An introduction to Lie group integrators -- basics, new developments and applications

We give a short and elementary introduction to Lie group methods. A selection of applications of Lie group integrators are discussed. Finally, a family of symplectic integrators on cotangent bundles of Lie groups is presented and the notion of discrete gradient methods is generalised to Lie groups

Modeling, simulation, and control of soft robots

2019 Fall.Includes bibliographical references.Soft robots are a new type of robot with deformable bodies and muscle-like actuations, which are fundamentally different from traditional robots with rigid links and motor-based actuators. Owing to their elasticity, soft robots outperform rigid ones in safety, maneuverability, and adaptability. With their advantages, many soft robots have been developed for manipulation and locomotion in recent years. However, the current state of soft robotics has significant design and development work, but lags behind in modeling and control due to the complex dynamic behavior of the soft bodies. This complexity prevents a unified dynamics model that captures the dynamic behavior, computationally-efficient algorithms to simulate the dynamics in real-time, and closed-loop control algorithms to accomplish desired dynamic responses. In this thesis, we address the three problems of modeling, simulation, and control of soft robots. For the modeling, we establish a general modeling framework for the dynamics by integrating Cosserat theory with Hamilton's principle. Such a framework can accommodate different actuation methods (e.g., pneumatic, cable-driven, artificial muscles, etc.). To simulate the proposed models, we develop efficient numerical algorithms and implement them in C++ to simulate the dynamics of soft robots in real-time. These algorithms consider qualities of the dynamics that are typically neglected (e.g., numerical damping, group structure). Using the developed numerical algorithms, we investigate the control of soft robots with the goal of achieving real-time and closed-loop control policies. Several control approaches are tested (e.g., model predictive control, reinforcement learning) for a few key tasks: reaching various points in a soft manipulator's workspace and tracking a given trajectory. The results show that model predictive control is possible but is computationally demanding, while reinforcement learning techniques are more computationally effective but require a substantial number of training samples. The modeling, simulation, and control framework developed in this thesis will lay a solid foundation to unleash the potential of soft robots for various applications, such as manipulation and locomotion

Geometric Pseudospectral Method on SE(3) for Rigid-Body Dynamics with Application to Aircraft

General pseudospectral method is extended to the special Euclidean group SE(3) by virtue of equivariant map for rigid-body dynamics of the aircraft. On SE(3), a complete left invariant rigid-body dynamics model of the aircraft in body-fixed frame is established, including configuration model and velocity model. For the left invariance of the configuration model, equivalent Lie algebra equation corresponding to the configuration equation is derived based on the left-trivialized tangent of local coordinate map, and the top eight orders truncated Magnus series expansion with its coefficients of the solution of the equivalent Lie algebra equation are given. A numerical method called geometric pseudospectral method is developed, which, respectively, computes configurations and velocities at the collocation points and the endpoint based on two different collocation strategies. Through numerical tests on a free-floating rigid-body dynamics compared with several same order classical methods in Euclidean space and Lie group, it is found that the proposed method has higher accuracy, satisfying computational efficiency, stable Lie group structural conservativeness. Finally, how to apply the previous discretization scheme to rigid-body dynamics simulation and control of the aircraft is illustrated

Stochastic Algorithms in Riemannian Manifolds and Adaptive Networks

The combination of adaptive network algorithms and stochastic geometric dynamics has the potential to make a large impact in distributed control and signal processing applications. However, both literatures contain fundamental unsolved problems. The thesis is thus in two main parts. In part I, we consider stochastic differential equations (SDEs) evolving in a matrix Lie group. To undertake any kind of statistical signal processing or control task in this setting requires the simulation of such geometric SDEs. This foundational issue has barely been addressed previously. Chapter 1 contains background and motivation. Chapter 2 develops numerical schemes for simulating SDEs that evolve in SO(n) and SE(n). We propose novel, reliable, efficient schemes based on diagonal Padé approximants, where each trajectory lies in the respective manifold. We prove first order convergence in mean uniform squared error using a new proof technique. Simulations for SDEs in SO(50) are provided. In part II, we study adaptive networks. These are collections of individual agents (nodes) that cooperate to solve estimation, detection, learning and adaptation problems in real time from streaming data, without a fusion center. We study general diffusion LMS algorithms - including real time consensus - for distributed MMSE parameter estimation. This choice is motivated by two major flaws in the literature. First, all analyses assume the regressors are white noise, whereas in practice serial correlation is pervasive. Dealing with it however is much harder than the white noise case. Secondly, since the algorithms operate in real time, we must consider realization-wise behavior. There are no such results. To remedy these flaws, we uncover the mixed time scale structure of the algorithms. We then perform a novel mixed time scale stochastic averaging analysis. Chapter 3 contains background and motivation. Realization-wise stability (chapter 4) and performance including network MSD, EMSE and realization-wise fluctuations (chapter 5) are then studied. We develop results in the difficult but realistic case of serial correlation. We observe that the popular ATC, CTA and real time consensus algorithms are remarkably similar in terms of stability and performance for small constant step sizes. Parts III and IV contain conclusions and future work

The Lie-group method based on radial basis functions for solving nonlinear high dimensional generalized Benjamin–Bona–Mahony–Burgers equation in arbitrary domains

The aim of this paper is to introduce a new numerical method for solving the nonlinear generalized Benjamin–Bona–Mahony–Burgers (GBBMB) equation. This method is combination of group preserving scheme (GPS) with radial basis functions (RBFs), which takes advantage of two powerful methods, one as geometric numerical integration method and the other meshless method. Thus, we introduce this method as the Lie-group method based on radial basis functions (LG–RBFs). In this method, we use Kansas approach to approximate the spatial derivatives and then we apply GPS method to approximate first-order time derivative. One of the important advantages of the developed method is that it can be applied to problems on arbitrary geometry with high dimensions. To demonstrate this point, we solve nonlinear GBBMB equation on various geometric domains in one, two and three dimension spaces. The results of numerical experiments are compared with analytical solutions and the method presented in Dehghan et al. (2014) to confirm the accuracy and efficiency of the presented method

Computational Geometric Mechanics and Control of Rigid Bodies.

This dissertation studies the dynamics and optimal control of rigid bodies from two complementary perspectives, by providing theoretical analyses that respect the fundamental geometric characteristics of rigid body dynamics and by developing computational algorithms that preserve those geometric features. This dissertation is focused on developing analytical theory and computational algorithms that are intrinsic and applicable to a wide class of multibody systems. A geometric numerical integrator, referred to as a Lie group variational integrator, is developed for rigid body dynamics. Discrete-time Lagrangian and Hamiltonian mechanics and Lie group methods are unified to obtain a systematic method for constructing numerical integrators that preserve the geometric properties of the dynamics as well as the structure of a Lie group. It is shown that Lie group variational integrators have substantial computational advantages over integrators that preserve either one of none of these properties. This approach is also extended to mechanical systems evolving on the product of two-spheres. A computational geometric approach is developed for optimal control of rigid bodies on a Lie group. An optimal control problem is discretized at the problem formulation stage by using a Lie group variational integrator, and discrete-time necessary conditions for optimality are derived using the calculus of variations. The discrete-time necessary conditions inherit the desirable computational properties of the Lie group variational integrator, as they are derived from a symplectic discrete flow. They do not exhibit the numerical dissipation introduced by conventional numerical integration schemes, and consequently, we can efficiently obtain optimal controls that respect the geometric features of the optimality conditions. The approach that combines computational geometric mechanics and optimal control is illustrated by various examples of rigid body dynamics, which include a rigid body pendulum on a cart, pure bending of an elastic rod, and two rigid bodies connected by a ball joint. Since all of the analytical and computational results developed in this dissertation are coordinate-free, they are independent of a specific choice of local coordinates, and they completely avoid any singularity, ambiguity, and complexity associated with local coordinates. This provides insight into the global dynamics of rigid bodies.Ph.D.Aerospace EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/60804/1/tylee_1.pd

Mecánica Discreta para Sistemas Forzados y Ligados

Tesis inédita de la Universidad Complutense de Madrid, Facultad de Ciencias Matemáticas, leída el 10/07/2019Geometric mechanics is a branch of mathematics that studies classical mechanics of particles and fields from the point of view of geometry and its relation to symmetry. One of its most interesting developments was bringing together numerical analysis and geometry by relating what is known as discrete mechanics with numerical integration. This is called geometric integration. In the last 30 years this latter field has exploded with researchfrom the purely theoretical to the strictly applied. Variational integrators are a type of geometric integrators arising naturally from the discretization process of variational principles in mechanics. They display some of the most salient features of the theory, such as symplecticity, preservation of momenta and quasi-preservation of energy. These methods also apply very naturally to optimal control problems, also based on variational principles. Unfortunately, not all mechanical systems of interest admit a variational formulation. Such is the case of forced and nonholonomic mechanical systems. In this thesis we study both of these types of systems and obtain several new results. By geometrizing a new technique of duplication of variables and applying it, we were able to definitely prove the order of integrators for forced systems by using only variational techniques. Furthermore, we could also extend these results to the reduced setting in Lie groups, leading us to a very interesting geometric structure, Poisson groupoids. In addition, we developed new methods to geometrically integrate nonholonomic systems to arbitrary order preserving their constraints exactly. These methods can be seen as nonholonomic extensions of variational methods, and we were able to prove their order, although not through variational means. These methods have a nice geometric interpretation and thanks to their closeness to variational methods, they can be easily generalized to other geometric settings, such as Lie group integration. Finally, we were able to apply these new methods to optimal control problems...La mecánica clásica es un campo tan fundamental para la física como la geometría lo es para las matemáticas. Ambos están interrelacionados y su estudio conjunto así como sus interacciones forman lo que hoy se conoce como la mecánica geométrica [véase, por ejemplo, AM78; Arn89; Hol11a; Hol11 b]. Hoy es bien sabido que el concepto de simetría tiene importantes consecuencias para los sistemas mecánicos. En particular, la evolución de los sistemas mecánicos suele mostrar ciertas propiedades de preservación en forma de cantidades conservadas del movimiento o preservación de estructuras geométricas. Ser capaces de capturar estas propiedades es vital para tener una imagen fiel, tanto en términos cuantitativos como cualitativos, de cara al estudio de estos sistemas. Esto tiene gran importancia en el campo teórico y también el aplicado, como en la ingeniería. La experimentación en laboratorios y la generación de prototipos son procesos costosos y que requieren de tiempo, y para determinad os sistemas pueden no ser siquiera factibles. Con la llegada el ordenador, simular y experimentar con sistemas mecánicos de forma rápida y económica se convirtió en una realidad . Desde sencillas simulaciones balísticas para alumnos de secundaria a simulaciones de dinámica molecular a gran escala; desde la planificación de trayectorias para vehículos autónomos a la estimación de movimientos en robots bípedos; desde costosas simulaciones basadas en modelos físicos para la industria de la animación a la simulación de sólidos rígidos y deformables en tiempo real para la industria del videojuego, el tratamiento numérico de sistemas de complejidad creciente se ha convertido en una necesidad. Naturalmente surgieron nuevos algoritmos capaces de conservar gran parte de las propiedades geométricas de estos sistemas, configurando lo que a hora se conoce como integración geométrica [véase SC94; HLW1O]. En los últimos 20 a 30 años se han dado grandes pasos en esta dirección, con el desarrollo de métodos que conservan energía, métodos simplécticos y multisimplécticos, métodos que preservan el espacio de configuración y más. Aún así, la investigación en esta área está todavía lejos de acabar. Por ejemplo , los sistemas sometidos a fuerzas externas y con ligaduras ofrecen ciertas dificultades que han de ser abordadas, y esta tesis se dedica a explorar estos dos casos ofreciendo nuevos desarrollos y resultados...Fac. de Ciencias MatemáticasTRUEunpu

A Unified Model for XVA, including Interest Rates and Rating

We start in Chapter 2 to investigate linear matrix-valued SDEs and the Itô-stochastic Magnus expansion. The Itô-stochastic Magnus expansion provides an efficient numerical scheme to solve matrix-valued SDEs. We show convergence of the expansion up to a stopping time τ and provide an asymptotic estimate of the cumulative distribution function of τ. Moreover, we show how to apply it to solve SPDEs with one and two spatial dimensions by combining it with the method of lines with high accuracy. We will see that the Magnus expansion allows us to use GPU techniques leading to major performance improvements compared to a standard Euler-Maruyama scheme. In Chapter 3, we study a short-rate model in a Cox-Ingersoll-Ross (CIR) framework for negative interest rates. We define the short rate as the difference of two independent CIR processes and add a deterministic shift to guarantee a perfect fit to the market term structure. We show how to use the Gram-Charlier expansion to efficiently calibrate the model to the market swaption surface and price Bermudan swaptions with good accuracy. We are taking two different perspectives for rating transition modelling. In Section 4.4, we study inhomogeneous continuous-time Markov chains (ICTMC) as a candidate for a rating model with deterministic rating transitions. We extend this model by taking a Lie group perspective in Section 4.5, to allow for stochastic rating transitions. In both cases, we will compare the most popular choices for a change of measure technique and show how to efficiently calibrate both models to the available historical rating data and market default probabilities. At the very end, we apply the techniques shown in this thesis to minimize the collateral-inclusive Credit/ Debit Valuation Adjustments under the constraint of small collateral postings by using a collateral account dependent on rating trigger