An Efficient Motion Planner Based on Random Sampling Jean-Claude Latombe Computer Science Department Stanford University

Main Collaborators u Lydia Kavraki (Rice U.) u David Hsu (U. of North Carolina, Chapel Hill) u Gildardo Sanchez (ITESM, Mexico) u James Kuffner (U. of Tokyo) u Rajeev Motwani (Stanford U.)

Goal of Motion Planning Answer queries about the connectivity of a space

Possible Constraints  Collision-free u Kino-dynamic u Stability u Visibility

The Beginning … Shakey (Nilsson, 1969): Visibility graph

Configuration Space Represent the robot as a point in a parameter space

Why Sampling-Based Planning? u Computing an explicit representation of the collision-free space is extremely time consuming and impractical u There exist fast collision-checking algorithms to test whether any given configuration or short path is collision-free, or not (0.001 sec or less)

Out line u General Approach u Specific Planner u Experimental Results u Other Applications

Probabilistic Roadmap (PRM) admissible space mbmbmbmb mgmgmgmg milestone [Kavraki, Svetska, Latombe,Overmars, 95]

Relation to Art-Gallery Problems [Kavraki, Latombe, Motwani, Raghavan, 95]

Narrow Passage Issue Easy Difficult

Probabilistic Completeness Under generally satisfied assumptions, if a solution path exists, the probability that a PRM planner fails to find one goes to 0 exponentially in the number of milestones. Full completeness  Too costly Heuristic  Too unreliable Probabilistic completeness  Fast and reliable

Key Techniques u Collision checking / Distance computation u Sampling strategies

Key Techniques u Collision checking / Distance computation  Hierarchical approach  Feature-based approach u Sampling strategies

Hierarchical Collision Checking

Three-Dimensional Case

Collision Checking

Performance u Collision checking takes between and.002 seconds for 2 objects of 500,000 triangles each on a 1-GHz Pentium III u Collision checking is faster when objects collide or are far apart, and gets slower when they get closer without colliding u Overall collision checking time grows roughly as the log of the number of triangles

Key Techniques u Collision checking / Distance computation u Sampling strategies  Multi-stage strategies  Obstacle-sensitive strategies  Multiple vs. single query strategies  Configuration vs. control sampling  Single vs. bi-directional sampling  Lazy collision checking  Probabilistic biases (e.g., medial axis transform)

Outline u General Approach u Specific Planner u Experimental Results u Other Applications

SBL Planner u S ingle-query Does not pre-compute a roadmap [Hsu, Latombe, Motwani, 1997] u B i-directional sampling Constructs a roadmap by growing two trees of milestones rooted at the input query configuration [Hsu, 2000] u L azy collision checking Postpone collision-checking operations until absolutely needed [Bohlin and Kavraki, 2000]

SBL Planner

m m is picked at random among the milestones with a probabilistic distribution inverse to the local density of sampling

SBL Planner

X

The collision-checking work is memorized

Why Postponing Collision Checking? u The a priori probability that a short edge be collision-free is rather large

Why Postponing Collision Checking? u The a priori probability that a short edge be collision-free is rather large u The test of an edge is most expensive when it is actually collision-free u Most edges of a roadmap do not end up in a solution path

Path Optimization u Problems –too few vertices: get stuck –too many vertices: slow u Remedy –remove as many vertices as possible –add vertices as needed

Outline u General Approach u Specific Planner u Experimental Results u Other Applications

Single-Robot Examples n rob = 5,000 and n obs = 21,000 n rob = 5,000; n obs = 83,000 n rob = 3,000 and n obs = 50,000 n rob = 3,000 and n obs = 100 n rob = 3,000; n obs = 50

Videos n robot =5,000; n obst = 21,000 T av = 0.6 s

Videos n robot =5,000; n obst = 83,000 T av = 4.42 s n robot =3,000; n obst = 50,000 T av = 0.17 s

Videos n robot =3,000; n obst = 50,000 T av = 4.45 s n robot =3,000; n obst = 100 T av = 6.99 s

Experimental Data on One Example (1 GHz Pentium III processor) n rob = 5,000 n obs = 21,000

Average Performance 1a 1b 1c 1d 1e (1GHz Pentium III processor) Averages over 100 runs

Convergence of SBL

Impact of Lazy Collision Checking Average performance with lazy collision checking Average performance without lazy collision checking

Multi-Robot Spot Welding

Typical Problem

Video

Average Running Times (1 GHz processor)

Centralized vs. Decoupled Planning Averages over 20 runs

Outline u General Approach u Specific Planner u Experimental Results u Other Applications

Design for Manufacturing/Servicing General Electric General Motors [Hsu, 2000]

Radio-Surgical Planning Cyberknife System (Accuray, Inc.) CARABEAMER Planner [Tombropoulos, Adler, and Latombe, 1997] Visibility constraints

Radio-Surgical Planning 2000 < Tumor < < B2 + B4 < < B4 < < B3 + B4 < < B3 < < B1 + B3 + B4 < < B1 + B4 < < B1 + B2 + B4 < < B1 < < B1 + B2 < < Critical < < B2 < 500 T C B1 B2 B3 B4 T

Radio-Surgical Planning 50% Isodose Surface 80% Isodose Surface Conventional system’s plan CARABEAMER’s plan

Contact Stanford Report Contact Stanford Report News Servic e News Servic e /Press Releas esPress Releas es Stanford Report, July 25, 2001 Patients gather to praise minimally invasive technique used in treating tumors By MICHELLE BRANDT When Jeanie Schmidt, a critical care nurse from Foster City, lost hearing in her left ear and experienced numbing in her face, she prayed that her first instincts were off. “I said to the doctor, `I think I have an acoustic neuroma (a brain tumor), but I'm hoping I'm wrong. Tell me it's wax, tell me it's anything,'” Schmidt recalled. It wasn't wax, however, and Schmidt – who wound up in the Stanford Hospital emergency room when her symptoms worsened – was quickly forced to make a decision regarding treatment for her tumor. On July 13, Schmidt found herself back at Stanford – but this time with a group of patients who were treated with the same minimally invasive treatment that Schmidt ultimately chose: the CyberKnife. She was one of 40 former patients who met with Stanford faculty and staff to discuss their experiences with the CyberKnife – a radiosurgery system designed at Stanford by John Adler Jr., MD, in 1994 for performing neurosurgeries without incisions. “I wanted the chance to thank everyone again and to share experiences with other patients,” said Schmidt, who had the procedure on June 20 and will have an MRI in six months to determine its effectiveness. “I feel really lucky that I came along when this technology was around.” The CyberKnife is the newest member of the radiosurgery family. Like its ancestor, the 33-year-old Gamma Knife, the CyberKnife uses 3-D computer targeting to deliver a single, large dose of radiation to the tumor in an outpatient setting. But unlike the Gamma Knife – which requires patients to wear an external frame to keep their head completely immobile during the procedure – the CyberKnife can make real-time adjustments to body movements so that patients aren't required to wear the bulky, uncomfortable head gear. The procedure provides patients an alternative to both difficult, risky surgery and conventional radiation therapy, in which small doses of radiation are delivered each day to a large area. The procedure is used to treat a variety of conditions – including several that can't be treated by any other procedure – but is most commonly used for metastases (the most common type of brain tumor in adults), meningomas (tumors that develop from the membranes that cover the brain), and acoustic neuromas. Since January 1999, more than 335 patients have been treated at Stanford with the CyberKnife. Cyberknife Systems

Modular Reconfigurable Robots Xerox, Parc Casal and Yim, 1999

Humanoid Robot [Kuffner and Inoue, 2000] (U. Tokyo) Stability constraints

Space Robotics air bearing gas tank air thrusters obstacles robot [Kindel, Hsu, Latombe, and Rock, 2000] Dynamic constraints

Total duration : 40 sec

Autonomous Helicopter [Feron, 2000] (AA Dept., MIT)

Interacting Nonholonomic Robots y1y1y1y1 x2x2x2x2 d x1x1x1x1 y2y2y2y2   (Grasp Lab - U. Penn)

Map Building [Gonzalez, 2000]

Next-Best View Computation

Map Building [Gonzalez, 2000]

Map Building [Gonzalez, 2000]

Graphic Animation of Digital Actors [Koga, Kondo, Kuffner, and Latombe, 1994] The Motion Factory

Prediction of Molecular Motions [Singh, Latombe, and Brutlag, 1999] Ligand-protein binding

Outline u General Approach u Specific Planner u Experimental Results u Other Applications u Conclusion

Conclusion u Probabilistic Roadmaps provide an efficient and reliable computational approach to motion planning u PRM planners are rather easy to implement u They have been experimented on very different problems

Remaining Issues u Relatively large standard deviation of planning time u No rigorous termination criterion when no solution is found u New challenging applications …

Optimal Touring of Multiple Goals

Surgical Planning with Soft Tissue

Planning Nice-Looking Motions A Bug’s Life (Pixar/Disney) Toy Story (Pixar/Disney) Tomb Raider 3 (Eidos Interactive)Final Fantasy VIII (SquareOne)The Legend of Zelda (Nintendo) Antz (Dreamworks)

1,000s of Degrees of Freedom Protein folding