1 Path Planning in Expansive C-Spaces D. HsuJ. –C. LatombeR. Motwani Prepared for CS326A, Spring 2003 By Xiaoshan (Shan) Pan

2 Structure of the Paper  1 st section Theoretical analysis of expansiveness  2 nd section ( will not be covered ) A new PRM planner

3 Issues of probabilistic roadmaps  Coverage  Connectivity

4 Is the coverage adequate?  It means that milestones are distributed such that almost any point of the configuration space can be connected by a straight line segment to one milestone BadGood

5 Connectivity  There should be a one-to-one correspondence between the connected components of the roadmap and those of F

6 Narrow Passage  Connectivity is difficult to capture when there are narrow passages. Characterize coverage & connectivity?  Expansiveness  a narrow passage is difficult to define.Easy Difficult

7 Notion: Visibility  All configurations in Free Space that can be seen by a free configuration p p

8 Notion: Є-good  Every free configuration “sees” at least a є fraction of the free space, є is in (0,1]. 0.5-good1-good F is 0.5-good

9 Notion: β-lookout of a subspace S  Subset of points in S that can see a β fraction of F\S, β is in (0,1]. S F\S 0.4-lookout of S This area is about 40% of F\S S F\S 0.4-lookout of S

10 S F\S F is ε-good  ε=0.5 Definition: (ε,α,β)-expansive  The free space F is (ε,α,β)-expansive if Free space F is ε-good For each subspace S of F, its β-lookout is at least α fraction of S. ε, α, β are in (0,1] β-lookout  β=0.4 Volume(β-lookout) Volume(S)  α=0.2 F is (ε, α, β)-expansive, where ε=0.5, α=0.2, β=0.4.

11 Why expansive?  ε, α, and β measure the expansiveness of a free space.  Bigger ε, α, and β  less cost of constructing a roadmap with good connectivity/coverage.

12 p3p3 pnpn P n+1 q p2p2 Definition: Linking sequence p p1p1 P n+1 is chosen from the lookout of the subset seen by p, p 1,…,p n Visibility of p Lookout of V(p)

13 Size of lookout set A C-space with larger lookout set has higher probability to construct a linking sequence with the same number of milestones. Small lookoutbig lookout

14 Theorem 1  Probability of achieving good connectivity increases exponentially with the number of milestones (in an expansive space).  If (ε, α, β) decreases  then need to increase a great amount of milestones (to maintain good connectivity)

15 Theorem 2  Probability of achieving good coverage, increases exponentially with the number of milestones (in an expansive space).

16 Probabilistic Completeness In an expansive space, the probability that a PRM planner fails to find a path when one exists goes to 0 exponentially in the number of milestones (~ running time). [Hsu, Latombe, Motwani, 97]

17 Summary  Main result If a C-space is expansive, then a roadmap can be constructed efficiently with good connectivity and coverage.  Limitation in implementation It does not tell you when to stop growing the roadmap. A planner stops when either a path is found or Max steps are reached.

18 Basic idea of the planner 1. Grow two trees from Init position and Goal position. 2. Randomly sample nodes around existing nodes. 3. Find linking sequence between Init and Goal by Incorporating new nodes that see a large portion of the free space  i.e., those that are likely in the lookout set Init Goal Expansion + Connection

19 Pick a node x with probability 1/w(x). Disk with radius d, w(x)=3 Algorithm: expansion root Randomly sample k points around x. For each sample y, calculate w(y), which gives probability 1/w(y)

20 Pick a node x with probability 1/w(x). Algorithm: expansion root Randomly sample k points around x. For each sample y, calculate w(y), which gives probability 1/w(y) /W(y1)=1/4

21 Pick a node x with probability 1/w(x). Algorithm: expansion root Randomly sample k points around x. For each sample y, calculate w(y), which gives probability 1/w(y) /W(y2)=1/1

22 Pick a node x with probability 1/w(x). Algorithm: expansion root Randomly sample k points around x. For each sample y, calculate w(y), which gives probability 1/w(y) /W(y3)=1/2

23 Pick a node x with probability 1/w(x). Algorithm: expansion root Randomly sample k points around x. For each sample y, calculate w(y), which gives probability 1/w(y), if y has bigger probability; 2. collision free; 3. can sees x then add y into the tree.

24 Algorithm: connection If a pair of nodes (i.e., x in Init tree and y in Goal tree)  distance(x,y)<L, check if x can see y Init Goal YES, then connect x and y x y

25 Algorithm: terminate condition The program iterates between Expansion and Connection, until 1. two trees are connected, or 2. max number of expansion & connection steps is reached Init Goal

26 Computed example

27 Future research Accelerate the planner by automatically generating intermediate configurations to decompose the free space into expansive components.

28 Future research Accelerate the planner by automatically generating intermediate configurations to decompose the free space into expansive components. Use geometric transformations to increase the expansiveness of a free space. E.g., widening narrow passages.

29 Future research Accelerate the planner by automatically generating intermediate configurations to decompose the free space into expansive components. Use geometric transformations to increase the expansiveness of a free space. E.g., widening narrow passages. Integrate the new planner with other planner for multiple-query path planning problems. Questions?