The equations of motion for a rigid flight vehicle, expressed as the translation of its center of mass and rotation
about the center of mass, decouple into the following two vector equations:

(2.1)** F** = m d**V**/dt

(2.2) **M** = d**h**/dt

where F is the applied force and M is the moment. (Recall bold quantities are vectors.)

**V** is the velocity vector: **V** = {U, V, W} and **h** is the angular momentum.

We can write h as the product of the inertia tensor and the angular velocity:

(2.3) **h** = [I] **w**, where **w** = {P,Q,R}

and [I] is the inertia tensor which is defined as follows:

The individual components of the inertia tensor are defined as:

and similarly for the other components.

The components of **h** are therefore:

(2.4a) h_{x} = I_{xx} P - I_{xy} Q - I_{xz} R

(2.4b) h_{y} = -I_{yx} P + I_{yy} Q - I_{yz} R

(2.4c) h_{z} = -I_{zx} P - I_{zy} Q + I_{zz} R

Since we want to write the equations of motion in a reference frame that is fixed to the body, we can write the
time derivatives in the inertial frame in terms of those in body axes as follows:

(2.5) d**p**/dt = *d***p***/dt* + **w** x
**p**

where the italicized *d/dt* represents a derivative in the rotating frame.

(Recall that a x b = {a_{y} b_{z} - a_{z} b_{y}, a_{z} b_{x} -
a_{x} b_{z}, a_{x} b_{y} - a_{y} b_{x}} )

Using (2.5) we can rewrite the equations of motion in (2.1) as follows:

(2.6) **F** = m *d***V***/dt* + m **w** x
**V**

or as components:

Fx = m [ *d*U*/dt* + QW - RV]

Fy = m [ *d*V*/dt* + RU - PW]

Fz = m [ *d*W*/dt* + PV - QU]

The moment equations (2.2) become:

**M** = d**h**/dt + **w** x **h**

or:

M_{x} = dh_{x}/dt + Q h_{z} - R h_{y}

M_{y} = dh_{y}/dt + R h_{x} - P h_{z}

M_{z} = dh_{z}/dt +P h_{y} - Q h_{x}

Using the definition of h in 2.4:

**M** = d**h**/dt + **w** x **h**

or

(2.7) **M** = [I] d**w**/dt + **w** x
([I]**w**)

Equations 2.6 and 2.7 may be rewritten for the purposes of simulation or control as:

*d***V***/dt* = **F** / m - **w** x **V**

*d***w***/dt* = [I]^{-1}**M + **[I]^{-1 }(**w**
x [I]**w**)

These nonlinear, coupled differential equations can be integrated in time, or linearized for use in control system
design. The problem then becomes estimating the force and moment vectors which are themselves complex functions
of the vehicle state.

We note that the above equations are simplified insofar as they ignore the angular momentum that may arise from
rotating propellers, for example. We have assumed six degrees of freedom here and actual problems may involve additional
states associated with control surface motion, structural deflections, or propulsion system dynamics. These additional
degrees of freedom may be added to form a more general dynamical equation that may be written in state vector form
as:

*d***X***/dt* = **f**(**X**)

WritingX= {U, V, W, P, Q, R, F, Q, Y}, we can write the complete equations of motion out in component form as: mdu/dt= Fx - m Q W + m R V mdv/dt= Fy - m R U + m P W mdw/dt= Fz - m P V + m Q U I_{xx}dP/dt- I_{xz}dR/dt- I_{xy}dQ/dt= L + I_{yz}(Q^{2}- R^{2}) + I_{xz}PQ - I_{xy}RP + (I_{yy}-I_{zz}) QR I_{yy}dQ/dt- I_{xy}dP/dt- I_{yz}dR/dt= M + I_{xz}(R^{2}- P^{2}) + I_{xy}QR - I_{yz}PQ + (I_{zz}-I_{xx}) RP I_{zz}dR/dt- I_{yz}dQ/dt- I_{xz}dP/dt= N + I_{xy}(P^{2}- Q^{2}) + I_{yz}RP - I_{xz}QR + (I_{xx}-I_{yy}) PQdF/dt= P + Q sinF tanQ + R cosF tanQdQ/dt= Q cosF - R sinFdY/dt= Q sinF secQ + R cosF secQ Or in matrix form: [A]X=f(X) Where: [A] = m 0 0 0 0 0 0 0 0 0 m 0 0 0 0 0 0 0 0 0 m 0 0 0 0 0 0 0 0 0 Ixx -Ixy -Ixz 0 0 0 0 0 0 -Ixy Iyy -Iyz 0 0 0 0 0 0 -Ixz -Iyz Izz 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 Andf(X) = { Fx - m Q W + m R V, Fy - m R U + m P W, Fz - m P V + m Q U, L + I_{yz}(Q^{2}- R^{2}) + I_{xz}PQ - I_{xy}RP + (I_{yy}-I_{zz}) QR, M + I_{xz}(R^{2}- P^{2}) + I_{xy}QR - I_{yz}PQ + (I_{zz}-I_{xx}) RP, N + I_{xy}(P^{2}- Q^{2}) + I_{yz}RP - I_{xz}QR + (I_{xx}-I_{yy}) PQ, P + Q sinF tanQ + R cosF tanQ, Q cosF - R sinF, Q sinF secQ + R cosF secQ }