Dynamics 101

1
Dynamics 101

• Jim Van Verth
• Red Storm Entertainment
• jimvv_at_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
• Cant 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

x(ti)
y
x'(ti)
y(t)
t
14
Differential Calculus Review

• Our function is position
• Derivative is velocity
• Derivative of velocity is acceleration

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 ( )

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 xi1
• Current frame velocity vi1

18
Linear Dynamics

• Could use Newtons equations
• Problem assumes F constant across frame
• Not always true
• E.g. spring force Fspring kx
• E.g. drag force Fdrag m?v

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 Eulers method

21
Linear Dynamics

• Eulers method

22
Linear Dynamics

• Another way use linear momentum
• Then

23
Linear Final Formulas

• Using Eulers method with time step h

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 Ri1 or qi1
• Current frame ang. velocity ?i1

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

v
r
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
Rotational Dynamics

• 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)

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

34
Computing New Orientation

• If have matrix R, then

where
35
Computing New Orientation

• If have quaternion q, then
• See Baraff or Eberly for derivation

where
36
Computing Angular Velocity

• Cant easily integrate angular velocity from
  angular acceleration
• Can no longer divide by I and do Euler step

37
Computing Angular Momentum

• Easier way use angular momentum
• Then

38
Using I in World Space

• Remember,
• I computed in local space, must transform to
  world space
• If using rotation matrix R, use formula
• If using quaternion, convert to matrix

39
Rotational Formulas
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 -gt acceleration -gt velocity -gt position
• Rotational simulation
• Torque -gt ang. mom. -gt ang. vel. -gt 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.