«IV-3'1]- 5,7: .4474 3 39,; 3'3“.“ ' "' £3,153!"- ' .' .,." 4).“ .‘r' < , ' ' .'- ‘ . .3?’§;‘7-. 3-,~ ' ',,"3‘I'i/"'I'{-g'n. / '4‘. $.nin‘..-3'.u.‘-JRL'W x1, vf, in l ‘dfihg' "F':‘ ‘l 7.. ':l' . n'. " “I '7'. «‘45:, “35.33.323.333", My: " o ' ‘ ’y. L.'. V I". ". ~ ‘3: “if L3,, “-‘ s 32'2' °“-n .."nn\'n L]! I . a "5- n'n -"" 'n: ‘3‘ (‘u'm’"' .v mi." 4- n. -_R . _R, \. J‘r 9W4" ‘1'?) K v ‘52}:‘3‘? Y ' X ‘ TIA?- 3“.”3193}. ,. {up a {An '0... ,z'm?’“ "‘8 3' ' . .3. .5» ‘ , (3? ’{v "$&'%R.. )‘i‘lfi’i': - ' ~‘{‘ ("in 4"“ mm 4< , we". W44: :92. «‘ 415' -n :1 _ , 3,! .1 : 3'5‘f"£oafn't""fi\‘i~€ ‘: . I 9' -)" Ly; hru:\\ ' '9‘ n" 4 i. .33}? i; 1'31”"‘1‘R‘45‘a. vh: u"? 1 \‘uf 1.2334: an!“ “' "' 9’- ' SE ““:I'~Q:I‘ 1' n?! «4' a ' ':""- . "if :‘J' 'V 'o ‘ .115 "v V’q-izv " 1' . . QI v AI:' ’3 v'}"“ up, n . e\$. .n' -'\ INN“ 31-‘(I'Jnf‘fn‘s' . ‘J (in/:5 ‘l‘i "Mi?“ “ 4:33? «12$. inf/13‘. 3:13.133” l .1413-3. £47,”; I F‘ {H‘- 3“: x'; 3"; fl}. "' if"; [$391 G’Hlfi‘.“ u - ‘- 3. AIA-Jsvn'4""" " 'Y' n, L '9‘ {:3';:¢)n2‘~‘_‘ 33:": . , , 3n, 4. . w“ 5': 1:} 5‘» 5!” n ." 1‘ " ‘q'n‘fi‘ I)”; ' R‘MH/H,‘ 333;. «m la nI-‘AK' L‘n' ;‘ If (fining. :3; 3W '," ' 4‘ ,nfl" ."n ;-3 .‘Ji. _‘5 fit 334‘ d£;_=< < .' _'.'";JL;.'(,1 wn'f‘n ,A‘ /':"‘n" .n-," p' "2': """3" :m‘" \"" ‘ ‘ 4 ’ ';"' '3“ ' 3,135,. "3 t n, 3.4.“... 13.» .K‘tl"a‘i‘#,"‘.3_.|k)‘..r.‘ .' 3w , 3, ",3 31 -. ' ,ni‘fi‘ktfi‘ffl -n 11.033" 'n i"! "EVE” 2;? a . .-:r' MW 13%: .‘k’ win (‘3': A")?! ":ij‘g‘af 3 n/' n"""".' _ . '3 1,3,3“, 33 5‘43‘. - ”’45. [:j: Pry-'1" 1:! ;‘~ Hufr‘ '61,. '5', f g 151%; : 3., , . . .L, ‘I'r‘. 43’;- ‘J‘X‘tj .‘ .nn‘g 3_I.J .3 ,3». ‘ ‘ .3 1.3214135" ?§ l‘y'fidf' - m. (1:. .5. .4» «44444.44». 3‘ All . . f. r. .‘ " 3 . 3 .135: . . LL“: 2“}; “WIRE; \ 5:13,?! . ,r;V‘-':'e“, '3'”: '(e‘ ' ":5‘ I’ o ‘f I nvfi‘uurcr. 23A)”. ...’4 {M 333‘"? (Mica; «nut Ma\{l Tin . 3‘3""? {1' 17“, $33M, .3 A: fit-C n (4", =(3!?‘:_‘ 3139;};"' f: 4.0,", "' . n ’R" 1433”,,“ y???» " .i‘ ' ,4 ' a?“ If 'n . Ycz‘. “("3 v r‘fl‘ . ,3 4.3% 3 ‘3 3‘ . 3 33:33:15.1;1‘ "§3"&';&1{"¥”: 33,3??? ‘I'u’l‘1,q._.," "34¢."1','-. 5." $15 "33'“ .,.. I v "s 7\ 3.36“.) 30 v MlCHIGAN STATE UNIVFRQHTY LIBRARIES mg; !l w I gllwixlgll‘llw l‘ I” :ng igl. ll J: M“?! .‘fil.’ l 3i :IH I” ll “m9 ll Mi 7 3 1293 00577 9966 LIBRARY l Michigan State University 1 This is to certify that the dissertation entitled A METHOD FOR COMPUTING THE THREE-DIMENSIONAL FORCES AND MOMENTS IN THE LOWER LIMB DURING LOCOMOTION presented by Mary C. Verstraete has been accepted towards fulfillment of the requirements for PhD Mechanics degree in Major professor Date June 10, 1988 MS U in an Affirmative Action/Equal Opportunity Institution 0-12771 -¢‘l PLACE IN RETURN BOX to remove thic checkout from your record. TO AVOID FINES return on or before date due. DATEQB'U-E DATE DUE DATE DUE 7336211399 OE?) U o 1992*. MSU Is An Affirmative Action/Equal Opportunity Institution A METHOD FOR COMPUTING THE THREE-DIMENSIONAL FORCES AND MOMENTS IN THE LOWER LIMB DURING LOCOMOTION BY Mary C. Verstraete A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Metallurgy, Mechanics and Material Science 1988 l 'JUI’TJ ABSTRACT A METHOD FOR COMPUTING THE THREE-DIMENSIONAL FORCES AND MOMENTS IN THE LOWER LIMB DURING LOCOMOTION BY Mary C. Verstraete The forces and moments internal to the human body may not be directly measured. Therefore, models are developed to estimate these reactions using the principles of rigid- body mechanics. In this study, the Newton-Euler equations of motion were written for rigid-body approximations of the foot, shank and thigh segments of the right lower limb and were solved for the three—dimensional components of the force and moment at the linkage centers of the ankle, knee and hip during stance phase of locomotion. An experimental protocol was also developed to provide the necessary input for the equations of motion. The experimental and analytical techniques available provided three-dimensional position data for targets placed on the limb segments, three-dimensional ground reaction forces and moments, and anthropometric measurements.' The literature was reviewed to predict the directions of the principal axes, the location of the center of mass, the principal moments of inertia and the mass of each segment. A numerical solution technique, based on the method of least squares, was developed to compute the angular velocities and accelerations of the three links. The forces and moments computed at the linkage centers of the right lower limb compared favorably with those presented in the literature. The choice of coordinate system in which these reactions were expressed was found to have an extreme effect on both the magnitudes and directions of the forces and moments. The static and dynamic components of the resultant forces and moments computed at each of the three joints were determined and their percent contribution to the total was calculated. The dynamic terms were found to provide a minimal contribution at the ankle, but showed a contribution of 0% - 80% of the total force at the knee and hip, and 0% - 50% of the total moment at these two joints. It was concluded that a complete analysis of the forces and moments at the linkage centers of the lower limbs produced during stance phase of locomotion must include the dynamic components of the equations of motion, as well as, the static components. To my wonderful family: for their neverending love, encouragement and patience. ii ACKNOWLEDGEMENTS The author wishes to express her sincerest appreciation and gratitude to the following: To Dr. Robert Wm. Soutas-Little, her major professor, for the past seven years of guidance and friendship. To Brooks Shoe, Inc., for their generous funding for this and other projects. To Jane Walsh, for her constant warm friendship, and for always listening and understanding when the going got rough. To V. Dianne Ulibarri, for her advice, her friendship, and for her wonderful sense of humor. To all of my friends and colleagues in the Department of Biomechanics at Michigan State University, for making the past seven years the most enjoyable years of my life. iii TABLE LIST OF TABLES . . . . . . LIST OF FIGURES . . . . . LIST OF NOMENCLATURE . . . INTRODUCTION . . . . . . . LITERATURE SURVEY . . . . OF CONTENTS I. RIGID-BODY MECHANICS . . . . . II. JOINT ANATOMY AND MOTION . . . III. LOCOMOTION . . . EXPERIMENTAL METHODS . . . THEORETICAL DEVELOPMENT . I. THE FOOT AND ANKLE . . . . . . II. THE SHANK AND KNEE . . . . . . III. THE THIGH AND HIP IV. SUMMARY . . . . RESULTS AND DISCUSSION . . CONCLUSIONS . . . . . . . APPENDIX A - THE METHOD OF APPENDIX B - THE CENTER OF BIBLIOGRAPHY . . . . . . . LEAST SQUARES PRESSURE EQUATIONS iv . ix. . 14. . 23. . 36. . 41. . 48. . 71. .100. .133. .138. .145. .147. Table Table Table Table Table Table LIST OF TABLES The Reported Location of the ' Center of Mass (CM) of the Foot Relative to Anatomical Landmarks Segment Weights Reported as Percentages of Total Body Weight The Reported Location of the Center of Mass (CM) of the Shank Relative to Anatomical Landmarks The Reported Location of the Center of Mass (CM) of the Thigh Relative to Anatomical Landmarks Target Locations . . . . . . . . Anatomical Landmarks . . . . . . 55. 59. 77. 89. 97. 98. Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure LIST OF FIGURES Filming Set-up with the Calibration Structure. . . . . . . . . . . . . Free Body Diagram of the Foot Segment. . . . . . . . . . . . . . Free Body Diagram of the Shank Segment. . . . . . . . . . . . . . Free Body Diagram of the Thigh Segment. . . . . . . . . . . . . . Targetting of the Foot. . . . . . . The Effect of Toe-off on the Posterior-Anterior Axis of the Foot. Geometrical Representation of the Angle of Rotation,<1. . . . . Geometrical Representation for the Center of Mass of the Foot. . . Model of the Foot Used for the Moment of Inertia Calculations. . . The Force Plate Origin and Coordinate Axes and the Ground Reactions. . . . . . . . . . The Lab and Force Plate Coordinate Systems and the Targetting of the Plate. . . . . . Targetting of the Shank. . . . . . Geometrical Representation of the Angle of Rotation,f3. . . . . . The Location of the Center of Mass of the Shank. . . . . . . vi 38. 43. 44. 45. 49. 50. 52. 56. 61. 66. 69. 74. 75. 80. Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. The Frustrum of an Inverted Right Circular Cone. . . . . . . . Targetting of the Thigh and the A518. . . . . . . . . . . . . . Geometrical Representation of the Angle of Rotation,£ . . . . . . Segmentation of the Thigh and the Flap. O O O O O O I O O O O O O Portion of the Thigh Used for Moment of Inertia Calculations. . . Orientation of the Three Local Coordinate Systems and the Lab Coordinate System. . . . . . . . . The Force at the Ankle for a Single Subject. . . . . . . . . . . The Moment at the Ankle for a Single Subject. . . . . . . . . . . The Components of the Force at the Ankle Joint. . . . . . . . . . . . The Components of the Moment at the Ankle Joint. . . . . . . . . . . . The Force at the Knee for a Single Subject. . . . . . . . . . . The Moment at the Knee for a Single Subject. . . . . . . . . . . The Components of the Force at the Knee Joint. . . . . . . . . . . . . The Components of the Moment at the Knee Joint. . . . . . . . . . . . . The Force at the Hip for a Single Subject. . . . . . . . . . . The Moment at the Hip for a Single Subject. . . . . . . . . . . The Components of the Force at the Hip Joint. . . . . . . . . . . . . The Components of the Moment at the Hip Joint. . . . . . . . . . . . . vii 82. 87. 88. 93. 94. 102. 104. 105. 106. 108. 110. 111. 112. 114. 116. 117. 119. 120. Figure 33. Figure 34. Figure 35. The x, y and 2 Components of the Forces at the Ankle, Knee and Hip. . . . . 124. The x, y and 2 Components of the Moments at the Ankle, Knee and Hip. . . . . 126. The Convergence of the Angular Velocity and Acceleration with an Increased Number of Vectors. . . . . . . 144. viii aCMF' acnsr aCMT CJA, CJK CMF, CMS, CMT cpx, cpy, cpz CoP FA' FK' FH F1, F2, F3 ‘IB A F 'Ef.'§§.'§f HJC A\ A\ A\ lFx' le' le fix . /k /\ 1F1r 1F2r iF3 LIST OF NOMENCLATURE Acceleration of the center of mass of the foot, shank and thigh. The location of the linkage centers of the ankle and knee joint. The location of the center of mass of the foot, shank and thigh. The location of the center of pressure in the lab coordinate system. The location of the center of pressure in the force plate coordinate system. The resultant force computed at the ankle, knee and hip. The components of the force in the local coordinate system of the distal limb segment. The ground reaction force. _A_unit vector in the direction of F De The gravity vector expressed in terms of the local foot, shank and thigh coordinate systems. The location of the linkage center of the hip joint. Unit vectors defining the three orthogonal directions for the force 'plate coordinate system. Unit vectors defining the three “ orthogonal directions for the local foot coordinate system. ix A\ A\ 151' 152' 133 A. At 1A 1T1' 1T2' 1T3 IFl' IFZ' IF3 151' Iszr IS3 ITl' ITZ' IT3 MA, MK, MH M1, M2, M3 'fif, RS, RT 3? 32, S3, 54 T1, T2, T3 wF, ws, wT 3%” Unit vectors defining the three orthogonal directions for the local shank coordinate system. Unit vectors defining the three orthogonal directions for the local thigh coordinate system. The principal moments of inertia for the foot. The principal moments of inertia for the shank. The principal moments of inertia for the thigh. The resultant moments computed at the ankle, knee and hip. The components of the moment in the local coordinate system of the distal limb segment. The ground reaction moment. The parallel component of M0. The mass of the foot, shank and thigh segments. Vectors from the center of mass of each segment to the distal linkage center. Vectors from the center of mass of each segment to the proximal linkage center. The ground reaction force expressed in the lab coordinate system. Designation for subjects 2, 3 and 4, the three male subjects. Designation for three different trials, of the same subject. The angular velocity of the foot, shank and thigh. The angular acceleration of the foot, shank and thigh. XCF . /\ xFF r YCPI h QFP' /Z\FP Coordinates of the center of pressure expressed in the force plate coordinate system. Unit vectors defining the directions of the force plate coordinate system in terms of the lab coordinate system. xi INTRODUCTION Walking is possibly the most common of all human activities, the process of locomotion occurring with little thought or exertion. Yet, this recurring process involves a complex interaction of bones, muscles, nerves and other biological factors. The skeletal system, connected together through ligaments, tendons and muscles, provides the structural support for the body. Walking, like most physical motions, is a learned activity. From a child‘s first attempt to support him/herself in a standing posture, the size, shape and level of development of a child's limbs continue to affect his/her style of gait well into the adult years. These personal characteristics, superimposed upon the basic pattern of bipedal locomotion, provide subtle differences in the gait patterns of the adult population. Although no two people walk identically alike, certain attributes of walking are prevalent, and these, therefore, define what researchers refer to as normal walking. In a clinical sense, normal is referred to as the uninjured, nonpathological state, and therefore, normal walking is characterized by the locomotion of subjects who show no indication of injury or pathology. 2 The process of normal walking is commonly characterized by a set of repetitive cycles. The fundamental cycle is labelled a stride and is generally defined as the interval between two successive initial contacts of the same foot. This interval is measured by the distance between the two points of contact and is designated as stride length. Each stride is further divided into two periods; stance phase, the amount of time the foot is in contact with the ground, and swing phase, the time period during which the foot leaves the ground and moves ahead of the body prior to contact. These two features, stride length and stance phase, define the distance and time parameters of a particular gait. Minor variations in these basic factors have been shown to cause large deviations in the dynamics of walking. Therefore, any analysis of normal walking should include these elementary measurements. The process of locomotion has been studied for many years by investigators from various fields of research. The demands for information regarding the mechanism of normal gait are numerous and diverse. Rapid advancements in technology have necessitated an increased understanding of the workings of the human body. The development of prosthetic limbs and artificial joints has required an extensive knowledge of the functions of normal limbs and joints, as well as the interactions between them. During the process of walking, large forces may develop across the joints. These forces create stresses which have been shown 3 to contribute to joint degeneration or loosening of prosthetic joint replacements. Determination of these forces and their resultant line of action is therefore a critical factor in the design of artificial mechanical joints. The rise in popularity of exercise and athletics has created increased demands on suppliers to develop better equipment and on physicians to provide improved care for various injuries. Shoes, braces and other equipment need to provide adequate support and protection during various athletic activities. Yet, the amount of support or protection necessary to prevent injury without limiting performance is unknown. Also, it is not known whether the amount of support or protection varies between individuals. In order to answer these questions, and others, an understanding of the motions and forces carried by the limbs and joints during various activities is needed. Many difficulties arise in identifying the numerous factors involved in locomotion. Static, dynamic and anthropometric quantities are necessary to completely describe the mechanism of walking. Several of these factors are difficult, if not impossible, to attain for living human subjects. Within the bounds of current technology, the forces and motions occurring within the joints of the human body are unobtainable. The presence of surrounding tissues creates an obstruction to measuring the actions and interactions of the articulating surfaces. It is also 4 difficult to determine the mass distribution properties of the lower limbs of living subjects. The geometries of the segments of the lower limbs are irregular and are therefore hard to model. The mass of each link has only been directly measured from cadaveric studies. Thus, prior to developing an adequate description of the dynamics of gait, several assumptions or estimations must be made to account for these unknown values. The human body may be viewed as a mechanical structure, composed of materials having rather unusual mechanical properties. In order to closely approximate the mechanisms involved in walking, the lower limbs are commonly modelled as a mechanical linkage system and the laws of rigid-body mechanics are utilized to govern the dynamics of locomotion. Bones form the structural organization for the body segments and these, therefore, become the basis for the dynamic analysis of motion. The solid nature of the bones allow for the limb segments to be represented by rigid links and the joints by three-dimensional articulations. By making these approximations, the mechanical linkage system of the lower limbs is assumed to obey Newton's laws of motion and the concepts of classical rigid-body mechanics. One very important consideration, often simplified in the literature, is the three-dimensional nature of this rigid-body system. Although the motions of the lower limbs during walking are principally in the sagittal plane, or the plane of progression, the forces and moments have components 5 in all three planes of motion allowed by the joints and these six degrees of freedom must be included in the analysis of human locomotion. This creates a more complex model and often requires rather lengthy calculations. Researchers have investigated the various aspects of locomotion. Previous studies have attempted to describe the individual factors involved in normal gait utilizing several methods and varying assumptions. Some authors omit various dynamic components involved in the solution for the forces and moments acting at the joints of the lower limbs in order to simplify the numerous computations. There still exists a need for a complete description of the three-dimensional dynamics of walking utilizing the results and methods of current research. The dynamics of the lower limbs during walking gait involve a complex relationship between three- dimensional displacements, velocities, accelerations, forces, moments, and inertial characteristics. The objective of this research was to formulate the equations of motion for the linkage system composed of the foot, shank and thigh, and to solve the inverse dynamics problem for the three-dimensional forces and moments at the joints of the lower limbs during walking gait. In doing so, it was also necessary to develop an experimental protocol to determine the individual components required for input to the Newton-Euler equations of motion, utilizingavailable experimental techniques. The contribution of each term to the sum of the terms in the equations of motion was also 6 determined. In the third chapter, general experimental methods will be presented to set guidelines for the theoretical development. Theoretical methods will then expand upon the specific procedures used for this study. LITERATURE SURVEY The analysis of human locomotion has been an ongoing task since the late 1800's. Numerous researchers have investigated the many separate components involved in the process of walking, as well as, the interactions between many of these factors. The mechanism of walking has been examined by biomechanists, engineers, and physicans alike, each utilizing the principles of their profession to more closely define the parameters of gait. Due to the diverse background necessary to completely describe the three- dimensional mechanics of walking, researchers must review the current methods and theories of several disciplines. The following survey of the literature is, therefore, divided into three major sections. The first area reviews current techniques utilized by engineers to characterize the dynamics of rigid-body linkages and mechanisms. The second section summarizes the physiolgical attributes of the lower limbs, including the description of the motions allowed at each of the joints, the structure of the articulations, and the mass distribution properties of the segments of the lower limbs. The final section reviews selected methods utilized in gait analysis, the effects of temporal factors, 8 . and two and three-dimensional dynamic analyses of walking gait. I. RIGID-BODY DYNAMICS Multi-rigid-body dynamics is the study of the relationship between the motions of a system of interconnected rigid bodies and the forces and moments, internal and external to the system. Such rigid-body dynamics problems have been generally classified as one of two types, specific to the input variables known for the system. The first, known as the direct dynamics problem, determines the resulting motions of a multi-rigid-body system given the external forces and moments acting on the system. The second type of problem is generated when the motions of the system are known and the forces and moments are desired. This is defined as the inverse dynamics problem and is commonly encountered in determining the forces necessary to produce certain motions in a mechanical device. The inverse dynamics problem has been solved for numerous mechanical and biomechanical systems, using various techniques. The type of problem developed is highly dependent upon the input variables known. The methods for formulating the equations of motion are also varied with the given input data. All of these solution methods stem from ‘the fundamentals of classical dynamics. The dynamic equations for rigid—body systems may be .formulated in a number of ways (29,30,50): Lagrange's 9 equations, Newton's equations, Kane's equations (Lagrange's form of D'Alembert's principle) and Euler's equations, as well as variations or combinations of these. The geometry of the system, whether complex or simple, plays an important role in determining the analysis method to be utilized. Typical mechanical systems investigated using these methods include : closed chain linkages - every link in the chain is connected to at least two other links, open chain linkages, and various spatial mechanisms - linkage systems having one link fixed and allowing three-dimensional motions between the links. Such analyses are important for the development of industrial and robotic systems. Many mechanical systems are composed of a number of rigid links connected by joints that allow only a single degree of freedom, either a rotation or a translation. Analysis of this type of linkage commonly utilizes a system of generalized coordinates to describe the relative motion between two adjoining links. As Denavit, et al. (25) showed in 1965, the relative velocity and acceleration between adjacent links may be simply computed as the first and second time derivatives of the generalized coordinate, respectively. These definitions and the application of matrix algebra to coordinate transformations allowed Denavit and coworkers to describe the motion of an arbitrary single- degree-of-freedom closed chain linkage. They also performed a static force analysis on the linkage, using the method of virtual work, such that, given the output force or moment, 10 they could predict the input force or moment, and the internal forces and moments required to maintain static equilibrium. The use of generalized coordinates facilitates the application of Lagrange's equations to describe the behavior of mechanical systems. Chace (13), in 1967, formulated Lagrange's differential equations of motion, with appropriate second order constraints, to analyze the time- dependence of highly-constrained multi-degree-of-freedom mechanical systems. He noted that the number of generalized coordinates necessary to specify the position of the mechanical network was smaller than the number of coordinates needed for a conventional analysis. Uicker (77), at the same time, developed similar expressions for the kinetic and potential energies of a multiloop, multidegree of freedom system, and included the possibility of springs or damping devices at any or all of its joints. Chace and Bayazitoglu (14), in 1971, expanded on the first author's earlier work by including a generalized D'Alembert's force analysis and constraint functions in terms of Lagrangian multipliers. These additions simplified the development of the differential equations by treating the mechanical system as a generalized pendulum and eliminating the possibility of closed loop conditions. The dynamics of open-chain mechanisms were also investigated by Stepanenko and Vukobratovic (74). The Icinetostatic approach for developing Lagrange's second order 11 differential equations was presented in their 1976 publication. This method was based on the known relative motions of the members of the chain at each joint, at time t1. The angular accelerations and the accelerations of each member's center of gravity were determined and the forces and moments due to the inertial forces were calculated. The equations of kinematic equilibrium were developed according to D'Alembert's principle and a system of linear algebraic equations for the unknown accelerations were obtained. Integration was then perfomed and values at (t1 + delta t) were computed and used as input conditions for t2. The authors described the calculations necessary for this type of approach to modelling an open-chain system for application to a wide range of control problems. A series of articles by Houston and Passerello (37,38,39), in the late 1970's, discussed a computer- oriented method for obtaining the dynamical equations of motion for multi-rigid-body systems, utilizing Lagrange's form of D'Alembert's principle. They initially modelled the mechanical system by a chain of interconnected rigid bodies, such that no closed loops were formed. Due to the authors' interest in formulating an algorithm applicable to computer usage, the majority of the three publications were devoted to the development of a routine for describing the system's geometry. The use of Euler parameters, defined by Kane and Likins (41), was advocated by these authors to describe the relative orientation of two bodies in space. These 12 researchers also noted that the advantage of utilizing Lagrange's form of D'Alembert's principle is that the inertial constraint forces between the bodies of the system were eliminated from the analysis and only the internal and external forces needed to be considered. Recently, Wang and Houston (79) applied a similar formulation to constrained multibody systems. During the early 1970's, a series of researchers suggested using dual vectors and screw calculus to dynamically analyze spatial mechanisms or systems of rigid bodies. Bagci (3,4) developed dual equilibrium equations for each link of a system and solved for the dual force components required to drive a mechanism. Yang (87) also used a dual dynamic equation to analyze inertial forces in spatial linkages. In 1971, Woo and Freudenstein (84) utilized screw coordinates to develop the equations of motion for a system of rigid bodies. They simplified the analysis by considering the instantaneous kinematics of the system. These analyses again considered joints possessing only a single degree of freedom. Another method for dynamic analysis of a system of rigid bodies, utilizes Newton's laws and the classical Newton-Euler equations. This type of analysis requires that the input information; motions, forces, moments and inertial properties, be written in terms of the center of mass and principal axes of each rigid body. This, in turn, necessitates the development of multiple coordinate systems 13 to describe the dynamics of the system. In 1968, Sherby and Chmielewski (68) set up equations for calculating the derivatives of relative motion vectors in terms of any moving reference frame in the multi-rigid-body system being examined. They derived Newton's equations of motion in terms of relative coordinates. Gupta (32) described the spatial orientation of a set of rigid bodies in terms of the four Euler-Rodriques parameters, as suggested by Roberson in 1968 (64). The Newton-Euler equations of motion were then developed for these relative coordinates and the inverse dynamics problem solved for the unknown reaction forces and moments at the connecting joints of a closed loop system of interconnected rigid bodies. Lagranga and Bartel (44), in 1975, also utilized Newtonian methods to develop an automated method for analyzing spatial linkages. They, in turn, utilized Euler angles as relative coordinates to describe the relative orientation of a given rigid body. These authors chose to analyze the direct dynamics problem for a linkage system, allowing for ball and socket joints or pin joints connecting the links. Vukobratovic (78), in a 1978, three part publication, expanded on his earlier work with Stepanenko (74), to analyze the dynamics of complex kinematic chains and to investigate the synthesis of artificial motions. As opposed to this previous work, Vukobratovic utilized Euler angles to define the relative motions of the ball and socket joints l4 connecting rigid members. The resultant equations of motion were written to describe the moments at the joints in terms of Euler angles and their derivatives. Vukobratovic's method for developing the equations followed the same steps as described in his earlier work. Orin, et al. (57), presented some improvements on this method by introducing a simplified notation and increasing computational efficiency. Much of the investigation into manipulator and mechanism dynamics stems from a need for system controlling routines, specifically computer algorithms. Therefore, one priority of these dynamical formulations is computational efficiency. Silver (70), in 1982, compared the efficiency of the Newton-Euler and Lagrangian formulations. Silver showed that there was no fundamental difference in computational efficiency between the two methods. The author also stated that, although the computation time is similar, the Newton-Euler formulation includes the effect of external forces and moments acting on the terminal link. To utilize the Lagrangian method, the external forces and moments must be included by assuming an additional link, attached to the terminal link, whose motion is defined so as to produce the necessary external effects. II. JOINT ANATOMY AND MOTION The concepts of rigid-body mechanics have been utilized to describe the dynamics of the human body during various activities. In order to incorporate these principles, the body has been divided into segments and these segments then 15 modelled as a system of interconnected, rigid links, allowing for various types of articulations. Although the segments of the lower limb are not rigid from a structural point of view, they are rigid in a mechanical sense, such that they do not deform under the action of tension or compression forces along their lengths. Numerous investigators have used such a rigid-body linkage system to analyze the dynamics of the lower limbs. In order to model the legs by this method, several simplifying assumptions must be made. Unlike traditional mechanical systems, the physical properties and geometries of the links composing the lower limbs vary along their length. The diameter, curvature, shape and density change from the proximal end to the distal end of each segment. Likewise, the articulating surfaces and joint structures are unlike the simple connections used in three-dimensional mechanical mechanisms and manipulators. The foot is considered to be the terminal link of the rigid-body system. Although composed of numerous bones allowing interrelated articulations, the foot is approximated by a single rigid segment. This body is connected to the shank segment by a complex articulation of several bony structures. The movements of the ankle joint are typically defined by the articulation of the talus and the distal tibia. Yet, the relative motion between the foot and the shank during locomotion involves the interaction of the ankle joint, as described, and the subtalar joint, 16 having three articulations between the superior surface of the calcaneous and the inferior surface of the talus. The foot undergoes three relative angular motions with respect to the shank : inversion/eversion, plantar/dorsiflexion, and medial/lateral rotation. A small amount of translation, or sliding, at the joints may also be possible. The anatomy and biomechanics of the ankle and subtalar joints have been discussed by several researchers (Wright, et al. (85), Procter and Paul (63), Perry (61), and Wynarsky and Greenwald (86)). Wynarsky and Greenwald mathematically modelled the ankle joint as two circular half-cylinders, the talus component posessing a varying sinusiodal radius, while the tibial portion maintained a constant radius. The geometry of this model allowed Wynarsky and Greenwald to predict the radial stress at the ankle for varying joint deflections. The other three studies (85,63,61) discussed the actions of both the ankle joint and the subtalar joint during stance phase of locomotion. These authors stated that the ankle joint acts as a simple oblique hinge, allowing for flexion and extension of the foot relative to the shank. The subtalar joint was defined to be responsible for a combination of motions, traditionally referred to as pronation and supination (i.e. pronation = dorsiflexion + abduction + eversion), but principally accounts for inversion and eversion of the hindfoot relativeto the talus. Wright, et al. (85) concluded that the subtalar and 17 ankle joints "are interdependent and in general act as a single mechanism during walking". The shank segment of the lower limb linkage system is connected to the thigh segment by the knee joint. The articular surfaces of the knee joint include the juncture of the tibia and the femur and the area between the femur and the patella, although the latter is typically omitted when modelling the knee joint. The articular surfaces of the tibia and femur include the medial and lateral femoral condyles and the plateaus of the medial and lateral tibial condyles. The action at the knee joint is often modelled as a simple hinge, due to the fact that the majority of the motion between the tibia and the femur is related to flexion and extension, although a minimal amount of abduction and adduction, as well as medial and lateral rotation, are also possible between these two segments. Moeinzadeh, Engin and Akkas (53) utilized such a planar assumption to model the action at the knee joint, in order to determine contact forces between the femur and the tibia and the forces carried by the ligaments, for varying loads. Although a two-dimensional analysis may closely approximate the motions of the knee, the loading conditions of this joint are three- dimensional and a sagittal approximation omits the components of the force and moment in the frontal and transverse planes. Wismans, et al. (82) developed a three- dimensional analytical model of the knee joint to predict ligament elongation and force, as well as contact points and 18 contact forces for a static loading condition. Both of the previously mentioned studies reported their results in terms of the flexion angle at the time of loading, again relating their results to the motion in the sagittal plane. The third major joint of the lower limb is the hip joint, formed by the articulation of the head of the femur and the acetabular cup of the pelvis. The hip joint is typically modelled as a ball and socket, yet the femoral head is not a perfect sphere and the rim of the acetabulum is approximately horseshoe-shaped. The hip joint is offset from the long axis of the femur by the femoral neck, which lies in a distal-lateral direction from the head of the femur. Rydell (66) reported that the cervico-diaphyseal angle, the angle between the ideal axis of the femur and the cervical axis of the femoral neck, was on the average of 126°. Denham (26), in 1959, demonstrated how this angular offset causes the hip joint to act as the fulcrum of a lever system, with the pelvis acting as the bar of the lever. His simple mechanical analogy showed that the forces transmitted through the hip joint during normal activity, such as walking, often exceeds the total weight of the body. The rotational motion of the hip joint includes flexion and extension, abduction and adduction, and medial and lateral rotation. The motions allowed by the joints of the lower limbs are three-dimensional by nature, and therefore, analysis of these linkages should include the three-dimensional 19 kinematics of the system. The action of a particular joint has been described by the relative motion between the articulating surfaces composing the joint. To represent the three-dimensional motion of a rigid-body, six degrees of freedom must be defined; traditionally, three are rotations and three are translations. Several authors (73,42,10,71,58) have utilized the concepts of a screw axis (helical axis or axis of rotation) and center of rotation to define the relative motion between two body segments. These same authors also discussed the possible errors in utilizing such an analysis. The theory supporting the screw axis states that the displacement of a moving body may be described by a rotation about, and a translation along, a unique axis. Kinzel, Hall and Hillberry (42) were among the first to utilize the concepts of a screw axis to describe the relative motion between two body segments. They presented the analytical basis for such an analysis and developed an electromechanical linkage system for measurement of the necessary screw parameters. Other investigators have approximated joint motion as a planar action and therefore, have used the concept of an instantaneous center of rotation to describe this action. The errors inherent in such an estimation have been throughly discussed (10,71,58,83). These authors showed that such a method is extremely sensitive to small measurement errors, that the results of a planar analysis may omit significant factors, and that such a description of 20 motion lacks clinical significance. Lewis and Lew (46) attempted to define an optimal "fixed" axis of rotation for the knee joint based upon a technique for predicting the length of in-vivo ligaments. Another method for defining three-dimensional joint motions, using Euler angles, was initially suggested by Chao and Morrey (17) in 1978 to describe the rotation of the elbow. Grood and Suntay (31) in 1983 applied these same ideas to describe the motion of the knee. They specified three spatial axes about which rotations at the knee occurred and defined three angular coordinates as the three rotational motions : flexion/extension, abduction/adduction, and medial/lateral rotation. One of the major advantages of this coordinate system, is that the order in which the rotations and translations occur are independant, unlike the rotations about the axes of a laboratory coordinate system. The joint coordinate system they developed allowed for a precise and clinically significant method for studying the motions at the knee. Soutas-Little et al. (72) applied a similar analysis to the ankle joint complex to investigate the motions of the foot relative to the shank during stance phase of running. In modelling the lower limbs as a series of three interconnected rigid bodies, the physical properties of each link must be known. Unlike traditional mechanical systems, the physical characteristics of each rigid—body of the leg are inconsistent and variable. The size and inertial 21 properties of the segments may vary between individuals, as well as within individuals over time. The effects of weight changes and conditioning commonly alter the geometry and mass distribution of the body. Methods for determination of the physical parameters of the body date back to the late 1800's. One of the most widely cited investigations was performed by Braune and Fischer in 1889 (8). They used four male, middle-aged cadavers to determine the position of the center of gravity of the entire body, as well as the center of gravity of each individual segment. In 1955, Dempster (23,24) reported on the weight, volume, center of mass and moments of inertia of the body segments of eight cadavers representing "individuals of the older segment of the population". Similar procedures were followed in both cadaveric studies, although Dempster utilized a slightly different technique for dismemberment. Results from these two investigations are widely used to date. More recently, research has been done, using similar cadaveric methods, to develop predictive models for determining mass distribution, location of segmental center of mass and principle moments of inertia for the living human body from simple anthropometric measurements. Hanavan (33) developed a 15 segment mathematical model of the human body. Each segment was represented by a simple geometric solid of uniform density, such as frustrums of right circular cones, elipsoidal cylinders, etc. Anthropometric measurements of living subjects were used to define segment 22 lengths and radii, while weights were taken from cadaver studies. In 1969, Clauser et al. (21) developed a set of multi-step regression equations for predicting weight, volume and the location of the center of mass of individual body segments from experimental data obtained from 13 male cadavers. Miller and Morrison (52) utilized these equations and the model developed by Hanavan to predict weights and principle moments of inertia for 30 adult male athletes. These investigators found varying discrepancies between the sum of the predicted segmental values and the total weight of the individual, yet supported Clauser, et al.'s equations and suggested that improvements needed to be made on the geometrical representation of the human body. Jensen (40), in 1976, developed a mathematical model to estimate the inertial properties of the body segments by assuming that each segment is composed of a series of two millimeter thick elliptical zones. The dimensions of the elliptical zones were obtained experimentally from digitized photographic records. This method followed the fluctuations in body shape more closely and subsequently, Jensen found an error of less than two percent for estimates of total body mass, when compared to direct measurements of total body mass. In 1975, Chandler, et al. (15) improved on Clauser's (21) methods for determining the weight, center of mass location and principal moments of inertia for cadaveric specimens. During the early 1980's the United States Air 23 Force sponsored a long range research program to determine the relationship between human body size and its mass distribution properties using the methods developed by Chandler, et al. In 1980, McConville, et al.(49) obtained anthropometric measurements from a group of 31 living adult male subjects and developed predictive equations for determining the moments of inertia of the body segments. Young and coworkers (88) developed a similar set of equations for the female population using a group of 46 adult female subjects. Such multi-step regression equations have allowed investigators to predict the mass and principle moments of inertia of the body segments of living subjects from simple anthropometric measurements. III. LOCOMOTION An analysis of the mechanics of locomotion involves a complex interaction of various methods, three of which have already been discussed : the rigid-body assumption used to define the segments of the lower limbs, the mass distribution properties of the limb segments and the range of motion of the articulating joints. By combining the information from these types of investigations and appropriate experimental and/or theoretical methods, researchers have attempted to describe the complete mechanism of walking. As with traditional rigid-body mechanics, gait research has utilized both the direct dynamics problem, as well as, the inverse dynamics approach. The former typically involves inputing forces into a 24 mathematical model of the linkage system to predict the motions generated during locomotion. The inverse problem utilizes experimentally recorded motions of the limbs to calculate resultant forces and moments at the joints of the lower limbs. This form of investigation has been the predominant method reported in the literature to analyze the mechanism of gait. The kinetics and kinematics involved in locomotion have been widely investigated since the 19th century. Much of the experimental work performed has been dependent upon the methods and instrumentation available at the time of the investigation. The efforts of Muybridge (55) and Marey (48) pioneered the use of photographic techniques in recording the motions of the body during various activities. Since their time, both still photography and cinematography have been used widely to document kinematic information. Typically, the location of markers, designating anatomical landmarks, are recorded over time by either one or two cameras. Then each frame of the film is projected onto a surface and the markers' positions are digitized to provide planar or, with appropriate transformations, three dimensional coordinates. This information may then be differentiated to obtain linear velocities and accelerations for each segment. More recently, opto-electronic devices have been developed to automatically yield position coordinates. 25 Other methods for determining relative motions between limb segments involve the use of externally attached exoskeletal structures. Such devices utiliize potentiometers and accelerometers to measure the degree of rotational motion at a joint and body segment accelerations, respectively. Chao (16), in 1980, reported on the development and use of triaxial goniometers in studying joint rotations. He also discussed the potential drawbacks in utilizing such equipment, including the error due to soft tissue motion at the point of attachment and the difficulties inherent in the placement of the apparatus. Cappozzo (11) discussed the use of exoskeletal linkages to define joint rotations, as well as other methods, in his 1985 article reviewing experimental techniques. Another important instrument utilized in gait analysis is the force plate. The force platform provides the three dimensional ground reaction forces and moments applied to the ground by a subject during stance phase of gait. This instrument is commonly used in conjunction with kinematic information to provide input data at the terminal end of the linkage system. Many authors have attempted to define "normal" properties of gait for large populations of subjects and have encountered various difficulties; the physical characteristics are extremely variable and the time-distance measurements, or temporal factors, have been found to greatly affect the mechanics of locomotion. Lamoreux (45) 26 defined normal walking as follows : "Normal walking is a process of human locomotion in which the erect moving body is supported alternately by one leg and then the other". The speed with which this process occurs varies from person to person. As a subject's walking speed varies, so too does that person's step length, stance duration and frequency. Lamoreux claimed that step length and duration can provide an initial comparison for the evaluation of gait mechanics. In 1977, Andriacchi, Ogle and Galante (1) found distinct relationships between walking speed and stride length, support time and swing time, as well as the amplitude of the ground reaction force. They determined that as velocity increased, the peak amplitudes of the components of the ground reaction force increased. Chao, Laughman, Schneider and Stauffer (19) extended this analysis by including variations in knee joint motion with variations in temporal distance factors. They also analyzed the changes in these factors with age and gender. It has been shown that the time-distance factors play an important role in determining the ground reaction forces and moments and therefore, these parameters must be included in the presentation of locomotion data. The motions observed during gait primarily occur in the sagittal plane. Therefore, many investigators have simplified their research by assuming a two-dimensional analysis of the dynamics of locomotion. One of the first investigators to apply the principles of engineering to 27 analyze gait in the sagittal plane was Manter (47). Although he analyzed the dynamics of walking for a cat, he successfully combined force plate data with cine techniques to calculate the planar muscle torques in the limbs necessary to produce the recorded forces and motions. Herbert Elftman (27) analyzed the stance phase of human walking gait utilizing a planar description. In 1939, Elftman constructed a force platform to record the three components of the ground reaction force and the point of contact between the ground and the foot. He computed the instantaneous horizontal and vertical forces and the resultant torque at the joints of the lower limb using D'Alembert's principle of effective force equilibrium. Elftman also analyzed the energy changes in the leg during both stance and swing phases of walking. Although the experimental methods utilized were rudimentary, Elftman's procedure for calculating the forces and moments has been widely used to date. By considering a planar analysis, the relative motion between limb segments may be simply described by the angle between two adjacent rigid bodies. Such a representation is simplified by the use of generalized coordinates and the use of Lagrange's equations of motion. The simplicity achieved by using this type of description, is that the angular velocity and acceleration are the first and second derivatives of the joint angles, respectively. In 1971, Chow and Jacobson (20) published an extensive report on the 28 application of Lagrangian dynamics to a seven link sagittal model of the human body in motion. Analysis of this mathematical model led to a series of five nonlinear, coupled, second order differential equations. After simplifying these equations, Chow and Jacobson used optimization to yield a predicted solution. Chao and Rim (18), in 1973, also utiliized the principles of optimization in determining the joint moments during gait. Their mathematical model dealt with only a single limb and therefore, included only four links. These researchers simplified the resultant Lagrangian equations of motion to form an initial value problem and used the method of steepest descent in optimization to obtain solutions for the approximate joint moments during gait from an experimentally derived displacement history. The direct dynamics problem was developed in two investigations utilizing a sagittal plane, Lagrangian analysis. Cappozzo, Leo and Pedotti (12) applied system theory to a three segment model in order to determine the overall displacement of the lower limb due to external forces and muscular moments. In 1980, Onyshko and Winter (56) developed a seven segment planar model of the lower limbs and trunk to predict the motions of the body during locomotion. During the same time, Winter (80) published a report on the principle of lower limb support during stance phase. He defined a support moment as the total extensor moment acting on the leg segments and stated that when this 29 moment is positive, the limb will not collapse, therefore, during stance phase of gait, the limb must posses a positive support moment. In yet a further study, Winter (81) showed that this was the case for approximately 90% of the stance phase of gait. In such a sagittal analysis though, no comment may be made about the effects of the other moments acting at the joints. Two experimental studies were performed to determine joint rotations in both a sagittal plane and a frontal plane from film data recorded perpendicular to these two directions. In 1972, Sutherland and Hagy (75) attempted to determine joint motions during gait by displaying images of subjects onto a plane and measuring the angles between projected line segments approximating the links of the leg. Although corrections for distortion were made, this method showed large discrepancies in measurements between observers. Boccardi, et al. (7) used a similar camera set- up to record position data from joint markers during locomotion. These investigators also utilized a force platform to simultaneously record ground reaction forces. Joint moments in the sagittal and frontal planes were then approximated by calculating the planar distance from the platform to the joint center and multiplying this by the ground reaction forces. Such an approximation ignores the effects of inertial and gravitational forces. The effects of these types of forces were included in a sagittal analysis performed by Gilbert, et al. (28) in 1984. 30 This group of researchers used a Newtonian formulation to develop equations for calculating the forces and moments at the knee and hip during walking. Experimental data was collected from a force plate, eight uniaxial accelerometers and a single goniometer for twelve normal subjects and nine lower limb amputees during stance phase of gait. Gilbert and coworkers found similar results as Winter (81) and agreed with his claim that the moment in the sagittal plane was the most important factor in determining the stability of stance. A sagittal plane analysis has also been used by researchers to describe the motion of the foot and leg during the swing phase of gait. Beckett and Chang (5) formulated a mechanical model to predict the motion of the lower limb, the moments at the hip and knee joints and the energy expended during swing phase. They also utilized Lagrangian dynamics to develop the equations of motion in terms of the joint angles. In 1981, Mena, Mansour and Simon (51) used a similar mathematical representation to predict the motion of the thigh, shank and foot during swing phase. They determined three necessary input variables to produce an adequate swing motion; initial thigh velocity, a knee . constraint and a moment applied to the foot. Phillips, Roberts and Huang (62) utilized a two segment model and Newtonian methods to examine the effects of proximal thigh motion on distal shank motion, assuming zero intersegmental 31 muscular forces or moments. They found that intersegmental reactions created large increases in distal segment speed. Bresler and Frankel (9) were the first to consider the more complex problem of determining the three-dimensional forces and moments in the lower limbs during walking. Their work described the fundamental methods used in three- dimensional gait analysis and has been widely cited by most researchers since 1950. Bresler and Frankel simultaneously recorded the horizontal and vertical ground reaction forces and the moment about the vertical axis, as well as the position of the leg in space, for four normal subjects. Using D'Alembert's principle of "dynamic equilibrium", and including the effects of gravity and the mass moments of inertia, they calculated the forces and moments at the joints of the leg for a complete stride. Their data was presented in terms of an inertial coordinate system located at the center of the force plate. These authors stressed the importance of the non-sagittal components of the joint forces in providing stability during stance phase, as well as their effect on the moments at the hip. Several researchers have utilized similar methods to those developed by Bresler and Frankel to analyze specific joints in the leg. Paul (60), in 1965, studied the forces acting on the hip joint during stance phase of gait and in 1968, Morrison (54) performed an analysis of the forces transmitted by the knee and the muscular force interaction. Both of these studies considered the portion of the limb 32 below the joint being analyzed as a free body and calculated the forces and moments by D'Alembert's principle of equilibrium of a body in motion. The effects of gravity and inertia of all segments below the joint were included in the force and moment summations. Unlike these authors, Harrington (35) claimed that inertial and gravity effects were minimal during stance phase of walking and omitted them from his investigation. He calculated moments at the knee by multiplying the ground reaction force by the distance from the force plate to the center of the knee joint. Harrington found that, upon comparison with Morrison's results, the moment at the knee could be successfully estimated by the cross-product of the radius vector and the ground reaction force. In 1980, Hardt and Mann (34) briefly described a five segment model of the human body (2 shanks, 2 thighs and 1 segment to represent the head, arms, trunk and pelvis). They defined a system of Newtonian equations of motion for each link to calculate joint dynamics during gait from position and orientation data, without input from a force plate. Their model showed similar results as previous investigators for single support phases of gait, but displayed difficulties when analyzing double leg stance. The authors acknowledged the possible errors produced by omitting information about foot position and ankle angle. Hayes, et al. (36) further simplified the analysis of the swing phase of walking by modelling the lower leg and foot 33 as a single rigid body undergoing pure rotation about a known fixed point at the initial time step. These authors advocated the use of multiaxial accelerometers to study limb motion and joint kinetics. Although they provided the theoretical development, Hayes and coworkers discussed the many limitations which must be overcome before such an analysis may be utilized. The sensitivity of the accelerometer to vibrations and impulse effects are only two of the drawbacks they encountered. More recently, Andriacchi and Strickland (2) reported on the three-dimensional moments at the joints of the lower limb during walking, utilizing similar methods as Bresler and Frankel. They tracked the motion of the joint centers during stance phase of gait for 29 normal adults over a range of walking speeds. These authors stressed the importance of analyzing all three components of the moment at all three joints and, in opposition to a sagittal analysis, found the magnitude of the abduction/adduction moment at the knee to be larger than the flexion/extension moment. Another approach taken in analyzing the dynamics of locomotion has involved the development of mathematical models to predict the ground reaction forces during gait. In 1975, Thornton-Trump and Daher (76) developed such a model for the human body based on kinematic and anthropometric data taken from earlier literature. Zarrugh (89) developed a similar model, assuming symmetry in fore 34 and aft shear forces during double support phase of walking. He then used the computed ground reaction forces to evaluate forces, moments and power at the joints of the lower limb. Siegler, Seliktar and Hyman (69) created a simpler mechanical model to simulate human locomotion. Their representation consisted of a lumped mass, supported by two elastic and viscous straight legs, each modelled by a spring and damper connected in parallel. They concluded from their results that, during stance phase of locomotion, the dynamic properties of gait are mainly the result of inertial effects. Several authors have extended the three-dimensional joint kinetics analysis by utilizing optimization techniques to distribute the resultant forces and moments at the joints to the internal load-bearing structures, such as muscles and ligaments. Serig and Arvikar (67) developed a mathematical model to predict the muscular load sharing in the lower limbs during walking. They selected an optimization criterion, based on an earlier study on posture, as the sum of all the muscle forces plus four times the sum of the joint moments. In 1978, Crowninshield, Johnston, Andrews and Brand (22) analyzed the dynamics of the human hip, incorporating muscle stress as an optimization objective function. Even though the hip was considered in three dimensions, only the flexion and extension components at the ankle and knee were included in the model. In a 1981 report, Patriarco, et a1. (59) evaluated different data 35 aquisition methods and optimization techniques in an effort to determine the significance of various factors which contributed to the solution of the dynamics problem. They utilized the human body model of Hardt and Mann (34) and minimized the total muscle force of the system. These authors found that the precision achieved in computing the moments at the joints had the largest influence on the muscle force distribution. Rohrle, et al. (65) also used the total force in the muscles as a cost factor for their optimization analysis of the lower limb. Again, the knee was modelled as a hinge and only the flexion and extension components were considered. EXPERIMENTAL METHODS A general description of the experimental methods and the techniques used for data reduction are presented in this chapter. Data collection methods were developed by researchers in the Department of Biomechanics and used equipment available through the Center for the Study of Human Performance and the A.E. Case Center for Computer Aided Design at Michigan State University. Three types of information were experimentally recorded for each subject: the kinematic activity of the lower limb segments: the kinetic ground reactions: and the anthropometric properties of the body as a whole, as well as anthropometric measurements of the individual limb segments. The experimental data were then reduced to forms consistent with theoretical constraints and input to the equations of motion. Position data were recorded by two high speed 16mm cine LOCAM cameras operating at 100 frames per second. The two cameras were placed anterior and lateral to the plane of motion at an included angle less than 90 degrees. The anterior and lateral camera placements were chosen due to the greater availability of boney landmarks on the anterior and lateral portions of the right lower limb. Timing 36 37 . lights, accurate to 1/1000 of a second, were placed in the field of view of each camrea so that both sets of two— dimensional data could be time-matched. A calibration structure, composed of twelve targets of known relative position, was filmed prior to subject filming. This structure defined the three-dimensional inertial coordinate system within which the motion would occur and provided matrix transformation constants necessary for the direct linear transformation technique. The filming set-up and calibration structure are shown in Figure 1. Once the position data for the targets placed on the lower limbs were recorded for the activity of interest, the film was processed and digitized. The digitization was performed on an Altek Datatab1 rear-projection digitizing system, coupled with an IBM personal computer which stored the data on floppy disks. The planar information from each camera were then combined into a single file and transferred to a Prime 9000 mainframe computer for data manipulation and analysis. The direct linear transformation technique was utilized, along with the digitized and known position data from the calibration structure, to provide three-dimensional position data for each target at every 1/100 of a second. These position data were transformed into the inertial reference frame defined by the targets of the calibration structure. For example, the position vector for a target i 1 Model ACT 23-3 RP, Altek Corporation, 2150 Industrial Parkway, Silver Springs, MD 20904 38 Figure 1. Filming Set-up with the Calibration Structure. at time t may be given by Bin) = xi(t) ’i\x + yi(t) Ty + zi(t) ’i\z where’ix, Ty and 92 are the unit vectors used to define the laboratory coordinate system. The three-dimensional position data were filtered using a two—pass Butterworth filter to reduce system noise and the effects of soft tissue motion, and then were differentiated to provide linear velocity and acceleration values for each of the limb targets. Ground reactions were measured during stance phase by an AMTI2 force platform mounted flush with the floor 2 Model OR6-3, Advanced Mechanical Technology, 141 California Street, Newton, MA 02158 39 surface. The force plate was interfaced with an IBM 9000 dedicated computer and recorded six channels of kinetic data: the forces in each of three mutually perpendicular directions and the three components of the resultant moment. One channel of time data was also recorded. The ground reactions were collected with respect to a reference frame embedded in the force platform. After storage onto floppy disk, the kinetic data were transferred to the Prime computer for analysis with the corresponding kinematic information. Prior to the filming session, anthropometric measurements of each subject were also obtained. The distances between targets and other anatomical landmarks were measured. The width and depth of the lower limb segments at various levels were also recorded. Other lengths and distances were recorded for use in calculating the mass distribution properties of the limb segments. These measurements and calculations will be further discussed in the next chapter. Photographic records were made of each subject from several planes of view to assist in these inertial calculations. Total body weight was also measured as part of the experimental protocol. The filming protocol initiated by filming the subject while standing on the force plate. This allowed for static information to be collected for each subject, the use of which will also be explained in the following chapter. Next, the subject was 40 asked to practice walking and/or running across the filming area, so that their right foot would fully contact the force plate during the locomotion. Data were collected for four subjects, three males and one female. Three seperate locomotion conditions were investigated: walking barefoot, walking in shoes and running in shoes. The shoes worn were identical, the Brooks Journey, a neutral running shoe which did not possess any type of motion control device. The original model was designed for an unshod foot, but since humans typically walk in shoes, the walking in shoes condition was added to the experimental protocol. Anthropometric measurements of the foot were recorded twice, once for the barefoot condition and again with the shoes on. The running in shoes condition was included to examine the validity of the model under higher velocities and greater ground reaction magnitudes. A minimum of three complete strides were then filmed for each subject, for each condition. THEORETICAL DEVELOPMENT The right lower limb was considered to be a series of three interConnected rigid bodies: the foot, the shank and the thigh. The translational equation of motion for a rigid body may be written as: F = m acm (l) where m is the mass of the body, acm is the acceleration of the center of mass of the body and F'is the sum of the external forces acting on the body. The equation for rotational motion may be written as: :I=-—;7+Vx (To?) =‘Fr (2) where.I'is the inertia tensor, 6 and 6 are the angular velocity and acceleration, respectively, of the body andlM is the sum of the moments about the center of mass. These equations are used to solve two types of problems. The first, known as the direct dynamics problem, is used when the forces and moments acting on a rigid body are known and the motions required to create these forces and moments are unknown. The second type of problem, typically encountered when studying the dynamics of the human body, occurs when the motions of a rigid body are known and the forces and moments required to create these motions are unknown. This is referred to as the indirect dynamics problem. In the 41 42 present analysis, the motions of the leg were recorded in terms of linear positions, velocities and accelerations of specified targets placed on each rigid body segment. The objective was to determine the unknown forces and moments acting at the joints. Therefore, an inverse dynamics problem was formed. The Newton-Euler equations of motion may be expressed with respect to the principal axes and center of mass of each rigid body as follows: F1=ma cml F3 = m acm3 1191 - w2w3(12 - I3) = M1 1202 - w3w1(13 - 11) = M2 (4) I393 - w1w2(I1 - 12) = M3 where 1, 2 and 3 refer to the principal directions of the body. This formulation was used to provide the most direct method for solving for the forces and moments at the joints. Other investigations have utilized the Lagrangian form of the equations of motion to solve this type of problem. In the Lagrangian method, sets of generalized coordinates are developed from the relative motions between body segments and the equations of motion are expressed in terms of these coordinates. The Lagrangian formulation is most often used in a two-dimensional analysis, since the angles between body segments, formed in the sagittal plane, provide a direct measure of the relative motion of the links and automatically create a set of generalized coordinates. Such 43 a set of relative coordinates facilitate the calculations of the relative angular velocities and accelerations. Yet, the calculations required to solve for the three-dimensional forces and moments in the lower limb during stance phase of locomotion, utilizing the Newtonian form of the equations, have been reported (69) to be the most direct and, with the input data in the form of linear position data, the least time consuming. A schematic representation of the three segments of the lower limb are shown in Figures 2, 3 and 4. Included in Figure 2. Free Body Diagram of the Foot Segment. 44 Figure 3. Free Body Diagram of the Shank Segment. 45 r9T1 ”72 Figure 4. Free Body Diagram of the Thigh Segment. 46 each figure are the principal axes of the segment, centered at the segment's center of mass, and the external forces and moments acting on the rigid body. The Newton-Euler equations of motion, (3) and (4), were written for each segment of the lower limb, with respect to that segment's center of mass and principal axes. The equations of motion for the fodt, shank and thigh were written as follows, assuming a positive force and moment convention: mFaCMFl = mFgFl + R1 + FA1 mFaCMFZ = mFng + R2 + FAZ mFaCMFB = mF9F3 + R3 + FA3 MR1 + MAl + (RFZFAB ‘ RFBFAZ) + (PF2R3 ’ PF3R2) ' IFlFFl ‘ wF2WF3 (IFz ‘ IF3) (5) MR2 + MA2 + (RF3FA1 ' RFlFAB) + (PF3R1 ‘ PF1R3) — IFzFFz ' wF3WF1 (IFB ‘ IFl) MR3 + MAB + (RFlFAZ ' RFZFAl) + (PFiRz ‘ PFZRl) — IF3‘7'F3 ‘ wFleZ (IFl ' IF2) mSaCMSl = mSgSl + FAl + FKl msacmsz = m$952 + FA2 + FK2 msacmsza = 1“$953 + FA3 + FK3 MAl + MKl + (RstK3 ' RSBFKZ) + (PSZFA3 ' PS3FA2) ‘ I$1831 ’ wszws3 (ISZ ’ 153) (6) MA2 + MKZ + (RSBFKl ' RSlFKB) + (PS3FA1 ' PSlFA3) — I$2852 ' W53W51 (133 ‘ I$1) MA3 + MKB + (R51FK2 ' RSZFKl) + (PS1FA2 ‘ PSZFAl) . — Is3ws3 ‘ wSleZ (131 ‘ 152) 47 mTaCMTl = ngTl + FKl + FHl mTaCMTZ = ngTz + FK2 + FH2 mTaCMT3 = ngT3 + FK3 + FHB MKl + MHl + (RTZFHB ‘ RT3FH2) + (PTzFK3 ‘ PT3FK2) - IT1‘7'T1 ' wT2WT3 (ITZ ’ 1T3) (7) MKZ + MHZ + (RT3FH1 ' RTlFH3) + (PT3FK1 ‘ PTlFK3) — IT2§T2 ‘ wT3WT1 (1T3 ' 1T1) MK3 + MHB + (RTlFHZ ’ RTZFHl) + (PTlFKZ ' PTzFK1)- . — IT3WT3 ' lewTZ (ITl ' 1T2) The vector, expressed as components RIJ' represents the relative position vector between the center of mass of the segment and the proximal linkage center. The vector, PIJ' represents the relative position between the center of mass and the distal linkage center. The solution of these three sets of equations was initiated at the foot, Equation (5). The external force and moment, R and MR, at the foot-floor interface were known from the force plate data, therefore, there existed six equations and six unknowns. These equations were then solved for the components of the unknown force and moment acting at the linkage center of the ankle joint. Once these values were calculated, their directions were reversed, in accordance with Newton's third law, and the forces and moments at the ankle were applied as the external forces and moments acting on the shank segment. This method of solution was then repeated to solve for the forces and moments at the knee, which were in turn reversed and applied to the thigh. Calculations of the other values 48 in each equation, not the reaction forces and moments, provided the difficulty in the solution of this inverse dynamics problem. For each segment, these values were: the location of the center of mass, the direction of the principal axes, the moments of inertia along the principal axes, the location of the linkage center of the joint and, for the foot, the location of the center of pressure and the forces and moments acting there. I. THE FOOT AND ANKLE The analysis began at the foot segment since one set of forces and moments acting on this segment could be determined prior to solution of the equations of motion. The foot segment, although assumed to be a single rigid body, is composed of a number of bones and joints. In this analysis, the motion of the rearfoot was used to define the actions of the entire foot, since during the toe-off portion of stance phase the forefoot follows a different set of motions, due to the flexion response of the toes. A targetting scheme for the foot was developed to locate the position of the center of mass, the directions of the principal axes and the center of the ankle joint, or linkage center. To minimize soft tissue effects, target placement was limited to boney landmarks visable to both the anterior and lateral cameras. Several investigators (33,40,49,88) have reported similar estimates for the directions of the principal axes of the foot. The vertical, or inferior-superior, axis has 49 been defined as an axis perpendicular to the plane of the ground, when the subject is standing. The horizontal, posterior-anterior, axis has been shown to lie in the direction of a line from the posterior point on the calcaneous to the tip of toe II, perpendicular to the inferior-superior axis. The lateral-medial axis has been defined as the cross product of the other two axes. Targetting of the foot initiated by placing two targets on the lateral calcaneous, approximating a vertical line perpendicular to the ground when the subject is standing (Figure 5). A unit vector, if3, in the inferior-superior direction was formed by these two targets. E. / IBM = ’i‘m. (8) Figure 5. Targetting of the Foot. 50 The posterior-anterior axis could be defined by placing targets on both the posterior calcaneous and toe II, yet two problems were encountered with this method. First, the posterior calcaneous target could not be seen by the anterior camera view and second, during toe off this axis would not coincide with the principal axis, due to the flexion of the toes, as shown in Figure 6. Therefore, a geometrical solution was used to develop this axis, based on data collected for a stationary standing posture. If the foot is viewed in a plane parallel to the sole of the foot, Axle From Posterlor Calcaneous to Toe ll ' osterlo Calcaneous Posterlor-Anterlor Axle ot the Foot Figure 6. The Effect of Toe-off on the Posterior-Anterior Axis of the Foot. 51 as shown in Figure 7, where B is the projection of the inferior target on the calcaneous, C is a target placed on the head of the metatarsal V and D is on the tip of toe II, the angles between various vectors may be calculated. It was assumed that the vector BC would remain constant for the duration of the activity, even during toe off. Therefore, this vector was chosen to be the base vector from which the posterior-anterior axis,‘ifl, was calculated. It was then necessary to determine the angle, a , through which BC must be rotated so that it would be parallel to the line from the posterior calcaneous to point D. Two measurements were included in the experimental protocal to assist in calculating this angle: 1) the width of the heel at the level of point B and 2) the length of the foot from posterior calcaneous to toe II. In Figure 7 these dimensions are referred to as WH and FL, respectively. The angle,Y’, between BC and BB, was calculated by vector analysis. Y = cos‘1(§E : 3'56 )/(|§E||EB|) (9). The angle between BB and the principal axis line, 6 , was calculated as e = sin"1 ( (wn/z) / (|§B|) ) (10). Once these two angles had been calculated, c: was simply expressed as ‘Y - 9. '|:l 52 D. \9. I c 2 3i 3'! 0‘ x l‘ B u Figure 7. Geometrical Representation of the Angle of Rotation, a . 53 Next, the component of BC along/if3 was eliminated so that the vertical and horizontal axes would be perpendicular, —*' A\ .A and a unit vector was formed /\ __ .__ Bcperp = Bcperp / IBCperpl (12). The principal axis in the posterior-anterior direction,’if1, /\ /\ was then formed by rotating the unit vector BC about if3 perp by the angle a . The final axis, in the lateral-medial direction, was formed by taking the cross product of if3 and 4,1. €T2 = 9,3 x 9T1 (13). Once a was computed from the standing data, the local foot coordinate system,’if1, sz and ik3 could be determined for any position of the foot using the above calculations, since this local coordinate system stays attached to the foot, regardless of the foot's position in space. It was then necessary to determine the location of the origin of this coordinate system, the center of mass of the foot. Previous researchers (8,21,23,49,88) have shown a high level of agreement on the location of the center of mass with respect to certain boney landmarks, and have reported the location of this point in terms of distances between these landmarks. Along the posterior-anterior axis of the foot, the center of mass has been shown to be an average of 56% of the foot length, FL in Figure 7, from the tip of toe II, point D. In the vertical direction, the 54 . center of mass lies approximately halfway between the tip of sphyrion, the most distal point on the medial malleolus, and the bottom of the sole. In the lateral-medial direction, the center of mass has been computed as 55.5% the width of the foot, measured between the head of metatarsal I and the head of metatarsal V. The information reported here was gathered from previous investigations and is summarized in Table 1. In order to locate the center of mass of the foot from the previously mentioned target locations, computations were made utilizing standing data and anthropometric measurements. The computations initiated by locating the point, DF, a distance 0.56 x FL from point D in the negative ifl-direction (see Figure 8). Next, the point, CF, was calculated as 0.555 x FW, the width of the foot as measured in the literature, from point C in the positive sz‘ direction. In the’ifiifz-plane, the planar components of the center of mass, CMFX and CMFy, were determined to be at the intersection of the right angle formed by DF, CF and the vector CFDF, as shown in Figure 8. The vector CFDF was calculated and the component parallel to the Til-axis was computed. ( CFDF along ”i‘fl ) = CFDF - (CFDF on2) Ta (14). The center of mass in the’iffifz-plane could then be computed as a point located a distance d from CF in the 55 Table 1. The Reported Location of the Center of Mass (CM) of the Foot Relative to Anatomical Landmarks (D-CM) (SPH-CM) (C-CM) FL SPH FW (%) (%) (%) Clauser, et al. 55.15 47.44 NA (1968) * Braune and Fischer 57.83 43.40 *** (1889) Dempster ‘ 57.00 NA NA (1955) ** McConville, et al. 56.00 NA 54.50 (1980) Young, et al. 56.00 52.56 56.50 (1983) Mean 56.40 47.80 55.50 * Measured from the ankle joint axis, not Sphyrion. ** Errors were found in the published data. *** Located at the anterior border of Navicular and the border of Cuneiform III. 56 D l l I CF C \ -—i> CFDF .1 COM ”F ‘ F Figure 8. Geometrical Representation of the Center of Mass of the Foot. 57 negative 1&1 direction, where d equals the magnitude of the vector CFDF parallel to‘ifl . The location of the vertical component of the center of mass of the foot, CMFz' has been shown to be one half the distance from the sole of the foot to the height of sphyrion. Since this measurement lies on the medial side of the leg, it could not be targetted and therefore, anthropometric measurement of sphyrion height was included in the experimental protocol. The magnitude of the vertical distance between target B (Figure 5) and the center of mass was then calculated as one half sphyrion height minus the vertical distance between target B and the ground, again utilizing standing data. The location of the center of mass of the foot was then computed by moving this distance in the positive TEB-direction from the center of mass lying in the ’Tfiifz-plane. In order to calculate the location of the center of mass for all position data, the position vector of the center of mass relative to target B was computed and expressed with respect to the foot local coordinate system, which remained fixed on the foot segment for all time. Previous investigators (61,63,85,86) have approached the idea of a joint center from the standpoint of motion analysis. Researchers have attempted to define rotation axes and centers of rotation for the ankle joint and/or subtalar joint. In the current analysis, the concept of the center of the ankle joint is used solely as a reference point of contact between the foot segment and the shank 58 segment of the lower limb during stance phase of locomotion. The ankle joint center, as it will be referred to here, was defined to lie on the contact surface of the talus and the tibia, at a point midway between the landmark sphyrion, on the medial malleolus, and a point on the lateral malleolus, at the same vertical distance from the ground. This location lies on the ankle axis, as defined by the authors referenced above. In order to compute this point from the position data, a target (F in Figure 5) was attached to the lateral malleolus, a distance above the ground equal to sphyrion height. A measurement of the width of the ankle at this level, AW, was also included in the experimental protocol. The location of the joint center was then computed for each frame of position data as ‘65; ='F_+ (AW/2) @152 (15), with respect to the inertial coordinate system. Once the location of this point was determined, the relative position of the joint center with respect to the center of mass of the foot was calculated. F} 963,, -EM—F (16). The ankle joint center, as approximated above, was used as the linkage center and as a transmission point for applying ankle forces and moments to the shank segment. The inertial properties of the segments of the human body, have been widely investigated in the past. As with the location of the center of mass and the principal axes, exact values for the mass and the principal moments of 59 inertia of each body segment have only been determined from cadaveric studies. Several researchers (33,40,49,52,88) have attempted to develop predictive equations, from both studies of cadavers and from geometric models, to compute these properties. A high level of agreement was shown in the literature regarding the mass of the foot as a proportion of the total mass of the body. Values are given in Table 2 for the weight of the foot, along with estimates for the weight of the shank and thigh, as a percentage of total body weight. In the current analysis, the weight of the foot was chosen to be 1.4% of the total weight of the subject, as reported by Dempster (23). Table 2. Segment weights Reported as Percentages of Total Body Weight Foot Wt. Shank Wt. Thigh Wt. (%) (*l (*l Braune and Fischer 1.70 4.80 10.75 (1889) Dempster 1.40 4.50 9.65 (1955) Clauser, et al. 1.47 4.35 10.27 (1968) * Hanavan 1.00(+0.75) 5.50(-0.95) 9.00(+1.6) (1964) * Originally reported by Barter (1957), WADC-Technical Report 57-260. The constants are added to or subtracted from the % x Weight computation. 60 Although the mass of the foot may be expressed as a simple proportion of the total body mass, the manner in which it is distributed about the principal axes is more closely related to the geometry or shape of the foot. The principal moments of inertia, with respect to the previously computed principal axes and center of mass, were therefore determined from the recorded anthropometric measurements. Two investigations (49,88) have used stereophotogrammetry to record and determine the volumes of various body segments of live subjects and the corresponding moments of inertia. Others (33,52) have modelled the foot as a geometric form and computed the moments of inertia as functions of the model's dimensions. In order to simplify the experimental protocol, this investigator chose to create a geometric model of the foot from the recorded geometry, and compute the principal moments of inertia from the model. The model chosen to represent the foot segment is shown in Figure 9 and consists of a block, approximating the portion of the foot posterior to the computed center of mass, and a wedge shaped section, to approximate the anterior part. This combination was chosen from observation of the shape of the foot and the measurements recorded during the experimental protocol. The dimensions of each section were determined by the anthropometric measurements. The principal axes of each part corresponded with, or were parallel to, the principal axes of the foot, as previously computed. The principal moments of inertia for each 61 ace—«2:030 Ste:— 3 EoEos. 2.: .2 can: soon. 05 to Enos. .m 0.53". 4....— $3.3. x V > _. / A/ x > VN lam 62 section, with respect to that section's center of mass, are given below. Block: Ix = m/12 [{(WH + FW)/2)2 + SPHZ] 1y = m/12 [(0.44FL)2 + SPHZ] (17) I2 = m/12 [(0.44FL)2 + {(WH + FW)/2}2] Wedge: IX = m/36 [2((WH + FW)/2)2 + 3(0.56FL)2] I = m/36 [3((WH + FW)/2)2 + 2(0.56FL)2] (18) I2 = m/18 [(0.56FL)2 + SPH2] Using the parallel axis theorem, the equations for the principal moments of inertia for each section with respect to the center of mass of the total volume were developed. The principal moments of inertia of the foot were then computed as the sum of those written for each part. IF1 = m/12[((WH +FW}/2)2 + SPHZ] + m/36[2({WH + FW)/2)2 + 3(0.56FL)2]’ + m[(SPH)2] IF2 = m/12[(0.44FL)2 + SPHz] + m/36[3({WH + FW}/2)2 + 2(0.56FL)2] (19) + m[((o.44FL)/2)2 + ((0.56FL}/3)2] IF3 = m/12[(0.44FL)2 + ((wn + FW}/2)2] + m/18[(0.56FL)2 + SPHZ] + m[((o.44FL)/2)2 + ({0.56FL}/3)2] Again, 1, 2 and 3 refer to the principal directions of the foot. 63 In rigid body kinematics, the relative position of any point on a body, with respect to any other point on the same body, remains constant with time and therefore, the magnitude of the relative position vector may be written as a constant, ie. l-EI/Zl =|"Ei -'E}| =lconstant| (20). The motion Of any point on a rigid body may be expressed as a rotation relative to any other point on the same body ‘51/2 ='Vi -'V§ ='§ x 51/2 (21), where'fi is the angular velocity of the rigid body. This equation creates a set of three linearly dependant scalar equations for the solution of the three components of'fil In the rigid-body analysis of human body motion, soft tissue motion and system noise often introduce errors into the position data used in Equation (21). A numerical method, based upon the method of least squares, was developed to calculate the angular velocity and acceleration of the foot, shank and thigh from the three—dimensional position data available. A general description of this method, as it applied to each segment, will be presented in this chapter. The exact equations for a general body segment are formulated in Appendix A. The method of least squares, which was developed from Legendre's principle, determines the solution of least error to a system of overdetermined equations, where the equations are not necessarily compatible with one another. From a previous investigation by this author (see Appendix A), it 64 was determined that four targets, creating six relative position vectors, placed on a lower limb segment, yielded a satisfactory solution for both the angular velocity and acceleration of the limb segment. For this reason, another target was added to the foot segment on the dorsal surface, just distal to the ankle. This sixth target (target E in Figure 5) was required due to the non-rigid body motion of target D during toe-off and the position of target F on the lateral malleolus of the fibula. Target E was only used for calculating the angular velocity and acceleration of the foot and therefore, no exact anatomical location was specified for its placement. The technique used to compute the angular velocity and acceleration was initiated by calculating the relative position vectors and relative linear velocity vectors for each combination of two vectors created by the four targets. The relative linear velocities were equated to the cross- product of the angular velocity,-W}, and the associated relative position vector, as shown in Equation (21). This procedure yielded a series of six vector equations, or 18 scalar equations, for the three unknown components of the angular velocity of the foot; wa' wa and sz.' These equations were solved by the least squares method to determine-Wk. A similar procedure was followed to compute the angular acceleration. The relative linear acceleration vectors of each target with respect to each of the other 65 targets were computed and equated to the cross-product of the angular acceleration and the cooresponding relative position vector, plus the triple vector product, as shown below for a general pair of targets, A and B: The angular acceleration of the foot,'Wf, was then computed as shown in Appendix A. Once the angular velocity and acceleration of the foot segment were known, the acceleration of the center of mass of the foot could be calculated by the following rigid-body equation: .3CMF ='38 +-FF X"'F-CMF/B +-FF x (FF x_f-crllF/B) (23): where.fbMF/B is the relative position vector between the center of mass and target B. The force plate measured the resultant force and moment vector acting on the platform by the foot as it makes contact during stance phase of a stride. The resultant force and moment vectors, Eb and M6, were recorded with respect to the origin and axes of the force plate coordinate system, as shown in Figure 10. In order to resolve this force system into an unique equivalent system acting at the surface of the plate, the resultant force vector and a parallel moment vector were formed. The moment vector was replaced by two vectors, fiperp and-M' A one perpendicular par' to the resultant force vector EB and one parallel to F6: __ __ / A Mpar = ( Mo - F ) F (24) where F ='Fb / ffb| (25) 66 Fz Figure 10. The Force Plate Origin and Coordinate Axes and the Ground Reactions. 67 and Mperp = M0 - Mpar (26). The intercept of the resultant force vector and the parallel moment vector with the surface of the force platform is defined as the center of pressure and was computed by CoP x F0 = Mperp (27) where «A /A IN is the vector from the origin of the force plate coordinate system to the center of pressure, expressed in terms of the force plate axes QFx' ify and 1&2, and h, the known vertical distance between the force plate origin and its surface. The values ch and yCp were then solved by the following two equations: ch = [ hF0x( Fo2 ) + M0xF0xF0y + MczFOyFoz - Moy( FOXZ + F022 ) 1 / [ Foz( F02 ) 1 (29) ch = [ hF0y( F02 ) - MoyFOXFOy ' MOZFOXFOZ + Mox( Foyz + F022 ) 1 / [ Foz( F02 ) 1 (30) where F02 = FOX2 + Foy2 + Foz2 (31). The complete derivation of these equations is given in Appendix B. The resultant force vector, the parallel moment vector and the center of pressure were all computed with respect to the coordinate system attached to the force plate. Therefore, a transformation matrix was formed to convert these values to the inertial, or lab, coordinate system. 68 For this purpose, three targets were placed on three corners of the plate and their positions, with respect to the lab coordinate system, were recorded during the filming of the static standing data. The three targets allowed the x and y-axes of the force plate to be computed with respect to the inertial axis system. The z-axis of the force plate was assumed to be equal to the negative of the inertial z—axis. The two coordinate systems and the three targets, T, S and R, are shown in Figure 11. Given the coordinates of T, S and R in the laboratory coordinate system, unit vectors were computed to define the x and y force plate directions: QFP ='S"fi/ IE'fil (32) 917p =ng/ IEI (33) where §§'= (RX — sx) 9k + (Ry - sy) 99 (34) 'T5 = (3x - TX) 9; + (sy - Ty) Ty (35), assuming that QFP = - 1;. The components of F6 and'Mbar were then transformed by the following matrix equations; r- ") '- 2 '- a Rx xFPx xFPy 0 FOX Ry = yFPx yFPy 0 F0y (36) R2 0 o 1 F02 and q - F -1 FMRx F Xpr Xpr 0 M0x MRz o o 1 MOz L. .J L _l L. .l 69 1 Figure 11. The Lab and Force Plate Coordinate Systems and the Targetting of the Plate. 70 These forces and moments represented the actions of the foot on the force plate. Thus, the directions of these values were reversed to provide the forces and moments the floor applied to the foot. RX = — RX (38) Ry = - Ry MRX = - MRX (39) MRy = - MRy Due to the direction of the z-axis of the force plate, with respect to the z-axis of the inertial coordinate system, the direction of the z-components were automatically reversed (see Figure 11). In order to transform the position vector of the center of pressure to the inertial coordinate system, the origin of the force plate was located with respect to the origin of the laboratory coordinate system and then, the coordinates of the center of pressure were transformed. - 1 1 1 - . _ 1 CPX Tx yFPx xFPx CPZ T2 0 o L. _i 1.. .. _ J )- _ l- ‘l ' '7 XFPX XFPy 0 xCF + Ypr Ypr 0 ch (4°)° 0 0 1 i 0 L _ 1 3 The first three terms on the right-hand side of Equation (40) located the origin of the center of the surface of the force platform from the origin of the laboratory coordinate system. The fourth term located the center of pressure from the surface of the plate. Once the location of the center 71 of pressure was determined, the position vector between the center of mass of the foot and the location of the point of application of the external force and moment vectors was computed. FF=FF-EF (41). The equations of motion for the foot, Equation (5), were written in terms of the principal axes, or the local coordinate system, of the foot. Therefore, it was necessary that all of the values input to these equations be expressed in terms of the fixed local coordinate system. A transformation matrix was formed from the direction cosines of the principal axes and each vector was multiplied by this matrix to provide appropriate principal component vectors for the input values. The six equations forming the inverse dynamics problem for the foot segment were solved for the six components of the force and moment acting at the ankle joint center, FA and MA. Again, the components of the force and moment vectors at the ankle were computed in terms of the principal axes of the foot. II. THE SHANK AND KNEE The bony component of the shank segment of the lower leg is formed by the tibia and the fibula, and their articulations with the talus, distally, and the femur and u patella, proximally. The shank segment was also considered to be a single rigid body. This approximation was more easily justified here than that for the foot, because the tibio-fibular joints, superior and inferior, are fewer in 72 number than the numerous boney articulations in the foot segment. Yet, the rigid body assumption was hindered in the shank by the increased amount of muscle mass and other soft tissues. The greatest amount of muscle tissue exists in the posterior portion of the shank and the bones of this segment lie closer to the anterior border. The effects of soft tissue motion on the analysis of position data was again minimized by limiting the number of targets attached to this link and restricting their placement to boney prominences visable in the anterior and lateral planes of view. The targetting scheme for the shank was established to facilitate locating the center of mass and the principal directions of this body segment. Once again, the literature was reviewed to determine the directions of the principal axes of the shank and the location of their origin. The vertical axis, or long axis, of the shank has been defined (49,88) as a line from the sphyrion landmark, on the medial malleolus of the tibia, to tibiale, the most superior point on the anterior and medial margin of the tibia. The lateral-medial axis has been computed to be parallel to a line from the lateral malleolus of the fibula to the vertical axis of the tibia, perpendicular to the long axis of the shank. The third axis, mutually perpendicular to the other two, has been calculated as the cross—product of the vertical and horizontal axes, as just defined, and is positive in a posterior to anterior direction. The inferior-superior axis of the shank has been defined by two 73 points lying on the medial side of the limb. These landmarks could not be targetted due to the anterior and lateral positions of the cameras. From observation of the tibia, the alignment of these two landmarks was seen to be parallel to the long axis of the shaft of the tibia. Thus, the vertical axis of the shank was defined by placing one target on the tuberosity of the tibia and a second target on the distal portion of the tibia, targets H and G in Figure 12. The position of target G was determined by palpating the distal tibia and locating the level at which the tendon of tibialis anterior crosses, from the foot, anterior and lateral to the tibia. Below this point, the motions of dorsiflexion and plantar flexion of the foot relative to the shank tend to influence the observed surface motions of the shank. Therefore, target G was placed just superior to this level. A unit vector, Th3, was formed in the direction of CH to define the inferior-superior principal axis of the shank segment. "1\S3=Efi/ IE?“ (42). The lateral—medial principal direction was formed at the proximal end of the shank segment, due to the position of the previously located targets. A third target, I in Figure 12, was placed on the superior and lateral border of the lateral condyle of the tibia and this position, along with the position of the target on the tibial tuberosity, were utilized to determine the igz-axis. The relative position vector between these two points was computed and (7 ( ) I: Figure 12. Targetting of the Shank. 75 the component parallel to 153 was removed. 16 = -I—H - (TH $283) 953 (43). This produced a vector in the plane of the surface of the condyles of the tibia, perpendicular to the vertical axis (Figure 13). The axis in the lateral-medial direction could then be formed by rotating I6 about the €g3-axis through the angle 8 .‘ The angle B was computed geometrically as follows: 8 = cos'1 [(KW/2)/|IEI 1 (44). where KW is the width of the knee at the level of the proximal surface of the tibia. The knee width at target I was thus included as an anthropometric measurement in the experimental protocol. The second principal axis for the 1+ Kw +1 Figure 13. Geometrical Representation of the Angle of Rotation, B . 76 shank was formed by first generating a unit vector in the direction of £5, ’i‘IQ =T6 / “IE (45), and then rotating this vector about 153 by the angle 8 . where [R0T]= 133x2 + (1 - is3x2) cos B iS3xiS3y(1 - cos B ) - 1532 sin 8 183xis3z(1 - cos B ) + i83y sin 8 iS3XiS3y(1 - cos B )-+ iSBZ sin 3 iSByz + (1 - iSByz) cos B (47) iSByiS3z(1 - cos B ) - 183x sin 8 i83xis3z(1 - cos B) - iSBy sin 8 iS3y183z(1 - cos (3) + 183x sin 8 i3322 + (1 - 18322) cos B The final principal axis, 1&1, directed from the posterior surface of the shank towards the anterior surface, was formed as the cross-product of ng and 1&3. 981 = 982 x /i\83 (48). The location of the center of mass of this segment has been described in the literature (8,21,23,49,88) with respect to the frontal plane, cut through the medial and lateral malleoli and the medial and lateral condyles. The boney landmark, tibiale, has commonly been used as a reference point for locating the center of mass. In the vertical direction, the center of mass has been shown to be u an average of 37.15% of the tibia length, measured from 77 tibiale to sphyrion. In the lateral-medial direction, the center of mass has been given as 60% of the knee width, with respect to tibiale. The values presented in the literature are summarized in Table 3. In this analysis, target I, lying on the superior border of the lateral condyle, was chosen to be the reference point for the calculations of the center of mass of the shank. The length of the shank was measured during the experimental protocol from the point of tibiale to the sphyrion landmark, parallel to the long axis of the shank, and was denoted as SL. The width of the knee at the level of the surface of the tibial condyles, KW, had been previously determined for the principal axis calculations. Table 3. The Reported Location of the Center of Mass (CM) of the Shank Relative to Anatomical landmarks (I-CM) (I-CM) (Ant. Aspect-CM) SL KW SW (96) (%) (33) Clauser, et al. 37.05 NA 42.47 (1968) Braune and Fischer 42.16 NA * (1889) McConville, et a1. 37.19 59.75 ** (1980) Young, et al. 37.20 61.75 ** (1983) Mean 38.40 60.75 42.47 * Located 1 cm posterior to the interosseous membrane. ** Located 1.25 cm posterior to Tibiale. 78 Therefore, to locate the center of mass of the shank in the frontal plane, with respect to target I, a point R was located as follows; "F"=—I'+ (0.40 x KW) Q52 - (0.3715 x SL) 933 (49) The position of the center of mass of the shank in the sagittal plane, with respect to posterior-anterior dimensions,'has been only breifly mentioned in the literature (Table 3). Braune and Fischer (8), in their cadaveric study of the mass distribution of the body, reported the center of mass of the shank of each of their subjects to be approximately "1 cm behind the middle of the interosseous membrane". McConville, et al. (49) and Young, et al. (88) both reported the center of mass to be an average of 1.25 cm posterior to tibiale, whereas, Clauser (21) stated it to be 42.47% of the depth of the shank at the level of the center of mass, measured from the anterior aspect of the shank. In order to resolve these inconsistancies, sagittal representations of the shank segments of several female subjects were measured. It was observed that an axis drawn parallel to the vertical axis of the shank, positioned at 42.47% of the depth of the shank at the level of the center of mass, fell approximately 1 cm posterior to the tibiale landmark. Therefore, it was assumed that the position of the center of mass in the sagittal plane was 1 cm posterior to the location of point I. The final position of the center of mass was determined 79 . . —- . . A to be 1 cm posterior to p01nt R, in the negat1ve 131- direction E315 ='E- 1931 (50). The location of the center of mass of the shank, with respect to point I, is shown graphically in Figure 14. The knee joint was assumed to consist solely of the proximal portion of the tibia and the distal end of the femur. The patello-femoral joint was omitted in this analysis. The convex medial and lateral condyles of the femur interact with the slightly concave medial and lateral condyles of the tibia, resulting in principally rotational motion during gait. A few investigators (46,71) have attempted to locate and/or model a rotational axis of the knee. Others (53,82) have attempted to describe the contact surfaces of the knee, allowing the forces transmitted by the tibia to be distributed over an area rather than a single point. In the current analysis, the forces and moments in the tibia were resolved with respect to a central linkage point defined on the proximal surface of the tibia. The center of the joint was assumed to be at the midpoint of the surface of the condyles, or tibial plateau, halfway between the tibiale landmark and the lateral border of the lateral condyle. The width of the knee at this level, the level of target I, has been previously defined as KW. From target I, the center of the knee joint was located as "“ - fi\ 80 0.3715SL Figure 14. The Location of the Center of Mass of the Shank. 81 The position vector between this point and the center of mass of the shank was then computed for input to the equations of motion. 'Eg = 53k — Efié (52). The mass of the shank segment, as with the foot, has been approximated as a proportion of the total body mass (Table 2).‘ Again, Dempster's estimate of 4.5% of the total body weight of the subject most closely approximated the other values, and was used to determine the mass of the shank. The mass distribution properties of this link were somewhat simplier to model, due to the more regular geometry of the shank. The shape of the shank is somewhat cylindrical, although the proximal circumference at the knee is typically larger than the distal circumference at the ankle. As Hanavan concluded in his 1964 article (33), the shank segment of the lower limb may be closely modelled by the frustrum of a right circular cone. This was determined to be the best geometrical model for approximating the moments of inertia for the shank. The geometry of the frustrum of an inverted right circular cone is shown in Figure 15. The moments of inertia for the frustrum of the cone are IX = Iy = 3m/20 ( R2 + 4h2 ) I2 = 3m/10 [( R5 - r5 )/(R3 - r3 )1 (53). These equations are written with respect to the principal axes and center of mass of the cone. It was assumed that the center of mass of the model for the shank closely 82 24 \ h/7 y» CO:/ L/ Figure 15. The Frustrum of an Inverted Right Circular Cone. 83 approximated the computed location of the center of mass of the shank. The radius values were replaced by the recorded anthropometric measurements and the following equations were generated for the principal axes of the shank: 151 = 3mS/20 ((KW/z)2 + 4SL2) - 3mS/20 ((KW/z)2 + 4SL2) (54) H m N I 153 = 3mS/10 [((KW/2>5 - (AW/2>5) / ((KW/2)3 - (AW/2)3)1 where 1, 2 and 3 refer to the principal directions of the shank. The angular velocity and angular acceleration of the shank were computed using the least squares procedure outlined in Appendix A. A fourth target was required for the solution of the least squares technique. The position of target F, placed on the lateral malleolus of the fibula, was used as the additional point. The relative positions, velocities and accelerations for each pair of targets were computed and the resultant two sets of 18 scalar equations were solved by the least squares method for-fig and'%;. From these angular values, the acceleration of the center of mass of the shank was computed relative to the acceleration of target I 'Ecms =‘51 +'Fé X ECMS/I +'Wé X (W8 X ECMS/I) (55)! where'EEMS/I is the relative position vector between the center of mass of the shank and target I. The external forces and moments acting on this link are applied at the proximal end, the knee, and the distal end, the ankle. The forces and moments at the center of the 84 ankle joint were computed for the rigid body approximation of the foot. For application to the shank segment, in accordance with Newton's third law, these forces of reaction have the same magnitude, same line of action and opposite direction. Therefore, the components of the force and moment computed for the proximal end of the foot segment were inverted. FA = -?A (56) EA = - EA (57) . The relative position vector between the center of mass of the shank and the ankle joint center was computed to define the location of the point of application of these reactions. 33 =33}, - 3171's (58). The equations of motion for the shank, Equation (6), were solved for the components of the force and moment at the knee joint center in terms of the principal axes. The input information computed for the shank segment of the lower limb was transformed into the fixed local coordinate system, expressing the vector components in terms of the three principal directions of the shank. The six scalar equations of motion were solved for the unknown components of the force and moment, Fk and Mk, at the center of the knee joint and were expressed with respect to the local coordinate system of the shank. III. THE THIGH AND HIP The final segment of the lower limb modelled in this analysis was the thigh segment, composed of a single bone, 85 the femur, and the articulations of the knee, distally, and the hip, proximally. Once again, the simplifications achieved by decreasing the number of internal boney articulations in the link, for the thigh this reduces to zero, are offset by the increased amount of soft tissues surrounding the bone itself. There exist only two directly palpable boney projections on the thigh that remain visible to lateral and anterior planes of vision, trochanterion and the lateral femoral epicondyle. Whereas three points are required to define the position of a rigid body in space and four targets are required for computing the angular velocity and acceleration, only two exact locations could be determined. The additional two targets were placed on the thigh segment with the understanding that soft tissue motion would affect their positions in space relative to the other targets. Placement of these two targets was also dictated by the information necessary to compute the principal directions and center of mass of the thigh. The long axis of the thigh has been reported (49,88) to lie along the long axis of the femur itself. This inferior- superior direction has been defined in the literature by a line between the lateral epicondyle of the femur and trochanterion, the most superior point of the greater trochanter at the proximal end of the femur. Upon inspection, this direction does not, in general, lie perpendicular to the floor when the subject is standing, as in the shank segment. The lateral-medial direction has been 86 computed as a normal from the long axis, as just defined, toward the medial epicondyle of the femur. The third axis, mutually perpendicular to the other two, directed in the posterior-anterior direction, has been computed as the cross-product of the inferior-superior and the lateral- medial axes. The inferior-superior principal axis of the thigh was computed for this analysis as defined in the literature. One target was placed on the lateral epicondyle of the femur and a second was attached to the skin overlying the superior/lateral projection of the greater trochanter. A unit vector was formed in the direction of the relative position vector between these two points, J and M in Figure 16. 9T3=3M/|FMI (59). The lateral-medial principal direction was formed in a similar manner as the TgZ-axis of the shank. An additional target was placed on the anterior/distal aspect of the thigh, superior to the patella. The vector between the target on the lateral epicondyle and the distal thigh target, J and K in Figure 16, was computed and the component of this vector parallel to the vertical axis was removed. 30 = 3‘12 - (fic’i‘m) 9.1.3 (60). This resulted in a vector lying in the plane of the femoral condyles, perpendicular to the long axis of the femur. The lateral-medial direction was determined by rotating this vector about 1&3 by the angle 5 . The angle, 6 , was 87 Figure 16. Targetting of the Thigh and the ASIS. 88 Figure 17. Geometrical Representation of the Angle of Rotation, g . computed geometrically, as shown in Figure 17, by a = cos“1 [(CW/2)/ JQ 1 (61). where CW is the width of the knee at the level of the femoral epicondyles. Measurement of the width of the knee at the level of target J was included in the experimental protocol. The ifz-axis was then formed by rotating the unit vector in the direction of JQ about the 1&3-axis as follows: 152 =‘36/IEEI [ROT] (62), where [ROT] is the same rotation matrix as given in equation (47), with 8 replaced by g . The third mutually orthogonal principal direction was then computed as the cross-product of the other two A A A 1T1 = 1T2 X 1T3 (63) 89 . and was directed from the posterior surface of the thigh towards the anterior surface. The dimensions of the thigh are highly dependant on the amount and level of development of the muscle mass of the subject, as well as the quantity of adipose tissue. Therefore, the location of the center of mass of the thigh has not been reported as a constant proportion of the femur dimensions, as in the other limb segments. In the vertical direction though, the center of mass has been reported as a proportion of the thigh length by several investigators, as shown in Table 4. Less agreement was seen between authors for these values due to the variations in gender and level of soft tissue development. These values were used to determine the approximate location of the center of mass in the vertical direction. An average value of 40% of the Table 4. The Reported Location of the Center of Mass (CM) of the Thigh Relative to Anatomical Landmarks (M-CM) (J-CM) (Ant. Aspect-CM) TL CW TW (is) (3;) (’3) Clauser, et al. 37.19 NA 53.35 (1968) Braune and Fischer 43.86 NA NA (1889) Dempster 43.30 NA NA (1955) McConville, et al. 42.86 51.90 NA (1980) Young, et al. 37.84 NA NA (1983) Mean 41.01 51.90 53.35 90 femur length, measured from trochanterion, was determined to be the nearest estimate. The length of the thigh, TL, was anthropometrically measured during the experimental protocol as the distance from the trochanterion landmark to the surface of the tibial plateau, at the level of target I. In a horizontal plane, perpendicular to the Q53-axis, the center of mass of the thigh lies at the midpoint of this cross-sectional area. The center of mass of the thigh was computed from the center of the knee joint as follows: 65.1. = 35,, + (0.6 TL) ’1‘.“ (64). The center of the hip joint, the linkage point between the head of the femur and the acetabulum of the pelvis, is located deep within the soft tissues of the hip. Therefore, estimating its location from external position data is difficult. Yet, the recent investigation by Bell, et al. (6) has produced a method for predicting the three- dimensional location of the hip joint center as a proportion of the distance between the right and left anterior superior iliac spine landmarks on the pelvis. They reported the hip joint center location in adults to be 30% distal, 14% medial and 22% posterior to the ASIS landmark, with respect to the ASIS-ASIS distance. These estimates were used to predict the relative position of the hip joint center with respect to the ASIS during static standing evaluation. The location of this point was then computed relative to trochanterion on the thigh to eliminate the effects of the relative motion between the thigh and pelvis during locomotion. The 91 distance between the two ASIS points was measured during the experimental protocol, designated as DAS, and an additional target was affixed to the right ASIS, target N in Figure 16. The relative position of the hip joint center with respect to the ASIS was computed from the recorded standing data utilizing the reported proportions. Iii—é =32? - (0.3 13145)?T3 + (0.14 DAS)/i\T2 - (0.22 DAS)/i\T1 (65). The relative position vector between the trochanterion, target M, and the H30, as given above, was then computed. The components of this vector were then transformed to the local coordinate system of the thigh for application to the position data recorded during locomotion. The relative position vector between the hip joint center and the center of mass of the thigh was computed for input to the equations of motion as follows: FT ="HFE - EM‘T (66). The mass of the thigh was computed utilizing the predictive equations determined by previous investigators (Table 2). The value of 10% of the total body mass was used to estimate the mass of the thigh for this analysis. The determination of the mass distribution properties of the thigh required that the boundaries of this segment be defined. When computing the point of force application, it was sufficient to define the proximal end of the link as the head of the femur. Yet, the soft tissue mass of the thigh surrounds the hip joint and shows no specific boundaries. Some investigators (49,88) have split the thigh into two 92 portions, one extending from a horizontal plane through the knee to a horizontal plane at the level of the gluteal furrow, and the second continuing from the level of the gluteal furrow to a plane paralleling the inguinal ligament, as shown in Figure 18. This latter portion has been referred to as the flap. Other investigators (33,40) have simply omitted the flap and approximated the thigh by the distal region. In the current analysis, the mass distribution of the thigh was modelled by the geometry of the frustrum of an inverted right circular cone, with the smaller diameter in the horizontal plane of the knee joint and the larger end at the level of the greater trochanter. This model underestimates the lateral portion of the thigh and overestimates the medial portion, by including half of the flap segment, as shown in Figure 19. The diameter of the distal portion of the cone was determined by the width of the knee at the level of the femoral condyle, CW, as previously described. The proximal diameter was determined from the circumference of the thigh at the level of the gluteal furrow. By assuming this section to be circular, the mean diameter, TD, was computed by dividing the circumference by Pi. The principal moments of inertia for the thigh segment were then computed as in equation (54) and were written as follows: 1T1 = 3mT/20 ((T0/2)2 + 4TL2) 1T2 = 3mT/20 ((TD/z)2 + 4TL2) (67). 1T3 = 3mT/1o [((TD/2)5 - (CW/2)5)/((TD/2)3 - (CW/2)3)1 93 Figure 18. Segmentation of the Thigh and Flap. 95 A fourth target, L, was placed on the anterior surface of the thigh segment superior to target K and inferior to the level of the gluteal furrow (see Figure 16). This provided the appropriate input, along with the other three targets, J, K and M, for the method of least squares. The relative vectors were calculated and the angular velocity and acceleration of the thigh,'w& and-$5, were computed for the rigid body approximation of the thigh segment as outlined in Appendix A. The acceleration of the center of mass, with respect to target M, was then determined as follows: .5CMT ='5M +36T x“130417111 +-WT x (7T x"FCMT/M) (68). where-EEMT/M is the relative position vector between the center of mass of the thigh and target M. External forces and moments were applied to the thigh segment at both the distal and proximal linkage endpoints. The distal components, acting at the knee joint center, were computed from the rigid body approximation of the shank segment. The relative position vector between the knee joint center and the center of mass of the thigh was computed 'FT = EEK +'Efif (69) and the directions of the components of the force and moment were reversed, in accordance with Newton's third law FK — FK (70) MK = - MK (71). 96 The components of each vector input to the equations of motion for the thigh (Equation (7)) were transformed into the local coordinate system of the thigh and the components of the force and moment acting at the hip joint center, Eh and-Mk, were determined. IV. SUMMARY The ferces and moments acting at each linkage center of the lower limb; the ankle, knee and hip, were determined for each 1/100 of a second for the stance portion of each stride. Table 5 lists the landmarks used for placement of each of the 14 targets on the right lower limb and Table 6 lists the anthropometric measurements recorded for each subject prior to filming. The three-dimensional position data were determined, filtered and differentiated. These linear quantities were then input to a computer program for solution of the inverse dynamics problem. Initially, the computer program utilized the anthropometric measurements to compute the principal moments of inertia of each segment and the mass of each link were determined as proportions of the total body mass. Standing data were analyzed next and used to determine: (1) the angle of rotation for the anterior-posterior principal axis of the foot, (2) the location of the center of mass of the foot relative to target B, and (3) the location of the hip joint center relative to target M. Then, for each frame of position data, initiating at the first instant of foot contact and continuing until the last instant of foot Table 5. Target 97 Target Locations Anatomical Landmark/Position A H530”! El U 0 OJ :2 3 t' 58 Q Superior/Lateral Calcaneous Inferior/Lateral Calcaneous Head of Metatarsal V Tip of Toe II Dorsal Surface of the Foot Lateral Malleolus, at the Level of Sphyrion Distal/Anterior Tibia, Superior to Tibialis Anterior Tibial Tuberosity Lateral Condyle of the Tibia, Superior Border Lateral Condyle of the Femur, Inferior Border Distal Femur, Superior to the Patella Proximal Femur, Inferior to the Gluteal Furrow Lateral/Superior Greater Trochanter Anterior Superior Iliac Spine 98 Table 6. Anthropometric Measurements Measurement Abbreviation Definition Foot Length - FL - The distance between the tip of Toe II and the most posterior point on the calcaneous, parallel to the floor. Heel Width - WH - The width of the heel at the ‘ level of Target B, parallel to the floor. Foot Width - FW - The distance between the head of Metatarsal I and the head of Metatarsal V. Sphyrion Height - SPH - The distance between the tip of the medial malleolus and the floor. Ankle Width - AW - The width of the ankle at the level of Target F, parallel to the floor. Shank Length - SL - The distance between target I and target F. Knee Width - KW - The width of the knee at the level of target I. Condyle Width - CW - The width of the knee at the level of target J. Thigh Length - TL - The distance between target M and target I. Thigh Circumference TC - The circumference of the thigh at the level of the gluteal furrow. ASIS-ASIS Distance DAS - The distance between the right and left ASIS. Limb Length - LL - The distance between target M and the floor. 99 contact, the following information was computed for each segment: (1) the directions of the principal axes, (2) the location of the center of mass, and subsequently the relative position vectors between the center of mass and both the proximal and distal joint centers, or the center of pressure, in the case of the foot, (3) the angular velocity and acceleration, and (4) the acceleration of the center of mass. Each vector, including the force and moment at the distal joint center and gravity, was transformed into the appropriate local segment coordinate system and the equations of motion, Equations (5), (6) and (7), were solved for the components of the force and moment acting at the proximal linkage centers RESULTS AND DISCUSSION The results of the kinematic and kinetic data collection were processed, as outlined in the experimental protocol chapter, and a computer program was written to compute the reaction forces and moments at the linkage centers, as described in the theoretical methods section. The resultant forces were normalized by total body weight and the moments were normalized by total body weight times limb length, measured as the distance from the greater trochanter to the floor. Total body weights ranged from 63.93 kg to 74.35 kg and limb lengths varied from 79.8 cm to 91 cm. Each frame of data was normalized to the total time of stance and thus, the forces and moments were presented as functions of percent stance phase, and not as functions of time. The majority of the data presented will examine the results computed for the walking barefoot locomotion condition for the three male subjects. Variations from these patterns with the other two conditions will be discussed in the text. The reaction forces computed at the three linkage centers, as defined in the previous section, represent the interactions between the proximal and distal limb segments which compose the joint. The reaction forces and moments 100 101 for each joint were computed in terms of the principal axes of the distal limb segment of that joint. The forces and moments acting at the ankle joint were expressed with respect to the local coordinate system of the foot:‘if1, QFZ"QF3' the forces and moments at the knee in the local coordinate system of the shank: €51! Qszr ig3, and the forces and moments at the hip in the local coordinate system of the thigh; Tm, 9T2, 9T3. Figure 20 depicts one possible alignment of the segments of the right leg during stance and the potential orientation of the three local coordinate systems, as well as the position of the lab coordinate system. These four coordinate systems, at any point in stance, may have notably different orientations and, therefore, the forces and moments computed at the different joints may not be compared in their present form. Initially, the forces and moments computed at the three joints in terms of the principal directions of the distal segment of each joint will be presented. The diversity of the axes and the comparison of forces and moments between the joints will be discussed later. Ground reaction forces during level walking have been shown to produce consistent patterns and magnitudes for repeated strides of the same individual for a constant velocity. A similar outcome was seen for the components of the forces computed at the linkage centers of the lower 102 T ‘L —’0( Figure 20. Orientation of the Three Local Coordinate Systems and the Lab Coordinate System. 103 limb. Figures 21 and 22 show the components of the force and moment, respectively, computed at the ankle joint for three seperate strides of a single subject. The variations in force between three strides of any subject were typically less than 10% body weight and differences in moments were consistently under 20% body weight limb length. Therefore, the following discussions for the ankle joint linkage center will include only a single stride of data for each subject. Figure 23 presents the three components of the reaction force computed at the ankle joint. The components were computed with respect to the local coordinate system of the foot, which travels with that segment throughout the entire stride. Discussions of the directions 1, 2 and 3 refer to the anatomical directions of the foot, as shown in Figure 2. Each graph in Figure 23 displays the force patterns for a typical barefoot stride of each of the three male subjects. The anterior/posterior force computed at the ankle joint linkage center, F1, followed a consistent pattern for all three subjects. There was an initial anteriorly directed force for the first 20% of stance, followed by an increasing posterior force for the remainder of stance. The posterior force achieved a maximum value of 55% - 70% body weight at 80% of stance phase. The other two components of the force at the ankle showed a larger amount of inter-subject variation, and yet, certain trends in the reaction forces were seen. A W-shaped curve was seen in the medial/lateral reaction force, F2, for all subjects, though the magnitudes % BODY WElGHT 104 40 20 \ J ‘60 0 20 40 60 30 100 % STANCE PHASE .Fl T‘eeeeeeee T2———— T3 10 o I g .l I- - " ' I 10 S? w . 3 -20 i 3: ~30 \ / .§ \0 .xr’”5?w -40 \ ’ {/ ... $ \\./ .."<-"° / ‘50 0 20 40 80 so 100 K STANCE PHASE F2 T‘........ T2_—_- T3 % BODY WElGHT _45 §K\\ /27f;;§\\\ V/i -125 0 20 4O 60 80 100 % STANCE PHASE F3 Figure 21. The Force at the Ankle for a Single Subject. $6 BODY WT. LlMB LENGTH 96 BODY WT. LIMB LENGTH 105 T‘oeeoe-oo T2———— T3 . . 4".“ 7— /' —~ m- —— -- -- ._ \ .- ° \ 1“” \ .......... \ \ . '-. x‘ ; a. ‘F /:-' «.1 3: 0 2° ‘° 5° 80 100 as STANCE PHASE M‘ T‘ .0000... T2_____ T3 20 g m __ /. ...... 4 .-' a’ m 10 .9", 2 , .- 3 /.-'V 5 E 4, g o .- . In / 3 x 0 WV) 0- -5 0 20 40 so a too ' xsr 71...“... T2_____ T3 ANCS2PHASE 2 -6 '-.\ \QQ/ \ -8 l o 20 4o 60 80 too 96 STANCE PHASE Figure 22. The Moment at the Ankle for a Single Subject. M3 106 $2 ........ $3-——_ 84 50 25 .\ 96 BODY WEIGHT I / / '50 ‘ / -75 ' NCE PHASE ' $31“ [:1 32 ........ . 33—-—— S4 0 /.. I 5' -1o -. 7 \ / \ \ ’/° 1- r (32: -2o \ \ I ' Ill \ I I 3 / \ .. \ \ ’ D -30 .’ LA ‘ 8 ‘ \ I .. ....... ... y * . ..O .0. 40 ‘1 l .0" J". / o: 1 "75 'o, / .' \ ..O .4 \ I '-..’- —so ‘ O 20 ‘0 so so 100 % STANCE PHASE F2 E - so $2 I.” 3 8 in __ ., , i ‘00 \ ... / \ / \ . / 450 o 20 40 so 80 ‘00 as STANCE PHASE F3 Figure 23. The Components of the Force at the Ankle Joint. 107 of the two peaks varied highly. This component of force exhibited a consistently lateral action at the ankle for all subjects. The superior/inferior force, F3, showed the typical W-shaped response, similar to the pattern of the vertical ground reaction force. The initial peak magnitude varied between 90% and 135% body weight and occurred at approximately 20% of stance. The second peak, at 90% body weight, and the force curve for the last 25% of stance phase showed a nearly identical pattern for all three subjects. The three components of the moment computed at the ankle joint may be loosely associated with the motions of the foot segment relative to the shank segment as follows: + / - M1 = inversion / eversion + / - M2 = plantar flexion / dorsiflexion + / - M3 = medial / lateral rotation Although, since a joint coordinate system was not used for the presentation of these moments, the above associations are understood to be approximations and not exact descriptions of the given motions. The moments computed about the principal axes of the foot segment, iFl, iF2 and iF3' are shown in Figure 24. The resultant moment about the posterior-anterior axis of the foot, M1, showed inconsistent results for the three subjects. The only similarity seen between them was a definite trend toward an eversion moment at approximately 80% of the stance, near the initiation of toe-off. The maximum magnitudes of this moment varied between 1% body weight limb length inversion to 7% body 96 BODY WT. LIMB LENGTH as soov WT. um LENGTH 1()8 52......" S3———— S4 2 \w—M\ G . \ ~.... ., .............. 100.. “.- \ \ / 1 \ \ ’ I '2 *I"/ ‘;k 4; 1% I...‘ \ k’ -4 A? r \ "a”! \ I -6 R ] \ I -8 2° 40 so so 100 96 STANCE PHASE M1 8200...... sa——-—— 8‘ 20 '5 .1 .>. IT: / \_.'1/ N11 .1 10 7 A. m - \ d . 5 ’ \ e I} ‘ . .- 5 T e 3 / / \\ >- '. o / \ .. O .- ID 0 “V31 _ 8 0 20 40. 80 so 1010 ' ssmmwmpwme $200000... sa____ 8‘ M2 2 y \ .. ..- x U x I .- \'«.. . l 5. -2 .\°'. 7 / - A‘ / i x " _4 N...”A7,. V "4 >5 -6 K\.:..\.// ‘8 Figure 24. 20 40 60 80 100 % STANCE PHASE ' M3 The Components of the Moment at the Ankle Joint. 109 weight limb length eversion. As would be expected, the moment acting about the lateral-medial axis of the foot, M2, maintained a constant plantar flexion moment during all but the first 10% of the stance, at heel strike. Peak magnitudes occurred at 75% of stance and achieved values between 14% and 16% body weight limb length. The moment corresponding to medial/lateral rotation, M3, showed a constant lateral action, except for the last 20% of stance for subject 3. As with the inversion/eversion component of the ankle moment, the magnitudes of this component ranged from 1% body weight limb length medial rotation to 7% body weight limb length lateral rotation. Figures 25 and 26 again demonstrate the small variations in the forces and moments computed for three strides of the same individual. The deviations in force at the knee joint were less than 10% body weight and were lower than 3% body weight limb length for the moment data computed for each subject. The individual commponents of the force calculated at the knee joint linkage center are presented in Figure 27 and are expressed in terms of the principal directions of the shank segment as shown in Figure 3. F1 corresponds to the force at the knee in the direction of 151, F2 represents the force in the iéz-direction and F3 is the component of the force parallel to ié3. The anterior/posterior force, F1, was found to be consistent, in both magnitude and direction, between the three subjects. F1 at the knee exhibited an initial anterior force of 10% - 96 BODY WEIGHT 96 BODY WEIGHT 11...... .. 110 25 -25 -50 - 100 20 T1 ......... 410 60 80 100 NOE PHASE % s_TA T‘OOQOOOQOD T2-—_— T3 F1 5 0 )- I S? w 3 >- - 5 o O a) a! - 10 -15 0 20 40 80 80 100 T2 — — — — T3-—— as STANCE PHASE ‘ ' ' ' F2 25 o /. l " 25 -. / \ / -100 .\>" .. I -125 20 40 60 BO 100 % STANCE PHASE F3 Figure 25. The Force at the Knee for a Single Subject. 9‘ BODY WT. LIMB LENGTH % BODY WT. LIMB LENGTH n ....... 111 .0 T2____ T3 -4 _ r " ‘\ 4 \\/-( ....... \ / -3 2. -8 0 20 4O 80 BO 100 % STANCEMPHASE T1......... T2———— T3 10.0 I 7 5 / p... 0 T X 5 A}u'f: ... V\ _J 50 ‘ ,- 7 r_ 4 / A i 26 \ ,° 3 l- . .- 5 0:0 : ‘ _.'. O :5 3' \ m 1.. .. 1: '. \, 3‘ -2.5 1': ~60 _ o 20 40 BO 80 100 % STANCE PHASE T10eeeeoeeo T2—___ T3 . M2 8 B . _ 1’. J ‘ _.-,' . / 1 ."/ 2 o a] \ q» ’- V '\ a '2' r /,-"-—" :7". -2 0 20 l 40 60 BO 100 x STANCE PHASE M3 Figure 26. The Moment at the Knee for a Single Subject. 16 BODY WEIGHT $6 BODY WEIGHT 112 $2 ......... 83— — — — S4 25 A 1,- I_:' i Dee" \/ -100 o 20 40 so so 100 % STANCE PHASE E‘ 820.000.... sau——__ S4 10 l\ I \ / \ L ‘ ’ 1 -/ \ 5 , ‘l .- E I :. \ / _ ~ ’ / \ 3 0 '1. .' ...... f?\ \ é \gwa7' “a, ~\_,4§ ID " _ : x ..O -5 ‘r . ~10 o 20 40 80 so 100 32... ..... _ 33___ __ S4-——-——- % STANCFEzPHASE 50 0 / / \. / -50 {A 7 \ ; V§\Q / \ 7 .0. 4’ .'.......' j \ .2. “K -.. ..-' -100 X a, ...... .r’ \ / \ / 1? -150 o 20 40 so so 100 $6 STANCE PHASE F3 Figure 27. The Components of the Force at the Knee Joint. 113 20% body weight during the first 20% of the stance, then switched direction, and decreased steadily until 80% of stance, where it reached a maximum posterior force of 75% - 90% body weight. The medial/lateral force at the knee, F2, maintained a very low magnitude throughout the entire stance, generally less than 10% body weight in either the lateral Cr medial direction. The inconsistency seen between subjects was amplified by the low peak magnitudes of this component of force. The superior/inferior force component, F3, computed at the knee again followed the expected W- shaped curve and showed variations in the first peak, 95% - 140% body weight, and in the mid-portion of stance. As in the superior/inferior force at the ankle, the last 25% of the stance showed nearly identical reactions for all three subjects, with the second peak reaching a consistent magnitude of 80% body weight. The resultant components of the moment computed at the linkage center of the knee joint are presented in Figure 28. The actions of these moments correspond to the following rotational motions of the shank segment relative to the thigh segment: + / - M1 = adduction / abduction + / - M2 extension / flexion + / - M3 = medial / lateral rotation These are also approximations, due to the orientation of the shank local coordinate system in which they were expressed. The moment acting about the posterior-anterior axis of the 96 BODY WT. LIMB LENGTH 114 32mm..- 83— —- — — S4 2 0 I \ .' .L Oo'\. I. -2 i=1. / I. \ . / .- \ 4’ -4 . 1- .. | ..... f \ "“‘~. I \ . ‘ \e.... -B ‘/c/ c/ -8 0 20 40 60 80 100 % STANCE FHASE $200.00.... 83— .__ _ 84 M1 10 /o°'/ \ E 5 a“;"/ 3 g / ‘... / ‘1 . . ’ / 3 0 ‘ 2‘ / i ’ \ g r .. .- / \ 1v. '3' 3 ‘ A ...-I.. .1'0 T—' \ ’ 3 ' 5 \ I § / on -10 \ / 3Q \ 1' -15 0 20 40 60 BO 100 % STANCE PHASE 820.00.000. sa—__ _ 8‘ M2 10.0 / i. E 7... / \ m / 30‘, J 1 .-' 1'- g 5.6 I ... ‘2. 3 / \'-. .-° 7 .-/\-. 3 2.5 / 7' 3. 8 “:2"; \_ / " ". ...--' ‘1. 3 0.0 .... {—Il/ ‘ \..' \'.. .' .e.l'y \ -2.5 O 20 4O 60 80 100 % STANCE PHASE M3 Figure 28. The Components of the Moment at the Knee Joint. 115 shank, M1, showed a distinct abduction action for 90% — 100% of stance, with peak magnitudes varying between 1% body weight limb length adduction and 6.5% body weight limb length abduction. The flexion/extension moment computed at the center of the knee joint demonstrated a similar pattern for all three subjects, although, the maximum magnitudes varied between the period of 5% to 60% of stance phase. A consistent flexion moment of 4% body weight limb length was produced during the first 5% of stance. The action of this component of the knee joint moment, M2, then decreased to an inconsistent extension moment until 30% of stance when the moment again acted to produce flexion of the knee. This flexion component then reached a maximum of 6% - 8% body weight limb length at 70% of stance and then decreased consistently to an extension moment during toe-off. The rotational component of the moment at the knee joint followed a uniform pattern for all three subjects, varying solely in peak magnitudes. M3, The moment about the inferior-superior axis, showed a slight lateral action, less than 1.5% body weight limb length, for the period of stance between 5% and 20%, crossed zero at 20% of stance, and then increased to produce a medial action for the remainder of stance, reaching a peak value of 3.5% to 8.5% body weight limb length at 80% stance phase. The variations between three strides for the forces and moments at the linkage center of the hip joint are displayed in Figures 29 and 30. Once again, the magnitudes differed 96 BODY WEIGHT 116 T1 ......... T2 _ _ _ _ Ta 40 .-"-'—‘J-. '/ \. 29 ./ FA. \\ : \ . 0 \ f1 \ Ir - 20 7 - 4O . <9] -50 0 20 40 60 30 100 96 STANCE PHASE T‘...".”. T2—-— _ _ T3 F1 20 15 I'- I S? 10 W 3 6 O 5 D a! 0 -5 0 20 40 60 BO 100 % STANCE PHASE F2 T1 0000000000 T2— —— — — T3 25 0 j :: Q - 25 .. III .' B : 6 _. O " 50 . m * \ / - 75 . -.AE\“ \J .....‘\Lfl/i . -100 . 20 96 STANCE PHASE Figure 29. 40 80 F3 50 100 The Force at the Hip for a Single Subject. T1 OOOOOOOOOO T2————— T3 .1JL7 ’10 96 BODY WT. LIMB LENGTH l u: T1.......... T2———— T3 20 o 20 40 so so 100 96 STANCE PHASE M1 10 (5 Z In _I In I; _J +3 s >. 0 O m a! -10 T1IOOOOOOOCO T2—-—- T3 15 ,1 E ,c A.- g / ‘3 / m 2 13 5 c 3: 6 / 8 o - " a: -5 o 20 40 60 80 100 % STANCE PHASE Figure 30. M3 4 D 60 as STANCE PHASE M2 30 100 The Moment at the Hip for a Single Subject. 118 less than 10% body weight and less than 7% body weight limb length for the components of the force and moment, respectively. Figure 31 shows the three components of force computed at the linkage center of the hip joint for a single stride of each of the three male subjects. The 1, 2 and 3 components refer to the principal directions of the thigh, as shown in Figure 4. The anterior/posterior force, F1, acting parallel to the Til-direction, exhibited an initial positive, or anterior, force with peak magnitudes of 20% to 60% body weight. This was followed by a posterior force which reached a maximum of 50% - 70% body weight at 80% of stance. The medial/lateral force, F2, was typically inconsistent between subjects, but unlike the force at the knee, showed a specific pattern. This force, F2, maintained a medial action for 80% of the stance, then changed direction and acted laterally at the hip for the next 10% - 15% of stance. The medial force at the hip reached a maximum value of 12% -15% body weight at 20% of stance. The superior/inferior force computed at the hip joint linkage center followed the same W-pattern as shown at the ankle and knee joint centers, with peak magnitudes of 70% - 115% body weight at the initial peak. The most consistent moment results were produced at the linkage center of the hip joint (Figure 32). These components were approximately related to the motions of the thigh with respect to the pelvis as follows: ‘0 .5 6 96 BODY WEIGHT 96 BODY WElGHT 119 82 eeeeeeeeee sa——_— 8‘ 100 50 - 50 -100‘ o 20 40 so 80 too 96$TANCE PHASE ' 82W 33____ 34 F1 20 15 _ H /‘~.-’ I {x ‘. 5: '° \ 1 “(NA 9 I .Q.’ a" k ..'6. 3 1:" ' \ 5 -' ‘V #3 A s .- \ \ .. o \ a" 9.. N {y m o \ .0 . * l ‘ \47 \ R, -5 \ ’2 -10 0 20 40 60 80 100 % STANCE PHASE 32.... ------ sa— — -— - 34—— n 50 / // G : , - 50 . 7 ._ ‘ / \ \ / F \ \ .4M\% -100 x ’ \ / -150 o 20 40 so so 100 96 STANCE PHASE ‘ F3 Figure 31. The Components of the Force at the Hip Joint. .% BODY WT. LIN LENGTH 96 BODY WT. LIME LENGTH 120 32 .......... '53—. _ __ __ 54 5 0 -5 /”\ WW w ‘ L." ,/ 3§>_f_q . - \ J N / ' aV I; ....\. \ / ..’. ‘15 0 20 410 80 80 100 % STANCE PHASE M‘ 820000.00... 33——- — s4 15 i E 10 7‘“ /\ 3 A .1 '. g \I ‘.\¢".. .. jrx o ".. j..' ~— ‘ é \ '-._. x / . 3 \ v ../ .-'.. m / ‘1 “an. °. \ / 3 '5 -10 a 20 40 60 80 100 $ 82 .......... sa— —. _ _. s‘ STANae PHASE , 2 15 . / \ / r}. '0 I A 1. / ' E \, 5 . \ 1 Ai , fi' \= G . .__/’4 ‘_-‘ \\+/ ...... y \3 ..... L" / -5 \ fl ,4 —10 O 20 40 60 SD 100 % STANCE PHASE Figure 32. The Components of the Moment at the Hip Joint. M3 121 adduction / abduction +/-Ml + / - M2 = extension / flexion + / - M3 = medial / lateral rotation The moment about the posterior—anterior axis of the thigh, M1, produced an abduction action throughout the entire period of stance, producing an W-shaped pattern of response. The maximum magnitudes ranged from 8.5% to 11.5% body weight limb length initially and to between 9% and 14% body weight limb length for the second peak. The flexion/extension component of the moment at the hip joint initiated with an erratic extension moment. This action then decreased to zero between 20% and 40% of stance and then produced a small, 1% - 4% body weight limb length, flexion moment near midstance. At 70% of stance, the moment at the hip joint center returned to an extension action. The moment corresponding to medial/lateral rotation of the thigh relative to the pelvis, M3, showed a slight initial medial rotation at 5% of stance which then decreased rapidly to produce a maximum lateral moment at approximately 15% of stance. This lateral action then decreased gradually and created a medial force at 40% stance, which reached a maximum value of 10% - 14% body weight limb length at 80% of the stance phase. The forces and moments computed at the joints for the walking in shoes locomotion condition exhibited parallel patterns and magnitudes as did the forces and moments determined for walking barefoot. The reaction forces at the 122 linkage centers of the joints during stance phase for the running in shoes condition were typically greater in magnitude than those found for walking. In the inferior- superior and posterior-anterior directions, the maximum values of the forces were, on the average, twice those incurred during walking. Yet, in the lateral-medial directions, the peak magnitudes were only slightly higher. The moments computed for the running in shoes condition were less consistent between subjects than those computed for the walking data. Similar patterns of action were developed, and as with the reaction forces, the peak magnitudes achieved were much greater. 1 Several investigators have attempted to describe the motions, forces and moments at the joints of the lower limbs in terms that have some anatomical significance. Some have developed joint axes (44,61,82) or joint coordinate systems (29,71) in order to relate the kinematics and dynamics of locomotion to the rotational motions of the joints. Yet, information computed in dissimilar coordinate systems is difficult to compare and may lead to inaccurate conclusions. Thus, many authors relate their results to fixed coordinate systems that correspond to the anatomical planes of the body. The vertical axis is typically chosen to be perpendicular to the plane of the ground, the horizontal axis is formed along the direction of motion, parallel to the sagittal plane of the body, and the lateral axis is chosen in the direction orthogonal to the other two. 123 In order to compare the results of this study with those published in the literature, the forces and moments computed at each joint, in terms of the moving principal axes system of the segment distal to the joint, were transformed into components relative to the fixed lab coordinate system. This set of coordinate axes was defined by the Calibration structure filmed during the experimental protocol and defined the z-axis as being perpendicular to the ground, positive in the superior direction. The x-axis was formed in the direction of motion, or the anterior direction, and the y-axis was computed perpendicular to the z and x-axes and was positive in a medial direction, with respect to the right leg. The x, y and 2 components of the force computed at the ankle, knee and hip joint linkage centers are presented in Figure 33 for a typical stride of a single Subject. The forces at each of the three joints followed similar distribution patterns, varying solely in peak magnitudes. The vertical force at each joint, Fz, followed the same W- shaped curve of the ground reaction force and reached peak magnitudes between 110% body weight and 97% body weight, the maximum force decreasing from the ankle to the hip. The horizontal force, Fx, varied between 15% and 10% body weight in the anterior direction and 35% - 25% body weight in the posterior direction, again decreasing from the ankle to the hip. Inversely, the peak magnitude of the medial/lateral force, Fy, increased from 5% body weight at the ankle to 25% 9‘ BODY WElGHT % BODY WElGHT 124 FA ........ FK— — —- — FH— 20 [ '° 7%:2-s\ 0 N . . fi ’30 ." 3 —4¢ o 20 40 so so 100 % STANCE PHASE Fx FAQOOOOC. FK-— — - FH—"" 5 O \ /"\ ._..h4 \\.‘_.~¥poo \ 'u.... ...' ’- ‘00-" g -5 m \ 3 \\ I § '10 \ m \ g _ \ 15 \L/ ‘20 u 20 40 80 80 100 as STANCE PHASE FA... ooooo FK— - - — m— FY 50 -50 J ... _g " 100 V, 5‘. - 150 0 20 410 60 80 100 ‘ $6 STANCE PHASE Fz Figure 33. The x, y and 2 Components of the Forces at the Ankle, Knee and Hip. 125 body weight at the hip joint. These magnitudes compared favorably with those reported in the literature. In 1950, Bresler and Frankel (9) reported values for the three-dimensional forces and moments in the leg during walking. The forces determined in their study showed similar magnitudes and directions of action as those presented in Figure 33. Bresler and Frankel computed peak y—forces of approximately 5% body weight at the three joints, peak x-forces of positive 16% body weight and negative 19% body weight and peak z-forces at 128% body weight. In a similar study by Zarrugh in 1981 (88), peak forces were computed in all three directions and exhibited similar magnitudes as those shown in this study. The components of the moments at each joint, computed in the lab coordinate system, Mx, My and M2, are presented in Figure 34. The moment at the ankle was dominated by the component acting in the sagittal plane, My, which reached a maximum value of 16% body weight limb length. This was expected, since the moment acting in the sagittal plane is the plantar flexion/dorsiflexion moment and accounts for the principle actions of the ankle during walking gait. The component of the moment at the knee corresponding to flexion/extension, My, also showed a large contribution to the moment at the knee. This component showed a peak value of approximately 7.5% body weight limb length. Unlike the ankle though, the component of the moment acting about the posterior-anterior axis, Mx, also exhibited a significant 126 96 BODY WT. LIMB LENGTH o 20 40 so 80 109 % STANCE PHASE Mx MA ........ MK- - - - 513+— 20 E w .."mfl 0 . '0 o. z ...O .0 3 1o ., g , , ~- _ A A .0 l d" "’ X o J 5 o ’ o . {V ’0. / \o F- \ 1 0. 3 c - ‘ .--"\ , ’ / - > M -. \ / v g -5 f z! -10 o 20 4o 60 00 100 % STANCE PHASE MA-u-"u MK- - -- MH My 96 BODY WT. UMB LENGTH ..- ’0 " - “'77.. .- ..... I ............. ? o 20 4o 60 so 100 as STANCE PHASE M2 Figure 34. The x, y and 2 Components of the Moments at the Ankle, Knee and Hip. 127 reaction at the knee, reaching a maximum of 6.5% body weight limb length at approximately 80% of stance. The moment acting about the vertical axis, Mz, created a peak reaction of approximately 2.5% body weight limb length. The components of the moment computed at the linkage center of the hip joint exhibited different magnitudes of reaction. The moment about the posterior-anterior axis displayed the highest value, at 17% body weight limb length, while the flexion/extension moment, My, showed a much lower maximum value of 5% - 6% body weight limb length. These values and patterns of action also compare favorably with the information found in the literature. In the study by Bresler and Frankel (9), the results showed identical magnitudes and patterns of distribution for the moments computed about the anterior/posterior axis and the vertical axis. The flexion/extension moment acting at the ankle, knee and hip resulted in higher peak magnitudes than those found in this study. But, this difference was expected, due to the fact that their vertical forces were also greater than those reported here. In 1985, Andriacchi and Strickland (2) published the results of a study on joint kinetics. The components of the joint moments in this investigation were presented in terms of % body weight times height, and therefore, the magnitudes were difficult to assess. Yet, their patterns of moment distributions followed the same variations as shown in Figure 34. 128 Some authors have omitted the dynamic terms in the equations of motion, stating that their contribution to the sum of the terms were insignificant. In order to check the validity of this assumption, the contribution of each term in each equation of motion was determined. The equations of motion for each of the three limb segments, Equations 5, 6 and 7, were divided into sets of terms in the following manner: Proximal force Term #1 = m acmi Term #2 = m 9i Distal force- Term #3 l Proximal moment We Term #1 = I 1 i w. Term #2 3 wk (Ij - Ik) Term #3 Distal momenti Term #4 = (fi x 5F)i Term #5 = (F x EF)i where i refers to the direction of the component, j and k are the two perpendicular directions, 5? is the proximal force acting on the segment and BF is the distal force. In the calculation of the proximal force, Term #1 accounts for the dynamic effects of the rigid body, while the sum of Terms #2 and #3 account for the static components. The proximal moment equation includes the effects of three static terms, Terms #3, #4 and #5. The dynamic influence is contained in Terms #1 and #2, Term #1 accounting for the 129 angular rotation in the plane of analysis, and Term #2 taking into account the coupled rotation in the other two directions. The absolute value of each term was determined for each component of the force and moment at each proximal linkage center for each frame of data. The corresponding terms were summed and the contribution of the dynamic terms and the static terms were then computed as a percentage of the total force or moment component. The three components of both the vertical and lateral forces computed at the ankle joint during the walking barefoot locomotion condition were found to be dominated by the influence of the static ground reaction force, Term #3, as would be expected due to the relatively small mass of the foot and the semi-fixed condition of the foot during the majority of stance phase. The contribution of the static terms to the vertical and lateral forces was typically 98% — 100% of the total, except for the initial and final 5% of stance, where the influence of the dynamic term increased slightly. The anterior/posterior force at the ankle exhibited a varying contribution from the dynamic effect of the acceleration of the center of mass of the foot in the posterior—anterior direction, which ranged from 0% - 43% of the total force. At the knee joint, the dynamic component of the equation, Term #1, significantly contributed to the total force in all three directions. During the first third of stance, the influence of this term accounted for 8% - 62% of 130 the total force in the posterior-anterior direction. The effect of the dynamic term in F1 decreased over the rest of the stance, until the last 10%, where it provided 10% - 35% of the total force. In the lateral-medial direction, Term #1 determined a variable 10% - 80% of the total force in this direction throughout the entire stance. The vertical force at the knee was again dominated by the static components of the equation of motion until the final 20% of stance, where the influence of the dynamic term increased from 0% to 58%, at toe-off. The vertical force at the hip showed a similar distribution of force between the static and dynamic terms as did the vertical force at the knee. The component of force at the hip in the posterior-anterior direction, F1, received a consistent 10% contribution from the dynamic term in the translational equation of motion, while the medial/lateral force at the hip showed a variable influence from Term #1, between 0% and 20% of the total force. The components of the moment computed at the ankle joint linkage center, as with the forces at this joint, showed a minimal contribution from the sum of the dynamic terms in the equation. Typically, this influence was less than 1% of the total moment computed at the ankle. Once again, a significant increase in dynamic effects was noted for the initial and final 10% of stance for all three components. At the knee joint, the effect of the dynamic component on the moment about the posterior-anterior axis, 131 M1, increased to account for 5% of the total abduction/adduction moment between 5% and 30% of stance and then increased further during the final 30% of stance to produce 10% - 30% of the total moment. The flexion/ extension moment at the knee received a 2% - 20% contribution from the dynamic terms for the first third of stance, then decreased to approximately 1% - 2% for the remainder of stance. The moment about the vertical axis of the shank, M3, was created solely by the influence of the sum of the static terms. This was partly due to the fact that Term #2 was identically zero for M3, since, for the shank segment, 182 was computed to be equal to 181' This was also the outcome for the moment at the hip joint acting about the vertical axis of the thigh, M3, where again, IT2 was equal to ITl' The moment about the posterior-anterior axis of the thigh showed a distinct contribution from the sum of the dynamic terms for the initial and final third of stance, where the effect of these terms accounted for 5% - 50% of the total moment. During the period of midstance, the contribution of these terms was minimal. The flexion/extension moment at the hip joint was heavily influenced by the dynamic terms in the rotational equation of motion. The contribution to this moment ranged from 0% to 40% of the total moment, again showing an increased influence during the initial and final third of stance phase. 132 In summary, the influence of the dynamic terms at the ankle joint have little, or no, effect on the forces and moments produced there during stance phase of gait. The same may not be said for the forces and moments at the other two linkage centers of the lower limbs. The forces acting at the knee joint and hip joint are significantly influenced by the contribution of the dynamic term consisting of the mass times the acceleration of the center of mass. The analysis of the contribution patterns of the moment terms also showed that the influence of the sum of the dynamic terms could not be omitted from the equations of rotational motion, as was previously assumed. Although the contribution of Term #2, when compared to Term #1, is minimal, typically less than 0.5% of the total moment, to ignore the dynamic effects of the body as they relate to the moments produced at the joints would be incorrect. Also, at the initiation and end of stance phase of gait, the dynamic terms showed a major contribution to the total forces and moments acting at all three of the joints of the lower limb. CONCLUSIONS The inability of researchers to directly measure the forces and moments internal to the segments of the human body restricts the determination of the exact interactions between the segments, at the joints connecting them. Therefore, estimations and models are necessary to closely approximate the dynamics of the body. The methods of rigid- body mechanics are often used to model the segments of the body, since rigid—bodies are confined to follow specific rules governing their actions and the rules which govern the actions of the human body are yet to be defined, or perhaps do not even exist. Estimation of the geometries and mechanical properties of the segments must be made to allow the limbs to be approximated by systems of interconnected rigid links. The goal of many biomechanists is to formulate the closest approximation of the actual kinetics and kinematics of the human body during various activities. The objective of this study was to create an approximation for the dynamics of the lower limbs during walking gait, one of the most common human activities. The Newton-Euler equations of motion, formulated for the rigid-body approximations of the foot, shank and thigh, were solved for the reaction forces and moments at the 133 134 . ankle, knee and hip during stance phase of locomotion. The methods developed in this study were highly dependent upon the experimental protocol that was formulated concurrently with the solution techniques. The targetting of the three segments of the right lower limb was limited by the number of external boney landmarks. The number of targets placed on each segment was also restricted by several requirements of the model; the analytical techniques used to approximate each segment, the method used for locating the center of mass of the segments and the linkage center of each of the joints, the requirements of the least squares method for solution of the angular velocities and accelerations, and the anterior/lateral position of the high speed cameras. The large variations in size, shape and mass distribution found in the human body required that the description of the mechanical properties of the body include these possible differences. Therefore, a number of anthropometric measurements were taken for each subject, in order to include these mass distribution properties in the model. Upon inspection of the results of this study, the methods used to compute the "known" variables in the equations of motion provided accurate results. The geometrical models used to determine the principal moments of inertia for each segment provided values that compared well with the information provided in the literature. The method of least squares proved to be an efficient way to solve the linearly dependant equations for the components of 135 the angular velocities and accelerations of the limb segments. The methods used to determine the locations of the center of mass of each segment and the linkage centers of each of the joints were developed from an accumulation of information presented in the literature, and therefore, were consistent with current anthropometric techniques. The'forces and moments computed for the rigid—body approximations of the lower limbs, as defined in this report, showed favorable results with those presented in the literature. Although, the choice of an appropriate coordinate system in which to represent the forces and moments at the joints of the lower limb, and potentially any of the joints in the human body, was found to have extreme effects on both the magnitudes and directions of these reactions. In this study, the Newton-Euler equations of motion required that the components of the equations were expressed in terms of the principal axes of the rigid body being analyzed. Thus, principal directions were determined for the lower limb segments and the forces and moments computed at the joints were solved and expressed in terms of the local coordinate system of the distal segment of each joint. This method of computation determined the reactions in terms relating to the general anatomy of the limb segments. Ideally, the components of these reactions must be transformed to a joint coordinate system, aligned with the axes of joint motion, to allow discussion of anatomically significant forces and moments. The non- 136 orthogonal orientation of these types of axes systems create difficulties in transforming the coordinates and were therefore left for future studies. The equations of motion written for any rigid body include the effects of both static and dynamic properties. Because of the difficulties in determining the dynamic terms in these equations, they are often omitted in the analysis of human motion, or are assumed to have minimal effect. The results of this study determined that these components of the resultant forces and moments acting at the linkage centers of the joints of the lower limbs had significant contributions to the reactions and therefore should not be omitted from the analysis. The effects of the dynamic components were found to increase significantly, at all three joints, during the initial and final portions of stance, as would be expected, since the ground reaction forces decrease at these periods of stance. During stance phase of walking gait, the effects of the dynamic terms were found to have minimal effect on the forces and moments computed at the ankle joint, but the forces and moments at the knee and hip joints showed distinct contributions from these components of the equations of motion. It is recommended that future analyses of human locomotion should include all of the dynamic contributions to the three-dimensional forces and moments acting at the joints. The techniques presented here were specifically formulated for the right lower limb, although, modifications 137 of this method are anticipated, to allow for analysis of the left leg. Improvements on this method may be made with further knowledge of the locations of the centers of the joints and techniques for locating the center of mass of various subjects. The complete analysis of locomotion must also include the swing portion of the stride, and not limit the analysis to stance phase. The determination of the three-dimensional forces and moments acting in the lower limbs can benefit various research interests. Muscle forces and actions may be computed utilizing optimization techniques, prosthetic limbs and joint replacements may be improved based on these results and the basic understanding of human body kinetics and kinematics may be increased. APPENDIX A APPENDIX A THE METHOD OF LEAST SQUARES The method of least squares was developed by Legendre in 1806 to derive from observed data the best results which that data were capable of providing. In experimental work, it is often found that the data which are collected are functions of some unknown quantities which are to be determined. Often, this results in more data being generated than would normally be required if an exact method of solution were available. The resulting overdetermined system of equations may be written in the matrix form A x = b . The equations of this system may not be strictly compatible with each other due to experimental inaccuracies. Therefore, a solution to this system is desired which minimizes the error. The error may be written as E = A x - b . The least squares solution technique determines the vector x which renders the sum of the square of the errors a minimum. E T E‘ = minimum. THE APPLICATION Four non-coplanar targets, A, B, C and D, placed on a rigid body have position vectors, at some time t, 138 A where 1x,‘A 1y [A and 12 UI (fl trl >I > ~> X H> X ~> X C O U! ~> XX are the inertial coordinate system. target, with respect to each of the other three targets may be written 75' 'XE 35' '56 '56 EB In a similar vectors, vectors, The relative velocity vectors are then equated to the (Bx (Cx (Dx (cX (DX (13,, manner, 0 m m 3’ 3’ 3’ X X X X X x £€>>F>’£€>£€>ufi>>:> + V + the 139 + Ayfy + AZQE + ByQy + 32?} + cyiy + c233 + Dyiy + DiTz unit vectors defining the The relative position of each (By — Ay)€& + (cy - Ay)i& + (Dy - AY)Q& + (cy - By)@& + (DY - By)€y + (Dy - Cy)€& + (32 (c2, (Dz (Cz (Dz (Dz for the same time t, may be computed. 0 on U! 3’ 3’ 3’ N N N N N N ofi>er> N~> N~> N~> §€> velocity vectors, relative velocity acceleration vectors and relative acceleration cross-product of the angular velocity vector and the cooresponding relative position vector. ‘VA/B R7A/c vA/D v19./c ‘Vé/o VC/D =3; x Em him VXEA/D ‘w'xEB/C an‘B/D w x RC/D three unknowns, Wx, matrix form as or in expanded form L. The error equations squared, ETE, lead to a set of three normal equations for the three unknowns. o 'rA/Bz rA/By o ‘rA/Cz rA/Cy o 'rA/Dz rA/Dy o 'rB/Cz rB/Cy 0 'rB/Dz rB/Dy. 0 "rC/Dz- rC/Dy rA/Bz o 'rA/Bx rA/Cz o 'rA/Cx rA/Dz o ‘rA/Dx rB/Cz o 'rB/Cx rB/Dz o 'rB/Dx rC/Dz 0 ‘rC/Dx W Y 140 In component form, this yields a set of 18 equations for the and w 2' which may be written in HDHVJ' 'rA/By rA/Bx o 'rA/cy rA/Cx o 'rA/Dy rA/Dx o ‘rB/cy rB/Cx o ’rB/Dy rB/Dx o ”re/Dy rC/Dx 0 equations have the form L_. vA/Bx VA/By vA/Bz VA/Cx vA/Cy vA/Cz vA/Dx vA/Dy vA/Dz vB/Cx vB/Cy vB/Cz vB/Dx vB/Dy vB/Dz vC/Dx vC/Dy vC/Dz These normal 141 (r1,r1) Wx + (r1,r2) WY + (r1,r3) Wz = (r1,v) (r2,r1) Wx + (r2,r2) WY + (r21r3) “Z = (r2,v) (r3rr1) Wx + (r3rr2) WY + (r3ur3) Wz (r3rV) where (ri,rj) is the scalar product of two column vectors of the matrix [:r:]. These equations may then be solved for Wx' wy and wz using a standard linear equation solution technique. Once the angular velocity has been found, the best solution for the angular acceleration may be computed. The relative angular acceleration vectors may be written as follows 3MB =‘éix-r‘A/B +37; x (‘w’ x zA/B) aA/C = w x rA/C + 'w—x (w x E'A/C) :A/D =OVx ?A/D + 37 x (_w'x ETA/D) gB/C =éz‘x—‘B/C + W x (w x ?B/C) 333/0 =‘51‘x—B/D +1; x a; x?B/D) EC/D =-w_x Ere/D + 37x (w x EEC/D) This again produces a set of 18 scalar equations for the three unknowns wx, wy and wz. These are rewritten so that only the unknowns appear on the right hand side of each 0 equation (ie.-aA/B -'w'x (fi'x'EA/B) ='w'x'§A/B). Again, these may be written in a matrix form, [ r][w]=[b]' where[ r] is the same matrix as before and[b] is given below F 142 The three normal equations may then be written aA/Bx wywxrA/By wxwer/Bz ‘ rA/Bx(wy2 W22) {1 aA/BY wywer/Bz wywxrA/Bx ’ rA/By(wx2 W22) ) aA/Bz wxwer/Bx wywer/By ' rA/Bz(wx2 wyz) ) aA/Cx wywxrA/Cy wxwer/Cz - rA/Cx(wy2 W22) ) aA/CY wszrA/Cz wywxrA/Cx ‘ rA/Cy(wx2 W22) ) aA/Cz waer/Cx wywer/Cy ‘ rA/Cz(wx2 Wyz) ) aA/Dx waXrA/Dy wxwer/Dz ' rA/Dx(wy2 W22) ) aA/DY wywer/Dz wywxrA/Dx ‘ rA/Dy(wx2 W22) ) aA/Dz wxwer/Dx wywer/Dy ' rA/Dz(wx2 Wyz) ) aB/Cx waXrB/Cy wxwer/Cz ' rB/CX(Wy2 W22) ) aB/CY wywer/Cz wywxrB/Cx ' rB/Cy(wx2 W22) ) aB/Cz waer/CX wywer/Cy ‘ rB/Cz("’x2 Wyz) ) aB/Dx wywxrB/Dy waer/Dz ’ rB/Dx(wy2 W22) ) aB/DY wywer/Dz wywxrB/Dx ' rB/DY(WX2 W22) ) aB/Dz wxwzrB/Dx wywer/Dy ‘ rB/Dz(wx2 Wyz) ) aC/DX waXrC/DY wXerC/Dz - rC/Dx(wy2 wzz) ) ac/DY wywer/Dz wywer/Dx ‘ rc/Dy(Wx2 W22) ) ac/Dz wszrC/Dx wywer/Dy ’ rC/Dz(wx2 Wyz) ) (r1,r1) wX + (r1,r2) wy + (r1,r3) wz = (r1,b) (r2,r1) &x + (r2,r2) &y + (r2,r3) wz = (r2,b) (r3,r1) éx + (r3,r2) &y + (r3,r3) wz = (r3,b) Again, 4x, wy and wz may be solved for through the use of a simple linear solution technique. The decision to use four targets, as opposed to three, the minimum number required to define the position of a rigid body in space, was based upon the results of a 143 previous study. In that study, the least squares method was applied to various combinations of vectors created by four targets. Every combination of two, three, four, five and six relative position vectors were used as input to the method and solutions were computed for the angular velocity and acceleration. The results for both'w'and'w'showed convergence as the total number of vectors, and hence, the level of overdeterminancy, increased (Figure 35). An even greater number of vectors would possibly yield a more accurate solution, yet the addition of more targets to the body segment would introduce additional innaccuracies due to soft tissue motion, since the number of honey landmarks available on certain segments are limited. 144 w w w 2 y x ‘ U"); x 0.00 0.90 0.25 . l wt . -0.t0- 0.00~ 0.20~ «l __ I e -0.20- 0.10- 0.15- U 3': ' a 3 . i — .30- 0.00~ 0.r0~ 0 1‘ -L :l f x -0.&0- 0.50- 0.0% L d -L ' -050" 0.40“ 0.33 l I I | I 2 3 t 5 6 Number 0| Veclors Used In Leasl Squares Solullan Angular Veloclly - Range and Mean Values ’ I e d, ml my 0.)" . x X . 53‘] 1.0-1 0.0T .1 my 0 w: I 0.0- 0.0- -1.0— _ x # 1 N8 -s.o--r.o—-2.o— i . on B E e —r0.0- -2.0— -3.0~ + -L —1s.0~ -J.0- 4.0- -20.0~LI..0-L—s.c l l l I 3 l. 5 6 Number 0| Veclors Used In Leasl Squares Solullon Angular Accelerallon — Range and Mean Values Figure 35. The Convergence of the Angular Velocity and Acceleration with an Increased Number of Vectors. APPENDIX B APPENDIX B CENTER OF PRESSURE EQUATIONS Given the resultant force vector and the resultant moment vector acting at the origin of the force platform, as shown in Figure 10, it is desired to locate the point at which the resultant force and a parallel moment vector pierces the surface of the force plate. Initially, a unit vector, in the direction of the resultant force, is formed, Q‘ ='F / {Fl where rEj = (sz + FY2 + Fzz)1/2. The parallel moment vector is then computed as ‘fipar = (fi0"€F) QT - = [(MOX/i‘ + Moy’j‘ + M02?) (PX/'1‘ + Fy/j\ + FZQ) /|F‘|] i‘F _ A . A ._ = [(MOXFX + MoyFy + MozevlF '1”in + Fy’g‘ + sz)/|F| ] = [ FX(MOXFX + MoyFy + MOZFZVIF‘IZJ ’i‘ + [ Fy(M0XFX + MoyFy + Moze)/|'§|2] {i‘ + [ Fz(M0xe + MoyFy + Moze)/|F| ] k. The perpendicular component is then -fiperp =-fi0 --fipar = (Moi? + M099 + M02?) - {([MOXFXZ + MOYFXFY + MozFszj/TFTZ)‘9 + ([MOXFXFY + MoyFyz + Moszsz/ffijz)‘9 + ([MOXFXFZ + MoyFsz + MOZFzz] /f§j2)’§}. 145 146 This reduces to the following form ~— _ 2 2 _ — 2 .A Mperp ‘ {[ M0x(Fy + F2 ) ' MOnyFy MOZFsz]/|Fl } 1 - .A + {[ M0y(Fx2 + F22) - MOYFXFY - MOZFsz]/|F|2} 3 2 2 -2 ‘A + {[ M020"x + FY ) - MOXFXFZ - MoyFsz]/|F| } k. To locate the position where the vector, p, intersects the surface of the plate, the following equation must be solved. Mperp = p x F A\ /\ /\ /\ 1 . A . . = (px + pyj + hk) x (in + ij + F29), where h is the vertical distance from the origin of the force plate coordinate system to the surface of the plate. This yields -— Ax .A /\ Mperp = (pyFz - hFY) 1 + (psz - hFx) j + (pxFy - pyFX) k. /\ A\ Now, equating the components for i, j and Q‘on each side of this equation yields the following three scalar equations 2 2 - 2 _ [ MOX(FY + F2 ) - MOYFXFY - MOZFXFZ]/ F — psz - hF Y 2 2 - 2 _ _ [ Moy(Fx + F2 ) - MOXFXFY - MOZFsz]/ F — hFx pXFz 2 2 - 2 _ _ [ M020“x + FY ) - MOXFXFZ - MoyFsz]/ F — pXFy pny. These three equations are combined to solve for pX and py px = [ hFX(FX2 + Fyz + F22) - MOY(FX2 + F22) 2 2 2 + MOXFxFy + MOZFYFZ ] / [ Fz(FX + F + F2 ) 1 Y 2) [ th(Fx2 + Fyz + F22) + M0X(FY2 + F ‘ 2 2 2 - MOYFXFY - MOZFXFZ ] / [ FZ(Fx + FY + F2 ) ]. py Z The values px and py represent the center of pressure at the surface of the force plate and are computed in terms of the force plate coordinate system. BI BLIOGRAPHY BIBLIOGRAPHY Andriacchi, T.P., J.A. Ogle and J.O. Galante, "Walking Speed as a Basis for Normal and Abnormal Gait Measurements." J. Biomechanics, v.10, No.4 (1977): 261- 68. Andriacchi, T.P. and A.B. Strickland, "Gait Analysis as a Tool to Assess Joint Kinetics." In, Biomechanics of Normal and Pathological Human Articulating Joints, ed. N. Berme, A.E. Engin and K.M. Correia da Silva, Martinus Nijhoff Pub. (1985): 83-102. Bagci, C., "Static Force and Torque Analysis Using 3 X 3 Screw Matrix and Transmission Criteria For Space Mechanisms." J. En . for Industr , v.93, No.1 (1971): 90-101. Bagci,C., "Dynamic Force and Torque Analysis of Mechanisms Using Dual Vectors and 3 X 3 Screw Matrix." J. Eng. for Industry, V.94, No.2 (1972): 738-45. Beckett, R. and K. Chang, "An Evaluation of the Kinematics of Gait by Minimum Energy." J. Biomechanics, V.4, No.2 (1968): 147-59. Bell, A.L., R.A Brand and D.R. Pedersen, "Prediction of Hip Joint Center Location from External Landmarks." Proceedings of the American Society of Biomechanics, (1987): 157-58. Boccardi, S., A. Pedotti, R. Rodano and G.C. Santambrogio, "Evaluation of Muscular Moments at the Lower Limb Joints by an On-Line Processing of Kinematic Data and Ground Reaction." J. Biomechanics, v.14, No.11 (1981): 35-45. Braune, W. and O. Fischer, On the Centre of Gravity of the Human Bodv. Translated by P.G. Maquet and R. Furlong. N.Y.: Springer-Verlag, 1985. Bresler, B. and J.P. Frankel, "The Forces and Moments in the Leg During Level Walking." Trans. of the ASME, January (1948): 27-36. 147 mi“ 3" 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 148 Bryant, J.T., H.W. Wevers and P.J. Lowe, "One Parameter Model for Error in Instantaneous Center of Rotation Measurements." J. Biomechanics, v.17, No.5 (1984): 317-23. Cappozzo, A., "Experimental Techniques, Data Aquisition and Reduction." In Biomechanics of Normgl and Pathological Human Articulating Joints, ed. N. Berme, A.E. Engin and K.N. Correia da Silva, Martinus Nijhoff Pub. (1985): 53-81. Cappozzo, A., T. Leo and A. Pedotti, “A General Computing Method for the Analysis of Human Locomotion." J. Biomechanics, v.8, No.5 (1975): 307-20. Chace, M.A., "Analysis of the Time-Dependence of Multi Freedom Mechanical Systems in Relative Coordinates." J. Eng. for Industry, v.89, No.1 (1967): 119-25. Chace, M.A. and Y.O. Bayazitoglu, "Development and Application of a Generalized D'Alembert Force for MultiFreedom Mechanical Systems." J. Eng. for Industry, v.93, No.1 (1971): 317-27. Chandler, R.F., C.E. Clauser, J.T. McConville, H.M. Reynolds and J.W. Young, "Investigation of Inertial Properties of the Human Body." U.S. Dept. of Transportation, Report No. DOT HS-801 430, Washington D.C. (1975). Chao, E.Y., "Justification of Triaxial Goniometer for the Measurement of Joint Rotation." Q; Biomechanics, v.13, No.12 (1980): 989-1006. Chao, E.Y. and B.F. Morrey, "Three-Dimensional Rotation of the Elbow." J. Biomechanics, v.11 (1978): 57-73. Chao, E.Y. and K. Rim, "Application of Optimization Principles in Determining the Applied Moments in the Human Leg Joints During Gait." gm Biomechanics, v.6, No.5 (1973): 497-510. Chao, E.Y., R.K. Laughman, E. Schneider and R.N. Stauffer, "Normative Data of Knee Joint Motion and Ground Reaction Forces in Adult Level Walking." Q; Biomechanics, v.16, No.3 (1983): 219-33. Chow, C.K. and D.H. Jacobson, "Studies of Human Locomotion Via Optimal Programming." Math. Biosci., v.10 (1971): 239-306. 21. 22. 23. 24. 25. 26. 27. 28. 29. 3o. 31. 32. 33. 149 Clauser, C.E., J.T. McConville and J.W. Young, "Weight, Volume and Center of Mass of Segments of the Human Body." AMRL Technical Report, WPAFB, Ohio (1969): 69-70. Crowninshield, R.D., R.C. Johnston, J.G. Andrews and R.A. Brand, "A Biomechanical Investigation of the Human Hip." J. Biomechanics, v.11, No.1/2 (1978): 75- 85. Dempster, W.T., "Space Requirements of the Seated Operator." WADC Technical Report, WPAFB, Ohio (1955): 55-159. Dempster, W.T., "The Anthropometry of Body in Action." Annals of the N.Y. Academy of Sci., v.63, Art.4 (1955): 559-85. Denavit, J., R.S. Hartenberg, R. Razi and J.J. Uicker, "Velocity, Acceleration and Static-Force Analyses of Spatial Linkages." J. Appl. Mechanics, December (1965): 903-910. Denham, R.A., "Hip Mechanics." J. Bone and Joint Surg., V.4lB, No.3 (1959): 550-57. Elftman, H., "Forces and Energy Changes in the Leg During Walking." Am. J. of Physiology, V.125 (1939): 339-356. Gilbert, J.A., G.M. Maxwell, J.H. McElhany and F.W. Clippinger, "A System to Measure the Forces and Moments at the Knee and Hip During Level Walking." J; of Orthopaedic Research, v.2 (1984): 281-88. Goldstein, H., Classical Mechanics, Addison-Wesley Publishing, Inc., Mass. 1950. Greenwood, D.T., Principles of Dynamics, Prentice- Hall, Inc., New Jersey, 1965. Grood, E.S. and W.J. Suntay, "A Joint Coordinate System for the Clinical Description of Three- Dimensional Motions : Application to the Knee." J; Biomedical Eng., V.105 (1983): 136-144. Gupta, V.K., "Dynamic Analysis of Multi-Rigid-Body Systems." J. Eng. for Industry, v.96 (1974): 886-892. Hanavan, E.P., "A Mathematical Model of the Human Body." AMRL Technical Report, WPAFB, Ohio (1964): 64- 102. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 150 Hardt, D. and R.W. Mann, "A Five Body - Three Dimensional Analysis of Walking." J. Biomechanics, v.13, No.5 (1980): 455-57. Harrington, I.J., "A Bioengineering Analysis of Force Actions at the Knee in Normal and Pathological Gait." J. Biomedical En ., v.11 (1976): 167-172. Hayes, W.C., J.D. Gran, M.L. Nagurka, J.M. Feldman and C. Oatis, "Leg Motion Analysis During Gait by Multiaxial Accelerometry : Theoretical Foundations and Preliminary Validations." J. Biomechanical Eng., V.105 (1983): 282-89. Houston, R.L., C.E. Passerello, and M.W. Harlow, "Dynamics of Multirigid-body Systems." J. Appl. Mechanics, v.45 (1978): 889-894. Houston, R.L. and C.E. Passerello, "On Multi-Rigid Body System Dynamics." Computers and Structures, v.10 (1979): 439-46. Houston, R.L. and C.E. Passerello, "Multibody Structural Dynamics Including Translation Between the Bodies." Computers and Structures, v.12 (1980): 713-20. Jensen, R.K., "Estimation of the Biomechanical Properties of Three Body Types Using a Photogrammetric Method." J. Biomechanics, V.11, No.8/9 (1978): 349-58. Kane, T.R. and P.W. Likins, "Kinematics of Rigid Bodies in Spaceflight." Appl. Mechanics, Report No. 204 (1971): 26-119. Kinzel, G.L., A.S. Hall and B.M. Hillberry, "Measurement of the Total Motion Between Two Body Segments - I. Analytical Development." J. Biomechanics, v.5, No.1 (1972): 93-105. Kinzel, G.L., B.M. Hillberry, A.S. Hall, D.C. Van Sickle and W.M. Harvey, "Measurement of the Total Motion Between Two Body Segments - II. Description of Application." J. Biomechanics, v.5, No.3 (1972): 283- 293. Lagrana, N.A. and D.L. Bartel, "An Automated Method for Dynamic Analysis of Spatial Linkages for Biomechanical Applications." J. End. for Industry, v.97, No.2 (1975): 565-574. Lamoreux, L.W., "Kinematic Measurements in the Study of Human Walking." Bull. of Prosthetics Research, v.10, No.15 (1971): 3-83. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 151 Lewis, J.L. and W.D. Lew, "A Note on the Description of Articulating Joint Motion." J; Biomechanics, v.10, No.11/12 (1977): 675-78. Manter, J.T., "The Dynamics of Quadrupedal Walking." J.of Experimental Biology, V.15 (1938): 522- 39. Marey, E.J., Movement, Translated by E. Pritchard, N.Y.:D. Appleton and Co., 1895. McConville, J.T., T.D. Churchill, I. Kaleps, C.E. Clauser and J. Cuzzi, "Anthropometric Relationships of Body and Body Segment Moments of Inertia." AFAMRL Technical Report, WPAFB, Ohio (1980): 80-119. McCuskey, S.W., An Introduction to Advanced Dynamics, Addison-Wesley Publishing Co., Inc., Mass., 1959. Mena, D., J. M. Mansour and S. R. Simon, "Analysis and Synthesis of Human Swing Leg Motion During Gait and Its Clinical Applications." J. Biomechanics, V. 14, No. 12 (1981): 823- 32. Miller, D.I. and W.E. Morrison, "Prediction of Segmental Parameters Using the Hanavan Human Body Model." Med. and Sci. in Sports, V.7, No.3 (1975): 207- 12. Moeinzadeh, M. H., A. E. Engin and N. Akkas, "Two Dimensional Dynamic Modelling of Human Knee Joint. " J. Biomechanics, V. 16, No. 4 (1983): 253- 64. Morrison, J. B., "Bioengineering Analysis of Force Actions Transmitted by the Knee Joint. " Biomed. Eng. V. 3, No. 4 (1968): 164- 70. Muybridge, E., The Human Figure in Motion, Dover Publications, Inc., New York, 1955. Onyshko, S. and D.A. Winter, "A Mathematical Model for the Dynamics of Human Locomotion." J. Biomechanics, V.13, No.4 (1980): 361-68. Orin, D.E., R. B. McGhee, M. Vukobratovic, and G. Hartoch, "Kinematic and Kinetic Analysis of Open-Chain Linkages Utilizing Newton-Euler Methods." Math. Biosci., V. 43 (1979):107-—30. Panjabi, M.M., "Centers and Angles of Rotation of Body Joints : A Study of Errors and Optimization." J; Biomechanics, V.12, No.12 (1979): 911-20. 59. 60. 61. 62. 63. 64. 65. 66. 67. 68. 69. 152 Patriarco, A.G., R.W. Mann, S.R. Simon and J.M. Mansour, "An Evaluation of the Approaches of Optimization Models in the Prediction of Muscle Forces During Human Gait." J. Biomechanics, v.14, No.8 (1981): 513-25. Paul, J.D., "Bio-Engineering Studies of the Forces Transmitted by Joints." In, Biomechanics and Related Bio-Engineering Topics, ed. R.M. Kenedi, 369-80. Oxford: Pergamon Press, Ltd., 1965. Perry, J., "Anatomy and Biomechanics of the Hindfoot." Clin. Ortho . and Related Res., No.177 (1983): 9-15. Phillips, S.J., E.M. Roberts and T.C. Huang, "Quantification of Intersegmental Reactions During Rapid Swing Motion." J. Biomechanics, V.16, No.6 (1983): 411-17. Procter, P. and J.P. Paul, "Ankle Joint Biomechanics." J. Biomechanics, v.15, No.9 (1982): 627- 34. Roberson, R.E., "Kinematical Equations for Bodies Whose Rotation is Described by the Euler-Rodrigues Parameters." AIAA Journal, v.6, No.5 (1968): 916-17. Rohrle, H.R., R. Scholten, C. Sigolotto and W. Sollbach, "Joint Forces in the Human Pelvis-Leg Skeleton During Walking." J Biomechanics, v.17, No.6 (1984): 409-24. Rydell, N., "Biomechanics of the Hip-Joint." Clin. Orthop., v.92 (1973): 6-15. Seireg, A. and R.J. Arvikar, "The Prediction of Muscular Load Sharing and Joint Forces in the Lower Extremities During Walking." J. Biomechanics, V.8, No.2 (1975): 89-102. Sherby, T.A. and J.F. Chmielewski, "Generalized Vector Derivatives for Systems with Multiple Relative Motion." J. of Applied Mechanics, V.90, No.1 (1968): 20-24. Siegler, S., R. Seliktar, and W. Hyman, "Simulation of Human Gait with the Aid of a Simple Mechanical Model." J. Biomechanics, V.15, No.6 (1982): 415-25. 70. 71. 72. 73. 74. 75. 76. 77. 78. 79. 80. 81. 153 Silver, W.M., "On the Equivalence of Lagrangian and Newton-Euler Dynamics for Manipulators." The International J. of Robotics Res., v.1, No.2 (1982): 60-70. Soudan, K., R. Van Audekercke and H. Martens, "Methods, Difficulties and Inaccuracies in the Study of Human Joint Kinematics and Pathokinematics by the Instant Axis Concept. Example : The Knee Joint. " J; Biomechanics, v.12, No.1 (1979): 27-33. Soutas- -Little, R. W., G. C. Beavis, M. C. Verstraete and T. L. Markus, "Analysis of Foot Motion During Running Using a Joint Coordinate System." Med. and Sci. in Sports and Exer., V. 19, No. 3 (1987): 285- -93. Spoor, C. W. and F. E. Veldpaus, "Rigid Body Motion Calculated from Spatial Coordinates of Markers." J. Biomechanics, V. 13, No. 4 (1980): 391- -93. Stepanenko, Y. U. and M. Vukobratovic, "Dynamics of Articulated Open-Chain Active Mechanisms." Math. Biosci., V. 28 (1976):137-70. Sutherland, D. H. and J. L. Hagy, "Measurement of Gait Movements from Motion Picture Film." J. Bone and Joint Surg. , V. 54-A, No. 4 (1972): 787- -9L Thornton-Trump, A. B. and R. Daher, "The Prediction of Reaction Forces from Gait Data." J. Biomechanics, V. 8, No. 3/4 (1975): 173- 78. Uicker, J. J., "Dynamic Behavior of Spatial Linkages. Part I- Exact Equations of Motion." J. Eng. for Industry, V. 91, No. 1 (1969): 251- 57. Vukobratovic, M., "Dynamics of Active Articulated Mechanisms and Synthesis of Artificial Motion." Mechanisms and Machine Theory, V.13 (1978): 1-39. Wang, J. T. and R. L. Houston, "Kane' 5 Equations with Undetermined Multipliers - Application to Constrained Multibody Systems." J. Applied Mech, Trans. ASME (1987). Winter, D. A., "Overall Principle of Lower Limb Support During Stance Phase of Gait. " J. Biomechanics, V. 13, No. 11 (1980): 923- -2L Winter, D. A., "Moments of Force and Mechanical Power in Jogging. " J. Biomechanics, V. 16, No. 1 (1983): 91- -97. 82. 83. 84. 85. 86. 87. 88. 89. 154 Wismans, J., F. Veldpaus and J. Janssen, "A Three Dimensional Mathematical Model of the Knee Joint." J; Biomechanics, v.13, No.8 (1980): 677-85. Woltring, H.J., R. Huskies, A. DeLange and F.E. Veldpaus, "Finite Centroid and Helical Axis Estimation from Noisy Landmark Measurements in the Study of Human Joint Kinematics." J. Biomechanics, v.18, No.5 (1985): 379-89. Woo, L.S. and F. Freudenstein, "Dynamic Analysis of Mechanisms Using Screw Coordinates." J. Eng. for Industry, v.93, No.1 (1971): 273-76. Wright, D.G., S.M. Desai and W.H. Henderson, "Action of the Subtalar and Ankle-Joint Complex During the Stance Phase of Walking." J. Bone and Joint Surq., V.46-A, No.2 (1964): 361-82. Wynarsky, G.T. and A.S. Greenwood, "Mathematical Model of the Human Ankle Joint." J. Biomechanics, V.16, No.4 (1983): 241-51. Yang, A.T., "Inertia Force Analysis of Spatial Mechanisms." J. Eng. for Industry, v.93, No.1 (1971): 27-32. ' Young, J.W., R.F. Chandler, C.C. Snow, K.M. Robinette, G.F. Zehner and M.S. Lofberg, "Anthropometric and Mass Distribution Characteristics of the Adult Female." FAA-AM Technical Report 83-16, FAA Civil Aeromedical Inst., Oklahoma (1983). Zarrugh, M.Y., "Kinematic Prediction of Intersegmental Loads and Power at the Joints of the Leg in Walking." J. Biomgchanics, v.14, No.10 (1981): 713- 25.