Closed Servoconstraints in Periodic Motion Planning for Underactuated Mechanical Systems

This paper is devoted to the servoconstraints approach in the problem of periodic motion planning for Euler – Lagrange systems with a single degree of underactuation. We focus on the case where the servoconstraint is not regular and thus leads to the appearance of isolated singularities in reduced dynamics. We demonstrate that, subject to supplementary conditions, the reduced dynamics possess smooth solutions that pass through the singular point and this can be utilized for finding trajectories of the original system. Building upon this outcome, we solve the problem of motion planning of the Pendubot system with an imposed eight-shaped servoconstraint. To verify the feasibility of the discovered trajectory, we present computer simulation results of the closed-loop system with feedback that enables orbital stabilization for the trajectory.