Regions of Feasible Point-to-Point Trajectories in the Cartesian Workspace of Fully-Parallel Manipulators

The goal of this paper is to define the n-connected regions in the Cartesian workspace of fully-parallel manipulators, i.e. the maximal regions where it is possible to execute pointto-point motions. The manipulators considered in this study may have multiple direct and inverse kinematic solutions. The N-connected regions are characterized by projection, onto the Cartesian workspace, of the connected components of the reachable configuration space defined in the Cartesian product of the Cartesian space by the joint space. Generalized octree models are used for the construction of all spaces. This study is illustrated with a simple planar fully-parallel manipulator. INTRODUCTION The Cartesian workspace of fully-parallel manipulators is generally defined as the set of all reachable configurations of the moving platform. However, this definition is misleading since the manipulator may not be able to move its platform between two prescribed configurations in the Cartesian workspace. This feature is well known in serial manipulators when the environment includes obstacles (Wenger, 91). For fully-parallel manipulators, point-to-point motions may be infeasible even in obstaclefree environments. For manipulators with one unique solution to their inverse kinematics (like Gough-platforms), one configuration of the moving platform is associated with one unique joint configuration and the connected-components of the singularityfree regions of the Cartesian workspace are the maximal regions of point-to-point motions (Chablat, 98a). Unfortunately, this result does not hold for manipulators which have multiple solutions to both their direct and inverse kinematics. For such manipulators which are the subject of this study, the singularity locus in the Cartesian workspace depends on the choice of the inverse kinematic solution (Chablat, 97) and the actual reachable space must be firstly defined in the Cartesian product of the Cartesian space by the joint space. The goal of this paper is to define the N-connected regions in the Cartesian workspace of fully-parallel manipulators, i.e., the maximal regions where it is possible to execute any point-to-point motion. The N-connected regions are characterized by projection, onto the Cartesian space, of the connected components of the manipulator configuration space defined in the Cartesian product of the Cartesian space by the joint space. Generalized Octree models are used for the construction of all spaces. This study is illustrated with a simple planar fullyparallel manipulator. 1 Preliminaries Some useful definitions are recalled in this section. 1.1 Fully-parallel manipulators Definition 1. A fully-parallel manipulator is a mechanism that includes as many elementary kinematic chains as the moving platform does admit degrees of freedom. In addition, every elementary kinematic chain possesses only one actuated joint (prismatic, pivot or kneecap). Besides, no segment of an elementary kinematic chain can be linked to more than two bodies (Merlet, 97). 1 Copyright  1999 by ASME In this study, kinematic chains, also called “leg” (Angeles, 97), will be always independent. 1.2 Kinematics The input vector q (the vector of actuated joint values) is related to the output vector X (the vector of configuration of the moving platform) through the following general equation :