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

In many applications we need to solve an orthogonal transformation tensorQ ∈ SO(3) from a tensorial equation Q̇ =WQ under a given spin historyW. 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. 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).


Introduction
Among many classical Lie groups, the three-dimensional rotation group (3) is the most widely used one.For its numerous engineering applications the development of simpler algorithms to calculate (3) 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 Q, connecting the spin-free and the fixed configurations due to the nonzero spin tensor denoted by W, satisfies the following tensorial differential equation: It does not lose any generality to assume that the initial condition of Q is an identity; that is, Q(0) = I 3 .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 Q 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  1 ,  2 , and  3 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 Q ∈ (3) 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 R 3 by a two-fold covering; see, for example, Goldstein [5].In the above, ‖q‖ denotes the Euclidean norm of q ∈ R 3 .Liu [8] has presented a four-dimensional Lie-algebra representation of the quaternion formulation of (3).
It is known that (2) is diffeomorphic to the threedimensional sphere S 2 and (3) is diffeomorphic to the quotient space of the three-dimensional sphere by the antipodal equivalence, hence diffeomorphic to the threedimensional projective space [9].According to [10] we can define the real projective space as follows.Let R  be the set of all straight lines through the origin in R  .a and b ∈ R  represent the same line if and only if there is a nonzero constant  ∈ R such that a = b, which constitutes an equivalence class denoted by In this paper, a simpler solution method of Q 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 R 3 .The local coordinate is an example of a globally defined nonsingular parameterization of rotations, which is suitable for treating the computations of large rotations.

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 twodimensional (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 Q.Since Q is orthogonal, it belongs to the special orthogonal group with dimensions three; that is, Q ∈  (3).Although (1) can be defined by nine simultaneous ordinary differential equations (ODEs), only three of them are independent.In geometry, Q, an element of (3), 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 F which applies to W and has the following form: where () := ∫  0 ‖()‖.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 Q can be decomposed into with Q 1 an unknown matrix belonging to  (3).Substituting it into (1) leads to where is a skew-symmetric matrix with The decomposition in (14) leads to a simpler spin matrix A for Q 1 in (16), with only two independent inputs u 1 and u 2 .For a given angular velocity ( 1 (),  2 (),  3 ()), it is easy to find the matrix F by (13) and (12).However, in order to obtain Q we still require to find Q 1 .In this paper, an analytic procedure will be developed to solve this problem for arbitrary inputs u 1 and u 2 generated from the angular velocity ( 1 (),  2 (),  3 ()).

A Projective Transformation
The system of ODEs deduced from A in (16) can be written as The initial values of  0 ,  1 , and  2 are assumed to be  0 (0),  1 (0), and  2 (0), respectively.So the determination of Q 1 () is now equivalent to searching a general solution of (18); that is, where Q 1 (0) = I 3 , X 0 = X(0), and Let be the homogeneous coordinates of R 3 .Then, the use of ( 18) implies where are the output and input of ( 22) and ( 23), respectively.The inner product of (23) with x and the use of ( 22) render Integrating (25) leads to By (21), it is equivalent to ‖X()‖ = ‖X 0 ‖; that is, the length of the vector X is preserved under the action of (3) group.
Obviously, X 0 cannot be a zero vector; otherwise, X() will be a zero vector for all  > 0.
By eliminating  0 , ( 22) and ( 23) can be combined into a nonlinear differential equations system for x: The transformation made in this section projects the threedimensional vector ( 0 ,  1 ,  2 ) T ∈ S 2 ‖X 0 ‖ , where S 2 ‖X 0 ‖ means a three-dimensional sphere with a constant radius ‖X 0 ‖, into a two-dimensional vector ( 1 ,  2 ) T in the topological space R 3 , which is correlated intimately with the two independent inputs of u .

Two Theorems.
In this section we are going to prove two main theorems.
Theorem 1.The solution of x governed by (27) with an initial condition x(0) = x 0 can be explicitly expressed in terms of a single variable : where v is a constant unit vector (with ‖ v ‖ = 1), and  is governed by a nonlinear ODE: under the initial condition (0) = 0.
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 v is a constant vector with norm ‖ v ‖ = 1 to be given, and the vector ẇ and the other two scalars  and  are allowed to be time-varying.We will determine v , ẇ, , and  below, under the assumptions  ̸ =  and  ̸ = 1.Substituting (31) and its integral into (23) we can obtain where v() :=  v .Upon defining equation ( 33) becomes the solution of which is where w 0 = w(0) = x 0 .The last term can be integrated by parts, leading to where is a time function.Under the conditions  ̸ = 1 and  ̸ = ,  is a well-defined nonzero function.It is remarkable that (37) expresses w in terms of a constant unit vector v .
(B) Governing Equations of w and .From ( 25) and (34) it follows that The inner product of w + v 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  > 0 is a time function to be determined; hence, (43) becomes Equations ( 35) and (45) are composed as the governing equations system for (w, ), with v being the input.
(C) Explicit Form of .Noting (37), (45) changes to Define and the relation between   and  is one-to-one, since  > 0. 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 (0) = 1 and   (0) = −x 0 ⋅ k are imposed.It is interesting that we have a closed-form solution of  in terms of .
(D) Explicit Form of x.Now, substituting (52) into (37) and integrating the resultant, the explicit form of x can be obtained as follows: where If  can be solved, the solution of x is obtained.Defining the vector z as given in (29) and substituting (52) for  into the above equation, we obtain (28).The square norm of x 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 x above.The issue to find  is very important as being given below.Using (35), (38), and (53) we have Substituting (56) for ẇ 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 ẏ =   ż being substituted into (57), renders In component form we have The above two equations can be used to solve  and ż .Eliminating ż 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 u 1 and u 2 .Multiplying (62) by  1 and (61) by  1 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 (0) = 0, 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 X are represented by

Numerical Tests.
In order to give a criterion to assess our numerical method we first derive a closed-form solution of Q 1 in the appendix under the angular velocities  1 = Ω − ,  2 = − sin Ω, and  3 = cos Ω, where Ω and  are parameters of angular frequencies.

Conclusions
Upon comparing with some different representations of the rotation group (3), 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 Q 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 R 3 .The new local coordinate is a globally defined nonsingular parameterization of rotations suitable for general solutions of large rotations.
[a] = {b | b ∼ a} with b ∼ a meaning that a = b for some  ̸ = 0. Note that R  = (R  − {0})/∼.The coordinates of any a ∈ R  such that b = [a] are called homogeneous coordinates for b.

Figure 1 :
Figure 1: Error between the exact and the numerical solution provided by the single-parameter method.