242-515 AGD: 12. Physics 11

• Objectiveo show how physics is used in games

programming

Animation and Games

Development242-515, Semester 1, 2014-2015

12. Basic Physics

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1. Why Physics?2. Point-based Physics3. Rigid Body Physics4. Other Types of Force5. The World of the Car6. JBullet Features7. A Game Physics Textbook

Overview

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• 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 expensiveo they may slow down the game too much

1. Why Physics?

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• Commercialo Havok (Havok.com)o Renderware (renderware.com)o NovodeX (novodex.com)

• Freeo Open Dynamic Engine (ODE) (ode.org)o PhysXo Bullet (Java version used in jME)o Box2D (2D)

Engines

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• Newtonian (this one for games)o Newton’s space and time are absolute

• Classicalo Einstein's space-time can be “changed” by matter

• Quantumo Deterministic causality questioned and multiple localities

in a single instance

Types of Physics

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• Timeo A measured duration

• Speedo The distance travelled given some time limit

• Directiono An indication of travel

• Velocityo The rate of change of position over time (has speed and direction)

• Accelerationo The rate of change of velocity

• Forceo Causes objects to accelerate (has direction as well as magnitude)

• Masso The weight of an object? (No! weight depends on gravitational pull)o The quantity of matter

Remember this stuff…..

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• 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"

Newton’s First Law of Motion

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• 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

Newton’s Second Law of Motion

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• For every action there is an equal and opposite reaction.

• If two objects bump into each other they will react by moving apart.

Newton’s 3rd Law of Motion

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As Cartoons

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• Two main types of Newtonian physics used in games: point-based, rigid body based

• Point-based physicso used in particle systems, bullet motion…o a point's mass is located inside the pointo a point does not rotate: it has no rotational orientation

Game Physics

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• 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

2. Point-based Physics

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)

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Standard Equations• v = u + a t• s = t (u + v)• v2 = u2 + 2 a s• s = u t + a t2

• u = initial velocity; v = final velocity• a = acceleration• t = time (seconds)• s = distance (meters)

Velocity

• Velocities are vectors:

Vtotal = √(Vx2 + Vy

2 + Vz2) for 3D

games

Vtotal = √(Vx2 + Vy

2) 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 + Fy

2 + Fz2) for 3D

games

Ftotal = √(Fx2 + Fy

2) for 2D games

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• Weight of the projectile (a point), W = mgo g: constant acceleration due to gravity

(9.81m/s2)

• Projectile equations of motion:

Example: 3D Projectile Motion

initinit ttt gVV )(

2

2

1)( initinitinitinit ttttt gVpp

p() is the position function

Target Practice

F = w e ig ht = m gTarget

Projectile LaunchPosition, pinit

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• 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.

3. Rigid Body Physics

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• 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:o F = m ao v = ∫a dto x = ∫v dto momentum is conserved

Rigid Body Translation

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center of mass= point

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• A rigid body has an orientation (rotated position in space)

• Rotation calculations are complicatedo 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

Rotation

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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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• The angular velocity ω describes the speed of rotation and the orientation of the axis about which the rotation occurs.

Angular Velocity, ω

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• Δθ = change in angular displacement• Δt = time

Angular Velocity Again

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• Δω = change in angular velocity• Δt = time

Angular Acceleration, α

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• 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?

Applying Force

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• 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

Torque, T

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• The same amount of torque can be applied with less force if the distance from the pivot (fulcrum) is increased.

Uses for Torque

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• 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.

Moment of Inertia I

shape effect

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axes effect

easy(torque is small)

hard(torque is large)

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• Another definition: moment of Inertia is an object’s resistance to rotating around an axis

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• Also called translational momentum: the product of the mass and velocity of an object

• p = m v• 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

Linear Momentum, p

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• 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 = l x ω

Angular momentum, L

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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 changeo useful for calculating rotational change when moments

of inertia change

L = I1 x ω1L = I2 x ω2

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Linear vs. AngularLinear Concept

Angular Version

velocity v angular velocity ω

acceleration a angular acceleration α

F = m a torque T = r x F' = I α

mass m moment of inertia I

momentump = m v

angular momentum L = I x ω

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• The maths for rigid bodies is simpler if we use a local coordinate system for each bodyo 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.

Changing Coordinate Systems

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• (Virtual) Springs• Damping• Friction• Aerodynamic Drag• …

4. Other Types of Force

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• 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 overlappingo cloth renderingo character animation

(Virtual) Springs

Linear Springs

dllkF restspring )(

Damping

ddVVcF epepdamping ))(( 12

Damping describes physical conditions such as viscosity, roughness, etc.

Static FrictionNot Sliding On the

BrinkSliding

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• 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

Kinetic Friction

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• Cars exist in an environment that exerts forces.

5. The World of the Car

Resistance

• A car driving down a road experiences two (main) types of resistance.o Aerodynamic drag, rolling resistance

rollingairtotal RRR

• Once resistance has been calculated, it is possible to calculate the amount of power a car needs to move.

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dpair CSVR 2)2/1(

Aerodynamic DragMass density of air

Speed of car

Projected frontal area of car normal to direction of V

Drag coefficient:0.29 – 0.4: sports cars;0.6 – 0.9: trucks

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)

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• 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 horsepowero 1 horsepower = 550 ft-lbs/s

Power

Horsepower• Horsepower needed to overcome total

resistance at a given speed (we are working in feet and lbs):

550/)( VRP total

horsepowerTotal 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.

rTF ww /

Force delivered by a wheel to the road to push the car along

Torque on thewheel

Radius of the wheel

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• 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.

Stopping

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)]sincos(2/[2 gvd s

Calculating Skidding Distance

Acceleration due to gravity Coefficient of

friction between wheels and the road.

Usually around 0.4

Initial speed of the car

Angle of the road

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• 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.

More Calculations Required?

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6. JBullet Features• Rigid Bodies

o Simple shapes, complex geometries

• Joints and Spring Constraints• Dynamics Modeling

o Integrating forces and torques

• Collision Detectiono Physical interactions between objects

• Ray Tracingo Range finders and optical flow

• Cloth, Soft (deformable) Bodies

56

what we've beentalking about inthis part

the next part

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• Physics for Game DevelopersDavid M Bourg, Bryan BywalecO'Reilly, April 2013, 2nd ed.

7. A Game Physics Textbook