Planning Smooth Paths among Obstacles using Minimum Curvature Variation B-splines

In this paper we study the problem of computing smooth planar paths in the presence of obstacles where we have an a priori knowledge about the environment. We investigate how the smoothness of a path and the total travel time required by the path are related for paths used by a four-wheel fourgear articulated vehicle. A path is considered smooth if the variation of its curvature, i.e., the integral of the square of the derivative of curvature along the path, is minimal. Paths are defined by quartic B-splines and obstacles are represented by polygonal chains. Quartic B-splines have a continuous derivative of curvature. Obstacle-avoidance is achieved by means of the envelope of the B-splines. We present a study of eight cases based on real-world application data from the Swedish mining company LuossavaaraKiirunavaara AB (LKAB). The results indicate that a minimum curvature variation B-spline path-planning algorithm we have developed yields paths that are substantially better than the ones used by LKAB today. Our simulations shows that the new paths are up to 39% faster to travel along than the paths currently in use. They even decrease the wear on the vehicle. Preliminary results from the production at LKAB show an overall 5-10% decrease in the total time. The total time includes both travel on the path and ore loading and unloading. Note to Practitioners – This article was motivated by the problem of how to automatically produce drive-paths of high quality for autonomous transportation vehicles in mines. The vehicles are heavy (> 100 tonnes) but are still expected to run at speeds up to 20 km/h to be productive. To reach these speeds without destroying the steering gear and the mechanics of the vehicles, their paths must be smooth. It turns out that visual inspection is often not sufficient to distinguish a smooth path from a harmful one. We suggest a method for computing paths, requiring an a priori knowledge about the environment, that minimizes the amount of steering needed during a transport. The computed paths are also safe in that they guarantee that the vehicle will not collide with a tunnel wall. We present a study of eight cases based on real-world application data from the Swedish mining company Luossavaara-Kiirunavaara AB (LKAB). Our simulations show that our paths are up to 39% faster to travel along than the paths currently in use while they cause no more wear on the vehicles. Tests performed in the production at LKAB show an overall 5-10% decrease in the total time if not only travel but also loading and unloading is included.

[1]  L. Dubins.  On Curves of Minimal Length with a Constraint on Average Curvature, and with Prescribed Initial and Terminal Positions and Tangents , 1957 .

[2]  M. Cox.  The Numerical Evaluation of B-Splines , 1972 .

[3]  C. R. Deboor,et al.  A practical guide to splines , 1978 .

[4]  Philip E. Gill,et al.  Practical optimization , 1981 .

[5]  M. Iri,et al.  Polygonal Approximations of a Curve — Formulations and Algorithms , 1988 .

[6]  Yutaka Kanayama,et al.  Smooth local path planning for autonomous vehicles , 1989, Proceedings, 1989 International Conference on Robotics and Automation.

[7]  Micha Sharir,et al.  Algorithmic motion planning in robotics , 1991, Computer.

[8]  Kazuo Tanie,et al.  Trajectory Design and Control of a Wheel-type Mobile Robot Using B-spline Curve , 1989, Proceedings. IEEE/RSJ International Workshop on Intelligent Robots and Systems '. (IROS '89) 'The Autonomous Mobile Robots and Its Applications.

[9]  John F. Canny,et al.  Planning smooth paths for mobile robots , 1989, Proceedings, 1989 International Conference on Robotics and Automation.

[10]  Henry P. Moreton.  Minimum curvature variation curves, networks, and surfaces for fair free-form shape design , 1993 .

[11]  Narendra Ahuja,et al.  Gross motion planning—a survey , 1992, CSUR.

[12]  Jean-Paul Laumond,et al.  Primitives for smoothing mobile robot trajectories , 1993, [1993] Proceedings IEEE International Conference on Robotics and Automation.

[13]  Jean-Daniel Boissonnat,et al.  A note on shortest paths in the plane subject to a constraint on the derivative of the curvature , 1994 .

[14]  Paul Dierckx,et al.  Curve and surface fitting with splines , 1994, Monographs on numerical analysis.

[15]  Gerald E. Farin,et al.  Curves and surfaces for computer-aided geometric design - a practical guide, 4th Edition , 1997, Computer science and scientific computing.

[16]  Gert Vegter,et al.  In handbook of discrete and computational geometry , 1997 .

[17]  Mark de Berg,et al.  Computational geometry: algorithms and applications , 1997 .

[18]  Jörg Peters,et al.  Smooth paths in a polygonal channel , 1999, SCG '99.

[19]  Jan Becker,et al.  Vehicle guidance for an autonomous vehicle , 1999, Proceedings 199 IEEE/IEEJ/JSAI International Conference on Intelligent Transportation Systems (Cat. No.99TH8383).

[20]  Motoji Yamamoto,et al.  Quasi-time-optimal motion planning of mobile platforms in the presence of obstacles , 1999, Proceedings 1999 IEEE International Conference on Robotics and Automation (Cat. No.99CH36288C).

[21]  Aurelio Piazzi,et al.  Global minimum-jerk trajectory planning of robot manipulators , 2000, IEEE Trans. Ind. Electron..

[22]  Paolo Gallina,et al.  A Technique to Analytically Formulate and to Solve the 2-Dimensional Constrained Trajectory Planning Problem for a Mobile Robot , 2000, J. Intell. Robotic Syst..

[23]  Ulrich Reif.  Best bounds on the approximation of polynomials and splines by their control structure , 2000, Comput. Aided Geom. Des..

[24]  Nancy M. Amato,et al.  Randomized motion planning for car-like robots with C-PRM , 2001, Proceedings 2001 IEEE/RSJ International Conference on Intelligent Robots and Systems. Expanding the Societal Role of Robotics in the the Next Millennium (Cat. No.01CH37180).

[25]  Håkan Jonsson,et al.  Automatic Generation of Smooth Paths Bounded by Polygonal Chains , 2001 .

[26]  Aurelio Piazzi,et al.  Quintic G2-splines for the iterative steering ofvision-based autonomous vehicles , 2002, IEEE Trans. Intell. Transp. Syst..

[27]  T. O. Espelid.  Doubly Adaptive Quadrature Routines Based on Newton–Cotes Rules , 2003 .

[28]  Mark H. Overmars,et al.  Motion Planning for Camera Movements in Virtual Environments , 2003 .

[29]  Håkan Jonsson,et al.  An obstacle-avoiding minimum variation B-spline problem , 2003, 2003 International Conference on Geometric Modeling and Graphics, 2003. Proceedings.

[30]  I. Söderkvist,et al.  Epi-convergence of minimum curvature variation B-splines , 2003 .

[31]  Thierry Fraichard,et al.  From Reeds and Shepp's to continuous-curvature paths , 1999, IEEE Transactions on Robotics.

[32]  G. Swaminathan.  Robot Motion Planning , 2006 .