Evaluation and Representation of the Theoretical Orientation Workspace of the Gough–Stewart Platform

The evaluation and representation of the orientation workspace of robotic manipulators is a challenging task. This work focuses on the determination of the theoretical orientation workspace of the Gough–Stewart platform with given leg length ranges $[ρimin,ρimax]$⁠. By use of the roll-pitch-yaw angles $(ϕ,θ,ψ)$⁠, the theoretical orientation workspace at a prescribed position $P0$ can be defined by up to 12 workspace surfaces. The defined orientation workspace is a closed region in the 3D orientation Cartesian space $Oϕθψ$⁠. As all rotations $R(x,ϕ)$⁠, $R(y,θ)$⁠, and $R(z,ψ)$ take place with respect to the fixed frame, any point of the defined orientation workspace provides a clear measure for the platform to, respectively, rotate in order around the $(x,y,z)$ axes of the fixed frame. An algorithm is presented to compute the size (volume) of the theoretical orientation workspace and intersectional curves of the workspace surfaces. The defined theoretical orientation workspace can be applied to determine a singularity-free orientation workspace.