Aerospace and Electronic Systems Magazine August 2017 - 25
Kim et al.
where the components are defined as follows:
UPDATE
C
Vehicle position1 is p = [x, y, z]T
C
Vehicle velocity is v = [υx, υy, υz]
C
Vehicle attitude (Euler angles) is ψ = [ϕ, θ, ψ]T
C
Accelerometer bias2 is b a = bxa , bya , bza
C
Gyroscope bias is b g = bxg , byg , bzg
C
Receiver clock bias is ctb, with c being the speed of light
C
Receiver clock drift is ctd
C
Map position is mi = [mix, miy, miz]T
C
Accelerometer measurement is f = [ fx, fy, fz]T
C
Gyroscope measurement is ω = [ωx, ωy, ωz]T
The measurement consists of either GNSS (pseudorange ρi and
pseudorange rate ρ i) from an ith satellite vehicle or range rkj,
bearing φkj, and elevation θ kj from a jth landmark feature as follows:
T
T
i i i T
zk = ρ k ρ k , for an ith satellite vehicle, or
zk =
T
zkj = rkj φkj θ kj , for a jth feature
T
The observation model of the pseudorange and the pseudorange
rate, respectively, is
ρ ki =
(X
i
−x
) + (Y
2
i
−y
) + (Z
2
i
−z
)
2
+ ctb + υ ρ
ρ ki = (Vxi − υ x ) lx + (Vyi − υ y ) l y + (Vzi − υ z ) lz + ctd + υ ρ ,
PREDICTION
The estimates are propagated using the state transition model,
which can be decoupled into a nonlinear part and a linear part
as
T
where [X i, Y i, Z i]T and Vxi ,Vyi ,Vzi are the ith satellite vehicle's
position and velocity, respectively, defined in the local NED coT
pˆ k + υˆk ΔT
pˆ k +1
n
n
ˆ
ˆ υˆk + Cb , k f k − ba , k + g ΔT
υ k +1
ψˆ k +1
n
ψˆ k + Eb , k ω k − bˆg , k ΔT
bˆa , k +1
bˆa , k
=
,
ˆ
bg , k +1
ˆ
bg , k
ctˆ
b , k +1
ctˆb , k + ctˆb , k ΔT
ˆ
ctd , k +1
ˆ
ctd , k
ˆ
m
k +1
ˆ
m
k
(
(
)
)
(7)
T
compute the relative position vector u j = u xj , u yj , u zj of the jth
feature from the vehicle and then transform it to the sensor-coordinate frame (refer to [32] for more details). The Cartesian-to-polar
coordinate transformation yields the observation
with ΔT being the prediction sampling time and gn being the gravitational vector [0, 0, −9.8 m/s2]T.
This is a simplified model of a strapdown inertial navigation
system targeting low-cost inertial sensors with accelerometer biases on the order of 2 mg and gyroscope biases on the order of
100°/h. Although the developed simulator (presented later in this
article) internally simulates Earth and the navigational frame rotation, they are not included in this model because of the low quality
of the sensor used. In addition, the effect of Earth's curvature on
the attitude compensation, called Schuler tuning, is ignored in the
model, because we are considering the small flight coverage in the
simulation.
We use the UKF to predict the state by propagating the sigma
points; for detailed implementation, refer to [21] and [31]. One of
key aspect is that the Jacobian matrix is inferred from the sigma
points rather than linearized, as discussed in our previous work
[22].
1
2
A NED navigation frame is used in this work.
A roll-pitch-yaw body frame is used in this work.
AUGUST 2017
ordinate system, and lxi , l yi , lzi are the line-of-sight vector of the
satellite vehicle from the user vehicle, with υρ and υ ρ being the
pseudorange and the pseudorange rate noise, respectively. In this
observation model, the ionospheric and tropospheric delay errors,
as well as any multipath error, are not simulated and thus are not
included in the filter model.
For the range, bearing, and elevation measurement, we first
(u ) + (u ) + (u )
= arctan ( u u ) + υ
rkj =
φ
j
k
j
x
2
θ kj = arctan
2
j
y
j
y
j
x
j
z
(u ) + (u )
j
x
+ υr
(8)
φ
u zj
2
2
j
y
2
+ υθ ,
with υr, υϕ, and υθ being the range, bearing, and elevation noise,
respectively.
At the update stage, a set of sigma points is selected from the
current state (typically, the control input is not related to the measurement and thus not included as the sigma points). The sigma
points are then propagated through the nonlinear observation models to predict the observations, thus generating the innovation and
the filter update. As in the prediction step, the Jacobian is inferred
and used to compute the cross-correlation term, which is required
to update the filter.
If a new (m + 1)th landmark is detected, the sigma points are
also used to predict the new feature position and the new crosscorrelation between the feature and the state [22], [31].
IEEE A&E SYSTEMS MAGAZINE
25
Table of Contents for the Digital Edition of Aerospace and Electronic Systems Magazine August 2017
No label
Aerospace and Electronic Systems Magazine August 2017 - No label
Aerospace and Electronic Systems Magazine August 2017 - Cover2
Aerospace and Electronic Systems Magazine August 2017 - 1
Aerospace and Electronic Systems Magazine August 2017 - 2
Aerospace and Electronic Systems Magazine August 2017 - 3
Aerospace and Electronic Systems Magazine August 2017 - 4
Aerospace and Electronic Systems Magazine August 2017 - 5
Aerospace and Electronic Systems Magazine August 2017 - 6
Aerospace and Electronic Systems Magazine August 2017 - 7
Aerospace and Electronic Systems Magazine August 2017 - 8
Aerospace and Electronic Systems Magazine August 2017 - 9
Aerospace and Electronic Systems Magazine August 2017 - 10
Aerospace and Electronic Systems Magazine August 2017 - 11
Aerospace and Electronic Systems Magazine August 2017 - 12
Aerospace and Electronic Systems Magazine August 2017 - 13
Aerospace and Electronic Systems Magazine August 2017 - 14
Aerospace and Electronic Systems Magazine August 2017 - 15
Aerospace and Electronic Systems Magazine August 2017 - 16
Aerospace and Electronic Systems Magazine August 2017 - 17
Aerospace and Electronic Systems Magazine August 2017 - 18
Aerospace and Electronic Systems Magazine August 2017 - 19
Aerospace and Electronic Systems Magazine August 2017 - 20
Aerospace and Electronic Systems Magazine August 2017 - 21
Aerospace and Electronic Systems Magazine August 2017 - 22
Aerospace and Electronic Systems Magazine August 2017 - 23
Aerospace and Electronic Systems Magazine August 2017 - 24
Aerospace and Electronic Systems Magazine August 2017 - 25
Aerospace and Electronic Systems Magazine August 2017 - 26
Aerospace and Electronic Systems Magazine August 2017 - 27
Aerospace and Electronic Systems Magazine August 2017 - 28
Aerospace and Electronic Systems Magazine August 2017 - 29
Aerospace and Electronic Systems Magazine August 2017 - 30
Aerospace and Electronic Systems Magazine August 2017 - 31
Aerospace and Electronic Systems Magazine August 2017 - 32
Aerospace and Electronic Systems Magazine August 2017 - 33
Aerospace and Electronic Systems Magazine August 2017 - 34
Aerospace and Electronic Systems Magazine August 2017 - 35
Aerospace and Electronic Systems Magazine August 2017 - 36
Aerospace and Electronic Systems Magazine August 2017 - 37
Aerospace and Electronic Systems Magazine August 2017 - 38
Aerospace and Electronic Systems Magazine August 2017 - 39
Aerospace and Electronic Systems Magazine August 2017 - 40
Aerospace and Electronic Systems Magazine August 2017 - 41
Aerospace and Electronic Systems Magazine August 2017 - 42
Aerospace and Electronic Systems Magazine August 2017 - 43
Aerospace and Electronic Systems Magazine August 2017 - 44
Aerospace and Electronic Systems Magazine August 2017 - 45
Aerospace and Electronic Systems Magazine August 2017 - 46
Aerospace and Electronic Systems Magazine August 2017 - 47
Aerospace and Electronic Systems Magazine August 2017 - 48
Aerospace and Electronic Systems Magazine August 2017 - 49
Aerospace and Electronic Systems Magazine August 2017 - 50
Aerospace and Electronic Systems Magazine August 2017 - 51
Aerospace and Electronic Systems Magazine August 2017 - 52
Aerospace and Electronic Systems Magazine August 2017 - 53
Aerospace and Electronic Systems Magazine August 2017 - 54
Aerospace and Electronic Systems Magazine August 2017 - 55
Aerospace and Electronic Systems Magazine August 2017 - 56
Aerospace and Electronic Systems Magazine August 2017 - 57
Aerospace and Electronic Systems Magazine August 2017 - 58
Aerospace and Electronic Systems Magazine August 2017 - 59
Aerospace and Electronic Systems Magazine August 2017 - 60
Aerospace and Electronic Systems Magazine August 2017 - 61
Aerospace and Electronic Systems Magazine August 2017 - 62
Aerospace and Electronic Systems Magazine August 2017 - 63
Aerospace and Electronic Systems Magazine August 2017 - 64
Aerospace and Electronic Systems Magazine August 2017 - 65
Aerospace and Electronic Systems Magazine August 2017 - 66
Aerospace and Electronic Systems Magazine August 2017 - 67
Aerospace and Electronic Systems Magazine August 2017 - 68
Aerospace and Electronic Systems Magazine August 2017 - 69
Aerospace and Electronic Systems Magazine August 2017 - 70
Aerospace and Electronic Systems Magazine August 2017 - 71
Aerospace and Electronic Systems Magazine August 2017 - 72
Aerospace and Electronic Systems Magazine August 2017 - 73
Aerospace and Electronic Systems Magazine August 2017 - 74
Aerospace and Electronic Systems Magazine August 2017 - 75
Aerospace and Electronic Systems Magazine August 2017 - 76
Aerospace and Electronic Systems Magazine August 2017 - 77
Aerospace and Electronic Systems Magazine August 2017 - 78
Aerospace and Electronic Systems Magazine August 2017 - 79
Aerospace and Electronic Systems Magazine August 2017 - 80
Aerospace and Electronic Systems Magazine August 2017 - Cover3
Aerospace and Electronic Systems Magazine August 2017 - Cover4
http://www.brightcopy.net/allen/aesm/34-2s
http://www.brightcopy.net/allen/aesm/34-2
http://www.brightcopy.net/allen/aesm/34-1
http://www.brightcopy.net/allen/aesm/33-12
http://www.brightcopy.net/allen/aesm/33-11
http://www.brightcopy.net/allen/aesm/33-10
http://www.brightcopy.net/allen/aesm/33-09
http://www.brightcopy.net/allen/aesm/33-8
http://www.brightcopy.net/allen/aesm/33-7
http://www.brightcopy.net/allen/aesm/33-5
http://www.brightcopy.net/allen/aesm/33-4
http://www.brightcopy.net/allen/aesm/33-3
http://www.brightcopy.net/allen/aesm/33-2
http://www.brightcopy.net/allen/aesm/33-1
http://www.brightcopy.net/allen/aesm/32-10
http://www.brightcopy.net/allen/aesm/32-12
http://www.brightcopy.net/allen/aesm/32-9
http://www.brightcopy.net/allen/aesm/32-11
http://www.brightcopy.net/allen/aesm/32-8
http://www.brightcopy.net/allen/aesm/32-7s
http://www.brightcopy.net/allen/aesm/32-7
http://www.brightcopy.net/allen/aesm/32-6
http://www.brightcopy.net/allen/aesm/32-5
http://www.brightcopy.net/allen/aesm/32-4
http://www.brightcopy.net/allen/aesm/32-3
http://www.brightcopy.net/allen/aesm/32-2
http://www.brightcopy.net/allen/aesm/32-1
http://www.brightcopy.net/allen/aesm/31-12
http://www.brightcopy.net/allen/aesm/31-11s
http://www.brightcopy.net/allen/aesm/31-11
http://www.brightcopy.net/allen/aesm/31-10
http://www.brightcopy.net/allen/aesm/31-9
http://www.brightcopy.net/allen/aesm/31-8
http://www.brightcopy.net/allen/aesm/31-7
https://www.nxtbookmedia.com