PlanetPhysics/Euler Angle Velocity

The [[../Velocity/|velocity]] of the Euler angles of a rotating coordinate frame can be derived from the angular velocity [[../Vectors/|vector]] of this frame with respect to a [[../CosmologicalConstant/|reference frame]]. The derivation is lengthy for each [[../EulerAngleSequence/|Euler angle sequence]] and is given in their respective entries. One useful result is acheived by combining the angular velocity vector, in terms of Euler angles, with [[../EulersMomentEquations/|Euler's moment equations]] yielding the [[../DifferentialEquations/|differential equations]] of [[../CosmologicalConstant/|motion]] that characterizes [[../RigidBody/|rigid body]] motion for a given sequence.

Notation: $cos(\theta )=c_{\theta },sin(\theta )=s_{\theta }$

[[../EulerAngleVelocityOf123Sequence/|Euler angle velocity of 123 Sequence]]:

$\left[\begin{array}{c}\dot{\varphi }\\ \dot{\theta }\\ \dot{\psi }\end{array}\right]=\left[\begin{array}{c}({\omega }_x{c}_{\psi }-{\omega }_y{s}_{\psi })/c_{\theta }\\ {\omega }_x{s}_{\psi }+{\omega }_y{c}_{\psi }\\ (-{\omega }_x{c}_{\psi }+{\omega }_y{s}_{\psi })s_{\theta }/c_{\theta }+{\omega }_z\end{array}\right]$

[[../EulerAngleVelocityOf321Sequence/|Euler angle velocity of 321 Sequence]]:

$\left[\begin{array}{c}\dot{\varphi }\\ \dot{\theta }\\ \dot{\psi }\end{array}\right]=\left[\begin{array}{c}({\omega }_y{s}_{\psi }+{\omega }_z{c}_{\psi })sec(\theta )\\ {\omega }_y{c}_{\psi }-{\omega }_z{s}_{\psi }\\ {\omega }_x+{\omega }_y{s}_{\psi }{t}_{\theta }+{\omega }_z{c}_{\psi }{t}_{\theta }\end{array}\right]$