Recursive motion planning using optimal vector smoothing splines with cross-coupled constraints

In this paper, we address an efficient motion planning method using a recursive design method of vector smoothing spline curves with equality and/or inequality constraints. The splines are constituted employing normalized uniform B-splines as the basis functions, and hence the central issue is to determine an optimal matrix of the so-called control points. Such a B-spline approach enables us to express various types of constraints as linear function of control points, including those coupled constraints among curves. Then, it is shown that the problem of constructing the constrained vector smoothing splines becomes a convex quadratic programming problem. Based on these results, we develop a recursive method for constructing the constrained splines. Such a method is useful in practice especially when some sets of data are observed one after another and we construct spline curves each time when a new set is given. In particular, we apply the proposed method to motion planning and replanning problems as shown in the field of robotics. The performance is examined by some numerical examples.