Minimum-Parameter Representations of N-Dimensional Principal Rotations

Minimum-Parameter Representations of N-Dimensional Principal Rotations Andrew J. Sinclair and John E. Hurtado Department of Aerospace Engineering, Texas A&M University, College Station, Texas, USA

Abstract Classic techniques have been established to characterize SO(N) using the N-dimensional Euler’s theorem and the Cayley transform. These techniques provide separate descriptions of N-dimensional orientation in terms of the constituent principal rotations or a minimum-parameter representation. The two descriptions can be linked by the canonical form of the extended Rodrigues parameters. This form is developed into a minimum-parameter representation that directly links to the principal rotations. The new representation can be solved analytically for N = 3 and numerically for higher dimensions. The new parameters represent a projection of the principal rotations onto the planes formed by the body coordinates.

Introduction In aerospace engineering the attitude of rigid bodies is described using Euler’s theorem and the Cayley transform. Euler’s theorem describes any given orientation in terms of a principal rotation. The Cayley transform provides a minimum-parameter representation of an orientation. Of course, the relationship between these two descriptions is well known, and an intuitive understanding of this link has been an important tool in the development of attitude determination and control techniques. The attitude of three-dimensional bodies, however, is a subset of N-dimensional isometries. This paper deals with proper linear isometries in ℜN as represented by the group of special (or proper) orthogonal matrices, SO(N). Euler’s theorem and the Cayley transform have both been generalized to describe SO(N). The generalization of the Cayley transform was first performed by Cayley himself [1], and the generalization of Euler’s theorem was developed by Schoute [2]. Recent treatments of these topics have been written by Bar-Itzhack[3], Bar-Itzhack and Markley[4], Mortari[5], and Bauer[6]. The relationship between the principal-rotation and Cayley-transform descriptions of three-dimensional orientations has been an important tool in the study of spacecraft attitude dynamics, control, and estimation. Recent studies have developed a representation of general mechanical-system dynamics as N-dimensional rotational motions based on the Cayley transform[7, 8, 9]. This motivates a desire for an improved understanding of the relationship between principal-rotation and Cayley-transform descriptions of SO(N). These concepts and N-dimensional isometries, in general,

have been extensively studied from the perspective of N-dimensional Euclidean geometry. The focus of this paper is to develop this relationship from an engineering perspective, rather than to add to the rigorous developments that have been achieved in N-dimensional Euclidian geometry. In the first section of the paper the descriptions of SO(N) using Euler’s theorem and the Cayley transform are reviewed and compared to the familiar concepts in spacecraft attitude. In the second section a new minimal parametrization of SO(N) is proposed that is directly related to the principal rotations. Two numerical examples are also provided for representative values of N.

Review of N-dimensional orientations Rotations in higher dimensions have important differences from rotations in threedimensions. For even-dimensioned spaces the dimensions are fully utilized in holding principal planes, and no principal axis exists. For spaces with odd dimension a principal axis does exist, but unlike the three-dimensional case, it will have more than one plane orthogonal to it. The even-dimensioned subspace orthogonal to the principal axis will hold several planes, and the rotation on each must be given to specify a particular orientation. The mathematical representation of this principal-rotation description, which forms the N-dimensional Euler’s theorem, comes from the eigenanalysis of the N-dimensional orientation matrix and was discussed by Mortari [5]. The transformation of an N-dimensional position vector, p , due to a rotation is given by C , a proper orthogonal matrix [10]. p′ = C p (1) The eigenvalues of a proper orthogonal matrix lie on the unit circle in the complex plane and are conjugate pairs. If the matrix is odd dimensioned, then the “left over” eigenvalue will be equal to 1 + i0. The eigenvector associated with this eigenvalue is the principal axis of the rotation and is the only vector untransformed by the rotation. th The eigenvectors associated with the ´ k conjugate pair of eigenvalues are themselves √ ³ (k) (k) a conjugate pair, 22 pˆ 1 ± i pˆ 2 . The normalized, real and imaginary parts of (k)

(k)

the pair, pˆ 1 and pˆ 2 , are orthogonal vectors that form a principal plane. The kth conjugate pair of eigenvalues are related to the angle of rotation in this plane. (C)

λk = cos φk ± i sin φk

(2)

Another characterization of N-dimensional orientation matrices is provided by the Cayley transform. C = (II − Q ) (II + Q )−1 = (II + Q )−1 (II − Q )

(3)

C ) (II +C C )−1 = (II +C C )−1 (II −C C) Q = (II −C

(4)

The upper-triangular elements of the skew-symmetric matrix Q are a minimumparameter representation of C and thus form an orientation representation. The number of independent elements in an N × N orthogonal matrix and the minimum number of parameters required to describe an N-dimensional orientation is M = N(N − 1)/2. In particular, for N = 3 the elements of Q are the Rodrigues parameters. For higher dimensions the elements of Q are referred to as the extended Rodrigues parameters (ERP’s)[4]. The Euler’s theorem and the Cayley-transform descriptions can be somewhat linked by comparing the eigenvalues and eigenvectors of C and Q . These matrices have the same eigenvectors, and their eigenvalues are related as shown below [5]. λ(C) =

1 − λ(Q) 1 + λ(Q)

;

λ(Q) =

1 − λ(C) 1 + λ(C)

(5)

This implies the following relationship between the eigenvalues of Q and the rotation angles. µ ¶ φk (Q) λk = ∓i tan (6) 2 Equation (5) shows that for odd N, the eigenvalue of C associated with the principal axis becomes a zero eigenvalue of Q . For N = 3 the relationship between the Rodrigues parameters and the principal rotation extends beyond the eigenvalues and eigenvectors of Q . This connection, however, makes intrinsic use of the special properties of N = 3 such as plane-vector equivalency and N = M. In the following section a canonical representation of the ERP’s will be considered that is directly related to the principal rotations. The canonical representation of a skew-symmetric matrix decomposes the matrix into a proper orthogonal matrix and a block-diagonal skew-symmetric matrix [6, 11]. The canonical representation of Q in terms of a proper orthogonal matrix R and a blockdiagonal skew-symmetric matrix Q ′ will be considered. Q′ = RQRT

;

Q = RT Q′ R

(7)

The elements of this new skew-symmetric matrix Q ′ are referred to as the “canonical ERP’s”. The similarity transformation enforces that Q and Q ′ share the same eigenvalues and their eigenvectors are related through R . In general the following form is chosen for Q ′ for even N.  0 Q′12 · · · 0 ′  −Q12 0 · · · 0 ¤  £ ′ .. .. ..  .. Q N even =  . . . .   0 0 ··· 0 0 0 · · · −Q′N−1,N

0 0 .. . Q′N−1,N 0

      

(8)

For odd N the form is similar with an appended row and column of zeros. Equation (7) implies the following interpretation for a general set of ERP’s, Q , which represent the orientation of a coordinate system with coordinate vectors {bb1 , . . . , b N } (i.e., a body frame) that is rotated relative to the reference coordinate system with coordinate vectors {ii1 , i 2 , . . . , i N } by the rotation matrix C . For any set of ERP’s there exists another coordinate system with coordinate vectors {cc1 , c 2 , . . . , c N } in which the principal rotation planes are aligned with the planes formed by these vectors. Because of this alignment these vectors are called principal coordinate vectors and compose a principal frame. The rotation viewed in this frame results in a matrix Q ′ of the blockdiagonal form. This explicitly decomposes the N-dimensional orientation problem into its constituent two-dimensional rotations. The mapping from the b i frame to the c i frame is given by R . For even N, Eq. (7) is rewritten as follows. Qi j even = Q′12 (R1i R2 j −R2i R1 j )+. . .+Q′N−1,N (RN−1,i RN, j −RN,i RN, j )

(9)

A similar expression is obtained for the odd N case. The component Rmi represents the projection of the mth principal-coordinate vector (ccm ) into the ith body vector (bbi ). Therefore the product Rmi Rn j can be considered as the projection of the (m, n) principal plane into the (i, j) body plane. Equation (9) shows that the (i, j) component of Q represents the projection of each of the principal rotations into the (i, j) body plane. This constitutes a physical interpretation because, while the entire N-dimensional space cannot be physically visualized, each principal rotation is simply a twodimensional rotation and is physically intuitive. It is a well known property of the Rodrigues parameters that a singularity is encountered if the magnitude of the principal rotation is equal to π rad. Equations (6) and (9) show that the ERP’s have a similar condition. The ERP’s encounter a singularity as the magnitudes of any of the principal angles approach π rad because one or more elements of Q ′ → ∞.

Minimal representations of principal rotations The canonical ERP set, Q ′ , has N/2 independent elements for the even case and (N − 1)/2 independent elements for the odd case. Additionally, due to the orthogonality constraints the N × N matrix R contains M independent elements. The general ERP set Q , however, has only a total of M independent elements. This implies that infinitely many values of R will perform the mapping shown in Eq. (7). A particular R can be considered by again applying the Cayley transform. R = (II − S) (II + S)−1 = (II + S)−1 (II − S)

(10)

S = (II − R ) (II + R )−1 = (II + R )−1 (II − R )

(11)

The Cayley transform defines a skew-symmetric matrix S , which is an M parameter representation of R . This representation thus describes the orientation of the c i frame relative to the b i frame. In this form the elements of S that lie in the principal planes defined by Q ′ are clearly arbitrary. The purpose of R is to describe the orientation of the b i frame with respect to the principal planes and axis, if one exists. The rotation in these planes is specified by Q ′ . This suggests an ansatz for S that depends on N. 

   [SS N even ] =    −S1,N−1 −S1,N 

0 0 .. .

0 0 .. .

··· ··· .. .

S1,N−1 S2,N−1 .. .

S1,N S2,N .. .

−S2,N−1 −S2,N

··· ···

0 0

0 0

0 0 .. .

0 0 .. .

··· ··· .. .

S1,N−2 S2,N−2 .. .

S1,N−1 S2,N−1 .. .

      

S1,N S2,N .. .

    [SS N odd ] =   −S1,N−2 −S2,N−2 · · · 0 0 SN−2,N   −S1,N−1 −S2,N−1 · · · 0 0 SN−1,N −S1,N −S2,N · · · −SN−2,N −SN−1,N 0

(12)

        

(13)

The off-diagonal elements that are non-zero in Q′ are arbitrarily set to zero in S , whereas the off-diagonal elements that are zero in Q′ are non-zero in S . The elements of these two matrices therefore combine to form a minimum-parameter (i.e. M) orientation representation in terms of the principal rotations. For any particular value of Q , the associated Q ′ and S can be found by substituting the assumed forms for these matrices (Eqs. (8) and (12) for N even and similar for N odd) into Eq. (7). This is referred to as the “assumed-form” representation of Q . Using Eq. (10) to expand Eq. (7) gives the following. Q ′ = (II − S ) (II + S )−1 Q (II − S )−1 (II + S )

(14)

Symbolically evaluating these equations and setting the appropriate elements of Q ′ to zero provides M − N/2 (for even N) or M − (N − 1)/2 (for odd N) equations for the non-zero elements of S . This process is straightforward but for large values of N the expanded product in Eq. (14) can clearly involve a large number of terms. The equations can be easily developed, however, using a symbolic manipulator such as Maple. Once obtained, the equations are solved for the non-zero elements of S corresponding to any particular value of Q . These elements can then be substituted into Eq. (14) to produce the non-zero elements of Q ′ . It will be seen that for N = 3 the solution for S can be obtained analytically, and preliminary results for higher dimensions indicate that a solution can be obtained numerically for general N. The equations for the elements of S for N = 3 are developed as follows. The Ro-

drigues parameters have the form shown below.  0 Q12 Q] =  −Q12 0 [Q −Q13 −Q23

 Q13 Q23  0

(15)

For the sake of generality in notation, the components of Q are retained in their matrix notation, instead of applying the vector notation specialized for N = 3. For this case Q ′ and S have the following forms.     0 Q′12 0 0 0 S13 £ ′¤ 0 0  ; [SS ] =  0 0 S23  Q =  −Q′12 (16) 0 0 0 −S13 −S23 0 From this form of S the Cayley transform is used to find R .   2 + S2 1 − S13 −2S13 S23 −2S13 23 1 2 − S2   −2S13 S23 R] = 1 + S13 −2S23 [R 23 2 +S2 1+S13 23 2S 2S 1 − S2 − S2 13

23

13

(17)

23

This, of course, is similar to the general expression for a rotation matrix in terms of a Rodrigues parameter set (with one identically-zero parameter). The elements of Q and S are related to the elements of Q ′ , and Eqs. (15) and (17) are substituted into Eq. (7). This product can be expanded to produce the elements of Q ′ . Setting the (1, 3) and (2, 3) elements of this matrix to zero yields the following two equations for the two unknown elements of S . 2 2 2Q12 S23 + Q13 S13 + 2Q23 S23 S13 − Q13 S23 + Q13 = 0

(18)

2 2 2Q12 S13 + Q23 S13 − 2Q13 S23 S13 − Q23 S23 − Q23 = 0

(19)

For this case of N = 3 the equations can be solved analytically. First for the situation Q12 = Q13 = Q23 = 0 infinitely many solutions exist: S13 ∈ ℜ and S23 ∈ ℜ. For this case there is no rotation, and any plane can be considered the principal plane. Next, for the situation Q12 6= 0 and Q13 = Q23 = 0 the equations admit a unique solution: S13 = S23 = 0. For this case the rotation is in the (1, 2) body plane, and the principal frame is aligned with the principal plane by definition. Two additional special cases can be solved directly³from Eqs. q (18) and´(19). For Q13 = 0 and Q23 6= 0 two real solutions exist: S13 = −Q12 ± Q212 +Q223 /Q23 and S23 = ³0. For q Q13 6= 0 and´Q23 = 0 there are also two real solutions: S13 = 0 and S23 = Q12 ±

Q212 + Q213 /Q13 .

The remaining general case of Q13 6= 0 and Q23 6= 0 can be solved by computing the Gr¨obner basis of Eqs. (18) and (19)[12]. The result of this computation with stronger

weight on S13 is a factorable, fourth-order polynomial in S23 and a second polynomial linear in S13 . ¤ ¢ 2 £¡ 2 − 2Q12 Q13 S23 − Q213 Q13 + Q223 S23 ¤ ¢ 2 £¡ (20) − 2Q12 Q13 S23 + Q212 + Q223 = 0 × Q213 + Q223 S23 ¢2 3 ¢ ¡ Q313 Q23 + Q13 Q323 + Q212 Q13 Q23 S13 + Q12 Q313 + Q213 + Q223 S23 ¢ ¢ ¢ 2 ¡ 2 ¡ 2 ¡ + Q12 2Q13 + Q223 − Q413 + Q423 S23 = 0 −3Q12 Q13 Q213 + Q223 S23 ¡

The real solutions of Eq. (20) are shown below. q Q12 Q13 ± Q13 Q212 + Q213 + Q223 S23 = Q213 + Q223

(21)

(22)

Associated with each real solution of S23 there is a unique solution for S13 from Eq. (21). Therefore Eqs. (18) and (19) have, in general, two real solutions with two exceptions for which there exists either a unique solution or infinitely many solutions. Numeric example for N = 4 The following numerical example will be considered for N = 4 [5].  0.1003 0.2496 −0.8894 −0.3697  0.9593 −0.0238 −0.0153 0.2810 C] =  [C  −0.1172 −0.8638 −0.3828 0.3059 −0.2366 0.4370 −0.2495 0.8311

   

(23)

From this rotation matrix a set of ERP’s is computed using the Cayley transform. Equations for the non-zero elements of S are then generated by setting the (1, 3), (1, 4), (2, 3), and (2, 4) elements of Eq. (14) to zero. However, the resulting equations are fairly extended and are not shown here. The equations, though, can be solved numerically and lead to the results shown below.   0 0 1.1900 −0.1273  0 0 1.0787 0.0854   (24) [SS ] =    −1.1900 −1.0787 0 0 0.1273 −0.0854 0 0 The matrix S is a set of ERP’s that relate the body axes to a principal frame. The associated rotation matrix, R is computed using the Cayley transform. The elements

of Q ′ are then computed from R and Q using Eq. (7).  0 −2.4396 0 £ ′ ¤  2.4396 0 0 Q =  0 0 0 0 0 −0.1132

 0  0  0.1132  0

(25)

From these canonical ERP’s the principal rotation angles are computed. φ1 = 2.3636 rad

φ2 = −0.2254 rad

;

(26)

Numeric example for N = 5 Finally, the assumed-form procedure can be used for an N = 5 example.  −0.5708 −0.2224 0.4317 −0.2972 −0.5917  0.6799 −0.6616 0.1856 −0.1815 −0.1806  C] =  0.6000 −0.0183 0.0554 −0.6758 [C  0.4241  −0.0505 −0.0280 −0.6987 −0.7067 −0.0955 −0.1719 −0.3899 −0.5392 0.6134 −0.3892

     

(27)

From this rotation matrix a set of ERP’s is again computed using the Cayley transform. Equations for the non-zero elements of S are found by setting the appropriate elements of Eq. (14) to zero. Again, the resulting equations are fairly extended and are not shown here. Numerical solution, however, produces the following results for S.   0 0 −0.3111 0.1449 −0.4123  0 0 0.0958 0.3897 −0.2318     0 0 −0.3816  [SS ] =  0.3111 −0.0958 (28)   −0.1449 −0.3897 0 0 0.0551  0.4123 0.2318 0.3816 −0.0551 0 The rotation matrix, R , associated with S is computed using the Cayley transform. Because the fifth axis of the principal frame was chosen to be aligned with the principal axis, the fifth column of R T (the fifth row of R ) is equal to the principal axis in body coordinates. From R and Q the elements of Q ′ can once again be computed using Eq. (7).   0 4.0839 0 0 0  0 0 0 0   £ ′ ¤  −4.0839  0 0 0 −2.8919 0 (29) Q =    0 0 2.8919 0 0  0 0 0 0 0 These canonical ERP’s give the following principal rotation angles. φ1 = −2.6613 rad

;

φ2 = 2.4758 rad

(30)

Discussion Historically, N-dimensional orientations have been described separately using principal rotation descriptions and minimum-parameter descriptions. The two concepts are linked by the canonical form of skew-symmetric matrices. This by itself, though, does not form a minimum-parameter orientation representation. The independent elements of Q ′ and S , however, do constitute an N-dimensional minimum-parameter orientation representation in terms of the principal rotations. These elements can be mapped to and from the principal rotations and a unique pair of Q and C . Perhaps most usefully, these matrices provide an interpretation for the ERP elements. Because these matrices form an orientation representation it should be possible to develop kinematic equations to directly relate their derivatives to the N-dimensional angular velocity. A result as elegant as the Cayley transform kinematic relations for the ERP rates, however, is not anticipated, and the idea is not pursued further in this paper. The evaluation that has been presented is similar to the eigenanalysis of C because both methods produce the principal angles, principal planes, and principal axis (if it exists) of any arbitrary orientation. The key difference between these two approaches comes in the representation of the principal planes. An important aspect of the canonical ERP’s is that they provide a principal frame in which the various principal rotations become geometrically decoupled. This is a useful tool for describing N-dimensional orientations and relating them to the more familiar two and three-dimensional rotations. In particular this could be applied in attempting to study the attitude dynamics and control of N-dimensional rigid bodies.

Acknowledgments The authors thank Robert Bauer, Jackie Schandua, and Michael Swanzy for their helpful comments and suggestions.

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