Cooperative Optimization and Navigation Problems

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Cooperative Optimization and Navigation Problems
Dimitrios Hristu-Varsakelis Mechanical
Engineering and Institute for Systems
Research University of Maryland, College
Park http//glue.umd.edu/hristu hristu_at_glue.umd.e
du Joint work with M. Egerstedt, S. B.
Andersson, C. Shao. P.R. Kumar, P. S.
Krishnaprasad,
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Outline

• Ensembles of autonomous vehicles operating on
  expansive terrain.
• Bio-inspired trajectory optimization
• Language-based navigation
• Report on Progress Event-driven communication

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Ensembles of Autonomous Systems

• Examples from biology (bees, ants, fish etc.)
• Ensembles can accomplish tasks that are
  impossible for an individual.
• Coordination requires thinking about
  control/communication interactions.

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Trajectory optimization without a map

• A group of vehicles traveling between a fixed
  pair of locations
• Terrain is unknown - no global map.
• On-board sensing provides local information
  about vehicles immediate surroundings

target
vehicle
Vn-1
V1
Vn
obstacle
control station
start
PROBLEM Given an initial path between a pair of
start and target locations, find the optimal
path connecting that pair, using local
interactions between vehicles.
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Trajectory optimization without a map

• A group of vehicles traveling between a fixed
  pair of locations
• Terrain is unknown - no global map.
• On-board sensing provides local information
  about vehicles immediate surroundings

target
Vn-1
vehicle
V1
obstacle
Vn
control station
start
PROBLEM Given an initial path between a pair of
start and target locations, find the optimal
path connecting that pair, using local
interactions between vehicles.
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Local pursuit A biologically-inspired algorithm
k1
k
...
K2
...
Target
...
Start
Initial path
path followed by the k-th vehicle,

Theorem (on ) The iterated paths
converge to a straight line as
Bruckstein, 92
(on a smooth manifold M) If vehicle separation
is sufficiently small, then the iterated paths
converge to a geodesic.
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Experimental results with Euclidean
metric

• A collection of mobile robots with
• Wireless communication between neighbors
• Sonar and odometry sensors

TARGET
START
Initial path length 7m Vehicle separation 1.5m
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Local Pursuit
location of k-th vehicle
M
Minimum-length geodesic connecting to

Idea Find optimal trajectory to leader and
follow it momentarily.
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Pursuit decreases vehicle separation
M
Minimum-length geodesic connecting a to b
location of k-th vehicle
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Local pursuit for more general optimal control
problems
Let
Given an initial trajectory
with
Find that minimizes
s.t.
The k-th vehicle moves as follows Wait at
until t?(?1) At time t, follow the optimal
trajectory from to
As , iterated trajectories
converge to a local min. for
Assumptions uniqueness, smoothness
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Simulation pursuit on
5m trajectory 0.7m separation
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A sub-Riemannian example
fixed
5m trajectory 1.5m separation
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Pursuit in vehicles with drift (minimum time
problem)

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Summary and Work in Progress

• A biologically-inspired trajectory optimization
  algorithm
• - local pursuit forms a string of vehicles
• - each vehicle uses local information and
• communicates with its closest neighbors
• Target state and optimal trajectory are unknown
• Local convergence
• Experiments
• Escaping local minima
• Comparison with gradient descent methods

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Control in a reasonably complex world

• The problem of specifying control tasks (e.g. go
  to the refrigerator and get the milk)
• Solving motion control problems of adequate
  complexity
• Many interesting systems evolve in environments
  that are not smooth, simply connected, etc.
• Using language primitives to navigate
• Specify control policies
• Represent the environment (what parts do we
  ignore?)

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Motion Description Languages
Atom
Evaluate
Evolve under
until
Concatenate, encapsulate atoms to form complex
strings (plans), e.g.
Def MDLe is the formal language defined by the
context free grammar
with production rules
N nonterminals T terminals S start symbol e
empty string
Fact MDLe is context free but not regular
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Symbolic Navigation
World
M
x

• Keep only interesting details about how to
  navigate the world
• Landmark L (M,x) M map patch, x
  coordinates
• Sensor signature L Li if s(t) si(t) for t in
  t0,T
• Navigation
• Local navigation on a given landmark Li
• Global navigation between landmarks

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A directed graph representation of a map

• Represent only interesting parts of the world.
• G L,E
• Li landmarks
• Eij i,j,Gij
• Gij an MDLe program
• Eij ? Eji
• Idea Replace details locally by a feedback
  program

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Experiment indoor navigation
Lab 1
Lab 2
Office
Partial floor plan of 2nd floor A.V. Williams
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Experiment, contd

• Goal Navigate between three landmarks

Front of lab
Rear of lab
Office
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Experiment Example MDLe plans

• Lab2toLab1Plan (bumper)
• (Atom (atIsection 0100) (goAvoid 90 40 20))
• (Atom (atIsection 0010) (go 0 0.36))
• (Atom (wait ) (align 7 9))
• (Atom (atIsection 1000) (goAvoid 0 40 20))
• (Atom (atIsection 0100) (go 0 0.36))
• (Atom (wait ) (align 3 5))
• (Atom (wait 7) (goAvoid 270 40 20))
• (Atom (atIsection 1000) (goAvoid 270 40 20))

Lab1toOfficePlan (bumper) (Atom (atIsection
1001) (goAvoid 90 40 20)) (Atom (atIsection
0011) (go 0 0.36)) (Atom (wait ) (align 11
13)) (Atom (atIsection 0100) (goAvoid 180 40
20)) (Atom (wait 10) (rotate -90))
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Experiment A typical run
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Incorporating Uncertainty

• Controllers (and MDLe plans) are not always
  successful.
• Environmental factors (moving obstacles)
• System uncertainty (e.g. actuator noise)
• Associate a probability density function with an
  MDLe plan
• Enumerate the MDLe strings associated with an
  environment graph
• G L,E,
• Define Prob.
  of arriving at by executing from
• Assumptions
• G is a good description of the world
• Sensor model

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A prototype navigation problem

• How do I get to a given landmark ?

Information at time k
Prob. density at time k, given observations up to
time k.
Probability after evaluating plan and making
a new observation
Maximize probability of arriving at a desired
landmark in N steps
Maximize prob. of arrival at a desired landmark
with minimum of steps
Maximize probability of arriving at desired
landmark in N steps
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A navigation example
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Example - data
with N(0,0.01) actuator noise
Example L2 to L3 (syntax (?,u))
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Example steer to a landmark in N steps
X0L1 , XFL2, N3 P001/3, 1/3, 1/3
Desired success probability set to 95
Evolution of probability density on G
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Example steer to a landmark in N steps
True and observed landmarks
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Example steer to a landmark in N steps
Executed Plans
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Summary and Ongoing Work

• Language-based Control
• The motion description language MDLe
• Landmarkinstruction-based descriptions of the
  world
• Optimal navigation via dynamic-programming
• Obtaining nominal densities for navigation
• Software

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• References
• S. Andersson and D. Hristu-Varsakelis,
  Stochastic Language-based Motion Control, to
  appear, CDC 2003.
• D. Hristu-Varsakelis, M. Egerstedt, P. S.
  Krishnaprasad, On the Structural Complexity of
  the Motion Description Language MDLe, to appear,
  CDC 2003.
• D. Hristu-Varsakelis and P. R. Kumar,
  Interrupt-based feedback control over a shared
  communication medium, IEEE CDC 2002.
• M. Egerstedt and D. Hristu-Varsakelis,
  Observability and Policy Optimization for Mobile
  Robots, CDC 2002

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• Event-Based Stabilization of
• Ensembles-Users of a Shared Network

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Dynamical systems as users of a shared network
plant
G1(s)
G2(s)
GN(s)

shared medium
K1
K2
KN
controller

• Control of collections of systems with limited
  communication
• A prototype problem in divided attention.
• Nnumber of systems in the ensemble
• nmax. number of feedback loops that can be
  closed at any time
• How much communication time must be devoted to
  each system
• to guarantee that the collection remains
  stable?
• Can the ensemble be stabilized?

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A feedback communication policy

• We would like to avoid having to specify the
  communication policy in advance
• (thus the need for memory, clocks)
• How much information is needed to implement an
  event-driven policy?
• Lets define a simple rule for deciding which
  system(s) should be allowed to use the network.
• Idea Close loops corresponding to states that
  are furthest from the origin

Ex. N3, n2
.
.
0
.
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A feedback communication policy
Definition An ensemble is d-captured if
for all i after some time
Lets define a simple rule for deciding which
system(s) should be allowed to use the
network. Idea Close loops corresponding to
states that are furthest from the origin
Ex. N3, n2
.
.
0
.
For each system, find a Lyapunov function V( )
such that
(feedback loop closed) (feedback loop open)
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Some possibilities for interrupt-based
communication (special case n1)
Policy (CLS-e) Let 1. At time ,
close the loop of the system , 2. When
, set . 3.
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A test run
e1
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A least conservative feedback communication
policy
Policy (MACLS e-t) 1. At time t, close the
loop of the system where 2. When
,
repeat from step 1.
Theorem the ensemble is captured using CLS-e-t
for t large enough, if
where otherwise there exists a choice of
dynamics with the same for which
there is no stabilizing communication sequence.
An alternative communication policy
Policy (Control Zone e-z) Pick egtzgt0. 1. At
time t, close the loop of the system
where 2. When
, repeat from step 1.
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A simulation example (N3, n1, e1, t0.2)
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Experiment stabilizing a pair of pendulums
Lengths 20cm, 45cm Communication 115Kbps ?0.8
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Experiment stabilizing a pair of pendulums
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Event-based feedback control - Summary and Work
in Progress

• A class of feedback communication policies
• - sampled Lyapunov functions
• - continuously monitored Lyapunov functions
• - continuously monitored state norms
• Sufficient condition for stability
• Stochasticity
• Performance analysis
• Effects of delays in the feedback loop