Physics simulation in a 3D

environment

Sylvain EUDIERMSCS Candidate

Union College

May 28th, Spring 2004

Agenda Why Physics Simulation? Where am I, where am I going? Start Coding Entry point: The Spring Model Extension to the Flag / Cloth simulation Introduction to Collision Detection Collision improvement: an example Conclusion

Why Physics Simulation? Getting more and more interest from the

game industry

How does it work behind the scenes?

Combines physics and CS

Where am I? Physics are used in many programs (CAD, games,

simulators…)

Commercial physics libraries exist

As well as open source

Evolution of the simulation models up to now

Evolution : Quake 2 – Doom 3

Where am I going? How precisely do we want to simulate the

world

How do we want to represent it

For what expected quality / expense

Start Coding – Define the rules Use of C++ Representation using the OpenGL API Game-like precision

Find a model for this problem (classes)

Starting point: Write a stable and easy-to-use CVector3D class

The CVector3D class 3 constructors : Default, copy, by

components Overloading of operators:

+, -, *, /, +=, -=, *=, /= Methods: length, normalize, unit,

crossProduct and dotProduct

What can I start with? The spring model

Simulate the behavior of a deformable objectunder certain constraints. Easy to implement (as a beginning) Gives convincing results rapidly

Allows me to test the architecture of my program

The Spring Model

To the basic formula, we add the inner friction (to stabilize it):

IfvelmassvelmassixkF .)().2().1(..

The Spring Model

These properties are the basics we can give to a mass.

Considered as a dot

The Spring Model

For the computation:

L: steady lengthx: actual length of the springu: unit vector between mass1 and mass2

21.)().2().1()..( FFIfvelmassvelmassuLxkF

Application to a rope The rope is made of several masses that

interact with each other

By changing the variables, the rope may be: stiffer, more / less extendable

We can create different kinds of extensible material

Demonstration Rope simulation 1 Rope simulation 2

Spring Model : First impressions (+) The result looks good enough for such

a simple simulation.

(-) The rope behaved differently on different machines (different speeds)

(-) The rope cannot be very stiff

Spring Model : Speed problem Need for a time regulation algorithm

Why? How?

After the first try, I had a slow and fast behavior… Due to the GetTickCount() function Use of the QueryPerformanceCounter()

Spring Model : Stiffness issue The stiffness problem:

Due to the Euler’s approximation method

0. vtav

amF .

0...2

1 2 postvtapos

Spring Model : Stiffness issue – Why?

Euler function stability comparison

Spring Model : Extensions The rope does not

include any bending information: Can be solved using

interleaved springs (explained later, cf. Flag)

Stiffness problem: Regarding the sources I

found, the Runge-Kutta algorithm should solve the problem

The Runge-Kutta Algorithm

Euler

Heun

Adams- Bashforth 2

Verlet

0

5000

10000

15000

20000

25000

18 20 22 24 26 28 30

Time (uSec)

Sp

rin

g S

tiff

nes

s

Runge-Kutta 4(55, 200000)

Spring Model : Flag simulation A flag is just a patch of springs

Create n*m masses Create (n-1)*(m-1) springs Connect the springs to the masses

Possibility to add a wind effect

Spring Model : Flag simulation Flag simulation

Flag simulation : Results (+) The mesh reacts well to the wind and

gravity

(-) The flag is harder to simulate because of the stiffness problem and the lack of bending factor

Flag simulation : Extensions Can simulate a flag

flexibility with interleaved springs…

…and add a universal repulsive force to every node

More complex and realistic simulation

High quality flag simulation Demonstration

Collision Detection Why?

How?

Dependencies: A strong math library: vectors, matrices, plane-

point collision, face-face collision… Possibility to work on predefined meshes

On the way to the collision Math library:

Matrices: Overloading of arithmetic operators (+, -, *, +=…) Overloading of input / output operators ([], <<) Matrices functions : determinant, multiplications,

inversions, transposition, identity Matrix-related functions : rotate, scale, translate…

Vectors Collision functions: PointToPlaneDistance,

IsPointInPolygon…

Importing 3DS files 3DS is a standard in the industry

I already had an importing class for 3DS files

.3DS files have several advantages: Face defined clockwise, Texture information, Normals information, And a lot more…

Into the collision Brute force algorithm:CheckForCollisions():

MakePreselection(Scene, Collisions)For all objects in the Collision Listif(this object collides with another one)

Find the collision pointApply the physics on the objects, at that point

But this will never work!!!

Buggy Collision Demonstration

Into the collision (2) New algorithm:

DoDo

ComputePhysics(NextTimeChunk);CheckForCollisions(Scene, Collisions);if(MaxPenetrationInAnObject < Limit)

Problem is solved;if(Problem NOT solved)

NextTimeChunk = PreviousTime / 2;CancelTheComputations();

elseValidateTheComputations();

While(Problem is NOT solved);proceed to the next time chunk;

While(TimeChunknotSimulated);

The rollback function

Collision improvement We can extend the sphere collision test to

a more general one.

Add a real collision and motion behavior (friction, rotation…)

The MakePreselection function can improve a lot the computation time

Improvements and trade-offs The vast majority of the program use an

aggressive MakePreselection algorithm to be able to deal with a lot of objects

Optimization without loss of information

AABB = Axis Aligned Bounding Box

OBB = Oriented Bounding Box

6-dop = Discrete Orientation Polytope

Convex Hull

Example of an approximation algorithmApproximation: Based on some assumptions over“insignificant” constraints of objects (=has to lookgood enough)

The Opposing Face Geometry algorithm: Algorithm in 8 steps, The pro… …And cons

Opposing Face Geometry algorithm 1. Preselection:

check collision between object A's bounding sphere and object B's bounding sphere.

2. Find the closest k faces of object A relative to object B.

O.F.G. algorithm 3. Calculate the

geometric center of the new selection and the bounding sphere radius.

4. Find the closest k faces of object B relative to object A's new selection of k faces.

5. Calculate the geometric center of object B's new selection of faces and their maximal bounding sphere radius.

The O.F.G. algorithm

6. PreSelection: check collision between spheres to determine if there is even a chance for face collisions.

7. Sort the two sets of faces by increasing distance

8. Test the two sets of faces against each other, starting with the closest pairs of faces.

Pro / Cons of such this algorithm (+) This is a lot faster. Runtime of O(k2)

Where k is usually between 4 and 8(k is a variable representing the number of faces we want to work on)

Brute force approach would be O(n*m)n and m could be 1000 of faces

(-) Cannot really work on concave polygonsThis is TRUE for most of today’s engines

The discrete Time problem Due to the intrinsic

nature of the simulation : Time-discrete based

If the dt variation is too big, an object might be “teleported” through another one

Solution: Extrude the silhouette of the object. Test this polygon for collisions

Summary Springs are the basis of a lot of models

and can be used for powerful simulations (i.e. any kind of elastic models)

Collision detection needs a robust design and math support. There is a lot to do about optimization and trade-offs

Physics simulation is a vast field where a lot of techniques are to be discovered

Selection of References “Computer Graphics with OpenGL”, third Edition by Hearn-

Baker, Prentice Hall

Huge source of information for game programming:www.gamedev.net

Chris Hecker’s famous columns about physics:http://www.d6.com/users/checker/dynamics.htm#articles

Everything you need to know about geometry:http://astronomy.swin.edu.au/~pbourke/geometry/

A lot about everything, from physics to light computations:http://freespace.virgin.net/hugo.elias/

Conclusion - Discussion Questions? Need Explanations?

What kind of extensions could we add to a physics simulator?

References Collision detection

Advanced Gamasutra - Features - "Advanced Collision Detection Techniques" [03.30.00] Advanced Collision Detection Techniques Advanced Collision Detection Techniques Chris Hecker's Rigid Body Dynamics Information DDJ

GameDev.net - Opposing Face Geometry GameDev.net - Simple Bounding-Sphere Collision Detection GameDev.net - Practical Collision Detection GameDev.net - General Collision Detection for Games Using Ellipsoids Collision Response: Bouncy, Trouncy, Fun Gamasutra - Features - "Crashing into the New Year" [02.10.00] Snooker simulation (+Formulaes) Rotation computation

HyperPhysics MODEL-BASED ANIMATION SIGGRAPH - Collision Detection ('88)

AIWisdom.com - Game Articles & Research Geometry Cours de Mécanique - Index The good-looking textured light-sourced bouncy fun smart and stretchy page Deformation

GameDev.net - Real time deformation of solids, Part 1 GameDev.net - Real time deformation of solids, Part 2 Gamasutra - Features - "Exploring Spring Models" [10.05.01]

Cloths Awesome paper on cloth simulation

ch06.pdf (application/pdf Object) Rotational Motion