The piecewise constant method in gait design through optimization

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Abstract：The objective of this paper is to introduce the piecewise constant method in gait design of a planar， under actuated， five-link biped robot model and to discuss the advantages and disadvantages. The piecewise constant method transforms the dynamic optimal control problem into a static problem.

Index Terms：gait design；the piecewise constant method；under actuated；biped robot

1 INTRODUCTION

Biped robots have been widely researched in many areas such as industry and medicine. The motion of biped robots is similar to human. The most important goal of the research of biped robots is to achieve its stable walking， because it is a complex multi-body system， which is difficult to guarantee the stability of the movement.

A planar biped walker is a robot that locomotes via alternation of two legs in the sagittal plane. The models for such robots are necessarily hybrid， consisting of ordinary differential equations to describe the motion of the robot when only one leg is in contact with the ground， and a map to model the impact when the second leg touches the ground[1].

The studied biped robot consists of a torso and two legs with knees but no feet， more precisely， under-actuated system. To produce a complete gait with minimized input energy， low-energy trajectories for biped robots are been researched in different methodologies. L. Roussel & A. Goswami[3] transform the dynamic optimal control problem into a static problem via a piecewise constant method.[5] gives a comparison of the piecewise constant method， Polynomial Approximation and Fourier Approximation. C. Chevallereau[2] [4]described the joint evolution by a polynomial function of s， where s is a monotonic function from 0 to 1 and presents the time conception.

Section 2 presents the dynamic model of the biped robots. Section 3 is devoted to the piecewise constant method for gait design. Section 4 gives the conclusions.

2 ROBOT MODEL

The robot is modelled on the sagittal plane. It consists of a torso， hips and two identical legs with knees， but no ankles. The knees and hips are degree of freedom rotational ideal joints. The cyclic gaits are composed of single support phases separated by impact phases. So the complete model of biped robot consists of two parts：

2.1 SINGLE SUPPORT PHASE MODEL

2.2 IMPACT MODEL

4 CONCLUSION

For the piecewise constant method， the continuous input torque is assumed as piecewise constant. This assumption transforms the dynamic optimal control problem into a static problem， i.e. the complex dynamic model is largely simplified by this method. Thus it’s relative easier to calculate the solution of optimal trajectory. And the time of convergence in the optimization process is reduced. But on the other side， the coupling relationships of the joints are ignored due to the simplification of dynamic model. This may lead to a unnatural gait.

[REFERENCE]

[1]Westervelt E R， Grizzle J W，Koditschek D E.Hybrid zero dynamics of planar biped walkers[J].Automatic Control，IEEE Transactions on，2003，48（1）：42-56.

[2]Djoudi D，Chevallereau C，Aoustin Y.Optimal reference motions for walking of a biped robot[C].Robotics and Automation， 2005.ICRA 2005.Proceedings of the 2005 IEEE International Conference on.IEEE，2005：2002-2007.

[3]Roussel L，Canudas-de-Wit C，Goswami A.Generation of energy optimal complete gait cycles for biped robots[C].Robotics and Automation，1998.Proceedings.1998 IEEE International Conference on.IEEE，1998，3：2036-2041.

[4]Chevallereau C，Aoustin Y.Optimal reference trajectories for walking and running of a biped robot[J].Robotica，2001， 19（5）：557-569.

[5]Roussel L，Canudas-de-Wit C，Goswami parative study of methods for energy-optimal gait generation for biped robots[J].published by st.petersburg institute for informatics and automation of the russian academy of sciences （spiiras）， 1997：1213.