**In Section 2.4,** the transformation of vectors from frame a to frame b is shown to involve an orthonormal matrix denoted byThe 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 defineorthonormality 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 asUsing 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

Let

whereare the ECEF coordinates of the origin of the local tangent plane. Thenis a vector from the local tangent plane origin to an arbitrary locationwith 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

**Figure 2.12: Variables for derivation of**

whereis the longitude of the pointThe second plane rotation is defined by

whereis the latitude of the point

The overall transformation for vectors from ECEF to tangent plane representation is thenwhere

The inverse transformation for vectors from tangent plane to ECEF is

Example 2.3 The angular rate of the ECEF frame with respect to the inertial frame represented in the ECEF frame isTherefore, the angular rate of the ECEF frame with respect to the inertial frame represented in the tangent frame is

Letdenote 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 locationis 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 is The latitude $ and longitude A define the position of the geographic frame origin (vehicle frame projection) on the reference ellipsoid. Second,

which is not the velocity vector for the vehicle. The Earth relative velocity vector represented in the ECEF frame isThis vector can be represented in the geographic frame as

**The vector****is not the** derivative of the geographic frame position vector The components of the Earth relative velocity vector represented in the geographic frame are named aswhich are the north, east, and down components of the velocity vector along the instantaneous geographic frame axes.

**The rotation matrix**has the exact same form asThe distinction is thatis computed using the latitude $ and longitudedefined by the position of the vehicle at the time of interest whereasis a constant matrix defined by the fixed latitude and longitude of the tangent plane origin. It should be clear thatwhilewhere andare discussed in eqns. (2.56) and (2.57), respectively.

**Eqns. (2.9-2.11)** provide the relationship betweenand which is repeated below

First, we note that

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

Therefore,

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

and