Validation of collision-free spheres of Stewart-Gough platforms for constant orientations using the Application Programming Interface of a CAD software

TL;DR

The paper addresses the challenge of ensuring collision-free operation of a spatial Stewart-Gough platform by validating the largest collision-free sphere (CFS) for a given moving-platform orientation. It adopts a capsule-based approximation for legs and derives a collision surface $S_3=d_{i,j}^2-4r_c^2=0$, with the CFS center constrained to the $Z$-axis at $oldsymbol{p}_0=[0,0,z_0]^T$ and radius $r_3$, tangent to $S_3=0$. Validation is implemented via an Autodesk Inventor API add-in that generates a shell $oldsymbol{S}'$ around the CFS using inner radius $r_3(1-oldsymbol{ abla})$ and outer radius $r_3(1+oldsymbol{ abla})$ (with $oldsymbol{ abla}=0.1$) and samples $N=N_rN_s$ MP-centers to check all leg-pair collisions with capsule elements. Numerical experiments on two scenarios, each with $N_s=2500$, show that unsafe poses lie outside $oldsymbol{S}_3$, confirming the CFS safety within the sampling resolution and offering a practical method to estimate CFS dimensions for other spatial PMs. The approach provides an automated, geometry-based independent validation tool that complements analytical results and enhances safe design and operation of parallel manipulators.

Abstract

This paper presents a method of validation of the size of the largest collision-free sphere (CFS) of a 6-6 Stewart-Gough platform manipulator (SGPM) for a given orientation of its moving platform (MP) using the Application Programming Interface (API) of a CAD software. The position of the MP is updated via the API in an automated manner over a set of samples within a shell enclosing the surface of the CFS. For each pose of the manipulator, each pair of legs is investigated for mutual collisions. The CFS is considered safe or validated iff none of the points falling inside the CFS lead to a collision between any pair of legs. This approach can not only validate the safety of a precomputed CFS, but also estimate the same for any spatial parallel manipulator.

Figures (7)

• Figure 1: Schematic of an SRSPM. The vertices of the fixed and moving platforms, $\boldsymbol{b}\xspace_i$ and $\boldsymbol{a}\xspace_i$, respectively, are represented in $\boldsymbol{O}\text{-}\boldsymbol{X}\xspace\boldsymbol{Y}\xspace\boldsymbol{Z}\xspace$.
• Figure 2: Geometry of the fixed and moving platforms. The vertices of the fixed and moving platforms, $\boldsymbol{b}\xspace_i$ and $\boldsymbol{t}\xspace_i$, are represented in $\boldsymbol{O}\text{-}\boldsymbol{X}\xspace\boldsymbol{Y}\xspace\boldsymbol{Z}\xspace$ and $\boldsymbol{o}\text{-}\boldsymbol{x}\xspace\boldsymbol{y}\xspace\boldsymbol{z}\xspace$, respectively.
• Figure 3: A geometrically equivalent CAD model of an SRSPM (refer to Table \ref{['tb:archparam']} for the architecture details), showing the CFS along with a zoomed view of a capsule
• Figure 4: An add-in to Autodesk Inventor for validating the CFS of the SRSPM. The values $r_3, \Delta\xspace r,$ and $N_\text{s}$ shown here correspond to Scenario 2 (see Tables \ref{['tb:testcase']}, \ref{['tb:paramscan']})
• Figure 5: Points corresponding to the unsafe poses fall outside of the collision-free sphere $\mathcal{S}_3$ for both Scenarios 1 and 2 (refer to Table \ref{['tb:testcase']})