In many applications we need to solve an orthogonal transformation
tensor

Among many classical Lie groups, the three-dimensional rotation group

The purpose of searching for a suitable spin tensor, in a word, is to find a reference configuration with zero spin throughout the whole motion, such that the constitutive equation for a rate-type material under large deformation can be objectively integrated. To characterize this spin-free reference configuration/corotational frame, an orthogonal transformation tensor

It should be noted that the history of

For an effective representation of the rotation matrix, it has led to the development of numerous techniques in the last several decades, and the review of the properties, advantages, and shortcomings of these parameterization techniques can be found in Ibrahimbegovic [

It is known that the spatial orientation

It is known that

In this paper, a simpler solution method of

We denote the spin matrix by

In what follows, we present a novel method to explore the general solution of

We consider a subset of (

It is cunning to presume that the solution of

The decomposition in (

The system of ODEs deduced from

Let

The inner product of (

By eliminating

In this section we are going to prove two main theorems.

The solution of

The proof of this theorem is quite lengthy, and we divide it into five parts.

Substituting (

Upon defining

From (

If

The above two equations can be used to solve

The solutions of

From (

From (

If we choose

First we calculate

In order to give a criterion to assess our numerical method we first derive a closed-form solution of

We calculate

Error between the exact and the numerical solution provided by the single-parameter method.

Now, let us turn to the case of a large rotation in Figure

Component-wise errors of rotation matrix (a)–(i) and error of orthogonality (j) produced by the single-parameter method.

Upon comparing with some different representations of the rotation group

For example, taking

Taking the differential of (

This paper is a purely academic research, and the author declares that there is no conflict of interests regarding the publication of this paper.

First, the author highly appreciates the constructive comments from anonymous referees, which improved the quality of this paper. Highly appreciated are also the project NSC-102-2221-E-002-125-MY3 and the 2011 Outstanding Research Award from the National Science Council of Taiwan and the 2011 Taiwan Research Front Award from Thomson Reuters. It is also acknowledged that the author has been promoted as being a Lifetime Distinguished Professor of National Taiwan University since 2013.