An obstacle-avoiding minimum variation B-spline problem

Abstract

We study the problem of computing a planar curve, restricted to lie between two given polygonal chains, such that the integral of the square of arc-length derivative of curvature along the curve is minimized. We introduce the Minimum Variation B-spline problem which is a linearly constrained optimization problem over curves defined by Bspline functions only.An empirical investigation indicates that this problem has one unique solution among all uniform quartic B-spline functions. Furthermore, we prove that, for any B-spline function, the convexity properties of the problem are preserved subject to a scaling and translation of the knot sequence defining the B-spline.

Obstacle-Avoiding Path Planning for High Velocity Wheeled Mobile Robots

“…Furthermore, works combining smooth path planning and obstacle-avoiding have so far barely been developed. Only 1 (Berglund et al, 2003) and (Scheuer, 1998) have treated this problem, the first one using an heuristic choice on the curve family (B-splines), the second one supported by Sussmann's results, the latter is closest to optimality.…”

Development and Implementation of Novel Parametrization Techniques for Multidisciplinary Design Initialization

“…Juhsz and Hoffmann 5 made use of the knot values of the B-spline to create a virtual envelope that the shape could not cross. A similar technique was presented by Berglund et al 6 These techniques are elegant, however, they only work for external constraints, whereas aircraft constraints are usually internal (e.g. fuel tanks, passenger cabin, bagage volume).…”

KAT‐5: Robust systems for autonomous vehicle navigation in challenging and unknown terrain

“…In addition, the curvature is kept continuous so that it is not necessary to stop the vehicle in order to turn the steering wheel to a new position before continuing. B-splines were chosen for use in the path planning algorithms primarily because of the ease in which the shape of their resulting curves can be controlled [Berglund et al, 2003].After an initial path is created that follows the center of the corridor, the path is checked against the obstacle repository to determine if it is a safe path. If the path is not safe, a simple algorithm generates and adjusts control points on the problem spots of the curve until the spline avoids all known obstacles while still containing valid maximum curvature.…”