Artificial Intelligence Lab Finding “Narrow Passages” with Probabilistic Roadmaps: The Small-Step Retraction Method Mitul Saha and Jean-Claude Latombe Hello/Hi, My name is Mitul and I am from Stanford AI lab. Research supported by NSF, ABB and GM Artificial Intelligence Lab Stanford University

Probabilistic Roadmaps (PRM) milestone local path Roadmap components goal configuration start configuration We have the problem of finding a collision free paths between start and goal configurations. The configuration space is usually unknown as it would take exponential amount of time to build one. Instead we construct a roadmap to capture/represent/approximate the freespace of the robot. Then paths are found by query-ing the roadmap. c-obstacle free-space Configuration-space components [Kavraki, Svetska, Latombe, Overmars, 1996]

PRM planners solve complicated problems Environment with Complex Geometry and robots with large number of DOFs Can find a path from a fraction of seconds to few minutes IF c-space is FREE from narrow passages Without explicitly computing the configuration space of the robot Complex geometries: obstacles: 43530 polygons Robot: 4053 polygons High dimensional

Main Issue: “Narrow Passages” low density of free samples high density of free samples narrow passage free samples It is difficult to capture the free space associated with the narrow passages Because in the narrow passage, volume associated with the free space is relatively low. colliding samples colliding local path The efficiency of PRM planners drops dramatically in spaces with narrow passages

Main Issue: “Narrow Passages” Problems with “narrow passages” are commonly encountered Narrow passage problems are not fictitious!! Narrow passages are quite common in path planning problems. For example this is a very Common industrial problem where the robot must place its bulky end-effector or welding gun Deep inside the car through the opening in the door window. A typical collision free path would Go through configurations which have very low clearance between the window edges and the welding Gun. In industrial scenarios consideration of compactness during design of workcells gives rise to narrow Passages in the configuration space of the operating robots.

Main Issue: “Narrow Passages” ? Proposed strategies: Filtering strategies, e.g., Gaussian sampling [Boor et al. ‘99] and bridge test [Hsu et al. ‘03]  rely heavily on rejection sampling Retraction strategies, e.g., [Wilmart et al. ‘99][Lien et al. ‘03]  waste time moving many configurations out of collision Narrow passages pose a difficult challenging to PRM based motion planning. The intriguing nature of “Narrow passages” has attracted a lot of attention in the PRM based motion planning community. A lot of interesting techniques have been proposed. They can be classified into two techniques: Filtering and Retraction strategies. Filtering strategies consist of over-sampling c-space and retaining much fewer collision-free Confs which are more likely to be narrow passages. Due to heavy rejection sampling, they have a significant overhead in running time. Hence they can be inefficient for problems without Difficult narrow passages. We do not reject samples, instead we increase the density of samples in narrow passages By retracting barely colliding confs into narrow passages. Retraction based methods retract colliding configurations into the free space. We have observed that Attempting to retracting deeply embedded confs can make the planner inefficient. We only retract barely colliding samples. They are relatively easier to retract than deeply embedded colliding confs And significantly improve the density of free samples in the narrow passages.

Motivating Observation easy narrow passages difficult planning time We observed that slightly widening difficult narrow passages dramatically improves the efficiency of PRM planning. We also observe not much gain for easy narrow passages, but anyway we do not add significant overhead. decreasing width of the narrow passage

Small-Step Retraction Method free space c-obstacle start goal Roadmap construction and repair fattened free space widened passage Fattening (1) (2 & 3) Slightly fatten the robot’s free space Construct a roadmap in fattened free space Repair the roadmap into original free space

Small-Step Retraction Method free space c-obstacle start goal Roadmap construction and repair Fattening widened passage fattened free space Free space can be “indirectly” fattened by reducing the scale of the geometries (usually of the robot) in the 3D workcell with respect to their medial axis -This can be pushed into the pre-processing phase

Small-Step Retraction Method free space c-obstacle start goal fattened free space widened passage Roadmap construction and repair Fattening start Repair during construction Pessimist Strategy Optimist Strategy goal Repair after construction fattened free space

Small-Step Retraction Method free space c-obstacle start goal fattened free space widened passage Roadmap construction and repair Fattening start Optimist may fail due to “false passages” but Pessimist is probabilistically complete Hence Optimist is less reliable, but much faster due to its lazy strategy Repair during construction Pessimist Strategy Optimist Strategy goal Repair after construction fattened free space

Small-Step Retraction Method free space c-obstacle start goal fattened free space widened passage Roadmap construction and repair Fattening start Repair during construction Pessimist Strategy Integrated planner: 1. Try Optimist for N time. 2. If Optimist fails, then run Pessimist Optimist Strategy goal Repair after construction fattened free space

Upto 3 orders of magnitude improvement in the planning time Quantitative Results Fattening “preserves” topology/ connectivity of the free space Fattening “alters” the topology/ connectivity of the free space (h) (a) (b) (c) Alpha 1.1 (g) Alpha 1.0 (d) (f) (e) Time SSRP (secs) SBL (g) 386 572 (h) 3365 >100000 Time SSRP (secs) SBL (a) 9.4 12295 (b) 32 5955 (c) 2.1 41 (d) 492 863 (e) 65 631 (f) 13588 >100000 Our planner A recent PRM planner SBL: an efficient and recent single query planner All times are in seconds Gains upto 3 order or magnitudes seen Upto 3 orders of magnitude improvement in the planning time was observed

Quantitative Results Test environments “without” narrow passages SSRP and SBL have similar performance In test environments without narrow passages, SSRP is as efficient as SBL. This is also a significant achievement because previous planners designed for problems with narrow passages have significant overheads, which makes them inefficient for problems without narrow passages. Recall that we repair only barely colliding configurations and hence do not have significant overhead. (i) (j) Time SSRP SBL (i) 1.68 1.60 (j) 2.59 2.40

Conclusion SSRP is very efficient at finding narrow passages and still works well when there is none The main drawback is that there is an additional pre-computation step

Finding “Narrow Passages” with Probabilistic Roadmaps: The Small-Step Retraction Method Hello/Hi, My name is Mitul and I am from Stanford AI lab.