Dynamic Simulations of Multibody Systems

Physically based modeling is increasingly gaining acceptance within the computer graphics and mechanical engineering industries as a way of achiev­ ing realistic animations and accurate simulations of complex systems. Such complex systems are usually hard to animate using scripts, and difficult to analyze using conventional mechanics theory, which makes them perfect candidates for physically based modeling and simulation techniques. The field of physically based modeling is broad. It includes everything from modeling a ball rolling on the floor, to a car engine working, to a hang­ ing shirt being moved by a gust of wind. The theory varies from precise mathematical methods to purpose-specific approximated solutions that are mathematically incorrect, but produce realistic animations for the partic­ ular situation being considered. Depending on the case, an approximated solution might serve the purpose, however, there are times when approx­ imations are not admissible, and the use of accurate simulation engines is a requirement. Developing and implementing physically based dynamic­ simulation engines that are robust is difficult. The main reason is that it requires a breadth of knowledge in a diverse set of subjects, each of them standing alone as a broad and complex topic. Instead of attempting to address all types of simulation engines available in the broad area of physically based modeling, this book provides in-depth coverage of the most common simulation engines. These simulation engines restrict the general case of physically based modeling to the particular case wherein the objects interacting are either particles or rigid bodies.

Table of Contents:
1 Computational Dynamics.- 2 Hierarchical Representation of 3D Polyhedra.- 3 Particle Systems.- 4 Rigid-Body Systems.- 5 Articulated Rigid-Body Systems.- A Useful 3D Geometric Constructions.- A.1 Introduction.- A.2 Projection of a Point on a Line.- A.3 Projection of a Point on a Plane.- A.4 Intersection of a Line Segment and a Plane.- A.5 Closest Point between a Line and a Line Segment.- A.6 Computing the Collision- or Contact-Local Frame from the Collision- or Contact-Normal Vector.- A.7 Representing Cross-Products as Matrix-Vector multiplication.- A.8 Suggested Readings.- B Numerical Solution of Ordinary Differential Equations of Motion.- B.1 Introduction.- B.2 Euler Method.- B.2.1 Explicit Euler.- B.2.2 Implicit Euler.- B.3 Runge-Kutta Method.- B.3.1 Second-Order Runge-Kutta Method.- B.3.2 Forth-Order Runge-Kutta Method.- B.4 Using Adaptive Time-Step Sizes to Speed Computations.- B.5 Suggested Readings.- C Quaternions.- C.1 Introduction.- C.2 Basic Quaternion Operations.- C.2.1 Addition.- C.2.2 Dot product.- C.2.3 Multiplication.- C.2.4 Conjugate.- C.2.5 Module.- C.2.6 Inverse.- C.3 Unit Quaternions.- C.4 Rotation-Matrix Representation Using Unit Quaternions.- C.5 Advantages of Using Unit Quaternions.- C.6 Suggested Readings.- D Rigid-Body Mass Properties.- D.1 Introduction.- D.2 Mirtich’s Algorithm.- D.2.1 Volume-Integral to Surface-Integral Reduction..- D.2.2 Surface-Integral to Projected-Surface-Integral Reduction.- D.2.3 Projected-Surface-Integral to Line-Integral Reduction.- D.2.4 Computing the Line Integrals from the Vertex Coordinates.- D.3 Suggested Readings.- E Useful Time Derivatives.- E.1 Introduction.- E.2 Computing the Time Derivative of a Vector Attached to a Rigid Body.- E.3 Computing the Time Derivative of a Contact-Normal Vector.- E.3.1Particle-Particle Contact.- E.3.2 Rigid Body-Rigid Body Contact.- E.4 Computing the Time Derivative of the Tangent Plane.- E.5 Computing the Time Derivative of a Rotation Matrix.- E.6 Computing the Time Derivative of a Unit Quaternion.- E.7 Suggested Readings.- F Convex Decomposition of 3D Polyhedra.- F.1 Introduction.- F.2 Joe’s Algorithm.- F.2.1 Determining Candidate Cut Planes.- F.2.2 Computing the Cut Face Associated with a Cut Plane.- F.2.3 Termination Conditions.- F.3 Suggested Readings.- G The Linear-Complementarity Problem.- G.1 Introduction.- G.2 Dantzig’s Algorithm: The Frictionless Case.- G.2.1 Termination Conditions.- G.3 Baraff’s Algorithm: Coping with Friction.- G.3.1 Static-Friction Conditions.- G.3.2 Dynamic Friction.- G.3.3 Termination Conditions.- G.4 Suggested Readings.- H Software Implementation.- References.