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Estimation with Applications to Tracking and Navigation-chapter1...
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Chapter 12
INTRODUCTION TO NAVIGATION
APPLICATIONS
12.1 INTRODUCTION
12.1.1 Navigation Applications - Outline
Navigation and target tracking are closely related.
0 Target tracking
is the estimation of the state of a moving object based
on remote observations of the object.
l
Navigation
is tracking of the “own-ship” - the platform on which the
sensor(s) is (are) located. It answers the question “Where am I?” by estimat-
ing one’s own state (position, velocity, attitude, and possibly acceleration).
Another problem, which incorporates both of the above, is
l
Guidance,
which is concerned with questions like “Which way leads to
the destination?” It dete~ines an appropriate sequence of actions to reach
a destination. Guidance is a con& problem based on the information
obtained from, typically, a navigation system and, when the destination
(target) is a moving object, a system that track this object.
The primary considerations for the development, choice, and design of a
navigation device include cost, high accuracy, self-containment, security, world-
wide capability, and high availability, among others. Certain navigation systems,
such as inertial systems, are capable of being a stand-alone means for navigation
mainly because they are self-contained. Other navigation devices can be used
only as navigation aids to improve the navigation accuracy.
Estimation and filtering techniques, especially Kalman filtering, have found
extensive application in navigation systems because they can significantly im-
prove their accuracy.
This chapter presents an introduction to the application of estimation and
filtering techniques in the Global Positioning System (GPS), inertial navigation
systems, and their integration.
491
492
12 INTRODUCTION TO NAVIGATION APPLICATIONS
12.1.2 Navigation Applications - Summary of Objectives
Describe complementary filtering for navigation systems using multiple sensors.
Discuss briefly Inertial Navigation Systems (INS)
l
State models
l
Sensor models
Describe the Global Positioning System (GPS)
l
The GPS principle
l
The GPS observables and their accuracies
l
State models
l
Measurement models
Example of IMM vs. KF for navigation with GPS.
Discuss integrated navigation and present several schemes for fusion of the
data from different sensors
l
Centralized estimation fusion
l
Decentralized estimation fusion
0 Use of complementary filtering for integration
12.2 COMPLEMENTARY FILTERING FOR NAVIGATION
A key estimation and filtering concept in the processing of navigation data is the
so-called
complementary filtering.
It is also one of the distinctive features
of the state estimators for navigation. The philosophy of complementary filtering
is best explained in the framework of Wiener filtering.
Given noisy measurements z(t) = s(t) + n(t) of a signal
s(t),
the problem
of Wiener filtering is to recover the signal s(t) from the measurement z(t)
optimally in the sense that the mean-square error between the recovered signal
and the true signal is minimized (see Section 9.5). One of the fundamental
assumptions of the Wiener filtering is that the signal
s(t)
is a stationary random
process with known power spectral density.
12.2.1 The Operation of Complementary Filtering
Consider the case
available,
namely,
where two independent measurements of the same signal are
xl (4
= s(t) + q(t)
(12.2.1-1)
22 (t>
= s(t) + n2(t)
(12.2.1-2)
If the signal
s(t)
can be modeled well as a random process with known
spectral characteristics, Wiener filtering techniques can be used, at least in
12.2 COMPLEMENTARY FILTERING FOR NAVIGATION
493
Figure 12.2.1-l: A form of complementary filter.
s(t) + w(t)
+-
* H(s) -
-
+
s(t) +
n2(t)
tg$
*
Y (4 = s(t)
Figure 12.2.2-l: An alternative form of complementary filter.
principle, to recover it optimally from the two noisy measurements x 1 (t) and
x2 w
If, however, the signal
s(t)
cannot be properly modeled as a random process
with known spectral characteristics, Wiener filtering techniques are not applica-
ble. Instead, a complementary filter can be used. This is illustrated in Fig.
12.2.1- 1. Note that with this architecture, the signal passes through the filter
without distortion regardless of what H(s) is, which can be designed to remove
the two noises as much as possible, if they have different spectral characteristics.
Specifically, H should attenuate
nl (t)
and let
n2(t)
pass as much as possible.
12.2.2 The Implementation of Complementary Filtering
Clearly, one has, from Fig. 12.2.1- 1,
y( >
S
=
S(s)
+ N(s)H(s) + N2(4[1 - H(s)]
(12.2.2-1)
= S(s) + N2(4 + [N(s) - N2(4]fqs)
(12.2.2-2)
An equivalent and more commonly used implementation of the complementary
filter is thus given in Fig. 12.2.2-1.
494
12 INTRODUCTION TO NAVIGATION APPLICATIONS
The complementary filtering satisfies an additional constraint important for
many applications: The signal is not distorted. Nevertheless, in the case where
the signal s(t) can be properly modeled as a random process with known spectral
characteristics, complementary filtering is a conservative philosophy. As a re-
sult, compared with Wiener filtering, its performance will suffer to some extent.
However, if the signal-to-noise ratio in the measurements is high (say, 20 dB),
the performance improvement of the Wiener filtering over the complementary
filtering is insignificant.
Navigation signals can rarely be well-modeled as random processes, not
to mention that their spectral characteristics, especially the signal-noise cross-
power spectra, would not be known in practice. In addition, these signals
usually have a much greater power than the noise and thus the performance
degradation due to the conservativeness of the complementary filtering is often
quite marginal.
Complementary filters are often used in two levels in navigation: (1) Within
an inertial navigation system, the filter is used to compensate the measurement
errors of the INS system; and (2) for the integration of the inertial system and
other navigation systems, the filter is used to estimate the estimation errors of
the primary navigation system.
Example
The heading $J of a vehicle is to be estimated based on measurements of a
fluxgate compass and a directional (rate) gyro, respectively, given in Laplace
domain by
+c (4
= $44 + TMS>
(12.2.2-3)
R,(s)
= R(s) + R,(s)
(12.2.2-4)
where
R(s) =
s+(s) is the heading change rate, and gn and
R,
are compass and
gyro measurement noise, respectively. It is known that the compass provides
reliable information at low frequencies only (i.e., the spectrum of $J~ is mostly
in the high-frequency region), while in the same low-frequency band the gyro
exhibits a bias and drift phenomenon and is thus useful at higher frequencies
(i.e., the spectrum of
R,
is mostly in the low-frequency region).
This is a typical example where complementary filtering is useful. Note first
that (12.2.2-3)-(12.2.2-4) can be converted into the standard form of (12.2.1-1)-
(12.2.1-2) by setting 21(s) = Q+(s) and 22(s) =
R,(s)/s.
It then follows from
(12.2.2-1) that the estimate of $J provided by a cbmplemc
form
G(s) = $J(s) + $hl(s)H(s) + (Rn(s)/s)[l - H(s)
Thus the estimation error is
4(s) = lM4H(4 + (Rn(4l4P - H(s)1
ntary filter has the
(12.2.2-5)
(12.2.2-6)
12.3 INERTIAL NAVIGATION SYSTEMS
495
Since
R,
and $, are significant in the low- and high-frequency regions, respec-
- tively, H(s) should be low pass to attenuate $J, while letting
R,
pass.
The simplest such filter is H(s) = A. The gain
k
is a design parameter
to meet a target break frequency dictated by the frequency characteristics of the
compass and gyro. Kalman filtering techniques can be used to determine the
optimal
k
value because a complementary filter can be treated in principle as a
simple Kalman filter.
In practice, this simple complementary filter structure is modified to meet
additional constraints, such as to reject the steady-state bias of the rate gyro.
12.3 INERTIAL NAVIGATION SYSTEMS
An inertial navigation system (INS) is a dead-reckoning system. It
provides a self-contained, nonjammable, and nonradiating means for navigation.
It can be used as a stand-alone system or can be combined with some other
navigation systems.
An INS has three basic functions: sensing, computing, and outputting. The
sensors are accelerometers and gyroscopes (gyros, for short), which mea-
sure linear acceleration and rotational rate (angular velocity) of the vehicle, re-
spectively. Estimation and filtering techniques are used in the computation of the
navigation solutions based on the measurements from these sensors. In order to
provide three-dimensional navigation for a vehicle with six degrees of freedom,
three gyros are positioned in such a way to define a three-dimensional coordi-
nate system and a triad of accelerometers are mounted on the three orthogonal
axes.
The principle of navigation by an INS system is simple. With a precise
knowledge of the initial position and attitude of the vehicle, at any time, single-
and double-integration of the linear accelerations give the velocity and position
of the vehicle, respectively, and the attitude is obtained by integrating rotational
rates from the gyro measurements. However, the presence of random errors
as well as unknown constant (or nearly comstant) errors (e.g., drift rates and
biases) makes the navigation problem quite complex.
Inertial navigation systems are either gimbaled or strapdown. In a gim-
baled system, also known as a
platform system,
a stabilized physical plat-
form is maintained through a servo system on which the accelerometers and
gyros are mounted and thus isolated from the rotations of the vehicle. This not
only greatly reduces the requirements on the inertial sensors but also greatly
simplifies the navigation computation since the attitude of the platform with
respect to a navigation reference frame is preserved. However, the need to have
a stabilized platform makes a gimbaled system more complicated, costly, and
less reliable than a strapdown system.
In a strapdown system, there is no stabilized platform - the gyros and
accelerometers are fixed (strapped down) to the vehicle body directly. The re-
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