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.

1.
Shoemake
,
K.
, 1985, “
Animating Rotation With Quaternion Curves
,”
SIGGRAPH Computer Graphics
,
19
(
3
), pp.
245
254
.
2.
Pletinckx
,
D.
, 1989, “
Quaternion Calculus as a Basic Tool in Computer Graphics
,”
Visual Comput.
0178-2789,
5
, pp.
2
13
.
3.
Dam
,
E. B.
,
Koch
,
M.
, and
Lillholm
,
M.
, 1998. “
Quaternions, Interpolation and Animation
,”
University of Copenhagen
, Technical Report No. DIKU-TR-98/5.
4.
Duff
,
T.
, 1985, “
Quaternion Splines for Animating Orientation
,”
Proceedings of the USENIX Association Second Computer Graphics Workshop
, Monterey CA, pp.
54
62
.
5.
Kim
,
M.-J.
,
Kim
,
M.-S.
, and
Shin
,
S. Y.
, 1995, “
A C2 Continuous B-Spline Quaternion Curve Interpolating a Given Sequence of Solid Orientations
,”
Proceedings of the Computer Animation
,
IEEE Computer Society
,
Washington, DC
, pp.
19
21
.
6.
Kim
,
M. S.
, and
Nam
,
K. W.
, 1995, “
Interpolating Solid Orientations With Circular Blending Quaternion Curves
,”
Comput.-Aided Des.
0010-4485,
27
(
5
), pp.
385
398
.
7.
Nielson
,
G.
, 1993, “
Smooth Interpolation of Orientations
,”
Computer Animation, Models and Techniques in Computer Animation
,
Springer
,
New York
, pp.
75
93
.
8.
Nielson
,
G. M.
, 2004, “
Nu-Quaternion Splines for the Smooth Interpolation of Orientations
,”
IEEE Trans. Vis. Comput. Graph.
1077-2626,
10
(
2
), pp.
224
229
.
9.
Wang
,
W.
, and
Joe
,
B.
, 1993, “
Orientation Interpolation in Quaternion Space Using Spherical Biarcs
,”
Proceedings of the Graphics Interface’93
,
Morgan-Kaufmann
,
San Francisco, CA
, pp.
24
32
.
10.
Barr
,
A. H.
,
Currin
,
B.
,
Gabriel
,
S.
, and
Hughes
,
J. F.
, 1992, “
Smooth Interpolation of Orientations With Angular Velocity Constraints Using Quaternions
,”
Comput. Graph.
0097-8930,
26
(
2
), pp.
313
320
.
11.
Ge
,
Q. J.
, and
Ravani
,
B.
, 1994, “
Computer-Aided Geometric Design of Motion Interpolants
,”
ASME J. Mech. Des.
0161-8458,
116
(
3
), pp.
756
762
.
12.
Ge
,
Q. J.
, and
Ravani
,
B.
, 1994, “
Geometric Construction of Bezier Motions
,”
ASME J. Mech. Des.
0161-8458,
116
(
3
), pp.
749
755
.
13.
Juttler
,
B.
, and
Wagner
,
M. G.
, 1996, “
Computer-Aided Design With Spatial Rational b-Spline Motions
,”
ASME J. Mech. Des.
0161-8458,
118
(
2
), pp.
193
201
.
14.
Purwar
,
A.
, and
Ge
,
Q. J.
, 2005, “
On the Effect of Dual Weights in Computer Aided Design of Rational Motions
,”
ASME J. Mech. Des.
0161-8458,
127
(
5
), pp.
967
972
.
15.
Purwar
,
A.
,
Chi
,
X.
, and
Ge
,
Q. J.
, 2008, “
Automatic Fairing of Two-Parameter Rational b-Spline Motion
,”
ASME J. Mech. Des.
0161-8458,
130
(
1
), p.
011003
.
16.
Röschel
,
O.
, 1998, “
Rational Motion Design: A Survey
,”
Comput.-Aided Des.
0010-4485,
30
(
3
), pp.
169
178
.
17.
Horsch
,
T.
, and
Juttler
,
B.
, 1998, “
Cartesian Spline Interpolation for Industrial Robots
,”
Comput.-Aided Des.
0010-4485,
30
(
3
), pp.
217
224
.
18.
Wagner
,
M.
, and
Ravani
,
B.
, 1996, “
Computer Aided Design of Robot Trajectories Using Rational Motions
,”
Recent Advances in Robot Kinematics
,
J.
Lenarcic
and
V.
Parenti- Castelli
, eds.,
Kluwer
,
Dordrecht
, pp.
151
158
.
19.
Jin
,
Z.
, and
Ge
,
Q. J.
, 2007, “
Computer Aided Synthesis of Piecewise Rational Motion for Planar 2r and 3r Robot Arms
,”
ASME J. Mech. Des.
0161-8458,
129
(
10
), pp.
1031
1036
.
20.
Bottema
,
O.
, and
Roth
,
B.
, 1979,
Theoretical Kinematics
,
North-Holland
,
Amsterdam
.
21.
McCarthy
,
J. M.
, 1990,
Introduction to Theoretical Kinematics
,
MIT
,
Cambridge, MA
.
22.
Ravani
,
B.
, and
Roth
,
B.
, 1984, “
Mappings of Spatial Kinematics
,”
ASME J. Mech., Transm., Autom. Des.
0738-0666,
106
(
3
), pp.
341
347
.
23.
Farin
,
G.
, 1996,
Curves and Surfaces for Computer-Aided Geometric Design: A Practical Guide
,
4th ed.
,
Academic
,
New York
.
24.
Hoschek
,
J.
, and
Lasser
,
D.
, 1993,
Fundamentals of Computer Aided Geometric Design
,
A. K. Peters
,
Wellesley, MA
.
25.
Piegl
,
L.
, and
Tiller
,
W.
, 1995,
The Nurbs Book
,
Springer
,
Berlin
.
26.
Ge
,
Q. J.
, and
Purwar
,
A.
, 2004, “
Spears, Oriented Screw Displacements, and Their Image Spaces
,”
Proceedings of the 11th World Congress in Mechanism And Machine Science IFToMM 2004
, Aug. 18-21, Tianjin, China.
27.
Bangert
,
C.
, and
Prautzsch
,
H.
, 1997, “
Circle and Sphere as Rational Splines
,”
Neural, Parallel and Scientific Computations
,
5
(
1-2
), pp.
153
162
.
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