Motion planning problems are characterized by huge search spaces and complex obstacle structures with no concise mathematical expression. The fixed-wing airplane application considered in this thesis adds differential constraints and point-wise bounds, i. e. an infinite number of equality and inequality constraints. An optimal trajectory planning approach is presented, based on the randomized Rapidly-exploring Random Trees framework (RRT*). The local planner relies on differential flatness of the equations of motion to obtain tree branch candidates that automatically satisfy the differential constraints. Flat output trajectories, in this case equivalent to the airplane's flight path, are designed using Bézier curves. Segment feasibility in terms of point-wise inequality constraints is tested by an indicator integral, which is evaluated alongside the segment cost functional. Although the RRT* guarantees optimality in the limit of infinite planning time, it is argued by intuition and experimentation that convergence is not approached at a practically useful rate. Therefore, the randomized planner is augmented by a deterministic variational optimization technique. To this end, the optimal planning task is formulated as a semi-infinite optimization problem, using the intermediate result of the RRT(*) as an initial guess. The proposed optimization algorithm follows the feasible flavor of the primal-dual interior point paradigm. Discretization of functional (infinite) constraints is deferred to the linear subproblems, where it is realized implicitly by numeric quadrature. An inherent numerical ill-conditioning of the method is circumvented by a reduction-like approach, which tracks active constraint locations by introducing new problem variables. Obstacle avoidance is achieved by extending the line search procedure and dynamically adding obstacle-awareness constraints to the problem formulation. Experimental evaluation confirms that the hybrid approach is practically feasible and does indeed outperform RRT*'s built-in optimization mechanism, but the computational burden is still significant. / Bewegungsplanungsaufgaben sind typischerweise gekennzeichnet durch umfangreiche Suchräume, deren vollständige Exploration nicht praktikabel ist, sowie durch unstrukturierte Hindernisse, für die nur selten eine geschlossene mathematische Beschreibung existiert. Bei der in dieser Arbeit betrachteten Anwendung auf Flächenflugzeuge kommen differentielle Randbedingungen und beschränkte Systemgrößen erschwerend hinzu. Der vorgestellte Ansatz zur optimalen Trajektorienplanung basiert auf dem Rapidly-exploring Random Trees-Algorithmus (RRT*), welcher die Suchraumkomplexität durch Randomisierung beherrschbar macht. Der spezifische Beitrag ist eine Realisierung des lokalen Planers zur Generierung der Äste des Suchbaums. Dieser erfordert ein flaches Bewegungsmodell, sodass differentielle Randbedingungen automatisch erfüllt sind. Die Trajektorien des flachen Ausgangs, welche im betrachteten Beispiel der Flugbahn entsprechen, werden mittels Bézier-Kurven entworfen. Die Einhaltung der Ungleichungsnebenbedingungen wird durch ein Indikator-Integral überprüft, welches sich mit wenig Zusatzaufwand parallel zum Kostenfunktional berechnen lässt. Zwar konvergiert der RRT*-Algorithmus (im probabilistischen Sinne) zu einer optimalen Lösung, jedoch ist die Konvergenzrate aus praktischer Sicht unbrauchbar langsam. Es ist daher naheliegend, den Planer durch ein gradientenbasiertes lokales Optimierungsverfahren mit besseren Konvergenzeigenschaften zu unterstützen. Hierzu wird die aktuelle Zwischenlösung des Planers als Initialschätzung für ein kompatibles semi-infinites Optimierungsproblem verwendet. Der vorgeschlagene Optimierungsalgorithmus erweitert das verbreitete innere-Punkte-Konzept (primal dual interior point method) auf semi-infinite Probleme. Eine explizite Diskretisierung der funktionalen Ungleichungsnebenbedingungen ist nicht erforderlich, denn diese erfolgt implizit durch eine numerische Integralauswertung im Rahmen der linearen Teilprobleme. Da die Methode an Stellen aktiver Nebenbedingungen nicht wohldefiniert ist, kommt zusätzlich eine Variante des Reduktions-Ansatzes zum Einsatz, bei welcher der Vektor der Optimierungsvariablen um die (endliche) Menge der aktiven Indizes erweitert wird. Weiterhin wurde eine Kollisionsvermeidung integriert, die in den Teilschritt der Liniensuche eingreift und die Problemformulierung dynamisch um Randbedingungen zur lokalen Berücksichtigung von Hindernissen erweitert. Experimentelle Untersuchungen bestätigen, dass die Ergebnisse des hybriden Ansatzes aus RRT(*) und numerischem Optimierungsverfahren der klassischen RRT*-basierten Trajektorienoptimierung überlegen sind. Der erforderliche Rechenaufwand ist zwar beträchtlich, aber unter realistischen Bedingungen praktisch beherrschbar.

<p>Numerical trajectory optimization is helpful not only for mission planning but also design</p> <p>space exploration and quantifying vehicle performance. Direct methods for solving the opti-</p> <p>mal control problems, which first discretize the problem before applying necessary conditions</p> <p>of optimality, dominate the field of trajectory optimization because they are easier for the</p> <p>user to set up and are less reliant on a forming a good initial guess. On the other hand,</p> <p>many consider indirect methods, which apply the necessary conditions of optimality prior to</p> <p>discretization, too difficult to use for practical applications. Indirect methods though provide</p> <p>very high quality solutions, easily accessible sensitivity information, and faster convergence</p> <p>given a sufficiently good guess. Those strengths make indirect methods especially well-suited</p> <p>for generating large data sets for system analysis and worth revisiting.</p> <p>Recent advancements in the application of indirect methods have already mitigated many</p> <p>of the often cited issues. Automatic derivation of the necessary conditions with computer</p> <p>algebra systems have eliminated the manual step which was time-intensive and error-prone.</p> <p>Furthermore, regularization techniques have reduced problems which traditionally needed</p> <p>many phases and complex staging, like those with inequality path constraints, to a signifi-</p> <p>cantly easier to handle single arc. Finally, continuation methods can circumvent the small</p> <p>radius of convergence of indirect methods by gradually changing the problem and use previ-</p> <p>ously found solutions for guesses.</p> <p>The new optimal control problem solver Giuseppe incorporates and builds upon these</p> <p>advancements to make indirect methods more accessible and easily used. It seeks to enable</p> <p>greater research and creative approaches to problem solving by being more flexible and</p> <p>extensible than previous solvers. The solver accomplishes this by implementing a modular</p> <p>design with well-defined internal interfaces. Moreover, it allows the user easy access to and</p> <p>manipulation of component objects and functions to be use in the way best suited to solve</p> <p>a problem.</p> <p>A new technique simplifies and automates what was the predominate roadblock to using</p> <p>continuation, the generation of an initial guess for the seed solution. Reliable generation of</p> <p>a guess sufficient for convergence still usually required advanced knowledge optimal contrtheory or sometimes incorporation of an entirely separate optimization method. With the</p> <p>new method, a user only needs to supply initial states, a control profile, and a time-span</p> <p>over which to integrate. The guess generator then produces a guess for the “primal” problem</p> <p>through propagation of the initial value problem. It then estimates the “dual” (adjoint)</p> <p>variables by the Gauss-Newton method for solving the nonlinear least-squares problem. The</p> <p>decoupled approach prevents poorly guessed dual variables from altering the relatively easily</p> <p>guess primal variables. As a result, this method is simpler to use, faster to iterate, and much</p> <p>more reliable than previous guess generation techniques.</p> <p>Leveraging the continuation process also allows for greater insight into the solution space</p> <p>as there is only a small marginal cost to producing an additional nearby solutions. As a</p> <p>result, a user can quickly generate large families of solutions by sweeping parameters and</p> <p>modifying constraints. These families provide much greater insight in the general problem</p> <p>and underlying system than is obtainable with singular point solutions. One can extend</p> <p>these analyses to high-dimensional spaces through construction of compound continuation</p> <p>strategies expressible by directed trees.</p> <p>Lastly, a study into common convergence explicates their causes and recommends mitiga-</p> <p>tion strategies. In this area, the continuation process also serves an important role. Adaptive</p> <p>step-size routines usually suffice to handle common sensitivity issues and scaling constraints</p> <p>is simpler and out-performs scaling parameters directly. Issues arise when a cost functional</p> <p>becomes insensitive to the control, which a small control cost mitigates. The best perfor-</p> <p>mance of the solver requires proper sizing of the smoothing parameters used in regularization</p> <p>methods. An asymptotic increase in the magnitude of adjoint variables indicate approaching</p> <p>a feasibility boundary of the solution space.</p> <p>These techniques for indirect methods greatly facilitate their use and enable the gen-</p> <p>eration of large libraries of high-quality optimal trajectories for complex problems. In the</p> <p>future, these libraries can give a detailed account of vehicle performance throughout its flight</p> <p>envelope, feed higher-level system analyses, or inform real-time control applications.</p>

<p dir="ltr">Reverse osmosis (RO) is emerging as the world’s leading desalination technology due to its superior energy efficiency and the shift towards renewable electrification. However, RO systems need to further improve efficiency, increase operating flux, reach higher salinities (>7.5% w.t.), and minimize component complexity. Treating RO as a dynamical system, this dissertation invents new processes for high-efficiency desalination that achieve milestones for low downtime and high final salinity. It also introduces modeling methods that include more detail (e.g. salt retention, time-varying salinity, concentration polarization, salt transport, temporal multi-staging, etc.) and the first use of certain optimization methods in RO.</p><p dir="ltr">Batch RO is an unsteady, pressure driven process that efficiently desalinates a saline feed volume over time by continuously recirculating the brine through the membrane module. A tank houses the concentrating feed and mediates the streams entering and leaving the membrane module. Most studies so far have concentrated on the high-pressure tank design that requires finite downtime at the end of each stroke. A scalable pressure exchanger batch RO (PX-BRO) configuration using atmospheric tanks that practically has zero downtime and produces permeate even while flushing is first described in this dissertation.</p><p dir="ltr">To achieve high recovery at nominal RO pressures, osmotically assisted RO processes have both sides of the membrane saline and the streams usually in counterflow. The first unsteady osmotically assisted process based on the high-pressure piston tank design, batch counterflow RO (BCFRO) is introduced which dramatically reduces the energy needs. To address the issue of high component count in spatial multi-staging, the first “temporally multi-staged” BCFRO process is also introduced. The new process uses the pressure ex- changer and atmospheric pressure tank design for scalability and operational flexibility.</p><p dir="ltr">For membranes with low salt rejection, it becomes imperative to integrate the salt trans- port dynamics for deciding operating and initial conditions. Trajectory optimization is used to match salinity and volume between stages of temporally multi-staged BCFRO. Treating the process as an optimal control problem, a framework for obtaining time varying flux pro- files that minimize the specific energy consumption is also developed. Both reduced order and discretized models are developed to analyze these new batch RO configurations.</p>

<p dir="ltr">The future of the global economy lies in space. As the economic and scientific benefits from space become more accessible and apparent to the public, the demand for more spacecrafts will only increase. However, simply using the current space architecture to sustain any major activities past low Earth orbit is infeasible. The limiting factor of relying on ground operators via the Deep Space Network will blunt future growth in cislunar space traffic as the bandwidth is insufficient to satisfy the needs of every spacecraft in this domain. For this reason, spacecrafts must begin to operate autonomously or semi-autonomously for operators to be able to manage more missions at a given time. This thesis focuses on the guidance and navigation policies that could help vehicles such as logistical or resupply spacecrafts perform their rendezvous autonomously. It is found that using GNSS signals and Moon-based optical navigation has the potential to help spacecrafts perform autonomous orbit determination in near-Moon trajectories. The estimations are high enough quality such that a stochastic controller can use this navigation solution to confidently guide the spacecraft to a target within a tolerance before proximity operations commence. As the reliance on the ground is shifted away, spacecrafts would be able to operate in greater numbers outside of Earth's lower orbits, greatly assisting humanity's presence in space. </p>

There is growing interest in commercial aircraft formation flight as a means of reducing both airspace congestion and the carbon footprint of air transportation. Wake vortex surfing has been researched extensively and proven to have significant fuel-saving benefits, however, commercial air transportation has yet to take advantage of these formation benefits due to understandable safety concerns. The realization of these formations requires serious consideration of formation contingencies and safety during closer-in maneuvering of large commercial aircraft. Formation contingency scenarios are much more complex than those of individual aircraft and have not yet been studied in depth. This thesis investigates the utility of optimization modeling in providing insight into generation of aircraft escape paths for formation contingency planning. Three high-altitude commercial aircraft formation scenarios are presented; formation join, formation emergency exit, and formation escape. The model-generated paths are compared with pilot-generated escape plans using the author’s pilot expertise. The model results compare well with pilot intuition and are useful in presenting solutions not previously considered, in evaluating separation requirements for improvement of escape path planning and in confirming the viability of the pilot-generated plans. The novel optimization model formulation presented in this thesis is the first model shown to be capable of generating escape paths comparable to pilot- generated escape plans and is also the first to incorporate avoidance of persistent and drifting wake turbulence within the formation.

Military helicopter operations require precise maneuverability characteristics for performance to be determined for the entire helicopter flight envelope. Historically, these maneuverability analyses are combinatorial in nature and involve human-interaction, which hinders their integration into conceptual design. A model formulation that includes the necessary quantitative measures and captures the impact of changing requirements real-time is presented. The formulation is shown to offer a more conservative estimate of maneuverability than traditional energy-based formulations through quantitative analysis of a typical pop-up maneuver. Although the control system design is not directly integrated, two control constraint measures are deemed essential in this work: control deflection rate and trajectory divergence rate. Both of these measures are general enough to be applied to any control architecture, while at the same time enable quantitative trades that relate overall vehicle maneuverability to control system requirements. The dimensionality issues stemming from the immense maneuver space are mitigated through systematic development of a maneuver taxonomy that enables the operational envelope to be decomposed into a minimal set of fundamental maneuvers. The taxonomy approach is applied to a helicopter canonical example that requires maneuverability and design to be assessed simultaneously. The end result is a methodology that enables the impact of design choices on maneuverability to be assessed for the entire helicopter operational envelope, while enabling constraints from control system design to be assessed real-time.

The main objective of this work is to demonstrate some new computational methods for estimation, optimization and modeling of dynamical systems that use automatic differentiation. Particular focus will be upon dynamical systems arising in Aerospace Engineering. Automatic differentiation is a recursive computational algorithm, which enables computation of analytically rigorous partial derivatives of any user-specified function. All associated computations occur, in the background without user intervention, as the name implies. The computational methods of this dissertation are enabled by a new automatic differentiation tool, OCEA (Object oriented Coordinate Embedding Method). OCEA has been recently developed and makes possible efficient computation and evaluation of partial derivatives with minimal user coding. The key results in this dissertation details the use of OCEA through a number of computational studies in estimation and dynamical modeling. Several prototype problems are studied in order to evaluate judicious ways to use OCEA. Additionally, new solution methods are introduced in order to ascertain the extended capability of this new computational tool. Computational tradeoffs are studied in detail by looking at a number of different applications in the areas of estimation, dynamical system modeling, and validation of solution accuracy for complex dynamical systems. The results of these computational studies provide new insights and indicate the future potential of OCEA in its further development.

[pt] Robôs híbridos equipados com pernas e rodas são uma solução promissora para uma locomoção versátil em terrenos acidentados. Eles combinam a velocidade e a eficiência das rodas com a capacidade das pernas de atravessar terrenos com obstáculos. Em geral, os desafios em locomoção para robôs híbridos envolvem planejamento de trajetória e sistemas de controle para o rastreamento da trajetória planejada. Esta tese se concentra, em particular, na tarefa de otimização de trajetória para robôs híbridos que navegam em terrenos acidentados. Para isso, propõe-se um algoritmo de planejamento que otimiza a posição e a orientação da base do robô e as posições e forças de contato nas rodas em uma formulação única, levando em consideração as informações do terreno e a dinâmica do robô. O robô é modelado como um único corpo rígido com massa e inércia concentrada no centro de massa, o que permite planejar movimentos complexos por longos horizontes de tempo e ainda manter uma baixa complexidade computacional para resolver a otimização de forma mais eficiente. O conhecimento do mapa do terreno permite que a otimização gere trajetórias para negociação de obstáculos de maneira dinâmica, em velocidades mais altas. Tais movimentos não podem ser gerados sem levar em consideração as informações do terreno. Duas formulações diferentes são apresentadas, uma que permite movimentos somente com as rodas, onde a negociação de obstáculos é permitida pelas pernas, e outra focada em movimentos híbridos dando passos e movendo as rodas, capazes de lidar com descontinuidades no perfil do terreno. A otimização é formulada como um NLP e as trajetórias obtidas são rastreadas por um controlador hierárquico que computa os comandos de atuação de torque para as juntas e as rodas do robô. As trajetórias são verificadas no robô quadrúpede ANYmal equipado com rodas não esterçáveis controladas por torque, em simulações e testes experimentais. O algoritmo proposto de otimização de trajetória permite que robôs com pernas e rodas naveguem por terrenos complexos, contendo, por exemplo, degraus, declives e escadas, enquanto negociam esses obstáculos com movimentos dinâmicos. / [en] Wheeled-legged robots are an attractive solution for versatile locomotion in challenging terrain. They combine the speed and efficiency of wheels with the ability of legs to traverse challenging terrain. In general, the challenges with wheeled-legged locomotion involve trajectory generation and motion control for trajectory tracking. This thesis focuses in particular on the trajectory optimization task for wheeled-legged robots navigating in challenging terrain. For this, a motion planning framework is proposed that optimizes over the robot’s base position and orientation, and the wheels’ positions and contact forces in a single planning problem, taking into account the terrain information and the robot dynamics. The robot is modeled as a single rigid-body, which allows to plan complex motions for long time horizons and still keep a low computational complexity to solve the optimization quickly. The knowledge of the terrain map allows the optimizer to generate feasible motions for obstacle negotiation in a dynamic manner, at higher speeds. Such motions cannot be discovered without taking into account the terrain information. Two different formulations allow for either purely driving motions, where obstacle negotiation is enabled by the legs, or hybrid driving-walking motions, which are able to overcome discontinuities in the terrain profile. The optimization is formulated as a Nonlinear Programming Problem (NLP) and the reference motions are tracked by a hierarchical whole-body controller that computes the torque actuation commands for the robot. The trajectories are verified on the quadrupedal robot ANYmal equipped with non-steerable torque-controlled wheels in simulations and experimental tests. The proposed trajectory optimization framework enables wheeled-legged robots to navigate over challenging terrain, e.g., steps, slopes, stairs, while negotiating these obstacles with dynamic motions.

Cette thèse porte sur l'utilisation de données de vols pour l'optimisation de trajectoires de montée vis-à-vis de la consommation de carburant.Dans un premier temps nous nous sommes intéressé au problème d'identification de modèles de la dynamique de l'avion dans le but de les utiliser pour poser le problème d'optimisation de trajectoire à résoudre. Nous commençont par proposer une formulation statique du problème d'identification de la dynamique. Nous l'interpretons comme un problème de régression multi-tâche à structure latente, pour lequel nous proposons un modèle paramétrique. L'estimation des paramètres est faite par l'application de quelques variations de la méthode du maximum de vraisemblance.Nous suggérons également dans ce contexte d'employer des méthodes de sélection de variable pour construire une structure de modèle de régression polynomiale dépendant des données. L'approche proposée est une extension à un contexte multi-tâche structuré du bootstrap Lasso. Elle nous permet en effet de sélectionner les variables du modèle dans un contexte à fortes corrélations, tout en conservant la structure du problème inhérente à nos connaissances métier.Dans un deuxième temps, nous traitons la caractérisation des solutions du problème d'optimisation de trajectoire relativement au domaine de validité des modèles identifiés. Dans cette optique, nous proposons un critère probabiliste pour quantifier la proximité entre une courbe arbitraire et un ensemble de trajectoires échantillonnées à partir d'un même processus stochastique. Nous proposons une classe d'estimateurs de cette quantitée et nous étudions de façon plus pratique une implémentation nonparamétrique basé sur des estimateurs à noyau, et une implémentation paramétrique faisant intervenir des mélanges Gaussiens. Ce dernier est introduit comme pénalité dans le critère d'optimisation de trajectoire dans l'objectif l'intention d'obtenir directement des trajectoires consommant peu sans trop s'éloigner des régions de validité. / This thesis deals with the use of flight data for the optimization of climb trajectories with relation to fuel consumption.We first focus on methods for identifying the aircraft dynamics, in order to plug it in the trajectory optimization problem. We suggest a static formulation of the identification problem, which we interpret as a structured multi-task regression problem. In this framework, we propose parametric models and use different maximum likelihood approaches to learn the unknown parameters.Furthermore, polynomial models are considered and an extension to the structured multi-task setting of the bootstrap Lasso is used to make a consistent selection of the monomials despite the high correlations among them.Next, we consider the problem of assessing the optimized trajectories relatively to the validity region of the identified models. For this, we propose a probabilistic criterion for quantifying the closeness between an arbitrary curve and a set of trajectories sampled from the same stochastic process. We propose a class of estimators of this quantity and prove their consistency in some sense. A nonparemetric implementation based on kernel density estimators, as well as a parametric implementation based on Gaussian mixtures are presented. We introduce the later as a penalty term in the trajectory optimization problem, which allows us to control the trade-off between trajectory acceptability and consumption reduction.

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