where: q = [q 1 ] T : vector of generalized coordinates and q and q are vectors of joint angular velocity and acceleration.

Transcription

1 J Mech Behav Mater 15; 4(1-): Maime Sare* Control of a legge robot Abstract: This paper eals with the control of hopping an running systems that interact intermittently with the environment. The control, base on a nonlinear energy reference moel, has the main task of conferring to the system, a perioic stable behavior. This approach may be use for gait generation, nominal stance stabilization, energy shaping, an optimization. eywors: control of limit cycle; hopping robots; robotics. DOI /jmbm Introuction The stuy of legge robots benefits an increasing attention an control of its ifferent gaits (hopping, running, walking, etc.) are the subject of some researcher s efforts. The stuy of such systems is ifficult, owing to the nonlinear nature of equations, phase transitions (contact phase, flight phase), an interaction with varying or unknown environments. In oitschek et al. [1], an analysis of a simplifie hopping robot has been given an return maps inuce by the oscillatory ynamics of the system have been establishe. In Vakakis et al. [], a strange attractor intervening in the ynamics of a hopping robot has been presente. This work is the continuation of our previous investigations [3 6], which present some aspects of the system moeling an control principle. The classical methos employe to obtain ynamic fast gaits have not performe [7, 8] ue to system compleity, the varying nature of the groun, an the constraints impose on the system. Accoring to the control approach propose in this article, the system operates perioic motions that can be characterize by a limit cycle in the phase plane. The control approach is base on energetic optimization that makes the system trajectory ten towars the limit cycle. The evaluation of the energy reference moel oes not necessitate a great eal of calculation. Therefore, the trajectory evaluation can be one *Corresponing author: Maime Sare, Department of Mechatronics, University of Versailles-Saint-Quentin-En-Yvelines, ISTY, 8 Boulevar Roger Salengro, 7871 Mantes-La-Ville, France, maime.sare@uvsq.fr online to allow fast ynamic gaits. The article is organize as follows: Section presents the ynamic moel of the robot an an approach of control base on controlle limit cycles (CLC) is introuce. Section 3 provies some simulation an eperimental results both for hopping an forwar motion. The final section is evote to iscussions an conclusions. Dynamic moel of the robot Figure 1 illustrates the robot structure. The robot possesses two legs, each one compose of two links. Two wheels ensure the stability of the robot an two joints are actuate pneumatically. In this paper, we consier only the motion of one leg. In the case of a bipe robot, the equation of motion of the secon leg can be obtaine from that of the first one by linear transformation. This can be generalize to a multi-legge robot. The robot is equippe with three sensors: one gyrometer for acquisition of vertical velocity, one potentiometer permitting measurement of joint angle q 4 (knee), an one optic encoer for measurement of joint angle q 3 (hip). The whole system (robot, control, an environment) with the interactions an constraints between ifferent system elements has been illustrate in Figure. The ynamic moel of the robot can be obtaine by Lagrange equations: T T T q q = τ + τ + τ - f e where: q = [q 1 q q 3 q 4 ] T : vector of generalize coorinates an q an q are vectors of joint angular velocity an acceleration. T( q, q ) = E -E T( q, q ) is the system s lagrangian, E k an E p, respectively esignate kinetic an potential energy: τ: control input vector, only the joints q 3 an q 4 are actuate: τ [,, τ τ ] T =, 3 4 τ ( q, q ): vector of friction forces; τ : vector of groun f e reaction; M(q): (4 4) generalize inertia matri. The ynamic equation of the system can be written: k p

2 54 M. Sare: Control of a legge robot l q 3 q l 3 l z q 1 q 4 r θ l 3 Figure 1: Legge robot actuate pneumatically Système à Pattes (SAP). Gait parameters u Interaction CONTROL ROBOT ENVIRONMENT q, q, q Reaction Constraints is illustrate in Figure 3. The classical methos o not cope with esire performances ue to the nonlinear nature of the system, phase transitions (contact an flying phases), an uncertainty of environment parameters. Figure 4 illustrates the principle of gait stabilization an energy optimization. The control is operate in two stages. The first stage consists of a nominal stance stabilization when the robot s leg is in contact with the groun, this coul be obtaine by means of a PD controller: F = P ( -) with v = an = [ r, z, φ ] T : esire cartesian positions vector Variations Figure : Control-robot-environment interactions. M( q) q+ C( q, q) q+ g( q)-τ f = τ + τ e C( q, qq ) : (4 1) vector of centripetal an Coriolis forces; g(q): (4 1) vector of gravitational forces. The groun reaction forces shoul be evaluate in cartesian coorinates, so we efine the cartesian variables as: = [ r, z, φ] T ; r : horizontal position; z: vertical position; an φ: platform orientation. The corresponing robot ynamic equation in cartesian space can be obtaine by the use of geometric moel: δl = L( q) = q= Jq δq where J is the Jacobian matri. For the sake of simplicity, we omit the friction term in the equation. Thus, the system s equation in the Cartesian space can be epresse: M ( ) + C (, ) + g ( ) = ( - ) + F (1) Definition of gaits Generation of trajectory Definition of nominal positions p e The groun reaction force may be moelize by a mechanical impeance. F = -Z z - z e e e where Z e, e an Δz esignate respectively groun mechanical impeance, stiffness an eformation. Equation (1) can be rewritten: M ( ) C (, ) - g + + = ( ) () p Constraints M ( ) + C (, ) + g ( ) = F+ Fe Evaluation of velocities Environment 3 Control strategy When ealing with the control of legge robots, the problem is how we can aopt a methoology [9]. This methoology Definition of behavior Definition of control objectives Figure 3: Methoology for robot control system esign.

3 M. Sare: Control of a legge robot 55 Energy reference an gait efinition Motion evaluation ξ(z-u) Nominal stance + _ Stabilizing control ROBOT Figure 4: Gait stabilization an energy optimization. is the equivalent stiffness of the robot, groun an the control, in serial connection [6]. When the groun is very stiff: p The secon stage consists of an energy reference optimization, to achieve a controlle limit cycle. This proceure involves ifferent energy transformations: kinetic energy-gravity potential energy-elastic potential energy accumulation-elastic potential energy restitution-kinetic energy, an so on. In our approach the environment interaction: friction an groun reaction forces are taken into account. The nonlinear control law is efine as follows: = - sign( VV - ) z where V an V esignate, respectively, the system s energy an energy reference. an enote respectively control gain an equivalent system stiffness. The energy of the system can be formulate by: 1 T V= M ( ) g ( ss ) + The general form of the reference energy in terms of esire velocities can be epresse: V 1 T ( ) ( ) M g = + ss (3) In the case, where we are solely intereste in hopping, V can be epresse in terms of maimum jump height z m or maimum touchown velocity ( ż)m: V = g m = m z z.5 ( z ) z m m The jump height shoul be given beforehan. This permits to efine energy reference. The control shoul act to obtain a trajectory in the phase plane ( z, ż ) with a constant energy. This is compatible with the system properties. In [1], it has been illustrate that a hopping robot s behavior coul be euce from a mass-spring-amper system. This is a secon orer system, in which the resonance frequency epens on mass an stiffness. The angular frequency of resonance is equal to: ω = / The oscillation frequency epens on either the system parameters or the control gain. The trajectory of such a system in the phase plane, with an appropriate control law correspons to a limit cycle. A limit cycle is a close orbit corresponing to a region of attraction. In fact, in the propose approach, the generation of the trajectory is implicit. In the literature, some authors propose eplicit trajectory generation [8, 11]. In [1], the trajectory generation is eecute by a trajectory planner compose of a trajectory memory unit an an aaptive unit. In these approaches, the control pursues a pre-establishe trajectory, in orer that the tracking error with respect to reference trajectory tens to zero. Put into practice, such methos encounter some ifficulties when the environment is varying or unknown. To overcome this problem, the control must be robust with regar to moel parameters errors, environment uncertainties, an isturbances. 3.1 Stability analysis In the case of mechanical systems performing a perioic gait, we shoul efine the orbital stability that iffers somewhat from the classical efinition of stability. Definition: the system trajectory in the phase space R is a stable orbit if ε : δ such that M z - Ω δ inf z( t)- p ε, t t p Ω This means that all trajectories starting in the vicinity of Ω approach it uring a finite time. Let us efine a Lyapunov function that is the same as the energy of the system: 1 T V= M ( ) g ( ss ) +

4 56 M. Sare: Control of a legge robot Differentiating this function, we obtain: T 1 T = M ( ) + M ( ) + g ( s ) (4) A Controlle limit cycle z=f (zp).7.6 Substituting M ( ), alreay calculate by Eq. (), in Eq. (4): T 1 T = (--C+ - g ( )) + M + g ( ) As M * -C * is a skew-symmetric matri [13], we obtain finally: = ( - ) To achieve an stabilize the esire limit cycle, the following conition must be impose: ( VV - ) (5) ( VV - ) T ( - ) This conition can be satisfie if the control is of one of the following form: = - ( VV - ) = - [sign( VV - )] = - [sat( VV - )] (6) Consequently, the inequality (5), with the control law as efine by (6) becomes: ( VV - ) T (-sign[( V- V)] ) If V V V : the control shoul furnish energy to the system to compensate energy wasting, the control has then an active contribution. If V V V : the jump height is greater than z m, the global energy of the system shoul be ecrease, the control has then a passive (issipative) action. As emonstrate above, the close orbit efine by Ω = {(z, z) R, V(, ) = V } is asymptotically orbitally stable. 4 Simulation an eperimental results Figure 5 illustrates the eperimental results which match the simulation results. The control gains are p = ; v = 1.5; = 1. Figure 5A represents the limit controlle cycle for a esire jump height z m =.6 m. The steay state error on the maimum height is ue to stiffness an inertial D.5 u z (m) B z (m) F Platform Enpoint zp (m/s) Controlle limit cycle z=f (t) Control u=f (t) p (m/s) Figure 5: Eperimental results. C zp (m/s) E (m) zp=f (t) Forwar motion Forwar motion velocity p=f (t) 1 parameters errors. This error can be reuce by use of an integral action. The platform position, as well as the leg enpoint position, is epicte in Figure 5B. Figure 5C

5 M. Sare: Control of a legge robot 57 presents the platform velocity. We remark that the perio is nearly.8 s (epening on control gain). Figure 5D represents the control function with the control being inactive in the flight phase. The contact phase uration is very short compare to the flight phase uration. Figure 5E illustrates the position variations in forwar motion for an initial leg orientation angle θ =.4161 ra. Figure 5F illustrates the average velocity of the forwar motion, the stabilize velocity being attaine within nearly 7 s. 5 Conclusion In this paper, we presente a control approach for realizing fast gaits in the case of hopping or running robots. The control has for tasks: stabilization of a nominal stance an optimization of energetic behavior of the system. The former uses a partial PD feeback an the latter an energy reference moel, associate to a feeback, which is nonlinear in regar to positions an velocities. This leas to controlle limit cycles that characterize a robot gait. This approach is appropriate for implicit online trajectory generation, gait stabilization, an energy optimization. The control is robust vs. moel parameter uncertainties an isturbances [14 18]. References [1] oitschek DA, Büler M. Int. J. Robot. Res. 1991, 1, [] Vakakis AF, Burick JW, Caughey T. Int. J. Robot. Res. 1991, 1, [3] Manamani N, M Siri N, Najar-Gautier N. Methoology for Control of Legge Robots with Fast Dynamics. RoManSy98: Paris, [4] M Siri N, Manamani N, Najar-Gautier N. Control approach for hopping robots: controlle limit cycles. Proc IEEE AVCS98, Amiens, [5] M Siri N, Manamani N, Najar-Gautier N. Controlle limit cycles approach for control of legge robots. Proc IFAC Motion Control 98, Grenoble, [6] M Siri N, Manamani N, El Ghanamani D. J. Intell. Robot. Syst., 7, [7] Espiau B, Gooswami A, ermane A. Limit cycles an their stability in a passive bipeal gait. Proc. IEEE Int. Conf. on Robotics an Automation, Minneapolis, MN, [8] Roussel L, Canuas-e Wit C, an Goswami A. Generation of energy optimal complete gait cycles for bipe robots. IEEE Int. Conf. Rob. Aut. 1998, 3, [9] Channon P, Hopkins S, Pham D. Derivation of optimal walking motions for a bipeal walking robot. Robotica 199, 1, [1] Raibert MH, Sutherlan IE. Sci. Am. 1983, 48, [11] Chevallereau C, Formal sky A, Perrin B, Gosselin F. In: Proceeings of Worl Automation Congress, International Symposium on Robotics an Manufacturing Systems. Ballistic motions for a quarupe robot. IEE: Piscataway, NJ, USA, 1996, Vol. 3, pp [1] Yi Y. Walking of a bipe robot with compliant ankle joints. Proc. IROS 97, 1997, pp [13] Arimoto S, Myazaki F. In Robotic Research: The First International Symposium. MIT Press: Cambrige, MA, 1984, pp [14] M Siri N, Manamani N, Najar-Gautier N. Methoology base on CLC for control of fast legge robots. IROS 98, 1998, pp [15] To DJ. In Walking Machines An Introuction to Legge Robots. ogan Page: Lonon, [16] Nijmeijer H, Vaner Schaft AJ. In Nonlinear Dynamical Control Systems. Springer Verlag: New York, NY, 199. [17] Cau S, Zapata R. Robotica 1999, 17, [18] Silva FM, Tereira Machao JA. inematics aspects of robotic bipe locomotion systems. Proc. IROS 97, 1997, pp