Nonlinear finite-element modeling and optimal tracking control of a rolling ball on an elastic simply supported beam with boundary torque actuation

This paper develops a control-oriented finite-element (FE) formulation for a rolling ball interacting with an elastic simply supported Euler–Bernoulli beam actuated by a boundary torque at the left support. The beam is discretized using cubic Hermite elements, while the ball is modeled as a moving load with rotational dynamics governed by a no-slip rolling constraint. To retain the moving-contact effects within a fixed FE state dimension, global position-dependent interpolation operators are introduced to evaluate the beam deflection, slope, curvature, and contact-point velocity at the ball location. Applying Lagrange’s equations yields a coupled nonlinear model with a configuration-dependent inertia matrix and mixed-velocity coupling terms. A continuum of static equilibria parameterized by the ball position is then computed by enforcing static force balance together with a zero-slope rest condition at the contact point, providing consistent operating points for linearization and a position-dependent feedforward bias torque. Based on frozen-time linear models, a gain-scheduled linear–quadratic–integral (LQI) tracking controller is constructed by interpolating local LQI gains and combining them with the equilibrium feedforward term, while explicitly accounting for actuator saturation through anti-windup logic. Numerical studies for a 0.60 m beam discretized by three FE elements demonstrate accurate set-point regulation from s(0) = 0.30 m to multiple targets under a torque limit of ± 4 N m, with well-damped transients and bounded control effort. The resulting framework provides a reproducible workflow for modeling and gain-scheduled optimal tracking control of underactuated flexible structures subject to rolling moving bodies using boundary actuation.