GPS and INS Integration with Kalman Filtering for Direct Georeferencing of Airborne Imagery

1 INTRODUCTION
Georeferencing can be defined as a process of obtaining knowledge about the origin of some event
in space-time. Depending on the sensor type, this origin needs to be defined by a number of
parameters such as time, position (location), attitude (orientation) and possibly also the velocity of
the object of the interest. When this information is attained directly by means of measurements
from sensors on-board the vehicle the term direct georeferencing is used [Skaloud, 1999].
For georeferencing the image data, the position (X0, Y0, Z0) and orientation (ω, ϕ, κ) of the sensor,
which also are called as exterior orientation elements, should be known. Then, the uncorrected
image vector is transformed to the corrected georeferenced position and the relation between the
local image coordinate system and the global object frame is solved. The traditional way of
georeferencing of airborne imagery is to use ground control points (GCPs) which counts a major
cost for photogrammetry projects. A number of different vehicles and methods can be used for
direct georeferencing of airborne imagery depending on the sensor and platform type. Today,
differential kinematic GPS positioning is a standard tool for determining the camera exposure
centres for aerial triangulation [Heipke et al, 2002]. Airborne GPS can greatly reduce, but not
completely eliminate the need for ground control.
Since the need for GCP and overlapping imagery cannot be eliminated with the use of GPS, the
integration of GPS and the inertial technology became a subject of research in this field. Inertial
navigation relies on knowing the initial position of the object, velocity and attitude, and thereafter
measuring the attitude rates and accelerations. An Inertial Navigation System (INS) consists of an
Inertial Measurement Unit (IMU) or Inertial Reference Unit (IRU), and navigation computers to
calculate the gravitational acceleration. However, in this report, the terms INS and IMU are used
for the same purpose. An IMU is composed of gyroscopes, which is used for determining the
rotation elements of the exterior orientation, and accelerometers, which provides the sensor
velocity and position. In this report, use of the term INS is preferred. In principle, a GPS/IMU
sensor combination can yield the exterior orientation elements of each image without aerial
triangulation.
The application of direct georeferencing to the image data provides some important advantages,
which can be summarized as:
♦ Direct georeferencing enables a faster acquisition of the exterior orientation, since the
computational burden for automatic aerial triangulation is higher compared to the effort for
GPS/inertial integration [Cramer, 1999].
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♦ Direct georeferencing removes limitations to the flight path during image acquisition.
Continuous absolute GPS trajectories, as obtainable by on-the-fly (OTF) methods, would in
principle permit an aerial triangulation without ground control points. For that purpose a
certain number of images has to be captured in the well-known photogrammetric block
configuration. However, this flight configuration can be disadvantageous, if only small areas
have to be captured or if a linear flight path is aspired for tasks like the supervision of power
lines or the image acquisition at coast lines [Cramer, 1999].
♦ Additional problems of image matching required for automatic aerial triangulation are avoided
if direct georeferencing is applied [Cramer, 1999].
Integrated systems will provide a system that has superior performance in comparison with either a
GPS, an INS, or vision-based stand-alone system. The main strengths and weakness of INS and
DGPS are summarized in figure 1.1.
Figure 1.1: Benefits of INS/DGPS Integration [Skaloud, 1999].
The overall performance of the direct orientation method is limited primarily by the following
components:
♦ Quality of the calibration of the integrated system:
INS
! high position velocity accuracy over
the short term
! accurate attitude information
! accuracy decreasing with time
! high measurement output rate
! autonomous
! no signal outages
! affected by gravity
DGPS
! high position velocity accuracy over
the long term
! noisy attitude information (multiple
antenna arrays)
! uniform accuracy, independent of
time
! low measurement output rate
! non-autonomous
! cycle slip and loss of lock
! not sensitive to gravity
INS/DGPS
! high position and velocity accuracy
! precise attitude determination
! high data rate
! navigational output during GPS signal
outages
! cycle slip detection and correction
! gravity vector determination
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− Imaging sensor modeling
− Lever arm between INS and GPS antenna
− Boresight transformation between INS and camera frames
♦ In-flight variation of the calibration components
♦ Rigidity of the imaging sensor/INS mount
♦ Quality of the IMU sensor
♦ Continuity of the GPS lock
♦ Kalman filter design [Grejner-Brzezinska, Toth, 2000].
Before using the position and orientation components (GPS antenna and IMU) for sensor
orientation, we must determine the correct time, spatial eccentricity, and boresight alignment
between the camera coordinate frame and IMU. The calibration of the GPS/IMU and the camera is
vital since minor errors will cause major inaccuracies in object point determination [Sanchez,
Hothem]. Direct georeferencing of airborne imaging data by INS/DGPS is schematically depicted
in figure 1.2.
Figure 1.2: Direct georeferencing of airborne imaging by INS/DGPS [Skaloud, 1999].
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Kalman Filter is an extremely effective and versatile procedure for combining noisy sensor outputs
to estimate the state of a system with uncertain dynamics. In GPS/INS integration case, noisy
sensors include GPS receivers and IMU components, and the system state include the position,
velocity, acceleration, attitude, and attitude rate of a vehicle. Uncertain dynamics include
unpredictable disturbances of the host vehicle and unpredictable changes in the sensor parameters
[Grewal et al., 2001]. Kalman filter optimally estimates position, velocity, and attitude errors, as
well as errors in the inertial and GPS measurements [Grejner-Brzezinska and Toth, 1998].
Main purposes of this report are to figure out basic system requirements of direct georeferencing of
airborne imagery using GPS and INS, to explain the fundamentals of GPS/INS integration with
Kalman Filtering and limitations of the integration, and to describe the common problems and
results reported in the literature.
In the second and third parts of this report, the fundamentals and basic error models of GPS and
INS are introduced, respectively. The theory of Kalman filtering for discrete and continuous
processes are explained in the fourth part. While the integration purposes and types of GPS and
INS are provided in the fifth part, a general overview of the applications seen in the literature is
given in sixth part. The conclusion and future work is placed at the and of this report.
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2 FUNDAMENTALS of the GPS
The GPS, officially also known as NAVSTAR (Navigation and Satellite Timing and Ranging), is
part of a satellite-based navigation system developed by the U.S. Department of Defense. GPS
belongs to a large class of radio navigation systems that allow the user to determine his range
and/or direction from a known signal transmitting station by measuring the differential time of
travel of the signal. The Global Orbiting Navigation Satellite System (GLONASS) developed by
Russia has almost an equivalent structure and used for satellite radio navigation purposes similar to
GPS. Terrestial radio navigation systems predate the satellite systems and include such as VOR
(VHF Omnidirectional Range) and DME (Distance Measuring Equipment) for civilian aviation; the
military equivalent, Tacan (Tactical Air Navigation); and low-frequency, long-range systems such
as LORAN (Long Range Navigation) and OMEGA (Jekeli, 2000).
The GPS comprises a set of orbiting satellites at known locations in space and their signals can be
observed on the Earth. Three distances to distinct satellites having known positions provide
sufficient information to solve the observer’s three-dimensional position. The system is designed so
that a minimum of four satellites is always in view anywhere in the world to provide continual
positioning capability. This is accomplished with 24 satellites distributed unevenly in six
symmetrically arranged orbital planes.
The applications of GPS range from military navigation, vehicle monitoring, to sporting activities.
For geodetic applications, the precise measurement of baselines (relative positioning) in static
mode of GPS is widely used. Static positioning involves placing the receiver at a fixed location on
the Earth and determining the position of that point. Opposite to this, kinematic positioning refers
to determining the position of a vehicle or a platform that is moving continually with respect to
Earth. It is also known as real-time positioning. The term navigation is used for real-time
processing of the positioning data. Differential GPS (DGPS) is a technique for reducing the error in
GPS-derived positions by using additional data from a reference GPS receiver at a known position.
The most common form of DGPS involves determining the combined effects of navigation
message ephemeris and satellite clock errors at a reference station and transmitting the pseudorange
corrections in real time, to a user’s receiver (Grewal et al., 2001).
The GPS is not without problems and limitations (Jekeli, 2000). It is not a self-contained,
autonomous system. The user must be able to “see” the GPS satellites. Satellite visibility may be
obstructed locally by intervening buildings, mountains, bridges, tunnels, etc. For kinematic
applications, the effects of electronic interference or brief obstructions may cause the receiver to
miss one or more cycles of the carrier wave. The frequency of the data output in most receivers is
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often 1 Hz. Most of the error in GPS positioning come from medium propagation effects that are
unpredictable to model such as atmospheric effects.
2.1 Clocks and Time
Each GPS satellite carries an atomic clock to provide timing information for the signals transmitted
by the satellites. The clocks are oscillating at a particular frequency. The relationship between the
phase φ, frequency ƒ, and the time is:
ƒ(t)=
dt
dφ (t)
(2.1)
where t represents true time. The phase of an oscillating signal is the angle it sweeps out over time
(0≤φ≤2π) and has the units of cycle. The frequency of the signal is the rate at which the phase
changes in time and has the units of Hertz (cycles per second).
= + ∫ƒ
t
to
φ (t) φ (t0) (t')dt' (2.2)
t0 is some initial time. τ denote the indicated time related to the phase:
τ(t)=
0
( ( ) 0)
f
φ t −φ
(2.3)
ƒ0 is some nominal (constant) frequency since the initial indicated time does not coincide with
initial phase (φ(t0) ≠ φ
0 ).
The oscillator clock time (τ) and the true time (t) differ from each other both in scale and in origin.
The true time reflects the atomic clock time in U.S., which also differs from Coordinated Universal
Time (UTC) by 2000 with 13 seconds. However, the GPS true time is calibrated by U.S. atomic
time. The true time reflects the fact that the times indicated on satellite and receiver clocks are not
perfectly uniform and must be calibrated by master clocks on the Earth. The relationship between
the phase-time, τ, and the true time, t, is:
τ(t)=t - t0 + τ(t0) + δτ(t), (2.4)
where,
∫ ƒ
ƒ
=
t
to
(t) 1 (t')dt'
0
δ δ
The abbreviated form of (2.4) is:
τ(t)=t - Δτ(t) (2.5)
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2.2 GPS Signals
The signal is a carrier wave (sinusoidal wave) modulated in phase by binary codes that represent
interpretable data. It can be represented mathematically by:
S(t)= AC(t)D(t)cos(2πƒt) (2.6)
where ƒ is the frequency of the carrier wave, and A is the amplitude of the signal. The code
sequence C(t) is a step function having values (1, -1), also known as chips or bits. D(t) represents a
data message.
Each satellite actually transmits two different codes, the C/A (coarse acquisition) code and the Pcode
(precision code). The P-code has 10 times higher chipping rate and wavelength in comparison
with C/A code. They are transmitted on two separate microwave regions, an L1 signal with carrier
frequency ƒ1 = 1545.72 MHz and with wavelength λ1=0.1903 m; and an L2 signal with carrier
frequency ƒ2 = 1227.6 MHz and with wavelength λ2=0.2442 m. The transmission on two
frequencies allows approximate computation of the delay of the signal due to ionospheric
refraction. The total signal transmitted by the satellite is given by the sum of three sinusoids, two
for the two codes on the L1 carrier and one for the P-code on the L2 carrier. The total signal
transmitted by a GPS satellite is given by
SP(t)=APPP(t)WP(t) DP(t)cos(2πƒ1t) + ACCP(t) DP(t)sin(2πƒ1t) + BPPP(t)WP(t)DP(t)sin(2πƒ2t) (2.7)
AP, AC, and BP represent amplitudes of the corresponding codes, C and P represent C/A and P
codes, D represents the data message, superscript P identifies a particular satellite, and W represents
a special code which is used to decrypt a military code.
The codes serve two operational purposes: determining the range between the satellite and receiver
and to spread the signal over a large frequency bandwidth, thus permitting small antennas on the
Earth to gather the transmitted signal. Both codes consist of unique sequences of binary states that
are generated using a pseudorandom noise (PRN) algorithmic process. The PRN for C/A code is
different for each satellite and repeats every millisecond. For P-code, it is much longer and repeats
only after 38 weeks. Each satellite uses only one distinct week’s worth of the code. The satellites
are distinguished by the codes rather than by frequency.
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2.3 GPS Receiver
Before the signal is processed by the receiver, it is pre-amplified and filtered at the antenna, and
subsequently down-shifted in frequency to a more manageable level for processing (Jekeli, 2000).
The mixed signal is given by
Sr(t) SP(t)=A cos(2πƒLOt) cos(2πƒSt + φ(t))
=
2
A cos(2π(ƒS-ƒLO)t + φ(t)) +
2
A cos(2π(ƒS+ƒLO)t + φ(t)) (2.8)
where Sr(t) is the pure signal sinusoid generated by the receiver oscillator, ƒLO is the local oscillator
frequency, SP(t) is the satellite signal with frequency ƒS, and A is an amplitude factor.
The satellite signal is then shifted to an intermediate frequency (IF), and appropriate filters are
applied to control the amplitude of the signal for subsequent processing. The signal then passes to
the main signal processing part of the receiver.
To calculate the distance between the satellite and the receiver, the time tag of the signal at the time
of transmission and the time of reception at the receiver is compared, and using the speed of the
light, the delay is converted to the distance. It is not the true range if the satellite and the receiver
clocks differ; and therefore, the calculated range is called pseudorange.
2.4 GPS Observables and Errors
A broad overview of GPS errors is provided in table 2.1. The largest error is due to the receiver
clock. The next significant error source is the medium in which the signal must travel. This
includes the Ionosphere, which has an altitude between 50 km to 1000 km and has many free
electrons; and the Troposphere, which is a non-dispersive medium and contains mostly electrically
neutral particles.
Table 2.1: Error sources in GPS positioning (Jekeli, 2000).
Error Source Typical Magnitude
Receiver clock error (synchronized) 1 μs (300 m)
Residual satellite clock error 20 ns (6m)
Satellite synchronization to UTC 100 ns (30 m)
Selective Availability (cancelled by 2001) 100 m
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Orbit error (precise, IGS) 20 cm
Tropospheric delay <30 m
Ionospheric delay <150 m
Multipath <5 m (P-code); <5 cm (phase)
Receiver Noise 1 m (C/A code); 0.1 m (P-code); 0.2 mm (L1 phase)
Other errors in GPS observables include the multipath error (the reflection of the GPS signal from
nearby objects prior to entering the antenna), equipment delays and biases, antenna eccentricities
(phase center variations), and the thermal noise of the receiver.
The pseudorange is formulated as
Sr
P(τr)=ρr
P(τr) + c( Δτr(t) - ΔτP(t- Δtr
P)) + ΔρP
iono,r + εP
p,r (2.9)
where ρr
P(τr) is the true range between the satellite and the receiver, Δtr
P is the time of transit,
ΔρP
iono,r is the ionospheric error for each satellite, c is the speed of the light, and εP
p,r represents
pseudorange observation error (different for each satellite). For simplicity, the Tropospheric delay,
the equipment and antenna offsets, the multipath error, and the time registration error due to the
receiver clock error are excluded.
The phase observable can be expressed as follows:
φr
P(τr)=
c
f 0 ρr
P(τr) + f 0 ( Δτr(t) - ΔτP(t- Δtr
P)) + φ0,r - φ0
P – Nr
P + ΔφP
iono,r + εP
φ,r (2.10)
where φ0,r and φ0
P are the arbitrary phase offsets Nr
P is the integer representing the unknown
number of full cycles and also called as carrier phase ambiguity, εP
φ,r is the phase observation error.
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3 FUNDAMENTALS of INERTIAL NAVIGATION
3.1 Basic Concepts of Inertial Navigation
Inertia is the propensity of bodies to maintain constant translational and rotational velocity, unless
disturbed by forces or torques, respectively (Newton’s first law of motion). An inertial reference
frame is a coordinate frame which Newton’s law of motion are valid. Inertial reference frames are
neither rotating nor accelerating (Grewal et al., 2001). Inertial sensors measure rotation rate and
acceleration by gyroscopes and accelerometers respectively. Accelerometers cannot measure
gravitational acceleration, which is an accelerometer in free fall or in orbit has no detectable input.
The input axis of an inertial sensor defines which vector component it measures. Multiaxis sensors
measure more than one component.
Inertial navigation uses gyroscopes and accelerometers to maintain an estimate of the position,
velocity, attitude, and attitude rates of the vehicle in or on which the INS is carried. An INS
consists of navigation computers, to calculate the gravitational acceleration and to integrate the net
acceleration, and an inertial measurement unit containing accelerometers and gyroscopes.
Literally, there are thousands of designs for gyroscopes and accelerometers. Not all of them are
used for inertial navigation. For example, gyroscopes are used for steering and stabilizing ships,
missiles, cameras and binoculars, etc. The acceleration sensors are also used for measuring gravity,
sensing seismic signals, leveling, and measuring vibrations.
Traditionally, inertial systems have been divided into three groups according to the free-running
growth of their position error (Skaloud, 1999):
• the strategic-grade instruments ( performance ≈ 100 ft/h)
• the navigation-grade instruments (performance ≈ 1 nm/h)
• the tactical-grade instruments (performance ≈ 10-20 nm/h)
In a further categorization, the inertial navigation systems are designed in two main groups: the
platform (or gimbaled) systems and the strapdown systems. In a gimbaled system the
accelerometer triad is rigidly mounted on the inner gimbal of three gyros (see figure 3.1.b). The
inner gimbal is isolated from the vehicle rotations and its attitude remains constant in a desired
orientation in space during the motion of the system. The gyroscopes on the stable platform are
used to sense any rotation of the platform, and their outputs are used in servo feedback loops with
gimbal pivot torque actuators to control the gimbals such that the platform remains stable. These
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systems are very accurate, because the sensors can be designed for very precise measurements in a
small measurement range.
In contrary, a strap-down inertial navigation system uses orthogonal accelerometers and gyro triads
rigidly fixed to the axes of the moving vehicle (figure 3.1.a). The angular motion of the system is
continuously measured using the rate sensors. The accelerometers do not remain stable in space,
but follow the motion of the vehicle.
Figure 3.1: Inertial measurement units (Grewal et al., 2001).
3.2 Common Sensor Error Models
Gyroscopes, which are used as attitude sensors in inertial navigation, are also called as inertial
grade. There are many types of gyroscope designs, such as momentum wheels, rotating
multisensor, laser gyroscopes, etc. Error models for gyroscopes are primarily used for two
purposes: predicting performance characteristics as function of gyroscope design parameters and
calibration and compensation of output errors. The common error sources for gyroscopes are output
bias, input axis misalignments, combined (clustered) three-gyroscope compensation, input/output
non-linearity, and acceleration sensitivity.
Depending on the purpose, acceleration sensors also have several designs such as, gyroscopic
accelerometers, pendulous accelerometers, strain-sensing accelerometers, etc. The main error
sources for accelerometers are biases, parameter instabilities (i.e., turn-on and drift), centrifugal
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acceleration effects due to high rotation rates, center of percussion, and angular accelerometer
sensitivity.
3.3 Initialization and Alignment
INS initialization is the process of determining initial values for system position, velocity, and
attitude in navigation coordinates. INS position initialization ordinarily relies on external sources
such as GPS or manual entry. INS velocity initialization can be accomplished by starting when it is
zero (i.e., the host vehicle is not moving) or by reference to the carrier velocity.
INS alignment is the process of aligning the stable platform axes parallel to navigation coordinates
(for gimbaled systems) or that of determining the initial values of the coordinate transformation
from sensor coordinates to navigation coordinates (for strapdown systems). There are four basic
methods for INS alignment (Grewal et al., 2001):
i) Optical alignment using either optical line-of-sight reference to a ground based direction or an
on board star tracker.
ii) Gyrocompass alignment of stationary vehicles, using the sensed direction of acceleration to
determine the local vertical and sensed direction of rotation to determine north.
iii) Transfer alignment in a moving host vehicle, using velocity matching with an aligned and
operating INS.
iv) GPS-aided alignment, using position matching with GPS to estimate the alignment variables.
3.4 System-Level Error Models
Since there is no single, standard design for an INS, the system-level error sources vary very much.
General error sources can be classified as:
i) initialization errors, comes from initial estimates of position and velocity;
ii) alignment errors, from period for initial alignment of gimbals or attitude direction cosines (for
strapdown systems) with respect to navigation axes;
iii) sensor compensation errors, occur due to the change in the initial sensor calibration over the
time;
iv) gravity model errors, is the influence of the unknown gravity modeling errors on vehicle
dynamics.