Parafoil Control for STRAPLEX

FACULDADEDE ENGENHARIADA UNIVERSIDADEDO PORTO

João Luís Granja da Costa

FOR JURY EVALUATION

Mestrado Integrado em Engenharia Eletrotécnica e de Computadores

Supervisor: Sérgio Reis Cunha

June 25, 2013 c João Luís Granja da Costa, 2013 Resumo

Esta dissertação está inserida no projeto STRAPLEX (STRAtospheric PLatform EXperiment), que é um programa da Faculdade da Engenharia da Universidade do Porto em parceria com a Agencia Espacial Europeia (ESA). Este projeto é constituído por uma plataforma que permite à comunidade científica enviar experiências para a estratosfera recorrendo a balões de hélio. Devido às condições extremas presentes na estratosfera, o balão rebenta e a plataforma inicia a sua fase de descida estabilizada por um pára-quedas circular. Uma vez que este pára-quedas não permite qualquer tipo de controlo, nesta dissertação sugere-se que seja utilizado um parapente no lugar deste. O objetivo principal desta dissertação é implementar um fiável algoritmo de controlo para este sistema. Este trabalho dá continuidade a uma dissertação desenvolvida anteriormente, que propôs uma estrutura mecânica e de hardware que possibilita a implementação do algoritmo de controlo para a descida da plataforma. O movimento descendente da plataforma é descrito por um fiável modelo matemático, que inclui os vários movimentos relativos entre os diferentes objetos da plataforma. Este modelo tam- bém engloba os distintos tipos de controlo existentes no sistema, possibilitando a implementação de um algoritmo de controlo. Este algoritmo foi concebido para permitir a aterragem da plataforma no local desejado, sob certas condições atmosféricas. Como suporte ao sistema de controlo, é efetuada uma análise do modelo quanto à sua esta- bilidade, controlabilidade e observabilidade. No fim, é realizada uma optimização de ganhos de controlo recorrendo à técnica da teoria de controlo pole placement. Durante o desenvolvimento do trabalho são ilustrados alguns resultados de simulação que foram obtidos recorrendo ao programa MATLAB.

i ii Abstract

This thesis is part of the project STRAPLEX (STRAtospheric PLatform EXperiment), which results from a partnership between the Faculty of Engineering of the University of Porto and the European Spatial Agency (ESA). This project is constituted by a platform that allows the scien- tiﬁc community to send experiments for educational purpose into the stratosphere, using balloons ﬁlled with helium. Due to the extreme conditions in the stratosphere, the balloon bursts and the platform begins to fall back into the atmosphere. This fall is stabilized by a round parachute. Since this type of parachute does not allow any kind of control, in this thesis is suggested the usage of a parafoil, instead of the round parachute. The main goal of this thesis is implementing a reliable control algorithm for this system. This work gives continuity to another one developed before, in which a mechanical structure for the control of the parafoil is proposed. This structure includes some suitable hardware that allows the implementation of the control algorithm. The downward movement of the platform is described by a reliable mathematical model which includes the various relative movements between each of the elements of the platform. This model also comprises the distinct types of control of the system, allowing the implementation of a control algorithm. Such algorithm was designed to allow the landing of the platform in the desired place, under some constrained atmospheric conditions. As a support to the control system, an analysis of the stability, controllability and observability of the mathematical model is made. At the end, an optimization of the control gains is developed by using the pole placement technique. Along the thesis, some simulation results are shown, which were obtained with the software MATLAB.

iii iv Agradecimentos

Ao meu orientador, Professor Sérgio Reis Cunha, pela sua constante motivação, ajuda e dedicação que sempre demonstrou ao longo de todo este trabalho. Aos meus amigos por todos os momentos, aprendizagens e experiências que me propor- cionaram. Em especial aos meus amigos de sempre porque sem dúvida contribuíram para a pessoa que sou hoje. Estou extremamente grato aos meus colegas e amigos Diogo Pernes, Hugo Cruz e Luís Pires pelo companheirismo e amizade ao longo do meu percurso académico. À minha família por serem sempre um pilar na minha vida. Em especial aos meus pais e irmãos pelo amor e incansável apoio que sempre demonstraram. À minha namorada, Joana, pelo acompanhamento, paciência e amor prestado durante estes últimos 5 anos.

João Luís Granja da Costa

v vi “The true sign of intelligence is not knowledge but imagination”

Albert Einstein

vii viii Contents

1 Introduction1 1.1 Motivation ...... 1 1.2 Objectives ...... 1 1.3 Thesis Outline ...... 2 1.4 Website ...... 2

2 STRAPLEX3 2.1 STRAPLEX project ...... 3 2.2 Actual status ...... 5 2.3 The STRAPLEX project’s next step ...... 7 2.4 State of the art ...... 8

3 System Overview 13 3.1 System Components ...... 13 3.2 Parafoil ...... 14 3.2.1 Dimensions ...... 15 3.2.2 Features ...... 16 3.3 Drone ...... 18 3.3.1 Dimensions ...... 18 3.3.2 Features ...... 19 3.4 Capsule ...... 19 3.4.1 Dimensions ...... 20 3.4.2 Features ...... 21 3.5 Transponder ...... 21 3.6 Control Input ...... 22 3.6.1 Main Lines Control ...... 22 3.7 Flight Stages ...... 24

4 Mathematical Model 25 4.1 Model Description ...... 25 4.1.1 Vector Notation and Frames ...... 29 4.1.2 Rotation Matrices ...... 30 4.2 Position Points and Vectors ...... 33 4.2.1 Position Points ...... 33 4.2.2 Position Vectors ...... 34 4.3 System Kinematics ...... 35 4.3.1 Rotation Kinematics ...... 35 4.3.2 Position Kinematics ...... 41

ix x CONTENTS

4.4 System Dynamics ...... 45 4.4.1 Forces ...... 46 4.4.2 Moments ...... 52 4.5 Equations of Motion ...... 56 4.6 Simulation Results ...... 58

5 Control Algorithm 67 5.1 Horizontal Control ...... 67 5.1.1 Heading ...... 67 5.1.2 Course ...... 68 5.1.3 One Point Localizer and Course or Two Points Localizer ...... 69 5.2 Vertical Control ...... 71 5.3 Spiral Mode ...... 72

6 Stability Analysis 77 6.1 Linear Model ...... 77 6.1.1 Lyapunov’s Linearization Method ...... 78 6.2 Stability Analysis ...... 79 6.2.1 Stability ...... 79 6.2.2 Controllability ...... 79 6.2.3 Observability ...... 80 6.3 Pole Placement ...... 80 6.4 Graphic User Interface ...... 80

7 Final Remarks 83 7.1 Conclusions ...... 83 7.2 Future Work ...... 84

A Appendix 85 A.1 Equation of Coriolis ...... 85 A.2 Poisson’s Kinematical Equation ...... 85 A.3 Derivative of a Vector ...... 86 A.4 Drone Rotation Kinematics ...... 88 A.5 Capsule Rotation Kinematics ...... 89 A.6 Transponder Rotation Kinematics ...... 89 A.7 Drone Position Kinematics ...... 90 A.8 Capsule Position Kinematics ...... 93 A.9 Transponder Position Kinematics ...... 95 A.10 Apparent Force ...... 96 A.11 Matrix M and B ...... 97 A.11.1 Matrix M ...... 97 A.11.2 Matrix N ...... 98 A.12 Matrix A and B ...... 99

References 101 List of Figures

2.1 Launch of STRAPLEX ...... 4 2.2 STX center ...... 7

3.1 STRAPLEX components ...... 14 3.2 Parafoil ...... 14 3.3 Aerodynamic angles (α and β) for an Airplane [1]...... 17 3.4 View of the Drone ...... 18 3.5 Capsule ...... 20 3.6 Transponder ...... 21 3.7 Drone displacement ...... 23 3.8 Flight Stages ...... 24

4.1 Platform front view ...... 26 4.2 Roll, Pitch and Yaw Directions for an Airplane ...... 27 4.3 Three dimensional Trajectory ...... 60 4.4 Kinematics of the Canopy ...... 61 4.5 Kinematics of the Drone ...... 62 4.6 Kinematics of the Capsule ...... 62 4.7 Kinematics of the Transponder ...... 63 4.8 Kinematics of the Capsule in a Spiral Motion ...... 63 4.9 Trajectory of the Platform in a Spiral Motion ...... 64 4.10 Kinematics of the Capsule for a launching on 35km ...... 64 4.11 Trajectory of the platform launched a high altitude ...... 65

5.1 One Point Localizer and Course [2]...... 69 5.2 Horizontal Control - One Point Localizer and Course ...... 71 5.3 Vertical Control ...... 73 5.4 Spiral Region Decision ...... 74 5.5 Block Diagram of the Spiral Control Algorithm ...... 75

6.1 Graphic User Interface ...... 81

xi xii LIST OF FIGURES List of Tables

3.1 Parafoil Dimensions ...... 16 3.2 Drone physical features ...... 19 3.3 Capsule physical features ...... 20 3.4 Transponder physic features ...... 22

4.1 Position Vectors Components ...... 58 4.2 Aerodynamic Coefﬁcients ...... 59

6.1 Equilibrium Point ...... 79

xiii xiv LIST OF TABLES Abreviaturas e Símbolos

AR Aspect Ratio AHRS Attitude and Heading Reference System CD Convergence Distance CPU Central Processing Unit DOF Degrees Of Freedom DTMF Dual-Tone Multi-Frequency ESA European Spacial Agency GFSK Gaussian Frequency Shift Keying GPS Global Positioning System GUI Graphic User Interface LTI Linear Time-Invariant PD Proportional and Derivative RF Radio Frequency STRAPLEX STRAtospheric PLatform EXperiment SVD Singular Value Decomposition TCAS Trafﬁc Collision Avoidance System

Chapter 1

Introduction

A parafoil is a light ﬂying vehicle and since its invention by Ms. Domina Jalbert in 1960, has been widely used in a variety of windsports and in space and military missions. The parafoil is a very stable paraglider with a large manoeuvrability. For those reasons it can be effectively controlled by an autonomous control system attached to the parafoil. The dynamic model of the parafoil system has been a topic of scientiﬁc research in order to reach a reliable and precise control algorithm. This work focuses on the dynamic analysis of the system along with the performance of a control algorithm for a system used on the STRAPLEX project. This chapter is split into four sections. In Section 1.1 it is described the motivation of this work. Section 1.2 characterizes in detail the main objectives of this work whereas an outline of this thesis is presented in Section 1.3. Finally, a description of the website support is detailed in Section 1.4.

1.1 Motivation

STRAPLEX project offers students the possibility to send experiments for educational purposes into the stratosphere. A controllable drone was developed by Mario Martins [2] to be incorporated on the project. That work provided improvements in performing a controllable descending ﬂight. Moreover, it opened a new perspective for further enhancements in control systems theory. A precision control algorithm can be designed to allow the missions to land on a desired land point, avoiding undesirable situations like the ones happened in some past tests. Along with the author’s enthusiasm in the control area, led the author to accept this exciting challenge.

1.2 Objectives

A parafoil along with the drone were employed on the STRAPLEX project to make the descending phase a controllable ﬂight. The drone had a strong hardware and software system together with a reliable communication architecture which allowed the execution of human commands from a ground station. Using a joystick, a user can fully maneuver the system and in this way control its

1 2 Introduction trajectory. The drone is also able to perform autonomous maneuvers allowing the control of the horizontal and vertical motion of the system. The main objective of this work is to develop an accurate and steady control algorithm to make the system able to steer itself to a desired trajectory. For that the dynamic motion of the whole system should be previously analysed. A mathematical model should be studied and modelled from the detail analysis of the system motion, deﬁned by the kinematic and dynamic equations. The control algorithm must be developed for the horizontal motion as well as the vertical motion to allow the control in all directions. Taken into account that the mathematical model is nonlinear, a linearization method should applied in order to analyse the stability of the model. Using the linear model by the pole placement method, the control system can be optimized. The ultimate target of the control system is the spiral approach strategy algorithm in order to, in certain wind conditions, be possible to make the system autonomously land on the launching point.

1.3 Thesis Outline

This thesis is organized in six chapters. On this one, Chapter1 it is described the motivation and main objectives of this work. In Chapter2, the STRAPLEX project is presented as well as its actual state. In that chapter it is also presented a background of the work. The third Chapter3 analyses the four components of the system and the two control types. Here, it is also presented a usual ﬂight of the STRAPLEX. A detailed mathematical model of the system is developed in Chapter4. The next Chapter5 is devoted to the detail of the control algorithm and at tuning of control loops for the system. Finally, Chapter6, after the linearization of the model, analyses the stability of the system . The conclusions and the future work are presented in Chapter7.

1.4 Website

Using the Wordpress blogging tool, a website was created as a support to this thesis. It contains a short description of the work developed as well as the STRAPLEX project. A short presentation of the mathematical model and the control algorithm is presented together with some simulation results. In Section "Reports", the weekly reports as well as the PDI report can be downloaded. Chapter 2

STRAPLEX

This chapter is organized as follows. Section 2.1 presents the STRAPLEX project and explains the several phases of a mission. The actual state of the project as well as the work developed by Mario Martins ([2]) are detailed in Section 2.2. Section 2.3 presents the next step for the STRAPLEX project. Finally Section 2.4 presents some existent scientiﬁc research about parafoil control.

2.1 STRAPLEX project

STRAPLEX (STRAtospheric PLatform EXperiment) is a programme by the University of Porto, Portugal in collaboration with the Education Projects Division of the European Space Agency (ESA) [3]. This programme began in 2005 and offers the scientific community the possibility to make experiences into the stratosphere. It makes it possible to do tests near to the specific space environment because the stratosphere has near-space conditions. The near-space conditions of the stratosphere are interpreted by the scientific community as the near vacuum conditions which are good conditions to make following experiments:

• Experiments related to Stratospheric Balloon design: Archimedes force, Balloon princi- ples, Ascent velocity, Parachute system, Helium utilisation, etc;

• Experiments related to the Atmosphere: Temperature environment, Pressure environment, Atmosphere density, Humidity, sound propagation, pollution, etc;

• Experiments related to radiation: Solar radiation ﬂux, solar energy, cosmic ray, etc;

• Experiments related to tele-detection: Albedo, colour photography, black and white photography, digital photography, video, data transmission, etc;

• Experiments related to biology;

• Landing systems;

• Detachable capsules (including speciﬁc localisation and recovery system).

3 4 STRAPLEX

Figure 2.1: Launch of STRAPLEX

The scientific community can then make experiments in a specific environment through STRAPLEX, in a low cost and flexible way. For that, the platform is essentially composed by five main objects: a capsule for payload accommodation, a parachute system, a helium balloon, a transponder and a cutdown system. The four first objects can be viewed on the Figure 2.1 which was taken on the last realized launch. The capsule is the main object that carries both the scientific experiment and the control system. It reaches a high altitude through means of the balloon filled with helium. Depending on the mass of the experiments, STRAPLEX can reach up to a remarkable 40 km altitude. The second system is a round parachute which is activated in the descending phase to stabilize the trajectory, decreasing significantly the sink rate. The cutdown system is used to safely separate the balloon from the platform, when it is no longer necessary. This enables the possibility to, in case of emer- gency, abort the mission. The STRAPLEX mission may be split into four distinct phases:

1. Launch;

2. Rising phase;

3. Descending phase;

4. Landing.

In the first phase, the balloon, made of a latex material, is filled with helium until its lower density makes the system go up. Thenceforth the platform attached to the balloon climbs to the stratosphere. During the rising phase, the air pressure decreases which leads to the helium expands until the latex reaches its rupture point, bursting the balloon. Depending of the platform mass, the balloon can climb up to 40 km altitude. After that, the platform starts the descending phase while the balloon is separated from the platform by the cutdown system. At such a high altitude the air mass is rarefied therefore the resistance force is very low and the platform can reach a 2.2 Actual status 5 very high sink rate (300 km/h). In order to reduce this velocity, a round parachute is inflated. Furthermore this parachute aides to stabilize the platform, but doesn’t allow an effective control of the trajectory. Finally the last phase, landing, is the most critical phase. It is strongly dependent on the wind conditions and until Mario Martins’ s work, there wasn’t any type of control. The main goal of this work was to add a control system to avoid an unwanted landing. Places like wooden areas and water bodies should be avoided to reduce the recovery operation time and the inherent risks. The solution developed by Mario is later analysed in Section 2.2. In all phases of the mission, the capsule is in constant communication with a fixed ground station through a redundant RF communications. Information such as air pressure, humidity, temperature (in and outside the capsule), position, velocity and other relevant information are measured by the capsule and sent to the ground station. On the ground, the station receives the flight information and stores it for future analysis. Moreover it can send some action for the capsule, e.g. the cutdown command. Through the performed measurements, the capsule can also calculate an estimated landing point. A mobile station receives this information and tracks the platform path in order to recover the platform as soon as it lands. The STRAPLEX launch is normally carried out in Évora. The moderate climate, low population rate and considerably flat region are good reasons for the success of the missions.

2.2 Actual status

As stated above, the descending phase is only aided by a round parachute which doesn’t allow any control of the trajectory. So the landing point is very influenced by wind conditions. The lack of control on these phases already led to some problems on the rescue operation in past missions. In one mission the platform transcended the border with Spain due to a strong west wind. It affected the Spanish traffic and complicated the recover operation. In another mission the platform sub- merged on the Alqueva dam, near Évora. This caused a large waste of time on the rescue operation and some equipment inside the main capsule to get damaged. In order to find a solution for this problem, the engineering student Mário Martins de Sousa ori- ented by the professor Sérgio Reis Cunha developed a system based on a controlled parafoil [?]. The control of the parafoil is assured by a special capsule, called Drone, which was built for this purpose. It was built over a strong mechanical structure and contains three servo-actuators attached to the lines of the parafoil. These actuators are able to change the shape of the wing like pilots do in windsport. The actuator’s position is managed by a control system supported by a navigation and measurement system. He implemented two types of flight modes: manual and autonomous. In the first one, the user on the ground station maneuvers the parachute using a joystick. The joystick movements are interpreted by the ground station software, named STX center, that sends the respective actions to Drone where the actuators act accordingly. This operation can be done with the platform in sight or using either the real time video of the capsule or the Google Earth interaction which was implemented on the STX center. In the autonomous mode, the control 6 STRAPLEX system directs the platform along a desired path resorting to an algorithm control implemented on the drone’s software. This algorithm can be split into four types:

• Heading;

• Course;

• One point localizer and course;

• Two points localizer.

These four types of control will be explained in detail in Chapter5. A series of tests of the parafoil-drone system were conducted to evaluate the functionality of the control system. In manual mode, the control system proved be a very maneuverable and responsive system. However in autonomous mode, the system didn’t reach the desired point. It was verified that some drawbacks greatly influence the landing point, mainly the wind effects. The navigation and measurement system inside of drone allow the measurement of relevant information which is used for control loop and further analysis. A GPS receiver and an attitude and heading reference system (AHRS) provide kinematic information. The first one provides information of the position and velocity on the three axis (North, East and Down). The AHRS system which is composed by a 3-axis accelerometer, magnetometer and a gyro, provides three orientation angles using a Kalman filter. Furthermore, the drone has a set of sensors to continuously track the external and internal parameters, such as temperature, humidity, pressure and power. The internal sensors have a significant role to ensuring the good processing and monitoring of the microcon- trollers inside the drone. A hand made pitot tube was built to calculate the airspeed. This method of air speed measurement employs a differential and an absolute pressure sensor . It relies on the simplified Bernoulli’s equation to define the airspeed expression in function of the differential pressure. All of these informations can be sent to the ground station by a strong communication protocol composed by redundant channels. These redundant channels are based on a GFSK, a 5DPSK and a DTMF modulation, so that if one of the system fails, a total blackout is avoided. Moreover it was implemented a more flexible communication method which allows the direct communication with the main capsule, with the drone, with the cutdown system and with the ground station. Moreover, an aviation transponder is installed on the platform to increase the air traffic control. With this device, the platform is continuously communicating with the air traffic controller or other nearby aircraft equipped with the traffic collision avoidance system (TCAS). The ground station has the main objective to monitor the platform during the flight. It is essentially composed by a computer which gathers the received flight information and transmits the control action through transceivers. Its interface with the operator is a MATLAB graphic user interface (GUI) named STX center, visible in the Figure 2.2. In the left part of this interface it is possible to see the most important flight information (coordinates, attitude velocity, course, etc), GPS information and some sensor status. The wind 2.3 The STRAPLEX project’s next step 7

Figure 2.2: STX center

forecast, some Drone options and the actuators position are shown in the center part. The right side is responsible for the autonomous control part where it is possible to download or upload important parameters to the control (heading/course, sink rate, etc). The last part is constituted by buttons which allow the user to see the platform trajectory on Google Earth and choose the operating control mode. It should be noted that Mario also improved the cutdown system making it more reliable and efﬁcient using a mechanical approach instead a pyrotechnic method. The cutdown system has the function of separating the helium balloon from the capsules. This device has a crucial function on the experiment’s success because if the balloon doesn’t release the capsule, during the descending phase, the balloon, already busted, will compromise the parafoil control. After some tests, it was proved that the new cutdown system successfully played its function.

2.3 The STRAPLEX project’s next step

The controlled parafoil developed by Mario provided to be a reliable solution with great stability and maneuverability. However during the ﬂight tests the parafoil-drone system didn’t reach the default destination because of the wing effects and, on some case, the double pendulum effect which isn’t taken into account on the control system. In order to overcome these issues, and therefore heading towards a desired path, a better control algorithm may be implemented. For example, an algorithm capable of adjusting the internal control gain based on wind effect and some other system features. Moreover a complete control algorithm may be achieved for the descending and landing phase, for example through a spiral motion. This trajectory would be chosen from a database depending on the wind direction and the desired landing point selected by 8 STRAPLEX the user. At the end, a full stratospheric ﬂight should be conducted in order to fully validate the control system.

2.4 State of the art

Since the appearance of the ram-air parafoils many researches and studies have been carried out proving its great advantages in autonomous ﬂights such as the stability, controllability and maneuverability. For a dynamic motion analysis of the parafoil system, the aerodynamic coefﬁcients

(lift CL, drag CD) should be known and therefore estimated. The articles [4],[5],[6] analyse these coefficients at small size models of the canopy up to 300 ft2 and at low-aspect ratio parafoil up to 3.0. At [7] the maximum obtained lift-drag ratios of the wings varied from about 1.9 to 2.7 and the maximum obtained lift coefficients ranged from 0.9 to 1.1. Moreover Burk analysed the canopy stability and concluded that the tested parafoils were statically longitudinally stable over the entire test angle-off-attack range of 0o to 70o. Nikolaides presented results of the aerodynamic coefficients and of the velocity in various direction obtained from ascending flights and manned jumps from aircraft tests [8]. The results are showed graphically or in tables for various aspect ratio (1.0-3.0). A comparison of the predicted flight performance of the parafoil with the measured flight performance of the parafoil, as obtained from ascending flight and glide tests and from jump tests, has been made concluding that the agreement between the predicted performance and the measured performance was good. Lingard discussed the performance and design of ram-air parachutes for the Precision Aerial Delivery System (PADS) [9]. He made a briefly general description of the ram-air parachute and analysed the aerodynamic characteristics of ram-air wings. The theoretical drag and lift coefficients and L/D ratio are plotted, versus incident angle for aspect ratios 2.0-4.0. Analysing the previous results it should be noted that the lift curve slope increases with increasing aspect ratio, but the drag coefficient at a given incidence varies little with the aspect ratio. In relation to the L/D ratio it improves with the increasing of the aspect ratio. A comparison of the experimental and theoretical drag and lift coefficients and L/D ratio is showed for the aspect ratio 3.0. Some little differences are observed mainly on the lift coefficient results where upper 10 degree of the incident angle, the experimental and theoretical plots are very different. For the analysis of the aerodynamic characteristics ram-air parachute, the effect of line length is verified on wing lift coefficient and on L/D ratio. He also presented the forces equation on the system horizontally and vertically in order to analyse the ram-air parachute flight performance. Its longitudinal static stability is analysed showing the effect of the trailing edge deflection on the pitching moment, drag and lift coefficients, L/D ratio, attitude and flight velocity for the standard system. The response of ram air parachute to trailing edge deflection measuring the attitude and flight velocity is plotted in way to analyse the longitudinal dynamics of ram-air parachutes for different mass ratio. The effect of a 7.5m step tail gust on the standard large system is also shown. A more complex mathematical model of the paragliger system is presented on the articles described below. The models mainly differ on the number of Degrees Of Freedom (DOF). Some 2.4 State of the art 9 models consider the paraglider-payload as a rigid system and therefore their dynamic analyse concern on the center of mass. Others consider the relative motion between the paraglider and payload, increasing the DoF number. The general modelling approach is to determine the system kinematics and dynamic equations. In the first one is present the position, velocity and acceleration equations of the system, while in the second it is determined the force and moment equations of the system. Toglia et al. [10] presents two models, one with nine DOF and another with six DOF. On the first one, she takes account the effect of the payload twisting and therefore uses three DOF to characterize the inertial position of the joint point and six DOF to describe the parafoil and payload attitude motion, using the three Euler angles. The forces applied on the system are: the aerodynamic force, the weight force and the reaction exerted at the joint point to the payload and the aerodynamic, the weight, the reaction and the apparent force to the parafoil. On the other model the relative motion is neglected and the analyse is made only on the global center of mass, being the inertial position and the attitude motion the six DOF. On this case the forces (total weight force, both aerodynamic forces and the apparent force) are applied on the center of mass. On both models, the control is applied on the flaps deflection with the symmetrical term δs and the asymmetrical term δa which is given by the differential flaps deflection. After setting the kinematics and dynamic equations, it is derived the global system motion equation with the effect of the flap deflection. Using the MATLAB tool, the two models were simulated to a free dynamic trajectory and a spiral motion, analysing the flap deflection effect and the payload twist effect (this only on the nine DOF model). Through the results we can concluded that the six DOF model presents a delay in turning to the spiral motion and has less oscillations due to the lack presence of relative motion caused by the payload influence. A control algorithm for autonomous paraglider was been proposed by [11]. The first step was to modulate the kinematic and dynamic equations of the paraglider and payload system considering six DOF (the relative motion of the parafoil and the payload were neglected) and the flap deflection as the control input. Really this model is similar of the six DOF model in [10]. In order to reduce the model complexity, a simplified model was proposed, neglecting some features of the system (apparent force, payload drag, etc). Through the dynamic equation of the simplified model, an equilibrium point is founded for a linear path in the XY plane and a input-output feedback control is designed. In way to test the performance of the control input, two path following task (a pure line and a polygonal path) are simulated. On the second path, which represents more interests for us, the system converges to the reference path for each segment in about 50 s. The stabilization of the system along a polygonal path was achieved using lateral directional control input within acceptable real values. A different point of view of a powered paraglider system is described on [12]. Beyond the payload has a propeller motor, the system dynamic equations are determined on the basis of the state variables. The system is analysed accounting the relative pitching and twisting motion of the payload which represent two DOF that more six DOF of the parafoil characteristic totals eight DOF of the system. Through the DOF of the system it was defined twelve state of variables: three 10 STRAPLEX canopy velocities, 3 canopy angular velocities, payload pitch and yaw angular velocities relative to the canopy, payload pitch and yaw angles relative to the canopy and canopy roll and pitch angles. Regarding the control variables, we have the propelling force, the symmetric and the asymmetric brake deflection of the canopy. A nonlinear state equation is achieved taking account the weight, aerodynamic, inertial and cable tensions forces. In order to simply the model, a state equation linearization is made and evaluated its effect through numerical simulation results applying an individual impulse of the control input. Gathering the output results of the simulations, it was acceptable conclude that the linearization effect is minimum. Watanabe and Ochi also implemented a state feedback control system using a Kalman filter for enhance the damping characteristics of the canopy and the payload oscillation due to wind disturbance. For this it is necessary an observer which requires the system to be observable and controllable. The controllability and the observability were proved separating the linear model on the longitudinal and lateral-directional models. After that a closed-loop system with a state feedback system and wind disturbance was described. Slegers and Costello analysed the effect of small brake deflections parafoil-payload on the directional control [13]. They presented the combined system of the parafoil canopy and payload modelled with nine DOF including three inertial position components of the joint as well as the three Euler orientation angles of the canopy and the payload. After the determination of the kinematic and dynamic equations for the payload and parafoil, the system of equations is solved using LU decomposition and the equations of motion are numerically integrated using a fourth- order Runge–Kutta algorithm to generate the trajectory of the system from its point of release. Although this procedure was omitted. Simulations under different conditions are performed so that the performance of the controllable parafoil and payload system can be evaluated. Simulating the effect of small brake deflections parafoil-payload, it was observed that the parafoil and payload systems exhibit two basic modes of directional control: skid and roll steering. The relative pitching and yawing motion of a payload with respect to a parafoil are studied using a eight DOF model for the parafoil-payload system [14]. The eight DOF included the three inertial position components of the joint as well as the three Euler orientation angles of the parafoil system and two Euler orientation angles of the payload with respect to the canopy. The kinematic and dynamic equations for the parafoil and payload systems were determined. Using estimated aerodynamic and apparent mass coefficients, it is presented the response to a constant brake deflection observing the angular rates and the system yaw, roll and pitch angles. It was shown that relative payload motion had little effect on the predicted ground track. However, using a turn rate controller common in precision placement algorithms it was demonstrated that relative yawing motion of the payload can result in persistent oscillations of the system. These oscillations can be eliminated by reduction of feedback gains, but the resulting tracking performance was poor. A different approach of the model development is presented in [15]. The article addresses a six DOF model of a low-aspect ratio controllable parafoil-based delivery system, where the model was derived using general equations of fluid dynamics. The simulation results are obtained recording through the MATLAB tool. In the first step it is simulated the effect of the angle of attack on the lift, drag and side-force coefficients and on the rolling moment, pitch moment, and yawing 2.4 State of the art 11 moment coefficients for some aspect ratios and some symmetrical flaps deflections. Next the model response of control inputs is presented, illustrating the longitudinal response. A parameter identification technique including employment of the multicriteria optimization and zero-order Hooke-Jeeves method was applied to tune the initial aerodynamic dependences and apparent mass terms as well. A comparisons of the flight test data with the tuned and non-tuned model are presented and discussed. Two mathematical models of the payload-parachute system are presented and detailed in [16]. The payload is considered as a rigid body with six DOF and the parachute canopy acts as a rigid body and they are connected by a single riser. The system of differential equations that describes the parachute-payload system can be broken into sections, six rigid body equations of motion, four quaternion equations, three position equations, and the velocity and displacement of the parachute. In this way the most simplest model is based on euler angles orientation beside the other is based on quaternions. A simulation software is used to observe the differences between these two models on the trajectory profile. The major discrepancy between the two models was the level of dynamic stability of the modal response. Some differences can also been observed between the second model simulation results and the flight tests. An identification problem consist to find the dimension of the system and the space state system matrices (A, B, C, D), up to similarity transforms. As mentioned in [17] to resolve this problem we can have two different type of data: a sequence of impulse responses of a discrete- time LTI system or an input-output data. In parafoil-payload system, generally we have only input- output data by sensors presented on the payload system. So, the identification problem for parafoil- payload system is through input-output data to identify the dimension of the system and the system matrices (A, B, C, D), up to similarity transforms. In order to solve this problem, Katayama presented two identification method. On both methods it is utilized the LQ decomposition and the SVD theory which also are explained in [17]. After method description, it is summarized an algorithm with steps to resolve the problem. Some examples where the algorithms are applied are illustrated. The system identification results for a commercial powered parafoil vehicle are described by Valasek and Hur [4]. They analysed the controllability and observability of the system and applied an observer/Kalman filter identification method for the system identification. Initially Markov parameters and observer Markov parameters are developed. These parameters are very useful to identify mathematical models, thus with them we can obtain the Hankel matrix and through this matrix it is possible to find the matrices (A, B, C, D). Identification results are showed and it is made a comparison between the identification model results and the nonlinear model simulations. The results demonstrated that the identification method can identify the dynamic system effectively and accurately. A more complex and realistic identification technique is presented in [5]. Unlike the previous identification technique, which is base on one single criterion, Yakimenko et al. studied a multicriteria parametrical identification technique. In addition to by their nature, applied identification problems are multicriteria problems, there are many several groups of parameters, with different 12 STRAPLEX nature, which should be identified. Therefore several adequacy criteria are suggested to be used. The identification problem is studied and differences between these two identification technique are showed. The multicriteria identification is developed from the multicriteria optimization and its results are discussed. The parafoil trajectories are showed with respect to the real drop and to different criteria. The criteria numbers are two describing the closeness of the horizontal and vertical projections of the trajectories, and three relating to the adequacy of the natural eigenvalues (power spectrum) for all channels (roll, pitch and yaw). Analysing the results it was concluded that not much difference was observed between the several criteria and the real drop. In order to improve the parafoil-payload system control, a model predictive control strategy is described [6]. The optimal input which minimize the cost function is presented using a discrete system described in state-space form. Identification of aerodynamic coefficients is performed using a recursive weighted least-squares method. The model predictive control strategy is described for a simplified six DOF model. Three autonomous flight tests showed that model predictive control is an effective way to control autonomously the trajectory of a parafoil-payload system. Chapter 3

System Overview

In this chapter it is presented an overview of the system which will be later analysed in a mathematical model design. Firstly the several components of the system that play an important role during the controllable descending phase are listed in Section 3.1. Each Section 3.2, 3.3, 3.4 and 3.5 describes in detail each component, beginning with the Parafoil, followed by the Drone, the Capsule and at the end the Transponder. In each of the previous section, it is explained the physical dimensions of the object and its main features during the ﬂight phase. Section 3.6 presents the control methods and their features as well their functionalities. Finally, the descending ﬂight is described in Section 3.7.

3.1 System Components

The descending phase can be divided in two distinct phases: uncontrollable and controllable phase. In the first one, due to the characteristics of the stratosphere where the air is rarefied, the round parachute is opened to decrease the sink rate. This type of parachute doesn’t allow any type of control, hence the phase type. The second phase begins when the platform reaches stable air mass. Here it is easier to control the trajectory so that the parachute is released and the controller parafoil is inflated. A controllable trajectory can now be followed manoeuvring the parafoil. In this phase the platform is composed by four important devices as it is possible to see in the Figure 3.1. The drone is attached, through a set of lines, to the inferior surface of the wing of the parafoil, also called canopy. The capsule is connected to the drone through three strong lines in the similar manner as the transponder is connected to the capsule. These connections are visible on the Figure 3.1. For reasons of simplification, the three connections will be treated as just one connection.

13 14 System Overview

Figure 3.1: STRAPLEX components

3.2 Parafoil

Currently there are many types of parachutes which are very used in windsports and military missions. One of them is the device called the ram-air parachute or parafoil. It was designed by Ms. Domina Jalbert in 1960 and proven to be a very stable device. It has a rectangular platform with its leading edge opened and the trailing edge closed, so that ram air pressure enters and inﬂates the canopy, while it is moving through the air. In this work it is used a parafoil which was adapted from a common recreational power kite (Figure 3.2).

Figure 3.2: Parafoil 3.2 Parafoil 15

3.2.1 Dimensions

The parafoil is constituted by a canopy and by a set of lines which connects the canopy with the drone. The canopy is considered to have a fixed shape and it should always be completely inflated during the descending phase. It has an ellipsoidal shape with a plane of symmetric and is made of a thin fabric composed of small sections called cells with no rigid members. The leading edge has cells opening, with 7 centimetres thickness, to allow the air to flow inside while the trailing edge is closed to keep the air inside and maintain the pressure for keeping the canopy inflated. It is very important that the air pressure inside the canopy will be high enough because otherwise there is the danger of the canopy collapsing. The connection between the canopy with the drone is essentially supported by the two main lines while the lines which are connected to the trailing edge of the canopy on both sides are the breaks. The two main lines have a cascade configuration and they are attached to the canopy in specific points in order to maintain the canopy inflated. Each main line is divided in seven lines that each composed by six smaller lines. The seven lines are grouped and tied in specific points, named canopy rotation points because during the flight, as it was possible to see in previous tests, the canopy has a rotation motion that resolves around these points. The breaks play a more passive role, only activated for control purposes. The functions of the main lines and breaks will be explained in detail in Section 3.6. In order to avoid the interlacing of the main lines, a wooden beam with 34.5 centimetres wide is used. The length and the position of the beam was analysed in [2] in order to decrease the arc of the canopy and insert an anti twist force maintaining the parafoil controllability. Taking into account the lines and wooden beam weights, the total weight of the parafoil is about 700 grams. The straight line which joins the leading and trailing edge of a canopy is called chord line, which for this canopy is 90 centimetres long. The span is the distance in a straight line between the two sides of the canopy. The span of this canopy is 3.3 meters. The canopy area is calculated as the product of the span times the chord which equals 3 square meters. Considering the current mass of the entire system is about 9 kg, the expected wing loading is 3 kg/m2. The wing loading is a very important parameter on the canopy choice since it defines the sink rate. The main lines are attached under the lower surface of the canopy so that it has a lateral camber. The height of the arc in the mid point is approximately 75 centimetres and it is a parameter with significant influence on the canopy’s aerodynamic. Another typical parameter of a canopy is the aspect ratio (AR) which is defined as the ratio of its length to its breadth. This parameter is a typical parameter on the decision of the appropriate canopy for a windsport. The medium value is approximately equal to 4, which is the value of this canopy [18]. The table 3.1 summarizes the physical features of the parafoil. 16 System Overview

Table 3.1: Parafoil Dimensions

Parameter Value Units Canopy Length 3.3 m Maximum wide 0.9 m Minimum wide 0.6 m Maximum thickness 0.07 m Area 3 m2 Main lines Length 3.2 m Beam Length 0.345 m

3.2.2 Features

This parafoil is the main control component of the platform. It proved to have high glide capability and controllability as well as ability to travel large distances with a payload. It proved to be able to manoeuvre the drone along a predefined path and to approach the target landing point. The realized tests in [2] demonstrated that the parafoil-drone system is able to achieve a forward velocity of 18 km/h and a down velocity of 10 km/h. This gives an approximately velocity ratio of 2:1. In the aerospace field, this parameter is normally named glide slope and it is a very important parameter because it defines the glide performance of an aircraft/parafoil. The glide slope is normally calculated by the lift force to drag force ratio. The lift force is defined to be perpendicular in the vertical plane to the vector of freestream velocity and is derived of the differential pressure between the top and the bottom surfaces. The drag force is the resistance force which act on the object when it is moving through the air. The direction of this force is the same direction as the vector of freestream velocity. The high controllability of the parafoil obtained because of its light weight and, more importantly, its small mass-to-volume ratio. Indeed, due to small mass- to-volume ratio, it is easier to change the configuration of the canopy and consequently the flight trajectory. The parafoil has the great advantage to control both horizontal motion and vertical motion. If the right side of the canopy is pulled down, the parafoil will turn to the right. With same analogy, a left curve can be done by pulling down the left side of the canopy. On the other hand the vertical control can be achieved by pulling down the trailing edge. For example, if the trailing edge is deflected, pulling down both brakes, the drag force increases, decreasing the glide slope. Managing these two types of control, the parafoil is able to follow a desired path. However this is only possible on certain wind conditions. If the wind velocity exceeds the parafoil airspeed, there is no chance of forward progression. In this way, the path control is very dependent of the wind effects. Mário made several flights with winds and verified that with winds stronger than 10 knots, the system is not able to progress forward in relation to the ground [2]. Moreover, it was verified that the parafoil has a natural behaviour to turn into the wind. This feature is very useful 3.2 Parafoil 17 because it prevents the parafoil from collapsing. A motion of an object through the air induces forces called aerodynamics forces. These forces are described mathematically in the model. Here follow the descriptions of a few aerodynamic terms that take part in any standard aerodynamic analysis..

3.2.2.1 Angle of Attack (α)

The angle of attack, usually denoted by α, is the vertical angle between the chord line vector and the vector of the freestream velocity 1. The angle of attack is positive downwards as seen in the Figure 3.3 for an airplane. This angle is a key point in what concerns the stability of the parafoil. On the one hand a parafoil ﬂying with a too high attack angle can collapse because the air can not enter the canopy decreasing its inside air pressure. On the other hand with a too negative attack angle the forces applied in the upper surface of the canopy can make the canopy deﬂate and therefore it can collapse.

3.2.2.2 Side Slip Angle (β)

When lateral motion is considered, there is another angle, named side slip angle (β), which is the angle between the chord line vector and the vector of the freestream velocity in the horizontal plane. A positive angle is shown in the Figure 3.3. The same deﬁnition applies to a parafoil.

Figure 3.3: Aerodynamic angles (α and β) for an Airplane [1]

1The freestream velocity is the relative velocity of a vehicle with respect to the wind. 18 System Overview

(a) Top view of Drone (b) Side view of Drone

Figure 3.4: View of the Drone

3.3 Drone

Drone was specially developed to perform control and navigation tasks on the platform. This device proved to be able to control a parafoil [2]. The ﬁgure 3.4 shows the exterior appearance of the drone, with a top and side view. The shape of drone was designed in detail to improve the aerodynamic features. Using the Solidworks tool, Mario designed a three-dimensional model of the drone taking into account aspects such as aerodynamics, impact robustness and the position of its center of mass. The tail was designed so that in motion it allows the drone to face the wind just like the parafoil. Moreover, the internal structure was built to support the constraint forces which will be applied in the drone during all the ﬂight phases.

3.3.1 Dimensions

The drone can be approximated to a parallelepiped shape. Due to the aerodynamic aspect it is larger and taller in the front face. This face a height of 34 centimetres and a width of 15.5 centimetres while the back face is 20 centimetres high and 9.2 centimetres wide. The center of mass was designed to be approximately 20 centimetres below the upper surface, 17 centimetres from the front part and in the middle of the width. These positions were designed to reduce the effort of the mechanical structure. Two important points are the ones where the main lines leave the control structure. The ratio of the distance between them and the main lines’ length should be such that the system will be as stable and controllable as possible. For a line length of 3.2 meters, the optimum distance between the points was calculated to be 2 centimetres. Another important 3.4 Capsule 19 feature of the drone is the cover material. Among others, it should dissipate the landing impact and isolate the outside temperature. For that, Styrofoam was used and proved to be an effective cover material. This is a light material so that the drone with all its components weighs about 1770 grams. The most important physical features of the drone are summarized in the Table 3.2. The area considered is the area of the front face of the drone which is the face in contact with the air. Table 3.2: Drone physical features

Parameter Value Units Length 0.365 m Maximum wide 0.155 m Minimum wide 0.92 m Maximum height 0.34 m Maximum height 0.20 m Area 0.1 m2 Mass 1.77 Kg

3.3.2 Features

The drone has a fundamental role during the descending phase. Besides the transmission tasks with the central ground station, it is responsible for doing important measurements for the control algorithm. Some of the most important data measured by the drone are its attitude, its position and its linear velocity. To achieve that, the drone uses an AHRS along with a GPS receiver. Through a pitot tube, the drone can also measure the airspeed. However this measurement has one limitation because the pitot tube can only measure in one direction (the drone’s direction). A support hardware and software implemented in the drone allow measurements with a rate of 4 measurements per second. The control actions are performed by servo-actuators. A differential servo-actuator is used to control the main lines with a limited response time. In addition to that, two smaller servo-actuator control the tailing edge of the canopy by the break lines. Each servo-actuator is connected to a micro-controller which along with a CPU processes the control action and drives the servos- actuators using pulse-width modulation.

3.4 Capsule

As it was stated in Chapter2, the main goal of STRAPLEX project is to offer the scientific community the possibility to make experiments into the stratosphere. These experiments will be carried inside of the main capsule, in this thesis it will be called capsule indistinctly. This special device was carefully built to assemble a set of electronic equipment that supports the experiments. The figure 3.5 shows the outside view of the capsule. The capsule has a hexagonal shape with the semi-hemispherical top surface and an antenna installed on the top, as seen in the previous figure. The top surface has a semi-hemispherical shape 20 System Overview

Figure 3.5: Capsule

and is covered by a special material in order to minimize the noise signal of the communication signals.

3.4.1 Dimensions

For theoretical results, the semi-hemispherical top surface and the antenna will be neglected, because its contribution to the capsule motion can be considered negligible. Each face of the hexagon is 20.5 centimetres long and 31.5 centimetres high. The contact area with the air can be approximated to 0.5 square meters. Such as the drone, this capsule is also covered by Styrofoam, so the total weight of the capsule is about 5 kg. The connection between the capsule and the drone is accomplished by three strong lines attached in three eye screws screwed in three speciﬁc points. Its length can vary but it’s usually 3 meters to maximize the stability of the relative motion between these two objects. The table 3.3 presents a summary of the capsule dimensions.

Table 3.3: Capsule physical features

Parameter Value Units Wide 0.41 m Length 0.41 m Height 0.315 m Contact Area 0.5 m2 Mass 5 Kg 3.5 Transponder 21

Figure 3.6: Transponder

3.4.2 Features

The internal structure of the capsule can be divided in four layers stacked with different functions. The top layer houses a set of antennas which provide communication with the base station and the others objects of the system. Below this layer are the experiments which will be performed in the Stratosphere. The next layer is the set of electronic equipment that supports the experiments. It consists on communication devices (receivers and transceivers), modems, data acquisition module and a processing unit. A special electronic device is a video camera installed on the faces which have a window. This camera ﬁlms the trajectory which the capsule makes during the ﬂight phases. The video can then be sent to the ground station so that it can be useful for the control tool in the manual mode which will be in Chapter5. The communication between the several components and between the several objects of the system and with the base station is based on a strong and redundant communication protocol. This minimizes the probability of a communication failure. The last layer houses the power unit to feed all electronic equipment.

3.5 Transponder

The last object of the system to analyse is the transponder. In the aerospace industry, this is a special electronic device incorporated in an aircraft that identifies the aircraft for the controller. Together with an altitude reporting equipment, they are significant elements for the safe operation in the national airspace system [19]. They enhance the air traffic control and collision avoidance system. With this same objective, the platform incorporates one transponder. It is installed into a small capsule (Figure 3.6) which has a structure similar to the main capsule. The transponder weighs 1.735 kilograms and has an approximated contact area of 0.1 square meters. The Table 3.4 shows the physical features of the transponder. Such as the connection between the capsule 22 System Overview and drone, also the transponder is connected to the capsule by three strong lines attached in three eye screws screwed in three specific points.

Table 3.4: Transponder physic features

Parameter Value Units Wide 0.32 m Length 0.32 m Height 0.15.8 m Contact Area 0.1 m2 Mass 1.735 Kg

3.6 Control Input

The majority research about parafoil control analyses the parafoil-payload system with one type of control, the breaks. They are two lines connected to two flaps located at the tail of the canopy, one on each side. This two lines allow an asymmetric and a symmetric break. When a symmetric break is applied, which corresponds to the two flaps deflected of the same angle, the lift and drag force increase, while the glide slope decreases. This control is very useful to control the sink rate of the system. If one flap is more deflected than other there is an asymmetric break which make the system turn to the side where the flap is more deflected. However, Mario Martins verified that when he applied an asymmetric break action, the parafoil turn rightly accomplish by a glide slope decreasing. This result can be unwanted and therefore a new control was designed using the two main lines. They allow lateral control with low influence in the glide slope.

3.6.1 Main Lines Control

The idea of the main lines control is to use the payload weight shift mechanism to control the parafoil orientation. This method is widely used for human skydivers in windsports. For that a differential servo-actuator can extend and/or shorten the main lines. The difference length of the main lines induces a lateral displacement of the drone with respect to the canopy. In this way, the weight of drone will be distributed in the main lines with different weight. Moreover as the lines should be always stretched, the canopy will deﬂated in order to the weight of drone will be distributed in the main lines in the same way. An example of a right displacement is shown on the Figure 3.7, where it was neglected the spacing between the main lines in the drone connection point. It should be note that it is only presented the lines (grey lines) which connects the wooden beam (brown beam) to the drone. 3.6 Control Input 23

Figure 3.7: Drone displacement

B is the beam length and L0 is the natural line length. It is important to note that this ﬁgure is an illustrative exhibition without the right objects dimensions. When a line displacement ∆L is performed (black lines), it induces a lateral displacement ∆D (dashed line) on the drone as it is possible to see in the Figure 3.7. There is also a vertical displacement on the drone but it is smaller than the lateral displacement and its effect is neglected. Using trigonometric equations, it is possible to write ∆D as function of ∆L, L0 and B. Really, taking on H as the least distance between drone and the beam:

 B 2 2 2  − ∆D + H = (L0 − ∆L) 2 (3.1) B 2 2 2  2 + ∆D + H = (L0 + ∆L)

By eliminating term H, it becomes

B 2 B 2 + ∆D − − ∆D = (L + ∆L)2 − (L − ∆L)2 (3.2) 2 2 0 0

Solving for ∆D,

L0 ∆D = B ∆L (3.3) 2

Being L0 and B constant parameters, the drone lateral displacement is directly proportional to the main lines displacement ∆L. 24 System Overview

3.7 Flight Stages

The descending flight can be divided in two major phases: uncontrollable and controllable phase. The first one starts after the balloon being released. In this altitude the air is rarefied and therefore the platform can reach high sink rate and behaves very oscillatory. In order to stabilize that oscillation, a round parachute is opened. This type of parachute is very used in the aviation because it can easily open. However this parachute isn’t controllable and therefore there isn’t any way to control the trajectory. As the air mass becomes increasingly stable as the platform close to the ground, the down velocity decreases until reaches a stable value. At this moment, the round parachute is released and the parafoil is inflated. As it was state before it has the great advantage to control the platform trajectory, which probably was affected by the wind effects. Applying a guidance strategy, it is possible, in certain wind conditions, to manage the platform toward the landing target. The guidance strategy can be divided in four distinct phases (Figure 3.8). In the first one, the control algorithm takes the system near to the land area following a desired direction. In the second phase, the platform describes a spiral motion in a specific area taking into account the wind effect. The loop number of the spiral should be that the platform reaches a specific altitude for the final approach. The third phase executes the final turn to the approach glide slope toward the landing point into the wind. The flight ends with the landing, controlling the glide slope for that the impact velocity is minimum.

Figure 3.8: Flight Stages Chapter 4

Mathematical Model

In this chapter, the platform motion in a descending phase is modelled in a mathematical model. This model is non linear and describes the system dynamic behaviour. The system is composed by four bodies and contains three types of control input, both analysed in the chapter3. Section 4.1 describes the dynamic model of the system and some assumptions performed. It will be also presented the vector notation used during the model modelling as the several frames and the several rotation matrices to switch among frame. The position points as well the position vectors are presented in Section 4.2. Section 4.3 analyses the kinematics of each component of the system. After the equations of motion are determined in Section System Dynamics 4.4.

4.1 Model Description

A schematic of the platform composed by four components (parafoil, drone, capsule and transponder) and their respective connection lines along with the control lines is shown in Figure 4.1. The system will be treated as a multi-rigid bodies with constraint forces of the lines between bodies (parafoil-drone, drone-capsule and capsule-transponder). In this way, the kinematics and dynamics of the system will be analysed separately for each component, taking in account the constraint forces and the existing relative motion between them. The nonlinear model will be obtained by the system equations of motion which will be organized as a set of simultaneous ﬁrst-order differential equations. Solving the equations for the derivatives, we can written symbolically the nonlinear equations as:

X˙ = f (X,U) (4.1)

where the state vector X is an (n × 1) column array of the n state variables, the control input vector U is an (m × 1) column array of the m control variables, and f is an array of nonlinear functions.

25 26 Mathematical Model

Figure 4.1: Platform front view

The state vector X as the name indicates describe the state of the system inside the vector U is composed by the control input which it will be used to perform the control algorithm. In the mechanical field, the state variables are normally defined as positions, orientation angles, linear and angular velocities. Where the positions and velocity vector are described respectively as x and u to forward, y and v to right and z and w to down. The three orientation angles are the roll (φ), theta (θ) and yaw (ψ) and its rates are respectively p, q, r. The three angles are normally the Euler Angles that describe a three dimensional orientation of a body and are typically denoted by: φ, θ and ψ. For the case of a plane, defining the longitudinal axis of a plane as the straight line drawn from the nose to the tail, the pitch angle is the angle made by the longitudinal axis and the ground plane. A positive quantity of this angle means that the nose is upper than the tail. A positive roll means a rotation of the plane about the longitudinal axis on the clockwise orientation. A horizontal rotation of the plane is quantified by the yaw angle, where a clockwise rotation means a positive angle. The directions of these angles can be interpreted as a right-handed rotation about the perpendicular axis of the plane, as shown on figure 4.2 for an airplane. 4.1 Model Description 27

Figure 4.2: Roll, Pitch and Yaw Directions for an Airplane

The number of the state variables is usually deﬁned by numbers of the degrees of freedom of a system plus its rates. This system will be modelled with 14 DOFs. Three DOFs to describe the three inertial position components of the canopy center of mass, three DOFs described the three Euler orientation angles of the canopy system, one DOF is used for the pitch orientation angles of the main lines with respect to the canopy system. The model uses more three DOFs for three Euler orientation angles of drone with respect to the main lines, two DOFs for the roll and pitch orientation angles of capsule with respect to drone and lastly two DOFs for the roll and pitch orientation angles of transponder with respect to capsule. In this way, the state vector of this system will be composed by:

I I C C D D CAP CAP X = [ RCMC/I, ΘC/I, VCMC/I, ωC/I,θL/C,qL/C, ΘD/L, ωD/L, ΘCAP/D, ωCAP/D,... T T T ... ΘT/CAP, ωT/CAP,4Control] (4.2) where

I T RCMC/I = xCMC/I,yCMC/I,zCMC/I I T ΘC/I = φC/I,θC/I,ψC/I C T VCMC/I = uCMC/I,vCMC/I,wCMC/I C T ωC/I = pC/I,qC/I,rC/I D T ΘD/L = φD/L,θD/L,ψD/L D T ωD/L = pD/L,qD/L,rD/L 28 Mathematical Model

CAP T ΘCAP/D = φCAP/D,θCAP/D CAP T ωCAP/D = pCAP/D,qCAP/D T T ΘT/CAP = φT/CAP,θT/CAP T T ωT/CAP = pT/CAP,qT/CAP 4Control = [4L,4ServoR,4ServoL]T

C Relative to the canopy body, the vector XCMC/I describes the three inertial position components C of the canopy center of mass, the vector ΘC/I defines the orientation of the canopy frame with C respect to the inertial frame, the vector VCMC/I describes the three linear velocity components of C the canopy center of mass and the the vector ωC/I describes the three angular velocity components of the canopy. The state variable θL/C and qL/C describe the orientation and its rate of the line frame D with respect to the canopy frame. Relative to drone, the vector ΘD/L defines the orientation D of the drone frame with respect to the line frame while the vector ωD/L describes the three CAP angular velocity components of drone with respect to the line frame. The vectors ΘCAP/D and CAP ωCAP/D describe the two relative Euler Angles of the capsule frame with respect to drone frame and the relative angular velocity of capsule with respect to drone, respectively. In the same way, T T the vectors ΘT/CAP and ωT/CAP describe the two relative Euler Angles of the transponder frame with respect to the capsule frame and the relative angular velocity of transponder with respect to capsule, respectively. The state vector 4Control regards to the control input. Although the control input isn’t a state vector, in this system, the control has a finite response time, in others words the velocity of the servo-actuators is limited. In order to describe this fact it is necessary to define more three state variables, one for each servo-actuator. Defining the three state variable

4L, 4ServoR and 4ServoL as the position of the respective servo-actuator and UL, USR and USL as the control input for these servo-actuator, the equations 4.3,4.4 and 4.5 describe the dynamic of these control.

4˙ L = KL (UL − 4L) (4.3)

4˙ ServoR = KSR (USR − 4ServoR) (4.4)

4˙ ServoL = KSL (USL − 4ServoL) (4.5) 4.1 Model Description 29

where KL, KSR and KSL are constants which define the velocity of the dynamic of the control. The larger value is, the faster the system will response to a control input. One can easily note that in the frequency domain, each equation will originate a pole situated in the open left-half plane of the C. The bigger the distance to the imaginary axis, the faster the control response will be. These state variables also have the great advantage to allow a quick analyse of the control input value. In the analysis of the motion of an aerospace vehicle, there is some aspects that should be considered. Indeed, in the modern aerospace industry, accurate simulations of high speed flight over large areas of the Earth’s surface requires an accurate model of the Earth’s shape, rotation and gravity as it is referred by Stevens and Lewis in [1]. However considering a parafoil’s flight cov- ering a small region of the Earth it is valid to neglect variation of Earth’s rotation and gravity and to assume a flat Earth. Another important parameter in the aerospace field is the air density which varies with the altitude. As the platform can start to down at 40 km, the variation of this parameter is taken into account.

4.1.1 Vector Notation and Frames

The motion of the bodies will be described by means of vectors in three dimensions and expressed in several frames, so it is important to choose a clear vector notation such as a description of the all frames which will be used. Using a similar notation applied in [1], it will be use the following notation:

• A right subscript will be used to designate the point of the vector. When it is used a "\ ", this means that the point is relative to an other point for a position vector or relative to a frame for a velocity or acceleration vector.

• A left superscript will specify a coordinate system. The vector is expressed in the coordinate system.

• A right superscript on a vector will specify the frame in which a derivative is taken, and a dot notation will indicate a derivative.

An example of the notation is:

A ˙B ~vC/I ≡ derivative taken in the frame B of the linear velocity vector v of the body C with respect to the inertial frame I. The resulting vector is expressed in the frame A.

The frames that will be used on the development of the equations of motion are followed presented:

• Inertial frame (I) - it is the reference frame and it will considered as the Earth frame using the north, east and down convention.

• Canopy frame (C) - is ﬁxed at the mass center of the canopy with the ~IC, J~C and K~ C axes illustrated in the Figure 4.1. 30 Mathematical Model

• Line frame (L) - is ﬁxed at the mass center of the canopy but with the orientation deﬁned by the orientation of the group of the two main lines.

• Drone frame (D) - is ﬁxed at the mass center of drone with the~ID, J~D and K~ D axes illustrated in the Figure ﬁg:Schematic.

• Capsule frame (CAP) - is ﬁxed at the mass center of capsule with the orientation with respect to the drone frame deﬁned by the orientation angles of capsule with respect to drone.

• Transponder frame (T) - is ﬁxed at the mass center of transponder with the orientation with respect to the capsule frame deﬁned by the orientation angles of transponder with respect to the capsule.

• Aerodynamic frame (A) - is ﬁxed to the canopy pressure center with its orientation deﬁned by the direction of the aerodynamic velocity.

The orientation of the aerodynamic velocity with respect to the canopy is deﬁned by the aerodynamic angles α and β. Where the aerodynamic angle α is called angle of attack and while β is the sideslip angle. They are the angles made by the chord line and the wind velocity vector as illustrated in the ﬁgure 3.3 for an airplane.

4.1.2 Rotation Matrices

A rotation matrix, also called a direction cosine matrix is a matrix that transforms the components of a vector from a frame to another frame. For example the rotation matrix TA/B allows to transform the components of a vector ~u from frame A to frame B.

B A ~u = TA/B ~u (4.6) where

 cos(ψ) sin(ψ) 0   TA/B = −sin(ψ) cos(ψ) 0 (4.7)   0 0 1

Where the frame B is the frame A rotated about the z-axis by a positive angle ψ. Some properties of this matrix are presented in [1] where one can highlight the following:

(i) Successive rotations can be described by the product of the individual rotation matrices;

(ii) Rotation matrices are orthogonal matrices; 4.1 Model Description 31

(iii) The determinant of a rotation matrix is unity.

From the properties of the rotation matrix, one can easily ﬁnd that:

−1 T TB/A = TA/B = TA/B (4.8)

The non-inertial frames (C, L, D, CAP, T) described above can be obtained relative to the inertial frame (I) by a three-dimensional coordinate rotations. These rotations can be built up as a sequence of plane rotations:

(1) Right-handed rotation about the z-axis (positive ψ)

(2) Right-handed rotation about the new y-axis (positive θ)

(3) Right-handed rotation about the new x-axis (positive φ)

Where each plane of rotations can be described by the following rotation matrices:

1 0 0    Tφ = 0 cos(φ) sin(φ) (4.9)   0 −sin(φ) cos(φ)

cos(θ) 0 −sin(θ)   Tθ =  0 1 0  (4.10)   sin(θ) 0 cos(θ)

 cos(ψ) sin(ψ) 0   Tψ = −sin(ψ) cos(ψ) 0 (4.11)   0 0 1

Following this sequence of plane rotations, the orientation of the canopy frame with respect to the inertial frame can be obtained by the three dimensional rotation matrix TI−C.

TI−C = TφC/I TθC/I TψC/I 32 Mathematical Model

    1 0 0 cos(θC/I) 0 −sin(θC/I) cos(ψC/I) sin(ψC/I) 0     TI−C = 0 cos(φ ) sin(φ ) 0 1 0 −sin(ψ ) cos(ψ ) 0  C/I C/I   C/I C/I  0 −sin(φC/I) cos(φC/I) sin(θC/I) 0 cos(θC/I) 0 0 1

Using the common shorthand notation for trigonometric functions (sin(α) ≡ sα , cos(α) ≡ cα and tan(α) ≡ tα ) The transformation matrix becomes:

  cθC/I cψC/I cθC/I sψC/I −sθC/I   TI−C = s s c − c s s s s + c c s c  (4.12)  φC/I θC/I ψC/I φC/I ψC/I φC/I θC/I ψC/I φC/I ψC/I φC/I θC/I 

cφC/I sθC/I cψC/I + sφC/I sψC/I cφC/I sθC/I sψC/I − sφC/I cψC/I cφC/I cθC/I

In the same way, the transformation matrices from the canopy to the line frame, from the line to the drone frame, from the drone to the capsule frame and from the capsule to the transponder frame are deﬁned in the equations 4.13,4.14,4.15,4.16and 4.17:

TC−L = TθL/C   cos(θL/C) 0 −sin(θL/C)   TC−L =  0 1 0  (4.13)   sin(θL/C) 0 cos(θL/C)

TL−D = TφD/L TθD/L TψD/L   cθD/L cψD/L cθD/L sψD/L −sθD/L   TL−D = s s c − c s s s s + c c s c  (4.14)  φD/L θD/L ψD/L φD/L ψD/L φD/L θD/L ψD/L φD/L ψD/L φD/L θD/L 

cφD/L sθD/L cψD/L + sφD/L sψD/L cφD/L sθD/L sψD/L − sφD/L cψD/L cφD/L cθD/L

TD−CAP = TφCAP/D TθCAP/D   cθCAP/D 0 −sθCAP/D   TD−CAP = s s c s c  (4.15)  φCAP/D θCAP/D φCAP/D φCAP/D θCAP/D 

cφCAP/D sθCAP/D −sφCAP/D cφCAP/D cθCAP/D 4.2 Position Points and Vectors 33

TCAP−T = TφT/CAP TθT/CAP   cθT/CAP 0 −sθT/CAP   TCAP−T = s s c s c  (4.16)  φT/CAP θT/CAP φT/CAP φT/CAP θT/CAP 

cφT/CAP sθT/CAP −sφT/CAP cφT/CAP cθT/CAP

TC−A = Tα Tβ  cos(α) 0 sin(α)cos(β) −sin(β) 0    TC−A =  0 1 0 sin(β) cos(β) 0    −sin(α) 0 cos(α) 0 0 1  cos(α)cos(β) −cos(α)sin(β) sin(α)   TC−A =  sin(β) cos(β) 0  (4.17)   −sin(α)cos(β) sin(α)sin(β) cos(α)

4.2 Position Points and Vectors

The position points and vectors that will be used in the model modelling will be presented using the notation described in the section 4.1. Some points presented can be view in the schematic 4.1.

4.2.1 Position Points

The points that will be used in the modelling of the mathematical model are:

• CMC - center of mass of the canopy/parafoil (the mass of the lines and of the wooden beam will be neglected);

• CPC - pressure center of the canopy;

• T1C - tension point on the canopy of the right main line;

• T2C - tension point on the canopy of the left main line;

• CMD - center of mass of drone;

• CPD - pressure center of drone;

• T1D - tension point on drone of the right main line;

• T2D - tension point on drone of the left main line;

• TCAPD - tension point on drone of the line which connects capsule to drone; 34 Mathematical Model

• CMCAP - center of mass of the capsule (it is also the tension points on capsule);

• CMT - center of mass of the transponder (it is also the tension point on transponder);

4.2.2 Position Vectors

Taking into account the points presented above, it will be deﬁned position vectors expressing in the respective frame.

C~ • RCPC/CMC - position vector of point CPC with respect to point CMC expressed in the canopy frame;

C~ • RT1C/CMC - position vector of point T1C with respect to point CMC expressed in the canopy frame;

C~ • RT2C/CMC - position vector of point T2C with respect to point CMC expressed in the canopy frame;

L~ • RT1D/T1C - position vector of point T1D with respect to point T1C expressed in the line frame;

L~ • RT2D/T2C - position vector of point T2D with respect to point T2C expressed in the line frame;

D~ • RCPD/CMD - position vector of point CPD with respect to point CMD expressed in the drone frame;

D~ • RT1D/CMD - position vector of point T1D with respect to point CMD expressed in the drone frame;

D~ • RT2D/CMD - position vector of point T2D with respect to point CMD expressed in the drone frame;

D~ • RTCAPD/CMD - position vector of point TCAPD with respect to point CMD expressed in the drone frame;

CAP~ • RCMCAP/TCAPD - position vector of point CMCAP with respect to point TCAPD expressed in the capsule frame;

T ~ • RCMT /CMCAP - position vector of point CMT with respect to point CMCAP expressed in the transponder frame;

In addition to these vectors, one can deﬁned others position vectors through the previous vectors. 4.3 System Kinematics 35

4.3 System Kinematics

In this section, it will be analysed the kinematic of the four bodies. This analysis can be divided in two major categories: rotation kinematics and position kinematics. In the first one, the orientation, the angular velocity and its derivative expressions of each body will be evaluated. In the position kinematics category, the position vector, the linear velocity and the linear acceleration of each body will be defined. All of the kinematic expressions will be expressed in the appropriate frame. In some cases, it will be used in parenthesis straight rotation matrices defined in Section 4.1.

4.3.1 Rotation Kinematics

The rotation kinematics is deﬁned by the orientation expression such as the angular velocity and the derivative of the angular velocity expressions of the body. It will be analysed the rotation kinematics of the four bodies, beginning with the canopy, followed by drone and capsule and ﬁnishing with the transponder. The variables will be always determined with respect to the inertial frame and expressed in the inertial frame except for the angular acceleration which will be determined with respect to the frame of the body under analysis for reasons which will be later perceived.

4.3.1.1 Canopy

• Euler Angles

The canopy Euler Angles are state variables ΘC/I = [φC/I;θC/I;ψC/I] which described the orientation of the canopy with respect to the inertial frame and are expressed in the inertial frame.

Rotating the inertial frame to the canopy frame through the rotational matrix TI−C, the Euler An- gles vector expressed in the canopy frame is given by:

C~ I~ ΘC/I = [TI−C] ΘC/I (4.18)

• Angular Velocity

The angular velocity expression is defined as the derivative of the Euler Angles vector. How- ever it is desired to express the angular velocity in the canopy frame and the state variable ΘC/I is expressed in the inertial frame. So to obtain the canopy angular velocity in order to the state variable ΘC/I it is necessary to multiply by rotational matrices presented in Section 4.1 in a specific sequence, which in the aerospace field is given by:

        pC/I φ˙C/I 0 0 C         ω~ C/I = q  =  0  + Tφ θ˙  + Tφ Tθ  0 [1]  C/I    C  C/I C C   rC/I 0 0 ψ˙C/I 36 Mathematical Model

Replacing the rotation matrices TφC and TθC by the equations 4.9 and 4.10, the previous equation becomes:

     pC/I 1 0 −sin(θC/I) φ˙C/I      q  = 0 cos(φ ) sin(φ )cos(θ )θ˙  (4.19)  C/I   C/I C/I C/I  C/I  rC/I 0 −sin(φC/I) cos(φC/I)cos(θC/I) ψ˙C/I

The Euler angles rates can be found using matrix inversion as

     φ˙C/I 1 sin(φC/I)tan(θC/I) cos(φC/I)tan(θC/I) pC/I      θ˙  = 0 cos(φ ) −sin(φ ) q  (4.20)  C/I   C/I C/I  C/I  ψ˙C/I 0 sin(φC/I)cos(θC/I) cos(φC/I)cos(θC/I) rC/I

• Derivative of the Angular Velocity

The derivative of the angular velocity for the canopy is equal to the derivative of the angular velocity taking in the canopy frame

d C Cω~˙ C = Cω~ (4.21) C/I dt C/I

4.3.1.2 Drone

• Euler Angles

The state variables ΘD/L = [φD/L;θD/L;ψD/L] defined the orientation of drone frame with respect to the main lines frame. Additionally the orientation of the main lines frame with respect to the canopy frame is given by the pitch angle θL/C. In this way the orientation of drone with respect to the inertial frame can be defined by the state variables ΘD/L, θL/C and the canopy orientation C~ ΘC/I defined in the equation 4.18.

D~ C~ L~ D~ ΘD/I = [TL−D][TC−L] ΘC/I + [TL−D] ΘL/C + ΘD/L (4.22) 4.3 System Kinematics 37

L~ where ΘL/C is given by:

 0  L~   ΘL/C = θ  (4.23)  L/C 0

• Angular Velocity

The angular velocity expression of drone expressed in the drone frame can be obtained deriving previous equation in the inertial frame

d I Dω~ = D~Θ D/I dt D/I

I I D d C~ d L~ d D~ ω~ D/I = [TL−D][TC−L] ΘC/I + [TL−D] ΘL/C + ΘD/L dt I dt dt

Applying the equation A.17 of Appendix, the previous equation becomes

d I d I d I Dω~ = [T ][T ] C~Θ + [T ] L~Θ + D~Θ ⇔ D/I L−D C−L dt C/I L−D dt L/C dt D/L D C L D ⇔ ω~ D/I = [TL−D][TC−L] ω~ C/I + [TL−D] ω~ L/C + ω~ D/L (4.24)

L where ω~ L/C is given by

 0   0  L     ω~ L/C = q  = θ˙  (4.25)  L/C  L/C 0 0

D and ω~ D/L can be deﬁned with respect to the Euler Angles rates φ˙D/L, θ˙D/L and ψ˙D/L in the same way as it was made to the canopy described in the equation 4.19

     pD/L 1 0 −sin(θD/L) φ˙D/L D      ω~ D/L = q  = 0 cos(φ ) sin(φ )cos(θ )θ˙  (4.26)  D/L  D/L D/L D/L  D/L  rD/L 0 −sin(φD/L) cos(φD/L)cos(θD/L) ψ˙D/L 38 Mathematical Model

• Derivative of the Angular Velocity

Deriving the angular velocity expression 4.24 in the drone frame, one obtains the derivative of the angular velocity for drone:

D ˙ D D C C ˙ C ω~ D/I = ω~ C/I × ω~ D/I + [TL−D][TC−L] ω~ C/I D D L ˙ L D ˙ D + ω~ L/C × ω~ D/L + [TL−D] ω~ L/C + ω~ D/L (4.27)

L ˙ L D ˙ D with a derivation provided in the Appendix A.4 and ω~ L/C and ω~ D/L are deﬁned in the equations 4.28 and 4.29 respectively.

 0  L L   ω~˙ = q˙  (4.28) L/C  L/C 0

  p˙D/L D D   ω~˙ = q˙  (4.29) D/L  D/L r˙D/L

4.3.1.3 Capsule

The rotation kinematics of the capsule will be determined in the same way as it was done to drone.

• Euler Angles

The Euler Angles vector expression for capsule can be described in function to the Euler CAP~ Angles vector of drone and the relative orientation angles ΘCAP/D

CAP~ D~ CAP~ ΘCAP/I = [TD−CAP] ΘD/I + ΘCAP/D (4.30)

CAP~ where the vector ΘCAP/D is given by the state variables φCAP/D and θCAP/D as follows

  φCAP/D CAP~   ΘCAP/D = θ  (4.31)  CAP/D 0 4.3 System Kinematics 39

• Angular Velocity

The angular velocity expression for capsule deﬁned in the capsule frame can be obtained deriving the Euler Angles vector of capsule deﬁned in the equation 4.30

d I CAPω~ = CAP~Θ ⇔ CAP/I dt CAP/I d I d I ⇔ CAPω~ = [T ]D~Θ + CAP~Θ CAP/I dt D−CAP D/I dt CAP/D

Using the equation A.17, the angular velocity expression for capsule can be written as

d I d I CAPω~ = [T ] D~Θ + CAP~Θ ⇔ CAP/I D−CAP dt D/I dt CAP/D CAP D CAP ⇔ ω~ CAP/I = [TD−CAP] ω~ D/I + ω~ CAP/D (4.32)

D CAP where ω~ D/I is deﬁned in the equation 4.24 and ω~ CAP/D is obtained from the equation 4.19 for a null yaw angle rate between the capsule and drone.

     pCAP/D 1 0 0 φ˙CAP/D CAP      ω~ CAP/D = q  = 0 cos(φ ) 0θ˙  (4.33)  CAP/D  CAP/D  CAP/D 0 0 −sin(φCAP/D) 0 0

• Derivative of the Angular Velocity

Deriving the angular velocity expression for capsule in the capsule frame, one obtain the expression for the derivative of the angular velocity

CAP ˙ CAP CAP CAP D ˙ D CAP ˙ CAP ω~ CAP/I = ω~ D/I × ω~ CAP/I + [TD−CAP] ω~ D/I + ω~ CAP/D (4.34)

CAP ˙ CAP with a derivation provided in the Appendix A.5 and ω~ CAP/D is deﬁned in the equation 4.35.

  p˙CAP/D CAP CAP   ω~˙ = q˙  (4.35) CAP/D  CAP/D 0 40 Mathematical Model

4.3.1.4 Transponder

Making the same procedure to obtain the rotation kinematics expressions for the transponder, one easily obtain the Euler Angles, angular velocity and the derivative of the angular velocity expressions deﬁned in the equations 4.36, 4.38 and 4.40 respectively.

• Euler Angles

T~ CAP~ T~ ΘT/I = [TCAP−T ] ΘCAP/I + ΘT/CAP (4.36)

T~ where the vector ΘT/CAP is deﬁned by the state variables φT/CAP and θT/CAP as follows

  φT/CAP T~   ΘT/CAP = θ  (4.37)  T/CAP 0

• Angular velocity

The angular velocity is deﬁned as the derivative of the Euler Angles vector whose expression is given by the equation 4.36

d I T ω~ = T~Θ ⇔ T/I dt T/I d I d I ⇔ T ω~ = [T ]T~Θ + T~Θ T/I dt CAP−T T/I dt T/CAP

Using the equation A.17, the angular velocity expression for transponder becomes

d I d I T ω~ = [T ] CAP~Θ + T~Θ ⇔ T/I CAP−T dt CAP/I dt T/CAP T CAP T ⇔ ω~ T/I = [TCAP−T ] ω~ CAP/I + ω~ T/CAP (4.38)

CAP T where ω~ CAP/I is deﬁned in the equation 4.32 and ω~ T/CAP is given by

     pT/CAP 1 0 0 φ˙T/CAP T      ω~ T/CAP = q  = 0 cos(φ ) 0θ˙  (4.39)  T/CAP  T/CAP  T/CAP 0 0 −sin(φT/CAP) 0 0 4.3 System Kinematics 41

• Derivative of the Angular Velocity

Deriving the previous angular velocity expression in the transponder frame, the derivative of the angular velocity can be written as

T ˙ T T T CAP ˙ CAP T ˙ T ω~ T/I = ω~ CAP/I × ω~ T/I + [TCAP−T ] ω~ CAP/I + ω~ T/CAP (4.40)

T ˙ T with a derivation provided in the Appendix A.5 and ω~ T/CAP is deﬁned in the equation 4.41.

  p˙T/CAP T T   ω~˙ = q˙  (4.41) T/CAP  T/CAP 0

4.3.2 Position Kinematics

In this sub section it will be analysed the position, linear velocity and linear acceleration expressions for four bodies. These three variables describe the translational motion of the body and in this work they will be always determined with respect to the inertial frame and expressed in the frame of the body under analysis. The canopy motion will be ﬁrst analysed followed by the drone and the capsule motion. Finally, it will be analysed the transponder motion.

4.3.2.1 Canopy

• Position

I The state variables RCMC/I = [xCMC ;yCMC ;zCMC ] described the position of the center of mass of the canopy with respect to the inertial frame and it is expressed in the inertial frame. The position vector expressed in the canopy frame can be obtained through the rotational matrix TI−C as follows:

C~ I~ RCMC/I = TI−C RCMC/I (4.42) 42 Mathematical Model

• Linear Velocity

The linear velocity is deﬁned as the derivative of the position vector, so the linear velocity of the canopy can be determined as:

  uCMC/I I C   d C ~VCM /I = v  = ~RCM /I (4.43) C  CMC/I  dt C

wCMC/I

Once the position vector of the center of mass of the canopy is expressed in the inertial frame, using the equation A.17 the previous equation can be written as:

I C d I I ˙I ~V = [TI−C] ~R = [TI−C] ~R (4.44) CMC/I dt CMC/I CMC/I

The position rates can be found using matrix inversion as

 ˙    XCMC/I uCMC/I   T   Y˙  = [TI−C] v  (4.45)  CMC/I   CMC/I  ˙ ZCMC/I wCMC/I

• Linear Acceleration

The linear acceleration is deﬁned as the derivative of the linear velocity in the inertial frame. In this way, the linear acceleration for canopy can be written as

d I C~a = C~V CMC/I dt CMC/I

Applying the equation of Coriolis A.1, the previous equation becomes

C C C d C ~aCM /I = ω~ CM /I × ~VCM /I + ~VCM /I ⇔ C C C dt C C ⇔ C~a = C~V˙ C + Cω~ × C~V (4.46) CMC/I CMC/I CMC/I CMC/I where C~V˙ C is deﬁned in the equation 4.47. CMC/I 4.3 System Kinematics 43

  u˙CMC/I C ˙ C   ~V = v˙  (4.47) CMC/I  CMC/I 

w˙CMC/I

4.3.2.2 Drone

• Position

As it was made for the Euler Angles of the drone, also the drone position vector will be described in function of others position vectors. Indeed the position vector of the drone expressed in the drone frame can be calculated as:

D~ C~ C~ L~ D~ RCMD/I = [TC−D] RCMC/I + [TC−D] RTC/CMC + [TL−D] RTD/TC + RCMD/TD (4.48)

C~ C~ C~ L~ L~ L~ D~ where RT /CM = RT1 /CM + RT2 /CM , RT /T = RT1 /T1 + RT2 /T2 and RCM /T = C C C C C C D C D C D C D D D~ D~ − RT1D/CMD + RT2D/CMD

• Linear Velocity

The linear velocity expression for the drone can be deﬁned as

D~ C~ D~ C~ D~ L~ VCMD/I = [TC−D] VCMC/I + ωC/I × [TC−D] RTC/CMC + ωL/I × [TL−D] RTD/TC L~ D C~ + [TL−D] VControl/I + ω~ D/I × RCMD/TD (4.49)

with a derivation provided in the Appendix A.7.

• Linear Acceleration

Deriving the previous linear velocity expression, the linear acceleration expression for the drone can be written as:

D~a = [T ]C~V˙ C + Dω~ × D~V + [T ]Cω~˙ C × D~R + Dω~ × Dω~ × D~R CMD/I C−D CMC/I C/I CMC/I C−D C/I TC/CMC C/I C/I TC/CMC L~˙ L D~ D~ D~ D~ D~ D~ + [TL−D] ωL/I × RTD/TC + ωL/I × ωL/I × RTD/TC + 2 ωL/I × VControl/I L~˙ L D~˙ D C~ D~ D~ D~ + [TL−D] VControl/I + ωD/I × RCMD/TD + ωD/I × ωD/I × RCMD/TD (4.50)

with a derivation provided in the Appendix A.7. 44 Mathematical Model

4.3.2.3 Capsule

• Position

In the same way, the capsule position vector expressed in the capsule frame can be described according to others position vector as one can see in the equation 4.51

CAP~ D~ D~ CAP~ RCMCAP/I = [TD−CAP] RCMD/I + [TD−CAP] RTCAPD/CMD + RCMCAP/TCAPD (4.51)

• Linear Velocity

Deriving the previous equation in the inertial frame the capsule linear velocity becomes:

CAP~ D~ CAP~ CAP~ CAP~ CAP~ VCMCAP/I = [TD−CAP] VCMD/I + ωD/I × RTCAPD/CMD + ωCAP/I × RCMCAP/TCAPD (4.52)

with a derivation provided in the Appendix A.8.

• Linear Acceleration

The linear acceleration was determined in the same way for the drone. It can be written as

CAP D D~˙ D CAP~ ~aCMCAP/I = [TD−CAP] ~aCMD/I + [TD−CAP] ωD/I × RTCAPD/CMD CAP~ CAP~ CAP~ CAP~˙ CAP CAP~ + ωD/I × ωD/I × RTCAPD/CMD + ωCAP/I × RCMCAP/TCAPD CAP~ CAP~ CAP~ + ωCAP/I × ωCAP/I × RCMCAP/TCAPD (4.53)

with a derivation provided in the Appendix A.8.

4.3.2.4 Transponder

The transponder is attached to the capsule in a similar way for a capsule with respect to drone. So one can previously note that the two pairs (capsule-drone and transponder-capsule) are similar therefore it is predictable that the transponder kinematics will be similar of the capsule kinematics. 4.4 System Dynamics 45

• Position

The transponder position vector expressed in the transponder frame can be written as

T ~ T ~ T ~ RCMT /I = RCAP/I + RCMT /CMCAP (4.54)

• Linear Velocity

The linear velocity of the transponder is deﬁned in the equation 4.55 with its derivation provided in the Appendix A.9.

T~ CAP~ T ~ T ~ VCMT /I = [TCAP−T ] VCMCAP/I + ωT/I × RCMT /CMCAP (4.55)

• Linear Acceleration

Finally, the linear acceleration expression for the transponder is deﬁned in the equation 4.56 with its derivation provided in the Appendix A.9.

T CAP T ~˙ T T ~ T ~ T ~ T ~ ~aCMT /I = [TCAP−T ] ~aCMCAP/I + ωT/I × RCMT /CMCAP + ωT/I × ωT/I × RCMT /CMCAP (4.56)

4.4 System Dynamics

In this section, it will be analysed the translation and rotation dynamic equations of motion of each body of the system. These equations describe the dynamic behaviour of the body. The translation dynamic equation of motion of a body B (equation 4.57) is provided by equating the time derivative of linear moment with the total forces applied in the body B. Whereas the rotation dynamic equation of motion of the body B (equation 4.58) is provided by equating the time derivative of angular moment with the total moment forces applied in the body B.

d I ~F = M B~V ⇔ ∑ B dt B B/I ~ B ⇔ ∑FB = MB ~aB/I (4.57) where MB is the mass of the body B.

d I M~ = [I ]Bω~ (4.58) ∑ B dt B B/I 46 Mathematical Model

where [IB] is the inertial matrix of the body B. Applying the Coriolis equation A.1, the previous equation can be written as

d B M~ = Bω~ × [I ]Bω~ + [I ]Bω~ ∑ B B/I B B/I dt B B/I

As the inertial matrix is constant, its derivative is null

d M~ = Bω~ × [I ]Bω~ + [I ] Bω~ ⇔ ∑ B B/I B B/I B dt B/I B ~ B B B ˙ B ⇔ ∑MB = ω~ B/I × [IB] ω~ B/I + [IB] ω~ B/I (4.59)

In order to perform the equations 4.57 and 4.59 for whole system, it will be analysed the forces and the moment forces applied on the system.

4.4.1 Forces

The forces applying on the system will be evaluated separately for each body and its expression will be deﬁned in each respective frame.

4.4.1.1 Canopy

C~ C~ In the canopy body there are applied 5 forces: weight force FwC, aerodynamic force FAC , ap- C C C parent force ~Fapp and the constraint forces ~T1D/C and ~T2D/C.

I. Weight Force

The weight force is the force due to the gravitational acceleration g that it will be considered invariant with altitude of the body. This force expressed in the canopy frame can be written as

0 C   ~FwC = MC[TI−C]0 ⇔   g   −sin(θC/I) C   ⇔ ~FwC = MCgsin(φ )cos(θ ) (4.60)  C/I C/I  cos(φC/I)cos(θC/I)

II. Aerodynamic Force 4.4 System Dynamics 47

In the aerodynamic field when a body immersed in a fluid moves through it, the relative motion between them induces a force on the body in the opposite direction to motion. This force is a function of the magnitude of the density of the fluid ρ, the airspeed of the body VA, the contact area of the body S and a dimensionless aerodynamic coefficients. In the aerospace field these coefficients are Lift coefficient CL, Drag coefficient CD and Sideforce coefficient CY whose expressions for a canopy will be modelled as in [14]

2 CD = CD0 +CDα2 α (4.61)

CL = CL0 +CLα α (4.62)

CY = CYβ β (4.63)

where the angle α and β are the aerodynamic angles referred previously. The aerodynamic force expressed in the aerodynamic frame can be deﬁned as

  CD A 1 2   ~FA = − ρV SC CY  (4.64) C 2 C/A   CL

q 2 2 2 where SC is the contact area of the canopy and VC/A = uC/A + vC/A + wC/A where uC/A, vC/A C and wC/A are the components of the canopy aerodynamic velocity ~VC/A that is given by:

C~ C~ I~ VC/A = VCMC/I − [TI−C] VA/I (4.65) wC/A vC/A The aerodynamic angles then become α = atan u and β = asin . The aerody- C/A VC/A namic forces expressed in the canopy frame can be written using the rotational matrix [TA−C].

  CD C 1 2   ~FA = − ρV SC[TA−C]CY  (4.66) C 2 C/A   CL

III. Apparent force

In the fluid mechanical field, the body with small mass-to-volume ratio, such as the parafoil, can experience large force exerted by the fluid on the body. This force, called apparent force is 48 Mathematical Model provided by fluid motion when a body moves and it is proportional to the amount of fluid displaced. This force can also be appear in body equations of motion as an apparent mass. The apparent mass is based of the formulas of Lissaman and Brown and it is calculated in the chapter4. In addition to the aerodynamic forces, when a body moves through a fluid there is an addi- tional force due to the accelerating fluid. This action can be modelled as an apparent mass and inertial which for a lightly loaded flight vehicle, such as parafoils, has a strong effect on the flight dynamics. Lissaman and Brown [20] have calculated the forces and moments from apparent mass and inertial by relating the kinetic energy of the fluid to resultant force and moment. The impor- tance of these force and moment, called apparent force and moment, is higher the smaller the wing loading, being the apparent mass contribution expressed in the canopy frame can be written as

C~ C C~ C~˙ C C I~ d I~ Fapp = ω~ C/I × [Mapp] VC/A − [Mapp] VCM /I + [Mapp] ω~ I/C × [TI−C] VA/I + [TI−C] VA/I ⇔ C dt I ⇔ C~F = Cω~ × [M ]C~V − [M ]C~V˙ C − [M ] Cω~ × [T ]I~V (4.67) app C/I app C/A app CMC/I app C/I I−C A/I

with a derivation provided in the Appendix A.10 and the apparent mass [Mapp] is a matrix of dimension 3 × 3

A 0 0   [Mapp] = 0 B 0 (4.68)   0 0 C where the terms A, B, C can be calculated using the following formulas given by Lissaman and Brown

π k = 0.848 A = k ρd2 b 1 + 8 a3 A 4 A C C 3 C π k = 0.339 B = k ρc d2 + 2a2 1 − d2 B 4 B C C C C AR π q k = C = k ρc2 b 1 + 2a2 1 − d2 C 1 + AR 4 C C C C C

The term AR is the aspect ratio and the parameters aC, bC, cC and dC are the dimensions of the the canopy presented in Chapter3.

IV. Constraint Forces

C C The constraint forces ~T1D/C and ~T2D/C are the pulling force from the main lines which for L~ L~ the canopy have the same directions of the position vectors RT1D/T1C and RT2D/T2C , respectively. The translation dynamic equation 4.57 for the canopy body can be written as 4.4 System Dynamics 49

C C~ C~ C~ C~ C~ MC ~aCMC/I = FwC + FAC + Fapp + T1D/C + T2D/C (4.69)

4.4.1.2 Drone

D Analysing the drone forces one should consider 5 forces: weight force ~FwD, aerodynamic force D~ D~ D~ D~ FAD , and the constraint forces T1C/D, T2C/D and TCAP/D.

I. Weight Force

The weight force of drone expressed in its frame can be written as

  −sin(θC/I) D   ~FwD = MDg[TC−D]sin(φ )cos(θ ) (4.70)  C/I C/I  cos(φC/I)cos(θC/I)

II. Aerodynamic Force

The derivation of the aerodynamic force for drone will be performed considering the drone shape as a cube so that the aerodynamic coefﬁcient in each axis is equal to CDD . In this way, the aerodynamic force expressed in the drone frame can be deﬁned as

  uD/A D 1   ~FA = − ρVD/ASDCD v  (4.71) D 2 D  D/A  wD/A

q 2 2 2 where SD is the contact area of the drone, VD/A = uD/A + vD/A + wD/A and uD/A, vD/A and D wD/A are the components of the drone aerodynamic velocity ~VD/A that can be written as:

  uD/A D   D I ~VD/A = v  = ~VCM /I − [TC−D][TI−C] ~VA/I (4.72)  D/A  D wD/A

III. Constraint Forces

C C D In the drone body, there are three constraint forces: ~T1C/D, ~T2C/D and ~TCAP/D. The ﬁrst C C ones have the same magnitude as ~T1D/C, ~T2D/C and expressing in the drone frame it can be written as: 50 Mathematical Model

D C D C ~T1C/D = −[TC−D] ~T1D/C// ~T2C/D = −[TC−D] ~T2D/C (4.73)

The third one is the pulling force of the line which connects the drone and capsule. In this way, the direction of this force is given by the vector CAP~R CMCAP/TCAPD The translation dynamic equation for the drone body can be written as

D D~ D~ D~ D~ D~ MD ~aCMD/I = FwD + FAD + T1C/D + T2C/D + TCAP/D (4.74)

4.4.1.3 Capsule

Analysing now the capsule, there are 4 forces which are applied in this body: weight force CAP~ CAP~ CAP~ CAP~ FwCAP, aerodynamic force FACAP , and the constraint forces TD/CAP and TT/CAP.

I. Weight Force

The weight force of capsule expressed in its frame can be deﬁned as

  −sin(θC/I) CAP   ~FwCAP = MCAPg[TD−CAP][TC−D]sin(φ )cos(θ ) (4.75)  C/I C/I  cos(φC/I)cos(θC/I)

II. Aerodynamic Force

The aerodynamic force expression for capsule is deﬁned in a similar manner to the equation 4.71,

  uCAP/A CAP 1   ~FA = − ρVCAP/ASCAPCD v  (4.76) CAP 2 CAP  CAP/A  wCAP/A

where SCAP is the contact area of the capsule, CDCAP is the aerodynamic coefﬁcient for the q 2 2 2 capsule, VCAP/A = uCAP/A + vCAP/A + wCAP/A and uCAP/A, vCAP/A and wCAP/A are the components CAP of the capsule aerodynamic velocity ~VCAP/A that can be written as: 4.4 System Dynamics 51

  uCAP/A CAP   CAP I ~VCAP/A = v  = ~VCM /I − [TD−CAP][TC−D][TI−C] ~VA/I (4.77)  CAP/A  CAP wCAP/A

III. Constraint Forces

The capsule is connected to the drone and to the transponder by a line as one can see in the CAP CAP schematic 4.1 therefore it is clear to consider two constraint forces: ~TD/CAP and ~TT/CAP. D The ﬁrst one has the same magnitude as ~TCAP/D and expressing in the capsule frame it can be written as:

CAP D ~TD/CAP = −[TD−CAP] ~TCAP/D (4.78)

The second one is the pulling force of the line which connects the capsule and the transponder. T ~ Such as for others constraint forces, the direction of this force is given by the vector, RCMT /CMCAP , deﬁned by the direction of the line. The translation dynamic equation for the capsule body is deﬁned in the equation 4.79

CAP CAP~ CAP~ CAP~ CAP~ MCAP ~aCMCAP/I = FwCAP + FACAP + TD/CAP + TT/CAP (4.79)

4.4.1.4 Transponder

T ~ T ~ In the transponder body only 3 forces are applied: weight force FwT , aerodynamic force FAT , T and the constraint force ~TCAP/T .

I. Weight Force

The weight force of transponder expressed in its frame is given by

  −sin(θC/I) T   ~FwT = MT g[TCAP−T ][TD−CAP][TC−D]sin(φ )cos(θ ) (4.80)  C/I C/I  cos(φC/I)cos(θC/I)

II. Aerodynamic Force

Applying the equation 4.71 for the transponder body, the aerodynamic force expression can be written as 52 Mathematical Model

  uT/A T 1   ~FA = − ρVT/ASTCD v  (4.81) T 2 T  T/A  wT/A

where ST is the contact area of the transponder, CDT is the aerodynamic coefﬁcient for the q 2 2 2 transponder, VT/A = uT/A + vT/A + wT/A and uT/A, vT/A and wT/A are the components of the T transponder aerodynamic velocity ~VT/A that can be deﬁned as:

  uT/A T   T I ~VT/A = v  = ~VCM /I − [TCAP−T ][TD−CAP][TC−D][TI−C] ~VA/I (4.82)  T/A  T wT/A

III. Constraint Force

T In the transponder there is only the constraint forces ~TCAP/T which has the same magnitude CAP as ~TT/CAP. This force expressed in the transponder frame can be written as:

T CAP ~TCAP/T = −[TCAP−T ] ~TT/CAP (4.83)

Finally, applying the translation dynamic equation 4.57 for the transponder body one has

T T ~ T ~ T ~ MT ~aCMT /I = FwT + FAT + TCAP/T (4.84)

4.4.2 Moments

In the analysis of the moments acting on the system, it should be note that the two bodies, capsule and transponder, are free to rotate in the z-axis of its frame. So the contribution of this rotation for the system motion can be neglected and in this way the total moment acting on the system only have contributions from the canopy and drone body. Therefore only the moments applied on the canopy and drone will be analysed.

4.4.2.1 Canopy

C ~ In the canopy body there are 5 moments applied on the center of mass: aerodynamic moment MAC , C ~ C~ C~ C~ apparent moment Mapp and the moment of forces FAC , T1D/C and T2D/C.

I. Aerodynamic Moment 4.4 System Dynamics 53

Such as the aerodynamic forces exists when a body moves through a fluid, a rotational motion of the body also induces an opposed rotational motions caused by the fluid. This opposed rotational motions are called aerodynamic moments. Such as the aerodynamic forces, the aerodynamic moments are defined in terms of the magnitude of the density of the fluid ρ, the airspeed of the body VA, the contact area of the body S and the dimensionless aerodynamic coefficients (rolling moment coefficient Cl, pitching moment coefficient Cm and yawing moment coefficient Cn) whose expressions for a canopy will be modelled in a similar manner as in [14]

2 bC ClC = ClpC pC/A (4.85) 2VAC 2 cC CmC = CmqC qC/A (4.86) 2VAC 2 bC CnC = CnrC rC/A + bCCnβC β (4.87) 2VAC

where pC/A, qC/A and rC/A are the components of the aerodynamic angular velocity which it C is equal to ω~ C/I and β is the aerodynamic angle defined previously. A special attention should be given to the parameter Cnβ . This parameter should be added on the yawing moment coefficient to account for the natural tendency of the canopy to turn against the wind direction. Taking into account the aerodynamic coefficients defined previously, the aerodynamic moments can be written as

  ClC C 1 2   M~ A = − ρV SC Cm  (4.88) C 2 C/A  C 

CnC

where SC is the contact area of the canopy and VC/A is the magnitude of the vector deﬁned in the equation 4.65.

II. Apparent Moment

The apparent moment has the same principle of the apparent force considering now the apparent inertial. Expressing in the canopy frame, it can be written as

d CM~ = − [I ]Cω~ ⇔ app dt app C/A I d ⇔ CM~ = − [I ]Cω~ app dt app C/I I 54 Mathematical Model

Applying the equation of Coriolis A.1, the apparent moment becomes

d C CM~ = −Cω~ × [I ]Cω~ − [I ]Cω~ ⇔ app C/I app C/I dt app C/I C C ˙ C C ⇔ M~ app = −[Iapp] ω~ C/I − ω~ C/I × [Iapp] ω~ C/I (4.89) where [Iapp] is the inertial matrix given by:

  IA 0 0   [Iapp] =  0 IB 0  (4.90)   0 0 IC where the terms IA , IB, IC can be calculated using the following formulas given by Lissaman and Brown

AR k∗ = 0.055 I = k∗ ρc2 b3 A 1 + AR A A C C AR k∗ = 0.0308 I = k∗ ρc4 b 1 + π (1 + AR)ARa2 d2 B 1 + AR B B C C 6 C C ∗ ∗ 2 3 2 kC = 0.0555 IC = kCρdCbC 1 + 8aC III. Moment of Force

The moment of force is deﬁned as the vectorial product between the force and the vector from the point where the moment of force is calculated to the point where force is applied. For the canopy, its center of mass is the point where the moment of force will be calculated. In this way C ~ C ~ C ~ the moment of force MFAC , MT1D/C and MT2D/C are deﬁned in the equation 4.91, 4.92 and 4.93 respectively.

C ~ C~ C~ MFAC = RCPC/CMC × FAC (4.91)

C ~ C~ C~ MT1D/C = RT1C/CMC × T1D/C (4.92)

C ~ C~ C~ MT2D/C = RT2C/CMC × T2D/C (4.93)

The rotation dynamic expression for canopy can then be written as 4.4 System Dynamics 55

C ˙ C C C C ~ C ~ C ~ C ~ C ~ [IC] ω~ C/I + ω~ C/I × [IC] ω~ C/I = MAC + Mapp + MFAC + MT1D/C + MT2D/C (4.94) where [IC] is the moment of inertial of the canopy which is given by the following diagonal matrix, deﬁned in [10]:

 2 2  bC + dC 0 0 MC  2 2  [Iapp] =  0 c + d 0  (4.95) 12  C C  2 2 0 0 bC + cC

4.4.2.2 Drone

D ~ In the drone body one can consider 4 moments: aerodynamic moment MAD and the moment of D ~ D ~ D forces T1C/D, T2C/D and ~TCAP/D.

I. Aerodynamic Moment

The aerodynamic moment can be deﬁned in a similar manner as it was made for aerodynamic moment of the canopy described in the equation 4.88.

  ClD D 1 2   M~ A = − ρV SD Cm  (4.96) D 2 D/A  D 

CnD where SD is the contact area of the canopy, VD/A is the magnitude of the vector deﬁned in the equation 4.72 and ClD , CmD and CnD are dimensionless aerodynamic coefﬁcients given by:

2 bD ClD = ClpD pD/A (4.97) 2VAD 2 cD CmD = CmqD qD/A (4.98) 2VAD 2 bD CnD = CnrD rD/A (4.99) 2VAD

where bD is the drone long, cD is the drone wide and pD/A, qD/A and rD/A are the components D of the aerodynamic angular velocity which it is equal to ω~ D/I.

II. Moment of Force 56 Mathematical Model

As it was developed for the canopy body, the moment of forces for the drone can be deﬁned as D D the vectorial product between the force and the position vector. In this way MT~ 1C/D, MT~ 2C/D D and MT~ CAP/D are deﬁned in the equations 4.100, 4.101 and 4.102 respectively.

D ~ D~ D~ MT1C/D = RCMD/T1D × T1C/D (4.100)

D ~ D~ D~ MT2C/D = RCMD/T2D × T2C/D (4.101)

D ~ D~ D~ MTCAP/D = RCMD/TCAPD × TCAP/D (4.102)

The rotation dynamic expression for drone can then be calculated as:

D ˙ D D D D ~ D ~ D ~ D ~ [ID] ω~ D/I + ω~ D/I × [ID] ω~ D/I = MAD + MT1C/D + MT2C/D + MTCAP/D (4.103)

4.5 Equations of Motion

In Sections 4.3 and 4.4 it was analysed the kinematics and dynamics equations which together with the equations 4.3, 4.4, 4.5 are the equations of motion of the system. A summary of these equations are presented below:

• I~R˙I = [T ]TC~V CMC/I I−C CMC/I      φ˙C/I 1 sin(φC/I)tan(θC/I) cos(φC/I)tan(θC/I) pC/I      • θ˙  = 0 cos(φ ) −sin(φ ) q   C/I   C/I C/I  C/I  ψ˙C/I 0 sin(φC/I)cos(θC/I) cos(φC/I)cos(θC/I) rC/I

C C~ C~ C~ C~ C~ • MC ~aCMC/I = FwC + FAC + Fapp + T1D/C + T2D/C

C~˙ C C~ C~ C ~ C ~ C ~ C ~ C ~ • [IC] ωC/I + ωC/I × [IC] ωC/I = MAC + Mapp + MFAC + MT1D/C + MT2D/C

• θ˙L/C = qL/C

   −1   φ˙D/L 1 0 −sin(θD/L) pD/L       • θ˙  = 0 cos(φ ) sin(φ )cos(θ ) q   D/L   D/L D/L D/L   D/L ψ˙D/L 0 −sin(φD/L) cos(φD/L)cos(θD/L) rD/L 4.5 Equations of Motion 57

D D~ D~ D~ D~ D~ • MD ~aCMD/I = FwD + FAD + T1C/D + T2C/D + TCAP/D

D~˙ D D~ D~ D ~ D ~ D ~ D ~ • [ID] ωD/I + ωD/I × [ID] ωD/I = MAD + MT1C/D + MT2C/D + MTCAP/D    −1   φ˙CAP/D 1 0 0 pCAP/D       • θ˙  = 0 cos(φ ) 0 q   CAP/D  CAP/D   CAP/D 0 0 −sin(φCAP/D) 0 0

CAP CAP~ CAP~ CAP~ CAP~ • MCAP ~aCMCAP/I = FwCAP + FACAP + TD/CAP + TT/CAP    −1   φ˙T/CAP 1 0 0 pT/CAP       • θ˙  = 0 cos(φ ) 0 q   T/CAP  T/CAP   T/CAP 0 0 −sin(φT/CAP) 0 0

T T ~ T ~ T ~ • MT ~aCMT /I = FwT + FAT + TCAP/T

• 4˙ L = KL (UL − 4L)

• 4˙ ServoR = KSR (USR − 4ServoR)

• 4˙ ServoL = KSL (USL − 4ServoL)

All the equations can be expressed compactly in matrix form as:

" # X˙ M = N (4.104) FC where X is the state vector described in the equation 4.2 and FC are set of the constraint forces FC = h iT T1D/C T2D/C TCAP/D TT/CAP . The matrix M has dimension 35 × 35. and it is composed by blocks of type Mi j with i = 1,2...15and j = 1,2...15. N is a vector matrix composed by Ni with i = 1,2...15. The blocks Mi j and Ni are derived from the equations of motion and are deﬁned in Appendix A.11. Although an algebraic manipulation of the equations of motion can eliminate the constraint forces, it is more easy to ﬁnd their values during the numerical solution along with the model states. But it should be emphasized that they aren’t state variables. Through an inversion matrix, the equation 4.104 can be written in function of the derivatives of the state vector and the constraint forces.

" # X˙ = M−1N (4.105) FC

The previous equation is of the type X˙ = f (X,U) where the set of nonlinear functions f is given by the product of matrix M−1N. Thus it is determined the nonlinear model for the system. 58 Mathematical Model

4.6 Simulation Results

The nonlinear model described in the equation 4.105 should be numerically integrated between a limited time to generate the trajectory described by the bodies of the system. A numerical solution methods is the Runge-Kutta method described by Stevens and Lewis. In the MATLAB environment, there is a set of functions to solve differential equations and one of them is the ode45 which integrates the system of differential equations X˙ = f (t,X) from time T0 to TFINAL with initial conditions X0. This function uses the Runge-Kutta algorithm and it will be used to simulate the model. The physical parameters of the four bodies are provided in Chapter3 while the position vectors and the aerodynamic coefficients are provided in Tables 4.1 and 4.2, respectively. The aerodynamic coefficients were defined somewhat similar to those used in [14] and tuned comparing the simulations results with the already tests results developed by Mario Martins in [2].

Table 4.1: Position Vectors Components

Position vector X(cm) Y(cm) Z(cm) C~ RCPC/CMC 15 0 -15 C~ RT1C/CMC 10 -85 60 C~ RT2C/CMC 10 85 60 L~ RT1D/T1C -8 85 255 L~ RT2D/T2C -8 -85 255 D~ RCPD/CMD -3 0 -2 D~ RT1D/CMD 7 -2.25 -17.5 D~ RT2D/CMD 7 2.25 -17.5 D~ RTCAPD/CMD 4.75 0 14.5 CAP~ RCMCAP/TCAPD 0 0 300 T ~ RCMT /CMCAP 0 0 500 4.6 Simulation Results 59

Table 4.2: Aerodynamic Coefﬁcients

Parameter Value

CD0 0.15

CDα2 1

CYβ 0.064

CL0 0.25

CLα 2.3

ClpC 0.755

CmqC 0.842

CnrC 0.09

CnβC -0.2

CDD 0.9

ClpD 0.655

CmqD 3.5

CnrD 0.2

CDCAP 1

CDT 1

An uncontrolled ﬂight was simulated using a MATLAB code for an interval range of [0,50] seconds. The simulation starts from 100 meters above sea level with a forward velocity of 3.3 m/s and a 1 m/s down velocity. Relative to the Euler Angles, θC/I is initialized with 6.5 degrees, θL/C with -9 degrees, θD/L with 20 degrees, θCAP/D with -7 degrees and θT/CAP with 5 degrees. Figure 4.3 shows the three dimensional trajectory of whole system. For a better understanding, it was used four points to represent the parafoil shape, three points for drone and one point for capsule and for transponder. The canopy is represented by three collinear points as it can be seen in the closest view. The other point of the parafoil represented the rotation point of the canopy. The others three points from top to bottom of the closest view represented the canopy-drone tension point, center of mass, and drone-capsule tension point of the drone body. The capsule and transponder body is represented by its center of mass. A more detailed description of the trajectory of each body is shown in Figure 4.4, 4.5, 4.6 and 4.7. They also shown the variation of the Euler Angles and as well each components of the linear and angular velocity. All parameters shown are expressed in the inertial frame and they are function of the time. 60 Mathematical Model

Figure 4.3: Three dimensional Trajectory

In the aerospace ﬁeld as well as in the paragliding sport, a special motion is the spiral motion. Indeed, if one pilot enters in a spiral motion, it can be very difﬁcult to take out the paraglider of the spiral. In this way, it is important to analyse this issue. Acting the control during only 10 seconds, one can see that after 5 seconds, the parafoil can alone to take out of the spiral motion, as it is possible to see in Figure 4.8, 4.9. This behaviour is very desired because the parafoil try alone to follow a stable motion. 4.6 Simulation Results 61

Figure 4.4: Kinematics of the Canopy

One issue to verify with simulations of the model is the behaviour of the system in very high altitude. Case the parafoil stabilizes the system and mainly can control its trajectory, it can replace the round paraglider. If yes, it is possible to neglect the paraglider decreasing the platform weight and its complexity. For a launching on 35km, one can see, in the Figure 4.10, an exponential increase of the down velocity of the canopy, followed by forward velocity. Moreover, in the initial ﬂight, there is an altitude droop. However, the platform quickly (approximately 50 seconds) reaches a stable position (Figure 4.11). The altitude droop is expected because the rareﬁed air at this altitude and therefore the resistance of the air is very slow. Not considering possible parameters at this altitude (e.g. possible collapse of the canopy when it has a great oscillation in pitch), the parafoil can stabilize the platform at very high altitude and therefore it can replace the round paraglider. 62 Mathematical Model

Figure 4.5: Kinematics of the Drone

Figure 4.6: Kinematics of the Capsule 4.6 Simulation Results 63

Figure 4.7: Kinematics of the Transponder

Figure 4.8: Kinematics of the Capsule in a Spiral Motion 64 Mathematical Model

Figure 4.9: Trajectory of the Platform in a Spiral Motion

Figure 4.10: Kinematics of the Capsule for a launching on 35km 4.6 Simulation Results 65

Figure 4.11: Trajectory of the platform launched a high altitude 66 Mathematical Model Chapter 5

Control Algorithm

The control algorithm performed for this system is analysed in this chapter. The main goal of the control algorithm is that the system follows a speciﬁed path. For that it is used two control input: main lines and symmetric breaks. The main lines allow to change the orientation of the system while the breaks control on the symmetric way allows to change the glide slope of the trajectory. In this way the control system can be divided on the Horizontal Control and Vertical Control. Sections 5.1 and 5.2 described these two types of control. Finally, the spiral mode control system is performed in the Section 5.3 taking into account the guidance strategy deﬁned in the Section 3.7.

5.1 Horizontal Control

The purpose of the horizontal control system is to command the system to follow a specified path controlling the main lines. A displacement of these lines, through the differential servo-actuators, makes the canopy rotate on the roll axis. An non-zero roll angle induces a yaw motion on the canopy, making the system turn to the deflected side of the canopy. For example, if the right line is pulled down, it makes a positive variation of the roll angle which induces the system turn to the right. The path to follow can be defined by three ways: a specified direction or a specified direction and one point or two points. Considering possible wind presence along the flight, three types of strategy control were defined: Heading, Course, One Point Localizer and Course or Two Points Localizer. These types of control are performed as a feedback control and they are a PD (Proportional and Derivative) controller. This type of control acts on the system with a proportional gain and a derivative gain on the error and error variation control, respectively.

5.1.1 Heading

The heading control allows to manage the system towards a direction with a speciﬁed heading, where heading refers to the direction that chord line of the canopy is pointing, it is equal to the yaw angle. The reference angle is the North and a positive variation means a clockwise rotation.

67 68 Control Algorithm

The control errors are given by the heading error HError and heading error of the variation HVError, deﬁned in the equations 5.1 and 5.2.

HError = ψC − H0 (5.1)

HVError = ψ˙C (5.2) where H0 is the desired heading of the path and its variation is null. The controller can now be written as

HU = −HK1HError − HK2HVError (5.3) where HK1 and HK2 are constant and determine the control responsiveness.

5.1.2 Course

The heading control can manage the system for a non-desired trajectory when there is the wind presence. Indeed, on the wind presence, the system can experience a side shift caused by the winds and describe a non-desired trajectory. To avoid this situation, the course control is performed where course refers to the direction of the path which the canopy describes. In others words, the course is the angle between the inertial velocity components uCMC/I and vCMC/I. In a null wind condition, the two control system are equal. Such as the heading the reference angle is the North and a positive variation means a clockwise rotation. The controller used is similar whose used on the Heading Control replacing the heading errors by the course error CError and course error of the variation CVError.

vCMC/I CError = arctan −C0 (5.4) uCMC/I

CVError = ψ˙C (5.5) where C0 is the desired course of the path and its variation is null. It should be noted that the velocity components uCMC/I and vCMC/I should be expressed in the inertial frame. Due to the lack of available data about the variation of the linear velocity, the error CVError is approximately to the 5.1 Horizontal Control 69

error HVError described in the equation 5.2. This assumption can be considered valid because the variation of the course is similar to the variation of the heading. On the same way for the heading control, the course controller can be written as

HU = −HK1CError − HK2CVError (5.6)

5.1.3 One Point Localizer and Course or Two Points Localizer

A more completed control strategy uses a specified course and a point localizer or if more desired, two points localizer. Without lost of propriety, the two types of control are equivalent and only it will be analysed the first one. In the aerospace field one of the most used navigation methods uses a spacial point (localizer) and an angle of arrival (Figure 5.1). The control allow to manage the system for a desired path which pass through the localizer. In this control, three different errors may be present: cross track error, course error and course error of variation. The cross track error, CTError is the distance between the actual point and its projection in the desired path as it is possible to see in the Figure 5.1. Through a trigonometric equations, the expression of this error is defined in the equation 5.7. In order to avoid a great overshoot on the control response and to ensure a shorter path toward the target, this control has two phases: approach and following phase. CD The first one corresponds to an cross track error higher than 5 , where CD is the value of the Convergence Distance register. In this phase, the system heads to a path at an angle α defined in the equation 5.8.

Figure 5.1: One Point Localizer and Course [2] 70 Control Algorithm

CTError = (yCMC/I − y0)cos(C0) − (xCMC/I − x0)sin(C0) (5.7) where x0 and y0 are the positions (x,y) of the localizer and C0 is the desired course of the path.

min(|CT |,CD) α = arctan Error sign(CT ) (5.8) CD Error where sign is the mathematical function that extracts the sign of a real number. Knowing the angle of the ideal path relative to the control path, the course error is equal to:

vCMC/I CError = arctan − (C0 − α) (5.9) uCMC/I

The course error variation CVError is equal to the Course Control, deﬁned in the equation 5.5. The control expression of this phase is so given by:

HU = −HK1CError − HK2CVError (5.10)

CD When the cross track error is under the threshold value 5 , the phase switches to the following phase. The main goal of this phase is to ensure that the system heads the desired path toward the target. The control expression of this phase can be written as:

HU = −HK1CError − HK2CVError − HK3CTError (5.11) where CError, CVError and CTError are deﬁned in the equations 5.4, 5.5 and 5.7. An example of the One Point Localizer and Course control application is illustrated in Figure 5.2. The localizer point chosen is (x=100, y=200) and the desired course is equal to 90 degrees. Initially, the platform follows a course of 45 degrees and only when approaching the line x = 100, the platform turns to the course of 90 degrees. After some time, it reaches the localizer. 5.2 Vertical Control 71

Figure 5.2: Horizontal Control - One Point Localizer and Course

5.2 Vertical Control

The main goal of the vertical control is to manage the system to a desired glide slope. A similar procedure of the One Point Localizer and Course control system is used to the vertical control. Two different types of control are used: altitude error and glide slope error. The altitude error

AError is the distance between the actual point and its perpendicular projection in the desired path. Through a trigonometric equations, the expression of this error is defined in the equation 5.12. Such as the One Point Localizer control system, also this control has two phases: approach and CD following phase. The first one is activated when the altitude error is higher than 5 . In the other hand, the following phase is activated. The approach phase manage the system to head a path at an angle β defined in the equation 5.13.

AError = (yCMC/I − y0)cos(θ)cos(C0) + (yCMC/I − y0)cos(θ)sin(C0) − (zCMC/I − z0)sin(θ) (5.12) where x0, y0 and z0 are the positions (x,y,z) of the localizer, C0 is the desired course of the path and θ is the angle of the path slope.

min(|A |,CD) β = arctan Error sign(A ) (5.13) CD Error

The glide slope error can then be calculated as: 72 Control Algorithm

wCMC/I GSError = − (GS0 − tan(β)) (5.14) uCMC/I

where uCMC/I should expressed in the canopy frame. GS0 is the desired glide slope. The control expression of this phase becomes:

VU = −VK1GSError (5.15)

CD When the altitude error is under the threshold value 5 , the phase switches to the following phase. Such as in the One Point Localizer and Course control system, the main goal of this phase is to ensure that the system heads the desired path toward the target. The control expression of this phase can be written as:

VU = −VK1GSError −VK2AError (5.16) where VK1 and VK2 are constants which determines the control responsiveness. An example of this control type is presented in Figure 5.3. One can see that the trajectory can be spilt into 2 phases. At the begin, the control system limits the glide slope of the platform. When it reaches a certain altitude (approximately 90 meters), the control system switches to the next phase. A smoother glide slope is followed by the platform in direction to the landing point.

5.3 Spiral Mode

(Not finished...) In Section 3.7 it was stated an usual controllable flight. After Heading or Course Control be applied to head the system to a desired direction, the Spiral Mode is activated. Before a explanation of this algorithm, a special description should be taken. The main goal of this control system is to ensure that the system lands safely on a desired place. However due to the constant wind presence on the troposphere layer, to down on a unique path can be undesirable. Indeed, the system may not be able to overcome the wind conditions. So a possible solution is system describes a spiral motion. Dividing the spiral motion on four different branch (like a square), the system can easily correct some wind perturbation by scaling the branches length. On the other hand, the system can flight on a slide spiral motion by the presence of wind. With these assumptions on mind, a control algorithm was built. The control strategy is divided in seven distinct phases. The first one, approximation mode, the One Point Localizer and Course control system is used to manage the system to a desired point, taking into account the wind presence, the desired land point and the actual position of the system. The localizer point is chosen according to the region where the 5.3 Spiral Mode 73

Figure 5.3: Vertical Control

system is. An example is showed on the figure 5.4, where the system flies over the green region. So, the control should manage the system for the right branch. The four next phases corresponds to each branch of the spiral. In each branch, it is used the One Point Localizer and Course control system to manage the system to follow the branch until a safe distance of the next phase. This distance should be enough to allow a appropriate turn. After the phase five be concluded the second phase restarts. The six phase corresponds to the final stage of the spiral motion and starts when the system is on the 5 phase at a certain altitude. This altitude is continuously computed during the phase two to five, changing the branches length. In the phase six, it is activated the One Point Localizer and Course and the Vertical Control control system to the desired land point. This phase is previously computed so that the system lands into the wind. The spiral control system finishes with the seven phase which controls the breaks in order to reduce the impact velocity. The block diagram (Figure 5.5) performs the spiral control algorithm designed in this work. 74 Control Algorithm

Figure 5.4: Spiral Region Decision 5.3 Spiral Mode 75

Figure 5.5: Block Diagram of the Spiral Control Algorithm 76 Control Algorithm Chapter 6

Stability Analysis

On the designing a given control system, one of the most important question which should be in mind is whether it is stable, because an unstable control system is usually dangerous and unwanted. Before the stability analysis, the non linear model is ﬁrstly linearised by the Lyapunov’s linearization method in Section 6.1.The stability of the linear model as well the control system presented in Chapter5 is analysed in Section 6.2. For the purposes of this work, it is analysed the internal stability, controllability and observability of the model. At the end, Section 6.4 presents a Graphic User Interface (GUI) specially developed for this work.

6.1 Linear Model

The equations of motion developed in Chapter4 are nonlinear differential equations which can be expressed compactly in matrix form:

x˙ = f (x,u) (6.1) where x and u are the state and input vector, respectively. x is a vector of n state variables while u has dimension p, number of control input. A linearization method allows to deﬁne the derivative of the state vectorx ˙ as a linear combi- nation of the state vector themselves x and the input vector u of the form

x˙ = Ax + Bu (6.2) where A is a square matrix of dimension n × n and it is usually called state matrix. B is the input matrix of dimension n × p. The equation 6.2 along with the equation 6.3 deﬁnes completely a linear system. This repre- sentation is called space state and it is very used on control theory.

77 78 Stability Analysis

y = Cx + Du (6.3) y is the output vector of dimension q, C is the output matrix with dimension q × n and D is the feedthrough matrix of dimension q × p.

6.1.1 Lyapunov’s Linearization Method

One linearization method is the Lyapunov’s linearization method developed by Lyapunov to apply on the stability theory. His method is a formalization of the intuition that a nonlinear system should behave similarly to its linearized approximation for small range motions [21]. The linearization method should be performed at an equilibrium point 1 [x∗,u∗]. On a small range region, the equation 6.1 can be represented as

∂ f ∂ f x˙ = x + u (6.4) ∂x (x=x∗,u=u∗) ∂u (x=x∗,u=u∗)

Comparing the equation 6.4 with 6.2, one can note that both are equivalent, taking

∂ f A = (6.5) ∂x (x=x∗,u=u∗) ∂ f B = (6.6) ∂u (x=x∗,u=u∗)

So, to linearise a nonlinear model consists in to derivative the n differential equations f in order to the n state variables to obtain the matrix A while B is obtained deriving the n differential equations f in order to the q input variables. The same procedure can be made in order to obtain the linear equation 6.3.

The equilibrium point of the platform in which it will be applied the Lyapunov’s linearization method is chosen observing the simulation results illustrated in Chapter4. In the uncontrolled ﬂight,the platform oscillates until reach a stable trajectory. This trajectory corresponds to an equilibrium trajectory, where the respective non-zero components of the equilibrium point are shown in Table 6.1. The values x and z can take any number in meters except z which should be negative. Where X, Z, u and w are the positions and linear velocity components of the canopy, respectively.

1An equilibrium point is a solution of the system equations in which the derivative of the state vector in this point is null 6.2 Stability Analysis 79

Table 6.1: Equilibrium Point

Parameter X Z u w θC/I θL/C θD/L θCAP/D θT/CAP Value x z 22.1m/s 11.0m/s −8.1deg −5.3deg 6.1deg −7.1deg 5.4deg

Using MATLAB code speciﬁcally developed to apply the Lyapunov’s linearization method through the equation 6.4, the nonlinear model 4.104 was linearised.

6.2 Stability Analysis

In the control theory, one of the most important parameter of a system is the stability. Qualitatively, a system is described as stable if starting the system somewhere near its desired operating point implies that it will stay around the point ever after [21]. The Lyapunov stability theory allows to easily analyse the stability of a model through the analysis of the state matrix A.

6.2.1 Stability

(Not ﬁnished...) According to the Lyapunov theorem, a linearized system or a matrix A is strictly stable, if all eigenvalues of A are strictly in the left-half complex plane [21]. Calculating the eigenvalues λ of the matrix A of the system, one obtains

λ = [0,0,0,0,−0.26 ± 1.53 j,−0.36 ± 1.64 j,−0.40 ± 0.93 j,−0.62,−0.74 ± 3.85 j,... − 0.91 ± 5.34 j,−1.16 ± 22.98 j,−1.17 ± 4.45 j,−1.32 ± 30.72 j,−1.88 ± 6.53 j,−2.03,−3.91 ± 5.50 j,... − 5.0,−5.0,−5.0. − 15.30 ± 10.47 j]T

As all eigenvalues of A are in the left-half complex plane (part real less than zero), the system is strictly stable.

6.2.2 Controllability

(Not finished...) As it is explained in [22], a system or the pair (A,B) is said to be controllable if for any initial state x(0) = x0 and any final state x f , there exists an input sequence that transfers the system from x0 to x f in a finite time. Mathematically, the controllability of a system can be check trough the controllability matrix which is given by

h i C(A,B) = B AB ... An−1B (6.7) 80 Stability Analysis where n is the number of the state variables. Second Chi-Tsong Chen [22], a necessary and sufﬁcient condition for the system to be controlled is that rank(C(A,B)) = n. If rank(C(A,B)) < n, so there exists n − rank(C(A,B)) uncontrollable state variables and rank(C(A,B)) controllable state variables. The system is said to be uncontrollable. Using MATLAB proprieties, it was veriﬁed that all state variables are controllable rank(C(A,B)) = 31 and therefore, the system is controllable.

6.2.3 Observability

(Not finished...) A similar analysis can be done for the concept of the observability which is dual to that of controllability. Observability studies the possibility of determining the state vector from the output. For that it is necessary to consider the state equations 6.2, 6.3. A system or the pair (A,C) is said to be observable if any unknown initial state x0 can be determined over a finite time t1 > 0 from the knowledge of the input u and the output y over the time t1. In a similar manner for the concept of the controllability, the verification of observability of a system can be performed trough the observability matrix which is given by

 C     CA  O(A,C) =   (6.8)  .   .  Cn−1A

Second Chi-Tsong Chen, a necessary and sufﬁcient condition for the system to be observable is that rank(O(A,C)) = n. If rank(O(A,C)) < n, so there exists n − rank(O(A,C)) unobserv- able state variables and rank(O(A,C)) observable state variables. The system is said to be un- observable. Using MATLAB proprieties, it was veriﬁed that all state variables are observable rank(O(A,C)) = 31 and therefore, the system is observable.

6.3 Pole Placement

(Not ﬁnished...)

6.4 Graphic User Interface

(Not ﬁnished...) The wish for an easy simulation of results led to the development of a graphic user interface 6.1. A simple and easy simulation of the model as well others simulations can be performed for data given by an user. This work was done on the GUI development environment. 6.4 Graphic User Interface 81

Figure 6.1: Graphic User Interface

This interface can be divided in three sub-windows. The first one, on the left side, is composed by five panels where the user can introduce data which are used in the mathematical model. On the left upper side is possible to change the aerodynamic coefficients of the four bodies of the platform as well of the breaks of the canopy. It should be note that these parameters have some restrictions on their value, once these should be realistic values. The panel Position provides the possibility to set different configuration of the system, in respect to the lines length and the most important points of the bodies on the mathematical model. The mass, area and dimension of the bodies can also be changed on the three lower left corner. On the center of the screen, the second sub-windows shows the different simulations which can be performed: Mathematical Model, Linearization and Estimation. In the first, it is simulated the mathematical model of the selected bodies and the data introduced on the others sub-windows. In the Linearization panel, the linearization task and its analysis is performed. Finally, the comparison of the theoretical with the experimental results can be done in the Estimation panel. In this panel, the user should introduce the file name ended by ".txt". On the right side of the GUI, a least sub-window is divided in three panels. On the right upper corner, several types of figures can be selected to better analysis of the simulation results. The control mode as well its parameters can be chosen by the user on the second panel. Information about the control mode selected can be introduced, furthermore the control system responsiveness can be analysed, changing the control gains. Finally, the initial conditions of the mathematical model, if necessary can be changed on the right lower corner. The button Simulate along the Simulation Time which can be changed allow to simulate the model. If it is desired, the default values of the parameters of the system can be reselected using the button Reset. 82 Stability Analysis Chapter 7

Final Remarks

The final conclusions as well the next steps in the framework of the parafoil control for the STRAPLEX project are presented in this chapter. Section 7.1 presents some general conclusions of this work. Some of them highlights the flight’s behaviour modelled by the mathematical model and the control algorithm developed for a specific purpose. In the Section 7.2, it is presented a few important steps forward in the STRAPLEX project.

7.1 Conclusions

A new and high fidelity mathematical model was developed for the STRAPLEX project in order to model the nonlinear dynamic motion of the platform. Through the kinematics and dynamics equation, a 14 degree of freedom mode was derived. This model predicted accurately some behaviours of a realistic flight, with emphasis for the pendulum effect between the bodies and the response to the wind effect. On development of the model, aerodynamic forces as well the apparent force were considered to make the model the most realistic possible. Moreover, two types of control input were introduced in the model: main lines and breaks. The model was simulated with realistic inputs, parameters and flight conditions using a code specially developed in MATLAB environment. The simulation results demonstrated that a stable flight was quickly achieved even when the platform was launched at high altitude. Here, high velocities were reached both down and forward direction, making a striking total velocity of 300 km/h. However the parafoil showed be able to stabilize the platform motion and after it reached a peak velocity, the velocity was decreasing until reached a stable value. This last result is very important once the parafoil can replace the paraglider rule thus reducing the complexity of the STRAPLEX project. Moreover the control system can be activated as soon as the platform reaches a stable motion. Another important conclusion was achieved when the platform enters in a spiral motion. As it is desired, the platform can take out of this motion without any type of control. One of main objective of this work was to design a robust control algorithm to land the platform in a specific landing point taking into account wind effects. This objective was accomplished and proved by simulations results. A virtual flight was simulated with certain wind conditions and for

83 84 Final Remarks a specific landing point. The control system was able to manage the platform to the landing point with a precision error considerably reduced. The approach followed in the design of the control algorithm was proved to be an innovation and reliable work with capacity to correct wind changes. A stability analysis of the system proved that the model only has stable poles, for a given equilibrium point. Moreover, the linear model, previously linearized, is controllable and observable. It allows to conclude that the system can follow any trajectory and any unknown state variable of the model can be determined knowing the data which the platform is able to measures during the flight. Although the complexity of the platform motion during the flight, the model developed is successfully able to predicted its motion. Furthermore it is a important basis for the control system design which was efficiently developed and simulated or for other type of control which can be design in future.

7.2 Future Work

Since it is the ﬁrst formal treatment of the platform motion in the STRAPLEX project, there are a lot of improvements to make in this project. A few important steps are outline below:

(1) Installation of more measuring device in Drone in order to available more information about the ﬂight; for example, an power meter strain installed in the differential servo-actuator to measure the tension forces in the main lines.

(2) More ﬂights tests may be performed in order to improve the value of some parameters of the system (e.g. aerodynamic coefﬁcients).

(3) The control strategies developed in this work should be tested in the STRAPLEX project. Flight tests should be made with different atmospheric conditions in order to validate the control system. Appendix A

Appendix

A.1 Equation of Coriolis

During the study of the wind currents, the French physicist Gaspard de Coriolis demonstrated that the Earth’s rotation had effect on atmospheric motions. This effect, named Coriolis effect, explains the different orientation of the winds in the Northern Hemisphere and Southern Hemisphere. This conclusion can be utilized to determine the vector derivatives in moving frames. The vector W is ﬁxed in a frame A, which is rotating with respect to the reference frame R. For the calculation of the derivative of the vector W in the frame R, one can utilize the derivation of the angular velocity vector found in Aircraft Control and Simulation [1], to obtain:

R ˙ A ˙ A R ~w = ~w + ωA/R × ~w (A.1)

A where ωA/R is the instantaneous angular velocity vector of the frame A with respect to the frame R, expressed in the frame A. From this result some conclusions can be draw:

• Calculate a vector derivative in a different frames just requires the angular velocity vector between the two frames;

R A • The angular velocity derivative is the same in either frame, ω˙A/R = ω˙A/R. This is made evident because the vectorial product of the same vector is null.

A.2 Poisson’s Kinematical Equation

A derivation of the poisson’s kinematical equation or, in inertial navigation, as the strapdown equation is made in [1]. It concludes that for a rotation matrix TR−A which transforms a vector A w from the frame R to the frame A and for an instantaneous angular velocity vector ωA/R, the derivative of the rotation matrix is given by:

85 86 Appendix

A T˙R−A = − ωA/R × TR−A (A.2)

This equation is an useful tool in developing equations of motion, mainly for the derivative of an array. Taking a vector W ﬁxed in the frame R, this vector can be expressed in the frame A by the rotation matrix TR−A:

A R ~w = [TR−A] ~w (A.3)

Differentiating the vector A~w in the Frame A is equivalent to:

d A d A A~w = [T ]R~w (A.4) dt dt R−A

Applying the product rule, the previous equation is equal to:

d d R A~w˙A = ([T ])A R~w + [T ] R~w (A.5) dt R−A R−A dt

d We can now use the Poisson’s kinematical equation A.2 to replace ([T ])A, dt R−A

A ˙A A R R ˙R ~w = − ωA/R × TR−A ~w + [TR−A] ~w (A.6)

Simplifying, one has:

A ˙A A A A ˙R ~w = − ωA/R × ~w + ~w (A.7)

A.3 Derivative of a Vector

The derivative of a vector~V expressed in a frame A is equal to the derivative of the vector expressed in other frame B, multiplied by the rotational matrix TB−A. The proof follows. The derivative of the vector to get is:

d I A~V (A.8) dt A.3 Derivative of a Vector 87

Applying the equation of Coriolis A.1, the equation A.8 can be written as:

d A A~V˙ I = Aω~ × A~V + A~V ⇔ A/I dt

d A ⇔ A~V˙ I = Aω~ × A~V + [T ]B~V (A.9) A/I dt B−A

Through the Poisson’s equation, it is possible to simplify the second term of the right side.

d B A~V˙ I = Aω~ × A~V + Aω~ × A~V + [T ] B~V ⇔ A/I B/A B−A dt

d I ⇔ A~V˙ I = Aω~ × A~V + Aω~ × A~V + [T ] Aω~ × B~V + B~V (A.10) A/I B/A B−A I/B dt

A Knowing that the angular velocity ω~ B/A is equal to:

A A A ω~ B/A = ω~ B/I − ω~ A/I (A.11)

The equation A.10 becomes:

A ˙ I A A A A A A B B ˙ I ~V = ω~ A/I × ~V + ω~ B/I − ω~ A/I × ~V + [TB−A] − ω~ B/I × ~V + ~V (A.12)

Using the commutative and distributive propriety of the vectorial product,

A ˙ I A A A A A A B A B ˙ I ~V = ω~ A/I × ~V + ω~ B/I × ~V − ω~ A/I × ~V + [TB−A] ~V × ω~ B/I + ~V (A.13)

The previous equation can be simpliﬁed to:

A ˙ I A A B A B ˙ I ~V = ω~ B/I × ~V + [TB−A] ~V × ω~ B/I + ~V (A.14)

Using now the matrix distributive propriety, 88 Appendix

A ˙ I A A A A B ˙ I ~V = ω~ B/I × ~V + ~V × ω~ B/I + [TB−A] ~V (A.15)

Using again the commutative propriety of the vectorial product,

A ˙ I A A A A B ˙ I ~V = ω~ B/I × ~V − ω~ B/I × ~V + [TB−A] ~V (A.16)

And ﬁnally, the derivative of the vector ~V can be written as

A ˙ I B ˙ I ~V = [TB−A] ~V (A.17)

A.4 Drone Rotation Kinematics

• Derivative of the Angular Velocity

The derivative of the angular velocity for drone expressed in the drone frame is given by

D ˙ D d D~ ω~ D/I = WD/I (A.18) dt D

Using the equation 4.24 and applying the equation A.17

d d d Dω~˙ D = [T ][T ]Cω~ + [T ]Lω~ + Dω~ ⇔ D/I dt L−D C−L C/I D dt L−D L/C D dt D/L D d d d ⇔ Dω~˙ D = [T ][T ] Cω~ + [T ] Lω~ + Dω~ (A.19) D/I L−D C−L dt C/I D L−D dt L/C D dt D/L D

Applying the Coriolis equation A.1, the previous equation becomes

d Dω~˙ D = [T ][T ] Dω~ × Cω~ + Cω~ D/I L−D C−L C/D C/I dt C/I C d + [T ] Dω~ × Lω~ + Lω~ + Dω~˙ D (A.20) L−D L/D L/C dt L/C L D/L

Finally, the derivative angular velocity expression for drone can be written as A.5 Capsule Rotation Kinematics 89

D ˙ D D C C ˙ C ω~ D/I = ω~ C/I × ω~ D/I + [TL−D][TC−L] ω~ C/I D D L ˙ L D ˙ D + ω~ L/C × ω~ D/L + [TL−D] ω~ L/C + ω~ D/L (A.21)

A.5 Capsule Rotation Kinematics

• Derivative of the Angular Velocity

The derivative of the angular velocity expression for the capsule in the capsule frame is obtained deriving the angular velocity expression in the capsule frame,

d CAPω~˙ CAP = CAPω~ ⇔ CAP/I dt CAP/I CAP d d ⇔ CAPω~˙ CAP = [T ]Dω~ + CAPω~ (A.22) CAP/I dt D−CAP D/I CAP dt CAP/D CAP

Applying the equation A.17, the previous equation can be written as

d d CAPω~˙ CAP = [T ] Dω~ + CAPω~ (A.23) CAP/I D−CAP dt D/I CAP dt CAP/D CAP

Applying the Coriolis equation A.1

d d CAPω~˙ CAP = [T ]CAPω~ × Dω~ + [T ] Dω~ + CAPω~ CAP/I D−CAP D/CAP D/I D−CAP dt D/I D dt CAP/D CAP (A.24)

Finally, the derivative of the angular velocity expression can be written as

CAP ˙ CAP CAP CAP D ˙ D CAP ˙ CAP ω~ CAP/I = ω~ D/I × ω~ CAP/I + [TD−CAP] ω~ D/I + ω~ CAP/D (A.25)

A.6 Transponder Rotation Kinematics

• Derivative of the Angular Velocity

The derivative of the angular velocity expression for the transponder is obtained deriving the angular velocity expression in the transponder frame, 90 Appendix

d T ω~˙ T = T ω~ ⇔ T/I dt T/I T d d ⇔ T ω~˙ T = [T ]CAPω~ + T ω~ (A.26) T/I dt CAP−T CAP/I T dt T/CAP T

Applying the equation A.17, the previous equation can be written as

d d T ω~˙ T = [T ] CAPω~ + T ω~ (A.27) T/I CAP−T dt CAP/I T dt T/CAP T

Applying the Coriolis equation A.1

d d T ω~˙ T = [T ]T ω~ × CAPω~ + [T ] CAPω~ + T ω~ T/I CAP−T CAP/T CAP/I CAP−T dt CAP/I CAP dt T/CAP T (A.28)

Finally, the derivative of the angular velocity of the transponder is given by

T ˙ T T CAP CAP ˙ CAP T ˙ T ω~ T/I = ω~ CAP/I × ω~ T/I + [TCAP−T ] ω~ CAP/I + ω~ T/CAP (A.29)

Using the equation 4.42, the linear velocity becomes:

A.7 Drone Position Kinematics

• Linear Velocity

The drone linear velocity vector can be calculated as:

D d D ~VCM /I = ~RCM /I (A.30) D dt D I

Using the equation of the drone position vector ??, the drone linear velocity vector can be written as:

D d C d C d L d D ~VCM /I = [TC−D] ~RCM /I + [TC−D] ~RT /CM + [TL−D] ~RT /T + ~RCM /T D dt C I dt C C I dt D C I dt D D I (A.31) A.7 Drone Position Kinematics 91

Applying the Coriolis and the Poisson equations A.1 and A.2, the previous equation can be developed to:

D D C d C D C ~VCM /I = ω~ D/I × [TC−D] ~XCM /I + [TC−D] ~XCM /I + ω~ D/I × [TC−D] ~RT /CM /I D C dt C D C C d C D L d L + [TC−D] ~RT /CM + ω~ D/I × [TL−D] ~RT /T /I + [TL−D] ~RT /T dt C C D D C dt D C D D C d D + ω~ D/I × ~RCM /T /I + ~RCM /T ⇔ (A.32) D D dt D D D

D D C D C d C ⇔ ~VCM /I = ω~ D/I × [TC−D] ~XCM /I + ω~ C/D × [TC−D] ~XCM /I + [TC−D] ~XCM /I D C C dt C C C ! d( ~RT /CM ) + Dω~ × [T ]C~R + Dω~ × [T ]C~R + [T ] C C D/I C−D TC/CMC C/D C−D TC/CMC C−D dt C D L D L d L + ω~ D/I × [TL−D] ~RT /T + ω~ L/D × [TL−D] ~RT /T + [TL−D] ~RT /T D C D C dt D C L D C d D + ω~ D/I × ~RCM /T + ~RCM /T (A.33) D D dt D D D

D~ C~ The vectors RCMD/TD and RTC/CMC are always constant, so their derivatives are null. Taking into account the equations [SSDS]-[sdS] of the chapter4 and the equation of Coriolis, the equation A.33 comes:

D D C D C d C ~VCM /I = ω~ C/I × [TC−D] ~XCM /I + [TC−D] ω~ I/C × ~XCM /I + ~XCM /I D C C dt C I D~ C~ D~ L~ + ωC/I × [TC−D] RTC/CMC + ωL/I × [TL−D] RTD/TC D L d L D D + [TL−D] ω~ I/L × ~RT /T + ~RT /T + ω~ D/I × ~RCM /T (A.34) D C dt D C I D D

Using the commutative and distributive propriety of the vectorial product, the previous is sim- pliﬁed to

D~ C~ D~ C~ D~ L~ VCMD/I = [TC−D] VCMC/I + ωC/I × [TC−D] RTC/CMC + ωL/I × [TL−D] RTD/TC

d L D D + [TL−D] ~RT /T + ω~ D/I × ~RCM /T (A.35) dt D C I D D

In the chapter3 it was demonstrated that a main line control input induces a horizontal dis- d L placement on the drone, therefore the term ~RT /T isn’t null. Naming the horizontal dis- dt D C I L placement rate as ~VControl/I, the linear velocity of the drone becomes: 92 Appendix

D~ C~ D~ C~ D~ L~ VCMD/I = [TC−D] VCMC/I + ωC/I × [TC−D] RTC/CMC + ωL/I × [TL−D] RTD/TC L~ D D~ + [TL−D] VControl/I + ω~ D/I × RCMD/TD (A.36)

• Linear Acceleration

By deﬁnition the linear acceleration is equal to the derivative of the linear velocity, so the linear acceleration of the drone equal to:

D d D ~aCM /I = ~VCM /I (A.37) D dt D I

Using the expression of the linear velocity of the drone described in the equation A.36, the linear acceleration of the drone is given by:

D d C d D C ~aCM /I = [TC−D] ~VCM /I + ω~ C/I × [TC−D] ~RT /CM D dt C I dt C C I d D L d L + ω~ L/I × [TL−D] ~RT /T + [TL−D] ~VControl/I dt D C I dt I d D D + ω~ D/I × ~RCM /T (A.38) dt D D I

Using the equation A.17 for the ﬁrst and fourth right terms and applying distributive property of the external product for others terms, the previous equation becomes:

D d C d D C D d C ~aCM /I = [TC−D] ~VCM /I + ω~ C/I × [TC−D] ~RT /CM + ω~ C/I × [TC−D] ~RT /CM D dt C I dt I C C dt C C I d D L D d L d L + ω~ L/I × [TL−D] ~RT /T + ω~ L/I × [TL−D] ~RT /T + [TL−D] ~VControl/I dt I D C dt D C I dt I d D C D d D + ω~ D/I × ~RCM /T + ω~ D/I × ~RCM /T (A.39) dt I D D dt D D I

Through the conclusion stated in Section A.1, the angular velocity derivative is the same in either frame, so:

D d C d D C D d C ~aCM /I = [TC−D] ~VCM /I + ω~ C/I × [TC−D] ~RT /CM + ω~ C/I × [TC−D] ~RT /CM D dt C I dt C C C dt C C I d D L D d L d L + ω~ L/I × [TL−D] ~RT /T + ω~ L/I × [TL−D] ~RT /T + [TL−D] ~VControl/I dt L D C dt D C I dt I d D C D d D + ω~ D/I × ~RCM /T + ω~ D/I × ~RCM /T (A.40) dt D D D dt D D I A.8 Capsule Position Kinematics 93

Once again applying the equation A.17, the Coriolis equation and some results obtained on Section A.7, the linear acceleration of drone is given by:

D D C d C d C C ~aCM /I = [TC−D] ω~ C/I × ~VCM /I + ~VCM /I + [TC−D] ω~ C/I × [TC−D] ~RT /CM D C dt C C dt C C C d + Dω~ × Dω~ × [T ]C~R + [T ] Lω~ × [T ]L~R C/I C/I C−D TC/CMC L−D dt L/I L L−D TD/TC D~ D~ L~ L~ D~ L~ + ωL/I × ωL/I × [TL−D] RTD/TC + [TL−D] VControl/I + [TL−D] ωL/I × VControl/I

d L d D C D D D + [TL−D] ~VControl/I + ω~ D/I × ~RCM /T + ω~ D/I × ω~ D/I × ~RCM /T dt L dt D D D D D (A.41)

Simplifying, the previous equation becomes:

D C~˙ D~ D~ C~˙ D~ D~ D~ D~ ~aCMD/I = [TC−D] VCMC/I + ωC/I × VCMC/I + [TC−D] ωC/I × RTC/CMC + ωC/I × ωC/I × RTC/CMC L~˙ D~ D~ D~ D~ D~ D~ + [TL−D] ωL/I × RTD/TC + ωL/I × ωL/I × RTD/TC + 2 ωL/I × VControl/I L~˙ D ˙ C~ D D D~ + [TL−D] VControl/I + ω~ D/I × RCMD/TD + ω~ D/I × ω~ D/I × RCMD/TD (A.42)

A.8 Capsule Position Kinematics

• Linear Velocity

The capsule linear velocity can be determined deriving thee position vector of the capsule expressed in the equation 4.51 of the Chapter4:

CAP d CAP ~VCM /I = ~RCM /I ⇔ CAP dt CAP I CAP d D d D d CAP ⇔ ~VCM /I = [TD−CAP] ~RCM /I + [TD−CAP] ~RTCAP /CM + ~RCM /TCAP CAP dt D I dt D D I dt CAP D I (A.43)

Using the equation A.17 and the Coriolis equation, the previous equation becomes:

CAP d D CAP D ~VCM /I = [TD−CAP] ~RCM /I + ω~ CAP/I × [TD−CAP] ~RTCAP /CM CAP dt D I D D d D CAP CAP d CAP + [TD−CAP] ~RTCAP /CM + ω~ CAP/I × ~RCM /TCAP + ~RCM /TCAP dt D D CAP CAP D dt CAP D CAP (A.44) 94 Appendix

CAP~ Applying the Poisson equation A.2 and knowing that the position vector RCMCAP/TCAPD is constant in the capsule frame and therefore its derivative is null, the linear velocity of the capsule is given by:

CAP~ D~ CAP~ CAP~ CAP~ D~ VCMCAP/I = [TD−CAP] VCMD/I + ωCAP/I × RTCAPD/CMD + ωD/CAP × [TD−CAP] RTCAPD/CMD

d D CAP CAP + [TD−CAP] ~RTCAP /CM + ω~ CAP/I × ~RCM /TCAP (A.45) dt D D D CAP D

D~ The position vector RTCAPD/CMD is also constant in the drone frame, so its derivative is null. Finally using the relations of the angular velocity expressed in the Section 4.1 of the Chapter4, the linear velocity of the capsule can be written as:

CAP~ D~ CAP~ CAP~ CAP~ CAP~ VCMCAP/I = [TD−CAP] VCMD/I + ωD/I × RTCAPD/CMD + ωCAP/I × RCMCAP/TCAPD (A.46)

• Linear Acceleration

Following the same method developed to the linear acceleration of the drone, linear acceleration of the capsule is deﬁned deriving the capsule linear velocity:

CAP d CAP ~aCM /I = ~VCM /I (A.47) CAP dt CAP I

Using the equation A.46, the linear acceleration is given by:

CAP d D d CAP CAP ~aCM /I = [TD−CAP] ~VCM /I + ω~ D/I × ~RTCAP /CM CAP dt D I dt D D I d CAP CAP + ω~ CAP/I × ~RCM /TCAP (A.48) dt CAP D I

Applying the equation A.17 and the distributive property of the external product, the previous equation becomes:

CAP d D d CAP CAP ~aCM /I = [TD−CAP] ~VCM /I + ω~ D/I × ~RTCAP /CM CAP dt D I dt I D D CAP d CAP d CAP CAP + ω~ D/I × ~RTCAP /CM + ω~ CAP/I × ~RCM /TCAP dt D D I dt I CAP D CAP d CAP + ω~ CAP/I × ~RCM /TCAP (A.49) dt CAP D I A.9 Transponder Position Kinematics 95

Applying now the Coriolis A.2 equation and some conclusions stated in Section A.1, the previous can be written as

d CAP~a = [T ]D~a + [T ]Dω~ × CAP~R CMCAP/I D−CAP CMD/I dt D−CAP D/I D TCAPD/CMD CAP d CAP d CAP CAP + ω~ D/I × ~RTCAP /CM + ω~ CAP/I × ~RCM /TCAP dt D D I dt CAP CAP D CAP d CAP + ω~ CAP/I × ~RCM /TCAP (A.50) dt CAP D I

Using some results derived of the Section A.8 and applying again the equation A.17, the linear acceleration of the capsule is equal to:

CAP D D~˙ CAP~ ~aCMCAP/I = [TD−CAP] ~aCMD/I + [TD−CAP] ωD/I × RTCAPD/CMD CAP~ CAP~ CAP~ CAP~˙ CAP~ + ωD/I × ωD/I × RTCAPD/CMD + ωCAP/I × RCMCAP/TCAPD CAP~ CAP~ CAP~ + ωCAP/I × ωCAP/I × RCMCAP/TCAPD (A.51)

A.9 Transponder Position Kinematics

• Linear Velocity

The linear velocity vector of transponder is deﬁned as the derivative of the position vector of the transponder (deﬁned in equation 4.54):

T d T ~VCM /I = ~RCM /I ⇔ T dt T I T d T d T ⇔ ~VCM /I = ~RCM /I + ~RCM /CM ⇔ T dt CAP I dt T CAP I T d CAP d T ⇔ ~VCM /I = [TCAP−T ] ~RCM /I + ~RCM /CM (A.52) T dt CAP I dt T CAP I

Applying the equation A.17 and the Coriolis equation A.1, the previous equation becomes:

T d CAP T T d T ~VCM /I = [TCAP−T ] ~RCM /I + ω~ T/I × ~RCM /CM + ~RCM /CM (A.53) T dt CAP I T CAP dt T CAP T

Once, in the transponder frame, the transponder is ﬁxed with respect to the capsule, the deriva- d T tive ~RCM /CM is null, so the linear velocity of the transponder can be written as dt T CAP T 96 Appendix

T~ CAP~ T ~ T ~ VCMT /I = [TCAP−T ] VCMCAP/I + ωT/I × RCMT /CMCAP (A.54)

• Linear Acceleration

Finally the linear acceleration of the transponder can be determined deriving the equation A.54 in order to time:

T d T ~aCM /I = ~VCM /I ⇔ T dt T I T d CAP d T T ⇔ ~aCM /I = [TCAP−T ] ~VCM /I + ω~ T/I × ~RCM /CM (A.55) T dt CAP I dt T CAP I

Applying the equation A.17 and the distributive property of the external product, the previous equation becomes:

T d CAP d T T T d T ~aCM /I = [TCAP−T ] ~VCM /I + ω~ T/I × ~RCM /CM + ω~ T/I × ~RCM /CM T dt CAP I dt I T CAP dt T CAP I (A.56)

d T Using some results of Section A.9 about the derivative ~RCM /CM and the conclusion dt T CAP I stated in Section A.1 about the derivative of a angular velocity, the linear acceleration expression of transponder can be written as

T CAP T ~˙ T ~ T ~ T ~ T ~ ~aCMT /I = [TCAP−T ] ~aCMCAP/I + ωT/I × RCMT /CMCAP + ωT/I × ωT/I × RCMT /CMCAP (A.57)

A.10 Apparent Force

The apparent force such as any force is equal to the time derivative of the linear momentum, where in this case the mass is the apparent mass matrix.

d C~Fapp C = − [Mapp] ~VC/A (A.58) dt I

Applying the Coriolis equation A.1 A.11 Matrix M and B 97

C C C d C ~Fapp = − ω~ C/I × [Mapp] ~VC/A + [Mapp] ~VC/A (A.59) dt C

C Replacing ~VC/A by equation 4.65, the previous equation becomes:

C C C d C I ~Fapp = ω~ C/I × [Mapp] ~VC/A − [Mapp] ~VCM /I − [TI−C] ~VA/I ⇔ dt C C C C C d C d I ⇔ ~Fapp = ω~ C/I × [Mapp] ~VC/A − [Mapp] ~VCM /I + [Mapp] [TI−C] ~VA/I (A.60) dt C dt C

Applying the Poisson’s equation A.2 and assuming a constant wind velocity

C~ C C~ C~˙ C C I~ d I~ Fapp = ω~ C/I × [Mapp] VC/A − [Mapp] VCM /I + [Mapp] ω~ I/C × [TI−C] VA/I + [TI−C] VA/I ⇔ C dt I ⇔ C~F = Cω~ × [M ]C~V − [M ]C~V˙ C − [M ] Cω~ × [T ]I~V (A.61) app C/I app C/A app CMC/I app C/I I−C A/I

A.11 Matrix M and B

A.11.1 Matrix M

M10 = eye(3); M21 = eye(3); M32 = (Mc ∗ eye(3) + MAPP); 0 MM35 = −1∗Tcl ∗[0;cos(atan2(lRTd1T 1(3),lRTd1T 1(2)));sin(atan2(lRTd1T 1(3),lRTd1T 1(2)))]; 0 MM36 = −1∗Tcl ∗[0;cos(atan2(lRTd2T 2(3),lRTd2T 2(2)));sin(atan2(lRTd2T 2(3),lRTd2T 2(2)))]; M43 = IC + IAPP; 0 MM45 = −cSRT1cmc∗(Tcl ∗[0;cos(atan2(lRTd1T 1(3),lRTd1T 1(2)));sin(atan2(lRTd1T 1(3),lRTd1T 1(2)))]); 0 MM46 = −cSRT2cmc∗(Tcl ∗[0;cos(atan2(lRTd2T 2(3),lRTd2T 2(2)));sin(atan2(lRTd2T 2(3),lRTd2T 2(2)))]); M54 = 1; M66 = eye(3);

M72 = Md ∗ eye(3) ∗ Tcd;

M73 = −Md ∗ eye(3) ∗ (dSRrcmc + dSRRdr + dSRcmdRd) ∗ Tcd;

M75 = (−Md ∗ eye(3) ∗ (dSRRdr + dSRcmdRd) ∗ Tld ∗ [0;1;0]);

M77 = (−Md ∗ eye(3) ∗ dSRcmdRd);

MM72 = Md ∗ eye(3) ∗ (Tld ∗ [0;−HK;0] + 2 ∗ SdwL ∗ Tld ∗ [0;1;0]);

MM75 = Tld ∗[0;cos(atan2(lRTd1T 1(3),lRTd1T 1(2)));sin(atan2(lRTd1T 1(3),lRTd1T 1(2)))];

MM76 = Tld ∗[0;cos(atan2(lRTd2T 2(3),lRTd2T 2(2)));sin(atan2(lRTd2T 2(3),lRTd2T 2(2)))]; 0 MM77 = −Tdcap ∗ [0;0;1];

M83 = ID ∗ Tcd; 98 Appendix

M85 = ID ∗ Tld ∗ [0;1;0]; M87 = ID;

MM85 = dSRTd1cmd ∗(Tld ∗[0;cos(atan2(lRTd1T 1(3),lRTd1T 1(2)));sin(atan2(lRTd1T 1(3),lRTd1T 1(2)))]);

MM86 = dSRTd2cmd ∗(Tld ∗[0;cos(atan2(lRTd2T 2(3),lRTd2T 2(2)));sin(atan2(lRTd2T 2(3),lRTd2T 2(2)))]); 0 MM87 = −dSRtcapcmp ∗ (Tdcap ∗ [0;0;1]); M98 = [10;01;00];

M102 = Mcap ∗ eye(3) ∗ Tdcap ∗ Tcd;

M103 = −Mcap ∗ eye(3) ∗ Tdcap ∗ (dSRrcmc + dSRRdr + dSRcmdRd) ∗ Tcd − Mcap ∗ eye(3) ∗

(capSRtcapcmd ∗ (Tdcap ∗ Tcd) + capSRcapt cap ∗ (Tdcap ∗ Tcd));

M105 = (−Mcap ∗ eye(3) ∗ Tdcap ∗ (dSRRdr + dSRcmdRd) ∗ Tld ∗ [0;1;0]) − Mcap ∗ eye(3) ∗

(capSRtcapcmd ∗(Tdcap∗Tld ∗[0;1;0]))−Mcap∗eye(3)∗(capSRcapt cap∗(Tdcap∗Tld ∗[0;1;0]));

M107 = (−Mcap∗eye(3)∗(Tdcap∗dSRcmdRd +capSRtcapcmd ∗Tdcap+capSRcapt cap∗Tdcap));

M109 = −Mcap ∗ eye(3) ∗ (capSRcapt cap) ∗ [10;01;00];

MM102 = Mcap ∗ eye(3) ∗ Tdcap ∗ (Tld ∗ [0;−HK;0] + 2 ∗ SdwL ∗ Tld ∗ [0;1;0]); MM107 = [0;0;1]; 0 MM108 = −Tcapt ∗ [0;0;1]; MM110 = [10;01;00];

M122 = Mt ∗ eye(3) ∗ Tcapt ∗ (Tdcap ∗ Tcd);

M123 = −Mt ∗eye(3)∗Tcapt ∗(Tdcap∗(dSRrcmc+dSRRdr +dSRcmdRd)∗Tcd)−Mt ∗eye(3)∗

Tcapt ∗(capSRtcapcmd ∗(Tdcap∗Tcd)+capSRcapt cap∗(Tdcap∗Tcd))−Mt ∗eye(3)∗tSRtcap∗

(Tcapt ∗ (Tdcap ∗ Tcd));

M125 = (−Mt ∗eye(3)∗Tcapt ∗(Tdcap∗(dSRRdr +dSRcmdRd))∗Tld ∗[0;1;0])−Mt ∗eye(3)∗

Tcapt ∗(capSRtcapcmd ∗(Tdcap∗(Tld ∗[0;1;0])))−Mt ∗eye(3)∗Tcapt ∗(capSRcapt cap∗(Tdcap∗

Tld ∗ [0;1;0])) − Mt ∗ eye(3) ∗ (tSRtcap) ∗ (Tcapt ∗ (Tdcap ∗ Tld ∗ [0;1;0]));

M127 = (−Mt ∗eye(3)∗Tcapt ∗(Tdcap∗dSRcmdRd +capSRtcapcmd ∗Tdcap+capSRcapt cap∗

Tdcap)) − Mt ∗ eye(3) ∗ (tSRtcap) ∗ Tcapt ∗ Tdcap;

M129 = −Mt ∗ eye(3) ∗ Tcapt ∗ (capSRcapt cap) ∗ [10;01;00] − Mt ∗ eye(3) ∗ (tSRtcap) ∗ Tcapt ∗ [10;01;00];

MM121 = −Mt ∗ eye(3) ∗ (tSRtcap) ∗ [10;01;00];

MM122 = Mt ∗ eye(3) ∗ Tcapt ∗ (Tdcap ∗ (Tld ∗ [0;−HK;0] + 2 ∗ SdwL ∗ Tld ∗ [0;1;0])); MM128 = [0;0;1]; MM132 = 1; MM143 = 1; MM154 = 1; And the remaining are zero matrices.

A.11.2 Matrix N

0 N1 = Tic ∗ cVC; N2 = TanglesC ∗ cWC;

N3 = −(Mc ∗ eye(3) ∗ ScwC ∗ cVC) − MAPP ∗ ScwC ∗ (Tic ∗VaI) − ScwC ∗ (MAPP ∗Vwind) + A.12 Matrix A and B 99 cFwc + cFA;

N4 = cSRCPccmc ∗ cFA + cMA − (ScwC ∗ ((IC + IAPP) ∗ cWC));

N5 = lqLC;

N6 = TanglesDL ∗ dWDL;

N7 = dFwd + dFd − Md ∗ eye(3) ∗ (SdwC ∗ (Tcd ∗ cVC) + SdwC ∗ (SdwC ∗ dRrcmc) + SdwL ∗

(SdwL ∗ dRRdr) + SdwD ∗ (SdwD ∗ dRcmdRd) − dSRRdr ∗ Tld ∗ (−SlwL ∗ lWC) − dSRcmdRd ∗

Tld ∗ (−SlwL ∗ lWC) − dSRcmdRd ∗ (−SdwD ∗ dWL));

N8 = −ID∗(−Tld ∗(SlwL∗lWC)−SdwD∗dWL)−(SdwD∗(ID∗dWD))+dSRCPdcmd ∗dFd + dMd;

N9 = [100;0cos(capphiCAPD)sin(capphiCAPD);0−sin(capphiCAPD)cos(capphiCAPD)]/capWCAPD;

N10 = capFwcap +capFcap −Mcap ∗ eye(3)∗ Tdcap ∗ (SdwC ∗ (Tcd ∗ cVC) +SdwC ∗ (SdwC ∗ dRrcmc)+SdwL∗(SdwL∗dRRdr)+SdwD∗(SdwD∗dRcmdRd)−dSRRdr ∗Tld ∗(−SlwL∗lWC)− dSRcmdRd ∗Tld ∗(−SlwL∗lWC)−dSRcmdRd ∗(−SdwD∗dWL))−Mcap∗eye(3)∗(−capSRtcapcmd ∗

(Tdcap∗(Tld ∗(−SlwL∗lWC)−SdwD∗dWL))+ScapwD∗(ScapwD∗capRtcapcmd)−capSRcapt cap∗

(Tdcap∗(Tld ∗(−SlwL∗lWC)−SdwD∗dWL)−ScapwCAP∗capWD)+ScapwCAP∗(ScapwCAP∗ capRcapt cap));

N11 = [100;0cos(t phiTCAP)sin(t phiTCAP);0 − sin(t phiTCAP)cos(t phiTCAP)]/tWTCAP;N12 = tFwt + tFT − Mt ∗ eye(3) ∗ Tcapt ∗ Tdcap ∗ (SdwC ∗ (Tcd ∗ cVC) + SdwC ∗ (SdwC ∗ dRrcmc) +

SdwL∗(SdwL∗dRRdr)+SdwD∗(SdwD∗dRcmdRd)−dSRRdr ∗Tld ∗(−SdwL∗lWC)−dSRcmdRd ∗

Tld ∗(−SdwL∗lWC)−dSRcmdRd ∗(−SdwD∗dWL))−Mt ∗eye(3)∗Tcapt ∗(−capSRtcapcmd ∗

(Tdcap∗(Tld ∗(−SdwL∗lWC)−SdwD∗dWL))+ScapwD∗(ScapwD∗capRtcapcmd)−capSRcapt cap∗

(Tdcap∗(Tld ∗(−SdwL∗lWC)−SdwD∗dWL)−ScapwCAP∗capWD)+ScapwCAP∗(ScapwCAP∗ capRcapt cap)) − Mt ∗ eye(3) ∗ (−tSRtcap) ∗ (Tcapt ∗ (Tdcap ∗ (Tld ∗ (−SdwL ∗ lWC) − SdwD ∗ dWL) − ScapwCAP ∗ capWD) − StwT ∗tWCAP) − Mt ∗ eye(3) ∗ (StwT ∗ (StwT ∗tRtcap)); N13 = HK ∗ (Hu − deltaL); N14 = VK ∗ (Vu − ServoR); N15 = VK ∗ (Vu − ServoL);

A.12 Matrix A and B

(Not ﬁnished...) 100 Appendix References

[1] Aircraft Control and Simulation. John Wiley & Sons, Inc, Second edition, 2003.

[2] Mário Martins de Sousa. Autonomous paraglider for straplex. Master’s thesis, Faculdade de Engenharia da Universidade do Porto, July 2012.

[3] User’s Manual.

[4] Gi-Bong Hur and John Valasek. System identification of powered parafoil-vehicle from flight test data. AIAA atmospheric flight mechanics conference and exhibit, August 2003.

[5] Oleg A. Yakimenko and Roman B. Statnikov. Multicriteria parametrical identiﬁcation of the parafoil-load delivery systen. 18th AIAA Aerodynamic Decelerator Systems Technology Conference and Seminar, 2005.

[6] Nathan Slegers and Mark Costello. Model predictive control of a parafoil and payload system. JOURNAL OF GUIDANCE, CONTROL, AND DYNAMICS Vol. 28, No. 4, July-August 2005.

[7] Jr. Sanger M. Burk and George M. Ware. Static aerodynamic characteristics of three ram- air-inﬂated low-aspect-ratio fabric wings. NASA TECHNICAL NOTE D-4182, September 1967.

[8] JOHN D. NICOLAIDES and MICHAEL A. TRAGARZ. Parafoil ﬂight performance. Tech- nical report, UNIVERSITY OF NOTRE DAME, June 1971.

[9] J.Stephen Lingar. Ram-air parachute design. 13th AIAA Aerodynamic Decelerator Systems Technology Conference, May 1995.

[10] Chiara Toglia and Marilena Vendittelli. Modeling and motion analysis of autonomous paragliders. Technical report, Università di Roma "La Sapienza", May 2010.

[11] Marilena Vendittelli Chiara Toglia and Leonardo Lanari. Path following for an autonomous paraglider. 49th IEEE Conference on Decision and Control, December 2010.

[12] Masahito Watanabe and Yoshimasa Ochi. Linear model of a powered paraglider and observer design. SICE Annual Conference 2008, August 2008.

[13] Nathan Slegers and Mark Costello. Aspects of control for a parafoil and payload system. JOURNAL OF GUIDANCE, CONTROL, AND DYNAMICS Vol. 26, No. 6, Novem- ber–December 2003.

[14] Nathan J. Slegers. Effects of canopy-payload relative motion on control of autonomous parafoils. JOURNAL OF GUIDANCE, CONTROL, AND DYNAMICS Vol. 33, No. 1, January- February 2010.

101 102 REFERENCES

[15] Vladimir N. Dobrokhodov Richard M. Howard Paul A. Mortaloni, Oleg A. Yakimenko. On the development of a six-degree-of-freedom model of a low-aspect-ratio parafoil delivery system. 17th AIAA Aerodynamic Decelerator Systems Technology Conference and Seminar, May 2003.

[16] Giorgio Gulieri. Parachute-payload system ﬂight dynamics and trajectory simulation. In- ternational Journal of Aerospace Engineering Volume 2012, Article ID 182907, 17 pages, March 2012.

[17] Subspace Methods for System Identiﬁcation (Communications and Control Engineering). Springer, 2005.

[18] Inc. Coastal Wind Sports. Choosing your ﬁrst traction kite, May 2013. URL: http:// www.coastalwindsports.com/ChoosingAKite.html.

[19] Federal Aviation Administration. Altitude reporting equipment and transponder system maintenance and inspection practices. Advisory Circulars, No. 43-6C, September 2012.

[20] P. Lissaman and G. Brown. Apparent mass effects on parafoil dynamics. AIAA Paper 93- 1236, May 1993.

[21] Slotine JJE Li W. Applied Nonlinear Control. Prentice Hall, 1st edition, 1991.

[22] Chi-Tsong Chen. Linear System Theory and Design. Oxford University Press, third edition edition, 1999.