CS 378: Computer Game Technologyfussell/courses/cs378/lectures/cs378-6.pdf · This equation: is a second order differential equation. Our solution method, though, worked on first

University of Texas at Austin CS 378 – Game Technology Don Fussell

CS 378: Computer Game Technology

Physics for Games Spring 2012

Game Physics – Basic Areas

!   Point Masses ! Particle simulation ! Collision response

!   Rigid-Bodies ! Extensions to non-points

!   Soft Body Dynamic Systems !   Articulated Systems and Constraints !   Collision Detection

Physics Engines

!   API for collision detection !   API for kinematics (motion but no forces) !   API for dynamics !   Examples

! Box2d ! Bullet ! ODE (Open Dynamics Engine) ! PhysX ! Havok ! Etc.

University of Texas at Austin CS 378 – Game Technology Don Fussell 3

University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 4

Particle dynamics and particle systems

!   A particle system is a collection of point masses that obeys some physical laws (e.g, gravity, heat convection, spring behaviors, …).

!   Particle systems can be used to simulate all sorts of physical phenomena:


Particle in a flow field

!   We begin with a single particle with:

! Position,

! Velocity,

!   Suppose the velocity is actually dictated by some driving function g:

g(x,t)

x

y

€

x =xy"

# $ %

& '

€

v = x•

=d x dt

=dx dtdy dt"

# $

%

& '

€

x•

= g( x ,t)


Vector fields

!  At any moment in time, the function g defines a vector field over x:

!  How does our particle move through the vector field?


Diff eqs and integral curves

!   The equation

is actually a first order differential equation. !   We can solve for x through time by starting at an initial

point and stepping along the vector field:

!   This is called an initial value problem and the solution is called an integral curve.

Start Here

€

x•

= g( x ,t)

Eulers method

!   One simple approach is to choose a time step, Δt, and take linear steps along the flow:

!   Writing as a time iteration:

!   This approach is called Euler’s method and looks like:

!   Properties: ! Simplest numerical method ! Bigger steps, bigger errors. Error ~ O(Δt2).

!   Need to take pretty small steps, so not very efficient. Better (more complicated) methods exist, e.g., “Runge-Kutta” and “implicit integration.”

€

x i+1 = x i + Δ t ⋅ v i

€

x (t + Δ t) = x (t) + Δ t ⋅ x(t) = x (t) + Δ t ⋅ g( x ,t)€

•


Particle in a force field

!   Now consider a particle in a force field f. !   In this case, the particle has:

! Mass, m ! Acceleration,

!   The particle obeys Newton’s law:

!   The force field f can in general depend on the position and velocity of the particle as well as time.

!   Thus, with some rearrangement, we end up with:

€

a ≡ x = d v dt

=d2 x dt 2

€

••

€

f = m a = mx

€

••

€

x = f ( x ,x,t)m

€

••

€

•


This equation:

is a second order differential equation.

Our solution method, though, worked on first order differential equations.

We can rewrite this as:

where we have added a new variable v to get a pair of coupled first order equations.

Second order equations

€

x = v

v = f ( x , v ,t)m

"

#

$ $

%

&

' '

€

•

€

•

€

x = f ( x ,x,t)m

€

••

€

•


Phase space

!  Concatenate x and v to make a 6-vector: position in phase space.

!  Taking the time derivative: another 6-vector.

!  A vanilla 1st-order differential equation. €

xv"

# $ %

& '

€

•

€

•

€

x v "

# $ %

& '

€

xv"

# $ %

& ' =

v f m"

# $

%

& '

€

•€

•


Differential equation solver

Applying Euler’s method:

Again, performs poorly for large Δt.

And making substitutions:

Writing this as an iteration, we have:

Starting with:

€

xv"

# $ %

& ' =

v f m"

# $

%

& '

€

•€

•

€

x (t + Δ t) = x (t) + Δ t ⋅ x(t)€

•

€

x(t + Δ t) = x(t) + Δ t ⋅ x(t)€

•

€

•

€

••

€

x (t + Δ t) = x (t) + Δ t ⋅ v (t)

€

•

€

x(t + Δ t) = x(t) + Δ t ⋅ f ( x ,x,t) m

€

•

€

•

€

x i+1 = x i + Δ t ⋅ v i

€

v i+1 = v i + Δ t ⋅ f i

m


Particle structure

m

! "# $# $# $# $% &

xvf

position velocity force accumulator mass

Position in phase space

How do we represent a particle?


Single particle solver interface

m

! "# $# $# $# $% &

xvf ! "# $

% &

xv

/m! "# $% &

vf

[ ]6getDim

derivEval

getState

setState


Particle systems

particles n time

In general, we have a particle system consisting of n particles to be managed over time:

€

x 1 v 1 f 1m1

"

#

$ $ $ $

%

&

' ' ' '

x 2 v 2 f 2m2

"

#

$ $ $ $

%

&

' ' ' '

…

x n v n f nmn

"

#

$ $ $ $

%

&

' ' ' '


Particle system solver interface

derivEval

get/setState getDim

For n particles, the solver interface now looks like:

particles n time

€

6n x 1 v 1 x 2 v 2 …

x n v n

v 1

f 1m1

v 2

f 2m2

… v n

f nmn


Particle system diff. eq. solver

We can solve the evolution of a particle system again using the Euler method:

€

x 1i+1

v 1i+1

x ni+1

v ni+1

"

#

$ $ $ $ $ $

%

&

' ' ' ' ' '

=

x 1i

v 1i

x ni

v ni

"

#

$ $ $ $ $ $

%

&

' ' ' ' ' '

+ Δt

v 1i

f 1i m1 v ni

f ni mn

"

#

$ $ $ $ $ $

%

&

' ' ' ' ' '


Forces

!  Each particle can experience a force which sends it on its merry way.

!  Where do these forces come from? Some examples: ! Constant (gravity) ! Position/time dependent (force fields) ! Velocity-dependent (drag) ! Combinations (Damped springs)

!  How do we compute the net force on a particle?


Particle systems with forces

particles n time forces

F2 Fnf

nf

!   Force objects are black boxes that point to the particles they influence and add in their contributions.

!   We can now visualize the particle system with force objects:

F1

€

x 1 v 1 f 1m1

"

#

$ $ $ $

%

&

' ' ' '

x 2 v 2 f 2m2

"

#

$ $ $ $

%

&

' ' ' '

…

x n v n f nmn

"

#

$ $ $ $

%

&

' ' ' '


Gravity and viscous drag

p->f += p->m * F->G

p->f -= F->k * p->v

The force due to gravity is simply:

Often, we want to slow things down with viscous drag:

€

f grav = m

G

€

f drag = −k

v


Recall the equation for the force due to a spring: We can augment this with damping: The resulting force equations for a spring between two particles become:

Damped spring

€

f = −kspring (Δ x − r)

€

f = − kspring (Δ x − r) + kdamp

v [ ]

€

f 1 = − kspring Δ

x − r( ) + kdampΔ v •Δ x Δ x

$

% &

'

( )

*

+ , ,

-

. / /

Δ x

Δ x

f 2 = −

f 1

r = rest length

€

p1 = x 1 v 1

"

# $ %

& '

€

p2 = x 2 v 2

"

# $

%

& '

€

Δ x = x 1 −

x 2


derivEval Clear forces

Loop over particles, zero force accumulators

Calculate forces Sum all forces into accumulators

Return derivatives Loop over particles, return v and f/m Apply forces

to particles

Clear force accumulators 1

2

3 Return derivatives

to solver

F2 F3 Fnf F1

€

x 1 v 1 f 1m1

"

#

$ $ $ $

%

&

' ' ' '

x 2 v 2 f 2m2

"

#

$ $ $ $

%

&

' ' ' '

…

x n v n f nmn

"

#

$ $ $ $

%

&

' ' ' '

€

x 1 v 1 f 1m1

"

#

$ $ $ $

%

&

' ' ' '

x 2 v 2 f 2m2

"

#

$ $ $ $

%

&

' ' ' '

…

x n v n f nmn

"

#

$ $ $ $

%

&

' ' ' '

€

v 1 f 1 m1

"

# $

%

& '

v 2 f 2 m2

"

# $

%

& ' …

v n f n mn

"

# $

%

& '


Bouncing off the walls

! Add-on for a particle simulator ! For now, just simple point-plane

collisions

N P

A plane is fully specified by any point P on the plane and its normal N.


Collision Detection

N

v

P

x

How do you decide when you’ve crossed a plane?


Normal and tangential velocity

N

v

P

Nv

v Tv

To compute the collision response, we need to consider the normal and tangential components of a particle’s velocity.

€

v N = N • v ( )

N

v T = v − v N


Collision Response

before after

Without backtracking, the response may not be enough to bring a particle to the other side of a wall. In that case, detection should include a velocity check:

€

v N

€

v T

€

v T

€

" v = v T − krestitution

v N

€

−krestitution v N

€

" v

€

v

CS 378: Computer Game Technologyfussell/courses/cs378/lectures/cs378-6.pdf · This equation: is a second order differential equation. Our solution method, though, worked on first

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