Motion Algorithms

Jean-Claude Latombeai.stanford.edu/~latombe

1

Basic QuestionCan two given points be connected by a path in a feasible space?

Feasible space

Forbidden space2

Configuration Space

Issues:• Space dimensionality• Geometric complexity

q1

q3

q0

qn

q4

3

4

q1

q3

q0

qn

q4

Any moving object maps to a point in its configuration space

6 D

~40 D

12 D

15 D

~65-120 D100’s D

feasible space

Probabilistic Sampling

5

F1 F2

Connectivity Issue

Lookout of F1

The β-lookout of a subset X of F is the set of all configurations in X that see a β-fraction of F\X

β-lookout(X) = {q ∈ X | μ(V(q)\X) ≥β×μ(F\X)}

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F1 F2

(ε,α,β)-Expansiveness of F

Lookout of F1

F is (ε,α,β)-expansive if it is ε-good and each one of its subsets X has a β-lookout whose volume is at least α×μ(X)

The β-lookout of a subset X of F is the set of all configurations in X that see a β-fraction of F\X

β-lookout(X) = {q ∈ X | μ(V(q)\X) ≥β×μ(F\X)}

7[Hsu et al., 1997]

Expansiveness only depends on volumetric ratiosIt is not directly related to the dimensionality of the configuration space

In 2-D the expansiveness of the free space can be made arbitrarily poor

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Theoretical Convergence of PRM Planning

Let F be (ε,α,β)-expansive, and s and g be two configurations in the same component of F. BasicPRM(s,g,N) with uniform sampling returns a path between s and g with probability converging to 1 at an exponential rate as N increases

γ = Pr(Failure) ≤

Experimental convergence

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Most narrow passages in F are intentional …

… but it is not easy to intentionally create complex narrow passages in F

Alpha puzzle

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Automatic robot programming

Assembly planning

Space robotics

Reconfigurablerobots

Crowd simulation(egress)

Building codeverification

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Surgicalplanning

Surgicalsimulation

Digital actors

Climbing and legged robots

Proteinmotion 12

Current projectsai.stanford.edu/~latombe/projects/projects-www.html

Development of a fully autonomous climbing robot

Study of the long-timescale dynamics of protein

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Autonomous Climbing Robots

Lemur Capuchin

Motion of a Protein

10-15

femtosec10-12

picosec10-9

nanosec10-6

microsec10-3

millisec100

seconds

Bond/atomic vibration

Waterdynamics

Helixforms

Fast anisotropicconf change

long MD run

where weneed to be

MDstep

where we’dlove to be

Slowconf change

one-day MD run

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Approaches1. Experimental approach: Model discrete

heterogeneity in proteins from X-Ray crystallography data

2. Geometric/kinematic approach: Explore reachable conformation space under geometric/kinematic constraints

3. Markov model approach: Model long timescale dynamics with Hidden Markov Models trained from Molecular Dynamics simulation trajectories

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X-Ray Crystallography

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Multiple states lead to ambiguous density

A323Hist

A316Ser

A

B

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Multi-Conformer Approach

1. Sample a huge set C of candidate conformers

2. Select the subset M that collectively best models the electron density map (EDM)

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Multi-Conformer Approach

1. Sample a huge set C of candidate conformers

2. Select the subset M that collectively best models the electron density map (EDM)

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Multi-Conformer Approach

1. Sample a huge set C of candidate conformers

2. Select the subset M that collectively best models the electron density map (EDM)

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Multi-Conformer Approach was Used to produce PDB ID: 3eo6

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