1. Introduction systems are used during actual space fight in order to compensate for errors from the precomputed trajectory. Mathematically, stability theory, dynamical systems, and especially the theory of functions of a complex variable, have had a strong influence on this approach. It is widely recognized today that these two broad lines of work deal just with different aspects of the same problems and we do not make an artificial distinction between them in this book Later on we shall give an axiomatic definition of what we mean by a"system or"machine. " Its role will be somewhat analogous to that played in mathematics oy the definition of"function"as a set of ordered pairs: not itself the object of study, but a necessary foundation upon which the entire theoretical development will rest. In this Chapter, however, we dispense with precise definitions and will use a very simple physical example in order to give an intuitive presentation of some of the goals, terminology, and methodology of control theory. The discussion here will be informal and not rigorous, but the reader is encouraged to follow it in detail, since the ideas to be given underlie everything else in the book. Without them, many problems may look artificial. Later, we often refer back to this Chapter for motivation 1.2 Proportional-Derivative Control One of the simplest problems in robotics is that of controlling the position of a single-link rotational joint using a motor placed at the pivot. Mathematically, this is just a pendulum to which one can apply a torque as an external force see Figure 1.1) Figure 1.1: Pendulum. e assume that friction is negligible, that all of the mass is concentrated at the end, and that the rod has unit length. From Newtons law for rotating objects, there results, in terms of the variable 0 that describes the counterclock wise angle with respect to the vertical, the second-order nonlinear differential me(t)+mg sin e(t)=u(t)2 1. Introduction systems are used during actual space flight in order to compensate for errors from the precomputed trajectory. Mathematically, stability theory, dynamical systems, and especially the theory of functions of a complex variable, have had a strong influence on this approach. It is widely recognized today that these two broad lines of work deal just with different aspects of the same problems, and we do not make an artificial distinction between them in this book. Later on we shall give an axiomatic definition of what we mean by a “system” or “machine.” Its role will be somewhat analogous to that played in mathematics by the definition of “function” as a set of ordered pairs: not itself the object of study, but a necessary foundation upon which the entire theoretical development will rest. In this Chapter, however, we dispense with precise definitions and will use a very simple physical example in order to give an intuitive presentation of some of the goals, terminology, and methodology of control theory. The discussion here will be informal and not rigorous, but the reader is encouraged to follow it in detail, since the ideas to be given underlie everything else in the book. Without them, many problems may look artificial. Later, we often refer back to this Chapter for motivation. 1.2 Proportional-Derivative Control One of the simplest problems in robotics is that of controlling the position of a single-link rotational joint using a motor placed at the pivot. Mathematically, this is just a pendulum to which one can apply a torque as an external force (see Figure 1.1). mg u mg sin θ θ Figure 1.1: Pendulum. We assume that friction is negligible, that all of the mass is concentrated at the end, and that the rod has unit length. From Newton’s law for rotating objects, there results, in terms of the variable θ that describes the counterclock￾wise angle with respect to the vertical, the second-order nonlinear differential equation m¨θ(t) + mg sin θ(t) = u(t), (1.1)