Accurate and Eecient Simulation of Rigid Body Rotations Revised Version

This paper introduces eecient 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 nally 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.