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Chapter 3 / Sensors / Gyroscopes
_ _
The configurations shown in Figure 3-26 and Figure 3-27 contain 6 of the
possible 12 sensor orientations (6 more are obtained by rotating each input
axis through 180°). The negative signs in these figures indicate that the
positive sensor axis is in the opposite direction to that shown.
For each sensor, let the (X,Y,Z) Missile Body axes be aligned with the
Input-Output-Spin (I,O,S) sensor axes such that a clockwise spinning rotor
about the Spin axis implies positive output and input angular rates. The
configurations in Figure 3-26 and Figure 3-27 minimise torque due to
missile lateral and longitudinal acceleration respectively. For long-range
missiles the spin axis of the roll gyroscope should be aligned with Z B since
roll motion is often restricted and pitch manoeuvres are rare, except when
terrain following. For the orientation shown in Figure 3-26 the mapping of
the case inputs to the (I,O,S) axes for mechanical RPY gyroscopes is,
Μ
R
D
: =
⎡ 1
⎢
⎢
0
⎢⎣
0
0
1
0
0 ⎤
0
⎥
⎥
1 ⎥⎦
;
Μ
P
D
: =
3.4-6
⎡ 0
⎢
⎢
0
⎢⎣
1
1
0
0
0 ⎤
1
⎥
⎥
0 ⎥⎦
;
Μ
Y
D
: =
⎡ 0
⎢
⎢
0
⎢⎣
−1
0
1
0
1 ⎤
0
⎥
⎥
0 ⎥⎦
Equation 3.4-3
For optical instruments the mappings are simply the identity matrix since
these sensors are insensitive to acceleration induced errors.
3.4.3.3 Mechanical Sensor Cross-Coupling Error
Unlike optical gyroscopes mechanical instruments are also affected by
cross-coupling errors. Imperfect and finite bandwidth re-balancing of the
float assembly supporting the rotor causes misalignment of the Spin and
Input axes. The angular displacement of the float about the Output axis (γ)
is usually < 0.5°. The angular rate experienced by each gyroscope is thus,
⎛ Ii
⎡ 1 0 − γ
⎞
i ⎤ ω
⎜ C,
B ⎟
⎛ 0 ⎞
i ⎢
⎥
⎜ ⎟
Oi
∀ i ∈ { R , P , Y } ⇒ ωC,
B : = ⎢ 0 1 0 ⎥ ⋅ ⎜ ωC,
B ⎟ + ⎜ γ&
i ⎟
⎢
⎥ ⎜ Si ⎟ ⎜ ⎟
⎣
γ i 0 1
⎦ ⎝
ωC,
B ⎠ ⎝ 0 ⎠
Equation 3.4-4
The error induced by the angular rate of the float assembly is small
compared with some of the other error sources. The float angle is obtained
from the typical torquer model shown in Figure 3-28 using the parameters:
Float inertia J 2.5x10 -6 kg m 2
Float angular momentum H 0.005 kg m 2 /s
Viscous torque coefficient C 0.004 Nm/(rad/s)
Pulse-Width-Modulator (PWM) gain GSGPM 500 (rad/s)/rad
Torquer bandwidth 4000 rad/s