Evaluating the Fault Tolerance of a Parallel Manipulator Based on Relative Manipulability Indices

In this article, the authors investigate the fault tolerance of manipulators in their nominal configuration. In this work, fault tolerance is measured in terms of the worst case relative manipulability index. While this approach is applicable to both serial and parallel mechanisms, it is especially applicable to parallel mechanisms with a limited workspace. It is first shown that the relative manipulability indices are characterized by the null space of the manipulator Jacobian. This motivates the problem of determining the class of manipulator Jacobians with a prescribed null space. This approach can be used to find optimally fault-tolerant manipulators. It is then shown through dimensional arguments that there are limits to the amount of redundancy for this problem to be solvable. The authors use these limits to prove that a previously derived inequality for the worst case relative manipulability index is generally not achieved for fully spatial manipulators and that the concept of optimal fault tolerance to multiple failures is more subtle than previously indicated. After presenting an example of a seven degreeof-freedom mechanism that is optimally fault-tolerant to single failure, the authors consider the problem of finding a manipulator Jacobian that is optimally fault tolerant to multiple failures. It is shown that optimal solutions cannot be equally fault tolerant.