# Symbolic computation for the Determination of the Minimal direct kinematics Polynomial and the Singular configurations of parallel manipulators

• J.-P. Merlet
Conference paper

## Abstract

In this paper we will address the problems of parallel manipulator’s direct kinematics (i.e. find the position and orientation of the mobile plate as a function of the articular coordinates) and the determination of their singular configurations. We will show how symbolic computation has been used to determine the minimal degree of a polynomial formulation of the direct kinematic problem together with this polynomial and to calculate all the necessary and sufficient conditions which are to be satisfied by the position and orientation of the mobile plate to get a singular configuration.

We consider a 6 d.o.f. manipulator in the case where the mobile plate is a triangle and the links lengths are time-varying. Using geometrical considerations and symbolic computation we are able to show that an upper-bound of the maximum number of assembly-modes and thus the maximum number of solutions of the direct kinematics problem, is 16. We describe then how symbolic computation has been used to reduce the direct kinematics problem as the solution of a sixteenth order polynomial in one variable. Using a numerical procedure we show that this polynomial may have 16 real roots and we exhibit one example for which the maximum number of assembly mode is reached.

Then we consider shortly the problem of singular configuration which can be reduced to determine some special geometric conditions to be fulfilled by the link’s lines. We show how a geometric package can be used to determine necessary and sufficient conditions for the position and orientation of the mobile plate so that the manipulator is in a singular configuration.

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