Regularity properties and deformation of wheeled robots trajectories

Abstract

Our contribution in this article is twofold. First, we identify the regularity properties of the trajectories of planar wheeled mobile robots. By regularity properties of a trajectory we mean whether this trajectory, or a function computed from it, belongs to a certain class C <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> (the class of functions that are differentiable n times with a continuous n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">th</sup> derivative). We show that, under some generic assumptions about the rotation and steering velocities of the wheels, any non-degenerate wheeled robot belongs to one of the two following classes. Class I comprises those robots whose admissible trajectories in the plane are C <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup> and piecewise C <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ; and class II comprises those robots whose admissible trajectories are C <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup> , piecewise C <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> and, in addition, curvature-continuous. Second, based on this characterization, we derive new feedback control and gap-filling algorithms for wheeled mobile robots using the recently-developed affine trajectory deformation framework.