Equilibrium Configurations of a Kirchhoff Elastic Rod under Quasi-static Manipulation

  • Timothy BretlEmail author
  • Zoe McCarthy
Part of the Springer Tracts in Advanced Robotics book series (STAR, volume 86)


Consider a thin, flexible wire of fixed length that is held at each end by a robotic gripper. The curve traced by this wire can be described as a local solution to a geometric optimal control problem, with boundary conditions that vary with the position and orientation of each gripper. The set of all local solutions to this problem is the configuration space of the wire under quasi-static manipulation. We will show that this configuration space is a smooth manifold of finite dimension that can be parameterized by a single chart. Working in this chart—rather than in the space of boundary conditions—makes the problem of manipulation planning very easy to solve. Examples in simulation illustrate our approach.


Optimal Control Problem Path Planning Smooth Manifold Integral Curve Coordinate Chart 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Agrachev, A.A., Sachkov, Y.L.: Control theory from the geometric viewpoint, vol. 87. Springer, Berlin (2004)zbMATHGoogle Scholar
  2. 2.
    Amato, N.M., Song, G.: Using motion planning to study protein folding pathways. Journal of Computational Biology 9(2), 149–168 (2002)CrossRefGoogle Scholar
  3. 3.
    Antman, S.S.: Nonlinear Problems of Elasticity, 2nd edn. Applied Mathematical Sciences, vol. 107. Springer, New York (2005)zbMATHGoogle Scholar
  4. 4.
    Asano, Y., Wakamatsu, H., Morinaga, E., Arai, E., Hirai, S.: Deformation path planning for manipulation of flexible circuit boards. In: IEEE/RSJ Int. Conf. Int. Rob. Sys. (2010)Google Scholar
  5. 5.
    Bell, M., Balkcom, D.: Knot tying with single piece fixtures. In: Int. Conf. Rob. Aut. (2008)Google Scholar
  6. 6.
    van den Berg, J., Miller, S., Goldberg, K., Abbeel, P.: Gravity-based robotic cloth folding. In: WAFR (2011)Google Scholar
  7. 7.
    Bergou, M., Wardetzky, M., Robinson, S., Audoly, B., Grinspun, E.: Discrete elastic rods. ACM Trans. Graph. 27(3), 1–12 (2008)CrossRefGoogle Scholar
  8. 8.
    Biggs, J., Holderbaum, W., Jurdjevic, V.: Singularities of optimal control problems on some 6-d lie groups. IEEE Trans. Autom. Control 52(6), 1027–1038 (2007)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Bloch, A., Krishnaprasad, P., Marsden, J., Ratiu, T.: The Euler-Poincaré equations and double bracket dissipation. Communications In Mathematical Physics 175(1), 1–42 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Chirikjian, G.S., Burdick, J.W.: The kinematics of hyper-redundant robot locomotion. IEEE Trans. Robot. Autom. 11(6), 781–793 (1995)CrossRefGoogle Scholar
  11. 11.
    Choset, H., Lynch, K., Hutchinson, S., Kanto, G., Burgard, W., Kavraki, L., Thrun, S.: Principles of Robot Motion: Theory, Algorithms, and Implementations. MIT Press (2005)Google Scholar
  12. 12.
    Clements, T.N., Rahn, C.D.: Three-dimensional contact imaging with an actuated whisker. IEEE Trans. Robot. 22(4), 844–848 (2006)CrossRefGoogle Scholar
  13. 13.
    Gopalakrishnan, K., Goldberg, K.: D-space and deform closure grasps of deformable parts. International Journal of Robotics Research 24(11), 899–910 (2005)CrossRefGoogle Scholar
  14. 14.
    Hoffman, K.A.: Methods for determining stability in continuum elastic-rod models of dna. Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 362(1820), 1301–1315 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Hopcroft, J.E., Kearney, J.K., Krafft, D.B.: A case study of flexible object manipulation. The International Journal of Robotics Research 10(1), 41–50 (1991)CrossRefGoogle Scholar
  16. 16.
    Inoue, H., Inaba, H.: Hand-eye coordination in rope handling. In: ISRR, pp. 163–174 (1985)Google Scholar
  17. 17.
    Jansen, R., Hauser, K., Chentanez, N., van der Stappen, F., Goldberg, K.: Surgical retraction of non-uniform deformable layers of tissue: 2d robot grasping and path planning. In: IEEE/RSJ Int. Conf. Int. Rob. Sys., pp. 4092–4097 (2009)Google Scholar
  18. 18.
    Javdani, S., Tandon, S., Tang, J., O’Brien, J.F., Abbeel, P.: Modeling and perception of deformable one-dimensional objects. In: Int. Conf. Rob. Aut., Shanghai, China (2011)Google Scholar
  19. 19.
    Kavraki, L.E., Svetska, P., Latombe, J.C., Overmars, M.: Probabilistic roadmaps for path planning in high-dimensional configuration spaces. IEEE Trans. Robot. Autom. 12(4), 566–580 (1996)CrossRefGoogle Scholar
  20. 20.
    Keshavarz, A., Wang, Y., Boyd, S.: Imputing a convex objective function. In: IEEE Multi-Conference on Systems and Control (2011)Google Scholar
  21. 21.
    Lamiraux, F., Kavraki, L.E.: Planning paths for elastic objects under manipulation constraints. International Journal of Robotics Research 20(3), 188–208 (2001)CrossRefGoogle Scholar
  22. 22.
    Langer, J., Singer, D.: The total squared curvature of closed curves. Journal of Differential Geometry 20, 1–22 (1984)zbMATHMathSciNetGoogle Scholar
  23. 23.
    Latombe, J.C.: Robot Motion Planning. Kluwer Academic Publishers, Boston (1991)CrossRefGoogle Scholar
  24. 24.
    LaValle, S.M.: Planning algorithms. Cambridge University Press, New York (2006)CrossRefzbMATHGoogle Scholar
  25. 25.
    Lee, J.M.: Introduction to smooth manifolds, vol. 218. Springer, New York (2003)Google Scholar
  26. 26.
    Lin, Q., Burdick, J., Rimon, E.: A stiffness-based quality measure for compliant grasps and fixtures. IEEE Trans. Robot. Autom. 16(6), 675–688 (2000)CrossRefGoogle Scholar
  27. 27.
    Marsden, J.E., Ratiu, T.S.: Introduction to mechanics and symmetry: a basic exposition of classical mechanical systems, 2nd edn. Springer, New York (1999)CrossRefzbMATHGoogle Scholar
  28. 28.
    McCarthy, Z., Bretl, T.: Mechanics and manipulation of planar elastic kinematic chains. In: IEEE Int. Conf. Rob. Aut. St. Paul, MN (2012)Google Scholar
  29. 29.
    Moll, M., Kavraki, L.E.: Path planning for deformable linear objects. IEEE Trans. Robot. 22(4), 625–636 (2006)CrossRefGoogle Scholar
  30. 30.
    Rucker, D.C., Webster, R.J., Chirikjian, G.S., Cowan, N.J.: Equilibrium conformations of concentric-tube continuum robots. Int. J. Rob. Res. 29(10), 1263–1280 (2010)CrossRefGoogle Scholar
  31. 31.
    Sachkov, Y.: Conjugate points in the euler elastic problem. Journal of Dynamical and Control Systems 14(3), 409–439 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  32. 32.
    Sachkov, Y.: Maxwell strata in the euler elastic problem. Journal of Dynamical and Control Systems 14(2), 169–234 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  33. 33.
    Saha, M., Isto, P.: Manipulation planning for deformable linear objects. IEEE Trans. Robot. 23(6), 1141–1150 (2007)CrossRefGoogle Scholar
  34. 34.
    Sánchez, G., Latombe, J.C.: On delaying collision checking in PRM planning: Application to multi-robot coordination. Int. J. Rob. Res. 21(1), 5–26 (2002)CrossRefGoogle Scholar
  35. 35.
    Schwarzer, F., Saha, M., Latombe, J.C.: Exact collision checking of robot paths. In: WAFR, Nice, France (2002)Google Scholar
  36. 36.
    Solomon, J.H., Hartmann, M.J.Z.: Extracting object contours with the sweep of a robotic whisker using torque information. Int. J. Rob. Res. 29(9), 1233–1245 (2010)CrossRefGoogle Scholar
  37. 37.
    Starostin, E.L., van der Heijden, G.H.M.: Tension-induced multistability in inextensible helical ribbons. Physical Review Letters 101(8), 084,301 (2008)Google Scholar
  38. 38.
    Takamatsu, J., Morita, T., Ogawara, K., Kimura, H., Ikeuchi, K.: Representation for knot-tying tasks. IEEE Trans. Robot. 22(1), 65–78 (2006)CrossRefGoogle Scholar
  39. 39.
    Tanner, H.: Mobile manipulation of flexible objects under deformation constraints. IEEE Trans. Robot. 22(1), 179–184 (2006)CrossRefGoogle Scholar
  40. 40.
    Wakamatsu, H., Arai, E., Hirai, S.: Knotting/unknotting manipulation of deformable linear objects. The International Journal of Robotics Research 25(4), 371–395 (2006)CrossRefGoogle Scholar
  41. 41.
    Walsh, G., Montgomery, R., Sastry, S.: Optimal path planning on matrix lie groups. In: IEEE Conference on Decision and Control, vol. 2, pp. 1258–1263 (1994)Google Scholar
  42. 42.
    Webster, R.J., Jones, B.A.: Design and kinematic modeling of constant curvature continuum robots: A review. Int. J. Rob. Res. 29(13), 1661–1683 (2010)CrossRefGoogle Scholar
  43. 43.
    Yamakawa, Y., Namiki, A., Ishikawa, M.: Motion planning for dynamic folding of a cloth with two high-speed robot hands and two high-speed sliders. In: Int. Conf. Rob. Aut., pp. 5486–5491 (2011)Google Scholar

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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of Aerospace EngineeringUniversity of Illinois at Urbana-ChampaignUrbanaUSA
  2. 2.Department of Electrical and Computer EngineeringUniversity of Illinois at Urbana-ChampaignUrbanaUSA

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