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Robust Combination of Local Controllers

Robust Combination of Local Controllers. Carlos Guestrin Dirk Ormoneit Stanford University. Planning. Planning is central in real-world systems; However, planning is hard: Motion planning is PSPACE-hard [Reif 79]; State and Action spaces are often continuous; Uncertainty is ubiquitous:

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Robust Combination of Local Controllers

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  1. Robust Combination of Local Controllers Carlos Guestrin Dirk Ormoneit Stanford University

  2. Planning • Planning is central in real-world systems; • However, planning is hard: • Motion planning is PSPACE-hard [Reif 79]; • State and Action spaces are often continuous; • Uncertainty is ubiquitous: • Imprecise actuators; • Noisy sensors.

  3. Global versus Local Controllers • Designing a global controller is hard, but… • Many real-world domains allow us to design good local controllers with no global guarantees: How can we combine local controllers to obtain a global solution ?

  4. Combining Local Controllers • Randomized algorithm: • Nonparametric combination of local controllers; • Generalizes probabilistic roadmaps: [Hsu et al.99] • stochastic domains; • Discounted MDPs; • Theoretical analysis: • Characterizing local goodness of controllers; • polynomial number of milestones is sufficient.

  5. Goal Start Motion Planning Case • Deterministic motion planning: • Given some start and goal configurations, find a collision free path; • Stochastic motion planning: • Given some start and goal configurations, find a high probability of success path. Path

  6. Nonparametric Combination of Local Controllers i j Use simulation to estimate quality of local controllers Quality: prob. controller reaches neighbor without collisions

  7. Nonparametric Combination of Local Controllers i pij j

  8. Finding a high success probability path • Sample milestones uniformly at random: • X1, …, XN-1 ; • Set start as X0 and goal as XN; • Simulation to estimate local connectivity: • Estimate pij for j in the K nearest neigbors of i; • Shortest path algorithm to find most probable path from X0 to XN: • Edge weights become –log pij .

  9. Example: Maximum Success Probability Path

  10. Example: Maximum Success Probability Path

  11. What About Costs ? • MDPs find path with lowest expected cost: • Implicit trade-off: cost of hitting obstacles and reward for goal; • In Robotics, a successful path often more important than a short path: • Robotic museum guide; • Manufacturing; • Thus, we make the trade-off explicit: • What is the lowest cost path with success probability of at least pmin ?

  12. Restricted Shortest Path • Lowest cost path with success prob. at least pmin: • Restricted shortest path problem; • NP-hard, however, FPAS algorithms [Hassin 92]; • Dynamic programming algorithm: • Discretize [pmin,1] into S+1 values; • q(s) = (pmin)s/S, s = 0, …, S; • V(s,xi): minimum cost-to-go starting at xi, reaching goal with success probability at least q(s).

  13. Success prob.: 0.51 Path length: 1.08 Success prob.: 0.99 Path length: 1.75 Examples:Restricted Shortest Paths

  14. Examples:Restricted Shortest Paths Success prob.: 0.51 Path length: 1.08 Success prob.: 0.99 Path length: 1.75

  15. Theoretical Analysis:Characterizing quality of local controllers • Probabilistic roadmaps (PRMs): [Hsu et al. 99] • Deterministic motion planning; • Characterize space as (,,)-good; • Bound number of milestones; • Extension to stochastic domains: • Characterize space and controller as (,,,p)-good. RX – points reachable using controller from X with probability of success  p X RX Space is (,p)-good if: Volume(RX)   . Volume(free space)

  16. Theorem • For any >0, a roadmap with N=28ln(8/)/+3/+2 milestones, with probability at least 1-, will contain a path between any two milestones in the same connected component and this path will have success probability of at least p 3/+1. In words: • Complete with probability at least 1-; • Number of milestones poly(ln(1/), 1/, 1/, 1/); • Final path has success probability of at least p 3/+1.

  17. Related Work • Macro actions in discrete discounted MDPs: • Hauskrecht et al. 1998, Parr 1998; • Probabilistic Roadmaps (PRMs) for deterministic motion planning: • Hsu et al. 1999; • Continuous state, discrete actions discounted MDPs: • Rust 1997.

  18. Centralized Control of Two Holonomic Robots

  19. Success prob.: 0.99 Total path length: 3.53 Success prob.: 0.13 Total path length: 1.53 Centralized Control of Two Holonomic Robots Success prob.: 0.54 Total path length: 2.79

  20. 5 dof Robot Arm Success prob.: 0.95 Path length: 10.07 Success prob.: 0.60 Path length: 7.81

  21. 7 dof Snake Shortest: Most Success Probaility: Success prob.: 0.11 Path length: 15.4 Success prob.: 0.96 Path length: 27.0

  22. Conclusions • Algorithm for planning in stochastic domains with continuous state and action spaces: • Nonparametric combination of local controllers; • Motion planning: • Theoretical analysis quantifies local quality of controllers; • Proposed alternative objective function; • Qualitative and quantitative properties demonstrated; • Also applicable for discounted MDPs: • Describe methods for robustly combining local controllers. http://robotics.stanford.edu/~guestrin/Research/RobustLocalControl/

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