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.

[1]  M. Egerstedt,et al.  Periodic smoothing splines , 2008, Autom..

[2]  Peter E. Crouch,et al.  Dynamic interpolation and application to flight control , 1991 .

[3]  H. Kano,et al.  Discrete-Time Control Systems Approach for Optimal Smoothing Splines , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[4]  Zhimin Zhang,et al.  Splines and Linear Control Theory , 1997 .

[5]  Clyde F. Martin,et al.  Monotone Smoothing Splines Using General Linear Systems , 2013, ArXiv.

[6]  Magnus Egerstedt,et al.  Optimal Control and Monotone Smoothing Splines , 2003 .

[7]  Magnus Egerstedt,et al.  B-splines and control theory , 2003, Appl. Math. Comput..

[8]  Claudio Melchiorri,et al.  Trajectory Planning for Automatic Machines and Robots , 2010 .

[9]  D. L. Russell,et al.  A discrete systems approach to cardinal spline Hermite interpolation , 2005 .

[10]  Hiroyuki Fujioka,et al.  Optimal smoothing and interpolating splines with constraints , 2007, 2007 46th IEEE Conference on Decision and Control.

[11]  Andrew Y. Ng,et al.  Task-space trajectories via cubic spline optimization , 2009, 2009 IEEE International Conference on Robotics and Automation.

[12]  D K Smith,et al.  Numerical Optimization , 2001, J. Oper. Res. Soc..

[13]  Magnus Egerstedt,et al.  Optimal trajectory planning and smoothing splines , 2001, Autom..

[14]  Hiroyuki Fujioka,et al.  Optimal smoothing spline with constraints on its derivatives , 2010, 49th IEEE Conference on Decision and Control (CDC).

[15]  G. Wahba.  Spline models for observational data , 1990 .