Rotating Reference Frames (GPS)

As discussed relative to Figure 2.1, reference frames may be free to rotate arbitrarily with respect to one another. Consider for example the body frame moving with respect to the ECEF frame. The following subsections are concerned with frames-of-reference rotating with respect to one another.

Direction Cosine Kinematics

Section 2.5 showed that the transformation of vectors between two coordinate systems could be represented by an appropriately defined direction cosine matrix. In subsequent sections, it is necessary to calculate derivatives of direction cosine matrices for coordinate systems experiencing relative rotation. Such derivatives are the subject of this present section.

The definition of the derivative of the rotation matrix from frame a to frame b is


For smalltmp20-485_thumbthe rotationtmp20-486_thumbcan be considered as the rotation from frame a to frame b at time t followed by the rotation from frame b at time t to frame b at timetmp20-487_thumb


Because the instantaneous angular velocitytmp20-492_thumbof frame b with respect frame a represented in frame b is finite and St will be approaching zero, the rotation matrixtmp20-493_thumbrepresents the small angle rotationtmp20-494_thumb which by eqn. (2.48) is


wheretmp20-499_thumbSubstituting eqns. (2.51) and (2.52) into eqn. (2.50) yields



Using the facts thattmp20-503_thumb can also be expressed as


Example 2.6 Let a represent the geographic frame and b represent the ECEF frame. Using the fact thetmp20-506_thumbwe can computetmp20-507_thumbas the vector form of tmp20-510_thumb

For tmp20-511_thumb as defined in Section 2.5.3, we can compute tmp20-512_thumb as defined in Section 2.5.3, we can compute


After algebra to simplify the result, this product yields


which gives


This, combined with



tmp20-517_thumbRotating coordinate frames.

Figure 2.17: Rotating coordinate frames.

Derivative Calculations in Rotation Frames

As shown in Figure 2.17, let p be the vector from the b frame origin to point P, p be the vector from the b frame origin to the a frame origin, and r be the vector from the a frame origin to point P. These vectors are related by tmp20-519_thumb

Iftmp20-520_thumbare known, then from Section 2.4, the representation of p in frame b can be computed:


If the a frame is rotating with respect to the b frame, the rate of change oftmp20-523_thumbcan be expressed as in eqn. (2.59):


The first term on the right accounts for the relative instantaneous linear velocity of the two reference frames. The second term is the instantaneous velocity of point P relative to the b frame due to the relative rotation of the a frame. The last term is the transformation to the b frame of the instantaneous velocity of point P relative to the origin of the a frame.

This equation can be considered as a special case of the following theorem, which is a statement of the Law of Coriolis.

Theorem 2.1 If two frames-of-reference experience relative angular rotationtmp20-526_thumbwithtmp20-527_thumbthen the time rate of change of the vector in the two coordinate systems are related by tmp20-530_thumb

Taking a second derivative of eqn. (2.59) gives


Note the following points regarding the derivation of eqn. (2.61): the equation is exact; the equation is applicable between any two coordinate systems; and, the equation is linear in the position and velocity vectors. This equation is the foundation on which a variety of navigation systems are built.

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