The problems on designing time-optimal motion profile in the motion system with elasticity are briefly reviewed. Related positioning error and robustness constraints are constructed based on the analytic envelope of residual vibration response which is formulated by the proposed energy-based approach. The time-optimal model under such constraints is applied in the proposed freeform S-curve profile, which is the superset of symmetric S-curve and AS-curve profiles, and to minimize the positioning time while keeping positioning accuracy under a neighborhood range of modeling frequency. The optimized motion profile shows that the proposed freeform S-curve profile allows the time-optimal model to simultaneously optimize the magnitude and phase terms in residual vibration response in order to obtain an optimal trade-off between shortening positioning time and keeping positioning accuracy. The experiment results show that the proposed optimization model can significantly reduce the positioning time compared with traditional S-curve under the same physical limits. The universality of the optimization model is verified by experiments with short distance and long distance cases. The robustness of the proposed optimization model is also validated by these experiments. Moreover, the analysis on the optimized freeform S-curve profile reveals that perfect waveform cancellation is not always necessary to limit the residual vibration.

It is well believed that S-curve motion profiles are able to reduce residual vibration, and are widely applied in the motion control fields. Recently, a new asymmetric S-curve (AS-curve) motion profile, which is able to effectively adjust the acceleration and deceleration periods, is proposed to enhance the performance of S-curve motion profile, and proved to be better than the traditional symmetric S-curve in many cases. However, most commercial motion controllers do not support the AS-curve motion profiles inherently. Special knowledge or expensive advanced controlling systems, such as dSPACE system, are required to generate the AS-curve motion command, which limits the applications of the AS-curve motion profile in many practical applications. In this paper, a generic method based on the Position-Velocity-Time (PVT) mode move supported by most commercial motion controllers is proposed to generate exact AS-curve motion command in real machines. The analytic polynomial functions of AS-curve motion profile are also derived to simplify the further application, and the effectiveness of the proposed method is verified by numerical simulation.

This study introduces a rigid–flexible coupled constant force actuator integrated with Active Disturbance Rejection Control (ADRC) to tackle the rigidity–compliance trade-off in precision force-sensitive applications. The actuator utilizes compliant hinges to decrease contact stiffness by three orders of magnitude (${10}^6\to {10}^3 \mathrm{N}/\mathrm{m}$), facilitating effective force management through millimeter-scale placement (0.1∼1 mm) and inherently mitigating high-frequency disturbances. The ADRC framework, augmented by an Extended State Observer (ESO), dynamically assesses and compensates for internal nonlinearities (such as friction hysteresis) and external disturbances without necessitating accurate system models. Experimental results indicate enhanced performance compared to PID control: under dynamic disturbances, force deviations are limited to $\pm 0.2 \mathrm{N}$ with a 98.5% reduction in mean absolute error, a 96.3% increase in settling speed, and 99% suppression of oscillations. The co-design of mechanical compliance with model-free control addresses the constraints of traditional high-stiffness systems, providing a scalable solution for industrial robots, compliant material processing, and medical device operations. Validation of the prototype under sinusoidal perturbations demonstrates reliable force regulation (settling time $<0.56 \mathrm{s}$, errors $<0.5 \mathrm{N}$), underscoring its relevance in dynamic situations. This study integrates theoretical innovation with experimental precision, enhancing intelligent manufacturing systems via adaptive control and structural synergy.