This paper is concerned with the integration of the differential equations for the Euler parameters, for the purpose of describing the orientation of a rigid body. This can be done using standard methods, but in some cases, such as in the presence of impulsive forces, the angular velocities are not continuous and methods based on high order continuity are not appropriate. In this paper, the use of the closed-form solution for piecewise constant angular velocity as the basis for a computational algorithm is studied. It is seen that if this solution is implemented in a leapfrog manner a method with second-order accuracy is obtained in the smooth case, while this method also makes sense in the discontinuous case.

Issue Section:
Technical Briefs
Keywords:
differential equations, integration, piecewise constant techniques, classical mechanics
Topics:
Algorithms, Computation, Differential equations, Errors, Classical mechanics
1.
Wittenburg, J., 1977, Dynamics of Systems of Rigid Bodies, Teubner, Stuttgart.
2.
Johansson
,
L.
, and
Klarbring
,
A.
,
2000
, “
Study of Frictional Impact Using a Non-Smooth Equations Solver
,”
ASME J. Appl. Mech.
,
67
, pp.
267
–
273
.
3.
Johansson
,
L.
,
1999
, “
A Linear Complementarity Algorithm for Rigid Body Impact with Friction
,”
European Journal of Mechanics
,
18
, pp.
703
–
717
.
4.
Johansson, L., “A Newton Method for Rigid Body Frictional Impact with Multiple Simultaneous Impact Points,” Comput. Methods Appl. Mech. Eng., to appear.
5.
Whitmore, S. A., Fife, M., and Logan, B., 1997, “Development of a Closed-Loop Strap Down Attitude System for an Ultralight Altitude Flight Experiment,” NASA Technical Memorandum 4775.
6.
Stevens, B. L., and Lewis, F. L., 1992, Aircraft Control and Simulation, Wiley, New York.
7.
Morton
,
H. S.
,
Junkins
,
J. L.
, and
Blanton
,
J. N.
,
1974
, “
Analytical Solutions for Euler Parameters
,”
Celest. Mech.
,
10
, pp.
287
–
301
.
8.
Lukes, D. H., 1982, Differential Equations: Classical to Controlled, Academic Press, New York.
9.
Byers
,
R. M.
, and
Vadali
,
S. R.
,
1993
, “
Qasi-Closed-Form Solution to the Time-Optimal Rigid Spacecraft Reorientation Problem
,”
J. Guid. Control Dyn.
,
16
, pp.
453
–
461
.
10.
Dahlquist, G, Bjo¨rk, A˚, and Anderson, N., 1974, Numerical Methods, Prentice-Hall, Englewood Cliffs.
11.
Vandergraft, J. S., 1978, Introduction to Numerical Computations, Academic Press, New York.
12.
Omelyan
,
I. P.
,
1998
, “
Algorithm for Numerical Integration of the Rigid-Body Equations of Motion
,”
Phys. Rev. E
,
58
, pp.
1169
–
1172
.
13.
Omelyan
,
I. P.
,
1998
, “
Numerical Integration of the Equations of Motion for Rigid Polyatomics: The Matrix Method
,”
Comput. Phys. Commun.
,
109
, pp.
171
–
183
.
14.
Vu-Quoc
,
L.
,
Zhang
,
X.
, and
Walton
,
O. R.
,
2000
, “
A 3-D Discrete-Element Method for Dry Granular Flows of Ellipsoidal Particles
,”
Comput. Methods Appl. Mech. Eng.
,
187
, pp.
483
–
528
.
Copyright © 2001
 by ASME
You do not currently have access to this content.