Attitude and Orbit Error in n-Dimensional Spaces ∗ Introduction

AAS 05-468

Attitude and Orbit Error in n-Dimensional Spaces

∗

Daniele Mortari† , Sante R. Scuro,‡ , and Christian Bruccoleri,§ Texas A&M University, College Station, Texas 77843-3141 Abstract This paper focuses the attention on theoretical aspects associated with the definition of the error for Special Orthogonal and for General Linear transformations, in any dimensional space, Real or Complex, and in both Euclidean and Riemannian spaces. In particular, the paper shows that the orbit error can be described by a complex number whose phase represents the orbit orientation error while the modulus describes the orbit shape error. This paper also shows that the angle between two quaternion has a specific geometrical meaning. In particular, the QR decomposition allows us to see a General Linear transformation as made of two subsequent effects: a dilation and a rigid rotation. This decomposition has been applied to the Lorentz transformations by highlighting the relativity effects of length change and coordinate axes bending.

Introduction The error in position can be easily described by the distance between two vectors, which can be intended, for instance, as an estimated and the true position, respectively. Unfortunately, the error on rotation (or orientation or attitude) cannot be so easily represented. The problem consists to define an equivalent distance between two orientations, that is, between two different proper orthogonal matrices (see Ref. [7]). To this end, let us indicate with T and C the true and the estimated orientations, respectively. Matrix C can even be randomly chosen, provided that the conditions C T C = CC T = I, and det(C) = +1, which assure C to be a rotation matrix, are satisfied. Let ∆ = T C T . Since ∆ is a product of rotation matrices, ∆ is a rotation matrix too. This matrix represents the corrective rotation that, applied to our estimated attitude C, changes/corrects the estimation C to the true attitude T , and viceversa. In fact, ∆ C = T C TC = T

and

∆T T = C T T T = C

(1)

Since ∆ is a rotation matrix it has it own principal axis ˆ e and principal angle εmax (see Ref. [12]). Since ˆ e is the principal axis of the corrective matrix ∆, its direction does not change during the correction and, therefore, its direction is error-free (∆ ˆ e=ˆ e), independently of the choice of the C matrix. ∗ Paper AAS 05-468 of the “Malcolm D. Shuster” Astrodynamics Symposium, June 13–15, 2005, Grand Island, Buffalo/Niagara Falls. † Associate Professor, Department of Aerospace Engineering, Texas A&M University, 741A H.R. Bright Building, 3141 TAMU, College Station, TX 77843-3141. Tel. (979) 845-0734, Fax (979) 845-6051. Email: [email protected] ‡ Research Assistant, George P. & Cynthia W. Mitchell Institute for Fundamental Physics, Texas A&M University, College Station, TX 77843-4242. Email: [email protected] § Research Assistant, Department of Aerospace Engineering, Texas A&M University, 620C H.R. Bright Building, 3141 TAMU, College Station, TX 77843-3141. Tel. (979) 458-0550. Email: [email protected]

1

Now let ˆ r be an arbitrary direction expressed in the IRF, that it is considered error-free. This means ˆ which represents the true direction that should be measured by an that the associated direction b, ideal attitude sensor on-board the spacecraft, is ˆ=Tˆ b r

(2)

Now, in the real noisy case, the true attitude matrix T will never be know or, better, it can just be estimated with a given, limited, approximation. The accuracy of this approximation depends on the accuracy precision of the attitude sensors and on their spatial distribution. The choice of the attitude determination algorithm is, therefore, important in terms of the way it is capable to filter the noise. Therefore, different filtering technique will results in different attitude estimation accuracies. This is not true, however, when you want to compare different single point algorithms fully complying with Wahba’s optimality criterion, such as, for instance, the q ˆ-Method [7], ESOQ-1 [6], ESOQ-2 [8], the SVD algorithm [3], FOAM [4], QUEST [13], and EAA [5], just to mention the most used and known. The estimated direction cosine matrix C, therefore, always differs from T . Due to this difference, ˆ m , where the errorfree direction ˆ r is expected to be observed, in the BRF, as the unit-vector b ˆm = C ˆ ˆ b r
= b

(3)

ˆm = T T b ˆ ˆ r = CTb

(4)

ˆ = T CTb ˆm = ∆ b ˆm b

(5)

Equations (2) and (3) allow us to write

and, consequently, This equation shows the important property of the corrective attitude matrix ∆ of being the operator correcting any estimated on board direction. Therefore, the matrix ∆ contains all the information on the direction error distribution that identifies the rotation error. Let us to summarize the details of the rotational correction mechanism given by Eq. (5). To this ˆm = C ˆ ˆ = Tˆ ˆ m , and b r. end, let us consider the isosceles spherical triangle of vertex ˆ e, b r = ∆b ˆ The unit-vector b represents any given direction (defined in BRF) displaced from ˆ e by the angle ˆ = cos θ). The effect of the correction ∆ is a pure rotation about the axis ˆ e and by the θ (ˆ eT b angle εmax (where ˆ e and εmax are the principal axis and angle of ∆). This corrective rotation brings ˆm to its correct direction b ˆ = ∆b ˆm . Note that, the rigid the error-affected measured-direction b ˆ ˆ e by the same angle θ. This corrective rotation of ∆ implies that b and bm are displaced from ˆ ˆ=b ˆ Tm ∆b ˆm is affected by the error ε, where cos ε = b ˆ Tm b ˆm . implies that the direction identified by b ˆ and b ˆm , allows us Now, spherical trigonometry applied to the spherical triangle with vertices ˆ e, b, to write the relationship (6) cos ε = (1 − cos εmax ) cos2 θ + cos εmax where ˆm = ˆ ˆ=ˆ ˆm ˆm = ˆ eT ∆b cos θ = ˆ eT b eT b eT T C T b

(7)

ˆ=b ˆ Tm ∆b ˆm ˆ Tm b ˆm = b ˆ Tm T C T b cos ε = b

(8)

and

Equation (6), which provides the attitude error distribution or the geometry of the rotation error, establishes that it exists a maximum value for the error associated with a given direction (εmax ). π This error is reached by the directions orthogonal to ˆ e, that is, those displaced from ˆ e by θ = , 2 while ε = 0 can be obtained for εmax = 0, independently from the value of θ (all directions, in this e, case, are error free). When εmax
= 0, then we have ε = 0 for the two directions aligned with ˆ 2

that is, for θ = 0 and for θ = π, regardless the value of C. This is an important result, because it implies that, if you consider a spacecraft that has the orientation described by the matrix C (with C any proper orthogonal matrix), while T is an attitude which allows a correct pointing of an on-board telescope, then in the lucky case that the mounting angle of our telescope coincides with the principal axis of ∆, then the telescope would point correctly! Another important consequence of Eq. (6) is that it is not possible to orient a spacecraft with all the onboard directions pointing to wrong directions. Two directions, and only two directions (those aligned with the principal axis of the corrective attitude matrix), are always error-free, no matter how accurate C is for T .

Rotation Error in 3-Dimensions The principal angle εmax of the corrective attitude matrix ∆ represents, therefore, the minimum angular distance between two attitude matrices (that is, the attitude error if you consider estimated and true attitude matrices). This angle can easily be evaluated from the propriety that the trace operator is invariant with respect to similar transformations. This means that the trace of a given matrix is the trace of its eigenvalue matrix. When the matrix is not defective, as in our problem, then the trace is equal the sum of the eigenvalues. In particular, for the 3 × 3 rotation matrix ∆, we have tr(∆) = (cos εmax + i sin εmax ) + (cos εmax − i sin εmax ) + 1 = 1 + 2 cos εmax

(9)

Equation (9) provides an easy way to evaluate the Rotation Error cos εmax =

tr(T C T ) − 1 tr(∆) − 1 = 2 2

(10)

that is representing the Maximum Direction Error. Let us now consider two arbitrarily attitude matrices, C (1) = [ x ˆ1 y ˆ1 ˆ z1 ]T and C (2) = [ x ˆ2 y ˆ2 ˆ z2 ]T . The angular distance between these two matrices can easily be evaluated by the following important relationship 1 T x x cos εmax = (ˆ ˆ2 + y ˆ1T y ˆ2 + ˆ zT1 ˆ z2 − 1) (11) 2 1 that can be written as 1 cos εmax = (cos ϑx + cos ϑy + cos ϑz − 1) (12) 2 where ϑx , ϑy , and ϑz , are the angular displacements between the corresponding attitude axes. Equations (11) and (12) can be demonstrated by evaluating the diagonal elements of the corrective attitude matrix ⎤ ⎡ (2) ⎤ ⎡ (1) (1) (1) (2) (2) C11 C21 C31 C11 C12 C13 ⎢ (1) (1) (1) ⎥ ⎢ (2) (2) (2) ⎥ (13) ∆ = ⎣ C21 C22 C23 ⎦ ⎣ C12 C22 C32 ⎦ (1)

C31

(1)

C32

(1)

(2)

C33

C13

(2)

C23

(2)

C33

and then summing them in order to obtain the trace of ∆ (1)

(2)

(1)

(2)

(1)

(2)

tr(∆) = +C11 C11 + C12 C12 + C13 C13 + (1) (2) (1) (2) (1) (2) +C21 C21 + C22 C22 + C23 C23 + (1) (2) (1) (2) (1) (2) +C31 C31 + C32 C32 + C33 C33

(14)

Note that Eq. (14) is just the mathematical propriety of the trace operator T

tr(A B ) =

n  n  i=1 j=1

3

aij bij

(15)

Error with the Euler-Rodrigues Symmetric Parameters Any direction cosine matrix C can be expressed in term of the Euler-Rodrigues symmetric parameters, by mean of ˜v C(ˆ q) = ( q42 − qTv qv ) I + 2 qv qTv − 2 q4 q (16) that we re-write here for simplicity ⎡ 2 q1 − q22 − q32 + q42 ⎣ 2(q1 q2 − q3 q4 ) C(ˆ q) = 2(q1 q3 + q2 q4 )

2(q1 q2 + q3 q4 ) −q12 + q22 − q32 + q42 2(q2 q3 − q1 q4 )

⎤ 2(q1 q3 − q2 q4 ) ⎦ 2(q2 q3 + q1 q4 ) 2 2 2 2 −q1 − q2 + q3 + q4

(17)

Substituting in Eq. (10) the explicit expressions of C and T in term of the associated two Eulerˆ2 , then the attitude error εmax (after some manipulation) has the simple Rodrigues vectors, q ˆ1 and q exact expression qT1 q ˆ2 )2 − 1 (18) cos εmax = 2 (ˆ From Eq. (18) we can derive two important consequences: 1. the angle between two Euler-Rodrigues symmetric parameters, q ˆ1 and q ˆ2 , is half of the principal angle εmax of the corrective matrix ∆. In fact, this is straightforwardly derived by comparing Eq. (18) with the trigonometric relationship cos εmax = 2 cos2 (εmax /2) − 1. Thus, we have ε max (19) ˆ2 = cos q ˆT1 q 2 Note the impressive connection between Eq. (19) and

 Φ 2 2 cos Φ = 2 q4 − 1 = 2 cos −1 (20) 2 2. the modulus of the difference between two Euler-Rodrigues symmetric parameters, q ˆ1 and q ˆ2 , has the expression |q ˆ1 − q ˆ2 | = (ˆ q1 − q ˆ2 )T (ˆ q1 − q ˆ2 ) =  2 − 2ˆ qT1 q ˆ2 = εmax (21) = 2 − 2 cos(εmax /2) = 2 sin 4 3. Equation (19) tells us that the attitude variations due to a pure spin is associated with a quaternion rotating on a big circle in 4-D. Therefore, a rigid body spinning with angular velocity ω in 3-D is described by a quaternion rotating with angular velocity ω/2 in 4-D. When q ˆ1 is sufficiently close to q ˆ2 , Eq. (21) can be linearized, giving |q ˆ1 − q ˆ2 | ≈

εmax 2

(22)

This results justifies the use of differences between Euler-Rodrigues quaternions in Kalman filtering. However, we outline here that when the linearization does not hold, then the exact solution for εmax , provided by Eq. (19), must be used.

Orbit Error Let us consider this problem: how to describe the error between two different orbits, for instance, the error between the nominal orbit, characterized by the orbital [at , et , Ωt , ωt , it ] or by the cartesian [rt , vt ] parameters, and the measured orbit, characterized by [am , em , Ωm , ωm , im ] or by [rm , vm ]. 4

The five orbital parameters identifying the orbit in space can be suitably split in two independent sets. One set consisting of the three orbital parameters, Ω, ω, and i, which identify the orbit attitude or orientation, while the remaining two parameters, a and e, identify the orbit shape on the orbit plane. Therefore, the orbit error can be seen as made of two independent errors, the attitude error and the shape error. As for the attitude error, we can introduce the two attitude matrices Ct = R3 (ωt ) R1 (it ) R3 (Ωt )

and

Cm = R3 (ωm ) R1 (im ) R3 (Ωm )

(23)

and, therefore, the associated attitude error between Ct and Cm , can be evaluated as previously T discussed using the corrective attitude matrix ∆ = Ct Cm .

 T −1 tr(Ct Cm ) − 1 (24) δ = cos 2 In the contrary, the shape error can be seen as a planar position error. In fact, the error between √ [at , et ] and [am , em ], can be seen as the error between [at , bt ] and [am , bm ], where b = a 1 − e2 represents the orbit semi-minor axis. By this substitution, the shape error simply becomes the positioning error, which can be quantified by the positioning distance (25) d = (at − am )2 + (bt − bm )2 which, written in term of a and e becomes  d=

a2t (2

−

e2t )

+

a2m (2

−

e2m )



 2 2 − 2at am 1 + (1 − et )(1 − em )

(26)

In conclusion, the orbit error is made of two parts: an attitude error identified by the angle δ and a position error identified by the distance d. Because of this dual nature (angle, distance), a polar representation is proposed to globally describe the orbit error. To this end let us introduce the complex orbit error vector, Eo , defined as Eo (d, δ) = (1 + d ) ei δ = (1 + d ) (cos δ + i sin δ)

(27)

which has modulus (1 + d ) and components (1 + d ) cos δ and i (1 + d ) sin δ, respectively. Figure 1 explains the orbit error.

Figure 1: Orbit Error Complex Representation

Rigid Rotation in n-D Euclidean Spaces In a 3-Dimensional space the matrix performing a rigid rotation has the expression eˆ eT + ˜ ˆ e sin Φ R(ˆ e, Φ) = I3 cos Φ + (1 − cos Φ) ˆ 5

(28)

where ˆ e represents the axis of rotation, and Φ the angle of rotation. In n-Dimensional space this equation cannot be extended as it is. The main reason is that the concept of Orientation (which is called General Rotation in Group theory) and the concept of Simple Rotation, which is characterized by a rotation angle and rotation mechanism, while coincident in 2-D and 3-D, become different in dimensional spaces higher than 3-D (see Ref. [9]). First of all the extension of the Simple Rotation to n-D is performed by characterizing the rotation mechanism that is not performed about something (axis in 3-D, plane in 4-D, volume in 5-D, etc.) but is performed on something, which is the plane of rotation, that is the null space of the axis in 3-D, plane in 4-D, volume in 5-D, etc. Therefore, in order to describe a simple rotations in n-D, the plane of rotation must be introduced. A plane of rotation can be completely identified by introducing, in the n-Dimensional Euclidean ˆ2 . These two unit-vectors identify the plane of rotation space, two orthogonal unit-vectors, p ˆ1 and p while the direction from p ˆ1 to p ˆ2 characterizes the rotation direction. Therefore, the n × 2 matrix . ˆ ] identifies completely the plane of rotation. Using this notation Ref. [9] has provided P = [p ˆ .. p 1

2

the mean to extend the skew-symmetric term in Eq. (28) to n-D. This has been done by providing the closed form expression of the “2-form” (see [11, 1], or the exterior product of two vectors (1-form) which, in the Grassmann algebra and in the matrix notation, can be written as

 . . 0 −1 ˆ 2 = P J2 P T = [ p ˆ 1 .. p [p ˆ 1 .. p (29) p ˆ1 ∧ p ˆ2 ] ˆ2 ]T = 1 0 where J2 is the 2 × 2 symplectic matrix (J2 J2T = I2 , J2 J2 = −I2 ). The expression given in Eq. (29) has allowed to write down the matrix performing a simple rigid rotation in n-Dimensional Euclidean complex spaces. The rotation matrix R, which is provided as expressed in term of the simple rotation parameters (rotation plane P and rotation angle Φ) is R(P, Φ) = In + (cos Φ − 1) P P † + P J2 P † sin Φ

(30)

In the n-Dimensional Euclidean space, any proper orthogonal matrix C describes an orientation or a general rotation. Reference [9] has shown that any proper orthogonal matrix C can be decomposed into a minimum set of simple rotations Rk performing rigid rotations on a set of fully orthogonal planes Pk (Pi† Pj = δij I2 ) and by the independent angles Φk , respectively. This decomposition is C=

np 

Rk (Pk , Φk ) =

k=1

np 

Rk (Pk , Φk ) − (np − 1) In

where

k=1

np =

n 2

(31)

Equation (31) implies that the orientation provided by the matrix C can be described by np different sequences of simple rotations, fully orthogonal, where the fully orthogonality is expressed by the conditions ⇐⇒ Ri Rj = Ri + Rj − In (32) (Ri − In ) (Rj − In ) = 0n where i, j = 1, 2, . . . , np , for i
= j.

Rotation Error in n-Dimensional Euclidean Real Spaces ˆ = 1) due to simple rotation error, is ˆ (where b ˆTb Now the error experienced by a direction b ˆ ˆ that is represented by the angle between b and the rotated (or corrected) vector R b, ˆ ˆT R b ˆ = 1 + (cos Φ − 1) b ˆT P P T b cos ε = b

(33)

ˆ = 0. As easy to verify, the product b ˆT P P T b ˆ represents the square of the ˆT P J2 P T b since b ˆ on the plane P . Therefore, we can write projection of b ˆT P P T b ˆ = cos2 α b 6

(34)

ˆ and the plane identified by P . The error will be where α identifies the angle between b cos ε = 1 + (cos Φ − 1) cos2 α = sin2 α + cos2 α cos Φ

(35)

Accordingly with the definition given in Eq. (31), the attitude error in n-D spaces experienced by ˆ = 1), is ˆ (where b ˆT b a direction b, ˆT C b ˆ = 1 − np + cos ε = b

np 

ˆT Rk b ˆ b

(36)

k=1

The summation term can be written as can be expressed as follows np 

ˆT Rk b ˆ = np + b

k=1

np 

ˆ = np + ˆT Pk PkT b (cos Φk − 1) b

k=1

np 

(cos Φk − 1) cos2 αk

(37)

k=1

ˆ and the kth rotation plane, identified by Pk . Therefore, the where αk identifies the angle between b rotation error in n-D is np  cos ε = 1 + (cos Φk − 1) cos2 αk (38) k=1

As example, let us consider the rotation error in 4-D Euclidean Real Spaces. In this dimensional space we have only np = 2 orthogonal plane, P1 and P2 , such that P1T P2 = 02 . In this specific case we have ˆT P1 P1T b ˆ = cos2 α1 ˆ = cos2 α2 ˆT P2 P2T b b and b (39) from which we can write that

cos2 α1 + cos2 α2 = 1

(40)

because ˆ T P1 P T b ˆ+b ˆT P2 P T b ˆ=b ˆ T In b ˆ = 1. b 1 2 Equation (40) implies that, specifically for 4-D space the error is described by the equation cos ε

= 1 + (cos Φ1 − 1) cos2 α1 + (cos Φ2 − 1) cos2 α2 = = cos Φ1 cos2 α1 + cos Φ2 cos2 α2

(41)

Rotation Error in n-D Euclidean Real Spaces The extension to n-dimensional spaces is immediate. In particular, in even dimensional spaces (n = 2, 4, · · · , 2m) the error is described by cos ε =

np 

cos Φk cos2 αk

where

k=1

np 

cos2 αk = 1

(42)

k=1

while in odd dimensional spaces (n = 3, 5, · · · , 2m + 1) it is ruled by the relationship 2

cos ε = cos α0 +

np 

2

cos Φk cos αk

where

k=1

2

cos α0 +

np 

cos2 αk = 1

(43)

k=1

ˆ and the direction p ˆ 0 associated with the real eigenvalue λ = +1 where α0 is the angle between b characteristic of proper orthogonal matrices describing orientation in odd dimensional spaces (e.g., in 3-D this direction is the axis of rotation ˆ e). The constraint for the αk given in Eqs. (42) and (43), associated with even and odd dimensional spaces, respectively, tell us that any direction in n-Dimensional space has associated, with a specific General Rotation, an intrinsic reduced reference frame made with n + 1/2 orthogonal axes. 7

Lie Algebra and Other Representations of SO(n) In order to study the group, for many purposes it is sufficient to look at elements in a neighborhood of the identity, i.e. of the form In + ε X, where ε is an infinitesimal parameter and X is a generator of the group. The local structure of the group is called the Lie algebra. The elements of the full group can be obtained by exponentiating the generators of the Lie algebra. For compact Lie groups G (i.e. the group space has finite volume), the generators are usually taken to be Hermitian matrices X† = X with = 1, 2, . . . ,Dim(G). Hence the group elements can be written as ei α
X
with α parameters. For the orthogonal group SO(n) we require the generators X to be imaginary and antisymmetric. There is a systematic way to construct such generators: we can write the ij-components of their n × n defining matrix representation by (Xmk )ij = −i (δmi δkj − δmj δki ). The notation Xmk , where the indices m and k span the dimensional space (from 1 to n), is used just to construct the X matrices, where spans the group dimension space [from 1 to Dim(G)]. Let us provide a couple of examples to show how the formalism works. Example 1: SO(3). Here n = 3 and the SO(3) ⎡ ⎤ ⎡ 0 −1 0 0 0 0 ⎦, X12 = i ⎣ 1 X13 = i ⎣ 0 0 0 0 1

group depends on = 3 parameters. ⎤ ⎡ ⎤ 0 −1 0 0 0 0 0 ⎦, X23 = i ⎣ 0 0 −1 ⎦ . 0 0 0 1 0

(44)

The elementary rotations are therefore associated to the exponentiation of such generators, (C = cos, S = sin) i.e. ⎡ ⎡ ⎡ ⎤ ⎤ ⎤ Cφ Sφ 0 Cψ 0 Sψ 1 0 0 1 0 ⎦ ei θ X23 = ⎣ 0 Cθ Sθ ⎦ (45) ei φ X12 = ⎣ −Sφ Cφ 0 ⎦ ei ψ X13 = ⎣ 0 0 −Sθ Cθ 0 0 1 −Sψ 0 Cψ Example 2: SO(4). ⎡ 0 −1 ⎢ 1 0 X12 = i ⎢ ⎣ 0 0 0 0 and

X23

⎡

0 ⎢ 0 ⎢ =i⎣ 0 0

0 0 1 0

Here n = 4 and the SO(4) group depends on = 6 parameters. ⎡ ⎡ ⎤ ⎤ ⎤ 0 0 0 0 −1 0 0 0 0 −1 ⎢ ⎢ 0 0 ⎥ 0 0 ⎥ 0 ⎥ ⎥ , X13 = i ⎢ 0 0 ⎥ , X14 = i ⎢ 0 0 0 ⎥ ⎣ 1 0 ⎣ 0 0 0 0 0 ⎦ 0 0 ⎦ 0 ⎦ 0 0 0 0 0 0 1 0 0 0

0 −1 0 0

⎤ 0 0 ⎥ ⎥, 0 ⎦ 0

⎡

X24

0 ⎢ 0 ⎢ =i⎣ 0 0

0 0 0 1

0 0 0 0

⎤ 0 −1 ⎥ ⎥, 0 ⎦ 0

⎡

X34

0 ⎢ 0 ⎢ =i⎣ 0 0

0 0 0 0

0 0 0 1

⎤ 0 0 ⎥ ⎥. −1 ⎦ 0

(46)

(47)

The separate exponentiation of these generators, eiαX , gives rise to six real matrices with angle descriptions in the same fashion as of SO(3). Now let us consider a complex example for the orthogonal group, i.e. SU (2). It has dimension three and therefore we need three generators that we can choose to be the Pauli matrices





 0 1 0 −i 1 0 σ1 = σ2 = σ3 = (48) 1 0 i 0 0 −1 which satisfy σj σk = δjk I2 + i jkm σm ,

where

j, k, m = 1, 2, 3,

(49)

and therefore the commutator relations are given by [ σj , σk ] ≡ σj σk − σk σj = 2 i jkm σm . 8

(50)

The group element of SU (2) can be represented by ˆ σ sin α ei α
σ
= I2 cos α + i n

(51)

where α = α n ˆ  , with n ˆT n ˆ = 1, parameterizes the elements of the group. In term of the Euler ψ ˆ is given by angles, we can set the parameter α = , while the unit vector n 2 ⎫ ⎧ ⎫ ⎧ ⎨ n1 ⎬ ⎨ sin θ cos ϕ ⎬ sin θ sin ϕ n2 n ˆ≡ = ⎭ ⎩ ⎭ ⎩ cos θ n3 making the connection with the classical construction of the Cayley-Klein parameters. Therefore, we used SU (2), with 2 × 2 complex matrix representations, to describe elements of the SO(3) group. A further digression into the structure of Lie algebra may transcend the purpose of the present discussion and therefore we skip it. Similarly, setting m = 2 n/2, we can now construct the SO(n) with 2n/2 × 2n/2 complex matrix representations via a spinor representation which is somehow the multi-dimensional analogous of the Cayley-Klein SU (2) parametrization for a SO(3) representation. In order to make explicit computations of a “spinor” representation of the rotation group SO(n), we need to choose a representation for the Dirac Γ-matrices whose appropriate antisymmetric combination gives rise to the generators which play a similar role as the Pauli matrices play for SO(3). The generators are a sort of Pauli matrix generalization for higher dimensions. There is systematic way to proceed. We can construct a representation of the Dirac Γ-matrices in any dimension by taking suitable tensor products of the Pauli matrices given in Eq. (48). In particular, we choose representations of the Dirac Γ-matrices as follows. In even dimensions we have Γ1 Γ2 Γ3 Γ4 Γ5 Γ6

= = = = = = .. .

σ2 σ1 I2 I2 I2 I2

⊗ ⊗ ⊗ ⊗ ⊗ ⊗ .. .

σ3 σ3 σ2 σ1 I2 I2

⊗ ⊗ ⊗ ⊗ ⊗ ⊗ .. .

σ3 σ3 σ3 σ3 σ2 σ1

⊗ ⊗ ⊗ ⊗ ⊗ ⊗ .. .

··· ··· ··· ··· ··· ··· .. .

⊗ ⊗ ⊗ ⊗ ⊗ ⊗ .. .

σ3 σ3 σ3 σ3 σ3 σ3

⊗ ⊗ ⊗ ⊗ ⊗ ⊗ .. .

Γm−1 Γm

= =

I2 I2

⊗ ⊗

I2 I2

⊗ ⊗

I2 I2

⊗ ⊗

··· ···

⊗ ⊗

I2 I2

⊗ σ2 ⊗ σ1

σ3 σ3 σ3 σ3 σ3 σ3

(52)

In odd dimensions we use the above construction for the Dirac Γ-matrices of m dimensions, and take Γm+1 as follows Γm+1

=

i σ3

⊗

σ3

⊗

σ3

⊗ ···

⊗

σ3

⊗

σ3

(53)

Other standard representations of the Γ-matrices in any dimension are constructed similarly, as in Refs. [10, 2]. The Γ-matrices so defined satisfy Clifford Algebra, i.e. { Γj , Γk } ≡ Γj Γk + Γk Γj = 2 δjk . The generators Xjk for the spinor representation of SO(n) can be constructed by anti-symmetrizing the Γ-matrices via the commutator and according to Xjk = − 4i [ Γj , Γk ] ≡ − 4i ( Γj Γk −Γk Γj ). Similarly to the 3-D case, here we have a complex representation of SO(n). Therefore the full group of SO(n) is spanned by the exponentiation of a linear combination of the generators. Let us provide a couple of examples to show how the formalism works. Example 3: SO(3). For n = 3 we have Γ1 = σ2 , Γ2 = σ1 , Γ3 = i σ3 , while Xjk = 12 jkm σm by using the commutator relations given in Eq. (50). Hence, X12 = 12 σ3 , X13 = − 21 σ2 and X23 = 12 σ1 , which is a similar representation to the original Pauli representation but with α = ψ in this case. 9

Example 4: SO(n). For n = 4 we have Γ1 = σ2 ⊗ σ3 , Γ2 = σ1 ⊗ σ3 , Explicitly we have ⎡ ⎤ ⎡ 0 0 −i 0 0 0 ⎢ 0 ⎥ ⎢ 0 0 0 i 0 ⎥, Γ1 = ⎢ Γ2 = ⎢ ⎣ i ⎣ 1 0 0 0 ⎦ 0 0 −i 0 0 0 −1 ⎡

and

0 −i 0 ⎢ i 0 0 Γ3 = ⎢ ⎣ 0 0 0 0 0 i

⎤ 0 0 ⎥ ⎥, −i ⎦ 0

⎡

0 ⎢ 1 Γ4 = ⎢ ⎣ 0 0

The six generators Mjk are given explicitly by ⎡ ⎡ ⎤ −1 0 0 0 0 0 0 ⎢ 0 ⎥ 1⎢ i 0 −1 0 0 0 1 ⎥ X13 = ⎢ X12 = ⎢ 0 1 0 ⎦ 2⎣ 0 2 ⎣ 0 −1 0 0 0 0 1 −1 0 0 and

X23

⎡

0 1⎢ 0 =− ⎢ 2⎣ 0 1

0 0 1 0

0 1 0 0

⎤ 1 0 ⎥ ⎥ 0 ⎦ 0

⎡

X24

0 i⎢ 0 = ⎢ 2⎣ 0 1

0 0 −1 0

0 1 0 0

⎤ 1 0 ⎥ ⎥ 0 ⎦ 0

⎤ −1 0 ⎥ ⎥ 0 ⎦ 0

1 0 0 0

Γ3 = I4 ⊗ σ2 , and Γ4 = I4 ⊗ σ1 . 1 0 0 0

⎤ 0 −1 ⎥ ⎥, 0 ⎦ 0

0 0 0 1

⎤ 0 0 ⎥ ⎥. 1 ⎦ 0 ⎡

X14

0 1⎢ 0 = ⎢ 2⎣ 0 −1 ⎡

X34

−1 1⎢ 0 = ⎢ 2⎣ 0 0

(54)

(55)

0 0 1 0

0 1 0 0

⎤ −1 0 ⎥ ⎥ (56) 0 ⎦ 0

0 1 0 0

0 0 −1 0

⎤ 0 0 ⎥ ⎥ (57) 0 ⎦ 1

The separate exponentiation of these generators, eiα
X
, gives rise to six complex matrices with angle descriptions in the same fashion as the Cayley-Klein parametrization of SU (2).

General Linear Transformations In order to understand how to methodically construct a higher dimensional rotation matrix in Group Theory, we need to discuss some of the basic properties and isomorphisms in Classical Groups. Classical Groups are introduced via the action of their elements on a vector space. Using tensor notation¶ , let’s consider an action that transforms from a set of basis vectors ek to another e k , i.e. e i = Aij ej ,

(58)

for some set of n2 quantities Aij , which we view as an n × n matrix with rows labelled by i and columns labelled by j. Of course, the inversion is guaranteed if the transformation matrices are non-singular, i.e. det(A)
= 0. As long as the set of basis vectors and the transformed basis vectors constitute two orthogonal sets, then Aij ∈ SO(n). This property does not hold anymore in the general case of non-orthogonal basis vector sets, which appear in the General Riemannian Spaces. For any curved space it is always possible to introduce a local tangent space that can be defined by a set of orthogonal basis vectors. However, for parametric representation of curved space there exist a coordinate induced basis that is not, in general, orthogonal. The linear transformation moving from these two sets of basis is called vielbeins. Equation (58) rules also the change of non orthogonal basis. The General Linear group of n × n matrices is defined as the group involved in performing arbitrary non-singular changes of basis of ¶ In

tensor notation repeated indices imply summation.

10

an n-Dimensional real, or complex, vector space. The notation used is respectively GL(n; IR) and GL(n; C). I The General Linear transformations, in general, do not preserve lengths. In fact, by transforming the unit-vector ˆ r, we obtain b = Gˆ r

=⇒

bT b = ˆ rT GT G ˆ r
= 1

(59)

where the transformed vector b is not, in general, a unit-vector. As long as the two unit-vector sets ˆk identifying non-orthogonal bases, are assigned, then the GL transformation is unique ˆ rk and b r1 , ˆ r2 , · · · , ˆ rn ] [ b1 , b2 , · · · , bn ] = B = G R = G [ ˆ

(60)

r2 , · · · , ˆ rn ] and B = [ b1 , b2 , · · · , bn ], have been introduced. where the two matrices, R = [ ˆ r1 , ˆ Let us apply the “QR” Orthogonal matrix triangularization (decomposition) to the transformation matrix G. We have rk (61) G = QT =⇒ QT bk = T ˆ where Q ∈ SO(n) is a proper (det(Q) = +1) orthogonal (QT Q = In ) matrix and T is an upper triangular matrix. Note that, in the QR orthogonal decomposition of square matrices, it is always possible to assume det(Q) = +1. In fact, if det(Q) = −1, then it is always possible to simultaneously change the sign to any odd number of columns of Q, which transform Q into a proper (det(Q) = +1) orthogonal matrix, and change the sign to the corresponding rows of T . The sign change leaves invariant the product Q T and demonstrates that the QR decomposition is not unique. Note that there exists three alternative equivalent orthogonal matrix triangularizations, “RQ”, “QL”, and “LQ”. Specifically, RQ performs the decomposition with the rigid rotation first and then the dilution, while QL and LQ works similarly to QR and RQ, respectively, but the dilution is described by the lower triangular matrix (L) rather than the upper triangular matrix (T ). Since Q is proper orthogonal, then its effect is just to rigidly rotate any vectors T ˆ rk to the final displacement bk . QR decomposition gives us the important result that the GL transformations can be seen/split into two subsequent transformations: the first transformation, provided by the upper triangular matrix T , performing the deformations (dilutions) of the GL transformation, while the second transformation, provided by the proper orthogonal matrix Q, performs the final rigid rotation correction. Therefore, the relative angular displacements and the lengths of the vectors bk in matrix B must be ¯k = T ˆ identical to those (b rk ) listed in matrix B¯ = T R. This means that we can write B T B = B¯T B¯ = RT T T T R

(62)

which gives us insights on the triangular matrix T . Let us map the hyper-unit-sphere identified by the unit-vector ˆ r into the hyper-ellipsoid identified ¯=Tˆ ¯ is by the transformation b r. The square modulus of b ¯ =ˆ ¯Tb rT T T Tˆ r 2 = b

(63)

Stationary condition, which identifies the locations of the principal axes of the hyper-ellipsoid, implies to perform the derivative of the augmented square modulus rT T T Tˆ r − λ (ˆ rTˆ r − 1) ∗2 = ˆ which implies

(64)

d ∗2 r − 2λ ˆ r=0 =⇒ T T Tˆ r = λˆ r (65) = 2 T T Tˆ dˆ r Therefore, the principal axes of the hyper-ellipsoid are identified by the eigenvectors of the matrix T T T . Since the T TT matrix is symmetric, then the eigenvectors are real and they constitute an orthogonal set. The lengths of these principal axes are √ √ √ rT T T Tˆ r= ˆ rT λ ˆ r= λ (66) = ˆ 11

A short digression into classical groups is introduced in appendix, which show the transformations in geometrical language.

Application: Lorentz Transformations There are two approaches for Minkowskian representation of the Lorentz transformations, one cartesian and complex, and another non-cartesian metric and real. In the former, we have a 2dimensional complex orthogonal representation of the Minkowski group, i.e. SO(2; C), I resembling the 2-dimensional rotations, i.e. SO(2; IR)

     cosh ξ i sinh ξ ict i c t , (67) = GC , with GC = −i sinh ξ cosh ξ x x satisfying GTC GC = I2 , where ξ = tanh−1

v c

=

1 ln 2



c+v c−v

 (68)

is the rapidity. Its interpretation is the simultaneous tilting of the two axes by the same angle ξ towards the asymptotic value of 45◦ (for v = c) in the first quadrant. It is useful to note that by sending the rapidity ξ to imaginary, we recover the special orthogonal transformation. In the non-cartesian metric case, we have a 2-dimensional real representation of the Minkowski group, i.e. SO(1, 1; IR), resembling the 2-dimensional rotations, i.e. SO(2; IR),

      cosh ξ sinh ξ ct ct = , (69) = G , with G R R sinh ξ cosh ξ x x

 −1 0 T satisfying GR η GR = η, where we set η = . We should bear in mind that this is a 0 1 non-compact representation of the group while the corresponding rotational group is compact. In the complex representation given by Eq. (67), the QR decomposition of the transformation matrix, GC = QC TC , yields to the matrices



 cosh ξ −i sinh ξ cosh(2ξ) i sinh(2ξ) QC = γ , and TC = γ (70) −i sinh ξ cosh ξ 0 1 where it has been set

1 (71) γ= cosh(2ξ) In the real representation given by Eq. (69), the QR decomposition of the transformation matrix, GR = QR TR , yields to the matrices



 cosh ξ − sinh ξ cosh(2ξ) sinh(2ξ) , and TR = γ (72) QR = γ sinh ξ cosh ξ 0 1 This means that the Lorentz transformation, provided either by the matrix GC or the matrix GR , can be seen as two subsequent effects: a dilation, provided by the triangular matrix T , and a subsequent rigid rotation, provided by the proper orthogonal matrix Q. The dilation effect, provided by TR and quantified by Eq. (66), has maximum and minimum effect along the eigenvector directions of the matrix TRT TR , which are toward the first/third and second/fouth quadrant √ bisectors, respectively. Matrix TRT TR has eigenvalues λ1,2 = cosh ξ ± sinh ξ, therefore 1,2 = cosh ξ ± sinh ξ. Figure 2 shows the variations of the lengths of the two principal axes of the hyper-ellipsoid. The bending effect of the space and the time axes (with respect to the orthogonal directions associated with v = 0 reference frames) is simply given by the rotation angle of matrix QR . This angle is given as a function of the v/c ratio in Fig. 3. This angle increases almost linearly with the v/c ratio and reaches the upper limit value of π/4 as v → c. 12

1

Principal axes lengths

10

0

10

−1

10

0

0.2

0.4

0.6

0.8

1

v/c ratio Figure 2: Principal axis lengths variations (dilution effect)

Conclusion This paper presents a theoretical research study on the error associated with Special Orthogonal SO(n) and with General Linear GL(n) transformations, in any dimensional space. After introducing the 3-D rotation error, the paper shows how it can be quantified using either, Direction Cosine Matrices and Euler-Rodrigues Symmetric Parameters. For the latter, the geometrical meaning of the angle between two quaternions is given. Then the paper shows that the orbit error can be completely described by a complex number whose phase quantify the orbit’s orientation error while the modulus represents the orbit shape error. The extension of the rotation error to n-dimensional spaces is provided by using the closed form expression of the orthogonal matrix performing a simple rotation as a function of the rotation angle and the rotation plane. In particular, the extension to n-dimensional space of the Principal Axis theorem is used to provide the expression for the error associated with proper orthogonal matrices representing General Rotations. This paper then completes the research study on the error by analyzing the General Linear GL(n) transformations that relate general non-orthogonal basis, as those introduced in parametric representations of Riemannian spaces. For these general coordinate transformations, the QR decomposition allows us to see them as made of two subsequent effects: a dilation (or deformation) and a rigid rotation. The paper concludes by applying the QR decomposition to the Lorentz transformations (Minkowski-orthogonal) by highlighting the effects and the meaning of the two involved matrices.

13

45 40

Bending angle (deg)

35 30 25 20 15 10 5 0

0

0.2

0.4

0.6

0.8

1

v/c ratio

Figure 3: Bending angle (rigid rotation effect) Acknowledgements The authors would like to dedicate this paper to Malcolm D. Shuster in the occasion of the Astrodynamics Symposium organized in his honor. The authors would like also to thank Prof. Christopher Pope and Dr. Arta Sadrzadeh for insights given to complete the connection with the Group Theory in describing General Rotations and Linear Transformations.

APPENDIX: A Short Digression into Group Theory More generally a vector space is seen as the space of linear maps defined by V : TM → IR, where TM is the tangent space of a n-Dimensional, real or complex, Riemannian manifold M , with TM GL(n; IR) or TM GL(n; C) I accordingly, where the notation “ ” means locally equivalent. We are commonly interested in a direct product of two or more copies of the same vector space. This notion of action can be further extended to the direct product of vector spaces. For instance, the space of bi-linear maps defined by T : TM ×TM → IR, endowed with the addition, as an internal operation, and a multiplication by a number, as an external operation, is a linear vector space called the tensor product of TM by itself and denoted by TM ⊗ TM . Therefore, any element u of this space, called a tensor, can be expanded as (73) u = uij ei ⊗ ej , where u ij are the tensor components in the {ek ⊗ el } basis of TM ⊗ TM . Similarly to Eq. (58), the action of the group is seen as a change from one set of basis vectors, ei ⊗ ej , to another, e i ⊗ e j , i.e. (74) u = uij e i ⊗ e j = uij Aik Ajl ek ⊗ el ≡ uij ei ⊗ ej ,

14

or uij = ukl Aik Ajl , where A ∈ TM . This can be further generalized to any multi-linear maps of tangent and co-tangent spaces of a n-Dimensional, real or complex, Riemannian manifold M , and therefore covariant and contravariant type of tensors should be considered. Of course this will require the distinction of upper and lower index notation and the introduction of vielbeins (i.e. transformations between non-coordinate bases and induced coordinate bases), but this transcends the purpose of this study and it is merely a topological definition of general tensor spaces. The remaining classical groups can be defined by introducing some further structure on the vector space: a metric, i.e. a tensor of the second rank over a Riemannian manifold. We have two types of metric tensors: bi-linear metrics or sesqui-linear metrics (the latter arises only for complex vector spaces). It is commonly denote by g = gij ei ⊗ ej . Hence, we demand that a certain subgroup of I matrices leaves the metric invariant, or TM GL(n; IR) or TM GL(n, ; C) Bilinear Metric gij = g kl Aki Alj Sesquilinear Metric gij = gkl A¯ki Alj

(75) (76)

where gij are the components of the metric tensor, A ∈ TM , and A¯ refers only to complex Riemannian manifolds. If a bi-linear metric is symmetric, the group associated with it is the Orthogonal group O(p, q), with n = p + q, real or complex, (e.g. the Rotation group with gij = δij , the Minkowski group in special relativity with gij = ηij , etc.). While, if it is anti-symmetric, the group associated with it is the Symplectic group Sp(2m), with n = 2m, even-Dimensional real or complex. A sesqui-linear metric gives rise to the Unitary group U (p, q), with n = p + q. The above discussion can be further extended to Riemannian manifolds over the field of Hamiltonianvalued numbers (or Quaternions), or one can consider appropriate intersections among groups. Also, some of the classical groups in low dimensions are easily seen to be isomorphic or homomorphic to others, e.g. SO(3) ∼ = SU (2), SO(4) ∼ = SU (2) ⊗ SU (2), SO(5) ∼ = U Sp(4), SO(6) ∼ = SU (4), etc. An important special case of tensor product is the the totally-antisymmetric direct product, called the wedge product. It is used to define geometrically a volume element Ω, i.e. Ω ≡ ∗1 =

1 i1 i2 ... in ε ei1 ∧ ei2 ∧ · · · ∧ ein = | det(g)| e1 ∧ e2 ∧ · · · ∧ en , n!

(77)

and this shows that under the action of the group elements this volume element resizes, i.e. Ω = det(A) Ω. Therefore, the subset of TM GL(n; IR) or TM GL(n; C) I matrices that preserve the volume element, i.e. for which Ω = Ω, are those for which det(A) = 1. These subsets correspond to the Special groups SL(n; IR) or SL(n; C) I respectively, and hence are regarded as volume-preserving groups.

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, Attitude determination from vector observations: A fast optimal matrix algorithm, Journal of the Astronautical Sciences 41 (1993), no. 2, 261–280. 15

[5] Daniele Mortari, Energy Approach Algorithm for Attitude Determination from Vector Observations, Journal of the Astronautical Sciences 45 (1997), no. 1, 41–55. [6]

, ESOQ: A Closed–Form Solution to the Wahba Problem, Journal of the Astronautical Sciences 45 (1997), no. 2, 195–204.

[7]

, Euler–q Algorithm for Attitude Determination from Vector Observations, Journal of Guidance, Control and Dynamics 21 (1998), no. 2, 328–334.

[8]

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[9]

, On the Rigid Rotation Concept in n–Dimensional Spaces, Journal of the Astronautical Sciences 49 (2001), no. 3, 401–420.

[10] Christopher N. Pope, A. Sadrzadeh, and Sante R. Scuro, Timelike Hopf Duality and Type IIA∗ String Solutions, hep-th/9905161. [11] Abraham R., J.E. Marsden, and T. Ratiu, Manifolds, tensor analysis, and applications, Applied Math Sciences, no. 75, Springer Verlag, Addison Wesley, 1983. [12] Malcolm D. Shuster, A Survey of Attitude Representations, Journal of the Astronautical Sciences 41 (1993), no. 4, 439–517. [13] Malcolm D. Shuster and S. D. Oh, Three–Axis Attitude Determination from Vector Observations, Journal of Guidance, Control and Dynamics 4 (1981), no. 1, 70–77.

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