A Geometric Method of Singularity Avoidance for Kinematically Redundant Planar Parallel Robots

Abstract

Methods for avoiding singularities of closed-loop robot mechanisms have been traditionally based on the value of the determinant or the condition number of the Jacobian. A major drawback of these standard techniques is that the closeness of a robot configuration to a singularity lacks geometric, physical interpretation, thus implying that it is uncertain how changes in the robot pose actually move further away the mechanism from such a problematic configuration. This paper presents a geometric approach of singularity avoidance for kinematically redundant planar parallel robots that eliminates the disadvantages of Jacobian-based techniques. The proposed method, which is based on the properties of instantaneous centres of rotation, defines a mathematical distance to a singularity and provides a reliable way of moving the robot further from a singular configuration without changing the pose of the end-effector. The approach is demonstrated on an example robot mechanism and the reciprocal of the condition number of the Jacobian is used to show its advantages.

A robust geometric method of singularity avoidance for kinematically redundant planar parallel robot manipulators

“…The robot manipulator's proximity to a singularity is measured by computing the in-circle radii of the triangles formed by these ICRs, and finding the minimum of these radii, r min . This is different to the method presented in [16] and used in [3], where the robot manipulator's proximity to a singularity is given by the minimum distance between two ICRs. The avoidance of singularities is carried out by determining the optimum value of the parameter which describes the degree of kinematic redundancy of the robot manipulator such that r min is maximised without passing through a singular configuration, that where r min =0; this is distinct from the method proposed in [16] in which the singularity avoidance instructs how the mechanism should be reconfigured such that the distance between two ICRs is increased.…”

On the False Positives and False Negatives of the Jacobian Matrix in Kinematically Redundant Parallel Mechanisms

“…A family of kinematically redundant parallel robots with non-serially connected actuators proposed in the literature. The architectures, from left to right, were first presented in [21], [13], and [14], respectively. actuators can be obtained by taking a non-redundant architecture and adding extra actuated joints to the existing limbs.…”

“…The architecture displayed on the left-hand side of Fig. 2, first presented in [21], is a planar mechanism that consists of four RPR legs, that is an actuated prismatic joint with a passive revolute joint at each of its ends, two of which join the end-effector to the base directly, and the the other two join the end-effector to a ternary link, which itself is connected to the base via a passive revolute joint. The second architecture, presented in [13], is also a planar mechanism that consists of four RPR legs.…”

“…After forming the vector loop equations along each of the four legs, and taking the derivative with respect to time for each of them, we obtain Fig. 2, that of the mechanism presented in [21].…”

“…6. Configuration of the robot proposed in [21] where the inverse of the condition number of the Jacobian is zero, suggesting that the configuration is singular, but the rigidity matrix is of full rank, indicating that the mechanism is not in a singularity.…”