txt,,1 !/CWD,'C:\Documents and Settings\amunoz4\' !*********************************************************************** FINISH /CLEAR /FILENAME, Knee /title,Knee /PREP7 Enter the pre-processor !*********************************************************************** !*************** Model Parameters **************************************** !*********************************************************************** WRITE=1 1= Write output files, Else= Don't Write PI=acos(-1.) hp=6 Posterior Thickness (mm) bp=17 Posterior Width (mm) ha=2 Anterior Thickness (mm) ba=12 Anterior Width (mm) hb=26 Bottom Width (mm) (approx.) bb=5 Bottom Width (mm) (approx.) hc=5 Compliant Geometry bc=1 K1=1e6 Joint Stiffness !***********************************************************************
105 Appendix I (Continued) !*************** Define Area ********************************************* !*********************************************************************** Ap = hp*bp Cross sectional area of posterior Izp= 1/12*bp*hp*hp*hp Second Moment of Area (aka Area Moment of Inertia) Ixp= 1/12*hp*bp*bp*bp Aa = ha*ba Cross sectional area of anterior links Iza= 1/12*ba*ha*ha*ha Second Moment of Area (aka Area Moment of Inertia) Ixa= 1/12*ha*ba*ba*ba Ab = hb*bb Cross sectional area of bottom link (approx.) Izb= 1/12*bb*hb*hb*hb Second Moment of Area (aka Area Moment of Inertia) Ixb= 1/12*hb*bb*bb*bb Ac = hc*bc Cross sectional area of compliant link Izc= 1/12*bc*hc*hc*hc Second Moment of Area (aka Area Moment of Inertia) Ixc= 1/12*hc*bc*bc*bc !*********************************************************************** !************** Define Keypoints ****************************************** !*********************************************************************** Create Keypoints: K(Point #, X-Coord, Y-Coord, Z-Coord) K,1,0,0,0, !(mm) K,2,23.62,0,0, !(mm) K,3,35.52,-85.58,0, !(mm) K,4,35.52,-85.58,0, !(mm) K,5,-12.74,-85.58,0, !(mm) K,6,-12.74,-85.58,0, !(mm) K,7,35.52,-85.58,1, !(mm) K,8,-12.74,-85.58,1, !(mm) K,9,-12.74,-85.58,0, !(mm)
106 Appendix I (Continued) !*********************************************************************** !*************** Create Links ********************************************* !*********************************************************************** L,2,3 Anterior link L,4,5 bottom link L,1,6 Posterior link L,6,8 Pin Joint Direction Line L,4,7 Pin Joint Direction Line !*********************************************************************** !************ Declare Element Type *************************************** !*********************************************************************** SECTYPE, 1, BEAM, RECT, 0 Defines BEAM188 Properties SECOFFSET, CENT SECDATA,1,5,1,1,0,0,0,0,0,0 Defines BEAM188 Geometry ET,1,BEAM4 Element Type 1 Rigid Links !KEYOPT,1,2,1 !KEYOPT,1,6,1 ET,2,COMBIN7,,1 Element Type 2 Pin Joints ET,3,BEAM188 Element Type 3 Compliant Link(BEAM188) !KEYOPT,3,2,1 !KEYOPT,3,6,1 !*********************************************************************** !************ Define Real Constants **************************************** !*********************************************************************** R,1,Ap,Ixp,Izp,hp,bp, Properties of Posterior Links R,2,K1,K1,K1,0,0,0 Prope rties of the pin joints R,3,Aa,Ixa,Iza,ha,ba, Properties of Anterior Links R,4,Ab,Ixb,Izb,hb,bb, Properties of Bottom Link !R,5,Ac,Ixc,Izc,hc,bc, Properties of compliant Link
107 Appendix I (Continued) !*********************************************************************** !********** Define Material Properties *************************************** !*********************************************************************** MP,EX,1,207000 Young's Modulus of Elasticity Steel (MPa) MP,PRXY,1,0.29 Poisson's ratio MP,EX,2, 1400 Young's Modulus of Elasticity Polypropylene (MPa) MP,PRXY,2,0.4103 Poisson's ratio !*********************************************************************** !******************** Mesh ********************************************* !*********************************************************************** type,1 Use element type 1 (Beam4) mat,1 use material property set 1 real,1 Use real constant set 1 LESIZE,ALL,,,2 LMESH,3,3 Mesh Posterior Link real,3 Use real constant set 3 LMESH,1,1 Mesh Anterior Link real,4 Use real constant set 4 LMESH,2,2 Mesh Bottom Link nx = 12.74 Initial x position for prestressed link (xdirections) ny = 85.58 Initial y position for prestressed link (ydir) n_abs = SQRT(12.74*12.74+85.58*85.58) Delta = 2 NUMBER OF INCREMENTAL CHANGES IN COMPLIANT LINK phi_max = PI/2 deleted /2
108 Appendix I (Continued) phi_incr = phi_max/Delta *DO,j,0,Delta,1 Begin th e Compliant Link Creation Loop !*********************************************************************** !************** DEFINE COMPLIANT ************************************* !*********************************************************************** phi = j*phi_incr De fines the "Arc Angle" *IF,j,EQ,0,THEN First iteration: Arc Length=link length R = 1000 L = n_abs *ELSE R = n_abs/(2*sin(phi/2)) L = R*phi *ENDIF DX = (L-n_abs)*nx/n_abs Defines Steps in x direction DY = (L-n_abs)*ny/n_abs Defines steps in y direction K,10,DX,DY,0 Defines keypoi nt (top of compliant link) L,9,10, Line #6 = Compliant Link type,3 Use element type 3 (Beam188) mat,2 !real,5 secnum,1 Makes BEAM188 active for Meshing LESIZE,6,,,32 LMESH,6,6 Mesh Compliant Link !*********************************************************************** !***************** GET NODES ****************************************** !*********************************************************************** ksel,s,kp,,1 nslk,s *get,nkp1,node,0,num,max Retrieves and stores a value as a scalar or part of an array ksel,s,kp,,2 nslk,s
109 Appendix I (Continued) *get,nkp2,node,0,num,max Retrieves and stores a value as a scalar or part of an array ksel,s,kp,,3 nslk,s *get,nkp3,node,0,num,max ksel,s,kp,,4 nslk,s *get,nkp4,node,0,num,max Retrieves and stores a value as a scalar or part of an array ksel,s,kp,,5 nslk,s *get,nkp5,node,0,num,max Retrieves and stores a value as a scalar or part of an array ksel,s,kp,,6 nslk,s *get,nkp6,node,0,num,max ksel,s,kp,,7 nslk,s *get,nkp7,node,0,num,max Retrieves and stores a value as a scalar or part of an array ksel,s,kp,,8 nslk,s *get,nkp8,node,0,num,max ksel,s,kp,,9 nslk,s *get,nkp9,node,0,num,max Retrieves and stores a value as a scalar or part of an array ksel,s,kp,,10 nslk,s *get,nkp10,node,0,num,max ALLSEL TYPE,2 mat,1 use material property set 1 REAL,2 E,nkp3,nkp4,nkp7 Defines an element by node connectivity. E,nkp5,nkp6,nkp8 E,nkp5,nkp9,nkp8
110 Appendix I (Continued) FINISH Finish pre-processing !*********************************************************************** !***************** SOLUTION ******************************************* !*********************************************************************** *get,date,active,,dbase,ldate *get,time,active,,dbase,ltime year=nint(date/10000) month=nint(nint(date-year*10000)/100) day=date-(nint(date/100))*100 hour=nint(time/10000-.5) minute=nint((time-hour*10000)/100-.5) KEYW,PR_SGUI,1 Suppresses "Soluti on is Done" text box /SOL Enter the solution processor /gst,off Turn off graphical convergence monitor ANTYPE,0 Analysis type, static NLGEOM,1 Includes large-deflect ion effects in a static or full transient analysis LNSRCH,AUTO ANSYS automatically switches line search on/off NEQIT,50 Set max # of iterations DELTIM,,0.0001 Set minimum time step increment !*********************************************************************** !*********** Define Displacement Constraints ******************************** !*********************************************************************** DK,1,,0,,,UX,UY,UZ,ROTX,ROTY Pin Joint at Keypoint 1 DK,2,,0,,,UX,UY,UZ,ROTX,ROTY Pin Joint at Keypoint 2 !*********************************************************************** !********** Pre-Stress Comp liant Member ************************************ !*********************************************************************** DK,10,,0,,,UZ,ROTX,ROTY, Constrains the top pin before prestressing preload_steps = 10 Applies Pres tress to Compliant Link in steps DK,1,ROTZ,0 Constrai ns KP1 while Prestressing
111 Appendix I (Continued) *DO,step, 1,preload_steps,1 DK,10,UX,-DX/preload_steps*step DK,10,UY,-DY/preload_steps*step FK,10,MZ,4 Apply a moment to direct the Compliant Link during "assembly" LSWRITE, step *ENDDO !*********************************************************************** !************ Displacement Load ****************************************** !*********************************************************************** FKDELE,10,MZ DK,10,ROTZ,phi/2 Maxrot = 191 Maximum rotation STABILIZE,CONSTANT,ENERGY,1e-5 A pplies a stabilization damping action during snap phenomena *DO,step,1,90,1 theta=step*PI/180 DK,1,ROTZ,theta LSWRITE,step+preload_steps *ENDDO *DO,step,90,0,-1 theta=step*PI/180 DK,1,ROTZ,theta LSWRITE,191-step *ENDDO LSSOLVE,1,Maxrot STABILIZE De-activat es the Stabilize command FINISH Finish the solution processor !*********************************************************************** !************** Postprocessor ********************************************* !***********************************************************************
112 Appendix I (Continued) /POST1 Enter the postprocessor PLDISP,1 Displays deformed & undeformed shape SET,LAST /REPLOT *DIM,ANTERIOR,TABLE,Maxrot,3 *DIM,POSTERIOR,TABLE,Maxrot,3 *DIM,COMPLIANT,TABLE,Maxrot,3 *Do,i,1,Maxrot SET,i Read data for step "i" *GET,rotz1,Node,nkp1,ROT,Z Assi gn ANTERIOR data to ANTERIOR table *SET,ANTERIOR(i,1),rotz1 *GET,fx1,Node,nkp1,RF,FX *SET,ANTERIOR(i,2),fx1 *GET,fy1,Node,nkp1,RF,FY *SET,ANTERIOR(i,3),fy1 *GET,rotz1,Node,nkp2,ROT,Z Assi gn POSTERIOR data to POSTERIOR table *SET,POSTERIOR(i,1),rotz1 *GET,fx1,Node,nkp2,RF,FX *SET,POSTERIOR(i,2),fx1 *GET,fy1,Node,nkp2,RF,FY *SET,POSTERIOR(i,3),fy1 *GET,mz,Node,nkp10,RF,MZ Assign COMPLIANT data to COMPLIANT table *SET,COMPLIANT(i,1),mz *GET,fx1,Node,nkp10,RF,FX *SET,COMPLIANT(i,2),fx1 *GET,fy1,Node,nkp10,RF,FY *SET,COMPLIANT(i,3),fy1 *ENDDO *IF,WRITE,EQ,1,THEN *cfopen,C:\DOCUME~1\aroetter\ANSY S_Results\index%j%,txt *vwrite,month,'-',day,'-',year,hour,':',minute %I %C %I %C %I %4.2I %C %2.2I
113 Appendix I (Continued) *vwrite,'ANTERIOR LINK' %C *vwrite,'ROTX (rad)','FX','FY' %-17C %-17C %-17C *vwrite,ANTERIOR(1,1),AN TERIOR(1,2),ANTERIOR(1,3) %16.8G %16.8G %16.8G *vwrite,'POSTERIOR LINK' %C *vwrite,'ROTX (rad)','FX','FY' %-17C %-17C %-17C *vwrite,POSTERIOR(1,1),POS TERIOR(1,2),POSTERIOR(1,3) %16.8G %16.8G %16.8G *vwrite,'COMPLIANT LINK, L = ',L %C %16.8G *vwrite,'MZ','FX','FY' %-17C %-17C %-17C *vwrite,COMPLIANT(1,1),COMPL IANT(1,2),COMPLIANT(1,3) %16.8G %16.8G %16.8G *cfclose *ENDIF FINISH !*********************************************************************** !********** DELETE COMPLIANT LINK *********************************** !*********************************************************************** /PREP7 LCLEAR,6,6 LDELE,6,6 KCLEAR,11
114 Appendix I (Continued) *ENDDO !ANTIME,45,0.1, ,1,1,0,0 Animate !*********************************************************************** !******************* FINISH ******************************************** !***********************************************************************
115 Appendix II: ANSYS Results File ( =/2) ANTERIOR LINK ROTX (rad) FX FY 0.0000000 -0.12275462 -0.25355748 0.0000000 -8.25826006E-02 2.21055171E-02 0.0000000 -6.57499172E-02 0.14110606 0.0000000 -5.61195668E-02 0.21184787 0.0000000 -4.97955059E-02 0.26050694 0.0000000 -4.53107617E-02 0.29694249 0.0000000 -4.19729793E-02 0.32580702 0.0000000 -3.94107317E-02 0.34960197 0.0000000 -3.74030535E-02 0.36981594 0.0000000 -3.58098291E-02 0.38739485 1.74532925E-02 1.16364284E-03 0.47859500 3.49065850E-02 4.37110341E-03 0.50367282 5.23598776E-02 6.51178243E-03 0.52845713 6.98131701E-02 7.60999646E-03 0.55293795 8.72664626E-02 7.68930672E-03 0.57710563 0.10471976 6.77245016E-03 0.60095079 0.12217305 4.88137068E-03 0.62446422 0.13962634 2.03725745E-03 0.64763689 0.15707963 -1.73942494E-03 0.67045996 0.17453293 -6.42890129E-03 0.69292461 0.19198622 -1.20120547E-02 0.71502223 0.20943951 -1.84703932E-02 0.73674423 0.22689280 -2.57860222E-02 0.75808212 0.24434610 -3.39416081E-02 0.77902742 0.26179939 -4.29203595E-02 0.79957175 0.27925268 -5.27059897E-02 0.81970670 0.29670597 -6.32826936E-02 0.83942389 0.31415927 -7.46351245E-02 0.85871497 0.33161256 -8.67483681E-02 0.87757156 0.34906585 -9.96079145E-02 0.89598528 0.36651914 -0.11319965 0.91394775 0.38397244 -0.12750980 0.93145053 0.40142573 -0.14252498 0.94848518 0.41887902 -0.15823210 0.96504324 0.43633231 -0.17461838 0.98111617 0.45378561 -0.19167134 0.99669543
116 Appendix II (Continued) 0.47123890 -0.20937879 1.0117724 0.48869219 -0.22772881 1.0263385 0.50614548 -0.24670970 1.0403851 0.52359878 -0.26631004 1.0539033 0.54105207 -0.28651862 1.0668844 0.55850536 -0.30732449 1.0793196 0.57595865 -0.32871688 1.0912000 0.59341195 -0.35068528 1.1025166 0.61086524 -0.37321935 1.1132605 0.62831853 -0.39630901 1.1234226 0.64577182 -0.41994434 1.1329938 0.66322512 -0.44411570 1.1419649 0.68067841 -0.46881362 1.1503267 0.69813170 -0.49402889 1.1580699 0.71558499 -0.51975255 1.1651851 0.73303829 -0.54597587 1.1716629 0.75049158 -0.57269040 1.1774937 0.76794487 -0.59988800 1.1826681 0.78539816 -0.62756082 1.1871762 0.80285146 -0.65570139 1.1910082 0.82030475 -0.68430260 1.1941544 0.83775804 -0.71335777 1.1966047 0.85521133 -0.74286068 1.1983490 0.87266463 -0.77280566 1.1993769 0.89011792 -0.80318759 1.1996782 0.90757121 -0.83400206 1.1992421 0.92502450 -0.86524539 1.1980580 0.94247780 -0.89691477 1.1961148 0.95993109 -0.92900836 1.1934013 0.97738438 -0.96152545 1.1899058 0.99483767 -0.99446662 1.1856167 1.0122910 -1.0278339 1.1805216 1.0297443 -1.0616312 1.1746080 1.0471975 -1.0958643 1.1678627 1.0646508 -1.1305413 1.1602720 1.0821041 -1.1656733 1.1518216 1.0995574 -1.2012746 1.1424963 1.1170107 -1.2373633 1.1322803 1.1344640 -1.2739624 1.1211564 1.1519173 -1.3111007 1.1091064 1.1693706 -1.3488138 1.0961105 1.1868239 -1.3871462 1.0821472 1.2042772 -1.4261532 1.0671926 1.2217305 -1.4659037 1.0512205
117 Appendix II (Continued) 1.2391838 -1.5064846 1.0342012 1.2566371 -1.5480058 1.0161009 1.2740903 -1.5906085 0.99688061 1.2915436 -1.6344765 0.97649452 1.3089969 -1.6798542 0.95488767 1.3264502 -1.7270747 0.93199262 1.3439035 -1.7766082 0.90772408 1.3613568 -1.8291483 0.88196996 1.3788101 -1.8857852 0.85457508 1.3962634 -1.9483987 0.82530731 1.4137167 -2.0207494 0.79377348 1.4311700 -2.1132177 0.75909882 1.4486233 -2.2875178 0.71674734 1.4660766 -0.57671990 0.16870011 1.4835299 -0.55434973 0.16075208 1.5009832 -0.53120905 0.15378854 1.5184364 -0.50747209 0.14770845 1.5358897 -0.48314117 0.14253714 1.5533430 -0.45821857 0.13830084 1.5707963 -0.43270667 0.13502674 1.5707963 -0.43266698 0.13501719 1.5533430 -0.45821527 0.13830003 1.5358897 -0.48313831 0.14253639 1.5184364 -0.50746965 0.14770776 1.5009832 -0.53120696 0.15378792 1.4835299 -0.55434796 0.16075152 1.4660766 -0.57689025 0.16857409 1.4486233 -0.59883148 0.17723195 1.4311700 -0.62016917 0.18670221 1.4137167 -0.64090084 0.19696273 1.3962634 -0.66102384 0.20799202 1.3788101 -0.68053548 0.21976931 1.3613568 -0.69943296 0.23227440 1.3439035 -0.71771337 0.24548772 1.3264502 -0.73537363 0.25939020 1.3089969 -0.75241062 0.27396335 1.2915436 -0.76882104 0.28918914 1.2740903 -0.78460149 0.30505004 1.2566371 -0.79974843 0.32152891 1.2391838 -0.81425819 0.33860906 1.2217305 -0.82812698 0.35627416 1.2042772 -0.84135089 0.37450828 1.1868239 -0.85392584 0.39329578 1.1693706 -0.86584766 0.41262139
118 Appendix II (Continued) 1.1519173 -0.87711207 0.43247013 1.1344640 -0.88771462 0.45282730 1.1170107 -0.89765076 0.47367846 1.0995574 -0.90691582 0.49500944 1.0821041 -0.91550501 0.51680633 1.0646508 -0.92341343 0.53905542 1.0471975 -0.93063604 0.56174321 1.0297443 -0.93716769 0.58485644 1.0122910 -0.94300316 0.60838206 0.99483767 -0.94813705 0.63230718 0.97738438 -0.95256388 0.65661912 0.95993109 -0.95627804 0.68130539 0.94247780 -0.95927383 0.70635370 0.92502450 -0.96154541 0.73175192 0.90757121 -0.96308681 0.75748813 0.89011792 -0.96389196 0.78355061 0.87266463 -0.96395464 0.80992782 0.85521133 -0.96326849 0.83660845 0.83775804 -0.96182703 0.86358142 0.82030475 -0.95962362 0.89083586 0.80285146 -0.95665143 0.91836119 0.78539816 -0.95290348 0.94614708 0.76794487 -0.94837259 0.97418352 0.75049158 -0.94305137 1.0024608 0.73303829 -0.93693221 1.0309697 0.71558499 -0.93000720 1.0597013 0.69813170 -0.92226821 1.0886471 0.68067841 -0.91370674 1.1177992 0.66322512 -0.90431396 1.1471504 0.64577182 -0.89408060 1.1766939 0.62831853 -0.88299697 1.2064238 0.61086524 -0.87105285 1.2363352 0.59341195 -0.85823740 1.2664240 0.57595865 -0.84453913 1.2966872 0.55850536 -0.82994574 1.3271233 0.54105207 -0.81444403 1.3577323 0.52359878 -0.79801975 1.3885159 0.50614548 -0.78065744 1.4194780 0.48869219 -0.76234016 1.4506248 0.47123890 -0.74304932 1.4819657 0.45378561 -0.72276431 1.5135134 0.43633231 -0.70146215 1.5452849 0.41887902 -0.67911698 1.5773024 0.40142573 -0.65569946 1.6095942
119 Appendix II (Continued) 0.38397244 -0.63117606 1.6421966 0.36651914 -0.60550797 1.6751555 0.34906585 -0.57864987 1.7085292 0.33161256 -0.55054811 1.7423922 0.31415927 -0.52113837 1.7768399 0.29670597 -0.49034223 1.8119958 0.27925268 -0.45806227 1.8480225 0.26179939 -0.42417476 1.8851377 0.24434610 -0.38851814 1.9236406 0.22689280 -0.35087392 1.9639559 0.20943951 -0.31093284 2.0067146 0.19198622 -0.26822884 2.0529145 0.17453293 -0.22199235 2.1042863 0.15707963 -0.17073312 2.1643744 0.13962634 -0.11064601 2.2427072 0.12217305 4.73122866E-03 0.62439373 0.10471976 6.77244703E-03 0.60095073 8.72664626E-02 7.68930383E-03 0.57710559 6.98131701E-02 7.60999253E-03 0.55293791 5.23598776E-02 6.51177769E-03 0.52845709 3.49065850E-02 4.37109710E-03 0.50367276 1.74532925E-02 1.16352861E-03 0.47859481 0.0000000 -3.13624507E-03 0.45323340 POSTERIOR LINK ROTX (rad) FX FY 4.14762417E-07 -1.03590297E-10 8.31092147E-10 4.16918034E-07 -1.06284760E-10 7.88110742E-10 4.20321322E-07 1.16419659E-10 -8.43510450E-10 4.24171207E-07 1.05138429E-10 -8.45079124E-10 4.28276799E-07 -2.36527072E-10 1.64390138E-09 4.32568497E-07 1.26457928E-12 1.75841798E-13 4.37012983E-07 1.11480755E-10 -8.29994968E-10 4.41596322E-07 -1.48313694E-11 2.63421695E-11 4.46310565E-07 1.09795665E-10 -8.16027036E-10 4.51152099E-07 -2.11737686E-10 1.63314613E-09 1.74750786E-02 -5.08725134E-09 3.18856876E-08 3.49929066E-02 -5.94695857E-09 3.34431501E-08 5.25540019E-02 -6.90012012E-09 3.51042395E-08 7.01585608E-02 -8.22974336E-09 3.82613477E-08 8.78067967E-02 -9.03186141E-09 3.84402078E-08 0.10549894 -1.05264652E-08 4.15939279E-08 0.12323525 -1.17756891E-08 4.32158845E-08 0.14101598 -1.33242437E-08 4.58023841E-08 0.15884142 -1.47434502E-08 4.73178646E-08
120 Appendix II (Continued) 0.17671189 -1.67372681E-08 5.05309114E-08 0.19462770 -1.83293541E-08 5.21717638E-08 0.21258921 -2.00185711E-08 5.37786626E-08 0.23059678 -2.27204650E-08 5.77588800E-08 0.24865079 -2.48944730E-08 6.01429103E-08 0.26675165 -2.81434577E-08 6.45700814E-08 0.28489980 -3.09190516E-08 6.76934251E-08 0.30309566 -3.46401040E-08 7.21830625E-08 0.32133973 -3.75724014E-08 7.45728286E-08 0.33963248 -4.07827653E-08 7.74493924E-08 0.35797443 -4.58312915E-08 8.33032594E-08 0.37636611 -5.10724535E-08 8.88296990E-08 0.39480808 -5.67189356E-08 9.46764176E-08 0.41330092 -6.27278000E-08 1.00366837E-07 0.43184523 -6.84125974E-08 1.05024158E-07 0.45044163 -7.65807111E-08 1.12908459E-07 0.46909077 -8.38427106E-08 1.18664615E-07 0.48779331 -9.21669777E-08 1.25360335E-07 0.50654995 -1.01732241E-07 1.33078322E-07 0.52536140 -1.12851553E-07 1.41960354E-07 0.54422840 -1.24736479E-07 1.50877541E-07 0.56315170 -1.38005818E-07 1.60473966E-07 0.58213209 -1.52153354E-07 1.70360813E-07 0.60117037 -1.69568438E-07 1.82744179E-07 0.62026736 -1.88162362E-07 1.95151794E-07 0.63942391 -2.06965971E-07 2.06511679E-07 0.65864090 -2.30071205E-07 2.20924511E-07 0.67791920 -2.56260256E-07 2.36718966E-07 0.69725973 -2.84561125E-07 2.52804581E-07 0.71666341 -3.16024578E-07 2.69900963E-07 0.73613120 -3.52218503E-07 2.89117421E-07 0.75566405 -3.90943761E-07 3.08205395E-07 0.77526296 -4.35214506E-07 3.29522251E-07 0.79492892 -4.85301318E-07 3.52727662E-07 0.81466294 -5.40763760E-07 3.76864436E-07 0.83446606 -6.02798595E-07 4.02458208E-07 0.85433931 -6.74050314E-07 4.30962778E-07 0.87428374 -7.51247862E-07 4.59474432E-07 0.89430042 -8.39558802E-07 4.90497508E-07 0.91439041 -9.37366354E-07 5.22637162E-07 0.93455477 -1.04974239E-06 5.57767411E-07 0.95479459 -1.17195401E-06 5.92723721E-07 0.97511094 -1.31261858E-06 6.30685831E-07 0.99550489 -1.46921558E-06 6.69328522E-07
121 Appendix II (Continued) 1.0159775 -1.64448127E-06 7.08878688E-07 1.0365298 -1.84096817E-06 7.49113707E-07 1.0571629 -2.06205856E-06 7.89723539E-07 1.0778778 -2.30911789E-06 8.30126251E-07 1.0986754 -2.58627865E-06 8.69414058E-07 1.1195569 -2.89800145E-06 9.07272204E-07 1.1405230 -3.24163069E-06 9.40879881E-07 1.1615748 -3.62923346E-06 9.71189915E-07 1.1827132 -4.05938368E-06 9.94932782E-07 1.2039388 -4.54150554E-06 1.01146057E-06 1.2252527 -5.07357900E-06 1.01726875E-06 1.2466554 -5.66598723E-06 1.01060704E-06 1.2681476 -6.32192634E-06 9.88028431E-07 1.2897300 -7.04716257E-06 9.46420623E-07 1.3114030 -7.84815566E-06 8.81513754E-07 1.3331672 -8.72976617E-06 7.88983039E-07 1.3550228 -9.70260304E-06 6.63999172E-07 1.3769701 -1.07625402E-05 5.00105303E-07 1.3990092 -1.19201500E-05 2.91386339E-07 1.4211403 -1.31827123E-05 3.10895031E-08 1.4433632 -1.45493575E-05 -2.88168021E-07 1.4656777 -1.60298310E-05 -6.74683821E-07 1.4880835 -1.76191819E-05 -1.13636846E-06 1.5105800 -1.93263039E-05 -1.68256701E-06 1.5331667 -2.11479135E-05 -2.32219033E-06 1.5558428 -2.30912406E-05 -3.06545070E-06 1.5786073 -2.51643888E-05 -3.92373613E-06 1.6014591 -2.73952180E-05 -4.91285048E-06 1.6243969 -7.36181232E-10 -2.12467544E-10 1.6474193 -2.38266363E-09 -5.18788189E-10 1.6705246 -3.56291520E-05 -8.78645618E-06 1.6937103 -3.61645686E-05 -9.98972057E-06 1.7169750 -3.85147803E-05 -1.16072207E-05 1.7403162 -4.08468308E-05 -1.33549578E-05 1.7637314 -4.31349029E-05 -1.52269867E-05 1.7872181 -4.53493010E-05 -1.72135802E-05 1.8107733 -4.74532170E-05 -1.92996512E-05 1.8107733 8.06359272E-10 4.35563040E-10 1.7872181 -4.13113288E-05 -1.56931109E-05 1.7637314 -3.96171975E-05 -1.39969930E-05 1.7403162 -3.78015765E-05 -1.23707171E-05 1.7169750 -3.58945854E-05 -1.08284464E-05 1.6937103 -3.39233111E-05 -9.38111152E-06 1.6705246 -3.19185494E-05 -8.03730769E-06
122 Appendix II (Continued) 1.6474197 -2.99024403E-05 -6.80120927E-06 1.6243976 -2.78977847E-05 -5.67517681E-06 1.6014600 -2.59281419E-05 -4.65983709E-06 1.5786084 -2.40092611E-05 -3.75255196E-06 1.5558440 -2.21540199E-05 -2.94924520E-06 1.5331681 -2.03691295E-05 -2.24448743E-06 1.5105814 -1.86741586E-05 -1.63305484E-06 1.4880850 -1.70642763E-05 -1.10742278E-06 1.4656793 -1.55485035E-05 -6.60769804E-07 1.4433649 -1.41326189E-05 -2.85874870E-07 1.4211420 -1.28125947E-05 2.46985831E-08 1.3990110 -1.15884815E-05 2.78039676E-07 1.3769719 -1.04562528E-05 4.80854611E-07 1.3550247 -9.41597270E-06 6.39660385E-07 1.3331691 -8.46602003E-06 7.60778542E-07 1.3114050 -7.59502068E-06 8.48916227E-07 1.2897320 -6.80384664E-06 9.09744676E-07 1.2681497 -6.08599727E-06 9.47538841E-07 1.2466574 -5.43497185E-06 9.65918119E-07 1.2252548 -4.84837587E-06 9.68791726E-07 1.2039410 -4.31952076E-06 9.59052325E-07 1.1827153 -3.84463431E-06 9.39207348E-07 1.1615770 -3.41781742E-06 9.11804718E-07 1.1405252 -3.03687751E-06 8.78885842E-07 1.1195591 -2.69870864E-06 8.42481836E-07 1.0986777 -2.39213995E-06 8.01872019E-07 1.0778800 -2.11967658E-06 7.59929844E-07 1.0571651 -1.87988102E-06 7.18141619E-07 1.0365321 -1.66523889E-06 6.75705605E-07 1.0159798 -1.47449070E-06 6.33786165E-07 0.99550716 -1.30471062E-06 5.92689893E-07 0.97511322 -1.15333609E-06 5.52550978E-07 0.95479687 -1.02223384E-06 5.15471285E-07 0.93455705 -9.04038768E-07 4.78989811E-07 0.91439268 -7.99182246E-07 4.44363846E-07 0.89430270 -7.07212862E-07 4.12018162E-07 0.87428602 -6.25885454E-07 3.81589592E-07 0.85434158 -5.52546181E-07 3.52150264E-07 0.83446832 -4.88168474E-07 3.24774024E-07 0.81466519 -4.31993646E-07 3.00117864E-07 0.79493116 -3.83363621E-07 2.77572497E-07 0.77526519 -3.38215503E-07 2.55271020E-07 0.75566627 -2.99778880E-07 2.35657561E-07 0.73613340 -2.65747348E-07 2.17453732E-07
123 Appendix II (Continued) 0.71666560 -2.34464185E-07 1.99459978E-07 0.69726190 -2.08466835E-07 1.84370744E-07 0.67792135 -1.83952469E-07 1.69303695E-07 0.65864303 -1.62206827E-07 1.55195387E-07 0.63942602 -1.43645950E-07 1.42767620E-07 0.62026945 -1.26662156E-07 1.30808351E-07 0.60117243 -1.12970323E-07 1.21289210E-07 0.58213413 -9.93648212E-08 1.10788646E-07 0.56315371 -8.71907857E-08 1.00957420E-07 0.54423038 -7.79921793E-08 9.38192826E-08 0.52536335 -7.01087904E-08 8.77331245E-08 0.50655187 -6.10875305E-08 7.96067981E-08 0.48779519 -5.38099820E-08 7.27375902E-08 0.46909261 -4.73315328E-08 6.64753522E-08 0.45044344 -4.15701943E-08 6.08042163E-08 0.43184699 -3.74674202E-08 5.70107299E-08 0.41330264 -3.32544575E-08 5.27598843E-08 0.39480976 -2.91313036E-08 4.81349651E-08 0.37636774 -2.59132607E-08 4.48412096E-08 0.35797601 -2.24924221E-08 4.06213356E-08 0.33963401 -2.05984129E-08 3.89202621E-08 0.32134121 -1.76991231E-08 3.48676539E-08 0.30309709 -1.45921288E-08 3.00940068E-08 0.28490116 -1.32133675E-08 2.85352336E-08 0.26675295 -1.17851906E-08 2.68157833E-08 0.24865202 -9.97614384E-09 2.37924154E-08 0.23059794 -9.02836419E-09 2.27840445E-08 0.21259029 -8.14118153E-09 2.18481453E-08 0.19462869 -6.14660881E-09 1.71525668E-08 0.17671278 -4.66600838E-09 1.38345413E-08 0.15884220 -2.36450556E-10 7.44723913E-10 0.14101660 1.53403203E-11 4.39759000E-12 0.12323525 -2.32018913E-10 8.09669424E-10 0.10549894 -3.56419321E-09 1.38678047E-08 8.78067967E-02 -2.94076450E-09 1.23316964E-08 7.01585608E-02 -2.81346042E-09 1.30400919E-08 5.25540019E-02 -1.97318858E-09 9.77909261E-09 3.49929066E-02 -2.03928196E-09 1.14398919E-08 1.74750786E-02 -1.20443953E-09 7.32510239E-09 3.39573733E-07 -1.06427490E-09 7.37626628E-09 COMPLIANT LINK, L = 96.102983 MZ FX FY 5.8959368 0.12273181 0.25367429 5.8959368 8.25794175E-02 -2.20957357E-02
124 Appendix II (Continued) 5.8959368 6.57280172E-02 -0.14105002 5.8959368 5.61188787E-02 -0.21184645 5.8959368 4.97954614E-02 -0.26050689 5.8959368 4.50273356E-02 -0.29645170 5.8959368 4.18012981E-02 -0.32553479 5.8959368 3.92950865E-02 -0.34943282 5.8959368 3.73187904E-02 -0.36970157 5.8959368 3.57448207E-02 -0.38731258 5.2817123 -1.16360698E-03 -0.47859513 4.5327583 -4.37110328E-03 -0.50367275 3.7955388 -6.51178398E-03 -0.52845708 3.0699153 -7.60999905E-03 -0.55293789 2.3557480 -7.68931040E-03 -0.57710557 1.6529045 -6.77245435E-03 -0.60095072 0.96125970 -4.88137600E-03 -0.62446415 0.28069578 -2.03726381E-03 -0.64763683 -0.38889782 1.73941754E-03 -0.67045988 -1.0476242 6.42889270E-03 -0.69292453 -1.6955793 1.20120449E-02 -0.71502215 -2.3328517 1.84703822E-02 -0.73674415 -2.9595227 2.57860092E-02 -0.75808203 -3.5756664 3.39415943E-02 -0.77902733 -4.1813498 4.29203439E-02 -0.79957165 -4.7766324 5.27059715E-02 -0.81970659 -5.3615668 6.32826742E-02 -0.83942379 -5.9361982 7.46351034E-02 -0.85871486 -6.5005648 8.67483445E-02 -0.87757145 -7.0546973 9.96078891E-02 -0.89598517 -7.5986197 0.11319962 -0.91394763 -8.1323482 0.12750977 -0.93145040 -8.6558922 0.14252495 -0.94848505 -9.1692536 0.15823206 -0.96504310 -9.6724268 0.17461833 -0.98111602 -10.165399 0.19167129 -0.99669528 -10.648150 0.20937874 -1.0117723 -11.120651 0.22772875 -1.0263384 -11.582867 0.24670963 -1.0403849 -12.034753 0.26630997 -1.0539031 -12.476258 0.28651855 -1.0668842 -12.907322 0.30732441 -1.0793194 -13.327875 0.32871679 -1.0911997 -13.737840 0.35068518 -1.1025163 -14.137130 0.37321924 -1.1132602 -14.525649 0.39630887 -1.1234223
125 Appendix II (Continued) -14.903290 0.41994420 -1.1329934 -15.269938 0.44411553 -1.1419645 -15.625464 0.46881344 -1.1503263 -15.969731 0.49402869 -1.1580695 -16.302587 0.51975232 -1.1651847 -16.623868 0.54597561 -1.1716624 -16.933399 0.57269011 -1.1774933 -17.230987 0.59988767 -1.1826675 -17.516426 0.62756046 -1.1871756 -17.789492 0.65570099 -1.1910076 -18.049944 0.68430214 -1.1941537 -18.297521 0.71335725 -1.1966040 -18.531943 0.74286010 -1.1983482 -18.752903 0.77280500 -1.1993761 -18.960072 0.80318685 -1.1996772 -19.153094 0.83400123 -1.1992411 -19.331579 0.86524446 -1.1980569 -19.495107 0.89691372 -1.1961136 -19.643217 0.92900717 -1.1933999 -19.775409 0.96152411 -1.1899044 -19.891133 0.99446511 -1.1856151 -19.989787 1.0278322 -1.1805199 -20.070707 1.0616293 -1.1746061 -20.133160 1.0958621 -1.1678606 -20.176332 1.1305389 -1.1602698 -20.199319 1.1656706 -1.1518191 -20.201106 1.2012715 -1.1424937 -20.180552 1.2373598 -1.1322774 -20.136365 1.2739585 -1.1211533 -20.067077 1.3110963 -1.1091030 -19.970997 1.3488088 -1.0961068 -19.846175 1.3871406 -1.0821430 -19.690329 1.4261469 -1.0671881 -19.500769 1.4658966 -1.0512155 -19.274279 1.5064766 -1.0341957 -19.006965 1.5479967 -1.0160947 -18.694029 1.5905980 -0.99687366 -18.329444 1.6344644 -0.97648644 -17.905460 1.6798399 -0.95487793 -17.411809 1.7270574 -0.93198018 -16.834386 1.7765859 -0.90770680 -16.152846 1.8291169 -0.88194284 -15.335871 1.8857339 -0.85452485 -14.330426 1.9482930 -0.82519101
126 Appendix II (Continued) -13.032101 2.0204384 -0.79340264 -11.155889 2.1132167 -0.75909681 -9.0192968 2.2874484 -0.71665331 -1.6532915 0.57681237 -0.16853089 -2.3557479 0.55432110 -0.16074229 -3.0699154 0.53117826 -0.15377864 -3.7955388 0.50743909 -0.14769851 -4.5327582 0.48310592 -0.14252724 -5.2817206 0.45818110 -0.13829107 -6.0425807 0.43266697 -0.13501718 -6.0425808 0.43266698 -0.13501719 -5.2817206 0.45818110 -0.13829107 -4.5327582 0.48310592 -0.14252724 -3.7955388 0.50743909 -0.14769851 -3.0699154 0.53117826 -0.15377864 -2.3557478 0.55432111 -0.16074229 -1.6529045 0.57686523 -0.16856495 -0.96125978 0.59880825 -0.17722298 -0.28069594 0.62014769 -0.18669345 0.38889793 0.64088104 -0.19695423 1.0476242 0.66100565 -0.20798382 1.6955792 0.68051883 -0.21976143 2.3328515 0.69941776 -0.23226686 2.9595228 0.71769952 -0.24548053 3.5756665 0.73536106 -0.25938338 4.1813497 0.75239923 -0.27395691 4.7766323 0.76881075 -0.28918308 5.3615669 0.78459221 -0.30504435 5.9361982 0.79974008 -0.32152359 6.5005647 0.81425070 -0.33860408 7.0546972 0.82812026 -0.35626953 7.5986198 0.84134488 -0.37450398 8.1323483 0.85392047 -0.39329179 8.6558922 0.86584288 -0.41261770 9.1692535 0.87710780 -0.43246672 9.6724269 0.88771082 -0.45282416 10.165399 0.89764739 -0.47367557 10.648150 0.90691283 -0.49500679 11.120651 0.91550236 -0.51680389 11.582867 0.92341108 -0.53905318 12.034753 0.93063396 -0.56174117 12.476258 0.93716586 -0.58485458 12.907321 0.94300154 -0.60838035 13.327875 0.94813562 -0.63230562
127 Appendix II (Continued) 13.737840 0.95256262 -0.65661771 14.137130 0.95627695 -0.68130411 14.525649 0.95927287 -0.70635253 14.903290 0.96154457 -0.73175086 15.269938 0.96308608 -0.75748717 15.625464 0.96389132 -0.78354974 15.969731 0.96395408 -0.80992704 16.302587 0.96326801 -0.83660775 16.623868 0.96182662 -0.86358078 16.933399 0.95962326 -0.89083529 17.230987 0.95665113 -0.91836068 17.516426 0.95290322 -0.94614663 17.789492 0.94837238 -0.97418312 18.049944 0.94305120 -1.0024605 18.297521 0.93693206 -1.0309694 18.531943 0.93000710 -1.0597010 18.752903 0.92226813 -1.0886469 18.960072 0.91370668 -1.1177990 19.153094 0.90431392 -1.1471502 19.331579 0.89408058 -1.1766938 19.495107 0.88299697 -1.2064237 19.643217 0.87105286 -1.2363352 19.775409 0.85823743 -1.2664239 19.891133 0.84453917 -1.2966872 19.989787 0.82994579 -1.3271233 20.070707 0.81444409 -1.3577324 20.133160 0.79801982 -1.3885160 20.176332 0.78065751 -1.4194781 20.199319 0.76234023 -1.4506249 20.201106 0.74304939 -1.4819658 20.180552 0.72276439 -1.5135135 20.136365 0.70146221 -1.5452850 20.067077 0.67911702 -1.5773024 19.970997 0.65569948 -1.6095942 19.846175 0.63117604 -1.6421965 19.690329 0.60550789 -1.6751554 19.500769 0.57864970 -1.7085290 19.274279 0.55054781 -1.7423918 19.006965 0.52113786 -1.7768392 18.694029 0.49034138 -1.8119946 18.329445 0.45806087 -1.8480205 17.905460 0.42417244 -1.8851343 17.411809 0.38851414 -1.9236348 16.834386 0.35086669 -1.9639453
128 Appendix II (Continued) 16.152847 0.31091875 -2.0066940 15.335871 0.26819827 -2.0528696 14.330426 0.22191417 -2.1041715 13.030323 0.17073315 -2.1643746 11.155582 0.11064468 -2.2427060 0.96163888 -4.89963670E-03 -0.62440516 1.6529045 -6.77245572E-03 -0.60095072 2.3557480 -7.68931150E-03 -0.57710557 3.0699153 -7.60999988E-03 -0.55293789 3.7955388 -6.51178452E-03 -0.52845707 4.5327583 -4.37110352E-03 -0.50367275 5.2817208 -1.16353473E-03 -0.47859479 6.0425805 3.13623875E-03 -0.45323340 Max Stress 6.1251658 7.7995846 9.1967479 10.386807 11.462996 12.454655 13.371542 14.229913 15.029086 15.803786 14.017332 14.037438 14.093132 14.144231 14.239578 14.331166 14.428180 14.556667 14.681731 14.803323 14.949454 15.105262 15.258134 15.408012 15.559058 15.739972 15.918394 16.094262 16.267512 16.438081
129 Appendix II (Continued) 16.606119 16.804978 17.001622 17.195985 17.388001 17.577605 17.764731 17.949315 18.131290 18.330078 18.538316 18.744389 18.948229 19.149772 19.348948 19.545689 19.739925 19.931585 20.120597 20.306885 20.490374 20.700464 20.909066 21.115349 21.319241 21.520667 21.719550 21.915809 22.109361 22.300120 22.487994 22.672890 22.854705 23.033336 23.208669 23.380586 23.579214 23.796170 23.995157 24.175361 24.342269 24.529522 24.713642 24.894495
130 Appendix II (Continued) 25.071937 25.245810 25.415938 25.582126 25.744154 25.925583 26.098868 26.244099 26.357657 26.549890 26.756104 26.927398 27.073487 27.255062 27.435807 27.617228 27.803307 28.046970
131 Appendix III: Matlab Code for Plotting Flexion and Extension Moments clf for index = 0:30 filename = ['index',num2str(index),'.txt'] string1 = 'C:\DOCUME~1\aroe tter\ANSYS_~1\'; % Directory fid1 = fopen([string1,filename]); % opens the file ABT = fread(fid1); % reads the file into variable ABT in machine code fclose(fid1); % closes the data file GBT = native2unicode(ABT)'; % changes data from machine code to text and writes it to GBT header_begin = findstr('FY', GBT); % finds end of first header anterior_end = findstr('POSTERIOR', GBT); posterior_end = findstr('COMPLIANT', GBT); compliant_end = length(GBT); ANTERIOR = str2num(GBT(header_b egin(1)+3:anterior_end-1)); % turns the data into a numerical matrix POSTERIOR = str2num(GBT( header_begin(2)+3:p osterior_end-1)); COMPLIANT = str2num(GB T(header_begin(3)+3:compliant_end)); h=figure(1) h1=plot(ANTERIOR(:,1) *180/pi,COMPLIANT(:,1),'*-') hold on h4 =gca set(h4,'FontSize',12) axis([0 90 -15 15]) % low x high x, low y high y %grid on h2=xlabel('Knee Angle (deg)') set(h2,'FontSize',12) h3=ylabel('BCEA Moment (N-m)')
132 Appendix III (Continued) set(h3,'FontSize',12) h=figure(2) h1=plot(ANTERIOR(101:191,1) *180/pi,COMPLIANT(101:191,1),'*-') hold on h4 =gca set(h4,'FontSize',12) axis([0 90 -15 15]) % low x high x, low y high y %grid on h2=xlabel('Knee Angle (deg)') set(h2,'FontSize',12) h3=ylabel('BCEA Moment (N-m)') set(h3,'FontSize',12) end %print BCEAMomentall -dtiff -r600 % MATLAB Code for Extension Data Graphing clf for index = 25:30 filename = ['index',num2str(index),'.txt'] string1 = 'C:\DOCUM E~1\aroetter\ANSYS_~1\Loadin~1\'; % Directory fid1 = fopen([string1,filename]); % opens the file ABT = fread(fid1); % reads the file into variable ABT in machine code fclose(fid1); % closes the data file GBT = native2unicode(ABT)'; % changes data from machine code to text and writes it to GBT header_begin = findstr('FY', GBT); % finds end of first header anterior_end = findstr('POSTERIOR', GBT); posterior_end = findstr('COMPLIANT', GBT); compliant_end = length(GBT);
133 Appendix III (Continued) ANTERIOR = str2num(GBT(header_begin( 1)+3:anterior_end-1)); % turns the data into a numerical matrix POSTERIOR = str2num(GBT( header_begin(2)+3:p osterior_end-1)); COMPLIANT = str2num(GB T(header_begin(3)+3:compliant_end)); h=figure(1) h1=plot(ANTERIOR(:,1) *180/pi,COMPLIANT(:,1),'*-') hold on h4 =gca set(h4,'FontSize',12) axis([0 90 -15 5]) % low x high x, low y high y h2=xlabel('Knee Angle (deg)') set(h2,'FontSize',12) h3=ylabel('BCEA Moment (N-m)') set(h3,'FontSize',12) end %print BCEAMomentall -dtiff -r600
134 Appendix IV: Matlab Code for Plotting Reaction Forces clf index = 30 filename = ['index',num2str(index),'.txt'] string1 = 'C:\DOCUME~1\aroe tter\ANSYS_~1\'; % Directory fid1 = fopen([string1,filename]); % opens the file ABT = fread(fid1); % reads the file into variable ABT in machine code fclose(fid1); % closes the data file GBT = native2unicode(ABT)'; % changes data from machine code to text and writes it to GBT header_begin = findstr('FY', GBT); % finds end of first header anterior_end = findstr('POSTERIOR', GBT); posterior_end = findstr('COMPLIANT', GBT); compliant_end = length(GBT); ANTERIOR = str2num(GBT(header _begin(1)+3:anterior_end1)); % turns the data into a numerical matrix %ANTERIORFX = str2num(GBT (header_begin(1)+19:anterior_end-1)); %ANTERIORFY = str2num(GBT (header_begin(1)+36:anterior_end-1)); POSTERIOR = str2num(GBT( header_begin(2)+3:posterior_end-1)); %POSTERIORFY = str2num(GBT (header_begin(2)+36:posterior_end-1)); ANTERIOR_MAG = (ANTERIOR( 10:100,2).^2+ANTERIOR(10:100,3).^2).^0.5; ANTERIOR_ANG = atan2(ANTER IOR(10:100,3),ANTERIOR(10:100,2))*180/pi; POSTERIOR_MAG = (POSTERIOR( 10:100,2).^2+POSTERIOR(10:100,3).^2).^0.5; POSTERIOR_ANG = atan2(POSTER IOR(10:100,3),POSTERIO R(10:100,2))*180/pi; h=figure(1) h1=plot(ANTERIOR(10:100,1)*180/pi,ANTERIO R(10:100,2),'*-') hold on h4 =gca set(h4,'FontSize',12) %axis([0 90 -15 15]) % low x high x, low y high y %grid on h2=xlabel('Knee Angle (deg)') set(h2,'FontSize',12) h3=ylabel('Fx-Anterior (N)') set(h3,'FontSize',12)
135 Appendix IV (Continued) h=figure(2) h1=plot(ANTERIOR(10:100,1)*180/pi,ANTERIO R(10:100,3),'*-') hold on h4 =gca set(h4,'FontSize',12) %axis([0 90 -15 15]) % low x high x, low y high y %grid on h2=xlabel('Knee Angle (deg)') set(h2,'FontSize',12) h3=ylabel('Fy-Anterior (N)') set(h3,'FontSize',12) h=figure(3) h1=plot(ANTERIOR_ANG,ANTERIOR_MAG,'*-') hold on h4 =gca set(h4,'FontSize',12) %axis([0 90 -15 15]) % low x high x, low y high y %grid on h2=xlabel('Direction (deg)') set(h2,'FontSize',12) h3=ylabel('|F|-Anterior (N)') set(h3,'FontSize',12) h=figure(4) h1=plot(ANTERIOR(10:100,1)*180/pi,ANTERIO R_MAG,'*-') hold on h4 =gca set(h4,'FontSize',12) %axis([0 90 -15 15]) % low x high x, low y high y %grid on h2=xlabel('Knee Angle (deg)') set(h2,'FontSize',12) h3=ylabel('|F|-Anterior (N)') set(h3,'FontSize',12) h=figure(5) h1=plot(ANTERIOR(10:100,1)* 180/pi,POSTERIOR(10:100,2),'*-') hold on h4 =gca
136 Appendix IV (Continued) set(h4,'FontSize',12) %axis([0 90 -15 15]) % low x high x, low y high y %grid on h2=xlabel('Knee Angle (deg)') set(h2,'FontSize',12) h3=ylabel('Fx-Posterior (N)') set(h3,'FontSize',12) h=figure(6) h1=plot(ANTERIOR(10:100,1)* 180/pi,POSTERIOR(10:100,3),'*-') hold on h4 =gca set(h4,'FontSize',12) %axis([0 90 -15 15]) % low x high x, low y high y %grid on h2=xlabel('Knee Angle (deg)') set(h2,'FontSize',12) h3=ylabel('Fy-Posterior (N)') set(h3,'FontSize',12) h=figure(7) h1=plot(POSTERIOR_ANG,POSTERIOR_MAG,'*-') hold on h4 =gca set(h4,'FontSize',12) %axis([0 90 -15 15]) % low x high x, low y high y %grid on h2=xlabel('Direction (deg)') set(h2,'FontSize',12) h3=ylabel('|F|-Posterior (N)') set(h3,'FontSize',12) h=figure(8) h1=plot(ANTERIOR(10:100,1)*180/pi,POSTERIO R_MAG,'*-') hold on h4 =gca set(h4,'FontSize',12) %axis([0 90 -15 15]) % low x high x, low y high y %grid on h2=xlabel('Knee Angle (deg)') set(h2,'FontSize',12) h3=ylabel('|F|-Posterior (N)') set(h3,'FontSize',12)
137 Appendix IV (Continued) h=figure(9) h1=plot(ANTERIOR(10:100,1)*180/pi,POSTERIO R_MAG,'.-') hold on h4 =gca set(h4,'FontSize',12) %axis([0 90 -15 15]) % low x high x, low y high y %grid on h2=xlabel('Knee Angle (deg)') set(h2,'FontSize',12) h3=ylabel('|F| (N)') set(h3,'FontSize',12) h5=plot(ANTERIOR(10:100,1)* 180/pi,ANTERIOR_MAG,'k+-') %print BCEAMomentall -dtiff -r600
138 Appendix V: Reaction Force Plots Figure A-1. Anterior Force in x-Direction vs. Knee Angle Figure A-2. Anterior Force in y-Direction vs. Knee Angle
139 Appendix V (Continued) Figure A-3. Magnitude of Anterior Force vs. Knee Angle Figure A-4. Magnitude of Anterior Force vs. Direction
140 Appendix V (Continued) Figure A-5. Posterior Force in x-Direction vs. Knee Angle Figure A-6. Posterior Force in y-Direction vs. Knee Angle
141 Appendix V (Continued) Figure A-7. Magnitude of Posterior Force vs. Knee Angle Figure A-8. Magnitude of Posterior Force vs. Direction
142 Appendix VI: COSMOSWorks Repo rt File Â– Socket and Knee Stress analysis of Residuum and Knee with BCEA Loads during Flexion Author: Adam D. Roetter Introduction File Information Materials Load & Restraint Information Study Property Contact Results Appendix 1. Introduction Summarize the FEM analysis on Top Half of Leg with Knee Bracket for Moment Application Note: Do not base your design decisions solely on the data presente d in this report. Use this information in conjunction with experimental data and practical expe rience. Field testing is mandatory to validate your final de sign. COSMOSWorks helps you reduce your timeto-market by reducing but not eliminating field tests.
143 Appendix VI (Continued) 2. File Information Model name: Top Half of Leg with Knee Bracket for Moment Application Model location: C:\Users\Adam\Documents\Soli dWorks Thesis\Top half of System (Residuum Stresses)\Top Half of Leg with Knee Bracket for Moment Application.SLDASM Results location: c:\users\adam\appdata\local\temp Study name: Moment Application Xdeg of Rotation (-Default-) 3. Materials No. Part Name Material Mass Volume 1 Disartic Knee Top Link Bracket With Socket Attachment-1 [SW]Titanium 0.151837 kg 3.30081e-005 m^3 2 Residuum-1 [SW]Rubber 8.65506 kg 0.00865506 m^3 3 Socket to fit top link with bracket-1 [SW]PE Low/Medium Density 0.536364 kg 0.000584911 m^3
144 Appendix VI (Continued) 4. Load & Restraint Information Restraint Restraint-1 on 1 Face(s) fixed. Description: Load Force-1 on 1 Face(s) apply force -20.201 N along circumferential. with respect to selected reference Face< 1 > using uniform distribution Force-2 on 1 Edge(s) apply force -1.2013 N normal to reference plane with respect to selected reference Edge< 1 > using uniform distribution Force-3 on 1 Edge(s) apply force -4.0594e-006 N normal to reference plane with respect to selected reference Edge< 1 > using uniform distribution Force-4 on 1 Edge(s) apply force 1.1425 N normal to reference plane with respect to selected reference Top Plane using uniform distribution
145 Appendix VI (Continued) 5. Study Property Mesh Information Mesh Type: Solid mesh Mesher Used: Standard Automatic Transition: On Smooth Surface: On Jacobian Check: 4 Points Element Size: 1.0754 in Tolerance: 0.053772 in Quality: High Number of elements: 13803 Number of nodes: 22393 Time to complete mesh(hh;mm;ss): 00:00:14 Computer name: INTELC2D Solver Information Quality: High Solver Type: FFEPlus Option: Include Thermal Effects Thermal Option: Input Temperature Thermal Option: Reference Temperature at zero strain: 298 Kelvin
146 Appendix VI (Continued) 6. Contact Contact state: Touching faces Bonded 7. Results 7a. Stress2 (von Mises) 7a. Strain1 (-Equivalent-)
147 Appendix VI (Continued) 7b. Default Results Name Type Min Location Max Location Stress2 VON: von Mises stress 2.49959 N/m^2 Node: 12454 (0.295982 in, 0.127197 in, 1.62194 in) 3.91105e+006 N/m^2 Node: 2652 (1.13819 in, -0.945371 in, 0.071128 in) Strain1 ESTRN: Equivalent strain 1.03548e008 Node: 5103 (0.946526 in, -2.36945 in, 0.071128 in) 0.00115734 Node: 21022 (0.941113 in, -1.36445 in, 2.35248 in) 8. Appendix Material name: [SW]Titanium Description: Material Source: Used SolidWorks material Material Library Name: SolidWorks Materials Material Model Type: Linear Elastic Isotropic
148 Appendix VI (Continued) Property Name Value Units Value Type Elastic modulus 1.1e+011 N/m^2 Constant Poisson's ratio 0.3 NA Constant Shear modulus 4.3e+010 N/m^2 Constant Mass density 4600 kg/m^3 Constant Tensile strength 2.35e+008 N/m^2 Constant Yield strength 1.4e+008 N/m^2 Constant Thermal expansion coefficient 9e-006 /Kelvin Constant Thermal conductivity 22 W/(m.K) Constant Specific heat 460 J/(kg.K) Constant Material name: [SW]Rubber Description: Material Source: Used SolidWorks material Material Library Name: solidworks materials Material Model Type: Linear Elastic Isotropic Property Name Value Units Value Type Elastic modulus 6.1e+006 N/m^2 Constant Poisson's ratio 0.49 NA Constant Shear modulus 2.9e+006 N/m^2 Constant Mass density 1000 kg/m^3 Constant Tensile strength 1.3787e+007 N/m^2 Constant Yield strength 9.2374e+006 N/m^2 Constant Thermal expansion coefficient 0.00067 /Kelvin Constant Thermal conductivity 0.14 W/(m.K) Constant
149 Appendix VI (Continued) Material name: [SW]PE Low/Medium Density Description: Material Source: Used SolidWorks material Material Library Name: solidworks materials Material Model Type: Linear Elastic Isotropic Property Name Value Units Value Type Elastic modulus 1.72e+008 N/m^2 Constant Poisson's ratio 0.439 NA Constant Shear modulus 5.94e+007 N/m^2 Constant Mass density 917 kg/m^3 Constant Tensile strength 1.327e+007 N/m^2 Constant Thermal conductivity 0.322 W/(m.K) Constant Specific heat 1842 J/(kg.K) Constant