$567.1 «7m...4.u!3 . 1.;r,.. . 93...... mum... mini 5.1.1.... ,, x3145. , . . V . .. . x .. 74.31 . . . "......iw . . a . . x . .. ... . . up... .A : I . A . . , , , , . . ...... ...: ,_ ...”: .... .33. . m1.“ EJ.U¥¢T?:W , ,. High”? . vi. .7. . t . MICHIGAN sure us m u l )m ,1 I ullllll: will a 3 1293 00887 6454 ll LIBRARY Mlchlgan State University This is to certify that the thesis entitled A METHOD FOR LOCATING HIP JOINT CENTER AND CALCULATING THREE-DIMENSIONAL QUASI- STATIC HIP JOINT MOMENTS AND POWER DURING GAIT. presented by DAVID MATTHEW MARCHINDA has been accepted towards fulfillment of the requirements for MS degree in Mechanics WEfiu‘w Major professor Date 27 August 1993 0-7639 MS U is an Affirmative Action/Equal Opportunity Institution PLACE IN RETURN BOX to remove this checkout from your record. TO AVOID FINES return on or before due due. DATE DUE DATE DUE DATE DUE I m l I ll JI——Il MSU le An Affin'netlve ActioNEqueI Opportunlty Institution omens-pt A METHOD FOR LOCATING HIP JOINT CENTER AND CALCULATING THREE-DIMENSIONAL QUASI-STATIC HIP JOINT MOMENTS AND POWER DURING GAIT By David Matthew Marchinda A THESIS Submimd to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Material Science and Mechanics 1993 ABSTRACT A METHOD FOR LOCATING HIP JOINT CENTER AND CALCULATING THREE-DIMENSIONAL QUASI-STATIC HIP JOINT MOMENTS AND POWER DURING GAIT By David Matthew Marchinda The purpose of this thesis was to calculate the three-dimensional moments at the hip joint due to external forces applied to the lower limb during ambulation. In order to accurately compute the moments at the hip, the precise location of hip joint center (HJC) had to be defined. An anatomical study of adult human cadaveric pelves was initiated to investigate the correlation between HIC and pelvic geometry. Pearson product-moment correlations revealed that HJC could be accurately located relative to the anterior superior iliac spline (A518) 14% of pelvic width medial, 34% of pelvic depth posterior, and 80% of pelvic height inferior. With the location of HIC defined, the quasi-static hip joint moments were calculated during the stance phase of gait. Power in the plane of progression and actual power at the hip joint were also defined and calculated during the stance phase of gait. In loving memory of my brother, Anthony. Semper Fidelis. ACKNOWLEDGMENTS To Dr. Robert Soutas-Ijttle, my sincere thanks for your guidance, for having faith in me (especially when I did not), for your knowledge, and for our friendship. To my committee members, Dr. Robert Soutas-Little, Dr. Charlie DeCamp, and Dr. Tom Pence, thank you for your insights and comments on my thesis. To Dr. Geoffrey Seidel, thank you for including me in your research, and for introducing me to the human body. To Patricia Soutas-Little, LeAnn Slicer, and Bob Wells, thank you for all the "little things" that provided me with the tools to complete my research. To Brooks Shoes for their generous funding of my research. A special THANKS to my fellow graduate students: Gordie Alder-ink, Terry Bemis, Cheng Cao, Yasin Dhaher, Kathy Hillmer, Brock Horsley, Kim Lovasik, Jim Patton, and Tammy Reid. Wow, what memories! To all the other workers and friends at BEL: Jennifer, Adam, Mark, N eha, and Shawn. Thanks for your support. To all my friends who have "been there" for me over these years of study. Thanks for everything! Most importantly, to my family, Mom, Dad, Ken, and Anna Maria, thank you for your love, support, and kindness. I could not have done it without you! iv TABLE OF CONTENTS LIST OF TABLES LIST OF FIGURES CHAPTERS I. INTRODUCTION II. SURVEY OF LITERATURE A. Hip Joint Center B. The kinetics of biomechanics 1. The basic kinetic equations 2. The history of kinetic analysis III. METHODS OF DATA COLLECTION A. Hip Joint Center Location B. Kinematic and kinetic data collection 1. Equipment 2. Walking data collection IV. ANALYTICAL METHODOLOGY A. Hip Joint Center B. Moments and Power 1 . Force plate calculations 2. Kinemch analysis 3. Combining motion and force data 4. Solution of moments at the hip expressed in lab coordinates 5. Joint coordinate system and angle calculations Page B‘Szfitfiifilfi 55 57 vi 6. Transformation of moments from lab space to joint coordinate space 7. Power V. RESULTS AND DISCUSSION A. Hip Joint Center B. Moments and power at the hip during stance VI. CONCLUSIONS LIST OF REFERENCES 96 98 LIST OF TABLES TABLE PAGE 1 2 Comparison of maximum moments (Ramakrishnan, 1990) Sample force plate data for walking Sample motion data Comparison of female and male pelvic measurements Mean pelvic measurements and interrater reliability (IRR) coefficients Pearson product-moment correlations of HJC related to pelvic geometry Errors of I-IJC—x location estimation Comparison of maximum moments (N -m/ kg) 35 39 66 66 67 70 LIST OF FIGURES FIGURE 10 11 12 13 14 Anatomy of femur and pelvis Hip motion definitions Newton-Euler schematic Quasi-static schematic Bresler and Frankel's free body diagram of the lower limb Joint moment results (Bresler and Frankel, 1950) Contribution of reactions for flex/ ext moment Hip resultant moments (kN-m) (Crowninshield et al., 1978) Boccardi's flexion/ extension and abduction / adduction hip moments due to ground reaction effects alone Projection of GRF up the body (Wells, 1981) Net joint moments, N-m (Wells, 1981) Flexion/ extension moments (Winter, 1980) Moments in joint coordinate system (Cappozzo, 1984) Hip moments of walking normalized to % body weight limb length (Ramakrishnan, 1990) PAGE 14 14 16 17 17 18 19 20 20 15 16 17 18 19 20 21 24 26 27 28' 29 31 32 ix Frontal and sagittal plane hip moments and power (Ounpuu et a1. 1991) Medial / lateral view of pelvic measurements Frontal view of pelvic measurements Calibration set-up with lab space (13) and force plate space (fps) coordinate systems Targeting scheme Vertical ground reaction force (Fz) of three trials EV3D stick figure of standing subject EV3D stick figure of walking subject Pelvic geometry of frontal plane Equivalent force system of force plate Local segment coordinate systems Position vector in lab space Joint coordinate system Scatter plots of HJC vs. pelvic measurements Splined HJC-z Hip angles during gait - subject 1 Hip angles during gait - subject 2 Forces in lab space - subject 1 Forces in lab space - subject 2 .29 31 33 36 38 41 3 49 57 69 73 75 76 35 36 37 38 39 41 6 X Applied quasi-static hip moments - subject 1 Applied quasi-static hip moments - subject 2 Stick figure depicting forces and moment arms in the sagittal plane - subject 1 Inertial and gravitational terms in moments (Boccardi, 1981) Applied hip moments with and without gravity terms - subject 1 Applied hip moments with and without gravity terms - subject 2 Flexion/ extension moments at hjc and greater trocanter (gt) Hip angular velocity during stance - subject 1 Hip angular velocity during stance - subject 2 Hip power during stance phase - subject 1 Hip power during stance phase - subject 2 79 80 84 86 90 91 92 93 I. INTRODUCTION Biomechanics can be defined as the science of the application of the laws of mechanics to biological systems. In this research, the laws of classical statics and dynamics are applied to the human body to study movement, particularly locomotion. This is commonly referred to as gait analysis. Gait analysis is performed on humans to achieve an objective description of the motion and forces associated with locomotion and to use these objective results to aid in the correction of dysfunctional gait patterns of people with neuromuscular abnormalities. These "abnormalities" can be caused, but are not limited to, cerebral palsy, spina bifida, arthritis, traumatic brain injury, Charcot-Marie Tooth disease, congenital and surgical amputations, traumatic musculo-skeletal injury, and Parkinson's disease. It is the goal of gait analysis to obtain an objective assessment of a subject's gait pattern in order to better define what surgical procedures, physical rehabilitation, and / or orthotic devices could improve the subject's gait and quality of life. The primary locomotive pattern of humans is, of course, walking. Walking has been described as a series of nearly missed catastrophes. During this motion the body is literally in a cyclic pattern of falling and catching itself. Although the process of walking appears very basic since it is done so often and without conscious thought, its intricacies l 2 are actually quite complicated. This act of walking involves the interactions of the brain, the spinal column, nerves, muscles, ligaments, tendons, bones, and joints. A person's stride is measured as the distance between two successive initial contacts of the same foot with the ground or floor. The gait cycle is divided into two phases - stance and swing. Stance occurs when the foot is in contact with the ground and swing occurs when the foot is airborne. Stance is divided into three phases: initial foot contact, when the foot first comes in contact with the ground; midstance, when the foot is firmly planted on the ground, and toe off, when the foot is leaving or pushing off from the ground. At normal walking speed, stance occurs approximately for 60-62% of the gait cycle and swing for 38-40% of the gait cycle. Gait analysis or evaluation includes three major components —- kinematics, kinetics, and electromyography. In biomechanical definitions, kinematics describe the motions of body limbs and joints. This includes angular ranges of motion, angular velocities and accelerations, and translational components. Kinetics describe the forces, moments of forces, and powers associated with movement of the limbs and joints. Electromyography (EMG) is the measurement of muscle activity. EMG is performed by using fine wire or surface electrodes. When a muscle contracts in response to impulses, depolarization occurs This gives rise to action potentials that are picked up by surface electrodes placed on the skin near the muscle's motor point. In order to understand the procedures involved and the results obtained by the kinematic and kinetic analysis, a general knowledge of anatomical terms is needed. Descriptions of the human body reference a standard anatomical position. This position is defined as standing erect with feet together, 3 looking straight forward, with arms at the sides of the body, and palms forward. From this position, the body is divided into three planes. The frontal or coronal plane divides the body into front and back segments. The transverse or horizontal plane divides the body into upper and lower segments. The sagittal plane divides the body into right and left segments. This plane is also called the plane of progression, since it is in this plane that most forward motion occurs during gait. In describing the positions of objects on the body or body segments in relationship to each other, various descriptive names are used. Media! is towards the midline (the line from head to floor, in the middle of the body) and lateral is towards the sides. Superior describes the upper positions, closer to the head and inferior describes the lower positions, closer to the feet. Anterior describes the position closer to the front and posterior describes the position closer to the back. Proximal describes an object as being towards something; whereas, distal describes an object as being away from something. The hip is the major place of support between the upper body and the lower limbs. The hip musculature is also responsible for the most ' positive work (in comparison to the knee and ankle joints) during walking (Ounpuu et al. 1991). Therefore, describing the function, movements, forces, and moments at the hip are vital to obtaining an accurate description of gait. The hip joints are the joints connecting the pelvis with the femurs (Figure 1). The hip joint is a ball and socket joint, formed by the articulation of the femoral head (extended by the femoral neck) with the acetabulum (Johnston, 1973). The acetabulum is the hemi-spherical socket of the joint imbedded in the pelvis. The center of the hip joint is difficult to accurately measure since its location is deep within the upper thigh. It is medial, anterior, and superior to the greater trochanter of the 4 femur, a distance that is subject specific. It is crucial to accurately define hip joint center (HJ C) in order to define the moment arms, and hence, the moments at this joint. Figure 1. Anatomy of the femur and pelvis Most of the muscles responsible for movement at the hip have origins on the pelvis and insertions on the femur. The motion at the hip is described by three angles: flexion / extension (sagittal plane motion); abduction / adduction(frontal plane motion); and extemal/ internal rotation (transverse plane metion) (Figure 2). Each of the muscles across the joint are responsible for one or more of these motions. The flexor group 5 includes the iliopsoas, rectus femoris, sartorius, and tensor fasciae latae. The extensors include the gluteus maximus, biceps femoris, semimembranosus, and semitendinous. The gluteus medius and minimus abduct the hip and the adductor magnus, brevis, and longus adduct the hip. The rotators include the piriformis, obturator internus and extemus, and the gemellus superior and inferior. Hip joint moments are described by the motions they induce. Hence, the flexion/ extension moment is the moment about the axis of flexion/ extension. This axis rests in the frontal plane and is perpendicular to the sagittal plane. The abduction/adduction moment axis rests in the sagittal plane and is perpendicular to the frontal plane. The external/ internal rotation moment axis rests in the sagittal plane and is perpendicular to the transverse plane. ‘ J T‘I Exaension Internal Rotation a . \ ... External Rotation r “ Figure 2. Hip motion definitions 6 In studying the different pathologies (the functional changes due to diseases or inflictions) of a gait analysis subject, it is important to separate the direct results of the primary abnormality from the secondary - compensatory results associated with the problem. Kinematics describe the movement of a body, such as, the range of angular motion and the angular velocities. Kinetics help to differentiate between the primary and secondary results of deformities since it describes the driving mechanisms of motion. The joint moments, in particular, indicate which muscle groups are active in order to produce a desired motion. Joint power defines the rate of energy absorption or generation which further enables a clearer description of the gait evaluation. The objective of this study was to accurately define a three-dimensional hip joint center on a subject specific basis and to compute three-dimensional hip joint moments and power during the stance phase of gait. It is behaved that these parameters will provide tools for an accurate gait analysis of research subjects and patients. II. SURVEY OF LITERATURE The literature survey is presented in two sections. The first section explains why the precise location of hip joint center for biomechanical studies is needed and how this location has been determined by previous investigators. The second section explains the methods of calculating the moments at the hip joint during the stance phase of gait and the history behind these methods. A. Hip Joint Center Hip center location is important in defining the kinematics and kinetics of the hip during gait. Small errors in locating HJC result in large errors in moment computation. Crowninshield et al. (1977) conducted an investigation on the effect of HJC location on the peak total muscle force (TMF) and peak joint contact forces (J CF) occurring during gait. Lateral displacement of HJC 2 cm resulted in 20% greater TMF and 15% greater JCF while a medial displacement of 2 cm reduced TMF by 35% and JCF by 25%. Superior displacement of HJC 2 cm resulted in 15% greater TMF and 10% greater JCF, while an inferior displacement of 2 cm reduced TMF 20% and JCF 15%. Posterior displacement of HJC 1 cm resulted in 20% greater JCF in early phase of gait. If HJC was displaced 2 cm laterally and 2 cm superiorly the abductor force increased to three times normal. Delp et a1. (1993) also studied the effects of HJC location on 7 the moment generating capacity of the muscles. They concluded that a 2 cm displacement of the hip center superiorly effected abduction muscle force up to 44%, the flexion force 27%, abduction moment up to 49%, and the flexion moment 22%. A 2 cm inferior displacement of HJC effected the abduction force by 20% and the abduction moment by 26% while also increasing the flexion force and moment. They found that anterior- posterior displacement of HJC by 2 cm effected the flexion/ extension muscle force up to 16% and the flexion/ extension moment capacity up to 36%, while a medial-lateral displacement of HJC by 2 cm affected the adduction muscle force by 20% and moment capacity by 40%. Many studies using different techniques have been conducted to predict the three dimensional hip joint center (HJC). Some investigators have used radiographs of subjects to locate H] C (Crowninshield et al., 1978; Bell et a1. 1989,1990). This procedure's accuracy depends on the ability to measure the hip center in three dimensions from two dimensional pictures. Magnification errors and alignments must be taken into account and the use of radiation on patients makes this an invasive technique. Ellis et al., 1979, obtained the center of rotation of hip joint center in two dimensions by the use of Moiré fringes. Other investigators have proposed the use of intercepts of helical axes to define joint centers (Blankevoort et al. 1990) or by estimating the HJC to be the center of a sphere described by the three dimensional rotation of a point on a rigid body, the thigh (Cappozzo, 1984). These approaches are not accurate for small angular movements and velocities that may be prevalent in many gait patterns. Also, it may not be feasible to conduct maximum range of motion trials prior to data collection to determine hip joint center on some patients (i.e. the elderly, arthritic, severely contracted). Estimating the position of HJC from bony palpable landmarks is another procedure used by researchers (Andriacchi et al. 1980; Tylkowski et al. 1982). Andriacchi et al. estimated HJC as 1.5-2.0 cm distal to the midpoint of a line between the pubic symphysis and the anterior superior iliac spine in the frontal plane and along a line directly medial to the greater trochanter in the sagittal plane . Andriacchi's group did not report how their approach was devised. Tylkowski et al. studied five museum cadaveric pelvic specimens of undetermined sex and age, and AP and lateral pelvic radiographs of 200 children followed in a growth study. Tylkowski's group predicted HJC by translations along a pelvic coordinate system in fixed percentages of anthropometric measurements. The distance of H] C medial and posterior to the anterior superior iliac spines (ASIS) was calculated as a constant proportion of the distance between the ASIS: 11% medial and 21 % posterior to the respective ASIS. The distance to HJC inferior from the ASIS was found to be 12% of the distance between the ASIS and the ipsilateral knee joint line (Tylkowski et al. 1982). Bell et al. (1989) compared Andriacchi's and Tylkowski's approaches to find HJC. Bell's group obtained AP pelvic radiographs of 39 children and 31 adults of known sex. HJC was defined in these groups as the center of a series of concentric circles matched to the size of the femoral head shadow. Measurements were taken of bony landmarks on the AP pelvic x-rays using a Graf/ Pen Sonic Digitizer. Since the ASIS and pubic tubercles (PT) were not radiographically distinct landmarks their locations had to be estimated. AP and right lateral biplanar radiographs were taken in a fixed reference frame of 20 adult dry pelves (unknown sex) with 10 radiopaque markers on bony prominences. These were used to establish the ASIS and PT locations on the aforementioned AP radiographs and were also pooled into the data of H] C location. For the 20 adult pelves, HJC was located at the center of a plastic half-sphere matched to the size of the acetabulum. In summary, Bell's group had AP x-rays of 51 adults and 39 children and right lateral x-rays of 20 adults to establish the location of HJC. HJC location was established by a percentage of pelvic width (the distance between ASIS), and they found HJC to be 14% medial, 30% distal, and 22% posterior to the ASIS in adults. The results of the location of HJC medial and posterior to the ASIS were different from those reported by Tylkowski et al. Comparison of the different approaches of H] C estimation was done by measuring the differences of actual HJC measured on the x-rays to the estimated location using Andriacchi's and Tylkowski's methods. In this study, Bell et al. concluded that the approach suggested by Andriacchi et al. was the most accurate method of predicting HJC in the frontal plane and the approach suggested by Tylkowski et al., using the figure of 2.2% of pelvic width, best predicted AP location of HJC in the sagittal plane. Bell et al. (1990) then compared the accuracy of Cappozzo's (1984) rotational method of HJC localization to Andriacchi's and Tylkowski's methods of HJC estimation. AP and lateral radiographs of seven healthy adult males with reflective skin markers over bony landmarks were taken in a reference frame to find the actual location of HJC and to check the estimation of HJC of Andriacchi's group and Tylkowski's group. Cappozzo's rotational method of HJC localization utilized the application of the least squares method of spherical analysis. To check this method, the seven healthy males were required to stand on one limb and then to ll continuously and sequentially flex, extend, and abduct the hip with the knee extended and then return to the normal position without rotating the femur during the three-second video data collection. Two to three trials were required to calculate the HJC for each side. In comparing the methods of the estimated HJC to the actual HJC, Bell et al. (1990) concluded that Cappozzo's method was the least accurate of the three and that a combination of Andriacchi's and Tylkowski's methods was most accurate to find HJC. However, Bell et al. (1990) reversed their previous conclusions (Bell et al. 1989) in this study by concluding that the method suggested by Andriacchi et al. most accurately predicted the AP HJC location and that Tylkowski's approach was better for the frontal plane location of HJC, but with the percentages of ASIS width based on their subjects' pelves (14% medial and 30% distal). The present study was designed to directly measure pelvic cadaveric anatomy utilizing a large sample size to establish the location of hip joint center accurately. The objectives were to find a correlation between HJC and pelvic geometry, to study if H] C could be estimated by taking percentages of pelvic width alone, and to investigate if there are any significant differences in locating HJC in males vs. females. B. The kinetics of biomechanics The studies of the kinetics of the body during locomotion has progressed over time with the increase in knowledge of the basic equations of motion and the advancement of technology. 12 1. The basic kinetic equations The objective of this study was focused on the kinetics of the hip, namely the hip moments and power, during the stance phase of gait. The computations of the moments at the hip joint requires the modeling of the human body as a set of rigid bodies. In classical mechanics, the dynamics problem involving rigid bodies is solved for the motions of the bodies, since all the driving forces, both internal and external to the body, are usually well defined. In biomechanics, however, the problem is reversed. The forges. and the moments at the joints are internal and cannot be experimentally measured; whereas, the motions are obtainable through the use of special targets on body segments and advanced camera systems. The equations of motion that define both of these problems are expressed in translational and rotational components. The translational (Newtonian) equation of the rigid body is cm where F is the sum of the external forces on the body, in is the mass of the body, and 5cm is the acceleration of the body at its center of mass. The rotational (Euler) equation of motion for a rigid body is where M is the sum of the moments of the body about its center of mass, I is the inertia tensor, (1') is the angular velocity of the body, and ii is the angular acceleration of the body. These equations can be simplified by 13 expressing them with respect to the principal axes of the body (Figure 3). These Newton-Euler equations can then be written as F = ma x cmx F = ma y cmy F = ma 2 cm2 Mx = Ixxmx + mymz (Iyy “122) where x, y, and z are the principal axes of the body (Greenwood,1988). In walking, the magnitude of the effects of the inertial terms of the lower limbs on the moments at the joints are questionable and often ignored. This allows the moments about the joints to be computed at each instance of time as a quasi-static solution (Figure 4). The equation for the moments at the hip then becomes M = prRF + GRT + Ix'W_T + stS where p is the moment arm between the ground reaction force (GR-F) and hip joint center (HJC), GRT is the ground reaction torque, I is the moment arm between the center of the gravity of the thigh and HJC, WT— is the l4 "t K car RF Figure 3. N ewton-Euler sshematic l TN... 6!“ Figure 4. Quasi-static schematic 15 weight of the thigh , ii is the moment arm between the center of the gravity of the shank and HJC, and WS is the weight of the shank. Whether or not the inertial and gravitational terms are needed to precisely calculate the moments at the hip during stance phase of walking is still a matter of debate in the literature. 2. The history of kinetic analysis As far back as Leonardo da Vinci, scientists have been investigating the movements, forces, and moments of the body. All of the early studies were done by the process of observation. In 1680, however, Borelli began to apply Galileo's scientific methods to the study of human locomotion. Muybridge (1887) and Marey (1885) advanced the observational kinematical studies of biomechanics with the use of photography. Braune and Fisher (1895) began to use the process of stereophotogrammetry (photographical representation of bodies in three dimensions) and were the first to study the human body as a classical mechanics problem of a system of rigid bodies. This was considered the start of modern biodynamics. As technology increased, the science of biomechanics advanced as well, and the process and accuracy of the threedimensional analysis of joints became more reliable. Early studies of ground reaction forces on the foot during walking were attempted by the use of subjects walking on plaster of Paris and finely spaded garden soil to study their foot imprints. This, however, only gave information on the shape of the subject's foot and not the amount of pressure distributed (Soutas-Little, 1987). Elftman (1939) used a force platform (developed by himself) that measured the magnitudes of the three components of force, from which he derived the point of application of the force of the foot on the ground. Elftrnan also captured motion by taking "cinematic" records at 92 exposures per second of a subject walking behind a rectangular grid. Timing was obtained by using a vibrating reed of known period placed in the photographic field. His force plate data, coupled with his cinematic records, enabled him to calculate "reversed effective torques" (rudimentary moments) at the joints. The most widely cited paper on the moments at the lower limb joints while walking was written by Bresler and Frankel (1950). They modeled the lower limb during walking with the free-body diagram shown in Figure 5. They used D'Alembert's principle of equilibrium of bodies in motion to express internal forces and moments at joints directly in terms of ground reaction forces, gravity forces, and inertial forces. Figure 5. Bresler and Frankel's free—body diagram of the lower limb 17 Figure 6 shows the results of their calculations. Bresler and Frankel were interested in the significance of the inertial and gravitational effects on the moment equations, and also calculated the moments without these terms. The results of the "fore-and-aft" (flexion/ extension) moments with and without inertial and gravitational terms are shown in Figure 7. ANKLL e Jv‘v K/uenunmm—v- 'i" astronomical!” a \/ . on e “ ’1 out W l. root venues (00! mos ' root ”W05 MOMENT '- south!"- I “OMEN! Comrarmox or Rue-nose: Foaa-axo-Arr Mariam: Figure 6. Joint moment results Figure 7. Contribution of reactions for (Bresler and Frankel, 1950) flex/ extension moment Bresler and Frankel concluded that ”the effect of gravity and inertia on the fore-and-aft moments is very small for the ankle and the knee moments, and relatively small on the hip moments throughout most of stance phase.” Crowninshield et al. (1978) also solved the inverse dynamics problem (calculating the forces and moments from position data) for the l'uip during walking. This was achieved by modeling the body as a system 18 of rigid links, experimentally measuring the locations of the body segments through time, and using the kinetic data from a force plate. Once the resultant moments and forces were calculated, they were transformed to local joint coordinates (Figure 8). 008‘ x COMPONENT test semence ‘t‘. - +1— oos .' *2 ----- 1:.f- I. .76.“..- 00.:- ; ‘~ ~. 1 “\- 002. “ t 't i. 001 o I O -002 ~001- 41on HIP RESULTANT 'MOMENTIkNm) x - abduction/adduction; y - internal/ external rotation; z - flexion/ extension Figure 8. Hip resultant moments (kN-m) (Crowninshield et al. 1978) Boccardi et al. (1981) studied moments at the lower limb joints by an on-line processing of kinematic data and ground reaction data. This was done by superimposing images of the resultant vector of force with images showing the position of the hip, knee, and ankle joints. They defined the muscular moment ”as the product of the force exerted by a muscle and the distance from its articular axis.” Specifically, the moment at the hip was calculated by the cross product of the perpendicular distance between the l9 projection of the ground reaction force and the joint center with the ground reaction force itself. This neglected gravitational and inertial effects. Boccardi et al. only studied the results in the sagittal (flexion / extension) and frontal (abduction/adduction) planes (Figure 9). ) > > 10 HIP 0 v— ..I/ Figure 9. Boccardi's flexion/ extension and abduction/ adduction hip moments due to ground reaction effects alone. ‘ 2o HIP S a“. std. -—t-—-o They concluded that "both the general shapes of the diagrams and the numerical maximum and minimum values are very similar to those obtained by other means." The magnitudes, however, appear to be half of what Crowninshield, 1978, reported. Although the hip joint moments had consistent shape and similar values with and without inertial and gravitational terms, they had some discrepancies during the end of stance phase compared to the other joints. This is due to the inertial components of deceleration and acceleration of the leg at the push-off part of stance. These results were also taken for subjects walking at a "relatively" high speed, and it was concluded that the patients or subjects with typically slower gait would have even less contributions from the inertial terms. Wells (1981) did a study on the projection of the ground reaction force as a predictor of joint moments. He claimed that computing '20 moments about joints as the cross product of the moment arm to the ground reaction force from joint center with the ground reaction force is an approximation and a potential source of error. As this method is used for joints more superior, the source of error is assumed to increase (Figure 10). Wells calculated the moments in a similar manner as Bresler and Frankel (1950) and compared these results with the moments calculated due to ground reaction force alone. He pointed out that the link segment model's use of gravitational and inertial terms requires the approximations of various anthropometric data and is a source of error. For slow cadence (1.4 m/s), the results of both methods are shown in Figure 11. Wells concluded that "the projection of the ground reaction force is a good predictor of net joint moments for slow walking (more typical of patients than of normals). Increasing the velocity of gait results in increasing errors, especially at the hip. It can be concluded that it is a useful estimate, but care must be taken when using the method in normal or faster speed walking, or for moments at the hip joint." Figure 10. Projection of GRF up the Figure 11. Net joint moments, N-m body (Wells 1981). (Wells 1981). 21 Winter (1981, 1984) performed comprehensive studies on the flexion/ extension forces and moments of the lower limb during gait. Winter (1981) states that the time history of the moments of force at the joints is ”one of the most valuable biomechanical variables to have for the assessment of any human movement." He also points out that the main function of the lower limb during stance is to resist collapse and then to extend for limb push-off. Support of the body is achieved by extensor moments at every joint of the lower limb during stance. He concludes that the variability of each joint moment (ankle, knee, and hip) forces the use of a new parameter for gait analysis - the support moment (Ms). This is simply the summation of the ankle, knee, and hip extensor moments. The pattern and magnitude of M5 was found to be consistent and useful for gait interpretation. The individual fleidon/ extension (sagittal plane) moments were calculated using Bresler and Frankel's (1950) equations, including the inertial and gravitational terms. Winter used a projection of the greater trochanter medially as his location of hip joint center for moment calculations. Figure 12 shows two examples of the flexion/ extension moments and the typical support moments. Cappozzo (1984) computed moments by Bresler and Frankel's (1950) approach. He also pointed out that when expressing joint moments, a point of application must be chosen. The functional significance of the moment depends upon its location and "this point should be defined either as the point through which the resultant articular surface contact force acts or, in the absence of friction, could be taken as the center of rotation" of the joint. An example of his joint coordinate system moments are shown in Figure 13. ”iii—H— as.» JOINT MOMENTS '- r U' I i 8?»;ré03885é03 if A a ‘3. “1:“. \ S‘ k 5:)- ' O! . gong: /fi\n_ W s ..1 .. ‘ - i'fA A .. 1.. .. -...” I ....I . ! .. Till! (as) $3! I SC .4 2 1 m 1 “Ci . h! C Figure 12. Flexion/ extension moments: ankle (Ma), knee (Mk), hip (Mr), and total support (Ms) (Winter, 1980) INTERSEG. COUPLE COMPONENTS [l dIV’- ID N-ni IN.ROT. EX.ROT. RDD. RBD. \ RTO 'Rro ans .1 ._ D 50 1 TIME [2 of cycle] Figure 13. Moments in joint coordinate system (dark solid line indicates hip moment) (Cappozzo, 1984) 23 In 1990, Ramakrishnan et al. did a study on lower extremity joint moments and ground reaction torques to establish a "normal" data base of adults. They used a VICON motion analysis syStem, a force plate, and foot switches to obtain the data needed to compute three-dimensional moments of the hip, knee, and ankle joints and ground reaction torques of 40 adults (18-40 yrs old) and 10 aged adults (55-70 yrs old). The moments were calculated using three-dimensional linear motion, angular motion, velocities, accelerations of body segments, and external ground reaction forces. The inertial components of the moments were calculated by the Euler equations of rotation about principal axes described earlier. Moments were normalized as percent body weight and limb length. Figure 14 displays sample results and Table 1 shows magnitude comparisons between their results, Winter's (1984) results and Cappozzo's (1984) results. Ramakrishnan attributed the difference in magnitudes to normalization techniques and different estimations of hip joint center locations. .4 l W 1 ‘I 1 a a 4. ” fl 1” X OAIT m Figure 14. Hip moments of walking normalized to % body weight limb length (Ramakrishnan, 1.990) 24 Table 1. Comparison of maximum moments (Ramakrishnan, 1990) Comparison of maximum moment with previous authors Present Winter 1 Cappozzo F No. of Subjects: N=40 N=16 N=1 (120 evaluations) Hip flexion 13.2(2.3) 6.5(2.9) 8.0 Kneeextension . 6.9(1.5) 5.5(2.5) 8.0 Ankle plantarflexion 13.5(1.2) 17.0(2.0) 14.0 Hip abduction 7.7(1.6) - 8.0 Knee abduction 3.2(1.4) - 2.3 Hip rotation (external) 0.9(0.4) - 3.1 Knee rotation (external) 0.4(0.1) - 1.5 Ankle rotation (external) 1.3(0.3) - 1.5 Ground reaction torque 0.32(0.10) 1 Normalized to unit leg length. Values shown are mean (standard deviation) and were interpolated from the published graphs. Ounpuu et al. (1991) established a data base for pediatric (ages 5 - 14) gait of three-dimensional lower extremity joint kinetics. They used Euler's equations to calculate the net joint moments relative to each body segment coordinate system. Ounpuu used Dempster's (1959) anthropometric estimates for adults because of the lack of data for children. They believed that this would not be a source of error because "the contribution of the inertial components to the joint kinetic data in the stance phase is small." They also calculated power defined to be the angular velocity of a particular segment times the moment of that segment. Graphs of hip moments and power for the sagittal and frontal plane are shown in Figure 15. Their results were similar to the adult data results of Bresler and Frankel (1950) and Ramakrishnan et al. (1990). From this, they inferred that ”children typically establish their mature gait as early as age 5 years." HIP 3.0 j" _ .Abdnotor n o ' a _ Intense: 1.: . 1.0 - Jotnt lonent o.s (l-nlng) ' o.o ' 1.0 Joint lenent . '. 0 0’ (hall!) 0.0‘ -o.t Addnotor [tenor -r.o 1 l I i -1 e. . I GOIOIOSAOI I a Generation .- Jetnt Fever I .. Joint Fever 1 ‘ (noticing) a .. "‘s ' "flit/kt) 0 WM «II 1" .‘Oo'.. Absorption 1 “2l I 1 -1 _ .. "o .’ '0 '. ‘00 Absorption -3 l I I 'l n enrr cross 0 as to_ rt ‘ . too N CA3? CYCLE ' Figure 15. Frontal and sagittal plane hip moments and power (Ounpuu et al. 1991) The methods of calculating moments at the hip are well established by using the laws of classical mechanics. The question still remains as to whether or not inertial terms are needed to obtain accurate results. Most authors (Bresler and Frankel (1950), Boccardi (1981 ), Wells (1981), and Ounpuu (1991)) stated that the inertial effects on joint moments were small but started to become significant at the hip joint, especially with a fast gait. However, with inertial terms, care must be taken to compute accurate angular velocities and accelerations about the axes of the moments studied. Since only displacement data of angular motion is directly obtained, differentiation must be applied twice to generate velocity and acceleration terms. Estimations of anthropometric data (body segments' inertias, weights, and center of gravity locations) are based upon average data, not individual subjects. Subjects seen in clinical gait laboratories are patients with constrained, slow gait, and the effect of inertial terms are minimal. For these reasons, the moments in this study were calculated without the inertial terms. However, since the weight of the lower limb is roughly one-sixth body weight, the gravitational effects were considered. Power is defined as the rate of doing work. Most authors express power at a joint during gait as the power associated with the individual planes of motion. They (Ounpuu, 1992; Winter 1992) compute this 'power' as the magnitude of the moment multiplied by the angular velocity of the planar angle in question. If the model being studied is planar, then this is logical, such as Winter's (1992) study of the biomechanics in the sagittal plane. However, when three-dimensional kinetics are being studied, the true definition of power is 191-6) and individual components cannot be separated in a scalar quantity. This planar 'pseudo-power' is, however, used as a clinical tool to help describe muscle activity. Therefore, in this research, power in the sagittal plane was calculated as the flerdon/ extension moment multiplied by the angular velocity of flexion/ extension; true power at the hip was calculated as M - 6‘). III. METHODS OF DATA COLLECTION This chapter is divided into two major sections. The first section is a description of the methods utilized to define the location of HJC based on pelvic geometry. This was completed through an extensive cadaver study performed by the author and Geoffrey Seidel, MD. of the Rehabilitation Institute of Michigan, Wayne State University, Detroit, Michigan. The second section of this chapter is a general description of the experimental methods used to collect the kinematic and kinetic data of a subject walking. This work was performed at the Biomechanics Evaluation Laboratory, Saint Lawrence Health Science Pavilion, East Lansing, Michigan. A. Hip Joint Center Location All human cadavers used for anatomical instruction at Wayne State University School of Medicine, Detroit, Michigan from April 1991 through January 1993 were examined after anatomical dissection. The Wayne State University mortuary acquires voluntarily donated cadavers from three counties of southeastern Michigan. Those cadavers with intact pelves were entered into the study group and available demographic data recorded. Sixty-four cadavers were examined, 35 female and 29 male. The ages ranged from 53 to 99 (mean 75.1, sd 12.6) representing skeletally 27 28 mature adults. Intact cadaveric pelvic specimens were excluded from the study if there was a pelvic fracture, acetabular surgery, severe degenerative changes with osteophytosis, and obvious pelvic asymmetry. If an orthopedic procedure was performed unilaterally, the other side was included in the study group. This left a total of 122 sides examined. Pelves were removed from the cadavers and defleshed to allow accurate measurement of bony landmarks. Each pelvis was placed on a flat surface with both ASIS and the pubis in contact with a data recording sheet. To establish a pelvic coordinate system, the frontal plane was defined as the plane passing through both ASIS and the pubic symphysis. The coordinate system was defined with: its origin at the respective ASIS side that was being measured; y-axis mediolateral (positive medial); z-axis superoinferior (positive inferior); and x-axis anteroposterior' (positive posterior). The point of contact of both the A518 and the pubis were traced on the data recording sheet to ensure the pelvis did not move during the pelvic measurement procedure. All measurements were made with a millimeter ruler or calipers from bony surfaces, not compressible soft tissue structures. Measurements were made of acetabulum depth and diameter to ensure that the acetabulum could be considered a hemisphere and HJC was defined as the center of the acetabular rim (labrum removed). HJC-x (depth measurement) was defined and measured as the perpendicular distance from the frontal plane to the center of the bony acetabular rim (Figure 16). The pelvis was lifted and the specific contact points of each A515 and the pubis were noted within the traced areas on the data recording sheet. The position of the projection of HJC-x on the frontal plane was recorded on the data sheet. 29 PD ’i HJC-z -— HJC-x PUBIC CENTER Figure 16. Medial / lateral view of pelvic measurement 30 Other pelvic measurements included pelvic width (ASIS to ASIS), pelvic height (perpendicular from pubic center to the inter-ASIS line), ASIS to pubic center, and ASIS and pubic center to the intersection of HJC-x with the frontal plane. Pelvic depth was measured with a caliper accurate to a millimeter in an oblique fashion from the ASIS to the posterior superior iliac spine (PSIS). With the ASIS location known, the HJC-x projection of the HJC onto the frontal plane, and the pubic center (PC) defined as the center of the pubic symphysis, the HJC-y (medial measurement) and the HJC-z (height measurement) were calculated (Figure 17). All measurements were made by Dr. Seidel, to within a millimeter. Interrater reliability was assessed with the assistance of two additional raters who were instructed in the measurement methodology and then each allowed to select a group of pelvic specimens at random to reproduce all measurements (rater 1: 20 sides and rater 2: 17 sides). Dr. Seidel's measurements were compared to those of each rater. Mean interrater differences for each measurement were less than one millimeter. An analysis of interrater differences with data plot analysis failed to reveal any significant systematic or random error. Interrater reliability correlation coefficients of the pelvic measurements were high for pelvic measurements indicating a reliable measurement procedure. Parametric analysis of interval measurement data and Pearson product-moment correlation analysis was performed with SPSS/ PC+ statistical package. No violations of normality were observed and outlier data were included in analysis. Power analysis established the sample sizes adequate for systematic analysis. The student's paired t-test was utilized to compare male and female data. Measured distances from the HJC-z PH é- Pubic Center Figure 17. Frontal View of pelvic measurements 32 ASIS to ipsilateral HJC along x, y, and z axes were divided by pelvic width (ASIS separation), pelvic depth, and pelvic height and were intercorrelated and compared to previously reported data. The absolute value of the differences between estimated and true HJC coordinate measurements were made and analyzed with Student's Host to assess the relative accuracy of HJC location. B. Kinemch and Kinetic Data Collection In order to calculate the moments at the hip joint during gait, the external forces acting on a subject while walking and the positions of the subject's body segments must be measured throughout the gait cycle. Therefore, both kinematic and kinetic data were collected of a subject walking. 1. Equipment The kinematic data were collected using four solid state, shuttered 60 Hz NEC video cameras and a Motion Analysis Corporation VP-320 model dynamic image processor. Prior to collecting subject walking data, a calibrated space of 150 x 100 x 160 cm centered over the force plate was defined by sixteen control points (Figure 18). The cameras were positioned to encompass the control points (Spherical targets covered with 3M retro- _ reflective tape) such that the entire space could be viewed by each camera. The targets were illuminated by incident light provided by flood lights mounted 2.0 cm away from the camera lens, and the target's light image was reflected back to the camera. The kinematic coordinate system is referred to as 'lab space' (18) and is also shown in Figure 18. The Expertvision three-dimensional (EV3D) digitizing program, utilizing 33 160 .. HO... w- 15 . ”gay row §/— #1:" 4\ I——...... ——l 4\ Figure 18. Calibration set-up with lab space (18) and force plate space (fps) coordinate systems 34 direct linear transformations, was used to compute the three-dimensional positions of the targets. The norm of residuals (the accuracy of the input location of a target compared to the computed position of the target) ranged from .38 to .47, indicating an accurately calibrated space. Stance phase ground reaction data were collected using an AMTI Biomechanics Platform Model OR6-6. This force plate simultaneously measures the foot-ground forces (Fx, Fy, F2) and the moments (Mx, My, and M2) about its instrument center at a collection rate of 1000 Hz. The forces and moments are measured through the use of strain gages attached to load cells at the four corners of the plate. Sample force data for a walking trial is shown in Table 2. The force plate is flush with the laboratory floor and the instrument center is 4.05 cm below the top surface. The orientation of the force plate coordinate system is different from that of lab space and is shown in Figure 18. 2. Walking data collection Kinemch and kinetic data were collected simultaneously for two ‘ subjects walking through lab space at comfortable stride. The subjects used for this test were 24 and 26 year old males with no known pathologies. The subjects signed an informed consent form approved by UCHRIS (IRB #89-559). Using hypo-allergenic tape, nine retro-reflective targets were attached to the subject (Figure 19) over the following bony landmarks: left anterior superior iliac spine (LASIS) right anterior superior iliac spine (RASIS) posterior superior iliac spine (PSIS) greater trochanter (GT) lateral femoral epicondyle (LATCOND) mmdmthNI-‘O sauna borne Fz(N) 45.596 60.363 86.945 122.388 163.738 213.949 267.113 311.416 340.952 358.673 367.534 361.627 355.720 346.859 335.045 329.138 320.277 317.323 314.370 311.416 311.416 311.416 314.370 317.323 317.323 323.231 326.184 332.091 335.045 337.998 340.952 340.952 340.952 340.952 337.998 335.045 332.091 332.091 326.184 326.184 323.231 323.231 323.231 323.231 320.277 Fx(N) 1.877 2.628 2.628 4.130 4.130 3.379 1.877 -2.628 -7.885 -13.141 -17.647 -22.904 -26.658 -29.662 -31.164 -32.666 -33.417 -34.167 -32.666 -33.417 -3l.915 -31.915 -31.164 -29.662 -28.911 -27.409 -25.907 -24.405 -22.904 -21.402 -19.149 -17.647 -16.145 -14.643 -13.141 -12.391 -10.889 -10.889 -8.636 -6.383 -6.383 -5.632 -4.881 -3.379 -2.628 35 Table 2. Sample force plate data for walking Fy(N) ~1.szs -3.328 -6.328 -8.578 -11.578 -15.328 -17.578 -1s.328 -19.s2s -20.578 —22.078 -23.578 -24.328 -27.328 -28.828 -30.328 -32.578 -34.078 -36.328 -37.828 -38.578 ~40.078 -41.578 -41.578 -42.328 -44.578 -44.578 -45.328 -46.828 -47.578 —48.328 -49.828 -49.828 -so.57s -52.s2s -53.578 -53.578 -53.578 -52.828 -52.s2s -52.828 -52.078 -51.328 -51.323 -50.578 Mz(N-m) Mx(N-m) 0.397 0.246 0.700 0.851 1.003 1.457 1.911 2.365 2.971 3.425 4.030 4.636 5.090 5.392 5.847 5.998 6.149 6.452 6.603 6.603 6.755 6.755 6.755 6.603 6.755 4.611 -6.080 8.137 11.076 14.897 19.599 23.420 26.947 29.004 30.180 30.180 29.592 28.710 27.828 26.653 25.771 25.183 24.596 24.008 24.008 23.420 23.714 23.714 24.008 24.008 23.714 24.008 24.596 24.889 24.889 24.889 24.596 24.596 24.302 24.008 23.420 23.420 23.420 23.420 22.538 22.538 22.244 22.244 22.244 21.950 My (N-m) 4.328 6.089 8.437 11.665 15.773 21.056 25.751 29.566 32.207 33.674 34.261 34.555 33.968 33.087 32.207 31.620 31.033 30.740 30.446 30.153 30.153 30.446 30.446 30.740 30.740 31.033 31.620 31.620 31.913 31.913 32.207 31.913 31.913 31.913 31.620 31.327 31.033 30.446 30.446 29.859 29.859- 29.566 29.272 29.272 29.272 36 asxs ASIS. 3”“ O O O t Lama: Medial .3 Condyle (:09th O Distal ' Shank Posterior ’ Shank Ptoxhnal ‘ . Shank \12: "I “ frontal lateral Figure 19. Targeting scheme 37 medial femoral epicondyle (MEDCOND) proximal anterior tibia (PROX) distal anterior tibia (DIST) distal posterior tibia (POST) At least three non-collinear targets were placed on each body segment studied so segment anatomical planes and axes could be formed. Anthropometric measurements were taken of the distance between the inter-ASIS line and the pubic symphysis (ps), of pubic depth (ASIS to P818) and of the shank length (LATCOND to calcaneus). The subjects were instructed to walk through the calibrated space at a "normal” cadence. Trials were saved if the subject successfully struck the force plate with the entire foot of the targeted side and nothing else. The reproducibility of the force data is shown for three trials in Figure 20. Data were collected for walking with both left and right sides targeted. In all, five trials of each condition were saved. Using EV3D, all trials were tracked and edited to establish the three-dimensional trajectories of the targets as the subject walked through lab space (Table 3). With the subject standing erect in the middle of lab space standing measurements were taken to correct for any misalignment of targets (Figure 21). A completed tracked and edited walking file is shown in Figure 22. 38 Biomechanics Evaluation Laboratory Stu”: Mp on. quiet static Protocol: calling saute! use: to super “or: 1 on. 01 um: o—z-u any awn: mam Sex: on]. i i i m . 2 g 3 a :2 z: 1“ ‘ H 0 H 8 8 8 d d d l- v- v- i S S N H H 2 a 5: t C ‘ 2 ‘3 ’3 c- . c- g a o 3 .. 8 .. 8 .. §=§céc .. I .. a .. a 8§I»3§ 5 5 E 5 a o- \ c- '- :‘a?a? ~i,:,: * :3 *- 3 r S x N z I. 2 N 5:?353 : g g 3 3‘ E A A 2 3 2'. a S: -m m m m m m su m m 71' (I506) Figure 20. Vertical ground reaction force (F2) of three trials SKILL 1: FOR WHICH THERE IS (ARE) TRIAL 1: INITIAL TIME TIME INCREMENT - CUTOFF FREQ’S: OBSERVATION LT ASIS RT ASIS PSIS PUBIC SYMPH GR TROC CONDYLE LAT CONDYLE MED PROX SHANK DIST SHANK POST SHANK OBSERVATION LT ASIS RT ASIS PSIS PUBIC SYMPH GR TROC CONDYLE LAT CONDYLE MED PROX SHANK DIST SHANK POST SHANK OBSERVATION LT ASIS RT ASIS PSIS PUBIC SYMPH GR TROC CONDYLE LAT CONDYLE MED PROX SHANK DIST SHANK POST SHANK OBSERVATION LT ASIS RT.ASIS PSIS PUBIC SYMPH GR TROC CONDYLE LAT anflTYLE‘MED PROX SHANK 39 Table 3. Sample motion data 1 TRIAL(S). - 0.000000 sacs. 16666*10-6 SECS. 1 son WHICH T - 0.000000 IS AT x - -53.7131 Y - 16.6629 IS AT x - -47.2895 Y - -6.6110 IS AT x - —72.0099 Y - -4.4761 15 AT x - -47.4855 Y - 6.3542 Is AT x - -53.2762 Y - -14.3585 IS AT x - -29.1295 Y - -8.287O IS AT x - -30.6517 Y - 3.1821 IS AT x - -24.1522 Y - -2.4788 15 AT x - -21.7351 Y - -0.9415 IS AT x - -35.1483 Y - -1.3715 2 FOR WHICH T - 0.016667 IS AT x - -50.0936 Y - 16.5952 15 AT x - -43.7977 Y - -6.8167 IS AT x - -68.5345 Y - -4.4377 Is AT x - -43.8352 Y - 6.1335 IS AT x - -49.7967 Y - -14.6527 IS AT x - -25.6819 Y - -8.1218 IS AT x - -27.5735 Y - 3.1890 IS AT x - -21.3871 Y - -2.3525 IS AT x - -19.8167 Y - -0.8387 IS AT x - -33.2452 Y - -1.5312 3 FOR WHICH T - 0.033333 IS AT X - -46.4443 Y - 16.4538 IS AT x - -40.4418 Y - -7.1014 IS AT x - -65.1557 Y - -4.3538 IS AT x - -40.1842 Y - 5.8039 IS AT x - -46.4512 Y - -14.9987 IS AT x - -22.4307 Y - -7.9684 IS AT x - -24.6236 Y - 3.1098 IS AT x - -18.8933 Y - -2.2087 Is AT x - -18.1619 Y - -0.7200 IS AT x - -31.5276 Y - -1.7496 4 FOR WHICH T - 0.050000 IS AT X - -42.8189 Y - 16.2669 Is AT x - -37.2354 Y - -7.4092 IS AT x - ~61.8818 Y - -4.2599 IS AT x - -36.6093 Y - 5.3889 IS AT x - -43.2420 Y - -15.3147 IS AT x - -19.5219 Y - -7.8853 13 AT x - -21.8326 Y - 2.9768 IS AT X - -16.6971 Y - -2.0532 NNNNNNNNNN NNNNNNNNNN NNNNNNNNNN NNNNNNNN 98.1406 97.7227 100.3528 86.7713 87.5132 50.5237 50.2506 39.2524 24.7071 22.3886 98.0593 98.0319 100.5000 86.9013 87.8699 50.7285 50.4429 39.1517 24.5000 22.7333 98.0612 98.4025 100.7019 87.1212 88.2897 50.8828 50.6384 39.0623 24.3399 23.1249 98.1380 98.7967 100.9201 87.3940 88.7205 50.8944 50.7670 38.9279 examle of walking barefoot right ted file / j § 463.39 . «£0.75 Marinate] Poultion (a) Figure 21. EVBD stick figure of standing subject 41 lthsls 0818 asis lt eater was is ocanter standing barefoot lt 3:35:13 lat condyle g ggfigflst shank 42188 +1547 Horizontal Position (ca) Figure 22. EVBD stick figure of walking subject IV. ANALYTICAL METHODOLOGY This chapter is presented in two major sections: the first describing the mathematical methods of calculating hip joint center and the correlations to pelvic geometry; and the second describing the analytical methods to calculate the hip moments and power during gait. A. Hip Joint Center Only the posterior measurement (HJC-x) of hip joint center could be obtained by direct measurements on the defleshed pelves. HJC-y (medial measurement) and HJC-z (inferior measurement) had to be calculated from the other measurements taken using trigonometric functions (Figure 23). The measured distances taken from the cadaver study were the distance between the ASIS and the midline of the inter-ASIS line (a), the pelvis height (b), the distance between the ASIS and pubic center (c), and the distances between the projection of HJC-x on the frontal plane and the ASIS (e) and the pubic center (d). These distances are related by the following trigonometric identities: d2 = 82 +02 - 280(0080) b2 = 82 +c2 - 280(0089) yielding, the following values of (p and 0 : 42 43 _. 2 2 2 _ 2 2 2 ¢ =cos'1[ d +e +c ] 9 =cos"1[ b +8 +0 ] 2ec 2ac where (b and 0 are intermediate angles needed to calculate the values: HJC- z = -esin(¢ + 8) ! HJC- y = -ecos(¢+ 0) u. asrer PW IASIS J Pubic cm Figure 23. Pelvic geometry of frontal plane The location of the H] C has now been determined from pelvic measurements. Pearson product-moment correlations were performed to evaluate the relationship between H] C values and pelvic anthropometric measurements. For example, the correlation (r) of HJC-y with PW was calculated as = glue-y - HJC-y)(PW-PV_V) N(SHJC _ y)(sm) where, _[2(HJC-N y- -HJC- y)2 S-HJCy and, —2 (PW -PW) S W =1]; N are the standard deviations of the HJC-y and PW measurements, with HJC - y equal to the mean HJC-y measurement, PW- equal to the mean PW measurement, and N equal to the number of pelvis sides measured. B. Moments and Power Calculating the moments and power at the hip joint during gait involves the interaction of the force plate data, the kinematic data, and anthropometric data. 1. Force plate calculations The forces and moments of the reaction between the foot and the ground create forces and moments at each joint throughout the body in the quasi-static model. The force plate data includes the magnitudes of the interactive forces and the moments about instrument center but does not directly give the position of those forces on the foot. The position of the resultant forces on the foot is needed to calculate the moment arm between 45 the GRF and the HIC. This can be solved as a system of general forces. The position of the resultant force on the force plate is called the center of pressure (COP): COP = xcopi + ycopj (in force plate space coordinates) In order to solve for the COP, a force and a couple about the vertical axis had to be formed and a point along the resultant line of action had to be defined. The point of interest was the intercept of the line of action with the floor surface. The force plate surface was 4.05 cm (zht) above the instrument center of the force plate. This problem can be solved by resolving the forces and moments into a single reaction force and a vertical torque, T. The force and moments about the instrument center are: R=in+Fy3+Fer M = Mxi + My]: + Mzk We want an equivalent force system M = 1'5 X R + Tzk where, f) is a vector from the instrument center to the intercept of the line of action with the force plate surface (Figure 24): f) = xcopi + ycopj + zhtk then, M = [{(YCOPXFZ) - (211003)}; - {(szxcop) - (zht)(Fx)}j j' .. {(XCOPXFY) - (YCOPXFXHK + Tzk and, by equating the vector components: M = (ycopXFz) - (zht)(Fy) My = (zht)(Fx) - (Fz)(xcop) : M2 = (xcopXFy) - (YCOPXFX) + T2 K).B.o) 3., (zht)(Fx) - My Mx + (zht)(Fy) xcop = Fz ycop = Fz T7, = M2 - xcop(Fy) + ycop(Fx) Figure 24. Equivalent force system of force plate 47 2. Kinematic analysis For each instance of time (At = l / 60 sec), the raw kinematic data is transformed from trajectory data to the form of three-dimensional positions of targets in lab coordinate space, where the origin is floor level in the center of the force plate (Figure 18). With at least three targets on each body segment between articulating joints, a local coordinate system could be defined for each segment. W (Figure 25): The y—axis of the pelvis coordinate system was defined as LASIS - RASIS = lyp where LASIS equals the position vector whoSe components are the coordinates of the target LASIS in lab space (Figure 26). The unit base vector is: The z-axis was formed by the cross product of the two vectors formed in the transverse plane of the pelvis: (LASIS - RASIS)X(PSIS- RASIS) = i zp i i=3 ”Iil 2p Figure 25. Local segment coordinate systems Figure 26. Position vector in lab space 50 The x-axis was formed as the final vector axis orthogonal to the y-axis and z-axis: lxpzinXizp WW (Figure 25)= The z-axis of the thigh coordinate system was defined as CT - LA—TC' O—ND = izt The x-axis was derived by the cross product of the two vectors defined in the frontal plane of the thiglu Right side: (m - W)x(fi-m) = ixt Left side: (LATCOND - W) x(-G_T- LATCOND) = ixt XI The y-axis was formed as the final vector axis orthogonal to the x-axis and the z-axis: 51 hank F t ml r ' m(Figure25): The z-axis of the shank/ foot coordinate system was defined as PROX - DIST = izs i = ‘zs zs Ii ZS The y-axis was formed by the cross product of the two vectors formed in the sagittal plane of the shank: (POST - DIST)X(PROX-DIST) = iys i iys = ’3 i ys The x-axis was formed as the final vector axis orthogonal to the y-axis and z-axis: i XS = i y s x i zs Using the pelvic coordinate base vectors, HJC was defined in lab space for both the right and left targeted sides: RHJC = RASIS - allxp + 2121yp - 831zp LHJC = LASIS - alrxp - 821yp - H31zp 52 where, al =.- .34|I>'§I‘s' - RAS' Is| 82 = .14lfi's'1‘s' - F518] a3 = .80|PS| (.34 is the percentage of pelvic depth to HJC, .14 is the percentage of pelvic width to HJC, and .80 is the percentage of pelvic height to HJC. These were the results of the cadaver study, and will be discussed later.) Using the thigh coordinate base vectors, the center of gravity of the thigh (CGT) in lab space was calculated for both the right and left targeted sides: RCGT = GT + blixt + b2iyt - b3izt LCGT = GT + blixt - b2iyt - b3izt where, b1 0.0 .SIMEDC" " '—0ND" - LA—Tc"o' FIB] 53 = .4341? - LATCONDI b2 (.433 is the percentage of the thigh length measure from CT to CGT as reported by Winter (1990), .5 is the estimated percentage of thigh width to CGT.) 53 Using the shank / foot coordinate base vectors, the center of gravity of the shank/ foot complex (CGS) was calculated for both the right and left targeted sides: RCGS = LATCOND + clixs + c2iy$ - c3izs LCGS = LATCOND + clixs - 02iys - c3iZS where, cl 0.0 02 .SIMEDCTND - LATCOND] c3 = .606ILATCOND - CALCANEUSI (.606 is the percentage of the shank length measure from LATCOND to C68 as reported by Winter (1990), .5 is the estimated percentage of shank width to CGS.) 3. Combining motion and force data In order to use the force plate data and kinematic data together, they both had to have the same coordinate systems. Therefore, the force data was transformed from force plate space (fps) to lab space (15): xcop(ls) = - ycop(fps) ycop(ls) = ~xcop(fps) and the applied forces on the force plate become reaction forces: Rx(ls) = Fy(fps) Ry(ls) = Fx(fps) Rz(ls) = Fz(fps) 54 The kinematic data were collected at 60 Hz and the force plate data at 1000 Hz. Therefore, the kinematic data had to be interpolated to be able to combine it with force data. A cubic spline was used to perform this interpolation. As defined by Cheney (1988), "a spline function is a function consisting of polynomial pieces on subintervals, joined together with certain smoothness conditions." In order to have a continuous second derivative of the interpolated data, a spline of degree three was chosen. From Cheney (1988), With data points of the form x = t1, t2, , tn and y = y1,y2, ..., yn a frmction S, was constructed with n-l cubic polynomial pieces (81,82, ...,Sn.1), with the following conditions: Interpolating conditions: S(ti) = yi (15 i Sn) Continuity conditions: lim 5mm: lirn S(k)(x) (05k52,2$iSn-1) x-st; x-Hi" In order to satisfy all of the degrees of freedom, it was decided to use a natural spline with, S"(t1) = S"(tn) = 0 The resulting cubic spline function was solved as, 55 -x)3 + Z. Z. - _1_T_1. - 3 _1 r i y. z. 11. y. z.h. 1+1 - 1 1 r _ _1 _ _l.__1 - [ h. 5 ](" ti) + [1,. 5 ](‘i+1 ’0 1 1 where, Y- -Y- Yo'Y- z. = 6[ 1+1 1 - —‘———‘—=-1-] (ZSiSn-l) t1+1 " ti ‘1 ‘ ti-r 21 is a n by n matrix that is a symmetric tridiagonal system and was solved by forward elimination and back substitution. 4. Solution of moments at the hip expressed in lab coordinates The quasi-static hip moments are composed of three parts. The first part is the moment at the H] C due to the ground reaction force and torque (MR). This was calculated by computing the cross product of the moment arm of the HIC to the COP with the GRF and then adding the CRT. MR = (COP-HJC) x R(1s) + 0' RT”('ls) The second part of the moment at HJC is due to gravity acting on the thigh at its center of gravity (MT). This was calculated by computing the cross 56 product of the moment arm from HJC to the thigh center (CGT) with the weight of the thigh. MT = (CGT -' HJ'"C") x (—-wT)iE where, WT = 0.1 (BW) (the thigh is approximately 10% of body weight (Winter, 1990)) The third part of the moment at HJC is due to gravity acting on the shank/ foot complex at its center of gravity (MS). This was calculated by computing the cross product of the moment arm from HJC to the shank / foot center (CGS) with the weight of the shank and foot. M8 = (cos- HJC) x (-WS )1} where, W5 = 0.061(BW) (the shank/ foot is approximately 6.1% of body weight (Winter, 1990)) Therefore, the total moments at hip joint center in lab space are HIPMOM(ls) = MR + MT + MS 57 5. Joint Coordinate System and angle calculations The hip moments are expressed in lab space coordinates but, it is more practical to express them in the joint coordinate system at the hip joint. This allows the definition of the moments to apply to body axes (i.e. moment about y-axis is the flexion/ extension moment). A joint coordinate system was derived between the pelvis and thigh as suggested by Grood and Suntay (1983). The joint coordinate system (JCS) is formed by choosing two body axes (one from each body) and taking the cross product of them to create the third axis of the coordinate system (Figure 27). The body axes were chosen as 82 = iyp (y axis unit vector of pelvis) 63 = in (z axis unit vector of thigh) The common perpendicular to both body axes, called the floating axis, was 8 X“ definedas 61: Figure 27. Joint coordinate system 58 These three axes form a non-orthogonal joint coordinate system from which the three angles of the joint motion could be computed at each instance of time: flexion / extension = sin'1 (81 o ipz) {flexion(+)/extension(-) ] external / internal rotation = sin'1(é‘:1 C ity) (ext. rot. (+) for right side} (ext. rot (+) for left side } abduction / adduction = sin'l(é2 063) {abduction(+) for right } [abduction(-) for left } 6. Transformation of moments from lab space to joint coordinate space To analyze the moments as a driving force that produce angular displacements at the hip, the lab space moments were transformed to the joint coordinate system. Again, the joint coordinate system was defined as ) e1 = el + e1y + elz e2 = e2x + e2y + e22 e3 = 63 + e3 + e3 and the moments in lab space (HIPMOM(ls)) as Mx, My, and M2. The relationship between moment components in joint coordinate space with the moments in lab space were derived as follows: 59 M = M e x + M282" 4» M3e3x x l 1 My = Mle1y + M2°2y + M383y M2 = Mlelz + M2°22 + M3632 where M1, M2, and M3 are the moment components in joint coordinate space. Inmatrixform, PM 1 ' q "M 7 x 61x 62x 63x 1 My = °ly °2Y 3y M2 .MZ. -612 322 e3z. .M3. or [M(ls)] = A[M(jcs)] then, [M(jcs)] = A'1 [M(ls)] This 3X3 system of linear equations was solved by Cramer's rule. A‘1 = {—L)Ade detA where, det A = e1x(e2ye32 -e22e3y) - e’melye3z -elz°3y) + G3x(¢1y322 -elze2y) 60 and Ade is the transpose of the matrix of cofactors (mg) of aij in A and an = (-1i +j)detAij Therefore, . _ __1_ . [Mow] - (detA)AdJA[M(ls)] then, M M(' ) detA1 = cs-x = 1 J detA M M(' ) detA2 = C8- = 2 J y detA M M(' detA3 = cs-z = 3 J ) detA where, Mi: 62): GBA clx Mx c:31: 61x 32x Mx A1 = My e2y e3y A2 = C1}, My e3y A3 = e1), e2” My _Mz 622 632. _elz M2 e32, .612 622 M2. 7. Power With the angles defined, and the moments in joint coordinate space, power in the plane of progression (sagittal, flexion/ extension plane) and the true mathematical power at the hip during stance were computed. Power is defined as the rate of doing work. In biomechanics applications, it equals the dot product of the moment and the angular velocity of the segment, when describing the power generated or absorbed by the muscle activity at the joint. If the contribution from internal/ external rotation and ”'2‘...‘.lc"_-.lo ' 61 abduction/adduction were considered to be negligible, the 'power' in the plane of progression as a function of time is solved as: Power(t) = M2(t)-(0)fe(t)) (flex / ext moment)(angular velocity of flex/ ext at the hip) The angular velocity of the flexion/ extension at the hip ((Dfe(t)) was computed from the time history of the angular displacement. This was done by forward, central, and backward differentiation of the angle file. 30601)) + 4(fe(t2)) - fe(t3) 2(12 - 11) 0)fe (t1) = fe(i+l) - fe(i-l) 2(t mf¢(t151