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 talk:
"Taylor Series Methods for Rigid Body Simulation and
Extensions to Lie Groups."
SIAM Conference on Geometrid Design and Computing, Sacramento, November
2001.
Download slides: postscript or PDF