This paper deals with the synthesis of a second order parametrically continuous (C2) motion that interpolates through a given set of configurations of an object. It derives conditions for blending two motion segments with C2 continuity and develops an algorithm for constructing a C2 composite Be´zier type motion that has similarities to Beta-splines in the field of Computer Aided Geometric Design. A criterion for evaluating the smoothness of a motion is established and is used to synthesize a “globally smooth” motion. The results have applications in trajectory generation in robotics, mechanical systems animation and CAD/CAM.

1.
Barr
A.
,
Curin
B.
,
Gabriel
S.
,
Hughes
J.
,
1992
, “
Smooth Interpolation of Orientations With Angular Velocity Constraints Using Quaternions
,”
Computer Graphics
, Vol.
26
, pp.
313
320
.
2.
Barsky
B. A.
, and
DeRose
T. D.
,
1990
, “
Geometric Continuity of Parametric Curves: Constructions of Geometrically Continuous Splines
,”
IEEE Computer Graphics and Applications
, Vol.
10
, No. (
1
), pp.
60
68
.
3.
Boehm, W., 1987, “Smooth Curves and Surfaces,” Geometric Modeling: Algorithms and New Trends, G. Farin, ed., SIAM, Philadelphia, PA, pp. 175–184.
4.
Bottema, O., and Roth, B., 1990, Theoretical Kinematics, Reprinted Dover Pub.
5.
Coxeter, H. S. M., 1957, Non-Euclidean Geometry, 3rd ed., University of Toronto Press, Toronto, 309 pp.
6.
Duff, T., 1986, “Quaternion Splines for Animating Orientation,” Technical report, AT&T Bell Laboratories.
7.
Farin, G., 1993, Curves and Surfaces for Computer Aided Geometric Design. 3rd ed., Academic Press, San Diego.
8.
Flanders, H., 1989, Differential Forms with Applications to the Physical Sciences, Dover Publ., New York, 205 pp.
9.
Ge
Q. J.
,
1994
, “
On Matrix Algebra Realizations of the Theory of Biquaternions
,”
Mechanism Synthesis and Analysis
, ASME Publication DE-Vol.
70
425
432
.
1.
Ge
Q. J.
, and
Ravani
B.
,
1991
, “
Computer aided Geometric Design of Motion Interpolants
,”,
Advances in Design Automation 1991
ASME Publ. DE-Vol.
32
-
2
:
33
41
,
2.
See also
ASME JOURNAL OF MECHANICAL DESIGN
, Vol.
116
, No.
3
, pp.
749
755
,
1994
.
1.
Ge, Q. J., and Ravani, B., 1993, “Computational Geometry and Motion Approximation,” Computational Kinematics, pp. 229–238, Kluwer Academic Publishers.
2.
Ge
Q. J.
, and
Ravani
B.
,
1994
, “
Geometric Construction of Be´zier Motions
,”
ASME JOURNAL OF MECHANICAL DESIGN
, Vol.
116
, No.
3
, pp.
756
762
.
3.
Ge
Q. J.
, and
Kang
D.
,
1995
, “
Motion Interpolation With G2 Composite Be´zier Motions
,”
ASME JOURNAL OF MECHANICAL DESIGN
, Vol.
117
, No.
4
, pp.
520
525
.
4.
Ju¨ttler, B., 1994, “Visualization of Moving Objects Using Dual Quaternion Curves,” Computers & Graphics, Vol. 18, No. 3.
5.
Ju¨ttler, B., and Wagner, M. G., 1994, “Computer aided Design of Rational Motions,” Technical Report No. 10, Institut fu¨r Geometric, Technische Universita¨t Wien.
6.
Kim
K.-S.
, and
Nam
K.-W.
,
1995
, “
Interpolating Solid Orientations With Circular Blending Quaternion Curves
,”
Computer-Aided Design
, Vol.
27
, No.
5
, pp.
385
398
.
7.
Kirson, Y., 1975, “Higher Order Curvature Theory in Space Kinematics,” Ph.D. Dissertation, University of California at Berkeley.
8.
McCarthy
J. M.
, and
Ravani
B.
,
1986
, “
Differential Kinematics of Spherical and Spatial Motions Using Kinematic Mapping
,”
ASME Journal of Applied Mechanics
, Vol.
53
, pp.
15
22
.
9.
McCarthy
J. M.
,
1987
, “
The Differential Geometry of Curves in an Image Space of Spherical Kinematics
,”
Mech. Mach. Theory
, Vol.
22
, No.
3
, pp.
205
211
.
10.
McCarthy, J. M., 1990, Introduction to Theoretical Kinematics. MIT Press, Cambridge, MA.
11.
Nielson, G., 1993, “Smooth Interpolation of Orientations,” Proc. Computer Animation ’93: Models and Techniques in Computer Animation, Springer-Verlag, pp. 75–93.
12.
Pletinckx
D.
,
1989
, “
Quaternion Calculus as a Basic Tool in Computer Graphics
,”
The Visual Computer
, Vol.
5
, pp.
2
13
.
13.
Shoemake
K.
,
1985
, “
Animating Rotation With Quaternion Curves
,”
ACM Siggraph
, Vol.
19
, No.
3
, pp.
245
254
.
14.
Schlag, J., 1991, “Using Geometric Constructions to Interpolate Orientation With Quaternions,” Graphics Gem II, Arvo, J., ed., Academic Press, pp. 377– 380.
15.
Veldkamp
G. R.
,
1967
, “
Canonical Systems and Instantaneous Invariants in Spatial Kinematics
,”
Journal of Mechanisms
, Vol.
2
, pp.
329
388
.
16.
Wagner, M. G., 1994, “A Geometric Approach to Motion Design,” Ph.D. Dissertation, Institut fu¨r Geometric, Technische Universita¨t Wien.
17.
Wang, W., and Joe, B., “Orientation interpolation in quaternion space using spherical biarcs,” Proc. Graphics Interface ’93, pp. 24–32.
This content is only available via PDF.
You do not currently have access to this content.