Specific Vector Transformations (GPS) Part 1

In Section 2.4, the transformation of vectors from frame a to frame b is shown to involve an orthonormal matrix denoted bytmp20-308_thumbThe elements of this matrix, called the direction cosine matrix, are the cosines of the angles between the coordinate axes of the two frames-of-reference. Although this appears to allow nine independent variables to definetmp20-309_thumborthonormality restrictions result in only three independent quantities. Section 2.5.1 introduces the concept of a plane rotation. Sections 2.5.2-2.5.5 will use plane rotations to define the transformations between specific pairs of reference systems.

In addition to the direction cosine and Euler angle representations of the relative orientations of two reference frames, various other representations exist [120]. Advantages of alternative representations may include efficient computation, lack of singularities, or compact representation. One popular representation of relative attitude is the quaternion. Quaternions offer accurate and efficient computation methods without singularities. Often quaternions are preferred over both direction cosine and Euler angle methods. Nonetheless, their discussion is a topic in and of itself. To maintain the flow of the topic. It is recommended that designers read the main body of the text first, to understand the role and issues related to attitude representation; however, they should understand and consider quaternions prior to implementation of their first system.

Plane Rotations

A plane rotation is a convenient means for mathematically expressing the rotational transformation of vectors between two coordinate systems where the second coordinate system is related to the first by a rotation of the first coordinate system by an angle x around a vector v. In the special case where the vector v is one of the original coordinate axes, the plane rotation matrix takes on an especially simple form. In the following, a rotation of the first coordinate system by radians1 around the i-th axis will be expressed astmp20-310_thumbUsing this notation,


Each of these plane rotation matrices is an orthonormal matrix. For a rotation of x radians about the i-th axis of the first coordinate system, the components of vector z in each coordinate system are related by


When two coordinate systems are related by a sequence of rotations, then the corresponding rotation matrices are multiplied in the corresponding order. For example, continuing from the last equation, if a third frame is defined by a rotation of y radians about the j-th axis of the second frame, then the representation of the vector z in this frame is


The order of the matrix multiplication is critical. Since matrix multiplication is not commutative, neither is the order of rotation. For example, a 90 degree rotation about the first axis followed by a 90 degree rotation about the resultant second axis results in a distinct orientation from a 90 degree rotation about the second axis followed by a 90 degree rotation about the resultant first axis. The following two sections use plane rotations to determine the Euler angle representations of a few useful vector transformations.

Transformation: ECEF to Tangent Plane



wheretmp20-319_thumbare the ECEF coordinates of the origin of the local tangent plane. Thentmp20-320_thumbis a vector from the local tangent plane origin to an arbitrary locationtmp20-321_thumbwith the vector and point coordinates each expressed relative to the ECEF axis.

The transformation of vectors from ECEF to tangent plane (TP) can be constructed by two plane rotations, as depicted in Figure 2.12. First, a plane rotation about the ECEF z-axis to align the rotated y-axis (denoted y’) with the tangent plane east axis; second, a plane rotation about the new y’-axis to align the new z-axis (denoted z”) with tangent plane inward pointing normal vector. The first plane rotation is defined by tmp20-325_thumbVariables for derivation of

Figure 2.12: Variables for derivation oftmp20-327_thumb

wheretmp20-329_thumbis the longitude of the pointtmp20-330_thumbThe second plane rotation is defined by tmp20-334_thumb

wheretmp20-335_thumbis the latitude of the pointtmp20-336_thumb

The overall transformation for vectors from ECEF to tangent plane representation is thentmp20-337_thumbwhere


The inverse transformation for vectors from tangent plane to ECEF is tmp20-342_thumbtmp20-343_thumb

Example 2.3 The angular rate of the ECEF frame with respect to the inertial frame represented in the ECEF frame istmp20-344_thumbTherefore, the angular rate of the ECEF frame with respect to the inertial frame represented in the tangent frame is


Lettmp20-347_thumbdenote the coordinates of the point P represented in the tangent plane reference system, then


Using eqn. (2.28), the transformation of the coordinates of a point from the tangent plane system to the ECEF system is


Example 2.4 For the ECEF position.


transforms vectors from the ECEF coordinate system to tangent plane coordinates. The point transform is defined as


where the origin locationtmp20-353_thumbis defined in eqn. (2.12).

The inverse transformations are easily derived from the preceding text.

Transformation: ECEF to Geographic

The geographic frame has a few points that distinguish it from the other frames. First, because the origin of the geographic frame moves with the vehicle and is the projection of vehicle frame origin onto the reference ellipsoid, the position of the vehicle in the geographic frame istmp20-355_thumb The latitude $ and longitude A define the position of the geographic frame origin (vehicle frame projection) on the reference ellipsoid. Second,tmp20-356_thumb

tmp20-357_thumbwhich is not the velocity vector for the vehicle. The Earth relative velocity vector represented in the ECEF frame istmp20-358_thumbThis vector can be represented in the geographic frame as


The vectortmp20-366_thumbis not the derivative of the geographic frame position vector tmp20-367_thumbThe components of the Earth relative velocity vector represented in the geographic frame are named astmp20-368_thumbwhich are the north, east, and down components of the velocity vector along the instantaneous geographic frame axes.

The rotation matrixtmp20-369_thumbhas the exact same form astmp20-370_thumbThe distinction is thattmp20-371_thumbis computed using the latitude $ and longitudetmp20-372_thumbdefined by the position of the vehicle at the time of interest whereastmp20-373_thumbis a constant matrix defined by the fixed latitude and longitude of the tangent plane origin. It should be clear thattmp20-374_thumbwhiletmp20-375_thumbwheretmp20-376_thumb andtmp20-377_thumbare discussed in eqns. (2.56) and (2.57), respectively.

Eqns. (2.9-2.11) provide the relationship betweentmp20-378_thumband tmp20-379_thumbwhich is repeated below


and will be used to relate tmp20-395_thumb

First, we note that


Next, using eqns. (2.78-2.78) it is straightforward to show that




With this expression and eqn. (2.34) it is straightforward to show that




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