Collision-free piecewise quadratic spline with regular quadratic obstacles (Q2825883)

The framework of the paper is related to the study of the conditions for collision-free piecewise quadratic paths with respect to a regular quadric, in the multidimensional Euclidian space. For the quadric, the authors consider a boundary domain of a complex object and illustrate the problem in a three-dimensional space. Based on two fixed points out of the quadric, in the same connected component, the authors find the set of all quadratic Bézier curves (parabolic arcs) connecting the points and having no intersection with the quadric. The system of the quadratic Bézier curves is represented by their admissible middle control points. Reducing the problem from 3D to 2D, the regular quadric is represented by a conic section. Usually, the algorithm for finding the collision-free path consists in obtaining a linear spline path generated by sample-based planning algorithms and the smoothing of the path to avoid slippage of wheels that can occur in mobile robotics on a non-smooth path. The method proposed is based on a direct analytical computation of all possible collision-free smooth paths, without using the sample-based planning algorithm. The obstacle is represented by a regular quadric and based on the given start/end position of the robot, the optimization of the sought path can be done by finding all quadratic Bézier curves containing the set of collision-free paths. When the scene with obstacles is too complicated, such that the smooth free-collision path cannot be found directly, the sample-based planning algorithm is used together with the path pruning algorithm (to remove redundant nodes randomly generated). In the proposed method a node can be removed if there is a quadratic Bézier collision-free path between two neighbor nodes. For each type of conic section, the set of the middle control point is found and the boundary is formed from the middle control points of the Bézier curves, touching the given conic section. Also, another application of the proposed method can be that of searching for pointwise space-like curves in the Minkowski space.