Time-dependent system kinematic reliability of robotic manipulators, referring to the probability of the end-effector’s pose error falling into the specified safe boundary over the whole motion input, is of significant importance for its work performance. However, investigations regarding this issue are quite limited. Therefore, this work conducts time-dependent system kinematic reliability analysis defined with respect to the pose error for robotic manipulators based on the first-passage method. Central to the proposed method is to calculate the outcrossing rate. Given that the errors in robotic manipulators are very small, the closed-form solution to the covariance of the joint distribution of the pose error and its derivative is first derived by means of the Lie group theory. Then, by decomposing the outcrossing event of the pose error, calculating the outcrossing rate is transformed into a problem of determining the first-order moment of a truncated multivariate Gaussian. Then, based on the independent assumption that the outcrossing events occur independently, the analytical formula of the outcrossing rate is deduced for the stochastic kinematic process of robotic manipulators via taking advantage of the moment generating function of the multivariate Gaussian, accordingly leading to achievement of the time-dependent system kinematic reliability. Finally, a six-degrees-of-freedom (6-DOF) robotic manipulator is used to demonstrate the effectiveness of the proposed method by comparison with the Monte Carlo simulation and finite-difference-based outcrossing rate method.

Kinematic reliability is an essential index that assesses the performance of the mechanism associating with uncertainties. This study proposes a novel approach to kinematic reliability analysis for planar parallel manipulators based on error propagation on plane motion groups and clipped Gaussian in terms of joint clearance, input uncertainty, and manufacturing imperfection. First, the linear relationship between the local pose distortion coming from the passive joint and that caused by other error sources, which are all represented by the exponential coordinate, are established by means of the Baker–Campbell–Hausdorff formula. Then, the second-order nonparametric formulas of error propagation on independent and dependent plane motion groups are derived in closed form for analytically determining the mean and covariance of the pose error distribution of the end-effector. On this basis, the kinematic reliability, i.e., the probability of the pose error within the specified safe region, is evaluated by a fast algorithm. Compared to the previous methods, the proposed approach has a significantly high precision for both cases with small and large errors under small and large safe bounds, which is also very efficient. Additionally, it is available for arbitrarily distributed errors and can analyze the kinematic reliability only regarding either position or orientation as well. Finally, the effectiveness and advantages of the proposed approach are verified by comparing with the Monte Carlo simulation method.