Video Game Dev - Univ Insubria 2017/2018
Marco Tarini
Game Physics - 2 1
Newtonian dynamics:
summary
Actual object location
Rate of change of
(d / dt )
“with mass”
(momentum)
What changes the rate of change (d2/ dt2)
“with mass”
Position 𝑝 𝑝 = (x,y,z)
Velocity 𝑣⃗
𝑣⃗ = 𝑝̇
( |𝑣⃗| = “speed” )
Momentum 𝑣⃗ 𝑚
Acceleration 𝑎⃗ = 𝑣⃗̇ = 𝑝̈
Force 𝑓⃗
𝑓⃗ = 𝑎⃗ 𝑚
Orientation (e.g. quaternion)
Angular velocity 𝜔 Angular momentum 𝜔 𝐼
𝐼 = moment of inertia (for axis)
(“rotational inertia”)
Angular acc. α Torque τ τ = 𝑎⃗ 𝐼
(“mechanic momentum”)
Change the state
(no memory)
state (is kept! inertia!)
(changes, but only continuously)
A few constants per object
A few quantities associated to each object
constants: they don’t (usually) change
input of the physical simulation, not output
Mass:
resistance to change of velocity
Moment of Inertia:
resistance to change of angular velocity
Barycenter:
the center of mass
Video Game Dev - Univ Insubria 2017/2018
Marco Tarini
Game Physics - 2 2
Mass
resistance to change of velocity
inertial mass
also, incidentally:
ability to attract every other object
gravitational mass
happens to be the same
what you measure with a scale
Unity of measure:
kg, g…
Moment of inertia
Resistance to change of angular velocity
(an object rotates around its barycenter)
high
low
Video Game Dev - Univ Insubria 2017/2018
Marco Tarini
Game Physics - 2 3
Moment of inertia
Scalar
moment of inertia
Resistance to change of angular velocity
Depends on the mass, and on its distribution
the farthest one sub-mass from the axis, the > the resistance
In 3D: its different for each axis of rotation
It can be computed for any axis, thanks to…
Moment of inertia as a 3x3 Matrix
a matrix A used to extract the scalar, for any given axis
given an axis a (a = unit vector), the moment of inertia is
a
TA a
matrix A can be computed once and for all for a rigid object
how: that’s beyond this course
in practice: use given formulas for common shapes
or sum the contributions for each sub-mass
Barycenter
Aka the center of mass
(a position)
In the discrete setting:
simply the weighted average of the positions of the subparts composing an object
(literally “weighted”: with their masses)
Does not necessarily coincide with
the origin of the local frame of that object
(but it can)
Video Game Dev - Univ Insubria 2017/2018
Marco Tarini
Game Physics - 2 4
State of an object
current
current rates of change
constants
updated by
physics
Point position
Rotation orientation Vector velocity
Rotation angular_velocity Scalar mass
Matrix moment_of_inertia Point barycenter
Scalar drag
…
setup at initialization, (rarely) changed e.g. by scripts
Note: acceleration/forces/torques are not part of the state frictions;
see later
In
part of Transform component
the RigidBody component
Point position
Rotation orientation Vector velocity
Rotation angular_velocity Scalar mass
Matrix moment_of_inertia Point barycenter
Scalar drag
…
Adding a “RigidBody” component to a Game Object is to say:
“please let the Phys. engine take care of this object”
Video Game Dev - Univ Insubria 2017/2018
Marco Tarini
Game Physics - 2 5
In (using Unity terminology)
part of Transform component
the RigidBody component
Vector3 position Quaternion rotation Vector3 velocity
Quaternion angularVelocity float mass
Vector3 centerOfMass float drag
…
note: speed = velocity.magnitude
moment of inertia matrix the Vector3 = a diagonal matrix D by rotating it RTDR the final matrix note: they are the components
of the global transformation!
the barycenter (in object space) Vector3 inertiaTensor
Quaternion inertiaTensorRotation
per second (not per frame)
State of a point-particle
not used !
Point position
Rotation orientation Vector velocity
Rotation angular_velocity Scalar mass
Matrix moment_of_inertia Point barycenter
Scalar drag
…
One trend in game phys engines is to simulate point-particles only.
Much simpler!
E.g. no rotation needed!
We will see later how to still get complex objects back (with “PBD”) For now,
we focus on this simpler case.
Video Game Dev - Univ Insubria 2017/2018
Marco Tarini
Game Physics - 2 6
Dynamics (Newtonian)
dt v
p p
dt a
v v
m f
a
p fun
f
0 0
/
,...) (
Dynamics (Newtonian)
dt v
p p
dt a v
v
m f
a
p fun f
0 0
/
,...)
(
forcesacceler.
velocity positions
Video Game Dev - Univ Insubria 2017/2018
Marco Tarini
Game Physics - 2 7
Dynamics (Newtonian)
C C
t C
t C
C C
C C
dt t
v p
t p
dt t
a v
t v
m t
f t
a
t p fun t
f
0 0
0 0
) ( )
(
) ( )
(
/ ) (
) (
),...) (
( )
(
pos, acc, vel, forces:
in function of current time
t
CAn (obvious) precisation
= virtual time != real time
e.g.:
game paused t costant.
Fast forward, replay,
rallenty, reverse change of speed/flow direction of t
occasionally,
gameplay exploit this difference in spectacular ways!
PoP – the sands of times serie (Ubisoft, 2003-…) Braid (Jonathan Blow, 2008)
t
CVideo Game Dev - Univ Insubria 2017/2018
Marco Tarini
Game Physics - 2 8
Computing physics evolution
Analytical solutions:
state = function( t )
Given force functions (and acc), find the functions (pos, vel,…) in the specified relations:
Numerical solutions:
1. state
( t = 0) init 2. state
( t + 1)
evolve( state
t) 3. goto 2
C C
t C
t C
C C
C C
dt t v p t
p
dt t a v
t v
m t f t a
t p funz t
f
0 0
0 0
) ( )
(
) ( )
(
/ ) ( ) (
),...) (
( )
(
Simple example:
analytical solution
y x
v v
0v
0 0 p
0
9 . 8
m 0 f
x
y
a is in this specific case:one constant,
does not depend on pos
«ballistic shooting»
of a mass,
in 2D, ignoring friction...
Video Game Dev - Univ Insubria 2017/2018
Marco Tarini
Game Physics - 2 9
Simple example:
analytical solution
Solving…
2 0
0 0
0
2 / 8 . 9 8
. 0 9
) 0 ( )
(
8 . 8 9
. 9 ) 0
(
8 . 9 / 0
) ( )
(
8 . 9 ) 0
(
C C
y
C x t
y x t
C
C y
x t
y x C
C C
C
t t
v
t dt v
t v
dt v t
v p
t p
t v
dt v v
t v v
m t
f t
a
m t
f
C C
C
C C
t C
t C
C C
C C
dt t v p t
p
dt t a v
t v
m t f t a
t p fun t
f
0 0
0 0
) ( )
(
) ( )
(
/ ) ( ) (
),...) (
( )
(
Simple example:
analytical solution
Final result:
2
2/ 8 . ) 9
(
8 . ) 9
(
8 . 9 ) 0
(
8 . 9 ) 0
(
C C
y
C x C
C y
x C
C C
t t
v
t t v
p
t v
t v v
t a
m t
f
x y
)
( t
Cp
Video Game Dev - Univ Insubria 2017/2018
Marco Tarini
Game Physics - 2 10
Some numerical methods
Forward Euler method
(simple and direct)
Leapfrog method
Verlet method
(position based dynamics)
Numerical method features
How efficient / expensive
must be at least soft real-time
(if from time to time computation delayed to next frame, ok)
How accurate
must be at least plausible
(if stays plausible, differences from reality are acceptable)
How robust
rare completely wrong results
(and never crash)
How generic
Which phenomena / constraints / object types is it able to recreate?
needs dependent from the context (ex: gameplay)
Video Game Dev - Univ Insubria 2017/2018
Marco Tarini
Game Physics - 2 11
Euler method integration
(1) Evaluate the force (on each particle)
as a function of position(even of other particles)
(2) acceleration
of each particle given by:
forceson it and its mass
(3) Update velocitywith acceleration
(4) Update positionwith velocity
(state / variables) , (temp variables)
For each step:
dt v
p p
dt a
v v
m f
a
p fun
f
0 0
/
,...) (
Euler methods
dt v
p p
dt a
v v
m f
a
p fun f
v p
/
...) ,
( ...
init
...
state
one
step
t t dt
Video Game Dev - Univ Insubria 2017/2018
Marco Tarini
Game Physics - 2 12
Simple example:
numerical solution
4 2
0
y x
v v v
0 0 p
0
1
m 0 f
x
y
constant (in this specific case notdependent from pos) Same phenomena of previous example
1 dt
here, for instance,
Simple example:
numerical solution
dt v p p
dt a v v
m f a
m f
/
1 0
init
step: 0 1 2 3 4 5 6 7 …
pos: (0,0) (2,3) (4,5) (6,6) (8,6) (10,5) (12,3) (14,0) … vel: (2,4) (2,3) (2,2) (2,1) (2,0) (2,-1) (2,-2) (2,-3) …
x y
0 1
2 3 4
5 6
7
step step step step step step step step
Video Game Dev - Univ Insubria 2017/2018
Marco Tarini
Game Physics - 2 13
Physics evolution computation
Analytical solutions:
Numerical solutions:
x y
0 1
2 3 4
5 6
7
x y
) ( C
y
x function t p
p
Physics evolution computation
Analytical solutions:
Super efficient!
Close form solution
Accurate
Only simple systems
formulas found case by case
(often not existing!)
NO
(but, for instance, useful to allow the AI to make
predictions)
Numerical solutions:
Expensive (iterative)
but interactive
Integration errors
Flexible
Generic
YES
Video Game Dev - Univ Insubria 2017/2018
Marco Tarini
Game Physics - 2 14
Errors (of integration)
Dependent from dt
Small dt = more steps (with equal virtual time)
= more computationally expensive, but less error, more accurate simulation
(less difference from analytical solution, exact)
dt = 1.0 sec / FPS of physics simulation
(recall: not necessarily same rendering frame rate)
How much is the error increasing when dt change?
«Order» of the simulation
Euler is 1st order: the error is O( dt1 ) (but usually much less)
Keep on accumulating with time
(dependent also from )
Forces
In general, function of current position
Gravity
Friction
(these can be simulated using damping – later)
Opposition of solid materials
(But these well simulated using constraints… later)
Wind, electrical, magnetic, Archimedes force, springs, shock waves (explosions), etc etc …
“False” / “Magic” control forces
(added for controlling the evolution, not justified)
...
...) ,
( ...
p
fun
f
Video Game Dev - Univ Insubria 2017/2018
Marco Tarini
Game Physics - 2 15
Forces: Springs
Simplified model for elastic forces
1 spring connects two particles Va and Vb
Characterized by:
Rest length L
Elastic constant K
Force:
counteract stretching
and compression
VaVb
Exerted on Va and on Vb:
Given by:
• direction: the spring one
• magnitude: ( distance(Va,Vb) – L )* K
• direction: the other endpoint if distance(Va,Vb) > L or the opposite otherwise
Forces: Springs
Simplified model for elastic forces
1 spring connects two particles Va and Vb
Characterized by:
Rest length L
Elastic constant K
Force:
counteract stretching
and compression
VaVb
b a
b b a
a
v v
v L v
v v
K
f
( )
)
(
Video Game Dev - Univ Insubria 2017/2018
Marco Tarini
Game Physics - 2 16
Forces: Springs
Useful for deformable objects, for instance cloth, ropes, hairs…
Extra springs, for resistance
to bending
Forces: Springs
Useful for deformable objects, for instance cloth, ropes, hairs…
img by msqrt (pauli kemppinen)
Video Game Dev - Univ Insubria 2017/2018
Marco Tarini
Game Physics - 2 17
Forces: Springs
Spring composition:
Elastic deformable objects
(tend to go back to original shape)
Idea: change rest length
(in response to compression/stretching)
Plastic deformable objects
(take permanently the deformed pose)
Rigid objects?
Similar concept, but using constraints, not springs
(“hard constraints”)
Later!
Forces: control
Example:
The player presses the forward button:
A forward force is applied to his/her avatar
(no justification)
(“Don’t ask questions, physics engine”)
According to some:
it’s better when that’s not done
Only physically justified forces (usually hard to obtain)
plausibility VS realism
Video Game Dev - Univ Insubria 2017/2018
Marco Tarini
Game Physics - 2 18
Continuity of pos and vel
In theory, in a physics system pos and vel can changed only continuously
In practice, sometimes is useful to insert discontinuous changes in both:
in positions: “teleportation”
in velocity: “impulses”
model very big forces exerted in a very small time interval
ex.: impacts, v. collision handling
(those are not necessary variations justified by forces)
Implulse
Force (continuous)
Continuous application
every frame
...
/ ...
dt m
f v
v
Impulse
Infinitesimal time
una tantum
...
/ ...
m i
v
v
direct change (discontinuous!) of state (velocity)
it models a very short and intense force (ex: impacts)
Video Game Dev - Univ Insubria 2017/2018
Marco Tarini
Game Physics - 2 19
Impulses VS Forces
Force :
determines acceleration
acc determines a change (continuous!) of vel
physically correct
Impulse :
(discontinuous!) change of vel
ex: implulse caused by the rebound of a tennis ball against a tennis racket
a force with:
application time dt approaching zero
but magnitude approaching infinity
physically correct?
approximation.
model phenomena that happen at dt much smaller than simulated ones
(what does truly happen when the ball is hit by the racket?)
very strong forces in very short time (but not instantaneous)
Euler pseudo code
Vec3 position = … Vec3 velocity = … void initState(){
position = … velocity = … }
void physicStep( float dt ) {
position += velocity * dt;
Vec3 acceleration = compute_force( position ) / mass;
velocity += acceleration * dt;
}
void main(){
initState();
while (1) do physicStep( 1.0 / FPS );
}
Video Game Dev - Univ Insubria 2017/2018
Marco Tarini
Game Physics - 2 20
Numerical methods:
integration steps
dt : delta of virtual time from last step
the “temporal resolution” of the simulation!
if small: less efficiency
more steps for simulating same amount of virtual time
if big: less accuracy
Specially with strong forces and/or high velocities
Common values: 1 / 60 sec -- 1 / 30 sec
note: not necess. same refresh rate of rendering task
order of the simulation:
how much the error is increasing when dt increase
Euler Method: limitations
bad ratio efficiency / accuracy
error accumulated over time = linear in dt
Even plausibility could be compromised
No energy conservation !
ex: divergent oscillationts!
patch: damping of velocity (at each step)
float damp = 1.0 – SMALL_CONST ; velocity *= damp;
Continuous loss of kinetic energy
Official excuse: “friction with… everything (air included)”
Video Game Dev - Univ Insubria 2017/2018
Marco Tarini
Game Physics - 2 21
Trick: velocity «Damping»
(or «Drag», resistance e.g. air resistance)
Vec3 position = … Vec3 velocity = … void initState(){
position = … velocity = … }
void physicStep( float dt ) {
Vec3 acceleration = force( positions ) / mass;
velocity += acceleration * dt;
position += velocity * dt;
}
void main(){
initState();
while (1) do physicStep( 1.0 / FPS );
}
velocity *= (1.0 – DRAG * dt);
Artificial reduction of kinetic energy:
• robustness
(avoids energy incr. due to num. errors) (ex. avoid divergent oscillations)
• Not modelled dispersions (various frictions)
• high damp = like moving in molasses
• aka DRAG (often: DRAG = 1-DAMP)
Sub problem:
If I want to decrease velocity of a percentage every second
ex 20%
DRAG = 0.2 (or DAMPING = 0.8)
how long do I have to multiply vel every dt ?
for enough small DRAG, well approx. by
Drag
dtv
v ( 1 ) ) 1
( dt Drag v
v
1/FPS sec
