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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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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