Solving the 6R inverse position problem using a generic-case solution methodology

Abstract

AlmU'net--This paper considers the computation of all solutions to the inverse position problem for general six-revolute-joint manipulators. Instead of reducing the problem to one highly complicated input-output equation, we work with a system of I I very simple polynomial equations. Although the total degree of the system is large (1024), using the "method of the generic case" we show numerically that the generic number of solutions is 16, in agreement with previous works. Moreover. we present an elEcient nun~rical method for finding all 16 solutions, based on coe~icient-parameter polynomial continuation. We present a set of 41 test problems, on which the algorithm used an average of le-~ than l0 s of CPU time on an IBM 370-3090 in double precision FORTRAN. The methodology applies equally well to other problems in kinematics that can be formulated as polynomial systems.

Inverse Displacement Analysis of a General 6R Manipulator Based on the Hyper-Chaotic Least Square Method

“…Therefore, in relation to these problems of the mechanism synthesis, N should first be taken as smaller when counted approximately, and then be selected as larger. In general, N is selected at (8)(9)(10)(11)(12) times the variable number). After an iterative calculation for 948s, Eq.…”

“…In 1989, M. Raghavan and B. Roth [4] made use of the nature of the rationale generated by using the separation elimination method and multi-variable equations, and calculated a 16th degree polynomial from the half-angle tangent formula of the joint variables. In 1991, C. Wampler and A. P. Morgan [5] proposed that all the cases involving the inverse displacement problem were only solved using the extension method, but the algorithm based on the extension method ran very slowly. From 1990 to 1992, M. Raghavan and B. Roth [6][7][8] first proposed a method solving the characteristic polynomial of the general 6-DOF manipulator, including all the special circumstances of 6R, 5R1P, 4R2P, 3R3P, and with the aim of solving all of the 6-DOF robot mechanism.…”

“…From 1990 to 1992, M. Raghavan and B. Roth [6][7][8] first proposed a method solving the characteristic polynomial of the general 6-DOF manipulator, including all the special circumstances of 6R, 5R1P, 4R2P, 3R3P, and with the aim of solving all of the 6-DOF robot mechanism. In 1991, C. W. Wampler and A. P. Morgan [9] completed the 6R inverse position problem using a generic-case solution methodology with 11 polynomial equations. In 1993, D. Kohli and M. Osvatic [10] used the product of the powers and the elimination method to obtain a set of equations containing 16 16  matrix coefficients, and to directly have 16th degree polynomials of no extraneous roots,since the solution to the problem can be simplified to the eigenvector problem.…”

Structure Based Classification and Kinematic Analysis of Six-Joint Industrial Robotic Manipulators

“…Finally, the resulting high-order equation is solved numerically. However, requiring a lot of polynomial manipulations, this approach is quite cumbersome (Wampler & Morgan, 1991;Raghavan & Roth, 1993). On the other hand, the approach presented in this chapter aims at obtaining the inverse kinematic solutions analytically by manipulating the trigonometric equations directly without converting them necessarily into polynomial equations.…”

Analytical inverse kinematics for 5-DOF humanoid manipulator under arbitrarily specified unconstrained orientation of end-effector

“…Therefore, solving the inverse kinematic problem for such manipulators has been intensively studied over several decades (e.g., refs. [1][2][3][4][5][6]). To date, various analytical as well as numerical methods for inverse kinematic computation have been developed.…”