Reference Frame Transformations (GPS)

This section presents methods for transforming points and vectors between rectangular coordinate systems. The axes of each coordinate system are assumed to be right-handed and orthogonal. Three dimensions are used throughout the discussion; however, the discussion is equally valid fortmp20-118_thumb

The Direction Cosine Matrix

Lettmp20-119_thumbrepresent a right-handed orthogonal coordinate system. Lettmp20-120_thumbbe a vector from the origintmp20-121_thumbof thetmp20-122_thumbframe to the point P. The representation of the vectortmp20-123_thumbwith respect to frametmp20-124_thumbis


axes and where


are unit vectors along the tmp20-134_thumb axes and


The vectortmp20-136_thumbcontains the coordinates of the point P with respect to the axes oftmp20-137_thumband is the representation of the vectortmp20-138_thumb with respect totmp20-139_thumbThe physical interpretations of the coordinates is that they are the projections of the vectortmp20-140_thumbonto thetmp20-141_thumbaxes. For the two-dimensional x — y plane, the discussion of this paragraph is depicted in Figure 2.9.

A vector v can be defined without reference to a specific reference frame. When convenient, as discussed above, the representation oftmp20-142_thumbwith respect to the axes of frametmp20-143_thumbis


Eqn. (2.14) is used in derivations later in this subsection.

Two dimensional representation of the determination of the coordinates of a point P relative to the originof reference frame.

Figure 2.9: Two dimensional representation of the determination of the coordinates of a point P relative to the origintmp20-154_thumbof reference frame.

With two distinct reference framestmp20-156_thumbthe same point can be represented by a different sets of coordinates in each reference frame.The remainder of this section discusses the important question of how to use the coordinates of a point in one frame-of-reference to compute the coordinates of the same point with respect to a different frame-of-reference. The transformation of point coordinates from one frame-of-reference to another will require two operations: translation and rotation.

From the above discussion,tmp20-157_thumbis the vector fromtmp20-158_thumbto P and

tmp20-159_thumbis the vector fromtmp20-160_thumbDefinetmp20-161_thumbas the vector fromtmp20-162_thumbTherefore, we have that tmp20-171_thumb

This equation must hold whether the vectors are represented in the coordinates of thetmp20-172_thumbframe or thetmp20-173_thumbframe.

Denote the components of vectortmp20-174_thumbrelative to thetmp20-175_thumbframe astmp20-176_thumb tmp20-177_thumbthe components oftmp20-178_thumbrelative to thetmp20-179_thumbframe astmp20-180_thumb tmp20-181_thumband the components oftmp20-182_thumbrelative to thetmp20-183_thumbframe astmp20-184_thumb tmp20-185_thumbAssume thattmp20-186_thumband the relative orientation of the two reference frames are known. Then, the position of P with respect to thetmp20-187_thumbframe can be computed as


Because it is the only unknown term in the right hand side, the present question of interest is how to calculatetmp20-208_thumbbased on the available information.

Definition of the coordinates of a point P with respect to two frames-of-reference

Figure 2.10: Definition of the coordinates of a point P with respect to two frames-of-referencetmp20-206_thumb


Lettmp20-209_thumbrepresent the unit vectors along the axes. As discussed relative to eqn. (2.14), vectorstmp20-210_thumbdefined as


represent the unit vectors in the direction of thetmp20-216_thumbcoordinate axes that are resolved in thetmp20-217_thumbreference frame. Sincetmp20-218_thumbare orthonormal, so aretmp20-219_thumbTherefore, the matrix tmp20-228_thumb is an orthonormal matrixtmp20-229_thumb

Each element oftmp20-230_thumbis the cosine of the angle between one oftmp20-231_thumb and one oftmp20-232_thumbTo see this, consider the element in the third row second column:


wheretmp20-238_thumbis the angle betweentmp20-239_thumband we have used the fact that tmp20-240_thumb

Definition ofin eqn. (2.16).

Figure 2.11: Definition oftmp20-245_thumbin eqn. (2.16).

Because each element oftmp20-246_thumbis the cosine of the angle between one of the coordinate axes oftmp20-247_thumband one of the coordinate axes oftmp20-248_thumbthe matrix tmp20-249_thumbis referred to as a direction cosine matrix:


Figure 2.11 depicts the angles .tmp20-256_thumbfortmp20-257_thumbthat define the first column oftmp20-258_thumbThetmp20-259_thumbangles are defined similarly. When the relative orientation of two reference frames is known, the direction cosine matrix tmp20-260_thumbis unique and known.

Although the direction cosine matrix has nine elements, due to the three orthogonality constraints and the three normality constraints, there are only three degrees of freedom.





Point Transformation

When eqn. (2.17) is substituted into eqn. (2.15) it yields the desired equation for the transformation of the coordinates of P with respect to frame 2, as represented bytmp20-274_thumbto the coordinates of P with respect to frame 1, as represented bytmp20-275_thumb


The reverse transformation is easily shown from eqn. (2.18) to be


where we have used the fact thattmp20-280_thumbwhere the last equality is true due to the orthonormality of RJ. Note that the point transformation between reference systems involves two operations: translation to account for separation of the origins, and rotation to account for non-alignment of the axis.

Vector Transformation

Consider two pointstmp20-281_thumbLet the vector v denote the directed line segment fromtmp20-282_thumbRelative totmp20-283_thumbv can be described as


Eqn. (2.20) is the vector transformation between coordinate systems. This relation is valid for any vector quantity. As discussed in detail in [95], it is important to realized that vectors, vector operations, and relations between vectors are invariant relative to any two particular coordinate representations as long as the coordinate systems are related through eqn. (2.20). This is important, as it corresponds to the intuitive notion that the physical properties of a system are invariant no matter how we orient the coordinate system in which our analysis is performed.

In the discussion of this section, the two frames have been considered to have no relative motion. Issues related to relative motion will be critically important in navigation systems and are discussed in subsequent sections.

Throughout the text, the notationtmp20-290_thumbwill denote the rotation matrix transforming vectors from frame a to frame b. Therefore,


Matrix Transformation

In some instances, a matrix will be defined with respect to a specific frame of reference. Eqns. (2.21-2.22) can be used to derive the transformation of such matrices between frames-of-reference.

Lettmp20-294_thumbbe a matrix defined with respect to frame a andtmp20-295_thumb tmp20-296_thumbbe two vectors defined in frame a. Let b be a second frame of reference. Iftmp20-297_thumbare related by tmp20-302_thumb then eqns. (2.21-2.22) show that






is the representation of the matrixtmp20-306_thumbwith respect to frame b.

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