Kinematic reliability is an essential index assessing the work performance of robotic manipulators. In general, the kinematic reliability of robotic manipulators is defined as the probability of the pose or position error falling into a specified tolerant region. Therefore, this work proposes an efficient method to conduct kinematic reliability analysis for robotic manipulators under rectangular and spherical allowable safe boundaries in terms of dimension and input uncertainties. First, based on the Baker–Campbell–Hausdorff formula and Lie group theory, the mean and covariance matrix of the distribution of the pose error are analytically determined. Then, the expectation propagation of the multivariate Gaussian and saddlepoint approximation method are employed to calculate the probabilities of kinematic reliability under the rectangular and spherical safe boundaries, respectively. The proposed method takes into account the boundness of the random error variable and is available for arbitrarily distributed errors. Finally, a spatial six degrees-of-freedom industrial robot is used as an example to demonstrate the effectiveness of the proposed method by comparison with other methods. The comparison results indicate that the proposed method has higher accuracy and efficiency.