Sensing Planning for Robotic Sensor Networks

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Sensing Planning for Robotic Sensor Networks

• Volkan Isler
• Rensselaer Polytechnic Institute

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Outline

• Robotic sensor networks operating in dynamic
  environments
• Todays focus representative problems
• Sensors selection
• Sensor placement
• Pursuit-evasion games
• Selected ongoing projects
• Robotic Routers
• Human-Robot Interaction

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A generic tracking scenario
Where am I?
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A simple camera model
X
x
Image Plane
Optical center
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Localization with perfect sensing
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A simple camera model with uncertainty
X
x
Image Plane
Optical center
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Target localization with uncertainty
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The sensing model

• Each sensor measurement corresponds to a convex,
  polygonal subset of the plane. (i.e.
  intersection of a finite number of half-planes)
• The true location is contained in all the sets
• Are there such sensors? Omni-directional cameras.
• Estimation ?intersections of measurementsUncertai
  nty ? area of the intersection

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The sensing model

• Note that more measurements never hurt in terms
  of uncertainty.
• There are cases where all sensors contribute to
  the estimation.
• Example cameras on a circle, target at the
  center

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The sensor selection problem

• Ideally use all sensors. May not be feasible!
• Given
• A rough estimate of the robots location
• locations of n sensors,
• select k sensors so as to minimize the estimation
  area
• Bicriteria Optimization Minimize cost (number
  of sensors), Maximize Utility (1/Uncertainty
  1/Area)

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Why care?
Bad/Good 612
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The sixth sensor theorem!

• Let S be the set of all sensors
• For any S and any given target location, there
  exists a subset S of S with S lt 6such that
  
• Uncertainty(S) / Uncertainty(S) lt 2

If you are happy with a factor 2 approximation,
you never need more than 6 active sensors!
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Overview of the proof

• The Minimum Enclosing Parallelogram
• Properties

C. Schwarz, J. Teich, A. Vainshtein, E. Welzl,
and B. L. Evans, Minimal enclosing parallelogram
with application, in SCG 95
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Overview of the proof

• Using these two properties, we can bound the area
  of the MEP.

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Sensor Selection Algorithm

• Let N be the number of halfplanes, N ? n
• Intersect all n measurements O(N log N)
• Compute the MEP O(N)

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How to use this result

• For n static sensors, we can partition the plane
  and compute a look-up table.
• Can be computed offline.

Isler, Bajcsy. Sensor Selection for Bounded
Uncertainty Sensing Models. IEEE TASE, 2006
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Recent Results

• It turns out that 4 sensors are enough for a
  2-approximation (the proof has a similar flavor
  but uses minimum enclosing triangles)
• Higher dimensions
• 3D 9 approximation with 8 sensors
• In d dimensions d(d-1) approximation with
  d(d1) sensors

Isler, Magdon-Ismail. Sensor Selection in
Arbitrary Dimensions. IEEE TASE, to appear
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Placement of stereo cameras

• In the previous problem, we took the placement as
  given
• How about a good placement?
• Lets start with stereo cameras
• That is, only 2 cameras are chosen
• Justification previous theorem does not utilize
  symmetry/cones experience
• Whats the advantage?

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Uncertainty in stereo

• Can be represented in closed from

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Placement Problem

• Given
• a planar workspace W
• an uncertainty threshold U
• Place minimum number of cameras such that
• For p ? W, let c1(p) and c2(p) be the best choice
  of cameras to track p
• WantU(p, c1(p), c2(p)) ? U for all p ? W

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Initial Model

• Cameras can be placed anywhere on the plane
• No occlusions in the workspace
• We make no assumptions about the shape of the
  workspace
• Application Fire watch towers on a (flat) forest.

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Result

• Let OPT be the optimal algorithm which achieves
  U uncertainty with a placement of k cameras.
• We present an algorithm that
• uses at most ?k cameras, and
• Guarantees ?U uncertainty
• ? and ? are two constants. There is a trade-off
  between them.
• Ill present the result for ?6 and ? 3.

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Placement Algorithm

• Has two phases. Phase 1

where
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SelectSensors illustration
2R
R
R
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Claim

• If the optimal algorithm, OPT, achieves U with k
  cameras, then
• (the number of centers) ? k
• Proof idea
• For each center, draw a disk of radius R around
  it
• Note that these disks are disjoint
• OPT must have at least one sensor in each disk

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Suppose not
R
R
OPTs uncertainty gt R R / sin ? ? R2 ? U A
contradiction!
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Phase II Place Cameras
2R
For each center, we place 3 cameras on the
vertices of an equilateral triangle. The total
number of cameras ? 3k
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More precisely
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Error Bound

• For every point inside a circle, we can find two
  cameras (out of the three we placed) such that
  the uncertainty is at most 6U

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So far

• If OPT achives U uncertainty with k cameras
• We can achive
• 6 U uncertainty with 3k cameras, or
• In general, it is possible to improve the
  uncertainty guarantee at the expense of using
  more cameras.

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Recent results

• Deal with occlusions
• a more strict error metric
• d1,d2 ? D
• ? ? ?? ?- ?
• hard constraints on the distance and the angle
  instead of a threshold on their product
• Our algorithm uses at most O(OPT log(OPT))
  sensors and guarantees
• d1,d2 ? D
• ?/2 ? ?? ?- ?/2
• And accommodates constraints on the set of
  candidate locations

Tekdas and Isler, ICRA 2007
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Pursuit-Evasion on Graphs

• Players restricted to vertices of a graph
• Can move from u to v iff (u,v) is an edge
• The goal is to capture the evader i.e. to be
  co-located at the same vertex
• Visibility Models
• e.g. A player located at v can see only N(v)
• Motion Models
• Players move simultaneously

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The role of information

• Global Visibility (Cops Robbers Game)
• Introduced in Nowakowski Winkler83
• Need unbounded of pursuers AignerFromme84
• Characterization for special cases
• 1-pursuer win dismantlable graphs
  BrightwellWinkler00
• No Visibility (Hunter Rabbit Game)
• One pursuer suffices
• O(nm2) with random walks Aleliunas et al.79
• O(n log(n)) algorithm Adler et al02

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Limited (Local) Visibility

• Players can see only their neighborhoods
• As mentioned earlier, one hunter (pursuer) is
  enough if the rabbit (evader) has no visibility
• Our case

A rabbit with local visibilitycan not be
captured by a single hunter
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How many hunters to capture?
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Two hunters always suffice

• Theorem On any graph, two hunters suffice for
  capturing a rabbit with local visibility in
  expected polynomial time.
• We present a randomized strategy.

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Randomized Strategies

• The hunters
• make decisions based on the outcome of coin
  tosses
• their strategy works against any rabbit strategy
• The rabbit
• knows the hunters strategy beforehand
• does not know the outcome of the hunters coin
  tosses

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Strategy Overview

• Three phases
• Phase 0 Locate the rabbit
• Phase 1 Chase the rabbit
• Phase 2 Set a trap and attack the rabbit
• Will show The probability of capture tends to 1.

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Phase 0 Locating the rabbit

• Divide into rounds of length n
• Guess the vertex v from where the rabbit will be
  visible at the end of the round
• Go to v and wait till the end of the round
• Probability of success is ? 1/n
• After n log(n) rounds
• Probability of success is ? 1-1/n
• Using (1x) lt exp(x)
• The hunters will locate the rabbit w.h.p.in no
  more than n2 log(n) time-steps

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Chasing the rabbit in phase 1
The rabbit
Does not see H2
H2 Chases H1
H1 Chases the rabbit, occasionally attacks
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Trapping the rabbit in phase 2

• If H1 chases the rabbit for n steps, the rabbit
  must revisit a vertex v.
• H2 stops at v and attacks when the rabbit enters
  N(v) for the first time through u.
• Can do this, because the rabbit never sees H2.
• Guess u, v and the revisit time.

u
v
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Capture Time

• At the end of the three phases
• Probability of capture 1/n3
• Total length of phases
• O(n2 log n) w.h.p.
• Overallthe rabbit will be captured in O(n6) time

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The big picture

• Key players in the evolution of information
  technology
• Sensing (cameras, biosensors, )
• Actuation (mobile nodes, pan-tilt cameras, )
• Robotic sensor networks
• Communication, sensing and actuation
• Progress in these areas have been fairly
  independent

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Robotic Sensor Networks

• I am interested in the interplay between
  communication, actuation and sensing
• Many challenges
• Succinct problem formulations
• Environment complexity, dynamic environments
• Coordination among many entities ? yields hard
  optimization problems
• Interaction with humans especially in
  health-care, elder-care, education
• Lets see some examples

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Robotic Routers
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A third robot can relay messages and ensure
connectivity
Idea use robotic routers to keep mobile
clients connected to a base-station
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Robotic Routers

• Keep mobile users connected to a base station
• In some cases, static deployment is wasteful
• Examples farming, military applications
• How should the routers move?

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How to model the user?

• Known trajectory robot whose trajectory is
  preprogrammed
• Adversarial trajectory no clue about the
  trajectory. Assume the worst case, the user tries
  to break the connection as quickly as possible
  (becomes a pursuit-evasion game)

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Does Mobility Help? Depends on the environment,
connectivity model and the speed of the routers
n/3 stationary nodes vs. 1 mobile router (which
is as fast as the target)
Visibility based communication two robots can
communicate if they can see each other
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Examples
One robotic router
Two robotic routers
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Results

• Can compute optimal (i.e max connection time)
  algorithms for the single user case
• Known trajectory ? dynamic programming
• Adversarial trajectory ? game theoretic solution
  (modified dynamic programming)
• Catch running time exponential in the number of
  robots

Isler and Tekdas, ICRA 2008
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A Recent Human Robot Interaction Project

• How to design human-friendly controllers?
• Friendly ? Not causing stress
• Incorporate stress measurements (from a Galvanic
  Skin Response Sensor) as feedback

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A crossing task

• Go Fast
• Go Slowly

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Control with GSR Feedback

• A representative task
• Track a person with a robot, in a way that does
  not cause distress
• Applications health-care monitoring, robotic
  shopping carts, mobile teleconferencing (?)
• Essentially a reinforcement learning task
• Challenge hard to obtain samples (human
  experiments)

Meisner, Isler, and Trinkle. Autonomous Robots,
2008
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Second HRI Project

• Identify underlying principles of interaction
• What makes a robot interactive?
• Prototype system

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GOAL
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Shadow puppets

• Current work
• Parse the video into basic tokens (nod, shake
  etc)
• Combine these into schemas scripts
  (initiation, greeting etc.) that exist in human
  interaction
• Can show that understanding the underlying
  context can help in prediction
• Work in progress demonstrate its utility in
  interaction

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Roadmap Ahead

• Many challenging problems at the intersection of
  sensing, actuation and communication
• Novel optimization problems ? theoretical results
• Application specific challenges ? proof of
  concept implementations and deployments
• Interactions with humans becoming increasingly
  common and important

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Thanks for your attention!

• Group
• Supported in part by NSF CCF-0634823 and NSF
  CNS-0707939.

Eric Meisner (HRI)
Onur Tekdas (Robotic Routers)
Nikhil Karnad (Pursuit-Evasion)
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Recent results

• Deal with occlusions
• a more strict error metric
• d1,d2 ? D
• ? ? ?? ?- ?
• hard constraints on the distance and the angle
  instead of a threshold on their product
• Our algorithm uses at most O(OPT log(OPT))
  sensors and guarantees
• d1,d2 ? D
• ?/2 ? ?? ?- ?/2
• And accommodates constraints on the set of
  candidate locations

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HRI Project 2

• How do people assign human attributes to other
  (non-human) objects?

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HRI Research Agenda

• Learn how people interact with each other for a
  given context
• Build a robot that explores the context and
  interacts with people

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What about other sensing models?

• Case I convex but non-polygonal

• In this case, we can approximate the uncertainty
  with a polygon efficiently.

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What about other sensing models?

• Case II non-convex uncertainty

Sometimes there is an efficient approximation
with a convex-shape.E.g. range and bearing
measurement
Sometimes not! E.g. range-only
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What about other sensing models?

• In this case one solution is to group small
  numbers of sensors and treat each group as a
  single sensor.

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An application of the planar result

• How to estimate the location of a target on a
  known plane (e.g. ground).
• Using cameras (whose parameters are also known).

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An application

• Experimental setup Ruzena Bajcsys lab at UC
  Berkeley. 48 calibrated cameras.

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Location of a target on a known plane
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Intersections with the plane
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Estimation Errors
Best
Worst
2
3
As good as it gets.
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Location of a target on a known plane
The chosen ones
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Adding more uncertainty

• So far, we assumed that our estimate of the
  targets location is a point.
• What if we have a region of uncertainty U?

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Online SSP

• Given an uncertainty region U, and a set of
  sensors
• We select a subset S
• An adversary picks1. the true location of the
  target (inside U) 2. sensors for the true
  location
• Performance? Competitive ratio

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Can we beat such an adversary?

• Meaning constant competitive ratio?

C
What is a good pair of cameras?
U
B
A
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Bounding the competitive ratio

• Define Z as furthest U gets from a sensord as
  closest U gets to a sensor
• Bad news The adversary can force an increase
  of O((Z/d)2) in our estimation area by changing
  the true location.
• MoralMust choose sensors before uncertainty U
  gets too big.

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Randomized vs. Deterministic
There is no deterministic strategy which
guarantees that the rabbit will be captured
regardless of its strategy.
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Characterization?

• A single hunter is not always enough
• What is the class of hunter-win graphs?
• We will present an algorithmic characterization