Variable-Gain Coupling Online Trajectory for Underactuated Mechanical Systems With Unactuated Output Constraints

Recently, online trajectory planning has become increasingly prominent as one of the mainstream studies for underactuated mechanical systems. However, most of the existing online trajectories are designed for a specific system, rather than for a class of systems with more general applications. Hence, the planning of universal online trajectory requires further investigation. In addition, most reported online trajectories fail to account for kinematic constraints, which are frequently encountered in practical mechanical systems. This article proposes a novel variable-gain coupling online trajectory, which satisfies unactuated output constraints for a class of underactuated systems, with the performance supported by rigorous Lyapunov-based proof. Specifically, a coupling term is constructed to make the closed-loop system effectively react to unactuated dynamics, thereby achieving asymptotic convergence of the unactuated states and keeping the unactuated outputs within preset constraints to ensure the safety of unactuated mechanisms. Furthermore, a variable gain is designed to integrate a proportional-integral (PI) regulator into the trajectory with theoretical justification, which ensures accurate convergence of the actuated outputs. Simulation is conducted by using the overhead crane model and the boom crane model. Experimental results are also provided to verify the constraint-enforcement capability of the proposed trajectory.