Dilip Kohli

A Gröbner-Sylvester Hybrid Method for Closed-Form Displacement Analysis of Mechanisms

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 Grobner basis form of a system of equations under degree lexicographic (dlex) term ordering of its monomials and Sylvesters Dialytic elimination method. Using the Grobner-Sylvester hybrid approach, a finitely solvable system of equations F is transformed into its reduced Grobner basis G using dlex term ordering. Next, using the entire or a subset of the set of generators in G, the Sylvesters matrix is assembled. The vanishing of the resultant, given as the determinant of Sylvesters 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 Sylvesters 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.

The Jacobian analysis of workspaces of mechanical manipulators

Abstract The workspace is considered to be the mapping of the m- dimensional manipulator space consisting of m independent coordinates denoting variables at m one degree of freedom joints onto n- dimensional space consisting of n coordinates of the rigid body attached to the last link of the manipulator. Several theorems for this class of mapping have been proved and algorithms for determining the boundaries of the workspace have been derived in the paper. The concept of orientational boundaries and dextrous workspace has been explored. The determination of maximum reach of the manipulator and cross section of the boundary with a specified plane has been investigated. Numerical examples are presented to illustrate the theory.

Closed-form displacement and coupler curve analysis of planar multi-loop mechanisms using Grobner bases

Abstract The displacement analysis problem for planar and spatial mechanisms can be written as a system of algebraic equations in particular as a system of multi-variate polynomial equations. Elimination theory based on resultants and polynomial continuation are some of the methods which have been used to solve this problem. This paper explores an alternate approach, based on Grobner bases, to solve the displacement analysis problem for planar mechanisms. It is shown that the reduced set of generators obtained using Buchbergers algorithm for Grobner bases not only yields the input–output polynomial for the mechanism, but also provides comprehensive information on the number of closures and the relationships between various links of the mechanism. Numerical examples illustrating the applicability of Grobner bases to displacement analysis of 10- and 12-link mechanisms, and determination of coupler curve equation for an 8-link mechanism are presented. It is seen that even though the Grobner bases method is versatile enough to handle finitely solvable as well as over-constrained systems of equations, it can run into computational problems due to rapidly growing numerical coefficients and/or the set of generators. The examples presented show how these difficulties can be overcome by artificially decoupling complex mechanisms to help facilitate their closed-form analysis.

Boundary surfaces and accessibility regions for regional structures of manipulators

Abstract The theory of workspace analysis developed in the companion paper is applied for the analysis of workspace of industrial robots whose last three orthogonal axes intersect at a common point called the wrist center. The Jacobian surfaces which separates, inacessible regions, two- and four-way regions have been analytically derived in both manipulator coordinates and cartesian coordinates. The distribution of the number of ways to reach a specified position in the workspace has been investigated. It is found that there are contiguous regions where there may be four ways or two ways and these regions are separated by the Jacobian surfaces. Also, a Jacobian volume index is introduced to reflect the capability of the robot in terms of numbers of ways in the workspace. Several numerical examples have been presented for illustration.

Closed-form displacement analysis of 8, 9 and 10-link mechanisms

Abstract This paper presents closed-form solutions to the displacement analysis problem of planar 8-link mechanisms with 1 degree-of-freedom (DOF). Using the successive elimination procedure presented herein, the degrees of the input–output (I/O) polynomials as well as the number of assembly configurations for all 71 mechanisms resulting from 16 8-link kinematic chains are presented. It is shown that the displacement analysis problems for these mechanisms can be classified into nine distinct structures each of which can be reduced into a univariate polynomial devoid of any extraneous roots. This univariate polynomial corresponds to the I/O polynomial of the mechanism. Three numerical examples illustrating the applicability of the successive elimination procedure to the displacement analysis of 8-link mechanisms are presented. The first example deals with the determination of I/O polynomial for an 8-link mechanism which does not contain any 4-link loops. The second and third examples address in detail some of the problems associated with the conversion of transcendental loop closure equations into an algebraic form using tangent half-angle substitutions. An application of the proposed approach to the displacement analysis of spherical 8-link mechanisms is also presented.

Synthesis of cam-link mechanisms for exact path generation

Abstract In this paper, it is shown that there are a significantly large number of cam-link mechanisms that may be used for generating an exact path or for guiding a body through exactly specified infinite positions. Using Grueblers mobility criterion, structural synthesis is performed to obtain several cam-link mechanisms for path and motion generation. A synthesis technique is also presented to design track-follower mechanisms. The technique utilizes the complex loop closure method and the envelope theory to find the centerline and the contact points of the track. The approach is general and may be applied to any cam-link mechanism used to generate exact paths or to guide a body through infinitely many positions. A method for finding the curvature of the track at the contact points is also given.

Generic maps of mechanical manipulators

Abstract General theory of generic maps for an n degrees of freedom (d.o.f.) manipulator is developed in this paper. Algorithms are provided to determine genericity of 6 d.o.f. manipulators and 7 d.o.f. redundant manipulators. The closed form equations are provided for determination of genericity of 3 R manipulators.

Closed-form analytic synthesis of a five-link spatial motion generator

Abstract In the synthesis of motion generating mechanisms we seek to determine the key dimensions of a preconceived type of single-degree-of-freedom mechanism which will guide one of its links through a sequence of finitely or infinitesimally separated arbitrarily prescribed positions. In space, each such finitely separated precision position can be specified by the radius vector of the origin and the Euler angles of orientation of a coordinate system, embedded in the guided link, with respect to a fixed coordinate system of reference. The motion generator mechanim considered here consists of two grounded S - S links, ¶ one grounded C - S link, a ternary S-S-S coupler and the R - R - C fixed frame. Motion of the coupler is to prescribed. Each equation of synthesis is written by expressing the closure of the vector polygon covering the starting and one displaced position of one dyad formed by a grounded link and the coupler. The resulting system can be solved for unknown vectors defining the dyad in its starting position, in closed form for up to three precision positions.

Closed-form displacement analysis of 8, 9 and 10-link mechanisms. Part II : 9-link 2-DOF and 10-link 3-DOF mechanisms

Abstract This paper presents closed-form polynomial solutions to the displacement analysis problem of planar 9-link two-degrees-of-freedom (2-DOF) and 10-link three-degrees-of-freedom (3-DOF) mechanisms. The successive elimination procedure developed in the companion paper (Part I) is used to solve the displacement analysis problems. The degrees of input–output polynomials as well as the number of possible assembly configurations for all mechanisms resulting from 35 9-link kinematic chains and 74 10-link kinematic chains with non-fractionated degrees of freedom are given. The computational procedure is illustrated through two numerical examples. Based on the results presented, the displacement analysis problem for all mechanisms resulting from 9-link 2-DOF and 10-link 3-DOF kinematic chains is completely solvable, in closed-form, devoid of any extraneous roots.

Synthesis of RSSR-SRR spatial motion generator mechanism with prescribed crank rotations for three and four finite positions

Abstract The mechanism considered here is an RSSR-SRR spatial mechanism for motion generation with prescribed crank rotations. It consists of an R - R - R ( R = revolute, S = spheric joint) fixed frame, two grounded R-S links, one grounded RRS dyad and a ternary S-S-S coupler. Motion of the coupler is to be prescribed for three and four finitely separated positions and to be correlated to the prescribed input rotations of the crank, the grounded R-R link of the RRS dyad. The synthesis equations obtained by using vector mathematics are linear in the unknowns up to three prescribed positions of the body in spatial motion. For four precision positions, the design procedure involves solving two cubic algebraic equations simultaneously for two unknowns for the R-S dyad synthesis, and solving one fourth-degree algebraic equation in one unknown for synthesis of the RRS group. The design procedure is described in detail and numerical examples are presented for illustration.