Inverse Optimal Feedback and Gain Margins for Unicycle Stabilization

The recent development of globally strict control Lyapunov functions (CLFs) for the challenging unicycle parking problem provides a foundation for pursuing optimality. We address this in the inverse optimal framework, thereby avoiding the need to solve the Hamilton$\unicode{x2013}$Jacobi$\unicode{x2013}$Bellman (HJB) equations, and establish a general result that is optimal with respect to a meaningful cost. We present several design examples that impose varying levels of penalty on the control effort, including arbitrarily bounded control. Furthermore, we show that the inverse optimal controller possesses an infinite gain margin thanks to the system being driftless, and leveraging this property, we extend the design to an adaptive controller that handles model uncertainty. Finally, we compare the performance of the non-adaptive inverse optimal controller with its adaptive counterpart.