On the Dynamics and State Dependent Multiple Equilibria of a Post-Buckled Ultra-Flexible Inverted Pendulum on a Rotating Hub

摘要

Compliant element systems with ultra-large deformation display rich nonlinear dynamics and pose challenging control problems, which, when solved, could enable enhancements in several mechatronics applications, such as soft robotics, MEMS, and biomedical applications. This paper considers post-buckled dynamic analysis of an inverted ultra-flexible pendulum actuated by a rotary hub. We first derive a complete set of equations capturing the dynamics of the system, essential for control development, using the assumed modes method framework, considering ultra-large deformations. Constrained Lagrange formulation is used for the same. In the perfect inverted configuration with zero hub angle, the buckled beam would display two symmetric stable equilibria and one unstable. However, as the hub angle changes on either side, the equilibrium positions shift, and eventually two of them vanish, and we are left with only one stable equilibrium. We use the dynamic equations to characterize this interesting phenomenon, demonstrating the continuous state dependence of multiple equilibria. Furthermore, experimental counterparts of the equilibrium results are meticulously obtained and discussed. Moreover, simulation results capture the nonlinear dynamics of this system. Overall, the work establishes a solid mathematical foundation with a control-amenable model for futuristic ultra-compliant mechatronic systems.