A Gro¨bner-Sylvester Hybrid Method for Closed-Form Displacement Analysis of Mechanisms

Abstract

The displacement analysis problem for planar and spatial mechanisms can be written as a system of multivariate polynomial equations. Elimination theory based on resultants and polynomial continuation are some of the methods that have been used to solve this problem. This paper presents a new approach to displacement analysis using the reduced Gro¨bner basis form of a system of equations under degree lexicographic (dlex) term ordering of its monomials and Sylvester’s Dialytic elimination method. Using the Gro¨bner-Sylvester hybrid approach, a finitely solvable system of equations F is transformed into its reduced Gro¨bner basis G using dlex term ordering. Next, using the entire or a subset of the set of generators in G, the Sylvester’s matrix is assembled. The vanishing of the resultant, given as the determinant of Sylvester’s matrix, yields the necessary condition for polynomials in G (as well as F) to have a common factor. The proposed approach appears to provide a systematic and rational procedure to the problem discussed by Roth, dealing with the generation of (additional) equations for constructing the Sylvester’s matrix. Three examples illustrating the applicability of the proposed approach to displacement analysis of planar and spatial mechanisms are presented. The first and second examples address the forward displacement analysis of the general 6-6 Stewart mechanism and the 6-6 Stewart platform, whereas the third example deals with the determination of the I/O polynomial of an 8-link 1-DOF mechanism that does not contain any 4-link loop. [S1050-0472(00)01204-6]

On the Direct Kinematics Problem of Parallel Mechanisms

“…We now have a system of three nonlinear equations, (17), (18), and (19), in three variables ( , , and ), which, in general, is uniquely not solvable. So, it is not possible to find a unique solution for the direct kinematics problem of -6 mechanisms by only measuring the linear actuators' orientations.…”

“…The first group consists of using the minimal number of sensors, in our case, six length sensors, and then including additional numerical procedures to uniquely identify the parallel mechanism's pose [11][12][13][14][15][16][17][18][19][20]. These procedures, however, are generally not real-time capable, require an initial estimate of the solution, and may exhibit convergence problems or even converge to a wrong solution.…”

Error Modeling and Accuracy of Parallel Industrial Robots

“…On the subject of forward kinematics, focus has been on finding closed form solutions based on various robotic configurations, and numerical solutions for difficult configurations of robots. It can be found in the work done by Dhingra A. K. et al 1998, 2000, Shi, X. & Fenton, R. G. 1994, Didrit, O. et al, 1998, Zhang, X.…”

Automated Generation of Linkage Loop Equations for Planar 1-DoF Linkages, Demonstrated up to 8-Bar

“…The same linkage was also evaluated in rational formulation by Nielsen and Roth [16]. Various other methods for solving the configuration of a linkage have been published such as the Gr€ obner-Sylvester method by Dhingra et al [17] and the linear relaxation method by Porta et al [18].…”