Derivative-Free Kalman Filtering-Based Control of Prosthetic Legs*

Transcription

1 7 Amercan Control Conerence Sheraton Seattle Hotel May 4 6, 7, Seattle, USA Dervatve-Free Kalman Flterng-Based Control o Prosthetc Legs* S Mahmoud Moosav, Seyed Abolazl Fakooran, Vahd Azm, Hanz Rchter and Dan Smon Abstract A dervatve-ree method or state estmatonbased control o a robot/prosthess system s presented The system s the combnaton o a test robot that emulates human hp and thgh moton, and a powered transemoral prosthetc leg The robot/prosthess combnaton s modeled as a three degreeo-reedom (DOF) robot: vertcal hp dsplacement, thgh angle, and knee angle We develop a dervatve-ree Kalman lter (DKF) or state estmaton-based control or an n-dof robotc system We then propose a method to make the DKF robust when the robot dynamcs nclude dsturbances In the robust DKF, we use two derent methods or dsturbance rejecton: PD and PI These dsturbance compensators are used or supervsory control to make the DKF robust n the presence o dsturbances The smulaton results show the advantages o the DKF and the robust DKF or the three-dof robot/prosthess system or state estmaton-based control I INTRODUCTION The desgn and development o transemoral leg prostheses has receved consderable attenton due to the ncreasng number o above-knee amputees Recent advances n mcroelectroncs and robotc technologes have led to the development o new powered prosthetc legs Transemoral amputees who use powered (actve) prosthetc legs expend less energy compared wth those who use passve prostheses snce powered prostheses are able to generate net power at the jonts [] Some recent research on the development o lower lmb prostheses wth powered jonts ncludes [], [3], [4], [] Robotc testng n the sagttal plane o transemoral prostheses s presented n [6], [7], where nonlnear control technques are proposed on a test robot combned wth a prosthetc leg Passvty-based robust control, robust adaptve mpedance control, hybrd control, and robust composte adaptve mpedance control are some recent attempts or the control o a three degree-o-reedom robot/prosthess system [7], [8], [9], [] The proposed methods have demonstrated good trackng perormance n the presence o parameter uncertantes and ground reacton orce (GRF) as an external dsturbance However, GRF s treated n the aorementoned research as an nput wth known dynamc propertes and known bounds Thus, trackng perormance may break down the system s aected by dsturbances In ths paper we propose observer-based prosthess controllers n order to address ths shortcomng Note that unknown *Ths work was supported by NSF Grant Department o Electrcal Engneerng and Computer Scence, Cleveland State Unversty, Cleveland, OH, USA Correspondng author e-mal address: smoosav7@csuohoedu Department o Mechancal Engneerng, Cleveland State Unversty, Cleveland, OH, USA nputs can also be consdered as dsturbances n the context o ths paper A dervatve-ree Kalman lter (DKF) s a new method or estmaton-based control loop whch uses exact eedback lnearzaton to control nonlnear systems [] The DKF s adapted and mplemented n ths paper or estmatonbased control o a class o nput-output lnearzable nonlnear systems The nonlnear system s lnearzed and controlled by exact eedback lnearzaton, and then the standard Kalman lter s appled to the lnearzed model to estmate the state vector However, unlke the extended Kalman lter (EKF), the DKF provdes state estmaton o the nonlnear system wthout on-lne dervatve and Jacoban calculatons The eedback lnearzaton step s nstead part o the problem ormulaton, and so on-lne dervatve calculatons are not necessary Thereore, the problems o the EKF whch result rom the local lnearzaton o the nonlnear system by rst-order Taylor seres expansons can be avoded n the DKF These EKF problems can aect the accuracy o the state estmaton and consequently the stablty o the state estmaton-based controller, and are allevated wth the DKF The man contrbuton o ths paper s the desgn o a DKF or a leg prosthess test robot that s used or transemoral prosthess desgn and test Ths research s the rst attempt or closed-loop estmaton-based control or the robot/prosthess system usng a DKF We rst consder an n-dof robot and desgn a DKF when the robot s model s known n parameters, nputs, and GRF dynamcs Then we examne the case n whch the robot model s perturbed wth dsturbances as parametrc uncertantes or GRF In both cases nose terms are modeled n an ane state space model or the robot The dsturbances are assumed to be bounded but ther bounds and dynamcs are unknown System perormance may degrade when the system s perturbed by dsturbances snce the DKF s based on eedback lnearzaton and the accuracy o exact eedback lnearzaton deterorates under dsturbances Thereore, the new contrbutons n ths paper nclude the use o two methods, namely PD and PI dsturbance compensators, as supervsory control terms or the rejecton o dsturbances The use o these methods or dsturbance rejecton n the DKF s proposed or the rst tme n ths paper These methods combne wth the DKF to ncrease robustness The combned DKF and dsturbance compensator s called the robust DKF snce t can tolerate dsturbances or estmaton-based control We consder a smulated three-dof prosthess test robot to show the results o the DKF; the DKF s used or onlne estmaton-based control o hp dsplacement, thgh angle, knee angle, and correspondng veloctes /$3 7 AACC

2 The paper s organzed as ollows In Secton II the DKF s desgned or estmaton-based control or general robotc systems, under some assumptons In Secton III methods or dsturbance rejecton are presented to make the DKF more robust n the presence o dsturbances Secton IV provdes smulaton results o the DKF on a three-dof robot/prosthess system and analyzes the perormance o the DKF n the presence o dsturbances Secton V concludes the paper and suggests uture work II DERIVATIVE-FREE KALMAN FILTERING FOR ROBOTS The Kalman lter apples drectly only to lnear systems However, we can lnearze a nonlnear system and then use lnear estmaton technques Applyng the standard Kalman lter to a nonlnear system through the transormaton o the nonlnear system to the observer canoncal orm s called dervatve-ree Kalman lterng (DKF) The lnearzaton transormaton o the nonlnear system s based on a deomorphsm and, unlke the EKF, does not nvolve the computaton o Jacobans The DKF can be used or estmaton-based control o robots the robot model s subjected to a lnearzaton transormaton or exact eedback lnearzaton control and then state estmaton s perormed on the lnearzed model wth a standard Kalman lter A dynamc model or a mult-nput mult-output (MIMO) n-dof robot s gven as M (q) q +C (q, q) q + J e T F e + g(q) = u () where q s the vector o jont dsplacements, M (q) s the mass matrx, C (q, q) s a matrx accountng or centrpetal and Corols eects, J e s the knematc Jacoban relatve to the pont o the applcaton o external orces F e, g(q) s the gravty vector, and u s the vector o control sgnals [6], [7], [], [] The dynamc model () can be wrtten n ane state space orm whle also ncludng nose terms n the process and measurement equatons as ollows: ẋ = (x,t) + g(x,t)u + ω(t) y = h(x,t) + ϖ(t) () where state vector x(t), nput u(t), and output y(t) are n R n, R p and R m respectvely; and the smooth vector elds () and g() are n R n and R n p respectvely The nose terms ω(t) and ϖ(t) are assumed to be bandlmted, zero-mean, and uncorrelated Note that the process nose s partally due to parameter uncertantes, unmodeled dynamcs, and nput uncertantes n the robot model, and the measurement nose corresponds to errors n the measurement equpment The nonlnear system () s subjected to a lnearzaton transormaton to obtan an observer canoncal orm n order to perorm dervatve-ree Kalman lterng (DKF) [] In ths secton the nput-output eedback lnearzaton s presented to lnearze the model n () Snce t s desred to obtan an nvertble decouplng matrx n the lnearzaton transormaton, we assume that the number o nputs s equal to the number o the outputs; that s, p = m The robot model s nput-output lnearzable under the condtons gven n [3, Chapter 6] The classcal nput-output lnearzaton method can be appled by derentatng each output uncton y ( =,,,m) untl the nputs appear Remark: We assume that the lat output o the system [4, Chapter ] s a lnear uncton o the state vector elements In ths case the appled transormaton does not have nonlnear eects on the nose sgnals We also assume that we know the mean and covarance o the nose so that we can calculate the mean and covarance o the derentated nose [, Chapter 9] Assume that r s the smallest nteger such that at least one o the nputs appears n y (r ) ; then y (r ) (t) = L r h (x) + m j= L g j L r h (x)u j (t) + γ (t) (3) where L h s the Le dervatve L h = ( h), wth L g j L r h It can be shown that γ (t) s nose wth zero mean: γ (t) = ω (t) + d(r ) ϖ (t) (4) dt Note that one o the man advantages o transormng the robot dynamcs () nto the ane state-space orm () s to allow treatment o the nose terms n the deomorphsm Based on (3) and exact eedback lnearzaton, the system model () (gnorng nose or now) can be transormed nto the ollowng orm [3, Chapter 6]: v (t) L r h (x) = + Γ(x)u(t) () v m (t) L r m h m (x) where Γ(x) s the m m nvertble decouplng matrx and the scalar r = r + r + + r m s called the total relatve degree o the system The vrtual control term v (t) wll be desgned later n ths paper The partal relatve degrees r are called well-dened L g j L r h In ths case, the nput transormaton v (t) L r u(t) = Γ (x) h (x) v m (t) L r m h m (x) (6) yelds m smple lnear systems rom (3) n new coordnates: Ż (t) = Z (t) Ż r (t) = y (r ) (t) = v (t) + γ (t) (7) or =,,,m Now we can wrte each decoupled subsystem n state space orm: Ż (t) = A Z (t) + B v (t) + M γ (t) y (t) = C Z (t) + ϖ (t) (8) 6

3 We can then wrte each state space equaton n canoncal orm as ollows: A Ż (t) {}}{ Z Ż (t) (t) {}}{{}}{ = Z (t) + v (t) + γ (t) Żr (t) Zr (t) y (t) = [ ] Z (t) + ϖ (t) (9) }{{} C Ater nput-output lnearzaton and model transormaton or each subsystem, a lnear control law (vrtual control) v can be desgned to control each transormed subsystem o (9) The controller s am s to make the system s output ollow a gven desred trajectory Z d The vrtual control s obtaned or each subsystem as v (t) = Z (r ) d (t) θ T e (t) () where the gan matrx θ T = [ θr,,θ] s desgned by pole placement and the error dynamcs are dened on the bass o the states e (t) = Ẑ Z d wth e (t) = [ e,ė,,e (r ) ] T such that the polynomal e (r ) B M +θ e(r ) + + θ r e s Hurwtz To obtan the estmate Ẑ o the state vector, we apply a standard Kalman lter to the lnearzed model o the robot (9) as ollows [6, Chapter 8]: Ẑ () = E[Z ()] [ (Z P () = E () Ẑ () )( Z () Ẑ () ) ] T K (t) = P (t)c T R (t) Ẑ (t) = A Ẑ (t) + B v (t) + K (t)(y (t) C Ẑ (t)) Ṗ (t) = A P (t) + P (t)a T (t) + Q (t) P (t)c T R (t)c P (t) () where K (t) and P (t) are the Kalman gan and estmaton error covarance matrx; and the covarances o the process and measurement nose are Q (t) = E [ γ (t)γ T (t) ] and R (t) = E [ ϖ (t)ϖ T (t) ], respectvely We see that a Kalman lter can be separately desgned or each decoupled subsystem, =,,,m These two steps transormaton by eedback lnearzaton, and state estmaton by standard Kalman lters comprse the DKF Remark: Measured values o all states are requred n eedback lnearzaton However, the Kalman lter can be used to estmate unmeasured states n the transormed system Thus, n the DKF, we can use the states to obtan the control nput u(t) III DERIVATIVE-FREE KALMAN FILTERING IN THE PRESENCE OF DISTURBANCES Parameter uncertantes and dsturbances aect the robot Parameter uncertantes are oten modeled as process nose (as dscussed n the prevous secton), whle determnstc dsturbances are modeled as external nputs In robotc systems, dsturbances can be an order o magntude or more greater than the process nose The DKF typcally handles process nose well, but s not desgned to handle dsturbances In ths secton we extend the DKF to handle dsturbances The dsturbances can arse due to the ollowng [4]: () unknown torques on the robot jonts, () unknown orces on the robot (eg, rcton) In the DKF we use eedback lnearzaton to lnearze the nonlnear model o the robot to apply the standard Kalman lter However, the perormance o the DKF may break down snce eedback lnearzaton s not robust to dsturbances The robot s dynamc model n ane state space orm () wth dsturbances can be wrtten as ẋ = (x,t) + g(x,t)u + ω(t) + E(x,t)d(x,t) y = h(x,t) + ϖ(t) () where d(x,t) R l s the dsturbance vector and E(x,t)d(x,t) comprses the eect o the dsturbances on each jont, where E(x,t) R n l can be calculated rom the dynamc model () It should be noted that the dsturbance and ts dervatves must be bounded or stablty reasons, but we do not assume any pror bounds and we do not assume any normaton or the dynamcs o the dsturbances We use the same approach here or the transormaton lnearzaton o the nonlnear model () as we dd n Secton II Exact eedback lnearzaton s used and the nput transormaton s obtaned as n (6), whch results n the transormaton o () nto the ollowng m decoupled subsystems: Ż (t) = Z (t) Ż r (t) = y (r ) (t) = v (t) + γ (t) + d (x,t) (3) or =,,,m Note that we stll have the assumpton about the number o nputs p beng equal to the number o outputs m, as n Secton II, or the exstence o u The components ( d, d,) whch represent the transormed eects o the dsturbances can be obtaned as ollows [7]: d (x,t) = d m (x,t) L r E d h + + dt d (r ) (LE d h ) L r m E d h m + + dt d (4) (r m ) (LE d h m ) The state space canoncal orm can be wrtten or every subsystem rom the transormed model (3) and the bass o (9): Ż (t) = A Z (t) + B v (t) + M γ (t) + N d (x,t) y (t) = C Z (t) + ϖ (t) () or =,,,m, where N = M In order to acheve state estmaton-based control o the robot wth the DKF, we apply the standard Kalman lter () to the lnearzed model () However, the DKF s not able to estmate the states o the robot n the presence o dsturbances So we need to desgn a vrtual control v as 7

4 n () wth dsturbance rejecton to stablze the transormed system dynamcs v (t) = Z (r ) d (t) θ T e (t) ˆ d (x,t) (6) where ˆ d (x,t) s the estmate o the dsturbance whch s used or the rejecton/compensaton o the dsturbance We propose two derent methods to obtan ˆ d (x,t) ) Use a PD dsturbance compensator or dsturbance rejecton; ˆ d (x,t) = L T E, wth the gan L T = [L P,L D ] and E = [e,ė ] T ) Use a PI dsturbance compensator or dsturbance rejecton; ˆ d (x,t) = J T Ξ, wth the gan J T = [J P,J I ] and Ξ = [ e, t e(τ)dτ] T The vrtual control v (6) ams to make the DKF robust aganst dsturbances The DKF or state estmaton and the proposed methods or dsturbance rejecton results n the augmented system [ ] [ ][ ] [ ] [ ] Ẑ (t) A N = Ẑ (t) B K (t)(y(t) C + v ˆ d (x,t) ˆ d (x,t) + Ẑ (t)) L T E or J T Ξ (7) We dene the state estmaton errors as ξ (t) = Ẑ (t) Z (t) and the dsturbance estmaton errors as ϕ(t) = ˆ d (x,t) d (x,t) I L P L D or J P J I, the estmaton errors ξ (s) F or ϕ(s) F converge to a postve small value ε s d (s) F s bounded, where F represents the Frobenus norm and s s the Laplace transorm varable The proposed method or dsturbance rejecton and the DKF or estmatonbased control o the robot s depcted n Fg IV SIMULATION RESULTS In ths secton, we show the eectveness o the DKF and the robust DKF or state estmaton-based control usng smulaton studes o the three-dof prosthess test robot Fg shows a schematc o the hp robot and prosthess combnaton, whch s modeled as a three-lnk robot The dynamc model s gven by (), and the vector o generalzed jont dsplacements s q = [q,q,q 3 ] T, where q s vertcal hp dsplacement, q s thgh angle, and q 3 s knee angle The state vector and outputs are gven as x = [q, q,q, q,q 3, q 3 ] T, y = [q,q,q 3 ] T We thus have three states or the generalzed dsplacements and three states or the generalzed veloctes Moreover, the number o nputs u = [u,u,u 3 ] T s the same as the number o outputs In the dynamcs o the prosthetc leg robot, the nput u wll appear ater two dervatves o the correspondng output y, =,,3 So the total relatve degree s r = r + r + r 3 = 6 as gven n (3), whch s equal to the system order n It should be noted that there are no nternal dynamcs assocated wth the nput-output lnearzaton o the robot/prosthess system snce r = n We assume that the robot model s known except or process nose In Subsecton A we assume that the GRF s modeled as n [8], whch results n the external orce vector F e n (), and that the GRF model s known to the controller In Subsecton B we make no assumptons about the controller s knowledge o the GRF or ts model, and we treat GRF as dsturbances We consder the perormance o the robot/prosthess system durng our steps o walkng at a normal speed, whch s approxmately our seconds The reerence data were provded by research partcpants at the Moton Studes Laboratory o the Cleveland Department o Veterans Aars Medcal Center [8] A The robot/prosthess system wthout dsturbances: As mentoned earler, the robot dynamcs o () s nput-output lnearzable Based on the assumpton o the equalty o the number o nputs and outputs, the robot dynamcs can be transormed nto three decoupled subsystems as [ ] [ ] [ ] Ż (t) = Z (t) + v + γ (t) y (t) = [ ] Z (t) + ϖ (t) (8) or =,,3, where Z (t) = [q, q ] T The unts o Z are m and m/s, the unts o Z are rad and rad/sec, and the unts o Z 3 are also rad and rad/sec We apply the DKF to each lnearzed subsystem (8) to obtan state estmaton-based control The covarances o the process and measurement nose are consdered as Q = 3 and R = 3 or =,, 3 These covarances are based on pror experence wth the accuracy o system dynamc modelng and measurement uncertanty Nose Dsturbances Exact Feedback lnearzaton Inputs u = [u,, u m] Robot model Outputs y (r) = [y (r),, y (rm) ] m decoupled subsystems through nput-output lnearzed model v = [v,, v m] Vrtual control term Dsturbances rejecton d Desred trajectores PD or PI dsturbance compensator State estmaton DKF Fg Block dagram o the proposed methods or dsturbance rejecton and or state estmaton-based control o the robotc system, (robust DKF) Fg Dagram o the robot/prosthess system wth rgd ankle jont 8

5 The ntal value o the state vector s gven as Z () = [9,93] T, Z () = [3,779] T, Z 3 () = [9,4] T We set the ntal value o the state vector to provde an arbtrary but nonzero ntal estmaton error: Ẑ () = [39,6] T, Ẑ () = [9,3] T, Ẑ 3 () = [3,884] T The gans θ n () are desgned by pole placement to provde good estmaton-based control perormance and are obtaned as θ T = [,] or =,,3 The results or the state estmaton-based control o the robot/prosthess system are shown n Fg 3 Although sgncant ntal estmaton errors are mposed on the jont dsplacements and veloctes, the DKF quckly converges to the states B The robot/prosthess system wth dsturbances: In the robot/prosthess system, parameters may change sgncantly rom ther nomnal values Also, the external orces F e nclude unmodeled GRF whch can be consdered as a dsturbance sensors are not avalable We consder these two sources o uncertanty as dsturbances d(x,t) n the robot dynamcs We assume that the ndependent dsturbances d,d,d 3 are ncluded n the robot dynamcs The dsturbances that we use n our smulaton are d = sn(t) N, d = N, and d 3 = sn(t + π) N These nputs aect each jont o the robot as gven by the transormaton lnearzaton (4); the transormed nputs are denoted as d, d, d 3 n the lnearzed model () The robot dynamcs are transormed nto three decoupled subsystems as shown n () The DKF can be desgned usng dsturbance rejecton as shown n Secton III The ntal state vector and estmate, and the covarances o the nose terms, are dentcal wth those n Subsecton A above In the ollowng, we rst use the PD compensator to estmate d or dsturbance rejecton, and then we use the PI compensator Fg 4 shows the perormance o the robust DKF usng PD compensaton or state estmaton-based control o the robot n the presence o these large dsturbances The dsturbances are as a sde benet o ths approach (see Fg (a)) We next use the PI compensator or dsturbance rejecton Estmaton results are shown n Fg (b) Table I compares the PD and PI compensators n terms o root mean square error (RMSE) o the state estmates, and RMS values o the control sgnals Table I shows that the PD compensator outperorms the PI compensator n the ollowng ways: () The PD compensator has aster dynamcs than the PI compensator due to the error dervatve term ė; ths provdes predcton whch mproves the compensator s bandwdth or dsturbance estmaton () The ntegral term n the PI compensator nvolves a pole at the orgn whch results not only n overshoot durng transents, but also slowness n dsturbance estmaton Snce the system has been lnearzed based on eedback lnearzaton, a pole at the orgn may cause oscllaton or nstablty n some cases V CONCLUSION AND FUTURE WORK We developed and appled a DKF to robotc systems or estmaton-based control We proposed a PD or PI dsturbance compensator as a supervsory control term or hp dsplasment (m) thgh angle (rad) knee angle (rad) 4 3 tme tme tme hp velocty (m/s) 4 4 angular velocty o thgh (rad/s) angular velocty o knee (rad/s) 4 4 Fg 3 State estmaton-based control o the robot/prosthess system usng dervatve-ree Kalman lterng n the absence o dsturbances hp dsplasment (m) thgh angle (rad) knee angle (rad) 4 3 tme tme tme hp velocty (m/s) 4 4 angular velocty o thgh (rad/s) angular velocty o knee (rad/s) 4 4 Fg 4 State estmaton-based control o the robot/prosthess system n the presence o dsturbances usng robust dervatve-ree Kalman lterng wth PD dsturbance compensaton 9

6 TABLE I RMSE AND RMS VALUES OF CONTROL SIGNALS FOR THE ROBUST DKF: COMPARISON BETWEEN PD DISTURBANCE COMPENSATOR AND PI x (m) x (m/s) DISTURBANCE COMPENSATOR x 3 (rad) x 4 (rad/s) x (rad) x 6 (rad/s) u (N) u (Nm) PD compensator PI compensator u 3 (Nm) dstrubance/unkown nput estmaton dstrubance/unkown nput estmaton dstrubance/unkown nput estmaton (a) dstrubance/unkown nput estmaton dstrubance/unkown nput estmaton dstrubance/unkown nput estmaton Fg (a) Estmates o dsturbances usng PD compensaton and (b) estmates o dsturbances usng PI compensaton dsturbance rejecton, whch makes the DKF robust A three- DOF robot/prosthess system or transemoral amputees was smulated to demonstrate the DKF or estmaton-based control o generalzed jont dsplacements and veloctes Results show that the DKF estmates quckly converge to the states, even wth sgncant nose and ntal condton errors In addton, we presented robust state estmatonbased control n the presence o dsturbances We showed that the robust DKF acheves smaller estmaton errors wth PD compensaton than wth PI compensaton For uture work, we wll analytcally prove the stablty o the DKF or the robot/prosthess system The DKF can be appled to an underactuated robot as well, whch wll requre an nvestgaton o the stablty o the nternal dynamcs Future work wll nclude expermental hardware mplementaton and human trals (b) [] Z Harvey, Z Benjamn, J Vandersea, and E Wol, Prosthetc advances, Journal o Surgcal Orthopedc Advances, vol, no, pp 8 64, [3] H Zhao, S Kolathaya, and A D Ames, Quadratc programmng and mpedance control or transemoral prosthess, n Internatonal Conerence on Robotcs and Automaton (ICRA), (Hong Kong), pp , 4 [4] R Gregg, T Lenz, L Hargrove, and J Sensnger, Vrtual constrant control o a powered prosthetc leg: rom smulaton to experments wth transemoral amputees, IEEE Transactons on Robotcs, vol 3, no 6, pp 4 47, 4 [] A J Young, A M Smon, and L J Hargrove, A tranng method or locomoton mode predcton usng powered lower lmb prostheses, IEEE Transactons on Neural Systems and Rehabltaton Engneerng, vol, no 3, pp , 4 [6] H Rchter, D Smon, W Smth, and S Samorezov, Dynamc modelng, parameter estmaton and control o a leg prosthess test robot, Appled Mathematcal Modellng, vol 39, no, pp 9 73, [7] H Rchter and D Smon, Robust trackng control o a prosthess test robot, Journal o Dynamc Systems, Measurement, and Control, vol 36, no 3, p 3, 4 [8] V Azm, D Smon, and H Rchter, Stable robust adaptve mpedance control o a prosthetc leg, n Proc o the ASME Dynamc Systems and Control Conerence, (Columbus, Oho), pp VT9A3 VT9A3, [9] D Ebegbe, D Smon, and H Rchter, Hybrd uncton approxmaton based control wth applcaton to prosthetc legs, n IEEE Int Systems Conerence, (Orlando, Florda), pp 6, 6 [] V Azm, D Smon, H Rchter, and S A Fakooran, Robust composte adaptve transemoral prosthess control wth non-scalar boundary layer trajectores, n Proc Amercan Control Conerence, (Boston, Massachusetts), 6 [] G G Rgatos, A dervatve-ree Kalman lterng approach to state estmaton-based control o nonlnear systems, IEEE Trans Ind Electron, vol 9, no, pp , [] S A Fakooran, D Smon, H Rchter, and V Azm, Ground reacton orce estmaton n prosthetc legs wth an extended Kalman lter, n IEEE Int Systems Conerence, (Orlando, Florda), pp , 6 [3] J-J E Slotne and W L, Appled Nonlnear Control Prentce-Hall Englewood Cls, NJ, 99 [4] G G Rgatos, Nonlnear Control and Flterng Usng Derental Flatness Approaches: Applcatons to Electromechancal Systems Sprnger, [] E C Rasmussen and C K I Wllams, Gaussan Processes or Machne Learnng The MIT press, 6 [6] D Smon, Optmal State Estmaton: Kalman, H-Innty, and Nonlnear Approaches John Wley & Sons, 6 [7] Y Lu and D Söker, Robust control approach or nputoutput lnearzable nonlnear systems usng hgh-gan dsturbance observer, Internatonal Journal o Robust and Nonlnear Control, vol 4, no, pp , 4 [8] G Khadem, H Mohammad, E C Hardn, and D Smon, Evolutonary optmzaton o user ntent recognton or transemoral amputees, n Proc Bomedcal Crcuts and Systems Conerence, (Atlanta, Georga), pp 4, REFERENCES [] F Sub and H A Varol, Upslope walkng wth a powered knee and ankle prosthess: Intal results wth an amputee subject, IEEE Trans Neural Systems and Rehabltaton Engneerng, vol 9, no, pp 7 78,