Energy-Based Closed-Form Solution of the Divergent Component of Motion for Nonlinear Inverted Pendulum Trajectories

The divergent component of motion (DCM) quantifies the unstable component of a robot's motion and is widely used for online motion planning and stabilization in robots. Although the DCM is derived from the linear inverted pendulum model, which admits a closed-form solution, extending it to nonlinear, height-varying pendulum dynamics complicates the derivation and typically requires iterative computation or piecewise linearization, resulting in a high computational cost. This letter introduces an energy-based derivation that enables the derivation of the DCM in closed form, even when the pendulum height varies, allowing for the flexible generation of curved paths that include both convex and concave shapes without requiring piecewise linearization. The proposed method is validated through comparative pendulum simulations and through its application to bipedal walking and a balance-assist motorcycle. The results confirm that the proposed method successfully enables the DCM to be expressed for nonlinear trajectories, allowing a conventional DCM-based stabilizer to be applied directly.