Reconfiguration of a Rolling Sphere: A Problem in Evolute-Involute Geometry

[+] Author and Article Information
Tuhin Das

Department of Mechanical Engineering, Michigan State University, East Lansing, MI 48824

Ranjan Mukherjee

Department of Mechanical Engineering, Michigan State University, East Lansing, MI 48824mukherji@egr.msu.edu

J. Appl. Mech 73(4), 590-597 (Oct 28, 2005) (8 pages) doi:10.1115/1.2164515 History: Received February 23, 2005; Revised October 28, 2005

This paper provides a new perspective to the problem of reconfiguration of a rolling sphere. It is shown that the motion of a rolling sphere can be characterized by evolute-involute geometry. This characterization, which is a manifestation of our specific selection of Euler angle coordinates and choice of angular velocities in a rotating coordinate frame, allows us to recast the three-dimensional kinematics problem as a problem in planar geometry. This, in turn, allows a variety of optimization problems to be defined and admits infinite solution trajectories. It is shown that logarithmic spirals form a class of solution trajectories and they result in exponential convergence of the configuration variables.

Copyright © 2006 by American Society of Mechanical Engineers
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Figure 4

The C-C′ pair for the Dual-Point Theorem

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Figure 5

RS and DPT maneuvers

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Figure 10

A counterclockwise logarithmic spiral motion of C

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Figure 11

Locus of C that allows partial reconfiguration

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Figure 1

(a) An arbitrary configuration of the sphere; (b) desired configuration of the sphere

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Figure 6

Motion of C and F due to ωx1 and ωy1

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Figure 7

Wrapping and unwrapping trajectories of the point C

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Figure 8

Reconfiguration with ψ<cos−1(1∕λ)

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Figure 9

Reconfiguration with ψ≈cos−1(1∕λ)

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Figure 12

Simulation showing complete reconfiguration using logarithmic spiral motion of C

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Figure 2

Actuations (A) and (B)

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Figure 3

An arbitrary configuration of the sphere




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