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

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. 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.