Pathfinding and Movement

Michael Moll

September 29, 2004

A Character Has to Move

• Amount of processing time per cycle for AI is low• While movement is obviously a vital part of a

game, it’s not exactly the most interesting task performed

• With A* as an example, plenty of algorithms to find paths, but probably less considered, but more important is

How will we represent space we move through??

Overview

• Alternative Search Space Representations– Grids– Graphs– Meshes…

• Ok, we know what the world looks like, how can we move through it?– A*– Precomputed Pathfinding with Navigation Sets– Potential Fields

Search Space Considerations

• Scope of articles in book and presentation are 2-D worlds

• Main consideration for search spaces are memory requirements and if they facilitate fast searches (using whatever algorithm we choose)

• Since all of these are basically graphs, bigger worlds will require more nodes, more edges more memory, longer searches

Path Optimality

More Considerations

• Representation must take into account way an character moves (size, etc)

• How will we create the search space?

• Can they be automatically generated?

• Scripting paths should use different structure

Regular Grids

Regular Grids

• How do we generate?• Advantages

– Random access lookup (O(1)) to determine what tile lies at any coordinate

– Complete

• Negatives– Usually requires large number of nodes to accurately

represent world– Path Quality

- Agent can only walk in four cardinal directions? That’s no fun.

- Let them walk diagonals! Still not much fun

Regular Grids

NWSE Diagonals

String-PullingCatmull-Rom Spline

Grids as Graphs

• Everything else we look at will be a graph

• Grids are graphs too– Each cell is a node and edges are adjoining

cells

• Maybe we should just handle grid as a graph – Undirected graph could tell us about

topography, etc, whereas array can’t

Grids as Graphs

Corner Graphs

• Waypoints around obstacles

Corner Graphs

• How do we generate?– Identify convex corners, can character walk in straight line

between? Add edge if possible

• Advantages– Less memory– Faster generation

• Negatives – Character will “walk on a rail” hugging edges of obstacles

instead of walking through open space– And what about different sized characters?– Lookup is O(n2), have to check every node in graph against

every other. Imagine a world with a boat load or corners!

Corner Graphs

Corner Graphs

Waypoint Graphs

• Place nodes in middle of rooms instead of at convex corners

• How do we generate? – Place nodes wherever we want (suits 3-D worlds

But requires hand tuning to be effective )

• Advantages– Reduce memory footprint from regular grids, reduce

wall hugging from corner graphs– Work well in “human” architectures

• Negatives– Still O(n2)– Path Quality vs. Simplicity– Works poorly in open areas

Waypoint Graphs

Waypoint Graphs

Circle-Based Waypoint Graphs

• Add radius parameter to indicate open space near waypoint

Circle-Based Waypoint

• Advantages– Only look at overlapping circles, alleviating

O(n2) problem from before– Easier to obtain optimal paths– Works well in open terrain

• Negatives– Doesn’t work as well in human architectures

that aren’t circle friendly

Space-Filling Volumes

• Use rectangles or 3-D Boxes instead of circles

Space-Filling Volumes

• How do we generate?– Seed and Grow– Make Grid and Merge

• Very similar to circle-based, but handles angles better

Navigation Meshes

• Enough with the graphs already!

• Let’s try and cover walk able surfaces with convex polygons

• Character can travel between adjoining polygons

Navigation Meshes

Navigation Meshes

• How do we generate? – By hand (time consuming)

• Automated tools to analyze and optimize geometry of world

• Too complex and not represented as a single polygon mesh, instead may be overlapping, etc

Navigation Meshes

• Advantages– Quickly find optimal paths independent of

character shapes and capabilities– Handle indoor and outdoor terrains well

• Negatives– Can become complex and expensive memory

wise– Difficult to generate

Problem with N-Sided Meshes

Interacting with Pathfinding

• What about dynamic objects in world?

• All the representations discussed have been static and obviously can’t handle a dynamic world directly

• These representations need to be able to provide information to system determining path– Waypoint Graphs and Corner

Graphs don’t illustrate walk able surfaces

– Meshes and Grids do map every walk able surface

– Space-filling volumes and Circle Waypoints provide some representation of walk able areas

Further Options for Representation

• Any other ideas of what we can do to make job of path finding algorithm easier?

• Prof. Munoz has mentioned this before….

• Hierarchical Representations– Choose most suitable

scheme for given world, and break up into more manageable pieces

• Any other ideas of what we can do to make job of path finding algorithm easier?

• Prof. Munoz has mentioned this before….

• Hierarchical Representations– Choose most suitable

scheme for given world, and break up into more manageable pieces

Path finding

• Now that we have world represented, how do we plan movement?– A*, Depth-First, Dijkstra

• Dynamic path finding is expensive

– Precompiled solutions can eliminate runtime cost, but memory expensive

• Article suggests technique of Navigation Set Hierarchy

Transition Table• Main element in pre computed solutions is a

lookup table• Each entry represents next step to take from

one node to some goal node

Transition Table

Transition Table

• Do not need full search of nodes at run time, just series of lookups

• Fast, but becomes memory expensive as size of world grows– How expensive? n2

• …solution to shrink transition tables?

Hierarchy!

Navigation Set

• Self-contained collection of nodes that requires no links to external nodes to complete a path

• Nodes can be members of more than one set

• Goal: Find someway to partition large Navigation Sets into smaller ones

Complete Hierarchy

Interface Nodes and Sets

• Need to account for paths that cross navigation sets

• Any node that connects to a node in another navigation set is an interface node

• Have a second layer of nodes in addition to navigation sets, called interface set

• Interface set itself is a navigation set– Therefore, can make transition table for it too

Complete Hierarchy

• 21 nodes– 1 Navigation Set = 441 Table Entries (21*21)– 4 Navigation Sets = 183 Table Entries (7*7 + 7*7 + 7*7 + 6*6)

Constructing the Hierarchy

Two goals to process– How many tables to create?

• Amount of data needed is 1/n of original size + interface set• As size of navigation sets increase, cost of interface set

becomes less a factor

– Where to place boundaries?• Keep interface nodes as low as possible

Constructing the Hierarchy

Path finding

• Determine best paths leading from source node to boundary of source set

• Determine best path from source set boundary to goal set boundary

• Determine best path from goal set boundary to goal node

• Compile list of complete paths and choose one with least cost

Cost

• Amount of searching is limited to number of interface nodes in goal and source sets only

• Cost of searching between sets does not scale up with increases in navigation set size– Dependent on number of interface nodes

Applications of Navigation Set Hierarchy

• Interfacing heterogeneous navigation regions

• Navigation data on demand

• Extending beyond two tiers

Potential Fields

• Used by Scared Little Girl in Robocode

• Reactive approach to path finding– Setup virtual potential or force field around

objects– Must define field and agents reaction to field

• Interactions in general are explicitly stated and movement “emerges”– Paths are not planned explicitly

Potential Fields

• Think of world as matrix– Each point tells has value to represent

strength of field under it– Possibly assign potential based on distance

from goal• Might get stuck behind obstacle

– Result/Goal: “Glide down gradient/path of least resistance”

Potential Fields

• Advantages?– Works in continuous space! No need to make

grid, place waypoints, etc

• Disadvantages?– Can be trapped in local minima– It’s emergent, not sure what it will do

Commercial Implementations

• How advanced is Half-Life 2's path finding? Is it still based around triangulations, or is it more focused on waypoints, or both... or?

• Steve Bond: I guess the best answer to this question is "A character who wants to get somewhere will find a way to get there". Half-Life 2's path finding supports swimming, flying, jumping, climbing ladders, and opening doors.

Commercial Implementations

• Unreal Tournament– Waypoints with pre computed paths

• Navigation Points placed throughout world

– Assumes world is generally static• Run local path finding on top of global path finding• Local path finder constants queries physics engine

to find dynamic objects

Commercial Implementations

• Area Awareness System– Designed for Quake 3, used in Doom 3– Does not use waypoints, instead 3-D bounded

hulls, called areas• Cost of moving from one point to another within a

hull (reachable area) is minimal• Reachability can also be determined if a hull

touches another

– http://www.kbs.twi.tudelft.nl/docs/MSc/2001/Waveren_Jean-Paul_van/thesis.pdf

Commercial Implementations

• Half Life 2– Waypoint Based

• Call of Duty– Based off of Quake 3 AAS – Path finding handled by Conduit Engine

Sources• AI Game Programming Wisdom 2• H. Munoz-Avila & T. Fisher, Strategic Planning for Unreal

Tournament Bots.• First Person Shooter Architecture AI,

http://wiki.beyondunreal.com/wiki/First_Person_Shooter_AI_Architecture

• Bryan Stout, Smart Moves: Intelligent Pathfinding, http://www.gamasutra.com/features/19970801/pathfinding.htm

• JMP van Wavern, The Quake III Arena Bot, http://www.kbs.twi.tudelft.nl/docs/MSc/2001/Waveren_Jean-Paul_van/thesis.pdf

• Stefan Baert, Motion Planning Using Potential Fields, http://www.gamedev.net/reference/programming/features/motionplanning/

• John Spletzer, Lecture Notes from An Introduction to Mobile Robotics, 9/16/03