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Über dieses Buch

The subject of this book is model abstraction of dynamical systems. The p- mary goal of the work embodied in this book is to design a controller for the mobile robotic car using abstraction. Abstraction provides a means to rep- sent the dynamics of a system using a simpler model while retaining important characteristics of the original system. A second goal of this work is to study the propagation of uncertain initial conditions in the framework of abstraction. The summation of this work is presented in this book. It includes the following: • An overview of the history and current research in mobile robotic control design. • A mathematical review that provides the tools used in this research area. • The development of the robotic car model and both controllers used in the new control design. • A review of abstraction and an extension of these ideas into new system relationship characterizations called traceability and -traceability. • A framework for designing controllers based on abstraction. • An open-loop control design with simulation results. • An investigation of system abstraction with uncertain initial conditions.

Inhaltsverzeichnis

Frontmatter

Introduction

Abstract
The subject of this book is model abstraction of dynamical systems. The primary goal of the work embodied in this book is to design a controller for the mobile robotic car using abstraction. Abstraction provides a means to represent the dynamics of a system using a simpler model while retaining important characteristics of the original system. A second goal of this work is to study the propagation of uncertain initial conditions in the framework of abstraction. The summation of this work is presented in this book. It includes the following:
  • An overview of the history and current research in mobile robotic control design.
  • A mathematical review that provides the tools used in this research area.
  • The development of the robotic car model and both controllers used in the new control design.
  • A review of abstraction and an extension of these ideas into new system relationship characterizations called traceability and ε-traceability.
  • A framework for designing controllers based on abstraction.
  • An open-loop control design with simulation results.
  • An investigation of system abstraction with uncertain initial conditions.
Patricia Mellodge, Pushkin Kachroo

Mathematical Preliminaries

Abstract
This chapter provides background information on material that will be used throughout this work. Material covered includes the differential geometric or coordinate-free description of control systems, control system properties, nonholonomic systems, and chained forms.
Patricia Mellodge, Pushkin Kachroo

Kinematic Modeling and Control

Abstract
This chapter describes the kinematic modeling of a car-like mobile robot. Kinematic modeling is often used because of its simplicity and accuracy in predicting the car’s behavior under normal driving conditions. This type of modeling uses the nonholonomic constraints of the system as described in Section 2.3.
After the modeling is discussed, the control design from 34 is described. This controller uses the path’s curvature as a parameter. Since the curvature must be determined, two estimation methods are described and simulation results are presented.
Patricia Mellodge, Pushkin Kachroo

Vision Based Modeling and Control

Image Dynamic Modeling
Much of the work that has been done in the area of vision based control uses the image to obtain information about the car’s location. For example, see [19], [20], and [58]. Often the car’s position, angle, etc. are extracted from the image data. The idea of the model presented here (from [35]) is to bypass the extraction phase and use parameters directly measurable from the image to control the car.
In this chapter, it is assumed that there is a camera mounted rigidly on the car. The camera frame, F c , is attached to the mobile robot as shown in Figure 4.1. This frame is chosen so that the origins of F m and F c are the same and that the y m -axis lies along the y c -axis. The camera is mounted at height h above the (x,y) plane. If, on the actual robot, the origins do not coincide, then a point in the actual camera frame can be transformed into F c by a simple translation. Also, it is assumed that the camera is tilted downward so that α, the angle between the x c -axis and x m -axis, is positive. It is assumed that h and α are known fixed values. Throughout this chapter, points in the car’s frame are denoted by (x m ,y m ,z m ) and points in the camera’s frame are denoted by (x c ,y c ,z c ).
Patricia Mellodge, Pushkin Kachroo

Abstraction

Abstract
This chapter reviews the concept of abstracted control systems and introduces the new concepts of traceability, ε-traceability, and ε-consistency. Abstracted systems are simplifications of more complex dynamical systems that retain some important information, such as controllability, about the original system. The notions of traceability, ε-traceability, ε-consistency deal with the relationship between two dynamical systems. This chapter discusses these concepts and applies them to the robotic car and unicycle systems. It starts with a review of several concepts used in the definition of abstraction. A full discussion of these concepts can be found in [47].
Patricia Mellodge, Pushkin Kachroo

Control Design

Abstract
This chapter focuses on control design using abstraction. Given a system and its abstraction, as defined in the previous chapter, a method is presented for transforming a controller in the abstracted system back to the original system. Conditions for the existence and uniqueness of transformation are provided and the relationship with traceability is given. Finally, if such a transformation does not exist, the existence of an arbitrarily close solution is investigated and its relationship to ε-traceability is studied.
Patricia Mellodge, Pushkin Kachroo

Open-Loop Control Design

Abstract
As discussed in the Chapter 5, the car and uncycle have very close relationship that allows controllability to propagate between the two systems. However, it was shown that the uniycle is not traceable by the car. This means that there exists a unicycle trajectory that does not the Φ-mapping of any car trajectory. The particular trajectory that causes problems is that of the rotating unicycle and addressing this problem in the open-loop setting is the focus of this chapter.
In this chapter, an open-loop optimal control algorithm is presented that utilizes the ε-traceability of the unicycle by the car. The algorithm is developed and simulation results are given for different initial car inputs. Their results are compared.
Patricia Mellodge, Pushkin Kachroo

Uncertainty Propagation in Abstracted Systems

Abstract
In this chapter, it is shown that given a system and its abstraction, the evolution of uncertain initial conditions in the original system is, in some sense, matched by the evolution of the uncertainty in the abstracted system. In other words, it is shown that the concept of Φ-related vector fields extends to the case of stochastic initial conditions where the probability density function (pdf) for the initial conditions is known. In the deterministic case, the Φ mapping commutes with the system dynamics. In this chapter, it is shown that in the case of stochastic initial conditions, the induced mapping, Φpdf, commutes with the evolution of the pdf according to the Liouville equation. It is also shown that a control system abstraction can capture the time evolution of the uncertainty in the original system by an appropriate choice of control input. Application of the convservation law results in a partial differential equation known as the Liouville equation, for which a closed form solution is known. The solution provides the time evolution of the initial pdf which can be followed by the abstracted system.
Patricia Mellodge, Pushkin Kachroo

Conclusion

Abstract
This work has investigated abstraction of dynamical systems. Abstraction deals with the representation of a system using a simpler model which captures the important behavior of the original system. The motivating example throughout this work was the robotic car.
After a review of the mathematical preliminaries in Chapter 2, details of the robotic car modeling and one particular controller were reviewed in Chapter 3. Also in that chapter, results of a curvature estimator implemented for the author’s Master’s thesis were given. In Chapter 4, another controller was reviewed. This controller was designed for the unicycle rather than the car, and so must be converted in some way before being implemented on the car. This conversion was the motivation for studying abstraction and working on the problem: Can the controller designed for the unicycle be converted to one for the car?
Patricia Mellodge, Pushkin Kachroo

Backmatter

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