1. Workspace-based Connectivity Oracle An Adaptive Sampling Strategy for PRM Planning Hanna Kurniawati and David Hsu Presented by Nicolas Lee and Stephen Russell

2. Outline Introduction/Motivation WCO Planner Constructing a component sampler Ensemble sampler Results

3. Introduction Standard Probabilistic Road Map (PRM) –Two phases: construction and query –Construction creates map, R, that tries to accurately model connectivity of C –Query tries to connect start/goal locations to R

4. Motivation Performance depends on quality of R –Coverage and connectivity Algorithm struggles with narrow passages in C Other sampling strategies: –Dynamic: Machine learning/adaptive hybrid –Workspace information: Identifying important regions in W e.g. Workspace Importance Sampling (WIS) focuses on regions with small local feature size

5. WCO Foundations Proposition: If two configurations q, q’ є C are connected by a path in F c, then for any point f in a robot, P f (q) and P f (q’), the projections of q and q’ in W, are connected by a path in F w

6. WCO Distinct components of R may in fact lie in the same connected component of F c Examine workspace paths for multiple feature points and construct sampler for each f Search for channels in W and adapt distribution to sample more densely in regions covered by these channels

7. Workspace Connectivity Decomposition T of F w into non- overlapping cells –Create adjacency grid G T of T Consider two milestones, m and m’, and projections onto W, P f (m) є t and P f (m’) є t’ Find workspace channel, λ : set of nodes in G T connecting t and t’ L f ( λ) suggests a region of F c for sampling

8. Example (a)Milestones projected to decomposed workspace (b)Adjacency graph G T (c)Channel graph G’

9. Component Sampler Algorithm 1.Given f, sample configuration q based on sampling distribution over T 2.If q is collision free, then 3.Insert q into R as new milestone m 4. N m, set of neighbors 5.for each m’ є N m do 6.if m є R i and m’ є R j, then 7.connect if possible 8.Project m to W 9.Update label sets for affected T 10.Delete paths in G’ connecting terminals with same label set 11.Let t є T containing P f (m). Perform breadth-first search and stop when reaching first terminal t’ ≠ t 12.Add path from t→t’ to G’ if they have different label sets 13.Update the sampling distribution

10. Ensemble Sampler Algorithm 1.Initialize p i = 1/K for i = 0, 1, …, K-1 2.for t = 1, 2, … do 3.Pick a component sampler s i with probability p i 4.Sample a new configuration q using the component sampler picked 5.If a new milestone m is added to the roadmap R then 6.Update the distribution for each component sampler s i 7.Update the probabilities p i

11. Probability Update Ensemble sampler performs almost as well as the best component sampler Kinematic constraints taken into account through higher probability in overlapping lifted channels

12. Choosing Feature Points Must be representative of the robot Use vertices of convex hull and centroid for each rigid link of a robot

13. Test Configurations

14. Comparison With Other Samplers WCO has better sampling in channel regions without too many samples elsewhere In many cases, run time is cut in half compared to the best of the other three samplers

15. Limitations - 2 Bars Example

16. Conclusion WCO is an adaptive sampling strategy for PRM planning Using AHS, combine information from workspace geometry and sampling history In trials, WCO outperformed strategies which only use workspace information OR dynamic sampling