Optimal Trajectories of a UAV Base Station Using Hamilton-Jacobi Equations

We consider the problem of optimizing the trajectory of an Unmanned Aerial Vehicle (UAV). Assuming a traffic intensity map of users to be served, the UAV must travel from a given initial location to a final position within a given duration and serves the traffic on its way. The problem consists in finding the optimal trajectory that minimizes a certain cost depending on the velocity and on the amount of served traffic. We formulate the problem using the framework of Lagrangian mechanics. We derive closed-form formulas for the optimal trajectory when the traffic intensity is quadratic (single-phase) using Hamilton-Jacobi equations. When the traffic intensity is bi-phase, i.e. made of two quadratics, we provide necessary conditions of optimality that allow us to propose a gradient-based algorithm and a new algorithm based on the linear control properties of the quadratic model. These two solutions are of very low complexity because they rely on fast convergence numerical schemes and closed form formulas. These two approaches return a trajectory satisfying the necessary conditions of optimality. At last, we propose a data processing procedure based on a modified K-means algorithm to derive a bi-phase model and an optimal trajectory simulation from real traffic data.Comment: 30 pages, 10 figures, 2 tables. arXiv admin note: substantial text overlap with arXiv:1812.0875

A Survey on Aerial Swarm Robotics

The use of aerial swarms to solve real-world problems has been increasing steadily, accompanied by falling prices and improving performance of communication, sensing, and processing hardware. The commoditization of hardware has reduced unit costs, thereby lowering the barriers to entry to the field of aerial swarm robotics. A key enabling technology for swarms is the family of algorithms that allow the individual members of the swarm to communicate and allocate tasks amongst themselves, plan their trajectories, and coordinate their flight in such a way that the overall objectives of the swarm are achieved efficiently. These algorithms, often organized in a hierarchical fashion, endow the swarm with autonomy at every level, and the role of a human operator can be reduced, in principle, to interactions at a higher level without direct intervention. This technology depends on the clever and innovative application of theoretical tools from control and estimation. This paper reviews the state of the art of these theoretical tools, specifically focusing on how they have been developed for, and applied to, aerial swarms. Aerial swarms differ from swarms of ground-based vehicles in two respects: they operate in a three-dimensional space and the dynamics of individual vehicles adds an extra layer of complexity. We review dynamic modeling and conditions for stability and controllability that are essential in order to achieve cooperative flight and distributed sensing. The main sections of this paper focus on major results covering trajectory generation, task allocation, adversarial control, distributed sensing, monitoring, and mapping. Wherever possible, we indicate how the physics and subsystem technologies of aerial robots are brought to bear on these individual areas

A Geometric Approach to Trajectory Planning for Underactuated Mechanical Systems

In the last decade, multi-rotors flying robots had a rapid development in industry and hobbyist communities thanks to the affordable cost and their availability of parts and components. The high number of degrees of freedom and the challenging dynamics of multi-rotors gave rise to new research problems. In particular, we are interested in the development of technologies for an autonomous fly that would al- low using multi-rotors systems to be used in contexts where the presence of humans is denied, for example in post-disaster areas or during search-and-rescue operations. Multi-rotors are an example of a larger category of robots, called \u201cunder-actuated mechanical systems\u201d (UMS) where the number of actuated degrees of freedom (DoF) is less than the number of available DoF. This thesis applies methods com- ing from geometric mechanics to study the under-actuation problem and proposes a novel method, based on the Hamiltonian formalism, to plan a feasible trajectory for UMS. We first show the application of a method called \u201cVariational Constrained System approach\u201d to a cart-pole example. We discovered that it is not possible to extend this method to generic UMS because it is valid only for a sub-class of UMS, called \u201csuper-articulated\u201d mechanical system. To overcome this limitation, we wrote the Hamilton equations of the quad- rotor and we apply a numerical \u201cdi- rect method\u201d to compute a feasible trajectory that satisfies system and endpoint constraints. We found that by including the system energy in the multi-rotor states, we are able to compute maneuvers that cannot be planned with other methods and that overcome the under-actuation constraints. To demonstrate the benefit of the method developed, we built a custom quad- rotor and an experimental setup with different obstacles, such as a gap in a wall and we show the correctness of the trajectory computed with the new method

Conflict resolution algorithms for optimal trajectories in presence of uncertainty

Mención Internacional en el título de doctorThe objective of the work presented in this Ph.D. thesis is to develop a novel method to address the aircraft-obstacle avoidance problem in presence of uncertainty, providing optimal trajectories in terms of risk of collision and time of flight. The obstacle avoidance maneuver is the result of a Conflict Detection and Resolution (CD&R) algorithm prepared for a potential conflict between an aircraft and a fixed obstacle which position is uncertain. Due to the growing interest in Unmanned Aerial System (UAS) operations, CD&R topic has been intensively discussed and tackled in literature in the last 10 years. One of the crucial aspects that needs to be addressed for a safe and efficient integration of UAS vehicles in non-segregated airspace is the CD&R activity. The inherent nature of UAS, and the dynamic environment they are intended to work in, put on the table of the challenges the capability of CD&R algorithms to handle with scenarios in presence of uncertainty. Modeling uncertainty sources accurately, and predicting future trajectories taking into account stochastic events, are rocky issues in developing CD&R algorithms for optimal trajectories. Uncertainty about the origin of threats, variable weather hazards, sensing and communication errors, are only some of the possible uncertainty sources that make jeopardize air vehicle operations. In this work, conflict is defined as the violation of the minimum distance between a vehicle and a fixed obstacle, and conflict avoidance maneuvers can be achieved by only varying the aircraft heading angle. The CD&R problem, formulated as Optimal Control Problem (OCP), is solved via indirect optimal control method. Necessary conditions of optimality, namely, the Euler-Lagrange equations, obtained from calculus of variations, are applied to the vehicle dynamics and the obstacle constraint modeled as stochastic variable. The implicit equations of optimality lead to formulate a Multipoint Boundary Value Problem (MPBVP) which solution is in general not trivial. The structure of the optimality trajectory is inferred from the type of path constraint, and the trend of Lagrange multiplier is analyzed along the optimal route. The MPBVP is firstly approximated by Taylor polynomials, and then solved via Differential Algebra (DA) techniques. The solution of the OCP is therefore a set of polynomials approximating the optimal controls in presence of uncertainty, i.e., the optimal heading angles that minimize the time of flight, while taking into account the uncertainty of the obstacle position. Once the obstacle is detected by on-board sensors, this method provide a useful tool that allows the pilot, or remote controller, to choose the best trade-off between optimality and collision risk of the avoidance maneuver. Monte Carlo simulations are run to validate the results and the effectiveness of the method presented. The method is also valid to address CD&R problems in presence of storms, other aircraft, or other types of hazards in the airspace characterized by constant relative velocity with respect to the own aircraft.L’obiettivo del lavoro presentato in questa tesi di dottorato è la ricerca e lo sviluppo di un nuovo metodo di anti collisione velivolo-ostacolo in presenza di incertezza, fornendo traiettorie ottimali in termini di rischio di collisione e tempo di volo. La manovra di anticollisione è il risultato di un algoritmo di detezione e risoluzione dei conflitti, in inglese Conflict Detection and Resolution (CD&R), che risolve un potenziale conflitto tra un velivolo e un ostacolo fisso la cui posizione è incerta. A causa del crescente interesse nelle operazioni che coinvolgono velivoli autonomi, anche definiti Unmanned Aerial System (UAS), negli ultimi 10 anni molte ricerche e sviluppi sono state condotte nel campo degli algoritmi CD&R. Uno degli aspetti cruciali per un’integrazione sicura ed efficiente dei velivoli UAS negli spazi aerei non segregati è l’attività CD&R. La natura intrinseca degli UAS e l’ambiente dinamico in cui sono destinati a lavorare, impongono delle numerose sfide fra cui la capacità degli algoritmi CD&R di gestire scenari in presenza di incertezza. La modellizzazione accurata delle fonti di incertezza e la previsione di traiettorie che tengano conto di eventi stocastici, sono problemi particolarmente difficoltosi nello sviluppo di algoritmi CD&R per traiettorie ottimali. L’incertezza sull’origine delle minacce, zone di condizioni metereologiche avverse al volo, errori nei sensori e nei sistemi di comunicazione per la navigazione aerea, sono solo alcune delle possibili fonti di incertezza che mettono a repentaglio le operazioni degli aeromobili. In questo lavoro, il conflitto è definito come la violazione della distanza minima tra un veicolo e un ostacolo fisso, e le manovre per evitare i conflitti possono essere ottenute solo variando l’angolo di rotta dell’aeromobile, ovvero virando. Il problema CD&R, formulato come un problema di controllo ottimo, o Optimal Control Problem (OCP), viene risolto tramite un metodo indiretto. Le condizioni necessarie di ottimalità, vale a dire le equazioni di Eulero-Lagrange derivanti dal calcolo delle variazioni, sono applicate alla dinamica del velivolo e all’ostacolo modellizato come una variabile stocastica. Le equazioni implicite di ottimalità formano un problema di valori al controno multipunto, Multipoint Boundary Value Problem(MPBVP), la cui soluzione in generale è tutt’altro che banale. La struttura della traiettoria ottimale viene dedotta dal tipo di vincolo, e l’andamento del moltiplicatore di Lagrange viene analizzato lungo il percorso ottimale. Il MPBVP viene prima approssimato con un spazio di polinomi di Taylor e successimvamente risolto tramite tecniche di algebra differenziale, in inglese Differential Algebra (DA). La soluzione del OCP è quindi un insieme di polinomi che approssima il controllo ottimo del problema in presenza di incertezza. In altri termini, il controllo ottimo è l’insieme degli angoli di prua del velivolo che minimizzano il tempo di volo e che tenendo conto dell’incertezza sulla posizione dell’ostacolo. Quando l’ostacolo viene rilevato dai sensori di bordo, questo metodo fornisce un utile strumento al pilota, o al controllore remoto, al fine di scegliere il miglior compromesso tra ottimalità e rischio di collisione con l’ostacolo. Simulazioni Monte Carlo sono eseguite per convalidare i risultati e l’efficacia del metodo presentato. Il metodo è valido anche per affrontare problemi CD&R in presenza di tempeste, altri velivoli, o altri tipi di ostacoli caratterizzati da una velocità relativa costante rispetto al proprio velivolo.El objetivo del trabajo presentado en esta tesis doctoral es la búsqueda y el desarrollo de un método novedoso de anticolisión con osbstáculos en espacios aéreos en presencia de incertidumbre, proporcionando trayectorias óptimas en términos de riesgo de colisión y tiempo de vuelo. La maniobra de anticolisión es el resultado de un algoritmo de detección y resolución de conflictos, en inglés Conflict Detection and Resolution (CD&R), preparado para un conflicto potencial entre una aeronave y un obstáculo fijo cuya posición es incierta. Debido al creciente interés en las operaciones de vehículos autónomos, también definidos como Unmanned Aerial System (UAS), en los últimos 10 años muchas investigaciones se han llevado a cabo en el tema CD&R. Uno de los aspectos cruciales que debe abordarse para una integración segura y eficiente de los vehículos UAS en el espacio aéreo no segregado es la actividad CD&R. La naturaleza intrínseca de UAS, y el entorno dinámico en el que están destinados a trabajar, suponen un reto para la capacidad de los algoritmos de CD&R de trabajar con escenarios en presencia de incertidumbre. La precisa modelización de las fuentes de incertidumbre, y la predicción de trayectorias que tengan en cuenta los eventos estocásticos, son problemas muy difíciles en el desarrollo de algoritmos CD&R para trayectorias óptimas. La incertidumbre sobre el origen de las amenazas, condiciones climáticas adversas, errores en sensores y sistemas de comunicación para la navegación aérea, son solo algunas de las posibles fuentes de incertidumbre que ponen en peligro las operaciones de los vehículos aéreos. En este trabajo, el conflicto se define como la violación de la distancia mínima entre un vehículo y un obstáculo fijo, y las maniobras de anticolisión se pueden lograr variando solo el ángulo de rumbo de la aeronave, es decir virando. El problema CD&R, formulado como problema de control óptimo, o Optimal Control Problem (OCP), se resuelve a través del método de control óptimo indirecto. Las condiciones necesarias de optimalidad, es decir, las ecuaciones de Euler-Lagrange que se obtienen a partir del cálculo de variaciones, son aplicadas a la dinámica de la aeronave y al obstáculo modelizado como variable estocástica. Las ecuaciones implícitas de optimalidad forman un problema de valor de frontera multipunto (MPBVP) cuya solución en general no es trivial. La estructura de la trayectoria de optimalidad se deduce del tipo de vínculo, y la tendencia del multiplicador de Lagrange se analiza a lo largo de la ruta óptima. El MPBVP se aproxima en primer lugar a través de un espacio de polinomios de Taylor, y luego se resuelve por medio de técnicas de álgebra diferencial, en inglés Differential Algebra(DA). La solución del OCP es un conjunto de polinomios que aproximan los controles óptimos en presencia de incertidumbre, es decir, los ángulos de rumbo óptimos que minimizan el tiempo de vuelo teniendo en cuenta la incertidumbre asociada a la posición del obstáculo. Una vez que los sensores a bordo detectan el obstáculo, este método proporciona una herramienta muy útil que permite al piloto, o control remoto, elegir el mejor compromiso entre optimalidad y riesgo de colisión con el obstáculo. Se ejecutan simulaciones de Monte Carlo para validar los resultados y la efectividad del método presentado. El método también es válido para abordar los problemas de CD&R en presencia de tormentas, otras aeronaves u otros tipos de obstáculos caracterizados por una velocidad relativa constante con respecto a la propia aeronave.Programa de Doctorado en Mecánica de Fluidos por la Universidad Carlos III de Madrid; la Universidad de Jaén; la Universidad de Zaragoza; la Universidad Nacional de Educación a Distancia; la Universidad Politécnica de Madrid y la Universidad Rovira i VirgiliPresidente: Carlo Novara.- Secretario: Lucia Pallotino.- Vocales: Manuel Sanjurjo Rivo; Yoshinori Matsuno; Alfonso Valenzuela Romer

Optimal trajectory generation with DMOC versus NTG : application to an underwater glider and a JPL aerobot.

Optimal trajectory generation is an essential part for robotic explorers to execute the total exploration of deep oceans or outer space planets while curiosity of human and technology advancements of society both require robots to search for unknown territories efficiently and safely. As one of state-of-the-art optimal trajectory generation methodologies, Nonlinear Trajectory Generation (NTG) combines with B-spline, nonlinear programming, differential flatness technique to generate optimal trajectories for modelled mechanical systems. While Discrete Mechanics and Optimal Control (DMOC) is a newly proposed optimal control method for mechanical systems, it is based on direct discretization of Lagrange-d\u27Alembert principle. In this dissertation, NTG is utilized to generate trajectories for an underwater glider with a 3D B-spline ocean current model. The optimal trajectories are corresponding well with the Lagrangian Coherent Structures (LCS). Then NTG is utilized to generate 3D opportunistic trajectories for a JPL (Jet Propulsion Laboratory) Aerobot by taking advantage of wind velocity. Since both DMOC and NTG are methods which can generate optimal trajectories for mechanical systems, their differences in theory and application are investigated. In a simple ocean current example and a more complex ocean current model, DMOC with discrete Euler-Lagrange constraints generates local optimal solutions with different initial guesses while NTG is also generating similar solutions with more computation time and comparable energy consumption. DMOC is much easier to implement than NTG because in order to generate good solutions in NTG, its variables need to be correctly defined as B-spline variables with rightly-chosen orders. Finally, the MARIT (Multiple Air Robotics Indoor Testbed) is established with a Vicon 8i motion capture system. Six Mcam 2 cameras connected with a datastation are able to track real-time coordinates of a draganflyer helicopter. This motion capture system establishes a good foundation for future NTG and DMOC algorithms verifications

Robots in Agriculture: State of Art and Practical Experiences

The presence of robots in agriculture has grown significantly in recent years, overcoming some of the challenges and complications of this field. This chapter aims to collect a complete and recent state of the art about the application of robots in agriculture. The work addresses this topic from two perspectives. On the one hand, it involves the disciplines that lead the automation of agriculture, such as precision agriculture and greenhouse farming, and collects the proposals for automatizing tasks like planting and harvesting, environmental monitoring and crop inspection and treatment. On the other hand, it compiles and analyses the robots that are proposed to accomplish these tasks: e.g. manipulators, ground vehicles and aerial robots. Additionally, the chapter reports with more detail some practical experiences about the application of robot teams to crop inspection and treatment in outdoor agriculture, as well as to environmental monitoring in greenhouse farming

Planification et commande d'une plate-forme aéroportée stationnaire autonome dédiée à la surveillance des ouvrages d'art

Today, the inspection of structures is carried out through visual assessments effectedby qualified inspectors. This procedure is very expensive and can put the personal indangerous situations. Consequently, the development of an unmanned aerial vehicleequipped with on-board vision systems is privileged nowadays in order to facilitate theaccess to unreachable zones.In this context, the main focus in the thesis is developing original methods to deal withplanning, reference trajectories generation and tracking issues by a hovering airborneplatform. These methods should allow an automation of the flight in the presence of airdisturbances and obstacles. Within this framework, we are interested in two kinds ofaerial vehicles with hovering capacity: airship and quad-rotors.Aujourd'hui, l'inspection des ouvrages d'art est réalisée de façon visuelle par des contrôleurs sur l'ensemble de la structure. Cette procédure est coûteuse et peut être particulièrement dangereuse pour les intervenants. Pour cela, le développement du système de vision embarquée sur des drones est privilégié ces jours-ci afin de faciliter l'accès aux zones dangereuses.Dans ce contexte, le travail de cette thèse porte sur l'obtention des méthodes originales permettant la planification, la génération des trajectoires de référence, et le suivi de ces trajectoires par une plate-forme aéroportée stationnaire autonome. Ces méthodes devront habiliter une automatisation du vol en présence de perturbations aérologiques ainsi que des obstacles. Dans ce cadre, nous nous sommes intéressés à deux types de véhicules aériens capable de vol stationnaire : le dirigeable et le quadri-rotors.Premièrement, la représentation mathématique du véhicule volant en présence du vent a été réalisée en se basant sur la deuxième loi de Newton. Deuxièmement, la problématique de génération de trajectoire en présence de vent a été étudiée : le problème de temps minimal est formulé, analysé analytiquement et résolu numériquement. Ensuite, une stratégie de planification de trajectoire basée sur les approches de recherche opérationnelle a été développée.Troisièmement, le problème de suivi de trajectoire a été abordé. Une loi de commande non-linéaire robuste basée sur l'analyse de Lyapunov a été proposée. En outre, un pilote automatique basée sur les fonctions de saturations pour un quadri-rotors a été développée.Les méthodes et algorithmes proposés dans cette thèse ont été validés par des simulations

Advanced Strategies for Robot Manipulators

Amongst the robotic systems, robot manipulators have proven themselves to be of increasing importance and are widely adopted to substitute for human in repetitive and/or hazardous tasks. Modern manipulators are designed complicatedly and need to do more precise, crucial and critical tasks. So, the simple traditional control methods cannot be efficient, and advanced control strategies with considering special constraints are needed to establish. In spite of the fact that groundbreaking researches have been carried out in this realm until now, there are still many novel aspects which have to be explored