1. 10 Chapter 10: Metric Path Planning a. Representations b. Algorithms

2. 10 Introduction to AI Robotics (MIT Press), copyright Robin Murphy 2000 Chapter 10: Metric Path Planning2 Objectives Define Cspace, path relaxation, digitization bias, subgoal obsession, termination condition Explain the difference between graph and wavefront planners Represent an indoor environment with a GVG, a regular grid, or a quadtree, and create a graph suitable for path planning Apply A* search Apply wavefront propagation Explain the differences between continuous and event-driven replanning

3. 10 Introduction to AI Robotics (MIT Press), copyright Robin Murphy 2000 Chapter 10: Metric Path Planning3 Navigation Where am I going? Mission planning What’s the best way there? Path planning Where have I been? Map making Where am I? Localization Mission Planner Carto- grapher Behaviors deliberative reactive How am I going to get there?

4. 10 Introduction to AI Robotics (MIT Press), copyright Robin Murphy 2000 Chapter 10: Metric Path Planning4 Spatial Memory What’s the Best Way There? depends on the representation of the world A robot’s world representation and how it is maintained over time is its spatial memory –Attention –Reasoning –Path planning –Information collection Two forms –Route (or qualitative) –Layout (or metric) Layout leads to Route, but not the other way

5. 10 Introduction to AI Robotics (MIT Press), copyright Robin Murphy 2000 Chapter 10: Metric Path Planning5 Representing Area/Volume in Path Planning In Quantitative or metric maps: –Rep: Many different ways to represent an area or volume of space Looks like a “bird’s eye” view, position & viewpoint independent –Algorithms Graph or network algorithms Wavefront or graphics-derived algorithms

6. 10 Introduction to AI Robotics (MIT Press), copyright Robin Murphy 2000 Chapter 10: Metric Path Planning6 Metric Maps Motivation for having a metric map is often path planning (others include reasoning about space…) Determine a path from one point to goal –Generally interested in “best” or “optimal” –What are measures of best/optimal? –Relevant: occupied or empty Path planning assumes an a priori map of relevant aspects –Only as good as last time map was updated

7. 10 Introduction to AI Robotics (MIT Press), copyright Robin Murphy 2000 Chapter 10: Metric Path Planning7 Metric Maps use Cspace World Space: physical space robots and obstacles exist in –In order to use, generally need to know (x,y,z) plus Euler angles: 6DOF Ex. Travel by car, what to do next depends on where you are and what direction you’re currently heading Configuration Space (Cspace) –Transform space into a representation suitable for robots, simplifying assumptions 6DOF3DOF

8. 10 Introduction to AI Robotics (MIT Press), copyright Robin Murphy 2000 Chapter 10: Metric Path Planning8 Major Cspace Representations Idea: reduce physical space to a cspace representation which is more amenable for storage in computers and for rapid execution of algorithms Major types –Meadow Maps –Generalized Voronoi Graphs (GVG) –Regular grids, quadtrees

9. 10 Introduction to AI Robotics (MIT Press), copyright Robin Murphy 2000 Chapter 10: Metric Path Planning9 Meadow Maps Example of the basic procedure of transforming world space to cspace Step 1 (optional): grow obstacles as big as robot

10. 10 Introduction to AI Robotics (MIT Press), copyright Robin Murphy 2000 Chapter 10: Metric Path Planning10 Meadow Maps cont. Step 2: Construct convex polygons as line segments between pairs of corners, edges –Why convex polygons? Interior has no obstacles so can safely transit (“freeway”, “free space”) –Oops, not necessarily unique set of polygons

11. 10 Introduction to AI Robotics (MIT Press), copyright Robin Murphy 2000 Chapter 10: Metric Path Planning11 Meadow Maps cont. Step 3: represent convex polygons in way suitable for path planning-convert to a relational graph –Is this less storage, data points than a pixel-by-pixel representation?

12. 10 Introduction to AI Robotics (MIT Press), copyright Robin Murphy 2000 Chapter 10: Metric Path Planning12 Problems with Meadow Maps Not unique generation of polygons Could you actually create this type of map with sensor data? How does it tie into actually navigating the path? –How does robot recognize “right” corners, edges and go to “middle”? –What about sensor noise?

13. 10 Introduction to AI Robotics (MIT Press), copyright Robin Murphy 2000 Chapter 10: Metric Path Planning13 Path Relaxation Get the kinks out of the path –Can be used with any cspace representation

14. 10 Introduction to AI Robotics (MIT Press), copyright Robin Murphy 2000 Chapter 10: Metric Path Planning14 Class Exercise Create a meadow map and relational graph using mid- point of line segments

15. 10 Introduction to AI Robotics (MIT Press), copyright Robin Murphy 2000 Chapter 10: Metric Path Planning15 Generalized Voronoi Graphs Imagine a fire starting at the boundaries, creating a line where they intersect, intersections of lines are nodes Result is a relational graph

16. 10 Introduction to AI Robotics (MIT Press), copyright Robin Murphy 2000 Chapter 10: Metric Path Planning16 Class Exercise Create a GVG of this space

17. 10 Introduction to AI Robotics (MIT Press), copyright Robin Murphy 2000 Chapter 10: Metric Path Planning17 Regular Grids Bigger than pixels, but same idea –Often on order of 4inches square –Make a relational graph by each element as a node, connecting neighbors (4-connected, 8-connected) –Moore’s law effect: fast processors, cheap hard drives, who cares about overhead anymore?

18. 10 Introduction to AI Robotics (MIT Press), copyright Robin Murphy 2000 Chapter 10: Metric Path Planning18 Class Exercise Use 4-connected neighbors to create a relational graph

19. 10 Introduction to AI Robotics (MIT Press), copyright Robin Murphy 2000 Chapter 10: Metric Path Planning19 Problems with GVG and Regular Grids GVG –Sensitive to sensor noise –Path execution: requires robot to be able to sense boundaries Grids –World doesn’t always line up on grids –Digitalization bias: left over space marked as occupied

20. 10 Introduction to AI Robotics (MIT Press), copyright Robin Murphy 2000 Chapter 10: Metric Path Planning20 Summary of Representations Metric path planning requires –Representation of world space, usually try to simplify to cspace –Algorithms which can operate over representation to produce best/optimal path Representation –Usually try to end up with relational graph –Regular grids are currently most popular in practice, GVGs are interesting –Tricks of the trade Grow obstacles to size of robot to be able to treat holonomic robots as point Relaxation (string tightening) Metric methods often ignore issue of –how to execute a planned path –Impact of sensor noise or uncertainty, localization

21. 10 Introduction to AI Robotics (MIT Press), copyright Robin Murphy 2000 Chapter 10: Metric Path Planning21 Algorithms For Path planning –A* for relational graphs –Wavefront for operating directly on regular grids For Interleaving Path Planning and Execution

22. 10 Introduction to AI Robotics (MIT Press), copyright Robin Murphy 2000 Chapter 10: Metric Path Planning22 Motivation for A* Single Source Shortest Path algorithms are exhaustive, visiting all edges –Can’t we throw away paths when we see that they aren’t going to the goal, rather than follow all branches? This means having a mechanism to “prune” branches as we go, rather than after full exploration Algorithms which prune earlier (but correctly) are preferred over algorithms which do it later. Issue -> the mechanism for pruning

23. 10 Introduction to AI Robotics (MIT Press), copyright Robin Murphy 2000 Chapter 10: Metric Path Planning23 A* Similar to breadth-first: at each point in the time the planner can only “see” it’s node and 1 set of nodes “in front” Idea is to rate the choices, choose the best one first, throw away any choices whenever you can f*(n) is the “goodness” of the path from Start to n g*(n) is the “cost” of going from the Start to node n H*(n) is the cost of going from n to the Goal –H is for “heuristic function” because must have a way of guessing the cost of n to Goal since can’t see the path between n and the Goal f*(n)=g*(n)+h*(n)

24. 10 Introduction to AI Robotics (MIT Press), copyright Robin Murphy 2000 Chapter 10: Metric Path Planning24 A* Heuristic Function G*(n) is easy: just sum up the path costs to n H*(n) is tricky –But path planning requires an a priori map –Metric path planning requires a METRIC a priori map –Therefore, know the distance between Initial and Goal nodes, just not the optimal way to get there –H*(n)= distance between n and Goal f*(n)=g*(n)+h*(n)

25. 10 Introduction to AI Robotics (MIT Press), copyright Robin Murphy 2000 Chapter 10: Metric Path Planning25 Example: A to E But since you’re starting at A and can only look 1 node ahead, this is what you see: AB D FE 1 1 1 1 1.4 AB D E 1

26. 10 Introduction to AI Robotics (MIT Press), copyright Robin Murphy 2000 Chapter 10: Metric Path Planning26 Two choices for n: B, D Do both –f*(B)=1+2.24=3.24 –f*(D)=1.4+1.4=2.8 Can’t prune, so much keep going (recurse) –Pick the most plausible path first => A-D-?-E AB D E 1 1.4 2.24

27. 10 Introduction to AI Robotics (MIT Press), copyright Robin Murphy 2000 Chapter 10: Metric Path Planning27 A-D-?-E –“stand on D” –Can see 2 new nodes: F, E –f*(F)=(1.4+1)+1=3.4 –f*(E)=(1.4+1.4)+0=2.8 Three paths –A-B-?-E >= 3.24 –A-D-E = 2.8 –A-D-F-?-D >=3.4 A-D-E is the winner! –Don’t have to look farther because expanded the shortest first, others couldn’t possibly do better without having negative distances, violations of laws of geometry… AB D E 1 1.4 F 1 1

28. 10 Introduction to AI Robotics (MIT Press), copyright Robin Murphy 2000 Chapter 10: Metric Path Planning28 Class Exercise Find the best path from A to E D FE 1 1 1 1 1.4 AB C

29. 10 Introduction to AI Robotics (MIT Press), copyright Robin Murphy 2000 Chapter 10: Metric Path Planning29 Wavefront Planners

30. 10 Introduction to AI Robotics (MIT Press), copyright Robin Murphy 2000 Chapter 10: Metric Path Planning30 Trulla

31. 10 Introduction to AI Robotics (MIT Press), copyright Robin Murphy 2000 Chapter 10: Metric Path Planning31 Interleaving Path Planning and Reactive Execution Graph-based planners generate a path and subpaths or subsegments Recall NHC, AuRA –Pilot looks at current subpath, instantiates behaviors to get from current location to subgoal What happens if blocked? What happens if avoid an obstacle and actually is now closer to the next subgoal? –Causes Subgoal obsession When does the robot think it has reached (x,y)? What about encoder error? –Need a Termination condition, usually +/- width of robot

32. 10 Introduction to AI Robotics (MIT Press), copyright Robin Murphy 2000 Chapter 10: Metric Path Planning32 Trulla Example

33. 10 Introduction to AI Robotics (MIT Press), copyright Robin Murphy 2000 Chapter 10: Metric Path Planning33 Two Example Approaches If compute all possible paths in advance, not a problem –Shortest path between pairs will be part of shortest path to more distant pairs D* –Tony Stentz –Run A* over all possible pairs of nodes Trulla –Ken Hughes –By-product of wave propagation style is path to everywhere

34. 10 Introduction to AI Robotics (MIT Press), copyright Robin Murphy 2000 Chapter 10: Metric Path Planning34 Summary Metric path planners –graph-based (A* is best known) –Wavefront Graph-based generate paths and subgoals. –Good for NHC styles of control –In practice leads to: Subgoal obsession Termination conditions Planning all possible paths helps with subgoal obsession –What happens when the map is wrong, things change, missed opportunities? How can you tell when the map is wrong or that’s it worth the computation?