fulltext01 - Multibody Syst Dyn (2010) 23: 3356 DOI...

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Multibody Syst Dyn (2010) 23: 33–56 DOI 10.1007/s11044-009-9175-1 Design of a walking cyclic gait with single support phases and impacts for the locomotor system of a thirteen-link 3D biped using the parametric optimization David Tlalolini · Yannick Aoustin · Christine Chevallereau Received: 6 March 2008 / Accepted: 13 August 2009 / Published online: 23 September 2009 © Springer Science+Business Media B.V. 2009 Abstract The development of an algorithm of parametric optimization to achieve optimal cyclic gaits in space for a thirteen-link 3D bipedal robot with twelve actuated joints is pro- posed. The cyclic walking gait is composed of successive single support phases and impul- sive impacts with full contact between the sole of the feet and the ground. The evolution of the joints are chosen as spline functions. The parameters to deFne the spline functions are determined using an optimization under constraints on the dynamic balance, on the ground reactions, on the validity of impact, on the torques, and on the joints velocities. The cost functional considered is represented by the integral of the torques norm. The torques and the constraints are computed at sampling times during one step to evaluate the cost func- tional for a feasible walking gait. To improve the convergence of the optimization algorithm the explicit analytical gradient of the cost functional with respect to the optimization para- meters is calculated using the recursive computation of torques. The algorithm is tested for a bipedal robot whose numerical walking results are presented. Keywords 3D Bipedal robot · Robot dynamics · ±ully actuated robot · Newton–Euler algorithm · Cyclic walking gait · Parametric optimization 1 Introduction The design of walking cyclic gaits for legged robots and particularly the bipeds has at- tracted the interest of many researchers for several decades. Due to the unilateral constraints of the biped with the ground and the great number of degrees of freedom, this problem is not trivial. Intuitive methods can be used to obtain walking gaits as in [ 1 ]. Using physical D. Tlalolini · Y. Aoustin ( B ) · C. Chevallereau IRCCyN, UMR 6597, École Centrale de Nantes, Université de Nantes, Nantes, ±rance e-mail: Yannick.Aoustin@irccyn.ec-nantes.fr D. Tlalolini e-mail: David.Tlalolini-Romero@irccyn.ec-nantes.fr C. Chevallereau e-mail: Christine.Chevallereau@irccyn.ec-nantes.fr
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34 D. Tlalolini et al. considerations, the authors of [ 1 ] deFned polynomial functions in time for an experimental planar biped. This method is efFcient. However, to build a bipedal robot and to choose the appropriate actuators or to improve the autonomy of a biped, an optimization algorithm can lead to very interesting results. In [ 2 ], the Pontryagin’s principle is used to design impact- less nominal trajectories for a planar biped with feet. However, the calculations are complex and difFcult to extend to the 3D case. ±urthermore, the adjoint equations are not stable and highly sensitive to the initial conditions [ 3 ]. As a consequence, a parametric optimization is a useful tool to Fnd optimal motion. ±or example, in robotics, basis functions as polynomial
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fulltext01 - Multibody Syst Dyn (2010) 23: 3356 DOI...

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