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Robust Combination of Local Controllers
Carlos Guestrin
Dirk OrmoneitStanford University
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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.
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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 ?
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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.
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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
Start Goal
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Nonparametric Combination of Local Controllers
i
j
Use simulation to estimate quality of local controllers
Quality: prob. controller reaches neighbor without collisions
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Nonparametric Combination of Local Controllers
i
jpi
j
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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 .
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Example: Maximum Success Probability Path
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Example: Maximum Success Probability Path
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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 ?
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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).
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Examples:Restricted Shortest Paths
Success prob.: 0.99Path length: 1.75
Success prob.: 0.51Path length: 1.08
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Examples:Restricted Shortest Paths
Success prob.: 0.99Path length: 1.75
Success prob.: 0.51Path length: 1.08
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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 (,,,pp)-
good.XRX
RX – points reachable using controller from X with probability of success pp
Space is (,pp)-good if:Volume(RX) . Volume(free space)
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Theorem For any >0, a roadmap with
N=28ln(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 pp
3/+1.
Complete with probability at least 1-; Number of milestones poly(ln(1/), 1/, 1/, 1/); Final path has success probability of at least pp
3/+1.
In words:
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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.
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Centralized Control of Two Holonomic Robots
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Centralized Control of Two Holonomic Robots
Success prob.: 0.99Total path length: 3.53
Success prob.: 0.13Total path length: 1.53
Success prob.: 0.54Total path length: 2.79
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5 dof Robot Arm
Success prob.: 0.95Path length: 10.07
Success prob.: 0.60Path length: 7.81
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7 dof Snake
Success prob.: 0.96Path length: 27.0
Success prob.: 0.11Path length: 15.4
Shortest: Most Success Probaility:
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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/