Energy-based swing-up of the acrobot and time-optimal motion

We present a control law for the swing-up of an acrobot with torque constraints on the actuator. The domain of the initial condition for the strategy to work is the entire manifold and we further guarantee that the system reaches a small neighbourhood about the upward equilibrium position. For a restricted domain of the initial condition, we observe that the control law is bang-bang in nature. This motivates us to verify the time-optimality of the control strategy. Necessary conditions for time-optimality are presented and these are subsequently verified numerically.