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In this article we address the problem of the direct kinematics of parallel manipulators and the corollary problem of their assembly modes (i.e., the different ways of assembling these mechanisms when their geometry are fixed).

As an example we consider first a 6-DOF manipulator with a trianglular mobile plate and variable links lengths. A geometric proof is presented to show that the number of assembly modes is at most 16. Then we show that solving the direct kinematics problem is equivalent to solve a l6th-order polynomial in one variable, which is presented. We exhibit one example for which there are effectively 16 assembly modes.

We extend then the results of this example to various architec tures of parallel manipulators with a triangular mobile plate (among them the famous Stewart platform). Unfortunately the method used in those cases cannot be extended to the most general parallel manipulators. We …