Journal of Applied Mathematics

Volume 2014 (2014), Article ID 671921, 9 pages

http://dx.doi.org/10.1155/2014/671921

Research Article

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

Chein-Shan Liu

Department of Civil Engineering, National Taiwan University, Taipei 106-17, Taiwan

Received 4 February 2014; Accepted 12 April 2014; Published 30 April 2014

Academic Editor: Song Cen

Copyright © 2014 Chein-Shan Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

In many applications we need to solve an orthogonal transformation tensor (3) from a tensorial equation 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 . 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).

1. Introduction

Among many classical Lie groups, the three-dimensional rotation group is the most widely used one. For its numerous engineering applications the development of simpler algorithms to calculate under a large rotation has received a considerable attention in the literature. A comprehensive review on the spacecraft attitude was given by Shuster [1] and on the solid mechanics by Atluri and Cazzani [2]. A framework of minimal parameterizations of the rotation matrix was proposed by Bauchau and Trainelli [3].

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 , connecting the spin-free and the fixed configurations due to the nonzero spin tensor denoted by , satisfies the following tensorial differential equation:

It does not lose any generality to assume that the initial condition of is an identity; that is, . Throughout this paper, a superimposed dot denotes the differential with respect to the current time . Computational techniques were proposed by Rubinstein and Atluri [4] for integrating (1), which required a constant rate of rotation for each time step.

It should be noted that the history of can be represented by the histories of three Euler's angles , , and as follows [5]:

and the corresponding differential equations are

Provided that the angular velocities , , and are given, the above nonlinear ordinary differential equations (ODEs) need to be integrated in a time-marching direction.

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 [6], Borri et al. [7], and Bauchau and Trainelli [3]. To represent the three-dimensional rotation, usually the number of parameters is three, like the Euler parameters, the Rodrigues parameters, and the modified Rodrigues parameters. However, these representations contain certain singularities, and their governing equations are highly nonlinear in nature. The procedures for finding the solutions of rotation matrix involving these nonlinear ODEs systems are usually very complicated.

It is known that the spatial orientation of a rigid body rotation can be expressed in terms of the unit quaternion [8]:

These parameters are obtained by using the stereographic projection of

onto by a two-fold covering; see, for example, Goldstein [5]. In the above, denotes the Euclidean norm of . Liu [8] has presented a four-dimensional Lie-algebra representation of the quaternion formulation of .

It is known that is diffeomorphic to the three-dimensional sphere and is diffeomorphic to the quotient space of the three-dimensional sphere by the antipodal equivalence, hence diffeomorphic to the three-dimensional projective space [9]. According to [10] we can define the real projective space as follows. Let be the set of all straight lines through the origin in . and represent the same line if and only if there is a nonzero constant such that , which constitutes an equivalence class denoted by with meaning that for some . Note that . The coordinates of any such that are called homogeneous coordinates for .

In this paper, a simpler solution method of is proposed by expressing the orientation equations in terms of local coordinates, yielding a scalar differential equation which can avoid the singularity. For a specified level of accuracy in numerical integration, a scalar equation requires less CPU time than an equivalent transcendental set of ODEs as shown in (3). To interpret the results of integration, the time evolution of orientation is presented as a curve with a single parameter in the three-dimensional topological space . The local coordinate is an example of a globally defined nonsingular parameterization of rotations, which is suitable for treating the computations of large rotations.

2. A Decomposition of Q

We denote the spin matrix by

and the corresponding angular velocity vector is

whose magnitude is denoted by

Also the instantaneous spin axis in the three-dimensional space is denoted by

When the spin axis is fixed, it can be viewed as a two-dimensional (2D) spin since the rotation only occurs on the plane which is perpendicular to a fixed axis. While the spin axis is varying with time, it is a three-dimensional (3D) spin.

In what follows, we present a novel method to explore the general solution of . Since is orthogonal, it belongs to the special orthogonal group with dimensions three; that is, . Although (1) can be defined by nine simultaneous ordinary differential equations (ODEs), only three of them are independent. In geometry, , an element of , represents a certain 3D algebraic surface in a real space of nine dimensions. It is unwise to find the analytical solution of (1) by solving these simultaneous ODEs. Here, the problem is solved by a judicious consideration based on a novel technique. For the sake of convenience, let us define a matrix operator which applies to and has the following form:

where .

We consider a subset of (6), by defining the following 2D spin matrix:

from which we have

where

It is cunning to presume that the solution of can be decomposed into

with an unknown matrix belonging to . Substituting it into (1) leads to

where

is a skew-symmetric matrix with

The decomposition in (14) leads to a simpler spin matrix for in (16), with only two independent inputs and . For a given angular velocity , it is easy to find the matrix by (13) and (12). However, in order to obtain we still require to find . In this paper, an analytic procedure will be developed to solve this problem for arbitrary inputs and generated from the angular velocity .

3. A Projective Transformation

The system of ODEs deduced from in (16) can be written as

The initial values of , , and are assumed to be , , and , respectively. So the determination of is now equivalent to searching a general solution of (18); that is,

where , , and

Let

be the homogeneous coordinates of . Then, the use of (18) implies

where

are the output and input of (22) and (23), respectively.

The inner product of (23) with and the use of (22) render

Integrating (25) leads to

By (21), it is equivalent to ; that is, the length of the vector is preserved under the action of group. Obviously, cannot be a zero vector; otherwise, will be a zero vector for all .

By eliminating , (22) and (23) can be combined into a nonlinear differential equations system for :

The transformation made in this section projects the three-dimensional vector , where means a three-dimensional sphere with a constant radius , into a two-dimensional vector in the topological space , which is correlated intimately with the two independent inputs of .

4. The Main Results

4.1. Two Theorems

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

Theorem 1. The solution of governed by (27) with an initial condition can be explicitly expressed in terms of a single variable :

where is a constant unit vector (with ), and is governed by a nonlinear ODE:

under the initial condition .

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

(A) Mixing the Input and Output. Consider the following transformations of variables:

where is a constant vector with norm to be given, and the vector and the other two scalars and are allowed to be time-varying. We will determine , , , and below, under the assumptions and .

Substituting (31) and its integral into (23) we can obtain

where .

Upon defining

equation (33) becomes

the solution of which is

where . The last term can be integrated by parts, leading to

where

is a time function. Under the conditions and , is a well-defined nonzero function. It is remarkable that (37) expresses in terms of a constant unit vector .

(B) Governing Equations of and . From (25) and (34) it follows that

The inner product of and (35) is

and by (31) we have

Thus (40) can be changed to

which upon using (39) becomes

Without losing any generality we may select as

where is a time function to be determined; hence, (43) becomes

Equations (35) and (45) are composed as the governing equations system for , with being the input.

(C) Explicit Form of . Noting (37), (45) changes to

Define

and the relation between and is one-to-one, since . Now, is viewed as a function of , such that

by (46) and (47).

From (48) we have

where denotes the differential with respect to , which is defined by

Taking the differential of (49) with respect to again, we can obtain

The solution of is

where and are imposed. It is interesting that we have a closed-form solution of in terms of .

(D) Explicit Form of . Now, substituting (52) into (37) and integrating the resultant, the explicit form of can be obtained as follows:

where

If can be solved, the solution of is obtained. Defining the vector as given in (29) and substituting (52) for into the above equation, we obtain (28). The square norm of is given by

(E) The Governing Equation of . It can be seen that the single parameter of variable plays the major role to express the solutions of and above. The issue to find is very important as being given below. Using (35), (38), and (53) we have

Substituting (56) for into (32) and using (38), one has

From (44) and (55) the term reads as

which together with

a result deduced from (50) and (47), and being substituted into (57), renders

In component form we have

The above two equations can be used to solve and . Eliminating from the above two equations we can obtain

where

It can be seen that is a function of and , and the latter is induced by the inputs and . Multiplying (62) by and (61) by and then subtracting them we obtain

After substituting (63) for into the above equation we can derive

It is a first order ODE for under the initial condition , the integration of which gives . With the aid of (64), (52), and (29) and through some manipulations the above equation leads to (30). This ends the proof of Theorem 1.

Theorem 2. The solutions of are represented by

Proof. From (63), (52), and (65) the history of can be obtained. Thus, from (58) and (59) the histories of and can be calculated, respectively, whereas through (26) and (55) the history of can be evaluated as follows:

Inserting (52) for into the above equation we can derive (67). The last two components and of can be obtained explicitly via (21), (67), and (28), from which we can derive (68) and (69). This ends the proof of Theorem 2.

From (67)–(69) it is obvious that ; that is, , where means a three-dimensional sphere with a constant radius . Therefore, the above mappings belong to , which preserves the invariant length of . The above result is significant upon recalling the number of parameters used in the Euler's angels is three and the governing equations are three nonlinear ODEs. Here, we only need a single parameter and a nonlinear ODE in (66).

4.2. The Choice of v

If we choose

we have by (63). In (29) we expand the two-dimensional vector in the plane by two oblique coordinates and , which are subjected to the constraint that they are not parallel; that is, ; otherwise, the denominators in (30) are zero, which would lead to an undefined differential equation for . In the whole process should hold. Upon , where is a given constant, at some time instant, say , the numerical integration process is restarted with new values of and given by

and at the same time we set , , and .

4.3. The Computation of Q

First we calculate by the above method. Select three independent initial values of ; for example,

The corresponding solutions are denoted by

and from (19) we obtain

Then, inserting the above equation for and (12) for into (14) we can obtain :

4.4. Numerical Tests

In order to give a criterion to assess our numerical method we first derive a closed-form solution of in the appendix under the angular velocities , , and , where and are parameters of angular frequencies.

We calculate by the above single-parameter method with . The initial value is and we fix , . We apply the Euler method to integrate (30) by using a stepsize . The errors between exact solutions and numerical solutions are plotted in Figure 1, whose maximum errors are , , and , respectively.

(a)

(b)

(c)

Figure 1: 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 2 up to , where and were used. We apply the Euler method to integrate (30) by using a stepsize and . Then the method in Section 4.3 is used to compute , whose componential errors are shown in Figures 2(a)–2(i), where the maximum error is smaller than . The error of orthogonality is defined as with calculated by the numerical method. From Figure 2(j) it can be seen that the present numerical method can preserve orthogonality almost exactly.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

Figure 2: Component-wise errors of rotation matrix (a)–(i) and error of orthogonality (j) produced by the single-parameter method.

5. Conclusions

Upon comparing with some different representations of the rotation group , including the Euler's angles representation, the Rodrigues parameters representation, and the modified Rodrigues parameters representation, we succeeded to develop a simpler mathematical procedure to find an analytical solution of through a single parameter, where we just need to solve a single nonlinear ODE. To interpret the results of the integration, the time evolution of orientation is presented as a curve with a single parameter in the topological space . The new local coordinate is a globally defined nonsingular parameterization of rotations suitable for general solutions of large rotations.

Appendix

For example, taking , , and in (6) we have

We attempt to compare the analytic solution constructed by the algorithm developed in the context to the closed-form solution given in the following. For the input (A.1) we have

where the superscript in denotes the third-order derivative of with respect to time. The solution of above equation is found to be

where

Taking the differential of (22) and using (23), and thus substituting (A.1) and (A.2) into those results and noting that (21), we obtain

Solution of the above two equations for and renders

Finally, substituting (A.3) and its differentials into the above equations we obtain

In the form of (19) the components of can be written as follows:

Conflict of Interests

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

Acknowledgments

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.

References

M. D. Shuster, “A survey of attitude representations,” Journal of the Astronautical Sciences, vol. 41, no. 4, pp. 439–517, 1993.

S. N. Atluri and A. Cazzani, “Rotations in computational solid mechanics,” Archives of Computational Methods in Engineering. State of the Art Reviews, vol. 2, no. 1, pp. 49–138, 1995.

O. A. Bauchau and L. Trainelli, “The vectorial parameterization of rotation,” Nonlinear Dynamics. An International Journal of Nonlinear Dynamics and Chaos in Engineering Systems, vol. 32, no. 1, pp. 71–92, 2003.

R. Rubinstein and S. N. Atluri, “Objectivity of incremental constitutive relations over finite time steps in computational finite deformation analyses,” Computer Methods in Applied Mechanics and Engineering, vol. 36, no. 3, pp. 277–290, 1983.

H. Goldstein, Classical Mechanics, Addison-Wesley, Reading, Mass, USA, 2nd edition, 1980.

A. Ibrahimbegovic, “On the choice of finite rotation parameters,” Computer Methods in Applied Mechanics and Engineering, vol. 149, no. 1–4, pp. 49–71, 1997.

M. Borri, L. Trainelli, and C. L. Bottasso, “On representations and parameterizations of motion,” Multibody System Dynamics, vol. 4, no. 2-3, pp. 129–193, 2000.

C.-S. Liu, “The computations of large rotation through an index two nilpotent matrix,” Computer Modeling in Engineering & Sciences, vol. 16, no. 3, pp. 157–175, 2006.

Y. Choquet-Bruhat, C. DeWitt-Morette, and M. Dillard-Bleick, Analysis, Manifolds and Physics, North-Holland, Amsterdam, The Netherlands, 1982.

R. S. Millman and G. D. Parker, Elements of Differential Geometry, Prentice-Hall, Englewood Cliffs, NJ, USA, 1977.