Properties of the Rotation Tensor

The orientation of a frame A is modeled by its base triad consisting of the three orthonormal base vectors «і, аг, and a (Sec. 3.1.1). Figure 4.1 shows the two frames A and В and their base triads. The orientation of frame В wrt frame A is established by the rotation tensor RBA, which maps the a, into the b, base vectors:

bi=RBAai, / = 1,2,3 (4.1)

Our first concern is whether RBA is a tensor. If we can show that it transforms like a tensor [see Eq. (2.4)], then it is a tensor.

Property 1

The rotation tensor RBA of frame В wrt frame A is a tensor, i. e., for any two allowable coordinate systems ]A and ]B with their transformation matrix T]LtA it transforms like a second-order tensor

Proof: Coordinate Eq. (4.1) by any two allowable coordinate systems, say ]4 and ]B,

[bt]A = [ЯМ]АЫЛ (4.2)

and

biL< = [ RBA ]z< I a,]R (4.3)

Because the base vectors themselves are tensors, they transform like first-order tensors, Eq. (2.3):

[ЬЛВ = [T]BA[bi]A (4.4)

and

[ai]B = [T]BA[ai]A (4.5)

Substitute Eq. (4.2) into Eq. (4.4) and replace [a, J‘4 by the transposed of Eq. (4.5) [bt]B = [T]BA[RBA]A[T]BA[ai]B Comparing with Eq. (4.3), we deduct

_ j-yjBAjAі’у’jBA

Because ]A and ]z< can be any allowable coordinate systems, RBA transforms like a second-order tensor and therefore is a tensor.

Property 2

Sequential rotations are obtained by multiplying the individual rotation tensors. For three frames A, B, and C and the rotation tensors RBA and RCB, the rotation tensor Rca of frame C wrt frame A is obtained from

2^ca_ rcbrba

Note the contraction of the superscripts: the adjacent B’s are deleted to form CA.

Proof: Let each frame A, В, and C be modeled by the triads a,-, bt, and c,, і = 1, 2, 3 and related by

bi = RBAat, a = Rc%, a = RCAdi, і = 1,2,3

Substituting the first into the second equality and comparing with the third one proves the property

d = RCBbi = RCBRBAaj => Rca = RCBRBA

Property 3

Rotation tensors, coordinated in preferred coordinate systems, are related to their transformation matrices. For any two triads at, bi, with the rotation tensor Rba, and the preferred coordinate systems ]A, Iй with the transformation matrix [T]BA, the following relationships hold:

[ЯЙА]А = = [T]BA (4.6)

Note first the surprising result that the rotation tensor has the same coordinates in both of its preferred coordinate systems. Furthermore, the rotation sequence is the reverse of the transformation sequel. This reversal becomes clear when we exchange the transpose sign of the TM for the reversal of the transformation order [RM]A = = уГ]АВ.

Proof: If the base vectors are coordinated in their respective preferred coordinate systems, they have the same coordinates:

[bi]B = [a,]A, / = 1,2,3 (4.7)

First substitute this equation into Eq. (4.2) and then replace [b, ]B by Eq. (4.4):

[bi]A = [RBA]A[ai]A = [RBA]A[bi]B = [Лм]л[Г]м[Ь,]Л

Because bi]A is arbitrary and certainly not zero, it follows that [Л“й4 ATUA = E and therefore [RBA]A = [T]BA. This completes the first part of the proof. Similarly, if we start with Eq. (4.3) and replace [&, ]B by Eq. (4.7) and then transpose Eq. (4.5) for substitution, we can prove the second relationship

[Rba ]B[ai)B = [bt]B = ША = [Т]ВА[щ]в =► [Rba]b = mBA

Combining both results delivers the proof

[^BA]A = = [f

Property 4

The rotation tensor is orthogonal.

Proof: It follows from Eq. (4.6) and the proof of Sec. 3.2, Property 1. Because the coordinate transformation is orthogonal, at least one matrix realization of the rotation tensor is orthogonal. But if one coordinate form is orthogonal, so are all and, therefore, the tensor is orthogonal.

Just as the determinant of the transformation matrix is ±1, so is that of the rota­tion tensor. Every such orthogonal linear transformation in Euclidean three-space preserves absolute values of vectors and angles between vectors. In addition, if the determinant is +1, it also preserves the relative orientation of vectors embedded in the frame. These are very useful properties, and therefore we limit ourselves to these rotations. The case with a “negative one” determinant is the reflection tensor, which we already encountered in Sec. 2.3.6.

Property 5

Transposing the rotation tensor reverses the direction of rotation RBA = RAB.

Proof: Exchanging the b and a, for both lower – and upper-case letters in Eq. (4.1) provides

dj — RABbj, і = 1,2,3 Substituting Eq. (4.1) yields

ai=RABRBAai, г = 1,2,3

Because a, is nonzero and the rotation tensor is orthogonal, RABRBA = £=> RllA = RAB.

We have established the rotation tensor as an absolute model of the mutual orientation of two frames. The nomenclature RBA expresses that relationship of frame В relative to frame A. You can read it as the orientation of frame В obtained from frame A, or just as the rotation of В wrt A.

No reference point is needed. Rotations are independent of points; they only engage frames. This independence becomes clear by an example. Suppose you stand on the east side of a runway and watch an airplane A take off and climb out at 10 deg. The airplane’s 10-deg rotation wrt to the runway R is modeled by Rar. On the next day you position yourself at the west end and watch the same airplane take off and climb out. The same Rar will give its orientation, although you changed your reference point from E to W. To define the airplane’s orientation, no reference points are needed.

Fig. 4.2 Earth displays: a) flat and b) round.

Example 4.1 Conversion from Flat to Round Earth

Problem. You build a three-dimensional visualization of a long-range air intercept missile. Vectors of polygons model the missile shape. The simulation calculates the missile attitude RBF wrt a flat Earth. You are required to display the missile orientation over a round Earth. How do you convert the vectors of the missile shape?

Solution. The missile frame (geometry) is related to the flat Earth by RBF (see Fig. 4.2a). As the missile flies toward the intercept, the Earth’s local level tilts wrt to the flat surface by Rrf (see Fig. 4.2b). The orientation of the missile wrt the local level is therefore

Rbr = RbfRrf

Any geometrical vector of the missile, say tt, is oriented wrt the flat Earth by RBFtt and wrt the local level by RBRti.