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Research Papers: Mechanisms and Robotics

Rational Motion Interpolation Under Kinematic Constraints of Spherical 6R Closed Chains

[+] Author and Article Information
Anurag Purwar, Zhe Jin, Q. J. Ge

Department of Mechanical Engineering, State University of New York, Stony Brook, NY 11794-2300

Designing a C2B-spline curve is a standard scheme in CAGD (see Farin (22), Hoschek and Lasser (23), and Piegl and Tiller (24)).

J. Mech. Des 130(6), 062301 (Apr 14, 2008) (9 pages) doi:10.1115/1.2898879 History: Received April 15, 2007; Revised October 11, 2007; Published April 14, 2008

The work reported in this paper brings together the kinematics of spherical closed chains and the recently developed free-form rational motions to study the problem of synthesizing rational interpolating motions under the kinematic constraints of spherical 6R closed chains. The results presented in this paper are an extension of our previous work on the synthesis of piecewise rational spherical motions for spherical open chains. The kinematic constraints under consideration are workspace related constraints that limit the position of the links of spherical closed chains in the Cartesian space. Quaternions are used to represent spherical displacements. The problem of synthesizing smooth piecewise rational motions is converted into that of designing smooth piecewise rational curves in the space of quaternions. The kinematic constraints are transformed into geometric constraints for the design of quaternion curves. An iterative algorithm for constrained motion interpolation is presented. It detects the violation of the kinematic constraints by searching for those extreme points of the quaternion curve that do not satisfy the constraints. Such extreme points are modified so that the constraints are satisfied, and the resulting new points are added to the ordered set of the initial positions to be interpolated. An example is presented to show how this algorithm produces smooth spherical rational spline motions that satisfy the kinematic constraints of a spherical 6R closed chain. The algorithm can also be used for the synthesis of rational interpolating motions that approximate the kinematic constraints of spherical 5R and 4R closed chains within a user-defined tolerance.

FIGURES IN THIS ARTICLE
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Copyright © 2008 by American Society of Mechanical Engineers
Topics: Motion , Chain , Interpolation
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Figures

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Figure 1

A 4R spherical closed chain

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Figure 2

A 5R spherical closed chain

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Figure 3

A 6R spherical closed chain

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Figure 4

Kinematic constraint surfaces obtained by intersection with the hyperplane Y4=1

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Figure 5

Unconstrained interpolation and the kinematic constraint surfaces obtained by intersection with the hyperplane Y4=1. The image curve is shown as a continuous curve, the two extreme points that violate kinematic constraints as (★) and the given positions as (◼).

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Figure 6

A zoomed-in and rotated view of Fig. 5: One extreme point with F2(8.74)=0.38 violates the kinematic constraint: 0.5⩽F2. Surfaces given by the limits of the other constraint are shown in light broken lines.

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Figure 7

A zoomed-in and rotated view of Fig. 5: One extreme point with F1(0.92)=1.0 violates the kinematic constraint: F1⩽0.98. Surfaces given by the limits of the other constraint are shown in light broken lines.

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Figure 8

Constrained interpolation (compare to Fig. 6). The labeled new point (o) is the point that replaces the labeled extreme point.

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Figure 9

Constrained interpolation (compare to Fig. 7). The labeled new point (o) is the point that replaces the labeled extreme point.

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