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CU-Boulder ASEN 3200 - CoordinateTransformations

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Coordinate TransformationsASEN 32002/22/04George H. BornIn engineering it is often necessary to express vectors in different coordinate frames. This requires the rotation matrix, which relates coordinates and basis (unit) vectors in one frame to those in another frame.Consider for example two frames with a common origin but separated by a positive rotation* about their common Z-axis through the positive angle θ.Figure 1 - Definitions of the A and B frames and a positive rotation, θ, about the Z-axisWe wish to express a vector given in the A frame coordinates in terms of B frame coordinates, i.e.,     AB B Ar T r. (1)A simple way to do this is to express the unit vectors of the B frame in terms of those in the A frame. Since we are rotating about the Z-axis let's look at a view down the Z-axis toward the origin:* a positive rotation means that it obeys the right hand rule, i.e., place the thumb of the right hand in the positive direction of the rotation axis and the fingers will indicate the positive direction of rotation.1ˆˆK, kθθÎîĴĵZ zĵĴîÎθyYxXBAIn order to do this we first write the projection of the Î and Ĵ vectors (or X and Y coordinates of a vector) on the î and ĵ vectors (or x and y axes). Note, for example, that the projection of Î on î is obtained by drawing a line from Î perpendicular to î. Hence,ˆ ˆˆcos sinˆ ˆˆsin cosˆˆ.i I Jj I Jk K    (2)In matrix form:ˆˆcos sin 0ˆˆsin cos 0ˆ ˆ0 0 1Iij JKk                   . (3)To transform a vector we replace the unit vectors with the coordinates of that vector, i.e., the coordinates of a vector in the ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆI, J, K frame, , transforms to the i, j, k frame asr XI YJ ZK  cos sin 0sin cos 00 0 1x Xy Yz Z                            . (4)2Similarly a transformation involving a positive rotation through the angle β about the Y-axis would be as follows (from a view down the Y-axis):ˆ ˆˆcos sinˆˆˆˆ ˆsin cosi I Kj Jk I K    orˆˆcos 0 sinˆˆ0 1 0ˆ ˆsin 0 cosIij JKk                  . (5)Finally, a rotation about the X-axis through the positive angle α is given by (draw in the projections for yourself)ˆˆ1 0 0ˆˆ0 cos sinˆ ˆ0 sin cosIij JKk                  . (6)If we wish to transform from the B frame to the A frame we could simply write (using Eq. 3)3ˆkˆKααĵĴÎ, îβîββˆˆ, J jΈk1ˆˆcos sin 0ˆˆsin cos 0ˆ ˆ0 0 1IiJ jKk                   . (7)But transformation matrices are orthogonal so that their inverse is equal to their transpose; hence,1ˆˆcos sin 0ˆˆsin cos 0ˆ ˆ0 0 1IiJ jKk                  . (8)If the transformation from one frame to another requires several rotations, the final rotation matrix is the product of the individual rotation (transformation) matrices.ExamplesTransformation from ˆˆ ˆ ˆ ˆ ˆI, J, K to P, Q, WThe transformation from the ECI frame with unit vectors ˆ ˆ ˆI, J, K to the PQW frame with unit vectors ˆˆ ˆP, Q, W is described below. Note that ˆPis in the orbit plane in the direction of perigee, ˆQ is (in the orbit plane) perpendicular to ˆP and in the direction of motion and ˆW lie along the angular momentum vector. If this were a retrograde orbit ˆQ and ˆW would be in the opposite direction to that depicted here.4ˆJˆKˆPˆQˆIˆWΩiiωorbitline of nodesThe transformation from the ˆ ˆ ˆI, J, K frame to ˆˆ ˆP, Q, W involves three rotations:1. A rotation about the ˆK axis through the RAAN (Ω) to align the X-axis of this intermediate frame (which we will call B ) with the line of nodes.2. A rotation about the B frame X-axis through the inclination, i, to place the X andY axis of this intermediate frame (called C ) in the orbit plane. Note that the Z-axis of the C frame is along the angular momentum vector and will require no further transformation.3. Finally a rotation about the C frame Z-axis to align its X-axis with the ˆP vector, and Y-axis with the ˆQ vector.Hence, the transformation equation given by            ˆ ˆˆ ˆ ˆ ˆˆˆ ˆC B APQW PQW C B AAPQW Ar T T T rT r(9)where  ABT is a rotation about Z through Ω. From Eq. (3) cos sin 0sin cos 00 0 1ABT         . (10)The second rotation is about the X-axis through i and is given by Eq. (6) 1 0 00 cos sin0 sin cosBCT i ii i     . (11)Finally we have another rotation about the Z-axis through the argument of perigee, ω, and from Eq. (3) ˆˆ ˆcos sin 0sin cos 00 0 1CPQWT        . (12)Multiplying these three matrices in the order given by Eq. (9) yields the final result which is the transpose of the matrix given by Eq. (2.6-14) in Bate, et. al. You should carry out this multiplication for yourself.5Transformation from ˆˆ ˆS, E, Z to ˆ ˆ ˆI, J, KThe transformation from the topocentric ˆˆ ˆS, E, Z frame to the ˆ ˆ ˆI, J, K (ECI) frame is developed next.The ˆˆ ˆS, E, Z has its origin at a tracking station and ˆS directed south, ˆE directed east and ˆZ directed up along the local vertical. Note that ˆZ is perpendicular to a tangent to the oblate earth at the location of the tracking station. Hence, the angle between ˆZ and the equatorial plane is the local geodetic latitude, . Also, because of the oblateness of the earth the ˆZ axis is not parallel with the station position vector, R. Because in all mathematical operational vectors maybe treated as free vectors we may move the origin of the ˆˆ ˆS, E, Z frame to the origin of theˆ ˆ ˆI, J, K frame in the center of the Earth. This will aid our visualization of the


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