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Research Papers

A Polynomial Formulation of Inverse Kinematics of Rolling Contact

[+] Author and Article Information
Lei Cui

Department of Mechanical Engineering,
Curtin University,
Kent Street,
Bentley, Western Australia 6102, Australia
e-mail: lei.cui@curtin.edu.au

Jian S. Dai

Centre for Robotics Research,
King's College London,
University of London,
Strand, London WC2R 2LS, UK
e-mail: jian.dai@kcl.ac.uk

1Corresponding author.

Contributed by the Mechanisms and Robotics Committee of ASME for publication in the JOURNAL OF MECHANISMS AND ROBOTICS. Manuscript received June 18, 2013; final manuscript received December 18, 2014; published online March 11, 2015. Assoc. Editor: Xilun Ding.

J. Mechanisms Robotics 7(4), 041003 (Nov 01, 2015) (9 pages) Paper No: JMR-13-1116; doi: 10.1115/1.4029498 History: Received June 18, 2013; Revised December 18, 2014; Online March 11, 2015

Rolling contact has been used by robotic devices to drive between configurations. The degrees of freedom (DOFs) of rolling contact pairs can be one, two, or three, depending on the geometry of the objects. This paper aimed to derive three kinematic inputs required for the moving object to follow a trajectory described by its velocity profile when the moving object has three rotational DOFs and thus can rotate about any axis through the contact point with respect to the fixed object. We obtained three contact equations in the form of a system of three nonlinear algebraic equations by applying the curvature theory in differential geometry and simplified the three nonlinear algebraic equations to a univariate polynomial of degree six. Differing from the existing solution that requires solving a system of nonlinear ordinary differential equations, this polynomial is suitable for fast and accurate numerical root approximations. The contact equations further revealed the two essential parts of the spin velocity: The induced spin velocity governed by the geometry and the compensatory spin velocity provided externally to realize the desired spin velocity.

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References

Bicchi, A., and Kumar, V., 2000, “Robotic Grasping and Contact: A Review,” IEEE International Conference on Robotics and Automation (ICRA '00), San Francisco, CA, Apr. 24–28, pp. 348–353. [CrossRef]
Ma, R. R., and Dollar, A. M., 2013, “Linkage-Based Analysis and Optimization of an Underactuated Planar Manipulator for In-Hand Manipulation,” ASME J. Mech. Rob., 6(1), p. 011002. [CrossRef]
Joshi, V. A., Banavar, R. N., and Hippalgaonkar, R., 2010, “Design and Analysis of a Spherical Mobile Robot,” Mech. Mach. Theory, 45(2), pp. 130–136. [CrossRef]
Yangsheng, X., and Au, S. K. W., 2004, “Stabilization and Path Following of a Single Wheel Robot,” IEEE/ASME Trans. Mechatronics, 9(2), pp. 407–419. [CrossRef]
Cui, L., and Dai, J. S., 2012, “Reciprocity-Based Singular Value Decomposition for Inverse Kinematic Analysis of the Metamorphic Multifingered Hand,” ASME J. Mech. Rob., 4(3), p. 034502. [CrossRef]
Bloch, A. M., Baillieul, J., Crouch, P., and Marsden, J., 2007, Nonholonomic Mechanics and Control, Springer, New York.
Ostrowski, J. P., Desai, J. P., and Kumar, V., 2000, “Optimal Gait Selection for Nonholonomic Locomotion Systems,” Int. J. Rob. Res., 19(3), pp. 225–237. [CrossRef]
Javadi, A. H., and Mojabi, P., 2004, “Introducing Glory: A Novel Strategy for an Omnidirectional Spherical Rolling Robot,” ASME J. Dyn. Syst., Meas., Control, 126(3), pp. 678–683. [CrossRef]
Armour, R., and Vincent, J., 2006, “Rolling in Nature and Robotics: A Review,” J. Bionic Eng., 3(4), pp. 195–208. [CrossRef]
Joshi, V. A., and Banavar, R. N., 2009, “Motion Analysis of a Spherical Mobile Robot,” Robotica, 27(3), pp. 343–353. [CrossRef]
Kaznov, V., and Seeman, M., 2010, “Outdoor Navigation With a Spherical Amphibious Robot,” IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), IEEE, Montreal, Canada, Oct. 18–22, pp. 5113–5118. [CrossRef]
Li, T., and Liu, W., 2011, “Design and Analysis of a Wind-Driven Spherical Robot With Multiple Shapes for Environment Exploration,” J. Aerosp. Eng., 24(1), pp. 135–139. [CrossRef]
Xu, Y., Brown, H. B., and Au, K. W., 1999, “Dynamic Mobility With Single-Wheel Configuration,” Int. J. Rob. Res., 18(7), pp. 728–738. [CrossRef]
Nakajima, R., Tsubouchi, T., Yuta, S., and Koyanagi, E., 1997, “A Development of a New Mechanism of an Autonomous Unicycle,” IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS '97), Grenoble, France, Sept. 7–11, pp. 906–912. [CrossRef]
Cole, A., Hauser, J. E., and Sastry, S., 1989, “Kinematics and Control of Multifingered Hands With Rolling Contact,” IEEE Trans. Autom. Control, 34(4), pp. 398–404. [CrossRef]
Cui, L., Cupcic, U., and Dai, J. S., 2014, “An Optimization Approach to Teleoperation of the Thumb of a Humanoid Robot Hand: Kinematic Mapping and Calibration,” ASME J. Mech. Des., 136(9), p. 091005. [CrossRef]
Cui, L., and Dai, J. S., 2011, “Posture, Workspace, and Manipulability of the Metamorphic Multifingered Hand With an Articulated Palm,” ASME J. Mech. Rob., 3(2), p. 021001. [CrossRef]
Dai, J. S., Wang, D., and Cui, L., 2009, “Orientation and Workspace Analysis of the Multifingered Metamorphic Hand—Metahand,” IEEE Trans. Rob., 25(4), pp. 942–947. [CrossRef]
Howard, B., Yang, J., and Ozsoy, B., 2013, “Optimal Posture and Supporting Hand Force Prediction for Common Automotive Assembly One-Handed Tasks,” ASME J. Mech. Rob., 6(2), p. 021009. [CrossRef]
Sarkar, N., Yun, X., and Kumar, V., 1997, “Control of Contact Interactions With Acatastatic Nonholonomic Constraints,” Int. J. Rob. Res., 16(3), pp. 357–374. [CrossRef]
Kiss, B., Lévine, J., and Lantos, B., 2002, “On Motion Planning for Robotic Manipulation With Permanent Rolling Contacts,” Int. J. Rob. Res., 21(5–6), pp. 443–461. [CrossRef]
Tsai, W. L., 1999, Robot Analysis: The Mechanics of Serial and Parallel Manipulators, Wiley, New York.
Gan, D., Liao, Q., Dai, J. S., and Wei, S., 2010, “Design and Kinematics Analysis of a New 3CCC Parallel Mechanism,” Robotica, 28(7), pp. 1065–1072. [CrossRef]
Kerr, J., and Roth, B., 1986, “Analysis of Multifingered Hands,” Int. J. Rob. Res., 4(4), pp. 3–17. [CrossRef]
Montana, D. J., 1988, “The Kinematics of Contact and Grasp,” Int. J. Rob. Res., 7(3), pp. 17–32. [CrossRef]
Marigo, A., and Bicchi, A., 2000, “Rolling Bodies With Regular Surface: Controllability Theory and Application,” IEEE Trans. Autom. Control, 45(9), pp. 1586–1599. [CrossRef]
Li, Z., Hsu, P., and Sastry, S., 1989, “Grasping and Coordinated Manipulation by a Multifingered Robot Hand,” Int. J. Rob. Res., 8(4), pp. 33–50. [CrossRef]
Sankar, N., Kumar, V., and Yun, X., 1996, “Velocity and Acceleration Analysis of Contact Between Three-Dimensional Rigid Bodies,” ASME J. Appl. Mech., 63(4), pp. 974–984. [CrossRef]
Cui, L., and Dai, J. S., 2010, “A Darboux-Frame-Based Formulation of Spin-Rolling Motion of Rigid Objects With Point Contact,” IEEE Trans. Rob., 26(2), pp. 383–388. [CrossRef]
Cartan, E., 2002, Riemannian Geometry in an Orthogonal Frame, World Scientific Press, Singapore.
Cartan, H., 1996, Differential Forms, Dover Publisher, New York.
Carmo, M. P., 1976, Differential Geometry of Curves and Surfaces, Prentice-Hall, Englewood Cliffs, NJ.
Cui, L., Wang, D., and Dai, J. S., 2009, “Kinematic Geometry of Circular Surfaces With a Fixed Radius Based on Euclidean Invariants,” ASME J. Mech. Des., 131(10), p. 101009. [CrossRef]
McCarthy, J. M., 2011, “Kinematics, Polynomials, and Computers—A Brief History,” ASME J. Mech. Rob., 3(1), p. 010201. [CrossRef]
Borras, J., and Di Gregorio, R., 2009, “Polynomial Solution to the Position Analysis of Two Assur Kinematic Chains With Four Loops and the Same Topology,” ASME J. Mech. Rob., 1(2), p. 021003. [CrossRef]
Cui, L., 2010, “Differential Geometry Based Kinematics of Sliding-Rolling Contact and Its Use for Multifingered Hands,” Ph.D. thesis, King's College London, University of London, London, UK.
Liu, H., Song, X., Bimbo, J., Althoefer, K., and Senerivatne, L., 2012, “Intelligent Fingertip Sensing for Contact Information Identification,” Advances in Reconfigurable Mechanisms and Robots I, J. S.Dai, M.Zoppi, and X.Kong, eds., Springer, London, UK, pp. 599–608.
Plant, M. W., and Quandt, R. E., 1989, “On the Accuracy and Cost of Numerical Integration in Several Variables,” J. Stat. Comput. Simul., 32(4), pp. 229–248. [CrossRef]
National Instruments, 2014, “NI LabVIEW MathScript RT Module,” National Instruments Corp., Austin, TX, http://sine.ni.com/nips/cds/view/p/lang/en/nid/207267
Bates, D. J., Hauenstein, J. D., Sommese, A. J., and Wampler, C. W., 2013, Numerically Solving Polynomial Systems With Bertini, Society for Industrial and Applied Mathematics, Philadelphia, PA.
Jenkins, M. A., and Traub, J. F., 1970, “A Three-Stage Variable-Shift Iteration for Polynomial Zeros and Its Relation to Generalized Rayleigh Iteration,” Numer. Math., 14(3), pp. 252–263. [CrossRef]
Hameiri, E., and Shimshoni, I., 2003, “Estimating the Principal Curvatures and the Darboux Frame From Real 3-D Range Data,” IEEE Trans. Syst. Man Cybern. Part B, 33(4), pp. 626–637. [CrossRef]
Desbrun, M., Meyer, M., Schroder, P., and Barr, A. H., 2003, “Discrete Differential-Geometry Operators for Triangulated 2-Manifolds,” Visualization and Mathematics III, H. C. H.Polthier, ed., Springer-Verlag, New York.

Figures

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Fig. 1

Three applications of rolling contact: spherical robot, single wheel robot, and multifingered hand

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Fig. 2

The three rotational DOFs of the moving object

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Fig. 3

The inputs and outputs of the inverse kinematics of rolling contact

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Fig. 4

The Darboux frame at the point M

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Fig. 5

The moving frame along a small circle L

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Fig. 6

The surface S2 rolling on the surface S1

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Fig. 7

A unit sphere B rolling on a plane A

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Fig. 8

The parameterization of a unit sphere

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Fig. 9

The geometry of the rolling direction φ and the rolling rate σ

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Fig. 10

An ellipsoid rolling on a plane

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Fig. 11

The parameterization of an ellipsoid

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Fig. 12

A ball rolling on a paraboloid

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Fig. 13

The admissible angular velocities at a contact point

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Fig. 14

Two frames related by a rotation angle φ about e3

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Fig. 15

Relationship between the curve L and the coordinate curves

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