From Nominal To Robust Planning: The Plate-Ball Manipulation System Giuseppe Oriolo ∗ Marilena Vendittelli ∗ Alessia Marigo ⋄ Antonio Bicchi † ∗ Dipartimento di Informatica e Sistemistica, Universit`a di Roma “La Sapienza” Via Eudossiana 18, 00184 Roma, Italy, {oriolo,vendittelli}@dis.uniroma1.it ⋄ IAC-CNR, Viale del Policlinico 137, 00161 Roma, Italy, marigo@iac.rm.cnr.it † Interdept. Research Center “Enrico Piaggio”, Universit`a di Pisa Via Diotisalvi 2, 56100 Pisa, Italy, bicchi@ing.unipi.it Abstract Robotic manipulation by rolling contacts is an appeal- ing method for achieving dexterity with relatively sim- ple hardware. While there exist techniques for plan- ning motions of rigid bodies in rolling contact under nominal conditions, an inescapable challenge is the de- sign of robust controllers of provable performance in the presence of model perturbations. As a preliminary step in this direction, we present in this paper an itera- tive robust planner of arbitrary accuracy for the plate- ball manipulation system subject to perturbations on the sphere radius. The basic tool is an exact geometric planner for the nominal system, whose repeated appli- cation guarantees the desired robustness property on the basis of the Iterative Steering paradigm. Simula- tion results under perturbed conditions show the effec- tiveness of the method. 1 Introduction Rolling manipulation has recently attracted the inter- est of robotic researchers as a convenient way to design dextrous hands with simplified hardware (see [1, 2, 3] and the references therein). Here, dexterity indicates the capability to relocate and reorient a manipulated object by maintaining a firm grasp on it. A first proto- type of a hand purposefully implementing rolling ma- nipulation was presented in [2]. The nonholonomic na- ture of rolling contacts between rigid bodies guarantees the generic controllability of rolling pairs, i.e., that any two surfaces (with the only exception of surfaces that are mirror images of each other) can be arbitrarily re- oriented and relocated by rolling. While such result is limited to smooth surfaces, the case of a polyhedral object to be manipulated was considered in [4]. The archetypal example of rolling manipulation is the plate-ball system [5, 6, 7, 8]: the ball (the manipu- lated object) can be brought to any contact configura- tion by maneuvering the upper plate (the first finger), while the lower plate (the second finger) is fixed. De- spite its mechanical simplicity, the planning and con- trol problems for this device already raise challenging theoretical issues. In fact, in addition to the well- known limitations due to nonholonomy (essentially, the lack of smooth stabilizability), the plate-ball sys- tem is neither flat nor nilpotentizable; therefore the classical techniques (e.g., see [9]) for planning and sta- bilization of nonholonomic systems cannot be applied. To this date, only the planning problem has been attacked with some success; e.g., see the algorithms in [1, 6]. Like for any planner based on open-loop con- trol, however, the successful execution of maneuvers is not preserved in the presence of perturbations — some sort of feedback is necessary to induce a degree of ro- bustness. This advancement appears to be mandatory in order to fulfill the promise of rolling manipulation of providing a reliable technological solution. The final objective of our research is to move from planning to robust stabilization by exploiting the mechanism of iteration as proposed in [10], i.e., sam- pling the system state and repeatedly applying the same planner at discrete instants. In addition to the simplicity of design, this general stabilization approach (IS, or Iterative Steering) has the advantage of driving the system along the predictable trajectories typical of the planner. Such feature is particularly useful in the presence of configuration space constraints, e.g., due to workspace obstacles. A first step in the above direction was presented in [11], where we considered the problem of rolling a ball whose radius was only known up to some measure- ment error, and designed a robust controller for this system by iterating an approximate planner based on a nilpotent approximation of the dynamics. While sim- Proceedings of the 2003 IEEE International Conference on Robotics & Automation Taipei, Taiwan, September 14-19, 2003 0-7803-7736-2/03/$17.00 ©2003 IEEE 3175