Distributed Consensus in
Multivehicle Cooperative Control:
Theory
y and Applications
pp
Wei Ren and Randal W. Beard
Springer
ISBN: 978-1-84800-014-8
Tutorial Slides Prepared by Wei Ren
Department of Electrical and Computer Engineering
Utah State University
[email protected]
http://www.neng.usu.edu/ece/faculty/wren/
Potential Applications for Autonomous Vehicles
Civil and Commercial:
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Automated Mining
Monitoring environment
Monitoring disaster areas
Communications relays
Law enforcement
Precision agriculture
Military:
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Special Operations: Situational
Awareness
Intelligence, surveillance, and
reconnaissance
Communication node
Battle damage assessment
Homeland Security:
y
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Intel
Border patrol
Surveillance
Rural/Urban search and rescue2
Epson
Cooperative/Coordinated Control
• Motivation:
While single vehicles performing solo missions will
yield some benefits
benefits, greater benefits will come from the
cooperation of teams of vehicles.
courtesy: SRI, http://www.ai.sri.com
• Common Theme:
Coordinate the movement of multiple vehicles in a
certain way to accomplish an objective.
- e.g. many small, inexpensive vehicles acting together can
achieve more than one monolithic vehicle.
vehicle
e.g., networked computers
Shifts cost and complexity from hardware platform to courtesy: Airforce Technology, http://www.airforce-technology.com
software and algorithms.
• Multi-vehicle Applications:
Space-based interferometers, future combat systems,
surveillance and reconnaissance,
reconnaissance hazardous material
handling, distributed reconfigurable sensor networks …
3
courtesy: NASA, http://planetquest.jpl.nasa.gov
Cooperative Control Categorization
• Formation Control
- Approaches: leader-follower,
leader follower behavioral,
behavioral virtual
structure/leader, artificial potential function, graph-rigidity
- Applications: mobile robots, unmanned air vehicles,
autonomous underwater
d
vehicles,
hi l satellites,
lli
spacecraft,
f
automated highways
• Task Assignment, cooperative transport, cooperative role
assignment, air traffic control, cooperative timing
- Cooperative search,
search reconnaissance,
reconnaissance surveillance (military,
(military
homeland security, border patrol, etc.)
- Cooperative monitoring of forest fires, oil spills, wildlife, etc.
- Rural search and rescue.
4
Cooperative Control: Inherent Challenges
• Complexity:
p
y
• Systems of systems.
• Communication:
• Limited bandwidth and connectivity.
• What? When? To whom?
• Arbitration:
• Team vs. Individual goals.
• Computational
C
i l resources:
• Will always be limited
5
Cooperative Control: Centralized vs Distributed Schemes
• Centralized Schemes
Assumptions: availability of global team knowledge,
centralized planning and coordination, fully connected
network
Practical Issues: sparse & intermittent interaction
topologies (limited communication/sensing range,
environmental factors)
• Distributed Schemes
Features: Local neighbor-to-neighbor interaction, evolve
in a parallel manner
Strengths: reduced communication/sensing requirement;
improved scalability, flexibility, reliability, and
robustness
b
6
Distributed Consensus Algorithms
• Basic Idea
p
its information state based
Each vehicle updates
on the information states of its local (possibly
time-varying) neighbors in such a way that the
final information state of each vehicle
converges to a common value.
• Extensions
R l ti state
Relative
t t deviations,
d i ti
incorporation
i
ti off other
th
group behaviors (e.g., collision avoidance)
• Feature
Only local neighbor-to-neighbor interaction
required
Boids: http://www.red3d.com/cwr/boids/
7
R
Vicsek’s Model
Consensus Algorithms – Literature Review
• Historical Perspective
biology, physics, computer science, economics, load
balancing in industry, complex networks
• Theoretic Aspects
algebraic graph theory, nonlinear tools, random network,
optimality and synthesis, communication delay,
asynchronous communication, …
• Applications
rendezvous, formation control, flocking, attitude
synchronization, sensor fusion, …
Weii R
W
Ren, R
Randal
d lW
W. B
Beard,
d Distributed
Di t ib t d C
Consensus iin M
Multi-vehicle
lti hi l Cooperative
C p ti Control,
C t l
Communications and Control Engineering Series, Springer-Verlag, London, 2008
(ISBN: 978-1-84800-014-8)
8
Modeling of Vehicle Interactions
A directed spanning tree
A undirected graph
that is connected
(i) Separated groups
A directed graph that
is strongly connected
A graph that has a spanning
tree but not strongly connected
9
(ii) Multiple leaders
Modeling of Vehicle Interactions (cont.)
adjacency matrix
10
Laplacian matrix
Outline
•
Part 1: Consensus for SingleSingle-integrator Kinematics –
Theory and Applications
•
Part 2: Consensus for Double-integrator
Double integrator Dynamics –
Theory and Applications
Part 3: Consensus for Rigid Body Attitude Dynamics
– Theory and Applications
Part 4: Synchronization of Networked Euler
EulerLagrange Systems – Theory and Applications
•
•
11
Consensus Algorithm for 1st-order Kinematics
12
Convergence Result (Fixed Graph)
The Laplacian matrix has a simple zero eigenvalue and all
th others
the
th have
h
positive
iti reall parts.
t
Directed graph
has a directed
spanning tree.
Consensus is
reached.
13
Sketch of the Proof
An inductive approach
• Step 1: find a spanning tree
that is a subset of the graph
• Step 2: show that consensus
can be achieved with the
spanning tree (renumber each
agent))
(1)
• Step 3: show that if
consensus can be achieved for
a graph, then adding more
links will still guarantee
consensus
(2)
14
(3)
Consensus Equilibrium (Fixed Topology)
The
h iinitial
i i l condition
di i off a node
d contributes
ib
to the
h equilibrium
ilib i
value
l
if and only if the node has a directed path to all the other nodes.
15
Convergence Result (Switching Graphs)
16
Sketch of the Proof
17
Examples - Consensus and Directed Spanning Trees
Consensus can be achieved:
Equilibrium determined by
vehicles 1 and 2
Cases when consensus cannot be achieved:
(i) Separated groups
(ii) Multiple leaders
18
Consensus can be achieved:
Union of (i) and (ii)
Rendezvous ‐ Experiments
Experiment was performed in CSOIS (joint work
with Chao, Bougeous, Sorensen, and Chen)
Switching Topologies
Union Topology
19
Rendezvous Demos ‐ Switching Topologies
Union Topology
Switching Topologies
20
Axial Alignment
21
Consensus with a Virtual Leader
22
Sketch of the Proof
23
Examples – Consensus Tracking
Subcase (a)
1.5
reference
1
ξ
i
0.5
0
−0.5
−1
−1.5
0
2
4
6
8
10
t (sec)
12
14
16
18
20
Subcase (b)
2
reference
1.5
ξ
i
1
0.5
0
−0.5
0
2
24
4
6
8
10
t (sec)
12
14
16
18
20
Example - Virtual Leader/Structure Based Formation Control (Centralized)
rjF
rj
rjd
(xvc,yvc,vc)
CF
Co
25
Example - Virtual Leader/Structure Based Formation Control (decentralized)
rjF
rjd
CFj
Co
26
A Unified Scheme for Distributed Formation Control
Communication Network
Consensus-based Formation
State Estimator Module #i
Group Leader
Follower
Consensus-based Formation
Control Module #i
Vehicle #i
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Formation State Estimation Level
28
Vehicle Control Level
29
Experimental Platform at USU
30
Mobile Robot Kinematic Model
v
y
(xh,yh)
d
θ
(x y)
(x,y)
o 31
x
Experimental Specification
32
Formation Control with a Simple Group Leader
33
Experimental Demonstration (formation control)
Four robots maintaining a square shape
Three robots in line formation
34
Outline
•
Part 1: Consensus for Single-integrator Kinematics –
Theory and Applications
•
Part 2: Consensus for DoubleDouble-integrator Dynamics –
Theory and Applications
•
Part 3: Consensus for Rigid Body Attitude Dynamics
– Theory and Applications
Part 4: Synchronization of Networked EulerLagrange Systems – Theory and Applications
•
35
Consensus Algorithm for Double-integrator Dynamics
36
Convergence Analysis (Relative Damping)
37
Example – with Relative Damping
1.4
ξ1
ξ2
ξ3
ξ4
1.2
1
ξ
i
0.8
0.6
0.4
0.2
0
0
1
2
3
4
5
t (s)
6
7
8
9
10
0.3
ζ1
ζ2
ζ3
ζ4
0.2
ζi
0.1
0
−0.1
−0.2
0
1
2
3
4
5
t (s)
6
38
7
8
9
10
Switching Topologies – Directed Case
39
Sketch of the Proof
40
Application - Cooperative Monitoring with UAVs
Single UAV Experiment
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Application – UAV Formation Flying
3D postflight analysis of the flight on July 4th, 2008.
(Leader and follower are set at different altitudes to
avoid possible collision.
collision The blue dots represent the
leader's positions and the red dots represent the
follower's positions.)
42
Actuator Saturation
43
Coupled Second-order Harmonic Oscillators
44
Convergence Analysis (Leadless Case)
45
Application - Synchronized Motion Coordination
46
Outline
•
•
Part 1: Consensus for Single-integrator Kinematics –
Theory and Applications
Part 2: Consensus for Double-integrator Dynamics –
Theory and Applications
•
Part 3: Consensus for Rigid Body Attitude Dynamics
– Theory and Applications
•
Part 4: Synchronization of Networked EulerLagrange Systems – Theory and Applications
47
Consensus for Rigid Body Attitude Dynamics
48
Sketch of the Proof
49
Unavailability of Angular Velocity Measurements
50
Sketch of the Proof
51
Application - Spacecraft Attitude Synchronization
Synchronized spacecraft rotations
Courtesy: NASA JPL
52
Experimental Demonstration (attitude control)
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Reference Attitude Tracking
54
Reference Attitude Tracking Control Law
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Sketch of the Proof
56
Application - Spacecraft Reference Attitude Tracking
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Outline
•
•
•
•
Part 1: Consensus for Single-integrator Kinematics –
Theory and Applications
Part 2: Consensus for Double-integrator Dynamics –
Theory and Applications
Part 3: Consensus for Rigid Body Attitude Dynamics
– Theory
eo y andd Applications
pp c o s
Part 4: Synchronization of Networked Euler
Euler-Lagrange Systems – Theory and Applications
58
Synchronization of Networked Euler-Lagrange Systems
59