Forward kinematics of a general Stewart-Gough platform by elimination templates

TL;DR

This paper introduces an efficient algebraic method for solving the forward kinematics of a Stewart-Gough platform, capable of finding all 40 solutions quickly and robustly using elimination templates and eigenvector computations.

Contribution

It presents a novel elimination template-based algorithm that simplifies and accelerates the forward kinematics solution for general Stewart-Gough platforms.

Findings

Numerically robust and computationally efficient algorithm

Successfully computes all 40 solutions including complex ones

Implementations available in MATLAB, Julia, and Python

Abstract

The paper proposes an efficient algebraic solution to the problem of forward kinematics for a general Stewart-Gough platform. The problem involves determining all possible postures of a mobile platform connected to a fixed base by six legs, given the leg lengths and the internal geometries of the platform and base. The problem is known to have 40 solutions (whether real or complex). The proposed algorithm consists of three main steps: (i) a specific sparse matrix of size 293x362 (the elimination template) is constructed from the coefficients of the polynomial system describing the platform's kinematics; (ii) the PLU decomposition of this matrix is used to construct a pair of 69x69 matrices; (iii) all 40 solutions (including complex ones) are obtained by computing the generalized eigenvectors of this matrix pair. The proposed algorithm is numerically robust, computationally efficient, and straightforward to implement - requiring only standard linear algebra decompositions. MATLAB, Julia, and Python implementations of the algorithm will be made publicly available.