formations in a robotic swarm - deakin universitydefinitions of swarm robotics “swarm robotics is...
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Background
Formation of a roboticSwarm
Swarming GuidedWeapons
Summary
Formations in a Robotic SwarmAn artificial force based approach
Samitha Ekanayake
Networked Sensing and Control LaboratorySchool of Engineering
Deakin University
October 1, 2009
Background
Formation of a roboticSwarm
Swarming GuidedWeapons
Summary
Outline
1 BackgroundOverviewNatural to Artificial Swarms
2 Formation of a robotic SwarmProblemAssumptionsMathematical ModelBehavior Analysis
3 Swarming Guided WeaponsMotivationSolution
4 Summary
BackgroundOverview
Natural to Artificial Swarms
Formation of a roboticSwarm
Swarming GuidedWeapons
Summary
Outline
1 BackgroundOverviewNatural to Artificial Swarms
2 Formation of a robotic SwarmProblemAssumptionsMathematical ModelBehavior Analysis
3 Swarming Guided WeaponsMotivationSolution
4 Summary
BackgroundOverview
Natural to Artificial Swarms
Formation of a roboticSwarm
Swarming GuidedWeapons
Summary
BackgroundDefinitions of Swarm Robotics
“Swarm Robotics is the study of how largenumber of relatively simple physically embodiedagents can be designed such that a desiredcollective behavior emerges from the localinteractions among agents and between theagents and the environment.”- E. Sahin, “Swarm robotics: From sources ofinspiration to domains of application,”
BackgroundOverview
Natural to Artificial Swarms
Formation of a roboticSwarm
Swarming GuidedWeapons
Summary
BackgroundDefinitions of Swarm Robotics
“A network of a number of loosely coupleddynamic units that collectively reach goals thatare difficult to achieve by an individual agent or amonolithic system.”- V. Gazi and B. Fidan, “Coordination and controlof multi-agent dynamic systems: Models andapproaches,”
BackgroundOverview
Natural to Artificial Swarms
Formation of a roboticSwarm
Swarming GuidedWeapons
Summary
BackgroundKeywords
Collective Behavior
Relatively simple agents
Loosely Coupled
Local interactions
BackgroundOverview
Natural to Artificial Swarms
Formation of a roboticSwarm
Swarming GuidedWeapons
Summary
BackgroundKeywords
Collective Behavior
Relatively simple agents
Loosely Coupled
Local interactions
BackgroundOverview
Natural to Artificial Swarms
Formation of a roboticSwarm
Swarming GuidedWeapons
Summary
BackgroundKeywords
Collective Behavior
Relatively simple agents
Loosely Coupled
Local interactions
BackgroundOverview
Natural to Artificial Swarms
Formation of a roboticSwarm
Swarming GuidedWeapons
Summary
BackgroundKeywords
Collective Behavior
Relatively simple agents
Loosely Coupled
Local interactions
BackgroundOverview
Natural to Artificial Swarms
Formation of a roboticSwarm
Swarming GuidedWeapons
Summary
Outline
1 BackgroundOverviewNatural to Artificial Swarms
2 Formation of a robotic SwarmProblemAssumptionsMathematical ModelBehavior Analysis
3 Swarming Guided WeaponsMotivationSolution
4 Summary
BackgroundOverview
Natural to Artificial Swarms
Formation of a roboticSwarm
Swarming GuidedWeapons
Summary
Natural to Artificial SwarmsNatural Swarms
Natural SwarmsAntsBeesFishBirds
The concept of a swarm roboticsis inspired by them :
powerful societies due toextensive group workthe behavior as a groupensure their survival forthousands of years
BackgroundOverview
Natural to Artificial Swarms
Formation of a roboticSwarm
Swarming GuidedWeapons
Summary
Natural to Artificial SwarmsNatural Swarms
Natural SwarmsAntsBeesFishBirds
The concept of a swarm roboticsis inspired by them :
powerful societies due toextensive group workthe behavior as a groupensure their survival forthousands of years
BackgroundOverview
Natural to Artificial Swarms
Formation of a roboticSwarm
Swarming GuidedWeapons
Summary
Natural to Artificial SwarmsNatural Swarms
Natural SwarmsAntsBeesFishBirds
The concept of a swarm roboticsis inspired by them :
powerful societies due toextensive group workthe behavior as a groupensure their survival forthousands of years
BackgroundOverview
Natural to Artificial Swarms
Formation of a roboticSwarm
Swarming GuidedWeapons
Summary
Natural to Artificial SwarmsNatural Swarms
Natural SwarmsAntsBeesFishBirds
The concept of a swarm roboticsis inspired by them :
powerful societies due toextensive group workthe behavior as a groupensure their survival forthousands of years
BackgroundOverview
Natural to Artificial Swarms
Formation of a roboticSwarm
Swarming GuidedWeapons
Summary
Natural to Artificial SwarmsNatural Swarms
Natural SwarmsAntsBeesFishBirds
The concept of a swarm roboticsis inspired by them :
powerful societies due toextensive group workthe behavior as a groupensure their survival forthousands of years
BackgroundOverview
Natural to Artificial Swarms
Formation of a roboticSwarm
Swarming GuidedWeapons
Summary
Natural to Artificial SwarmsNatural Swarms
Natural SwarmsAntsBeesFishBirds
The concept of a swarm roboticsis inspired by them :
powerful societies due toextensive group workthe behavior as a groupensure their survival forthousands of years
BackgroundOverview
Natural to Artificial Swarms
Formation of a roboticSwarm
Swarming GuidedWeapons
Summary
Natural to Artificial SwarmsNatural Swarms
Natural SwarmsAntsBeesFishBirds
The concept of a swarm roboticsis inspired by them :
powerful societies due toextensive group workthe behavior as a groupensure their survival forthousands of years
BackgroundOverview
Natural to Artificial Swarms
Formation of a roboticSwarm
Swarming GuidedWeapons
Summary
Natural to Artificial SwarmsNatural Swarms
Natural SwarmsAntsBeesFishBirds
The concept of a swarm roboticsis inspired by them :
powerful societies due toextensive group workthe behavior as a groupensure their survival forthousands of years
BackgroundOverview
Natural to Artificial Swarms
Formation of a roboticSwarm
Swarming GuidedWeapons
Summary
Natural to Artificial SwarmsArtificial Swarms
Small or Miniature robots
Working for a collaborative goalAdvantages over monolithic system
Low unit costDisposableSmall unit sizeScalability, Flexibility and Robustness
BackgroundOverview
Natural to Artificial Swarms
Formation of a roboticSwarm
Swarming GuidedWeapons
Summary
Natural to Artificial SwarmsArtificial Swarms
Small or Miniature robots
Working for a collaborative goalAdvantages over monolithic system
Low unit costDisposableSmall unit sizeScalability, Flexibility and Robustness
BackgroundOverview
Natural to Artificial Swarms
Formation of a roboticSwarm
Swarming GuidedWeapons
Summary
Natural to Artificial SwarmsArtificial Swarms
Small or Miniature robots
Working for a collaborative goalAdvantages over monolithic system
Low unit costDisposableSmall unit sizeScalability, Flexibility and Robustness
BackgroundOverview
Natural to Artificial Swarms
Formation of a roboticSwarm
Swarming GuidedWeapons
Summary
Natural to Artificial SwarmsArtificial Swarms
Small or Miniature robots
Working for a collaborative goalAdvantages over monolithic system
Low unit costDisposableSmall unit sizeScalability, Flexibility and Robustness
BackgroundOverview
Natural to Artificial Swarms
Formation of a roboticSwarm
Swarming GuidedWeapons
Summary
Natural to Artificial SwarmsArtificial Swarms
Small or Miniature robots
Working for a collaborative goalAdvantages over monolithic system
Low unit costDisposableSmall unit sizeScalability, Flexibility and Robustness
BackgroundOverview
Natural to Artificial Swarms
Formation of a roboticSwarm
Swarming GuidedWeapons
Summary
Natural to Artificial SwarmsArtificial Swarms
Small or Miniature robots
Working for a collaborative goalAdvantages over monolithic system
Low unit costDisposableSmall unit sizeScalability, Flexibility and Robustness
BackgroundOverview
Natural to Artificial Swarms
Formation of a roboticSwarm
Swarming GuidedWeapons
Summary
Natural to Artificial SwarmsArtificial Swarms
Small or Miniature robots
Working for a collaborative goalAdvantages over monolithic system
Low unit costDisposableSmall unit sizeScalability, Flexibility and Robustness
BackgroundOverview
Natural to Artificial Swarms
Formation of a roboticSwarm
Swarming GuidedWeapons
Summary
Natural to Artificial SwarmsResearch Questions - Coordination and Control
Coordination and controlPattern formationCoordinated movementObstacle avoidanceForagingSelf-deployment activities
BackgroundOverview
Natural to Artificial Swarms
Formation of a roboticSwarm
Swarming GuidedWeapons
Summary
Natural to Artificial SwarmsResearch Questions - Coordination and Control
Coordination and controlPattern formationCoordinated movementObstacle avoidanceForagingSelf-deployment activities
BackgroundOverview
Natural to Artificial Swarms
Formation of a roboticSwarm
Swarming GuidedWeapons
Summary
Natural to Artificial SwarmsResearch Questions - Coordination and Control
Coordination and controlPattern formationCoordinated movementObstacle avoidanceForagingSelf-deployment activities
BackgroundOverview
Natural to Artificial Swarms
Formation of a roboticSwarm
Swarming GuidedWeapons
Summary
Natural to Artificial SwarmsResearch Questions - Coordination and Control
Coordination and controlPattern formationCoordinated movementObstacle avoidanceForagingSelf-deployment activities
BackgroundOverview
Natural to Artificial Swarms
Formation of a roboticSwarm
Swarming GuidedWeapons
Summary
Natural to Artificial SwarmsResearch Questions - Coordination and Control
Coordination and controlPattern formationCoordinated movementObstacle avoidanceForagingSelf-deployment activities
BackgroundOverview
Natural to Artificial Swarms
Formation of a roboticSwarm
Swarming GuidedWeapons
Summary
Natural to Artificial SwarmsResearch Questions - Coordination and Control
Coordination and controlPattern formationCoordinated movementObstacle avoidanceForagingSelf-deployment activities
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
Outline
1 BackgroundOverviewNatural to Artificial Swarms
2 Formation of a robotic SwarmProblemAssumptionsMathematical ModelBehavior Analysis
3 Swarming Guided WeaponsMotivationSolution
4 Summary
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
Shape Formation ProblemIntroduction
Populate a large number of robots.
Into a geographical location defined by a closedcontour.
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
Shape Formation ProblemIntroduction
Populate a large number of robots.
Into a geographical location defined by a closedcontour.
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
Shape Formation ProblemIntroduction
Populate a large number of robots.
Into a geographical location defined by a closedcontour.
Desired Target Contour
M embers of the robotic group
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
Shape Formation ProblemIntroduction
Populate a large number of robots.
Into a geographical location defined by a closedcontour.
Desired Target Contour
M embers of the robotic group
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
Shape Formation ProblemDeviation from other approaches
Comparison with existingapproaches
The robots/agents are placedalong the contour. Important inenclosing a target i.e. inescorting tasks.Di-graph based formationstrategies, which are based ongraph theory basedapproaches.Each robot is assigned with aspecific position in theformation
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
Shape Formation ProblemDeviation from other approaches
Comparison with existingapproaches
The robots/agents are placedalong the contour. Important inenclosing a target i.e. inescorting tasks.Di-graph based formationstrategies, which are based ongraph theory basedapproaches.Each robot is assigned with aspecific position in theformation
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
Shape Formation ProblemDeviation from other approaches
Comparison with existingapproaches
The robots/agents are placedalong the contour. Important inenclosing a target i.e. inescorting tasks.Di-graph based formationstrategies, which are based ongraph theory basedapproaches.Each robot is assigned with aspecific position in theformation
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
Shape Formation ProblemDeviation from other approaches
Comparison with existingapproaches
The robots/agents are placedalong the contour. Important inenclosing a target i.e. inescorting tasks.Di-graph based formationstrategies, which are based ongraph theory basedapproaches.Each robot is assigned with aspecific position in theformation
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
Shape Formation ProblemResearch Goals and Applications
Research GoalsGather a group of robots into a pre-defined shapein 2D spaceDecentralized controllerScalabilityObstacle avoiding capabilitiesCollision avoidance
Potential ApplicationsUAV / UGV controlMultiple Weapon control (Cluster bombs, MLRSetc.)Search and rescue robots / De-mining robotsSensor network deployment with air bornesystemLand exploration
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
Shape Formation ProblemResearch Goals and Applications
Research GoalsGather a group of robots into a pre-defined shapein 2D spaceDecentralized controllerScalabilityObstacle avoiding capabilitiesCollision avoidance
Potential ApplicationsUAV / UGV controlMultiple Weapon control (Cluster bombs, MLRSetc.)Search and rescue robots / De-mining robotsSensor network deployment with air bornesystemLand exploration
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
Shape Formation ProblemResearch Goals and Applications
Research GoalsGather a group of robots into a pre-defined shapein 2D spaceDecentralized controllerScalabilityObstacle avoiding capabilitiesCollision avoidance
Potential ApplicationsUAV / UGV controlMultiple Weapon control (Cluster bombs, MLRSetc.)Search and rescue robots / De-mining robotsSensor network deployment with air bornesystemLand exploration
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
Shape Formation ProblemResearch Goals and Applications
Research GoalsGather a group of robots into a pre-defined shapein 2D spaceDecentralized controllerScalabilityObstacle avoiding capabilitiesCollision avoidance
Potential ApplicationsUAV / UGV controlMultiple Weapon control (Cluster bombs, MLRSetc.)Search and rescue robots / De-mining robotsSensor network deployment with air bornesystemLand exploration
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
Shape Formation ProblemResearch Goals and Applications
Research GoalsGather a group of robots into a pre-defined shapein 2D spaceDecentralized controllerScalabilityObstacle avoiding capabilitiesCollision avoidance
Potential ApplicationsUAV / UGV controlMultiple Weapon control (Cluster bombs, MLRSetc.)Search and rescue robots / De-mining robotsSensor network deployment with air bornesystemLand exploration
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
Shape Formation ProblemResearch Goals and Applications
Research GoalsGather a group of robots into a pre-defined shapein 2D spaceDecentralized controllerScalabilityObstacle avoiding capabilitiesCollision avoidance
Potential ApplicationsUAV / UGV controlMultiple Weapon control (Cluster bombs, MLRSetc.)Search and rescue robots / De-mining robotsSensor network deployment with air bornesystemLand exploration
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
Shape Formation ProblemResearch Goals and Applications
Research GoalsGather a group of robots into a pre-defined shapein 2D spaceDecentralized controllerScalabilityObstacle avoiding capabilitiesCollision avoidance
Potential ApplicationsUAV / UGV controlMultiple Weapon control (Cluster bombs, MLRSetc.)Search and rescue robots / De-mining robotsSensor network deployment with air bornesystemLand exploration
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
Shape Formation ProblemResearch Goals and Applications
Research GoalsGather a group of robots into a pre-defined shapein 2D spaceDecentralized controllerScalabilityObstacle avoiding capabilitiesCollision avoidance
Potential ApplicationsUAV / UGV controlMultiple Weapon control (Cluster bombs, MLRSetc.)Search and rescue robots / De-mining robotsSensor network deployment with air bornesystemLand exploration
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
Shape Formation ProblemResearch Goals and Applications
Research GoalsGather a group of robots into a pre-defined shapein 2D spaceDecentralized controllerScalabilityObstacle avoiding capabilitiesCollision avoidance
Potential ApplicationsUAV / UGV controlMultiple Weapon control (Cluster bombs, MLRSetc.)Search and rescue robots / De-mining robotsSensor network deployment with air bornesystemLand exploration
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
Shape Formation ProblemResearch Goals and Applications
Research GoalsGather a group of robots into a pre-defined shapein 2D spaceDecentralized controllerScalabilityObstacle avoiding capabilitiesCollision avoidance
Potential ApplicationsUAV / UGV controlMultiple Weapon control (Cluster bombs, MLRSetc.)Search and rescue robots / De-mining robotsSensor network deployment with air bornesystemLand exploration
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
Shape Formation ProblemResearch Goals and Applications
Research GoalsGather a group of robots into a pre-defined shapein 2D spaceDecentralized controllerScalabilityObstacle avoiding capabilitiesCollision avoidance
Potential ApplicationsUAV / UGV controlMultiple Weapon control (Cluster bombs, MLRSetc.)Search and rescue robots / De-mining robotsSensor network deployment with air bornesystemLand exploration
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
Outline
1 BackgroundOverviewNatural to Artificial Swarms
2 Formation of a robotic SwarmProblemAssumptionsMathematical ModelBehavior Analysis
3 Swarming Guided WeaponsMotivationSolution
4 Summary
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
Shape Formation ProblemAssumptions
For the analysis, the following were assumed
Agents/members have identical physicalproperties (such as mass, mobility etc.)
Agents/members are point masses i.e. withoutany physical dimensions and demonstrate pointmass dynamics
Agents/members have instantaneous and errorfree localization capabilities
The communication network of the members cantransmit data to all the members within the groupinstantaneously, i.e. without delay
Agents/members operate on a 2D plane withoutobstacles
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
Shape Formation ProblemAssumptions
For the analysis, the following were assumed
Agents/members have identical physicalproperties (such as mass, mobility etc.)
Agents/members are point masses i.e. withoutany physical dimensions and demonstrate pointmass dynamics
Agents/members have instantaneous and errorfree localization capabilities
The communication network of the members cantransmit data to all the members within the groupinstantaneously, i.e. without delay
Agents/members operate on a 2D plane withoutobstacles
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
Shape Formation ProblemAssumptions
For the analysis, the following were assumed
Agents/members have identical physicalproperties (such as mass, mobility etc.)
Agents/members are point masses i.e. withoutany physical dimensions and demonstrate pointmass dynamics
Agents/members have instantaneous and errorfree localization capabilities
The communication network of the members cantransmit data to all the members within the groupinstantaneously, i.e. without delay
Agents/members operate on a 2D plane withoutobstacles
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
Shape Formation ProblemAssumptions
For the analysis, the following were assumed
Agents/members have identical physicalproperties (such as mass, mobility etc.)
Agents/members are point masses i.e. withoutany physical dimensions and demonstrate pointmass dynamics
Agents/members have instantaneous and errorfree localization capabilities
The communication network of the members cantransmit data to all the members within the groupinstantaneously, i.e. without delay
Agents/members operate on a 2D plane withoutobstacles
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
Shape Formation ProblemAssumptions
For the analysis, the following were assumed
Agents/members have identical physicalproperties (such as mass, mobility etc.)
Agents/members are point masses i.e. withoutany physical dimensions and demonstrate pointmass dynamics
Agents/members have instantaneous and errorfree localization capabilities
The communication network of the members cantransmit data to all the members within the groupinstantaneously, i.e. without delay
Agents/members operate on a 2D plane withoutobstacles
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
Outline
1 BackgroundOverviewNatural to Artificial Swarms
2 Formation of a robotic SwarmProblemAssumptionsMathematical ModelBehavior Analysis
3 Swarming Guided WeaponsMotivationSolution
4 Summary
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
Mathematical Model
We use complex plane instead of cartesian plane.
Swarm consisting of N number of identicalmembers.
Operating in two dimensional Euclidean space.
A simple closed contour γ defined in the complexplane.
Desired Target Contour
M embers of the robotic group
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
Mathematical Model
We use complex plane instead of cartesian plane.
Swarm consisting of N number of identicalmembers.
Operating in two dimensional Euclidean space.
A simple closed contour γ defined in the complexplane.
Desired Target Contour
M embers of the robotic group
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
Mathematical Model
We use complex plane instead of cartesian plane.
Swarm consisting of N number of identicalmembers.
Operating in two dimensional Euclidean space.
A simple closed contour γ defined in the complexplane.
Desired Target Contour
M embers of the robotic group
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
Mathematical Model
We use complex plane instead of cartesian plane.
Swarm consisting of N number of identicalmembers.
Operating in two dimensional Euclidean space.
A simple closed contour γ defined in the complexplane.
Desired Target Contour
M embers of the robotic group
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
Mathematical Model
The state of the member i is described by
Xi =
⎡⎣ zi
zi
⎤⎦ , (1)
where zi ∈ C, represents the position of the i th
member in 2D complex plane.
Let z be a point on γ, i.e. z ∈ γ.
Before stating the swarm model, we defineα =
[1 0
]and β =
[0 1
].
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
Mathematical Model
The state of the member i is described by
Xi =
⎡⎣ zi
zi
⎤⎦ , (1)
where zi ∈ C, represents the position of the i th
member in 2D complex plane.
Let z be a point on γ, i.e. z ∈ γ.
Before stating the swarm model, we defineα =
[1 0
]and β =
[0 1
].
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
Mathematical Model
The state of the member i is described by
Xi =
⎡⎣ zi
zi
⎤⎦ , (1)
where zi ∈ C, represents the position of the i th
member in 2D complex plane.
Let z be a point on γ, i.e. z ∈ γ.
Before stating the swarm model, we defineα =
[1 0
]and β =
[0 1
].
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
Mathematical ModelDynamical Model
Then the state of the whole swarm,x =
[X1 ... XN
]Tis determined by the
continuous time dynamic model described by,
x = Ax + Bu, (2)
where
A = diag(
A)
N×N, (3)
B =1m
diag(
B)
N×N(4)
with
A =
[0 10 0
]& B =
[01
]. (5)
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
Mathematical ModelDynamical Model
Then the state of the whole swarm,x =
[X1 ... XN
]Tis determined by the
continuous time dynamic model described by,
x = Ax + Bu, (2)
where
A = diag(
A)
N×N, (3)
B =1m
diag(
B)
N×N(4)
with
A =
[0 10 0
]& B =
[01
]. (5)
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
Mathematical ModelDecentralized Controller
The control input u in (2) consists of,
u =[
u1 u2 u3 ... uN]T
(6)
ui = Fi ,a + Fi ,r + Fi ,m − Fi ,f . (7)
Fi,a : Attraction force (from the contour)Fi,r : Repulsion force (from the contour)Fi,m : Repulsion force (from the other members)Fi,f : Friction like damping force
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
Mathematical ModelDecentralized Controller
The control input u in (2) consists of,
u =[
u1 u2 u3 ... uN]T
(6)
ui = Fi ,a + Fi ,r + Fi ,m − Fi ,f . (7)
Fi,a : Attraction force (from the contour)Fi,r : Repulsion force (from the contour)Fi,m : Repulsion force (from the other members)Fi,f : Friction like damping force
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
Mathematical ModelDecentralized Controller
The control input u in (2) consists of,
u =[
u1 u2 u3 ... uN]T
(6)
ui = Fi ,a + Fi ,r + Fi ,m − Fi ,f . (7)
Fi,a : Attraction force (from the contour)Fi,r : Repulsion force (from the contour)Fi,m : Repulsion force (from the other members)Fi,f : Friction like damping force
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
Mathematical ModelDecentralized Controller
The control input u in (2) consists of,
u =[
u1 u2 u3 ... uN]T
(6)
ui = Fi ,a + Fi ,r + Fi ,m − Fi ,f . (7)
Fi,a : Attraction force (from the contour)Fi,r : Repulsion force (from the contour)Fi,m : Repulsion force (from the other members)Fi,f : Friction like damping force
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
Mathematical ModelDecentralized Controller
The control input u in (2) consists of,
u =[
u1 u2 u3 ... uN]T
(6)
ui = Fi ,a + Fi ,r + Fi ,m − Fi ,f . (7)
Fi,a : Attraction force (from the contour)Fi,r : Repulsion force (from the contour)Fi,m : Repulsion force (from the other members)Fi,f : Friction like damping force
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
Mathematical ModelDecentralized Controller
The control input u in (2) consists of,
u =[
u1 u2 u3 ... uN]T
(6)
ui = Fi ,a + Fi ,r + Fi ,m − Fi ,f . (7)
Fi,a : Attraction force (from the contour)Fi,r : Repulsion force (from the contour)Fi,m : Repulsion force (from the other members)Fi,f : Friction like damping force
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
Mathematical ModelCauchy Winding Number
n(γ, zi ) represents the Cauchy Winding Numberof γ about zi ∈ C
n(γ, zi ) =1
2πi
∫γ
dzz − zi
n(γ, αXi ) =
{1 when member i is inside γ0 when member i is outside γ
,
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
Mathematical ModelCauchy Winding Number
n(γ, zi ) represents the Cauchy Winding Numberof γ about zi ∈ C
n(γ, zi ) =1
2πi
∫γ
dzz − zi
n(γ, αXi ) =
{1 when member i is inside γ0 when member i is outside γ
,
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
Mathematical ModelCauchy Winding Number
n(γ, zi ) represents the Cauchy Winding Numberof γ about zi ∈ C
n(γ, zi ) =1
2πi
∫γ
dzz − zi
n(γ, αXi ) =
{1 when member i is inside γ0 when member i is outside γ
,
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
Mathematical ModelAttraction force from the contour
Fi ,a, attraction force on the i th member from theshape
Fi ,a := ka (1 − n(γ, zi ))
∫γ(z − zi) ‖dz‖ . (8)
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
Mathematical ModelAttraction force from the contour
Fi ,a, attraction force on the i th member from theshape
Fi ,a := ka (1 − n(γ, zi ))
∫γ(z − zi) ‖dz‖ . (8)
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
Mathematical ModelAttraction force from the contour
Fi ,a, attraction force on the i th member from theshape
Fi ,a := ka (1 − n(γ, zi ))
∫γ(z − zi) ‖dz‖ . (8)
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
Mathematical ModelRepulsion force from the contour
Fi ,r , artificial repulsion force on the i th memberfrom the shape
Fi ,r := kr n(γ, zi )
∫γ
[(zi − z)
‖zi − z‖3
]‖dz‖ . (9)
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
Mathematical ModelRepulsion force from the contour
Fi ,r , artificial repulsion force on the i th memberfrom the shape
Fi ,r := kr n(γ, zi )
∫γ
[(zi − z)
‖zi − z‖3
]‖dz‖ . (9)
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
Mathematical ModelRepulsion force from the contour
Fi ,r , artificial repulsion force on the i th memberfrom the shape
Fi ,r := kr n(γ, zi )
∫γ
[(zi − z)
‖zi − z‖3
]‖dz‖ . (9)
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
Mathematical ModelAttraction force - Behavior
Fi ,a, attraction force on the i th member from theshape
Fi ,a := ka (1 − n(γ, zi))
∫γ(z − zi) ‖dz‖ .
−600 −400 −200 0 200
−300
−200
−100
0
100
200
300
X−Coordinate [m]
Y−
Coo
rdin
ate
[m]
Target contour
Path of motion
−600 −500 −400 −300 −200 −100 0−200
−100
0
100
200
300
400
Travel in X direction [m]
For
ce M
agni
tude
/Ang
le
Magnitude x 10 NAngle / (deg)
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
Mathematical ModelRepulsion force - Behavior
Fi ,r , artificial repulsion force on the i th memberfrom the shape
Fi ,r := kr n(γ, zi )
∫γ
[(zi − z)
‖zi − z‖3
]‖dz‖ .
−600 −400 −200 0 200
−300
−200
−100
0
100
200
300
X−Coordinate [m]
Y−
Coo
rdin
ate
[m]
Target contour
Path of motion
−600 −500 −400 −300 −200 −100 0−200
−100
0
100
200
300
Travel in X direction [m]
For
ce M
agni
tude
/Ang
le
Magnitude x 10 NAngle / (deg)
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
Mathematical ModelRepulsion force from the other members - Collision Avoidance
Fi ,m in refers to the resultant force acting on thei th member from the remaining members of theswarm (inter member repulsion force)
Fi ,m := km
⎡⎣ N∑
j=1,j �=i
(zi − zj
)‖zi − zj‖3
⎤⎦ . (10)
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
Mathematical ModelRepulsion force from the other members - Collision Avoidance
Fi ,m in refers to the resultant force acting on thei th member from the remaining members of theswarm (inter member repulsion force)
Fi ,m := km
⎡⎣ N∑
j=1,j �=i
(zi − zj
)‖zi − zj‖3
⎤⎦ . (10)
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
Mathematical ModelRepulsion force from the other members - Collision Avoidance
Fi ,m in refers to the resultant force acting on thei th member from the remaining members of theswarm (inter member repulsion force)
Fi ,m := km
⎡⎣ N∑
j=1,j �=i
(zi − zj
)‖zi − zj‖3
⎤⎦ . (10)
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
Mathematical ModelArtificial Friction Force
Fi ,f is the artificial friction force on member i ;
Fi ,f = kf zi , (11)
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
Mathematical ModelArtificial Friction Force
Fi ,f is the artificial friction force on member i ;
Fi ,f = kf zi , (11)
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
Mathematical ModelArtificial Friction Force
Fi ,f is the artificial friction force on member i ;
Fi ,f = kf zi , (11)
−600 −400 −200 0 200
−300
−200
−100
0
100
200
300
X−Coordinate [m]
Y−
Coo
rdin
ate
[m]
Target contour
Path of motion
−600 −500 −400 −300 −200 −100 0−200
−100
0
100
200
300
Travel in X direction [m]F
orce
Mag
nitu
de/A
ngle
Magnitude x 10 NAngle / (deg)
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
Outline
1 BackgroundOverviewNatural to Artificial Swarms
2 Formation of a robotic SwarmProblemAssumptionsMathematical ModelBehavior Analysis
3 Swarming Guided WeaponsMotivationSolution
4 Summary
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
Behavior AnalysisDefinitions
Center of mass of the swarm,
zcm =
N∑i=1
(zi)
N
Length of the contour
l(γ) =
∫γ‖dz‖
Center of mass of the contour
zc =
∫γ
z ‖dz‖l(γ)
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
Behavior AnalysisDefinitions
Center of mass of the swarm,
zcm =
N∑i=1
(zi)
N
Length of the contour
l(γ) =
∫γ‖dz‖
Center of mass of the contour
zc =
∫γ
z ‖dz‖l(γ)
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
Behavior AnalysisDefinitions
Center of mass of the swarm,
zcm =
N∑i=1
(zi)
N
Length of the contour
l(γ) =
∫γ‖dz‖
Center of mass of the contour
zc =
∫γ
z ‖dz‖l(γ)
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
Behavior AnalysisX Swarm definition
A swarm S is defined as “X swarm”, if there existspositive constants Δ, δ that satisfy the followingconditions simultaneously for all i , j ∈ S and i �= j .
1 dij ≥ δ + Δ,
2
∥∥∥∥zi − zicm
zi − zcm
∥∥∥∥ <
(1 +
Δ
δ
)3
,
where
dij = ‖zi − zj‖ and zicm =
N∑j=1;j �=i
zj
N − 1.
zicm is the center of mass of the swarm without the i th
member.
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
Behavior AnalysisX Swarm definition
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
Behavior AnalysisLemma 1
Using the definition of “X Swarm” we derive that theinter member repulsion force (Fi ,m) on any member ofthe swarm is bounded, as presented in followinglemma.
Lemma
For a member of a “X Swarm”, the magnitude of theartificial inter-member repulsion force is less thankm(N − 1)
δ3 ‖zi − zcm‖,
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
Behavior AnalysisLemma 1
Proof.
Fi ,m = km
N∑j=1,j �=i
(zi − zj)
d3ij
. (12)
Using the condition 1 of the “X Swarm”, we have
‖Fi ,m‖ <km (N − 1)
(δ + Δ)3 ‖zi − zicm)‖. (13)
Then using the condition 2, the following can bederived;
‖Fi ,m‖ <km(N − 1)
δ3 ‖zi − zcm‖ . (14)
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
CohesivenessAll the Members Outside the Shape
In our analysis, the swarm is considered as one objectin which the motion is governed by the resultantartificial force (R),
R = Ra − Rf + Rm. (15)
With this, the equation of motion of the whole swarmcan be described by,
m ε + kf ε + ka l(γ) ε = 0. (16)
where ε = (zcm − zc), with ε = zcm, ε = zcm.
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
CohesivenessAll the Members Outside the Shape
Rm, which represents the resultant inter member forceis zero as,
Rm =N∑
i=1
N∑j=1,j �=i
(zi − zj
)‖zi − zj‖3 = 0.
Ra, the resultant artificial attraction force from thecontour, is expressed as,
Ra = ka
N∑i=1
∫γ(z − zi) ‖dz‖ ,
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
CohesivenessAll the Members Outside the Shape
Using definitions for zc , zcm and l(γ), the above can bestated as:
Ra = l(γ)Nka (zc − zcm) .
Rf , represents the resultant artificial damping (friction)force, and is in the form of,
Rf = Nkf (zcm) .
Therefore, the net resultant force on the swarm (thisforce is applied on the center of mass of the swarm) is,
R = Nl(γ)ka (zc − zcm) − Nkf (zcm) ,
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
CohesivenessMotion of the center of mass of the swarm
Proposition
Consider the swarm model described by (16), motionof zcm is in the direction of decreasing ε (i.e. towardzc).
Proof.
If we select a Lyapunov function candidate as
Vcm =12
mεεT +12
kal(γ)εεT , then the derivative Vcm is
bounded by,Vcm ≤ −kf‖ε‖2.
Since kf‖ε‖2 > 0,∀ ˙‖ε‖ �= 0, the only invariant point isthe origin (i.e. ε = ε = 0), thus using extended versionof Lyapunov’s method we can state that the system isasymptotically stable at the origin.
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
CohesivenessMotion of the center of mass of the swarm - Summary
The motion of the center of mass of the completeswarm
is toward the center of mass of the contourregardless of the motions of members withrespect to zcm
This proposition does not hold, if any member ofthe swarm moves into the shape.Using the properties of second order ODEs
smooth motion of the swarm toward the targetcontour (zc)if the conditions m, ka, kf > 0 andkf ≥ 2
√m ka l(γ)
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
CohesivenessMotion of the center of mass of the swarm - Summary
The motion of the center of mass of the completeswarm
is toward the center of mass of the contourregardless of the motions of members withrespect to zcm
This proposition does not hold, if any member ofthe swarm moves into the shape.Using the properties of second order ODEs
smooth motion of the swarm toward the targetcontour (zc)if the conditions m, ka, kf > 0 andkf ≥ 2
√m ka l(γ)
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
CohesivenessMotion of the center of mass of the swarm - Summary
The motion of the center of mass of the completeswarm
is toward the center of mass of the contourregardless of the motions of members withrespect to zcm
This proposition does not hold, if any member ofthe swarm moves into the shape.Using the properties of second order ODEs
smooth motion of the swarm toward the targetcontour (zc)if the conditions m, ka, kf > 0 andkf ≥ 2
√m ka l(γ)
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
CohesivenessMotion of the center of mass of the swarm - Summary
The motion of the center of mass of the completeswarm
is toward the center of mass of the contourregardless of the motions of members withrespect to zcm
This proposition does not hold, if any member ofthe swarm moves into the shape.Using the properties of second order ODEs
smooth motion of the swarm toward the targetcontour (zc)if the conditions m, ka, kf > 0 andkf ≥ 2
√m ka l(γ)
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
CohesivenessMotion of the center of mass of the swarm - Summary
The motion of the center of mass of the completeswarm
is toward the center of mass of the contourregardless of the motions of members withrespect to zcm
This proposition does not hold, if any member ofthe swarm moves into the shape.Using the properties of second order ODEs
smooth motion of the swarm toward the targetcontour (zc)if the conditions m, ka, kf > 0 andkf ≥ 2
√m ka l(γ)
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
CohesivenessMotion of the center of mass of the swarm - Summary
The motion of the center of mass of the completeswarm
is toward the center of mass of the contourregardless of the motions of members withrespect to zcm
This proposition does not hold, if any member ofthe swarm moves into the shape.Using the properties of second order ODEs
smooth motion of the swarm toward the targetcontour (zc)if the conditions m, ka, kf > 0 andkf ≥ 2
√m ka l(γ)
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
CohesivenessMotion of the center of mass of the swarm - Summary
The motion of the center of mass of the completeswarm
is toward the center of mass of the contourregardless of the motions of members withrespect to zcm
This proposition does not hold, if any member ofthe swarm moves into the shape.Using the properties of second order ODEs
smooth motion of the swarm toward the targetcontour (zc)if the conditions m, ka, kf > 0 andkf ≥ 2
√m ka l(γ)
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
CohesivenessMotion of the center of mass of the swarm - Summary
−500 −400 −300 −200 −100 0 100 200 300 400−300
−200
−100
0
100
200
300
400
500
X−Coordinate [m]
Y−
Coo
rdin
ate
[m]
−400 −300 −200 −100 0 100 200 300 400−300
−200
−100
0
100
200
300
400
X−Coordinate [m]
Y−
Coo
rdin
ate
[m]
−400 −300 −200 −100 0 100 200 300 400−300
−200
−100
0
100
200
300
400
X−Coordinate [m]
Y−
Coo
rdin
ate
[m]
0 50 100 150 200−50
0
50
100
150
200
250
300
350Over damped (Figure (c))Critically damped (Figure (b))Under damped (Figure (a))
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
CohesivenessMotion of a member outside the shape
If we define the error between zi and zcm asυi = (zi − zcm).The motion of any member outside the contour can bedescribed by,
zi =1m
(ka× l(γ)(zc −zi)+km
N∑j=1,j �=i
(zi − zj)
‖zi − zj‖3 −kf zi
)
(17)
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
CohesivenessMotion of a member outside the shape
Proposition
Consider a member i of a “X Swarm” staying outside
the desired shape at any given time, ifka
km>
(N − 1)
δ3 × l(γ),
then the motion of that member is in the direction ofdecreasing ‖υ‖ (i.e. toward the center of the swarmzcm).
Member of a “X Swarm” and staying outside thecontour
No restriction on the positions of the othermembers of the swarm or the position of zcm
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
CohesivenessMotion of a member outside the shape
Proposition
Consider a member i of a “X Swarm” staying outside
the desired shape at any given time, ifka
km>
(N − 1)
δ3 × l(γ),
then the motion of that member is in the direction ofdecreasing ‖υ‖ (i.e. toward the center of the swarmzcm).
Member of a “X Swarm” and staying outside thecontour
No restriction on the positions of the othermembers of the swarm or the position of zcm
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
CohesivenessMotion of a member outside the shape
Proposition
Consider a member i of a “X Swarm” staying outside
the desired shape at any given time, ifka
km>
(N − 1)
δ3 × l(γ),
then the motion of that member is in the direction ofdecreasing ‖υ‖ (i.e. toward the center of the swarmzcm).
Member of a “X Swarm” and staying outside thecontour
No restriction on the positions of the othermembers of the swarm or the position of zcm
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
Behavior AnalysisMotion of a member in side the contour
Proposition
Consider a member i inside a given contour γ(θ) withthe following properties:
1 γ(θ) = γ∗(2π − θ),
2 �(zi) = 0.
Then, �(Fi ,r) = 0.
Proposition
Consider a given contour γ(θ) = r(θ)eiθ with twosymmetric axes. Then, for a member i staying on theintersection of the symmetrical axes, the artificialrepulsion force from the contour (Fi ,r ) is zero.
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
Behavior AnalysisMotion of a member in side the contour
800 900 1000 1100 1200 1300
−1700
−1650
−1600
−1550
−1500
−1450
−1400
−1350
−1300
X−Coordinate [m]
Y−
Coo
rdin
ate
[m]
600 700 800 900 1000 1100 1200 1300 1400
−1800
−1700
−1600
−1500
−1400
−1300
−1200
X−Coordinate [m]
Y−
Coo
rdin
ate
[m]
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
Discussion on AnalysisSummary
For a X SwarmIf all the members are outside the shape, themotion of the center of mass of the swarm (zcm) istoward the center of mass of the contour (zc).If conditions on the proposition remain true, themotion of a robot outside the contour will betoward the center of mass of the swarm
Inside a symmetrical shape :A member will have a stable equilibrium point onthe symmetrical axis.The stable equilibrium point lies on theintersection of the axes.When all the members are inside the shape
the motion of the center of mass of the swarm(zcm) satisfies above condition
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
Discussion on AnalysisSummary
For a X SwarmIf all the members are outside the shape, themotion of the center of mass of the swarm (zcm) istoward the center of mass of the contour (zc).If conditions on the proposition remain true, themotion of a robot outside the contour will betoward the center of mass of the swarm
Inside a symmetrical shape :A member will have a stable equilibrium point onthe symmetrical axis.The stable equilibrium point lies on theintersection of the axes.When all the members are inside the shape
the motion of the center of mass of the swarm(zcm) satisfies above condition
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
Discussion on AnalysisSummary
For a X SwarmIf all the members are outside the shape, themotion of the center of mass of the swarm (zcm) istoward the center of mass of the contour (zc).If conditions on the proposition remain true, themotion of a robot outside the contour will betoward the center of mass of the swarm
Inside a symmetrical shape :A member will have a stable equilibrium point onthe symmetrical axis.The stable equilibrium point lies on theintersection of the axes.When all the members are inside the shape
the motion of the center of mass of the swarm(zcm) satisfies above condition
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
Discussion on AnalysisSummary
For a X SwarmIf all the members are outside the shape, themotion of the center of mass of the swarm (zcm) istoward the center of mass of the contour (zc).If conditions on the proposition remain true, themotion of a robot outside the contour will betoward the center of mass of the swarm
Inside a symmetrical shape :A member will have a stable equilibrium point onthe symmetrical axis.The stable equilibrium point lies on theintersection of the axes.When all the members are inside the shape
the motion of the center of mass of the swarm(zcm) satisfies above condition
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
Discussion on AnalysisSummary
For a X SwarmIf all the members are outside the shape, themotion of the center of mass of the swarm (zcm) istoward the center of mass of the contour (zc).If conditions on the proposition remain true, themotion of a robot outside the contour will betoward the center of mass of the swarm
Inside a symmetrical shape :A member will have a stable equilibrium point onthe symmetrical axis.The stable equilibrium point lies on theintersection of the axes.When all the members are inside the shape
the motion of the center of mass of the swarm(zcm) satisfies above condition
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
Discussion on AnalysisSummary
For a X SwarmIf all the members are outside the shape, themotion of the center of mass of the swarm (zcm) istoward the center of mass of the contour (zc).If conditions on the proposition remain true, themotion of a robot outside the contour will betoward the center of mass of the swarm
Inside a symmetrical shape :A member will have a stable equilibrium point onthe symmetrical axis.The stable equilibrium point lies on theintersection of the axes.When all the members are inside the shape
the motion of the center of mass of the swarm(zcm) satisfies above condition
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
Discussion on AnalysisSummary
For a X SwarmIf all the members are outside the shape, themotion of the center of mass of the swarm (zcm) istoward the center of mass of the contour (zc).If conditions on the proposition remain true, themotion of a robot outside the contour will betoward the center of mass of the swarm
Inside a symmetrical shape :A member will have a stable equilibrium point onthe symmetrical axis.The stable equilibrium point lies on theintersection of the axes.When all the members are inside the shape
the motion of the center of mass of the swarm(zcm) satisfies above condition
Background
Formation of a roboticSwarmProblem
Assumptions
Mathematical Model
Behavior Analysis
Swarming GuidedWeapons
Summary
Discussion on AnalysisSummary
For a X SwarmIf all the members are outside the shape, themotion of the center of mass of the swarm (zcm) istoward the center of mass of the contour (zc).If conditions on the proposition remain true, themotion of a robot outside the contour will betoward the center of mass of the swarm
Inside a symmetrical shape :A member will have a stable equilibrium point onthe symmetrical axis.The stable equilibrium point lies on theintersection of the axes.When all the members are inside the shape
the motion of the center of mass of the swarm(zcm) satisfies above condition
Background
Formation of a roboticSwarm
Swarming GuidedWeaponsMotivation
Solution
Summary
Outline
1 BackgroundOverviewNatural to Artificial Swarms
2 Formation of a robotic SwarmProblemAssumptionsMathematical ModelBehavior Analysis
3 Swarming Guided WeaponsMotivationSolution
4 Summary
Background
Formation of a roboticSwarm
Swarming GuidedWeaponsMotivation
Solution
Summary
Application Case StudyWide Spread Targets
In any Military operation
Neutralizing a target at onceWide Spread Targets
Air fieldsCommand CentersWeapon StorageVehicle convoyNaval fleet
Background
Formation of a roboticSwarm
Swarming GuidedWeaponsMotivation
Solution
Summary
Application Case StudyWide Spread Targets
In any Military operation
Neutralizing a target at onceWide Spread Targets
Air fieldsCommand CentersWeapon StorageVehicle convoyNaval fleet
Background
Formation of a roboticSwarm
Swarming GuidedWeaponsMotivation
Solution
Summary
Application Case StudyWide Spread Targets
In any Military operation
Neutralizing a target at onceWide Spread Targets
Air fieldsCommand CentersWeapon StorageVehicle convoyNaval fleet
Background
Formation of a roboticSwarm
Swarming GuidedWeaponsMotivation
Solution
Summary
Application Case StudyWide Spread Targets
In any Military operation
Neutralizing a target at onceWide Spread Targets
Air fieldsCommand CentersWeapon StorageVehicle convoyNaval fleet
Background
Formation of a roboticSwarm
Swarming GuidedWeaponsMotivation
Solution
Summary
Application Case StudyWide Spread Targets
In any Military operation
Neutralizing a target at onceWide Spread Targets
Air fieldsCommand CentersWeapon StorageVehicle convoyNaval fleet
Background
Formation of a roboticSwarm
Swarming GuidedWeaponsMotivation
Solution
Summary
Application Case StudyWide Spread Targets
In any Military operation
Neutralizing a target at onceWide Spread Targets
Air fieldsCommand CentersWeapon StorageVehicle convoyNaval fleet
Background
Formation of a roboticSwarm
Swarming GuidedWeaponsMotivation
Solution
Summary
Application Case StudyWide Spread Targets
In any Military operation
Neutralizing a target at onceWide Spread Targets
Air fieldsCommand CentersWeapon StorageVehicle convoyNaval fleet
Background
Formation of a roboticSwarm
Swarming GuidedWeaponsMotivation
Solution
Summary
Application Case StudyWide Spread Targets
In any Military operation
Neutralizing a target at onceWide Spread Targets
Air fieldsCommand CentersWeapon StorageVehicle convoyNaval fleet
Background
Formation of a roboticSwarm
Swarming GuidedWeaponsMotivation
Solution
Summary
Application Case StudyHow to Neutralize a Wide Spread Target
Some methods used:Nuclear AttackMultiple Launch RocketSystemCluster BombsSerial Bombing
Drawbacks:Uncontrolled destructionIncreased collateral damage
Civilian casualtiesProperty damages
Background
Formation of a roboticSwarm
Swarming GuidedWeaponsMotivation
Solution
Summary
Application Case StudyHow to Neutralize a Wide Spread Target
Some methods used:Nuclear AttackMultiple Launch RocketSystemCluster BombsSerial Bombing
Drawbacks:Uncontrolled destructionIncreased collateral damage
Civilian casualtiesProperty damages
Background
Formation of a roboticSwarm
Swarming GuidedWeaponsMotivation
Solution
Summary
Application Case StudyHow to Neutralize a Wide Spread Target
Some methods used:Nuclear AttackMultiple Launch RocketSystemCluster BombsSerial Bombing
Drawbacks:Uncontrolled destructionIncreased collateral damage
Civilian casualtiesProperty damages
Background
Formation of a roboticSwarm
Swarming GuidedWeaponsMotivation
Solution
Summary
Application Case StudyHow to Neutralize a Wide Spread Target
Some methods used:Nuclear AttackMultiple Launch RocketSystemCluster BombsSerial Bombing
Drawbacks:Uncontrolled destructionIncreased collateral damage
Civilian casualtiesProperty damages
Background
Formation of a roboticSwarm
Swarming GuidedWeaponsMotivation
Solution
Summary
Application Case StudyHow to Neutralize a Wide Spread Target
Some methods used:Nuclear AttackMultiple Launch RocketSystemCluster BombsSerial Bombing
Drawbacks:Uncontrolled destructionIncreased collateral damage
Civilian casualtiesProperty damages
Background
Formation of a roboticSwarm
Swarming GuidedWeaponsMotivation
Solution
Summary
Application Case StudyHow to Neutralize a Wide Spread Target
Some methods used:Nuclear AttackMultiple Launch RocketSystemCluster BombsSerial Bombing
Drawbacks:Uncontrolled destructionIncreased collateral damage
Civilian casualtiesProperty damages
Background
Formation of a roboticSwarm
Swarming GuidedWeaponsMotivation
Solution
Summary
Application Case StudyHow to Neutralize a Wide Spread Target
Some methods used:Nuclear AttackMultiple Launch RocketSystemCluster BombsSerial Bombing
Drawbacks:Uncontrolled destructionIncreased collateral damage
Civilian casualtiesProperty damages
Background
Formation of a roboticSwarm
Swarming GuidedWeaponsMotivation
Solution
Summary
Application Case StudyHow to Neutralize a Wide Spread Target
Some methods used:Nuclear AttackMultiple Launch RocketSystemCluster BombsSerial Bombing
Drawbacks:Uncontrolled destructionIncreased collateral damage
Civilian casualtiesProperty damages
Background
Formation of a roboticSwarm
Swarming GuidedWeaponsMotivation
Solution
Summary
Application Case StudyHow to Neutralize a Wide Spread Target
Some methods used:Nuclear AttackMultiple Launch RocketSystemCluster BombsSerial Bombing
Drawbacks:Uncontrolled destructionIncreased collateral damage
Civilian casualtiesProperty damages
Background
Formation of a roboticSwarm
Swarming GuidedWeaponsMotivation
Solution
Summary
Application Case StudyHow to Neutralize a Wide Spread Target
Some methods used:Nuclear AttackMultiple Launch RocketSystemCluster BombsSerial Bombing
Drawbacks:Uncontrolled destructionIncreased collateral damage
Civilian casualtiesProperty damages
Background
Formation of a roboticSwarm
Swarming GuidedWeaponsMotivation
Solution
Summary
Outline
1 BackgroundOverviewNatural to Artificial Swarms
2 Formation of a robotic SwarmProblemAssumptionsMathematical ModelBehavior Analysis
3 Swarming Guided WeaponsMotivationSolution
4 Summary
Background
Formation of a roboticSwarm
Swarming GuidedWeaponsMotivation
Solution
Summary
SolutionBounded Targets and Swarming Weapons
Most Wide-spread targetsBoundedClear geographical boundariesTarget can be distinguished
Impose a closed contour aroundthe target
Apply formation algorithmSome modification
Background
Formation of a roboticSwarm
Swarming GuidedWeaponsMotivation
Solution
Summary
SolutionBounded Targets and Swarming Weapons
Most Wide-spread targetsBoundedClear geographical boundariesTarget can be distinguished
Impose a closed contour aroundthe target
Apply formation algorithmSome modification
Background
Formation of a roboticSwarm
Swarming GuidedWeaponsMotivation
Solution
Summary
SolutionBounded Targets and Swarming Weapons
Most Wide-spread targetsBoundedClear geographical boundariesTarget can be distinguished
Impose a closed contour aroundthe target
Apply formation algorithmSome modification
Background
Formation of a roboticSwarm
Swarming GuidedWeaponsMotivation
Solution
Summary
SolutionBounded Targets and Swarming Weapons
Most Wide-spread targetsBoundedClear geographical boundariesTarget can be distinguished
Impose a closed contour aroundthe target
Apply formation algorithmSome modification
Background
Formation of a roboticSwarm
Swarming GuidedWeaponsMotivation
Solution
Summary
SolutionBounded Targets and Swarming Weapons
Most Wide-spread targetsBoundedClear geographical boundariesTarget can be distinguished
Impose a closed contour aroundthe target
Apply formation algorithmSome modification
Background
Formation of a roboticSwarm
Swarming GuidedWeaponsMotivation
Solution
Summary
SolutionBounded Targets and Swarming Weapons
Most Wide-spread targetsBoundedClear geographical boundariesTarget can be distinguished
Impose a closed contour aroundthe target
Apply formation algorithmSome modification
Background
Formation of a roboticSwarm
Swarming GuidedWeaponsMotivation
Solution
Summary
SolutionBounded Targets and Swarming Weapons
Most Wide-spread targetsBoundedClear geographical boundariesTarget can be distinguished
Impose a closed contour aroundthe target
Apply formation algorithmSome modification
Background
Formation of a roboticSwarm
Swarming GuidedWeaponsMotivation
Solution
Summary
SolutionTwo-stage controller
The controller for any member i ,
ui =
{Ui ,1 ; while |zi − zc | > rc for any iUi ,2 ; active after the first stage elapsed
(18)
Ui,1
Ui,2
Yes
No
Cc := if |zi-zc|>rc for any i
0 500 1000 1500 2000
1000
1200
1400
1600
1800
2000
2200
2400
2600
2800
3000
X−Coordinate [m]
Y−
Coo
rdin
ate
[m]
rc
zc
Background
Formation of a roboticSwarm
Swarming GuidedWeaponsMotivation
Solution
Summary
SolutionTwo-stage controller
Horizontal motionThe first stage controller
Ui,1 = Fi,A1 + Fi,M − Fi,F
Does not have the repulsion componentFi,A1 does not vanishes upon entering the targetcontour.
Fi,A1 := kA1
�γ
(z − αXi) ‖dz‖ , (19)
The Second stage controllerUi,2 = Fi,A2 + Fi,R2 + Fi,M − Fi,F
Same as the controller described before.
Vertical motionFree fall motiongravitational accelerationresisting drag force on the weapon
Background
Formation of a roboticSwarm
Swarming GuidedWeaponsMotivation
Solution
Summary
SolutionTwo-stage controller
Horizontal motionThe first stage controller
Ui,1 = Fi,A1 + Fi,M − Fi,F
Does not have the repulsion componentFi,A1 does not vanishes upon entering the targetcontour.
Fi,A1 := kA1
�γ
(z − αXi) ‖dz‖ , (19)
The Second stage controllerUi,2 = Fi,A2 + Fi,R2 + Fi,M − Fi,F
Same as the controller described before.
Vertical motionFree fall motiongravitational accelerationresisting drag force on the weapon
Background
Formation of a roboticSwarm
Swarming GuidedWeaponsMotivation
Solution
Summary
SolutionTwo-stage controller
Horizontal motionThe first stage controller
Ui,1 = Fi,A1 + Fi,M − Fi,F
Does not have the repulsion componentFi,A1 does not vanishes upon entering the targetcontour.
Fi,A1 := kA1
�γ
(z − αXi) ‖dz‖ , (19)
The Second stage controllerUi,2 = Fi,A2 + Fi,R2 + Fi,M − Fi,F
Same as the controller described before.
Vertical motionFree fall motiongravitational accelerationresisting drag force on the weapon
Background
Formation of a roboticSwarm
Swarming GuidedWeaponsMotivation
Solution
Summary
SolutionTwo-stage controller
Horizontal motionThe first stage controller
Ui,1 = Fi,A1 + Fi,M − Fi,F
Does not have the repulsion componentFi,A1 does not vanishes upon entering the targetcontour.
Fi,A1 := kA1
�γ
(z − αXi) ‖dz‖ , (19)
The Second stage controllerUi,2 = Fi,A2 + Fi,R2 + Fi,M − Fi,F
Same as the controller described before.
Vertical motionFree fall motiongravitational accelerationresisting drag force on the weapon
Background
Formation of a roboticSwarm
Swarming GuidedWeaponsMotivation
Solution
Summary
SolutionTwo-stage controller
Horizontal motionThe first stage controller
Ui,1 = Fi,A1 + Fi,M − Fi,F
Does not have the repulsion componentFi,A1 does not vanishes upon entering the targetcontour.
Fi,A1 := kA1
�γ
(z − αXi) ‖dz‖ , (19)
The Second stage controllerUi,2 = Fi,A2 + Fi,R2 + Fi,M − Fi,F
Same as the controller described before.
Vertical motionFree fall motiongravitational accelerationresisting drag force on the weapon
Background
Formation of a roboticSwarm
Swarming GuidedWeaponsMotivation
Solution
Summary
SolutionTwo-stage controller
Horizontal motionThe first stage controller
Ui,1 = Fi,A1 + Fi,M − Fi,F
Does not have the repulsion componentFi,A1 does not vanishes upon entering the targetcontour.
Fi,A1 := kA1
�γ
(z − αXi) ‖dz‖ , (19)
The Second stage controllerUi,2 = Fi,A2 + Fi,R2 + Fi,M − Fi,F
Same as the controller described before.
Vertical motionFree fall motiongravitational accelerationresisting drag force on the weapon
Background
Formation of a roboticSwarm
Swarming GuidedWeaponsMotivation
Solution
Summary
SolutionTwo-stage controller
Horizontal motionThe first stage controller
Ui,1 = Fi,A1 + Fi,M − Fi,F
Does not have the repulsion componentFi,A1 does not vanishes upon entering the targetcontour.
Fi,A1 := kA1
�γ
(z − αXi) ‖dz‖ , (19)
The Second stage controllerUi,2 = Fi,A2 + Fi,R2 + Fi,M − Fi,F
Same as the controller described before.
Vertical motionFree fall motiongravitational accelerationresisting drag force on the weapon
Background
Formation of a roboticSwarm
Swarming GuidedWeaponsMotivation
Solution
Summary
SolutionTwo-stage controller
Horizontal motionThe first stage controller
Ui,1 = Fi,A1 + Fi,M − Fi,F
Does not have the repulsion componentFi,A1 does not vanishes upon entering the targetcontour.
Fi,A1 := kA1
�γ
(z − αXi) ‖dz‖ , (19)
The Second stage controllerUi,2 = Fi,A2 + Fi,R2 + Fi,M − Fi,F
Same as the controller described before.
Vertical motionFree fall motiongravitational accelerationresisting drag force on the weapon
Background
Formation of a roboticSwarm
Swarming GuidedWeaponsMotivation
Solution
Summary
SolutionTwo-stage controller
Horizontal motionThe first stage controller
Ui,1 = Fi,A1 + Fi,M − Fi,F
Does not have the repulsion componentFi,A1 does not vanishes upon entering the targetcontour.
Fi,A1 := kA1
�γ
(z − αXi) ‖dz‖ , (19)
The Second stage controllerUi,2 = Fi,A2 + Fi,R2 + Fi,M − Fi,F
Same as the controller described before.
Vertical motionFree fall motiongravitational accelerationresisting drag force on the weapon
Background
Formation of a roboticSwarm
Swarming GuidedWeaponsMotivation
Solution
Summary
SolutionVertical Motion
Drag force (Fd ) on an object moving in a fluid isgiven by,
Fd =12
ρArCdV 2 (20)
Ar -Reference area for the drag forceCd -Drag constantρ-Density of the fluidVo is the speed of the object
Vertical motion dynamics of a weapon
dVt
dt+
ρArCd
2mV 2
t = g, (21)
Vt - vertical velocity of the weapong - gravitational acceleration
Background
Formation of a roboticSwarm
Swarming GuidedWeaponsMotivation
Solution
Summary
SolutionVertical Motion
Drag force (Fd ) on an object moving in a fluid isgiven by,
Fd =12
ρArCdV 2 (20)
Ar -Reference area for the drag forceCd -Drag constantρ-Density of the fluidVo is the speed of the object
Vertical motion dynamics of a weapon
dVt
dt+
ρArCd
2mV 2
t = g, (21)
Vt - vertical velocity of the weapong - gravitational acceleration
Background
Formation of a roboticSwarm
Swarming GuidedWeaponsMotivation
Solution
Summary
SolutionVertical Motion
Drag force (Fd ) on an object moving in a fluid isgiven by,
Fd =12
ρArCdV 2 (20)
Ar -Reference area for the drag forceCd -Drag constantρ-Density of the fluidVo is the speed of the object
Vertical motion dynamics of a weapon
dVt
dt+
ρArCd
2mV 2
t = g, (21)
Vt - vertical velocity of the weapong - gravitational acceleration
Background
Formation of a roboticSwarm
Swarming GuidedWeaponsMotivation
Solution
Summary
SolutionVertical Motion
Drag force (Fd ) on an object moving in a fluid isgiven by,
Fd =12
ρArCdV 2 (20)
Ar -Reference area for the drag forceCd -Drag constantρ-Density of the fluidVo is the speed of the object
Vertical motion dynamics of a weapon
dVt
dt+
ρArCd
2mV 2
t = g, (21)
Vt - vertical velocity of the weapong - gravitational acceleration
Background
Formation of a roboticSwarm
Swarming GuidedWeaponsMotivation
Solution
Summary
SolutionVertical Motion
Drag force (Fd ) on an object moving in a fluid isgiven by,
Fd =12
ρArCdV 2 (20)
Ar -Reference area for the drag forceCd -Drag constantρ-Density of the fluidVo is the speed of the object
Vertical motion dynamics of a weapon
dVt
dt+
ρArCd
2mV 2
t = g, (21)
Vt - vertical velocity of the weapong - gravitational acceleration
Background
Formation of a roboticSwarm
Swarming GuidedWeaponsMotivation
Solution
Summary
SolutionTime of Convergence
Uses the stage one controllerMotion of the entire weapon system:
m ε + kF ε + kA1 l(γ) ε = 0,
ε = (zcm − zc)ε = zcm, ε = zcm.
Motion of a single weapon with maximuminter-member repulsion
m υi + kF υi + kA1 l(γ) υi − kM(N − 1)
δ3 υ = 0,
Background
Formation of a roboticSwarm
Swarming GuidedWeaponsMotivation
Solution
Summary
SolutionTime of Convergence
Uses the stage one controllerMotion of the entire weapon system:
m ε + kF ε + kA1 l(γ) ε = 0,
ε = (zcm − zc)ε = zcm, ε = zcm.
Motion of a single weapon with maximuminter-member repulsion
m υi + kF υi + kA1 l(γ) υi − kM(N − 1)
δ3 υ = 0,
Background
Formation of a roboticSwarm
Swarming GuidedWeaponsMotivation
Solution
Summary
SolutionTime of Convergence
Uses the stage one controllerMotion of the entire weapon system:
m ε + kF ε + kA1 l(γ) ε = 0,
ε = (zcm − zc)ε = zcm, ε = zcm.
Motion of a single weapon with maximuminter-member repulsion
m υi + kF υi + kA1 l(γ) υi − kM(N − 1)
δ3 υ = 0,
Background
Formation of a roboticSwarm
Swarming GuidedWeaponsMotivation
Solution
Summary
SolutionTime of Convergence
Uses the stage one controllerMotion of the entire weapon system:
m ε + kF ε + kA1 l(γ) ε = 0,
ε = (zcm − zc)ε = zcm, ε = zcm.
Motion of a single weapon with maximuminter-member repulsion
m υi + kF υi + kA1 l(γ) υi − kM(N − 1)
δ3 υ = 0,
Background
Formation of a roboticSwarm
Swarming GuidedWeaponsMotivation
Solution
Summary
SolutionTime of Convergence
Uses the stage one controllerMotion of the entire weapon system:
m ε + kF ε + kA1 l(γ) ε = 0,
ε = (zcm − zc)ε = zcm, ε = zcm.
Motion of a single weapon with maximuminter-member repulsion
m υi + kF υi + kA1 l(γ) υi − kM(N − 1)
δ3 υ = 0,
Background
Formation of a roboticSwarm
Swarming GuidedWeaponsMotivation
Solution
Summary
SolutionTime of Convergence
Lemma
Consider a second order ordinary differential equationin the form of, φ + b1φ + b2φ = 0 having a generalsolution in the form of φ(t) = cφ,1eλφ,1t + cφ,2eλφ,2t ,with the following properties;(i) λφ,1, λφ,2 < 0,(ii) λφ,1 < λφ,2
Let φ(td ) = φd and φ(0) = φ0.For such a system, the following statement holds;
td <
ln(
φd
cφ,1
)λφ,1
,
Background
Formation of a roboticSwarm
Swarming GuidedWeaponsMotivation
Solution
Summary
SolutionTime of Convergence
1 Time for zcm to move into a circle around zc
having radius εd ≈ 0
t(εd ) <
ln(
εd
c1
)λ1
.
2 Time for the most distant weapon to move into acircle around zcm, with radius rc
t(rc) <
ln(
rc
c3
)λ3
.
tc < t(εd ) + t(rc)
Background
Formation of a roboticSwarm
Swarming GuidedWeaponsMotivation
Solution
Summary
SolutionMinimum Release Height
Proposition
All the weapons will converge into the givengeographical boundary (γ), if the release height of theweapons (hrel ) satisfy the following condition,
hrel >2m
ρAr Cdlog
(cosh
(√gρArCd
2mtm
))
where,
tm =
ln(
rc
c3
)λ3
+
ln(
εd
c1
)λ1
Background
Formation of a roboticSwarm
Swarming GuidedWeaponsMotivation
Solution
Summary
SolutionSimulations
Simulation Results
Background
Formation of a roboticSwarm
Swarming GuidedWeaponsMotivation
Solution
Summary
Swarming Guided WeaponsDiscussion
A multiple weapon control systemEliminate/minimize the collateral damageDeliver maximum fire power on the target
Mathematical analysis forA lower-bound of the release height
Background
Formation of a roboticSwarm
Swarming GuidedWeaponsMotivation
Solution
Summary
Swarming Guided WeaponsDiscussion
A multiple weapon control systemEliminate/minimize the collateral damageDeliver maximum fire power on the target
Mathematical analysis forA lower-bound of the release height
Background
Formation of a roboticSwarm
Swarming GuidedWeaponsMotivation
Solution
Summary
Swarming Guided WeaponsDiscussion
A multiple weapon control systemEliminate/minimize the collateral damageDeliver maximum fire power on the target
Mathematical analysis forA lower-bound of the release height
Background
Formation of a roboticSwarm
Swarming GuidedWeaponsMotivation
Solution
Summary
Swarming Guided WeaponsDiscussion
A multiple weapon control systemEliminate/minimize the collateral damageDeliver maximum fire power on the target
Mathematical analysis forA lower-bound of the release height
Background
Formation of a roboticSwarm
Swarming GuidedWeaponsMotivation
Solution
Summary
Swarming Guided WeaponsDiscussion
A multiple weapon control systemEliminate/minimize the collateral damageDeliver maximum fire power on the target
Mathematical analysis forA lower-bound of the release height
Background
Formation of a roboticSwarm
Swarming GuidedWeapons
Summary
Summary
A Formation algorithm multiple robot system wasintroduced.
Mathematical analysis of the behaviorComputer simulations of the behavior
An application of the algorithm was introduced.Mathematical analysis for the minimum releaseheightComputer simulations of the behavior
Background
Formation of a roboticSwarm
Swarming GuidedWeapons
Summary
Summary
A Formation algorithm multiple robot system wasintroduced.
Mathematical analysis of the behaviorComputer simulations of the behavior
An application of the algorithm was introduced.Mathematical analysis for the minimum releaseheightComputer simulations of the behavior
Background
Formation of a roboticSwarm
Swarming GuidedWeapons
Summary
Summary
A Formation algorithm multiple robot system wasintroduced.
Mathematical analysis of the behaviorComputer simulations of the behavior
An application of the algorithm was introduced.Mathematical analysis for the minimum releaseheightComputer simulations of the behavior
Background
Formation of a roboticSwarm
Swarming GuidedWeapons
Summary
Summary
A Formation algorithm multiple robot system wasintroduced.
Mathematical analysis of the behaviorComputer simulations of the behavior
An application of the algorithm was introduced.Mathematical analysis for the minimum releaseheightComputer simulations of the behavior
Background
Formation of a roboticSwarm
Swarming GuidedWeapons
Summary
Summary
A Formation algorithm multiple robot system wasintroduced.
Mathematical analysis of the behaviorComputer simulations of the behavior
An application of the algorithm was introduced.Mathematical analysis for the minimum releaseheightComputer simulations of the behavior
Background
Formation of a roboticSwarm
Swarming GuidedWeapons
Summary
Summary
A Formation algorithm multiple robot system wasintroduced.
Mathematical analysis of the behaviorComputer simulations of the behavior
An application of the algorithm was introduced.Mathematical analysis for the minimum releaseheightComputer simulations of the behavior
AppendixReferences
References
Ekanayake, S.W. and Pathirana, P.N.Formations of Robotic Swarm - An Artificial ForceBased ApproachInternational Journal of Advanced RoboticSystems, 6(1):7–24, 2009.
Ekanayake, S.W. and Pathirana, P.N.Two stage architecture for navigating multipleguided weapons into a widespread targetIEEE Aerospace conference, 2008.