JOURNAL OF APPLIED BIOMECHANICS. 2004, 20, 25-37 © 2004 Human Kinetics Publishers, Inc. A Computer Model to Simulate the Swing Phase of aTransfemoral Prosthesis Brendan Burkett^•^, James Smeathers\ and Timothy M. Barker-^ ''^Queensland University of Technology ^University ofthe Sunshine Coast For amputees to perform an everyday task, or to participate in physical exercise, it is crucial that they have an appropriately designed and functional prosthesis. Past studies of transfemoral amputee gait have identified several limitations in the perfonnance of amputees and in their prosthesis when compared with able-bodied walking, such as asymmetrical gait, slower walking speed, and higher energy demands. In particular the different inertial characteristics ofthe prosthesis relative to the sound limb results in a longer swing time for the prosthesis. The aim of this study was to detennine whether this longer swing titne could he addressed by modifying the alignment ofthe prosthesis. The following hypothesis was tested: Can the inertia! characteristics of the prosthesis be improved by lowering the prosthetic knee joint, thereby producing a faster swing time? To test this hypothesis, a simple 2-D mathematical model was developed to simulate the swing-phase motion ofthe prosthetic leg. The model applies forward dynamics to the measured hip moment ofthe amputee in conjunction with the inertial characteristics of prosthetic components to predict the swing phase motion. To evaluate the model and measure any change in prosthetic function, we conducted a kinematic analysis on four Paralympic runners as they ran. When evaluated, there was no significant difference (p > 0.05) between predicted and measured swing time. Of particular interest was how swing time was affected by changes in the position of the prosthetic knee axis. The model suggested that lowering the axis ofthe prosthetic knee could reduce the longer swing time. This hypothesis was confinned when tested on the amputee mnners. Key Words: mathematical modeling, biomechanics, gait The many differences between the atnputee's prosthetic limb and anatomical limb often result in a unique pattem of gait. Past studies on transfemoral amputee gait have identified several limitations in the overall function of amputees and ' School of Human Movement Studies, and -^School of Mechanical, Manufacturing & Medical Engineering, Queensland Univ. of Technology, Brisbane, Australia; ^ Faculty of Science, Univ. ofthe Sunshine Coast, Maroochydore DC, Queensland 4558, Austraha. 25 26 Burkett, Smeathers, and Barker prostheses when compared with able-bodied walking, such as asymmetrical gait, slower walking speed, and higher energy demands (Czerniecki, 1996; Macfarlane. Nielson, & Shurr, 1997; van der Linden, Solomonidis, Spence, Li, & Paul, 1999). Some of these limitations have been reduced by improved prosthetic components, such as the documented faster walking velocity and an improved swing-phase symmetry index that results when the amputee wears a pneumatic rather than a mechanical swing-phase control knee unit (Boonstra, Schrama, Eisma, Hof, & Fidler). However, despite the improvements in components, the transfemoral prosthesis was still found to limit the amputee's functional performance (Blumentritt, Scherer, Wellershaus, & Michael, 1997). Further studies on walking gait have shown that one of the most important limitations is the longer swing time of the prosthetic knee, which can be as much as 36% greater than for the anatomical knee {Boonstra, Schrama, Fidler, & Eisma, 1994; Hale, 1990; Jaegers. Arendzen, & de Jongh, 1995). During running, the amputee overloads the prosthesis, and any limitations in function become more obvious. Several studies of transtibial amputee running gait have foutid that as speed of gait increased, so did the interlimb asymmetry {Isakov, Burger, Krajnik, Gregoric, & Marincek, 1996). Ofthe few studies on transfemoral amputee running gait, it was found that when amputees transitioned from walking to running, the longer swing time of their prosthesis caused them to "hop-skip" when attempting to run (Mensch & Ellis, 1986). Transfemoral amputee gait has been found to he sensitive to the mass and mass moment of inertia of the prosthesis; this has become evident with advances in prosthetic design that have led to lighter prosthetic components (Gitter, Czerniecki, & Meinders, 1997). However, there is a paradox in that the inertial characteristics of the prosthesis can be improved by adding an appropriate mass rather than making the prosthesis lighter {Gitter et al., 1997; Selles, Bussmann, Wagenaar, & Stani, 1999). In other studies, greater mass has been added to the prosthesis to the extent that the mass and mass moment of inertia of the prosthesis approximate the intact limb (Mattes, Martin, & Royer, 2000). However, this approach has resulted in greater gait asymmetry and energy expenditure for the amputee. These contradictions highlight the need to correctly match the inertial characteristics of the prosthesis to the amputee. In attempting to achieve this, mathematical models have been developed to study the inertial characteristics of the prosthesis. In a study by Bach (1994), a mathematical model was developed to optimize the inertia ofthe prosthesis hy adding varying masses at several locations along the prosthesis. To match the amputee with his or her prosthesis, there is little that can be done to modify the amputee's residual limb. That is, the length ofthe residual limb is fixed, although hypertrophy or atrophy of the remaining musculature may occtir depending on the loading (Sanders & Daly, 1993). On the other hand, the prosthesis is an extemal mechanical device that can be lengthened, shortened, and otherwise modified. Traditionally, the axis of the prosthetic knee has been located on aesthetic criteria to provide anatomical symmetry. That is, the prosthetic knee unit has been designed so that when the amputee sits down, the artificial knee coincides with the anatomical knee. This focus on aesthetics is understandable, but there is no way of knowing whether such positioning of the prosthetic knee will automatically elicit optimal walking or running functional perfonnance of the prosthesis. For highly active Computer Model ofa Prosthesis 27 amputees it is becoming more common to develop a dedicated prosthesis for specific applications, such as the Sprint Flex®, to improve a transtibial amputee's running performance (Buckley, 2000). The inertial characteristics of such prostheses have been manipulated to enable and enhance the amputee's ability to function in specific physical activities. The aim of this study was to determine whether the longer swing time of the transfemoral prosthesis could be addressed by modifying the alignment of the prosthesis. It was hypothesized that this longer swing time could be remedied by adjusting the inertial characteristics ofthe prosthesis by moving the knee axis distally. Due to the considerable amount of time it can take to relocate the knee axis of the prosthesis, allowing the amputee enough time to adapt followed by reevaluation of this new alignment, the use of a mathematical model was considered the best way to explore optimum knee axis location. The development of such a model is the focus of this manuscript. Once the model identified the optimum knee axis location, the alignment of the prosthesis was modified accordingly and tested on amputees. In total, three modified knee alignments were physically tested while the amputees ran. Methods The participants were four Paralympic unilateral transfemoral amputees {see Table I), Ethics approval and informed consent was ohtained prior to testing. The study was divided into three stages: (a) baseline biomechanical measurement ofthe swing 1 Characteristics of Participants (all male) Particip. 1 Particip. 2 Particip. 3 Particip. 4 Age Height Weight Years since amputation Amputated side 33 yrs 1.83 m 70 kg 26 yrs 1.83 m 78 kg 35 yrs 1.85 m 76 kg 24 yrs 1.90 m 82 kg lyr Left 4 yrs Right 32 yrs Left 3yrs Socket type Rigid ischial ramal. suction suspensioti Otto Bock 3R44 Flexible ischial ramal. suction suspension Rigid ischial gluteal. suction suspension Otto Bock 3R20 ICEROSS ischial ramal. suction suspension Otto Bock 3R55 M + IND SLF135 Flex-Foot modular III Ktiee unit Foot type M-FIND SLF135 USMC 24800Black Max Flex-Foot Air-Rex VSP Right Note: Participants' average age = 29.5 ± 3.9 yrs; height = 1.85 ± 0.03 m; weight = 76.5 ± 5.0 kg; years since amputation = 12.5 ± 13.5 yrs. 28 Burkett, Smeathers, and Barker phase of the prosthesis when running; (b) development of a mathematical model; and (c) the intervention phase which involved modifying the prosthetic alignment as per the model's guidance and repeating the biomechanical measurements. The kinematic data were collected at 200 Hz using four high-speed 8-mm lens calibrated cameras (Motion Analysis Corp., Santa Rosa. CA). Reflective markers were placed on the following anatomical or similar prosthetic landmarks: antedor superior iliac spine, fifth lumber vertebra, greater trochanter, lateral femoral epicondyle. lateral malleolus, and distal end ofthe second metatarsal. From these markers the angular position, velocity, and acceleration of the hip and knee joint were derived. Temporal data were gathered using foot-switches (MIE Medical Research Ltd, Leeds, England) attached under each foot. Kinematic and temporal data collection were synchronized via a trigger beam placed on the runway in the biomechanics laboratory. For each participant, five trials were conducted for the prosthetic limb and five for the anatomical limb. From each of these trials the mean and standard deviation were calculated. This analysis allowed the model's predicted swing time to be compared and evaluated with the measured swing time. A r-test for matched pairs (p = 0.05) statistically compared the predicted with the measured swing time. The iower-limb model was developed using a combination of MATLAB software (Version 4) and Robotic Toolbox subroutines (Corke, 1996). Figure I identifies the flow diagram ofthe subroutines for the model. A multilink pin-jointed model using a manipulator matrix to define the arrangement of the link segments was developed to demonstrate how the various links move in relation to one another over time (Figure 2). The four links in this model identify the thigh, shank, and foot segments, as well as an orientation link that locates the model in space (this fourth link is an essential requirement of the software). Using forward dynamics, and Equation I shown below, the model integrates the matrix ofthe kinematics and dynamics to determine the position, velocity, and acceleration for each link in the serial model. q = M{qr'[t-C{q,q)q^Giq)-¥iq)] (1) where q is the vector of joint coordinates describing the pose of the manipulator; / = vector of joint torques; q = vector of joint velocities; q:= vector of joint accelerations; M = manipulator inertia tensor; C describes Coriolis and centripetal effects; F describes viscous and Coulomb friction; and G describes gravity loading. Development of the model also required the following subsections: (a) defining the assumptions ofthe model, (b) measuring the input data, and (c) evaluating the model. The model was based on a number of assumptions. Listed below are the general assumptions: • The location of each segment's center of mass remains fixed (with respect to the end of the limb) during movement. • The joints are considered to be hinge joints. • The length of each segment and the mass moment of inertia of each segment about its mass center remain constant during movement. • The only inputs to the model are extemally applied forces and internally generated forces. 29 Computer Model of a Prosthesis T0RQAPP3 DYN Contains manipulator matrix, defined using DenavitHartenberg convention Calculates the applied torque for a given joint angle and velocfty. Logmodel Defines initial link conditions, position and velocity. Instigates modelling. Accel Computes manipulator forward dynamics, retums • joint accelerations from actuator torque. FRICTION FDYN Applies frictional resistance of knee unit, defined in dyn matrix, when knee angle is less than four degrees. Integrates equation 1 using forward dynamics • Ankle acceleration is set at zero Retums time (t), position(q) vekwity (qd) Odetim Using Runge-Kutta formula solves equation 1 • Modification of ODE45 integrator routine • Uses constant increment of 0.1 Figure 1 — Flow chart for the major programs in the model. Burkett, Smeathers, and Barker 30 Unki - orientatbn link Unk2-thigh +) for knee flexion m.l Link 4-foot Figure 2 — Link model diagram, where mt, It, m^, I^, and mf, If are the respective mass and mass moment of inertia for the thigh, shank, and foot segments; M), and Mj^ are hip and knee moments, respectively. Link 1, orientation link, establishes the reference coordinate system. Link 2, thigh, includes the residual limb and the prosthetic socket from the brim of the unit through to the connection at the knee. Link 3. shank, includes the prosthetic knee unit and the i^hin connector through to the ankle. Link 4, foot, includes the foot unit and the participant's running shoe. The philosophy for developing the model was to start simple and increase the complexity until suitable predictions could be obtained. In doing so, the specific assumptions were as follows: • To prevent the prosthesis from hyperextending at the knee joint, the model applied a corrective flexion moment at this joint. This was used to represent the action of the stopper in the knee joint which prevents excessive extension. The value of this moment was such that it maintained the angle of the knee within 4° of full extension. • For each participant, an instantaneous hip moment was assumed to be applied hy the residual limb and surrounding musculature, regardless ofthe position of the knee axis. The rationale for this approach was based on the compound pendulum theory of human movement (Bach, 1994; Yang, Solomonidis, Spence, & Paul, 1991). Confiputer Model of a Prosthesis 31 • The axis of rotation of the prosthetic knee was assumed to remain constant, regardless of whether the knee was a polycentric or a single-axis design. It was considered unnecessary to incorporate the polycentric knee axis of rotation within the model, as the difference in axis of rotation of the single-axis knee when compared to the polycentric knee (Otto Bock 3R55) was under 10 mm. • The initial velocity for all links at toe-off was assumed to be zero, and toe clearance was not factored into the model. • Modeling was only in two dimensions, and the location and position ofthe hip remained fixed. This was considered a reasonahle assumption as the prosthetic knee essentially moves in two dimensions as it rotates ahout one axis. Movement in the third dimension, abduction and adduction ofthe hip, was small compared to the other planes. • The mass moment of inertia for each link remained constant, regardless of knee axis location, but the length of each segment and ofthe center ofmass changed accordingly. This was a reasonahle assumption since the mass of the connecting pipe used to alter the knee axis position accounts for less than 2% of the total mass of each link. However, the distance from the distal point of the link to the center of mass for that link was adjusted for each altered knee axis position. • The foot-shank link (ankle) remained fixed in a neutral position of 90° to the shank. The prosthetic foot does have a small range of dorsi- and plantar-flexion depending on the rigidity of the foot, hut outside of the flexion at the start of the swing phase, the ankle essentially remains fixed at 90°. This small range of movement was not considered necessary for this simple model. • The clinical definition of swing time is from toe-off to heel-strike. This research modeled the swing ofthe prosthesis and measured the swing time as the time taken for Link 2 (thigh) to swing from the extended start position to a defined position of flexion; and for Link 3 (shank) to reach a position where the flexion angle of the knee was within a zone of stability, as defined in the manufacturer's specifications. (The zone of stability is the region where the knee can flex while the prosthesis maintains support in the stance phase.) Limits were placed on maximum flexion and extension for the hip to coincide with the physiological range of movement. The input data were of three types: (a) the initial environment conditions, i.e., position in space; (b) the actuator moments, i.e., hip moment and the resistive prosthetic knee moment; and (c) the inertial matrix data, i.e., length, mass, distance to center of mass, and mass moment of inertia. The initial environment conditions were obtained from the biomechanical analysis for each participant, such as the start and finish position of the prosthetic hip. To measure the hip moment, the participant lay supine on the bed of an isokinetic dynamometer, set for concentric/concentric isokinetic movement at a maximum speed of l50°/sec (KinCom, Model AP, Chattanooga, TN). The participants rotated (flexed and extended) their residual limb at a cadence similar to when running, and the hip moment was recorded. This moment was then applied about the hip joint of the dynamic model to drive it forward from the initial angular position. To measure the damping moment ofthe prosthetic knee, we attached the proximal end ofthe knee to the bed ofthe dynamometer, and the distal end ofthe knee to the loading arm. The dynamometer was set in the passive movement mode, and the 32 Burkett, Smeathers, and Barker resisting moment of the knee was recorded as the knee moved from zero to 70° of flexion. This moment was then applied about the knee joint ofthe model. The inertial data included the mass, length, distance to center of mass, and moment of inertia of the individual prosthetic components. As all prostheses were of endoskeleton modular fabrication, each component was weighed and the center of mass was determined using a knife edge. The inertia for each component was measured using the trifilar suspension system (McKerran, 1991). To determine the inertial properties ofthe residual limh. we measured the volume ofthe socket. The mass, center ofmass, and mass moment of inertia were determined after assuming the limb to be a cylinder, with a soft tissue density of 1.05 kg/L (Dempster. 1955). The model calculated the following: swing time (sec), angular position (deg), and angular velocity (radians per second) for hoth the hip and knee. The simulations started at the standard knee-axis location, and the knee axis was progressively lowered by 10-mm increments until it was physically impossible to lower the knee axis any further. The model was evaluated in two ways. First the predicted swing time, with the knee axis in the standard prosthetic alignment, was compared to the measured swing time. Second, hased on the location suggested by the model, each participant's prosthetic knee axis was lowered to the position recommended by the model. Following a period of adaptation to this new axis location, the participants repeated the biomechanical analysis and the data were compared to the standard alignment. Two additional alignments were also tested: 50 mm above and 50 mm below the initial starting point suggested hy the model; these locations may vary slightly depending on the physical dimensions ofthe prosthetic components. Results An example of the input data {Table 2) as well as the predicted and measured swing time at the standard knee axis location for all four participants (Table 3) is presented. Using a /-test for matched pairs, we found no significant difference, p > 0.05, between predicted and measured swing times. A typical example (Participant 3) of the predicted effect that lowering the knee axis had on the pattem of knee flexion is shown in Figure 3. As the knee axis was lowered, the pattem of the predicted knee flexion angle shifted toward the left, or a faster swing time. For this participant the faster swing time came at the expense of developing an unwanted "bounce" at the +150-nim and +200-mm locations. A similar pattem was observed for all four participants, though the location of the bounce varied. Model simulation predicted a unique amount of axis lowering for each participant to produce the fastest swing time (Figure 4). Figure 4 indicates the fastest swing time for Participant I would occur when the knee axis was lowered 0.100 m. Any further lowering of the knee axis caused an increase in predicted swing time until the simulation was completed at 0.200 m. Based on this information from the model, the knee axis of Participant Vs prosthesis was lowered 0.100 m, and the biomechanical testing was repeated. Similarly, the fastest swing time for Participants 2, 3, and 4 was predicted to occur when the knee axis was lowered by 0.125,0.200, and 0.175 m, respectively. The mean measured anatomical and prosthetic swing time, as well as the model's predicted swing time at the standard and three different knee axis locations, are listed in Table 4. As seen in the table, the swing time for all participants improved as the knee axis was lowered. Computer Mode! of a Prosthesis 33 Table 2 Typical Data Input Variables Participant & Link A a Link Link twist (radians) length (m) Participant 1 1 2 3 4 Participant 2 1 2 3 4 Participant 3 1 2 3 4 Participant 4 1 2 3 4 m Mass (kg) Tx Izz Distance Inertia to COM (m) (kg-m2) Tc Friction momemt (Nm) ^72 0 0 0 0 0.410 0.515 0.240 0 3.350 0.785 1.030 0 -0.284 -0.260 -0.140 0 0.0264 0.0161 0.0059 0 0 10 0 ji/2 0 0 0 0 0.435 0.520 0.245 0 5.612 0.905 0.845 0 -0.165 -0.255 -0.145 0 0,0744 0,0039 0,0073 0 0 9 0 n/2 0 0 0 0 0.440 0.540 0,240 0 4.137 0.558 1.020 0 -0,335 -0,280 -0.140 0 0.0241 0.0033 0.0060 0 0 5 0 Ji/2 0 0.450 0.530 0.220 0 7.905 1.061 0.459 0 -0.310 -0.290 -0.170 0 0.0795 0.0171 0,0030 0 0 L5 0 0 0 0 Table 3 Model Swing Time vs. Measured Swing Time for Standard Knee Axis Location (± SD) Model prediction (s) Participant 1 Participant 2 Participant 3 Participant 4 0.420 0.495 0,420 0.450 Measured (s) 0.463 0.385 0.496 0.444 ±0.031 ±0,047 ±0,046 ±0.033 34 Burkett, Smeathers, and Barker 20 std +50 mm std+100 mm std +150 mm \V\\ \\;A 14 3 std +200 mm A - V \ 0) o \ 'x 3 • ] ^ ' ^—^ zone of slability (< 6) "^.'-fe-- 1 00 1 0.3 Time (s) Figure 3 — Predicted knee flexion as the axis of the knee is moved distally and the thigh rotates through its predeflned range of motioo. Participant 3. *> •- B -A -, -X- - Participant 1 Participant 2 Participant 3 Participant 4 Lowering of knee axis (m) Figure 4 — Predicted swing time for each participant as the knee axis was lowered. Computer Model of a Prosthesis 35 Tahle 4 Predicted Swing Time (sec) and Measured Anatomical and Prosthetic Swing Time at Different Knee Axis Locations Knee location Participant 1 Prosthetic Model Anatomical Std 0.463 0.420 0.499 Std -K).O5O m 0,423 0,405 0.462 Std+0.100 m 0,410*t 0.375 0.419* Std+0.150 m 0.418 0.465 0.452 Participant 2 Pro.sthetic Model Anatomical Std 0.385 0.495 0.332 Std 40.080 m 0,361 0,465 0,313 Std+0.125 m 0,314* 0.450 0.336* Std+0,180 m O.28Ot 0.480 0.303 Participant 3 Prosthetic Model Anatomical Std 0.496 0.420 0.478 Std+0.140 m 0,502 0.405 0.450 Std +0.200 m 0.428* 0.380 0.441* Std +0.240 m 0.383t 0.440 0.478 Participant 4 Prosthetic Modei Anatomical Std 0.444 0.450 0.396 Std +0,090 m 0.417 0.435 0.390 Std+0.130 m 0.411* 0.435 0.403* Std+0,175 m 0.349t 0.420 0.381 Note: Bold indicates predicted fastest swing time; * Best prosthetic/anatomical swing time symmetry; f Fastest measured prosthetic swing time. Discussion Development ofthe model required several simplifying assumptions, the rationale for which was discussed in the Methods section. Although relatively simple, the mathematical model proved useful in simulating the swing phase ofthe prosthesis, as there was no significant difterence between predicted and measured swing times at the standard knee axis location. We recognize that since there were only four participants in the study, this can reduce the power of the statistical analysis. When comparing the predicted and measured location for the fastest prosthetic swing (Tahle 4), the model accurately predicted the fastest location for Participants 1 and 4. For the other two participants, the measured fastest knee axis location was slightly lower than that predicted by the model. This difference in result is attributed to the simplicity and to the assumptions of the model, but based on the complete list of results, the model is considered useful and as having served its purpose. The development ofthe model, in conjunction with the measured results on the participants, produced two important findings. First, during running, the longer swing time of the prosthesis was reduced by lowering the axis of the knee; and second, the amputees were able to adapt and run on this unique prosthetic alignment. Together these findings can help establish future criteria for developing prostheses for specific applications, in this case running. To our knowledge, there have 36 Burkett, Smeathers, and Barker been no other documented studies on prosthetic research and development that have investigated this new alignment configuration. It should be noted that this prosthetic design is specific to running, as there are cosmetic restrictions. In addition, while the participants found the new design better for running, some daily activities such as sitting down on a chair proved more difficult due to the longer length of the prosthetic thigh. Additional features for future development couid include provision for the hip extension moment at the end ofthe swing phase, toe clearance, and incorporation of an initial velocity of the prosthesis at the start of swing (particularly due to the energy release ofthe prosthetic foot). Some of these modifications have been included in other studies, but no single model has incorporated all of these features. Another area for future research could combine the findings from other studies with those from this current research. For example, in this study the inertial characteristics ofthe prosthesis were optimized by lowering the axis ofthe knee. Other studies (Bach, 1994) have optimized the prosthesis by adding mass at a determined optimal mass location. It would seem logical to explore the effects of simultaneously moving the knee axis distally while adding appropriate mass to the prosthesis to see if this further optimizes the inertial characteristics ofthe prosthesis. Because of the differences in residual limb morphology, it would be extremely difficuit to match the inertial characteristics for each manufactured prosthesis to each individual amputee. However, the use of mathematical models may make it possible to group the characteristics of the amputees to a set of prosthetic components. Due to the success of lowering the knee for running, future developments could focus on designing a system that enables onsite adjustment of the prosthesis depending on the activity being pursued. In this case the amputee could have a prosthesis that is aligned in the standard configuration for walking and sitting, but which also allows the alignment to be changed onsite should he or she want to run. Such a design would not only avoid the cost of manufacturing two prosthetic limbs but also alleviate the need to carry a spare limb. The biomechanical model that was developed to simulate the swing phase of the transfemoral amputee prosthesis showed that lowering the location ofthe prosthetic knee axis reduced the swing time of the prosthesis. The model used in this research was simple and served the purpose. This confirmed our hypothesis when implemented and tested on four Paralympic amputee runners. References Bach, T. (1994). ,4 computer simulation of swing phase in human walking with application to optimisation ofinerllal characteristics of transfemoral prosthetics loads. Unpublished PhD dissertation. La Trobe University, Melboume. Blumentritt, S,. Scherer, H,W.. Wellershaus, U,. & Michael, J,W. (1997). Design principles. biomechanical data and clinical experience with a polycentric knee offering controlled stance phase knee flexion: A prehminary report. Joumal of Prosthetics and Orthotics.9. 18-24, Boonstra. A.M.. Schrama. J,M,. Eisma, W.H.. Hof, A.L.. & Fidier. E,V. (1996). 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