Page 184 - Innovations in Intelligent Machines

176 R.W. Beard
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χ = tan −1 V a sin ψ cos γ + w e .
V a cos ψ cos γ + w n
The kinematic evolution of the Euler angles are given by [22]
˙
⎛ ⎞ ⎛ ⎞ ⎛ ⎞
φ 1sin(φ) tan(θ)cos(φ) tan(θ) p
θ = ⎝ 0 cos(φ) − sin(φ) , (3)
⎝ ˙⎠ ⎠ ⎝ q ⎠
ψ ˙ 0sin(φ) sec(θ)cos(φ) sec(θ) r
and the navigational equations of motion are given by
˙ r n = V a cos ψ cos γ + w n = V g cos χ cos γ (4)
˙ r e = V a sin ψ cos γ + w e = V g sin χ cos γ (5)
˙
h = V a sin γ. (6)

2 Sensor Models

This section derives mathematical models for sensors typically found on small
and micro UAVs. In particular, we discuss rate gyros, accelerometers, pressure
sensors, and GPS sensors.

2.1 Rate Gyros

A MEMS rate gyro contains a small vibrating lever. When the lever undergoes
an angular rotation, Coriolis effects change the frequency of the vibration, thus
detecting the rotation. A brief description of the physics of rate gyros can be
found in Ref [9, 15, 23].
The output of the rate gyro is given by
y gyro = k gyro ω + β gyro (T)+ η gyro ,
where y gyro is in Volts, k gyro is a gain, ω is the angular rate in radians per
second, β gyro is a temperature dependent bias term, and η gyro is a zero mean
Gaussian process with known variance. The bias term β gyro (T) is a function of
the temperature T and can be effectively determined by use of a temperature
chamber before flight.
If three rate gyros are aligned along the x, y,and z axes of the UAV, then
the rate gyros measure the angular body rates p, q,and r as follows:

y gyro,x = k gyro,x p + β gyro,x (T)+ η gyro,x
y gyro,y = k gyro,y q + β gyro,y (T)+ η gyro,y
y gyro,z = k gyro,z r + β gyro,z (T)+ η gyro,z .
We will assume that k gyro,∗ , β gyro,∗ (T), and the covariance of η gyro,∗ have
been determined a priori and are known in-flight. MEMS gyros are analog
devices that are sampled by the on-board processer. We will assume that the
sample rate is given by T s . As an example, the Procerus Kestrel autopilot
samples its rate gyros at approximately 120 Hz.