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Technical Briefs

Properties of Higher Order Instant Center: A Case Study of Classical Motions

[+] Author and Article Information
Amit Kulkarni

R&D SW Engineer
Automation Solutions Business,
Agilent Technologies,
5301 Stevens Creek Boulevard,
Santa Clara 95014, CA
e-mail: Amit_Kulkarni@Agilent.com

Delbert Tesar

Director
Robotics Research Group,
Professor and Carol Cockrell Curran Chair in Engineering,
5301 Stevens Creek Boulevard,
Santa Clara 95014, CA
e-mail: Tesar@mail.UTexas.edu

1Corresponding author.

Contributed by the Mechanisms and Robotics Committee of ASME for publication in the Journal of Mechanisms and Robotics. Manuscript received October 25, 2010; final manuscript received December 28, 2012; published online March 26, 2013. Assoc. Editor: Qiaode Jeffrey Ge.

J. Mechanisms Robotics 5(2), 024501 (Mar 26, 2013) (14 pages) Paper No: JMR-10-1148; doi: 10.1115/1.4023555 History: Received October 25, 2010; Revised December 28, 2012

A generalized algebraic formulation using instant centers (IC) was developed for the motion description of a general point in the rigid body under a planar 3-DOF (degrees-of-freedom) motion for up to the fifth order kinematics. This motion theory is being applied to planar wheeled mobile platforms. Though the first order and second order instant centers have been previously studied, the properties of higher order instant centers are yet to be understood. Also the expressions for third and higher order motion are highly coupled and more complex than the first and second order motion descriptions. To this effect, this paper studies some special case scenarios of planar rigid body motion that involve well documented 1-DOF motions such as a circle (cylinder /disk/wheel) rolling on a straight line (plane/flat, smooth surface), a circle rolling inside another circle, a circle rolling on another circle, etc. This study will help us understand the physical nature of the kinematic formulation using instant centers. Otherwise, numerical specifications for the higher order properties will have little known physical reference as to the meanings of their magnitudes.

Copyright © 2013 by ASME
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Figures

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Fig. 1

Different classical motion scenarios: (a) a circle rolling on a straight line without slipping, (b) a circle rolling on another circle without slipping, (c) a straight line rolling on a circle without slipping, and (d) a circle rolling inside another circle without slipping

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Fig. 2

Kinematic description of a circle rolling on another circle without slipping: (a) Circle 1 rolls the inside of circle 2 and (b) circle 1 rolls on the outside of circle 2

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Fig. 3

The locus of velocity IC for the general motion of a circle rolling on another circle: Case (a) circle 1 rolls on the inside of circle 2, and case (b) circle 1 rolls on the outside of circle 2

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Fig. 4

The locus of acceleration IC for the general motion of a circle rolling on another circle: Case (a) circle 1 rolls on the inside of circle 2, and case (b) circle 1 rolls on the outside of circle 2

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Fig. 5

General locus of the acceleration IC for circle rolling on the outside of another circle (negative λ) for different scenarios: (a) circle 2 (the fixed circle) is a straight line (r2 = ∞, thus λ = ∞) (Sec. 3.2), (b) circle 2 is larger than or equal to circle 1 (|λ|≥1), (c) circle 2 is smaller than circle 1 (|λ| < 1), and (d) circle 1 (the rolling circle) is a straight line (r = ∞, thus λ = 0)

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Fig. 6

The acceleration IC locus radius for rolling circles, three special cases: (a) circle rolling on a straight line, (b) circle rolling on the inner side of another circle, and (c) circle rolling on the outer side of another circle

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Fig. 7

A circle rolling on another circle: The acceleration IC location when α≠0,ω=0: Case (a) circle 1 rolls on the inside of circle 2, and case (b) circle 1 rolls on the outside of circle 2

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Fig. 8

The acceleration IC location when α=0,ω≠0 for a circle rolling on another circle: Case (a) circle 1 rolls on the inside of circle 2, and case (b) circle 1 rolls on the outside of circle 2

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Fig. 9

The locus of the third order IC for the general motion of a circle rolling on another circle when (a) (λ > 0) circle 1 rolls on the inside of circle 2, and (b) (λ < 0) circle 1 rolls on the outside of circle 2

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Fig. 10

The third order IC location when α·≠0,ω=0 for a circle rolling on another circle: Case (a) circle 1 rolls on the inside of circle 2, and case (b) circle 1 rolls on the outside of circle 2

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Fig. 11

The third order IC location when α·=0,α=0,and ω≠0 for a circle rolling on another circle: Shown here is the case when circle 1 rolls on the inside of circle 2

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Fig. 12

The locus of the third order IC for α·=0,α≠0,ω≠0 for a circle rolling on another circle: Shown here is the case when circle 1 rolls on the inside of circle 2

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Fig. 13

A circle rolling on another circle: The third order IC location when α·≠0,α=0,ω≠0.

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Fig. 14

A circle rolling on another circle: The locus of the fourth order IC location for general motion

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Fig. 15

The fourth order IC location when α≠0,α·=α==0 for a circle rolling on another circle: Case (a) circle 1 rolls on the inside of circle 2, and case (b) circle 1 rolls on the outside of circle 2

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Fig. 16

The fourth order IC location when ω≠0,α=α·=α=0 for a circle rolling on another circle: Shown here is the case when circle 1 rolls on the inside of circle 2

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Fig. 17

The fourth order IC location when ω≠0,α·≠0,α=α=0 for a circle rolling on another circle: Shown here is the case when circle 1 rolls on the inside of circle 2

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Fig. 18

The fourth order IC location when α··=α·=ω=0,α≠0 for a circle rolling on another circle: Shown here is the case when circle 1 rolls on the inside of circle 2

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Fig. 19

The fourth order IC location when ω≠0,α··≠0,α·=α=0 for a circle rolling on another circle: Shown here is the case when circle 1 rolls on the inside of circle 2

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Fig 20

Description of general J wheeled mobile platform with all active caster wheels

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Fig. 21

The loci of the fourth order IC for a general J wheeled mobile platforms

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