Minimum-Energy Trajectory Planning for an Underactuated Serial Planar Manipulator

Underactuated robotic systems are appealing for industrial use due to their reduced actuator number, which lowers energy consumption and system complexity. Underactuated systems are, however, often affected by residual vibrations. This paper addresses the challenge of generating energy-optimal trajectories while imposing theoretical null residual (and yet practical low) vibration in underactuated systems. The trajectory planning problem is cast as a constrained optimal control problem (OCP) for a two-degree-of-freedom revolute–revolute planar manipulator. The proposed method produces energy-efficient motion while limiting residual vibrations under motor torque limitations. Experiments compare the proposed trajectories to input shaping techniques (ZV, ZVD, NZV, NZVD). Results show energy savings that range from 12% to 69% with comparable and negligible residual oscillations.