Variable step-size adaptive feedforward control for end-effector micro-vibration suppression in Delta-type fused deposition modeling systems

Transverse micro-vibration at the nozzle end of Delta-type fused deposition modeling systems can be amplified by branch-chain coupling, structural compliance, and multi-frequency disturbances, thereby degrading deposition accuracy. Existing model-based compensation and command-shaping methods often rely on accurate dynamic models or modal information, while conventional filtered-x least mean squares with fixed or manually tuned step size struggle to balance fast convergence and low steady-state residual vibration. To address this problem, this paper proposes a deep deterministic policy gradient-filtered-x least mean squares adaptive feedforward vibration suppression method, in which the deep deterministic policy gradient learns a state-dependent step-size regulation policy, and filtered-x least mean squares remains the lower-level secondary-path-compensated adaptive controller. The policy is trained offline and deployed online only through Actor-network forward inference. ADAMS/MATLAB-Simulink co-simulation is performed under single-/two-degree-of-freedom and single-/dual-frequency disturbances. Compared with filtered-x least mean squares, normalized least mean squares, variable step-size least mean squares, and variable-forgetting-factor filtered-x recursive least-squares, the proposed method achieves the lowest ITAE in all tested cases. Relative to conventional filtered-x least mean squares, the average integral of time-weighted absolute error (ITAE) is reduced by 95 . 1 % , and the convergence time is shortened by approximately 89 . 5 % – 90 . 3 % . Robustness tests under noise, delay, secondary-path mismatch, actuator saturation, and combined non-ideal conditions show bounded closed-loop responses. The results demonstrate that deep deterministic policy gradient–based step-size regulation improves the convergence, average residual-vibration suppression, and robustness of filtered-x least mean squares–based nozzle-end micro-vibration control.