Motion planning for flat systems using positive B-splines: An LMI approach

In this paper, the motion planning problem is studied for nonlinear differentially flat systems using B-spline parameterization of the flat output history. In order to satisfy the constraints continuously in time, the motion planning problem is transformed into a B-spline positivity problem. The latter problem is formulated as a convex semidefinite programming problem by means of a non-negative piecewise polynomial function description based on sum of squares decomposition. The contribution of the paper is thus a one-step design procedure for motion planning that satisfies constraints continuously in time where usual B-spline and collocation techniques need post-analysis. Finally, an example of flexible link manipulator motion is presented to illustrate the overall approach.