In this study, we present a fast computing algorithm for the inverse kinematic analysis of a dexterous robot inspired by muscular hydrostats. First, the forward kinematics based on piecewise constant curvature (PCC) theory was established. Second, an inverse kinematic algorithm based on a chain-like connection comprising rigid link elements with revolute joints is proposed. The inverse kinematic algorithm starts with a solution derived from the Euler-Bernoulli beam equation and then performs an iterative constrained optimization with an embedded integral controller. The algorithm can reach the desired positions and orientations of the end effector with theoretically “zero” error. The simulation results showed that the algorithm can reach all positions and orientations in the continuous workspace with a small number of iterations.