Dynamic analysis and control of a double flexible pendulums mechanism

Abstract

This thesis presents the modelling, control and validation of a double flexible inverted pendulum (DFIP) on a cart: a seven-DOF, fourteen-state, single-input mechanism comprising two flexible beams in series, each discretised into three rigid sub-links with torsional springs and dampers. The nonlinear equations of motion are derived via Euler-Lagrange, yielding the mass matrix, Coriolis vector (via Christoffel symbols), gravity and spring-damper vectors without small-angle approximation. Linearisation about upright reveals two unstable eigenvalues and confirms full controllability. An exact partial feedback linearisation of the cart channel cancels all nonlinearities for any configuration where the unactuated mass matrix is invertible.

Three controllers are designed and compared: LQR with feedback linearisation; full-state PID via constrained pole placement; and cascaded PID (outer-position/inner-angle) tuned by ISE minimisation. All are evaluated on the full nonlinear model for settling time, peak force, energy and robustness. The analytical model is validated against an independent Simscape Multibody model, with agreement to within a fraction of a degree and a few millimetres over a 30 s transient.