This chapter provides an overview of matrices. Basic matrix operations are introduced first, such as addition, multiplication, transposition, and so on. Determinants and matrix inverses are then defined. The rank and Kruskal rank of matrices are defined and explained. The connection between rank, determinant, and invertibility is elaborated. Eigenvalues and eigenvectors are then reviewed. Many equivalent meanings of singularity (non-invertibility) of matrices are summarized. Unitary matrices are reviewed. Finally, linear equations are discussed. The conditions under which a solution exists and the condition for the solution to be unique are also explained and demonstrated with examples.

This chapter discusses the Fourier series representation for continuous-time signals. This is applicable to signals which are either periodic or have a finite duration. The connections between the continuous-time Fourier transform (CTFT), the discrete-time Fourier transform (DTFT), and Fourier series are also explained. Properties of Fourier series are discussed and many examples presented. For real-valued signals it is shown that the Fourier series can be written as a sum of a cosine series and a sine series; examples include rectified cosines, which have applications in electric power supplies. It is shown that the basis functions used in the Fourier series representation satisfy an orthogonality property. This makes the truncated version of the Fourier representation optimal in a certain sense. The so-called principal component approximation derived from the Fourier series is also discussed. A detailed discussion of the properties of musical signals in the light of Fourier series theory is presented, and leads to a discussion of musical scales, consonance, and dissonance. Also explained is the connection between Fourier series and the function-approximation property of multilayer neural networks, used widely in machine learning. An overview of wavelet representations and the contrast with Fourier series representations is also given.

This chapter examines discrete-time LTI systems in detail. It shows that the input–output behavior of an LTI system is characterized by the so-called impulse response. The output is shown to be the so-called convolution of the input with the impulse response. It is then shown that exponentials are eigenfunctions of LTI systems. This property leads to the ideas of transfer functions and frequency responses for LTI systems. It is argued that the frequency response gives a systematic meaning to the term “filtering.” Image filtering is demonstrated with examples. The discrete-time Fourier transform (DTFT) is introduced to describe the frequency domain behavior of LTI systems, and allows one to represent a signal as a superposition of single-frequency signals (the Fourier representation). DTFT is discussed in detail, with many examples. The z-transform, which is of great importance in the study of LTI systems, is also introduced and its connection to the Fourier transform explained. Attention is also given to real signals and real filters, because of their additional properties in the frequency domain. Homogeneous time-invariant (HTI) systems are also introduced. Continuous-time counterparts of these topics are explained. B-splines, which arise as examples in continuous-time convolution, are presented.

This chapter discusses many interesting properties of bandlimited signals. The subspace of bandlimited signals is introduced. It is shown that uniformly shifted versions of an appropriately chosen sinc function constitute an orthogonal basis for this subspace. It is also shown that the integral and the energy of a bandlimited signal can be obtained exactly from samples if the sampling rate is high enough. For non-bandlimited functions, such a result is only approximately true, with the approximation getting better as the sampling rate increases. A number of less obvious consequences of these results are also presented. Thus, well-known mathematical identities are derived using sampling theory. For example, the Madhava–Leibniz formula for the approximation of π can be derived like this. When samples of a bandlimited signal are contaminated with noise, the reconstructed signal is also noisy. This noise depends on the reconstruction filter, which in general is not unique. Excess bandwidth in this filter increases the noise, and this is quantitatively analyzed. An interesting connection between bandlimited signals and analytic functions (entire functions) is then presented. This has many implications, one being that bandlimited signals are infinitely smooth.

This chapter discusses the Fourier series representation for continuous-time signals. This is applicable to signals which are either periodic or have a finite duration. The connections between the continuous-time Fourier transform (CTFT), the discrete-time Fourier transform (DTFT), and Fourier series are also explained. Properties of Fourier series are discussed and many examples presented. For real-valued signals it is shown that the Fourier series can be written as a sum of a cosine series and a sine series; examples include rectified cosines, which have applications in electric power supplies. It is shown that the basis functions used in the Fourier series representation satisfy an orthogonality property. This makes the truncated version of the Fourier representation optimal in a certain sense. The so-called principal component approximation derived from the Fourier series is also discussed. A detailed discussion of the properties of musical signals in the light of Fourier series theory is presented, and leads to a discussion of musical scales, consonance, and dissonance. Also explained is the connection between Fourier series and the function-approximation property of multilayer neural networks, used widely in machine learning. An overview of wavelet representations and the contrast with Fourier series representations is also given.

This chapter introduces the discrete Fourier transform (DFT), which is different from the discrete-time Fourier transform (DTFT) introduced earlier. The DFT transforms an N-point sequence x[n] in the time domain to an N-point sequence X[k] in the frequency domain by sampling the DTFT of x[n]. A matrix representation for this transformation is introduced, and the properties of the DFT matrix are studied. The fast Fourier transform (FFT), which is a fast algorithm to compute the DFT, is also introduced. The FFT makes the computation of the Fourier transforms of large sets of data practical. The digital signal processing revolution of the 1960s was possible because of the FFT. This chapter introduces the simplest form of FFT, called the radix-2 FFT, and a number of its properties. The chapter also introduces circular or cyclic convolution, which has a special place in DFT theory, and explains the connection to ordinary convolution. Circular convolution paves the way for fast algorithms for ordinary convolution, using the FFT. The chapter also summarizes the relationships between the four types of Fourier transform studied in this book: CTFT, DTFT, DFT, and Fourier series.

Ever since we began to build software systems that interacted with humans, there have ethical concerns about the ways in which we interact with them. In [830], for example, Weizenbaum observes of the world’s first chatterbot that “ELIZA shows, if nothing else, how easy it is to create and maintain the illusion of understanding, hence perhaps of judgment deserving of credibility. A certain danger lurks there.”2 Fast forward more than 60 years, and this observation that a “certain danger lurks there” has emerged as a range of different concerns about the ways in which software (and hardware) systems are developed and deployed, and the range of data that modern data-driven systems rely upon. The space of machine ethics is vast, and a large number of texts, papers, and policy documents now exist on the subject.

Sensing is a key requirement for any but the simplest mobile behavior. In order for Robot to be able to warn the crew of Lost in Space that there is danger ahead, it must be able to sense and reason about its sensor responses. Sensing is a critical component of the fundamental tasks of pose estimation – determining where the robot is in its environment; pose maintenance – maintaining an ongoing estimate of the robot’s pose; and map construction – building a representation of the robot’s environment.

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 toward assisting with later system integration.

The ability to navigate purposefully through its environment 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, interactions among multiple complex components can lead to large-scale emergent behaviors that may be hard to predict.

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.