Dynamics 101

Jim Van VerthRed Storm Entertainment

jimvv@redstorm.com

2

Talk Summary

Going to talk about: A brief history of motion theory Newtonian motion for linear and

rotational dynamics Handling this in the computer

3

Physically Based-Motion

Want game objects to move consistent with world

Match our real-world experience But this is a game, so… Can’t be too expensive (no atomic-level interactions)

4

History I: Aristotle

Observed: Push an object, stop, it stops Rock falls faster than feather

From this, deduced: Objects want to stop Motion is in a line Motion only occurs with action Heavier object falls faster

Note: was not actually beggar for a bottle

5

History I: Aristotle

Motion as changing position

6

History I: Aristotle

Called kinematics Games: move controller, stop on a

dime, move again Not realistic

7

History II: Galileo

Observed: Object in motion slows down Cannonballs fall equally

Theorized: Slows due to unseen force: friction Object in motion stays in motion Object at rest stays at rest Called inertia Also: force changes velocity, not position Oh, and mass has no effect on velocity

8

History II: Galileo

Force as changing velocity Velocity changes position

Called dynamics

9

History III: Newton

Observed: Planet orbit like continuous falling

Theorized: Planet moves via gravity Planets and small objects linked Force related to velocity by mass Calculus helps formulate it all

10

History III: Newton

Sum of forces sets acceleration Acceleration changes velocity Velocity changes position g

11

History III: Newton

Games: Move controller, add force, then drift

12

History III: Newton

As mentioned, devised calculus (concurrent with Leibniz)

Differential calculus: rates of change

Integral calculus: areas and volumes antiderivatives

13

Differential Calculus Review

Have position function x(t) Derivative x'(t) describes how x

changes as t changes (also written dx/dt, or )

x'(t) gives tangent vector at time t

y(t)

y

t

x(ti)

x'(ti)€

˙ x

14

Differential Calculus Review

Our function is position: Derivative is velocity:

Derivative of velocity is acceleration

)(tx

xx

xv &==′=dtd

tt )()(

xx

x

vv

va

&&

&

==′′=

==′=

2

2

)(

)()(

dtd

t

dtd

tt

15

Newtonian Dynamics Summary

All objects affected by forces Gravity Ground (pushing up) Other objects pushing against it

Force determines acceleration (F = ma)

Acceleration changes velocity ( ) Velocity changes position ( )

av=

dtd

vx=

dtd

16

Dynamics on Computer

Break into two parts Linear dynamics (position) Rotational dynamics (orientation)

Simpler to start with position

17

Linear Dynamics

Simulating a single object with: Last frame position xi

Last frame velocity vi

Mass m Sum of forces F

Want to know Current frame position xi+1

Current frame velocity vi+1

18

Linear Dynamics

Could use Newton’s equations

Problem: assumes F constant across frame

Not always true: E.g. spring force: Fspring = –kx

E.g. drag force: Fdrag = –mv

€

a = F /m

x i+1 = x i + vit +1/2at 2

vi+1 = vi + at

19

Linear Dynamics

Need numeric solution Take stepwise approximation of

function

20

Linear Dynamics

Basic idea: derivative (velocity) is going in the right direction

Step a little way in that direction (scaled by frame time h)

Do same with velocity/acceleration Called Euler’s method

21

Linear Dynamics

Euler’s method

€

x i+1 = x i + hvi

a = F(x i,vi) /m

vi+1 = vi + ha

22

Linear Dynamics

Another way: use linear momentum

Then

€

P = mv˙ P = F

€

x i+1 = x i + hvi

Pi+1 = Pi + hF(x i,vi)

vi+1 = Pi+1 /m

23

Linear: Final Formulas

Using Euler’s method with time step h

€

F = Fk (k

∑ x i,vi)

a i =F

mx i+1 = x i + hvi

vi+1 = vi + ha i

24

Rotational Dynamics

Simulating a single object with: Last frame orientation Ri or qi

Last frame angular velocity i

Inertial tensor I Sum of torques

Want to know Current frame orientation Ri+1 or qi+1

Current frame ang. velocity i+1

25

Rotational Dynamics

Orientation Represented by

Rotation matrix R Quaternion q

Which depends on your needs Hint: quaternions are cheaper

26

Rotational Dynamics

Angular velocity Represents change in rotation How fast object spinning 3-vector

Direction is axis of rotation Length is amount of rotation (in radians) Ccw around axis (r.h. rule)

27

Rotational Dynamics

Angular velocity Often need to know linear velocity at

point Solution: cross product

r

v

€

×r = v

28

Moments of Inertia

Inertial tensor I is rotational equivalent of mass 3 x 3 matrix, not single scalar factor

(unlike m) Many factors - rotation depends on shape Describe how object rotates around

various axes Not always easy to compute Change as object changes orientation

29

Rotational Dynamics

Computing I Can use values for closest box or cylinder Alternatively, can compute based on

geometry Assume constant density, constant mass at

each vertex Solid integral across shape See Mirtich,Eberly for more details Blow and Melax do it with sums of tetrahedra

30

Rotational Dynamics

Torque Force equivalent Apply to offset from center of mass –

creates rotation Add up torques just like forces

31

Computing torque Cross product of vector r (from CoM to

point where force is applied), and force vector F

Applies torque ccw around vector (r.h. rule)

Rotational Dynamics

r F

€

=r × F

32

Rotational Dynamics

Center of Mass Point on body where applying a force acts just

like single particle “Balance point” of object Varies with density, shape of object

Pull/push anywhere but CoM, get torque Generally falls out of inertial tensor calculation

33

Rotational Dynamics

Have matrix R and vector How to compute ? Convert to give change in R

Convert to symmetric skew matrix Multiply by orientation matrix Can use Euler's method after that

€

Ri+1 = Ri + hωi

34

Computing New Orientation

If have matrix R, then

€

˜ ω =

0 −ω3 ω2

ω3 0 −ω1

−ω2 ω1 0

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

€

Ri+1 = Ri + h ˜ ω iRi

where

35

Computing New Orientation

If have quaternion q, then

See Baraff or Eberly for derivation

€

wi = (0,ωi)

iiii

hqwqq

21 +=+

where

36

Computing Angular Velocity

Can’t easily integrate angular velocity from angular acceleration:

Can no longer “divide” by I and do Euler step €

=Iα

= I ˙ ω + ω × Iω

37

Computing Angular Momentum

Easier way: use angular momentum

Then

€

L = Iω˙ L = τ

€

Li+1 = Li + hτ (Ri,ωi)

ωi+1 = Ii+1−1 Li+1

38

Remember, I computed in local space, must

transform to world space If using rotation matrix R, use

formula

If using quaternion, convert to matrix

Using I in World Space

€

=I−1L

iTiiii LRIRLI 1

01 −− =

39

Rotational Formulas

€

Ω= rk × Fk

k

∑

Ri+1 = Rn + h ˜ ω iRi

Li+1 = Li + hΩ

Ii+1−1 = Ri+1I0

−1Ri+1T

ωi+1 = Ii+1−1 Li+1

40

Impulses

Normally force acts over period of time

E.g., pushing a chair

F

t

41

Impulses

Even if constant over frame

sim assumes application over entire time

F

t

42

Impulses

But if instantaneous change in velocity? Discontinuity!

Still force, just instantaneous Called impulse - good for collisions/constraints

F

t

43

Summary

Basic Newtonian dynamics Position, velocity, force, momentum

Linear simulation Force -> acceleration -> velocity ->

position Rotational simulation

Torque -> ang. mom. -> ang. vel. -> orientation

44

Questions?

45

References

Burden, Richard L. and J. Douglas Faires, Numerical Analysis, PWS Publishing Company, Boston, MA, 1993.

Hecker, Chris, “Behind the Screen,” Game Developer, Miller Freeman, San Francisco, Dec. 1996-Jun. 1997.

Witken, Andrew, David Baraff, Michael Kass, SIGGRAPH Course Notes, Physically Based Modelling, SIGGRAPH 2002.

Eberly, David, Game Physics, Morgan Kaufmann, 2003.