A Kinematic Analysis and Evaluation of Planar Robots Designed From Optimally Fault-Tolerant Jacobians Khaled M. Ben-Gharbia, Student Member, IEEE,

Abstract—It is common practice to design a robot’s kinematics from the desired properties that are locally specified by a manipulator Jacobian. In this work, the desired property is fault tolerance, defined as the postfailure Jacobian possessing the largest possible minimum singular value over all possible locked-joint failures. A mathematical analysis based on the Gram matrix that describes the number of possible planar robot designs for optimally fault-tolerant Jacobians is presented. It is shown that rearranging the columns of the Jacobian or multiplying one or more of the columns of the Jacobian by ±1 will not affect local fault tolerance; however, this will typically result in a very different manipulator. Two examples, one that is optimal to a single joint failure and the second that is optimal to two joint failures, are analyzed. This analysis shows that there is a large variability in the global kinematic properties of these designs, despite being generated from the same Jacobian. It is especially surprising that major differences in global behavior occurs for manipulators that are identical in the working area. Index Terms—Fault-tolerant robots, robot kinematics, redundant robots. I

Trajectory Optimization for Velocity Jumps Reduction considering the Unexpectedness Characteristics of Space Manipulator Joint-Locked Failure

Aiming at reducing joint velocity jumps caused by an unexpected joint-locked failure during space manipulator on-orbit operations without shutting down manipulator, trajectory optimization strategy considering the unexpectedness characteristics of joint-locked failure is proposed in the paper, which can achieve velocity jumps reduction in both operation space and joint space simultaneously. In the strategy, velocity in operation space concerning task completion directly is treated as equality constraints, and velocity in joint space concerning motion performance is treated as objective function. Global compensation vector which consists of coefficient, gradient of manipulability, and orthogonal matrix of null space is constructed to minimize the objective function. For each particular failure time, unique optimal coefficient can be obtained when the objective function is minimal. As a basis, a method for optimal coefficient function fitting is proposed based on a priori failure information (possible failure time and the corresponding optimal coefficient) to guarantee the unexpectedness characteristics of joint-locked failure. Simulations are implemented to validate the efficiency of trajectory optimization strategy in reducing velocity jumps in both joint space and operation space. And the feasibility of coefficient function is also verified in reducing velocity jump no matter when joint-locked failure occurs

Identifying the failure-tolerant workspace boundaries of a kinematically redundant manipulator

Includes bibliographical references (page 4523).In addition to possessing a number of other important properties, kinematically redundant manipulators are inherently more tolerant to locked-joint failures than nonredundant manipulators. However, a joint failure can still render a kinematically redundant manipulator useless if the manipulator is poorly designed or controlled. This paper presents a method for identifying a region of the workspace of a redundant manipulator for which task completion is guaranteed in the event of a locked-joint failure. The existence of such a region, called a failure-tolerant workspace, will be guaranteed by imposing a suitable set of artificial joint limits prior to a failure. Conditions are presented that characterize end-effector locations in this region. Based on these conditions, a method is presented that identifies the boundaries of the failure-tolerant workspace. Optimized failure-tolerant workspaces for a three degree-of-freedom planar robot are presented

Kinematic design and motion planning of fault tolerant robots with locked joint failures

2019 Summer.Includes bibliographical references.The problem of kinematic design and motion planning of fault tolerant robots with locked joint failure is studied in this work. In kinematic design, the problem of designing optimally fault tolerant robots for equal joint failure probabilities is first explored. A measure of local fault tolerance for equal joint failure probabilities has previously been defined based on the properties of the singular values of the Jacobian matrix. Based on this measure, one can determine a Jacobian that is optimal. Because these measures are solely based on the singular values of the Jacobian, permutation of the columns does not affect the optimality. Therefore, when one generates a kinematic robot design from this optimal Jacobian, there will be 7! robot designs with the same locally optimal fault tolerant property. This work shows how to analyze and organize the kinematic structure of these 7! designs in terms of their Denavit and Hartenberg (DH) parameters. Furthermore, global fault tolerant measures are defined in order to evaluate the different designs. It is shown that robot designs that are very similar in terms of DH parameters, e.g., robots generated from Jacobians where the columns are in reverse order, can have very different global properties. Finally, a computationally efficient approach to calculate the global pre- and post-failure dexterity measures is presented and used to identify two Pareto optimal robot designs. The workspaces for these optimal designs are also shown. Then, the problem of designing optimally fault tolerant robots for different joint failure probabilities is considered. A measure of fault tolerance for different joint failure probabilities is defined based on the properties of the singular values of the Jacobian after failures. Using this measure, methods to design optimally fault tolerant robots for an arbitrary set of joint failure probabilities and multiple cases of joint failure probabilities are introduced separately. Given an arbitrary set of joint failure probabilities, the optimal null space that optimizes the fault tolerant measure is derived, and the associated isotropic Jacobians are constructed. The kinematic parameters of the optimally fault tolerant robots are then generated from these Jacobians. One special case, i.e., how to construct the optimal Jacobian of spatial 7R robots for both positioning and orienting is further discussed. For multiple cases of joint failure probabilities, the optimal robot is designed through optimizing the sum of the fault tolerant measures for all the possible joint failure probabilities. This technique is illustrated on planar 3R robots, and it is shown that there exists a family of optimal robots. After the optimally fault tolerant robots are designed, the problem of planning the optimal trajectory with minimum probability of task failure for a set of point-to-point tasks, after experiencing locked joint failures, is studied. The proposed approach first develops a method to calculate the probability of task failure for an arbitrary trajectory, where the trajectory is divided into small segments, and the probability of task failure of each segment is calculated based on its failure scenarios. Then, a motion planning algorithm is proposed to find the optimal trajectory with minimum probability of task failure. There are two cases. The trajectory in the first case is the optimal trajectory from the start configuration to the intersection of the bounding boxes of all the task points. In the other case, all the configurations along the self-motion manifold of task point 1 need to be checked, and the optimal trajectory is the trajectory with minimum probability of task failure among them. The proposed approach is demonstrated on planar 2R redundant robots, illustrating the effectiveness of the algorithm