A new closed-form solution to the forward displacement analysis of a 5–5 in-parallel platform

• Published in: Mechanism and machine theory Vol. 52; pp. 47 - 58
• Main Authors: Zhang, Ying, Liao, Qizheng, Su, Hai-Jun, Wei, Shimin
• Format: Journal Article
• Language: English
• Published: Kidlington Elsevier Ltd 01.06.2012; Elsevier
• Subjects: Applied sciences; Displacement; Drives; Exact sciences and technology; Exact solutions; Forward displacement analysis; General 5–5 in-parallel platform; Joints; Linkage mechanisms, cams; Mathematical analysis; Mathematical models; Mechanical engineering. Machine design; Platforms; Strategy; Sylvester resultant; Vector elimination; Vectors (mathematics); Forward displacement analysis; General 5–5 in-parallel platform; Vector elimination; Sylvester resultant; Closure; General 5-5 in-parallel platform; Polynomial equation; Parallel mechanism; Modeling; Exact solution; Direct problem; Kinematics; Linkage mechanism; Lie group; Computer system; Computer algebra; Novelty
• ISSN: 0094-114X; 1873-3999
• DOI: 10.1016/j.mechmachtheory.2012.01.003

This paper presents a new solution procedure for the forward displacement analysis of a general 5–5 in-parallel platform, i.e., the connection points are not restricted to lie in planes. This paper is the continuation of Ref. [1] and its novelty lies in elimination strategy of the basic closure equations which were already derived in Ref. [1]. First of all, three kinematics constraint equations in two variables are derived from the basic closure equations by the vector algebraic elimination technique. Then, the greatest common divisor (GCD) of two constraint equations is symbolically factored out by using computer algebra system Mathematica 7.0. Finally, a 24th-degree univariant polynomial equation is reduced from this GCD polynomial together with the third constraint equation by constructing a 10×10 Sylvester resultant matrix. At last, a numerical example is deployed to verify the procedure. ► A new method for the forward displacement analysis of a 5–5 in-parallel platform. ► Three constraint equations in two variables using vector elimination derived. ► The greatest common divisor (GCD) of two constraint equations in symbolic form. ► A 10×10 matrix using Sylvester resultant and a 24th-degree input–output equation. ► A numerical example with twelve real solutions given.