Skip to main content
SpringerLink
Log in

The distance between convex sets with Minkowski sum structure: application to collision detection

  • Published:
Computational Optimization and Applications Aims and scope Submit manuscript

Abstract

The distance between sets is a long-standing computational geometry problem. In robotics, the distance between convex sets with Minkowski sum structure plays a fundamental role in collision detection. However, it is typically nontrivial to be computed, even if the projection onto each component set admits explicit formula. In this paper, we explore the problem of calculating the distance between convex sets arising from robotics. Upon the recent progress in convex optimization community, the proposed model can be efficiently solved by the recent hot-investigated first-order methods, e.g., alternating direction method of multipliers or primal-dual hybrid gradient method. Preliminary numerical results demonstrate that those first-order methods are fairly efficient in solving distance problems in robotics.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

Notes

  1. The Moore–Penrose pseudo-inverse exists and is unique for any \(A\in \mathbb {C}^{m\times n}\). Let \(\hbox {rank}(A)=r\) and \(A=U\Sigma V^*\) be the singular value decomposition with \(U\in \mathbb {C}^{m\times r}\), \(\Sigma \in {{\mathbb {R}}}^{r\times r}\) and \(V\in \mathbb {C}^{m\times r}\). Then \(A^\dagger =V\Sigma ^{-1}U^*\). Particularly, \(A^\dagger =(A^*A)^{-1}A^*\) when A is of full column rank, and \(A^\dagger =A^*(AA^*)^{-1}\) when A is of full row rank.

  2. Let \(\mathrm {bd}({\mathcal {C}})\) and \(N_{\mathcal {C}}(x)\) denote the topological boundary and normal cone (see also Example 2.8 for definition) of a set \({\mathcal {C}}\subset {{\mathbb {R}}}^n\), respectively. The \({\mathcal {C}}\) is called normally smooth if, for any \(x\in \mathrm{bd}({\mathcal {C}})\), there exists an \(a_x\in {{\mathbb {R}}}^n\) such that \(N_{\mathcal {C}}(x)=\mathrm {cone}\{a_x\}\). The \({\mathcal {C}}\) is said to be round if \(N_{\mathcal {C}}(x)\ne N_{\mathcal {C}}(y)\) for any x, \(y\in \mathrm{bd}({\mathcal {C}})\) and \(x\ne y\).

  3. If an obstacle \({\mathcal {O}}_i\) has nonconvex feature, it can be approximate by convex decomposition (see Definition 2.1). Some literature on deriving convex decomposition of a given set can be referred to, e.g., [5, 28, 41].

References

  1. Aliyu, M.D.: A vertex algorithm for collision detection. Eur. J. Oper. Res. 120, 174–180 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, New York (2011)

    Book  MATH  Google Scholar 

  3. Bertsekas, D.P.: Convex Analysis and Optimization. Athena Scientific, Belmont (2003)

    MATH  Google Scholar 

  4. Bertsekas, D.P., Tsitsiklis, J.N.: Parallel and Distributed Computation: Numerical Methods. Prentice-Hall, Upper Saddle River (1989)

    MATH  Google Scholar 

  5. Böröczky, K., Reitzner, M.: Approximation of smooth convex bodies by random circumscribed polytopes. Ann. Appl. Probab. 14, 239–273 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  6. Boyd, S., Parikh, N., Chu, E., Peleato, B., Eckstein, J.: Distributed optimization and statistical learning via the alternating direction method of multipliers. Found. Trends Mach. Learn. 3, 1–122 (2011)

    Article  MATH  Google Scholar 

  7. Bronstein, E.M.: Approximation of convex sets by polytopes. J. Math. Sci. 153, 727–762 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cameron, S.: Enhancing GJK: computing minimum and penetration distances between convex polyhedra. In: IEEE International Conference on Robotics and Automation, pp. 3112–3117 (2002)

  9. Chambolle, A., Pock, T.: A first-order primal-dual algorithm for convex problems with applications to imaging. J. Math. Imaging Vis. 40, 120–145 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chambolle, A., Ehrhardt, M.J., Richtárik, P., Schönlieb, C.: Stochastic primal-dual hybrid gradient algorithm with arbitrary sampling and imaging applications. SIAM J. Optim. 28, 2783–2808 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  11. Combettes, P.L., Pesquet, J.C.: Proximal splitting methods in signal processing. In: Bauschke, H.H., et al. (eds.) Fixed-Point Algorithms for Inverse Problems in Science and Engineering, pp. 185–212. Springer, Berlin (2011)

    Chapter  MATH  Google Scholar 

  12. Condat, L.: Fast projection onto the simplex and the \(\ell _1\) ball. Math. Program. 158, 575–585 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  13. Dax, A.: The distance between two convex sets. Linear Algebra Appl. 416, 184–213 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  14. Eckstein, J.: Augmented Lagrangian and alternating direction methods for convex optimization: a tutorial and some illustrative computational results. RUTCOR Research Report, (2012)

  15. Eckstein, J.: Splitting methods for monotone operators with applications to parallel optimization. Ph.D. Thesis, MIT (1989)

  16. Esser, E., Zhang, X.Q., Chan, T.F.: A general framework for a class of first order primal-dual algorithms for convex optimization in imaging science. SIAM J. Imaging Sci. 3, 1015–1046 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  17. Fogel, E., Halperin, D., Wein, R.: CGAL Arrangements and Their Applications. Springer, Berlin (2012)

    Book  MATH  Google Scholar 

  18. Gabay, D., Mercier, B.: A dual algorithm for the solution of nonlinear variational problems via finite element approximations. Comput. Math. Appl. 2, 17–40 (1976)

    Article  MATH  Google Scholar 

  19. Gilbert, E.: An iterative procedure for computing the minimum of a quadratic form on a convex set. SIAM J. Control 6, 61–80 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  20. Gilbert, E., Foo, C.: Computing the distance between general convex objects in three-dimensional space. IEEE Trans. Robot. Autom. 6, 53–61 (1990)

    Article  Google Scholar 

  21. Gilbert, E., Johnson, D., Keerthi, S.: A fast procedure for computing the distance between complex objects in three-dimensional space. IEEE Trans. Robot. Autom. 4, 193–203 (1988)

    Article  Google Scholar 

  22. Glowinski, R.: On alternating direction methods of multipliers: a historical perspective. Model. Simul. Optim. Sci. Technol. Comput. Methods Appl. Sci. 34, 59–82 (2014)

    MathSciNet  MATH  Google Scholar 

  23. Glowinski, R., Marrocco, A.: Sur l’approximation paréléments finis d’ordre unet larésolution parpénalisation-dualité d’une classe deproblèmes de Dirichlet non linéaires. Revue Fr. Autom. Inform. Rech. Opér., Anal. Numér 2, 41–76 (1975)

    Google Scholar 

  24. He, B.S., Yuan, X.M.: Convergence analysis of primal-dual algorithms for a saddle-point problem: from contraction perspective. SIAM J. Imaging Sci. 5, 119–149 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  25. He, B.S., Yuan, X.M.: On the \(\cal{O}(1/n)\) convergence rate of the Douglas–Rachford alternating direction method. SIAM J. Numer. Anal. 50, 700–709 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  26. He, B.S., You, Y.F., Yuan, X.M.: On the convergence of primal-dual hybrid gradient algorithm. SIAM J. Imaging Sci. 7, 2526–2537 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  27. Kavraki, L.E., Svestka, P., Latombe, J.C., Overmars, M.H.: Probabilistic roadmaps for path planning in high-dimensional configuration space. IEEE Trans. Robot. Autom. 12, 566–580 (1996)

    Article  Google Scholar 

  28. Keil, J.M.: Decomposing a polygon into simpler components. SIAM J. Comput. 14, 799–817 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  29. Latombe, J.C.: Robot Motion Planning. Kluwer Academic Publishers, Boston (1991)

    Book  MATH  Google Scholar 

  30. Lewis, A.S., Malick, J.: Alternating projections on manifolds. Math. Oper. Res. 33, 216–234 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  31. Lin, M., Canny, J.: A fast algorithm for incremental distance calculation. In: IEEE International Conference on Robotics and Automation, pp. 266–275 (1991)

  32. Liu, Z., Fathi, Y.: An active index algorithm for the nearest point problem in a polyhedral cone. Comput. Optim. Appl. 49, 435–456 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  33. Mayer, A., Zelenyuk, V.: Aggregation of Malmquist productivity indexes allowing for realloction of resources. Eur. J. Oper. Res. 238, 774–785 (2014)

    Article  MATH  Google Scholar 

  34. Mitchell, B.F., Dem’Yanov, V.F., Malozemov, V.N.: Finding the point of a polyhedron closest to the origin. SIAM J. Control 12, 19–26 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  35. Németh, A., Németh, S.: How to project onto an isotone projection cone. Linear Algebra Appl. 433, 41–51 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  36. Nesterov, Y.: A method for unconstrained convex minimization problem with the rate of convergence \(O(1/k^2)\). Dokl. Akad. Nauk SSSR 269, 543–547 (1983)

    MathSciNet  Google Scholar 

  37. Nesterov, Y.: Smooth minimization of non-smooth functions. Math. Program. 103, 127–152 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  38. O’Connor, D., Vandenberghe, L.: On the equivalence of the primal-dual hybrid gradient method and Douglas–Rachford splitting. Math. Program. 179, 85–108 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  39. Ong, C.J., Gilbert, E.: Fast versions of the Gilbert–Johnson–Keerthi distance algorithm: additional results and comparisons. IEEE Trans. Robot. Autom. 17, 531–539 (2001)

    Article  Google Scholar 

  40. Qin, X.L., An, N.T.: Smoothing algorithms for computing the projection onto a Minkowski sum of convex sets. Comput. Optim. Appl. 74, 821–850 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  41. Rosin, P.L.: Shape partitioning by convexity. IEEE Trans. Syst. Man, Cybern. A 30, 202–210 (2000)

    Article  Google Scholar 

  42. Ryu, E., Boyd, S.: A primer on monotone operator methods. Appl. Comput. Math. 15, 3–43 (2016)

    MathSciNet  MATH  Google Scholar 

  43. Sekitani, K., Yamamoto, Y.: Recursive algorithm for finding the minimum norm point in a polytope and a pair of closest points in two polytopes. Math. Program. 61, 233–249 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  44. Serra, J.: Image Analysis and Mathematical Morphology. Academic Press, Cambridge (1983)

    Google Scholar 

  45. Smid, P.: CNC Programming Handbook. Industrial Press, New York (2008)

    Google Scholar 

  46. Spingarn, J.E.: Applications of the method of partial inverses to convex programming: decomposition. Math. Program. 32, 199–223 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  47. Sra, S.: Fast projections onto \(\ell _{1, q}\)-norm balls for grouped feature selection. In: Machine Learning and Knowledge Discovery in Databases, pp. 305–317 (2011)

  48. Sussman, G.J., Wisdom, J.: Structure and Interpretation of Classical Mechanics. MIT Press, Cambridge (2002)

    MATH  Google Scholar 

  49. van den Berg, E., Friedlander, M.P.: Probing the Pareto frontier for basis pursuit solutions. SIAM J. Sci. Comput. 31, 890–912 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  50. Wolfe, P.: Finding the nearest point in a polytope. Math. Program. 11, 128–149 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  51. Zheng, Y., Yamane, K.: Generalized distance between compact convex sets: algorithms and applications. IEEE Trans. Robot. 31, 988–1003 (2015)

    Article  Google Scholar 

  52. Zhu, X.Y., Tso, S.K.: A peudodistance function and its applications. IEEE Trans. Robot. 20, 344–352 (2004)

    Article  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the anonymous referees for their valuable comments, which helped improving the paper substantially. X. Wang was supported by NSFC 11871279 and STCSM 19ZR1414200. W. Zhang was supported by NSFC 11971003.

Author information

Authors and Affiliations

  1. School of Computer Science and Technology, East China Normal University, Shanghai, 200062, China

    Xiangfeng Wang

  2. School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, 611731, China

    Junping Zhang & Wenxing Zhang

Authors
  1. Xiangfeng Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Junping Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Wenxing Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wenxing Zhang.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, X., Zhang, J. & Zhang, W. The distance between convex sets with Minkowski sum structure: application to collision detection. Comput Optim Appl 77, 465–490 (2020). https://doi.org/10.1007/s10589-020-00211-0

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10589-020-00211-0

Keywords

Search

Navigation