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- 1. 11/04/03 CS679 - Fall 2003 - Copyright Univ. of Wisconsin Last Time • AI – Fuzzy Logic – Neural networks
- 2. 11/04/03 CS679 - Fall 2003 - Copyright Univ. of Wisconsin Today • Path Planning – Intro – Waypoints – A* Path Planning
- 3. 11/04/03 CS679 - Fall 2003 - Copyright Univ. of Wisconsin Path Finding • Very common problem in games: – In FPS: How does the AI get from room to room? – In RTS: User clicks on units, tells them to go somewhere. How do they get there? How do they avoid each other? – Chase games, sports games, … • Very expensive part of games – Lots of techniques that offer quality, robustness, speed trade-offs
- 4. 11/04/03 CS679 - Fall 2003 - Copyright Univ. of Wisconsin Path Finding Problem • Problem Statement (Academic): Given a start point, A, and a goal point, B, find a path from A to B that is clear – Generally want to minimize a cost: distance, travel time, … • Travel time depends on terrain, for instance – May be complicated by dynamic changes: paths being blocked or removed – May be complicated by unknowns – don’t have complete information • Problem Statement (Games): Find a reasonable path that gets the object from A to B – Reasonable may not be optimal – not shortest, for instance – It may be OK to pass through things sometimes – It may be OK to make mistakes and have to backtrack
- 5. 11/04/03 CS679 - Fall 2003 - Copyright Univ. of Wisconsin Search or Optimization? • Path planning (also called route-finding) can be phrased as a search problem: – Find a path to the goal B that minimizes Cost(path) – There are a wealth of ways to solve search problems, and we will look at some of them • Path planning is also an optimization problem: – Minimize Cost(path) subject to the constraint that path joins A and B • State space is paths joining A and B, kind of messy – There are a wealth of ways to solve optimization problems • The difference is mainly one of terminology: different communities (AI vs. Optimization) – But, search is normally on a discrete state space
- 6. 11/04/03 CS679 - Fall 2003 - Copyright Univ. of Wisconsin Brief Overview of Techniques • Discrete algorithms: BFS, Greedy search, A*, … • Potential fields: – Put a “force field” around obstacles, and follow the “potential valleys” • Pre-computed plans with dynamic re-planning – Plan as search, but pre-compute answer and modify as required • Special algorithms for special cases: – E.g. Given a fixed start point, fast ways to find paths around polygonal obstacles
- 7. 11/04/03 CS679 - Fall 2003 - Copyright Univ. of Wisconsin Graph-Based Algorithms • Ideally, path planning is point to point (any point in the world to any other, through any unoccupied point) • But, the search space is complex (space of arbitrary curves) • The solution is to discretize the search space – Restrict the start and goal points to a finite set – Restrict the paths to be on lines (or other simple curves) that join points • Form a graph: Nodes are points, edges join nodes that can be reached along a single curve segment – Search for paths on the graph
- 8. 11/04/03 CS679 - Fall 2003 - Copyright Univ. of Wisconsin Waypoints (and Questions) • The discrete set of points you choose are called waypoints • Where do you put the waypoints? – There are many possibilities • How do you find out if there is a simple path between them? – Depends on what paths you are willing to accept - almost always assume straight lines • The answers to these questions depend very much on the type of game you are developing – The environment: open fields, enclosed rooms, etc… – The style of game: covert hunting, open warfare, friendly romp, …
- 9. 11/04/03 CS679 - Fall 2003 - Copyright Univ. of Wisconsin Where Would You Put Waypoints?
- 10. 11/04/03 CS679 - Fall 2003 - Copyright Univ. of Wisconsin Waypoints By Hand • Place waypoints by hand as part of level design – Best control, most time consuming • Many heuristics for good places: – In doorways, because characters have to go through doors and straight lines joining rooms always go through doors – Along walls, for characters seeking cover – At other discontinuities in the environments (edges of rivers, for example) – At corners, because shortest paths go through corners • The choice of waypoints can make the AI seem smarter
- 11. 11/04/03 CS679 - Fall 2003 - Copyright Univ. of Wisconsin Waypoints By Grid • Place a grid over the world, and put a waypoint at every gridpoint that is open – Automated method, and maybe even implicit in the environment • Do an edge/world intersection test to decide which waypoints should be joined – Normally only allow moves to immediate (and maybe corner) neighbors • What sorts of environments is this likely to be OK for? • What are its advantages? • What are its problems?
- 12. 11/04/03 CS679 - Fall 2003 - Copyright Univ. of Wisconsin Grid Example • Note that grid points pay no attention to the geometry • Method can be improved: – Perturb grid to move closer to obstacles – Adjust grid resolution – Use different methods for inside and outside building • Join with waypoints in doorways
- 13. 11/04/03 CS679 - Fall 2003 - Copyright Univ. of Wisconsin Waypoints From Polygons • Choose waypoints based on the floor polygons in your world • Or, explicitly design polygons to be used for generating waypoints • How do we go from polygons to waypoints? – Hint: there are two obvious options
- 14. 11/04/03 CS679 - Fall 2003 - Copyright Univ. of Wisconsin Waypoints From Polygons ! Could also add points on walls
- 15. 11/04/03 CS679 - Fall 2003 - Copyright Univ. of Wisconsin Waypoints From Corners • Place waypoints at every convex corner of the obstacles – Actually, place the point away from the corner according to how wide the moving objects are – Or, compute corners of offset polygons • Connects all the corners that can see each other • Paths through these waypoints will be the shortest • However, some unnatural paths may result – Particularly along corridors - characters will stick to walls
- 16. 11/04/03 CS679 - Fall 2003 - Copyright Univ. of Wisconsin Waypoints From Corners • NOTE: Not every edge is drawn • Produces very dense graphs
- 17. 11/04/03 CS679 - Fall 2003 - Copyright Univ. of Wisconsin Getting On and Off • Typically, you do not wish to restrict the character to the waypoints or the graph edges • When the character starts, find the closest waypoint and move to that first – Or, find the waypoint most in the direction you think you need to go – Or, try all of the potential starting waypoints and see which gives the shortest path • When the character reaches the closest waypoint to its goal, jump off and go straight to the goal point • Best option: Add a new, temporary waypoint at the precise start and goal point, and join it to nearby waypoints
- 18. 11/04/03 CS679 - Fall 2003 - Copyright Univ. of Wisconsin Getting On and Off
- 19. 11/04/03 CS679 - Fall 2003 - Copyright Univ. of Wisconsin Best-First-Search • Start at the start node and search outwards • Maintain two sets of nodes: – Open nodes are those we have reached but don’t know best path – Closed nodes that we know the best path to • Keep the open nodes sorted by cost • Repeat: Expand the “best” open node – If it’s the goal, we’re done – Move the “best” open node to the closed set – Add any nodes reachable from the “best” node to the open set • Unless already there or closed – Update the cost for any nodes reachable from the “best” node • New cost is min(old-cost, cost-through-best)
- 20. 11/04/03 CS679 - Fall 2003 - Copyright Univ. of Wisconsin Best-First-Search Properties • Precise properties depend on how “best” is defined • But in general: – Will always find the goal if it can be reached – Maintains a frontier of nodes on the open list, surrounding nodes on the closed list – Expands the best node on the frontier, hence expanding the frontier – Eventually, frontier will expand to contain the goal node • To store the best path: – Keep a pointer in each node n to the previous node along the best path to n – Update these as nodes are added to the open set and as nodes are expanded (whenever the cost changes) – To find path to goal, trace pointers back from goal nodes
- 21. 11/04/03 CS679 - Fall 2003 - Copyright Univ. of Wisconsin Expanding Frontier
- 22. 11/04/03 CS679 - Fall 2003 - Copyright Univ. of Wisconsin Definitions • g(n): The current known best cost for getting to a node from the start point – Can be computed based on the cost of traversing each edge along the current shortest path to n • h(n): The current estimate for how much more it will cost to get from a node to the goal – A heuristic: The exact value is unknown but this is your best guess – Some algorithms place conditions on this estimate • f(n): The current best estimate for the best path through a node: f(n)=g(n)+h(n)
- 23. 11/04/03 CS679 - Fall 2003 - Copyright Univ. of Wisconsin Using g(n) Only • Define “best” according to f(n)=g(n), the shortest known path from the start to the node • Equivalent to breadth first search • Is it optimal? – When the goal node is expanded, is it along the shortest path? • Is it efficient? – How many nodes does it explore? Many, few, …? • Behavior is the same as defining a constant heuristic function: h(n)=const – Why?
- 24. 11/04/03 CS679 - Fall 2003 - Copyright Univ. of Wisconsin Breadth First Search
- 25. 11/04/03 CS679 - Fall 2003 - Copyright Univ. of Wisconsin Breadth First Search • See Game Programming Gems for another example: – On a grid with uniform cost per edge, the frontier expands in a circle out from the start point – Makes sense: We’re only using info about distance from the start
- 26. 11/04/03 CS679 - Fall 2003 - Copyright Univ. of Wisconsin Using h(n) Only (Greedy Search) • Define “best” according to f(n)=h(n), the best guess from the node to the goal state – Behavior depends on choice of heuristic – Straight line distance is a good one • Have to set the cost for a node with no exit to be infinite – If we expand such a node, our guess of the cost was wrong – Do it when you try to expand such a node • Is it optimal? – When the goal node is expanded, is it along the shortest path? • Is it efficient? – How many nodes does it explore? Many, few, …?
- 27. 11/04/03 CS679 - Fall 2003 - Copyright Univ. of Wisconsin Greedy Search (Straight-Line-Distance Heuristic)
- 28. 11/04/03 CS679 - Fall 2003 - Copyright Univ. of Wisconsin A* Search • Set f(n)=g(n)+h(n) – Now we are expanding nodes according to best estimated total path cost • Is it optimal? – It depends on h(n) • Is it efficient? – It is the most efficient of any optimal algorithm that uses the same h(n) • A* is the ubiquitous algorithm for path planning in games – Much effort goes into making it fast, and making it produce pretty looking paths – More articles on it than you can ever hope to read
- 29. 11/04/03 CS679 - Fall 2003 - Copyright Univ. of Wisconsin A* Search (Straight-Line-Distance Heuristic)
- 30. 11/04/03 CS679 - Fall 2003 - Copyright Univ. of Wisconsin A* Search (Straight-Line-Distance Heuristic) • Note that A* expands fewer nodes than breadth-first, but more than greedy • It’s the price you pay for optimality • See Game Programming Gems for implementation details. Keys are: – Data structure for a node – Priority queue for sorting open nodes – Underlying graph structure for finding neighbors
- 31. 11/04/03 CS679 - Fall 2003 - Copyright Univ. of Wisconsin Heuristics • For A* to be optimal, the heuristic must underestimate the true cost – Such a heuristic is admissible • Also, not mentioned in Gems, the f(n) function must monotonically increase along any path out of the start node – True for almost any admissible heuristic, related to triangle inequality – If not true, can fix by making cost through a node max(f(parent) + edge, f(n)) • Combining heuristics: – If you have more than one heuristic, all of which underestimate, but which give different estimates, can combine with: h(n)=max(h1(n),h2(n),h3(n),…)
- 32. 11/04/03 CS679 - Fall 2003 - Copyright Univ. of Wisconsin Inventing Heuristics • Bigger estimates are always better than smaller ones – They are closer to the “true” value – So straight line distance is better than a small constant • Important case: Motion on a grid – If diagonal steps are not allowed, use Manhattan distance • General strategy: Relax the constraints on the problem – For example: Normal path planning says avoid obstacles – Relax by assuming you can go through obstacles – Result is straight line distance is a bigger estimate than
- 33. 11/04/03 CS679 - Fall 2003 - Copyright Univ. of Wisconsin Non-Optimal A* • Can use heuristics that are not admissible - A* will still give an answer – But it won’t be optimal: May not explore a node on the optimal path because its estimated cost is too high – Optimal A* will eventually explore any such node before it reaches the goal • Non-admissible heuristics may be much faster – Trade-off computational efficiency for path-efficiency • One way to make non-admissible: Multiply underestimate by a constant factor – See Gems for an example of this
- 34. 11/04/03 CS679 - Fall 2003 - Copyright Univ. of Wisconsin Project Stage 4 • AI – Make your world interact with the player – Use any technique you want – that’s part of the challenge
- 35. 11/04/03 CS679 - Fall 2003 - Copyright Univ. of Wisconsin Todo • Thurs Nov 6, Midterm, in class – Everything up to and including lecture 15 – Sample exams online, but not all material appears in those exams • By Monday, Nov 10, Stage 4 goals • By Monday, Nov 24, Stage 4 demo • By Monday, Dec 1, Final goals • By Wednesday, Dec 10, Final Demo