The Optimal Swing-Up of the Double Pendulum

A classic control problem that appeared in different variants is the inverted pendulum, that serves as a concept for a number of practical applications. A challenging variant is the inversion of a double pendulum mounted on a sliding cart. The problem consists in erecting (‘swinging-up’) the pendulum from the ‘straight-down’ to a ‘straight-up’ position in the minimum time. In this paper a nonlinear optimal control approach is employed to investigate the structure of the optimal swing-up. A bang-bang control, possibly with a singular arc, is revealed. In addition, the same straight-up position can be reached with different strategies, namely with the pendulum links undergoing one or more revolutions during the manoeuvre. The different strategies call for different control structure, with a different manoeuvre time required to perform the inversion. Interesting enough, the optimal solution is not the one involving the shortest path in terms of rotations of the links, since dynamic effects can be leveraged to obtain faster swing-ups.