10.7: Basis for Simulation
- Page ID
- 50380
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Except for external forces and moments \(\vec{F}\) and \(\vec{M}\), we now have the necessary terms for writing a full nonlinear simulation of a rigid body, in body coordinates. There are twelve states, comprising the following components:
- \(\vec{v}_o\), the vector of body-reference velocities
- \(\vec{\omega}\), body rotation rate vector
- \(\vec{x}\), location of the body origin, in inertial space
- \(\vec{E}\), Euler angle vector
The derivatives of body-referenced velocity and rotation rate come from our equations for linear and angular momentum, with some coupling that generally requires a \(6 \times 6\) matrix inverse. The Cartesian position propagates according to
\[ \dot{\vec{x}} \, = \, R^T (\vec{E}) \vec{v}_o, \]
while the Euler angles follow: \[ \dot{\vec{E}} \, = \, \Gamma (\vec{E}) \vec{\omega}. \]