Kinematic Isotropy and Optimal Kinematic Design of Planar Manipulators and a 3-DOF Spatial Manipulator

Abstract

Robot operation near isotropic configurations, in which the condition number of the Jacobian matrix reaches unity, is desirable from several points of view. However, determination of all such configurations, given arbitrary robot geometry, is a rather complex problem. In this article all the isotropic configurations of planar manipulators with two, three, and four degrees of freedom are determined, together with that of a 3- DOF spatial manipulator. The solutions are obtained in the form of a fourth-order polynomial for the 3-DOF robot and an eighth-order polynomial for the 4-DOF planar manipulator, resulting in a maximum of eight sets of solutions. The condition numbers are obtained as explicit analytic functions of joint coordinates and link lengths ratios. The optimal length of the links is determined by minimizing the criterion that the condition number increase most slowly with joint angles in the vicinity of isotropic configurations.

Kinematic Design Optimization of Planer-Link Mechanism Based Manipulator

“…As seen in this figure, the = π/4 position of the input link and = 0.707107 value for the end-effector position equal local mobility index to one in such a parallel manipulator. These values obtained for the manipulator with parallelogram mechanism are the same with results for the planar two link manipulator obtained in the studies of optimal kinematic design by Gosselin and Angeles [8], Singh and Rastegar (1995), Kircanski (1996) and Lee et al (1993). design variables graph (dashed line) are derived.…”

“…Doel and Pai [11] introduced formalism for the systematic construction of performance measures of robot manipulators in a unified framework based on differential geometry. Kircanski [12] determined the isotropic configurations of planar and spatial manipulators in the form of polynominal. Zanganeh and Angeles [13] introduced a set of conditions for the submatrices of the Jacobians under which a parallel manipulator can attain an isotropic configuration.…”

“…The local mobility index of the planar manipulator shown in Fig. 1 can be derived from the generalized Jacobian matrix as (12) The expression LMI in Eq. (12) defines a fivevariable function depending on the input link position ρ as seen in Eqs.…”

Discriminant functions for isotropic configurations in robot manipulators

“…Now, for a fixed taskspace dimensionality m, it will be established how large must be a configuration space to have a chance to contain isotropic configurations. (10) have to be satisfied to get isotropic configurations. Proof: There is (m − 1)m/2 conditions to cancel elements out of the main diagonal in a symmetric (m × m) manipulability matrix M M M, cf.…”

Methodology for the kinematical selection of a manipulator for a specified task

“…To deal with the symbolic expressions in a simple manner, it is convenient to consider the Jacobian 2 J that corresponds to task coordinates expressed in the second coordinate frame located at the end-effecter of the manipulator (Kircanski, 1996).…”