2014 Volume 7 Issue 2 Pages 104-111
We consider the problem of optimal design of smoothing splines with constraints. Main concern is on constraints on derivatives over intervals as arising in monotone splines. The splines are constructed using normalized uniform B-splines as the basis functions. The authors show that the l-th derivative of the spline of degree k is obtained by using B-splines of degree k-l with the control points computed as the l-th difference of original control point sequence. This yields systematic treatment of equality and inequality constraints over intervals on derivatives of arbitrary degree, and the problem is formulated as convex quadratic programming. Thus the method is useful in various applications. 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.