On the Hybrid Minimum Principle On Lie Groups and the Exponential Gradient HMP Algorithm

This paper provides a geometrical derivation of the Hybrid Minimum Principle (HMP) for autonomous hybrid systems whose state manifolds constitute Lie groups $(G,\star)$ which are left invariant under the controlled dynamics of the system, and whose switching manifolds are defined as smooth embedded time invariant submanifolds of $G$. The analysis is expressed in terms of extremal (i.e. optimal) trajectories on the cotangent bundle of the state manifold $G$. The Hybrid Maximum Principle (HMP) algorithm introduced in \cite{Shaikh} is extended to the so-called Exponential Gradient algorithm. The convergence analysis for the algorithm is based upon the LaSalle Invariance Principle and simulation results illustrate their efficacy

A survey on low-thrust trajectory optimization approaches

In this paper, we provide a survey on available numerical approaches for solving low-thrust trajectory optimization problems. First, a general mathematical framework based on hybrid optimal control will be presented. This formulation and their elements, namely objective function, continuous and discrete state and controls, and discrete and continuous dynamics, will serve as a basis for discussion throughout the whole manuscript. Thereafter, solution approaches for classical continuous optimal control problems will be briefly introduced and their application to low-thrust trajectory optimization will be discussed. A special emphasis will be placed on the extension of the classical techniques to solve hybrid optimal control problems. Finally, an extensive review of traditional and state-of-the art methodologies and tools will be presented. They will be categorized regarding their solution approach, the objective function, the state variables, the dynamical model, and their application to planetocentric or interplanetary transfers

Hybrid multi-objective trajectory optimization of low-thrust space mission design

Mención Internacional en el título de doctorThe overall goal of this dissertation is to develop multi-objective optimization algorithms for computing low-thrust trajectories. The thesis is motivated by the increasing number of space projects that will benefit from low-thrust propulsion technologies to gain unprecedented scientific, economic and social return. The low-cost design of such missions and the inclusion of concurrent engineering practices during the preliminary design phase demand advanced tools to rapidly explore different solutions and to benchmark them with respect to multiple conicting criteria. However, the determination of optimal low-thrust transfers is a challenging task and remains an active research field that seeks performance improvements. This work contributes to increase the efficiency of searching wide design spaces, reduce the amount of necessary human involvement, and enhance the capabilities to include complex operational constraints. To that end, the general low-thrust trajectory optimization problem is stated as a multi-objective Hybrid Optimal Control Problem. This formulation allows to simultaneously optimize discrete decisionmaking processes, discrete dynamics, and the continuous low-thrust steering law. Within this framework, a sequential two-step solution approach is devised for two different scenarios. The first problem considers the optimization of low-thrust multi-gravity assist trajectories. The proposed solution procedure starts by assuming a planar shape-based model for the interplanetary trajectory. A multi-objective heuristic algorithm combined with a gradient-based solver optimize the parameters de_ning the shape of the trajectory, the number and sequence of the gravity assists, the departure and arrival dates, and the launch excess velocity. In the second step, candidate solutions are deemed as initial guesses to solve the Nonlinear Programming Problem resulting from applying a direct collocation transcription scheme. In this step, the sequence of planetary gravity assists is known and provided by the heuristic search, dynamics is three-dimensional, and the steering law is not predefined. Operational constraints to comply with launch asymptote declination limits and fixed reorientation times during the transfer apply. The presented approach is tested on a rendezvous mission to Ceres, on a yby mission to Jupiter, and on a rendezvous mission to Pluto. Pareto-optimal solutions in terms of time of ight and propellant mass consumed (or alternatively delivered mass) are obtained. Results outperform those found in the literature in terms of optimality while showing the effectiveness of the proposed methodology to generate quick performance estimates. The second problem considers the simultaneous optimization of fully electric, fully chemical and combined chemical-electric orbit raising transfers between Earth's orbits is considered. In the first step of the solution approach, the control law of the electric engine is parameterized by a Lyapunov function. A multi-objective heuristic algorithm selects the optimal propulsion system, the transfer type, the low-thrust control history, as well as the number, orientation, and magnitude of the chemical firings. Earth's shadow, oblateness and Van-Allen radiation effects are included. In the second step, candidate solutions are deemed as initial guesses to solve the Nonlinear Programming Problem resulting from applying a direct collocation scheme. Operational constraints to avoid the GEO ring in combination to slew rate limits and slot phasing constraints are included. The proposed approach is applied to two transfer scenarios to GEO orbit. Pareto-optimal solutions trading of propellant mass, time of ight and solar-cell degradation are obtained. It is identified that the application of operational restrictions causes minor penalties in the objective function. Additionally, the analysis highlights the benefits that combined chemical-electric platforms may provide for future GEO satellites.El objetivo principal de esta trabajo es desarrollar algoritmos de optimización multi-objetivo para la obtención de trayectorias espaciales con motores de bajo empuje. La tesis está motivada por el creciente número de misiones que se van a beneficiar del uso de estas tecnologías para conseguir beneficios científicos, económicos y sociales sin precedentes. El diseño de bajo coste de dichas misiones ligado a los principios de ingeniería concurrente requieren herramientas computacionales avanzadas que exploren rápidamente distintas soluciones y las comparen entre sí respecto a varios criterios. Sin embargo, esta tarea permanece como un campo de investigación activo que busca continuamente mejoras de rendimiento durante el proceso. Este trabajo contribuye a aumentar la eficiencia cuando espacio de diseño es amplio, a reducir la participación humana requerida y a mejorar las capacidades para incluir restricciones operacionales complejas. Para este fin, el problema general de optimización de trayectorias de bajo empuje se presenta como un problema híbrido de control óptimo. Esta formulación permite optimizar al mismo tiempo procesos de toma de decisiones, dinámica discreta y la ley de control del motor. Dentro de este marco, se idea un algoritmo secuencial de dos pasos para dos escenarios diferentes. El primer problema considera la optimización de trayectorias de bajo empuje con múltiples y-bys. El proceso de solución propuesto comienza asumiendo un modelo plano y shape-based para la trayectoria interplanetaria. Un algoritmo de optimización heurístico y multi-objetivo combinado con un resolvedor basado en gradiente optimizan los parámetros de la espiral que definen la forma de la trayectoria, el número y la secuencia de las maniobras gravitacionales, las fechas de salida y llegada, y la velocidad de lanzamiento. En el segundo paso, las soluciones candidatas se usan como estimación inicial para resolver el problema de optimización no lineal que resulta de aplicar un método de transcripción directa. En este paso, las secuencia de y-bys es conocida y determinada por el paso anterior, la dinámica es tridimensional, y la ley de control no está prefinida. Además, se pueden aplicar restricciones operacionales relacionadas con las declinación de la asíntota de salida e imponer tiempos de reorientación fijos. Este enfoque es probado en misiones a Ceres, a Júpiter y a Plutón. Se obtienen soluciones óptimas de Pareto en función del tiempo de vuelo y la masa de combustible consumida (o la masa entregada). Los resultados obtenidos mejoran los disponibles en la literatura en términos de optimalidad, a la vez que reflejan la efectividad de la metodología a propuesta para generar estimaciones rápidas. El segundo problema considera la optimización simultanea de transferencias entre órbitas terrestres que usan propulsión eléctrica, química o una combinación de ambas. En el primer paso del método de solución, la ley de control del motor eléctrico se parametriza por una función de Lyapunov. Un algoritmo de optimización heurístico y multi-objetivo selecciona el sistema propulsivo óptimo, el tipo de transferencia, la ley de control del motor de bajo empuje, así como el número, la orientación y la magnitud de los impulsos químicos. Se incluyen los efectos de la sombra y de la no esfericidad de la Tierra, además de la radiación de Van-Allen. En el segundo paso, las soluciones candidatas se usan como estimación inicial para resolver el problema de optimización no lineal que resulta de aplicar un método de transcripción directa. El método de solución propuesto se aplica a dos transferencias a GEO diferentes. Se obtienen soluciones óptimas de Pareto con respecto a la masa de combustible, el tiempo de vuelo y la degradación de las células solares. Se identifican que la aplicación de las restricciones operacionales penaliza mínimamente la función objetivo. Además, los análisis presentados destacan los beneficios que la propulsión química y eléctrica combinada proporcionarían a los satélites en GEO.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 Virgili.Presidente: Rafael Vázquez Valenzuela.- Secretario: Claudio Bombardelli.- Vocal: Bruce A. Conwa

Comprehensive and accurate estimation of lower body movement using few wearable sensors

Human pose estimation involves tracking the position and orientation (i.e., pose) of body segments, and estimating joint kinematics. It finds application in robotics, virtual reality, animation, and healthcare. Recent miniaturisation of inertial measurement units (IMUs) has paved the path towards inertial motion capture systems (MCS) suitable for use in unstructured environments. However, commercial inertial MCS attach one-sensor-per-body-segment (OSPS) which can be too cumbersome for daily use. A reduced-sensor-count configuration, where IMUs are placed on a subset of body segments, can improve user comfort while also reducing setup time and system cost. This work aims to develop pose estimation algorithms that track lower body motion using as few sensors as possible, towards developing a comfortable MCS for daily routine use. Such a tool can facilitate interactive rehabilitation, performance improvement, and the study of movement disorder progression to potentially enable predictive diagnostics. This thesis presents pose estimation algorithms that utilise biomechanical constraints, additional distance measurements, and balance assumptions to infer information lost from using less sensors. Specifically, it presents a novel use of Lie group representation of pose alongside a constrained extended Kalman filter for estimating pelvis, thigh, shank, and foot kinematics using only two or three IMUs. The algorithms iteratively use linear kinematic equations to predict the next state, leverage indirect observations and assumptions (e.g., pelvis height, zero-velocity update, flat-floor assumption, inter-IMU distance), and enforce biomechanical constraints (e.g., constant body segment length, hinged knee joints, range of motion). The algorithm was comprehensively evaluated on nine healthy subjects who performed free walking, jogging, and other random movements within a 4x4 m2 room using benchmark optical and inertial (i.e., OSPS) MCS. In contrast to existing benchmark datasets, both direct kinematics (e.g., Vicon plug-in gait commonly used in gait analysis) and inverse kinematics (used in robotics and musculoskeletal modelling) pose reconstruction, along with the corresponding measurements, are shared publicly. The mean position root-mean-square error relative to the mid-pelvis origin was 5.3±1.0 cm, while the sagittal knee and hip joint angle correlation coefficients were 0.85±0.05 and 0.89±0.05 indicating promising performance for joint kinematics in the sagittal plane

Characterization, Verification and Control for Large Quantum Systems

Quantum information processing offers potential improvements to a wide range of computing endevaors, including cryptography, chemistry simulations and machine learning. The development of practical quantum information processing devices is impeded, however, by challenges arising from the apparent exponential dimension of the space one must consider in characterizing quantum systems, verifying their correct operation, and in designing useful control sequences. In this work, we address each in turn by providing useful algorithms that can be readily applied in experimental practice. In order to characterize the dynamics of quantum systems, we apply statistical methods based on Bayes' rule, thus enabling the use of strong prior information and parameter reduction. We first discuss an analytically-tractable special case, and then employ a numerical algorithm, sequential Monte Carlo, that uses simulation as a resource for characterization. We discuss several examples of SMC and show its application in nitrogen vacancy centers and neutron interferometry. We then discuss how characterization techniques such as SMC can be used to verify quantum systems by using credible region estimation, model selection, state-space modeling and hyperparameterization. Together, these techniques allow us to reason about the validity of assumptions used in analyzing quantum devices, and to bound the credible range of quantum dynamics. Next, we discuss the use of optimal control theory to design robust control for quantum systems. We show extensions to existing OCT algorithms that allow for including models of classical electronics as well as quantum dynamics, enabling higher-fidelity control to be designed for cutting-edge experimental devices. Moreover, we show how control can be implemented in parallel across node-based architectures, providing a valuable tool for implementing proposed fault-tolerant protocols. We close by showing how these algorithms can be augmented using quantum simulation resources to enable addressing characterization and control design challenges in even large quantum devices. In particular, we will introduce a novel genetic algorithm for quantum control design, MOQCA, that utilizes quantum coprocessors to design robust control sequences. Importantly, MOQCA is also memetic, in that improvement is performed between genetic steps. We then extend sequential Monte Carlo with quantum simulation resources to enable characterizing and verifying the dynamics of large quantum devices. By using novel insights in epistemic information locality, we are able to learn dynamics using strictly smaller simulators, leading to an algorithm we call quantum bootstrapping. We demonstrate by using a numerical example of learning the dynamics of a 50-qubit device using an 8-qubit simulator

Wearable Movement Sensors for Rehabilitation: From Technology to Clinical Practice

This Special Issue shows a range of potential opportunities for the application of wearable movement sensors in motor rehabilitation. However, the papers surely do not cover the whole field of physical behavior monitoring in motor rehabilitation. Most studies in this Special Issue focused on the technical validation of wearable sensors and the development of algorithms. Clinical validation studies, studies applying wearable sensors for the monitoring of physical behavior in daily life conditions, and papers about the implementation of wearable sensors in motor rehabilitation are under-represented in this Special Issue. Studies investigating the usability and feasibility of wearable movement sensors in clinical populations were lacking. We encourage researchers to investigate the usability, acceptance, feasibility, reliability, and clinical validity of wearable sensors in clinical populations to facilitate the application of wearable movement sensors in motor rehabilitation

Mathematical Methods, Modelling and Applications

This volume deals with novel high-quality research results of a wide class of mathematical models with applications in engineering, nature, and social sciences. Analytical and numeric, deterministic and uncertain dimensions are treated. Complex and multidisciplinary models are treated, including novel techniques of obtaining observation data and pattern recognition. Among the examples of treated problems, we encounter problems in engineering, social sciences, physics, biology, and health sciences. The novelty arises with respect to the mathematical treatment of the problem. Mathematical models are built, some of them under a deterministic approach, and other ones taking into account the uncertainty of the data, deriving random models. Several resulting mathematical representations of the models are shown as equations and systems of equations of different types: difference equations, ordinary differential equations, partial differential equations, integral equations, and algebraic equations. Across the chapters of the book, a wide class of approaches can be found to solve the displayed mathematical models, from analytical to numeric techniques, such as finite difference schemes, finite volume methods, iteration schemes, and numerical integration methods