Spline Trajectory Planning for Path with Piecewise Linear Boundaries

We consider a problem of trajectory planning for path with piecewise linear boundaries. The trajectory is constructed as smoothing splines using normalized uniform B-splines as the basis functions. The boundary constraints are treated as a collection of inequality pairs by right and left boundary lines, and are formulated as linear inequality constraints on the so-called control point vector. Smoothing splines are constructed as an approximation of a piecewise linear centerline of the given path, where the given entire time interval is divided into subintervals according to the centripetal distribution rule. Other constraints as initial and terminal conditions on the trajectory can be included easily, and the problem reduces to convex quadratic programming problem where very efficient numerical solvers are available. The effectiveness of the proposed method is confirmed by an example of fairly complex path with piecewise linear boundaries. Also an example is included to demonstrate its usefulness for trajectory planning in an environment with obstacles.