Feedback Synthesis for Controllable Underactuated Systems using Sequential Second Order Actions

Abstract

Abstract-This paper derives nonlinear feedback control synthesis for general control affine systems using second-order actions-the needle variations of optimal control-as the basis for choosing each control response to the current state. A second result of the paper is that the method provably exploits the nonlinear controllability of a system by virtue of an explicit dependence of the second-order needle variation on the Lie bracket between vector fields. As a result, each control decision necessarily decreases the objective when the system is nonlinearly controllable using first-order Lie brackets. Simulation results using a differential drive cart, an underactuated kinematic vehicle in three dimensions, and an underactuated dynamic model of an underwater vehicle demonstrate that the method finds control solutions when the first-order analysis is singular. Moreover, the simulated examples demonstrate superior convergence when compared to synthesis based on first-order needle variations. Lastly, the underactuated dynamic underwater vehicle model demonstrates the convergence even in the presence of a velocity field.

Feedback synthesis for underactuated systems using sequential second-order needle variations

“…This paper expands the work presented in Mamakoukas et al (2017) by including the derivations of the MIH, the proofs that guarantee descent, and extensive simulation results that include comparisons with alternative feedback algorithms. Further, we extend the results to account for obstacles and prove the algorithm finds collision-free solutions for the controllable systems considered, including simulations of obstacle avoidance for static and moving obstacles.…”