Geometry‐Inspired Control of the Brockett‐Integrator in Cylindrical Coordinates

The so‐called Brockett‐Integrator is a famous control system. On the one hand it has a very simple structure (second order polynomial, three state components, two affine inputs, no drift) and fulfills the Lie‐Algebra‐Rank‐Condition for (strong) local accessibility. Nevertheless, it has the challenging control property, that it does not admit a continuously differentiable control law, which stabilizes the origin. In this contribution we consider the system in cylindrical coordinates. If the first time derivatives of the radius and the orientation are chosen as the new inputs, the transformed system dynamics is even simpler (in terms of expression length) than in usual Cartesian form. Moreover, the system can be decomposed into two subsystems and a differentially flat output can be easily read off. Based on this partially decoupled system dynamics we formulate a simple three‐staged control law which drives the system into the origin by aiming for a minimal arc length in the (original) state space.