Animation and Games Development
242-515, Semester 1, 2014-2015
12. Basic Physics
• Objective
o show how physics is used in games
programming
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Overview
1. Why Physics?
2. Point-based Physics
3. Rigid Body Physics
4. Other Types of Force
5. The World of the Car
6. JBullet Features
7. A Game Physics Textbook
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1. Why Physics?
• Physics can make game worlds appear
more natural / realistic
• But...
• Often games are not realistic
o e.g. cars travelling at 200 mph, double jumps of Mario
• Physics engines are computationally expensive
o they may slow down the game too much
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Engines
• Commercial
o Havok (Havok.com)
o Renderware (renderware.com)
o NovodeX (novodex.com)
• Free
o
o
o
o
Open Dynamic Engine (ODE) (ode.org)
PhysX
Bullet (Java version used in jME)
Box2D (2D)
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Types of Physics
• Newtonian (this one for games)
o Newton’s space and time are absolute
• Classical
o Einstein's space-time can be “changed” by matter
• Quantum
o Deterministic causality questioned and multiple localities in
a single instance
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Remember this stuff…..
•
•
•
•
•
•
•
Time
o A measured duration
Speed
o The distance travelled given some time limit
Direction
o An indication of travel
Velocity
o The rate of change of position over time (has speed and direction)
Acceleration
o The rate of change of velocity
Force
o Causes objects to accelerate (has direction as well as magnitude)
Mass
o The weight of an object? (No! weight depends on gravitational pull)
o The quantity of matter
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Newton’s First Law of Motion
• Every body (object) continues in its state of rest, or
uniform motion in a straight line, unless there is an
external force acting on it.
• Often called "The Law of Inertia"
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Newton’s Second Law of Motion
• The rate of change of momentum of a body is
proportional to the force acting on it, and takes
place in the direction of that force.
• Simplified – An object’s change in velocity is
proportional to an applied force.
• An object's mass m, its acceleration a, and the
applied force F canbe represented by the formula:
F = ma
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Newton’s 3rd Law of Motion
• For every action there is an equal and opposite
reaction.
• If two objects bump into each other they will react
by moving apart.
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As Cartoons
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Game Physics
• Two main types of Newtonian physics used in
games: point-based, rigid body based
• Point-based physics
o used in particle systems, bullet motion…
o a point's mass is located inside the point
o a point does not rotate: it has no rotational orientation
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2. Point-based Physics
• Solve Newtonian equations to get the position of a
point at time t:
o Force applied to the point, F(t), causes
acceleration
o Acceleration, a(t), causes a change in the point's
velocity
o Velocity, V(t) causes a change in the point's
position
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Calculation Examples
•
•
•
Calculate a new position based on a constant velocity:
xt = x0 + (vx * t)
yt = y0 + (vy * t)
zt = z0 + (vz * t)
Velocity is a Vector, (vx, vy, vz), or (vx, vy) in 2D
Example, point at (1, 5), velocity of (4, -3)
Standard Equations
•
•
•
•
v=u+at
s = ½ t (u + v)
v 2 = u2 + 2 a s
s = u t + ½ a t2
•
•
•
•
u = initial velocity; v = final velocity
a = acceleration
t = time (seconds)
s = distance (meters)
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Velocity
•
Velocities are vectors:
Vtotal = √(Vx2 + Vy2 + Vz2) for 3D games
Vtotal = √(Vx2 + Vy2) for 2D games
Acceleration
•
Calculate a new velocity (vt) based on a
constant acceleration:
vt = dx/dt = v0 + (a * dt)
•
Each of these calculations are in one dimension
– you must perform similar calculations on y & z axes
Forces
• Forces are vectors
Ftotal = √(Fx2 + Fy2 + Fz2) for 3D games
Ftotal = √(Fx2 + Fy2) for 2D games
Example: 3D Projectile Motion
• Weight of the projectile (a point), W = mg
o g: constant acceleration due to gravity (9.81m/s2)
• Projectile equations of motion:
V(t )  Vinit  gt  t init 
1
2
p(t )  p init  Vinit t  tinit   gt  tinit 
2
p() is the position function
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Target Practice
V
ini
t
Projectile Launch
Position, pinit
F = weight = mg
Target
3. Rigid Body Physics
• The shape (body) has a mass that occupies
volume.
• We assume that the body never changes shape
• The orientation of the body can change over time.
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Rigid Body Translation
• Good news: the maths of rigid body translation is
the same as for points, since we can treat the
center of mass of the body as a point.
• This means we can reuse the maths:
oF=ma
o v = ∫a dt
o x = ∫v dt
o momentum is conserved
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center of mass
= point
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Rotation
• A rigid body has an orientation (rotated position in
space)
• Rotation calculations are complicated
o one reason is that the order of rotation
operations is important:
o x-axis rotation then y-axis rotation
y-axis rotation then x-axis rotation
≠
• Rotation is not commutative
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• When a body can rotate the Newtonian motion
equations must be extended.
o new equations are needed for rotational (angular) mass,
velocity, acceleration, force, momemtum, etc.
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Angular Velocity, ω
• The angular velocity ω describes the speed of
rotation and the orientation of the axis about which
the rotation occurs.
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Angular Velocity Again
• Δθ = change in angular displacement
• Δt = time
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Angular Acceleration, α
• Δω = change in angular velocity
• Δt = time
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Applying Force
• What happens when you push on a rotating body?
• There are two things to consider: translation and
rotation.
• For translation, we can use F=ma, because the
body's center of mass can be treated like a point.
• But how does the force affect the body's
orientation?
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Torque, T
• Torque is a measure of how much a force acting on
an object causes that object to rotate.
• T = r x F = r F sin(θ)
• Torque is the cross product (x) between the
distance vector from the pivot point (O) to the point
where force vector F is applied
• θ is the angle between r and F
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Uses for Torque
• The same amount of torque can be applied with
less force if the distance from the pivot (fulcrum) is
increased.
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Moment of Inertia I
• Moment of inertia is the mass property of a rigid
body that determines the torque needed for a
desired angular acceleration about an axis of
rotation.
• Moment of inertia depends on the shape of the
body and may be different around different axes of
rotation.
shape effect
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easy
(torque
is small)
hard
(torque
is large)
axes effect
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• Another definition: moment of Inertia is an object’s
resistance to rotating around an axis
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Linear Momentum, p
• Also called translational momentum: the product of
the mass and velocity of an object
• p=mv
• Linear momentum is a conserved quantity:
o if a closed system is not affected by external forces, then its
total linear momentum cannot change
o useful for calculating velocity change after collisions
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Angular momentum, L
• Angular momentum is the rotational version of linear
momentum.
o e.g. a tire rolling down a hill has angular momentum
• Defined as the cross product of the moment of
inertia I and the angular velocity ω.
• L=lxω
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• Angular momentum is a conserved quantity:
o if a closed system is not affected by external torque, then
its total angular momentum cannot change
o useful for calculating rotational change when moments of
inertia change
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L = I1 x ω1
L = I2 x ω2
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Linear vs. Angular
Linear
Concept
velocity v
Angular Version
acceleration a
angular acceleration α
F=ma
mass m
torque T = r x F'
=Iα
moment of inertia I
momentum
p=mv
angular momentum
L=Ixω
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angular velocity ω
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Changing Coordinate Systems
• The maths for rigid bodies is simpler if we use a local
coordinate system for each body
o e.g. place the origin at the centre of mass of the body
• But, we also need to transform any global forces
into each body's local coordinate system
• We also have to transform any local motion back
into global coordinates.
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4. Other Types of Force
•
•
•
•
•
(Virtual) Springs
Damping
Friction
Aerodynamic Drag
…
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(Virtual) Springs
• Even if you don't see very many actual springs
in a game, there are likely to be many invisible
virtual springs at work.
• Virtual springs are useful for implementing
constraints between objects:
o preventing objects overlapping
o cloth rendering
o character animation
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Linear Springs
Fspring  k (l  lrest )d
Damping
Damping describes physical conditions such as
viscosity, roughness, etc.
Fdamping  c((Vep 2  Vep1 )  d )d
Static Friction
Not Sliding
On the
Brink
Sliding
Kinetic Friction
• Once static friction is overcome and the object is
moving, friction continues to push against the
relative motion of the two surfaces.
o called kinetic friction
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5. The World of the Car
• Cars exist in an environment that exerts forces.
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Resistance
• A car driving down a road experiences two
(main) types of resistance.
o Aerodynamic drag, rolling resistance
Rtotal  Rair  Rrolling
• Once resistance has been calculated, it is
possible to calculate the amount of power a car
needs to move.
Aerodynamic Drag
Mass density of
air
Projected frontal area of car
normal to direction of V
Rair  (1 / 2)V S pCd
2
Speed of
car
Drag coefficient:
0.29 – 0.4: sports cars;
0.6 – 0.9: trucks
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Rolling resistance
• Tires rolling on a road experience rolling resistance.
This is not friction and has a lot to do with wheel
deformation. Simplifying, we can say:
Rrolling  Cr
Coefficient of rolling resistance:
Cars ≈ 0.015;
trucks ≈ 0.006 – 0.01
Weight of car
(assuming four
identical wheels)
Power
• Power is the measure of the amount of work done
by a force, or torque, over time.
• Mechanical work done by a force is equal to the
force * distance an object moves under the
action of that force.
• Power is usually expressed in units of horsepower
o 1 horsepower = 550 ft-lbs/s
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Horsepower
• Horsepower needed to overcome total
resistance at a given speed (we are working in
feet and lbs):
P  ( RtotalV ) / 550
horsepower
Total resistance
corresponding to
a car’s speed (V)
Engine output
 The previous equation relates the power delivered
to the wheels to reach speed V.
 The actual power required will be higher due to
mechanical loss.
 Power is delivered to a wheel in the form of torque.
Fw  Tw / r
Force delivered by a
wheel to the road to
push the car along
Torque on the
wheel
Radius of the
wheel
Stopping
• Stopping distance depends on the braking system
and how hard the driver breaks.
o the harder the brakes are applied the shorter the stopping
distance
• If a car skids then the stopping distance depends
on the frictional force between the tyres and the
road.
• If travelling uphill the stopping distance will be
shorter.
o gravity plays a role
• If travelling downhill the stopping distance will be
greater.
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Calculating Skidding Distance
Initial speed of
the car
d s  v /[ 2 g (  cos   sin  )]
2
Acceleration due
to gravity
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Coefficient of
friction between
wheels and the
road.
Usually around
0.4
Angle of the road
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More Calculations Required?
• In a real driving game a lot more parameters
play a role.
o e.g. suspension, variable engine power, road surface
• Usually it is easier to use a physics engine, but
this still requires an understanding of the forces
that you want to model.
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6. JBullet Features
• Rigid Bodies
o Simple shapes, complex geometries
• Joints and Spring Constraints
• Dynamics Modeling
what we've been
talking about in
this part
o Integrating forces and torques
• Collision Detection
the next part
o Physical interactions between objects
• Ray Tracing
o Range finders and optical flow
• Cloth, Soft (deformable) Bodies
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7. A Game Physics Textbook
• Physics for Game Developers
David M Bourg, Bryan Bywalec
O'Reilly, April 2013, 2nd ed.
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