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Game Physics

- Not trying to build a perfect physical model
- Most things can be approximated assuming Newtonian physics and rigid bodies
- Use discrete simulation (constant step) techniques
- Just worry about center of mass for most things

Why Physics?

- The Human Experience
- Real-world motions are physically-based
- Physics can make simulated game worlds appear more natural
- Makes sense to strive for physically-realistic motion for some types of games
- Emergent Behavior
- Physics simulation can enable a richer gaming experience

Why Physics?

- Developer/Publisher Cost Savings
- Classic approaches to creating realistic motion:
- Artist-created keyframe animations
- Motion capture
- Both are labor intensive and expensive
- Physics simulation:
- Motion generated by algorithm
- Theoretically requires only minimal artist input
- Potential to substantially reduce content development cost

High-level Decisions

- Physics in Digital Content Creation Software:
- Many DCC modeling tools provide physics
- Export physics-engine-generated animation as keyframe data
- Enables incorporation of physics into game engines that do not support real-time physics
- Straightforward update of existing asset creation pipelines
- Does not provide player with the same emergent-behavior-rich game experience
- Does not provide full cost savings to developer/publisher

High-level Decisions

- Real-time Physics in Game at Runtime:
- Enables the emergent behavior that provides player a richer game experience
- Potential to provide full cost savings to developer/publisher
- May require significant upgrade of game engine
- May require significant update of asset creation pipelines
- May require special training for modelers, animators, and level designers
- Licensing an existing engine may significantly increase third party middleware costs

High-level Decisions

- License vs. Build Physics Engine:
- License middleware physics engine
- Complete solution from day 1
- Proven, robust code base (in theory)
- Most offer some integration with DCC tools
- Features are always a tradeoff

High-level Decisions

- License vs. Build Physics Engine:
- Build physics engine in-house
- Choose only the features you need
- Opportunity for more game-specific optimizations
- Greater opportunity to innovate
- Cost can be easily be much greater
- No asset pipeline at start of development

Position and Velocity

- Where is object at time t (using pixels)?
- Equations

player_x(t) = t * x_velocity + x_initial

player_y(t) = t * y_velocity + y_initial

- Computation

player_x = player_x + x_velocity

player_y = player_y + y_velocity

Acceleration

- Computation

x_velocity = x_velocity + x_acceleration

y_velocity = y_velocity + y_acceleration

- Use piecewise linear approximation to continuous functions

Gravity

- Force of attraction between objects

F = G * (M1 * M2) / D2

G = gravitational constant

D = distance between objects

- Free falling objects on Earth
- Equation

V(t) = 1/2 * g * t2

g = 9.8 m/sec2

- Computation

x_velocity = x_velocity + 0

y_velocity = y_velocity + gravity

X = x + Vx + W

Y = y + Vy

Vxi = cos(A) * Vi

Vyi = sin(A) * Vi

Vx = Vx - WR(Vx)

Vy - WR(Vy) + G

W = wind

A = inclination angle

Vi = initial velocity

WR = wind resistance

G = gravity

Projectile MotionFriction

- Conversion of kinetic energy into heat
- Equation
- Frictional Force = C * G * M
- C = force required to maintain constant speed
- G = gravity
- M = mass
- Computation

while (velocity > 0)

velocity = velocity - friction

Collision Detection

Complicated for two reasons

1. Geometry is typically very complex, potentially requiring expensive testing

2. Naïve solution is O(n2) time complexity, since every object can potentially collide with every other object

Collision Detection

Two basic techniques

1. Overlap testing

- Detects whether a collision has already occurred

2. Intersection testing

- Predicts whether a collision will occur in the future

Overlap Testing

- Facts
- Most common technique used in games
- Exhibits more error than intersection testing
- Concept
- For every simulation step, test every pair of objects to see if they overlap
- Easy for simple volumes like spheres, harder for polygonal models

Overlap Testing:Useful Results

- Useful results of detected collision
- Time collision took place
- Collision normal vector

Overlap Testing:Collision Time

- Collision time calculated by moving object back in time until right before collision
- Bisection is an effective technique

Overlap Testing:Limitations

- Fails with objects that move too fast
- Unlikely to catch time slice during overlap
- Possible solutions
- Design constraint on speed of objects
- Reduce simulation step size

Intersection Testing

- Predict future collisions
- When predicted:
- Move simulation to time of collision
- Resolve collision
- Simulate remaining time step

Intersection Testing:Swept Geometry

- Extrude geometry in direction of movement
- Swept sphere turns into a “capsule” shape

Intersection Testing:Sphere-Sphere Collision

- Smallest distance ever separating two spheres:
- If

there is a collision

Intersection Testing:Limitations

- Issue with networked games
- Future predictions rely on exact state of world at present time
- Due to packet latency, current state not always coherent
- Assumes constant velocity and zero acceleration over simulation step
- Has implications for physics model and choice of integrator

Dealing with Complexity

Two issues

1. Complex geometry must be simplified

2. Reduce number of object pair tests

Dealing with Complexity:Simplified Geometry

- Approximate complex objects with simpler geometry, like this ellipsoid

Dealing with Complexity:Bounding Volumes

- Bounding volume is a simple geometric shape
- Completely encapsulates object
- If no collision with bounding volume, no more testing is required
- Common bounding volumes
- Sphere
- Box

Dealing with Complexity:Achieving O(n) Time Complexity

One solution is to partition space

Collision Resolution:Examples

- Two billiard balls strike
- Calculate ball positions at time of impact
- Impart new velocities on balls
- Play “clinking” sound effect
- Rocket slams into wall
- Rocket disappears
- Explosion spawned and explosion sound effect
- Wall charred and area damage inflicted on nearby characters
- Character walks through wall
- Magical sound effect triggered
- No trajectories or velocities affected

Collision Resolution:Prologue

- Collision known to have occurred
- Check if collision should be ignored
- Other events might be triggered
- Sound effects
- Send collision notification messages

Collision Resolution:Collision

- Place objects at point of impact
- Assign new velocities
- Using physics or
- Using some other decision logic

Collision Resolution:Epilogue

- Propagate post-collision effects
- Possible effects
- Destroy one or both objects
- Play sound effect
- Inflict damage
- Many effects can be done either in the prologue or epilogue

Collision Resolution:Resolving Overlap Testing

1. Extract collision normal

2. Extract penetration depth

3. Move the two objects apart

4. Compute new velocities

Collision Resolution:Extract Collision Normal

- Find position of objects before impact
- Use two closest points to construct the collision normal vector

Collision Resolution:Extract Collision Normal

- Sphere collision normal vector
- Difference between centers at point of collision

Collision Resolution:Resolving Intersection Testing

- Simpler than resolving overlap testing
- No need to find penetration depth or move objects apart
- Simply

1. Extract collision normal

2. Compute new velocities

Simple Collision Handling

- Detect that collision has occurred

(bounding box)

- Determine the time of the collision

(may need to back up to point of collision)

- Determine where objects are when they touch
- Determine the collision normal

(angle of incidence = angle of reflection)

- Determine the velocity vectors after the collision
- Determine any changes in motion

Simple Kinematics

P(x, y)

- Forward kinematic problem
- find position of P from theta1, theta2, L1, L2
- use the 2D translation and rotation matrices
- (TL2*Rtheta2)* (TL1*Rtheta1)
- generalizes to any number of links

theta2

L2

L1

theta1

Particle System Explosions

- Start with lots of small objects (1 to 4 pixels)
- Initialize particles with random velocities based on velocity of exploding object
- Apply gravity
- Transform color intensity as a function of time
- Destroy objects when they collide or after a fixed amount of time

Generalized Rigid Bodies

- Key Differences from Particles
- Not necessarily spherical in shape
- Position, p, represents object’s center-of-mass location
- Surface may not be perfectly smooth
- Friction forces may be present
- Experience rotational motion in addition to translational (position only) motion

Generalized Rigid Bodies – Simulation

- Angular Kinematics
- Orientation, 3x3 matrix R or quaternion, q
- Angular velocity, w
- As with translational/particle kinematics, all properties are measured in world coordinates
- Additional Object Properties
- Inertia tensor, J
- Center-of-mass
- Additional State Properties for Simulation
- Orientation
- Angular momentum, L=Jw
- Corresponding state derivatives

Generalized Rigid Bodies - Simulation

- Torque
- Analogous to a force
- Causes rotational acceleration
- Cause a change in angular momentum
- Torque is the result of a force (friction, collision response, spring, damper, etc.)

Collision Response

- Why?
- Performed to keep objects from interpenetrating
- To ensure behavior similar to real-world objects
- Two Basic Approaches
- Approach 1: Instantaneous change of velocity at time of collision
- Benefits:
- Visually the objects never interpenetrate
- Result is generated via closed-form equations, and is perfectly stable
- Difficulties:
- Precise detection of time and location of collision can be prohibitively expensive (frame rate killer)
- Logic to manage state is complex

Collision Response

- Two Basic Approaches (continued)
- Approach 2: Gradual change of velocity and position over time, following collision
- Benefits
- Does not require precise detection of time and location of collision
- State management is easy
- Potential to be more realistic, if meshes are adjusted to deform according to predicted interpenetration
- Difficulties
- Object interpenetration is likely, and parameters must be tweaked to manage this
- Simulation can be subject to numerical instabilities, often requiring the use of implicit finite difference methods

Final Comments

- Instantaneous Collision Response
- Classical approach: Impulse-momentum equations
- See text for full details
- Gradual Collision Response
- Classical approach: Penalty force methods
- Resolve interpenetration over the course of a few integration steps
- Penalty forces can wreak havoc on numerical integration
- Instabilities galore
- Implicit finite difference equations can handle it
- But more difficult to code
- Geometric approach: Ignore physical response equations
- Enforce purely geometric constraints once interpenetration has occurred

Final Comments

- Simple Games
- Closed-form particle equations may be all you need
- Numerical particle simulation adds flexibility without much coding effort
- Collision detection is probably the most difficult part of this
- Generalized Rigid Body Simulation
- Includes rotational effects and interesting (non-constant) forces

Time-Based Modeling

- Time t is used in all kinematic equations that move objects (to avoid discontinuities caused by “slower” frame rates)
- This involves scaling dx and dy based on elapsed time (rather than a virtual clock)
- This allow constant movement regardless of processor slow downs

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