Multibody Syst Dyn (2010) 23: 33–56
DOI 10.1007/s1104400991751
Design of a walking cyclic gait with single support phases
and impacts for the locomotor system of a thirteenlink
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 thirteenlink 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 define 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
·
Fully 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 (
)
·
C. Chevallereau
IRCCyN, UMR 6597, École Centrale de Nantes, Université de Nantes, Nantes, France
email:
[email protected]
D. Tlalolini
email:
[email protected]
C. Chevallereau
email:
[email protected]
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34
D. Tlalolini et al.
considerations, the authors of [
1
] defined polynomial functions in time for an experimental
planar biped. This method is efficient. 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 difficult to extend to the 3D case. Furthermore, 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 find optimal motion. For example, in robotics, basis functions as polynomial
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 Spring '10
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 Acceleration, Optimization, Velocity, Ground reaction force, David Tlalolini, single support

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