An approximate characterisation of the set of feasible trajectories for constrained flat systems

We study differentially flat systems, subject to constraints on the input, the state and their derivatives. We are interested in providing an approximate characterisation of the set of trajectories satisfying both the equations of the system and the constraints. The flatness property and the choice of a polynomial parametrisation in terms of Bézier functions enable a finite dimension formulation of the problem in the parameter space of the Bézier control points. Such formulation provides a unified method to express a large variety of relevant constraint types in the state and the inputs. We present a simple and effective inner approximation of the aforementioned set of trajectories obtained by simple algebraic manipulations of the original constraints formulation. The symbolic nature of the proposed approximation allows a certain degree of adaptation to changes in the constraints without the need for a complete re-computation.