X2z22 p2z22E2d

0 = &TCcos{z/(x2+y2 + z2),/2} =arctan(p/z) O<0O8O° (E-2e)

<*>=arctan(^/*) =<t> 0<^.<360° (E-2f)

The correct quadrant for 4» in Eq. (E-2f) is obtained from the relative signs of x andy.

EL2 Transformations Between Cartesian Coordinates

If r and r' are the cartesian representations of a vector in two different cartesian coordinate systems, then they are related by r' = /lr + a (E-3)

where a represents the translation of the origin of the unprimed system in the primed system and the matrix A represents a rotation. For most attitude work, a=0.

The transformation matrix A (called the attitude matrix or direction cosine matrix in this book) can be obtained by forming the matrix product of matrices for successive rotations about the three coordinate axes as described in Section 12.1. The elements of matrix A are direction cosines of the primed axes in the unprimed system and satisfy the orthogonality condition. Because A is an orthogonal matrix, its inverse transformation matrix will be its transposed matrix; symbolically, r ¿"'-/T (E-4)

For many applications, the definition of the direction cosine matrix in terms of the orthogonal coordinate unit vectoins in the two coordinate systems,

is useful for computations. As an example, let the primed coordinate system have its coordinate axes aligned with the spacecraft-to-Earth vector R, the component of V perpendicular to R, and the orbit normal vector RXV/|RXV|, where V is the spacecraft velocity:

Then, substituting Eq. (E-5b) into, tq. (E-5a) gives ap expression for A which does not require the evaluation of trigonometric functions.

Euler's Theorem. Euler's theorem states that any finite rotation of a rigid body can be expressed as a rotation' through1 sdine angle about some fixed axis. Therefore, the most general transformation matrix A is a rotation by some angle,

,i tg fyiV&tst -'J-Hi »*im> ♦ f about some fixed axis, k. The axis k is unaffected by the rotation and, therefore, must have the same components in both the primed and the unprimed systems. Denoting the components of e by e„ e2, and e3, the matrix A is cos$+efO-cos$) i|i2(l-cosi)+ij8iii4 (|(3(l-cas4)-«2sint e,e](l-cos$)-«38111$ cos$+e§(l-cosi) ejCjO-cos^+^siirt (|()(l-cos4)+eism4 ejtj(l-cos4)-e,smi cosO+efO -cosi>)

In this case, the inverse transformation matrix may be obtained by using Eq. (E-4) or by replacing $ by - i>, in Eq. (E-6), that is, a rotation by the same amount in the opposite direction about the axis e.

Euler Symmetric Parameters. The Euler symmetric parameters, qx through 94, used to represent finite rotations are defined by the following equations:

94=COSy

Clearly,

The transformation matrix A in terms of Euler symmetric parameters is

0 0

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