Newtonian and Eulerian Mechanics

The kinematic synthesis of manipulators pointedly ignores one important aspect – size. It must be said that size matters! The parameter that is probably one of the best measures of size is inertia. The power required to drive a manipulator largely depends on the inertia of the manipulator. Thus the forces and moments that must be generated to ensure that the manipulator is not only mobile but performs the motions it is expected to take on a special significance. Yet there is one feature of the kinetics of mechanisms that sets it apart from the kinetics of free bodies. In the kinetic analysis of free bodies one is provided with the forces and moments acting on the body, and the objectives of the analysis are to determine the ensuing motions of the bodies. In the case of mechanisms the situation is just the opposite. The motions of all the kinematic links may be determined first, and it is then necessary to determine the forces and moments acting on each one of them. However, the basic principles of kinetic analysis remain the same. They all form a part of the field of dynamics, the study of the action of forces and moments on bodies and their relationship of the resulting motions. The study of the action of the forces and moments and the motions they generate relates to the kinetics whereas the relationships among the motion attributes, position, velocity, and acceleration relate to the kinematics.

Probably the primary underpinning principles of kinetic analysis are enshrined in Sir Isaac Newton's three laws of motion, which were enunciated in 1687 in the Principia Mathematica.

Introduction

The concept of biomimetic control, i.e., control systems that mimic biological animals in the way they exercise control, rather than just humans, has led to the definition of a new class of biologically inspired robots that exhibit much greater robustness in performance in unstructured environments than the robots that are currently being built. It is believed that there is a duality between engineering and nature that is based on optimum use of energy or an equivalent scarce resource, particularly in exercising control over actions and over interactions with the immediate environment. Biomimesis is generally based on this concept, and it is believed that by mimicking animals that are most capable of performing certain specialized actions, such as the lobster on the seabed, insects and birds in flight, and cockroaches in locomotion, one could build robots that can surpass any other in performance, agility, and dexterity.

A key feature of biomimetic robots is their capacity to adapt to the environment and ability to learn and react fast. However, a biomimetic robot is not just about learning and adaptation but also involves novel mechanisms and manipulator structures capable of meeting the enhanced performance requirements. Thus biomimetic robots are being designed to be substantially more compliant and stable than conventionally controlled robots and will take advantage of new developments in materials, microsystems technology, as well as developments that have led to a deeper understanding of biological behavior.

Roboticists have a lot to learn from animals. Birds have a superior flying machine with multielement “aerofoils” capable of controlling the flow around them quite effortlessly.

This book is intended for a first course in a robotics sequence. There are indeed a large number of books in the field of robotics, but for the engineer or student interested in unmanned aerial or underwater vehicles there are very few books. From the point of view of an aerospace engineer (flight simulation, unmanned aerial vehicles, and space robotics) three-dimensional and parallel mechanisms take on an added significance.

The fundamentals of robotics, from an aerospace perspective, can be very well presented by considering just the field of robot mechanisms. Biomimetic robot mechanisms are fundamental to manipulators and to walking, mobile, and flying robots. A distinguishing feature of the book is that a unified and integrated treatment of biomimetic robot mechanisms is offered.

This book is intended to prepare a student for the next logical module in a course on robotics: practical robot-control design. Throughout the book, in every chapter, are introduced the most important and recent developments relevant to the subject matter. Although the book's primary focus is on understanding principles, computational procedures are also given due importance as a matter of course. Students are encouraged to use whatever computational tools they consider appropriate to solve the examples in the exercises.

When writing this text, I included the topics in my course and some more for the enthusiastic reader with the initiative for self-learning.

Thus, Chapter 1 gives an overview of robotics, with a focus on generic robotic mechanisms, whereas Chapter 2 is about biomimetic mechanisms. Chapter 3 draws attention to the integrated representation of displacement and rotation.

Definition of Direct or Forward Kinematics

Kinematics is the study of the motion attributes of a collection of bodies without any reference to the forces or moments that may have caused it. In the context of robotics in general and robot manipulators in particular, it is the geometry of the mechanism that plays a pivotal role in the determination of the kinematics and the associated kinematic relations. Of primary importance are the relations expressing the position and orientation coordinates of the end-effector in terms of the DOFs at each joint. The objective is to establish parameters characterizing each link in a general coordinate system for satisfying the kinematic requirements of a manipulator. To find the location of the end-effector, only the DOFs at the joints, the joint variables, are given, and the problem is to express the position and orientation of the end-effector, the pose of the end-effector, in terms of the given variables. This is the direct or forward kinematics problem. For serial manipulators the direct kinematics is a unique relation, i.e., the location of the end-effector may be uniquely represented in terms of the joint variables.

Although the joint variables may be defined in an arbitrary fashion, a method of frame assignment proposed by Denavit and Hartenberg has emerged as a standard. The advantage of this systematic labeling of joint variables is that the associated homogeneous transformations may be decomposed into a product form that was discussed in the last chapter.

The Denavit–Hartenberg Convention

Robot manipulators are mechanisms involving a sequence of links connected together at the joints.

Pottery was the first man-made material. The oldest known pottery is dated at 27,000 to 23,000 B.C. Pottery is made by shaping clay, drying it, and then firing it at an elevated temperature, which changes its chemical structure.

Clay

There are several forms of clay. All are aluminosilicates. Many contain other elements. Figure 16.1 shows the plate-like structure of kaolinite, Al2(Si2O5)(OH)4. Water absorbed between platelets allows them to slide easily over one another.

Typical clays used for ceramics often contain other materials including ground quartz (SiO2) and a flux such as a feldspar, which is an aluminosilicate containing Na+, K+, and Ca2+.

Processing of Clay Products

Clay is shaped either by pressing it into the desired shape or by slip casting. Slip is a suspension of clay in water made by adding a small amount of a deflocculant (often sodium silicate or soda ash), which allows the clay to be suspended in water. The slip is poured into a plaster of paris mold that absorbs water and causes the clay near the surface of the plaster to become solid. When this solid reaches the desired thickness, the mold is emptied and the resulting shape removed. Figure 16.2 illustrates the process.

The shape is fired only after it is allowed to air dry. Some shrinkage occurs during air drying. After air drying, the shape is said to be green and must be handled with care because it has little strength. Next it is fired usually between 900 and 1400°C.

Fracture

Typically as the yield strength of a material is increased, its ductility and toughness decrease. Toughness is the energy absorbed in fracturing. If a material has a high yield strength, it can be subjected to stresses high enough to cause fracture before there has been much plastic deformation to absorb energy. Factors that inhibit plastic flow lower ductility as schematically indicated in Figure 5.1. These factors include decreased temperatures, increased strain rates, and the presence of notches.

Engineers should be interested in ductility and fracture for two reasons. Ductility is required to form metals into useful parts by forging, rolling, extrusion, or other plastic working processes. Also, some plastic deformation is necessary to absorb energy so as to prevent failure in service.

Fractures can be classified several ways. A fracture may be described as being ductile or brittle, depending on the amount of deformation that precedes it. Failures may also be described as intergranular or transgranular, depending on the fracture path. The terms cleavage, shear, void coalescence, etc., are used to identify failure mechanisms. These descriptions are not mutually exclusive. A brittle fracture may be intergranular, or it may occur by cleavage.

The failure in a tensile test of a ductile material occurs well after the maximum load is reached and a neck has formed. In this case, fracture usually starts by nucleation of voids in the center of the neck where the hydrostatic tension is the greatest.

Bulk Forming

Processes for forming solid metals can be classified as bulk forming in which the forces are mostly compressive, and sheet forming in which the metal is stretched in tension. Bulk forming processes may be further classified as either hot working or cold working, depending on whether the work material leaves in a recrystallized state.

Figure 10.1 illustrates several important bulk forming processes. Compression is clearly the dominant force in rolling, extrusion, and forging. For rolling, extrusion, and forging, the reduction is usually limited by the capacity of the machinery to deliver forces. Although drawing involves pulling a rod or wire through a die, the pressure the die exerts on the rod or wire is the dominant force. In rod and wire draw-ing, the maximum reduction per pass is limited by the possibility of tensile failure of the drawn wire. Typically a maximum strain per pass is about ε = ln(A1/A2) = 0.65, which corresponds to a diameter reduction of 38%. Obviously many passes are required to make fine wire. Many rolling passes are used to make sheets and shapes. Forging uses repeated blows to achieve a final shape. In contrast, rods, tubes, and other shapes are extruded in a single operation to their final shape because high reductions are possible.

Fibers

Fibers of nylon, polyester, and other thermoplastics are made by extruding molten material through tiny holes in a spinneret. The resulting fibers are cooled before coiling. The strengths of nylon, polyester, polypropylene, and high-density polyethylene fibers are increased greatly by stretching them by 400 to 500% in tension (drawing) to orient the molecules parallel to the fiber axis.

Fiber strength is often quoted in terms of grams force per denier (g force/denier). A denier is defined as grams mass per 9000 m of fiber. Figure 19.1 shows the relative strengths of various fibers. Kevlar fibers are much stronger, having strengths of about 22 g force/denier.

Example Problem 19–1:

Develop an equation for converting tenacity in g force/denier to MPa.

Solution: Let T be the tenacity in g force/denier. Then T[g force/(g mass/9000 m)]ρ(g/cm3)(100 cm/m)(980.7 × 10-7N/g-force)(1002cm2/m2) = 883 × 103 (Tρ) Pa. Tensile strength is 0.883Tρ MPa.

Fabrication of Porous Foams

Natural cellular materials include sponges and wood. Foams of polymers, metals, and ceramics can be made by numerous methods. Many foams are produced by gas evolvution. Inert gasses such as CO2 and N2 may be dissolved under high pressure and released by decreasing the pressure. Gas bubbles may also be formed by chemical decomposition or chemical reaction. Polyurethane foam is made by reacting isocyanate with water to form CO2. Mechanical beating also will produce foams. Foamed materials such as styrofoam can be formed by bonding together spheres that have been previously foamed.

Two-dimensional Photolithography

The intricate circuits formed on semiconductor chips are patterned by photolithography. A photosensitive material (photoresist) is applied to a semiconductor surface and baked at a low temperature. The surface is then exposed to intense ultraviolet light. The exposed regions of the photoresist become soluble in a developer and are removed. The surface is then baked at a somewhat higher temperature to harden the remaining photoresist. The exposed regions are then acid etched.

The etched regions may then be treated differently from the rest of the surface. They may be doped by exposure to plasma containing n- or p-type impurities, or they may be plated to form conducting circuits. Integrated circuits are composed of many overlapping layers (Figure 24.1), each defined by photolithography. Dopants are diffused into the substrate, and additional ions are implanted into some layers. Metal and polycrystalline silicon are deposited on others to form conductive circuits.

Transistors can be formed where there are n-p-n or p-n-p diffusion layers. Resistors are formed by meandering stripes of varying lengths. Capacitors are formed by parallel conductors separated by insulating material. The most common integrated circuits and those with the highest density are random access memories.

Photo-stereolithography

Three-dimensional parts can be made by photo-stereolithography using a liquid that polymerizes on exposure to ultraviolet light. A substrate platform is lowered into a bath and a desired portion of the surface of the liquid is exposed to a beam of ultraviolet light, causing it to polymerize.

The importance of materials to civilization is attested to by the names we give to various eras (Stone Age, Bronze Age, and Iron Age). We do not consider present times in terms of one specific material because so many are vital, including steel. The computer age would not be possible without silicon in computer chips.

Most introductory texts on materials science and engineering start with topics that are not of great interest to most engineers: atomic bonding, crystal structures, Miller indices. This introductory materials text differs from others because it is written primarily for engineers. It is shorter than most other materials texts so that it can easily be covered in one term. Emphasis is on mechanical and electrical properties of interest to most engineers. Thermal, optical, and magnetic behaviors are also covered. In addition, processing is treated in some detail.

Topics like X-ray diffraction, Miller indices, dislocations and coordination in compounds, surfaces, average molecular weights, Avrami kinetics, and Weibull analysis, which are of great interest to materials scientists but of little interest to most engineers, are covered only in the appendices. There is also an appendix on wood. There is no treatment of crystal systems, the Hall effect, or ferroelectricity.

After an introductory chapter, the text starts with phases and phase diagrams. This is followed by a chapter on diffusion, which treats diffusion in multiphase as well as single-phase systems. The next several chapters on mechanical behavior and failure should be of particular interest to mechanical engineers.

Edge and Screw Dislocations

Plastic deformation of crystalline solids occurs primarily by slip, which is the sliding of atomic planes over one another. An entire plane does not slide over another at one time. Rather it occurs by the motion of imperfections called dislocations. Dislocations are line imperfections in a crystal. The lattice around a dislocation is distorted so the atoms are displaced from their normal lattice sites. The lattice distortion is greatest near a dislocation and decreases with distance from it. One special form of a dislocation is an edge dislocation that is sketched in Figure A5.1. The geometry of an edge dislocation can be visualized by cutting into a perfect crystal and then inserting an extra half-plane of atoms. The dislocation is the bottom edge of this extra half-plane. An alternative way of visualizing dislocations is illustrated in Figure A5.2.

An edge dislocation is created by shearing the top half of the crystal by one atomic distance perpendicular to the end of the cut (Figure A5.3B). This produces an extra half-plane of atoms, the edge of which is the center of the dislocation. The other extreme form of a dislocation is the screw dislocation. This can be visualized by cutting into a perfect crystal and then shearing half of it by one atomic distance in a direction parallel to the end of the cut (Figure A5.3C). The end of the cut is the dislocation.

Carbon can occur in several different forms including diamond, graphite, amorphous carbon, and fullerenes. None of these forms fit into the classification of materials as metals, ceramics, or polymers. Figure 18.1 shows the equilibrium between graphite, diamond, and liquid.

Diamond

Each carbon atom in diamond is covalently bonded to four other carbon atoms as shown in Figure 18.2. Very strong bonding makes diamond the hardest material known (10,000 kg/mm2). Diamond is used for cutting very hard materials. Diamond has an extremely high Young's modulus (1,050 GPa) and a very low coefficient of thermal expansion (1 × 10−6/K). It has the highest thermal conductivity of all materials (2 kW/m-K compared with 401W/m-K for copper), making it useful for dissipating heat. Its density (3.52 Mg/m3) is considerably greater than that of graphite (2.25 Mg/m3).

The first synthetic diamonds were made by subjecting carbon to very high pressures at high temperatures. Diamond can also be grown by chemical vapor deposition (CVD) under low pressure (1 to 27 kPa). Gasses include a carbon source and typically hydrogen heated in a pressurized chamber and broken down, depositing diamond on exposed surfaces. Large areas (> 150 mm2) can be coated on a substrate. This allows CVD diamond films to be used as heat sinks in electronics and to be used in wear-resistant surfaces.

Graphite

The structure of graphite consists of sheets of carbon atoms arranged in a hexagonal pattern (Figure 18.3). The bonding in the hexagonal sheets is like that in a benzene ring.