The proposed project is the simulation of a system to search for air vehicles which have splashed-down in the ocean. The system comprises a group of 10+ autonomous underwater vehicles, which cooperate in order to locate the aircraft. The search algorithm used in this system is based on a quadratic Newton method and was developed at Sandia National Laboratories. The method has already been successfully applied to several two dimensional problems at Sandia. The original 2D algorithm was converted to 3D and tested for robustness in the presence of sensor error, position error and navigational error. Treating the robots as point masses, the system was found to be robust for all such errors. Several real-life adaptations were necessary. A round-robin communication strategy was implemented on the system to properly simulate the dissemination of information throughout the group. Time to convergence is delayed but the system still functioned adequately. Once simulations for the point masses had been exhausted, the dynamics of the robots were included. The robot equations of motion were described using Kane's equations. Path-planning was investigated using optimal control methods. The Variational Calculus approach was attempted using a line search tool "fsolve" found in Matlab and a Genetic Algorithm. A dynamic programming technique was also investigated using a method recently developed by Sandia National Laboratories. The Dynamic Programming with Interior Points (DPIP) method was a very effcient method for path planning and performed well in the presence of system constraints. Finally all components of the system were integrated. The motion of the robot exactly matched the motion of the particles, even when subjected to the same robustness tests carried out on the point masses. This thesis provides exciting developments for all types of cooperative projects.

Time-optimal solutions provide us with the fastest means to regulate a system in presence of input constraints. This advantage of time-optimal control solutions is offset by the fact that their real-time implementation involves computationally intensive iterative techniques. Moreover, time-optimal controls depend on the initial state and have to be recalculated for even the slightest perturbation. Clearly time-optimal controls are not good candidates for online regulation. Consequently, the search for alternatives to time-optimal solutions is a very active area of research. The work described here is inspired by the simplicity of optimal-aim concept. The "optimal-aim strategies" provide online regulation in presence of bounded inputs with minimal computational effort. These are based purely on state-space geometry of the plant and are inherently adaptive in nature. Optimal-aim techniques involve aiming of trajectory derivative (or the state velocity vector) so as to approach the equilibrium state in the best possible manner. This thesis documents the efforts to develop an online regulation algorithm for systems with input constraints. Through a number of hypotheses focussed on trying to reproduce the exact time-optimal solution, the diffculty associated with this task is demonstrated. A modification of optimal-aim concept is employed to develop a novel regulation algorithm. In this algorithm, aim directions are chosen in a special manner to generate the time-optimal control approximately. The control scheme thus developed is shown to be globally stabilizing for systems having eigenvalues in the CLHP (closed left half-plane). It is expected that this method or its modifications can be extended to higher dimensional systems as a part of future research. An alternative control algorithm involving a simple state-space aiming concept is also developed and discussed.

<p>The rolling isolation system (RIS) studied in this dissertation functions on the principle of a rolling pendulum; an isolated object rests on a steel frame that is supported at its corners by ball-bearings that roll between shallow steel bowls, dynamically decoupling the floor motion from the response of the object. The primary focus of this dissertation is to develop predictive models that can capture experimentally-observed phenomena and to advance the state-of-the-art by proposing new isolation technologies to surmount current performance limitations. To wit, a double RIS increases the system's displacement capacity, and semi-active and passive damped RISs suppress the system's displacement response.</p><p>This dissertation illustrates the performance of various high-performance isolation strategies using experimentally-validated predictive models. Effective modeling of RISs is complicated by the nonholonomic and chaotic nature of these systems which to date has not received much attention. Motivated by this observation, the first part of this dissertation addresses the high-fidelity modeling of a single, undamped RIS, and later this theory is augmented to account for the double (or stacked) configuration and the supplemental damping via rubber-coated bowl surfaces. The system's potential energy function (i.e. conical bowl shape) and energy dissipation model are calibrated to free-response experiments. Forced-response experiments successfully validate the models by comparing measured and predicted peak displacement and acceleration responses over a range of operating conditions.</p><p>Following the experimental analyses, numerical simulations demonstrate the potential benefits of the proposed technologies. This dissertation presents a method to optimize damping force trajectories subject to constraints imposed by the physical implementation of a particular controllable damper. Potential improvements in terms of acceleration response are shown to be achievable with the semi-active RIS. Finally, extensive time-history analyses establish how the undamped and damped RISs perform when located inside biaxial, hysteretic, multi-story structures under recorded earthquake ground motions. General design recommendations, supported by critical-disturbance spectra and peak-response distributions, are prescribed so as to ensure the uninterrupted operation of vital equipment.</p> / Dissertation

Continuous and discrete-time Successive Backward Sweep (SBS) methods for solving nonlinear optimal control problems involving terminal and control constraints are proposed in this dissertation. They closely resemble the Neighboring Extremals and Differential Dynamic Programming algorithms, which are based on the successive solutions to a series of linear control problems with quadratic performance indices. The SBS methods are relatively insensitive to the initial guesses of the state and control histories, which are not required to satisfy the system dynamics. Hessian modifications are utilized, especially for non-convex problems, to avoid singularities during the backward integration of the gain equations. The SBS method requires the satisfaction of the Jacobi no-conjugate point condition and hence, produces optimal solutions. The standard implementation of the SBS method for continuous-time systems incurs terminal boundary condition errors due to an algorithmic singularity as well as numerical inaccuracies in the computation of the gain matrices. Alternatives for boundary error reduction are proposed, notably the aiming point and the switching between two forms of the sweep expansion formulae. Modification of the sweep formula expands the domain of convergence of the SBS method and allows for a rigorous testing for the existence of conjugate points. Numerical accuracy of the continuous-time formulation of the optimal control problem can be improved with the use of symplectic integrators, which generally are implicit schemes in time. A time-explicit group preserving method based on the Magnus series representation of the state transition is implemented in the SBS setting and is shown to outperform a non-symplectic integrator of the same order. Discrete-time formulations of the optimal control problem, directly accounting for a specific time-stepping method, lead to consistent systems of equations, whose solutions satisfy the boundary conditions of the discretized problem accurately. In this regard, the second-order, implicit mid-point averaging scheme, a symplectic integrator, is adapted for use with the SBS method. The performance of the mid-point averaging scheme is compared with other methods of equal and higher-order non-symplectic schemes to show its advantages. The SBS method is augmented with a homotopy- continuation procedure to isolate and regulate certain nonlinear effects for difficult problems, in order to extend its domain of convergence. The discrete-time SBS method is also extended to solve problems where the controls are approximated to be impulsive and to handle waypoint constraints as well. A variety of highly nonlinear optimal control problems involving orbit transfer, atmospheric reentry, and the restricted three-body problem are treated to demonstrate the performance of the methods developed in this dissertation.

No description available.

<p>Aggressive fiscal and monetary policies by governments of countries and central banks in developed markets could somehow push inflation to some very high level in the long run. Due to the decreasing of pension fund benefits and increasing inflation rate, pension companies are selling inflation-linked products to hedge against inflation risk. Such companies are seriously considering the possible effects of inflation volatility on their investment, and some of them tend to include inflationary allowances in the pension payment plan. In this dissertation we study the management of pension funds of the defined contribution type in the presence of inflation-recession. We study how the fund manager maximizes his fund&rsquo / s wealth when the salaries and stocks are affected by inflation. In this regard, we consider the case of a pension company which invests in a stock, inflation-linked bonds and a money market account, while basing its investment on the contribution of the plan member. We use a benchmarking approach and martingale methods to compute an optimal strategy which maximizes the fund wealth.</p>

Modern industrial control systems use a multitude of spatially distributed sensors and actuators to continuously monitor and control physical processes. Information exchange among control system components is traditionally done through physical wires. The need to physically wire sensors and actuators limits flexibility, scalability and reliability, since the cabling cost is high, cable connectors are prone to wear and tear, and connector failures can be hard to isolate. By replacing some of the cables with wireless communication networks, costs and risks of connector failures can be decreased, resulting in a more cost-efficient and reliable system. Integrating wireless communication into industrial control systems is challenging, since wireless communication channels introduce imperfections such as stochastic delays and information losses. These imperfections deteriorate the closed-loop control performance, and may even cause instability. In this thesis, we aim at developing design frameworks that take these imperfections into account and improve the performance of closed-loop control systems. The thesis first considers the joint design of packet forwarding policies and controllers for wireless control loops where sensor measurements are sent to the controller over an unreliable and energy-constrained multi-hop wireless network. For a fixed sampling rate of the sensor, the co-design problem separates into two well-defined and independent subproblems: transmission scheduling for maximizing the deadline-constrained reliability and optimal control under packet losses. We develop optimal and implementable solutions for these subproblems and show that the optimally co-designed system can be obtained efficiently. The thesis continues by examining event-triggered control systems that can help to reduce the energy consumption of the network by transmitting data less frequently. To this end, we consider a stochastic system where the communication between the controller and the actuator is triggered by a threshold-based rule. The communication is performed across an unreliable link that stochastically erases transmitted packets. As a partial protection against dropped packets, the controller sends a sequence of control commands to the actuator in each packet. These commands are stored in a buffer and applied sequentially until the next control packet arrives. We derive analytical expressions that quantify the trade-off between the communication cost and the control performance for this class of event-triggered control systems. The thesis finally proposes a supervisory control structure for wireless control systems with time-varying delays. The supervisor has access to a crude indicator of the overall network state, and we assume that individual upper and lower bounds on network time-delays can be associated to each value of the indicator. Based on this information, the supervisor triggers the most appropriate controller from a multi-controller unit. The performance of such a supervisory controller allows for improving the performance over a single robust controller. As the granularity of the network state measurements increases, the performance of the supervisory controller improves at the expense of increased computational complexity. / <p>QC 20150504</p>

What does it mean to do the disco? Or perform a cheerleading routine? Or move in a style appropriate for a given mode of human interaction? Answering these questions requires an interpretation of what differentiates two distinct movement styles and a method for parsing this difference into quantitative parameters. Furthermore, such an understanding of principles of style has applications in control, robotics, and dance theory. This thesis present a definition for “style of motion” that is rooted in dance theory, a framework for stylistic motion generation that separates basic movement ordering from its precise trajectory, and an inverse optimal control method for extracting these stylistic parameters from real data. On the part of generation, the processes of sequencing and scaling are modulated by the stylistic parameters enumerated: an automation that lists basic primary movements, sets which determine the final structure of the state machine that encodes allowable sequences, and weights in an optimal control problem that generates motions of the desired quality. This generation framework is demonstrated on a humanoid robotic platform for two distinct case studies – disco dancing and cheerleading. In order to extract the parameters that comprise the stylistic definition put forth, two inverse optimal control problems are posed and solved -- one to classify individual movements and one to segment longer movement sequences into smaller motion primitives. The motion of a real human leg (recorded via motion capture) is classified in an example. Thus, the contents of the thesis comprise a tool to produce and understand stylistic motion.

Musculoskeletal dynamics is a branch of biomechanics that takes advantage of interdisciplinary models to describe the relation between muscle actuators and the corresponding motions of the human body. Muscle forces play a principal role in musculoskeletal dynamics. Unfortunately, these forces cannot be measured non-invasively. Measuring surface EMGs as a non-invasive technique is recognized as a surrogate to invasive muscle force measurement; however, these signals do not reflect the muscle forces accurately. Instead of measurement, mathematical modelling of the musculoskeletal dynamics is a well established tool to simulate, predict and analyse human movements. Computer simulations have been used to estimate a variety of variables that are difficult or impossible to measure directly, such as joint reaction forces, muscle forces, metabolic energy consumption, and muscle recruitment patterns. Musculoskeletal dynamic simulations can be divided into two branches: inverse and forward dynamics. Inverse dynamics is the approach in which net joint moments and/or muscle forces are calculated given the measured or specified kinematics. It is the most popular simulation technique used to study human musculoskeletal dynamics. The major disadvantage of inverse dynamics is that it is not predictive and can rarely be used in the cause-effect interpretations. In contrast with inverse dynamics, forward dynamics can be used to determine the human body movement when it is driven by known muscle forces. The musculoskeletal system (MSS) is dynamically under-determinate, i.e., the number of muscles is more than the degrees of freedom (dof) of the system. This redundancy will lead to infinite solutions of muscle force sets, which implies that there are infinite ways of recruiting different muscles for a specific motion. Therefore, there needs to be an extra criterion in order to resolve this issue. Optimization has been widely used for solving the redundancy of the force-sharing problem. Optimization is considered as the missing consideration in the dynamics of the MSS such that, once appended to the under-determinate problem, \human-like" movements will be acquired. \Human-like" implies that the human body tends to minimize a criterion during a movement, e.g., muscle fatigue or metabolic energy. It is commonly accepted that using those criteria, within the optimization necessary in the forward dynamic simulations, leads to a reasonable representation of real human motions. In this thesis, optimal control and forward dynamic simulation of human musculoskeletal systems are targeted. Forward dynamics requires integration of the differential equations of motion of the system, which takes a considerable time, especially within an optimization framework. Therefore, computationally efficient models are required. Musculoskeletal models in this thesis are implemented in the symbolic multibody package MapleSim that uses Maple as the leverage. MapleSim generates the equations of motion governing a multibody system automatically using linear graph theory. These equations will be simplified and highly optimized for further simulations taking advantage of symbolic techniques in Maple. The output codes are the best form for the equations to be applied in optimization-based simulation fields, such as the research area of this thesis. The specific objectives of this thesis were to develop frameworks for such predictive simulations and validate the estimations. Simulating human gait motion is set as the end goal of this research. To successfully achieve that, several intermediate steps are taken prior to gait modelling. One big step was to choose an efficient strategy to solve the optimal control and muscle redundancy problems. The optimal control techniques are benchmarked on simpler models, such as forearm flexion/extension, to study the efficacy of the proposed approaches more easily. Another major step to modelling gait is to create a high-fidelity foot-ground contact model. The foot contact model in this thesis is based on a nonlinear volumetric approach, which is able to generate the experimental ground reaction forces more effectively than the previously used models. Although the proposed models and approaches showed strong potential and capability, there is still room for improvement in both modelling and validation aspects. These cutting-edge future works can be followed by any researcher working in the optimal control and forward dynamic modelling of human musculoskeletal systems.

The focus of this thesis is optimal path planning. The path planning problem is posed as an optimal control problem, for which the viscosity solution to the static Hamilton-Jacobi-Bellman (HJB) equation is used to determine the optimal path. The Ordered Upwind Method (OUM) has been previously used to numerically approximate the viscosity solution of the static HJB equation for direction-dependent weights. The contributions of this thesis include an analytical bound on the convergence rate of the OUM for the boundary value problem to the viscosity solution of the HJB equation. The convergence result provided in this thesis is to our knowledge the tightest existing bound on the convergence order of OUM solutions to the viscosity solution of the static HJB equation. Only convergence without any guarantee of rate has been previously shown. Navigation functions are often used to provide controls to robots. These functions can suffer from local minima that are not also global minima, which correspond to the inability to find a path at those minima. Provided the weight function is positive, the viscosity solution to the static HJB equation cannot have local minima. Though this has been discussed in literature, a proof has not yet appeared. The solution of the HJB equation is shown in this work to have no local minima that is not also global. A path can be found using this method. Though finding the shortest path is often considered in optimal path planning, safe and energy efficient paths are required for rover path planning. Reducing instability risk based on tip-over axes and maximizing solar exposure are important to consider in achieving these goals. In addition to obstacle avoidance, soil risk and path length on terrain are considered. In particular, the tip-over instability risk is a direction-dependent criteria, for which accurate approximate solutions to the static HJB equation cannot be found using the simpler Fast Marching Method. An extension of the OUM to include a bi-directional search for the source-point path planning problem is also presented. The solution is found on a smaller region of the environment, containing the optimal path. Savings in computational time are observed. A comparison is made in the path planning problem in both timing and performance between a genetic algorithm rover path planner and OUM. A comparison in timing and number of updates required is made between OUM and several other algorithms that approximate the same static HJB equation. Finally, the OUM algorithm solving the boundary value problem is shown to converge numerically with the rate of the proven theoretical bound.