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Tracking Control for Electromechanical Systems Using Robust Discrete Time H∞*
Proceedings of the 20th World Congress Proceedings of the 20th World Congress The International Federation of Congress Automatic Control Proceedings of the 20th World The International Federation of Congress Automatic Control Proceedings of the 20th9-14, World Toulouse, France, July 2017 Available online at www.sciencedirect.com The International Federation of Automatic Control Toulouse, France,Federation July 9-14, 2017 The International of Automatic Control Toulouse, France, July 9-14, 2017 Toulouse, France, July 9-14, 2017
ScienceDirect
IFAC PapersOnLine 50-1 (2017) 11361–11366
Tracking Control for Tracking Control for Tracking Control for Tracking Control for Systems Using Robust Systems Using Robust Systems Using Robust Systems Using Robust ∗
Electromechanical Electromechanical Electromechanical Electromechanical Discrete Time H Discrete Time H∞ ∞ Discrete Time Discrete Time∗ H H∞ ∞
L. a ∗ , H. Caballero-Barrag´ L. P. P. Osuna-Ibarra Osuna-Ibarra Caballero-Barrag´ an n ∗∗ ,, ∗ ∗ ,, H. L. a A. G. Loukianov Bayro-Corrochano ∗ H. A.P. G.Osuna-Ibarra Loukianov ∗∗ and and E.Caballero-Barrag´ Bayro-Corrochano L. P. Osuna-Ibarra , H.E. Caballero-Barrag´ an n ∗∗,, A. G. Loukianov ∗ and E. Bayro-Corrochano ∗ A. G. Loukianov and E. Bayro-Corrochano ∗ ∗ CINVESTAV del IPN, Unidad Guadalajara, Av. del Bosque CINVESTAV del IPN, Unidad Guadalajara, Av. del Bosque ∗ IPN, Unidad Guadalajara, Mexico. ∗ CINVESTAV del CP. 45019, Jalisco, Mexico.Av. CINVESTAV del CP. IPN,45019, UnidadJalisco, Guadalajara, Av. del del Bosque Bosque CP. 45019, Jalisco, Mexico. CP. 45019, Jalisco, Mexico.
1145 1145 1145 1145
Abstract: Abstract: In In this this work work we we propose propose a a robust robust controller controller to to do do tracking tracking using using the the sub-optimal sub-optimal Abstract: In this work we propose a robust controller to do tracking using the sub-optimal H technique with the approach of differential game theory. The problem is solved in H∞ technique with the approach of differential game theory. The problem is solved in two two Abstract: In this work we propose a robust controller to do tracking using the sub-optimal ∞ technique with the approach of differential game theory. The problem is solved in two H steps using the Block Control technique. The controller is designed in discrete time and it is ∞ steps using the Block Control technique. The controller is designed in discrete time and it is technique with the approach of differential game theory. The problem is solved in two H ∞ steps using the Block Control technique. The controller is designed in discrete time and it synthesized for electromechanical systems which are modeled by means of the Euler-Lagrange synthesized for electromechanical systems The which are modeled by means of the Euler-Lagrange steps using the Block Control technique. controller is designed in discrete time and it is is synthesized electromechanical systems which means of formulation. Making use Hamilton-Jacobi-Isaacs equation the discrete formulation. Making use of of the the discrete discrete Hamilton-Jacobi-Isaacs equation and the Euler-Lagrange discrete Riccati Riccati synthesized for for electromechanical systems which are are modeled modeled by by meansand of the the Euler-Lagrange formulation. use the equation and equation the control is The control to 6-DOF equation the Making control law law is derived. derived. TheHamilton-Jacobi-Isaacs control law law is is then then applied applied to aa continuous-time continuous-time 6-DOF formulation. Making use of of the discrete discrete Hamilton-Jacobi-Isaacs equation and the the discrete discrete Riccati Riccati equation the control law is derived. The control law is then applied to a continuous-time bipedal robot model in order to track the walking pattern references for each link. The system bipedal model law in order to track walking each link. The 6-DOF system equationrobot the control is derived. Thethe control lawpattern is then references applied to for a continuous-time 6-DOF bipedal robot model in order to track the walking pattern references for each link. The along with the control law is simulated, where the system is subjected to a disturbance that along themodel control simulated, thepattern systemreferences is subjected to a disturbance that bipedalwith robot in law orderis to track thewhere walking for each link. The system system along with the control law is simulated, where the system is subjected to a disturbance that emulates the action of a group of external unknown bounded forces over the links of the bipedal emulates thethe action of alaw group of external where unknown over the of the bipedal along with control is simulated, the bounded system isforces subjected to links a disturbance that emulates the action of a group of external unknown bounded forces over the links of the bipedal robot. The simulation results are shown displaying robustness against the disturbance, torques robot. The shown displaying against emulates thesimulation action of aresults groupare of external unknown robustness bounded forces overthe thedisturbance, links of the torques bipedal robot. simulation results are shown displaying robustness against the required from the are since the input optimized the within required from the motors motors are plot plot and since the control control input was was optimized the values values lie lietorques within robot. The The simulation results are and shown displaying robustness against the disturbance, disturbance, torques required from the motors are plot and since the control input was optimized the values lie within a reasonable bound. Furthermore, this work is compared to a similar approach that uses H a reasonable Furthermore, thissince work compared to was a similar approach that lie uses H∞ required frombound. the motors are plot and theiscontrol input optimized the values within ∞ a reasonable bound. Furthermore, this work is compared to a similar approach that uses technique in continuous time. technique in continuous time. a reasonable bound. Furthermore, this work is compared to a similar approach that uses H H∞ ∞ technique in time. technique in continuous continuous © 2017, IFAC (Internationaltime. Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Keywords: Tracking Tracking Control, Control, Robust Robust Control, Control, Robot, Robot, Sub-Optimal Sub-Optimal Control, Control, Block Block Control. Control. Keywords: Tracking Control, Robust Control, Robot, Sub-Optimal Control, Keywords: Tracking Control, Robust Control, Robot, Sub-Optimal Control, Block Block Control. Control. 1. This 1. INTRODUCTION INTRODUCTION This said, said, the the optimization optimization of of the the control control law law is is a a hot hot 1. INTRODUCTION This said, the optimization of the control law is topic in control research. Optimal control deals with the topic in control research. Optimal the 1. INTRODUCTION This said, the optimization of thecontrol controldeals law with is a a hot hot topic in control research. Optimal control deals with design of controllers subjected to a measure function, One of the most attractive features in the study of eleccontrollers subjected a measure topic inofcontrol research. Optimaltocontrol deals function, with the the One of the most attractive features in the study of elec- design design of subjected aa measure function, the performance index. Its is One attractive in of tromechanical systems is unification of concepts and performance index. Its main mainto goal is to to determine determine a design of controllers controllers subjected togoal measure function,a tromechanical systems is the thefeatures unification of study concepts and the One of of the the most most attractive features in the the study of elecelecthe main is control signal causes aa process to the tromechanical systems is the concepts and mathematical The majority electromesignal that thatindex. causesIts process to satisfy satisfy the physical physicala the performance performance index. Its main goal goal is to to determine determine a mathematical tools. The vast majority of ofof the electrometromechanical tools. systems is vast the unification unification ofthe concepts and control control that causes aa process to the restrictions of to or the mathematical tools. The vast of the chanical systems can using of the the system and to minimize minimize or maximize maximize the control signal signal thatsystem causesand process to satisfy satisfy the physical physical chanical systems can be be modeled using the the Euler-Lagrange mathematical tools. Themodeled vast majority majority of Euler-Lagrange the electromeelectrome- restrictions restrictions of and desired performance index. chanical can using Euler-Lagrange formalism (see and (2002)). Once performance index. restrictions of the the system system and to to minimize minimize or or maximize maximize the the formalism (see Egeland Egeland and Gravdahl Gravdahl (2002)). Once havhav- desired chanical systems systems can be be modeled modeled using the the Euler-Lagrange desired performance index. formalism (see Egeland and Gravdahl (2002)). Once having the dynamical model one can analyze the system performance index. the design of the controller ing the dynamical model can analyze system formalism (see Egeland and one Gravdahl (2002)).the Once hav- desired Yet, for perturbed systems, Yet, for perturbed systems, the design of the controller ing dynamical model can analyze the and design control that allow electromechanical and design control laws laws thatone allow the electromechanical ing the the dynamical model one canthe analyze the system system Yet, the the ceases to be if are (most ceases to perturbed be optimal optimal systems, if the the disturbances disturbances areof unknown (most Yet, for for perturbed systems, the design design ofunknown the controller controller and design control laws that allow the electromechanical system to have a desired performance. system to have a desired performance. and design control laws that allow the electromechanical ceases to be optimal if the disturbances are unknown (most cases), in this case, the controller turns sub-optimal and cases), in this case, the controller turns sub-optimal and ceases to be optimal if the disturbances are unknown (most system to performance. system to have have aa desired desired performance. cases), in this case, the controller turns sub-optimal and can reject perturbations that are bounded. Controllers The classification of the controllers can be done by its reject perturbations that are turns bounded. Controllers cases), in this case, the controller sub-optimal and The classification of the controllers can be done by its can can reject perturbations that are bounded. Controllers that use the H norm have the advantage of ensuring The classification of the controllers can be done by its independent time variable, which can be expressed in con∞ use the H∞ norm have ensuring can reject perturbations that the are advantage bounded. of Controllers independent time variable, which can be The classification of the controllers canexpressed be done inbyconits that that the use the norm have advantage transfer function matrix interest independent time can be expressed in continuous (Chavez-Guzm´ a al., time ∞ the peaks in transfer function matrixof ofensuring interest use peaks the H Hin norm have the the advantage ofof ensuring tinuous (Chavez-Guzm´ an n et etwhich al., 2015) 2015) or discrete time (Osindependent time variable, variable, which can or be discrete expressed in (Oscon- that the ∞ the that the peaks in the transfer function matrix of are knocked down (see Ball and Cohen (1987)). tinuous (Chavez-Guzm´ a n et al., 2015) or discrete time (Osuna et al., 2015). Strictly speaking, all the controllers are down (seetransfer Ball andfunction Cohen (1987)). thatknocked the peaks in the matrix of interest interest una et al., 2015). Strictly speaking, controllers are are tinuous (Chavez-Guzm´ an et al., 2015)allorthe discrete time (Osare down (see Ball Cohen una Strictly speaking, all are implemented in since are programmed are knocked knocked down (see Ball and and Cohen (1987)). (1987)). implemented in discrete discrete time, since they they arecontrollers programmed una et et al., al., 2015). 2015). Strictlytime, speaking, all the the controllers are Robots are a wide selection of interest cases within Robots are a wide selection of interest cases within elecelecimplemented in time, they are in a nevertheless, if period in a microcontroller, microcontroller, nevertheless, if the the sampling period is is Robots implemented in discrete discrete time, since since theysampling are programmed programmed are aa wide selection of interest cases within electromechanical systems, and among robots, bipedal robots tromechanical systems, and among robots, bipedal robots Robots are wide selection of interest cases within elecin a microcontroller, nevertheless, if the sampling period is small enough can controller to small enough the the system system can consider consider the controller to be be in a microcontroller, nevertheless, if thethe sampling period is tromechanical systems, and among robots, bipedal robots are particularly interesting. One of the major challenges are particularlysystems, interesting. One ofrobots, the major challenges tromechanical and among bipedal robots small enough the system can consider the controller to be continuous in time from its perspective of time. However, continuous in the timesystem from its of controller time. However, small enough canperspective consider the to be are particularly interesting. the challenges for robots research the of for bipedal robots research is isOne the of synthesis of controllers controllers are bipedal particularly interesting. One ofsynthesis the major major challenges continuous in from of some low-budget may the sampling some low-budget microprocessor may limit limit the However, sampling for continuous in time timemicroprocessor from its its perspective perspective of time. time. However, bipedal robots research is the synthesis of controllers that allow to perform walking routines. that allow torobots perform walking routines. for bipedal research is the synthesis of controllers some low-budget microprocessor may limit the sampling period, as well as some sensors that work at frequencies period, as well asmicroprocessor some sensors that some low-budget maywork limit at thefrequencies sampling that allow to perform walking routines. that bipedal allow to perform walking routines. period, well some sensors that work frequencies relatively In situation the relatively slow. In these these cases, the situation demands the The The bipedal robot robot walking walking process process can can be be divided divided into into two two period, as asslow. well as as some cases, sensorsthe that work at atdemands frequencies relatively slow. In these cases, the situation demands the The bipedal robot walking process can divided into analysis to be in discrete time, because it represents better main tasks: the path generation, and the analysis be inIn discrete time, because it represents better tasks: the path generation, and thebe reference tracking relativelytoslow. these cases, the situation demands the main The bipedal robot walking process can bereference divided tracking into two two analysis to discrete because represents tasks: generation, the behavior couple formed by for each link of the the behavior ofin the coupletime, formed by the theit electromechanical for each linkthe of path the robot. robot. analysis to be beof inthe discrete time, because itelectromechanical represents better better main main tasks: the path generation, and and the the reference reference tracking tracking the of couple system and controller. system and the the controller. the behavior behavior of the the couple formed formed by by the the electromechanical electromechanical for for each each link link of of the the robot. robot. Regarding Regarding path path generation generation there there exist exist different different algorithms algorithms system and the controller. system and the controller. Regarding path generation there exist algorithms One of the main performance indices of a control law in the literature. There are the ones that the One of the main performance indices of a control law in the literature. There are theexist onesdifferent that consider consider the Regarding path generation there different algorithms One of the main performance indices of a control law in the literature. There are the ones that consider the is its robustness against perturbations. It is required for entire dynamics of the robot precise and require precise is its of robustness perturbations. for entire the robot precise precise One the mainagainst performance indices It of isa required control law in the dynamics literature.ofThere are the onesand thatrequire consider the is its robustness against perturbations. It is required for entire dynamics of the robot precise and require precise the control law to be able to reject the effect of this knowledge of the robot mass, location of center of mass the control law to be able to reject the effect of this of theofrobot mass, precise locationand of center mass is its robustness against perturbations. It is required for knowledge entire dynamics the robot requireofprecise the to to effect robot center mass perturbations while it maintains itself confined to the and inertia each link to able generate walking perturbations while it able maintains itselfthe confined to this the knowledge and inertiaof ofthe each link mass, to be belocation able to to of generate walking the control control law law to be be able to reject reject the effect of of this knowledge ofof the robot mass, location of center of of mass perturbations while it maintains itself confined to the and inertia of each link to be able to generate walking physical restrictions imposed by the configuration and patterns, such as Yamaguchi et al. (1999), Hirai et physical restrictions imposed by the such as Yamaguchi al. to (1999), Hiraiwalking et al. al. perturbations while it maintains itselfconfiguration confined to and the patterns, and inertia of each link to beetable generate physical imposed the such Yamaguchi et Hirai et al. features the (1998), Kagami et (2000), al. And, features ofrestrictions the electromechanical electromechanical system. (1998), Kagami et al. al. (2000), Huang Huang et(1999), al. (2001). (2001). And, on physical of restrictions imposed by bysystem. the configuration configuration and and patterns, patterns, such as as Yamaguchi et al. al.et (1999), Hirai et on al. features of the electromechanical system. (1998), Kagami et al. (2000), Huang et al. (2001). And, on the other hand, there are the algorithms that use limited the other hand, there are the algorithms that use limited This work was electromechanical supported by CONACYT, M´ exico, under grant features of the system. (1998), Kagami et al. (2000), Huang et al. (2001). And, on This work was supported by CONACYT, M´ exico, under grant the the that limited knowledge of robots dynamics e.g. of total 300959, 301068 andsupported 252405. by CONACYT, M´ work was exico, under grant knowledge of the thethere robotsare dynamics e.g. location location of the the total the other other hand, hand, there are the algorithms algorithms that use use limited This 300959, 301068 andsupported 252405. by CONACYT, M´ This work was exico, under grant knowledge of the robots dynamics e.g. location of the 300959, 301068 and 252405. knowledge of the robots dynamics e.g. location of the total total
300959, 301068 and 252405. Copyright © 2017, 2017 IFAC 11853Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2017 IFAC 11853 Copyright ©under 2017 responsibility IFAC 11853 Peer review of International Federation of Automatic Control. Copyright © 2017 IFAC 11853 10.1016/j.ifacol.2017.08.1698
Proceedings of the 20th IFAC World Congress 11362 L.P. Osuna-Ibarra et al. / IFAC PapersOnLine 50-1 (2017) 11361–11366 Toulouse, France, July 9-14, 2017
center of mass, center of gravity, zero moment point, etc, like Sugihara et al. (2002), Kajita et al. (2003), Wieber (2006), Williams et al. (2013). In order to obtain the references for each link of the robot it is necessary the use of methods such as the analytic solution of inverse kinematics (Kajita et al., 2014). Once having the trajectories to track, it is designed a control law in order to do the tracking, which is required to be robust against disturbances. These performance can be achieved by means of a sub-optimal control law using the H∞ norm. In this work we present the development of a sub-optimal discrete time control law using H∞ norm for electromechanical systems modeled by Euler-Lagrange. This law is applied to a bipedal robot in order to do reference tracking for each link actuator. The main contribution of this paper is the development of a sub-optimal discrete-time H∞ controller for electromechanical systems, unlike Loukianov et al. (2009) that works in continuous time. This controller is applied to a bipedal robot to do the tracking for joint walking references. In Section 2 the mathematical background is presented, this includes a brief description of the H∞ control technique, including differential game theory and the development of the discrete time Hamilton-Jacobi-Isaacs equation. In Section 3 the main problem to solve is stated and the conditions for the system and the disturbance are asserted. Section 4 deals with the synthesis of the robust discrete time H∞ controller for a perturbed electromechanical system modeled by Euler-Lagrange formulation. Section 5 shows the application of the H∞ control to a bipedal robot in order to track the references for each joint in a walking routine. In Section 6 there is made a comparison of our proposed controller with a controller from the literature that uses H∞ in continuous time, the simulation results are shown and the Mean Squared Error is also calculated and displayed. Finally, the conclusions and future work are presented in Section 7. 2. MATHEMATICAL BACKGROUND 2.1 H∞ -Control and Differential Game Theory. The role of game theory in the design of robust (minimax) controllers is stated in Basar and Bernhard (2008). The goal is to minimize a given performance index under worst possible perturbations or parameter variations (which maximize the same performance index). This gametheoretic approach requires the setting of differential games, which seems to be the most natural. This is so because the original H∞ -optimal control problem is a minimax optimization problem, and hence a zero-sum game, where the controller is viewed as the minimizing player 1 and the perturbation as the maximizing player 2. Consider the following system: e¯i,k+1 =Ai e¯i,k + Bi υi,k + Bi f¯i,k zi,k =Ci e¯i,k + Di υi,k , with the finite horizon cost N J(υi,k , f¯i,k ) = ( zi,k − γi2 f¯i,k ), k=0
(1)
∗ which will be minimized by the control υi,k = υi,k ∗ (player1) and maximized by the perturbation f¯i,k = f¯i,k (player2), respectively; where zi,k is the unknown output to be controlled, i = 1, 2, CiT Di = 0 and γ is a positive real number γ > 0.
Suppose there exists a positive definite matrix Pi , and a positive function V (¯ eik ) = 12 e¯Ti,k Pi e¯i,k . Then, the discrete time Hamilton-Jacobi-Isaacs equation results ∗ ∗ + Bi f¯i,k ) − V (¯ ei,k )+ Hi =V (Ai e¯i,k + Bi υi,k 1 ∗ ∗ + ( Ci e¯i,k + Di υi,k − γi2 f¯i,k )<0 2 ∂ 2 Hi Ri,11 Ri,12 (2) Ri = = = Ri,21 Ri,22 ∂(υi,k , f¯i,k )2 T BiT Pi Bi B P B + DiT Di = i i Ti , B i Pi B i BiT Pi Bi − γi2 I such that L1) Ri,11 > 0 and Ri,22 − Ri,21 Ri,11 −1 Ri,12 < 0.
L2) The following discrete algebraic Riccati equation holds, i.e., (3) Pi = ATi Pi Ai + CiT Ci − T T GT + I, where T Bi Pi Ai , T = BiT Pi Ai ¯ ¯ iT Pi Bi − γi2 I)−1 G −G(B G= ¯ T Pi Bi − γ 2 I)−1 G ¯ −1 (BiT Pi Bi − γi2 I)−1 , −G(B i i ¯ = (BiT Pi Bi + DiT Di )−1 . G
L3) The control gain is given by (4) Fi = −(Bi Pi Bi + DiT Di )−1 BiT Pi Ai . The proof of stability can be found in Lin and Byrnes (1996). 3. STATEMENT OF PROBLEM Consider an electromechanical system modeled via EulerLagrange formulation for n-DOF x1,k+1 =x1,k + δx2,k (5) x2,k+1 =x2,k + δM (x1,k )−1 (−C(x1,k , x2,k )x2,k − − G(x1,k ) + τk + fk ), where x1,k ∈ Rn is the position vector, x2,k ∈ Rn is the velocity vector, fk ∈ Rn is a bounded external perturbation, τk ∈ Rn is torque control, M , C and G are the matrices of the system and δ is a constant sample period. Furthermore it is supposed that the following assumptions are fullfiled by system (5): Assumption 1. The system (5) is fully actuated and the M (x1,k ) matrix has full rank and its inverse M (x1,k )−1 exists ∀x1,k . x Assumption 2. The state vector xk = 1,k is available x2,k for the measurement. The objective of this paper is to synthesize a robust suboptimal discrete time controller using H∞ with the gametheoretic approach for the system (5).
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4. ROBUST TRACKING CONTROL USING H∞ In this section is developed a tracking control using the Block Control (BC) technique (Loukianov, 2002) and H∞ described in Section 2. Step 1. The state x1,k of the system (5) will track the reference x1ref . In this work we assume the references to be unknown and to fill the following condition: dx1ref (t) ≤ d2 , (6) x1ref (t) ≤ d1 , dt
where x1ref (t) is the reference, d1 > 0 is a constant and d2 > 0 is a constant. These bounds are assumed to be known. The tracking error is presented as e1,k = x1,k − x1ref,k .
The dynamics for error e1,k is obtained using (5).The introduction of the integral term e01,k+1 results in e01,k+1 =e01,k + δe1,k e1,k+1 =e1,k + δx2,k + f¯1,k ,
(7)
where f¯1,k is the derivative of the reference, considered as an unknown unmatched perturbation. Following the BC technique, the variable x2,k , considered in (7) as a virtual control, is selected to compensate the known terms and to introduce the new desired dynamics (8) δx2,k = −e1,k + L0 e01,k + L1 e1,k + υ1,k , where υ1,k is an auxiliary virtual control, that will be designed to reject the unknown perturbation f¯1,k . Thus, substituting (8) in (7), yields e (9) e¯1,k+1 = 01,k+1 = A1 e¯1,k + B1 (υ1,k + f¯1,k ), e1,k+1 with
A1 =
In δIn 0 , B1 = n . L0 L1 In
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Step 2. Using the transformation (12), a straightforward algebraic manipulation reveals e2,k+1 = g(¯ e1,k , e2,k ) + δ 2 M (·)−1 τk + f¯2,k , (15) where g(·) = − [δM (·)−1 C(·) + (L1 + F1 − In )](L0 + F0 )e01,k − − [δM (·)−1 C(·)(L1 + F1 − In ) + δ(L0 + F0 )+ + (L1 + F1 − In )2 ]e1,k − [δM (·)−1 C(·)+ + (L1 + F1 − 2In )]e2,k − δ 2 M (·)−1 G(·), f¯2,k = − (L1 + F1 − In )f¯1,k + δ 2 M (·)−1 fk . Let us now choose the control τk in (15) to cancel the dynamical term g(¯ e1,k , e2,k ) and introduce the new dynamics L2 e02,k + L3 e2,k , i.e., δ 2 M (·)−1 τk = −g(¯ e1,k , e2,k )+L2 e02,k +L3 e2,k +υ2,k , (16) where υ2,k is a auxiliary control and L2 and L3 are design parameters used to alleviate solving (3). Introducing the integral term e02,k+1 = e02,k + δe2,k , and substituting (16) in (15) results in e02,k+1 = A2 e¯02,k + B2 (υ2,k + f¯2,k ), (17) e¯2,k+1 = e2,k+1
where
A2 =
In δIn 0 , B2 = n . L2 L3 In
Using the solution F2 of (4), the auxiliary control υ2,k is defined as υ2,k = F2 e¯2,k . (18) Now, substituting (18) in (17) yields e2,k + B2 f¯2,k . e¯2,k+1 = (A2 + B2 F2 )¯
(19)
The dynamical error system in close-loop (see (14) and (19)) is presented as follows e¯ ek + B f¯k , e¯k+1 = 1,k+1 = A¯ (20) e¯2,k+1
To give robustness to subsystem (9) against the perturbation f¯1,k , the optimal robust control technique from Lin and Byrnes (1996) is applied. The desired value υ1d,k of the virtual control υ1,k in (9) is thus chosen in the form (10) υ1d,k = F1 e¯1,k = F0 e01,k + F1 e1,k , where F1 result of (4) (see Subsection 2.1).
where
The design parameters L0 and L1 are additionally used to alleviate solving the Riccati equation (3).
From the sub-optimal H∞ control methodology (Lin and Byrnes, 1996), it follows that the feedback, which is designed based on proper solutions of the Riccati equations (3), attenuates the influence of the external disturbances fk on the output tracking error e1,k , e1,k = x1,k − x1ref,k .
If the new variable (11) e2,k = υ1,k − υ1d,k , is introduced, then substituting υ1,k = δx2,k + e1,k − L0 e01,k −L1 e1,k , (8) and (10) in (11) results in the following transformation: e2,k = δx2,k − (L0 + F0 )e01,k − (L1 + F1 − In )e1,k , (12) which is equivalent to δx2,k = (L0 + F0 )e01,k + (L1 + F1 − In )e1,k + e2,k . (13) By substituting (13) in (7) or its equivalent (9), one derives e1,k + B1 (e02,k + f¯1,k ). (14) e¯1,k+1 = (A1 + B1 F1 )¯
f¯ (A1 + B1 F1 ) A12 f¯k = ¯1,k , A = , 02n (A2 + B2 F2 ) f2,k 0n 0n B1 02n×n , B= . A12 = 0n In 02n×n B2
5. APPLICATION TO A 6-DOF BIPEDAL ROBOT The control designed in Section 4 is applied to a bipedal robot presented in Osuna et al. (2015). In Fig. 1 the scheme of this bipedal robot is presented. The parameters used for the simulation were δ = 0.01, the 06 06 T design parameter γi = 3.2, Ci = , Di = [06 I6 ] , 06 I6
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1.2 1 0.8
x1(rad)
0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 0
2
4
6
8 10 Time(s)
12
14
16
18
Fig. 2. Simulation results of tracking control using H∞ (16) applied to system (5) (the position in rad).
1
Fig. 1. Scheme of the bipedal robot. 0.95
= L2 = L3 = I6 with i = 1, 2 and 0 0 0 . 0 0 1
The control gain results now Fi = [λ0 I6 λ1 I6 ], i = 1, 2, λ0 = −1.4916, λ1 = −0.6203.
The reference for each link was obtained from a path generator that uses limited knowledge of the plant (Harada et al., 2004) and the analytic inverse kinematics solution (Kajita et al., 2014). Having the references for the ZMP and CoG the use of the analytic solution of the inverse kinematic proposed in (Kajita et al., 2014) gave us the set of references for each link of the bipedal robot. The results are show in Figs. 2, 3, 4, 5 and 6.
Fig. 2 shows the position vector x1,k and the reference vector x1ref,k ; the tracking error is presented in Fig. 4. In Fig. 3 is presented a zoom of Fig. 2. The control torque is shown in the Fig. 5 and the perturbation fk is presented in the Fig. 6, the perturbation is defined for simulation as follows, [0.1 0.2 0.3 0.3 0.2 0.1]T if 0 ≤ t < 5 s if 5 s ≤ t < 10 s [0 0 0 0 0 0]T fk = T [0.1 0.2 0.3 0.3 0.2 0.1] if 10 s ≤ t < 15 s [0 0 0 0 0 0]T if t ≥ 15 s. The perturbation fk emulates a pushing force in the bipedal robot. 6. COMPARISON WITH CURRENT CONTINUOUS TIME H∞ CONTROLLERS In the literature there can be found controllers for electromechanical systems (robots) that were designed using the H∞ control technique in continuous time (see Chavez-
x1(3)(rad)
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Fig. 3. Zoom of Fig. 2, the position tracking of q3 (rad). 0.6 0.4 0.2 e1(rad)
and, L0 = 100 0 1 0 0 0 1 I6 = 0 0 0 0 0 0 000
0 −0.2 −0.4 −0.6 −0.8 0
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Fig. 4. The tracking error for bipedal robot position. Guzm´an et al. (2015) and Becerra-Vargas and MorgadoBelo (2012)). The aim of these controllers is to do tracking in each link of a manipulator robot, while rejecting the effect of external perturbations over the plant. The work by Chavez-Guzm´an et al. (2015) introduces a controller for a n-DOF manipulator robot in continuous time. Once obtained, the control law is applied to a 3DOF manipulator (shown in Fig. 7). The main difference is that the controller proposed in our work is designed in
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Fig. 5. The input torque during tracking control (N m). 0.35
Fig. 7. The 3-DOF robot manipulator.
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Fig. 6. The disturbance fk (N m). discrete time for the discretized model and applied to the continuous time system. In order to compare the performance of the controller proposed in this work with the controllers in the literature, it was applied to the 3-DOF manipulator under the perturbation πcos(2πt) ω(t) = πcos(2π2t) N m πcos(2π3t) which is the one used in Chavez-Guzm´ an et al. (2015) with an amplitude 10 times bigger. The tracking for each joint reference is presented in the Fig. 8. The dotted line is the tracking performed by the work of Chavez-Guzm´ an et al. (2015) The continuous line is the tracking using our proposed controller, the dashed line is the reference for each joint given in Rad. From Fig. 8 is observed that our proposed controller reaches the reference faster and that the disturbance is better rejected, since the effect of it is more noticeable when the controller proposed by Chavez-Guzm´ an et al. (2015) is applied. The reference vector is the following: π x1ref = 3π/4 rad. 5π/9 The Fig. 9 shows the Mean Squared Error (MSE) for each tracking control. The dashed line is the MSE for the tracking control using the controller proposed by ChavezGuzm´ an et al. (2015) and the solid line is the MSE using our controller.
0.5
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2
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Fig. 8. Comparison of our work against continuous time H∞ controller from Chavez-Guzm´an et al. (2015). 2.5
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Fig. 9. Mean squared error. 7. CONCLUSIONS In this work a robust discrete-time H∞ tracking control for electromechanical systems modeled using the EulerLagrange formulation was presented. The controller was derived making use of the discrete equations of HamiltonJacobi-Isaacs, and, the Riccati equations. The proposed control law was designed in two steps by means of the Block Control technique. The control law is robust and capable of rejecting unknown external disturbances while maintaining acceptable control signals.
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Although it is necessary for the derivative of the reference to be bounded, the controller does not need to know the value of this bound, meaning that the derivative of the reference may be unknown. The H∞ discrete-time controller was formulated from the perspective of the differential game theory, considering the control signal as one player, and the disturbance as the second player, the role of one player is to maximize a given performance index, while the other player tries to minimize this index. The controller was applied in simulation to a discretetime bipedal robot model. The aim of the controller was to do the tracking for reference signals for each link of the biped. These signals were obtained from a path generator and the use of the analytic solution of the inverse kinematics. The system was simulated under an external force as disturbance. The results displayed the successful disturbance rejection, the plots showed the controller to be ultimately bounded for non-vanishing disturbances. Finally, the performance of the discrete-time controller was compared with the performance of a continuous-time H∞ controller from the literature. The comparison was made using the model of a 3-DOF manipulator. From the simulation it was observed that the controller proposed in this work reached the reference faster with a smaller error, even though the controllers were both applied to a continuous-time system. The application to robots with a higher number of DOF and the implementation in real time is left as future work, as well as an extension of the control law in order to do adaptive control, in order to apply it to plants with unknown parameters. REFERENCES Ball, J.A. and Cohen, N. (1987). Sensitivity minimization in an H∞ norm: parametrization of all suboptimal solutions. International Journal of Control, 46. Basar, T. and Bernhard, P. (2008). H∞ Optimal Control and Related Minimax Design Problems: A Dynamic Game Approach. Birkhuser Basel. Becerra-Vargas, M. and Morgado-Belo, E. (2012). Application of H∞ theory to a 6 dof flight simulator motion base. Journal of the Brazilian Society of Mechanical Sciences and Engineering, 34(2), 193–204. Chavez-Guzm´ an, C.A., Aguilar-Bustos, L.T., and M´eridaRubio, J.O. (2015). Analysis and synthesis of global nonlinear H∞ controller for robot manipulators. Mathematical Problems in Engineering, vol. 2015, doi:10.1155/2015/410873. Egeland, O. and Gravdahl, J.T. (2002). Modeling and simulation for automatic control, volume 76. Marine Cybernetics Trondheim, Norway. Harada, K., Kajita, S., Kaneko, K., and Hirukawa, H. (2004). An analytical method on real-time gait planning for a humanoid robot. In Humanoids. Hirai, K., Hirose, M., Haikawa, Y., and Takenaka, T. (1998). The development of honda humanoid robot. Proceedings of the 1998 IEEE International Conference on Robotics and Automation, 2, 1321–1326. Huang, Q., Yokoi, K., Kajita, S., Kaneko, K., Arai, H., Koyachi, N., and Tanie, K. (2001). Planning walking
patterns for a biped robot. IEEE Transactions on Robotics and Automation, 17, 280–289. Kagami, S., Nishiwaki, K., Kitagawa, T., Sugihiara, T., Inaba, M., and Inoue, H. (2000). A fast generation method of a dynamically stable humanoid robot. Proceedings of the IEEE International Conference on Humanoid Robots. Kajita, S., Kanehiro, F., Kaneko, K., Fujiwara, K., Harada, K., Yokoi, K., and Hirukawa, H. (2003). Biped walking pattern generation by using preview control of zeromoment point. Proceedings of the IEEE International Conference on Robotics and Automation, ICRA 03, 2, 1620–1626. Kajita, S., Hirukawa, H., Harada, K., and Yokoi, K. (2014). Introduction to Humanoid Robotics. Springer-Verlag Berlin Heidelberg. Lin, W. and Byrnes, C.I. (1996). H∞ -control of discretetime nonlinear system. IEEE Transactions on Automatic Control, 41(4), 494–510. Loukianov, A.G. (2002). Robust block decomposition sliding mode control design. Math. Probl. Eng., 8(4-5), 349–365. Loukianov, A.G., Rivera, J., Orlov, Y.V., and MoralesTeraoka, E.Y. (2009). Robust trajectory tracking for an electrohydraulic actuator. IEEE Transactions on Industrial Electronics, 56(9), 3523–3531. Osuna, L., Caballero, H., Loukianov, A., CarbajalEspinosa, O., and Bayro-Corrochano, E. (2015). Continuous and discrete time robust control for bipedal robot assuming minimal knowledge of the plants. 2015 IEEE-RAS 15th International Conference on Humanoid Robots (Humanoids), 959–964. Sugihara, T., Nakamura, Y., and Inoue, H. (2002). Realtime humanoid motion generation through ZMP manipulation based on inverted pendulum control. Proceedings of the IEEE International Conference on Robotics and Automation, ICRA 02, 2, 1404–1409. Wieber, P.B. (2006). Trajectory free linear model predictive control for stable walking in the presence of strong perturbations. 2006 6th IEEE-RAS International Conference on Humanoid Robots, 137–142. Williams, M.M., Loukianov, A., and Baryro-Corrochano, E.J. (2013). ZMP based pattern generation for biped walking using optimal preview integral sliding mode control. Proceedings of 2013 13th IEEE-RAS International Conference on Humanoid Robots (Humanoids), 100–105. Yamaguchi, J., Soga, E., Inoue, S., and Takanishi, A. (1999). Development of a bipedal humanoid robotcontrol method of whole body cooperative dynamic biped walking. Proceedings of the 1999 IEEE International Conference on Robotics and Automation, 1, 368– 374.
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IFAC PapersOnLine 50-1 (2017) 11361–11366
Tracking Control for Tracking Control for Tracking Control for Tracking Control for Systems Using Robust Systems Using Robust Systems Using Robust Systems Using Robust ∗
Electromechanical Electromechanical Electromechanical Electromechanical Discrete Time H Discrete Time H∞ ∞ Discrete Time Discrete Time∗ H H∞ ∞
L. a ∗ , H. Caballero-Barrag´ L. P. P. Osuna-Ibarra Osuna-Ibarra Caballero-Barrag´ an n ∗∗ ,, ∗ ∗ ,, H. L. a A. G. Loukianov Bayro-Corrochano ∗ H. A.P. G.Osuna-Ibarra Loukianov ∗∗ and and E.Caballero-Barrag´ Bayro-Corrochano L. P. Osuna-Ibarra , H.E. Caballero-Barrag´ an n ∗∗,, A. G. Loukianov ∗ and E. Bayro-Corrochano ∗ A. G. Loukianov and E. Bayro-Corrochano ∗ ∗ CINVESTAV del IPN, Unidad Guadalajara, Av. del Bosque CINVESTAV del IPN, Unidad Guadalajara, Av. del Bosque ∗ IPN, Unidad Guadalajara, Mexico. ∗ CINVESTAV del CP. 45019, Jalisco, Mexico.Av. CINVESTAV del CP. IPN,45019, UnidadJalisco, Guadalajara, Av. del del Bosque Bosque CP. 45019, Jalisco, Mexico. CP. 45019, Jalisco, Mexico.
1145 1145 1145 1145
Abstract: Abstract: In In this this work work we we propose propose a a robust robust controller controller to to do do tracking tracking using using the the sub-optimal sub-optimal Abstract: In this work we propose a robust controller to do tracking using the sub-optimal H technique with the approach of differential game theory. The problem is solved in H∞ technique with the approach of differential game theory. The problem is solved in two two Abstract: In this work we propose a robust controller to do tracking using the sub-optimal ∞ technique with the approach of differential game theory. The problem is solved in two H steps using the Block Control technique. The controller is designed in discrete time and it is ∞ steps using the Block Control technique. The controller is designed in discrete time and it is technique with the approach of differential game theory. The problem is solved in two H ∞ steps using the Block Control technique. The controller is designed in discrete time and it synthesized for electromechanical systems which are modeled by means of the Euler-Lagrange synthesized for electromechanical systems The which are modeled by means of the Euler-Lagrange steps using the Block Control technique. controller is designed in discrete time and it is is synthesized electromechanical systems which means of formulation. Making use Hamilton-Jacobi-Isaacs equation the discrete formulation. Making use of of the the discrete discrete Hamilton-Jacobi-Isaacs equation and the Euler-Lagrange discrete Riccati Riccati synthesized for for electromechanical systems which are are modeled modeled by by meansand of the the Euler-Lagrange formulation. use the equation and equation the control is The control to 6-DOF equation the Making control law law is derived. derived. TheHamilton-Jacobi-Isaacs control law law is is then then applied applied to aa continuous-time continuous-time 6-DOF formulation. Making use of of the discrete discrete Hamilton-Jacobi-Isaacs equation and the the discrete discrete Riccati Riccati equation the control law is derived. The control law is then applied to a continuous-time bipedal robot model in order to track the walking pattern references for each link. The system bipedal model law in order to track walking each link. The 6-DOF system equationrobot the control is derived. Thethe control lawpattern is then references applied to for a continuous-time 6-DOF bipedal robot model in order to track the walking pattern references for each link. The along with the control law is simulated, where the system is subjected to a disturbance that along themodel control simulated, thepattern systemreferences is subjected to a disturbance that bipedalwith robot in law orderis to track thewhere walking for each link. The system system along with the control law is simulated, where the system is subjected to a disturbance that emulates the action of a group of external unknown bounded forces over the links of the bipedal emulates thethe action of alaw group of external where unknown over the of the bipedal along with control is simulated, the bounded system isforces subjected to links a disturbance that emulates the action of a group of external unknown bounded forces over the links of the bipedal robot. The simulation results are shown displaying robustness against the disturbance, torques robot. The shown displaying against emulates thesimulation action of aresults groupare of external unknown robustness bounded forces overthe thedisturbance, links of the torques bipedal robot. simulation results are shown displaying robustness against the required from the are since the input optimized the within required from the motors motors are plot plot and since the control control input was was optimized the values values lie lietorques within robot. The The simulation results are and shown displaying robustness against the disturbance, disturbance, torques required from the motors are plot and since the control input was optimized the values lie within a reasonable bound. Furthermore, this work is compared to a similar approach that uses H a reasonable Furthermore, thissince work compared to was a similar approach that lie uses H∞ required frombound. the motors are plot and theiscontrol input optimized the values within ∞ a reasonable bound. Furthermore, this work is compared to a similar approach that uses technique in continuous time. technique in continuous time. a reasonable bound. Furthermore, this work is compared to a similar approach that uses H H∞ ∞ technique in time. technique in continuous continuous © 2017, IFAC (Internationaltime. Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Keywords: Tracking Tracking Control, Control, Robust Robust Control, Control, Robot, Robot, Sub-Optimal Sub-Optimal Control, Control, Block Block Control. Control. Keywords: Tracking Control, Robust Control, Robot, Sub-Optimal Control, Keywords: Tracking Control, Robust Control, Robot, Sub-Optimal Control, Block Block Control. Control. 1. This 1. INTRODUCTION INTRODUCTION This said, said, the the optimization optimization of of the the control control law law is is a a hot hot 1. INTRODUCTION This said, the optimization of the control law is topic in control research. Optimal control deals with the topic in control research. Optimal the 1. INTRODUCTION This said, the optimization of thecontrol controldeals law with is a a hot hot topic in control research. Optimal control deals with design of controllers subjected to a measure function, One of the most attractive features in the study of eleccontrollers subjected a measure topic inofcontrol research. Optimaltocontrol deals function, with the the One of the most attractive features in the study of elec- design design of subjected aa measure function, the performance index. Its is One attractive in of tromechanical systems is unification of concepts and performance index. Its main mainto goal is to to determine determine a design of controllers controllers subjected togoal measure function,a tromechanical systems is the thefeatures unification of study concepts and the One of of the the most most attractive features in the the study of elecelecthe main is control signal causes aa process to the tromechanical systems is the concepts and mathematical The majority electromesignal that thatindex. causesIts process to satisfy satisfy the physical physicala the performance performance index. Its main goal goal is to to determine determine a mathematical tools. The vast majority of ofof the electrometromechanical tools. systems is vast the unification unification ofthe concepts and control control that causes aa process to the restrictions of to or the mathematical tools. The vast of the chanical systems can using of the the system and to minimize minimize or maximize maximize the control signal signal thatsystem causesand process to satisfy satisfy the physical physical chanical systems can be be modeled using the the Euler-Lagrange mathematical tools. Themodeled vast majority majority of Euler-Lagrange the electromeelectrome- restrictions restrictions of and desired performance index. chanical can using Euler-Lagrange formalism (see and (2002)). Once performance index. restrictions of the the system system and to to minimize minimize or or maximize maximize the the formalism (see Egeland Egeland and Gravdahl Gravdahl (2002)). Once havhav- desired chanical systems systems can be be modeled modeled using the the Euler-Lagrange desired performance index. formalism (see Egeland and Gravdahl (2002)). Once having the dynamical model one can analyze the system performance index. the design of the controller ing the dynamical model can analyze system formalism (see Egeland and one Gravdahl (2002)).the Once hav- desired Yet, for perturbed systems, Yet, for perturbed systems, the design of the controller ing dynamical model can analyze the and design control that allow electromechanical and design control laws laws thatone allow the electromechanical ing the the dynamical model one canthe analyze the system system Yet, the the ceases to be if are (most ceases to perturbed be optimal optimal systems, if the the disturbances disturbances areof unknown (most Yet, for for perturbed systems, the design design ofunknown the controller controller and design control laws that allow the electromechanical system to have a desired performance. system to have a desired performance. and design control laws that allow the electromechanical ceases to be optimal if the disturbances are unknown (most cases), in this case, the controller turns sub-optimal and cases), in this case, the controller turns sub-optimal and ceases to be optimal if the disturbances are unknown (most system to performance. system to have have aa desired desired performance. cases), in this case, the controller turns sub-optimal and can reject perturbations that are bounded. Controllers The classification of the controllers can be done by its reject perturbations that are turns bounded. Controllers cases), in this case, the controller sub-optimal and The classification of the controllers can be done by its can can reject perturbations that are bounded. Controllers that use the H norm have the advantage of ensuring The classification of the controllers can be done by its independent time variable, which can be expressed in con∞ use the H∞ norm have ensuring can reject perturbations that the are advantage bounded. of Controllers independent time variable, which can be The classification of the controllers canexpressed be done inbyconits that that the use the norm have advantage transfer function matrix interest independent time can be expressed in continuous (Chavez-Guzm´ a al., time ∞ the peaks in transfer function matrixof ofensuring interest use peaks the H Hin norm have the the advantage ofof ensuring tinuous (Chavez-Guzm´ an n et etwhich al., 2015) 2015) or discrete time (Osindependent time variable, variable, which can or be discrete expressed in (Oscon- that the ∞ the that the peaks in the transfer function matrix of are knocked down (see Ball and Cohen (1987)). tinuous (Chavez-Guzm´ a n et al., 2015) or discrete time (Osuna et al., 2015). Strictly speaking, all the controllers are down (seetransfer Ball andfunction Cohen (1987)). thatknocked the peaks in the matrix of interest interest una et al., 2015). Strictly speaking, controllers are are tinuous (Chavez-Guzm´ an et al., 2015)allorthe discrete time (Osare down (see Ball Cohen una Strictly speaking, all are implemented in since are programmed are knocked knocked down (see Ball and and Cohen (1987)). (1987)). implemented in discrete discrete time, since they they arecontrollers programmed una et et al., al., 2015). 2015). Strictlytime, speaking, all the the controllers are Robots are a wide selection of interest cases within Robots are a wide selection of interest cases within elecelecimplemented in time, they are in a nevertheless, if period in a microcontroller, microcontroller, nevertheless, if the the sampling period is is Robots implemented in discrete discrete time, since since theysampling are programmed programmed are aa wide selection of interest cases within electromechanical systems, and among robots, bipedal robots tromechanical systems, and among robots, bipedal robots Robots are wide selection of interest cases within elecin a microcontroller, nevertheless, if the sampling period is small enough can controller to small enough the the system system can consider consider the controller to be be in a microcontroller, nevertheless, if thethe sampling period is tromechanical systems, and among robots, bipedal robots are particularly interesting. One of the major challenges are particularlysystems, interesting. One ofrobots, the major challenges tromechanical and among bipedal robots small enough the system can consider the controller to be continuous in time from its perspective of time. However, continuous in the timesystem from its of controller time. However, small enough canperspective consider the to be are particularly interesting. the challenges for robots research the of for bipedal robots research is isOne the of synthesis of controllers controllers are bipedal particularly interesting. One ofsynthesis the major major challenges continuous in from of some low-budget may the sampling some low-budget microprocessor may limit limit the However, sampling for continuous in time timemicroprocessor from its its perspective perspective of time. time. However, bipedal robots research is the synthesis of controllers that allow to perform walking routines. that allow torobots perform walking routines. for bipedal research is the synthesis of controllers some low-budget microprocessor may limit the sampling period, as well as some sensors that work at frequencies period, as well asmicroprocessor some sensors that some low-budget maywork limit at thefrequencies sampling that allow to perform walking routines. that bipedal allow to perform walking routines. period, well some sensors that work frequencies relatively In situation the relatively slow. In these these cases, the situation demands the The The bipedal robot robot walking walking process process can can be be divided divided into into two two period, as asslow. well as as some cases, sensorsthe that work at atdemands frequencies relatively slow. In these cases, the situation demands the The bipedal robot walking process can divided into analysis to be in discrete time, because it represents better main tasks: the path generation, and the analysis be inIn discrete time, because it represents better tasks: the path generation, and thebe reference tracking relativelytoslow. these cases, the situation demands the main The bipedal robot walking process can bereference divided tracking into two two analysis to discrete because represents tasks: generation, the behavior couple formed by for each link of the the behavior ofin the coupletime, formed by the theit electromechanical for each linkthe of path the robot. robot. analysis to be beof inthe discrete time, because itelectromechanical represents better better main main tasks: the path generation, and and the the reference reference tracking tracking the of couple system and controller. system and the the controller. the behavior behavior of the the couple formed formed by by the the electromechanical electromechanical for for each each link link of of the the robot. robot. Regarding Regarding path path generation generation there there exist exist different different algorithms algorithms system and the controller. system and the controller. Regarding path generation there exist algorithms One of the main performance indices of a control law in the literature. There are the ones that the One of the main performance indices of a control law in the literature. There are theexist onesdifferent that consider consider the Regarding path generation there different algorithms One of the main performance indices of a control law in the literature. There are the ones that consider the is its robustness against perturbations. It is required for entire dynamics of the robot precise and require precise is its of robustness perturbations. for entire the robot precise precise One the mainagainst performance indices It of isa required control law in the dynamics literature.ofThere are the onesand thatrequire consider the is its robustness against perturbations. It is required for entire dynamics of the robot precise and require precise the control law to be able to reject the effect of this knowledge of the robot mass, location of center of mass the control law to be able to reject the effect of this of theofrobot mass, precise locationand of center mass is its robustness against perturbations. It is required for knowledge entire dynamics the robot requireofprecise the to to effect robot center mass perturbations while it maintains itself confined to the and inertia each link to able generate walking perturbations while it able maintains itselfthe confined to this the knowledge and inertiaof ofthe each link mass, to be belocation able to to of generate walking the control control law law to be be able to reject reject the effect of of this knowledge ofof the robot mass, location of center of of mass perturbations while it maintains itself confined to the and inertia of each link to be able to generate walking physical restrictions imposed by the configuration and patterns, such as Yamaguchi et al. (1999), Hirai et physical restrictions imposed by the such as Yamaguchi al. to (1999), Hiraiwalking et al. al. perturbations while it maintains itselfconfiguration confined to and the patterns, and inertia of each link to beetable generate physical imposed the such Yamaguchi et Hirai et al. features the (1998), Kagami et (2000), al. And, features ofrestrictions the electromechanical electromechanical system. (1998), Kagami et al. al. (2000), Huang Huang et(1999), al. (2001). (2001). And, on physical of restrictions imposed by bysystem. the configuration configuration and and patterns, patterns, such as as Yamaguchi et al. al.et (1999), Hirai et on al. features of the electromechanical system. (1998), Kagami et al. (2000), Huang et al. (2001). And, on the other hand, there are the algorithms that use limited the other hand, there are the algorithms that use limited This work was electromechanical supported by CONACYT, M´ exico, under grant features of the system. (1998), Kagami et al. (2000), Huang et al. (2001). And, on This work was supported by CONACYT, M´ exico, under grant the the that limited knowledge of robots dynamics e.g. of total 300959, 301068 andsupported 252405. by CONACYT, M´ work was exico, under grant knowledge of the thethere robotsare dynamics e.g. location location of the the total the other other hand, hand, there are the algorithms algorithms that use use limited This 300959, 301068 andsupported 252405. by CONACYT, M´ This work was exico, under grant knowledge of the robots dynamics e.g. location of the 300959, 301068 and 252405. knowledge of the robots dynamics e.g. location of the total total
300959, 301068 and 252405. Copyright © 2017, 2017 IFAC 11853Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2017 IFAC 11853 Copyright ©under 2017 responsibility IFAC 11853 Peer review of International Federation of Automatic Control. Copyright © 2017 IFAC 11853 10.1016/j.ifacol.2017.08.1698
Proceedings of the 20th IFAC World Congress 11362 L.P. Osuna-Ibarra et al. / IFAC PapersOnLine 50-1 (2017) 11361–11366 Toulouse, France, July 9-14, 2017
center of mass, center of gravity, zero moment point, etc, like Sugihara et al. (2002), Kajita et al. (2003), Wieber (2006), Williams et al. (2013). In order to obtain the references for each link of the robot it is necessary the use of methods such as the analytic solution of inverse kinematics (Kajita et al., 2014). Once having the trajectories to track, it is designed a control law in order to do the tracking, which is required to be robust against disturbances. These performance can be achieved by means of a sub-optimal control law using the H∞ norm. In this work we present the development of a sub-optimal discrete time control law using H∞ norm for electromechanical systems modeled by Euler-Lagrange. This law is applied to a bipedal robot in order to do reference tracking for each link actuator. The main contribution of this paper is the development of a sub-optimal discrete-time H∞ controller for electromechanical systems, unlike Loukianov et al. (2009) that works in continuous time. This controller is applied to a bipedal robot to do the tracking for joint walking references. In Section 2 the mathematical background is presented, this includes a brief description of the H∞ control technique, including differential game theory and the development of the discrete time Hamilton-Jacobi-Isaacs equation. In Section 3 the main problem to solve is stated and the conditions for the system and the disturbance are asserted. Section 4 deals with the synthesis of the robust discrete time H∞ controller for a perturbed electromechanical system modeled by Euler-Lagrange formulation. Section 5 shows the application of the H∞ control to a bipedal robot in order to track the references for each joint in a walking routine. In Section 6 there is made a comparison of our proposed controller with a controller from the literature that uses H∞ in continuous time, the simulation results are shown and the Mean Squared Error is also calculated and displayed. Finally, the conclusions and future work are presented in Section 7. 2. MATHEMATICAL BACKGROUND 2.1 H∞ -Control and Differential Game Theory. The role of game theory in the design of robust (minimax) controllers is stated in Basar and Bernhard (2008). The goal is to minimize a given performance index under worst possible perturbations or parameter variations (which maximize the same performance index). This gametheoretic approach requires the setting of differential games, which seems to be the most natural. This is so because the original H∞ -optimal control problem is a minimax optimization problem, and hence a zero-sum game, where the controller is viewed as the minimizing player 1 and the perturbation as the maximizing player 2. Consider the following system: e¯i,k+1 =Ai e¯i,k + Bi υi,k + Bi f¯i,k zi,k =Ci e¯i,k + Di υi,k , with the finite horizon cost N J(υi,k , f¯i,k ) = ( zi,k − γi2 f¯i,k ), k=0
(1)
∗ which will be minimized by the control υi,k = υi,k ∗ (player1) and maximized by the perturbation f¯i,k = f¯i,k (player2), respectively; where zi,k is the unknown output to be controlled, i = 1, 2, CiT Di = 0 and γ is a positive real number γ > 0.
Suppose there exists a positive definite matrix Pi , and a positive function V (¯ eik ) = 12 e¯Ti,k Pi e¯i,k . Then, the discrete time Hamilton-Jacobi-Isaacs equation results ∗ ∗ + Bi f¯i,k ) − V (¯ ei,k )+ Hi =V (Ai e¯i,k + Bi υi,k 1 ∗ ∗ + ( Ci e¯i,k + Di υi,k − γi2 f¯i,k )<0 2 ∂ 2 Hi Ri,11 Ri,12 (2) Ri = = = Ri,21 Ri,22 ∂(υi,k , f¯i,k )2 T BiT Pi Bi B P B + DiT Di = i i Ti , B i Pi B i BiT Pi Bi − γi2 I such that L1) Ri,11 > 0 and Ri,22 − Ri,21 Ri,11 −1 Ri,12 < 0.
L2) The following discrete algebraic Riccati equation holds, i.e., (3) Pi = ATi Pi Ai + CiT Ci − T T GT + I, where T Bi Pi Ai , T = BiT Pi Ai ¯ ¯ iT Pi Bi − γi2 I)−1 G −G(B G= ¯ T Pi Bi − γ 2 I)−1 G ¯ −1 (BiT Pi Bi − γi2 I)−1 , −G(B i i ¯ = (BiT Pi Bi + DiT Di )−1 . G
L3) The control gain is given by (4) Fi = −(Bi Pi Bi + DiT Di )−1 BiT Pi Ai . The proof of stability can be found in Lin and Byrnes (1996). 3. STATEMENT OF PROBLEM Consider an electromechanical system modeled via EulerLagrange formulation for n-DOF x1,k+1 =x1,k + δx2,k (5) x2,k+1 =x2,k + δM (x1,k )−1 (−C(x1,k , x2,k )x2,k − − G(x1,k ) + τk + fk ), where x1,k ∈ Rn is the position vector, x2,k ∈ Rn is the velocity vector, fk ∈ Rn is a bounded external perturbation, τk ∈ Rn is torque control, M , C and G are the matrices of the system and δ is a constant sample period. Furthermore it is supposed that the following assumptions are fullfiled by system (5): Assumption 1. The system (5) is fully actuated and the M (x1,k ) matrix has full rank and its inverse M (x1,k )−1 exists ∀x1,k . x Assumption 2. The state vector xk = 1,k is available x2,k for the measurement. The objective of this paper is to synthesize a robust suboptimal discrete time controller using H∞ with the gametheoretic approach for the system (5).
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4. ROBUST TRACKING CONTROL USING H∞ In this section is developed a tracking control using the Block Control (BC) technique (Loukianov, 2002) and H∞ described in Section 2. Step 1. The state x1,k of the system (5) will track the reference x1ref . In this work we assume the references to be unknown and to fill the following condition: dx1ref (t) ≤ d2 , (6) x1ref (t) ≤ d1 , dt
where x1ref (t) is the reference, d1 > 0 is a constant and d2 > 0 is a constant. These bounds are assumed to be known. The tracking error is presented as e1,k = x1,k − x1ref,k .
The dynamics for error e1,k is obtained using (5).The introduction of the integral term e01,k+1 results in e01,k+1 =e01,k + δe1,k e1,k+1 =e1,k + δx2,k + f¯1,k ,
(7)
where f¯1,k is the derivative of the reference, considered as an unknown unmatched perturbation. Following the BC technique, the variable x2,k , considered in (7) as a virtual control, is selected to compensate the known terms and to introduce the new desired dynamics (8) δx2,k = −e1,k + L0 e01,k + L1 e1,k + υ1,k , where υ1,k is an auxiliary virtual control, that will be designed to reject the unknown perturbation f¯1,k . Thus, substituting (8) in (7), yields e (9) e¯1,k+1 = 01,k+1 = A1 e¯1,k + B1 (υ1,k + f¯1,k ), e1,k+1 with
A1 =
In δIn 0 , B1 = n . L0 L1 In
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Step 2. Using the transformation (12), a straightforward algebraic manipulation reveals e2,k+1 = g(¯ e1,k , e2,k ) + δ 2 M (·)−1 τk + f¯2,k , (15) where g(·) = − [δM (·)−1 C(·) + (L1 + F1 − In )](L0 + F0 )e01,k − − [δM (·)−1 C(·)(L1 + F1 − In ) + δ(L0 + F0 )+ + (L1 + F1 − In )2 ]e1,k − [δM (·)−1 C(·)+ + (L1 + F1 − 2In )]e2,k − δ 2 M (·)−1 G(·), f¯2,k = − (L1 + F1 − In )f¯1,k + δ 2 M (·)−1 fk . Let us now choose the control τk in (15) to cancel the dynamical term g(¯ e1,k , e2,k ) and introduce the new dynamics L2 e02,k + L3 e2,k , i.e., δ 2 M (·)−1 τk = −g(¯ e1,k , e2,k )+L2 e02,k +L3 e2,k +υ2,k , (16) where υ2,k is a auxiliary control and L2 and L3 are design parameters used to alleviate solving (3). Introducing the integral term e02,k+1 = e02,k + δe2,k , and substituting (16) in (15) results in e02,k+1 = A2 e¯02,k + B2 (υ2,k + f¯2,k ), (17) e¯2,k+1 = e2,k+1
where
A2 =
In δIn 0 , B2 = n . L2 L3 In
Using the solution F2 of (4), the auxiliary control υ2,k is defined as υ2,k = F2 e¯2,k . (18) Now, substituting (18) in (17) yields e2,k + B2 f¯2,k . e¯2,k+1 = (A2 + B2 F2 )¯
(19)
The dynamical error system in close-loop (see (14) and (19)) is presented as follows e¯ ek + B f¯k , e¯k+1 = 1,k+1 = A¯ (20) e¯2,k+1
To give robustness to subsystem (9) against the perturbation f¯1,k , the optimal robust control technique from Lin and Byrnes (1996) is applied. The desired value υ1d,k of the virtual control υ1,k in (9) is thus chosen in the form (10) υ1d,k = F1 e¯1,k = F0 e01,k + F1 e1,k , where F1 result of (4) (see Subsection 2.1).
where
The design parameters L0 and L1 are additionally used to alleviate solving the Riccati equation (3).
From the sub-optimal H∞ control methodology (Lin and Byrnes, 1996), it follows that the feedback, which is designed based on proper solutions of the Riccati equations (3), attenuates the influence of the external disturbances fk on the output tracking error e1,k , e1,k = x1,k − x1ref,k .
If the new variable (11) e2,k = υ1,k − υ1d,k , is introduced, then substituting υ1,k = δx2,k + e1,k − L0 e01,k −L1 e1,k , (8) and (10) in (11) results in the following transformation: e2,k = δx2,k − (L0 + F0 )e01,k − (L1 + F1 − In )e1,k , (12) which is equivalent to δx2,k = (L0 + F0 )e01,k + (L1 + F1 − In )e1,k + e2,k . (13) By substituting (13) in (7) or its equivalent (9), one derives e1,k + B1 (e02,k + f¯1,k ). (14) e¯1,k+1 = (A1 + B1 F1 )¯
f¯ (A1 + B1 F1 ) A12 f¯k = ¯1,k , A = , 02n (A2 + B2 F2 ) f2,k 0n 0n B1 02n×n , B= . A12 = 0n In 02n×n B2
5. APPLICATION TO A 6-DOF BIPEDAL ROBOT The control designed in Section 4 is applied to a bipedal robot presented in Osuna et al. (2015). In Fig. 1 the scheme of this bipedal robot is presented. The parameters used for the simulation were δ = 0.01, the 06 06 T design parameter γi = 3.2, Ci = , Di = [06 I6 ] , 06 I6
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1.2 1 0.8
x1(rad)
0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 0
2
4
6
8 10 Time(s)
12
14
16
18
Fig. 2. Simulation results of tracking control using H∞ (16) applied to system (5) (the position in rad).
1
Fig. 1. Scheme of the bipedal robot. 0.95
= L2 = L3 = I6 with i = 1, 2 and 0 0 0 . 0 0 1
The control gain results now Fi = [λ0 I6 λ1 I6 ], i = 1, 2, λ0 = −1.4916, λ1 = −0.6203.
The reference for each link was obtained from a path generator that uses limited knowledge of the plant (Harada et al., 2004) and the analytic inverse kinematics solution (Kajita et al., 2014). Having the references for the ZMP and CoG the use of the analytic solution of the inverse kinematic proposed in (Kajita et al., 2014) gave us the set of references for each link of the bipedal robot. The results are show in Figs. 2, 3, 4, 5 and 6.
Fig. 2 shows the position vector x1,k and the reference vector x1ref,k ; the tracking error is presented in Fig. 4. In Fig. 3 is presented a zoom of Fig. 2. The control torque is shown in the Fig. 5 and the perturbation fk is presented in the Fig. 6, the perturbation is defined for simulation as follows, [0.1 0.2 0.3 0.3 0.2 0.1]T if 0 ≤ t < 5 s if 5 s ≤ t < 10 s [0 0 0 0 0 0]T fk = T [0.1 0.2 0.3 0.3 0.2 0.1] if 10 s ≤ t < 15 s [0 0 0 0 0 0]T if t ≥ 15 s. The perturbation fk emulates a pushing force in the bipedal robot. 6. COMPARISON WITH CURRENT CONTINUOUS TIME H∞ CONTROLLERS In the literature there can be found controllers for electromechanical systems (robots) that were designed using the H∞ control technique in continuous time (see Chavez-
x1(3)(rad)
L1 00 00 00 10 01 00
0.9
0.85
0.8 0
2
4
6
8 10 Time(s)
12
14
16
18
Fig. 3. Zoom of Fig. 2, the position tracking of q3 (rad). 0.6 0.4 0.2 e1(rad)
and, L0 = 100 0 1 0 0 0 1 I6 = 0 0 0 0 0 0 000
0 −0.2 −0.4 −0.6 −0.8 0
2
4
6
8 10 Time(s)
12
14
16
18
Fig. 4. The tracking error for bipedal robot position. Guzm´an et al. (2015) and Becerra-Vargas and MorgadoBelo (2012)). The aim of these controllers is to do tracking in each link of a manipulator robot, while rejecting the effect of external perturbations over the plant. The work by Chavez-Guzm´an et al. (2015) introduces a controller for a n-DOF manipulator robot in continuous time. Once obtained, the control law is applied to a 3DOF manipulator (shown in Fig. 7). The main difference is that the controller proposed in our work is designed in
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15 10
Torque(Nm)
5 0 −5 −10 −15 −20 0
2
4
6
8
10 Time(s)
12
14
16
18
Fig. 5. The input torque during tracking control (N m). 0.35
Fig. 7. The 3-DOF robot manipulator.
fk(Nm)
0.3 0.25
4
0.2
3.5 3
0.15
2.5 Rad
0.1
2
0.05 1.5
0 0
2
4
6
8 10 Time(s)
12
14
16
18
1 0.5 0
Fig. 6. The disturbance fk (N m). discrete time for the discretized model and applied to the continuous time system. In order to compare the performance of the controller proposed in this work with the controllers in the literature, it was applied to the 3-DOF manipulator under the perturbation πcos(2πt) ω(t) = πcos(2π2t) N m πcos(2π3t) which is the one used in Chavez-Guzm´ an et al. (2015) with an amplitude 10 times bigger. The tracking for each joint reference is presented in the Fig. 8. The dotted line is the tracking performed by the work of Chavez-Guzm´ an et al. (2015) The continuous line is the tracking using our proposed controller, the dashed line is the reference for each joint given in Rad. From Fig. 8 is observed that our proposed controller reaches the reference faster and that the disturbance is better rejected, since the effect of it is more noticeable when the controller proposed by Chavez-Guzm´ an et al. (2015) is applied. The reference vector is the following: π x1ref = 3π/4 rad. 5π/9 The Fig. 9 shows the Mean Squared Error (MSE) for each tracking control. The dashed line is the MSE for the tracking control using the controller proposed by ChavezGuzm´ an et al. (2015) and the solid line is the MSE using our controller.
0.5
1
1.5 Time(s)
2
2.5
3
Fig. 8. Comparison of our work against continuous time H∞ controller from Chavez-Guzm´an et al. (2015). 2.5
MSE our work MSE Chavez−Guzmán et al.
2
1.5
1
0.5
0 0
0.5
1
1.5 Time(s)
2
2.5
3
Fig. 9. Mean squared error. 7. CONCLUSIONS In this work a robust discrete-time H∞ tracking control for electromechanical systems modeled using the EulerLagrange formulation was presented. The controller was derived making use of the discrete equations of HamiltonJacobi-Isaacs, and, the Riccati equations. The proposed control law was designed in two steps by means of the Block Control technique. The control law is robust and capable of rejecting unknown external disturbances while maintaining acceptable control signals.
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Although it is necessary for the derivative of the reference to be bounded, the controller does not need to know the value of this bound, meaning that the derivative of the reference may be unknown. The H∞ discrete-time controller was formulated from the perspective of the differential game theory, considering the control signal as one player, and the disturbance as the second player, the role of one player is to maximize a given performance index, while the other player tries to minimize this index. The controller was applied in simulation to a discretetime bipedal robot model. The aim of the controller was to do the tracking for reference signals for each link of the biped. These signals were obtained from a path generator and the use of the analytic solution of the inverse kinematics. The system was simulated under an external force as disturbance. The results displayed the successful disturbance rejection, the plots showed the controller to be ultimately bounded for non-vanishing disturbances. Finally, the performance of the discrete-time controller was compared with the performance of a continuous-time H∞ controller from the literature. The comparison was made using the model of a 3-DOF manipulator. From the simulation it was observed that the controller proposed in this work reached the reference faster with a smaller error, even though the controllers were both applied to a continuous-time system. The application to robots with a higher number of DOF and the implementation in real time is left as future work, as well as an extension of the control law in order to do adaptive control, in order to apply it to plants with unknown parameters. REFERENCES Ball, J.A. and Cohen, N. (1987). Sensitivity minimization in an H∞ norm: parametrization of all suboptimal solutions. International Journal of Control, 46. Basar, T. and Bernhard, P. (2008). H∞ Optimal Control and Related Minimax Design Problems: A Dynamic Game Approach. Birkhuser Basel. Becerra-Vargas, M. and Morgado-Belo, E. (2012). Application of H∞ theory to a 6 dof flight simulator motion base. Journal of the Brazilian Society of Mechanical Sciences and Engineering, 34(2), 193–204. Chavez-Guzm´ an, C.A., Aguilar-Bustos, L.T., and M´eridaRubio, J.O. (2015). Analysis and synthesis of global nonlinear H∞ controller for robot manipulators. Mathematical Problems in Engineering, vol. 2015, doi:10.1155/2015/410873. Egeland, O. and Gravdahl, J.T. (2002). Modeling and simulation for automatic control, volume 76. Marine Cybernetics Trondheim, Norway. Harada, K., Kajita, S., Kaneko, K., and Hirukawa, H. (2004). An analytical method on real-time gait planning for a humanoid robot. In Humanoids. Hirai, K., Hirose, M., Haikawa, Y., and Takenaka, T. (1998). The development of honda humanoid robot. Proceedings of the 1998 IEEE International Conference on Robotics and Automation, 2, 1321–1326. Huang, Q., Yokoi, K., Kajita, S., Kaneko, K., Arai, H., Koyachi, N., and Tanie, K. (2001). Planning walking
patterns for a biped robot. IEEE Transactions on Robotics and Automation, 17, 280–289. Kagami, S., Nishiwaki, K., Kitagawa, T., Sugihiara, T., Inaba, M., and Inoue, H. (2000). A fast generation method of a dynamically stable humanoid robot. Proceedings of the IEEE International Conference on Humanoid Robots. Kajita, S., Kanehiro, F., Kaneko, K., Fujiwara, K., Harada, K., Yokoi, K., and Hirukawa, H. (2003). Biped walking pattern generation by using preview control of zeromoment point. Proceedings of the IEEE International Conference on Robotics and Automation, ICRA 03, 2, 1620–1626. Kajita, S., Hirukawa, H., Harada, K., and Yokoi, K. (2014). Introduction to Humanoid Robotics. Springer-Verlag Berlin Heidelberg. Lin, W. and Byrnes, C.I. (1996). H∞ -control of discretetime nonlinear system. IEEE Transactions on Automatic Control, 41(4), 494–510. Loukianov, A.G. (2002). Robust block decomposition sliding mode control design. Math. Probl. Eng., 8(4-5), 349–365. Loukianov, A.G., Rivera, J., Orlov, Y.V., and MoralesTeraoka, E.Y. (2009). Robust trajectory tracking for an electrohydraulic actuator. IEEE Transactions on Industrial Electronics, 56(9), 3523–3531. Osuna, L., Caballero, H., Loukianov, A., CarbajalEspinosa, O., and Bayro-Corrochano, E. (2015). Continuous and discrete time robust control for bipedal robot assuming minimal knowledge of the plants. 2015 IEEE-RAS 15th International Conference on Humanoid Robots (Humanoids), 959–964. Sugihara, T., Nakamura, Y., and Inoue, H. (2002). Realtime humanoid motion generation through ZMP manipulation based on inverted pendulum control. Proceedings of the IEEE International Conference on Robotics and Automation, ICRA 02, 2, 1404–1409. Wieber, P.B. (2006). Trajectory free linear model predictive control for stable walking in the presence of strong perturbations. 2006 6th IEEE-RAS International Conference on Humanoid Robots, 137–142. Williams, M.M., Loukianov, A., and Baryro-Corrochano, E.J. (2013). ZMP based pattern generation for biped walking using optimal preview integral sliding mode control. Proceedings of 2013 13th IEEE-RAS International Conference on Humanoid Robots (Humanoids), 100–105. Yamaguchi, J., Soga, E., Inoue, S., and Takanishi, A. (1999). Development of a bipedal humanoid robotcontrol method of whole body cooperative dynamic biped walking. Proceedings of the 1999 IEEE International Conference on Robotics and Automation, 1, 368– 374.
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