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A Coalition Game-Based Algorithm for Autonomous Self-Reconfigurations in Modular Self-Reconfigurable Robots. Zach Ramaekers Computer Science University of Nebraska at Omaha Advisor: Dr. Raj Dasgupta. Modular Self-Reconfigurable Robots (MSRs) – What and Why. - PowerPoint PPT Presentation
Citation preview
1
A Coalition Game-Based Algorithm for
Autonomous Self-Reconfigurations in
Modular Self-Reconfigurable Robots
Zach RamaekersComputer Science
University of Nebraska at OmahaAdvisor: Dr. Raj Dasgupta
2
Modular Self-Reconfigurable Robots (MSRs) – What and WhyAn MSR is a type of robot that is composed of
identical modulesThe modules connect together to form larger
robots capable of performing complex tasksWhy MSRs?
Inexpensive and SimpleHighly Adaptable
Three main types of MSRs: Chain, Lattice, Hybrid
3
ModRED (Modular Robot for Exploration and Discovery)
Novel 4 degrees of freedom designGives improved dexterityAllows maneuver itself and
get out of tight spaces’ModRED sensors and
actuators Arduino processor (for doing
computations) IR sensors (for sensing
how far obstacles are) Compass (which direction am
I heading) Tilt sensor (what is my
inclination) XBee radio (for wireless
comm.)
Designed by Dr. Nelson’s group, Mechanical Engineering, UNL
CAD diagram of
robot
Simulated robot in Webots
4
ModRED Movements in Fixed Configuration
All these movements are in a fixed configuration
5
Problem Addressed: Dynamic Self-Reconfiguration by MSRs
Why does an MSR need to reconfigure
dynamically?
Problem Statement: How can an MSR that needs reconfigure (e.g., after encountering an obstacle) determine 1. which other modules
to combine with, and2. the best configuration
to form with those modules?
... in an autonomous manner.
6
Coalition Games for Dynamic MSR Reconfiguration We propose a novel, coalition game theory
based approach to address the problem of MSR self-reconfiguration
A coalition game gives a set of rules that determine what teams a group of humans (or robots) will form between themselvesTeams are guranteed to be stable: once
teams are formed no one will want to change teams
Game TheoreticLayer
(Coalition Game)
ControllerLayer (Gait-
tables)
Mediator
Dynamic self- reconfiguration
Movement in Fixed Configuration
7
Coalition Games for Dynamic MSR Reconfiguration We propose a novel, coalition game theory
based approach to address the problem of MSR self-reconfiguration
A coalition game gives a set of rules that determine what teams a group of humans (or robots) will form between themselvesTeams are guranteed to be stable: once
teams are formed no one will want to change teams
Game TheoreticLayer
(Coalition Game)
ControllerLayer (Gait-
tables)
Mediator
Dynamic self- reconfiguration
Movement in Fixed Configuration
For our scenario: Each module of an MSR is provided with
embedded software called an agent that does the coalition
game related calculations
8
Coalition Games for Dynamic MSR Reconfiguration We propose a novel, coalition game theory
based approach to address the problem of MSR self-reconfiguration
A coalition game gives a set of rules that determine what teams a group of humans (or robots) will form between themselvesTeams are guranteed to be stable: once
teams are formed no one will want to change teams
Game TheoreticLayer
(Coalition Game)
ControllerLayer (Gait-
tables)
Mediator
Dynamic self- reconfiguration
Movement in Fixed Configuration
For our scenario: Each module of an MSR is provided with
embedded software called an agent that does the coalition
game related calculations
Our problem: How can we determine these teams or partitions or coalitions for our MSR problem?
9
Determining the PartitionsEnumerate all possible partitions that includes
all the agents - called coalition structures
{1} {2}{3}{4}
{1, 2}{3}{4} {1} {2 ,3}{4}Examplecoalition
structureswith 4 agents
Each coalition structure is associated with a value V(CSi) = sum of utilities of each coalition in CSi
All the possible coalition structures are represented as a coalition structure graph
V(CSi) = u(1,2) + u(3) + u(4)
V(CSi) = u(1) + u(2,3) + u(4)
V(CSi) = u(1) + u(2) + u(3) +
u(4)
10
Coalition Structure Graph (CSG)Possible partitions of 4 modules (agents)
Problem: Find the
node (coaliltion structure) that has
the highest value in
this graph
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Coalition Structure Graph (CSG)
Finding the optimal coalition structure node in this graph is not easy!CSG has w(nn) nodes, the search problem is hard (NP-complete)
Approximation algorithm used to find the optimal CSG node (Sandholm 1999, Rahwan 2007, etc.): in exponential time.
Possible partitions of 4 modules (agents)
Problem: Find the
node (coaliltion structure) that has
the highest value in
this graph
12
Dynamic Reconfiguration under Uncertainty
I need to form
another configurati
on
We are here
And we are here
Which modules should I join
with?
Ridge in planned path
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Modeling Uncertainty in Coalition Formation
{1} {2}{3}{4}
{1, 2}{3}{4} {1} {2 ,3}{4}PossibleCoalitionStructures
Conventional CSG
V(CSi) = u(1,2) + u(3) + u(4)
(additive reward value)
{1, 2}{3}{4}
CSG with uncertainty
V(CSi) < u(1,2) + u(3) + u(4) (subadditive)
V(CSi) = u(1,2) + u(3) + u(4) (additive)
V(CSi) > u(1,2) + u(3) + u(4) (superadditive)
{1, 2}{3}{4}
14
Modeling Uncertainty in Coalition Formation
Dealing with uncertainty: Markov Decision Process (MDP) provide a mathematical model for robots or agents to determine their actions in the presence of uncertainty
{1} {2}{3}{4}
{1, 2}{3}{4} {1} {2 ,3}{4}PossibleCoalitionStructures
Conventional CSG
V(CSi) = u(1,2) + u(3) + u(4)
(additive reward value)
{1, 2}{3}{4}
CSG with uncertainty
V(CSi) < u(1,2) + u(3) + u(4) (subadditive)
V(CSi) = u(1,2) + u(3) + u(4) (additive)
V(CSi) > u(1,2) + u(3) + u(4) (superadditive)
{1, 2}{3}{4}
15
MDP-Based CSG Solution
Conventional CSG without uncertainty
Modified CSG with
uncertainty
16
Algorithm to find optimal coalition structure
Pruning – used to reduce the number of nodes that are searched by the algorithm in the coalition structure graphThree strategies explored: Keep the optimal and two least
optimal children; keep the optimal, median, and least optimal children; keep three random children
Set of modules information
Generate Coalition Utility Values
Generate Coalition Structure Graph
Run Value Iteration and Determine Policy
Run MDP Traversal to Find Optimal CS
Optimal Coalition Structure
17
Simulation Results
1 2 3 4 5 6 7 8 9 10111213141516
Average Number of Nodes in the CS Graph
Full CS Graph
Number of Agents
Num
ber
of
Nodes
in t
he C
S
Gra
ph (
base
10)
1 2 3 4 5 6 7 8 9 101112131415160
5000
10000
15000
20000
25000
30000
35000
Average Number of Nodes in the CS Graph
Pruned Graph Size
Sandholm
Number of Agents
Tota
l N
odes
in t
he C
S G
raph
3 4 5 6 7 8 9 100.0001
0.001
0.01
0.1
1
10
100
1000
Average Time to Compute CS Graph
Full CS Graph
Aver-age Prun-ing CS Graph
Number of Agents
Avera
ge R
unnin
g T
ime (
sec)
3 4 5 6 7 8 9 100.2
0.25
0.3
0.35
0.4
0.45
0.5Average Optimal Reward Values
Full CS Graph
Prun-ing Method 1
Prun-ing Method 2
Number of Agents
Avera
ge R
ew
ard
Valu
e
18
Conclusions and Future WorkDeveloped coalition game theory based algorithm for MSR
self-reconfigurationValidated to work on accurate model of MSR called
ModRED within Webots robotic simulator To the best of our knowledge
First application of game theory to MSR self-reconfiguration problem
First attempt at combining planning under uncertainty (MDP) with coalition games*
Future workInvestigate distributed models of planning under uncertainty
(MMDPs, DEC-MDPs, etc.)Simulation of exploration and coverage on realistic terrainsIntegrate with hardware of ModRED robot
*: Suijs et al.(1999) defined a framework called stochastic coalition game, but using a different model involving agent types, which wasn’t validated empirically.
19
ModRED Self-Reconfiguration Simulation Demo