Journal of Global Optimization

, Volume 62, Issue 3, pp 507–527 | Cite as

Optimal curvature and gradient-constrained directional cost paths in 3-space

  • Alan J. Chang
  • Marcus BrazilEmail author
  • J. Hyam Rubinstein
  • Doreen A. Thomas


In the design of underground tunnel layout, the development cost is often dependent on the direction of the tunnel at each point due to directional ground fracturing. This paper considers the problem of finding a minimum cost curvature-constrained path between two directed points in 3-space, where the cost at every point along the path depends on the instantaneous direction. This anisotropic behaviour of the cost models the development cost of a tunnel in ground with faulting planes that are almost vertical. The main result we prove in this paper is that there exists an optimal path of the form \(\mathcal {C}\mathcal {S}\mathcal {C}\mathcal {S}\mathcal {C}\mathcal {S}\mathcal {C}\) (or a degeneracy), where \(\mathcal {C}\) represents a segment of a helix with unit radius and \(\mathcal {S}\) represents a straight line segment. This generalises a previous result that in the restriction of the problem to the horizontal plane there always exists a path of the form \(\mathcal {C}\mathcal {S}\mathcal {C}\mathcal {S}\mathcal {C}\) or a degeneracy which is optimal. We also prove some key structural results which are necessary for creating an algorithm which can construct an optimal path between a given pair of directed points in 3-space with a prescribed directional cost function.


Computational geometry Path optimization Curvature-constrained paths Anisotropic costs 



This research is supported by a grant from the Australian Research Council.


  1. 1.
    Ayala, J., Brazil, M., Rubinstein, J.H., Thomas, D.A.: A geometric approach to shortest bounded curvature paths (In preparation)Google Scholar
  2. 2.
    Boscain, U., Piccoli, B.: Optimal Syntheses for Control Systems on 2-d Manifolds, vol. 43. Springer, Berlin (2004)zbMATHGoogle Scholar
  3. 3.
    Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)zbMATHCrossRefGoogle Scholar
  4. 4.
    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. Min Technol 117(2), 93–99 (2008)CrossRefGoogle Scholar
  5. 5.
    Brazil, M., Thomas, D.A.: Network optimization for the design of underground mines. Networks 49(1), 40–50 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Chang, A.J.: Optimal curvature and gradient-constrained paths with anisotropic costs, Ph.D. thesis, Department of Mechanical Engineering, The University of Melbourne, Australia, Dec. 2011.Google Scholar
  7. 7.
    Chang, A.J., Brazil, M., Rubinstein, J.H., Thomas, D.A.: Optimal curvature-constrained paths for general directional-cost functions. Optim. Eng. 14, 395–416 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Chang, A.J., Brazil, M., Rubinstein, J.H., Thomas, D.A.: Constructing optimal curvature-constrained directional-cost paths. Eng Optim (submitted Jan. 2012)Google Scholar
  9. 9.
    Chang, A., Brazil, M., Rubinstein, J., Thomas, D.: Curvature-constrained directional-cost paths in the plane. J. Glob. Optim. (2011), pp. 1–19, doi: 10.1007/s10898-011-9730-1
  10. 10.
    Dubins, L.E.: On curves of minimal length with a constraint on average curvature, and with prescribed initial and terminal positions and tangents. Am. J. Math. 79, 497–516 (1957)zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Fuchs, M., Gehring, K.: Quantification of rock mass influence on cuttability with roadheaders, 28th ITA (International Tunnelling Association) General Assembly and World Tunnel Congress. Sydney, 2002Google Scholar
  12. 12.
    Laubscher, D.H.: A geomechanics classification system for the rating of rock mass in mine design. J. S. Afr. Inst. Min. Metal 90(10), 257–273 (1990)Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Alan J. Chang
    • 1
  • Marcus Brazil
    • 2
    Email author
  • J. Hyam Rubinstein
    • 3
  • Doreen A. Thomas
    • 1
  1. 1.Department of Mechanical EngineeringThe University of MelbourneMelbourneAustralia
  2. 2.Department of Electrical and Electronic EngineeringThe University of MelbourneMelbourneAustralia
  3. 3.Department of Mathematics and StatisticsThe University of MelbourneMelbourneAustralia

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