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

IF 4.5 1区 工程技术 Q1 ENGINEERING, MECHANICAL
Evgeniy Martyushev
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引用次数: 0

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 four main steps: (i) a specific sparse matrix of size 293 × 362 (the elimination template) is constructed from the coefficients of the polynomial system describing the platform’s kinematics; (ii) the QR decomposition of this matrix is used to construct a pair of 69 × 69 matrices; (iii) 69 candidate solutions (including complex ones) are obtained by computing the generalized eigenvectors of this matrix pair; (iv) 29 spurious solutions are eliminated through back substitution. 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.
基于消去模板的通用Stewart-Gough平台正运动学分析
本文提出了通用Stewart-Gough平台正运动学问题的一种有效的代数解法。这个问题涉及确定一个移动平台的所有可能的姿势,该平台通过六条腿连接到一个固定的基座上,给定腿的长度以及平台和基座的内部几何形状。这个问题已知有40个解(无论是实数还是复数)。提出的算法包括四个主要步骤:(i)从描述平台运动学的多项式系统的系数中构造一个大小为293 × 362的特定稀疏矩阵(消除模板);(ii)对该矩阵进行QR分解,构造一对69 × 69矩阵;(iii)通过计算该矩阵对的广义特征向量得到69个候选解(包括复解);(四)通过反向替代消除了29个伪解。该算法在数值上具有鲁棒性,计算效率高,且易于实现,只需要标准的线性代数分解。该算法的MATLAB、Julia和Python实现将公开提供。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Mechanism and Machine Theory
Mechanism and Machine Theory 工程技术-工程:机械
CiteScore
9.90
自引率
23.10%
发文量
450
审稿时长
20 days
期刊介绍: Mechanism and Machine Theory provides a medium of communication between engineers and scientists engaged in research and development within the fields of knowledge embraced by IFToMM, the International Federation for the Promotion of Mechanism and Machine Science, therefore affiliated with IFToMM as its official research journal. The main topics are: Design Theory and Methodology; Haptics and Human-Machine-Interfaces; Robotics, Mechatronics and Micro-Machines; Mechanisms, Mechanical Transmissions and Machines; Kinematics, Dynamics, and Control of Mechanical Systems; Applications to Bioengineering and Molecular Chemistry
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