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and origin at the rotor mass center. The angular velocities in equation (2.20)
are interpreted as those of the rotor relative to the airplane body axes [15]. If the
resultant relative angular momentum of all rotors is called h0, with components h0
x,
h0
y, and h0
z in FB, which are assumed to be constant, the total angular momentum of
an airplane with spinning rotors can be obtained by simply adding h0 to the angular
momentum of the airframe:
h = I · W + h0 (2.22)
The additional terms in the angular momentum cause certain extra terms, known as
gyroscopic couples, to appear in the moment equations, as we will see later.
2.1.4 Resulting general equations of motion for a rigid body
When we choose a reference frame fixed to the body (OXYZ = OXBYBZB) the inertial
moments and products from the equations (2.19) become constants. The reference
frame itself then rotates with angular velocity W. For an arbitrary position vector r
with respect to the body reference frame we can then write:
˙r = ¶r
¶t
+ W × r (2.23)
Applying equation (2.23) to the general force and moment equations for a rigid body,
(2.6) and (2.12), we find:
F = m
¶Vc.g.
¶t
+ W × Vc.g.
(2.24)
and:
Mc.g. = ¶h
¶t
+ W × h = ¶
¶t
I · W + h0
+ W ×
I · W + h0
=
= ¶
¶t
(I · W) + W × (I · W) + W × h0 (2.25)
The last term in this equation, W × h0, contains the gyroscopic couples, which take
into account the effect of spinning rotors. Notice that this derivation assumes the
resulting angular momentum of the rotors to be constant.
2.1. General rigid-body equations of motion 15
symbol definition
|I| Ixx Iyy Izz − 2Jxy Jxz Jyz − Ixx Jyz
2 − Iyy Jxz
2 − Izz Jxy
2
I1 Iyy Izz − Jyz
2
I2 Jxy Izz + Jyz Jxz
I3 Jxy Jyz + Iyy Jxz
I4 Ixx Izz − Jxz
2
I5 Ixx Jyz + Jxy Jxz
I6 Ixx Iyy − Jxy
2
Pl I1 / |I|
Pm I2 / |I|
Pn I3 / |I|
Ppp −(Jxz I2 − Jxy I3) / |I|
Ppq (Jxz I1 − Jyz I2 − (Iyy − Ixx)I3) / |I|
Ppr −(Jxy I1 + (Ixx − Izz)I2 − Jyz I3) / |I|
Pqq (Jyz I1 − Jxy I3) / |I|
Pqr −((Izz − Iyy)I1 − Jxy I2 + Jxz I3) / |I|
Prr −(Jyz I1 − Jxz I2) / |I|
Ql I2 / |I|
Qm I4 / |I|
Qn I5 / |I|
Qpp −(Jxz I4 − Jxy I5) / |I|
Qpq (Jxz I2 − Jyz I4 − (Iyy − Ixx)I5) / |I|
Qpr −(Jxy I2 + (Ixx − Izz)I4 − Jyz I5) / |I|
Qqq (Jyz I2 − Jxy I5) / |I|
Qqr −((Izz − Iyy)I2 − Jxy I4 + Jxz I5) / |I|
Qrr −(Jyz I2 − Jxz I4) / |I|
Rl I3 / |I|
Rm I5 / |I|
Rn I6 / |I|
Rpp −(Jxz I5 − Jxy I6) / |I|
Rpq (Jxz I3 − Jyz I5 − (Iyy − Ixx)I6) / |I|
Rpr −(Jxy I3 + (Ixx − Izz)I5 − Jyz I6) / |I|
Rqq (Jyz I3 − Jxy I6) / |I|
Rqr −((Izz − Iyy)I3 − Jxy I5 + Jxz I6) / |I|
Rrr −(Jyz I3 − Jxz I5) / |I|
Table 2.2: Definition of inertia coefficients
16 Chapter 2. Rigid body equations of motion
The vector-equations (2.24) and (2.25) form the basis for the development of the general
rigid-body dynamic model. In order to make these equations usable for control
system design and analysis, flight simulation, system identification, and other analytical
tasks, they need to be rewritten in nonlinear state-space format. This can be
achieved by moving the time-derivatives of the linear and angular velocities to the
left-hand side of these equations:
¶Vc.g.
¶t
= F
m
− W × Vc.g. (2.26)
¶W
¶t
= I−1
Mc.g. − W × I · W − W × h0
(2.27)
These equations can be written-out into their components along the body-axes, using
the following definitions for Vc.g., W, F, and Mc.g.:
Vc.g. = i u + j v + k w
W = i p + j q + k r
F = iFx + jFy + kFz
Mc.g. = iL + jM + kN
This yields:
u˙ = Fx
m
− qw + rv
v˙ =
Fy
m
+ pw − ru (2.28)
w˙ = Fz
m
− pv + qu
and:
p˙ = Ppp p2 + Ppq pq + Ppr pr + Pqqq2 + Pqrqr + Prrr2 + PlL + PmM+ PnN + p˙0
q˙ = Qpp p2 + Qpq pq + Qpr pr + Qqqq2 + Qqrqr + Qrrr2 + QlL + QmM+ QnN + q˙0
˙ r = Rpp p2 + Rpq pq + Rpr pr + Rqqq2 + Rqrqr + Rrrr2 + RlL + RmM+ RnN + ˙ r0
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