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A new software tool was needed for flight planning of a high altitude, low speed unmanned aerial vehicle which would be flying in winds close to the actual airspeed of the vehicle. An energy modeled NLP formulation was used to obtain results for a variety of missions and wind profiles. The energy constraint derived included terms due to the wind field and the performance index was a weighted combination of the amount of fuel used and the final time. With no emphasis on time and with no winds the vehicle was found to fly at maximum lift to drag velocity, V_md. When flying in tail winds the velocity was less than V_md, while flying in head winds the velocity was higher than Vmd. A family of solutions was found with varying times of flight and varying fuel amounts consumed which will aid the operator in choosing a flight plan depending on a desired landing time. At certain parts of the flight, the turning terms in the energy constraint equation were found to be significant. An analysis of a simpler vertical plane cruise optimal control problem was used to explain some of the characteristics of the vertical plane NLP results. / Master of Science

Demonstrating strength and agility on virtual and real humanoids has been an important goal in computer graphics and robotics. However, developing physics- based controllers for various agile motor skills requires a tremendous amount of prior knowledge and manual labor due to complex mechanisms of the motor skills. The focus of the dissertation is to develop a set of computational tools to expedite the design process of physics-based controllers that can execute a variety of agile motor skills on virtual and real humanoids. Instead of designing directly controllers real humanoids, this dissertation takes an approach that develops appropriate theories and models in virtual simulation and systematically transfers the solutions to hardware systems. The algorithms and frameworks in this dissertation span various topics from spe- cific physics-based controllers to general learning frameworks. We first present an online algorithm for controlling falling and landing motions of virtual characters. The proposed algorithm is effective and efficient enough to generate falling motions for a wide range of arbitrary initial conditions in real-time. Next, we present a robust falling strategy for real humanoids that can manage a wide range of perturbations by planning the optimal contact sequences. We then introduce an iterative learning framework to easily design various agile motions, which is inspired by human learn- ing techniques. The proposed framework is followed by novel algorithms to efficiently optimize control parameters for the target tasks, especially when they have many constraints or parameterized goals. Finally, we introduce an iterative approach for exporting simulation-optimized control policies to hardware of robots to reduce the number of hardware experiments, that accompany expensive costs and labors.

Master of Arts / Department of Economics / Steven P. Cassou / This report shows the mathematics behind the solution to continuous time optimization problems. It shows how to specify the Hamiltonian function, how to use the Hamiltonian to obtain the optimal conditions for a typical economic optimal control problem and applies these techniques to several optimal control problems commonly encountered in macroeconomics. An appendix shows how to set up the optimal conditions for the case in which the state and co-state variables are both vectors. A second appendix shows how to approach the control situation for a system of optimal control problems where the co-state variable for the first sub-optimal control problem is the state variable for the second sub-optimal control problem.

For robotic systems to continue to move towards ubiquity, robots need to be more autonomous. More autonomy dictates that robots need to be able to make better decisions. Control theory and machine learning are fields of robotics that focus on the decision making process. However, each of these fields implements decision making at different levels of abstraction and at different time scales. Control theory defines low-level decisions at high rates, while machine learning defines high-level decision at low rates. The objective of this research is to integrate tools from both machine leaning and control theory to solve higher dimensional, complex problems, and to optimize the decision making process. Throughout this research, multiple algorithms were created that use concepts from both control theory and machine learning, which provide new tools for robots to make better decisions. One algorithm enables a robot to learn how to optimally explore an unknown space, and autonomously decide when to explore for new information or exploit its current information. Another algorithm enables a robot to learn how to locomote with complex dynamics. These algorithms are evaluated both in simulation and on real robots. The results and analysis of these experiments are presented, which demonstrate the utility of the algorithms introduced in this work. Additionally, a new notion of “learnability” is introduced to define and determine when a given dynamical system has the ability to gain knowledge to optimize a given objective function.

An enhanced method in optimization rooted in direct collocation is formulated to treat the finite set optimal control problem. This is motivated by applications in which a hybrid dynamical system is subject to ordinary differential continuity constraints, but control variables are contained within finite spaces. Resulting solutions display control discontinuities as variables switch between one feasible value to another. Solutions derived are characterized as optimal switching schedules between feasible control values. The methodology allows control switches to be determined over a continuous spectrum, overcoming many of the limitations associated with discretized solutions. Implementation details are presented and several applications demonstrate the method’s utility and capability. Simple applications highlight the effectiveness of the methodology, while complicated dynamic systems showcase its relevance. A key example considers the challenges associated with libration point formations. Extensions are proposed for broader classes of hybrid systems. / text

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Pure mathematics is often classified as continuous or discrete, that is into topology and combinatorics. Classical topology is the study of spaces in the small, modern topology or homology theory is the study of their large scale structure. The latter and its applications to General Systems Theory and implications on computer programming are the subject of our investigations. A general homology theory includes boundary and adjoint operators defined over a graded category. Singular homology theory describes the structure of high dimensional Simplicial complexes, and is the basis of Kron's tearing of electrical networks. De ~ham Cohomology Theory describes the structure of exterior differential forms used to ~nalyse distributed fields in high dimensional spaces. Likewise optimal control ~roblems can be described by abstract homology theories. Ideas from tensor theory are ~sed to identify the homological structure of Leontief's economic model as a real ~xample of an optimal control system. The common property of each of the above ~ystems is that of optimisation or equivalently the mapping of an error to zero. The ~~iterion may be a metric in space, or energy in an electrical or mechanical network ~~ system, or an abstract cost function in state space or money in an economic system ~~d is always the product of a covariant and a contravariant variable. ~e axiomatic nature of General Homology Theory depends on the definition of an ~~missable category, be it group, ring or module structure. Similarly real systems ~~e analysed in terms of mutually recursive algebras, vector, matrix or polynomial. ~~rther the group morphisms or mode operators are defined recursively. An orthogonal ~~mputer language, Algo182, is proposed which is capable of manipulating the objects ~~scribed by homological systems theory, thus alleviating the tedium and insecurity t~curred in iDtplementing computer programs to analyse engineering systems.

The goal of this paper is to extend the results of Bayraktar and Young (2006) on minimizing an individual's probability of lifetime ruin; i.e. the probability that the individual goes bankrupt before dying. We consider a scenario in which the individual is allowed to invest in both a domestic bank account and a foreign bank account, with the exchange rate between the two currencies being modeled by geometric Brownian motion. Additionally, we impose the restriction that the individual is not allowed to borrow money, and assume that the individual's wealth is consumed at a constant rate. We derive formulas for the minimum probability of ruin as well as the individual's optimal investment strategy. We also give a few numerical examples to illustrate these results.