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In defining the scope of our subject, we have to establish the genealogy of robotic mechanical systems. These are, obviously, a subclass of the much broader class of mechanical systems . Mechanical systems, in turn, constitute a subset of the more general concept of dynamic systems . In the end, we must have an idea of what, in general, a system is.

First and foremost, the study of motions undergone by robotic mechanical systems or, for that matter, by mechanical systems at large, requires a suitable motion representation. Now, the motion of mechanical systems involves the motion of the particular links comprising those systems, which in this book are supposed to be rigid. The assumption of rigidity, although limited in scope, still covers a wide spectrum of applications, while providing insight into the motion of more complicated systems, such as those involving deformable bodies.

The purpose of this chapter is to lay down the foundations of the kinetostatics and dynamics of rigid bodies, as needed in the study of multibody mechanical systems. With this background, we study the kinetostatics and dynamics of robotic manipulators of the serial type in Chapters 5 and 7, respectively, while devoting Chapter 6 to the study of trajectory planning. The latter requires, additionally, the background of Chapter 4. A special feature of the current chapter is the study of the relations between the angular velocity of a rigid body and the time-rates of change of the various sets of rotation invariants introduced in Chapter 2. Similar relations between the angular acceleration and the second time-derivatives of the rotation invariants are also recalled, the corresponding derivations being outlined in Appendix A.

This chapter is devoted to the displacement analysis of robotic manipulators of the serial type, which we call the geometry of serial robots . The study is limited to decoupled robots , to be defined below, the inverse displacement analysis of general six-axis robots being the subject of Chapter 9. These robots serving mainly to perform manipulation tasks , they are also referred to as manipulators .

Kinetostatics is understood here as the study of the interplay between the feasible twists of and the constraint wrenches acting on the various rigid bodies of a mechanical system, when the system moves under static, conservative conditions . The feasible twists of the various rigid bodies, or links, are those allowed by the constraints imposed by the robot joints. The constraint wrenches are, in turn, the reaction forces and moments exerted on a link by the links to which that link is coupled by means of joints. The subject of this chapter is the kinetostatics of serial robots, with focus on six-axis manipulators. By virtue of the duality between the kinematic and the static relations in the mechanics of rigid bodies, as outlined in Section 3.7, the derivation of the kinematic relations is discussed in detail, the static relations following from the former.

The motions undergone by robotic mechanical systems should be, as a rule, as smooth as possible; i.e., abrupt changes in position, velocity, and acceleration should be avoided. Indeed, abrupt motions require unlimited amounts of power to be implemented, which the motors cannot supply because of their physical limitations. On the other hand, abrupt motion changes arise when the robot collides with an object, a situation that should also be avoided. While smooth motions can be planned with simple techniques, as described below, these are no guarantees that no abrupt motion changes will occur. In fact, if the work environment is cluttered with objects, whether stationary or mobile, collisions may occur. Under ideal conditions, a flexible manufacturing cell is a work environment in which all objects, machines and workpieces alike, move with preprogrammed motions that by their nature, can be predicted at any instant. Actual situations, however, are far from being ideal, and system failures are unavoidable. Unpredictable situations should thus be accounted for when designing a robotic system, which can be done by supplying the system with sensors for the automatic detection of unexpected events or by providing for human monitoring. Nevertheless, robotic systems find applications not only in the well-structured environments of flexible manufacturing cells, but also in unstructured environments such as exploration of unknown terrains and systems in which humans are present.

The main objectives of this chapter are ( i ) to devise an algorithm for the real-time computed-torque control and ( ii ) to derive the system of second-order ordinary differential equations (ODE) governing the motion of an n-axis manipulator. We will focus on serial manipulators, the dynamics of a much broader class of robotic mechanical systems, namely, parallel manipulators and mobile robots, being the subject of Chapter 12. Moreover, we will study mechanical systems with rigid links and rigid joints and will put aside systems with flexible elements, which pertain to a more specialized realm.

The motivation for this chapter is twofold. On the one hand, the determination of the angular velocity and angular acceleration of a rigid body from point-velocity measurements is a fundamental problem in kinematics. On the other hand, the solution of this problem is becoming increasingly relevant in the kinematics of parallel manipulators, to be studied in Chapter 10. Moreover, the estimation of the attitude of a rigid body from knowledge of the Cartesian coordinates of some of its points is sometimes accomplished by time-integration of the velocity data. Likewise, the use of accelerometers in the area of motion control readily leads to estimates of the acceleration of a sample of points of a rigid body, which can be used to estimate the angular acceleration of the body, and hence, to better control its motion.

Current serial robots, encountered not only in research laboratories but also in production or construction environments, include features that deserve a chapter apart. We will call here general serial robots all non-redundant serial robots that do not fall in the category of those studied in Chapter 4. Thus, the chapter is devoted to manipulators of the serial type that do not allow a decoupling of the positioning and the orientation problems. The focus of the chapter is, thus, the inverse displacement problem (IDP) of general six-revolute robots. While redundant manipulators of the serial type fall within this category as well, we will leave these aside, for their redundancy resolution calls for a more specialized background than what we have either assumed or given here.

The study of robotic mechanical systems has focused, so far, on serial manipulators. These are the most common systems of their kind, but nowadays by no means the majority. In recent years, other kinds of robotic mechanical systems have been developed, as outlined in Chapter 1. Under alternative robotic mechanical systems we understand here: i ) parallel robots; ( ii ) multi-fingered hands; ( iii ) walking machines; and ( iv ) rolling robots. A class that is increasingly receiving attention, humanoids , portrays an architecture inspired from the human musculo-skeletal system. This class deserves a study on its own because of the host of control problems that it poses to the roboticist; its kinematics, however, can be studied with the tools developed in this chapter for the first three kinds of systems listed above. For this reason, a section on humanoids is not included here.