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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