Trajectory planning using vector smoothing splines with coupled derivative constraints

In this paper, we present a trajectory planning method using optimal vector smoothing splines. The multiple curves are designed simultaneously with equality and/or inequality derivative constraints, cross-coupled among the element curves. For constituting the vector splines, we employ normalized uniform B-splines as the basis functions. It is shown that we can express the derivatives of vector splines as linear function of the so-called control points. Such an expression enables us to treat equality and inequality cross-coupled constraints over intervals on derivatives of splines. Pointwise constraints on the vector splines can also be incorporated. The problem of optimal vector smoothing splines with cross-coupled derivative constraints reduce to convex quadratic programming problems. The results are applied to the trajectory planning problem as seen in the field of robotics, and the performance is examined by some numerical examples.