IPAM, 3 Mar 06R. M. Murray, Caltech1 Cooperative Control of Multi-Vehicle Systems Richard M. Murray...
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Transcript of IPAM, 3 Mar 06R. M. Murray, Caltech1 Cooperative Control of Multi-Vehicle Systems Richard M. Murray...
IPAM, 3 Mar 06 R. M. Murray, Caltech 1
Cooperative Control of Multi-Vehicle Systems
Richard M. Murray
Control and Dynamical Systems
California Institute of Technology
Domitilla Del Vecchio (U Mich) Bill Dunbar (UCSC) Alex
Fax (NGC) Eric Klavins (U Wash)
Reza Olfati-Saber (Dartmouth)
Vijay Gupta Zhipu Jin Demetri Spanos
Abhishek Tiwari Yasi Mostofi
Cedric Langbort (CMI/UIUC)
IPAM, 3 Mar 06 R. M. Murray, Caltech 2
Arbiter
computersfor each vehicle
Humans(2-3 per team)
Example: RoboFlag (D’Andrea, Cornell)
Robot version of “Capture the Flag”• Teams try to capture flag of opposing
team without getting tagged• Mixed initiative system: two humans
controlling up to 6-10 robots• Limited BW comms + limited sensing
D’Andrea & MACC 2003
IPAM, 3 Mar 06 R. M. Murray, Caltech 3
QuickTime™ and aCinepak decompressor
are needed to see this picture.
RoboFlag Demonstration
Integration of computer science, communications, and control• Time scales don’t allow standard abstractions to isolate disciplines• Example: how do we maintain a consistent, shared view of the field?
Higher levels of decision making and mixed initiative systems• Where do we put the humans in the loop? what do we present to them?• Example: predict “plays” by the other team, predict next step, and react
Hayes et alACC 2003
Red Team view
Flagcarrier
Taggedrobot (blue)
Obstacle
IPAM, 3 Mar 06 R. M. Murray, Caltech 4
Outline
I. Cooperative Control subproblems
II. Information Flow for Consensus and Formation Control
III. Performance and Robustness
IV. Distributed receding horizon control
V. Protocols for cooperative control
IPAM, 3 Mar 06 R. M. Murray, Caltech 5
Context
Cooperative control of large collections of vehicles• Vehicles linked by task rather than dynamics• Distributed computation “consensus” required• Point to point communications; limited BW, noisy links• Dynamic and uncertain environments
Avenues of research• Stability and performance of interconnected systems• Robustness and reconfiguration of vehicle teams• Control over networks (“packet-based control theory”)• Higher-level decision making and autonomy
Model system for other applications• Air traffic control• Electric power networks• Congestion and router control
Integrated control,communications,
computation
IPAM, 3 Mar 06 R. M. Murray, Caltech 6
RoboFlag Subproblems
Goal: develop systematic techniques for solving subproblems• Cooperative control and graph Laplacians• Distributed receding horizon control• Verifiable protocols for consensus and control
1. Formation control
• Maintain positions to guard defense zone
2. Distributed estimation
• Fuse sensor data to determine opponent location
3. Distributed consensus
• Assign individuals to tag incoming vehicles
Implement and testas part of annual RoboFlag competition
IPAM, 3 Mar 06 R. M. Murray, Caltech 7
Information Flow in Vehicle Formations
Example: satellite formation• Blue links represent sensed
information• Green links represent
communicated information
Sensed information• Local sensors can see some subset of nearby
vehicles• Assume small time delays, pos’n/vel info only
Communicated information• Point to point communications (routing OK)• Assume limited bandwidth, some time delay• Advantage: can send more complex
information
Topological features• Information flow (sensed or communicated)
represents a directed graph• Cycles in graph information feedback loops
Question: How does topological structure of information flow affectstability of the overall formation?
IPAM, 3 Mar 06 R. M. Murray, Caltech 8
Sample Problem: Formation Stabilization
Goal: maintain position relative to neighbors• “Neighbors” defined by graph• Assume only sensed data for now• Assume identical vehicle dynamics, identical
controllers?
Example: hexagon formation• Maintain fixed relative spacing between left and
right neighbors
Can extend to more sophisticated “formations”• Include more complex spatia-temporal constraints
( )i
i j i j ijj N
e w y y h∈
= − −∑relativeposition
weightingfactor
offset
1 2
3
45
6
1 2
3
45
6
IPAM, 3 Mar 06 R. M. Murray, Caltech 9
Graph Laplacian
Construction of (weighted) Laplacian
A = adjacency matrix
D = diagonal matrix, weighted by outdegree
Properties of Laplacian• Row sum equal 0 (stochastic matrix)• All eigenvalues are non-negative, with at
least one zero eigenvalue (from row sum)• Multiplicity of 0 as an eigenvalue is equal
to the number of strongly connected compo-nents of the graph
• All eigenvalues lie in a circle of radius one centered at 1 + 0i (Perron Frobenius)
• For bidirectional (eg, undirected) graphs, eigenvalues are all real, in [0,2]
1 11 0 0 0
2 20 1 0 0 1 0
1 1 10 0 1
3 3 30 0 0 1 1 0
1 1 10 0 1
3 3 31 1
0 0 0 12 2
L
⎡ ⎤− −⎢ ⎥⎢ ⎥
−⎢ ⎥⎢ ⎥
− − −⎢ ⎥⎢ ⎥=
−⎢ ⎥⎢ ⎥⎢ ⎥− − −⎢ ⎥⎢ ⎥
− −⎢ ⎥⎢ ⎥⎣ ⎦
1L I D A−= −
IPAM, 3 Mar 06 R. M. Murray, Caltech 10
Mathematical Framework
Analyze stability of closed loop
• Interconnection matrix, L, is the Laplacian of the graph
• Stability of closed loop related to eigenstructure of the Laplacian
ˆ ( )K s ˆ( )P s
yL I⊗
y
h
e u
IPAM, 3 Mar 06 R. M. Murray, Caltech 11
Stability Condition
Theorem The closed loop system is (neutrally) stable iff the Nyquist plot of the open loop system does not encircle -1/i(L), where i(L) are the nonzero eigenvalues of L.
Example
ˆ ( )K s ˆ( )P s
yL I⊗
y
h
e u
2( )
seP s
s
τ−
= ( ) d pK s K s K= +
Fax and MurrayIFAC 02, TAC 04
IPAM, 3 Mar 06 R. M. Murray, Caltech 12
Spectra of Laplacians
0,1 =
Unidirectionaltree
2 ( 1) /1 i j Ni e π −= −
Cycle
[0,2] ∈
Undirected graph
1 0, 2N = =
Periodicgraph
IPAM, 3 Mar 06 R. M. Murray, Caltech 13
Example Revisited
Example
• Adding link increases the number of three cycles (leads to “resonances”)• Change in control law required to avoid instability• Q: Increasing amount of information available decreases stability (??)• A: Control law cannot ignore the information add’l feedback inserted
2( )
seP s
s
τ−
= ( ) d pK s K s K= +
x
x
x
x
IPAM, 3 Mar 06 R. M. Murray, Caltech 14
QuickTime™ and aMicrosoft Video 1 decompressorare needed to see this picture.
Improving Performance through Communication
Baseline: stability only• Poor performance due to interconnection
Method #1: tune information flow filter• Low pass filter to damp response• Improves performance somewhat
Method #2: consensus + feedforward• Agree on center of formation, then move• Compensate for motion of vehicles by
adjusting information flow
QuickTime™ and aMicrosoft Video 1 decompressorare needed to see this picture.
QuickTime™ and aMicrosoft Video 1 decompressorare needed to see this picture.
Fax and MurrayIFAC 02
IPAM, 3 Mar 06 R. M. Murray, Caltech 15
Special Case: Consensus
Consensus: agreement between agents using information flow graph• Can prove asymptotic convergence to single value if graph is connected
• If wij = 1/(in-degree) + graph is balanced (same in-degree for all nodes) all agents converge to average of initial condition
Variations and extensions (Jadbabaie, Leonard, Moreau, Morse, Olfati-Saber, Xiao, …)
• Switching (packet loss, dropped links, etc)• Time delays, plant uncertainty• Nearest neighbor graphs, small world networks, optimal weights• Nonlinear: potential fields, passive systems, gradient systems
-1
x
x
x
x
IPAM, 3 Mar 06 R. M. Murray, Caltech 16
Open Problems: Design of Information Flow (graph)
How does graph topology affect location of eigenvalues of L?• Would like to separate effects of topology from agent dynamics
• Possible approach: exploit for of characteristic polynomial
-1
x
x
x
x
IPAM, 3 Mar 06 R. M. Murray, Caltech 17
Performance
Look at motion between selected vehicles
Jin and MurrayCDC 04
G1 - Control G2 - Performance
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
IPAM, 3 Mar 06 R. M. Murray, Caltech 18
Robustness
What happens if a single node “locks up”
Different types of robustness (Gupta, Langbort & M)• Type I - node stops communicating (stopping failure)• Type II - node communicates constant value• Type III - node computes incorrect function (Byzantine failure)
Related ideas: delay margin for multi-hop models (Jin and M)• Improve consensus rate through multi-hop, but create sensitivity to
communcations delay
• Single node can change entirevalue of the consenus
• Desired effect for “robust”behavior: xI = /N
x1(0) = 4
x2(0) = 9
x3(0) = 6 x4(t) = 0
X5(t) = 6
x6(t) = 5
Gupta, Langbort and MurrayCDC 06
IPAM, 3 Mar 06 R. M. Murray, Caltech 19
RoboFlag Subproblems
Goal: develop systematic techniques for solving subproblems• Cooperative control and graph Laplacians• Distributed receding horizon control• Verifiable protocols for consensus and control
1. Formation control
• Maintain positions to guard defense zone
2. Distributed estimation
• Fuse sensor data to determine opponent location
3. Distributed consensus
• Assign individuals to tag incoming vehicles
Implement and testas part of annual RoboFlag competition
IPAM, 3 Mar 06 R. M. Murray, Caltech 20
Optimization-Based Control
Task:• Maintain equal
spacing of vehicles around circle
• Follow desired trajectory for center of mass
Parameters:• Horizon: 2 sec• Update: 0.5 sec
Local MPC + CLF• Assume neighbors follow
straight lines
Global MPC + CLF
Dunbar and MIFAC 02
QuickTime™ and aMicrosoft Video 1 decompressorare needed to see this picture.
IPAM, 3 Mar 06 R. M. Murray, Caltech 21
Individual optimization:
Theorem. Under suitable assumptions, vehicles are stable and converge to global optimal solution.
Pf Detailed Lyapunov calculation (Dunbar thesis)
Main Idea: Assume Plan for Neighbors
stat
e
timet0 t0+d
z3(t0)
z3*(t;t0) z3
k(t)
What 2 assumes
What 3 does
Compatibility constraint:• each vehicle transmits
plan to neighbors• stay w/in bounded path
of what was transmitted
IPAM, 3 Mar 06 R. M. Murray, Caltech 22
Example: Multi-Vehicle Fingertip Formation
1
3
2
4
qref
d31
IPAM, 3 Mar 06 R. M. Murray, Caltech 23
Simulation Results
IPAM, 3 Mar 06 R. M. Murray, Caltech 24
RoboFlag Subproblems
Goal: develop systematic techniques for solving subproblems• Distributed receding horizon control• Packet-based, distributed estimation• Verifiable protocols for consensus and control
1. Formation control
• Maintain positions to guard defense zone
2. Distributed estimation
• Fuse sensor data to determine opponent location
3. Distributed consensus
• Assign individuals to tag incoming vehicles
Implement and testas part of annual RoboFlag competition
IPAM, 3 Mar 06 R. M. Murray, Caltech 25
P(k1,k2) := { initializers guard1:rule1
guard2:rule2
...
}
S(k1,k2):=P(k1,k2)+C(k1+1) sharing y,u
"soup" of guarded commands
composition = union
non-shared variables remain local to component programs
CCL: Computation and Control LanguageFormal Language for Provably Correct Control Protocols
CCL Interpreter
Formal programming lang-uage for control and comp-utation. Interfaces with libraries in other languages.
Automated Verification
CCL encoded in the Isabelle theorem prover; basic specs verified semi-automatically. Investigating various model checking tools.
Formal Results
Formal semantics in transition systems and temporal logic. RoboFlag drill formalized and basic algorithms verified.
CCL Protocol forDecentralized
Target Allocation
IPAM, 3 Mar 06 R. M. Murray, Caltech 26
QuickTime™ and aMicrosoft Video 1 decompressorare needed to see this picture.
Example: RoboFlag Drill KlavinsCDC, 03
Things that we can prove (so far)• Semi-automated proofs (Isabelle)• Avoidance: no two robots collide• Self-stabilization: if attackers are
far enough away, defenders self-stabilize before attackers arrive
Next steps• Implement CCL on MVWT• Improved reasoning toolbox
Sample drill• N on N attack, w/
replenishment• Random initial
assignments• Switching proto-
col to avoid collisions
IPAM, 3 Mar 06 R. M. Murray, Caltech 27
Observation of CCL Programs
Del Vecchio & Klavins, CDC'03
Problem: Determine state of communications protocol used by a group of robots given their physical movements.
Assumptions: Protocol and motion control are described in CCL like language.
Results: • Defns of observability, etc. for CCL programs• Construction and analysis of observer that
converges when the system is "weakly" observable
• Construction of an efficient observer for Roboflag drill in particular
• Everything specified in CCL
IPAM, 3 Mar 06 R. M. Murray, Caltech 28
Summary: Cooperative Control of Multi-Vehicle Systems
1. Formation control
• Maintain positions to guard defense zone
2. Distributed estimation
• Fuse sensor data to determine opponent location
3. Distributed consensus
• Assign individuals to tag incoming vehicles
Integration of computer science, communications, and control• Mixture of techniques from computer science, communications, control• Increased need for reasoning at higher levels of abstraction (strategy)