where i j denotes the j-th column of J, and (3) The reduced manipulator Jacobian i J then determines the kinematic properties of the degraded system. In this article, a local measure of fault tolerance is defined that measures the performance of the degraded system relative to the original system.In the next section, necessary and sufficient conditions are derived for determining whether a manipulator with a single degree of redundancy is in a configuration for which the reduced system is singular. Using this condition, one can then develop strategies that will avoid such configurations. Section III discusses how the manipulability of a manipulator is affected by a joint failure. Once this has been determined, configurations can be identified for which the manipulability is reduced by a minimum amount due to any joint failure. The case of multiple joint failures is considered in Section IV. Section V discusses the application of these results to motion planning and Section VI presents a fully general spatial example. Finally, conclusions appear in Section VII.
II. FAULT INTOLERANT CONFIGURATIONSAs mentioned earlier, a joint failure can essentially result in a manipulator being in a singular configuration, even if the original Jacobian is of full rank. It is easy to show, using column space arguments, that the rank of the reduced Jacobian satisfies (1) Abstract-When a manipulator suffers a joint failure, its performance can be significantly affected. If the failed joint is locked, the resulting manipulator Jacobian is given by the original Jacobian, except that the column associated with the failed joint is removed. The rank of the resulting Jacobian then determines if the manipulator still has the ability to perform arbitrary end-effector motions. Unfortunately, even at an operating configuration that has a relatively high manipulability index, a joint failure may still result in a singular Jacobian. This work examines the problem of determining the reduced manipulability of a manipulator after one or more joint failures. Configurations that result in a minimal reduction of the manipulability index for any set of joint failures are determined.