2.2 Equations of motion

2.2.1 Derivation

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 dV/dt

(2.2) M = dh/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) hx = Ixx P - Ixy Q - Ixz R
(2.4b) hy = -Iyx P + Iyy Q - Iyz R
(2.4c) hz = -Izx P - Izy Q + Izz 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) dp/dt = dp/dt + w x p

where the italicized d/dt represents a derivative in the rotating frame.
(Recall that a x b = {ay bz - az by, az bx - ax bz, ax by - ay bx} )

Using (2.5) we can rewrite the equations of motion in (2.1) as follows:
(2.6) F = m dV/dt + m w x V

or as components:
Fx = m [ dU/dt + QW - RV]
Fy = m [ dV/dt + RU - PW]
Fz = m [ dW/dt + PV - QU]

The moment equations (2.2) become:
M = dh/dt + w x h
or:
Mx = dhx/dt + Q hz - R hy
My = dhy/dt + R hx - P hz
Mz = dhz/dt +P hy - Q hx

Using the definition of h in 2.4:
M = dh/dt + w x h
or
(2.7) M = [I] dw/dt + w x ([I]w)

Equations 2.6 and 2.7 may be rewritten for the purposes of simulation or control as:
dV/dt = F / m - w x V
dw/dt = [I]-1M + [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:
dX/dt = f(X)

2.2.2 Expanded Versions of 6DOF EOM's in Component Form

Writing X = {U, V, W, P, Q, R, F, Q, Y}, we can write the complete equations of motion out in component form as:

m du/dt = Fx - m Q W + m R V
m dv/dt = Fy - m R U + m P W
m dw/dt = Fz - m P V + m Q U

Ixx dP/dt - Ixz dR/dt - Ixy dQ/dt = L + Iyz (Q2 - R2) + Ixz PQ - Ixy RP + (Iyy-Izz) QR
Iyy dQ/dt - Ixy dP/dt - Iyz dR/dt = M + Ixz (R2 - P2) + Ixy QR - Iyz PQ + (Izz-Ixx) RP
Izz dR/dt - Iyz dQ/dt - Ixz dP/dt = N + Ixy (P2 - Q2) + Iyz RP - Ixz QR + (Ixx-Iyy) PQ

dF/dt = P + Q sinF tanQ + R cosF tanQ
dQ/dt = Q cosF - R sinF
dY/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


And f(X) = {
Fx - m Q W + m R V,
Fy - m R U + m P W,
Fz - m P V + m Q U,

L + Iyz (Q2 - R2) + Ixz PQ - Ixy RP + (Iyy-Izz) QR,
M + Ixz (R2 - P2) + Ixy QR - Iyz PQ + (Izz-Ixx) RP,
N + Ixy (P2 - Q2) + Iyz RP - Ixz QR + (Ixx-Iyy) PQ,

P + Q sinF tanQ + R cosF tanQ,
Q cosF - R sinF,
Q sinF secQ + R cosF secQ
}