构建交互3D应用中刚体动力学的统一框架

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2 1 Introduction and Overview

and other fields is necessary. Most research focuses on different selected methods and use

different formalisms, which makes them difficult to understand and hard to compare.

1.1 Goals and Contributions

The goal of this thesis is to build a framework for the simulation of rigid body dynamics. The

desired framework should exhibit these qualities:

• It should be suitable for usage in interactive 3d applications on standard PCs. So only fast

simulation methods that qualify for interactive and real-time applications are examined.

• The simulation results should be believable. Physical correctness is not the main goal, as long

as the results are felt to be realistic by a user of the application. This requires the

simulation to be stable and robust. Here, stability means that unrealistic motion due to

simulation errors is avoided. Robustness means that the simulation is able to recover from

user- or simulation-induced errors.

• The framework should be easy to use. The effort to understand and integrate rigid body

dynamics in an application should be low.

• And the framework should be independent of existing simulation methods or specific

physical effects – for several reasons: Without being an expert in this field it is impossible

to decide which simulation method is superior. An independent design facilitates the

understanding and comparison of different simulation methods. And hopefully it leads to

an improved design, which can be extended easily with new simulation methods and

physical effects.

With this goal at the horizon, the general contributions of this thesis can be summarized as:

• Basics – The fundamental and advanced concepts of mechanics, necessary to understand

and implement rigid body dynamics simulation, are summarized.

• Big Picture – A modular overview of the simulation process and an overview of state of the

art simulation methods are given.

• Detailed View – The important simulation methods are explained.

• Implementation – A design of a unified (simulation method independent) framework is

suggested.

1.2 Chapter Overview

This thesis is divided into the following chapters:

• Chapter 2, “Basics”, summarizes the basics of physics, which are necessary to simulate

rigid bodies. Additionally the linear complementarity problem is introduced, which is a key

problem in rigid body dynamics.

• Chapter 3, “Simulation Overview”, shows a modular overview of the simulation process.

2 Basics

“When I use a word, it means just what I choose it to mean - neither

more nor less.”

– Humpty Dumpty, in “Through the Looking Glass”

This chapter will tackle the first hurdle in building a rigid body simulation: the understanding of

the fundamentals of physics and mathematics.

Not all books on the basics handle all important issues for simulation of rigid body dynamics.

Most papers only present knowledge of a specific simulation method, often expecting that the

basics are well understood. Learning from these sources leaves a novice in a vast landscape of

inconsistent formalisms – making the introduction to this field a tedious task. This chapter

presents the relevant terms and fundamental knowledge in a consistent manner, together with all

formulas used in the implementations and with derivations, which are important for

understanding.

First it will be shown how rotations can be parameterized. Then the relevant dynamics equations

for particles will be derived. Next, systems of particles and rigid bodies as special systems of

particles will be discussed. A rigid body is a system of particles in which the distance between the

particles is constant – or let's say “rigid”. The following section explains how systems of rigid

bodies can be described. Different types of forces are discussed afterwards. The chapter closes

with a crucial problem of mathematical programming: the linear complementarity problem.

A solid understanding of these concepts is important. A good introduction into classical

mechanics in general is given in Goldstein et al. [GoPS02], or Kuypers [Kuyp03] for German

readers. Literature especially for computer scientists is Baraff [Bara97a] and Eberly [Eber04].

The notation that is used in this thesis is summarized in Appendix A, “Notation”.

2.1 Scope of this Work

This thesis deals with rigid body dynamics, a problem that has a variety of names in the literature: rigid

6 2.1 Scope of this Work

body simulation, rigid body physics, dynamic simulation of multi-body systems, physically based modeling, etc.

Likewise a piece of software that deals with this problem is known as physics engine, dynamics engine,

dynamic simulation SDK, etc. Confusion of notions makes it necessary to have a look at the

important terms and their distinction. The definitions given here follow the definitions given in

Encarta [Enca04] and Wikipedia [Wiki05].

Physics is the science of the natural world in the broadest sense. In more detail: Physics is the natural

science of the inanimate world, its structure and its laws which are accessible via measurement,

experimental verification and mathematical representation. It is difficult to clearly distinct physics

from other natural sciences, like biology, chemistry, geology, etc. Sometimes physics is said to be the

fundamental science because all other natural sciences deal with systems that obey the laws of

physics.

Classical Physics may be divided into subfields like acoustics, mechanics, thermodynamics, electricity,

magnetism, etc. Modern physics deals with astrophysics, cosmology, quantum theory, the general and special

theory of relativity, and more.

Mechanics as a subfield of physics can be split into classical mechanics and quantum mechanics. This work

is concerned with classical mechanics, which is the study of the motions of bodies and the forces that

cause these motions. Classical mechanics not only deals with solid bodies, therefore it may be

divided into solid mechanics and fluid dynamics (including hydrodynamics, aerodynamics, pneumatics, etc.).

Classical mechanics may also be separated into kinematics, statics and dynamics.

Kinematics studies the motion of bodies without regard to its causes. That means, it studies the

relation between position, speed, and acceleration without considering forces that cause the

motion of bodies.

Statics is concerned with bodies in static equilibrium. A body is in static equilibrium if no force is

acting on the body, or all forces cancel each other. Bodies in static equilibrium are at rest or

moving at constant velocity. Statics is important in architectural and structural engineering to

calculate quantities like stress and pressure.

Dynamics is the study of motion of bodies and the forces that cause and affect this motion.

Another term for dynamics is kinetics. Some sources (like Goldstein et al. [GoPS02]) use the term

dynamics synonymously with classical mechanics.

Figure 2.1: Term overview.

PhysicsBiology Chemistry Geology

...

Classical Physics Modern Physics

Acoustics Mechanics Thermodynamics

...

StaticsKinetics Dynamics

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