# IEEE Systems, Man and Cybernetics Magazine - October 2020 - 26

```Figure 1. The solid lines were implemented in the research,
the dotted lines are to be implemented in future research.

Ω btb = Ω bib - Ω bie . Propagating the PVA via this set of equations will result in drift due to the sensors' errors.
Thermal effects, quantization, and other imperfections
cause errors in the inertial sensors. These may be categorized as bias, scale factor, misalignment as well as noise
errors. In summary, a medium-fidelity accelerometer measurement model might be expressed as

Inertial navigation aims at furnishing the system with
PVA by processing measurements from accelerometers
and gyroscopes. Accelerometers provide a measurement
of specific force from which acceleration can be computed, and gyroscopes provide a measurement of angular
(2)
fuibb = f bib + Df bib = b a + (I + M a) f ibb + w a
velocity. Gyroscope measurements are used recursively to
compute the system's attitude that, in turn, is used to
and a gyroscope model as
resolve the acceleration in the proper direction and evenu bib = ~ bib + D~ bib = b g + (I + M g) ~ bib + G g f bib + w g,
tually compute the position. As those sensors measure
(3)
~
forces acting on the system with respect to inertial, it is
critical to understand the various coordinate frames
where fu bib denotes a measurement of true specific
involved and eventually describe the PVA with respect to
force f ibb cor r upt ed by mea su rement er ror Df bib .
the coordinate frame of interest.
Similarly, for gyroscopes, ~u bib
Figure 2 presents an illustration of
denotes a measurement of true
the four pr incipal coordinate
angular velocity ~ bib corrupted
frames of interest in the context of
by measurement error D~ bib . The
Model validity is
term M captures both scale facaccomplished by
In Figure 2, the Ear th-centor and misalignment errors, w
tered inertial (ECI) coordinate
represents the noise terms, b is
finding the composite
frame has its origin at the center
bias errors, and G g represents
difference between
of mass of Earth with the z-axis
the gyro sensitivity due to gravthe models.
^ z i h pointing along the nominal
ity (g-sensitivity effects). This
model is used to create a realisaxis of rotation of Earth (i.e.,
tic representation of the senpoints to true north), the x-axis
^ x i h is defined by the intersecs or s' per for ma nce w ith i n the
simution environment.
tion of the equatorial plane and Earth-Sun orbital plane at
Errors in the specific force and angular velocity meathe vernal equinox, and the y axis ^y ih completes a rightsurements
result in errors in PVA. These errors can be
hand coordinate system.
expressed as
The EC Earth-fixed (ECEF) frame (e-frame) shares the
same origin as the ECI frame but is fixed with respect to Earth.
J . tN
R - Ωt
The ECI and ECEF frames differ only by a rotation about their
0 3 x 3 0 3 x 3VWJd} ttbN
ie
Kd} tbO
S
K
O
t
t
b
common z-axis at a rate of approximately 15°/h ^~ ie h .
K dvo tb O = S- [Ct b ft ib #] - 2Ω tie dg tbWK dv ttb O
K
O
K
O
S
W
t
t
The tangential frame (t-frame) is a convenient, locally
0 3x3
I 3 x 3 0 3 x 3 L dr tb P
L dro tb P
TR
X
tV
t
level nor t hea st - dow n coord i n at e s y st em a nd i s
S 0 C bW df bib
(4)
+ SCt tb 0 Wd b n,
S 0 0 W d~ ib
the e-frame). Finally, the body frame, attached to the
T
X
platform of interest, shares a common origin with the
tangential frame and has its x-axis ^x b h pointing "forwhere the error of gravity is [3]
ward" and its z-axis pointing loosely "up," as illustrated
e
t
in Figure 2.
t te 2g 0 (L b) rteeb 2 (rt eeb) T C
t et dr ttb,
dg tb . C
(5)
e
t
t
|
r
b |
e
Equation 1 represents the navigation equations in the
r eS (L b)
t-frame:
and df bib and d~ bib are the estimates of the errors in the
R ro t V R
V
t
v
tb
accelerometers and the gyroscopes, respectively. These
S tb W S
W
Svo ttbW = SC tb f ibb + g tb - 2Ω tie v ttbW,
(1)
errors are typically modeled as a combination white GaussS o tW S
W
Cb
C tb Ω btb
ian noise and first-order Markov [15], [16]. As such, the
T X T
X
state model is augmented to model these additional errors
t
t
[3], [14]; in this article, closed-loop loosely coupled configuwhere r tb and v tb are the position and velocity of the
ration [17]. In this configuration, the measurements used in
body, respectively, with respect to the tangential frame.
the Kalman filter are the difference between nondrifting
Also, w ibb = w ibe + w tbb and Ω form the skew-symmetric
t
sensor measurements, e.g., RSSI measurements and GPS
matrix from ~, C b is the direction cosine matrix of the
position, and the drifting PVA estimation from the INS.
body w ith respect to the ta ngentia l fra me, a nd
26

IEEE SYSTEMS, MAN, & CYBERNETICS MAGAZINE O ctober 2020

```

# IEEE Systems, Man and Cybernetics Magazine - October 2020

## Table of Contents for the Digital Edition of IEEE Systems, Man and Cybernetics Magazine - October 2020

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