Forward Dynamics for Gait Analysis as an Intermediate Step to Motion Prediction

Transcription

1 Forwar Dynamics for Gait Analysis as an Intermeiate Ste to Motion Preiction J. Cuarao, U. Lugris Lab. Ingenieria Mecanica, Escuela Politecnica Suerior University of La Coruña Ferrol, Sain R. Pamies-Vila, J.M. Font-Llagunes Det. of Mech. Eng. an Biomeical Eng. Research Centre Universitat Politecnica e Catalunya Barcelona, Sain rosa.amies@uc.eu, jose.m.font@uc.es Abstract Human motion reiction through comutational simulation can serve as a tool to anticiate the result of surgery or to hel in the esign of rosthetic/orthotic evices. The latter is the motivation in a roject being run by the authors, evote to the esign of an active stance-control knee-ankle-foot orthosis (SCKAFO) as an assistive evice for the gait of incomlete sinal cor injure (SCI) subjects. Otimization is a well-suite technique to tackle the human motion reiction roblem, an several aroaches have been roose in the literature. However, no matter which is the use aroach, the imlementation of these methos reresents a great challenge in terms of both convergence an efficiency. Therefore, the authors inten to firstly aress the analysis of a certain measure motion through forwar ynamics, which can be consiere as an intermeiate ste towars the reiction roblem, since it requires ynamical consistency too, but oes not suffer from the same high amount of uncertainty. Consequently, a systematic stuy of the ifferent alternatives to obtain, through forwar ynamics, the rive efforts at joint level that rouce a certain known motion is starte in this aer. Three moel-base control methos have been imlemente for the gait of a healthy subject, an their erformances have been comare. Keywors gait analysis; forwar ynamics; motion simulation; moel-base control I. INTRODUCTION Many sinal cor injure subjects with no ability to control their knee an ankle joints can walk with the hel of assistive evices, such as crutches an orthoses. The knee ankle foot orthoses (KAFO) commonly use by these atients lock the knee joint flexion extension in orer to bear their weight uring stance. The roblem of locking the knee is that the hi must be lifte in orer to leave enough clearance for the leg to comlete the swing, thus leaing to an energetically inefficient gait. There are several orthoses an knee articulation moules in the market which allow to unlock the knee flexion uring leg swing to imrove gait. However, they o not rovie the necessary torque to erform the flexion extension motion, an in many cases they require the atient to lock/unlock the knee manually at each ste, by means of a remote control evice. The eveloment of an active KAFO [1, ] with an actively controlle knee joint can greatly reuce the metabolic cost, which woul encourage the atients to choose walking instea of using wheelchairs, thus imroving their rehabilitation. Fig. 1. Authors ongoing roject: eveloment of active SCKAFO. The authors ongoing roject is evote to the eveloment of such an active stance-control knee-anklefoot orthosis (SCKAFO) [3], shown in Fig. 1. The gait of atients wearing their assive orthoses is analyze by means of an exerimental setu for measuring gait ata, which is fe into a multiboy comutational moel that calculates the joint motor torques through inverse ynamics [4]. The obtaine kinematic an kinetic information is useful for the esign of the SCKAFO controllers. However, a most owerful tool woul be that caable of answering the following question: how woul this atient walk if he wore the SCKAFO we have eveloe with a certain esign of the controller? Or, in other wors, a tool caable of roviing the atient s motion reiction. With such a tool, the SCKAFO controllers coul be tune once an again until a satisfactory behavior of the atient s comutational moel woul be achieve in the simulation. The roblem of human motion reiction has been aresse in the literature, an is currently a toic of intensive research. Otimization has been the referre technique, where the cost function is usually base on hysiological criteria. Three main basic aroaches have been use in the literature [5]: a) To consier the arameters efining the motion as the esign variables, obtaining the rive efforts that 353

2 originate the corresoning motion through inverse ynamics [6]. b) To consier the arameters efining the rive efforts as the esign variables, obtaining the motion that results from the alie efforts through forwar ynamics [7]. c) To consier both the arameters efining the motion an the rive efforts as the esign variables, an incluing the relations among them (equations of motion) as constraints of the otimization roblem [8]. However, no matter which is the use aroach, the imlementation of these methos reresents a great challenge in terms of both convergence an efficiency. On the other han, the analysis of a real cature motion can be aresse by means of either inverse or forwar ynamics [9]. While inverse ynamics looks at each iscretize instant of time searately, forwar ynamics imlies by nature the ynamically consistent solution over the full erio of motion, which is in better agreement with the oeration rules of the musculoskeletal system. Therefore, the analysis of a certain measure motion through forwar ynamics can be consiere as an intermeiate ste towars the reiction roblem, since it requires ynamical consistency too, but oes not suffer from the same high amount of uncertainty. Consequently, a systematic stuy of the ifferent alternatives to obtain, through forwar ynamics, the rive efforts at joint level (not muscular yet) that rouce a certain known motion is starte in this aer. The stuy focuses on gait, since this is the relevant motion for the authors ongoing roject, as exlaine before. The literature review showe that both control an otimization aroaches have been use. Here, three moel-base control methos have been imlemente for a healthy subject an their erformances have been comare. The remaining of the aer is organize as follows. Section escribes the exeriment, the multiboy moel of the healthy subject an the signal rocessing technique alie to the cature ata. Section 3 gathers the multiboy ynamics formulation. Section 4 exlains the three methos that have been use to erform the forwar ynamics analysis, shows the obtaine results, an rocees to their iscussion. Section 5 raws the conclusions of the work an suggests the following research to be aresse. II. EXPERIMENT AND MODEL The subject selecte to erform the exeriment is a healthy ault male, 34 years ol, mass 85 kg an height 1.8 m. He walks on a walkway featuring two embee force lates (AMTI, AccuGait samling at 1 Hz). The motion is cature by 1 otical infrare cameras (Natural Point, OtiTrack FLEX:V1 also samling at 1 Hz) that comute the osition of 37 otical markers. The human boy is moele as a 3D multiboy system forme by rigi boies, as shown in Fig.. It consists of 18 anatomical segments: two hinfeet, two forefeet, two shanks, two thighs, elvis, torso, neck, hea, two arms, two forearms an two hans. The segments are linke by ieal sherical joints, thus efining a moel with 57 egrees of freeom. The global axes are efine as follows: x axis in the antero osterior irection, y axis in the meio lateral irection, an z axis in the vertical irection. The comutational moel is efine with 8 mixe (natural + angular) coorinates. The subset of natural coorinates comrises the three Cartesian coorinates of oints, an the three Cartesian comonents of 36 unit vectors, thus making a total of 174 variables. The oints correson to the ositions of all the sherical joints (white ots in Fig. ), along with oints of the five istal segments -hea, hans an forefeet- (black ots in Fig. ). Each one of the 18 boies is efine by its roximal an istal oints, lus two orthogonal unit vectors aligne at the antero osterior an meio lateral irections, resectively, when the moel is in a staning osture. The remaining 54 variables are the 18 sets of 3 angles that efine the orientation of each boy with resect to the inertial frame. The geometric an inertial arameters of the moel are obtaine, for the lower limbs, by alying correlation equations from a reuce set of measurements taken on the subject, following the roceures escribe in [1]. For the J 7 J 4 J 5 J 6 J 8 J 1 J 1 J 11 J 9 J J3 J 15 J 1 Z Y X J 16 J 13 J 14 J 17 Fig.. (Color online) Human multiboy moel. 354

3 uer art of the boy, ata from stanar tables [9] is scale accoring to the mass an height of the subject. In orer to ajust the total mass of the subject, a secon scaling is alie to the inertial arameters of the uer art of the boy. The kinematic information of the motion is obtaine from the trajectories of the 37 markers attache to the subject s boy (re ots in Fig. ), which are cature at 1 Hz frequency by means of the 1 infrare cameras. Position ata are filtere using an algorithm base on Singular Sectrum Analysis (SSA) an the natural coorinates of the moel are calculate using algebraic relations. Afterwars, a minimization roceure ensures the kinematic consistency of the natural coorinates. From that information, the histories of a set of 57 ineenent coorinates -as many as the system egrees of freeomforme by the Cartesian coorinates of the osition vector of the lumbar joint an the 18 x 3 angles that efine the absolute orientation of each boy, are kinematically obtaine an aroximate by using B-sline curves. Analytical ifferentiation yiels the corresoning velocity an acceleration histories. More etail about the treatment of the cature ata can be foun in [4]. III. MULTIBODY FORMULATION The ynamics of a multiboy system can be escribe by the constraine Lagrangian equations, Mq ɺɺ Φ λ Q T + q = Φ= which constitute a set of ifferential-algebraic equations (DAE), where M is the ositive semiefinite mass matrix, qɺɺ the accelerations vector, Φ the constraints vector, Φ the Jacobian matrix of the constraints, λ the Lagrange multiliers vector, an Q the forces vector. In this work, the formulation in minimum number of coorinates roose in [11] an calle matrix-r formulation has been use. The starting oint is to establish the following relation between eenent qɺ an ineenent zɺ velocities: (1) qɺ = Rzɺ () In the resent case, vector q is forme by all the 8 roblem variables, while vector z is forme by the subset of 57 ineenent coorinates (Cartesian coorinates of the osition vector of the lumbar joint an orientation angles of all the anatomical segments) alreay mentione in the revious section. Relation () can always be foun: each column of matrix R is nothing but the system velocities qɺ when a unit value is given to one of the ineenent velocities zɺ an the others are set to zero. The system velocity analysis can be carrie out by means of the time erivative of the constraints equation, Φqqɺ =. Differentiating () with resect to time yiels, q qɺɺ = Rz ɺɺ + Rz ɺ ɺ (3) where vector Rz ɺ ɺ can be obtaine by means of a system acceleration analysis with all the ineenent accelerations zɺɺ set to zero. The system acceleration analysis can be carrie out through the secon time erivative of the constraints equation, Φ qɺɺ = Φɺ qɺ. q T Now, substituting (3) into (1), remultilying by R, an taking into account that ΦqR =, the following system of orinary ifferential equations (ODE) is achieve, or, in a more comact form, q ( ɺ ) ɺɺ = ɺ (4) T T R MRz R Q MRz Mz ɺɺ = Q (5) where M an Q are, resectively, the mass matrix an force vector rojecte to the minimum set of coorinates z. Therefore, the result is that the DAE system (1) exresse in eenent coorinates q has been converte into the ODE system (5) exresse in ineenent coorinates z. In the resent work, the ODE system (5) has been numerically integrate in time by means of the single ste, fixe time ste, traezoial rule. IV. TESTS AND DISCUSSION As exlaine in the Introuction, a systematic stuy of the ifferent alternatives to obtain, through forwar ynamics, the rive efforts at joint level that rouce a certain known gait motion is starte in this aer. Initially, the gait of the healthy subject escribe at the beginning of Section is cature with the infrare cameras, an the corresoning groun contact forces are measure through the forces lates. An inverse ynamics analysis (IDA) is then carrie out to etermine the reactions an joint torques that rouce the motion. The measurements coming from the force lates are only use to overcome the reaction ineterminacy arising uring the ouble-suort hase: the measure reactions are correcte so that their force an moment resultants are coincient with those rovie by the IDA, the iscreancy being slit roortionally to their absolute values. To rovie a certain valiation, Fig. 3 shows the net torques in the lower limb joints (sagittal lane) obtaine from the IDA, an comares them with the average torque (ashe line) boune by the stanar eviation (grey area) resente by Winter in [1]. The recore motion contains more than one cycle. It starts at the heel strike of the right foot (% of gait cycle), inclues the next heel strike of the same foot (1%), an finishes at the toe-off of the left foot belonging to the next cycle (arox. 116%). In what follows, the elvis is consiere as the base boy of the kinematic chain an, then, the reactions are the 355

4 Ankle Moment [Nm/kg], Ankle Moment [Nm/kg], 1, 1,,, 1, 1, Knee Moment [Nm/kg], Knee Moment [Nm/kg], 1, 1,,, 1, 1, Hi Moment [Nm/kg], Hi Moment [Nm/kg], 1, 1,,, 1, , Winter average moment ± St. Desv. IDA Fig. 3. (Color online) Comarison with Winter s results. force an moment acting on the elvis, while the rive torques are the absolute torques unergone by the remaining anatomical segments. Such results can be transforme, whenever necessary, to the reaction in the suorting foot (or to the reactions in the two suorting feet uring ouble suort, as exlaine above) an the actual rive torques at the joints. Once the IDA has been carrie out an the reaction force an moment have been etermine along with the rive torques, the objective is to obtain the same reaction an rive torques by means of a forwar ynamics analysis (FDA). For this urose, the three following moel-base control methos have been imlemente: a) The first metho consists of using as inuts of the FDA the reaction an torques obtaine from the IDA. Ieally, the solution shoul be coincient with the original cature motion but, as ointe out in the literature, it is not ue to the unstable character of human gait an to the integration errors. Initially, a time ste of 1 ms was aote for the FDA, but the simulation was comletely unstable. Then, the time-ste size was reuce to 1 ms. Since the IDA ha been erforme at 1 Hz, aitional oints ha to be generate, which was straightforwar as B-slines ha been ajuste to the aquire motion, as exlaine in Section. Using the time ste of 1 ms, the FDA was able to rerouce the motion until the 9% of the gait cycle an then rifte away, as illustrate in Fig. 4, where the ankle, knee an hi angles in the sagittal lane are lotte for both legs. b) The secon metho consists of using as inuts of the FDA the reaction an torques obtaine from the IDA (as in the first metho), but inclues a roortional-erivative (PD) control of reaction an torques so as to follow the measure motion an avoi instabilities. The reference signals of the controllers are the measure time evolution of the ineenent coorinates z, that is, the osition of the lumbar joint an the three angles efining the absolute orientation of each segment. The error an the error time erivative of each controller have the following exressions: e = z z, eɺ = zɺ zɺ, i = 1,...,57 (6) ref ref i i i i i i 356

5 Ankle Joint Angle [eg] Knee Joint Angle [eg] Hi Joint Angle [eg] Ankle Joint Angle [eg] Knee Joint Angle [eg] Hi Joint Angle [eg] IDA FDA Fig. 4. (Color online) IDA results as inuts: FDA rifts away at 9% of gait cycle. Therefore, an actuator associate to each ineenent coorinate is inclue for the FDA, its generalize force being, f, 1,...,57 i = kp iei + kd ieɺ i i = (7) The gains k Pi an k Di for each actuator were ajuste by trial an error, an the results were shown to be quite sensitive to the values of the gains. Table 1 gathers the selecte gain values. It can be seen that all the gains have been exresse as functions of two basic arameters, K an K. Moreover, the gains are roortional to the mass of the corresoning link, m j. As it haene for the first metho, if a time ste of 1 ms is use the simulation is comletely unstable. However, using again a time ste of 1 ms, the FDA was able to rerouce the entire motion. Fig. 5 shows the ifference between the motion yiele by the FDA an the measure motion. The uer lot is evote to the three Cartesian coorinates of the osition vector of the lumbar joint 7 (osition errors are in the orer of 1 m), while the bottom lots refer to the thigh, shank, hinfoot an forefoot of the left an right legs, resectively (angular 5 errors are in the orer of 1 ra). Analogously, Fig. 6 shows the ifferences at force/torque level. In this case, errors in the reaction force at the lumbar joint are in the orer of 1 N, while errors in the torques of the lower 4 limbs are in the orer of 1 Nm. TABLE I. SELECTED GAINS FOR THE PD CONTROLLERS kpi K = 35 ; K = 1 m Lumbar joint force Segment torques x y z K 6K 8K j k Di m K K 8K.9K.3K c) The thir metho consists of using only, as inut of the FDA, the actuation rovie by the so-calle comute torque control (CTC) [13]. The reference signals of the controllers are the same as those alreay exlaine for the revious metho. In this case, the equations of motion (5) can be written as, j 357

6 5 1 7 x y z Lumbar joint [m] [ra] [ra] 4 Hinfoot Hinfoot Fig. 5. (Color online) Differences between FDA an IDA results at osition level (n metho)...1 x y z Lumbar joint [N] [Nm] [Nm] 1.5 Hinfoot Hinfoot Fig. 6. (Color online) Differences between FDA an IDA results at force/torque level (n metho). Mz ɺɺ = Q+ Q (8) where the vector of generalize forces ue to the controllers takes the form, CTC being c an ref ( ɺ ) Q = M ɺɺ z + c e + c e Q (9) CTC c iagonal matrices containing the gains z, e an eɺ the associate to each ineenent coorinate i 358

7 5 1 6 Lumbar joint [m] x y z [ra] [ra] Hinfoot Fig. 7. (Color online) Differences between FDA an IDA results at osition level (3r metho). Hinfoot 1.5 x y z Lumbar joint [N] [Nm].5 Hinfoot [Nm] Hinfoot Fig. 8. (Color online) Differences between FDA an IDA results at force/torque level (3r metho). vectors of error an error time erivative, resectively, as ref efine in (6), an zɺɺ the accelerations corresoning to the measure motion. As exlaine in [13], the error ynamics of this control metho is reresente by a system of secon orer ifferential equations, having c an c as coefficients of the roortional an first erivative terms, resectively. Therefore, if critical aming is esire, the following relation between the gains shoul be imose, c = c (1) i i 359

8 which means that only one of the two sets of coefficients must be ajuste. In this work, the c gains were consiere as ineenent, while the obtaine by alication of (1). c gains were Unlike the revious metho, which require the articular tuning of each gain (as shown in Table 1), the CTC metho allows giving the same value to all the c i elements, since each of them is affecte by the corresoning generalize mass, as inicate in (9). Moreover, the metho showe to be extremely robust, keeing the simulation stable for the wie range of gain values teste ( c i was set to values of ifferent magnitue orers, ranging from to 1 ). This time the controller was able to comlete the FDA analysis with a time ste of 1 ms, which further confirms the robustness of the metho. The obtaine results were quite accurate at osition level, but showe relevant errors at force/torque level, i.e. the reaction an torques yiele by the FDA were not in goo agreement with those rovie by the IDA. Using a time ste of 1 ms, the FDA was carrie out for the ifferent values of c i inicate above. It was observe that the accuracy of the CTC metho increases linearly with the value of c i (some noise aeare in the results for values of c i greater than 3 1 ). Setting 3 c i = 1, Fig. 7 shows the ifference between the motion yiele by the FDA an the measure motion, while Fig. 8 shows the ifferences at force/torque level. The format of these figures is the same as that followe for the revious metho in orer to streamline comarison. As it can be seen in the figures, osition errors are in 6 the orer of 1 m an angular errors are in the orer of 4 1 ra, while errors in the reaction force at the lumbar joint are in the orer of 1 N an errors in the torques of the 1 lower limbs are in the orer of 1 Nm. Although this accuracy is lower than that rovie by the revious metho, the accuracy of the CTC metho can be increase by simly increasing the value of c i, as ointe out before. V. CONCLUSIONS AND FUTURE RESEARCH Three methos have been teste in this aer to obtain, through forwar ynamics, the rive efforts at joint level that rouce a certain known gait motion. The first one consists of using as inuts of the forwar ynamics analysis (FDA) the reaction an torques obtaine from a revious inverse ynamics analysis (IDA). This metho is not able to rerouce the motion ue to the integration errors an the unstable character of human gait, which make the simulation rift. The secon metho as a PD controller to the IDA reaction an torques, allowing to simulate the comlete motion after a careful tuning of the PD gains. The last metho oes not require the IDA results, but imlements a comute torque controller instea. In this case, the arameters to be tune are reuce to one single gain, an the metho is robust with resect to both the gain an the time ste values, roviing higher accuracy for increasing gain values. Therefore, it is erceive as an excellent metho for the ursue objective. The next stes in this research, which has the final objective of eveloing motion reiction methos, will be oriente in two irections. On the one han, control-base methos shoul be teste for the case of incluing footgroun contact moels in the simulation. On the other han, otimization-base methos, in which the arameterize inut forces are the esign variables an the measure motion is the cost function, shoul be teste in orer to get exerience for the motion reiction challenge. ACKNOWLEDGMENT The suort of this work by the Sanish Ministry of Economy an Cometitiveness (MINECO) uner roject DPI C3-1, confinance by the Euroean Union through EFRD funs, is greatly acknowlege. REFERENCES [1] A.M. Dollar an H. Herr, Lower extremity exoskeletons an active orthoses: Challenges an state of the art, IEEE Transactions on Robotics, vol. 4, no. 1, , 8. [] T. Yakimovich, E.D. Lemaire, an J. Kofman, Engineering esign review of stance control knee ankle foot orthoses. Journal of Rehabilitation Research an Develoment, vol. 46, no.,. 57, 9. [3] J.M. Font-Llagunes, R. Pamies-Vila, F.J. Alonso, U. Lugris, Simulation an esign of an active orthosis for an incomlete sinal cor injure subject, IUTAM Symosium on Human Boy Dynamics: From Multiboy Systems to Biomechanics, Proceia IUTAM, vol., , Waterloo, Canaa, June 5-8, 11. [4] U. Lugris, J. Carlin, A. Luaces, J. Cuarao, Gait analysis system for sinal cor injure subjects assite by active orthoses an crutches, Journal of Multi-boy Dynamics, in ress. [5] Y. Xiang, J.S. Arora, K. Abel-Malek, Physics-base moeling an simulation of human walking: a review of otimization-base an other aroaches, Structural an Multiiscilinary Otimization, vol. 4,. 1-3, 1. [6] L. Ren, R.K. Jones, D. Howar, Preictive moelling of human walking over a comlete gait cycle, Journal of Biomechanics, vol. 4, , 7. [7] F.C. Anerson, M.G. Pany, Dynamic otimization of human walking, Journal of Biomechanical Engineering, vol. 13, , 1. [8] Y. 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