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Exponential solutions of this equation similar to Eq. (164) can be written, but are not used as frequently.
The kinematic equations of motion for the Gibbs vector, g, can be derived from Eq. (1219). For infinitesimal A/ we have, from Eq. (1216), g'
= e tan^« j« at where g' is the Gibbs vector representing the infinitesimal rotation between times t and /+A/, so that d t
The kinematic equations of motion for the Euler angles (<J>,0,<//) can be derived by a different technique. Consider the 313 sequence of rotations as illustrated in Fig. 123 as an example. The rotations involved are <> about z, 9 about x', and about w. If <> were the only angle changing, the angular velocity would be <j>z. Similarly, if only 9 or only xp were changing, the angular velocity would be Ox' or (¿ft, respectively. When all three angles are changing, the angular velocity is the vector sum of these three contributions:
Taking components of <o along the body axes û, v, w gives w„=4>zû + 9k'h (1610a)
Comparison with Eqs. (122) and (1220) gives zû = /4,3=sin0sin^
The inner products of x' with the body axes are elements of the matrix giving the orientation of the û, V, w triad relative to the y", z" triad:
A'aAJtytJg)'
cos^ cos 0 sin ^ sin 0 sin ^ —sinif» cos 0 cos ^ sin 0 cos 0 sin® cos 9
Thus,
Combining these results gives a3a=0cos\(>+<frsin$sai\f> «„= 0siri^+^sin0cos^
Equation (1611) can now be solved for 4, and <f> to yield the kinematic equations of motion for the 313 Euler angle sequence:
ê^uu cos\f> — uv sinip <¿>=■((0. sin + <o0cos </<) / sin 0 ^=<dw—(«„ sin^+<^ cos^) cot®
The lack of uniqueness in the specification of $ and when 9 is a multiple of 180 deg shows up as a singularity in the kinematic equations of motion, Eqs. (1612b) and (1612c), when sinfl*»0. This is a serious disadvantage of Euler angle formulations for numerical integration of the equations of motion.
For many applications, it is convenient to have expressions for the components of the angular velocity vector, <o, along the reference axes as functions of the Euler angle rates. These are given by
where A3X3(<p,0,\f>), the transpose of the matrix of Eq. (1220), is the matrix that transforms vector components from the body frame to the reference frame. The result of this matrix multiplication is
«1 

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