Optimal smoothing spline with constraints on its derivatives

This paper considers the problem of designing optimal smoothing spline with constraints on its derivatives. The splines of degree k are constituted by employing normalized uniform B-splines as the basis functions. We then show that the l-th derivative of the spline can be obtained by using B-splines of degree k-l with the control points computed as l-th difference of original control points. This yields systematic treatment of equality and inequality constraints over intervals on derivatives of arbitrary degree. Also, pointwise constraints can readily be incorporated. The problem of optimal smoothing splines with constraints reduce to convex quadratic programming problems. The effectiveness is demonstrated by numerical examples of approximations of probability distribution function and concave function, and trajectory planning with the constraints on velocity and acceleration.