Copyright © IFAC Robot Control, Nantes, France, 1997 PERIODIC STABILIZATION OF A I-DOF HOPPING ROBOT ON NONLINEAR COMPLIANT SURFACE C. Canudas· , L. Roussel· , A. Goswami·· • Laboratoire d 'A utomatique de Grenoble UMR-CNRS 5528, ENSIEG-INPG B.P. 46, 38402, St Martin d'Heres, France E-mail: canudas@lag.ensieg.inpg.fr •• INRIA Rhone-Alpes ZIRST, 655 ave. de l 'Europe 38330 Montbonnot St Martin, France Abstract: This paper deals with the problem of characterizing and stabilizing periodic orbits of a I-DOF hopping robot moving over a nonlinear compliant surface. It is shown, through implicit calculations of the Poincare map, that globally stable periodic motions are possible via the injection of non linear damping. Keywords: periodic stabilization, walking and hopping robots. 1. INTRODUCTION The system considered is the classic simple point mass bouncing on a surface or equivalently, an idealized I-dof hopping robot . The objective of this work is to introduce compliance in the robot/ground interface in order to control the interaction. The model can be seen as a rigid robot bouncing on a compliant surface or a compliant robot bouncing on a rigid surface. The current work stems from our efforts in the modeling and control of legged robots. While modeling a legged robot it is important to recall its principal differences with a conventional ma- nipulator arm, the differences which are of funda- mental importance but are sometimes overlooked and , in our opinion, not sufficiently emphasized. While a conventional manipulator arm is perma- nently rigidly fixed to the ground, a legged robot is not. More importantly the "constraint" between the robot foot and the ground is unilateral. Thus given appropriate joint torques the robot is free to leave the ground, which is an impossibility for a conventional arm . There are two common lines of approach in treat- ing the interaction of such a robot with the ground. The first is a rigid body approach where both the robot and the ground are idealized rigid bodies. A second approach, the one adopted here, is the modeling of the interaction as a compliant phenomenon. In this approach, we typically place 385 a spring/damper module on one of the two inter- acting objects. The advantage of this model is that it has continuous dynamics governed completely by differential equations and can be used to sim- ulate a variety of interacting surfaces by changing the spring/damper coefficients. In case of interactions involving non-zero rela- tive velocities between the interacting objects a linear damper produces a discontinuous jump in the interaction force during the first contact, a phenomenon which cannot be physically justified. Also, traditional spring-damper elements, which are active both in extension and compression, cannot correctly model the ground contact of ei- ther a bouncing ball or a legged robot since they would generate artificial tension forces pulling the interacting objects towards each other. In order to rectify the two first problem, we adopt the use of non-linear spring-damper elements fol- lowing (?) and (Marhefka and Orin., April 1996). The authors of both of these articles focus on the performance of the non-linear models in terms of their dynamic behavior. We use the model, in addition, to formulate simple control laws to stabilize a simple one-degree of freedom hopping robot about a desired periodic orbit . The char- acteristics of the walking robot is preserved here through the restriction of allowing only positive feedback. Given the height of jump altitude , the employed control law tries to bring the robot to