Adaptive GND–ZND Fusion Using LSTM with Projection Constraints for Trajectory Tracking

Time-varying nonlinear mechanical systems, such as robotic manipulators and chaotic systems, face significant challenges in trajectory tracking due to strong nonlinearities, parameter perturbations, external disturbances, and physical constraints. Conventional gradient neural dynamics (GND) and Zhang neural dynamics (ZND) methods cannot simultaneously achieve global search capability and local tracking precision. To address this issue, this paper proposes an adaptive GND–ZND fusion framework driven by a long short-term memory (LSTM) network, integrated with a smooth hyperbolic tangent projection constraint to realize high-precision, robust, and constraint-satisfying trajectory tracking. The core innovation lies in an LSTM-based online weight adaptation mechanism that learns multi-timescale error features and system states, dynamically balancing the global exploration of GND and the local optimization of ZND. Furthermore, an infinitely differentiable smooth projection function is designed to eliminate gradient discontinuities and chattering, ensuring that the system always operates within safe physical bounds. A Lyapunov analysis establishes exponential convergence of the neural dynamics residual in the nominal stationary case and uniform ultimate boundedness in the presence of bounded time variation and disturbances. Comparative simulations are conducted on a six-degree-of-freedom (DOF) PUMA560 manipulator and the Lorenz chaotic system under variable payloads, Gaussian noise, periodic disturbances, and impulse interference. Results demonstrate that, compared with conventional GND, ZND, and fixed-weight fusion methods, the proposed approach reduces the trajectory tracking root mean square error (RMSE) to 0.032 mm, shortens the convergence time to 1.2 s, achieves a 98.7% success rate over 100 repeated trials, and improves the comprehensive performance index by 40.3%. The proposed framework provides a feasible and effective intelligent solution for high-performance constrained trajectory tracking of complex time-varying nonlinear mechanical systems.