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CSE 381 – Advanced Game Programming Collision Detection

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Remember This?

- We spent a lot of time on this in the spring
- Remember what we learned
- What will we add?
- Sweep and Prune algorithm (today)
- GJK algorithm (Monday)

What is collision detection?

- Determining if, when, & where two objects intersect
- What objects?
- Linear components, planes, triangles, rectangles, oriented boxes, spheres, capsules, lozenges, cylinders, ellipsoids, etc.
- Hugely important part of a 3D game engine. Why?
- done continuously every frame
- involves huge amounts of computations
- Entire textbooks are written on this subject alone
- Real-Time Collision Detection by Christer Ericson
- Collision Detection in Interactive 3D Environments by Gino van den Bergen

For what is collision detection used?

- In 3D gaming:
- prevent players/monsters from walking through walls
- prevent players/monsters from walking/falling through terrain
- react to players/monsters colliding with each other
- react to players/monsters colliding with game objects
- i.e., pick up ammo
- react to projectiles colliding with players/monsters
- i.e., take health away for being hit by a bullet
- ragdoll physics

Why so important?

- Because if done improperly, it can ruin a game
- collision detection problems can still be found in state of the art games
- Ever get stuck in a wall?
- Ever go behind a wall you’re not supposed to?
- Ever have a collision that doesn’t look like a collision?
- Because if done inefficiently, it will ruin gameplay

Collision Detection & Graphics

- Work together in a game engine
- Should be developed together
- Both share:
- geometric data
- timing elements
- So, pool resources for their algorithms
- e.g., scene graphs, spatial partitioning (octrees, bsps…)

Game Engine Collision Detection

- 2 phases
- Broad phase: determine which pairs of shapes need to be tested for collision
- Narrow phase: determine collision results for each pair identified in broad phase
- All game objects maintain collision sets
- data for collisions calculations
- i.e., bounding volumes

Types of Collisions

- Object versus plane (navigation or culling)
- Object versus object (general collision)
- Linear component versus object (picking)

Imagine a complex world

- Hundreds of interior structures, each with rooms
- Complex external terrain environment
- Water with underwater structures (interiors)
- Rather than testing these items serially, we can do so hierarchically
- much faster
- continuously reduce the problem

How should collision data be organized?

- One of the most important questions in designing a collision system
- Game world might contain a large number of interacting objects
- an exhaustive comparison is too exhaustive
- Objects should be organized into collision groups
- e.g., rooms, partitions, etc.

Rooms as collision groups

- Scene graphs & collision sets are not static, they change with the game
- as players, NPCs, or other game objects move/change
- When a player enters a room, the scene graph is reconfigured
- player is now part of room collision group
- Only objects moving within a room are tested against one another for collisions

Collision Detection Game Physics

- This is a huge topic
- Game physics is rapidly changing

Collision Detection & Response

- Collision Detection
- detecting what game objects are colliding with each other
- Collision Response
- providing a programmed response to collisions that fits with the game’s design & custom laws of physics

Static vs. Dynamic Objects

- Static objects never move
- Dynamic objects move
- Collisions may be between:
- Static objects & dynamic objects (fast & easy)
- Dynamic objects & dynamic objects (harder)

What types of collisions might we care about?

- Main character and static objects:
- terrain, floor, tiles, walls, furniture, buildings, game objects (i.e. power-ups, ammo), etc.
- Main character & dynamic objects:
- enemies, projectiles (i.e. bullets, arrows, etc.), particles (expensive), etc.
- Other dynamic objects and:
- other static objects
- other dynamic objects

Collisions in Pairs

- In collision detection, we always compare pairs of objects. Easier to:
- understand
- design & implement solutions
- A naïve approach:
- one pair at a time, compare all game objects in a game world against all other game objects in a game world

Collision Detection Calculations

- What data are we looking for?
- Do these two objects potentially collide?
- Do these two objects collide?
- When did these two objects collide?
- Where did these two objects collide?
- where on geometry of objects, points of impact
- These 4 questions get progressively:
- more computationally expensive to implement
- more complex to implement (more math)

Phases of Collision Detection

- Spatial Partitioning Algorithms
- problem reduction
- only perform additional phases on pairs of object on same “islands”
- Broad Phase – early rejection tests
- Do the coarse Bounding Volumes of two objects collide?
- Narrow Phase
- What are the contact points on object geometries?
- Done down to the last triangle in 3D games

Bounding Volumes

- The base geometry used for all collision tests
- instead of the shape’s geometry, which is too expensive
- Properties of desirable BVs:
- inexpensive intersection tests
- tight fitting
- inexpensive to compute
- easy to rotate and transform
- use little memory
- Ref: [1]

Assumptions you might make

- All collideable game objects:
- have rectangular Bounding Volumes
- don’t rotate, or
- if they do rotate, we don’t care about them colliding with stuff
- Thus:
- no polytope collision detection equations
- no rotation equations
- this makes life much easier (no narrow phase)

Common Bounding Volumes

- Note that the space craft has been rotated
- Ref: [1], Figure 4.2
- This semester, we like AABBs and/or Spheres
- axis-aligned bounding boxes

How about a big, complicated object?

- We can have 2 AABBs
- Don’t test them against each other

Spatial Partitioning

- First, reduce the problem
- only perform additional phases on pairs of object on same “islands”
- Common Solutions:
- Octree Algorithms (quadtrees for 2D)
- also: Portals (ala Quake), BSP trees (Binary Space Partitions), Spatial Hashing, etc.

Octrees

- Used to divide a 3D world into “islands”
- 2D divided via Quadtrees
- Why?
- to make rendering more efficient
- to make collision detection more efficient
- What would be the data for these nodes?
- region coordinates for cell
- though a smart implementation might eliminate this too
- AABBs

Octree

- Source: http://en.wikipedia.org/wiki/Image:Octree2.png

Octree Drawbacks

- Objects cross islands
- octree has to be constantly updated
- not so bad
- Objects straddle islands
- collision detection may involve objects from multiple islands
- can be a headache

Uniform Grids

- Fast mechanism for reducing the problem
- used for 2D & 3D games
- Steps:
- divide the world into a grid of equal sized cells
- associate each object with the cells it overlaps
- only objects sharing a cell are compared
- Serves 2 purposes:
- reduces the problem for object-object collision detection
- provides solution for easy object-terrain collision detection

Calculating Grid Position

- What cells does our object currently occupy?
- How would you calculate that?
- For min/max tile rows/columns:
- object.X/terrainCellW
- object.Z/terrainCellL
- (object.X+aabb.Width)/terrainCellW
- (object.Z+aabb.Height)/terrainCellL
- For this guy, cells (0,y,0), (0,y,1), (1,y,0), & (1,y,1)
- only test against:
- other objects in those cells
- collidable tiles in those cells (treat like objects)
- don’t let sprites occupy “collidable cells”

Grid Cell Collision & Walkable Surfaces

- Side-scrollers simulate gravity
- not necessarily accelerated (constant velocity)
- move all affected sprites down by dY
- This way, characters can fall
- We must detect when sprites are colliding with a floor/platform
- Easy solution: make a tile with a walkable surface at the very top of the image
- Collision system will handle response

Grid Cell Collision Advantages/Disadvantages

- Advantages
- easy to implement
- very fast (computationally inexpensive)
- Only good for certain types of predicatable collisions
- Physics simulations require more complex approaches

Again

- Spatial Partitioning
- Octrees
- Portals, BSPs, etc.
- Uniform Grids
- What purpose do they serve?
- reduce the problem of collision detection
- How?
- reduce the number of object-object tests

Object Tests Premise

- Think of all collision tests as between pairs of collidable objects
- In our games this means:
- object – to – object
- In a single game loop, for each object, we may have to check against
- other moving objects (tricky)
- other stationary objects (easier)

Broad Phase

- Do the coarse AABBs of two objects collide?
- Common solution:
- separating axis algorithms
- including temporal coherence
- For efficiencies use:
- Sweep & Prune
- an extension of separating axis, more efficient for many elements

Narrow Phase

- What are the contact points on object geometries?
- for 3D might be convex hulls
- Two Steps
- determine potentially colliding primitives (ex: triangles) of a pair of objects
- AABB tree algorithms
- determine contact between primitives
- GJK algorithms
- Ref[3]

Position Calculations

- New Position with 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, object at (1, 5), velocity of (4, -3)

Velocity Vectors

- Velocities have direction, their vector

Vtotal = √(Vx2 + Vy2 + Vz2) for 3D games

Vtotal = √(Vx2 + Vy2) for 2D games

Acceleration

- Rate of change of velocity
- New Velocity (vt) with Acceleration:

vt = dx/dt = v0 + (a * dt)

- NOTE – each of these calculations are in one dimension
- You would perform similar calculations on y & z axes
- NOTE: we will avoid mid-frame acceleration
- everybody does it, so will we

Trajectory Assumption

- In a single frame, if no force is exerted upon an object, it will have a constant velocity for that frame

Forces and vectors

- F = ma
- Collisions produce forces on both colliding objects
- Forces have direction, their vector
- Forces in x, y, & z axes
- Ftotal = √(Fx2 + Fy2 + Fz2) for 3D games
- Ftotal = √(Fx2 + Fy2) for 2D games

Forces can be summed

- Done axis by axis

Fx = F1x + F2x + F3x

Fy = F1y + F2y + F3y

Fz = F1z + F2z + F3z

Momentum

- (P) – A property that objects in motion have

P = m * v, measured in kg * m/s

- Force equation can be reduced to:

F = dP/dt

Momentum & Collisions

- Note: Ignore friction for now
- Momentum transfer – if 2 objects collide:
- A perfectly elastic collision: no loss of energy
- Momentum is conserved
- An imperfect elastic collision: some energy is converted into heat, work, & deformation of objects
- Momentum is reduced after collision

Rigid Bodies

- Note: we are only dealing with rigid bodies
- No deformation due to collisions

Calculating new velocities

- Note:
- ignore rotation/angular velocities for now
- ignore centers of mass for now
- 2 moving blocks collide:
- Block A: mA & vAi
- Block B: mB & vBi
- Question, if they collide, what should their velocities be immediately after the collision?
- vAf & vBf

Why are we interested in final velocities?

- When a collision precisely happens, we want to:
- move our objects to that precise location
- change their velocities accordingly

Calculating vAf & vBf

- If momentum is conserved after collision:

PAi + PBi = PAf + PBf

(mA * vAi) + (mB * vBi) = (mA * vAf) + (mB * vBf)

- Problem has 2 unknowns (vaf & vbf)
- we need a second equation (Kinetic energy equation)
- ke = (m * v2)/2, where ke is never negative Joules (J)
- insert the ke equation into our momentum equation

vAf = ((2 * mB * vBi) + vAi * (mA – mB))/(mA + mB)

vBf = ((2 * mA * vAi) – vBi * (mA – mB))/(mA + mB)

Example

- 2 blocks moving
- Block A:
- mass is 10, initial velocity is (4, 0)
- Block b:
- mass is 10, initial velocity is (-4, 1)
- Results:
- Block A final velocity is (-4, 1)
- Block B final velocity is (4, 0)
- See my ElasticCollisionsVelocityCalculator.xls

Continuous Collision Detection

- For each moving object, compute what grid cells it occupies and will cross & occupy if its current velocity is added
- Find the first contact time between each object pair
- Sort the collisions, earliest first
- Move all affected objects to time of first collision, updating positions
- Update velocities of collided objects
- Go back to 1 for collided objects only
- If no more collisions, update all objects to end of frame
- Ref[3]

Time in between frames

- We will make calculations and update velocities and positions at various times
- When?
- In response to:
- input at start of frame
- AI at start of frame
- collisions/physics when they happen
- likely to be mid-frame

Axis Separation Tests

- How do we know if two AABBs are colliding?
- if we can squeeze a plane in between them
- i.e. if their projections overlap on all axes

NO COLLISION

COLLISION

A

C

B

D

Be careful of Tunneling

- What’s that?
- An object moves through another object because collision is never detected

A note about timing

- If we are running at 30 fps, we are only rendering 1/30th of the physical states of objects
- Squirrel Eiserloh calls this Flipbook syndrome [2]
- More things happen that we don’t see than we do

- We likely won’t see the ball in contact with the ground

One way to avoid tunneling

- Swept shapes
- Potential problem: false positives
- Good really only for early rejection tests

Better way to avoid tunneling

- Calculate first contact times
- Resolve contacts in order of occurrence

Times as %

- You might want to think of your times as % of the frame
- Then correct positions according to this %
- For example, if I know my object is supposed to move 10 units this frame (we’ve locked the frame rate), if a collision happens ½ way through, move all affected objects ½ distance of their velocity and recompute collisions data for collided data
- Re-sort collisions and find next contact time

% Example

- A at (1, 5,0), velocity of (4, -3,0)
- B stationary at (4, 4,0)
- When will they collide?
- When A goes one unit to the right
- How long will that take?
- If it takes 1 frame to go 4 units, it will take .25 frames to go 1

(0, 0, 0)

+ x

+ y

What about 2 moving objects?

- Solution, give the velocity of one to the other for your calculations
- make one of them stationary
- then do the calculation as with grid tiles
- make sure you move both objects
- make sure you update the velocities of both objects

So how do we calculate the when?

- We can do it axis by axis
- What’s the first time where there is overlap on all 3 axes?
- For each axis during a frame, what is the time of first and last contact?
- Time of first contact is

Again, for 2 moving bodies, freeze one

- Object A
- (xA, yA, zA, vxA, vyA, vzA)
- Object B
- (xB, yB, zB, vxB, vyB, vzB)
- Give velocity of B to A, now A is:
- (xA, yA, zA, vxA-vxB, vyA-vyB, vzA-vzB)

First time of contact

- For object A, in x-axis
- (xA, yA, zA, vxA-vxB, vyA-vyB, vzA-vzB)
- Time = distance/velocity
- Since velocity = distance/time

if ((xB-(Bwidth/2)) > (xA+(Awidth/2)))

{

tx_first_contact = (xB-(Bwidth/2) – (xA + (Awidth/2)))/(vxA-vxB);

tx_last_contact = ((xB + (Bwidth/2)) – (xA-(Awidth/2)))/vxA-vxB);

if (tx_first_contact > 1) no collision

}

What if ((xB-(Bwidth/2)) < (xA+(Awidth/2)))?

Time Calculation Example

- Awidth = 2, Bwidth = 2
- A at (1, 1, 0), velocity of (4, 3, 0)
- B at (4, 2, 0), velocity of (-1, 0, 0)
- When will they start colliding on x-axis?
- if ((xB-(Bwidth/2)) > (xA+(Awidth/2)))
- {
- tx_first_contact = (xB-(Bwidth/2) – (xA + (Awidth/2)))/(vxA-vxB);
- tx_last_contact = ((xB + (Bwidth/2)) – (xA-(Awidth/2)))/vxA-vxB);
- if (first_contact > 1) no collision
- }

(0, 0, 0)

tx_first_contact = 1/5 t

tx_last_contact = 1t

Physics & Timing

- Christer Ericson & Erin Catto’s recommendations:
- physics frame rate should be higher resolution than rendering
- minimum of 60 frames/second for physics calculations
- makes for more precise, and so realistic interactions
- How do we manage this?
- tie frame rate to 60 fps, but don’t render every frame
- don’t worry about this for now
- Note: these operations might commonly be done in different threads anyway (have their own frame rates)

Sweep & Prune

- Also called Sort & Sweep
- A broad phase algorithm
- This is an efficient way of managing data for method of separating axis (MSA) tests
- Baraff, D. (1992), Dynamic Simulation of Non-Penetrating Rigid Bodies, (Ph. D thesis), Computer Science Department, Cornell University

Improvements on MSA

- It greatly reduces the number of object-object comparisons.
- How?
- it only compares objects near each other on one of the 3 axes
- How?
- sorting

Sweep & Prune approach

- Think of each axis separately, for each object considered:
- Calculate position range on x-axis
- extract (minX, maxX)
- Place all (minX, maxX) pairs into array
- Sort array by minX
- now, it’s easy to see what overlaps in X axis
- an O(N) problem, not O(N2). Why?
- don’t compare all objects to one another, only ones “near” each other
- note, we’ll have sorting costs, of course
- For x-axis collided pairs, repeat for Y & Z axes
- Pairs that overlapped on all 3 tests collided

Combining our Algorithms

- Sweep and Prune will tell us if a collision might happen
- Continuous collision detection will tell us when it happens
- it might also tell us S & P gave us a false positive

Picking

- Selecting a 3D object from its 2D projection on the screen by pointing and clicking with a mouse
- Why is this useful?
- players selecting game objects/targets
- i.e., which enemy to attack
- bullets traveling through game world
- determining which player/NPC it hits

How is picking done?

- Build a ray (like a vector)
- Where’s the origin?
- the camera
- a gun
- What direction is the ray?
- from camera to a world point that projects onto the screen at the screen location
- Find those objects that are intersected by the ray

Testing for picking intersections

- Traverse through scene graphs to find those with triangles that intersect the ray
- Gather data about these intersections as needed
- Point of intersection
- Normal vector at intersection
- Surface attributes (color, texture coordinate)
- Etc.
- Find the closest triangle to the ray’s origin

How do we test intersections with linear components?

- Depends on what it’s intersecting
- sphere
- box
- triangle
- Each use different types of calculations
- e.g., sphere collision equations based on use of quadratic equation

Recursively finding triangles pseudocode

void FindPickTriangles(Node node, Ray ray, PickResults results)

{

if (ray intersects node.boundingVolume)

{

if (node is a leaf)

{

for each triangle of node do

{

if (ray intersects triangle)

add intersection information to results;

}

}

else

{

for each child of node do

DoPick(child, ray, results);

}

}

}

Scene Graphs & Picking

- What’s a node?
- represents a visual object in the game
- has a bounding volume
- may have child nodes
- If a ray intersects a node, it may intersect a child of that node
- If a ray does not intersect a node, it will not intersect any of its child nodes
- So what?
- this will greatly reduce the problem at hand

References

- [1] Real Time Collision Detection by Christer Ericson
- [2] Physics for Game Programmers by Squirrel Eiserloh
- [3] Collision Detection in Interactive 3D Environments by Gino van den Bergen

References

- [1] Real Time Collision Detection by Christer Ericson
- [2] Physics for Game Programmers by Squirrel Eiserloh
- [3] Collision Detection in Interactive 3D Environments by Gino van den Bergen

References

- Advanced Collision Detection Techniques
- http://www.gamasutra.com/features/20000330/bobic_01.htm
- N-Tutorials: Collision Detection & Response
- http://www.harveycartel.org/metanet/tutorials/tutorialA.html
- Physics in BSP Trees: Collision Detection
- http://www.devmaster.net/articles/bsp-trees-physics/
- 3D Game Engine Design by David H Eberly

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