Singularity Equations of Gough–Stewart Platforms Using a Minimal Set of Geometric Parameters

So far, in the derivation of the singularity equations of Gough–Stewart platforms, all researchers defined the mobile frame by making its origin coincide with the considered point on the platform. One problem can be that the obtained singularity equation contains too many geometric parameters and is not convenient for singularity analysis, especially not convenient for geometric optimization. Another problem can be that the obtained singularity equation cannot be used directly in practice. To solve these problems, this work presents a new approach to derive the singularity equation of the Gough–Stewart platform. The main point is that the origin of the mobile frame is separated from the considered point and chosen to coincide with a special point on the platform in order to minimize the geometric parameters defining the platform. Similarly, by defining a proper fixed frame, the geometric parameters defining the base can also be minimized. In this way, no matter which practical point of the platform is chosen as the considered point, the obtained singularity equation contains only a minimal set of geometric parameters and becomes a solid foundation for the geometric optimization based on singularity analysis.