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The Tricopter: an Agile UAV 117 6.1 Design of the Tricopter ECOLE DOCTORALE UNIVERSITÉ PARIS-SUD ÉCOLE DOCTORALE : Sciences et Technologie de l’Information, des Télécommunications et des Systèmes Laboratoire des Signaux et Systèmes (L2S) Centre de Robotique MinesParisTech (CAOR) DISCIPLINE : Physique THÈSE DE DOCTORAT Présentée et soutenue publiquement par Étienne SERVAIS le 18 septembre 2015. TRAJECTORY PLANNING AND CONTROL OF COLLABORATIVE SYSTEMS: APPLICATION TO TRIROTOR UAVS. Directeurs de thèse : Brigitte d’ANDRÉA-NOVEL Professeur (Mines ParisTech) Hugues MOUNIER Professeur (Université Paris XI) Composition du jury : Rapporteurs : Tarek HAMEL Professeur (Université de Nice Sophia Antipolis) Miroslav KRSTIC´ Professeur (Université de Californie à San Diego) Examinateurs : Jean-Michel CORON Professeur (Université Pierre et Marie Curie) Joachim RUDOLPH Professeur (Université de la Sarre) Claude SAMSON Directeur de recherche (INRIA) Membres invités : Silviu-Iulian NICULESCU Directeur de recherche (CNRS) Arnaud QUADRAT Ingénieur (Sagem-DS) Planification de trajectoire et contrôle d’un système collaboratif : Application à un drone trirotor. Résumé : L’objet de cette thèse est de proposer un cadre complet, du haut niveau au bas niveau, de génération de trajectoires pour un groupe de systèmes dynamiques indé- pendants. Ce cadre, basé sur la résolution de l’équation de Burgers pour la génération de trajectoires, est appliqué à un modèle original de drone trirotor et utilise la platitude des deux systèmes différentiels considérés. La première partie du manuscrit est consacrée à la génération de trajectoires. Celle-ci est effectuée en créant formellement, par le biais de la platitude du système considéré, des so- lutions à l’équation de la chaleur. Ces solutions sont transformées en solution de l’équation de Burgers par la transformation de Hopf-Cole pour correspondre aux formations voulues. Elles sont optimisées pour répondre à des contraintes spécifiques. Plusieurs exemples de trajectoires sont donnés. La deuxième partie est consacrée au suivi autonome de trajectoire par un drone trirotor. Ce drone est totalement actionné et un contrôleur en boucle fermée non-linéaire est pro- posé. Celui-ci est testé en suivant, en roulant, des trajectoires au sol et en vol. Un modèle est présenté et une démarche pour le contrôle est proposée pour transporter une charge pendulaire. Mots clés : Commande par platitude, génération de trajectoires, équation de Burgers’, systèmes multi-agent, optimisation par essaims particulaires, drone, rotors inclinables, commande non-linéaire, transport de charge pendulaire. Trajectory planning and control of collaborative systems: Application to trirotor UAVs. Abstract: This thesis is dedicated to the creation of a complete framework, from high- level to low-level, of trajectory generation for a group of independent dynamical systems. This framework, based for the trajectory generation, on the resolution of Burgers equa- tion, is applied to a novel model of trirotor UAV and uses the flatness of the two levels of dynamical systems. The first part of this thesis is dedicated to the generation of trajectories. Formal so- lutions to the heat equation are created using the differential flatness of this equation. These solutions are transformed into solutions to Burgers’ equation through Hopf-Cole transformation to match the desired formations. They are optimized to match specific requirements. Several examples of trajectories are given. The second part is dedicated to the autonomous trajectory tracking by a trirotor UAV. This UAV is totally actuated and a nonlinear closed-loop controller is suggested. This controller is tested on the ground and in flight by tracking, rolling or flying, a trajectory. A model is presented and a control approach is suggested to transport a pendulum load. Keywords: Differential flatness, trajectory generation, Burgers’ equation, multi-agent systems, particle swarm optimization, UAV, tilting rotors, non-linear control, pendulum load transportation. License This work is, unless otherwise noticed, the sole work of its author and is released, unless otherwise noticed, under the license Creative Commons Attribution-Non Commercial 4.0 International (CC BY-NC 4.0). As a consequence, you are free to: Share — copy and redistribute this dissertation in any medium or format. Adapt — remix, transform, and build upon the material released under this license. As long as you conform to the three following terms: Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor, i.e. the author, endorses you or your use. NonCommercial — You may not use the material for commercial purposes. Respect of other licenses — You must conform to the licenses applicable to contents of this dissertation created by other authors. Please refer to this work as: Servais, Étienne. Trajectory planning and control of collaborative systems: applications to trirotor UAVs. PhD thesis, 2015, Université Paris-Sud (France). Contents License i Contents iii List of Figures vi Remerciements ix Introduction xi Résumé étendu en français 1 I Trajectory planning for multi-agent systems 15 1 Motion planning for multi-agent systems, an overview 17 1.1 Different methods of managing a multi-agent system . 19 1.1.1 Behavioral methods ................................. 20 1.1.2 Methods based on Particle Swarm Optimization . 22 1.1.3 Potential methods .................................. 24 1.1.4 Graph based methods ................................ 27 1.1.5 Methods based on Partial Differential Equations . 28 1.2 Different problems of collaborative systems ....................... 31 1.2.1 Deployment problems ................................ 31 1.2.2 Cooperative transportation of a swinging load . 35 1.2.3 Collision avoidance on determined paths .................... 38 2 Trajectory generation for PDE systems, existing works and available tools 41 2.1 Existing works: an overview ................................ 41 2.2 Differential flatness and Gevrey functions ........................ 47 2.2.1 Gevrey functions, definition and examples ................... 47 2.2.2 Efficient computation of the derivatives of Φσ . 49 3 Formal solutions to the heat equation 53 3.1 Rewriting the heat equation with formal differential operators of infinite order . 53 3.2 The heat equation with controls on both sides ..................... 55 3.2.1 Objectives ....................................... 55 3.2.2 Formal derivation .................................. 55 3.2.3 Computational implementation .......................... 61 Contents 3.3 Various properties of the Tx operator ........................... 62 3.3.1 Polynomial states and controls .......................... 62 3.3.2 Application to the Weierstrass approximation theorem . 66 3.3.3 Convergence of the Tx operator .......................... 67 3.3.4 Application of Tx to exponential functions ................... 69 3.3.5 The operator Tx and product of functions .................... 73 3.4 The Hopf-Cole transformation and Gevrey functions . 79 4 PDE-based motion planning framework 81 4.1 Generating solutions to the heat equation ........................ 81 4.2 Generating solutions to Burgers’ equation ........................ 85 4.2.1 Optimization of the trajectories .......................... 85 4.2.2 Leaders and followers ................................ 87 4.2.3 Transition between successive steps ....................... 87 4.3 Combining solutions to create trajectories ........................ 89 4.4 Conclusion and perspectives ................................ 91 II Modeling and control of a trirotor UAV 93 5 Unmanned Aerial Vehicles: a brief review 95 5.1 Flight dynamics ........................................ 95 5.1.1 Basics of flight .................................... 95 5.1.2 The roll-pitch-yaw convention and aircrafts’ centers . 97 5.1.3 Fixed-wing aircrafts ................................. 98 5.1.4 Rotary-wing aircrafts ................................100 5.1.5 Convertible airplanes and other VTOL aircrafts . 104 5.2 A usual model, the quadrotor ................................105 5.2.1 Design and basic model of a quadrotor helicopter . 106 5.2.2 Control and applications for a quadrotor UAV . 108 5.3 Tilting rotor multirotor UAVs ................................115 6 The tricopter: an agile UAV 117 6.1 Design of the tricopter ....................................118 6.1.1 Mechanics design ..................................118 6.1.2 Electronics design ..................................120 6.1.3 The motion capture platform . 122 6.2 Mechanical model of the tricopter .............................122 6.2.1 Formalism and assumptions ............................122 6.2.2 Screws acting on the tricopter . 126 6.2.3 Equations of motion .................................130 6.3 Introduction to flatness based control . 130 7 Various applications of the tricopter 133 7.1 Simulation platform, test trajectory and control approach . 133 7.1.1 The simulation platform ..............................133 7.1.2 The test trajectory ..................................135 iv Revision: a466fdc (2015-10-14 23:06:42 +0200) Contents 7.1.3 The control approach ................................136 7.2 The rolling tricopter .....................................138 7.2.1 Motivations and mechanical modifications . 138 7.2.2 Controlling the rolling tricopter, a flatness-based control approach . 139 7.2.3 Admissible trajectories and open-loop simulations . 142 7.2.4 Experiments
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