Consider a thin, flexible wire of fixed length that is held at each end by a robotic gripper. Any curve traced by this wire when in static equilibrium is a local solution to a geometric optimal control problem, with boundary conditions that vary with the position and orientation of each gripper. We prove that the set of all local solutions to this problem over all possible boundary conditions is a smooth manifold of finite dimension that can be parameterized by a single chart. We show that this chart makes it easy to implement a sampling-based algorithm for quasi-static manipulation planning. We characterize the performance of such an algorithm with experiments in simulation.
AgarwalPGuibasLNguyenARusselDZhangL (2004) Collision detection for deforming necklaces. Computational Geometry28(2–3): 137–163.
2.
AgrachevAASachkovYL (2004) Control Theory from the Geometric Viewpoint. Berlin: Springer.
3.
AmatoNMSongG (2002) Using motion planning to study protein folding pathways. Journal of Computational Biology9(2): 149–168.
4.
AntmanSS (2005) Nonlinear Problems of Elasticity. New York: Springer.
5.
AsanoYWakamatsuHMorinagaEAraiEHiraiS (2010) Deformation path planning for manipulation of flexible circuit boards. In: IEEE/RSJ international conference on intelligent robots and systems (IROS ‘10), Taipei, Taiwan, 18–22 October 2010, pp. 5386–5391. Piscataway: IEEE Press.
6.
BalkcomDJMasonMT (2008) Robotic origami folding. International Journal of Robotics Research27(5): 613–627.
7.
BellMBalkcomD (2008) Knot tying with single piece fixtures. In: IEEE international conference on robotics and automation (ICRA ‘08), Pasadena, CA, 19–23 May 2008, pp. 379–384. Piscataway: IEEE Press.
8.
BergouMWardetzkyMRobinsonSAudolyBGrinspunE (2008) Discrete elastic rods. ACM Transactions on Graphics27(3): 1–12.
9.
BiggsJHolderbaumWJurdjevicV (2007) Singularities of optimal control problems on some 6-D Lie groups. IEEE Transactions on Automation and Control52(6): 1027–1038.
10.
BlochAKrishnaprasadPMarsdenJRatiuT (1996) The Euler–Poincaré equations and double bracket dissipation. Communications in Mathematical Physics175(1): 1–42.
11.
BretlTMcCarthyZ (2012) Equilibrium configurations of a Kirchhoff elastic rod under quasi-static manipulation. In: 10th international workshop on the algorithmic foundations of robotics (WAFR ‘12), Cambridge, USA, 13–15 June 2012.
12.
CamarilloDBMilneCFCarlsonCRZinnMRSalisburyJK (2008) Mechanics modeling of tendon-driven continuum manipulators. IEEE Transactions on Robotics24(6): 1262–1273.
13.
ChirikjianGSBurdickJW (1995) The kinematics of hyper-redundant robot locomotion. IEEE Transactions on Robotics and Automation11 (6): 781–793.
14.
ChosetHLynchKHutchinsonSKantoGBurgardWKavrakiLPrinciples of Robot Motion: Theory, Algorithms, and Implementations. Cambridge: MIT Press.
15.
ClementsTNRahnCD (2006) Three-dimensional contact imaging with an actuated whisker. IEEE Transactions on Robotics22(4): 844–848.
16.
GopalakrishnanKGoldbergK (2005) D-Space and deform closure grasps of deformable parts. International Journal of Robotics Research24 (11): 899–910.
17.
GottschalkSLinMManochaD (1996) OBB-tree: A hierarchical structure for rapid interference detection. Computational Graphics30: 171–180.
18.
HendersonMENeukirchS (2004) Classification of the spatial equilibria of the clamped elastica: Numerical continuation of the solution set. International Journal of Bifurcation & Chaos in Applied Sciences & Engineering14 (4): 1223–1239.
19.
HirschMW (1976) Differential Topology. Berlin: Springer-Verlag, pp. 34–35.
20.
HoffmanKA (2004) 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 Sciences362 (1820): 1301–1315.
21.
HopcroftJEKearneyJKKrafftDB (1991) A case study of flexible object manipulation. International Journal of Robotics Research10(1): 41–50.
22.
InabaM.Hand eye coordination in rope handling. J. of Robotics Society Japan, 3(6): 32–41, 1985.
23.
IveyTASingerDA (1999) Knot types, homotopies and stability of closed elastic rods. Proceedings of the London Mathematical Society79(2): 429–450.
24.
JansenRHauserKChentanezNvan der StappenFGoldbergK (2009) Surgical retraction of non-uniform deformable layers of tissue: 2D robot grasping and path planning. In: IEEE/RSJ international conference on intelligent robots and systems (IROS ‘09), St. Louis, MO., 10–15 October 2009, pp. 4092–4097. Piscataway: IEEE Press.
25.
JavdaniSTandonSTangJO’BrienJFAbbeelP (2011) Modeling and perception of deformable one-dimensional objects. In: IEEE international conference on robotics and automation (ICRA ‘11), Shanghai, China, 9–13 May 2011, pp. 1607–1614. Piscataway: IEEE Press.
26.
JurdjevicV (2005) Integrable Hamiltonian Systems on Complex Lie Groups. Providence: American Mathematical Society.
27.
KavrakiLESvetskaPLatombeJ-COvermarsM (1996) Probabilistic roadmaps for path planning in high-dimensional configuration spaces. IEEE Transactions on Robotics and Automation12(4): 566–580.
28.
KeshavarzAWangYBoydS (2011) Imputing a convex objective function. In: IEEE multi-conference on systems and control, Denver, CO., 28–30 September 2011, pp. 613–619. Piscataway: IEEE Press.
29.
LamirauxFKavrakiLE (2001) Planning paths for elastic objects under manipulation constraints. International Journal of Robotics Research20 (3): 188–208.
30.
LangerJSingerD (1984) The total squared curvature of closed curves. Journal of Differential Geometry20: 1–22.
31.
LangerJSingerDA (1996) Lagrangian aspects of the Kirchhoff elastic rod. SIAM Review38(4): 605–618.
LaValleSM (2006) Planning Algorithms. New York: Cambridge University Press.
34.
LeeJM (2003) Introduction to Smooth Manifolds. New York: Springer.
35.
LinQBurdickJRimonE (2000) A stiffness-based quality measure for compliant grasps and fixtures. IEEE Transactions on Robotics and Automation16(6): 675–688.
36.
MarsdenJERatiuTS (1999) Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems. New York: Springer.
37.
McCarthyZBretlT (2012) Mechanics and manipulation of planar elastic kinematic chains. In: IEEE international conference on robotics and automation (ICRA ‘12), St. Paul, USA, 14–18 May 2012, pp. 2798–2805. Piscataway: IEEE Press.
38.
MollMKavrakiLE (2006) Path planning for deformable linear objects. IEEE Transactions on Robotics22(4): 625–636.
39.
MurrayRLiZXSastryS (1994) A Mathematical Introduction to Robotic Manipulation. Boca Raton: CRC Press.
40.
NeukirchSHendersonM (2002) Classification of the spatial equilibria of the clamped elastica: Symmetries and zoology of solutions. Journal of Elasticity68(1): 95–121.
41.
PontryaginLSBoltyanskiiVGGamkrelidzeRVMishchenkoEF (1962) The Mathematical Theory of Optimal Processes. New York: Wiley.
42.
RuckerDCWebsterRJChirikjianGSCowanNJ (2010) Equilibrium conformations of concentric-tube continuum robots. International Journal of Robotics Research29(10): 1263–1280
43.
SachkovY (2008a) Conjugate points in the Euler elastic problem. Journal of Dynamical and Control Systems14(3): 409–439.
44.
SachkovY (2008b) Maxwell strata in the Euler elastic problem. Journal of Dynamical and Control Systems14(2): 169–234.
45.
SahaMIstoP (2007) Manipulation planning for deformable linear objects. IEEE Transactions on Robotics23(6): 1141–1150.
46.
SánchezGLatombeJ-C (2002) On delaying collision checking in PRM planning: Application to multi-robot coordination. International Journal of Robotics Research21(1): 5–26.
47.
SchwarzerFSahaMLatombeJC (2005) Adaptive dynamic collision checking for single and multiple articulated robots in complex environments. IEEE Transactions on Robotics21(3): 338–353.
48.
ShampineLThompsonS (2000) Event location for ordinary differential equations. Computers & Mathematics with Applications39(5–6): 43–54.
49.
ShiYHearstJE (1994) The Kirchhoff elastic rod, the nonlinear Schrödinger equation, and DNA supercoiling. Journal of Chemical Physics101(6): 5186–5200.
50.
ShiYHearstJEBishopTCHalvorsonHR (1998) Erratum: “the Kirchhoff elastic rod, the nonlinear Schrödinger equation, and DNA supercoiling” [J. Chem. Phys. 101, 5186 (1994)]. Journal of Chemical Physics109(7): 2959–2961.
51.
SolomonJHHartmannMJZ (2010) Extracting object contours with the sweep of a robotic whisker using torque information. International Journal of Robotics Research29(9): 1233–1245.
52.
SouéresPBoissonnatJ (1998) Optimal trajectories for nonholonomic mobile robots. In: LaumondJ-P (ed) Robot Motion Planning and Control. Berlin: Springer, vol. 229, ch. 3, pp. 93–170.
53.
StarostinELvan der HeijdenGHM (2008) Tension-induced multistability in inextensible helical ribbons. Physical Review Letters101(8): 084301.
54.
TakamatsuJMoritaTOgawaraKKimuraHIkeuchiK (2006) Representation for knot-tying tasks. IEEE Transactions on Robotics22 (1): 65–78.
55.
TannerH (2006) Mobile manipulation of flexible objects under deformation constraints. IEEE Transactions on Robotics22(1): 179–184.
56.
van den BergJMillerSGoldbergKAbbeelP (2011) Gravity-based robotic cloth folding. Algorithmic Foundations of Robotics IX Springer Tracts in Advanced Robotics Volume 68, 2011, pp 409–424 Editors: HsuDavidIslerVolkanLatombeJean-ClaudeLinMing C. http://dx.doi.org/10.1007/978-3-642-17452-0_24 Springer Berlin Heidelberg
57.
WakamatsuHAraiEHiraiS (2006) Knotting/unknotting manipulation of deformable linear objects. International Journal of Robotics Research25 (4): 371–395.
58.
WalshGMontgomeryRSastryS (1994) Optimal path planning on matrix Lie groups. In: IEEE conference on decision and control, Lake Buena Vista, FL., 14–16 December 1994, vol. 2, pp. 1258–1263. Piscataway: IEEE Press.
59.
WebsterRJJonesBA (2010) Design and kinematic modeling of constant curvature continuum robots: A review. International Journal of Robotics Research29(13): 1661–1683
60.
WestM (2004) Variational integrators. PhD Thesis, California Institute of Technology, USA.
61.
YamakawaYNamikiAIshikawaM (2011) Motion planning for dynamic folding of a cloth with two high-speed robot hands and two high-speed sliders. In: IEEE international conference on robotics and automation (ICRA ‘11), Shanghai, China, 9–13 May 2011, pp. 5486–5491. Piscataway: IEEE Press.
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