Parallel robots adopt exact computation through computer algebra
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| Six articulated legs moving a milling tool on a chuck tray
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Inédit : You work in collaboration with Jean-Pierre Merlet (project SAGA, INRIA) and engnçois Wildenberg, CEO of CMW (Constructions Mécaniques des Vosges-Marioni) in Epinal. What kind of problems do you study ?
F. R. : CMW is a new parallel manipulator manufacturer for ultra high speed milling. The goal is to maneuver a very large robot (approximately one meter high) and also heavy (weighing a few hundred kilograms) with an accuracy of a few microns. The main difficulty concerns movement control, since the moving tray position and orientation must be determined based on the length of the robot's six motorised legs. This is a complex geometric problem which can have up to 40 solutions. Now, in the best case, the existing computational methods used in digital control can only find one solution. They cannot always identify the right one or distinguish trajectories one from another when they are getting too close. The robot can therefore bifurcate on a different and hazardous trajectory. Our goal is to identify and correct the errors generated by the digital control components and to help in devising "driving" strategies that are able to minimize such errors.
Inédit : : In which way does your specialty, symbolic computation, bring a solution to these problems ?
F. R. : Symbolic computation avoids error accumulation during the computation course by retaining formal expressions as long as possible. For example, the square root of 2 is expressed as "the positive root of X2-2" instead of a necessarily incorrect numerical approximation. The core of our work in collaboration with Jean-Charles Faugère (University of Paris 6) consists in producing results in effective algebraic geometry which can be used to simplify polynomial equation systems and to allow for their solution study. In the case at hand, we do model the region in which we cannot have any robot control by a tube surrounding the robot's current trajectory. The tube diameter is made a function of the errors generated by the various control components, the user-imposed constraints and the trajectory control strategy. Our simulator can be used to minimize this error region by adjusting the robot's parameters and the numerical algorithms used to control it.
