Determination of 6D workspaces of Gough-type parallel manipulator and comparison between different geometries

We consider in this paper a Gough-type parallel robot whose leg length values are constrained to lie within some fixed ranges and for which there may be mechanical limits for the motion of the passive joints. The purpose of this paper is to present algorithms to determine: the constant orientation workspace: all the possible locations of the center of the platform that can be reached with a fixed orientation the total orientation workspace: all the possible locations of the center of the platform that can be reached with any orientation in a set defined by three ranges for the orientation angles (the dextrous workspace is an example of total orientation workspace case, the three ranges being [0, 360] degree(1)) the inclusive orientation workspace: all the possible locations of the center of the platform that can be reached with at least one orientation among a set defined by three ranges for the orientation angles (the maximal or reachable workspace is an example of inclusive orientation workspace, the three ranges being [0, 360]) Most of these algorithms are based on a basic method: approximation of the result by a set of 3D or 6D boxes obtained from an initial estimation through a bisection process. The boxes in the result will either fully or partially lie inside the workspace: the bisection stops as soon as all the boxes that do not lie fully inside the workspace have a size that is lower than a fixed threshold. A companion algorithm enables to verification that any 6D workspace (i, e., a continuous set of poses) lies within the reachable workspace of the robot. The paper includes a comparison between the workspace volumes of four different robot geometries, which shows that for robots of similar dimensions the joints layout has a large influence on the workspace volume.