Parameter Estimation in Chaotic Systems

This report examines how to estimate the parameters of a chaotic system given noisy observations of the state behavior of the system. Investigating parameter estimation for chaotic systems is interesting because of possible applications for high-precision measurement and for use in other signal processing, communication, and control applications involving chaotic systems. In this report, we examine theoretical issues regarding parameter estimation in chaotic systems and develop an efficient algorithm to perform parameter estimation. We discover two properties that are helpful for performing parameter estimation on non-structurally stable systems. First, it turns out that most data in a time series of state observations contribute very little information about the underlying parameters of a system, while a few sections of data may be extraordinarily sensitive to parameter changes. Second, for one-parameter families of systems, we demonstrate that there is often a preferred direction in parameter space governing how easily trajectories of one system can "shadow'" trajectories of nearby systems. This asymmetry of shadowing behavior in parameter space is proved for certain families of maps of the interval. Numerical evidence indicates that similar results may be true for a wide variety of other systems. Using the two properties cited above, we devise an algorithm for performing parameter estimation. Standard parameter estimation techniques such as the extended Kalman filter perform poorly on chaotic systems because of divergence problems. The proposed algorithm achieves accuracies several orders of magnitude better than the Kalman filter and has good convergence properties for large data sets.

Modeling, Estimation, and Control of Robot-Soil Interactions

This thesis presents the development of hardware, theory, and experimental methods to enable a robotic manipulator arm to interact with soils and estimate soil properties from interaction forces. Unlike the majority of robotic systems interacting with soil, our objective is parameter estimation, not excavation. To this end, we design our manipulator with a flat plate for easy modeling of interactions. By using a flat plate, we take advantage of the wealth of research on the similar problem of earth pressure on retaining walls. There are a number of existing earth pressure models. These models typically provide estimates of force which are in uncertain relation to the true force. A recent technique, known as numerical limit analysis, provides upper and lower bounds on the true force. Predictions from the numerical limit analysis technique are shown to be in good agreement with other accepted models. Experimental methods for plate insertion, soil-tool interface friction estimation, and control of applied forces on the soil are presented. In addition, a novel graphical technique for inverting the soil models is developed, which is an improvement over standard nonlinear optimization. This graphical technique utilizes the uncertainties associated with each set of force measurements to obtain all possible parameters which could have produced the measured forces. The system is tested on three cohesionless soils, two in a loose state and one in a loose and dense state. The results are compared with friction angles obtained from direct shear tests. The results highlight a number of key points. Common assumptions are made in soil modeling. Most notably, the Mohr-Coulomb failure law and perfectly plastic behavior. In the direct shear tests, a marked dependence of friction angle on the normal stress at low stresses is found. This has ramifications for any study of friction done at low stresses. In addition, gradual failures are often observed for vertical tools and tools inclined away from the direction of motion. After accounting for the change in friction angle at low stresses, the results show good agreement with the direct shear values.

Teaching an Old Robot New Tricks: Learning Novel Tasks via Interaction with People and Things

As AI has begun to reach out beyond its symbolic, objectivist roots into the embodied, experientialist realm, many projects are exploring different aspects of creating machines which interact with and respond to the world as humans do. Techniques for visual processing, object recognition, emotional response, gesture production and recognition, etc., are necessary components of a complete humanoid robot. However, most projects invariably concentrate on developing a few of these individual components, neglecting the issue of how all of these pieces would eventually fit together. The focus of the work in this dissertation is on creating a framework into which such specific competencies can be embedded, in a way that they can interact with each other and build layers of new functionality. To be of any practical value, such a framework must satisfy the real-world constraints of functioning in real-time with noisy sensors and actuators. The humanoid robot Cog provides an unapologetically adequate platform from which to take on such a challenge. This work makes three contributions to embodied AI. First, it offers a general-purpose architecture for developing behavior-based systems distributed over networks of PC's. Second, it provides a motor-control system that simulates several biological features which impact the development of motor behavior. Third, it develops a framework for a system which enables a robot to learn new behaviors via interacting with itself and the outside world. A few basic functional modules are built into this framework, enough to demonstrate the robot learning some very simple behaviors taught by a human trainer. A primary motivation for this project is the notion that it is practically impossible to build an "intelligent" machine unless it is designed partly to build itself. This work is a proof-of-concept of such an approach to integrating multiple perceptual and motor systems into a complete learning agent.

Interaction and Intelligent Behavior

We introduce basic behaviors as primitives for control and learning in situated, embodied agents interacting in complex domains. We propose methods for selecting, formally specifying, algorithmically implementing, empirically evaluating, and combining behaviors from a basic set. We also introduce a general methodology for automatically constructing higher--level behaviors by learning to select from this set. Based on a formulation of reinforcement learning using conditions, behaviors, and shaped reinforcement, out approach makes behavior selection learnable in noisy, uncertain environments with stochastic dynamics. All described ideas are validated with groups of up to 20 mobile robots performing safe--wandering, following, aggregation, dispersion, homing, flocking, foraging, and learning to forage.

Certified Rapid Solution of Parametrized Linear Elliptic Equations: Application to Parameter Estimation

We present a technique for the rapid and reliable evaluation of linear-functional output of elliptic partial differential equations with affine parameter dependence. The essential components are (i) rapidly uniformly convergent reduced-basis approximations — Galerkin projection onto a space WN spanned by solutions of the governing partial differential equation at N (optimally) selected points in parameter space; (ii) a posteriori error estimation — relaxations of the residual equation that provide inexpensive yet sharp and rigorous bounds for the error in the outputs; and (iii) offline/online computational procedures — stratagems that exploit affine parameter dependence to de-couple the generation and projection stages of the approximation process. The operation count for the online stage — in which, given a new parameter value, we calculate the output and associated error bound — depends only on N (typically small) and the parametric complexity of the problem. The method is thus ideally suited to the many-query and real-time contexts. In this paper, based on the technique we develop a robust inverse computational method for very fast solution of inverse problems characterized by parametrized partial differential equations. The essential ideas are in three-fold: first, we apply the technique to the forward problem for the rapid certified evaluation of PDE input-output relations and associated rigorous error bounds; second, we incorporate the reduced-basis approximation and error bounds into the inverse problem formulation; and third, rather than regularize the goodness-of-fit objective, we may instead identify all (or almost all, in the probabilistic sense) system configurations consistent with the available experimental data — well-posedness is reflected in a bounded "possibility region" that furthermore shrinks as the experimental error is decreased.