A posteriori error analysis and adaptive non-intrusive numerical schemes for systems of random conservation laws

In this article we consider one-dimensional random systems of hyperbolic conservation laws. We first establish existence and uniqueness of random entropy admissible solutions for initial value problems of conservation laws which involve random initial data and random flux functions. Based on these results we present an a posteriori error analysis for a numerical approximation of the random entropy admissible solution. For the stochastic discretization, we consider a non-intrusive approach, the Stochastic Collocation method. The spatio-temporal discretization relies on the Runge--Kutta Discontinuous Galerkin method. We derive the a posteriori estimator using continuous reconstructions of the discrete solution. Combined with the relative entropy stability framework this yields computable error bounds for the entire space-stochastic discretization error. The estimator admits a splitting into a stochastic and a deterministic (space-time) part, allowing for a novel residual-based space-stochastic adaptive mesh refinement algorithm. We conclude with various numerical examples investigating the scaling properties of the residuals and illustrating the efficiency of the proposed adaptive algorithm

Control Of Rigid Robots With Large Uncertainties Using The Function Approximation Technique

This dissertation focuses on the control of rigid robots that cannot easily be modeled due to complexity and large uncertainties. The function approximation technique (FAT), which represents uncertainties as finite linear combinations of orthonormal basis functions, provides an alternate form of robot control - in situations where the dynamic equation cannot easily be modeled - with no dependency on the use of model information or training data. This dissertation has four aims - using the FAT - to improve controller efficiency and robustness in scenarios where reliable mathematical models cannot easily be derived or are otherwise unavailable. The first aim is to analyze the uncertain combination of a test robot and prosthesis in a scenario where the test robot and prosthesis are adequately controlled by different controllers - this is tied to efficiency. We develop a hybrid FAT controller, theoretically prove stability, and verify its performance using computer simulations. We show that systematically combining controllers can improve controller analysis and yield desired performance. In the second aim addressed in this dissertation, we investigate the simplification of the adaptive FAT controller complexity for ease of implementation - this is tied to efficiency. We achieve this by applying the passivity property and prove controller stability. We conduct computer simulations on a rigid robot under good and poor initial conditions to demonstrate the effectiveness of the controller. For an n degrees of freedom (DOFs) robot, we see a reduction of controller tuning parameters by 2n. The third aim addressed in this dissertation is the extension of the adaptive FAT controller to the robust control framework - this is tied to robustness. We invent a novel robust controller based on the FAT that uses continuous switching laws and eliminates the dependency on update laws. The controller, when compared against three state-of-the-art controllers via computer simulations and experimental tests on a rigid robot, shows good performance and robustness to fast time-varying uncertainties and random parameter perturbations. This introduces the first purely robust FAT-based controller. The fourth and final aim addressed in this dissertation is the development of a more compact form of the robust FAT controller developed in aim~3 - this is tied to efficiency and robustness. We investigate the simplification of the control structure and its applicability to a broader class of systems that can be modeled via the state-space approach. Computer simulations and experimental tests on a rigid robot demonstrate good controller performance and robustness to fast time-varying uncertainties and random parameter perturbations when compared to the robust FAT controller developed in aim 3. For an n-DOF robot, we see a reduction in the number of switching laws from 3 to 1

Dynamics and controls for robot manipulators with open and closed kinematic chain mechanisms

This dissertation deals with dynamics and controls for robot manipulators with open and closed kinematic chain mechanisms;Part I of this dissertation considers the problem of designing a class of robust algorithms for the trajectory tracking control of unconstrained single robot manipulator. The general control structure consists of two parts: The nominal control laws are first introduced to stabilize the system in the absence of uncertainties, then a class of robust nonlinear control laws are adopted to compensate for both the structured uncertainties and the unstructured uncertainties by using deterministic approach. The possible upper bounds of uncertainties are assumed to be known for the nonadaptive version of robust nonlinear controls. If information on these bounds is not available, then the adaptive bound of the robust controller is presented to overcome possible time-varying uncertainties (i.e., decentralized adaptive control scheme);Part II of the dissertation presents the efficient methodology of formulating system dynamics and hybrid position/force control for a single robot manipulator under geometric end-effector constraints. In order to facilitate dynamic analysis and control synthesis, the original joint-space dynamics (or a set of DAEs) is transformed into the constraint-space model through nonlinear transformations. Using the transformed dynamic model, a class of hybrid control laws are presented to manipulate the position and contact force at the end-effector simultaneously and accurately: the modified computed torque method, the robust adaptive controller, and the adaptive hybrid impedance controller;Part III of the dissertation deals with a mathematical modeling and coordinated control of multiple robot manipulators holding and transporting a rigid common object on the constraint surfaces. First, the kinematics and dynamics of multiple robot systems containing the closed-chain mechanisms are formulated from a unified viewpoint. After a series of model transformations, a new combined dynamic model is derived for dynamic analysis and control synthesis. Next, a class of hybrid position/force controllers are developed. The control laws can be used to simultaneously control the position of the object along the constraint surfaces and the contact forces (the internal grasping forces and the external constraint forces)

Survey of robust control for rigid robots

Current approaches to the robust control of the motion of rigid robots are surveyed, and the available literature is summarized. The five major design approaches discussed are the linear-multivariable approach, the passivity approach, the variable-structure approach, the saturation approach, and the robust-adaptive approach. Some guidelines for choosing a method are offered