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Distributed Manipulation
Todd Murphey, Joel W. Burdick

Abstract.This research analyzes the stability of distributed manipulation control schemes. A commonly proposed method for designing a distributed actuator array control scheme assumes that the system's control action can be approximated by a continuous vector force field. The continuous control vector field idealization must then be adapted to the physical actuator array. However, we have shown that when one takes into account the discreteness of actuator arrays and realistic models of the actuator/object contact mechanics, the controls designed by the continuous approximation approach can be unstable. For this analysis we introduce and use a ``power dissipation'' method that captures the contact mechanics in a general but tractable way. We show that the quasi-static contact equations have the form of a multi-model hybrid system. We introduce a discontinuous feedback law can produce stability which is robust with respect to variations in contact state.

Research Results. A distributed manipulation system consists of an (roughly planar) array of actuators that can reposition an object by the movements of its array elements. In the future, arrays of this type should be useful for industrial assembly operations where small parts must be robustly transported and precisely positioned. This research considers the design of manipulation control strategies for such distributed systems. We focus on autonomous controllers that stabilize an object to a precise configuration equilibrium on the array.

First we show that when one takes into account the discrete nature of real actuator arrays and a fairly general model of the actuator-to-object contact mechanics, the control systems designed by the continuous approximation method can often be unstable when deployed on the actual array. This is not unexpected, as the programmable vector field approach is based on the restrictive assumption that the continuous vector field abstraction is a good approximation to the array's actual physical characteristics. This instability result has been previously shown for specific array geometries. This research generalizes work done by Luntz et al in that where they considered a specific model with smooth dynamics, we have instability results which take into account contact mechanics.

Secondly, we discuss a power dissipation methodology for modeling the array/object contact. This method, which is adapted from the work of Alexander and Maddocks in the area of wheeled vehicles, is based on the principle that an object will move in the direction that minimizes the power dissipation associated with moving. This method applies to fairly general types of array/object contact, and it results in tractable models. We formalized this approach, showing that actuator/object contact models take the form of a multi-model hybrid control system. This observation allows tools from non-smooth analysis and the study of differential inclusions to be applied to this problem.

Finally, we introduce a control scheme to stabilize an object on a distributed array, and use our power dissipation model to prove the scheme's stability in the quasi-static case. Our recent results have been primarily stability results for distributed systems. We have been able to show that locally, feedback is necessary in order to overcome the problems caused by changes in contact states. On the other hand, we have also been able to show that feedback is only required locally around an equillibrium point, which has practical implementation implications. Lastly, we have shown that global exponential stability (the holy grail of most control theory) can be achieved even with only local feedback.

Future Directions. We are currently building an experimental set up in our lab that has 18 actuators and uses a CCD camera for visual feedback. The purpose of the set up is to test the theories we have developed on an actual, reasonably simple, system. This test bed is nearly finished now.

Publications
Issues in Controllability and Motion Planning for Overconstrained Wheeled Vehicles. Conference on the Mathematical Theory of Networks and Systems, 2000. Perpginan, France.

On the Stability and Design of Distributed Manipulation Control Systems. International Conference on Robotics and Automation, 2001. Seoul, Korea.

Global Stability for Distributed Systems with Changing Contact States. International Conference on Intelligent Robotis and Systems, 2001. Maui, Hawaii, USA.