Sandipan Bandyopadhyay

This paper presents the computation of singularity-free regions in the position and orientation workspaces of Stewart platform manipulators (SPMs). Notwithstanding several previous publications in this regard, certain issues persist, such as the unavailability of analytical solutions and the consequent lack of accurate estimates of the numbers of solutions in various cases; the improper use of an incompatible metric while finding such regions inside SO(3) or SE(3). In this paper, the singularity-free regions have been envisaged as spheres in either R3 or SO(3). An analytical formulation for identifying such singularity-free spheres (SFSs) is presented. Several algebraic methods are explored to solve the resulting set of polynomial equations, leading to valuable insights into the numbers of (finite, complex) solutions to this problem, in both the position and the orientation workspaces. A physically meaningful and mathematically sound metric is used while computing the SFS in the orientation workspace. The formulation and computational algorithms have been demonstrated by identifying the SFSs (in R3 as well as SO(3)) for multiple architectures of the SPM through implementations in the C language.

This paper presents a method to compute the largest possible cylindrical volume within the translational workspace of the semi-regular Stewart platform manipulator (SRSPM), which would be free of gain-type singularities. An analytical approach is used in finding the singularity-free regions rather than discretising the workspace into small singularity-free volumes. Comparison with another convex shape, i.e., the sphere, is performed to demonstrate the relative importance and usefulness of using the cylindrical geometry for finding the singularity-free spaces.

This paper presents a novel geometric solution to the problem of finding singularity-free paths joining two arbitrary points in the constant orientation workspace of a semi-regular Stewart platform manipulator. The formulation builds upon the known closed-form expression for the gain-type singularity surface of the manipulator. Using a rational parametrisation of the surface, it computes the geodesic curve on this surface, connecting the projections of the two given points on this surface. A sequence of spheres is then constructed in such a manner that each sphere is tangential to a previous one as well as the singularity surface, at a point on the said geodesic curve. Thus the geodesic curve acts as a guide, over which the singularity-free sphere is rolled, till it reaches its destination. Multiple methods for computing such sequences of spheres are presented and compared with the help of a numerical example. Finally, a sequence of line segments connecting the centres of the spheres is constructed, which connects the two given points via a provably singularity-free path.

This paper presents a method to compute the largest sphere inside the position-workspace of a semi-regular Stewart platform manipulator, that is free of gain-type singularities. The sphere is specific to a given orientation of the moving platform, and is centred at a designated point of interest. The computation is performed in two parts; in the first part, a Computer Algebra System (CAS) is used to derive a set of exact symbolic expressions, which are then used further in a purely numerical manner for faster computation. The method thus affords high computation speed, while retaining the exactness and generic nature of the results. The numerical results are validated against those obtained from an established numerical algebraic geometry tool, namely, Bertini, and are illustrated via an example.