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Chapter 3 / Sensors / Accelerometers
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3.5.3.3 Sensor Cross Coupling Error
Imperfect and finite bandwidth re-balancing of the pendulum causes
misalignment of the Pendulum and Input axes. The angular displacement of
the pendulum about the Hinge axis (γ) is kept as small as possible
commensurate with pick-off scale factor. The specific acceleration
experienced by each accelerometer is thus,
∀
∈
{ R , P , Y }
⇒
⎛ Ii
⎡ 1 0 − γ ⎞
i ⎤ f
⎜ r, u ⎟
⎛ 0 ⎞
i ⎢
⎥
⎜ ⎟
Hi
f r, u : = ⎢ 0 1 0 ⎥ ⋅ ⎜ fr,
u ⎟ + ⎜ γ&
⎟
⎢
⎥ ⎜ Pi ⎟ ⎜ ⎟
⎣
γ i 0 1
⎦ ⎝
fr,
u ⎠ ⎝ 0 ⎠
i i
3.5-5
Equation 3.5-3
The pendulum forcer is similar in principle to the gyroscope torquer, with
typical PWM re-balancing loop bandwidths of 80-150 Hz and a damping of
0.6-0.8. The forcer can be reduced to the 2 nd order lag with range limiting
and quantisation described later. Applying a masking matrix [AXM] to
deal with the different Hinge axis orientations, and relating the Hinge axis to
individual sensor RPY axes,
f
U
r,
u
⎛ XR
f ⎞
⎜ r, u ⎟ ⎛ ⎡ YR XP XY ⎤ ⎞
XP
⎜
fr,
u fr,
u fr,
u
: = ⎜ f ⎟ − γ ⊗ [ ] ⋅
⎟
r, u
AXM
AC ⎜
⎢ ZR ZP YY ⎥
⎜ ⎟
⎟
XY
⎝ ⎢⎣
fr,
u fr,
u fr,
u ⎥⎦
⎠
⎝
fr,
u ⎠
AC
For the configuration selected,
γ
( ) T
γ , γ ,
: =
γ
AC
R
[ AXM ]
: =
AC
P
⎡ 0
⎢
⎢
1
⎢⎣
−1
1
0
0
⎤
⎥
⎥
⎥⎦
AC
Y
Equation 3.5-4
Equation 3.5-5
Equation 3.5-6
The cross-coupling misalignment angle (γ) is determined from the 2 nd order
PWM dynamics activated by setting bit 3 of MS_AC_ER.
3.5.4 Measured Specific Acceleration and Velocity Increments
Ignoring biases caused by pendulum angular velocity and acceleration,
Savage’s model of a single axis mechanical accelerometer expressed in
(I,H,P) axes reduces to,