Singularity Representation and Workspace Determination of a Special Class of the Gough-Stewart Platforms

The closed-loop nature of the Gough-Stewart platform generates complex singular configurations inside the workspace and makes its workspace smaller compared to the serial mechanism. It is desirable to obtain a non-singular workspace based on describing the constraint workspace and representing the singularities inside the constraint workspace. Some algorithms have been proposed by researchers to find the workspace boundary, but cannot locate the voids inside the workspace and are not applicable to the generation of a workspace with more than one zone. In this paper, the position-singularity expression in R3 and the orientation-singularity expression in SO(3) are obtained, respectively. The new algorithms of the two types of constraint workspace in R3 and in SO(3) are developed considering the limitations of the kinematic pairs. It can be shown that the singularities may exist inside the constraint workspace. Based on the singularity representation and the abovementioned constraint workspace determination, the two types of procedures of the non-singular workspace in R3 and in SO(3) are further addressed, respectively. When the moving platform translates inside the non-singular position-workspace in R3 for a constant-orientation or rotates inside the non-singular orientation-workspace in SO(3) for a given position, the mechanism is not singular. The two types of non-singular workspace representations in R3 and SO(3) can help the designers to explore the singularity-free path planning, on which our next work will be focused. The novel method of workspace determination of the mechanism can also be used for the workspace analysis of the other types of parallel mechanisms.