Page 185 - Innovations in Intelligent Machines

State Estimation for Micro Air Vehicles 177
2.2 Accelerometers
A MEMS accelerometer contains a small plate attached to torsion levers. The
plate rotates under acceleration which changes the capacitance between the
plate and the surrounding walls. The change in capacitance is proportional to
the linear acceleration [1, 23].
The output of the accelerometers is given by
y acc = k acc a + β acc (T)+ η acc ,

where y acc is in Volts, k acc is a gain, a is the acceleration in meters per second
squared, β acc is a temperature dependent bias term, and η acc is zero mean
Gaussian noise with known variance.
Accelerometers measure the specific force in the body frame of the vehicle.
A physically intuitive explanation is given in [22, p. 13–15]. An additional
explanation is given in [19, p. 27]. Mathematically we have
⎛ ⎞
a x
1
⎝ a y ⎠ = (F − F gravity )
m
a z
1
= ˙ v + ω × v − F gravity .
m
In component form we have
a x =˙u + qw − rv + g sin θ
a y =˙v + ru − pw − g cos θ sin φ
a z =˙w + pv − qu − g cos θ cos φ.
The output of an accelerometer is usually in units of [g], therefore k acc =
1/g. The output of the accelerometers are therefore given by
˙ u + qw − rv + g sin θ
y acc,x = + β acc,x (T)+ η acc,x
g
˙ v + ru − pw − g cos θ sin φ
y acc,y = + β acc,y (T)+ η acc,y (7)
g
˙ w + pv − qu − g cos θ cos φ
y acc,z = + β acc,z (T)+ η acc,z . (8)
g
As with the rate gyros, we will assume that the biases and noise statistics are
known and available in-flight. MEMS accelerometers are analog devices that
are sampled by the on-board processer. We will assume that the sample rate
is given by T s .

2.3 Pressure Sensors

Small autopilots typically have two pressure sensors: a static pressure sensor
which is used to measure altitude, and a dynamic pressure sensor which is
used to measure airspeed. These sensors will be discussed in the following two
sections.