DEFORMABLE GROUND CONTACT MODELING: AN OPTIMIZATION APPROACH. Jeff Reinbolt. EML 5595 Mechanics of the Human Locomotor System Final Project Report

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1 DEFORMABLE GROUND CONAC MODELING: AN OPIMIZAION APPROACH Jeff Renbolt EML 9 Mechancs of the Human Locomotor System Fnal Project Report INRODUCION One of the most mportant choces made n creatng a mult-body dynamc model of human movement nvolves how to handle the foot-ground nterface. Ground contact models are essental to modelng changes n ground contact condtons (.e., sngle vs. double support and dfferent movement tasks (.e., gat vs. jumpng. here are two man optons for modelng ground contact: constrantbased models and 2 deformable models. Constrant-based methods confne the feet to the ground usng revolute or weld jonts. hs approach requres separate models for dfferent ground contact condtons. Constrant-based models permt the determnaton of constranng reacton forces and torques and facltate analyss of nduced acceleraton and nduced power. Deformable ground contact methods use sprngdamper elements, or unts, as constrants. hs approach allows the use of a sngle model regardless of changng ground contact condtons. Deformable models permt determnaton of dscrete forces that prevent the foot from excessvely penetratng the floor. However, these models complcate the nduced acceleraton and nduced power analyses. hs fnal project report presents the development of a deformable ground contact modelng approach to reproduce expermental ground reactons gven the expermental moton. Moreover, optmzaton technques were mplemented to determne a set of optmal ground contact model parameters. MEHODS he approach was based upon an exstng generc, parametrc three-dmensonal full-body knematc model constructed wth Autolev (Onlne Dynamcs, Inc., Sunnyvale, CA as a 4 segment, 27 degree-offreedom lnkage joned by a set of gmbal, unversal, and pn jonts (Appendx - Fgure. he foot ncluded a hndfoot segment lnked to a toes segment wth a weld jont. he foot-ground nterface was modeled wth four sprng-damper unts at the corners of the hndfoot and one unt at the dstal end of the toes (Appendx - Fgure 2, where these locatons are smlar to the Anderson and Pandy (999 model. he postons and veloctes of each sprngdamper attachment pont were computed as functons of the model s generalzed coordnates and patentspecfc model parameters (.e., segment lengths, jont postons, and jont orentatons. Gven the poston of a partcular attachment pont,, was known n the foot coordnate system, a seres of transformaton matrx multplcatons (Appendx Equaton was carred out to determne the attachment pont poston n the fxed coordnate system, N. Subsequently, the velocty of each attachment pont was computed n N as well. he poston and velocty nformaton was used to compute normal and tangental forces actng on each attachment pont. Comparable to Hunt and Crossley s (97 model for contact force between two colldng objects, each normal force was computed by a nonlnear equaton (Appendx Equaton 2. In addton, the tangental forces were computed usng Appendx Equaton 3, or Coulomb frcton, appled opposte to the drecton of sldng. he ndvdual attachment pont forces were replaced by an equvalent set of forces and torques actng at the electrcal center of the correspondng force plate for each foot model n N. he resultng three forces were smply the sum of each attachment pont force component (Appendx Equatons 4-6. he three torques produced about the force plate s electrcal center were also computed to complete the replacement (Appendx Equatons 7-9. An optmzaton framework was formulated wth Matlab (he MathWorks, Inc., Natck, MA to reproduce ground reactons gven the model s moton. he objectve functon was to mnmze the error between recorded and smulated ground reactons. he desgn varables were the ground contact model parameters. As opposed to a nonlnear least squares algorthm to solve for sprng stffness, k, sprng dsplacement rased to the power, n, dampng coeffcent, c, smultaneously n Appendx Equaton 2, a lnear least squares approach was used to solve for k alone, where n and c 0. hs smplfcaton allowed the optmzaton framework to be developed wthn the scope of the course project. Appendx Equatons 4-9 were combned nto a large lnear system of equatons (ncludng each tme frame of data to solve for k by nvertng the coeffcent matrx (rght-hand sde of Appendx Equatons 4-9 factored by k and post-multplyng by the recorded ground reactons to be matched. he ground contact optmzaton was used to solve fve example applcatons. o demonstrate the computatonal model was well desgned and wrtten, synthetc attachment pont postons and ground

2 reactons were mplemented wth unform and subsequently random k values for each unt. As an ntermedate step, random numercal nose was added to each synthetc data set. Fnally, expermental data recorded from a normal subject s gat was appled. Drvng the model through a moton smlar to gat generated four sets of synthetc ground reactons. By usng a unform k,974 N/m, one set of noseless synthetc data was created. By usng random k values wthn + 0% of the unform k, a second set of noseless data was produced. he thrd and fourth sets of synthetc data resulted from randomly varyng the attachment pont postons wthn + mm. he ablty of the ground contact optmzaton to recover the orgnal model parameter (synthetc only used to generate synthetc data and the recorded ground reactons was assessed wth normalzed (.e., untless root-mean-square (RMS errors for each category. he k RMS errors were normalzed by the unform k,974 N/m. he force RMS errors were normalzed by the body weght (BW 689 N. he torque RMS errors were normalzed by the product of body weght and heght (BW*H,73 Nm. RESULS he current ground contact optmzaton solutons decreased n accuracy wth ncreased nose ntroduced nto the data. For the noseless synthetc data cases, each ground contact optmzaton precsely recovered the orgnal model parameters and the recorded ground reactons to wthn an arbtrarly tght tolerance on the order of 2e-3% for k and 4e-4% for force and torque RMS errors (Appendx able frst and second data sets. For synthetc data wth nose cases, the RMS errors ncreased for both unform k and random k cases (Appendx able thrd and fourth data sets. For the expermental data case, the RMS errors were categorcally larger compared to the synthetc data cases (Appendx able ffth data set. Although t was not possble to assess the accuracy of the expermental k parameters, the k value of the posteror lateral unt was negatve for each foot (Appendx able 2. he largest RMS errors occurred for the superor force (20% BW and medolateral torque (7% BW*H, whch are much larger n recorded values compared to ther orthogonal counterparts. DISCUSSION hs fnal project report has presented the development of a deformable ground contact modelng approach to reproduce expermental ground reactons gven the expermental moton. hrough smplfcaton of a nonlnear sprng-damper model, a lnear least squares approach was used to determne ndvdual sprng stffness values for fve unts dstrbuted under each foot. Untng the foot-ground nterface model wth a full-body model wll facltate forward dynamc smulatons of human movement. Gven the scope of the course project, there are several lmtatons nherent to the current model. Fve out of ten expermental sprng stffness values were comparable to 40,000 N/m reported by Glchrst and Wnter (996. However, the other values were roughly twce ths magntude or negatve. he noseless synthetc data results were promsng. However, ntroducton of postonal nose (on the order of + mm sgnfcantly affected the recovery of sprng stffness values and ground reactons. In addton, the expermental RMS errors were consderably hgh. Knematc nose certanly affects the computed ground reactons; as a result, the k values found for expermental data may not be relable. Future work s necessary to answer several relevant questons related to model fdelty and relablty. It s necessary to determne the accuracy of the foot sole locaton and shape based upon skn markers. he bottom of the model foot segment s flat and does not deform as a true foot. he optmal number and locaton of sprng-damper unts s not known. Obvously, fve unts place on the outsde edge of the foot segment s not optmal. Excessve nterpenetraton of the foot edge wth the ground lowers the assocated sprng stffness, such as wth both current posteror lateral unts. Moreover, a smaller number of unts rase the resultng sprng stffness values compared to many more unts. In concluson, the current deformable ground contact model approach needs further development to determne trustworthy model parameters and ground reactons. REFERENCES. Anderson, F.C. and Pandy, M.G. (999 Computer Methods n Bomechancs and Bomedcal Engneerng 2, Glchrst, L.A. and Wnter, D.A. (996 Journal of Bomechancs 29, Hunt, K.H. and Crossley, F.R.E. (97 ASME Journal of Appled Mechancs, ACKNOWLEDGMENS hs study was funded by Whtaker Foundaton and NIH Natonal Lbrary of Medcne (R03 LM grants to Professor B.J. Fregly and t was motvated by the EML 9 course grade and Research Assstantshp to the author. he author gratefully recognzes Professor Fregly for hs project nput and support.

3 APPENDIX N Pelvs N Femur Pelvs ba Femur alus ba Foot alus Foot ( x { [( pz ( pz Fy + [( py ( py Fz } Equaton. he transformaton matrx multplcaton to compute the poston of sprng-damper unt attachment pont,, n the fxed coordnate system, N. F normal k x n ( + c x Equaton 2. he nonlnear sprng-damper normal force equaton as a functon of sprng stffness, k, sprng dsplacement, x, rased to the power, n, dampng coeffcent, c, and frst tme dervatve of sprng dsplacement, x. F tangent µ F normal Equaton 3. he Coulomb frcton tangental force equaton as a functon of frcton coeffcent, μ, and normal force. ( F x ( F Equaton 4. he sum of the forces for ndvdual attachment ponts,, n the x-drecton (anteror replaced by the equvalent force actng on the electrcal center,, of the correspondng force plate. x Equaton 7. he sum of the torques about the x- drecton (anteroposteror at the electrcal center,, of the correspondng force plate produced by forces (superor and lateral actng on ndvdual attachment ponts,. he poston of s gven by (p x, p y, p z and the poston of each s gven by (p x, p y, p z n the fxed coordnate system. ( y { [( pz ( pz Fx [( px ( px Fz } Equaton 8. he sum of the torques about the y- drecton (longtudnal at the electrcal center,, of the correspondng force plate produced by forces (anteror and lateral actng on ndvdual attachment ponts,. he poston of s gven by (p x, p y, p z and the poston of each s gven by (p x, p y, p z n the fxed coordnate system. ( z { [( py ( py Fx + [( px ( px Fy } Equaton 9. he sum of the torques about the z- drecton (medolateral at the electrcal center,, of the correspondng force plate produced by forces (anteror and superor actng on ndvdual attachment ponts,. he poston of s gven by (p x, p y, p z and the poston of each s gven by (p x, p y, p z n the fxed coordnate system. ( F y ( F y Equaton. he sum of the forces for ndvdual attachment ponts,, n the y-drecton (superor replaced by the equvalent force actng on the electrcal center,, of the correspondng force plate. ( F z ( F z Equaton 6. he sum of the forces for ndvdual attachment ponts,, n the z-drecton (lateral replaced by the equvalent force actng on the electrcal center,, of the correspondng force plate.

4 Fgure. he three-dmensonal, 4 segment, 27 degree-of-freedom full-body knematc model lnkage joned by a set of gmbal, unversal, and pn jonts.

5 Fgure 2. he foot coordnate system (top defned by skn-based markers placed over the heel, toe, and besde the toe jont. he smulated sole of the foot (mddle wth fve sprng-damper attachment ponts: posteror lateral (PL; posteror medal (PM; anteror lateral (AL; anteror medal (AM; and dstal toe tp (. he skn-based markers, the foot coordnate system, and each attachment pont are known n the laboratory fxed frame, N (bottom.

6 able. Root-mean-square (RMS errors between recorded and smulated ground contact model sprng stffness, k, and ground reactons for fve example applcatons: synthetc data wthout nose generated wth unform k values; 2 synthetc data wthout nose generated wth random k values; 3 synthetc data wth nose generated wth unform k values; 4 synthetc data wth nose generated wth random k values; and expermental data collected from a normal subject s gat wth unknown k values. Data ype Synthetc wthout nose wth unform k Synthetc wthout nose wth random k Synthetc wth nose wth unform k Synthetc wth nose wth random k k (% Unform k Anteror Force (% BW Superor Force (% BW RMS Error Lateral Force (% BW Anteroposteror orque (% BW*H Longtudnal orque (% BW*H Medolateral orque (% BW*H 2.3e-3 2.e e-4 2.2e e-.0e-6.89e e e e e e- 7.8e-7.48e Expermental n/a

7 able 2. Expermental sprng stffness, k, values determned by the ground contact optmzaton. Sde Left Rght Unt Locaton Posteror Lateral (PL k (N/m Posteror Medal (PM Anteror Lateral (AL Anteror Medal (AM oe p ( 63.2 Posteror Lateral (PL Posteror Medal (PM Anteror Lateral (AL Anteror Medal (AM oe p ( 36.6