Journal article:
Samuel R. Buss. "Accurate and efficient simulations of rigid body rotations," Journal of Computational Physics, 164 (2000) 377-406.
Abstract: This paper introduces efficient and
accurate algorithms for simulating the rotation of a three-dimensional rigid object and
compares them to several prior methods. The paper considers algorithms which exactly
preserve angular momentum and either closely preserve or exactly conserve energy.
First, we introduce a second-order accurate method that incorporates a
third-order correction; then a third-order accurate method; and finally a fourth-order
accurate method. These methods are single-step and the update operation is only a single
rotation. The algorithms are derived in a general Lie group setting.
Second, we introduce a near-optimal energy-correction method which allows exact
conservation of energy. This algorithm is faster and easier to implement than implicit
methods for exact energy-conservation. Our third-order method with energy
conservation is experimentally seen to act better than a fourth-order accurate method.
These new methods are superior to naive Runge-Kutta or
predictor-corrector methods, which are only second-order accurate for sphere-valued
functions. They are also superior to the explicit methods of Simo-Wong. The
second-order symplectic McLachlan-Reich methods are observed to be excellent at
approximate energy-conservation for extended periods of time, but are not as good at
long-term accuracy as our best methods. Finally we present comparisons with fourth-order
accurate symplectic methods, which have good accuracy but higher computational cost.
Download postscript or PDF.
Related talks:
"Taylor Series Methods for Rigid Body Simulation and
Extensions to Lie Groups."
SIAM Conference on Geometrid Design and Computing, Sacramento, November
2001.
"Taylor Series Methods for Rigid Body Simulation and
Extensions to Lie Groups."
Clifford meeting, Tulane University, March 2003.
"Accurate Simulation of Rigid Body Rotation and
Extensions to Lie Groups."
Department of Computer Science, Swansea University, March 2011.
Download slides: PDF