{"title":"Forward kinematics of a general Stewart–Gough platform by elimination templates","authors":"Evgeniy Martyushev","doi":"10.1016/j.mechmachtheory.2025.106170","DOIUrl":null,"url":null,"abstract":"<div><div>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.</div></div>","PeriodicalId":49845,"journal":{"name":"Mechanism and Machine Theory","volume":"215 ","pages":"Article 106170"},"PeriodicalIF":4.5000,"publicationDate":"2025-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mechanism and Machine Theory","FirstCategoryId":"5","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0094114X25002599","RegionNum":1,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"ENGINEERING, MECHANICAL","Score":null,"Total":0}
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.
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