Curvature-constrained directional-cost paths in the plane
This paper looks at the problem of finding the minimum cost curvature-constrained path between two directed points where the cost at every point along the path depends on the instantaneous direction. This generalises the results obtained by Dubins for curvature-constrained paths of minimum length, commonly referred to as Dubins paths. We conclude that if the reciprocal of the directional-cost function is strictly polarly convex, then the forms of the optimal paths are of the same forms as Dubins paths. If we relax the strict polar convexity to weak polar convexity, then we show that there exists a Dubins path which is optimal. The results obtained can be applied to optimising the development of underground mine networks, where the paths need to satisfy a curvature constraint and the cost of development of the tunnel depends on the direction due to the geological characteristics of the ground.
KeywordsCurvature constraint Dubins paths Path optimization Directional cost Anisotropic velocity Pontryagin’s minimum principle
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- 2.Brazil M., Grossman P.A., Lee D.H., Rubinstein J.H., Thomas D.A., Wormald N.C.: Decline design in underground mines using constrained path optimisation. Trans. Inst. Min. Metall. A 117(2), 93–99 (2008)Google Scholar
- 3.Bui, X.N., Soueres, P., Boissonnat, J.D., Laumond, J.P.: The shortest path synthesis for non-holonomic robots moving forwards. INRIA, Nice-Sophia-Antipolis, Research Report 2153, (1993)Google Scholar
- 4.Dolinskaya, I.S.: Optimal path finding in direction, location and time dependent environments. Ph.D. thesis, Industrial and operations engineering, The University of Michigan (2009)Google Scholar
- 6.Gehring, K., Fuchs, M.: Quantification of rock mass influence on cuttability with roadheaders, 28th ITA (International Tunnelling Association) General assembly and World tunnel congress, Sydney, (2002)Google Scholar
- 7.Laubscher D.H.: A geomechanics classification system for the rating of rock mass in mine design. J. S. Atr. Inst. Min. Metal 90(10), 257–273 (1990)Google Scholar
- 8.McGee, T.G., Spry, S., Hedrick, J.K.: Optimal path planning in a constant wind with a bounded turning rate. In: Proceedings of the AIAA conference on guidance, navigation and control, Ketstone, Colorado (2005)Google Scholar
- 9.Pontryagin, L.S.: The mathematical theory of optimal processes. vol. 4, Interscience, Translation of a Russian book. (1962)Google Scholar
- 10.Reeds J.A., Shepp L.A.: Optimal paths for a car that goes both forwards and backwards. Pac. J. Math. 145(2), 367–393 (1990)Google Scholar
- 11.Sanfelice, R.G., Frazzoli, E.: On the optimality of dubins paths across heterogeneous terrain. In: Egerstedt, M., Mishra, B. (eds.) Hybrid Systems: Computation and Control, vol. 4981 pp.457–470. Springer, Heidelberg (2008)Google Scholar