A semi-analytical approach for computing the largest singularity-free spheres of a class of 6-6 Stewart-Gough platforms for specified orientation workspaces

TL;DR

The study tackles the problem of identifying the largest singularity-free sphere (SFS) for a class of semi-regular Stewart-Gough platforms over a specified orientation workspace. It employs a semi-analytical approach: analytically compute the SFS for a fixed MP orientation via a cubic singularity surface $e(x,y,z)=0$ with $g(x,y,z)$, then extend to the orientation workspace by sampling the Rodrigues-parameter sphere $ ext{WO}$ and selecting the minimum SFS radius $r_2$ across samples. The method is demonstrated on four SRSPM architectures, revealing how $r_2$ varies with the orientation bound and architecture, and showing total runtimes around 9 minutes for $N_S=299{,}940$ samples. The framework provides a practical tool for design comparison and path planning in SRSPMs, balancing analytic precision with a scalable sampling strategy in orientation space.

Abstract

This article presents a method for computing the largest singularity-free sphere (SFS) of a 6-6 Stewart-Gough platform manipulator (SGPM) over a specified orientation workspace. For a fixed orientation of the moving platform, the SFS is computed analytically. This process is repeated over a set of samples generated within the orientation workspace, and the smallest among them is designated as the desired SFS for the given orientation workspace. Numerical experiments are performed on four distinct architectures of the SGPM to understand their relative performances w.r.t. SFS volumes over the same orientation workspace. This study demonstrates the potential utility of the proposed computational method both in analysis and design of SGPMs.

Figures (6)

• Figure 1: Schematic of a semi-regular Stewart-Gough platform manipulator
• Figure 2: Geometry of the fixed and moving platforms
• Figure 3: Singularity surface and the corresponding largest SFS centred at $\boldsymbol{p}\xspace_0=[0,0,2.5000]^\top$ for a given orientation, $\boldsymbol{c}\xspace = [0.0639,0.1107,0.2597]^\top$, of the MP of the SRSPM 1 (see Table \ref{['tb:archdata']})
• Figure 4: Fixed platforms of the four SRSPMs (see Table \ref{['tb:archdata']})
• Figure 5: Moving platforms of the four SRSPMs (see Table \ref{['tb:archdata']})