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Toward Optimal Configuration Space Sampling. Presented by: Yan Ke. Sampling Problem. Tool: Sample points. Target: Construct a roadmap representing the complete connectivity of the configuration space. More Points ≠ Better Sampling. How to Sample Smartly?. - PowerPoint PPT Presentation
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Toward Optimal Toward Optimal Configuration Space Configuration Space
SamplingSampling
Presented by: Presented by: Yan KeYan Ke
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Sampling Problem
Tool: Sample points.
Target: Construct a roadmap representing the complete connectivity of the configuration space.
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More Points ≠ Better Sampling
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How to Sample Smartly?
Complete knowledge of configuration Space (usually unavailable).
Using information from past experience (our approach).
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Modeling Modeling Configuration Configuration
SpaceSpace
Section Section 11
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Build a Model from Past Exp.
Machine learning is concerned with how to automate learning from experience.
An existing obstructed node indicates being his neighbors, you are also likely to be obstructed.
And vise versa.
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Probability for a single node
P(q=i | M)q – newly sampled point i – 1(free) or 0 (obstructed)M– Model built from past experience
We are learning P base on M.
We want : P(q=1 | M)↑
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Basic Idea
Model configuration space as binary classification: C(p) = (0,1)
If q is p’s neighbor,C(p) = 1 P(q=1 | M)↑C(p) = 0 P(q=1 | M)↓
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Approximation Function
Denote Ĉ(q) = P(q=1 | M)
Obviously Ĉ(q) [0,1]
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K-nearest Neighbors
Q = { qi | i = 1,2……n}
N(q,k) – The function provides the k-nearest neighbors in Q.
Ĉ(q) = ),(
)(kqN
iiqC
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A Screen Shot from the Paper
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Probabilities
P(q=1 | M) = Ĉ(q)
P(q=0 | M) = 1 - Ĉ(q)
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Utility FunctionUtility Function
Section 2Section 2
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Utility Function Purpose: Characterize the relevance of a
configuration to successfully guide sampling.
Relevance of a configuration: Unexplored regions near to existing roadmap
components? maximally distance from existing components in
unexplored regions of configuration space?
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Utility Function
U(q=i , R)q – newly sampled point i – 1(free) or 0 (obstructed)R– the roadmap
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Information Gain IG(S,K) = H(S) – H(S|K)
S – some system K – new knowledge H() – entropy function
As S getting more information, H(S)↓
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Utility Function
U(q=i , R) = IG (R,q) = H(R) – H(R|q)
We claim that an obstructed sample doesn’t provide us any IG
i.e. U(q=i , R) = 0
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Another Screen Shot
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How to get around it? Return to our very basic goal: Full Connectivity
We restrict our current roadmap to be a set of disjoint component. The maximal IG is likely to appear near the middle point of two large
disjoint components.
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Utility-Guided Utility-Guided SamplingSampling
Section 3Section 3
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Utility-Guided Sampling
),1()|1(),0()|0(),1()|1(
),()|(
)|(
)1,0(
exp
RqUMqPRqUMqPRqUMqP
RiqUMiqP
MqU
i
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Algorithm:
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Experiment
Environment: Two workspaces with robots of varying degrees of freedom.
Each robot – 3-4 links. Each joint – 3 degrees of
freedom. Total – 9 or 12 DOF
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Result: Faster
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Conclusion
Utility-Guided Sampling
Guiding sampling to more relevant configurations.
Experimentally proved to be efficient