This text presents a uniform and comprehensive treatment of the theory and use of homogeneous coordinates and transformation matrices in the kinematic and dynamic design analysis and the numeric simulation of mechanisms and multibody systems.

The following observations, originally set down by Reuleaux in 1875, are every bit as true today, and it would be difficult to state them better.

Introduction

In Chapter 3 we defined the words position and posture as the terms that tell “where” an item is. Depending on the “item,” we find it convenient to use a Cartesian coordinate system as a global reference and we choose homogeneous coordinates to define the position of a point. We use the (4 × 4) transformation matrix T0b to represent the posture of a rigid body, and we refer to “posture” to emphasize that we include both the orientation of the body as well as the location of a reference point. The posture of a mechanism or multibody system can usually be described by a vector of generalized coordinates ψ equal in number to the mobility of the system. However, because it is sometimes possible that a system can be assembled in more than one way for identical values of the generalized coordinates, we choose to represent the posture of a system by the vector φ that explicitly includes all of the joint variables.

In Chapter 3 we defined the term displacement as the change in position or posture of a point, a joint variable, a rigid body, or a system. Then, in Chapter 6, we showed how the concept of differential displacement leads naturally to the derivative of position or posture. We defined the very powerful derivative operator matrices, Qh and Dh, to make the process of numeric differentiation both easy and precise. However, a quick review shows that derivatives are taken first with respect to a changing joint variable value because these are the variables on which the joint transformation matrices explicitly depend. Little is said about the fact that, in most mechanisms or multibody systems, many joint variables change simultaneously.

Introduction

In Chapter 5 we studied how the postures of some mechanisms and multibody systems can be found analytically using hand calculations to find closed-form solutions. Typically, this requires forming the necessary transformation matrices, and ensuring that all dependent position variables are made consistent with the constraints expressed by the loop-closure equations. In Chapter 5 we solved several example problems, in both 2-D and 3-D, to illustrate the process, but we also found that the calculations quickly became burdensome, even for problems with only a few unknown joint variables. In principle the methods look powerful, but in practice they quickly reach a limit on practicality.

Does this mean that the methods are not adequate? Not exactly; rather, it means that we are in need of a better means of calculating. Perhaps these tedious computations should be automated for solution by numeric methods using a computer.

Let us reflect on the nature of the problem of posture analysis of a multibody system. In general, the number of bodies (ℓ) is usually reasonably small, typically limited by cost and the desire for simplicity and reliability to tens of moving parts or less. The number of joints (n) is of the same order. The number of closed loops (NL) is usually much smaller. The number of joint variables (φ) is of the same order as the number of joints. However, the number of independent variables (ψ) is almost always very small. After all, the whole point of our multibody system is to control the movements of the parts to only those required for proper function of the system. Thus, the mobility (f) is often only one, and is very rarely as many as ten.

Introduction

A mechanism or a multibody system consists of several bodies or links that move together in a coordinated fashion based on the nature of the connections between them. The individual bodies or links are usually attached through joints such as in robot manipulators, biomechanical systems, mechanisms and machines, or other clever devices such as in aerospace systems. As a system moves, its posture changes, including displacements of the individual bodies while maintaining the connections through the joints.

The classical formulations of kinematics of rigid bodies discussed in Chapter 3 can be adapted to multibody systems. In order to do this, however, we must keep track of all bodies and their interconnections and make sure that their displacements and motions are described in a fashion that allows us to track the posture of the entire mechanism or multibody system. The matrix method presented in this and subsequent chapters provides a systematic method that allows such a development with no ambiguities. When combined with the methods for topological examination of mechanical systems from Chapter 2, the overall approach provides a powerful tool for computer-aided analysis of mechanisms and multibody systems and for development of general-purpose software tools for such applications.

Introduction

Before formulating a numeric method for design analysis of mechanisms and multibody systems, let us first consider the essential characteristics of the problem being addressed. What are the chief difficulties encountered in the design analysis of a mechanism or multibody system? It is not the laws of mechanics as such that cause difficulty. It is the fact that, once a problem has been formulated, it is often too formidable algebraically to be easily solved. This complexity does not arise from static and dynamic force relationships, but from the kinematics – the changing geometry. The basic constraint equations that govern the motions within a machine or multibody system come from the fact that rigid bodies hold their respective joint elements in constant spatial relationships to one another. This type of constraint invariably leads to a set of highly nonlinear simultaneous algebraic equations.

Because the difficulties in an analytic approach to mechanism and multibody system analysis stem from the geometry, it is wise to choose a mathematical formulation suited to this type of problem. One such formulation is based on the use of matrices to represent the transformation equations between strategically located coordinate systems fixed in successive bodies. This approach has been developed into an extremely general and powerful technique for mechanism and multibody system analysis, and the next several chapters are devoted to its presentation. Before this can be presented effectively, however, we must become familiar with a number of basic operations that render matrix algebra so useful in performing coordinate transformations. The purpose of this chapter, therefore, is to develop this foundation.

Introduction

Throughout earlier chapters we have carefully formulated our equations in a very general, multi-degree of freedom form. In fact, our only two limiting assumptions so far have been: (1) that all bodies of our system are totally rigid, allowing no deformation or deflection, and (2) that all joints act precisely as described by their mathematical models shown in section 4.6, exhibiting no effects such as backlash or clearances. Indeed, our efforts have produced a kinematic model of our system that is extremely general and powerful. Even though its solution may be tedious for hand calculation, we recognize that evaluation is intended by digital computation and we hope to continue this generality and precision throughout our work in dynamics.

Lagrange's Equation

Although it may be possible to formulate the equations of motion for a general dynamic system by sketching free-body diagrams, assigning sign conventions and notation, and applying Newton's laws, such an approach is not used here because we are interested in complex and diversified three-dimensional mechanisms and multibody systems and our focus is on developing methods that can be coded for computation in a general setting. An approach based on energy and Lagrange's equation is adopted here, which results in a very general form and minimizes the potential for errors in formulation. Before we discuss the method, however, let us review a very brief history of energy methods in mechanics.

Introduction

In the very beginning of this text, section 1.1, we observed that the science of mechanics is composed of two parts called statics and dynamics, first distinguished by Euler in 1765. His advice is, perhaps, worth repeating here [1]:

We also noted that dynamics is made up of two major disciplines, later recognized as the distinct sciences of kinematics and kinetics, which treat the motion and the forces producing it, respectively.

This work is intended to supplement and complement standard texts on continuum mechanics by drawing attention to physical assumptions implicit in continuum modelling. Particular attention is paid to linking continuum concepts, fields, and relations with underlying molecular behaviour via local averaging in both space and time. The aim is to clarify physical interpretations of concepts and fields and in so doing provide a sound basis for future studies. The contents should be of interest to engineers, mathematicians, and physicists who study macroscopic material behaviour.

The contents are the result of a long-standing study of formal and axiomatic presentations of continuum mechanics. Some of the issues were first addressed in courses delivered under the auspices of CISM (Udine, 1986, 1987), University of Cairo (1994, 1996), and AMAS (Warsaw, 2002; Bydgoszcz, 2003), and other topics treated in published papers. Here the opportunity has been taken to elaborate upon and extend earlier works and to present a unified, more readily accessible treatment of the subject matter.

Given the differing backgrounds of the intended readership, two extensive appendices have been included which develop relevant mathematical concepts and results. In particular, the use of direct (i.e., co-ordinate-free) notation is explained and related to that of Cartesian tensors.

No work exists in isolation: the author is above all indebted to his teachers Mort Gurtin and Walter Noll who introduced him to the mathematical precision and clarity of exposition to be found in modern continuum mechanics.