Mechanics is the study of the behavior of matter under the action of internal and external forces. In this introductory treatment of continuum mechanics, we accept the concepts of time, space, matter, energy, and force as the Newtonian ideals. Here our objective is the formulation of engineering problems consistent with the fundamental principles of mechanics. To paraphrase Professor Y. C. Fung–there are generally two ways of approaching mechanics: One is the ad hoc method, in which specific problems are considered and specific solution methods are devised that incorporate simplifying assumptions, and the other is the general approach, in which the general features of a theory are explored and specific applications are considered at a later stage. Engineering students are familiar with the former approach from their experience with “Strength of Materials” in the undergraduate curriculum. The latter approach enables them to understand an entire field in a systematic way in a short time. It has been traditional, at least in the United States, to have a course in continuum mechanics at the senior or graduate level to unify the ad hoc concepts students have learned in the undergraduate courses. Having had the knowledge of thermodynamics, fluid dynamics, and strength of materials, at this stage, we look at the entire field in a unified way.

Concept of a Continuum

Although mechanics is a branch of physics in which, according to current developments, space and time may be discrete, in engineering the length and time scales are orders (and orders) of magnitude larger than those in quantum physics and we use space coordinates and time as continuous.

The time-independent, permanent deformation in metals beyond the elastic limit is described by the term plasticity. As shown in Fig. 14.1, the elastic part of the uniaxial stress–strain curve OA is reversible. When we unload from any point beyond point A, corresponding to the zero-stress state, there is a permanent deformation in the specimen. Several theories describing the plastic behavior of metals have been proposed, but none is totally satisfactory. Ideally, the stress σ, separating the elastic part OA from the inelastic part, is called the yield stress σ0 of the material. However, it is difficult to obtain this special point accurately from experiments, and it often depends on the history of the loading the specimen has undergone.

When a specimen is unloaded from a point C to zero stress and reloaded, permanent deformation usually begins at a stress level σC. In other words, the yield stress is higher after the specimen has undergone a certain amount of permanent deformation. This is known as work hardening or strain hardening.

Another feature of metal deformation is that, from a strained state, beyond the yield point, reversal of loading causes the compressive yield stress to be different from the tensile yield stress. It is usually lower in magnitude than the tensile yield stress. This effect is called the Bauschinger effect.

This text is based on a one-semester course I have been teaching at the Illinois Institute of Technology for about 30 years. Graduate students from mechanical and aerospace engineering, civil engineering, chemical engineering, and applied mathematics have been the main customers. Most of the students in my course have had some exposure to Newtonian fluids and linear elasticity. These two topics are covered here, neglecting the large number of boundary-value problems solved in undergraduate texts. On a number of topics, it becomes necessary to sacrifice depth in favor of breadth, as students specializing in a particular area will be able to delve deeper into that area with the foundation laid out in this course. Space and time constraints prevented the inclusion of classical topics such as hypoelasticity and electromagnetic effects in elastic and fluid materials and a more detailed treatment of nonlinear viscoelastic fluids.

I have included a small selection of exercises at the end of each chapter, and students who attempt some of these exercises will benefit the most from this text. Instructors may add reading assignments from other sources.

Instead of placing all the references at the end of the book, I have given the pertinent books and articles relevant to each chapter at the end of that chapter. There are some duplications in this mode of presentation, but I hope it is more convenient.

Robotics: An Introduction

The concept of a robot as we know it today evolved over many years. In fact, its origins could be traced to ancient Greece well before the time when Archimedes invented the screw pump. Leonardo da Vinci (1452–1519) made far-reaching contributions to the field of robotics with his pioneering research into the brain that led him to make discoveries in neuroanatomy and neurophysiology. He provided physical explanations of how the brain processes visual and other sensory inputs and invented a number of ingenious machines. His flying devices, although not practicable, embodied sound principles of aerodynamics, and a toy built to bring to fruition Leonardo's drawing inspired the Wright brothers in building their own flying machine, which was successfully flown in 1903. The word robot itself seems to have first appeared in 1921 in Karel Capek's play, Rossum's Universal Robots, and originated from the Slavic languages. In many of these languages the word robot is quite common as it stands for worker. It is derived from the Czech word robitit, which implies drudgery. Indeed, robots were conceived as machines capable of repetitive tasks requiring a lower intelligence than that of humans. Yet today robots are thought to be capable of possessing intelligence, and the term is probably inappropriate. Nevertheless it is in use. The term robotics was probably first coined in science fiction work published around the 1950s by Isaac Asimov, who also enunciated his three laws of robotics. It was from Asimov's work that the concept of emulating humans emerged.

Introduction

In this chapter we consider some important benchmark problems involving biomimetic robots, such as the dynamics and balance of walking biped robots, dynamics and control of four-legged robotic vehicles, dynamics and control of robotic manipulators in free flight, dynamics and control of flapping propulsion in aerial vehicles, and the underwater propulsion of aquatic vehicles.

Dynamics and Balance of Walking Biped Robots

The dynamics and control of walking bipeds provides much insight into biomimetic robots. We consider a relatively simple model of a walking model that is capable of capturing the principal features of the kinematics and dynamics of coordinating walking that resemble a human gait. Generally a human can be modeled to operate in two-dimensional space, with the model having a head, a pair of arms attached by a shoulder joint to a torso, and two identical legs with knees as well as two ankles and feet. Such a model is capable of demonstrating uniform walking based on the stance and swing mechanism, dynamic balance of the torso and the head, the role of the ankle and feet in providing rolling contact with ground, and the role of the human arms that act as “stabilizers” while the robot is walking forward with uniform forward velocity. Such a planar model is illustrated in Figure 12.1.

Dynamic Model for Walking

To understand the basic kinematics of coordinated walking, a simplified seven-DOF model involving the head, torso, and legs is adequate. The legs by themselves form a classical five-bar linkage, and when the constraint of a single-leg support is included, the number of DOFs is reduced to just five.

Differential Motion

In Chapters 4 and 5, direct kinematics and inverse kinematics of manipulators, relating the position of the end-effector in base coordinates to the joint coordinates and vice versa, were considered. These relationships represent transformations from one set of coordinates to the other. However, in the context of forces acting at the joints, the definition of these transformations is incomplete. Although the work done by the joint forces is a scalar, it may be expressed as a surface or volume integral in the space defined by the coordinates. Thus it is important to relate the volume of an element in the Cartesian frame to the volume of an element in the frame defined by the joint coordinates. This relationship was first demonstrated by Carl Gustav Jacob Jacobi (1804–1851). Jacobi, who hailed from a family of Prussian bankers, worked at the University of Königsberg. He arrived there in May 1826 and pursued an academic career in pure mathematics. There he worked on, among other topics, determinants and studied the functional determinant now called the Jacobian. Although Jacobi was not the first to study the functional determinant that now bears his name, as it appears that the functional determinant was mentioned in 1815 by Cauchy, Jacobi wrote a paper on De determinantibus functionalibus in 1841 devoted to this determinant. He proved, among many other things, that if a set of n functions in n variables is functionally unrelated or independent, then the Jacobian cannot be identically zero.

Fundamentals of Trajectory Following

The motion of the end-effector of a manipulator is described by a trajectory in multidimensional space. It refers to a time history of the position, the velocity, and the acceleration of each DOF. Trajectory control is a fundamental problem in robotics and involves two distinct steps. The first of these steps is the planning of the desired trajectory or path of the end-effector. The second is the design and implementation of a suitable controller so as to ensure that the robot does indeed follow the planned path. This step is known as the trajectory or path tracking and is essentially a feedback-control problem. The path tracker is responsible for making the robot's actual position and velocity match the desired values of position and velocity provided to it by the path planner.

Path planning is one of the most vital issues to ensure autonomy of a robot, whether it is a robot manipulator or a mobile robot. It can be viewed as finding a safe and optimum path through an obstacle-filled environment from starting point to some destination in a collision-free manner. Once the desired path of an endeffector is planned there is also concern about the representation of the path for subsequent computational purposes. In the case of manipulators, a continuous time path either in the joint variable space or in a Cartesian base frame is essential. The trajectory controller then synthesizes an appropriate path following the control law based on this continuous time path.

Path planning in mobile robot applications is usually different from the pathplanning problem for manipulators.

Dynamical Systems of the Liouville Type

William Rowan Hamilton was a remarkable dynamicist who was considered to be the greatest mathematician, after Sir Isaac Newton, of the English-speaking world. Born in Dublin in 1805 and named after an outlawed Irish patriot, Archibald Hamilton Rowan, William R. Hamilton died in 1865 as the Astronomer Royal of Ireland. In the span of 60 years he was responsible for a remarkable number of new concepts, including the calculus of the quaternion, a generalization of a complex number to three-dimensional space. Hamilton's discovery not only led to Arthur Cayley's application, in 1854, of the quaternion to the representation of spatial rotations but also led to the development of new algebras, including the theory of matrices and the algebra associated with biquaternions, defined in 1873 by William Kingdon Clifford. In 1834, when he was just 29 years of age, Hamilton wrote to his uncle: “It is my hope and purpose to remodel the whole of dynamics, in the most extensive sense of the word, by the idea of my characteristic function.” He proceeded to apply this principle to the motion of systems of bodies, and in the following year expressed these equations of motion in a form that established a duality between the components of momentum of a system of bodies and the coordinates of their positions. He largely achieved his objectives and, in the process, set out on a quest for simplicity that yielded remarkable dividends.

Before Hamilton's general equations of motion are introduced, it is instructive to consider a restricted class of dynamical systems associated with name of Liouville.