Constrained Reachability and Trajectory Generation for Flat Systems

We consider the problem of trajectory generation for constrained differentially flat systems. Based on the topological properties of the set of admissible steady state values of a flat output we derive conditions which allow for an a priori verification of the feasibility of constrained set-point changes. We propose to utilize this relation to generate feasible reference trajectories. To this end, we suggest to split the trajectory generation problem into two stages: (a) the planning of geometric reference paths in the flat output space combined with (b) an assignment of a dynamic reference motion to these paths. This assignment is based on a reduced optimal control problem. The unique feature of the approach is that due to the specific construction of the reference paths the optimal control problem to be solved is guaranteed to be feasible. To illustrate our results we consider a Van de Vusse reactor as an example

Minimum-time trajectory generation for constrained linear systems using flatness and B-splines

In this article, we discuss minimum-time trajectory generation for input-and-state constrained continuous-time LTI systems in the light of the notion of flatness and B-splines parametrisation. Flat systems have the useful property that the input and the state trajectories can be completely characterised by the (so-called) flat output. We propose a splines parametrisation for the flat output, and the corresponding parametrisations for the performance output, the states and the inputs. Using this parametrisation the problem of minimum-time constrained trajectory planning is cast into a feasibility-search problem in the splines control-point space, in which the constraint region is characterised by a polytope. A close approximation of the minimum-time trajectory is obtained by systematically searching the end-time that makes the constraint polytope to be minimally feasible