Robust Automotive Positioning: Integration of GPS and Relative Motion Sensors

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Robust Automotive Positioning:

Integration of GPS and Relative Motion

Sensors

Examensarbete utfört i Reglerteknik vid Tekniska Högskolan i Linköping

av Jon Kronander

Reg nr: LiTH-ISY-EX-3578-2004 Linköping 2004

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Robust Automotive Positioning:

Integration of GPS and Relative Motion

Sensors

Examensarbete utfört i Reglerteknik vid Tekniska Högskolan i Linköping

av Jon Kronander

Reg nr: LiTH-ISY-EX-3578-2004

Supervisor: Peter Hall Thomas Schön Examiner: Fredrik Gustafsson Linköping 13th December 2004.

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Avdelning, Institution Division, Department

Institutionen för systemteknik

581 83 LINKÖPING

Datum Date 2004-10-20 Språk

Language Rapporttyp Report category ISBN

Svenska/Swedish

X Engelska/English Licentiatavhandling X Examensarbete ISRN LITH-ISY-EX-3578-2004

C-uppsats

D-uppsats Serietitel och serienummer _{Title of series, numbering} ISSN

Övrig rapport

____

URL för elektronisk version

http://www.ep.liu.se/exjobb/isy/2004/3578/

Titel

Title Robust fordonspositionering: Integration av GPS och sensorer för relativ rörelse Robust Automotive Positioning: Integration of GPS and Relative Motion Sensors

Författare

Author Jon Kronander

Sammanfattning

Abstract

Automotive positioning systems relying exclusively on the input from a GPS receiver, which is a line of sight sensor, tend to be sensitive to situations with limited sky visibility. Such situations include: urban environments with tall buildings; inside parking structures; underneath trees; in tunnels and under bridges. In these situations, the system has to rely on integration of relative motion sensors to estimate vehicle position. However, these sensor measurements are generally affected by errors such as offsets and scale factors, that will cause the resulting position accuracy to deteriorate rapidly once GPS input is lost.

The approach in this thesis is to use a GPS receiver in combination with low cost sensor equipment to produce a robust positioning module. The module should be capable of handling situations where GPS input is corrupted or unavailable. The working principle is to calibrate the relative motion sensors when GPS is available to improve the accuracy during GPS intermission. To fuse the GPS information with the sensor outputs, different models have been proposed and evaluated on real data sets. These models tend to be nonlinear, and have therefore been processed in an Extended Kalman Filter structure.

Experiments show that the proposed solutions can compensate for most of the errors associated with the relative motion sensors, and that the resulting positioning accuracy is improved accordingly.

Nyckelord

Keyword

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Abstract

Automotive positioning systems relying exclusively on the input from a GPS re-ceiver, which is a line of sight sensor, tend to be sensitive to situations with limited sky visibility. Such situations include: urban environments with tall buildings; in-side parking structures; underneath trees; in tunnels and under bridges. In these situations, the system has to rely on integration of relative motion sensors to esti-mate vehicle position. However, these sensor measurements are generally aﬀected by errors such as oﬀsets and scale factors, that will cause the resulting position accuracy to deteriorate rapidly once GPS input is lost.

The approach in this thesis is to use a GPS receiver in combination with low cost sensor equipment to produce a robust positioning module. The module should be capable of handling situations where GPS input is corrupted or unavailable. The working principle is to calibrate the relative motion sensors when GPS is available to improve the accuracy during GPS intermission. To fuse the GPS information with the sensor outputs, diﬀerent models have been proposed and evaluated on real data sets. These models tend to be nonlinear, and have therefore been processed in an Extended Kalman Filter structure.

Experiments show that the proposed solutions can compensate for most of the errors associated with the relative motion sensors, and that the resulting positioning accuracy is improved accordingly.

Keywords: Positioning, Vehicle Navigation, Sensor Fusion, Extended Kalman Filtering, Robustness

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Acknowledgments

First of all I would like to thank my supervisor at NIRA Dynamics, Peter Hall. Peter has very patiently provided help and support with both the theoretical and the practical problems that have arisen during the progress of this project.

I would also like to thank Professor Fredrik Gustafsson and Urban Forssell for their valuable input and encouragement. Special thanks to my supervisor at LiTH, Thomas Schön, for the interesting discussions and fruitful feedback during the past months. Christian Sahlén also deserves special mention for his swift solution to the voltage problem.

Finally, I would like to thank all of the staﬀ at NIRA Dynamics AB in Linköping for making my time there rewarding and enjoyable.

Linköping, September 2004 Jon Kronander

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Notation

Symbols

ax Longitudinal acceleration

ay Lateral acceleration

br Distance between rear wheels

C Vehicle roll constant

E The RT90 easting coordinate

hCG Distance from ground to centre of gravity

K Kalman gain matrix

M Mass of car

N The RT90 northing coordinate

N₃ Normal force on rear left wheel

N4 Normal force on rear right wheel

P State estimate error covariance

r Average eﬀective radius of rear axle

r3 Eﬀective radius of rear left wheel

r4 Eﬀective radius of rear right wheel

R Distance from ICM to the middle of the rear axle

R₃ Distance from ICM to the rear left wheel

R₄ Distance from ICM to the rear right wheel

s Longitudinal slip

T System sample time

v₃ Velocity of the centre of the rear left wheel

v₄ Velocity of the centre of the rear right wheel

vx Longitudinal velocity

x True state vector ˆx Estimate of state vector

δturn

dif f Radius diﬀerence during steady state turning

δG Gyro oﬀset

η Roll moment distribution between front and rear wheels

ω₃ Angular velocity of the rear left wheel

ω₄ Angular velocity of the rear right wheel v

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Ψ Yaw or heading angle ˙Ψ Yaw rate

˙Ψm

Measured yaw rate (from gyro) ¨Ψ Yaw angular acceleration

σ Standard deviation

Operators and functions

E Expected value

f Nonlinear state dynamics h Nonlinear measurement function

Abbreviations

ABS Anti-lock Braking System DGPS Diﬀerential GPS

DOP Dilution of Precision

DR Dead Reckoning

ECEF Earth Centred and Earth Fixed

EGNOS European Geostationary Navigation Overlay Service EKF Extended Kalman Filter

GNSS Global Navigation Satellite System GPS Global Positioning System

ICM Instantaneous Centre of Motion INS Integrated Navigation System

KF Kalman Filter

MEMS Micro Electro-Mechanical Systems PDF Probability Density Function

RF Radio Frequency

RMS Root Mean Square

RT90 Rikets Koordinatsystem 1990 SAE Society of Automotive Engineers SBAS Satellite Based Augmentation Systems SDR Simple Dead Reckoning

SV Satellite Vehicle

WGS84 World Geodetic System 1984

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Introduction

1.1 Background

1.1.1 Satellite Aided Absolute Positioning

Navigation and positioning are critical to many human activities, but the process has always been quite problematic. It was only during the latter half of the 20th century that navigation took a big leap with the advent of the Global Positioning System (GPS). The GPS satellites – there are approximately 24 of them – are positioned in six orbital planes around the earth. Once every second, each satellite transmits a radio signal containing information about its location. With the aid of a relatively low cost GPS receiver, these satellites make it possible to determine the receiver position to within a few metres, almost anywhere on earth.

Upon first hearing about GPS, most people feel that the conundrum of nav-igation and positioning has been solved once and for all. However, despite the magnificence of GPS, it is not without problems. To attain a position fix, radio messages from at least four satellites have to reach the receiver uncorrupted. How-ever, interference from obstacles can be quite noticeable, and sometimes completely invalidates the GPS position fix. Such situations include urban environments with high rise buildings, underneath trees, inside parking structures, in tunnels and under bridges.

1.1.2 Relative Positioning

A simple technique used to determine relative displacement is dead reckoning. Dead reckoning1, or simply DR, involves knowing the relative direction of travel and distance covered. Combined with knowledge of the initial position, DR can be a practical method of navigation. Before celestial navigation was developed in the late ﬁfteenth century, DR was the primary navigational method used by sailors, including Columbus. In modern automotive applications, individual wheel angular

1_{Originally called "ded-reckoning", an abbreviation for deduced reckoning.}

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2 Introduction −4000 −3000 −2000 −1000 0 1000 2000 −1000 −500 0 500 1000 1500 2000 2500 3000 3500 4000 Easting [m] Northing [m] GPS DR DR (no gyro)

Figure 1.1. Trajectories calculated from DR sensor input compared to GPS reference. velocity information, and optionally a heading angle gyroscope, combined with a known initial position and orientation can be used to provide a DR estimate.

Inherent problems with the DR technique are the errors that all sensors have to some degree. Gyros for instance, tend to have a noticeable oﬀset, that if uncor-rected will make a DR estimate useless after a relatively short period of time. In addition, the wheel angular velocity has to be converted into a metric velocity by multiplication with the radius of the wheel, which tend to vary with pressure and wear and tear.

As an example, Figure 1.1 illustrates the difference between two DR trajectories and the corresponding "true" GPS trajectory. The first of the DR trajectories relies on a gyro to provide heading angle information, the second does not use the gyro, but instead estimates the heading angle change by comparing the velocities of the left and right wheels (how this is done will be covered in Section 6.1). Wheel speeds and gyro signals were used without filtration, and the wheel radii were set to the same nominal value.

1.1.3 Integrated Positioning

The DR position estimate has an unbounded error over longer periods of time, but will remain accurate over shorter intervals. In contrast, the GPS estimate is bounded, but suﬀers from high frequency disturbances like multi path propagation, and as mentioned earlier, can become unavailable during signiﬁcant periods of time. The DR sensor signals, on the other hand, are practically always available. Because of these complementary characteristics, DR and GPS should together be able to provide a more robust position estimate.

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1.2 NIRA Dynamics 3

1.1.4 Map Aided Positioning

In automotive applications, a potential source of additional positioning information is a digitized map of the local road network. By assuming that the vehicle is travelling on roads included in the map, a position estimate can be obtained by matching the trajectory obtained from the DR sensors with the spatial structure of the road network. The work presented in [12] shows that such a system, using only angular velocities and a digital map, is able to provide an absolute position with precision equal to or better than the GPS. The only external input required is an initial position estimate to limit the region of the map that needs to be searched for a matching road.

1.2 NIRA Dynamics

The project that this master thesis is based on has been performed at NIRA Dy-namics AB. NIRA DyDy-namics is a research and development company focusing on safety enhancing products for the automotive industry. It has oﬃces in Linköping and Gothenburg, and altogether there are around 16 employees. For more infor-mation about the company, visit www.niradynamics.se.

1.3 Problem Speciﬁcation

The general objective of this thesis is to ﬁnd a robust solution to the positioning problem using sensors available in automotive environments, such as:

• GPS

• ABS (individual wheel angular velocities) • Yaw rate gyroscope

Of special interest is the performance in scenarios where conventional positioning methods do not apply, or have a general degradation of performance, e.g., when the vehicle is not driving on the road network of the map (matching and map-aided positioning do not apply). Another issue is situations where the GPS input is temporarily blocked, or corrupted in some other way. Examples of such situations are:

• Urban environments with tall buildings (”Urban canyons”) • Inside parking structures

• In tunnels

• Under heavy foliage • Under bridges

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4 Introduction

Therefore, important requirements on the solution are that it cannot rely on map data, and that it should be able to handle troublesome situations such as the ones mentioned above.

The yaw rate gyro is a comparatively expensive sensor, and not as ubiquitous within the automotive industry as the ABS sensors. Therefore, it is interesting to investigate if a position estimate can be delivered in the situations mentioned above without using any gyro input.

1.4 Objectives

The main purpose of this master thesis is to develop and implement an integrated positioning module, that fuses information from the GPS receiver with signals from DR sensors to provide a more robust positioning solution. To achieve the main objective, the following issues have to be considered:

• Models of the vehicle suitable for the sensor fusion have to be proposed. • Diﬀerent positioning algorithms must be evaluated on real measurement data

from relevant driving scenarios.

• The most promising solution, or solutions, should be implemented in ANSI C.

1.5 Limitations

The positioning system in this thesis only handles planar positioning. One reason for this is that the DR sensors used, wheel angular velocities and optionally a yaw rate gyroscope, only produce information about the relative changes in an assumed plane of motion. Additionally, most automotive GPS receivers have very poor precision in the vertical coordinate, with a drift that is usually higher than the actual vertical movement of the vehicle.

1.6 Thesis Outline

Chapter 2 introduces the theory used for the sensor fusion, starting with some basics of observer design and concluding with the Extended Kalman Filter. Chapter 3 describes the Global Positioning System, with focus on diﬀerent error factors and a possible method of reducing one of the them. The GPS receiver used in the thesis is also presented here. Chapter 4 explains the DR sensors, and Chapter 5 is an overview of some issues of integration of GPS with such sensors. Chapter 6 introduces the basic vehicle equations and some of the phenomena that will aﬀect the positioning accuracy. Finally, Chapter 7 and 8 present various models used to solve the positioning problem along with relevant results. The conclusions are summarized in Chapter 9, along with suggestions for future work.

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Chapter 2

Sensor Fusion

The basic principle behind a robust positioning system is the fusion of information from several diﬀerent sensors, that are in some way related to the position of the vehicle. To be able to perform this fusion, a model relating the vehicle to the sensor outputs is necessary.

2.1 Linear State Space Models

When approximating reality with a model, it is important to choose a level of ap-proximation that makes the model computationally manageable, but still providing meaningful description of the properties that are of interest. A model structure that is commonly used is the linear state space description:

xk+1 = Fkxk+ Gkwk (2.1a)

yk = Hkxk+ ek (2.1b)

where xk is the state vector, yk the measurable system output, and wk and ek are

white noise processes with the following statistical properties:

E(wk) = 0 (2.2a) E(ek) = 0 (2.2b) E(wkwkT) = Qk (2.2c) E(ekeTk) = Rk (2.2d) E(wkwkT+i) = 0, i= 0 (2.2e) E(ekeTk+i) = 0, i= 0 (2.2f) E(ekwiT) = 0, ∀ k, i (2.2g)

The initial state vector x₀ is usually considered to be a random vector:

E(x₀) = ¯x₀ (2.3a)

E((x₀− ¯x₀)(x₀− ¯x₀)T) = Π₀ (2.3b) 5

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6 Sensor Fusion

In many applications, it is of great interest to know the value of the state vector of a system model. A problem is that the state variables are usually not directly measurable. Instead, the measurements consist of noisy linear (or nonlinear) com-binations of the state variables. Determining the states of a linear system, given access to its known inputs and measurable outputs, is referred to as the observer

design problem [13].

2.2 Basics of Observer Design

As stated in the previous section, the task of the observer is to provide an estimate ˆxk of the true state xk. In principle, such an estimate could be calculated in the

following way:

ˆx0 = x0 (2.4)

ˆxk+1 = Fkˆxk (2.5)

It is obvious that this observer is suboptimal; it relies only on the initial state esti-mate as input, and it does not use the information contained in the measurements

yk. To utilize the measurements, a feedback term can be added to the observer

equation:

ˆxk+1 = Fkˆxk+ Kk (2.6)

k = yk− Hkˆxk (2.7)

To see the beneﬁt of this, consider the special case of a time invariant state space model with no process or measurement noise:

xk₊₁ = F xk (2.8)

yk = Hxk (2.9)

The observer error,˜xk= x∆ k− ˆxk, will now obey the equation:

˜xk+1= F ˜xk− K(yk− H ˆxk) = (F − KH)˜xk (2.10)

This implies that

˜xk = (F − KH)k˜x0 (2.11)

Thus, if we can ﬁnd K = 0 so that F − KH has any desired eigenvalues, then the error ˜xk can be made to go to zero at any desired rate as k increases. If the

system is observable, K can be chosen to give F− KH arbitrary eigenvalues [11]. To return to the more general model (2.1a), the corresponding observer error can be written

˜xk₊₁= (Fk− KkHk)˜xk+ Gkwk− Kkek (2.12)

Now the choice of K is not as obvious as in the case without process or measurement noise.

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2.3 The Kalman Filter 7

2.3 The Kalman Filter

where Pk_|k is the covariance of the state estimate error. Equations (2.14) and

(2.15) are commonly referred to as the measurement update and the time update, respectively.

Although the matrices Qk and Rk are meant to represent the true covariances

of the process and measurement noise, respectively, they are often viewed as tuning parameters of the filter. For instance, the filter can be tuned to follow rapid changes in a certain state variable more directly by increasing the corresponding diagonal value of the Q matrix. In the same way, the effect of noisy measurements can be reduced by increasing the diagonal values of the R matrix.

In practical applications, the values of Q and R are usually set to some ad hoc value that is believed to reﬂect the properties of the underlying signal, and then during simulations tuned to achieve the desired compromise between the ability of the ﬁlter to track state changes and suppress measurement noise.

2.4 The Extended Kalman Filter

Linear Kalman ﬁlter theory has proved to be useful also when working with

non-linear models. The nonnon-linear state space model is non-linearized around a nominal

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8 Sensor Fusion

Kalman equations to be applied, cf. Section 5.1. Consider the nonlinear continu-ous system with discrete time observations:

˙xt = f(xt) + wt (2.16a) yk = h(xk) + ek (2.16b) E(wt) = 0 (2.16c) E(wtwTt) = Qt (2.16d) E(ek) = 0 (2.16e) E(ekeTk) = Rk (2.16f)

2.4.1 Time Update

To perform the time update step of the ﬁlter, the nonlinear continuous model can either be linearized and then discretized, or the other way around. The discretized

linearization has the following time update equations, see [10]:

ˆxk+1|k = ˆxk|k+ T 0 ef _(ˆx k|k)τ_dτ f(ˆxk|k) (2.17a) Pk_+1|k= ef(ˆx_k|k)T_P k_|k ef(ˆx_k|k)TT _{+ ¯}_Q k (2.17b)

where f is the jacobian of f , and ¯Qk is the covariance of the discretized version

of wt in (2.16a). There are several alternative ways of calculating ¯Qk from Qt,

depending on the assumptions made about wt. For its simplicity it is here assumed

that wt is white noise such that its total inﬂuence during one sample interval is

¯

Qk = T Qt. Other more elaborate assumptions on wt have also been evaluated,

but they did not result in any noticeable beneﬁt.

2.4.2 Measurement Update

A linear measurement equation can be created by a Taylor expansion of (2.16b) around the predicted state:

yk ≈ Hkˆxk|k−1+ ek (2.18a) Hk = dh(x) dx x_=ˆx_k|k−1 (2.18b) With this approximation, it is straightforward to apply the standard Kalman ﬁlter measurement update equations (2.14a) to (2.14d), with one exception: in (2.14c), the calculation of the residual ek= (yk− Hkˆxk_|k−1) should be performed using the

nonlinear measurement equation:

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Chapter 3

The Global Positioning System

The performance of the filters proposed in this thesis depend heavily upon the quality of the position fixes supplied by the GPS receiver. Therefore, this chapter will describe the basic functionality of satellite based positioning systems, along with the most significant error sources. Before GPS is handled, some coordinate systems commonly used in positioning are described, including the thesis specific body fixed reference frame.

3.1 Coordinate Systems

When working with the position of an object in space, it becomes necessary to deﬁne that position relative to some known frame of reference. In this thesis the Earth Centred Earth Fixed frame is used together with a local tangent plane frame. A body ﬁxed vehicle frame is also introduced.

3.1.1 Earth Centred Earth Fixed Frame

Earth Centred Earth Fixed frames (ECEF) have their origin at the centre of the earth and rotate around the earth’s spin axis. Two coordinate systems are used in the ECEF frame: rectangular and geodetic.

Rectangular coordinates

The x-axis of this system extends through the prime meridian (0◦ longitude) and the equator (0◦latitude), see Figure 3.1. The z-axis extends through the north pole, parallel to the earth’s spin axis. The y-axis completes the right-handed coordinate system.

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10 The Global Positioning System

Prime Meridian

z

y

x

_Equator

Figure 3.1. ECEF rectangular coordinates

Geodetic Coordinates

The earth is often approximated with an ellipsoid of revolution about its minor axis. Such an ellipsoid is determined by its major (a) and minor (b) axes, as described in Figure 3.2. Note that the surface normal vector P does not intersect the origin of the xyz-axes.

Most position information from navigation tools are expressed in geodetic co-ordinates, where λ denotes latitude, φ longitude and h altitude (h is the elevation above the ellipsoid along the surface normal). There are several ellipsoid models of earth. Perhaps the most widely used, mainly because GPS is based on it, is the World Geodetic System 1984 (WGS84). In addition to a major and minor axis, WGS84 speciﬁes a gravitational constant and an angular velocity (rotation of the earth).

3.1.2 Vehicle Body Fixed Frame

This frame is suitable for description of basic vehicle dynamics and derivation of related equations of motion, and is depicted in Figure 3.3 along with the numbering of the wheels. The x-axis, pointing in the rear-to-front direction, is referred to as the longitudinal axis. Similarly, the y-axis is referred to as the lateral axis and points in the left-to-right direction. The z-axis completes the right hand system. This is the vehicle ﬁxed coordinate system used by the Society of Automotive Engineers (SAE) [8].

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3.2 Global Navigation Satellite Systems 11

about z

b

a

P

y

x

P

z

rotation

λ

φ

Figure 3.2. ECEF geodetic coordinates

x y z 4 2 1 3

Figure 3.3. Body ﬁxed frame and wheel numbering

3.1.3 Planar Coordinates

Instead of describing the vehicle position in WGS84 coordinates, a simpler planar system will be used during development, see Figure 3.4. There are several such planar approximations, one is the Swedish RT90 system, which will be used in this thesis. Basically, such planar approximations are tangent planes to the WGS84 ellipsoid, and they are only useful in a limited geographical area where the deviation between the plane and the ellipsoid remains small.

The vehicle’s position in a planar system is speciﬁed by two coordinates called

easting and northing. Since the navigation system described in this thesis only

concerns planar positioning, the altitude component is excluded. To integrate DR and GPS signals, an absolute heading angle,Ψ is deﬁned. Ψ will from now on be referred to as the yaw angle.

3.2 Global Navigation Satellite Systems

The Global Positioning System (GPS) falls into the category of navigation systems referred to as Global Navigation Satellite Systems (GNSS). A GNSS is basically a network of satellites. Each satellite regularly, and in synchronization with the other satellites in the system, transmits high frequency radio signals containing

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N

E

vx

Ψ

Figure 3.4. Navigation coordinates

time data. A receiver able to pick up these signals from several satellites will be able to determine its position on Earth with an accuracy of about 10 to 15 metres. Currently (May 2004) there are two independent GNSS in operation: the U.S. Global Positioning System (GPS) and the Russian Global Navigation Satellite Sys-tem (GLONASS). A third GNSS is being developed by the European Space Agency, called GALILEO. It is planned to become fully operational in 2008, providing higher standard of both accuracy and integrity. There are other satellites in orbit that transmit signals used for navigation, but these only ﬁll a supportive role to GPS and/or GLONASS, rather than providing stand alone navigation solutions.

In this thesis a GPS receiver is used. This is mainly because GPS is the most widely used GNSS today, which has resulted in relatively low cost receiver technol-ogy, and the future of GLONASS appears uncertain. For example, the questionable ﬁnances of the Russian agency responsible for maintenance has resulted in system degradation. As of January 2002 there are only 6 functional GLONASS satel-lites, which is far from the approximately 24 satellites needed for continuous global coverage [14].

The actual GPS messages are complex, but the basic principle fairly straight-forward. A GPS receiver needs signals from a minimum of two satellites to be able to estimate its location. The receiver calculates the distance to each of the satellites by measuring the travel time of the signal (from satellite to receiver an-tenna). These measurements deﬁne two spheres, centred on each satellite with the radii equal to the respective satellite-receiver distance. The intersection of these spheres constitute a circle of possible locations of the antenna. Assuming that the receiver is on the surface of the earth, a third sphere (actually an ellipsoid) enters the equation. The receiver location ambiguity is now reduced to two points where all three spheres intersect. In most practical cases it is obvious which of these is the location, since they are usually very far apart.

The more general situation when the assumption of a location on the surface of the Earth does not hold, a third satellite is needed to provide a solvable set of equations. In general, the more satellites used the greater the accuracy. Many modern receivers have the ability to receive signals from at least 12 satellites in

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3.3 Error Sources 13

parallel.

Another problem is that the receiver clock is not synchronized with the satellite clocks. A consequence of this is that the measurement is not really a range from satellite to receiver, but some other distance depending on the receiver clock bias. It is therefore called a pseudorange. Therefore, to be able to estimate the three unknown position coordinates and the clock bias, four satellites have to be in view.

3.3 Error Sources

The main error sources in GPS are listed in Table 3.1. These errors can be divided into two categories [5]: common and non common. Common errors are approxi-mately the same for receivers operating within a limited geographic region. Non common errors are unique to each receiver and depend on the receiver type and multipath mitigation technique being used (if any). The point of this classification is that Differential GPS – a GPS augmentation system, discussed in Section 3.5 – can effectively remove the common errors. A further description of each error source follows.

Source Standard deviation (m)

Common

Ionosphere 7.0

Clock and ephemeris 3.6

Troposphere 0.7

Non common

Receiver noise 0.1-0.7

Multipath 0.1-5.0

Table 3.1. GPS error sources and their approximate σ.

3.3.1 Receiver clock bias

The receiver measures travel times by comparing time marks imprinted on the satellite signals with the time recorded on the receiver’s clock. The time marks are generated by high precision atomic clocks on board each satellite. Atomic clocks are too expensive to incorporate in the GPS receiver. Instead, standard quartz oscillators are used. Quartz oscillators are very accurate when measuring times of less than a few seconds, but tend to drift over longer periods. All satellite clocks are synchronized by the control segment. Therefore, if four simultaneous satellite signals are available, both the position and the receiver clock bias can be estimated.

3.3.2 Satellite clock bias

The control segment continuously monitor the on board atomic clocks of each GPS satellite. A comparison between the control centre atomic clock and the

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individual satellite clocks results in corrective parameters that are sent to the user via the standard GPS messages. Satellite clock errors aﬀect all users, independent of location, in the same way. This means that diﬀerential corrections are valid even for users very far from the DGPS station, more on this in section 3.5. Even though there is a detectable drift in these clocks, they are still extremely accurate (1 second drift in 100 million years), and the resulting gain obtained with the correction parameters is very small.

3.3.3 Atmospheric delay

The so called troposphere is the lower part of the atmosphere, extending from 8 to 40 km above earth’s surface. Weather changes will cause changes in temperature, pressure and humidity in the troposphere. These variables will in turn aﬀect the speed of light, resulting in errors in the measured range. Tropospheric delays can be considerable (20 m) for satellites at low elevations (low angle above the horizon). Tropospheric delays are normally divided into a wet component and a dry component. The wet component refers to the delay caused by water vapour conditions. The dry component concerns the larger distribution of gasses in the troposphere. The wet component is diﬃcult to model because of local water-vapour variations. However, the dry component, which is estimated to make up 90 % of the total delay, can be modelled fairly well. Most receivers are able to reduce the tropospheric errors to below 1 m.

The upper part of the atmosphere is called the ionosphere. The ionosphere is the layer above 50 km that consists of ionized air. Changes in the level of ionization aﬀect the refractive indices of the various layers and therefore aﬀect the travel time of GPS signals. There are models of the ionospheric delay based on parameters broadcast by the GPS satellites. These models can compensate up to 50 % of the delay.

3.3.4 Ephemeris errors

The position of each satellite is determined by the control segment by using four monitoring stations. Because the locations of these stations are known precisely, the position of the satellite can be calculated by an ”inverted” GPS solution (regarding the satellite as the user). Position data is then relayed via the space segment to the ﬁnal user. Ephemeris errors are the deviations of the satellites from their transmitted positions. The ephemeris error can be divided into three components with respect to orbit: radial, tangential and cross track. The mean values of these errors are in the range of a 1-2 meteres. Generally, the radial error will increase slowly with time since the last control segment correction.

3.3.5 Multipath

The GPS receiver determines the GPS signal transit time by correlating an inter-nally generated version of the satellite signal with the received signal. The internal

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3.4 Dilution of Precision 15

version can be shifted in time until maximum correlation occurs. The time cor-responding to maximum correlation minus the known time at which the satellite generated the signal is the measured transit time. Multipath errors occur when the GPS signal is reﬂected on surfaces close to the receiver that shift the correlation peak. This will result in erroneous pseudorange measurements. Multipath errors can be expected to be less than 5 m [5] in most cases.

3.3.6 Receiver noise

This error depends upon receiver implementation factors, such as antenna design, method for A/D conversion and the correlation processes.

3.4 Dilution of Precision

As was discussed in Section 3.3, the pseudorange measurements are uncertain. Con-sidering a fixed pseudorange uncertainty, the geometry of the satellites included in the solution will determine how the range ambiguity translates into a final position uncertainty. A measurement of this effect is called dilution of precision (DOP), see Figure 3.5. In the case of four available satellites, minimum DOP values are

ob-SV2

SV1

SV2

SV1

Figure 3.5. Low and high DOP conﬁguration

tained when three of the satellites are spread equally over the horizon,120◦apart, and the fourth directly over the receiver.

There are three types of DOP values: Position DOP, Horizontal DOP and Vertical DOP (PDOP, HDOP and VDOP, respectively). These are measurements of how the geometric situation aﬀects the uncertainty in the respective directions. For the horizontal applications in this thesis, a threshold on HDOP can possibly be used to invalidate certain GPS ﬁx-points that otherwise appear valid.

3.5 Diﬀerential GPS

Two GPS receivers operating within 100-200 km of each other will be subject to approximately the same atmospheric errors. In addition, ephemeris and satellite

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clock errors will be identical as long as the same set of satellites is used. If it was possible to estimate the errors affecting one of the receivers, corrections could be sent to the other receiver to remove (or mostly remove) these errors. This is the operating principle in Differential GPS (DGPS). By comparing the precisely known location of a stationary GPS receiver, a reference receiver, with the calculated GPS fix for that location, the common errors can be estimated. Accuracies around 1 m are common.

DGPS corrections are usually transmitted from the reference receiver to the user via radio link, and requires additional hardware and sometimes also a subscription to the correction messages.

3.6 Satellite Based Augmentation Systems

SBAS basically work the same way as DGPS, but the corrections are transmitted via geostationary satellites. In 2004, an SBAS system covering most parts of Eu-rope will become operational: EuEu-ropean Geostationary Navigation Overlay Service (EGNOS). EGNOS will complement GPS by transmitting correction and integrity messages. EGNOS will when operational consist of three geostationary satellites, 34 reference stations spread across Europe and three control centres. Accuracy is expected to be better than 5 m 95 % of the time. Similar systems to EGNOS are available, or under development, in both North America and Asia.

3.7 Carrier Phase GPS

Centimeter level accuracy is possible, relative to a reference station, by utilizing information about the carrier phase. The method consists of determining the num-ber of phases between the receiver and the individual SVs. The position can be determined to fractions of a wavelength (≈ 20 cm). As of this writing (October 2004), carrier phase GPS requires expensive equipment and is a computationally complex problem, and therefore it seems unlikely that it will be used in standard automotive applications in the near future.

3.8 Detecting Erroneous Position Fixes

Assuming that an estimate of the current position is available, contained in the state vector ˆx, it might in some situations be possible to draw the conclusion that an incoming GPS ﬁx deviates so much from the estimated position, that it is likely subject to multipath disturbances. If a Kalman ﬁlter is used, the calculated innovation, e= y − ˆy, along with its estimated covariance Sk from (2.14a), make

up the components of a test that can be performed to decide whether the new position measurement is too unlikely, and therefore should not be included in the measurement update [17]. The test is based on chi-square distribution theory.

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3.8 Detecting Erroneous Position Fixes 17

3.8.1 Chi-Square Distributed Random Variables

Consider the n-dimensional Gaussian random vector z, with mean ¯z and covari-ance Pz. The scalar random variable deﬁned by the quadratic form:

q= (z − ¯z)TPz−1(z − ¯z) (3.1)

can be shown to be the sum of the squares of n independent zero-mean, unity variance Gaussian random variables [2]. This sum is said to have a chi-squared distribution with n degrees of freedom:

q ∈ χ2n (3.2a)

E(q) = n (3.2b)

V ar(q) = 2n (3.2c)

When using a Kalman ﬁlter and a proper model, the parameter estimation error:

˜x = x − ˆx (3.3)

is approximately Gaussian distributed, with covariance given by:

P = E[˜x˜x∆ T] (3.4)

where P is assumed invertible (rank(P ) = dim(x). The normalized estimation

error squared (NEES) for the parameter x, deﬁned as

x= ˜x∆ TP−1˜x (3.5)

is chi-square distributed with dim(x) degrees of freedom. Let g be such that

P{x≤ g2} = 1 − Q (3.6)

where Q is a small ”tail” probability. The conﬁdence region for the parameter x follows from (3.5) and (3.6) as the inside of the g-sigma ellipsoid, deﬁned by:

(x − ˆx)T

P−1(x − ˆx) = g2 (3.7)

where g is chosen for the desired gate probability. The fact that (3.7) is an ellip-soid is a consequence of the positive deﬁniteness of the covariance matrix P. This region is the probability concentration ellipsoid, obtained by cutting the tail of the multivariate Gaussian density.

In this thesis, the theory presented above will be used primarily for two-dimensional Gaussian vectors. The probability density function (PDF) of a chi-square distrib-ution with n= 2 is:

p() = 1 2exp −1 2 (3.8) Thus, to obtain an expression for an ellipse with conﬁdence level q, the equation:

P{x≤ g2} = g2 0 1 2exp −1₂ d= q ⇒ g2= −2 ln (1 − q) (3.9)

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is used together with (3.7). More speciﬁcally, to test the incoming measurement vector yk, Equation (3.7) would be modiﬁed into:

(yk− ˆyk)TSk−1(yk− ˆyk) = g2 (3.10)

The value of Q that determines g2, would have to be a compromise between ability to detect multipath errors above certain magnitudes and still allowing the ﬁlter to function properly, i.e., not disconnect the GPS ﬁx because of high innovations caused by other errors not included in the model.

3.9 Receiver Used in Experiments

The receiver used in this thesis is the Garmin GPS 35LP. It is specified to deliver position fixes with an error below 15 m, 95% of the time. It is connected to the measurement platform via a serial cable, and the GPS data is transmitted using the standard NMEA 0183 protocol. The 35LP is a basic automotive receiver, and does not use any carrier phase observables. Therefore, even though the receiver outputs signals such as velocity and heading, they are all derived from position fixes, i.e., the heading estimate is simply the angle from true north to the vector between the last two fixes.

Position estimates are delivered once every second along with information spec-ifying how the estimate was obtained. From this additional data it is possible to deduce if the ﬁx is invalid (in the sense that no new estimate has been calculated), if it is a DGPS (when hardware available) or non-DGPS ﬁx, and also if it has been

extrapolated by the receiver (a primitive form of DR internal to the receiver using

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Chapter 4

Dead Reckoning Sensors

The DR performed in this thesis use data from individual wheel angular velocity sensors and optionally a yaw rate gyro. These signals are collected from a measure-ment platform developed by NIRA Dynamics, along with other data such as GPS signals and vehicle status information, e.g., ﬂags indicating events like braking and reverse gear activation. The sampling rate, fs, has been limited to 10 Hz due to

real-time constraints of the platform.

4.1 Wheel Angular Velocities

Most modern cars are equipped with a Anti-lock Braking System (ABS). The ABS system depends on accurate measurements of the angular velocities of each indi-vidual wheel. Within the automotive industry, the prevalent method of measuring the angular velocities involves a cogged ferrous disc mounted on the wheel axle. When the wheel rotates, a variable-reluctance or a hall-eﬀect sensor will be able to detect the passage of individual cogs; typically there are 48 cogs. These sensors have particular characteristics, discussed below, that aﬀect their applicability in a positioning system.

4.1.1 Variable-reluctance Sensor

The variable-reluctance sensor is composed of a permanent magnet with a coil of wire around it, see Figure 4.1. The magnet is mounted in such a way that the cogs of the ferrous wheel pass between the poles. In doing so, the cogs will alter the reluctance of the magnetic circuit. Reluctance can be thought of as magnetic ”resistance” and the magnetic ﬂux is inversely proportional to it. When a cog approaches the gap between the magnetic poles, the reluctance and the magnetic ﬂux (Φ) will change, which can be detected by the induced electromotoric force in the coil:

EM F = −dΦ

dt (4.1)

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20 Dead Reckoning Sensors

At low angular velocities, the induced EMF will not be strong enough to be de-tected by the complementary circuitry. This makes the variable reluctance sensor diﬃcult to use in positioning systems involving low speeds, typically at or below 1.6-4.8 km/h [20]. Other drawbacks are a variable voltage envelope with varying speed, degradation of signal to noise ratio because of vibrations and resonance, and sensitivity to the gap between the sensor and the disc.

Figure 4.1. Conﬁguration of ferrous wheel and variable reluctance circuit

4.1.2 Hall-eﬀect Sensor

Another way of detecting the passing of cogs is by taking advantage of the Hall-effect. Without going into the details of the Hall-effect, suffice it to say that sensors relying on this effect require a supply of external voltage. One of the advantages of this sensor is that once a cog lines up with the sensor there will be a peak in the waveform that does not depend on the speed of the cog, hence it will be reliable even at very low speeds. Whereas the variable-reluctance sensor delivers a sinusoidal wave with varying amplitude, the Hall-effect sensor produces a constant envelope square waveform.

In recent years, the use of Hall-eﬀect sensors has increased. The primary test car used in this work, a Volvo V70 from 2001, is equipped with such sensors. This will eliminate part of the problem with reliable wheel angular velocities at low vehicle speeds.

4.1.3 Wheel Angular Velocity Calculations

There are principally two diﬀerent ways of calculating the wheel angular velocity based on cog passage sensors.

The ﬁrst method consists of marking each cog passage with a time stamp, and comparing it to the time stamp of the previous cog. This is an event triggered operation that works well as long as the vehicle is moving at speeds where several cogs will pass during each sample interval of 100 ms. However, since one goal of the positioning module proposed in this thesis is to enhance the performance in diﬃcult situations, e.g., inside parking structures, it becomes important to have

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4.1 Wheel Angular Velocities 21 0 20 40 60 80 100 120 140 160 81 82 83 84 85 86 87 [sample] [rad/s] cog diff. cog count

Figure 4.2. Comparison between angular velocity from cog diﬀerentiation and direct 100 ms interval cog counts.

reliable wheel velocities even during very low speeds. Due to the decrease in event rate during such driving conditions, the diﬀerentiation of time stamps can become invalid because of overﬂows in the timer register. Another problem is that, at some point after waiting for the next cog event, a decision needs to be made that the vehicle is in fact standing still. This method has been used in previous po-sitioning related work at NIRA Dynamics [18, 12]. The data extracted from the platform every Ts seconds consists of the last value obtained from cog

diﬀerenti-ation. Note that multiple diﬀerentiations will be performed within each sample interval at higher vehicle speeds.

The second method is to simply count the cog passages in each 0.1 s sample interval, and can be seen as quantized measure of traversed distance, multiplied by a scale factor (the wheel radius and the number of radians per cog). This is the approach taken in [3], albeit with a lower quantization error (using 100 cogs/wheel instead of 48). One advantage with this method is that no ”artiﬁcial” velocities need to be deﬁned if no cog events have occurred during the last sample interval. A drawback is the quantization error, which is inversely proportional to the number of cogs per wheel.

A comparison between angular velocity derived from cog differentiaion and cog counts is presented in Figure 4.2. As can be seen, the quantization error in the cog count signal is much more prominent. However, the consequence of this quantiza-tion error will depend on how the velocities are included in the model. For instance, if the angular velocities from cog counts are used to drive first order differential equations, the integration will reduce the quantization errors (they are correlated between sample intervals). In some cases, where a momentary measurement of the angular velocity is required, cog differentiation is preferable.

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22 Dead Reckoning Sensors

4.2 Yaw-rate Gyroscope

To measure the angular rate around an axis, there are several phenomena that can be used; gyroscopes can be mechanical, optical or vibrational. In automotive applications, the vibrational gyro has become standard due to its low cost and small size. A vibrational gyroscope relies on the Coriolis force, ﬁrst described by French mathematician G. G. Coriolis in 1835. This force is the apparent force acting on a particle that moves in a rotating frame of reference. Such gyros are commonly implemented as MEMS (Micro ElectroMechanical System) devices.

Here we are mostly concerned with the potential errors associated with the gyro measurement. The error model of a gyro measurement, rm, of the true angular rate, r0, usually contains a scale factor and an oﬀset:

rm= a(t)r0+ b(t) (4.2)

Both the scale factor and the oﬀset can vary with time and temperature. Some modern gyros, including the one in the test vehicle, have built in standstill correc-tions of the oﬀset error.

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Chapter 5

Integration of DR and GPS

5.1 Modelling Alternatives

To use the complementary characteristics of dead-reckoning and GPS in an inte-grated positioning system is not a new approach. For instance, the automotive positioning systems proposed in [3, 16, 1, 15, 17] use some combination of GPS, an odometer and heading sensors. All of these use some form of Kalman ﬁlter.

When implementing an integrated positioning system with a Kalman ﬁlter, two choices need to be made: whether to use a complete vehicle or error state model, and if the integration should be done loosely or tightly. These are general terms used in integrated navigation literature [5, 9].

5.1.1 Complete Vehicle or Error State Model

Within the aviation industry, it has for long been customary to try to apply the linear KF by using a linearized error state model [19]. Put simply, the state repre-sentation includes only error states: position error, attitude error, etc. The dynam-ics of the error states are obtained by linearizing the vehicle and sensor dynamdynam-ics around a nominal state, where the nominal state trajectory is the one calculated from the measurements of the DR sensors. These measurements have to be highly accurate, otherwise the linearization will no longer be valid. The observations used as input to the KF are the diﬀerences between the GPS output and the nominal trajectory. The error states are then used to correct the DR output to improve its precision.

A slightly diﬀerent error state model approach involves a feedback term of the estimated error states to the DR unit, and in eﬀect ”reinitializing” it after each new Kalman cycle. In other words, the calculated nominal trajectory from the DR unit, used as the point of linearization, is recalculated after the measurement update in the KF in order to keep the nominal trajectory closer to reality. Fusing the information this way makes it possible to use DR sensors with lesser accuracy, without invalidating the linearization.

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24 Integration of DR and GPS

The intuitively most appealing approach is to describe the vehicle, position, orientation and various sensor errors in a single ”complete” state vector. This approach generally results in nonlinear state dynamics. Since this way of modelling the system is the most robust alternative concerning inexact DR measurements [19], it is the approach chosen in this thesis. Furthermore, the authors of [6] tried to use an error state model of a car, using the feedback term, but reached the conclusion that the automotive gyro was too imprecise to keep the linearization valid.

5.1.2 Loosely or Tightly Coupled

If the calculated positions (and in some cases also the velocity) from the GPS receiver unit are used as measurements, the integration is said to be done loosely. This is a slightly suboptimal approach, since the processing inside the receiver causes a loss of information and undesirable cross correlations between the output signals.

Instead, using the tightly coupled method involves a modiﬁcation of the mea-surement equations to accommodate the input of raw pseudo range meamea-surements (and in some cases pseudo range rate). Of course, this is only possible if the re-ceiver can actually deliver the raw range measurements, and this is generally not the case since pseudo range observables are not part of the standardized NMEA 0183 protocol. Therefore, the loosely coupled approach is the one used in this work.

5.2 Factors of Accuracy

5.2.1 Fix Precision

The work presented in [1] shows that the positioning accuracy of the integrated navigation system (INS) is roughly the same as that of the GPS position fixes themselves. This is perhaps not so surprising, noting that the GPS fixes are the only definite connection between the vehicle and the coordinates of the reference frame. Moreover, if some kind of DGPS is used, the parameter estimations improve significantly. As was mentioned in Section 3.9, the receiver used in this project only utilizes the standard positioning service.

5.2.2 Synchronization

During periods of good GPS coverage, a new fix will be delivered by the receiver at the start of each GPS second (synchronized with the satellite network), and stored in a specific register on the measurement platform. All input signals, including the GPS register, are sampled at 10 Hz. Since the sampling instant is not synchronized with the GPS satellites, it is possible that a new fix will be delivered to the platform register from the receiver, but not sampled until another 0.1 s (at most) have passed. How this will impact the filter is not obvious. What is certain is that a higher sampling frequency would allow for a better temporal match between GPS and

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5.2 Factors of Accuracy 25

DR data, but the probable, though perhaps not signiﬁcant, gains of this have not been possible to investigate in this thesis due to the sampling frequency limitation.

5.2.3 Yaw Rate Bias

When the GPS fixes become unavailable, the factor that has the strongest influence on the growth rate of the positioning error in the dead-reckoning unit, is the set of sensors utilized [1]. More specifically, the bias in the yaw rate sensor will constitute, by far, the largest contribution to the final position error. This can be explained by the fact that the bias in the yaw rate sensor is two integrations removed from position. Therefore, a constant bias in the yaw rate will cause the position error to grow approximately as the square of time. As a comparison, the offset in the longitudinal velocity measurement from the wheel angular velocities is only one integration removed from position, implying that the error caused by a constant offset will grow linearly with time. Hence, special care should be taken to try to compensate for as much as possible of the yaw rate bias. In this work that translates into estimating the yaw rate offset of the gyro, and/or the difference of the wheel radii on the same axis.

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Chapter 6

Vehicle Dynamics

6.1 Basic Vehicle Equations

For our purposes, viewing the vehicle as a rigid body is a suitable level of approx-imation. When a rigid body moves in a plane, basic mechanics deﬁnes a point at every instant, around which the body rotates, called the Instantaneous Centre of Motion (ICM). Figure 6.1 describes the ICM of a vehicle undergoing a turn. The longitudinal velocity of the rear axle centre, vx, and the hub velocities of the rear

wheels, v₃ and v₄, can be expressed using the notation from Figure 6.1:

vx = R ˙Ψ (6.1a) v₃ = R₃˙Ψ = vx+ ˙Ψ br 2 (6.1b) v₄ = R₄˙Ψ = vx− ˙Ψ br 2 (6.1c)

To obtain an expression for the angular velocities of the rear wheels, ω3 and ω4, the hub velocities are divided by the eﬀective wheel radii, r3 and r4:

ωi=

vi

ri

i= 3, 4 (6.2)

Combining (6.1) with (6.2) yield:

vx = r₃ω₃+ r₄ω₄ 2 (6.3a) ˙Ψ = r3ω3− r4ω4 br (6.3b) ω3 = 1 r₃ vx+ ˙Ψ br 2 (6.3c) ω₄ = 1 r4 vx− ˙Ψ br 2 (6.3d) where vx is the longitudinal velocity of the vehicle.

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28 Vehicle Dynamics ICM R3 R ˙Ψ R4 br

Figure 6.1. Geometry of a turn without side slip

6.2 Side Slip

Side slip occurs when the velocity vectors of the wheels are no longer parallel to their longitudinal axes. Such lateral velocity components are usually present when the vehicle is turning at relatively high speed and the friction between the road and the tyres is poor. A quantitative measure of side slip is the side slip angle, deﬁned as the angle between the longitudinal axis of the wheel and the velocity vector of the same wheel. During normal driving at relatively low speeds and high tyre-road friction, the side slip angle is very small and can be neglected. Furthermore, even if the side slip angle is nonzero, it is shown in [4] that (6.3a) to (6.3d) will still hold.

6.3 Longitudinal Slip

The SAE deﬁnition of longitudinal slip is:

si= −

vi− riωi

vi

, i= 1, 2, 3, 4 (6.4)

where viis the velocity of the hub of the wheel, riis the eﬀective wheel radius and

ωithe angular velocity. When the car is accelerating, the slip of the driving wheels

(front wheels in test vehicle) will tend to be positive. When braking, which aﬀects all wheels, the slip will be negative. The slip during normal driving rarely exceeds 2 % [7].

6.4 Tyre Compression

When the vehicle is turning, a lateral acceleration will cause a shift in the normal forces acting on the wheels; the outside tires are compressed and the inside tyres are extended. Therefore, the outside wheels will rotate faster and the inside wheels

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6.4 Tyre Compression 29 ICM vx ˙Ψ R ˆz(b) ˆx(b) ˆy(b) ˆz(e) ˆx(e) ˆy(e)

Figure 6.2. Reference frames during steady state turning

slower, than if the radii had been fixed. To obtain an expression of this lateral acceleration, consider two reference frames: one body fixed (b) and one earth fixed (e), as shown in Figure 6.2. The vehicle longitudinal velocity can be expressed in the body frame by the vector:

v(b)x = R ˙Ψˆx(b) (6.5)

The body ﬁxed frame rotates with respect to the earth ﬁxed frame: Ω = ⎛ ⎝0₀ ˙Ψ ⎞ ⎠ (6.6)

The lateral acceleration, ay, expressed in the body ﬁxed frame now follows from

Coriolis’ equation: a(b)y = d dt(v (b) x )(e)+ Ω × v(b)x = ⎛ ⎝0₀ 0 ⎞ ⎠ + ⎛ ⎝00 ˙Ψ ⎞ ⎠ × ⎛ ⎝˙ΨR₀ 0 ⎞ ⎠ (b) = ⎛ ⎝_˙Ψ02_R 0 ⎞ ⎠ (b) (6.7)

If not accounted for, the contraction/expansion caused by the lateral acceleration will result in an overestimation of the actual heading change when using (6.3b).

Once a vehicle has started turning, and is subject to a nearly constant yaw rate and longitudinal velocity, the roll moment will also be approximately constant. By using the notation deﬁned in Figure 6.3, a steady state relationship between the lateral acceleration and the normal forces is obtained:

N₃− N₄= 2 br F hCG= F = ηMay ay = vx˙Ψ =2ηMhCG br vx˙Ψ (6.8)

where η is the ratio of roll moment resisted by the undriven axle to the total roll moment and hCG the height of the centre of gravity. Under the assumption that

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30 Vehicle Dynamics CG N3 N4 F hCG vx ay br

Figure 6.3. Vehicle forces during steady state turning (into paper, to the right) the tyres behaves like a linear spring in the radial direction, with the same spring constant k, the resulting deformation can be expressed as:

δturndif f =

2ηMhCG

brk

vx˙Ψ∆= vx˙Ψ (6.9)

Using (6.9), the wheel radii might instead be modelled:

r₃ = r3− Cay= r3− Cvx˙Ψ (6.10a)

r₄ = r4+ Cay= r3+ Cvx˙Ψ (6.10b)

where the value of C (hereafter referred to as the vehicle roll constant ) can be determined by assigning approximate values to the variables in (6.9). However, the parameters will vary with the load proﬁle of the car, the tyres used, the inﬂation pressure, etc. A more attractive alternative – if possible – is therefore to estimate the value of C online when GPS is available (more on this in Section 8.2).

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Chapter 7

Models and Experiments:

With Yaw Rate Gyro

This chapter presents some of the models used to solve the positioning problem when using GPS, angular velocities and a yaw rate gyro. Results from relevant experiments are also included. Altogether, over 20 diﬀerent models have been eval-uated. Most of them are very similar, and only a few representatives are described in the following.

Data sets used for ﬁlter evaluation were collected from various driving scenarios with the primary test vehicle used in the project, a front wheel driven Volvo V70. Descriptions and overviews of these scenarios can be found in Appendix A.

7.1 Model M1

If a yaw rate gyro is available, one can choose to integrate it with the rest of the sensors in several different ways. Perhaps the most straightforward approach is to only use the gyro measurement, possibly corrected for bias and scale errors, to drive the filter in the lateral direction, i.e., no yaw rate information from the ABS is utilized. This might seem contrary to the principle that the more measurements one has to fuse, the better the result. However, a Kalman filter sensor fusion algorithm has a performance that is based on the validity of the underlying model of the system. As will be evident in Chapter 8, the yaw rate estimate from differential wheel angular velocities has some properties that are difficult to model correctly.

7.1.1 State Space Model

This basic model uses the following state vector to describe the system (for the sake of brevity the time index will be suppressed, i.e., E(t) is represented by E and so on):

x=E N Ψ r δG

T

(7.1) 31

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32 Models and Experiments: With Yaw Rate Gyro

where E and N are the absolute coordinates expressed in the RT90 system (see Figure 3.4); r is the average radius of the rear wheels; δG is the gyro oﬀset. The

nonlinear state dynamics are:

˙x = ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ ˆvxsin Ψ ˆvxcos Ψ ˆ˙Ψ 0 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠+ w = f(x) + w (7.2)

whereˆvx and ˆ˙Ψ are the calculated inputs:

ˆvx =

r

2(w3+ w4) (7.3a)

ˆ˙Ψ = ˙Ψgyro_{+ δ}

G (7.3b)

The time discrete observations at time kT (zk = z(kT )) of the sensor

measure-ments, when GPS is available, are the absolute RT 90 coordinates:

zk = E_kgps N_kgps (7.4) These observations are modelled as direct measurements of the corresponding states, aﬀected by white noise:

yk= Ek Nk + ek (7.5)

When GPS is unavailable, the ﬁlter will remain in prediction mode (no measure-ment update is performed). The discretized linearization of this model can be found in Appendix B.1.

When the vehicle is braking, there will be a negative longitudinal slip on all wheels. If the ﬁlter is allowed to continue estimating the wheel radius in such situ-ations, the estimated radius will increase. To avoid this, the radius is frozen during periods of braking. This freezing is done by setting the corresponding process noise variance to 0, along with the 4th column and 4th row of P . When braking is com-pleted, the Q values are restored to their previous settings. The 4th column and row of P are also reset to the values they had just before braking. In eﬀect, this is equivalent to using a reduced model alternative x=E N Ψ δG

T

during braking.

7.1.2 Filter Tuning

The tuning parameters used during evaluation on real data are presented in Ta-ble 7.1 and 7.2. Except for the measurement covariances, which were set to ap-proximate the errors speciﬁed in the receiver manual, these parameters were more or less tuned in an ad hoc manner. The ﬁlter did not seem very sensitive to the process noise of E and N , the values below appeared to work well.

Robust Automotive Positioning: Integration of GPS and Relative Motion Sensors

Robust Automotive Positioning:

Integration of GPS and Relative Motion

Sensors

Robust Automotive Positioning:

Integration of GPS and Relative Motion

Sensors

Institutionen för systemteknik

581 83 LINKÖPING

Abstract

Acknowledgments

Notation

Symbols

Operators and functions

Abbreviations

Contents

Chapter 1

Introduction

1.1

Background

1.1.1

Satellite Aided Absolute Positioning

1.1.2

Relative Positioning

1.1.3

Integrated Positioning

1.1.4

Map Aided Positioning

1.2

NIRA Dynamics

1.3

Problem Speciﬁcation

1.4

Objectives

1.5

Limitations

1.6

Thesis Outline

Chapter 2

Sensor Fusion

2.1

Linear State Space Models

2.2

Basics of Observer Design

2.3

The Kalman Filter

2.4

The Extended Kalman Filter

2.4.1

Time Update

2.4.2

Measurement Update

Chapter 3

The Global Positioning System

3.1

Coordinate Systems

3.1.1

Earth Centred Earth Fixed Frame

Prime Meridian

z

y

x

Equator

3.1.2

Vehicle Body Fixed Frame

about z

b

a

P

y

x

P

z

z

rotation

λ

φ

3.1.3

Planar Coordinates

3.2

_Equator