simultaneous trajectory optimization and target estimation...

Master of Science Thesis in Automatic Control

Department of Electrical Engineering, Linköping University, 2016

Simultaneous Trajectory

Optimization and Target

Estimation Using RSS

Measurements to Land a

UAV

Jonathan Stenström

Master of Science Thesis in Automatic Control

Simultaneous Trajectory Optimization and Target Estimation Using RSS

Measurements to Land a UAV

Jonathan Stenström

LiTH-ISY-EX–16/4988–SE

Supervisor: Clas Veibäckisy, Linköpings universitet

Jonas EkskogCombitech AB

Jacob SundqvistCombitech AB

Examiner: Fredrik Gustafssonisy, Linköpings universitet

Division of Automatic ControlDepartment of Electrical Engineering

Linköping UniversitySE-581 83 Linköping, Sweden

Copyright © 2016 Jonathan Stenström

Sammanfattning

Användningen av drönare har etablerat sig på en allt mer expanderande mark-nad. Detta examensarbete fokuserar på landningssekvensen med huvudbegräns-ningen att inga visuella hjälpmedel får användas. Istället använder detta systemsig av radiosignaler. Detta gör att landningsbasen kan gömmas från luften ochlandningen kan göras i dåliga väderförhållanden samt i mörker.

För att kunna landa utan att initialt veta vara landningsplattan är behövs ettlokaliseringssystem. I detta examensarbete används ett Extended Kalman filter(EKF) för att göra lokaliseringen. Det finns två huvudmål för landningsproce-duren. Det ena är att landa med så hög precision som möjligt och det andraär att göra detta så fort som möjligt. Dessa två mål kombineras i ett simultanttrajektorieoptimerings- och målestimeringsproblem som kan lösas under flyg-ning. Det optimala resultatet blir den trajektoria som genererar den bästa lokali-seringen samt får UAV:n att landa vid den estimerade landningsplatsen. I dettaexamensarbete visas det att trajektorian som alltid styr mot estimatet inte kom-mer vara den optimala, däremot visas det att en spiral kommer uppfylla målenpå ett tillfredsställande sätt.

iii

Abstract

The use of autonomous UAV’s is a progressively expanding industry. This thesisfocuses on the landing procedure with the main goal to be independent of visualaids. That means that the landing site can be hidden from the air, the landingcan be done in bad weather conditions and in the dark. In this thesis the use ofradio signals is investigated as an alternative to the visual sensor based systems.

A localization system is needed to perform the landing without knowing wherethe landing site is. In this thesis an Extended Kalman Filter (EKF) is derived andused for the localization, based on the received signal strength from a radio bea-con at the landing site. There are two main goals that are included in the landing,to land as accurate and as fast as possible. To combine these two goals a simulta-neous trajectory optimization and target estimation problem is set up that can bepartially solved while flying. The optimal solution to this problem produces thepath that the UAVwill travel to get the best target localization while still reachingthe target. It is shown that trying to move directly towards the estimated landingsite is not the best strategy. Instead, the optimal trajectory is a spiral that jointlyoptimizes the information from the sensors and minimizes the arrival time.

v

Acknowledgments

I would like to thank everyone at Reality Labs at Combitech AB for their hospotal-ity and support during the execution of this thesis project. Special thanks goesout to my supervisors at Combitech, Jonas Ekskog and Jacob Sundqvist, who havehelped with ideas and comments to the project.

I also would like to thank my supervisor at the university, Clas Veibäck, whom Ihave had lots of insightful discussions with trying to formulate the optimizationproblem. I have learned a lot from our discussions. I also want to reach out athanks to my examiner Fredrik Gustafsson for taking time and interest in mythesis project.

Lastly but not least I like to thank all of the people I have been around duringmy years at the university. You are the best! I would also like to display myspecial gratitude to Emma that have really kept my spirit up and supported methroughout this project.

Linköping, August 2016Jonathan Stenström

vii

Contents

Notation xi

1 Introduction 11.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Scope and Delimitations . . . . . . . . . . . . . . . . . . . . . . . . 31.4 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.5 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Localization 52.1 Sensor Fusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Radio Signal Properties . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2.1 WLAN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2.2 Signal-to-noise Ratio . . . . . . . . . . . . . . . . . . . . . . 7

2.3 Measurement Models . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3.1 Free-Space Propagation . . . . . . . . . . . . . . . . . . . . . 72.3.2 Log-Normal Shadow Model . . . . . . . . . . . . . . . . . . 82.3.3 Received Signal Strength . . . . . . . . . . . . . . . . . . . . 82.3.4 Global Positioning System . . . . . . . . . . . . . . . . . . . 82.3.5 Pressure Altitude . . . . . . . . . . . . . . . . . . . . . . . . 8

2.4 State Space Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.4.1 Altitude Models . . . . . . . . . . . . . . . . . . . . . . . . . 112.4.2 Landing Site Models . . . . . . . . . . . . . . . . . . . . . . 11

2.5 Estimation Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.5.1 The Extended Kalman Filter . . . . . . . . . . . . . . . . . . 132.5.2 The Particle Filter . . . . . . . . . . . . . . . . . . . . . . . . 15

2.6 Means of Improving the Localization Results . . . . . . . . . . . . 162.6.1 Hardware Analysis . . . . . . . . . . . . . . . . . . . . . . . 162.6.2 Adaptive Variance . . . . . . . . . . . . . . . . . . . . . . . . 182.6.3 Multiple Measurement Mean Sampling . . . . . . . . . . . 18

3 Trajectory Generation 193.1 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 19

ix

x Contents

3.1.1 Fisher Information Matrix . . . . . . . . . . . . . . . . . . . 213.1.2 Using FIM in an Optimization Problem . . . . . . . . . . . 223.1.3 Matrix to a Scalar . . . . . . . . . . . . . . . . . . . . . . . . 22

3.2 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.2.1 Solving the Optimization Problem . . . . . . . . . . . . . . 25

3.3 Monte Carlo Simulations . . . . . . . . . . . . . . . . . . . . . . . . 26

4 Experiments and Results 294.1 Data Collection Application . . . . . . . . . . . . . . . . . . . . . . 294.2 Measurement Model Validation . . . . . . . . . . . . . . . . . . . . 304.3 Solving the Optimization Problem . . . . . . . . . . . . . . . . . . 31

4.3.1 Gradient Search . . . . . . . . . . . . . . . . . . . . . . . . . 314.3.2 Elliptical Trajectory . . . . . . . . . . . . . . . . . . . . . . . 334.3.3 The Information Cone, Spiral Approach . . . . . . . . . . . 354.3.4 Following the Confidence Ellipsoid, Spiral Apprach . . . . 36

4.4 Experiment with Real Measurements . . . . . . . . . . . . . . . . . 384.5 Analysis of the Experimental Results . . . . . . . . . . . . . . . . . 414.6 Suggested Main Algorithm . . . . . . . . . . . . . . . . . . . . . . . 41

4.6.1 Algorithm : Confidence Ellipsoid . . . . . . . . . . . . . . . 424.6.2 Tuning Parameters and Results . . . . . . . . . . . . . . . . 424.6.3 Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.6.4 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

5 Conclusions and Future Work 495.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5.2.1 Hardware . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505.2.2 The Particle Filter . . . . . . . . . . . . . . . . . . . . . . . . 505.2.3 Solve the Optimization Problem . . . . . . . . . . . . . . . . 50

Bibliography 51

Notation

Abbreviations

Abbreviation Description

crlb Cramer Raó Lower Bounddoa Distance Of Arrivalekf Extended Kalman Filterfim Fisher Information Matrixgps Global Positioning Systemicao International Civil Aviation Organisationisa International Standard Atmospherelnsm Log-Normal Shadow Modelpf Particle Filterrss Received Signal Strengthrssi Received Signal Strength Indicationsnr Signal-to-Noise Ratiotdoa Time Difference Of Arrivaltoa Time Of Arrivaluav Unmanned Aerial Vehiclewlan Wireless Local Area Networkwsn Wireless Sensor Network

xi

1Introduction

1.1 Background

The use of Unmanned Aerial Vehicles (UAV) is ever more in the interest of mil-itary purposes, but in the private sector as well. This is because of the widespectrum of applications for the UAV, everything from surveillance to playingwith it. It has also been a platform for many researchers that are trying to makethe world ever more autonomous.

When the UAV is used in different applications, an interesting idea is the usageof the UAV all around the clock. There is a market for UAVs that can start, plana route, land and recharge autonomously and continuously send information toa user. To be able to do this you need to take different situations into account inthe sense of weather, precision, availability and, of course, price.

1.2 Problem Formulation

This thesis will investigate the landing procedure for an autonomous UAV. Thegoal is to make an autonomous landing of the UAV with the amendment that novisible aids will be used, neither from the ground nor from the air. That meansthat a camera gear mounted on the UAV and a mark on the landing site is takenaway. Instead, radio beacon signals will be used to localize and track the positionof the landing site. Based on that information, a trajectory will be generated thatwill drive the UAV to execute the landing with as high precision and as quicklyas possible.

When using radio beacons for localization, the signal strength is measured to

1

2 1 Introduction

10

5

x

0

-5

-10-10

-5

0

y

5

6

7

3

1

2

0

4

5

10

z

Figure 1.1: Illustration of the problem. The problem is to find the trajectoryin the figure represented by the black curve so that the landing site can beestimated in an optimal way.

extract the desired information, these kinds of measurements are called RSS mea-surements, where RSS is the abbreviation for Received Signal Strength.

To solve the mentioned problem, the material in this thesis will suggest whatkind of radio signal that should be used, how the data should be sampled andhow to use the data to be able to land the UAV. Another interesting thought is ifthe system could function only using smart phones. The main task of the thesisis to do simultaneous trajectory generation and target estimation, which expandsthe problem into finding the best trajectory to be able to estimate the landing siteas fast as possible.

The main questions to answer in this thesis:

• Can autonomous landing of a UAV be done using RSS measurements? If so,how accurate and how fast?

• Is it possible to only use a smart phone with the associated sensors to con-struct the system?

In Figure 1.1 the problem is illustrated, given a starting point the trajectory mustbe decided. The problem will lie in finding the trajectory, in the figure repre-sented by the black curve, that will aid the localization system to estimate thelanding site as fast and as accurate as possible.

1.3 Scope and Delimitations 3

1.3 Scope and Delimitations

A scope to this thesis needs to be mentioned to give a baseline of preconditionsand limitations. The objective is to find and reach the target, in this case thelanding site. To be able to do this, a GPS, a WLAN beacon and receiver and apressure sensor are used. The WLAN beacon and receiver are used for locatingthe position of the landing site, the pressure sensor to determine the altitude ofthe UAV and the GPS to determine the position of the UAV. These kinds of sensorsare chosen because they are all included in a standard smart phone.

Another part of the scope to this thesis is the choice of using a GPS and pressuresensor for positioning. This is because of the availability of these sensors in acommon smart phone, here a Samsung Galaxy S5 has been used as a reference toa common smart phone. This drives the project to investigate the localization forthe landing site in a coordinate system relative to the UAV.

A delimitation to this scope is how the altitude of the landing site is situated. Inthis thesis the altitude will be measured using the ambient pressure and that willnot in all cases generate the altitude above the ground. This is because the alti-tude calculation is based on knowing a reference pressure which does not has tobe the same as the pressure on the ground. However, in this thesis a presumptionis that the ground is flat and that the reference pressure always is the pressure atthe landing site. This means that the altitude of the UAV will always be the sameas the height that the UAV is above the landing site. To translate the system to areal scenario the pressure at the landing site needs to be known or that the alti-tude of the landing site is at a known altitude. To fully deal with this problem thealtitude of the landing site would have to be estimated, however that implies thatthe calculations will be more complicated and that problem will not be furtheranalyzed in this thesis.

1.4 Related Work

There has been much work conducted in this area of autonomously landing aUAV. The difficulty lies within finding the landing site and previous work usesdifferent methods to do so. The most widespread method to land autonomouslyis to use visual aids and a camera to localize the landing site, by that, the precisionof the landing can be completed down to centimeters. For example, in [18] and[22] a camera is used to find a moving target.

There has also been some diversion in the field on which kind of sensor thatshould be used for localization. In [3], the writers land the UAV at a movingtarget using a rope sensor. The system senses the tension and direction of therope and through that guide the UAV to a landing.

In this thesis the localization will be done using radio beacons. This angle ofinterest comes from looking at different indoor localization techniques. The maindifference, when using radio signals for indoor respectively outdoor localization,

4 1 Introduction

is the presence of obstacles or walls. In [17], the writers discuss the variation inthe signals and the overall use of Wi-Fi signals for localization.

Regarding the optimization problem, in [19] the basic observer trajectory plan-ning problem is defined. The difference to the problem in this report is the sensorused, where [19] uses a bearing-only sensor. Another difference is that [19] hasno focus on reaching the estimated target, which in this thesis is an essential goalto achieve.

1.5 Thesis Outline

The outline of this thesis is as follows:

• Chapter 2: Localization - In this chapter the different sensors used are pre-sented. The models that are used and the estimation methods are derivedand explained.

• Chapter 3: Trajectory Generation - In this chapter the optimization problemof simultaneous trajectory generation and target estimation is derived andsome alternatives to the solution are given.

• Chapter 4: Experiments and Results- The different experiments that havebeen set up to test the solutions and validate the models are given. The re-sults from the experiments are also given, and a suggested main algorithmis presented and discussed.

• Chapter 5: Conclusions and Further Work

2Localization

This chapter will consider one of the major problems of autonomously landinga UAV. To be able to land, the UAV needs to know the location of itself and thelocation of the target, in this case the landing site. In the considered scenario, theposition of the UAV itself will be measured with the help of GPS satellites, butthe location of the landing site is unknown. There is therefore a need to localizethe landing site with the help of transmitters on the ground. How this all is puttogether is discussed and explained in this chapter, but first a short introductionto the subject sensor fusion is given.

2.1 Sensor Fusion

Sensor fusion combines different sources of information to generate a better re-sult than using the sources individually. In this thesis, sensor fusion is used forlocalization [11].

There are many different techniques to perform localization within a sensor net-work. The main presumption is that there is access to at least one receiver andone transmitter. The receiver samples observations from the transmitter and thena computer evaluates the information gathered. These observations, that the re-ceiver samples, are said to differ among three different types. Waveform obser-vation compares the phases between sensors that have a distance between eachother of half a wavelength. By that the system can calculate the direction of ar-rival (DOA). Timing observation, where the timestamps from each measurementare compared and time of arrival (TOA) and time difference of arrival (TDOA)can be calculated. The last is power observations, where the sensors can mea-sure the received signal strength (RSS) and by that calculate the distance that the

5

6 2 Localization

signal has traveled. RSS is the approach used in this thesis.

When performing localization, a set of models are needed to identify the mea-surements and to update the estimation, from one set of measurements to thenext. The typical nonlinear state space system with Gaussian noise is written as

xk+1 = f (xk , vk) (2.1a)

yk = h(xk , ek) (2.1b)

where xk is the state vector and yk is the measurements at time step k. The modelf is the dynamic model and describes how the estimates moves to be able to pre-dict how the estimate at the next time step, xk+1. The model h is the measurementmodel. This model is used to interpret the received signals or measurements tobe able to derive what the state at that time instant is. The system contains bothprocess and measurement noise, these noises are defined above as vk and ek re-spectively. The process noise is added to the model to be able to compensate forerrors in the model. Errors can arise from the inability to fully describe the re-ality in a mathematical manner. That means that the size of the process noisecovariance indicates how well the model captures the reality. The same conclu-sions about the size of the noise factor can be said about the measurement noisebut instead of mainly compensate for the model error it tries to model the actualnoise that is included in the transmitted signal. This means that it can be hardto fully capture all the information with the model. All of the noises used in thisthesis are seen as zero-mean Gaussian distributed noises.

The remainder of this chapter features the different sensors used and how the setup described in Equation (2.1) is derived.

2.2 Radio Signal Properties

The main source of information in the localization algorithm consists of radiosignals emitted from a hotspot by a smart phone on the ground. The hotspotgenerates a wireless local area network (WLAN) and is normally used to provideinternet access for WLAN devices. In this thesis, the signals are used to look atthe degrading factor of the signal and from that derive a distance, the derivationfrom signal strength to distance is further discussed in Section 2.3.3.

2.2.1 WLAN

WLAN, or the commercial brand name Wi-Fi, has both great advantages andgreat losses when using it as a localization unit. One of the major advantages ofusing WLAN signals for localization is the deployment practicability. For exam-ple, localization works both indoor and outdoor, and the availability of transmit-ters and receivers is great. In [9], the use of WLAN for indoor localization and itsadvantages and disadvantages are discussed. Almost every smart phone has thecapability to create a WLAN network and is able to connect to one.

However, there are some problems when using WLAN. Because of issues in the

2.3 Measurement Models 7

hardware, the variance of the signal is shifting, which causes the RSS to varygreatly at the same measuring point. Even the same chipset may vary from onetransmission to another, this problem makes it hard to model the signal strengthand to use it for localization [5]. The shifting variance makes the signals appearvery noisy, and irregular. Some ways to solve this problem is to map the differentbeacons in the system at known positions and create a table for localization [21].Another is to adaptively change the variance, during flight, for the noise in themeasurement model based on an initially estimated distance [23].

2.2.2 Signal-to-noise Ratio

Signal-to-noise ratio (SNR) is the relation between the energy of the signal andthe noise in the transmission [14]. The relationship is described below, in Equa-tion (2.2), assuming that the noise is white random noise. The signal-to-noiseratio can be used to calculate the RSSI in different WLAN devices, and is mostlyused in this thesis as a measure when designing the particle filter. The equationto calculate the SNR ratio is defined as

SNRdB = 10 log10(Esignal

Enoise), (2.2)

where SNRdB is the signal-to-noise ratio measured in decibels. The parametersEsignal and Enoise are defined as the energy of the signal and the noise respectively.

2.3 Measurement Models

In this thesis, where radio signals will be used for localization, the RSS is ofgreat importance. There are some different ways of modeling the signal propa-gation. According to [23], the signal strength propagation in a wireless sensornetwork (WSN) can be described by three different models; the free-space model,the ground bidirectional model and the log-normal shadow model. The bidirec-tional model is not explained further here since it is designed to deal with prob-lems that occur when the signals are traveling over long distances and with longantennas. The other models are explained in the following sections.

2.3.1 Free-Space Propagation

The free-space model is applicable in scenarios when the transmission distanceis much larger than the antenna size and the signal carrier wavelength, λ. Themodel is also applicable when there are no obstacles between the transmittersand the receivers. The model can be written as

Pr (d) =PtGtGrλ

2

(4π)2d2L, (2.3)

where Pr (d) is the received signal strength based on distance d, Pt is the transmit-ted signal strength, Gt and Gr are the antenna gains at the transmitter and thereceiver and L is the system loss factor.

8 2 Localization

2.3.2 Log-Normal Shadow Model

The log-normal shadow model (LNSM) is a general propagation model which issuitable for both indoor and outdoor environments. The formula is formulatedas

P(d) = P(d0) − 10η log(d

d0) + Xσ , (2.4)

where P(d) is the signal strength at distance d from the transmitter, P(d0) is thesignal strength received at distance d0 and is used as a known reference value inthe model, η is the path loss index and Xσ is the zero-mean Gaussian randomvariable [23].

2.3.3 Received Signal Strength

The received signal strength (RSS) plays a very crucial role in this thesis. RSS is ameasure of how strong the signal is at the receiving device. The RSS depends onthe distance between the transmitter and the receiver, but it can also vary due toair variations and between chipsets among other things [5]. Measuring the RSS isdone by looking at the received signal strength indicaton (RSSI), which translatesto the RSS value. No standard method exists to calculate the RSSI which leads tothat the results may differ, even for similar chipsets from the same manufacturer.The most common techniques of calculating the RSSI are done by using measuresof signal energy or SNR [8].

2.3.4 Global Positioning System

Global Positioning System (GPS) is a worldwide system that can determine theposition of a receiver with high accuracy; a standard GPS system has a resolutionof about ± 1 meter. GPS consists of three segments; the space, the user and thecontrol segment. The space segment consists of satellites orbiting earth, the usersegment is the receiver of the signals emitted from the satellites and the controlsegment keeps track of the satellites. GPS uses the TDOA technique as a methodfor localization, by sending out timestamps from the satellites the time can becompared with the timestamps of the receiver and a position can be calculatedthrough trilateration. Since the distance that the signals are traveling is longand the signals are traveling at speeds close to the speed of light the theory ofrelativity must be regarded. Further explanations about how the GPS systemworks can be found in [12].

2.3.5 Pressure Altitude

The lower part of the atmosphere is the part that mainly determines how theweather will alter. One key factor of how the weather changes is caused by thepressure, and how the pressure varies from one place to another. For aviationpurposes, the International Civil Aviation Organization (ICAO) has set up somestandard models to describe the atmosphere. This standard is called the Inter-national Standard Atmosphere (ISA) and includes everything from the standardtemperature at mean sea level to the acceleration of gravity and also how the

2.3 Measurement Models 9

ambient pressure depends on the altitude, which is of importance to this thesis.For more examples and models see [4]. According to the ISA, the pressure willdecline with the altitude as

ρ(h) = ρ0(1 − 0.0065h

T0)5.2561, (2.5)

where ρ0 is the comparison pressure. In the application of this thesis this pres-sure will be the pressure at the landing site. The variable ρ(h) is the pressureat altitude h, which is measured in meters and T0 will be the temperature in de-grees Kelvin. The comparison pressure, ρ0, and the temperature, T0, is knownvariables that are determined through calibration of the system upon take-off. Inthis thesis they will be seen as known constants to the system.

A nice feature with this model is that it is easily linearized with a high preci-sion at low altitudes. As can be seen in Figure 2.1, the relationship between thepressure and the altitude is almost a straight line at low altitudes. The small cur-vature, which arise from the nonlinear model, is negligible in comparison withthe noise. The plot in Figure 2.1 is generated using the above equation with theconstants ρ0 = 1013 mbar and T0 = 293◦ K, which are also defined in the ISA asthe standard pressure and the standard temperature at mean sea level.

1,011 1,012 1,0130

5

10

15

20

Pressure, ρ [bar]

Altitude,

h[m

]

Illustration of how the pressure alters with the altitude

Figure 2.1: As can be seen the curvature of the function appears close toa straight line which gives the function small information losses when lin-earized.

The above equation is one way of estimating the altitude and it will be the modelused in this thesis. There are alternative ways of finding the altitude besidesmeasuring the ambient pressure. For example, sonar sensors can be used to esti-mate the altitude. The resolution performances between these two methods arespread. A sonar sensor can have a resolution down to a few centimeters and apressure sensor indicates the altitude down to a few decimeters. In a simulatedenvironment, the true altitude is used to give measurements to the system andthe uncertainty in those measurements are in the range of ±2 decimeters of thetrue altitude. This uncertainty will replicate the same result as when using abarometric sensor. Some examples of the mentioned sensor types can be found

10 2 Localization

in [2] and [13].

2.4 State Space Models

In this section, the models are put together to form the state space set-up, de-scribed in Equation (2.1). These models will then be used in the estimation pro-cess.

As discussed above, one clear fact is that the overall accuracy in the RSS measure-ments is bad due to the influence of noise. On the other hand, the barometricpressure has a much greater accuracy in comparison. This means that the modelscan be split, converting the localization problem into a two respectively one di-mensional problem. This allows the estimate of the altitude to be included in thelanding site models. It also stops the calculations for the landing site estimateto influence on the altitude estimate, which could affect the altitude estimationbadly. The indexation to separate the two set-ups will be given by (z) for the es-timation of the altitude and (H) for the estimation of the helipad, or as it is alsocalled, the landing site.

The state vectors that are defined and used in the models, is for the altitude esti-mate

x(z)k = [pzk pzk]

T (2.6)

where pzk is the altitude, and pzk is the change in altitude for the UAV. For thelanding site estimation the state vector is

x(H)k = [sxk s

yk ]

T (2.7)

where s here is the position of the landing site. The height of the landing site,sz , is seen as zero in this thesis and is not included as a state. The indexation krepresent the time, which is used, although the true landing site is not moving, tofollow a indexation standard in the models and to easily compare the true stateswith the estimates. The position of the UAV is given by the GPS with independentnoise from the measurements

pk = [pxk pyk ]

T + [ex,GPSk e

y,GPSk ]T (2.8)

where ex,GPSk and e

y,GPSk are distributed as zero mean Gaussian noise

ex,GPSk ∼ N (0, Px,GPS

k ) and ey,GPSk ∼ N (0, P

y,GPSk ) (2.9)

where Px,GPSk and P

y,GPSk are the covariance in each direction given from the GPS.

Normally, the measurements from the GPS are correlated in time from one mea-surement to the next. Here, this correlation is neglected and the measurementsare approximated to be independent, this is to simplify the calculations.

2.4 State Space Models 11

2.4.1 Altitude Models

The first set of models will measure the atmospheric pressure and then estimatethe altitude of the UAV, defined in Equation (2.5). The measurement model willbe

h(z)k (x

(z)k , e

(z)k ) = ρ0(1 − 0.0065

pzkT0

)5.2561 + e(z)k , (2.10)

where pz is the altitude state that will be estimated. The noise is considered

Gaussian zero mean distributed as e(z)k ∼ N (0, R

(z)k ), at time step k. The covariance

matrix R(z)k can be seen as a small value since this model is believed to be a good

representation of the reality. Note that h(z) is the sensor model for the pressuremeasurement, and pzk is the altitude here, not pressure, cf Equation (2.5).

The barometric measurements that the above model is trying to capture is noted

as y(z)k and the set up can be compared with Equation (2.1b). This measurement is

given by the barometric sensor. This measurement will be used in the estimationfilter.

When estimating the altitude, the velocity of the altitude can be of interest to esti-mate, this to be able to suppress fluctuations and by that improve the estimationof the altitude. For that reason we incorporate the change in altitude as a state. Asimple kinematic constant velocity model is used to capture the dynamics of thesystem,

x(z)k+1 =

(

1 T0 1

)

x(z)k + v

(z)k (2.11)

where T in the model is the time update constant. The noise is assumed to be

Gaussian zero mean distributed as v(z)k ∼ N ([0 0]T , Q

(z)k ), where Q

(z)k is the covari-

ance matrix for this dynamic model, for each time step k. The state vector x(z)k

consists of the altitude and the vertical speed, cf Equation (2.6).

This model is different from the one described in Equation (2.1a) since it is linear.

The notation for the linear state space system is written as x(z)k+1 = F

(z)k x

(z)k + v

(z)k ,

where in this case

F(z)k =

(

1 T0 1

)

. (2.12)

2.4.2 Landing Site Models

The secondmeasurement model will measure the RSSmeasurements transmittedby the WLAN transmitters. The model is based on the LNSM model definedin Equation (2.4). The signal strength is decreased by the increase of distancebetween the transmitting beacon and the UAV. The position of the UAV in the XY-plane will be given by the GPS with some noise as stated in Equation (2.8), thealtitude can be estimated using the equations derived in the previous section andthe state pz will then be estimated using a filter which will be described later in

12 2 Localization

this chapter. The altitude state can then be described similar to the GPS positionas

pzk = pzk + epzk (2.13)

where epzk ∼ N (0, Pz

k ) and here, the noise covariance is given by Pzk . The altitude

estimate is, just as for the GPS measurement, correlated in time from one mea-surement to another. This correlation comes from the filtering algorithm, whereprevious information about the state is used to improve the estimate. This corre-lation is, just as it is for the GPS measurements, neglected and the estimates areapproximated as independent estimates to simplify the calculations.

With this information, the measurement model can the be written as

h(H)k (x

(H)k , pk , p

zk , ek) = P(d0) −

10

2η log(

d2k (pk , pzk , ek)

d20) + e

(H)k (2.14)

where

d2k (pk , pzk , ek) =(s

xk − (p

xk + ex,GPS

k ))2 + (syk − (p

yk + e

y,GPSk ))2

+(szk − (pzk + e

pzk ))2,

(2.15)

where ek = [e(x,GPS)k e

(y,GPS)k e

pzk e

(H)k ]T is the noise vector, where the first three

noises in ek are called non additive noise to the model. The last term is the addi-

tive noise, e(H)k , to the measurement model and is distributed as e

(H)k ∼ N (0, R

(H)k ),

where R(H)k is the covariance for the measurements. In the equation szk is seen as

zero for all k and will be seen as zero throughout this thesis, but the parametercan be changed or estimated in other applications. The denominator 2 in themodel must be used to cancel the squared distance that is used in the logarithmicfunction.

In this thesis, the transmitting beacon is known to be located at the landing site,and only one beacon is used. In Chapter 4, the path loss constant, η, and the ref-erence signal strength, P(d0), are experimentally estimated using real measure-ments.

The RSS measurements that the above model is trying to capture is noted as y(H)k

and the set up can be compared with Equation (2.1b). The measurements aresampled using the ability to connect the smart phone to a WLAN and extract theRSS value. This measurement will be used in the estimation filter.

The dynamic model to this system can seem to be a bit unnecessary since thelanding site will not be moving. But when adding some process noise it can helphandling faults in the measurement model and help with the information losscaused by the linearization. It can however be desirable to let this uncertaintychange over time. When the estimate will be starting to converge, the uncertaintyin themovement can be lowered to not influence on the calculations asmuch. Thedynamic model that will be combined with Equation (2.14) is as follows,

x(H)k+1 = x

(H)k + v

(H)k (2.16)

2.5 Estimation Process 13

where the noise is assumed to be Gaussian zero mean distributed asv(H)k ∼ N ([0 0]T , Q

(H)k ), where Q

(H)k , the process noise covariance, can be changed

over time. Just as in Equation (2.12) this model is linear and here F(H)k = I2, where

I2 is the two dimensional unity matrix.

2.5 Estimation Process

To be able to interpret the information gathered from the above derived equa-tions, there has to be some sort of estimation algorithm. The nonlinear filteringproblem can be solved in many different ways and with different techniques. Inthis thesis the extended Kalman filter is used, but for future work and to visualizean interesting phenomena the particle filter is briefly described.

2.5.1 The Extended Kalman Filter

The extended Kalman filter (EKF) is an extension to the regular Kalman filter.The Kalman filter is one of the most popular estimation algorithms [16]. It is arecursive estimating method that is optimal for linear, Gaussian, time-invariantdynamic system. This quality means that the estimate meets the Cramér-RaoLower Bound (CRLB). CRLB is the lower bound to the variance of an estimator[11]. The major difference from the Kalman filter to the EKF is the capabilityto handle nonlinearities in the models. To handle the nonlinearity problem themodels are linearized and for that reason the optimality insurance is lost. Theconvergence guarantee is also lost and the filter is sensitive to how it is initial-ized. This problem can be illustrated by a phenomena that here is called mul-tiple modes. This phenomena with multiple modes is illustrated in Figure 2.2and the reason for this behaviour will be further discussed in Section 2.5.2 whendiscussing another filtering method called a particle filter.

The extended Kalman filter is summarized in Algorithm 1 and shows the generalEKF with the additions to fulfill the purpose of this thesis. The filter is used whennonlinearities are present, these nonlinearities can be present in both the mea-surement model as well in the dynamic model. In this thesis, it is only present inthe measurement model and because of that linearization for the dynamic modelis not needed. There is also an addition that will handle the non-additive noisepresent in the landing site measurement model. In the algorithm when ek = 0 itmeans that the uncertainties for the stochastic variables are zero and the systemis considered free from noise.

The result from the extended Kalman filter is given by an estimate of the states,and with that state a covariance matrix. The estimate is written with a hat abovethe variable to show that it is an estimate of the true position. In this thesis, the

states that are estimated from the EKF is the variables in the state vectors, x(z)k and

x(H)k for each time step k, which is fully presented in Section 2.4. The covariance is

often used when presenting the results of how the estimate is changing, by usingthe eigenvalues of the covariance matrix a confidence ellipsoid can be displayed

14 2 Localization

Algorithm 1 The extended Kalman filter

1. Measurement update:Linearize measurement function:

Hk =δhk (x,e)

δx

∣

∣

∣

∣

xk|k−1,e=0

To handle the noise:

Mk =δhk (x,e)

δe

∣

∣

∣

∣

xk|k−1

Compute Kalman gain and update:

Kk = Pk|k−1HTk S−1k

Sk = (HkPk|k−1HTk +MkRkM

Tk )

ǫk = yk − hk(xk|k−1, 0)xk|k = xk|k−1 + KkǫkPk|k = Pk|k−1 − Pk|k−1H

Tk S−1k HkPk|k−1

2. Time update:xk+1|k = Fk xk|kPk+1|k = FkPk|kF

Tk + Qk

around the estimate. The eigenvalues are used to define the axis of the ellipsoid.The ellipsoid is often written with a confidence percentage, which means thatthe target is within the ellipsoid with a certainty of that percentage, how thisellipsoid is calculated in the two dimensional case is further explained in [20].

Linearization

To handle the nonlinearity in the models the EKF uses a Taylor expansion tolinearize the models as

w = g(x) = g(x) + g ′(x)(x − x) + r(x; x; g ′′(ξ)), (2.17)

where g ′(x) denotes the Jacobian of the function g(x) and r is the rest term. Thisimplementation is applicable if the rest term is small in relation with the esti-mation error and degree of nonlinearity of g . This rest term is the reason whythe EKF loses the optimality in comparison with the linear Kalman filter [11]. InEquation (2.17), the general case of linearization is presented, that means that inthis case the measurement noise in the linearization point is not accounted for.How the non-additive noise is handled can be seen in Algorithm 1. The Jacobians

2.5 Estimation Process 15

to this thesis can be analytically calculated as

H(z)k = −0.0342ρ0

(1 −0.0065pzk

T0)4.2561

T0(2.18a)

H(H)k =

10

2η

[

sx−(pxk+ex,GPSk )

d2k (pk ,pzk ,ek )

sy−(pyk+e

y,GPSk )

d2k (pk ,pzk ,ek )

]

(2.18b)

M(z)k = 1 (2.18c)

M(H)k = −

10

2η

[

sx−(pxk+ex,GPSk )

d2k (pk ,pzk ,ek )

sy−(py,k+ey,GPSy )

d2k (pk ,pzk ,ek )

−(pzk+epzk )

d2k (pk ,pzk ,ek )

1

]

(2.18d)

where d2k (pk , pzk , ek) is the distance, described in Equation (2.15). That implicates

that the third element in H(H)k is eliminated when the altitude is considered the

true altitude. Since the altitude model does not include any non-additive noise

M(z)k is equal to one.

The analytically calculated Jacobian shall always be used in implementations ifit is easily calculated. However sometimes it can be ineffective to always haveto calculate the Jacobians by hand. If the models changes often during testing, anumerical method could be more effective, these changes can include testing dif-ferent models or different state set ups. In this thesis the complex-step derivativeapproximation method has been used for this purpose to estimate the derivativein the linearization. In [15], it is stated that this estimate is suitable for numericalcomputing and has been shown to be accurate, robust and have a reasonably lowcomputational cost. The method is described in Algorithm 2.

Algorithm 2 Complex-Step Derivative Approximation

Let g(w) be an analytical function, let x0 be a point on the real axis and let h bea real parameter. Set w = x0 + ih and expand in a Taylor series, where i is theimaginary unit.

g(x0 + ih) = g(x0) + ihg ′(x0) + . . .

Take the imaginary part of both sides and divide by h.

g ′(x0) =Im(g(x0+ih))

h + O(h2)

By choosing h small the derivative of g(x0) is given.

2.5.2 The Particle Filter

Another way to estimate the landing site with the models, is to use a particlefilter. The particle filter uses a stochastic grid of particles to try to approximatethe posterior distribution of the states. Briefly described, the filter uses a givendistribution to deplete a given number of possible locations, these locations arecalled particles. Then all particles are compared to each other and the particles

16 2 Localization

are given a specific weight by assigning the particles that is most likely in thecorrect position a higher weight, and the less likely a lower or a negligible weight.The estimate is then given by adding all the particles with the associated weight.The particle filter is fully derived and explained in [10].

In Figure 2.2 a circle is drawn as a dashed line around the UAV. The circle indi-cates the correct distance to the landing site but since the system does not knowin what direction the landing site is positioned the possible locations can be illus-trated by a circle around the UAV. That means that given only one measurementthe particles should be distributed around this circle. As can be seen, some par-ticles are distributed far away from this circle or in gatherings around the circleand this is mainly because of noise and the design of the particle distributionfunction. This leads to the problem that the estimate of the landing site is placedclose to the UAV, as can be seen in the figure. To deal with this problem the par-ticles can be sorted in to different modes distinguished by the gatherings of theparticles. These modes have an interesting potential to be individually investi-gated to find the best estimate. The EKF does not include the property of beingable to find these modes, and theoretically it would need more measurements togive the same results which speaks for the particle filter to be a more secure esti-mation method. The main disadvantage the particle filter has to the EKF is thecomputational load that is much higher for the particle filter.

In this thesis the particle filter has not been implemented to the estimation algo-rithms due to the complexity and lack of time. To be able to converge to the truelanding site the problem with multiple possible locations needs to be handled inthe main algorithm.

2.6 Means of Improving the Localization Results

In this section, some different algorithms that theoretically should improve thelocalization, are presented. Some different methods have been found during aliterature research process and these have been studied to be used in a futureapplication. In a real application, some of these methods might need to be con-sidered for the system to work properly.

2.6.1 Hardware Analysis

The performance of most software algorithms is directly dependent on whichhardware that is used. The deciding factor when choosing the hardware is in de-ciding which price is reasonable for which performance, and what is needed forthe desirable result. The hardware simulated in the simulation environment isdesigned to replicate the sensors and transmitters that are found in a commonsmart phone. In this thesis, a Samsung Galaxy S5 with the Qualcomm Snap-dragon 801 chipset has been considered. In Table 2.1 some specifications aboutthe sensors in the phone are given.

Since a phone is not firstly constructed to aid UAV’s to autonomously land, the

2.6 Means of Improving the Localization Results 17

−40 −20 0 20 40

−40

−20

0

20

40

x

y

The particle filter with identified modes in the particle cloud

Particles

UAV

Landing Site

Current estimate

Modes

Figure 2.2: The phenomena of multiple modes during estimation.

above hardware is not the best for this particular scenario. There is a vast amountof different sensors that can be used in the application, all with different prop-erties. In an upsized scale of this project where the UAV itself is much moreexpensive some more costly sensors might be well motivated to purchase for abetter performance of the system. The GPS update frequency is altered depend-

Sensor Sample frequency Resolution

Wi-Fi ∼ 0.25 Hz Integer numbers in dBmBarometer ∼ 5 Hz ± 2 dm

GPS 0.25 to 1 Hz ± 1m

Table 2.1: The sensor used from the phone Samsung Galaxy S5 and somespecifications used in the simulation environment.

ing on how long the GPS has been on. In the simulation environment the GPS issampled at the same frequency as the WLAN signals.

An alternative to the WLAN signals used in this thesis is to use UWB equipment,UWB stands for ultra wideband. This technique can produce positioning withan accuracy down to a few centimeters and is based on TOA. For more informa-tion about this localization technique refer to [24]. There are smart phones calledspoon phones that have the UWB equipment built in, however in the referencesmart phone used in this thesis this equipment is not available and this alterna-tive is not further investigated.

18 2 Localization

2.6.2 Adaptive Variance

Another way of improving the results is to progressively compensate for the vari-ance at different distances. During experimentation the variance of each individ-ual transmitter is measured at different distances. The observation made is thatthe variance changes with the distance the UAV is apart from the transmitter. Byusing these observations, the variance for e(H) can be set to the same variance asfor the measurements. The variance for the measurements are given as

V ar{P(di )} = gi (di ), (2.19)

where P(di ) is the measured signal strength, gi (di ) is the function that calculatesthe variance depending on the distance di , where i indicates the different trans-mitters in the cases where multiple transmitters are used. By using the last esti-mated position an approximated distance from each transmitter is given and thevariance can be calculated. This method is dependent on earlier measurementsthat are gathered in a lookup table. This method is further discussed in [23]. Aproblem with this method is if the system has noise independent from the dis-tance this method will not improve the system at all.

2.6.3 Multiple Measurement Mean Sampling

Since the indicated signal strength from the WLAN includes a lot of noise, aver-aging multiple measurements could improve the results. By collecting multiplemeasurements a mean of these measurements can be made and used in the EKF.This method of sampling the measurements should be able to suppress some ofthe noise in the measurements. This method behaves just as a low pass filterwhere the high frequency terms to the measurements are taken away, and thesame results should be achievable with a well tuned EKF. One big disadvantageto this method is that it gives the system a lower update frequency since it needsseveral measurements before the EKF can update the estimate.

3Trajectory Generation

In this chapter, the trajectory generation problem is considered. The trajectoryhas two major purposes in the landing procedure. One aspect is, of course, tonavigate the UAV down to the landing site but the other is to also navigate so thatthe landing site estimate can be as accurate as possible. This chapter will handlehow the trajectory will be generated from different observations and try to definewhat is desirable in an optimization problem.

3.1 Problem formulation

The problem as introduced above is to both navigate the UAV so that the targetcan be estimated with as high accuracy as possible but also navigate towards thatestimated target. These two goals are not surely coherent to each other and renderthe same trajectory. To get an understanding on how this trajectory would looklike an optimization problem is preferable to set up. An important factor to thelanding procedure that has to be considered is the time. The time is importantbecause the UAV can not fly for an infinite amount of time, however the longerthe UAV is in the air the better the precision of the estimate will be.

Since the trajectory planner has no access to the true target state, the best that canbe done is to use a priori distribution on the target location and perform the targetestimation and trajectory optimization simultaneously. There are obvious limi-tations to this approach, and since combining the estimation and optimizationusually leads to a highly nonlinear problem, convergence cannot be guaranteed[16]. However when using the localization methods described in Chapter 2 it canbe given some initial conditions that preserve the convergence. That means thatthe adoption that an initial assumption of where the target is located is known to

19

20 3 Trajectory Generation

the system when it is started.

For this chapter, the notation is based on the three dimensional position of theUAV and the landing site. This is introduced to get a simple notation when de-scribing the distance between the UAV and the estimated landing site. The UAVposition is noted pQ = [px py pz], which is known from the GPS and altitude esti-mation, described in Chapter 2. The landing site is noted xH = [sx sy sz], wheresx and sy is given from the landing site estimation and sz = 0. The estimate of thelanding site is noted as xH .

Time has been introduced as an important factor, and that the ambition is to landas close to the landing site as possible. That gives that a natural starting pointis to take the accuracy and time into account. To be able to optimize on accu-racy and time, there has to be some sort of measures that can be investigated.Time and accuracy can easily be measured by looking at the end result where theelapsed time and where the UAV landed in the end. These measures are desiredto be minimized by the landing algorithm. But as mentioned, the system has noaccess to the true position of the landing site and must therefore do the optimiza-tion during the flight as the position is estimated. This is called simultaneoustrajectory generation and target estimation.

Optimization Goals

Time and the accuracy, to be able to optimize on these goals while in the air theyneed to bemeasured in a goodmanner. The first mentioned goal is time. Minimiz-ing the time consumed can be done by either increasing the velocity or decreasingthe distance or both. In this thesis, the distance will be minimized since the dis-tance can be easily calculated with the estimated target and the position of theUAV. The distance can be described using the norm

∥

∥

∥pQ,k − xH,k

∥

∥

∥

2(3.1)

where k = 1, ..., N and N is the final sample time.

When the norm described in Equation (3.1) is within a given threshold that trans-lates to the desired performance, it says nothing about how far the UAV haslanded from the true landing site. The quality of the estimate is unknown inthis context and that is a problem since it is desired to land with a high accu-racy. However when using the EKF, a common measurement for the quality ofthe estimate is given directly from the calculations. The measure of performancefor the estimate is called the error covariance, which represents the uncertaintyassociated with the estimate. The covariance should be as low as possible forthis purpose. As mentioned, the minimum covariance threshold is defined bythe CRLB. It is based on the physical properties of the system and the geometrydefined in the estimation problem. Its inverse is called the Fisher informationmatrix (FIM) and provides a measure of the amount of information contained ina given set of measurements [16]. This can be written as

P(H)k|k = E{(xH,k|k − xH,k)(xH,k|k − xH,k)

T } ≥ J−1k (3.2)

3.1 Problem formulation 21

where xH,k|k is the estimate at time k given the measurements obtained up to that

time, P(H)k|k is the error covariance for the landing site estimate at time k given

all the measurements up to that time, and Jk is the Fisher information matrix.In the following section the FIM is looked at in more detail. It has been shownin previous work that it is of very high interest when performing this kind ofoptimization [16], [19].

3.1.1 Fisher Information Matrix

As shown, the Fisher information matrix can be used to determine the perfor-mance or accuracy of the estimate. The FIM is calculated recursively as

JFIMk+1 = JFIMk + HTk (xk , ek)R

−1k Hk(xk , ek) (3.3a)

H(xk , ek) = ∇xh(xk , ek) (3.3b)

where JFIMk is the FIM at time step k and R is the covariance for the measurement

model. The above equation is initialized as J0 = P−10 . Equation (3.3b) is theJacobian of the measurement model, h. In this thesis, the information will becalculated for the landing site estimation system. In the above function, the noiseis included in the Jacobian and there is no special adjustments to handle thenosies in ek . This is an approximation where the uncertainties are included inthe Jacobian. This means that the larger the noise is the lower the informationprovided by the measurements is. In the general case, numerical methods areneeded to do these calculations, [11]. In this thesis the CSDA is used, furtherdescribed in Algorithm 2. However in this context the noise is too small to give asignificant effect to the results.

To illustrate how the information varies around the true landing site the positionis altered around the landing site and the trace of the information matrix in everyposition is plotted. In Figure 3.1, the distribution of where the most informationis gained is shown, in this illustration ek = 0 to be able to draw some conclusionabout the pure system. Calculating the trace is done by taking the sum of thediagonal values in the matrix. It is important to remember that the Fisher infor-mation matrix does not depend on the measurements but solely on the geometryof the problem which depends on the estimate, the position of the UAV, and themeasurement model.

Figure 3.1 shows a donut shaped distribution, the highest information gain can befound at the radius that is the same as the current altitude. Altering the altitudeand looking at the distributions such as the one shown in Figure 3.1 it can beseen that for altitudes below 10 meters the highest gain is at a radius equal tothe altitude. At higher altitudes the radius of the highest information gain getswider and wider. This appearance is because of the measurement model functionderived in Equation (2.14). This means that the highest gain of information canbe found in a cone around the estimate. Refer to Figure 3.2 for an illustration.


−20 −15 −10 −5 0 5 10 15 20−20

−10

0

10

20

x

y

Gain in Fisher information around the beacon

Figure 3.1: How the Fisher information varies around the landing site lo-cated at the origin. The brighter color indicates a higher gain of information.It is plotted when the UAV is 4 meters up in the air.

3.1.2 Using FIM in an Optimization Problem

The FIM has a strong connection to how well the target is estimated and in Equa-tion (3.2) it can be seen that the CRLB limits the error covariance of the estimate.This means that the larger the FIM is the lower the bound is, which means thatthe information is desired to be maximized.

In the design of this problem, it is desired to find the two-dimensional positionof the landing site. This means that when calculating the FIM it will be a 2-by-2 matrix. When performing optimization, a loss function is most often used todefine what it is that needs to be minimized or maximized. Using FIM in theoptimization problem means that it is desired to maximize a matrix, which is notintuitively defined. Because of this an alternative representation of the matrixis needed. In [16] and [19] they use a function that transform the informationmatrix in to a scalar.

3.1.3 Matrix to a Scalar

Since the FIM will be a 2-by-2 matrix and it is not possible to directly optimizea matrix, some sort of scalar function based on the FIM is needed. In [16] theproblem is discussed and some different methods are discussed.

The problem with going from matrix to scalar, is that it will result in a loss or

3.1 Problem formulation 23

−10−5

05

10−10

0

10

0

5

10

xy

z

Figure 3.2: Illustration of where the highest gain in information is dis-tributed.

a compression of information, which will lead to a weaker optimization. Forexample, the loss of directional information will be lost when transforming the2-by-2 matrix to a scalar.

In [16], four different functions based on the FIM are presented. They are allbased on the fact that the FIM should be maximized, however the functions arederived in a manner so that the loss function will be minimized by using thenegative scalar value or the inverse matrix of the FIM. The functions are

fD(JFIM ) = − log(det{JFIM }) (3.4a)

fE(JFIM ) = max

i{eig((JFIM )−1)} (3.4b)

fA(JFIM ) = tr{(JFIM )−1} (3.4c)

fB(JFIM ) = −tr{JFIM } (3.4d)

that all uses different qualities of the matrix. The function fD uses the determi-nant of the matrix which is given by multiplication of the eigenvalues. Optimiz-ing on fD results in a minimization of the volume of the uncertainty ellipsoid.The function fE uses the biggest eigenvalue to optimize on. This leads to the min-imization of the largest axis of the uncertainty ellipsoid. The next function fAis designed to minimize the average variance of the estimate and lastly the func-tion fB is called the sensitivity function and is most often used to initialize theoptimization.

The above mentioned functions can be used for optimization. The criterion theFisher information is supposed to fulfill has to do with the accuracy of the land-ing. The accuracy of the landing site estimate is directly linked to the shapeand volume of the confidence ellipsoid, since it is constructed from the covari-


ance matrix of the estimate. The decision stands between the functions fD andfE . Function fD is said to minimize the volume of the confidence ellipsoid, thedrawback to only looking at the volume is that it will not take any regard to thedirection of the uncertainty. The local minimum could result in a low uncertaintyin one direction but high in the other. The other function, fE , strives to get theconfidence ellipsoid as roundly shaped as possible. From that it is much easierto narrow the search area and it is believed to have a higher accuracy in bothdirections. Because of that, fE is believed to be the best suitable function for thisthesis.

3.2 Optimization

In [16] and [19], only the information is used to do the optimization and not thetime nor distance to the target. This is because of the problem formulation inthose articles is formulated in a way that there is no intention to reach the target.Here, the distance to the estimate term is added to the loss function to be able tohave a problem that aims to reach the target in time.

The approach to formulating the optimization problem starts by looking at thegoals defined in equations (3.1) and (3.2), where the later of these equations leadto the goal of minimizing the function fE(J

FIMk ). It can be preferable to first fulfill

one of these goals to a given threshold before looking at achieving the other goal.These goals can be defined as constraints that can be used in a main algorithm.The constraints would be

fE(JFIMk ) < ǫx

∥

∥

∥pQ,k − xH,k

∥

∥

∥ < ǫp(3.5)

where ǫx and ǫp are the thresholds that show when each constraint is fulfilledand can be used as design variables.

These constraints will not be constraints to the optimization problem but thegoals will formulate the loss function. A loss function with more than one termis usually formulated with weights which can give one term more significanceover the other. These weights are set as design variables to the optimization whensolving the problem. The weights can also be used if one term is faster to achievethen the other and by that giving that term a higher, unwanted significance. Thelater is the case in this thesis where the distance is more easily shortened than theinformation is gained.

The loss function for the problem to this thesis can be formulated as

VL = ωFIM fE(JFIMk ) + ωp

∥

∥

∥pQ,k − xH,k

∥

∥

∥ (3.6)

where ωFIM and ωp are the weights to each goal or term.

3.2 Optimization 25

From this the optimization problem can be stated as

argminpQ,1:N

ωFIM fE(JFIMk ) + ωp

∥

∥

∥pQ,k − xH,k

∥

∥

∥

such that∥

∥

∥pQ,k − pQ,k−1

∥

∥

∥ < v ·∆t

k = 1, ..., N

(3.7)

where pQ,1:N is the trajectory or all the positions that the UAV has had up to timeN . The movement speed is an essential constraint that does not allow the UAV tomove too fast from one point to another. The parameters v and ∆t are the speedand the time between two measurements. The time horizon is set by the variableN . Suppose that the algorithm continuously minimizes the loss function. Thiswill imply that the higher the variable N is, the smaller the loss function will be.However, this also implies that the time elapsed during the landing is increased.This variable will because of that be seen as a measurement of performance forthe complete system. Another measure of performance is the final distance theUAV have to the true landing site.

3.2.1 Solving the Optimization Problem

In [6] Cochran wrote, "You tell me the value of θ, and I promise to design the bestexperiment for estimating θ". It is a good quote to have in mind when lookingfor a solution to this problem.

As been mentioned, the solution lies in finding an optimal trajectory that willgenerate the best estimate and to reach that estimate in a reasonable time space.In the following subsections some different theories drawn from the problemdefined in Equation (3.7) are discussed.

Gradient Search

The gradient search method can be used to find a numerical solution to the prob-lem, and is used in [19]. Here the weighted loss function defined in Equation(3.6) can be used to find the steepest gradient. One important factor is that theweights need to be chosen with caution. The choosing of different weights givesone term more significance over the other which will effect the gradient of theloss function. The easiest way of determine the weights is by testing.

The method of looking at the gradient at each time step will in some meaningproduce the optimum trajectory in each time step. That means that the optimiza-tion problem defined in Equation (3.7) is solved for each position update ratherthen over the full trajectory. Doing this might lead to the problem of finding alocal extreme which will cause the system to get stuck in that point.

The algorithm for gradient search begins with considering the loss function VL(pQ),defined in Equation (3.6). Most numerical minimization methods are using an it-erative procedure

pQ,k+1 = pQ,k + αkq(pQ,k) (3.8)

where αk > 0 is the step size and q is the search direction. The step size is allowed


to be altered in every time step. This is to not overshoot a local extreme [1]. Inthis thesis the step size is proportional to the distance between the landing siteestimate and the UAV but also constrained by the maximum velocity of the UAVso the step size remains realistic. There is some different methods on how todetermine the search direction. One example is the steepest descent algorithmwhich is determined by the negative gradient

q(pQ,k) = −δVL(pQ,k)

δpQ,k. (3.9)

The steepest descend method can be divided into two groups, depending onwhich information that is available. If the loss function is known and differen-tiable the gradient can be derived analytically, or the opposite that the gradientis not directly available, an approximation of the gradient has to be calculatedfrom measurements of the loss function [19]. In this thesis, the gradient will beapproximated with the numerical derivative defined in Algorithm 2.

Heuristic Approaches

The problem can also be solved by using heuristic approaches. By looking at dif-ferent experiments and results presented in the theory, a way of finding a solutionmight be found. The solution might not be optimal, but can be good enough forthe purpose of the system. The inspiration to these methods have been found inthe theory presented in this thesis, but also from previous work presented in [16]and [19].

The common denominator to the heuristic approaches is to gain informationabout the estimate and generate trajectories that will do so. For example the in-formation cone presented in Figure 3.2 is shown to produce a gain in informationwhen traveling on this cone. By generating a trajectory that has the same altitudeas distance to the target that would lead to an increase in FIM. Another methodis generating a trajectory that uses the uncertainty in the landing site estimate. Ifthe uncertainty is higher in one direction it means that the UAV should travel inthat direction to get a bigger angular separation which will lead to a lower uncer-tainty. To generate the trajectory, the confidence ellipsoid to the estimate couldbe used.

3.3 Monte Carlo Simulations

Monte Carlo simulations can be used to solve a parametric problem that other-wise would be hard to solve. By altering the parameter that is desired and run-ning several simulations then analyzing the results from the different iterationsand identifying preferable trends and relations the parameter can be estimated.When altering the parameter, it is usually done by some stochastic distribution[7].

During the experimental phase, these kinds of simulation will be done. When try-ing different algorithms with different parameters and design variables the algo-

3.3 Monte Carlo Simulations 27

rithms can be ran several times and from these simulations a good representativemeasurement that in some manner measures the performance can be analyzed.If a design parameter needs to be set, the algorithm can be ran several times andin between simulations the variable is changed. The results from these simula-tions can then be compared and the best design variable can be determined. Bydoing this, conclusions to which method that is the best one can be motivated.In this thesis these simulations will be called Monte Carlo simulation due to thesimilarity between the methods.

One important thing to remember when running these simulations is that theresults may not be the optimum. It does not even have to be close. However,using intuitive methods when generating or altering the parameters the resultsfrom those simulations can be well motivated and compared.

4Experiments and Results

In this chapter, the experiments and the results from the experiment are beingpresented. The simulation environment has been implemented in Matlab and allof the gathered data has been processed in Matlab.

4.1 Data Collection Application

When designing the algorithms for localization, some real data was needed, forexample to validate the models and to be able to easily retrieve the data presentedin Table 2.1. To do this, a simple Android application was developed in AndroidStudio and used a simple interface. The Android application could then easily bedownloaded to the smart phone used and collect the data needed.

The application sampled the data given below.

• Wi-Fi signal strength (RSSI)

• GPS-signals

• Barometric pressure

• Acceleration data

• Orientation of the smart phone

This data was then pushed to a cloud database from where the data could bedownloaded and used in simulations. The acceleration and orientation are notused in this thesis, but as a future work the acceleration can be used to track theUAV between measurements and for this to work, in all attitudes, the orientation

29

30 4 Experiments and Results

have to be regarded when sampling the acceleration data to compensate for thedifferent coordinate systems.

4.2 Measurement Model Validation

To be able to simulate the system to be as close to the reality as possible, someparameter estimation is needed. To estimate the parameters for the WLAN sig-nals, an experiment was set up. The transmitting WLAN-beacon was laid on theground andmeasurements was sampled at different distances. In this experimentmeasurement was collected at 1,3,5,7,10, 13 and 15 meters.

0 5 10 15

−80

−60

Distance [m]

Measu

redRSS[dBm]

Data collection

(a) The data collected at differentdistances from the beacon. Thisplot also shows the variance in theRSS signals.

0 5 10 15

−80

−60

Distance [m]

Measu

redRSS[dBm]

η = 3.59

(b) The model fitted to the meansof the collected data

Figure 4.1: Experiment to validate and set parameters to the RSS measure-ment models.

In Figure 4.1a all the data is collected and presented at the different known dis-tances, what can be seen is that the variance of the signals is quite large. Themeasurement model uses the LNSM, presented in equation (2.14), to replicatethis behavior. To find the parameters to the measurement model the means ofthe data in each data point is used, the parameters that needs to be defined are

the reference power P(d0), the covariance R(H)k which is used to try to describe

the signal noise and the path loss constant, η. When P(d0) is defined the leastsquare method can be used to define η, the algorithm is written as

ǫ =P(d) − P(d0)

K =10 log(d)

η =(KTK)−1KT ǫ

(4.1)

and gives an estimate that is approximated so that the function is fitted to thedata points as good as possible. This is illustrated in Figure 4.1b.

4.3 Solving the Optimization Problem 31

In Table 4.1 the different parameters used in the LNSMmodel are presented. Thestandard deviation is very different for each distance. This is further discussedin Section 2.6.2, but in this thesis it is approximated to 8 dBm. This value isalso used when generating the signals and this number has been used in similarapplication such as in [8]. P(d0) is taken as the mean value of the measurementsat d0 = 1 m.

P(d0) η Standard deviation, R(H)k

-49 dBm 3.6 ∼ 8 dBm

Table 4.1: Table of the parameters used in the measurement model.

4.3 Solving the Optimization Problem

As explained in Chapter 3, the optimization problem to solve which trajectorythat gives the best estimate and shortest travel time, is complicated. Experimentsto test different theories need to be set up to be able to validate the results. Inthis thesis two main alternatives have been discussed, firstly the gradient searchmethod, and secondly using a heuristic method. The heuristic methods is de-rived using both inspiration from the theory in this thesis and inspiration fromprevious work such as [16], [19] and [7].

This section will investigate different methods of solving the optimization prob-lem. First the experiments are explained and motivated, then the results fromthose experiment is presented together with a discussion and analysis about thepresented results. At the end of this chapter, a summarized analysis of the exper-imental results is presented together with a suggested main algorithm.

4.3.1 Gradient Search

Experiment

From the derivation in Chapter 3, it is stated that the gradient of the loss function,presented in equation (3.6), is one way of finding the optimum way at each timestep. If the gradient is followed in each point that would generate a trajectory.

The gradient will be calculated numerically in the two dimensional space. Here,it is interesting how the weighting of the different terms in the loss function,described in Equation (3.6), matter and to see how the UAV will travel in compar-ison with the estimate. The UAV is kept at constant height of ten meters duringthis experiment.

Result and Discussion

The results from this experiment can be found in Figure 4.2.

Figure 4.2 presents some different combinations of the weights to the loss func-tion. The UAV had the same starting point in all of the configurations at [-15 -15],


−20 0 20−20

−10

0

10

x

y

(a) The gradient search methodwith the weights ωFIM = 1 andωp = 0.001.

−50 0

−80

−60

−40

−20

0

x

y

(b) The gradient searchmethodwith the weights ωFIM = 1 andωp = 1.

−50 0 50 100−100

−50

0

50

x

y

(c) The gradient search methodwith the weights ωFIM = 1 andωp = 0.

Figure 4.2: Here the trajectory that was generated while following the gra-dient is shown. The red line is the trajectory and the blue is the estimateposition at each time step. The black circle is the true position of the land-ing site, which is positioned at [0 10]. The different figures show differentcombinations of the weighting of the loss function defined in equation (3.6)

and the landing site is situated at [0 10].

It is interesting that the weighting seems to be very important. In Figure 4.2b,both terms in the loss function are equally important to the gradient and theresult is that the distance to the estimate takes the overhand. It is not until theend of the shown trajectory that the estimate is reached, but as can be seen theestimate has not converged to the true landing site.

In Figure 4.2c, the distance to the estimate is not important at all and that causesthe UAV to travel in a spiral away from the estimate, but the estimate seems toconverge to the true landing site with a low uncertainty. It is not time effectivebecause the UAV travels very far away in a short time. One dilemma here canbe the range of the WLAN beacon, and in a real application the maximum rangewould eventually be reached.


Lastly in Figure 4.2a, a spiral shape is shown that converges towards the estimate,which is a preferable look and is easily reproducible. It also gives the best overallestimate when comparing the three configurations.

These result gives an intuition of what the optimal trajectory may look like. Thiscan be used when building themain algorithm for the system. First it seems that aspiral shape is the best when trying to solve the gradient search problem, wherethe distance constraint is undermined to the gain in information. This meansthat it could be a good idea to gather some information before focusing on thedistance. When traveling directly towards the estimate, it does not converge. This

can be solved by altering Q(H)k and making it smaller with each k but by doing

this too soon could lead to a bad estimate because the landing site estimate willnot move as much in between updates. If the estimate is reached, the UAV willnot move and because of that the estimate will not improve since the movementof the UAV is essential for the localization system to get a good separation in themeasurements.

The problem here is that there is no control of the altitude. That means thatthis method is not a complete method. Another problem is the velocity of theUAV, in the experiment above the RSS and GPS-measurements are received eachtime update, if the real update frequency would be used, it will lead to the UAVbehaving in a non satisfactory manner going back and forth because the optimumis overshot. In theory, this would be solved by reducing the velocity so that theUAV travels extremely slow, but then the time aspect for the landing sequencewould not be satisfactory and there might be a better way to solve the problem.Because of that some heuristic methods need to be tested.

4.3.2 Elliptical Trajectory

Experiment

The thought behind this experiment is to both validate the system but also to gaininsight in how the system works when traveling in the vicinity of the landingsite in different elliptical trajectories. For this experiment a hundred differenttrajectories were predefined, a sample of these trajectories is shown in Figure 4.3.One key factor to investigate is the gain in information and how fast the estimatewill converge to the correct landing site.

Results and Discussion

Monte Carlo simulations were usedwith the different trajectories shown in Figure4.3, by traveling each of these trajectories a hundred times and then check howmany measurements were needed before the filter converges, the convergencetime could be calculated. Convergence in this context means that the filter hada prediction error lower than a threshold value and the time is the convergencetime. The prediction error is calculated in the simulation environment as thenorm of the difference between the estimate and the true position of the landingsite. In Figure 4.4 the experimental results are presented.


−20 −10 0 10 20−20

−10

0

10

20

1

60

100

x

y

Elliptical trajectories

Figure 4.3: The trajectories traveled during this experiment. The small greencircle shows the position of the true landing site. The numbers 1, 60 and 100are the trajectory numbers used as reference in the text.

What can be seen from this experiment is that the convergence time is decreasedwhen the UAV is traveling over the transmitting beacon, located at the landingsite. Another observation is that the more the UAV is traveling in different di-rections the faster the convergence time is. For example comparing trajectorynumber 1 with 100 the convergence time is much lower in the later. In Figure4.4b similar results as in Figure 4.4a are shown, the prediction error is decreas-ing faster when the trajectory is over the landing site, and much better if thetrajectory has a large coverage in the X and Y-axis. It can also be seen that thefilter does not converge while traveling on trajectory number 1. This is becausethe initial guess is placed at the origin making it a fifty-fifty chance that the filterconverges to the mirror point. In Figure 4.4a, it can be seen that for trajectorynumber one, the filter does converge since the convergence time is lower than250 seconds, which is the maximum allowed time for each iteration. This is be-cause of several Monte Carlo simulations that pushes the mean time below themaximum. The difference in time between the figures most likely arise becausethe results in Figure 4.4a depends on several separate Monte Carlo simulations.

In Figure 4.5, the Fisher information is presented when traveling along trajectorynumber 100. This means that, while traveling around the vicinity of the landingsite will result in a steady increase of information, which can be a useful initial-ization to the landing process.


0 20 40 60 80 100100

150

200

Trajectory number

Tim

eto

convergen

ce

Convergence time

(a) The time before convergencefor each trajectory

0 20 400

10

20

30

Time [s]

Pred.error

The prediction error for xH using the EKF

1

60

100

(b) The prediction error for trajectories 1, 60and 100 when the EKF is used.

Figure 4.4: Experiment to see how the different trajectories influence thespeed of convergence and the prediction error.

4.3.3 The Information Cone, Spiral Approach

Experiment

This method is based on the knowledge that a spiral will render the best resultas could be seen from the gradient search method. The idea is to glide down theinformation cone in a spiral and by that continuously gain information and getcloser to the estimate. This method is inspired by the distribution of the Fisherinformation, see Section 3.1.1. The altitude is continuously decreasing and theUAV is following the radius of the information cone. By always descending itwill minimize the time consumed in the air and different rate of descends will betested.


In Figure 4.6 the main results from this experiment is shown. What can be seen


0 100 200 300 400 5000

10

20

30

40

Number of measurements

Fisher

Inform

ation

Fisher Information of Trajectory number 100

X information

Y information

Figure 4.5: The Fisher information when traveling along trajectory number100.

is that the time elapsed is controlled with the rate of descend, which is a trivialresult. The overall result when landing is not that good and the distance is notwithin the desired accuracy. This is mostly because there is not any control onthe accuracy. In Figure 4.6b it can be seen that a lower rate of descent gives asignificant impact on the end result. This shows that it is essential that the UAVis not taken down towards the ground too fast.

4.3.4 Following the Confidence Ellipsoid, Spiral Apprach

Experiment

This method is motivated by looking at the confidence ellipsoid generated fromthe covariance matrix from the EKF estimation. The ellipsoid gives a direct trans-lation on how the information about the estimate is distributed and in whichdirections. By following the confidence ellipsoid it should give more informationin the direction that has a higher uncertainty. For example, if the ellipsoid isshaped widely in the X-direction and compact in the Y-direction it means thatthere is not enough information on how the landing site is positioned on the X-direction, the covariance is bigger in the X-direction than in the Y-direction. Byusing the ellipsoid and travel along it, this will result in a wider coverage in thedirection where the larger uncertainty is.

After some time, the certainty in the direction traveled will increase and that will


010

0

10

0

5

10

XY

Altitude

Trajectory

(a) The resulting trajectory whentraveling along the informationcone.

0.2 0.4 0.6 0.8 12

4

6

8

Rate of descend [m/Ts]

|xH−pQ|w

hen

landed

[m] Distance vs Time

0.2 0.4 0.6 0.8 10

100

200

300

Tim

eelap

sed[s]

(b) Comparison for the rate of descendin terms of time elapsed and distancefrom the true landing site when reach-ing the ground

Figure 4.6: Experimental results from traveling in a spiral around the infor-mation cone with a constant rate of descend.

impose that the confidence interval becomes smaller. This will result in that theUAV is traveling in a spiral trajectory. This method is also believed to be a goodfit with the chosen function to transform the FIM to a scalar value, discussed inSection 3.1.3.

This experiment will also look at different update frequencies for the confidenceellipsoid. The frequency will alter from updating the reference with every newestimate, to the case where the reference is followed a longer time with a lowerfrequency, getting a higher coverage in the more uncertain direction.


In Figure 4.7, the results from these experiments are given.

In Figure 4.7b, the update frequency of how often the confidence ellipsoid shouldbe changed is shown. It shows that the more often it is changed the better. Thisresult has been given trough Monte Carlo simulations and the result is used inthe other simulations producing the other figures in Figure 4.7 that are producedfrom one simulation. As predicted the final trajectory that were generated hasthe shape of a spiral when the confidence ellipsoid is continuously reducing insize. In Figure 4.7d, the Fisher information is presented. Here, it can be seenthat the information gained is exponentially increasing the closer the UAV is tothe transmitting beacon. In Figure 4.7c the estimate is shown for the differenttime steps, it can be seen that the estimate converges towards the true value. Asample of the confidence ellipsoids that was used as reference are also shown inthis figure.

This method shows good possibilities to work in the main algorithm, it uses the


spiral trajectory and has a steady gain in information. The estimate converges tothe correct landing site with a decreasing covariance. It is also a good combina-tion with the matrix to scalar function used for the FIM, since the minimizationof the function strives to minimize the largest axis in the confidence ellipse whichgives a natural converging spiral.

4.4 Experiment with Real Measurements

Experiment

The analysis of this thesis will be mostly based and built around simulated mea-surements. However, an experiment was set up to test how well the localiza-tion system performs with real measurements. The measurements were sampledwhile walking in a circle stopping at each node, the nodes and the experimentalset up is shown in Figure 4.8.


In Figure 4.9, the result of the estimation is presented when real measurementswere used. The result looks quite good for a small collection of measurement.The final estimate is off with about 1.5 meters.

4.4 Experiment with Real Measurements 39

−10 0 10−20

−10

0

10

20

x

y

Spiral Trajectory

(a) The resulting trajectorywhen traveling along the differ-ent confidence ellipsoids.

0 100 200 300

170

180

190

200

Update timeTim

eto

convergen

ce

Convergence time

(b) The time before convergencefor each trajectory when alteringthe update time, measurementsper new generated trajectory, of theconfidence ellipsoid to follow.

−20 −10 0−10

0

10

x

y

Estimation of the landing site

(c) How the estimate of the landingsite is converging during one simula-tion.

0 10 20 300

10

20

30

Number of measurements

Fisher

Inform

ation

Fisher Information

X information

Y Information

(d) The Fisher information dur-ing one simulation.

Figure 4.7: Experimental results from traveling along the confidence ellip-soid. The green circle in Figure 4.7a and 4.7c is the landing site. Figure4.7a, 4.7c and 4.7d show the results of one simulation. Figure 4.7b show theresults from Monte Carlo simulations with 100 iterations for each updatefrequency.


xH t = nTs − Ts

t = nTs

t = 2nTs

t = 3nTs

3m

Figure 4.8: How the measurements were sampled, starting at the east dotwalking and stopping at the north, west and south dot every measurementsample. Where n = 1, ..., 18 and Ts is the time between samples.

−4 −2 0 2 4−4

−2

0

2

4

6

8

10

x

y

Estimation of the landing site using real measurements

Figure 4.9: Plot that shows how the estimate converges towards the landingsite. This experiment was done using real measurements gathered with thedata collection application.

4.5 Analysis of the Experimental Results 41

4.5 Analysis of the Experimental Results

The gradient search shows a few different aspects to this problem that will im-prove and worsen the results. To go directly towards the estimate, shows thatthe estimate will, with a high probability, not converge to the true landing site.Another observation is that when all the focus lies on gathering information, theUAV will vanish from the landing site and eventually get out of range. The bestconvergence and time effective way seems to be when the UAV is traveling in aspiral towards the estimate and closing in after a while. This is a behavior that isdesired in a main algorithm.

The main problem with the gradient search method is the measurement updatefrequency for the GPS and RSS measurements. The update time is too slow toensure a good gradient descend and still minimizing the flight time. In a realscenario the gradient can not be calculated at the rate that is desired to get a goodresult.

Another aspect mentioned in Chapter 2, is the occurrence of multiple modeswhile estimating. If the filter converges to the wrong mode, the amount of in-formation is not significant. Because of this, a good initial estimate is neededto ensure a correct convergence. Another solution to this phenomena could besolved with a particle filter, but that method has not been fully implemented.However by circling the vicinity of the landing site, it seems by the experimentalresults that it will converge towards the right estimate and also gather a steadyincrease of information. This does not solve the problem of getting closer to thelanding site, but as mentioned before, as a possible solution it will partly fulfillthe information gain in the loss function before advancing towards the estimate.

Approaching the estimate can be done in some different ways, either just circlingthe estimate in the information cone where the most information gain is sup-pose to be, but the estimate is constantly changing leaving the information gainlacking anyway. The other method is to use the error covariance in the X and Y-direction and create a reference using those two variables which is the case whenfollowing the confidence ellipsoid.

In the next section a main algorithm is suggested and tested in the simulationenvironment.

4.6 Suggested Main Algorithm

In this section, a suggested main algorithm will be presented and how it works.The theoretical aspects of how the problem will be discussed as well. It will beusing the method where the confidence ellipsoid is followed, and to initialize theprocedure circle the vicinity of the landing site. In the end of this section, someperformance variables and tuning parameters are presented.


4.6.1 Algorithm : Confidence Ellipsoid

The algorithm is divided into four different stages. These stages are presented inthe list below.

• Stage 1 - In this initial stage, the vicinity that is initially known is circled tofind a reasonable estimate. When a certain amount of information has beencollected, the algorithm proceeds to stage 2.

• Stage 2 - The system now generates a trajectory that is placed around thecurrent estimates confidence ellipsoid. It is changed with every new mea-surement. The descent towards the landing site is initialized in this stage.

• Stage 3 - When the information is high enough the system locks on thecurrent estimate and descend further.

• Stage 4 - The UAV travels towards the locked on estimate and land.

In Algorithm 3, the procedure of switching between the different stages is pre-sented. It should be noted that in this algorithm, the altitude is managed sepa-rately and is a design parameter, here, it is divided in two altitude steps of 4 and 2meters. These numbers are not static but changeable depending on the situation.The ideal landing is commenced from 10 meters in the air, and that is what thisalgorithm is built upon.

Algorithm 3Main Algorithm : Confidence Ellipsoid

Design parameter is λinf o. Where λinf o is the threshold for when enough infor-

mation has been collected. Here βk = fE(JFIMk ) for each time step k.

Stage 1: Travel around the perimeterwhen: βk ≤ λinf o

switch: to stage 2Stage 2: Travel with the confidence ellipse and start descending

if: βk ≤ λinf o/2 and z < 4 mswitch: to stage 3

Stage 3: Descend further and lock on targetwhen: βk ≤ λinf o/4 and z < 2 m

switch: to stage 4Stage 4: Descend to ground level until locked target is reached.

4.6.2 Tuning Parameters and Results

There are many different parameters that needs to be set to get the desired results.The parameters have been decided trough different methods. The filter parame-ters have been set both by analyzing the real measurements but also using ad hocmethods to see directly what works best. The design threshold parameters forthe main algorithm have been set through Monte Carlo simulations. In Table 4.2,the parameter names and values used in the suggested algorithm are presented.

In Figure 4.10 the results from the Monte Carlo simulations are presented when

4.6 Suggested Main Algorithm 43

Parameter ValueT 0.25 s

R(z) 0.01

R(H) diag{1 1 0.1 8}

Q(z) 0.2I2Q(H) 0.1I2λinf o 0.75

Table 4.2: Table of design parameters for the filter and for the main algo-rithm. In this table I2 is the unit matrix in two dimensions. The diag-function means that the variables are put in the diagonal of a full matrixwith zeros around, in this case R(H) is a 4-by-4 matrix.

altering the design parameter λinf o. The weighting compromise is well illus-trated in the figure that shows the relation between the time elapsed and theaccuracy of the landing.

In Figure 4.11, the result distribution when running the algorithm 10000 timeswith the parameters in Table 4.2, is presented. The distance on the Y-axis showsthe distance measured when the UAV has landed.

4.6.3 Performance

In Figure 4.12, the generated trajectory as a whole is shown. It can be seen thatinitially the vicinity is circled to gain information about where the landing sitecould be. Then after a while, the UAV will start descending and because of thereference generated by the confidence ellipsoid, and the separate altitude controlin the algorithm, the UAV will do a spiral towards the ground.

If the measurement model for the RSS measurement was changed to handle mul-tiple beacons, it would give an impact on the end result. The changes that needsto be done is putting in an extra variable that deals with the separation betweenthe beacons and the landing site. In that case the true distance from the UAV tothe landing site would be

d2k =∥

∥

∥(xH,k + τi ) − pQ∥

∥

∥ (4.2)

where τi is the three dimensional separation from the landing site for beacon i,where i > 0. In Table 4.3, the average results are presented for different number ofbeacons. It can be seen that the results are improved with an increase in beaconsused.

4.6.4 Estimation

To present the performance of the filters the estimates with its covariance isshown in Figure 4.13. It can be seen that the estimates converge to the true val-ues, and the confidence interval is steadily decreasing. These results show thatthe filter is working in satisfactory manner and gives the desired results after a


0 1 2 3 4 50

10

20

30

λinf o

Distance

from

truelandingsite

atlanding

Experimental results when altering λinf o

0 1 2 3 4 50

500

1,000

1,500

Tim

eelap

sedbefore

landing

Figure 4.10: Monte Carlo simulations when altering the design parameterλinf o. The precision of the landing is compared with the time elapsed beforethe landing was completed.

few measurements. The velocity of the altitude is not presented in the figuresince there has been no good way to measure the true value and compare it to.


0 2 4 6 8 10 12 140

50

100

150

200

250

Distance [m]

Number

oflandings

Distance from true landing site when touched the ground

Figure 4.11: Distribution of the distance from the landing site after landing

Number of beacons, i Average results

1 1.97 m2 1.94 m4 1.79 m10 1.55 m100 0.66 m

Table 4.3: The average distance from the landing site after landing withmultiple beacons.


−20−10

010

20

−20

−10

010

200

5

10

XY

Altitude

Trajectory when landing

Figure 4.12: The trajectory generated during one landing procedure.


0 5 10 15 20

−10

0

10

Time [s]

X-planeposition

xH,x with 90% confidence interval

(a) The landing site estimate in theX-direction. The blue line is the esti-mate and the dashed blue line is the truevalue.

0 5 10 15 20

−10

0

10

Time [s]

Y-planeposition

xH,y with 90% confidence interval

(b) The landing site estimate in theY-direction. The red line is the esti-mate and the dashed blue line is the truevalue.

0 5 10 15 200

5

10

15

Time [s]

Altitude

z with 90% confidence interval

(c) The altitude estimate (blue line)compared with the true value (redline). It is oscillating because theUAV is regulated around the esti-mated altitude.

Figure 4.13: The estimates presented with a confidence interval of 90% dur-ing a run of the final algorithm. The black dashed lines in all of the graphsshow the confidence intervals.


4.7 Discussion

It needs to be stressed that the results and system are built in a simulation envi-ronment and have not been tested in a real-time scenario. As briefly discussedthe variance of the signals is not consistent but varying with the distance. Animplementation of that system has not been needed in the simulations.

The suggested algorithm is as said, a suggested algorithm, and there are multipleother ways of building a main algorithm. Most certainly many different waysthat also will present a better result. What can be said however, is that a spirallytrajectory instead of a straight line towards the estimate will be a better way tohandle this problem.

The suggested algorithm shows that the problem can be solved, the discussion ishowever if the result is good enough for the purpose that the system will be usedfor.

5Conclusions and Future Work

5.1 Conclusions

In this thesis, the theoretical aspects to land a UAV using RSS measurementshave been handled. The problem expanded from just localization of the landingsite to the optimization problem of generating a trajectory allowing the systemto perform better. The intuitive method of just traveling towards the solution asfast as possible is shown not to be the most efficient way.

The system has been implemented in a simulated environment, in that environ-ment external factors are kept at a minimum. In a real-life outdoor scenariothose factors are not controllable or in some cases even predictable. The systemperformance in ideal conditions prove to be working to some degree. Howeverthe problem formulation includes that the UAV should be able to land all aroundthe clock which may include that the visibility is limited. That may also includeadverse weather conditions such as strong winds and/or heavy rains.

Can autonomous landing be done using RSS-measurements? The use of WLANas localization equipment might not be the most efficient way. It is shown thatthe measurement noise is relatively large and not that consistent. When WLANwas constructed it had no purpose to be used as a localization aid. It is howevershown in this thesis that in theory, in a controlled environment, it can be done.That does not imply that the system can be used for a high precision landing in areal application. Alterations in the final algorithm might improve the results butfor a more robust performance the hardware would have to be replaced.

Can this system be done using a smart phone? All aspects in this thesis regardingthe sensors and transmitting hardware have been taken from the SamsungGalaxyS5, so with the same conclusion as above, yes it can be done with some limitations.

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50 5 Conclusions and Future Work

One aspect to this that has not been considered is the computational qualities inthe smart phone. That study is left for the future to implement the system in thesmart phone and to see if that is a barrier, but with the smart phones today andwith themethods used in this thesis that are constructed to put the computationalload to a minimum, it should not be a problem.

5.2 Future Work

There are several extensions that could be made to this thesis. The followingsection gives suggestions on areas that need more attention before implementingthe system on a real UAV.

5.2.1 Hardware

The hardware simulated in this thesis does not include the best possible sensorsthat could be used to perform the landing. The use of a different kind of radiosignal, i.e. using Bluetooth instead of WLAN which has a lower signal noise butalso a lower range. The WLAN signals can also be replaced with UWB radio anduse that technology to improve the localization.

Another aspect of the hardware is the altitude sensor, as mentioned the height ofthe landing site is said to lay on the ground but that gives some problems withthe pressure sensor. The use of a sonar sensor gives the result used in this thesiswithout the initial information about the calibration pressure.

5.2.2 The Particle Filter

In the final algorithm the EKF is the filter that is used but the theory about theparticle filter shows that it might be a better way to go. The particle filter cancompensate for the multiple modes that are created when estimating. By analyz-ing the modes and the trajectories to each mode and calculating the informationgained it could lead to a faster convergence and landing procedure.

5.2.3 Solve the Optimization Problem

The solutions discussed in this thesis are enough to complete the landing, but if itis optimal is not determined. To find this optimum trajectory is not an easy thingto do, at least with the resources used in this project. If this was done it could leadto another trajectroy that might be better then the suggested algorithm presentedin this thesis.

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