Application of selected methods of Kalman Filtration to estimation of state and model parameters of the flying robot

to estimation of state and model parameters of the flying robot

Zastosowanie wybranych metod filtracji Kalmana do estymacji stanu i parametrów bezzałogowego robota lataj ˛ acego

by

Jarosław Go´sli ´nski

Doctoral Dissertation

Institute of Control and Information Engineering Faculty of Electrical Engineering

Pozna ´n University of Technology

Instytut Automatyki i In ˙zynierii Informatycznej Wydział Elektryczny

Politechnika Pozna ´nska

Advisor: Andrzej Królikowski, Ph.D. D.Sc. Prof.

Pozna ´n, Poland, June 27, 2016

In the dissertation an issue of multirotor UAV state estimation is raised. Various models of multirotor are derived and the control algorithms as well as estimation algorithms are proposed. Among models two can be determined: the classic one - Euler/Tait-Bryan and quaternion-based. The instructions on how the model can be tailored to a given mul- tirotor is provided and additional schemes of control in different coordinate frames are shown. The model identification procedure is presented and, based on Square Root Un- scented Kalman Filter, model parameter observer is derived. Necessary modifications of the proposed algorithm are introduced and two schemes of identification are explained:

for open and closed loop models. In the further part of the work a comprehensive theory on state observers is given. The theory covers linear observers, various nonlinear Kalman Filters and a novel method. The results include classic state and dual estimation schemes.

Both schemes are used and prove their effectiveness. All of the presented methods of pa-

rameter and state estimation are optimized by heuristic method. In conclusion the errors

analysis is shown and the methods are compared. Based on the presented data, gen-

eral recommendations for observers applications in multirotors are stated. In the work,

both aspects i.e. theoretical and experimental are covered. The results prove the thesis

statement which is as follows: "Application of Adaptive Square-Root Unscented Kalman

Filter gains the robustness of the UAV quadrotor measurement system in case of data

distortion and sensor fault or failure".

W pracy doktorskiej poruszany jest temat estymacji wektora stanu bezzałogowych ro- botów lataj ˛ acych typu wielowirnikowego. Wyprowadzono ró ˙zne typy modeli oraz al- gorytmów sterowania i estymacji. W´sród zaproponowanych modeli wyró ˙znia si˛e dwa:

pierwszy zwi ˛ azany jest z k ˛ atami Eulera/Tait-Bryana oraz drugi, który bazuje na kwater- nionach. W pracy opisana jest metoda dopasowania podanych modeli do konkretnego robota wielowirnikowego. Dodatkowo zaprezentowano metody sterowania robotem w układzie lokalnym i globalnym. Przedstawiona jest równie ˙z procedura identyfikacji parametrycznej podanych modeli, opieraj ˛ aca si˛e na metodzie Square Root Unscented Kalman Filter. Metoda ta została poddana modyfikacji, prowadz ˛ acych do zwi˛ekszenia jej odporno´sci. Wprowadono dwa schematy w opisie identyfikacji. Pierwszy, dla mo- deli bez sterowania oraz drugi, dla modeli z pełnym sterowaniem (sprz˛e ˙zeniem zwro- tnym). W dalszej cz˛e´sci pracy przedło ˙zono teori˛e dotycz ˛ ac ˛ a obserwatorów stanu. Teo- ria ta pokrywa kwestie obserwatorów liniowych oraz nieliniowych tj. skupia si˛e na fil- trach Kalmana oraz na metodzie autorskiej. Wyniki pracy s ˛ a przedstawione dla metod estymacji: klasycznej oraz dualnej. Wszystkie metody estymacji s ˛ a optymalizowane przy u ˙zyciu metody heurystycznej. W wynikach przedstawiona jest analiza bł˛edów i porównanie obserwatorów. Podano równie ˙z zalecenia dotycz ˛ ace stosowania obser- watorów wektora stanu i parametrów modelu, maj ˛ ace zastosowanie w robotach lataj ˛ a- cych. Otrzymane rezultaty potwierdzaj ˛ a słuszno´s´c tezy pracy, która brzmi nast˛epuj ˛ aco:

"Zastosowanie filtracji dualnej opartej na metodzie Adaptacyjnego, Bez´sladowego Filtru

Kalmana typu Square Root zwi˛eksza odporno´s´c systemu pomiarowego robota lataj ˛ acego

na bł˛edy oraz uszkodzenia jego sensorów".

The list defines the symbols used within this document. Some of the symbols are ex- plained explicitly within the body of the text. Boldface is used to represent vectors and matrices. The first two time derivatives of an arbitrary variable x may be represented as x, ¨ ˙ x. The estimated variable x is denoted by adding a hat i.e. ˆ x. Any given vector x that is extended with additional dimension is represented as ˘ x.

p

, ν

– position and linear velocity of the robot in the body frame (body coordinate system)

p

, ν

– position and linear velocity of the robot in the reference frame (reference coordi- nate system)

ν

– planar velocity of the robot in the body frame ω

, ω – planar velocity of the robot in the reference frame L – robot angular momentum

L

– rotor-propeller angular momentum φ, θ, ψ – roll, pitch, and yaw angles of the robot Θ ˙ – Euler rates

ˆ

ω

– angular velocity estimated by AHRS (Attitude and Heading Reference System) ˆ

q

– quaternion estimated by AHRS ˆ

q – quaternion estimated by state observer (classic/dual) BF – coordinate system attached to the robot

EF – reference coordinate system

R

– rotation matrix, transforming given vector from the BF into the EF R

– rotation matrix, transforming given vector from the EF into the BF τ – input torque

τ – friction drag

f

– friction drag f

– gravitational force f

– friction force f

– aerodynamic drags

I, ˘ I – robot inertia tensor in R

and R

I

, ˘ I

– propeller inertia tensor in R

and R

A

, B

– aerodynamic flapping effects matrices w

– robot model parameters at k-th sample

x

– robot state vector at k-th sample

d

– output vector in parameter observers at k-th sample y

– output vector in state observers at k-th sample

Q

– covariance matrix of the process noise in parameter and state observer at k-th sam- ple

R

– covariance matrix of the measurement noise in parameter and state observers at k-th sample

T – sampling period

g

– position of the most fitted particle in the whole swarm (PSO) p

– the best position of i-th particle of the PSO

present

– position of i-th particle of the PSO

C

– fitness function

List of abbreviations 1

1 Introduction 4

1.1 Problem statement . . . . 4

1.2 Related work . . . . 9

2 Mathematical models of a multirotor UAV 13 2.1 Related work . . . . 13

2.2 Multirotor modeling conceptions . . . . 15

2.2.1 Location of the BF coordinate frame . . . . 18

2.2.2 Location of the EF coordinate frame . . . . 19

2.3 The Euler-based quadrotor model . . . . 19

2.4 The quaternion-based model . . . . 24

2.5 UAV models comparison . . . . 30

2.5.1 Overall control scheme . . . . 30

2.5.2 Coaxial quadrotor control and optimization . . . . 38

2.5.3 Classic quadrotor control and optimization . . . . 46

2.6 UAV models summary and conclusions . . . . 48

3 Model parameter estimation 51 3.1 Related work . . . . 51

3.2 General model parameter estimation - the SRUKF method . . . . 56

3.3 The coaxial quadrotor parameter estimation . . . . 59

3.3.1 Coaxial model assumptions and restrictions . . . . 59

3.3.2 The experiment on coaxial multirotor - gathered data . . . . 61

3.3.3 The results of coaxial model identification . . . . 62

3.4 The classic quadrotor parameter estimation . . . . 67

3.4.1 Classic quadrotor model assumptions and restrictions . . . . 67

3.4.2 The results of classic model identification . . . . 70

3.5 Discussion and conclusions on model parameter estimation . . . . 77

4 Multirotors state estimation 82

4.1 Related work . . . . 82

4.1.1 State estimation and attitude observers . . . . 82

4.1.2 Altitude estimation . . . . 86

4.1.3 Estimation of the horizontal position and velocity . . . . 88

4.2 Concepts of state estimation . . . . 89

4.3 Theory and construction of observers . . . . 94

4.3.1 Unscented Kalman Filter in a classic approach . . . 107

4.3.2 Tailored filters - UAV case . . . 111

4.4 State estimation . . . 115

4.4.1 Attitude observers - sensory-based filters . . . 115

4.4.2 Model-based state estimation . . . 120

4.5 Dual estimation . . . 126

5 Conclusions 135 5.1 Summary . . . 136

5.2 Prospects for future research . . . 139

A Appendix 140 A.1 Rotation formalisms 3-2-1 . . . 140

A.2 Euler rates in 3-2-1 notation . . . 140

A.3 Angular velocity in body-fixed and reference frames . . . 142

Bibliography 143

I would like to express my gratitude to my supervisor Andrzej Królikowski. Over the past five years, I have been taught by a person who I truly admire. Right from the begin- ning of this dissertation his knowledge in the filed of estimation and control engineering played the most important role in my research. As a result of abundance of hours spent on discussions and reviews, many articles and one patent application were brought into being. I would like to express my gratefulness to the head of the Institute of Control and Information Engineering, Andrzej Kasi ´nski who supported my work since I started my Bachelor Thesis. Without him, many works could not have been finished.

I wish to thank my good colleague and my auxiliary supervisor Mr. Wojciech Giernacki, with whom I work in the Polish Aerial Robotic Team. Also I would like to show appreci- ation to Mr. Stanisław Gardecki for sharing the UAV robots, which were made under his leadership at our team. I would like to express my gratitude to Mr. Xiang He from the University of Utah, for successful cooperation in the field of robotics and for providing me with necessary flight data for numerous studies in the dissertation.

Finally I would like to thank my wife Jagoda, for her patience, understanding, encour- agement and support over the last years of my research. I owe my parents and family a debt of gratitude for many years of support, especially through my childhood - the time when my dreams about robots were born.

The financial support of the Poznan University of Technology is gratefully acknowl- edged. I was also awarded twice with a scholarship of the project "Scholarship support for PhD students specializing in majors strategic for Wielkopolska’s development", Sub- measure 8.2.2 Human Capital Operational Programme, co-financed by European Union under the European Social Fund.

Jarosław Go´sli ´nski

Pozna ´n

June 27, 2016

Introduction

1.1 Problem statement

Autonomous estimation of state in unmanned flying robots is of key importance in the field of UAVs (Unmanned Aerial Vehicles). These robots have a wide range of applica- tions and offer tremendous potential [1], [2], [3]. Ever since the first UAV came out, it was clearly the option for a new kind of airborne observations, surveillance and recon- naissance (ISR) missions. For many different tasks such as border patrol, it could not even be possible for human or ground aerial vehicles (UAGs) to repeatedly control it at high speed. Although the UAVs are intentionally designed to fly without the operator, their autonomous operation is strongly limited. It is adequate and provides safety, for instance: during aerial photography or industrial inspection it is required to control the UAV, deciding when the shoots should be taken or to avoid obstacles arising in front.

Apparently it is desirable for UAV to have a person in charge, but in many missions the individual is not able to work continuously, not to mention their reliability and dexterity in the course of the task. It can be seen in surveillance or rescue mission where for an example UAVs work for hours and the human operator should be consistently good in quality of his work, while searching for missing people. Human operator suffers from at- tentional fatigue and boredom, while the UAVs are limited only by computational power and energy. In many applications where the UAVs are used, the operator has to control the state of the machine so it will not run out of the battery or fall onto trees. These are only trivial problems and they do not reflect the real issue regarding autonomous flight.

UAVs are suitable for most of the surveillance missions, the ability to work in the air gives them lots of possibilities including long-distance flights over the floods or forests.

With increasing power of processing units it is possible to handle all of the operations and

algorithms on-board without transferring data to the operator. However, the operator

still plays a key role in decision-making process. Unfortunately, human introduces im-

perfection into the entire system [4]. Although it might be controversial, it is true that the

human-based decision systems are slow, and reduce the UAVs capabilities. During a typ-

ical mission, the operator checks the maps and controls video signal, and consequently

he may enforce the UAV to go back, capture additional information or land. Focusing

on details may be time-consuming but essential in rescue or surveillance missions. Hav-

ing all of the processing units on-board it can be done without operator’s decision, but

it is not yet a probable and real case [5]. As the MIT reports show (Human/Unmanned

Systems Collaboration), nowadays the most popular in UAVs is the teleoperation, while

fully-autonomous systems are used only in laboratory environment.

two major groups: the first group is fixed-wing, tactical UAVs, interesting mainly for de- fense, surveillance and reconnaissance purposes. The second group is of vertically take- off and landing (VTOL) UAVs. The body of the fixed-wing UAV is similar to an aircraft, with fixed wings, reduced mass and limited dimension (without carrying passengers or heavy loads, the maximum payloads are limited to the total mass of on-board equip- ment). On the contrary, VTOLs are multirotors with two to dozen and more rotors. The basic difference lies in the purpose they are used for. The fixed-wing UAVs take an advan- tage of the lift force generated on the wings during the horizontal flight. That is why, they have been applied to high altitudes, primarily for surveillance and reconnaissance [1], [6]

(for instance Global Hawk or the Predator). VTOLs can operate at low altitude (generally less then 1000m) and in most scenarios they are used in narrow range [7], [8], [9]. Ac- cording to the report on UAS

, the UAVs market is growing very fast, and its role crosses various industries. Surprisingly, the report also shows that already existing jobs related to drones will triple by the end of the year 2020. This data covers only the drone pilots jobs and does not even mention the staff hired for the maintenance or consulting service.

Although it does not provide clear information about the autonomous UAVs, which can operate without human supervision, it can be deduced that this field is not heading for full autonomous drones or they are already sufficiently independent, so the further steps are not necessary. As it was said, for many types of missions, an autonomy is not the goal.

During the typical mission, the basic scheme assumes that the operator sets the position via Global Positioning System (GPS). Then the dron’s execution system heads towards the target, simultaneously providing visual data and its odometry data. We should take notice of the fact that fixed-wing UAVs are in most scenarios thousands of meters above the ground. Their speed is more than 100km/h, which results in the little probability of the obstacle presence. On the contrary, VTOLs are in general smaller [10], intended to work in the vicinity of human. VTOLs have more than one main rotor (compared with helicopters) and are equipped with fixed angle rotors driven by Brushless DC (BLDC) motors [11]. The mechanical structure is simple and optimized towards rigid body, built as a cross-like frame (in most cases, but not limited to more diverse shapes). Their ability to vertical take off and landing, in combination with minimal maintenance enhance their advantages. It also allows to use in a wide range of applications. The basic concept of the first quadrotor VTOL was developed in the twenties of the XX century [12]. In spite of the fact, that it was a manned vehicle, it could not be properly controlled, as it required simultaneous control of all rotors. It took more than 70 years to create first unmanned quadrotors. Nowadays, thanks to microprocessors and small sensors (MEMS technol- ogy), quadrotors are available in "ready-to-use" form for a small amount of money. The most popular commercial quadrotors are radio controlled or provide Wi-Fi link to smart- phones, tablets or computers [13]. Although it is still the beginning of the quadrotors era, there are already plenty of companies that produce drones, mainly for film-making purposes, to serve as a mobile cameras or for surveillance and reconnaissance missions

1

Unmanned Aircraft Systems Report, North Central Texas Regional General Aviation and Heliport Sys-

tem Plan

(AeroVironment, Aeryon Scout, AirRobot, Align T-Rex, AR.Drone Parrot, DJI, among others [13]).

As it can be seen, the fixed-wings UAVs are definitely different when compared to VTOLs. One can conclude that these two groups of UAVs are considered to have different structure, on-board systems, application and use-related problems. The great majority of the UAVs are the fixed-wing UAVs and are used in the military. Main roles drones play in these fields are: shadowing enemy fleets, decoying missiles by the emission of artificial signatures, reconnaissance, surveillance of enemy activity, monitoring of nuclear, biolog- ical or chemical contamination (NBC), radar system jamming and destruction, airfield damage assessment, elimination of unexploded bombs, strike and attack [14]. UAVs are also interesting for civil purposes: policing duties, traffic surveillance, pipeline survey, agricultural operations, border patrol, aerial mapping and meteorology, disaster and cri- sis management of the search and rescue, etc. Most of those needs can be covered by fixed-wing UAVs but only micro and mini VTOLs can operate in human-shared environ- ment. Due to their small size and ability to hover they are perfectly fitted to this area. As it was already said, VTOL UAVs application is not limited to open spaces and therefore can operate in houses, offices, warehouses and any other places where robots bring us new capabilities.

Micro and mini multirotor UAVs are considered both easy to build and trivial in control.

Even though it is true, small dimensions and low payload constraints dictate numerous

restrictions on creators and designers. It is still challenging to build small multirotor UAV

which is able to work autonomously. It stems from the processing units needs to cover all

worst-case scenarios especially due to environmental changes. The author would like to

indicate low processing power and lack of information just to point out some of the prob-

lems related with autonomous operation. For more clear comprehension, brief informa-

tion about multirotors architecture must be provided. Multirotors have simple mechan-

ical structure. In case of four-rotor UAVs, this structure and the basis of the propulsion

system are the simplest (among many other multirotors). Similarly to the aircraft, one

can distinguish roll, pitch and yaw angles (for instance: Tait-Bryan angels in the intrinsic

notation Z-Y -X). These three parameters describe the orientation. In order to change the

yaw, pitch or roll angle, speed of the appropriate rotor has to be changed (to be discussed

in Chapter 2). The position is expressed in the reference frame (chosen by the user, which

is not necessarily the same for the orientation) in the Cartesian coordinate system (X, Y ,

Z). Multirotors are characterized by simple and alike architecture. The main processing

unit (or control unit) is a single board computer or just a microprocessor. Sensors form

a sensing unit which communicates with the processing unit. The actuators are speed

controllers and the BLDC motors with mounted propellers. Based upon the orientation

and/or position the control unit sets new inputs for speed controllers. This configura-

tion forms a core system of the UAV (required to fly or hover). It is open to additional

subsystems like estimators, parameter observers, path planners, vision-based algorithms

and many others [15], [16]. The control algorithm is located in the main processing

unit and may be based on different types of control methods. The most common are

Proportional-Integral-Derivative (PID) [17], [18] and Linear-Quadratic (LQ) controllers

regulators [21]. In the closed loop, similarly to orientation, the position can be controlled with assumption that the orientation loop is always of the highest priority (in practice this is realized as the inner and outer loop - details are given in Chapter 2). The overall characteristics (briefly named dynamics) as well as the vertical speed are set by user or automatically by the throttle parameter. In brief, throttle is an absolute value of a speed added to each and every motor. The control of the speed is based on the characteristic of the motor and its controller and is often accomplished with PID or Coefficient Dia- gram Method (CDM) control techniques [22]. The basic concept has to include the main three blocks: control systems, sensors and navigation (expendable to vision systems and auxiliary algorithms) [23].

One of the most challenging task to be covered in quadrotors is a fully-autonomous operation. Over the last decade there has been a growing number of publications regard- ing this topic (mainly UAV). In many cases, papers dealt with monitoring, mapping or searching [2], reconnaissance [24] and surveillance issues [3]. Although the contribu- tion in the field of autonomous UAVs is substantial, its autonomy is not well-defined and does not constitute any class of UAVs. As it is clearly seen, mostly the term "autonomy"

refers to specific part of the operation of the UAV, for instance: landing, obstacle avoid- ance, mapping, reaching the target. In order to refine this definition it is obligatory to introduce the Autonomous Control Level (ACL) metrics [25]. The ACL refers to the level of autonomous control of UAVs. It has 10 degrees in the autonomy scale from remotely guided (1) to fully-autonomous swarms (10). On the second and third level the real- time health diagnostic and ability of adoption to failures are placed. Based on the scale, one can define the autonomy as a level of quality of being independent. In 2016 we are still not encountering the inundation of the fully-autonomous robots, especially UAVs.

According to [25], the situation is slowly improving, yet there are some basic problems between the second and the third level.

The hierarchical control architecture of autonomous control for UAV [25] is divided

into three main parts. The first one is the lowest in the hierarchy and is called the execu-

tion level. Within this tier, the most trivial tasks are being performed. The main control

loops as well as trajectory control are located here. This level has a direct influence on the

hardware. The execution level inherits the global information from upper parts of the

architecture. The second is the coordination level, which is responsible for estimation,

navigation, identification, planning, verification, data validation and inspection. It is in

charge of the processes that are important for the control loops of the execution level. For

UAVs, which are radio-controlled, it is often the highest level - sufficient to operate in the

air. This tier seems to be flexible, since it does not require any of the listed above elements

(subsystems), apart from the navigation. Finally, the last part of the hierarchical control

architecture is the organization level, responsible for the decisions and management. For

many UAVs with communication link, this layer is moved to base-computer [26], [5]. All

three layers are linked, but to some extend they may be independent. For instance: the

execution level needs to adopt restrictions and limitations for control parameters due to

stability or dynamic states.

Along with the level the intelligence increases. On the second level the fault detections, diagnostic and isolation are performed. Although for radio controlled UAVs this is un- necessary, the fault detection and proper isolation with reconfiguration is a must in case of an autonomous flight. It is analogous to biological system, where all errors and faults are detected and reported. Furthermore, in such a system relevant subsystems are called in order to repair, change into a different state and, what is crucial, to counteract the cause. Similar systems are desirable in fully-autonomous UAV. All the components from the third level have to lead the overall mission, so that its aim could be reached. It is also required to control the state of the machine. Without the proper mechanisms, the UAV cannot guarantee that it will fulfill the mission objectives. The third level, or simply the highest in the control hierarchy, has the ability to plan and manage the mission. It also changes the path in case of emergency or interruption. The mission target should be given in advance (human operator or another robot in a swarm) and the robot cannot change it, however it can plan how to reach it via some thought-out trajectory. It simply reflects a human brain, when a senior gives orders to subordinate. The UAV has to report whether the mission is being processed or has already finished. It also sends the current progress and the encountered problems. In the worse-case scenarios, the human operator is eligible to stop or decide about mission’s next steps.

The real issue may occur on lower levels i.e. on second and first level. When the UAV operates, the high speed control loops have to be fed with uncorrupted data. It means, that in order to properly control the UAV, the coordination level must work constantly and reliably, while dealing with unexpected conditions. To achieve this, the Fault Toler- ant Systems (FTS) and Fault Tolerant Control (FTC) are introduced into the UAVs [27].

The aim of the FTS is to control the correctness of the sensory system data as well as to involve improving mechanisms whenever distortion of these data occur. This task is cru- cial in UAV control architecture and might be executed in different ways. One of the most common approach implies the use of the Fault Tolerant algorithm as the reconfigurable controller [28]. In this dissertation a modified approach is being presented. The state estimation methods are introduced along with different models of the quadrotor UAV to gain robustness of the second and first level of the hierarchical control architecture. The main thesis of the dissertation is stated as follows:

Application of Adaptive Square-Root Unscented Kalman Filter gains the robust- ness of the UAV quadrotor measurement system in case of data distortion and sensor fault or failure.

Five more precise theses are stated to support the main theorem:

• the use of model parameter estimation reduces the model errors;

• the use of optimization methods for the observer settings improve its performance;

• observers decomposition efficiently reduces their execution time;

• closed loop models are better suited for state and model parameter estimation;

model parameter estimation) supported by artificial intelligence algorithms yield near-optimal solution.

1.2 Related work

In the field of state and model parameters estimation for UAV, plenty of works have been published. Basically, two major purposes of state and model parameter estimation can be distinguished. The first one is to create the model suitable for simulations, so that the trajectory and control algorithms might be evaluated prior to tests on the real object.

The second purpose is to create models which are used in control or state observers algo- rithms. Since the computational power in stationary and onboard computers is different, the models complexity shall also differ. In case of stationary simulations the overall test may take days while onboard systems cycle must be finished within milliseconds. In the dissertation, models for onboard computers are under consideration. It limits the com- plexity, but also challenges for precise, though simple quadrotor model. This model is used by the author for the estimation tasks.

In normal UAV systems, the most trivial and necessary is the Inertial Measurement Unit (IMU). The IMU provides such sensors like: accelerometers, gyroscopes or magnetome- ters (though, magnetometer is not an inertial sensor). The simplest way is to combine the data collected from them and estimate the necessary state vector elements e.g. Eu- ler angles, quaternions, angular velocities (for the orientation estimation only). The state vector elements should be gathered based on the control algorithm needs. This is what most of the inexpensive quadrotors UAV do [11], [10], [23]. In case of the position es- timation, the additional sensors are used: Global Positioning System (GPS), barometers and vision-based algorithms [23]. In real-life scenario, the IMU-based and GPS-based estimators cannot preserve and guarantee their expected parameters (accuracy). Due to data distortion, GPS-denied environment or even faults this poor system may be inad- equate and not reliable enough [29]. The most popular IMU sensors in a ready-to-use form are Attitude and Heading Reference Systems (AHRS). They are based on different fusing methods like Extended Kalman Filter [30], [29], Particle Filter or Complementary Filter [31], supported by different algorithms like TRIAD [32], Least Squares, Gradient descent [33] and others (to be discussed in further chapters). These systems may form a part of the coordination level of the UAV. However, when the AHRS is a stand-alone, it cannot exploit the data from the object, for an example the angular velocity of the UAV rotors. In case of such systems, the control data comes only from the sensors read- ings and estimation based on measurement model. When one of the crucial components of the AHRS fails, the estimation process will degrade, resulting in control failure. In [34], the author compared different approaches with redundant IMU sensors. Also the method for choosing the most reliable measurements was proposed. Although this may be used in reliable UAV control system, the redundant IMU sensors are influenced by the same noise and, in the worst-case scenario, the redundant systems are inefficient (e.g.

magnetometers suffer from the interfering magnetic fields overlap). In order to make the

AHRS reliable, other sources of knowledge must be used. The common approach is the utilization of object mathematical model (shortly: model). In [35], the authors proposed the estimator based on Luenberger observer as well as on Extended Kalman Filter, that incorporates model and sensory data. For this purpose, linear model for pitch angle, and nonlinear one for the heading (yaw) and roll angles were used. The main contribution of this article was a combination of two different observers, which can handle with the UAV models. The research was narrowed to the Euler or Tait-Bryan notation, while in the experiment the AHRS was used. One may consider it interesting to clarify, what the bounds of distortions which will not affect the observers are. Also it is highly important to disclose the nature of the distortions, e.g. the magnetometers may suffer from the Earth magnetic field distortions caused by motor currents, the UAV frame or simply by tem- perature growth (some of them may still be compensated). In [36], authors described the method of state estimation applied to commercial AR Drone quadrotor. Authors have accomplished their task with EKF state estimation with simple prediction model. The EKF observer was prepared so it could incorporate data from IMU sensors and visual system. The robustness of the proposed method was proved based on two possible sce- narios. First one: the inertial sensors noises are compensated with the visual data and the second one: visual data errors (such as bad illumination) are neglected with the usage of inertial sensors. For such a system, by introducing this particular drone model, one may obtain better results. Yet in the research, the problem of delays, as well as the issues related with the mismatch between of the visual and inertial data timestamps, may have the biggest influence on estimation errors (same for dithering). In different work [37], world-famous robotics engineers R. Mahony, V. Kumar and P. Corke present the overall framework for quadrotor modeling, estimation and control. The authors have provided detailed information on how an onboard software algorithms should be aligned in or- der to successfully accomplish the control task in a quadrotor. In the work one will find drone model derivation based on Euler notation, improved by additional drag-like ef- fects, often ignored in research problems. Provided information on estimation procedure may be covered as follows: authors have assumed the basic model of the MEMS sen- sor (accelerometer, gyroscope and magnetometer), they also have given the basics of the observer structure, namely the complementary filter, which was clearly opposed to well- known EKF observers (estimators). In the further part, method for translational velocity estimation was shown. None of the presented estimators were based on UAV model. In work [38], authors have introduced the group of filters for a flight path reconstruction.

Prior to that, the kinematic model of the fixed-wing sailplane was presented. The whole research refers to the manned machine, however it can be simply transferred to UAV field. The results which involved different observers for a flight path reconstruction are shown in the paper. The presented methods are: EKF, UKF and their smoother versions:

EKS and UKS. In conclusion the authors have remarked, that the best performance was

achieved with the UKF/UKS method. What is significant, the presented methods were

compared in terms of the total Root-Mean-Square error (RMSE). Based on this, authors

have proved that the IEKF (Iterated Extended Kalman Filter) was not better than EKF,

and both of them were worse than UKF. In another example [39], authors have presented

cated at ETH in Zurich. The platform has been developed since 2007, and it has recently been deployed in a mobile version. In short, the platform is equipped with plenty (from 8 to 20) of Vicon cameras, which along with specialized vision software are capable of capturing markers pose. The markers are placed on quadrotors and thanks to that both their position and orientation can be measured. In the article, the authors presented the overall framework for the research as well as the control algorithms, used in the flying robots. The estimation processes are performed on-board and off-board. These tasks are done via the complementary filter, which, by the authors, is simple in tunning and does not need the additional calculations of Jacobians, like in the EKF. The onboard estima- tion fuses data from the IMU and provides quaternion representation of the robot body orientation, referenced in global coordinate frame. Then it is transfered to the off-board computer, where along with vision pose estimation is finally fused and used in control algorithms. The estimation does not use a dynamic model of the quadrotor. The Flying Machine Arena authors have also revealed information on simple model representation, exploited for the control or simulation purposes. The model has basic structure and links the thrust force with the inertial rigid body dynamics, it does not include drag-like ef- fects or frictions. The model is not explicitly used for crucial parts of the control scheme.

In different work [40], authors propose three different models for fixed-wing UAV and the EKF-based estimators. The core of each model constitutes set of the Euler angles:

pitch roll and yaw. The value of the work lies in model parameter estimation, which was

performed for seven of them. The results were satisfactory, though the conclusion is con-

troversial. The authors stated, that the linear model gave similar results in comparison

with two nonlinear models. This can be maintained only in case of closeness to the point

of linearization, which is clearly not the case for multirotor UAVs. Nevertheless it might

be interesting in terms of future research. The work seems to be only a short report on the

results of a much bigger project. One will not be able to figure out how the parameters

estimation was performed or what are the strengths and weaknesses of the presented

method. It is worth to mentioning that the work shows original results obtained during

the state and model parameter estimation in UAV. In [41], authors have shown the results

from the FT system applied to UAV, based on Adaptive Thau Observer (ATO). The work

refers to the fault tolerant system able to detect a fault in one of the four rotor propeller

system (used in quadrotor UAV). It can also isolate the detected fault, so the observer

will preserve its characteristics. Authors have introduced the basic model of the quadro-

tor (rigid body model) with additional drag effect, which were not added to the state

function of the observer. The proposed ATO was formed as a combination of linear and

nonlinear models. The nonlinear model was later assumed to be a part of the error of the

observer. The observer was fed with data from sensory system and the control signals,

similarly to ordinary filters. Based on the model, the observer could estimate the offsets

in control that were correlated with the offsets in control signals for rotors. The study of

the residuals could give the straight answer to the question about which rotor, precisely

a propeller, has a fault. The work shows research on the ATO. It also provides important

derivation of the Thau Observer equations, suited to the quadrotor model. From imple-

mentation prospective, one would be interested in the procedure of the identification of drag effect coefficient. It would be also justifiable to prove whether those coefficients can be assumed to be constant during the experiment. The results of the proposed method are satisfactory and give opportunities for ATO to be applied in quadrotors.

To author’s best knowledge, there are only several works, where the pose was estimated based on sensory system and model excited by the control signals. Mostly, authors have faced the same problem: insufficiency and inadequacy of model representation. The problem stems from the bad modeling procedure and lack of model parameter identifi- cation.

In the dissertation, quadrotor models are derived (Chapter 2). Further, required estima-

tion of their parameters is provided along with comprehensive research on identification

schemes and their optimization (Chapter 3). Based on the models, estimation techniques

for multirotors are presented and tested in different scenarios (Chapter 4). The studies

on state and parameters estimation are based on heuristic method of optimization. Final

conclusions and summary are provided in Chapter 5. Some additional descriptions and

studies are presented, mainly because of some new methods, which were developed in

the research process.

2.1 Related work

In UAVs, both the precise control algorithms and estimation methods are crucial for proper operation in the air. These are not limited to standard algorithms but extend by custom methods added to onboard computer, yielding better performance. One is able to find different techniques in scientific papers, articles or in commerce. Most of them are demanding, it means they utilized simple and extended models. In a case of full state estimation techniques, the model is a must. Therefore it is important to focus on already available models or develop a new one. The quadrotor model can be simply divided into position and orientation parts. Both are important and will be discussed in details.

The first quadrotor models appeared at the beginning of the 2000s, however the multiro- tor manned machines had been modeled before then. The first, ready to use helicopter model was introduced by Heffley and Mnich [42] at NASA. It was a six degrees of free- doms (DOFs) model with roll, pitch and yaw angles. Although it is claimed that the model has minimum complexity structure, the authors included also aerodynamic drags, such as flapping. As a result of the fact, the paper is one of the most widespread cited technical work.

The presented model was suitable for helicopters, but also had a common part with the multirotor UAVs: the rigid body dynamics, which is the model’s core. For all models described in the literature of the subject the assumptions of the rigid frame structure are held. It means, that frame distortions or vibrations which certainly exist are not mod- eled and are assumed to be negligible. The rigid body dynamics refers to mechanical structure, which are the same for quadrotors and helicopters. Additionally, gyroscope moments must be included to fully define the model. Various works on modeling of multirotor UAVs may be found in the literature of the subject. Hamel et al [43] presented simple UAV model for X4 - flyer. Authors focused mainly on control law design. Al- though the model was simple (without aerodynamic effects), it reflected the rigid body dynamics as well as additional torques due to propeller and motor gyroscopic effects.

Further, the model was used in an attitude stabilization, based on Euler angles. That

work has revealed interesting nature of a quadrotor mechanics. Later on, many other

papers referred to [43]. In [8], McKerrow has provided interesting insight into inertia

tensors used in quadrotor model as well as gyroscopic torques, which were intensively

explained. These two papers are recognized as significant and precursory in terms of

quadrotor modeling. In a different work [44], authors have presented similar to [8] model

of a VTOL micro quadrotor. The basic concept was in control law design and simulation

in a special test bench. Another important work in the area of UAV quadrotors was given by Pounds et al [45].

ν

= R

ν

, (2.1)

R ˙

= R

[Ω]

, (2.2)

m ˙ ν

= −mω × ν + mgR

e

+

n

X

i=1

f

, (2.3)

I ˙ ω = −ω × Iω +

n

X

i=1

τ

. (2.4)

In the paper, the commonly known model given in Eqns. (2.1)-(2.4) was shown. In the model, the R

refers to rotation matrix, which transfers any given vector from the local to the global reference system (R

: BF → EF), [Ω]

is a skew symmetric matrix based on angular velocity ω, ν is a linear velocity and I stands for inertia tensor. In addition, authors shortly explained blade flapping effect and gave insight into the pitch and roll damping or simply induced drags. The work concluded with control law design and promising flight test results. More comprehensive study on quadrotor dynamics and aerodynamic drags was presented by Mahony et al [37]. The paper was formed as a tutorial for researchers and designers. The authors have given theory on modeling, es- timation and trajectory control. The model has been shown as a combination of drags and basic rigid body dynamics. What is worth mentioning, such effects as induced drag or blade flapping were explained and associated theory was presented. The paper [37]

was released in 2012, and since then, most of the model-based articles refer to it. What should be mentioned is that the presented models were Euler-based, so they had some inherited disadvantages due to angles singularities. It may strongly influence the model output when simulating or in real time operation in control with model as a reference.

The given rotation matrix R protects the model from singularities, however when Euler angles are used one will notice that for pitch angle close to ±π/2 the roll and yaw change arbitrarily (Gimbal lock). Even though, the R directly depends on angles, these will not affect the rotation matrix. However, in case of angle-dependent closed loop control sys- tem, it will partially or completely spoil the process. In order to avoid these side effects, researchers focus on different model representations. The most popular, singularity-free is quaternion-based representation. In case of quaternions, the model has different struc- ture, and most of the elements such as inertia tensor or rotation matrix are in R

. There are many works where multirotors with quaternions were used, however most of them are limited to the final form, without explanation of the temporary terms used in deriva- tion. What is more, great majority of the papers limit the derivation of "quaternion-based model" to estimation procedure in AHRS unit. This simply covers the estimation pro- cess which is the only quaternion-based one. Very often these estimated quaternions are calculated back to Euler or Tait-Bryan representation inheriting the singularities. In [46]

authors have presented the idea of full quaternion-based control of a quadrotor, with

simplified control process. Authors have converted the quaternion elements to fit to X,

Y and Z axis by providing product of reference quaternion and its measured equivalent.

multiplication yields error vector and scalar tending to one

. One would be interested in a theory and explanations for how the axial parameters are related with subsequent er- ror quaternion elements (geometrically or analytically). In different work ([47]), authors have presented three different models. The first one was a classic version with additional friction drag coefficients. The second was a quaternion-based model and the third one was a near-hover model. All these models are of great importance as long as they provide proper data. Here , author would like to stress, that the models must be properly derived and have to be consistent with the theory, especially when quaternions are used.

Most of the other papers, where the modeling task was included, are not within the scope of modeling. Rather than that, they stem from the control theory with the major objective of providing optimal control law. The presented state of the art gives quite an overview of multirotors models. It can be noticed, that the classic approach is popular and pro- vides necessary information, focusing on position and orientation parts of the model.

Furthermore, the drag-like effects are well understood and described. However, one is not able to find an universal quaternion-based model for multirotor. The missing part is mainly located in its derivation, where the necessary calculation and theory must be given. Moreover, the most interesting advantages and disadvantages of the classic and quaternion models and their comparison are not yet documented.

In the dissertation, author will focus on different models for classic quadrotor and coaxial quadrotor. The work will simply answer the question why and when should one choose the quaternion-based representation. In the further section of this chapter, the models will be shortly described with all parameters, that in normal operation must be identi- fied via suitable method. In the presented solution, the aim is to derive the model for estimation process, with minimal complexity, in clear and optimal form.

2.2 Multirotor modeling conceptions

A quadrotor model has to truly imitate drone’s dynamics. In order to achieve this, the Newton-Euler or Euler-Lagrange equations are used. In most cases, researchers use the Newton-Euler (NE) formalism [45], [43], [37]. The NE provides simple description of the rigid-body dynamics and can be extended with additional forces or torques (2.5).

"

f τ

#

=

"

mI

0

0 I

# "

˙ ν

˙ ω

# +

"

o ω × Iω

#

(2.5)

In the given NE description, the I

stands for identity matrix, while f and τ denotes acting forces and torques, respectively.

The model given in Eqn. (2.5) is expressed in a body-fixed frame and in order to be used in control or estimation process, one is forced to make necessary calculations, starting with choosing the notation and then multiplying the initial model by proper rotation ma- trices. The basic quadrotor model has six DOFs (when only zero-order derivatives are

1

to be proved in the further part of the chapter

Figure 2.1: Skeleton of the classic quadrotor, propellers marked by navy blue circles have clockwise rotation (in the BF ).

Figure 2.2: Skeleton of the coaxial quadrotor, propellers marked by navy blue circles have clockwise rotation (in the BF ).

included). First three DOFs describe position, the rest denotes orientation. The basic no-

tations are Euler and Tait-Bryan angles. The notation allows to present relation between

the local (BF ) and global (EF ) reference system (Fig. 2.1, 2.2). The Euler and Tait-Bryan

angles are most popular, mainly because of their intuitive meaning. In all cases, the three

angles are used in rotation in a proper sequence, e.g. the rotation matrix R

is a re-

about axis Z. These rotations are made about axes of the inertial frame (extrinsic rota- tions). Whenever the rotations are made about axes of new frames, the rotation matrix is denoted by additional apostrophes e.g. R

(intrinsic rotations). The difference between Euler or Tait-Bryan angles is trivial. In case of Euler angles, the first and the third rotation is about the same axis, e.g. R

while in Tait-Bryan representation, rota- tions are in three different axes (e.g. R

). The notations differ significantly, however in many technical papers, the Tait-Bryan notation is considered as one of the Euler rep- resentations. In the dissertation the most popular Tait-Bryan or Euler representation will be used. The notation is called 3 − 2 − 1, which refers to axes sequence. The result is given as rotation matrix: R

. This sequence assumes that during rotations the following steps are taken:

• Rotation Yaw about angle ψ in the Z axis of the global coordinate system EF,

• Rotation Pitch about angle θ in the Y axis of the new coordinate system EF

,

• Rotation Roll about angle φ in the X axis of the new local coordinate system EF

, yielding BF as a product.

This sequence can be depicted as in the Fig. 2.3 (note, that for sake of clarity the coordi- nate frames were translated. Originally their origins were in the same place). Under this assumption, transformation matrix may be written as (proof given in appendix A.1):

R

= R

(φ)R

(θ)R

(ψ) =







cθcψ cθsψ −sθ

sφsθcψ − cφsψ sφsθsψ + cφcψ sφcθ cφsθcψ + sφsψ cφsθsψ − sφcψ cφcθ





 , (2.6) where s and c stands for sin and cos, respectively, while R

(φ), R

(θ) and R

(ψ) refer to basic rotation matrices:

R

(φ) =







1 0 0

0 cosφ sinφ 0 −sinφ cosφ





 , (2.7)

R

(θ) =







cosθ 0 −sinθ

0 1 0

sinθ 0 cosθ





 , (2.8)

R

(ψ) =







cosψ sinψ 0

−sinψ cosψ 0

0 0 1





 . (2.9)

The rotation matrix R

= R

, transfers any given vector v

from the EF to the BF , yielding v

∈ BF :

v

= R

v

. (2.10)







x y

z

R^T_z() {EF}

{EF’}

{EF’’}

{BF}

w

w'

w’’

w'''

R^T_x()

R^Ty()

Figure 2.3: "3-2-1" rotation sequence

In the model derivation the R

is needed and since the R

is an orthonormal matrix, it can be written, that:

R

=  R



= 

R



(2.11)

2.2.1 Location of the BF coordinate frame

In the dissertation two different quadrotors are used (on modeling and experimental level). The first one is an ordinary structured quadrotor equipped with four propellers.

The second one is eight-propeller quadrotor mounted coaxially on its frame. Compared

to the classic quadrotor, this leads to the similar mechanical structure with increased

payload. In some works, it is called octarotor, though it may be confusing since normally

octarotors have planar mechanical structure. In order to simplify the issue, the author

will use the coaxial quadrotor as a name of that sort of robots. In addition, the author

would like to point out that planar octarotors are more efficient, but they are larger and

consume more space. Coaxial quadrotors provide a good trade-off between size and pay-

load, especially when applied indoors. These two quadrotors, namely classic and coaxial

are similar. Although the model of the coaxial quadrotor has additional terms which

locations of the body-fixed frame (BF ). The first one (associated with classic quadrotor) is set across with the cross-like mechanical frame, with Z pointing up (x configuration).

The second one (used in the coaxial quadrotor) is chosen, in such a way that it lies be- tween the main elements of the mechanical cross-like frame, with Z pointing down (also x-configuration). The BF s are shown in Fig. ??. The choice of BF location has an explicit influence on the final model. In addition it has a great impact on the control algorithm.

Also the choice of the x/+ configuration has influence on control and estimation. The x configuration is assumed to be more stable, while + (axes of the BF are situated along the mechanical construction) is considered an acrobatic configuration [23]. Their pros and cons will not be discussed, as this leads to different topic and diverges from the main thesis.

2.2.2 Location of the EF coordinate frame

The EF reference frame is not the Earth-fixed frame, but it comes down to simple orthog- onal frame, which has a certain orientation and position (in reference to the Earth). It is important to note, that it does not change during the mission or flight, however, it may change between missions. In order to understand how the EF is chosen, the geographic coordinates must be used. For an example, if one decides to use the NED (North, East, Down) EF , the X axis will point the North, the Y will be aligned with East direction and Z will point down. It means that it will change naturally with latitude and longitude (when referring to the global reference that is attached to the Earth in its center). In case of ENU (East, North, Up), the X is aligned with East direction, the Y is heading to North and the Z is pointing up. The NED and ENU are two most popular reference frames. In the work, the ENU will be used as the EF in case of classic quadrotor, however in case of coaxial quadrotor the NED will be utilized. The choice of EF does not have any in- fluence on the quality of control or overall performance. However it must be consistent with assumptions and has to be taken into account when controlling or estimating.

2.3 The Euler-based quadrotor model

One of the model simplifications is the presumption of a perfect stifness of the robot’s structure. On the basis of this simplification, the description of angular acceleration of the robot may be determined the same way as for the rigid body in space. As already stated, mathematical description for the movement is possible thanks to Euler equation of a rigid body motion. The general form of the Euler equation is given in Eqn. (2.5), however it stems from two Euler’s laws. The first law says about linear momentum Eqn.

(2.12), while the second describes the angular momentum Eqn. (2.13).

f = m dν

dt , (2.12)

τ = dL

dt . (2.13)

In the case of classic as well as coaxial quadrotor the inertia tensor does not change

, what finally transfers Eqn. (2.13) to the following:

L = Iω, (2.14)

dL

dt = ˙Iω + I ˙ ω

= I ˙ ω.

(2.15)

Based on the orientation model given above and the formula of the linear momentum, one is able to form the overall quadrotor model:

˙

ω = I

(τ − ω × (Iω) − τ

− τ

− τ

) , (2.16)

˙ ν = 1

m (f + f

− f

− f

− f

) , (2.17) where τ

is a gyroscopic torque due to propellers and motors, τ

stems from aerody- namic torques and τ

is a friction, f

, f

, f

and f

are gravitational, coriolis, drag and friction force, respectively.

Note that the overall force equation does not contain either Euler force or centripetal force, described by Eqn. (2.18) and Eqn. (2.19). It is due to inaccessibility of robot turning radius r

, though this would be an interesting extension of the model.

f

= ˙ ω × r

, (2.18)

f

= ω × (ω × r

) . (2.19)

The model presented in Eqns. (2.16)-(2.17) is given with reference to the body-fixed frame BF . In control tasks or estimation procedures, the model must be represented in the inertial frame (EF ). In order to achieve it, basic transformations need to be performed:

˙

ω = I

(τ − ω × (Iω) − τ

− τ

− τ

) , (2.20)

˙

ω

= R

ω, ˙ (2.21)

˙

ν

= R

1

m (f + f

− f

− f

− f

) , (2.22)

Θ = P ˙

ω. (2.23)

The Equation (2.20) describes angular acceleration in the body-fixed frame, while the Eqn. (2.23) transforms its lower derivative into the inertial reference, i.e. EF . The ˙ Θ is the Euler rate vector and the P

is a spatial rotation matrix for Euler rates:

Θ = ˙

h φ ˙ θ ˙ ψ ˙ i

, (2.24)

P

=







1 sinφtanθ cosφtanθ

0 cosφ −sinφ

0





 . (2.25)

2

valid only in case when drone has the same mass and all elements are within the fixed positions

b

P

is tailored for "3-2-1" convention (flight notation) with roll (bank), pitch (elevation) and yaw (heading) angles. Detailed derivation of the P

can be found in [48] as well as in the appendix A.2.

The general model given in Eqns. (2.20)-(2.23) has terms which depend on the chosen type of the robot structure (classic or coaxial). However some particular torques and forces are the same for all structures and can be given in advance:

τ

= h

ω + b

diag(sign(ω)) diag(ω)ω + c

diag(ω)

ω, (2.26)

τ

= ω × L

, (2.27)

L

= I

ω

, (2.28)

f

= −2mω × ν, (2.29)

f

= R

mge

, (2.30)

f

= b

diag(ν)ν, (2.31)

where e

= h

0 0 1 i

refers to the unit versor of Z axis, I

is the inertia tensor of motor and propeller and ω

is the sum of the propeller velocities. In case when the quadrotor is in a steady state, the propeller velocities are equal and the net is zero, which results in τ

= 0. The h

, b

, c

and b

are friction coefficients. The drag τ

acting on the body frame of the quadrotor stems from the aerodynamic effects. The total drag-like effect can be given only when based on the certain structure of the quadrotor, while the drag force f

has an explicit formulation. As it can be found [49], there are few types of drag-like effects: blade flapping, induced drag, translational drag, profile drag and parasitic drag (which may be ignored fot speeds up to 10m/s [49]). All these drag forces act on i-th propeller and depend on its thrust force:

f

=

n

X

i=1

f

, (2.32)

f

= kf

k (d

+ d

+ d

+ d

) , (2.33) d

= A

ν

ω

+ B

ω ω

, (2.34)

d

= k

ν

, (2.35)

d

=

( k

ν

, forkν

k 6 10m/s,

k

(−ν

+ ν

)

ν

, forkν

k > 10m/s, (2.36)

d

= k

ν

, (2.37)

where:

A

=







−A

A

0

−A

−A

0

0 0 0





 1

r , (2.38)

B

=







−B

B

0 B

−B

0

0 0 0





 , (2.39)

ν

=







1 0 0 0 1 0 0 0 0





 ν

, (2.40)

ν

= ν + ω × p

, (2.41)

where the n stands for number of quadrotor’s propellers, while p

is a position of i-th propeller given in the BF .

Blade flapping (d

) effect is ubiquitous in all rotor vehicles. It occurs when the rotor undergoes translational motion. It simply changes the rotor’s tip path plane, deflecting it by the flapping angle β

. Here the explanation will be narrowed to the final form of the force generated by the flapping, but one can find a full explanation in [49]. Another effect is called induced drag (d

). It may occur in small quadrotors, where rotor blades are rigid or semirigid which cause an imbalance in flapping forces. This effect can be sim- ply described as a product of planar velocity of i-th and a scalar parameter Eqn. (2.35).

Similarly to induced drag, it is also possible to distinguish translational drag (d

). This effect occurs, when the air is redirected form any translational movement to the down- ward of the BF . In accordance with [49], this effect depends on speed in the air. For slow movements, the translational drag is given, which is similar to a case of induced drag, while for fast maneuvers, it is necessary to incorporate the air velocity v

induced by i-th rotor Eqn. (2.36). The last two drag effects are profile drag (d

) and parasitic drag (d

).

Since the parasitic drag comes from non-lifting surfaces of quadrotor and occurs only for high speeds (> 10m/s) it may be ignored [49]. Profile drag is gained with the transverse velocity of the rotor blades in the air and it has a form given in Eqn. (2.37), where k

is a lumped parameter. The general form of drag force Eqn. (2.33) is multiplied by the thrust force (f

) generated by the i-th propeller. This force, can be expressed as a product of scalar parameter and second power of rotor’s angular velocity:

f =

n

X

i=1

f

, (2.42)

f

= h

0 0 b

ω

i

. (2.43)

The forces are generated in Z axis of the local frame BF . This assumption is only partially

true, however the influence of the semi-rigid structure of the mechanical frame can be

treated as an external disruption. Finally, having terms of the position motion model

Eqn. (2.17), the input torque τ as well as drag torque τ

can be given.

d

position in the BF , while the input torque τ is a sum of torques, due to subsequent propellers. The i-th torque is a cross product of the i-th force and the position of the i-th propeller in respect to the local frame.

τ

= p

× f

, (2.44)

same rule operates drag torques:

τ

= p

× f

. (2.45)

These equations are different for different type of quadrotor mechanical construction and can be derived after choosing the location of the BF reference.

• The model of classic quadrotor with x configuration

In the classic four-rotor construction with the x configuration the axes of the BF are virtually situated between the mechanical cross frame, with Z pointing up. Based on the Fig. ?? one is able to derive the following notation:

p

= h

−l

∓l

h i

, (2.46)

p

= h

l

±l

h i

, (2.47)

where l

=

√ 2

2

l, and l stands for arm length in the quadrotor cross frame and h is a distance between propeller’s plane and the arm in respect to Z axis (note that the origin of the BF is placed in the center of mass).

Now the Equation (2.44) covers the torques generated in X and Y axis in the BF , however the Z axis will also suffer from the reaction torques, given as:

τ

= d

(ω

+ ω

) − d

(ω

+ ω

). (2.48) In Eqn. (2.48) d

refers to reaction torque gains. Two of them were introduced, in order to cover different reactions of propellers system. Propeller 1 and 3 work in clockwise rotation and the rest are counter-clockwise (clockwise and counter- clockwise are referred to the BF ).

• The model of coaxial quadrotor with x configuration

Similar to the classic configuration, definition of the p

for coaxial quadrotor with x configuration must be introduced:

p

= h

±l

l

∓h i

, (2.49)

p

= h

−l

l

∓h i

, (2.50)

p

= h

−l

−l

∓h i

, (2.51)

p

= h

l

−l

∓h i

. (2.52)