| Abstract: |
| A control system is called non-holonomic when it cannot move instantaneously in all directions of its state space: at each configuration, only a lower-dimensional set of admissible velocities is available. A canonical example is the unicycle, which can roll forward and backwards and rotate, but cannot slide sideways, so lateral motion is not directly actuated. A similar constraint arises in the classical parallel-parking manoeuvre.
From a dynamical viewpoint, the missing directions can be generated through Lie brackets of the control vector fields. Traditionally, motion along Lie bracket directions is achieved using carefully designed switching or oscillatory controls, such as bang-bang strategies or sinusoidal averaging.
In this work, we took another approach. We introduce explicit Koch-type fractal controls that generate targeted motions along Lie-bracket directions using short-time inputs. To establish well-posedness for control systems driven by such fractal signals, we apply tools from rough path theory.
As model problems, we illustrate our approach through
bracket-induced lateral motion for the unicycle and a higher-order steering mechanisms in coupled trailer-type systems. |
|