Reduction of Velocity-Dependent Terms in Total Energy Shaping Approach

Abstract: Total energy shaping through interconnection and damping assignment passivity-based control (IDA-PBC) provides a powerful and systematic framework for stabilizing underactuated mechanical systems. Despite its theoretical appeal, incorporating actuator limitations into total energy shaping remains a largely open problem, with only limited results reported in the existing literature. In practice, the closed-loop behavior of energy-shaping controllers is strongly affected by the kinetic energy shaping terms. In this paper, a simultaneous IDA-PBC (SIDA-PBC) framework is employed to systematically attenuate the kinetic energy shaping terms by exploiting generalized forces, without altering the matching partial differential equations (PDEs). The free component of the generalized forces is derived analytically via an $\ell_\infty$-norm optimization formulation. Although a reduction in kinetic energy shaping terms does not necessarily guarantee a decrease in the overall control effort, the proposed approach effectively suppresses kinetic energy shaping components and achieves a reduced control magnitude whenever such a reduction is structurally feasible. Unlike existing approaches based on gyroscopic terms, which require multiple actuators, the proposed method is applicable to mechanical systems with a single actuator. Simulation and experimental results are provided to validate the effectiveness of the proposed approach.