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Path Planning in Expansive C-Spaces D. Hsu J. C. Latombe R. Motwani presented by Niloy J. Mitra 1 Overview Problem statement Expansive Configuration Spaces Definition Probabilistic analysis Path planning algorithm using expansive configuration space 2 Issues with Path Planning Problem Path planning: hard problem Complete planner Exponential in the number degrees of the configuration space Probabilistic roadmap good for multiple path-planning query Probabilistic completeness Algorithm for the single-path query problem 3 Probabilistic Roadmaps 4 Coverage Bad Good 5 Coverage Bad Good almost any point of the configuration space can be connected by a straight line segment to some milestone 6 Connectivity Bad Good 7 Connectivity Bad Good 1-1 correspondence between the connected components of the roadmap and those of F 8 Narrow Passages Connectivity is difficult to capture when there are narrow passages. a narrow passage is difficult to define. Difficult Easy How to characterize coverage/connectivity? Expansiveness 9 Visibility All configurations in Free Space that can be seen by a free configuration p p 10 Є-good Every free configuration “sees” at least a є fraction of the free space, є is in (0,1]. 0.5-good 1-good F is 0.5-good 11 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]. 0.4-lookout of S 0.4-lookout of S S F\S S F\S This area is about 40% of F\S 12 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] S F\S F is ε-good ε=0.5 β-lookout β=0.4 Volume(β-lookout) α=0.2 Volume(S) F is (ε, α, β)-expansive, where ε=0.5, α=0.2, β=0.4. 13 Why Expansive? ε, α, and β measure the expansiveness of a free space. Bigger ε, α, and β Easier to construct a roadmap with good connectivity/coverage. 14 Linking sequence Lookout of V(p) Visibility of p p1 p2 p p3 q pn Pn+1 Pn+1 is chosen from the lookout of the subset seen by p, p1,…,pn 15 Size of lookout set BETTER Small lookout big lookout A C-space with larger lookout set has higher probability to construct a linking sequence with the same number of milestones. 16 Theorem 1 Probability of achieving good connectivity increases exponentially with the number of milestones (in an expansive space). As (ε, α, β) decrease the number of milestones needs to be increased (to maintain good connectivity). 17 Theorem 2 Probability of achieving good coverage, increases exponentially with the number of milestones (in an expansive space). 18 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] 19 Summary Main result If a C-space is expansive, then a roadmap can be constructed efficiently with good connectivity and coverage. Limitation in implementation No theoretical guidance about the stopping time. A planner stops when either a path is found or Max steps have been taken. 20 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 21 Algorithm: Expansion • Pick a node x with probability 1/w(x). • Randomly sample k points around x. • For each sample y, calculate w(y), which gives probability 1/w(y) Disk with radius d, w(x)=3 root 22 Algorithm: Expansion • Pick a node x with probability 1/w(x). • Randomly sample k points around x. • For each sample y, calculate w(y), which gives probability 1/w(y) 1/W(y1)=1/4 root 1 2 3 23 Algorithm: Expansion • Pick a node x with probability 1/w(x). • Randomly sample k points around x. • For each sample y, calculate w(y), which gives probability 1/w(y), if y 1. has bigger probability; 2. collision free; 3. can sees x then add y into the tree. root 1 2 3 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 YES, then connect x and y y Goal Init x 25 Algorithm: Terminate • The program iterates between Expansion/Connection, until two trees are connected, OR 2. max number of expansion/connection steps has been reached 1. Init Goal 26 Computed Example 27 Comments Estimate complexity as a function of geometric attributes Other geometric qualities or properties may be exploited Dimension of narrow passages 28