This book has some of its genesis in the, possibly apocryphal, story that at an acoustics conference in the late 1980s a certain distinguished professor, tiring of the proceedings, turned to the assembled researchers and announced

Certainly it has become easy to think of linear acoustics as essentially completed. After all, classic texts such as Morse and Feshbach (1953) give admirably thorough expositions of very general techniques, particularly those based on Green's functions. Cases described by coordinate systems in which the governing equations are separable are extensively tabulated and admit analytic solutions. The alternative is to employ numerical methods, many of them also based on Green's functions, which work in arbitrarily complex geometries. There is perhaps a perception that notwithstanding a host of important applied problems, there are no fundamental issues remaining in linear acoustics. Increased understanding of the richness and complexity of nonlinear problems with the explosion of interest in chaos only serves to make linear systems seem “done and dusted” in comparison.

And yet this picture is overly dismissive. A solution of a linear differential equation depends nonlinearly on its coefficients and the shape of the boundary. The dependence is all the richer if those coefficients are random or if boundary reflections are defocusing. Developments in physics throughout the last four decades, often equally applicable to both quantum and linear acoustic problems, but overwhelmingly more often expressed in the language of the former, have explored this.

A full treatment of Markov models is beyond the limit of space available here. A number of introductory texts and tutorials on Markov models are given at the end of this chapter. The tutorial provided here follows that given in [520].

Discrete Markov Process

Suppose that an autonomous agent moves about its environment in such a way that transitions from one location to another are as sketched in Figure C.1. The robot's environment has 13 distinct locations, and at any one time the robot is in one of these. Identify these N = 13 distinct locations (or states) as S1, S2, …, Sn. At each time step the robot can move from one state to another, including possibly remaining in the same location, according to a set of probabilities associated with each state. Then the robot's environment can be represented as a directed graph with the nodes of the graph corresponding to the distinct locations and the directed edges of the graph corresponding to possible transitions for each location. Let t denote the time instant associated with robot motions (state changes), and denote the actual location (state) of the robot at time t as qt. Suppose that the robot at time t was in location Sk, at time t − 1 the robot was in location Sj, …, and at time 1 the robot was in location Si; then what is the probability that the robot is in location Sl at time t + 1?

The research problems and solutions discussed in this book leave many issues unresolved. Given the current state of mobile robotics, what can be considered essentially solved and what tasks remain? It is clear that for restrictive environments and for limited tasks, autonomous systems can be readily developed. Tasks such as parts delivery in a warehouse, materials transport in hospitals, limited autonomous driving, and so on can all be “solved” for tight definitions of the task and provided that restrictive assumptions can be made concerning the environment. If the environment can be populated with unique markers or if a guidewire can be buried in the floor and dynamic features (including people) can be removed, then most of the problems involved in point-to-point navigation can be completely solved.

Unfortunately, systems that require such engineering of the environment are fragile and prone to failure if the key landmarks fail or are somehow obscured. Perhaps more important, the inflexibility inherent in constraining the environment makes the use of robot systems less appealing for both financial and logistic reasons. In addition to imposing environmental constraints, the use of existing robot systems implies a restricted and well-defined task specification.

In ray approximations of boundary integral kernels, unitary transfer operators arise. The existence of transfer operators implies trace formulas for the fluctuating part of the spectral density and due to unitarity a reduction of the number of periodic orbits used in these formulas. We introduce the transfer operator method and discuss its applications in elasticity in the case of plate vibrations.

Introduction

In the construction of fluctuating spectral densities (Gutzwiller 1990, Brack & Bhaduri 1997, Stöckmann 1999), the physicist Gutzwiller took a trace of a unitary evolution operator, the Green's function in path integral form. The resulting trace formulas are expressed as a sum over periodic orbits. Later, in the setting of resonators it was discovered that the more conventional boundary integral kernels (Kitahara 1985, Bonnet 1995) could do the same job and that these kernels could also be made unitary.

One such approach is the transfer matrix of Bogomolny (Bogomolny 1992, Boasman 1994). Such ray kernels contain the essence of semiclassical approximations and can be used for numerics even without periodic orbits. Furthermore, these operators are unitary, leading to a truncation in the trace formula (called resurgence) of the infinitely many periodic orbits of the system in question (Berry & Keating 1990, Georgeot & Prange 1995). Another unitary approach is inside–outside duality based on scattering (Smilansky 1995, Smilansky & Ussishkin 1996).

One of the fascinating aspects of the field known colloquially as quantum chaos is the immense variety of physical contexts in which it appears. In the late 1980s it was recognized that ocean acoustics was one such context. It was discovered that the internal state of the ocean leads to multiple scattering of sound as it propagates and leads to an underlying ray dynamics which is predominantly unstable, that is, chaotic. This development helped motivate a resurgence of interest in extending dynamical systems theory suitably for applying ray theory in its full form to a “chaotic” wave mechanical propagation problem. A number of theoretical tools are indispensable, including semiclassical methods, action-angle variables, canonical perturbation theory, ray stability analysis and Lyapunov exponents, mode approximations, and various statistical methods. In the current work, we focus on these tools and how they enter into an analysis of the propagating sound.

Introduction

Acoustic wave propagation through the ocean became a topic of immense physical interest in the latter half of the twentieth century. Beyond the evident sonar applications, acoustic waves offer a means with which to probe the ocean itself. It is possible to monitor bulk mean ocean temperatures over time, which gives important information for studying global warming, and to obtain other information about the internal state of the ocean, that is, currents, eddies, internal waves, seafloor properties, and the like (Flatté et al. 1979, Munk et al. 1995).

The ability to navigate purposefully is fundamental to most animals and to every intelligent organism. In this book we examine the computational issues specific to the creation of machines that move intelligently in their environment. From the earliest modern speculation regarding the creation of autonomous robots, it was recognized that regardless of the mechanisms used to move the robot around or the methods used to sense the environment, the computational principles that govern the robot are of paramount importance. As Powell and Donovan discovered in Isaac Asimov's story “Runaround,” subtle definitions within the programs that control a robot can lead to significant changes in the robot's overall behavior or action. Moreover, the interactions among multiple complex components can lead to large-scale emergent behaviors that may be hard to predict.

The modeling of field distributions by Gaussian random wavefields is reviewed. Basic properties of Gaussian random functions are discussed and applied to the specific examples of randomly scattered fields (speckle patterns) and random eigenfunctions in chaotic enclosures, according to Berry's ergodic mode hypothesis. Sabine's law for reverberation time is derived for an ergodic random mode.

Introduction

The fact that deterministic wavefields in complex geometries can be represented and analyzed by statistical methods is an important assumption in many physical systems. In acoustics, such random-seeming deterministic fields can arise in scattering from a rough surface or in the spatial structure of the eigenfunctions of a perfectly resonant ergodic cavity. The random fields in the former case originate from approximating the surface roughness with randomly placed secondary sources, whose superposition gives rise to a random “speckle pattern,” typical for a wide range of wave scattering systems, especially laser light (Goodman 1985, 2007), and long studied in acoustics (Morse & Bolt 1944, Ebeling 1984). More surprising is Berry's hypothesis (Berry 1977), originally proposed in the context of quantum chaos, that a typical mode of an irregular cavity itself strongly resembles a typical Gaussian random function for high frequencies (i.e., in the semiclassical limit); again, this notion has existed in some sense for a long time in acoustics (Morse & Bolt 1944).

Probabililty

Probability finds application throughout many scientific fields, but finds particular use in robotics in the modelling of uncertainty in terms of a vehicle's pose, interpretating sensor data, and mapping. This appendix reviews some of the fundamental aspects of probability and relates these aspects to problems in the mobile robotics domain. Although this appendix attempts to be reasonably complete in this regard, it cannot hope to provide an exhaustive review of the subject. See the end of the appendix for additional readings.

One problem encountered when modelling real-world systems is that few systems exhibit constant behavior. Observations vary depending on what appears to be random or chance variations. Probability theory provides a general technique for describing these variations. Suppose there is an experiment whose outcome depends on chance. The outcome of the experiment is described as a random variable X. The sample space Ω is the space from which X can be drawn. If the sample space is finite or countably infinite, then the random variable and sample space are said to be discrete; otherwise they are said to be continuous.

Probability of Discrete Events

The discrete probability function P(X) scores evidence that a discrete event X, where X is drawn from sample space Ω, has occurred.

Robotic systems, and in particular mobile robotic systems, are the embodiment of a set of complex computational processes, mechanical systems, sensors, user interface, and communications infrastructure. The problems inherent in integrating these components into a working robot can be very challenging. Overall system control requires an approach that can properly handle the complexity of the system goals while dealing with poorly defined tasks and the existence of unplanned and unexpected events. This task is complicated by the non-standard nature of much robotic equipment. Often the hardware seems to have been built following a philosophy of “ease of design” rather that with an eye towards assisting with later system integration. For example, at the hardware level the robot's wheels rotate in encoder counts, sonar sensors return clock ticks to the first returned echo, and the hardware interfaces can be very specialized. Even if the devices provide “standard” interfaces such as through a standard system bus, serial connections, or via some other standard protocol, the actual command stream can be very device specific. These factors impede abstraction, portability, rapid prototyping, modularity, and pedagogy. Software developers must write layers of code to divorce the low-level device interface from higher-level software functions.

Before delving into the harsh realities of real robots, it is worthwhile exploring some of the computational tasks that are associated with an autonomous system. This chapter provides a taste (an amuse bouche, if you will) of some of the computational problems that will be considered in later chapters. Here these problems are considered in their simplest form, and many of the realities of autonomous systems are ignored. Rest assured, the full complexity of the problems are considered in later chapters.

Perhaps the simplest theoretical abstraction of an autonomous robot is the point robot. A point robot abstracts the robot as a single point operating in some environment, typically a continuous Cartesian plane. Within this formalism, the robot can be represented as a point (x, y) ∈ ℝ2. The domain of operation of the robot is the plane. The point (x, y) fully describes the state of the robot and is also known as the robot's pose or configuration.

Moving the robot involves changing its state from one value (a, b) to another (c, d). The robot operates on a plane, but not all of this domain is necessarily available to the robot. The set of valid poses of the robot are known as its free space. Some states are not valid; rather, they correspond to obstacles or other states that the robot cannot occupy.

A mobile robot is a combination of various physical (hardware) and computational (software) components. In terms of hardware components, a mobile robot can be considered as a collection of subsystems for:

Locomotion: How the robot moves through its environment

Sensing: How the robot measures properties of itself and its environment

Reasoning: How the robot maps these measurements into actions

Communication: How the robot communicates with an outside operator

Later chapters consider the algorithms and representations that make these capabilities possible, while this chapter concentrates on the underlying hardware, with special emphasis on locomotion for wheeled robots.

Locomotion

locomotionn. (Power of) motion from place to place.

Locomotion is the process by which an autonomous robot or vehicle moves. In order to produce motion, forces must be applied to the vehicle. The study of motion in which these forces are modelled is known as dynamics, while kinematics is the study of the mathematics of motion without considering the forces that affect the motion. That is, kinematics deals with the geometric relationships that govern the system, while dynamics includes the energies and speeds associated with these motions. Here we consider the locomotive strategies of various classes of mobile robots. For wheeled mobile robots, these strategies can often be well described in terms of their kinematics. For legged, space, aquatic, and flying robots, it is usually necessary to consider the locomotive strategies in terms of their dynamics.

Introduction

The propagation of waves through heterogeneous media occurs in many forms, including acoustic, electromagnetic, and elastic. As these waves propagate, the wave front is altered because of spatial variations in properties. The result of the interaction with the medium is that the incident energy is dispersed in many directions – the input energy is said to be scattered. If the scattering is strong and one waits long enough, the signal received will become complex because of multiple scattering effects. Understanding this process is necessary for locating an object within a scattering medium and/or for quantifying the properties of the medium itself. The focus here is on the use of diagrams than can aid in analysis of the multiple scattering process.

Multiple scattering has been discussed by theorists since the time of Rayleigh (1892, 1945). Systems with distributions of discrete inclusions (scatterers) in a homogeneous background were studied by Foldy (1945), Lax (1951, 1952), Waterman and Truell (1961), and Twersky (1977) in terms of assumed exact descriptions of scattering by isolated inclusions. This approach may be contrasted with a model of the heterogeneous medium as having continuously varying properties. This approach entails stochastic operator theory and includes the work of Karal and Keller (1964), Frisch (1968), McCoy (1981), Stanke and Kino (1984), and Hirsekorn (1988). Both approaches seek the wave speed and attenuation of an ensemble average field, although the connection to measurements in a single sample is not always obvious.

Given a robot that can move and sense its environment, the remaining task is to plan intelligent motions.

Chapter 6 looks at some of the fundamental computational tasks that must be addressed in order for a mobile robot to move intelligently through its environment: how the environment should be represented, how the robot should be represented, and how to plan, given these representations.

The software environment within which the control programs of modern mobile robots execute can be likened to mobile operating systems. The various tasks that provide overall robotic control must compete for scarce computational and sensor resources and must meet both soft and hard real-time constraints. In order to meet these requirements, various architectures and strategies have been proposed to provide a framework within which the controlling processes exist. Chapter 7 examines how the various computational components that make up a mobile robot can be put together: the software environment or operating system within which the computational tasks compete for the scarce resources both on and off the robot.

Chapter 8 considers pose maintenance, a necessary task in many robotic systems. This allows a robot to use and construct maps, topics covered in Chapter 9.

Chapter 10 looks at the problems involved in having two or more robots operating in the same environment.

We give an overview of wave scattering in complex geometries, where the corresponding rays are typically chaotic. In the high-frequency regime, a number of universal (geometry-independent) properties that are described by random matrix theory emerge. Asymptotic methods based on the underlaying rays explain this universality and are able to go beyond it to account for geometry-specific effects. We discuss in this context statistics of the scattering matrix, scattering states, the fractal Weyl law for resonances, and fractal resonance wavefunctions.

Introduction

Our purpose here is to give an introductory overview of wave scattering in complex geometries, where the corresponding rays are typically chaotic. For simplicity, we focus our discussion on domains with lossless walls, inside which the wave speed is constant. The rays then are straight, with specular reflections at the boundaries. This situation is often encountered in experiments (Stöckmann 1999, Kuhl et al. 2005, Tanner & Søndergaard 2007) and in acoustic applications. However, many of the features we shall identify occur much more generally. Indeed, most recent developments in the subject have taken place in the context of quantum wave scattering, where the underlying rays are the classical trajectories of Newtonian mechanics. Much of this review will be devoted to translating quantum results into the language of classical wave scattering.

One of the main observations we wish to make is that many of the essential mathematical features of wave scattering in complex geometries can be found in certain very simple discrete models, which we here call wave maps.

Chapter 1 provides the background, both the model equations and some of the mathematical transformations, needed to understand linear elastic waves. Only the basic equations are summarized, without derivation. Both Fourier and Laplace transforms and their inverses are introduced and important sign conventions settled. The Poisson summation formula is also introduced and used to distinguish between a propagating wave and vibration of a bounded body. A general survey of books and collections of papers that bear on the contents of the book are discussed at the end of the chapter.

A linear wave carries information at a particular velocity, the group velocity, which is characteristic of the propagation structure or environment. It is this transmission of information that gives linear waves their special importance. In order to introduce this aspect of wave propagation, we discuss propagation in one-dimensional periodic structures. Such structures are dispersive and therefore transmit information at a speed different from the wavespeed of their individual components.

Model equations

The equations of linear elasticity consist of:

1. (1) the conservation of linear and angular momentum; and

2. (2) a constitutive relation relating force and deformation.

In the linear approximation the density ρ is constant. The conservation of mechanical energy follows from (1) and (2). The most important feature of the model is that the force exerted across a surface, oriented by the unit normal nj, by one part of a material on the other is given by the traction ti = τjinj, where τji is the stress tensor.