Forward displacement analysis of the general 6–6 Stewart mechanism using Gröbner bases Dongming Gan a,b, * , Qizheng Liao a , Jian S. Dai b , Shimin Wei a , L.D. Seneviratne b a Beijing University of Posts and Telecommunications, Beijing 100876, China b Department of Mechanical Engineering, School of Physical Sciences and Engineering, King’s College London, University of London, Strand, London WC2R2LS, UK article info Article history: Received 13 March 2008 Received in revised form 10 December 2008 Accepted 10 January 2009 Available online 10 February 2009 Keywords: General 6–6 Stewart mechanism Forward displacement analysis Gröbner basis abstract A new algorithm for the solution of the forward displacement analysis of the general 6–6 Stewart mechanism is presented in this paper. Gröbner basis theory is used and the prob- lem of the forward displacement is reduced to a 40th-degree polynomial equation in a sin- gle unknown from a constructed 13  13 Sylvester’s matrix which is relatively small in the size, from which 40 different locations of the moving platform can be derived. Numerical examples confirm the efficiency of the procedure. Ó 2009 Elsevier Ltd. All rights reserved. 1. Introduction The general 6–6 Stewart mechanism, proposed in 1965 by Stewart [1], has found a central status in the literature on par- allel manipulators, due to it’s high performance and the fact that the studies of parallel manipulators started with the con- ception of this manipulating structure [2]. The forward displacement analysis of the general Stewart mechanism has also been considered a very difficult problem, which leads naturally to a system of highly nonlinear algebraic equations with multiple solutions. Although a numerical iterative procedure can find forward kinematics solutions, it is not suitable to a Stewart mechanism as it leads to a heavy computational burden and requires a good initial value. A closed-form forward kinematics solution will provide more information about the geometry and kinematic behaviour of a parallel mechanism. Furthermore, the input–output closed-form univariate polynomial equations derived by using the closed-form methods have highly theoretical values based on which many kinematic problems will be solved easily. Gröbner basis theory is a famous elimination method, which can be used to get the closed-form solutions to the non- linear algebraic equations derived from the spatial mechanism problems. The computation of Gröbner bases is based on the concept that the ease of computation of solutions to a system of polynomial equations can drastically improve if the given system of equations is transformed into another system that has the same solutions but is much easier to solve [3]. In the past, many scholars have applied various methods to solve the forward displacement problem of the general Stew- art mechanism, which has 40 solutions in the complex domain, firstly proved numerically by Raghavan [4]. After that, Zhang and Song [5] presented the first closed-form solution to this problem without any spurious solutions. As mentioned in [6], Husty [7] produced a 40th-degree univariate equation by finding the greatest common divisor of the intermediate 0094-114X/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmachtheory.2009.01.008 * Corresponding author. Address: Department of Mechanical Engineering, School of Physical Sciences and Engineering, King’s College London, University of London, Strand, London WC2R2LS, UK. Tel.: +44 07826452222. E-mail address: gandong64@sina.com.cn (D. Gan). Mechanism and Machine Theory 44 (2009) 1640–1647 Contents lists available at ScienceDirect Mechanism and Machine Theory journal homepage: www.elsevier.com/locate/mechmt