Discrete Geometric Control of Planar Flexible Link Manipulators

Abstract

Discrete control laws for flexible beam models obtained using structure-preserving discretization procedures are few in the existing literature so far and many control laws, including distributed ones, exist for continuous time models. Much of the effort has focussed on the discretization of the flexible structure and the fidelity of such models. The work in this paper examines the feasibility of a discretized version of a continuous time distributed control strategy, with a few approximations, to the discrete variational integrator of a flexible beam, in our case, a flexible single-link manipulator (FLM). The nonlinear control strategy is inspired by a potential shaping method in the literature based on the Cosserat continuum framework. We provide stability proof for this continuous-time control and approximate it to a discrete-time geometric framework. We use a geometrically exact beam model in which the manipulator's configuration space lies on an infinite-dimensional Lie group. The model-based discrete-time control uses a Lie Group Variational Integrator obtained by applying the variational principle on the Lagrangian of the FLM. Results show that the discrete closed-loop system stabilizes the FLM at setpoints during point-to-point rotational maneuvers. The controller demands fine time-sampling and we acknowledge the associated challenges for real-time implementation of the control scheme and intend to pursue it as our future work.