提出高效算法求解斯图尔特-盖夫平台正向运动学，可一次性获得40个解。
Forward kinematics of a general Stewart-Gough platform by elimination templates
- 基于稀疏矩阵消元模板构建求解框架
- 通过矩阵对广义特征分解得到全部40个解
- 代码开源，适合机器人与机构设计者使用
本文提出一种高效的代数方法求解通用斯图尔特-盖夫平台的正向运动学问题。该问题旨在已知六条连杆长度及平台与基座内部几何结构的情况下，确定移动平台所有可能的姿态。该问题已知存在40个解（实数或复数）。所提算法包含三个步骤：(i) 从描述平台运动学的多项式系统系数构造一个293×362的稀疏矩阵（消元模板）；(ii) 利用该矩阵的PLU分解生成一对69×69的矩阵；(iii) 通过计算该矩阵对的广义特征向量，得到全部40个解（含复数解）。该算法数值稳定、计算高效且实现简单，仅需标准线性代数分解。算法的MATLAB、Julia和Python实现将公开发布。
原文摘要 · Abstract (English)
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.
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