The Solution of SO(3) through a Single Parameter ODE

Abstract

In many applications we need to solve an orthogonal transformation tensor $\mathit{Q}\in SO$(3) from a tensorial equation $\dot{\mathbf{Q}}\mathbf{\hspace{0.17em}}\mathbf{=}\mathbf{\hspace{0.17em}}\mathbf{W}\mathbf{Q}$ under a given spin history W. In this paper, we address some interesting issues about this equation. A general solution of Q is obtained by transforming the governing equation into a new one in the space of $ℝP^3$. Then, we develop a novel method to solve Q in terms of a single parameter, whose governing equation is a single nonlinear ordinary differential equation (ODE).