Skip to main content
SpringerLink
Log in

Efficient Contact Mode Enumeration in 3D

  • Conference paper
  • First Online:
Algorithmic Foundations of Robotics XIV (WAFR 2020)

Part of the book series: Springer Proceedings in Advanced Robotics ((SPAR,volume 17))

Included in the following conference series:

Abstract

In a hybrid dynamical system with multiple rigid bodies, the relative motions of the contact points on two colliding bodies may be classified as separating, sticking (moving together), or sliding. Given a physical contact model, the active contact modes determine the dynamic equations of motion. Analogously, the set of all possible (valid) contact mode assignments enumerates the set of all possible dynamical flows of the hybrid dynamical system at a given state. Naturally, queries about the kinematics or dynamics of the system can be framed as computations over the set of possible contact modes. This paper investigates efficient ways to compute this set.

To that end, we have developed the first efficient 3D contact mode enumeration algorithm. The algorithm is exponential in the degrees of freedom of the system and polynomial in the number of contacts. The exponential term is unavoidable and an example is provided. Prior work in this area has only demonstrated efficient contact mode enumeration in 2D for a single rigid body. We validated our algorithm on peg-in-hole, boxes against walls, and a robot hand grasping an ellipse. Our experimental results indicate real-time contact mode enumeration is possible for small to medium sized systems. Finally, this paper concludes with a discussion of possible related application areas for future work. Ultimately, the goal of this paper is to provide a novel computational tool for researchers to use to simulate, analyze, and control robotic systems that make and break contact with the environment.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 229.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 299.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 299.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Avis, D., Fukuda, K.: Reverse search for enumeration. Discrete Appl. Math. 65(1), 21–46 (1996). https://doi.org/10.1016/0166-218X(95)00026-N. http://www.sciencedirect.com/science/article/pii/0166218X9500026N

  2. Baraff, D.: Coping with friction for non-penetrating rigid body simulation. In: Proceedings of the 18th Annual Conference on Computer Graphics and Interactive Techniques, SIGGRAPH 1991, pp. 31–41. Association for Computing Machinery, New York (1991). https://doi.org/10.1145/122718.122722

  3. Barber, C.B., Dobkin, D.P., Huhdanpaa, H.: The quickhull algorithm for convexhulls. ACM Trans. Math. Softw. 22(4), 469–483 (1996). https://doi.org/10.1145/235815.235821

  4. Bicchi, A.: On the closure properties of robotic grasping. Int. J. Robot. Res. 14(4), 319–334 (1995)

    Article  Google Scholar 

  5. Cottle, R.W., Pang, J.S., Stone, R.E.: The linear complementarity problem. Classics in Applied Mathematics, Society for Industrial and Applied Mathematics (2009). https://doi.org/10.1137/1.9780898719000

  6. Delos, V., Teissandier, D.: Minkowski sum of polytopes defined by their vertices. J. Appl. Math. Phys. 03(01), 62 (2015). https://doi.org/10.4236/jamp.2015.31008. http://www.scirp.org/journal/PaperInformation.aspx?PaperID=53577&#abstract

  7. Edelsbrunner, H.: Algorithms in Combinatorial Geometry. Springer, Berlin (2005). OCLC: 890383639

    Google Scholar 

  8. Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, Heidelberg (2007)

    Google Scholar 

  9. Geyer, T., Torrisi, F.D., Morari, M.: Efficient mode enumeration ofcompositional hybrid systems. Int. J. Control 83(2), 313–329 (2010). https://doi.org/10.1080/00207170903159285

  10. Greenfield, A., Saranli, U., Rizzi, A.A.: Solving models of controlled dynamic planar rigid-body systems with frictional contact. Int. J. Robot. Res. 24(11), 911–931 (2005). https://doi.org/10.1177/0278364905059056

  11. Haas-Heger, M., Papadimitriou, C., Yannakakis, M., Iyengar, G., Ciocarlie, M.: Passive static equilibrium with frictional contacts and application to grasp stability analysis. In: Proceedings of Robotics: Science and Systems, Pittsburgh, Pennsylvania (2018). https://doi.org/10.15607/RSS.2018.XIV.064

  12. Hou, Y., Jia, Z., Mason, M.T.: Manipulation with shared grasping (2020, under review)

    Google Scholar 

  13. Johnson, A.M., Burden, S.E., Koditschek, D.E.: A hybrid systems model for simple manipulation and self-manipulation systems. Int. J. Robot. Res. 35(11), 1354–1392 (2016)

    Article  Google Scholar 

  14. Kaibel, V., Pfetsch, M.E.: Computing the Face Lattice of a Polytope from its Vertex-Facet Incidences. arXiv:math/0106043 (2002)

  15. Lloyd, J.: Fast implementation of Lemke’s algorithm for rigid body contact simulation. In: Proceedings of the 2005 IEEE International Conference on Robotics and Automation, pp. 4538–4543 (2005). ISSN 1050-4729. https://doi.org/10.1109/ROBOT.2005.1570819

  16. Lynch, K.M., Mason, M.T.: Stable pushing: mechanics, controllability, and planning. Int. J. Robot. Res. 15(6), 533–556 (1996)

    Article  Google Scholar 

  17. Mason, M.T.: Mechanics of Robotic Manipulation. MIT Press, Cambridge (2001)

    Book  Google Scholar 

  18. Miller, A.T., Allen, P.K.: Graspit! A versatile simulator for robotic grasping. IEEE Robot. Autom. Mag. 11, 110–122 (2004)

    Article  Google Scholar 

  19. Murray, R.M., Sastry, S.S., Li, Z.: A mathematical introduction to robotic manipulation (1994)

    Google Scholar 

  20. Murty, K.G., Yu, F.T.: Linear complementarity, linear and nonlinear programming, vol. 3. Citeseer (1988)

    Google Scholar 

  21. Potočnik, B., Mušič, G., Zupančič, B.: A new technique for translating discrete hybrid automata into piecewise affine systems. Math. Comput. Model. Dyn. Syst. 10(1), 41–57 (2004). https://doi.org/10.1080/13873950412331318062

  22. QHull Library: QHalf Notes (2020). http://www.qhull.org/html/qhalf.htm#notes

  23. Smith, B., Kaufman, D.M., Vouga, E., Tamstorf, R., Grinspun, E.: Reflections on simultaneous impact. ACM Trans. Graph. (Proc. SIGGRAPH 2012) 31(4), 106:1–106:12 (2012)

    Google Scholar 

  24. Stewart, D.E., Trinkle, J.C.: An implicit time-stepping scheme for rigid body dynamics with inelastic collisions and coulomb friction. Int. J. Numer. Meth. Eng. 39(15), 2673–2691 (1996)

    Article  MathSciNet  Google Scholar 

  25. Tonge, R., Benevolenski, F., Voroshilov, A.: Mass splitting for jitter-free parallel rigid body simulation. ACM Trans. Graph. (TOG) 31(4), 105:1–105:8 (2012). https://doi.org/10.1145/2185520.2185601

  26. Vouga, E., Smith, B., Kaufman, D.M., Tamstorf, R., Grinspun, E.: All’s well that ends well: guaranteed resolution of simultaneous rigid body impact. ACM Trans. Graph. (TOG) 36(4), 151:1–151:19 (2017). https://doi.org/10.1145/3072959.3073689

  27. Ziegler, G.M.: Lectures on Polytopes. Graduate Texts in Mathematics. Springer, New York (1995). https://doi.org/10.1007/978-1-4613-8431-1. https://www.springer.com/gp/book/9780387943299

Download references

Acknowledgements

This material is based upon work supported by the National Science Foundation under Grants No. IIS-1909021, IIS-1813920, and IIS-1637908.

Author information

Authors and Affiliations

  1. Carnegie Mellon University, Pittsburgh, PA, 15213, USA

    Eric Huang, Xianyi Cheng & Matthew T. Mason

Authors
  1. Eric Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xianyi Cheng
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Matthew T. Mason
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Eric Huang .

Editor information

Editors and Affiliations

  1. Center for Ubiquitous Computing, University of Oulu, Oulu, Finland

    Steven M. LaValle

  2. Brendan Iribe Center for Computer Science and Engineering, University of Maryland, College Park, MD, USA

    Ming Lin

  3. Center for Ubiquitous Computing, University of Oulu, Oulu, Finland

    Timo Ojala

  4. Department of Computer Science and Engineering, Texas A&M University, College Station, TX, USA

    Dylan Shell

  5. Department of Computer Science, Rutgers University, Piscataway, NJ, USA

    Jingjin Yu

Rights and permissions

Reprints and permissions

Copyright information

© 2021 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Huang, E., Cheng, X., Mason, M.T. (2021). Efficient Contact Mode Enumeration in 3D. In: LaValle, S.M., Lin, M., Ojala, T., Shell, D., Yu, J. (eds) Algorithmic Foundations of Robotics XIV. WAFR 2020. Springer Proceedings in Advanced Robotics, vol 17. Springer, Cham. https://doi.org/10.1007/978-3-030-66723-8_29

Download citation

Publish with us

Policies and ethics

Search

Navigation