september 30, probabilistic modeling
DESCRIPTION
Multi-Robot SystemsTRANSCRIPT
Multi-Robot Systems
CSCI 7000-006Monday, September 30, 2009
Nikolaus Correll
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
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
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
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
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:
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?
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
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!
Interaction times
• Probabilistic• Deterministic• Non-Parametric
Distribution
Systematic experiments with 1 or 2 robots.
Deterministic Time-Out“180o turn avoidance”
Agent-based simulation
Simulationegocentric
Simulationallocentric
A
S
S
Example 1a: Collision AvoidanceProbabilistic Delay
Search Avoidance
pR
1/Ta
Example 1b: Collision AvoidanceDeterministic Delay
Search
pR
1
AvoidanceTa
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
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
Example 2a: CollaborationProbabilistic Delay
Search Wait
psNs(k)
1/Tw
ps(M0-Nw(k))
Example 3a: CollaborationDeterministic Delay
Search Wait
psNs(k)
ps(M0-Nw(k))
ps(M0-Nw(k-Tw))G(k;k-Tw)
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?
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”
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?
Master-Equations for Deliberative Systems: Greedy algorithm
AA AB
BBBA
Why does this system asymptotically converge to AB or BA?
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
Expected Time to Completion
greedy
collaborative
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
Upcoming
Formal approaches to obtain model parametersHow to model systems with large state space?