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.

1.
Shoemake
,
K.
, 1985, “
Animating Rotation With Quaternion Curves
,”
Proceedings of the 12th Annual Conference on Computer Graphics and Interactive Techniques
,
ACM
, pp.
245
254
.
2.
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, Proceedings of the Computer Animation
, IEEE Computer Society, pp.
19
21
.
3.
Nielson
,
G.
, 1993, “
Smooth Interpolation of Orientations
,”
Computer Animation, Models and Techniques in Computer Animation
,
Springer
,
New York
, pp.
75
93
.
4.
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
.
5.
Wang
,
W.
, and
Joe
,
B.
, 1993, “
Orientation Interpolation in Quaternion Space Using Spherical Biarcs
,”
Graphics Interface ’93
, Canadian Information Processing Society, pp.
24
32
.
6.
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
.
7.
Ge
,
Q. J.
, and
Ravani
,
B.
, 1994, “
Computer-Aided Geometric Design of Motion Interpolants
,”
ASME J. Mech. Des.
1050-0472,
116
(
3
), pp.
756
762
.
8.
Ge
,
Q. J.
, and
Ravani
,
B.
, 1994, “
Geometric Construction of Bezier Motions
,”
ASME J. Mech. Des.
1050-0472,
116
(
3
), pp.
749
755
.
9.
Juttler
,
B.
, and
Wagner
,
M. G.
, 1996, “
Computer-Aided Design With Spatial Rational B-Spline Motions
,”
ASME J. Mech. Des.
1050-0472,
118
(
2
), pp.
193
201
.
10.
Wagner
,
M. G.
, 1994, “
A Geometric Approach to Motion Design
,” Ph.D. thesis, Technische Universität Wien.
11.
Purwar
,
A.
, and
Ge
,
Q. J.
, 2005, “
On the Effect of Dual Weights in Computer Aided Design of Rational Motions
,”
ASME J. Mech. Des.
1050-0472,
127
(
5
), pp.
967
972
.
12.
Röschel
,
O.
, 1998, “
Rational Motion Design: A Survey
,”
Comput.-Aided Des.
0010-4485,
30
(
3
), pp.
169
178
.
13.
Horsch
,
T.
, and
Juttler
,
B.
, 1998, “
Cartesian Spline Interpolation for Industrial Robots
,”
Comput.-Aided Des.
0010-4485,
30
(
3
), pp.
217
224
.
14.
Ge
,
Q. J.
, and
Larochelle
,
P. M.
, 1999, “
Algebraic Motion Approximation With Nurbs Motions and Its Application to Spherical Mechanism Synthesis
,”
ASME J. Mech. Des.
1050-0472,
121
(
4
), pp.
529
532
.
15.
Purwar
,
A.
,
Zhe
,
J.
, and
Ge
,
Q. J.
, 2006, “
Computer Aided Synthesis of Piecewise Rational Motions for Spherical 2R And 3R Robot Arms
,”
Proceedings of IDETC∕CIE 2006, ASME 2006 International Design Engineering Technical Conferences
, Paper No. DETC2006-99650.
16.
Bottema
,
O.
, and
Roth
,
B.
, 1979,
Theoretical Kinematics
,
North-Holland
,
Amsterdam
.
17.
McCarthy
,
J. M.
, 1990,
Introduction to Theoretical Kinematics
,
MIT
,
Cambridge, MA
.
18.
Gfrerrer
,
A.
, 1999, “
Rational Interpolation on a Hypersphere
,”
Comput. Aided Geom. Des.
0167-8396,
16
(
1
), pp.
21
37
.
19.
Wang
,
W.
, and
Joe
,
B.
, 1997, “
Interpolation on Quadric Surfaces With Rational Quadratic Spline Curves
,”
Comput. Aided Geom. Des.
0167-8396,
14
(
3
), pp.
207
230
.
20.
Ravani
,
B.
, and
Roth
,
B.
, 1984, “
Mappings of Spatial Kinematics
,”
ASME J. Mech., Transm., Autom. Des.
0738-0666,
106
(
3
), pp.
341
347
.
21.
Müller
,
H. R.
, 1962,
Sphärische Kinematik
,
Deutscher Ver-lag der Wissenschaften
,
Berlin
.
22.
Farin
,
G.
, 1996,
Curves and Surfaces for Computer-Aided Geometric Design: A Practical Guide
, 4th ed.,
Academic
,
New York
.
23.
Hoschek
,
J.
, and
Lasser
,
D.
, 1993,
Fundamentals of Computer Aided Geometric Design
,
A. K. Peters
,
Wellesley, MA
.
24.
Piegl
,
L.
, and
Tiller
,
W.
, 1995,
The Nurbs Book
,
Springer
,
Berlin
.
25.
Ge
,
Q. J.
, 1990, “
Kinematics Constraints as Algebraic Manifolds in the Clifford Algebra of Projective Three Space
,” Ph.D. thesis, University of California, Irvine.
26.
Schröcker
,
H.-P.
,
Husty
,
M. L.
, and
McCarthy
,
J. M.
, 2007, “
Kinematic Mapping Based Assembly Mode Evaluation of Planar Four-Bar Mechanisms
,”
ASME J. Mech. Des.
1050-0472,
129
(
9
), pp.
924
929
.
27.
Schröcker
,
H.-P.
, and
Husty
,
M. L.
, 2007, “
Kinematic Mapping Based Assembly Mode Evaluation of Spherical Four-Bar Mechanisms
,”
IFToMM 2007, the 12th World Congress in Mechanism and Machine Science
,
J.-P.
Merlet
and
M.
Dahan
, eds., Vol.
129
, pp.
924
929
.
28.
Ravani
,
B.
, and
Roth
,
B.
, 1983, “
Motion Synthesis Using Kinematic Mappings
,”
ASME J. Mech., Transm., Autom. Des.
0738-0666,
105
(
3
), pp.
460
467
.
29.
Bodduluri
,
R. M. C.
, and
McCarthy
,
J. M.
, 1992, “
Finite Position Synthesis Using Image Curve of a Spherical Four-Bar Motion
,”
ASME J. Mech. Des.
1050-0472,
114
(
1
), pp.
55
60
.
You do not currently have access to this content.