On splines and polynomial tools for constrained motion planning

This paper discusses an open-loop trajectory planning problem for linear systems and polynomial systems with constraints. Flat systems have the nice property that the input and the state can be completely characterised by the so-called flat output. We propose a spline parameterisation for the flat output, the performance output, the states, and the inputs. Using this parameterisation the problem of constrained trajectory planning can be cast into a simple quadratic programming problem. An important result is that the B-spline parametrisation used gives exact results for constrained linear continuous-time system. The result is exact in the sense that the constrained signal can be made arbitrarily close to the boundary without having intersampling issues (as one would have in sampled-data systems). Numerical results are outlined, involving the generation of rest-to-rest trajectories.