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Research Papers: Mechanisms and Robotics

Computer Aided Synthesis of Piecewise Rational Motions for Spherical 2R and 3R Robot Arms

[+] Author and Article Information
Anurag Purwar

Department of Mechanical Engineering, Stony Brook University, Stony Brook, NY 11794-2300anurag.purwar@stonybrook.edu

Zhe Jin

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

Q. J. Ge

Department of Mechanical Engineering, Stony Brook University, Stony Brook, NY 11794-2300qiaode.ge@stonybrook.edu

Designing a C2B-spline curve is a standard scheme in CAGD (Farin (23), Hoschek and Lasser (24), and Piegl and Tiller(25)).

J. Mech. Des 130(11), 112301 (Sep 23, 2008) (9 pages) doi:10.1115/1.2976444 History: Received October 24, 2006; Revised July 16, 2008; Published September 23, 2008

This paper deals with the problem of synthesizing smooth piecewise rational spherical motions of an object that satisfies the kinematic constraints imposed by a spherical robot arm with revolute joints. This paper brings together the kinematics of spherical robot arms and recently developed freeform rational motions to study the problem of synthesizing constrained rational motions for Cartesian motion planning. The kinematic constraints under consideration are workspace related constraints that limit the orientation of the end link of robot arms. This paper extends our previous work on synthesis of rational motions under the kinematic constraints of planar robot arms. Using quaternion kinematics of spherical arms, it is shown that the problem of synthesizing the Cartesian rational motion of a 2R arm can be reduced to that of circular interpolation in two separate planes. Furthermore, the problem of synthesizing the Cartesian rational motion of a spherical 3R arm can be reduced to that of constrained spline interpolation in two separate planes. We present algorithms for the generation of C1 and C2 continuous rational motion of spherical 2R and 3R robot arms.

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Copyright © 2008 by American Society of Mechanical Engineers
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References

Figures

Grahic Jump Location
Figure 1

A Spherical 2R robot arm

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Figure 2

A Spherical 3R robot arm

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Figure 3

An unconstrained C2 cubic B-spline interpolation of a set of points in the s3s4 plane

Grahic Jump Location
Figure 4

A constrained C2 cubic B-spline interpolation of a set of points in the s3s4 plane

Grahic Jump Location
Figure 5

An unconstrained C2 cubic B-spline interpolation of a set of points in s1s2 plane; r1(u)=|w sin((α−β)/2)|, r2(u)=|w sin((α+β)/2)|, w2=Q12(u)+Q22(u)+Q32(u)+Q42(u)

Grahic Jump Location
Figure 6

An unconstrained C2 cubic B-spline interpolation of a set of points in s3s4 plane; r3(u)=|w cos((α+β)/2)|, r4(u)=|w cos((α−β)/2)|, w2=Q12(u)+Q22(u)+Q32(u)+Q42(u)

Grahic Jump Location
Figure 7

A constrained C2 cubic B-spline interpolation of a set of points in s1s2 plane; r1(u)=|w sin((α−β)/2)|, r2(u)=|w sin((α+β)/2)|, w2=Q12(u)+Q22(u)+Q32(u)+Q42(u)

Grahic Jump Location
Figure 8

A constrained C2 cubic B-spline interpolation of a set of points in s3s4 plane; r3(u)=|w cos((α+β)/2)|, r4(u)=|w cos((α−β)/2)|, w2=Q12(u)+Q22(u)+Q32(u)+Q42(u)

Grahic Jump Location
Figure 9

A Spherical 3R robot arm

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