September 30, Probabilistic Modeling
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Transcript of September 30, Probabilistic Modeling
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Multi-Robot Systems
CSCI 7000-006Monday, September 30, 2009
Nikolaus Correll
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So far
• Reactive and deliberative distributed algorithms
• Formal models describing sub-sets of the systems
• Deterministic models for deliberative algorithms
• Convex cost functions and feedback control for reactive systems
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Problems so far
• How to model– Sensor uncertainty (localization, vision, range)– Communication uncertainty– Actuation uncertainty (e.g. wheel-slip)
• Deterministic algorithms break• Reactive algorithm become unpredictable
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Problem Statement
• Predict the performance of a system given– Problem– Algorithms– Capabilities/Uncertainty
• Find most suitable coordination scheme / set of resources
4
Robot 1 Robot 2
Task A
Task B
Problem
Algorithms
Random Deliberative
CentralizedDecentralized
Collaborative Greedy
N. Correll. Coordination schemes for distributed boundary coverage with a swarm of miniature robots: synthesis, analysis and experimental validation. EPFL PhD thesis #3919, 2007.
Capabilities / Probabilistic Behavior
Navigation Localization Communication
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5
Master equations and Markov Chains
• The system state ({robot states} X {environment state}) is finite• Non-deterministic elements of the system follow a known statistical distribution
: Conditional probability to be in state w when in state w’ a time-step before
Transition probability from w’ to w in a Markov Chain
: Probability for the system to be in state w at time k
w’ w
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6
From Master to Rate Equations: probability to be in state x at time k x can be a robot’s or a system’s state
: Total number of robots
Average number of robots in state x:
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Example 1: Collision Avoidance
Two states, search and avoidN0 robots
State duration of avoid– Probabilistic– Deterministic
Possible implementations:Obstacle
“Proximal”
Obstacle
“180o turn”
What are the parameters of this system and what are their distributions?
How to get them?
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Parameters
Encountering probability pR
– Probability to encounter another robot per time step
Interaction time Ta
– Average time a collision lasts– Geometric distribution or Dirac pulse
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Interaction time
Average time Ta constant regardless whether probabilistic or deterministicDistribution Ta is different depending on– Controller– Model abstraction level
Model is only an approximation!
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Interaction times
• Probabilistic• Deterministic• Non-Parametric
Distribution
Systematic experiments with 1 or 2 robots.
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Deterministic Time-Out“180o turn avoidance”
Agent-based simulation
Simulationegocentric
Simulationallocentric
A
S
S
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Example 1a: Collision AvoidanceProbabilistic Delay
Search Avoidance
pR
1/Ta
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Example 1b: Collision AvoidanceDeterministic Delay
Search
pR
1
AvoidanceTa
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Example 2: Collaboration
Two states: search and waitN0 robots M0 collaboration sites
State duration of wait – probabilistic: robots wait a random time– deterministic: robots wait a fixed time
robot
site
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Parameters
Encountering probability ps
– Probability to encounter one site
Interaction time Tw
– (Average) time a robot waits for collaboration before moving on
Robot-Robot collisions are ignored in this example
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Example 2a: CollaborationProbabilistic Delay
Search Wait
psNs(k)
1/Tw
ps(M0-Nw(k))
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Example 3a: CollaborationDeterministic Delay
Search Wait
psNs(k)
ps(M0-Nw(k))
ps(M0-Nw(k-Tw))G(k;k-Tw)
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Summary: Memory-less systems
Systems with no or little memory (Time-outs), essentially reactiveMaster equation for a single robot allows estimation of population dynamics
How to deal with deliberative systems
that use memory?
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Example 3: Task Allocation
Scenario: 2 robots, 2 tasks A and BRobots prefer task A over task BGlobal metric requires solution of both tasksTask evaluation subject to noise, robots choose the wrong task with probability p
A B
1-p p
A B
1-p 1-p
“Greedy” “Coordinated”
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Example 3a: Task AllocationNon-Collaborative, Greedy
Both robots will go for task A, then BExpected time: 2 time-stepsNoise! Effective outcomes might be AA, AB, BB, BAThere is a possibility to complete in one time-step (due to noise): AB or BA
What is the state transition diagram of this system?
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Master-Equations for Deliberative Systems: Greedy algorithm
AA AB
BBBA
Why does this system asymptotically converge to AB or BA?
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Example 3b: Task AllocationCollaborative
Robots will allocate the tasks among themRobot 1 will go for task A, robot 2 go for task BIf only one task is left, both try to accomplish itEffective outcomes AA, AB, BB, BAExpected time to completion 1 time-step for: AB and BA
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Expected Time to Completion
greedy
collaborative
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Summary
Master equation: change of probability to be in state xEnumerate all possible states of a systemCalculate all possible state transition probabilitiesSolve difference equations (numerically, analytically, Lyapunov, …)Useful for analyzing dominant collaboration dynamics of a system
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Upcoming
Formal approaches to obtain model parametersHow to model systems with large state space?