Control theoretic B-spline smoothing with constraints on derivatives

In this paper, we develop a method for designing optimal smoothing spline with constraints on its derivatives. A linear control system is used as a spline generator. Employing the results developed in the B-spline approach, we show that equality or inequality constraints on spline and its derivative over interval can be expressed as constraint on the control input and initial state of the linear system. Such constraints are useful in trajectory planning problem and in the shape preserving splines as convex splines. Pointwise constraints can easily be incorporated into the control problem. The problem of optimal smoothing splines with such constraints reduce to convex quadratic programming problems. We demonstrate the effectiveness and usefulness by numerical examples of trajectory planning with the constraints on velocity, acceleration and control input.