Reciprocal Zeroing Neurodynamics for Inverse-Free Manipulator Kinematic Control with Unknown Jacobian

The kinematic control of manipulators has extensive applications in modern industry. Precise control enables robotic manipulators to perform diverse complex tasks, boost production efficiency/quality, and cut labor costs and safety risks. Kinematic control of manipulators centers on solving time-variant nonlinear equations. Traditional methods for this typically require a known Jacobian or matrix inversion. However, in practical applications, exact model parameters are typically unavailable, and computational resources are constrained. Moreover, the computational cost associated with matrix inverse is prohibitive. We propose an algorithm based on reciprocal zeroing neurodynamics for solving the time-variant nonlinear equation system of manipulator kinematic control with an unknown Jacobian matrix that is also inverse-free, which operates under the assumptions of continuous time formulation and perfect measurement conditions. We also provide theorems guaranteeing bounded convergence of Jacobian estimation error under noisy inputs. The proposed algorithm implicitly learns the unmeasurable kinematic model through online estimation of the Jacobian matrix based on real-time input−output relationships and approximates the solutions of the nonlinear equation system, thereby enabling precise control of manipulator kinematics in an inverse-free manner. Furthermore, the effectiveness and correctness of the proposed algorithm are ensured through the rigorous stability and convergence guarantees established through Lyapunov-based analysis. These analyses are conducted using concepts, lemmas, theorems, and remarks, laying a solid foundation for evaluating the proposed algorithm’s performance. Simulation-based and physical experiments exhibit tracking error constrained within the sub-millimeter to micrometer level, confirming the algorithm’s efficacy and highlighting its robustness in handling unknown Jacobians without inverse matrix computation.