Dexterity Analysis of Planar Parallel Manipulators

Abstract

Currently, two methods exist for obtaining constrained, dimensionally homogeneous Jacobian matrices for planar manipulators. These two methods differ in that a set of Cartesian velocities is expressed in either the moving or global frame which is then used to represent the end effector velocity. Therefore, the resulting Jacobian matrices are different depending on the method used. However, both methods formulate the Jacobian matrices based on an arbitrary choice of six potential Cartesian variables to define the dimensions of the task space.

This paper will obtain all six possible Jacobian matrices based on the six choices of available task space variables in the global frame. As an example, the Jacobian matrices of the 3-RRR planar parallel mechanism are formulated. These are then tested by comparing the resulting actuator velocities for a defined end effector path and velocity profile.

Each of the six Jacobian matrices is numerically unique and therefore have different singular values and condition numbers. Dexterity measures focusing on only one of the six matrices are potentially bias. Therefore, four proposed strategies for measuring the manipulator dexterity are then discussed. These are based on the singular values and condition number of the six constrained, dimensionally homogenous Jacobian matrices instead of a single matrix.