Definition sets for the Direct Kinematics of Parallel Manipulators

A well known feature of parallel manipulators is the existence of multiple solutions to the direct kinematic problem. That is, the mobile platform can admit several positions and orientations (or configurations) in the workspace for one given set of input joint values [Merlet 90]. The dual problem arises in serial manipulators, where several input joint values correspond to one given configuration of the end-effector. To cope with the existence of multiple inverse kinematic solutions in serial manipulators, the notion of aspects was introduced [Borrel 86]. The aspects were defined as the maximal singularity-free domains in the joint space. For usual industrial serial manipulators, the aspects were found to be the maximal sets in the joint space where there is only one inverse kinematic solution. Many other serial manipulators, referred to as cuspidal manipulators, were shown to be able to change solution without passing through a singularity, thus meaning that there is more than one inverse kinematic solution in one aspect. New uniqueness domains have been characterized for cuspidal manipulators [Wenger 92], [El Omri 96]. It is also of interest to be able to characterize the uniqueness domains for parallel manipulators, in order to separate and to identify, in the workspace, the different solutions to the direct kinematic problem. To the authors knowledge, the only work concerned with this issue is that of [Chételat 96], which proposes a generalization of the implicit function theorem. Unfortunately, the hypothesis of convexity required by this new theorem is still too restrictive. This paper is organized as follows. Section 2 describes the planar 3-RPR parallel manipulator which will be used all along this paper to illustrate the new theoretical results. Section 3 restates the notion of aspect for parallel manipulators. New surfaces, the characteristic surfaces, are defined in section 4, which, together with the singular surfaces, further divide the aspects into smaller regions, called basic regions. Finally, the uniqueness domains are defined in section 5. The workspace, the aspects, the characteristic and singular surfaces, and the uniqueness domains are calculated for the planar 3-RPR parallel manipulator using octrees. The images in the joint space of the uniqueness domains are also calculated. It is shown that the joint space is composed of several subspaces with different numbers of direct kinematic solutions.

This work deals with those parallel manipulators which have only one inverse kinematic solution. In addition, the passive joints will always be assumed unlimited in this study. For more legibility, a planar manipulator will be used as illustrative example all along this paper. This is a planar 3-DOF manipulator, with 3 parallel RPR chains (Figure 1). The input joint variables are the three prismatic actuated joints. The output variables are the positions and orientation of the platform in the plane. This manipulator has been frequently studied, in particular by [Merlet 90], [Gosselin 91] and [Innocenti 92].

The kinematic equations of this manipulator are

The dimensions of the platform are the same as in [Merlet 90] and in [Innocenti 92] :

• A 1 = (0.0, 0.0) B 1 B 2 = 17.04

• A 2 = (15.91, 0.0) B 2 B 3 = 16.54

• A 3 = (0.0, 10.0) B 3 B 1 = 20.84

The limits of the prismatic actuated joints are those chosen in [Innocenti 92] :

The passive revolute joints are assumed unlimited. The octree models of the joint space (in the space ρ 1 , ρ 2 , ρ 3 ) and of the workspace (in the space x, y et φ) of the 3-RPR parallel manipulator are shown in figures 2 and 3. The joint space is not a complete parallelepiped, since any joint vector can’t lead to an assembly configuration of the manipulator.

The vector of input variables and the vector of output variables for a n-DOF parallel manipulator are related through a system of non linear algebraic equations which can be written as :

( )

where 0 means here the n-dimensional zero vector.

Differentiating (4) with respect to time leads to the velocity model :

where A and B are n × n Jacobian matrices. Those matrices are functions of q and X :

These matrices are useful for the determination of the singular configurations [Sefrioui 92].

These singularities occur when det(B) = 0.

For the planar manipulator, this condition can be satisfied only when ρ 1 = 0 or ρ 2 = 0 or ρ 3 = 0.

In practise, the type-1 singularities are attained when one of the actuated prismatic joints reaches its limit [Gosselin 90]. The corresponding configurations are located at the boundary of the workspace.

For parallel manipulators which may have more than one inverse kinematic solutions, type-1 singularities are configurations where two solutions to the inverse kinematic problem meet. By hypothesis, type-1 singularities will be always associated with joint limits in this paper.

They occur when det(A) = 0. Unlike the preceding ones, such singular configurations occur inside the

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