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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

Joint work with:

M. Egerstedt, S. B. Andersson, C. Shao.

P.R. Kumar, P. S. Krishnaprasad,

Ensembles of autonomous vehicles operating on “expansive” terrain.

Bio-inspired trajectory optimization

Language-based navigation

Report on Progress – Event-driven communication

OutlineEnsembles 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.

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 vehicle’s 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.

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 vehicle’s 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.

Local pursuit: A biologically-inspired algorithm

k+1

k

...

K+2

...

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.

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

: location of k-th vehicle

M

: Minimum-length

geodesic connecting to

Idea: Find optimal trajectory to leader and follow it momentarily.

Pursuit decreases vehicle separation

M

: Minimum-length

geodesic connecting a to b

: location of k-th vehicle

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=Δ(κ+1)

At time t, “follow the optimal trajectory” from to

As , iterated trajectories

converge to a local min. for

Assumptions:

uniqueness,

smoothness

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

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?)

Motion Description Languages

Atom:

Evolve under

until

Evaluate

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

ε:empty string

Fact: MDLe is context free but not regular

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

Symbolic NavigationWorld

M

x

Represent only “interesting parts” of the world.

G = {L,E}

Li : landmarks

Eij : {i,j,Gij}

Γij: an MDLe program

Eij Eji

Idea: Replace details locally by a feedback program

A directed graph representation of a map{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))

}

Experiment: Example MDLe plans{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))

}

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:

Incorporating UncertaintyHow do I get to a given landmark ? A prototype navigation problem

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”

Example: steer to a landmark in N “steps”

X0=L1 , XF=L2, N=3

P0|0=[1/3, 1/3, 1/3]

Desired success probability set to 95%

Evolution of probability density on G

Example: steer to a landmark in N “steps”

True and observed landmarks

Example: steer to a landmark in N “steps”

Executed Plans

- Language-based Control
- The motion description language MDLe
- “Landmark+instruction”-based descriptions of the world
- Optimal navigation via dynamic-programming
- Obtaining “nominal” densities for navigation
- Software

- 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

Event-Based Stabilization of

Ensembles-Users of a Shared Network

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”.
- N=number of systems in the ensemble
- n=max. 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?

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?
- Let’s 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.: N=3, n=2

.

.

0

.

A feedback communication policy

Definition: An ensemble is δ-captured if for all i after some time

Let’s 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.: N=3, n=2

.

.

0

.

For each system, find a Lyapunov function V( ) such that:

(feedback loop closed)

(feedback loop open)

2’. When , set .

feedback

Policy: (open loop CLS):

1’’. At time , close the loop of the system ,

2’’. When , set .

Some possibilities for interrupt-based communication (special case n=1)

Policy: (CLS-e): Let

1. At time , close the loop of the system ,

2. When , set .

3.

e=1

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 e>z>0.

1. At time t, close the loop of the system where

2. When , repeat from step 1.

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

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