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

Abstract

Abstract-It is common practice to design a robot's kinematics from the desired properties that are locally specified by a manipulator Jacobian. 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. In this work, a mathematical analysis that describes the number of possible planar robot designs for optimally fault-tolerant Jacobians is presented. Two examples, one that is optimal to a single joint failure and the second that is optimal to two joint failures, are discussed. The paper concludes by illustrating some of the large variability in the global kinematic properties of these designs, despite being generated from the same Jacobian.

Robustly fault tolerant Gough-Stewart platforms

“…Since the sets of failed singular values of robustly fault tolerant manipulators are the same no matter which strut failed, all these (identical) sets possess the same minimum singular value, thus it is the largest (and only) possible value over all failures. Thus if a manipulator meets our new definition of robust fault tolerance, then it also is considered to be optimally fault tolerant by [2] and [9].…”

“…Since RFT manipulators keep not only the product of singular values, but all singular values invariant with respect to failure, they are also optimally fault tolerant according to this earlier definition. More recently, a manipulator has been defined to be optimally fault tolerant if it possesses the largest possible minimum singular value over all possible failures [2]. Since the sets of failed singular values of robustly fault tolerant manipulators are the same no matter which strut failed, all these (identical) sets possess the same minimum singular value, thus it is the largest (and only) possible value over all failures.…”

“…These newly discovered "Robustly Fault Tolerant" machines have simple formulas describing their Jacobian's local singular values, thus allowing optimal analytic design solutions. The designs will tolerate "free swinging" failures [2] of any single strut so these machines allow for breakage in their least reliable components, the actuators.…”

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

“…These analyses have been performed both on the local properties associated with the 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.…”

“…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.…”