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1
An Extended Kalman Filter algorithm for Integrating GPS and low-cost Dead reckoning system
data for vehicle performance and emissions monitoring
L. Zhao, W.Y. Ochieng, M.A. Quddus and R.B. Noland
Centre for Transport Studies, Department of Civil and Environmental Engineering
Imperial College London
Abstract
This paper describes the features of an extended Kalman filter algorithm designed to support the navigational
function of a real-time vehicle performance and emissions monitoring system currently under development. The
Kalman filter is used to process global positioning system (GPS) data enhanced with dead reckoning in an
integrated mode, to provide continuous positioning in built-up areas. The dynamic model and filter algorithms
are discussed in detail, followed by the findings based on computer simulations as well as a limited field trial
carried out in the Greater London area. The results of using the extended Kalman filter algorithm demonstrate
that the integrated system employing GPS and low cost dead reckoning devices is capable of meeting the
required navigation performance of the device under development.
1 INTRODUCTION
Problems posed by the environmental impact of transport are serious, growing and constitute a major challenge
to policy makers at all levels (DETR, 1999). The current array of technological, institutional and planning tools
available to deal with these problems are inadequate and need urgently to be upgraded. A key feature of the
problems is that they arise from the interaction of human behavioural systems and physical systems. Thus, to
improve the understanding of environmental and health problems associated with vehicle emissions it is
necessary to combine data on both travel and traffic behaviour with environmental data. There are currently no
such databases available.
A research and development project is currently underway which aims to contribute to the realisation of these
data requirements by developing and applying state-of-the art environmental monitoring, positioning,
communications, data mining and warehousing technologies to create and demonstrate the capabilities of an
accurate, reliable and cost effective real time data collection device, the vehicle performance and emissions
monitoring system (VPEMS). The VPEMS will be fitted on vehicles to monitor vehicle and driver performance
and, the level of emissions and concentrations.
The navigation function of the VPEMS is responsible for the derivation of all spatial, temporal and spatio-
temporal data about the vehicle including location in 3-D space, time, slope, speed and acceleration. The level of
positioning accuracy for VPEMS has been specified at 50m (95%) and 100m (99.9%). The availability of the
positioning system has been specified at 99% (corresponding to an outage of 14 minutes of 24 hours) [Sheridan
and Ochieng, 2000]. To achieve this level of performance in built-up areas, where the impact of pollution from
traffic is most serious, the navigation function cannot be supported by the global positioning system (GPS)
alone. A solution under consideration is the integrated use of data from GPS with dead reckoning (DR) and map
matching.
1.1 The Global Positioning System
The Global Positioning System (GPS) provides 24-hour, all-weather 3-D positioning and timing all over the
world, with a predicted horizontal accuracy of 22m (95%) [US DoD, 2001]. However, because the system
suffers from signal masking and multipath errors in areas such as urban canyons, densely treed streets, and
tunnels, navigation with GPS requires a level of augmentation to achieve the RNP. A recent study to characterise
the performance of GPS in a typical urban area showed that the required accuracy was available 90 percent of
the time, based on a 4-hour trip in the Greater London area [Ochieng, 2002]. The implication of the outage
involved here (i.e. 10%) is a potential loss of navigation capability during a crucial period.
2
1.2 Dead Reckoning
Dead Reckoning is a positioning technique based on the integration of an estimated or measured displacement
vector. It is not subject to signal masking or outages. However, its positioning errors accumulate with time, so
that external calibration or augmentation with other positioning devices is usually required. Generally, DR is
composed of two or more sensors that measure the heading and displacement of a vehicle. Usually gyroscopes
are used to measure heading-rate (i.e. rate of change of heading) and the odometer is used to measure
displacement (and speed). A variety of gyroscopes have emerged in recent years (mechanical, optic, electrostatic
and the relatively new micro-electro-mechanical system), with differences in accuracy, stability and cost. The
gyroscope is a core element used to measure the rate of rotation in inertial navigation.
With regard to the odometer, the wheel rotation sensor is used. The wheel revolutions from the sensor are then
transformed into the distance travelled. Given the corresponding timing information, the speed or velocity of the
vehicle can also be determined. In this case, the speed sensor is achieved at no additional cost.
For vehicle location and navigation applications, sensors must be chosen that add only minor costs to the
production or modification of a vehicle, while delivering continuous position availability. In the VPEMS project,
a standard built-in odometer and a low-cost gyroscope have been selected. The gyroscope is a piezoelectric
vibrating type based on micro-electro-mechanical system (MEMS) technology. The odometer outputs the
distance the vehicle has travelled as a pulse converted to distance by a scale factor, while the gyroscope produces
the heading rate of the vehicle, in mv/rad/s (microvolts per radian per second).
The integrated GPS/DR system adopted is based on the concept of loose coupling (described below) and uses an
extended Kalman filter. The system can calibrate for odometer and gyroscope errors in real-time when GPS
works well and uses the calibration data to produce a better position fix by Dead Reckoning when GPS suffers
signal masking. This type of integration is potentially suitable to the built-up urban environments where GPS
signal attenuation and masking are potential problems.
The rest of the paper presents a discussion of GPS/DR integration issues, the details of the mathematical models
and the results obtained using simulated and real field data.
2 GPS/DR INTEGRATION ISSUES
A crucial element needed for the establishment of an integrated (GPS/DR) navigation system model and a
Kalman filter structure is an understanding of the navigation errors involved. With the removal of the effects of
selective availability (SA) in May 2000, GPS positioning accuracy has improved from 100m (95%) to 22m
(95%). The remaining errors are mainly due to orbital instability, atmospheric propagation, multipath and
receiver noise. For the integration process, the modelling of the errors depends on whether the loose coupling
concept (in this case comparing the position solution from GPS with that obtained by DR) or the tight coupling
approach (involving integrating the raw measurements from each system into a single solution with appropriate
weighting of the various measurements) is adopted. With the loose coupling approach the errors in the derived
position, velocity and heading can be modelled as white noise with desired characteristics (note that the accuracy
of velocity and heading degrade significantly at low speed). For tight coupling, errors in the pseudorange and
pseudorange rate should be considered.
With regards to the DR the factors that affect the odometer output accuracy include the scale factor error, status
of the road and pulse truncation. The scale factor error, which is the difference between the true scale coefficient
and the calibrated one, is the most significant as it affects the distance measurement as long as the vehicle is
moving. It is caused by calibration error, tyre wear and tear, tyre pressure variation and vehicle speed. Although
the scale factor will vary during a period of travel, the change should not be significant over a short time. Hence,
a reasonable model for this could be either a random constant or a first order Gauss-Markov process with a long
3
autocorrelation time (Gelb, 1979). The latter was adopted in the study presented here as it captures the real
conditions slightly better.
Errors associated with the low cost piezoelectronic vibrating rate gyroscope can be caused by gyro bias drift,
gyro scale factor error, installation misalignment, and some other factors, such as temperature, vibration and
electro-mechanical properties of the operational environment. Of these the most significant are the bias drift and
scale factor error. The bias drift can be explained by the existence of an output (rate of change of heading) with
no corresponding input (e.g. a causal factor such as vehicle turning). The drift depends on the type of gyroscope
(i.e. the manufacturing process and quality) and can be as much as 10 degrees/second. It is usually an unknown
random constant and will affect the measurements cumulatively irrespective of whether the vehicle is moving or
not. It is reasonable to model the bias drift as a first-order Gauss-Markov process, because the unknown bias
changes with temperature and vibration. The scale factor error is caused by calibration errors and electro-
mechanical properties of the operational environment. Unlike the bias drift, the scale factor error affects the
measurements only when the vehicle makes a turn. Although the scale factor error will vary with time, the
change should not be significant over a short period of time. Hence, the scale factor error has been modelled as a
random constant disturbed by white noise with a low covariance.
3 EXTENDED KALMAN FILTER
Kalman Filtering is widely used in various system state estimations and predictions. It is a kind of linear
minimum mean-square error (MMSE) filtering process using state-space methods. The two main features of
Kalman formulation and problem solution are: vector modelling of the dynamic process under consideration, and
recursive processing of the noisy measurement data. In some applications of Kalman filtering, due to the non-
linearity of the dynamic and/or measurement equations, the corresponding models have to be linearised. One of
the linearisation methods, known as Extended Kalman Filtering (EKF), is to linearise about a trajectory that is
continually updated with the state estimates resulting from the measurements.
3.1 State Equation
The dynamic model is based on the knowledge of how the vehicle is expected to move. In order to establish the
integrated navigation system, either the vehicle dynamic model or the navigation state error model, must be
specified. Usually, the error model is used only in the case where there is dominant navigation equipment, such
as an inertial navigation system (INS), whose state errors need to be estimated and calibrated by a feedback
mechanism. For satellite-based vehicle navigation applications, although GPS can be thought of as the dominant
equipment, two scenarios always emerge. On the one hand, GPS positioning accuracy is good enough so that
there is no requirement for calibration and; on the other, there is a loss of navigation capability due to signal
masking making calibration meaningless. In this case, choosing a vehicle dynamic model is more appropriate.
Hence the vehicle dynamic model was established as state equations. The following states were selected:
T
Gvv
]KSaHvne[x
εδδω
= (1)
Where
e = terrestrial easting position, in meters
n = terrestrial northing position, in meters
v
v
= forward velocity of the vehicle, in m/s, with forward being positive
H
v
= heading of the vehicle, in radian, with north being zero and clockwise being positive
a = the acceleration of the vehicle, in
2
/ sm
ω
= the rate of the heading, in rad/s
S
δ
= the odometer scale factor error, in m/pulse
K
δ
= the rate gyro scale factor error, in mv/rad/s
G
ε
= the bias drift of the rate gyro, in rad/s
The dynamic equations can be written as:
4
ï
ï
ï
ï
ï
ï
þ
ï
ï
ï
ï
ï
ï
ý
ü
+−=
=
=
+−=
=
+=
+=
+⋅=
+⋅=
9GgG
8
7
6
5
4v
3v
2vv
1vv
w
wK
wS
w
wa
wH
wav
wHcosvn
wHsinve
εβε
δ
δ
ωβω
ω
ω
(2)
Or written as vector form:
w)t),t(x(fx +=
(3)
Where
[]
T
987654321
wwwwwwwwww = is the dynamic noise.
w
β
,
g
β
are the skew correlation
times (i.e. inverse of correlation times). It can be seen that this is a non-linear set of equations because there are
two items that are non-linear.
3.2 Measurement Equations
From the GPS receiver, the information on position (
GPS
ϕ
,
GPS
λ
), velocity
GPS
v and heading
GPS
H of the
vehicles can be derived. From the odometer and rate gyroscope, the pulses
odo
N
∆
during a time interval t
∆
,
representing the displacement travelled, and direct current (DC) voltage output
RG
V , representing the heading-
rate of the vehicle respectively, can be acquired. So the observation variables are chosen as follows:
T
RGodoGPSGPSGPSGPS
]VNHv[z
∆ϕλ
= (4)
And the measurement equations are defined as:
ï
ï
ï
þ
ï
ï
ï
ý
ü
+++=
+⋅−⋅+⋅⋅=⋅
+=
+=
+=
+⋅=
6GRG
5odov
2
odo
4vGPS
3vGPS
1GPS
2GPSGPS
v))(KK(V
vNStvta21NS
vHH
vvv
vR/n
vcosR/e
εωδ
∆δ∆∆∆
ϕ
ϕλ
(5)
Where S and K are the nominal scale factor for the odometer and gyroscope respectively;
t∆ the interval time; R
the radius of the earth. These equations can be rewritten in matrix form as:
v)t),t(x(hz +=
(6)
Where
[]
T
654321
vvvvvvv = is the observation noise. It can be seen that the measurement equations
are also non-linear because of the last measurement equation.
3.3 Kalman Filter Design
The Kalman filter can be used to produce optimal estimates of the state vectors listed above with well-defined
statistical properties. For convenience of computer calculation, discrete recursive algorithms are usually adopted.