Singularity-Free Inverse Dynamics for Underactuated Systems with a Rotating Mass
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Underactuated systems consist of passive bodies/joints that don't have any active torque. To control these systems through inverse dynamics, the inertial coupling creates certain singularities where they are mainly related to constraints that passive bodies impose. In this study, we propose that modeling the trajectory of the rotating mass with multiple phase-shifted sinusoidal curves removes the singularity. At first, we derive the modified non-linear dynamics of a considered rolling system with active torque on the rotating mass by the trajectories of the combined-waves. Also, the inverse dynamics are derived and singularity regions for this underactuated system are demonstrated. Then, we propose a theory that designing the parameters of the combined phase-shifted waves under certain conditions removes the singularity. We obtain this parametric condition from the positive definiteness of the inertia matrix in the inverse dynamics. Finally, the simulation results are confirmed by using a prescribed Beta function on the specified states of the rolling carrier. Because our algebraic method is integrated into the non-linear dynamics, the proposed solution has a great potential to be extended to the Lagrangian mechanics with multiple degrees-of-freedom.
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