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Although in digital simulations, this exact number has an almost zero probability of
occurrence, this may still cause significant computational errors in the vicinity of this
singularity. Second, the Euler angles may integrate up to values outside the normal
±90 range of pitch and the normal ±180 range of roll and yaw angles, which may
make it difficult to determine the attitude uniquely. And finally, the equations are
linear in p, q, and r, but nonlinear in terms of the Euler angles themselves [33].
There are several other ways besides the Euler angles to represent the orientation
of a rotated coordinate frame. These methods, which involve four, five, or even six
variables instead of the three Euler angles, aim to avoid the singularity of the Euler
angle representation, and maximize the speed of computer processing in navigation
applications. The most common of these methods is the so-called quaternion representation,
which uses four variables. For a detailed discussion about this method, refer
to ref.[33]. In this report, we will limit ourselves to the Euler angle representation,
because it is still the most commonly used method for aircraft simulations.
2.5. Assembling the state equations 25
2.5 Assembling the state equations
To build the nonlinear state-space model we start with the six differential equations
that were derived from the basic force and moment equations:
˙V
= 1
m
􀀀
Fx cos a cos b + Fy sin b + Fz sin a cos b


˙a
= 1
V cos b

1
m
(−Fx sin a + Fz cos a)

+ q − (p cos a + r sin a) tan b
˙b
= 1
V

1
m
(−Fx cos a sin b + Fy cos b − Fz sin a sin b)

+ p sin a − r cos a
p˙ = Pppp2 + Ppqpq + Ppr pr + Pqqq2 + Pqrqr + Prrr2 + PlL + PmM+ PnN + p˙0
q˙ = Qppp2 + Qpqpq + Qpr pr + Qqqq2 + Qqrqr + Qrrr2 + QlL + QmM+ QnN + q˙0
˙ r = Rppp2 + Rpqpq + Rpr pr + Rqqq2 + Rqrqr + Rrrr2 + RlL + RmM+ RnN + ˙ r0
(2.63)
The variables V, a, b, p, q, and r, which represent the linear and angular velocities
of the aircraft can be regarded as the state variables from this model. The state-space
model would already be complete if the body-axes components of the external forces
and moments could be treated as independent input variables, but unfortunately,
the external forces and moments themselves depend on the motion variables of the
aircraft. Because of this, the state variables need to be coupled back into the force
and moments equations.
As explained in section 2.4, the attitude of the airplane and its altitude are needed
to determine the gravitational, aerodynamical, and propulsive forces and moments.
This means that the model needs to be extended with the equations for the Euler
angles and the altitude. The aircraft’s horizontal coordinates relative to the surface
of the Earth are not needed to solve the equations of motion, but they are included
for practical purposes. This yields an additional six state equations:
˙y
= q sin j + r cos j
cos q
˙q
= q cos j − r sin j
˙j
= p + (q sin j + r cos j) tan q = p + ˙y sin q
x˙e = {ue cos q + (ve sin j + we cos j) sin q} cos y − (ve cos j − we sin j) sin y
y˙e = {ue cos q + (ve sin j + we cos j) sin q} sin y + (ve cos j − we sin j) cos y
˙H
= ue sin q − (ve sin j + we cos j) cos q
(2.64)
with new state variables y, q, j, xe, ye, and H. These twelve state variables are
combined in the state vector x:
x = [ V a b p q r y q j xe ye H ] T (2.65)
and the resulting equations are combined in a single vector equation:
˙x = f0 (x, Ftot(t),Mtot(t)) (2.66)
26 Chapter 2. Rigid body equations of motion
If we also collect any external control inputs in the input vector u, and any external
disturbances in the vector v, we can express the external forces and moments as
nonlinear functions of x(t), u(t), and v(t). In addition, the forces and moments may
possibly depend directly on time itself (e.g. the fuel-burn decreases the weight of the
airplane in time). Hence:
Ftot(t) = g1 (x(t), u(t), v(t), t)
Mtot(t) = g2 (x(t), u(t), v(t), t) (2.67)
Substituting in equation (2.66) yields the nonlinear state-space system:
x˙ (t) = f (x(t), u(t), v(t), t) (2.68)
In some cases, the forces and/or moments not only depend on the state vector x but
also on its time-derivative x˙ , which causes equation (2.68) to become implicit, as x˙
now appears on both sides of the equation:
x˙ (t) = f(x(t), x˙ (t), u(t), v(t), t) (2.69)
Luckily, this implicit relation can often be written as:
x˙ (t) = f1(x(t), u(t), v(t), t) + f2(x˙ (t), t) (2.70)
　
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