An Illustration of Generating Robots from Optimal Fault-Tolerant Jacobians

Abstract

The local kinematic properties of a robotic manipulator's configuration can be described by its corresponding Jacobian matrix. Conversely, one can determine a manipulator that possesses certain desirable kinematic properties by specifying the required Jacobian. In this work, design criteria that require a manipulator to function in a configuration that is optimal under normal operation and after an arbitrary single joint fails and is locked in position is first described. Specifically, the desired Jacobian matrix must be isotropic, i.e., possess all equal singular values prior to a failure, and have equal minimum singular values for every possible single column being removed. Then a simple planar three degree-of-freedom example is used to illustrate how one can identify all of the possible manipulator designs that possess the desired local properties described by the required structure of the Jacobian matrix. This paper concludes by showing that despite having identical local properties, the resulting manipulator designs have significantly different global kinematic properties that can be used to match a design to additional application-specific performance criteria.

Examples of spatial positioning redundant robotic manipulators that are optimally fault tolerant

“…The exact values of the DH parameters for a given set of β i 's can be computed using the algorithm that is presented in [36]. Clearly, there is an infinite family of robots that correspond to (5).…”

“…In previous work [1], [36], it was shown that there exist multiple different physical manipulators that correspond to the same optimally fault tolerant Jacobian. This is due to the fact that permutation of the columns of the Jacobian (or multiplying by ±1) does not affect its fault tolerant properties, however, it does significantly impact the resulting physical manipulator.…”

Examples of planar robot kinematic designs from optimally fault-tolerant Jacobians

“…These analyses have been performed both on the local properties associated with the manipulator Jacobian [28]- [31] as well as the global This work was supported in part by the National Science Foundation under Contract IIS-0812437. 1 Sections I and II are very similar to those in [1], and are included here to provide the background to make this paper self-contained. characteristics such as the resulting workspace following a particular failure [32]- [35].…”

“…It has been recently shown that multiple different physical robot kinematic designs can be obtained from (essentially) a single Jacobian that has desirable fault tolerant properties [1]. Fault tolerance in this case is defined as the post-failure Jacobian possessing the largest possible minimum singular value over all possible locked-joint failures.…”