Transition Control of a Rotary Double Inverted Pendulum Using Direct Collocation ^†

The rotary double inverted pendulum system is characterized by one stable equilibrium point and three unstable equilibrium points due to its kinematic properties. This paper defines the transition control problem between these equilibrium points to extend the conventional swing-up control problem and proposes an implementation method using a laboratory-developed rotary double inverted pendulum. To minimize energy consumption during the transition process while satisfying the boundary conditions of different equilibrium points, a two-point boundary value optimal control problem is formulated. The feedforward trajectory required for feedforward control is computed offline by solving this problem. The direct collocation method is employed to convert the constrained continuous optimal control problem into a nonlinear optimization problem. Furthermore, a time-varying linear–quadratic (LQ) controller is utilized as a feedback controller to accurately track the generated feedforward trajectory during real-time control, compensating for uncertainties in the feedforward control process. The proposed transition control strategy is experimentally implemented, and its effectiveness and practicality are validated through the successful tracking of 12 transition trajectories.