Challenges and Techniques for Efficient Contour Covering using - - PowerPoint PPT Presentation

Challenges and Techniques for Efficient Contour Covering using Collaborating Mobile Sensors

Sumana Srinivasan

Advisor : Dr. Krithi Ramamritham Co-advisor : Dr. Purushottam Kulkarni Department of Computer Science and Engineering Indian Institute of Technology Bombay, Mumbai, INDIA

Pre-synopsis Seminar, March 26th, 2010

1 / 94

A spill and its “wake”(up call)!!

• FIG. 1: Exxon spill, 1989 — Millions
• f marine life forms destroyed.
• FIG. 2: Statfjord spill, 2007

◮ Inefficient way of determining extent and mitigating spread!! ◮ Could the unprecedented adverse impact of this disaster been

diminished by a quicker and more efficient response?

2 / 94

Cyber Physical Systems to the rescue!!

• FIG. 3: Applications of CPS
• FIG. 4: Applications of

CPS

Can advances in CPS be used to determine extent and mitigation of disasters?

3 / 94

Challenges for CPS

◮ Can CPS determine the exact location and extent of the

most hazardous region accurately?

◮ Can the region be covered energy efficiently to contain the

spill and perform remedial actions making best use of information that is available?

◮ Can the system achieve best coverage under energy

constraints?

◮ Can they adapt to limitations of the system such as —

limited sensing, communication range and energy available?

4 / 94

How do you determine extent?

Contour estimation — determining extent of “hazardous region” and sources of contamination.

• FIG. 5: A light field with light intensity

contours.

Parameters

◮ Temperature ◮ Pressure ◮ Depth ◮ Light intensity ◮ Concentration ◮ Salinity ◮ Density ◮ A contour is defined as a set of points of equal value of a

specified parameter.

5 / 94

Outline

◮ Motivation ◮ Challenges and Contributions ◮ System model, Problem formulation and Evaluation metrics ◮ Evolution of strategies based on Information Utilization ◮ Our Solution — Ingredients and evaluation ◮ Conclusions

6 / 94

Existing techniques for contour estimation

Techniques Accuracy Minimize Physically Perform Adaptive depends cover the remedial to spatio- upon contour actions temporal dynamics Remote Image Error No No Yes sensing Resolution Static High Communication No No No sensor node cost networks density Mobile Number of Movement Yes Yes Yes sensor samples cost networks

7 / 94

Mobile sensors - State of the art

◮ SOTAB1 — image sensors, viscosity sensors, wind and depth meter,

water thermometers.

◮ Dorado - Used for mapping the sea bed equipped with sonars. ◮ Robotic carp — tiny chemical sensors in the port of Gijon in northern

Spain to find sources of pollutants in the water. Energy consumption due to movement — COSTLY!!!

8 / 94

Contour Covering Problem

CCP

Using a network of mobile sensors, can we determine the location and extent of a hazardous contour in a bounded sensing region and cover it accurately in an energy efficient manner?

Objectives

◮ Maximize the coverage of contour ⇒ hazardous region of

the spill is contained.

◮ Minimize the coverage of nonrelevant regions ⇒ better

resource utilization.

◮ Minimize energy consumption due to movement and

response time ⇒ increase lifetime and timely action.

9 / 94

System Model and Assumptions

2D Model

Notations.

Assumptions

◮ Target contour

continuous and static w.r.t. sensors.

◮ Sensors are location

aware.

◮ No odometry errors. ◮ Sensing measurements

are smoothened by averaging.

10 / 94

Fundamental Challenges of CCP

• 1. Which direction should the sensor move ?

◮ How does the sensor make a nontrivial decision of

choosing one direction over others given the constraints of the system?

◮ What factors does this decision depend upon? ◮ Are there any trade-offs in choosing one direction over any

• ther?
• 2. What information is needed to determine direction? How is

the information gathered and utilized?

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Contributions

• 1. A novel formulation to the Contour Covering Problem

(CCP) with maximizing coverage, minimizing coverage error and energy consumption.

• 2. A proof of hardness for the optimal latency for CCP

.

• 3. Motion planning algorithms based on information

exploitation

◮ Baseline:Gradient Descent G and Surround S ◮ Minimizing Centroid Distance MCD [Srinivasan and

Ramamritham, 2006]

◮ Centralized Periodic Udate Adaptive Contour Estimation

ACE [S.Srinivasan et al., 2008]

◮ Energy-aware Distributed and Adaptive Contour Covering E

• 4. Extensive performance evaluation of the algorithms

◮ deployments, contours, energy constraints, number of sensors,

communication range and sensing error.

• 5. Comparison with optimal latency and previous mobility

strategies..

12 / 94

Evaluation Metrics

• Coverage. C and Coverage Error,

ℵ

C = Area of (Cest ∩ Cact) Area of Cact ℵ = Area of ([Creg − Cact] ∩ Cest) Area of [Creg − Cact]

Latency, L

L = max(T1, T2, . . . , Tm) (1)

Precision, φ and F-measures, Fβ

Φ = Area of (Cest ∩ Cact) Area of Cest Fβ = (1 + β2) × Φ × C β2 × Φ + C

GO 13 / 94

Contour Covering Problem CCP Definition

Given N mobile sensors s1, · · · sN,

◮ each with energy, einiti ◮ deployed in a 2D bounded sensing region R ≡ [l × l], ◮ defined by a scalar field f such that f(x, y) ∈ [L, U] (where

L and U are lower and upper bounds of the field value) and

◮ value of the target contour τ ∈ [L, U],

the goal is to determine the points on the estimated level set in

• rder to
• 1. Maximize coverage, C

subjected to constraints,

• 1. Minimize coverage error, ℵ
• 2. Minimize latency, L

14 / 94

Movement Phases and Termination Condition

Movement Phases

◮ Converge Phase: Movement towards the contour until the

sensor hits the contour

◮ Coverage Phase: Movement along the contour until

termination.

Termination Conditions

◮ Condition I: When none of the sensors have energy to

move.

◮ Condition II: When the contour is fully estimated or every

point on the contour is visited by at least one sensor (i.e., C = 100% and ℵ = 0%).

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Type of Information

16 / 94

Strategies for CCP based on information utilization

Strategy 1 — Complete Information

◮ Information: Locations of contour points and locations of other sensors

are known to a sensor

◮ Technique: Determination of optimal latency for CCP (OPTCCP) is NP

• Complete. MTSP reduces to OPTCCP

.

Strategy 2 — No information

◮ Information: Measurement at current location only known to a sensor ◮ Techniques:

• 1. Exhaustive search, based on space filling curves [Spires

and Goldsmith, 1998] to locate the contour — worst case O(l2)

• 2. Random search, based on random walk to locate the

contour — worst case O((l2)3) (Brightwell Winkler Theorem [G. and P ., 1990])

17 / 94

Strategies for CCP , contd.

Strategy 3 — Overlap information

◮ Information: Location and field values from other sensors are known to

a sensor and overlap w.r.t. contour.

◮ Techniques:

• 1. All sensors inside, move in the direction of maximizing hull

area.

• 2. All sensors outside and overlapping, move in the direction
• f minimizing hull area,
• 3. If no overlap, how should they determine the direction of

movement?

18 / 94

Strategies for CCP , contd.

Strategy 4 — Gradient information

∇f = ( ∂f ∂x , ∂f ∂y ) ∂f ∂x = f(x + h) − f(x) h

◮ Information: Location and field value from all sensors. ◮ Techniques:

• 1. Clustered deployments to minimize h, spatially correlated

readings to determine ∇f [Marthaler and Bertozzi, 2004, Zhang and Leonard, 2005].

• 2. Use exporation to determine samples for approximating

gradient.

• 3. Can sensors that can sense beyond their location be used?

19 / 94

Are measurable gradients present within rs?

Sensors rs Concentration/Slick Thickness Height Laser fluorosensor up to 50m submicrons 3m SlickSleuth several meters 0.001% 1.5 - 5m ◮ Take home: If deployed close to the spill (within 100kms) quickly (within 0.5d), steep gradients (0-60ppb) can be found within a spatial resolution of 1m which can be detected by sensors today.

OIL MODEL 20 / 94

Gradient Descent Algorithm, G

Information

Uses only measurements within rs to determine direction of approach.

Technique

Define artificial potential gradient function, gf(x, y) =

• (1 − f(x,y)

τ

)2 if f(x, y) ≤ τ (1 −

τ f(x,y))2

if f(x, y) > τ (2)

◮ G moves in the direction that minimizes gf or moves in the

direction of steepest gradient.

◮ If f is a continuous convex function, then gf is convex with

a minimum at f = τ.

◮ Perform exploration in the absence of gradients (spiral

search)

21 / 94

Limitations of G

◮ When sensors are clustered —

does not maximize coverage under energy constraints

◮ Higher latency in the absence of

energy constraints.

◮ Sensors can get into local

minima in a non-uniform field.

• FIG. 6: Gradient

descent under clustered deployment

Can sensors move in a direction that approaches as well as spread out with respect to contour?

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Surrounding the contour

Information

◮ Inputs are

centroid of the contour, measurements within rs and location information from

• ther sensors.

Technique

Angular distribution around centroid.

◮ Compute and assign target angles such that the total

angular difference is minimized, the Hungarian algorithm [H.W.Kuhn, 1955].

23 / 94

Surround Algorithm, S

◮ Define an artificial potential spread function that

determines the proximity of the sensor to its target angle.

Spread function

sf(x, y) = (θd π )2, θd = |θt − θ| (3)

◮ Spread function sf(x, y) is convex. ◮ In S, at every step, the sensor sorts all the points in rs

based on the spread function and amongst the top three chooses the one whose field value is closest to τ.

24 / 94

Limitations of S

◮ The main limitation is the sensitivity to the input centroid.

Can the centroid be dynamically estimated using information from other sensors?

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Just approach or surround, not enough!!

• FIG. 7: Gradient good, Surround bad
• FIG. 8: Gradient bad, Surround good

Parameters

◮ Distance from contour ◮ Size of contour ◮ Extent of spread of sensors ◮ Can the sensors adaptively decide when to approach or surround based on the parameters?

26 / 94

Combining approach and surround, a preliminary study

Proposed and compared three strategies: Greedy Algorithm (GA), Simulated Annealing (SA) and Minimizing Centroid Distance Algorithm (MCD).

MCD Algorithm [Srinivasan and Ramamritham, 2006]

ci(xi, yi) = α ∗ gf(xi, yi)

• approach

+(1 − α) ∗ sf(xi, yi)

• spread

(4)

◮ Assumes centroid of the contour to be input to the

algorithm.

◮ α is a preset constant. α = 0.5 in converge phase ◮ Coverage phase begins only after a preset number of

sensors converge on the contour.

27 / 94

Lessons learnt from MCD

◮ Sensitive to centroid. ◮ Latency is sensitive to choice of α ◮ Latency is sensitive to number of sensors on contour when

coverage phase begins. Can α be chosen adaptively, centroid be estimated dynamically, converge and coverage phases be overlapped?

28 / 94

Energy-aware, Adaptive, Distributed Contour Covering Algorithm, E.

• FIG. 9: Ingredients of the solution

29 / 94

Estimating Cpred(t)

Contour estimate Cpred(t) is computed by collaboration.

◮ IECP → set of all points on contour where non-converged

sensors are expected to converge,

◮ ITP → set of all points visited by converged sensors where

f(x, y) = τ, then the global estimate of the contour,

◮ Cpred(t) is a convex hull bounding the points in

{ITP IECP}.

◮ Convex hull computed using the Graham Scam

Algorithm [Graham, 1972] How to compute the estimated convergence point for sensor si?

30 / 94

Estimating convergence point, (ˆ x, ˆ y)

Information

Location and field value information of all the points traversed by the sensors so far — history.

Technique

◮ Fit history with a model f

′, an approximation of the sensing

field, f if available, or use a generic model and compute the coefficients [Nelder and Mead, 1965] e.g., zi = p0 + p1e−p2xi + p3e−p4yi (5)

◮ Determine the ECP for sensor si, (ˆ

xi, ˆ yi), where (f

′ − τ) = 0 by root finding.

◮ Multiple roots? - Choose ECP that is on the contour

closest to the target angle of the sensor

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Modeling bias factor α as a regression tree

◮ α decreases with extent of spread of sensors ◮ α increases with the increase distance from the contour ◮ α increases with decreasing size of contour

32 / 94

Estimating bias factor, α

Information

Location of other sensors and Cpred(t) and current location.

Technique

◮ A sigmoid function that satisfies these conditions is positive tanh

function. α = tanh( di Rpred(t) × Pext(t) Ppred(t)) where the radius of Cpred(t), denoted as Rpred(t)

33 / 94

Combining approach and surround using α

Approaching using Cpred(t).

Information

Current location and Cpred(t).

Technique

An artificial potential df(x, y) ∈ (0, 1), that minimizes the distance to the contour. df(x, y) = Distance to Cpred(t) Maximum distance to the contour in sensing region R (6) where, normalization factor is √ 2 × l − Minimum diameter of Cpred(t)

◮ Cpred(t) absent??, use the gf for approach.

34 / 94

Surrounding the contour using Cpred(t)

Information

Current location of the sensor, locations of other sensors and centroid of Cpred(t).

Technique

◮ Current angle θ(x, y) using centroid and current location

information.

◮ Target angle using centroid and location information of

• ther sensors by collaboration.

◮ Spread function sf(x, y) using

sf(x, y) = (θd π )2, θd = |θt − θ| (7)

35 / 94

Combining Approach and Surround

Adaptive spread function

An artificial potential asf(x, y) — a weighted combination of the distance and the spread functions. asf(x, y) = α × df(x, y) + (1 − α) × sf(x, y) (8) where α ∈ [0, 1] is the bias factor.

◮ α = 1 ⇒ the sensor moves directly towards the contour ◮ α = 0, the sensor moves towards its target angle. ◮ For all other 0 < α < 1, the sensor combines the directions

• f moving towards the contour and spreading.

36 / 94

Coverage Phase — Wall moving Algorithm

• FIG. 10: Wall Following Algorithm
• FIG. 11: Movement a

sensor performing wall following

37 / 94

Determination of termination in coverage phase

A converged sensor si with eremi > 0, terminates when

◮ Approach I: It reaches a point on the contour already covered by

another sensor. Does not maximize coverage under energy constraints.

◮ Approach II: It returns to its starting point on the contour. Overlaps

paths with other sensors and therefore inefficient.

◮ Approach III: If (∀si ∈convergederemi (t) > Ppred(t)) perform I else perform II.

— Used in E.

38 / 94

Centralized, Periodic Update Algorithm, ACE [S.Srinivasan et al., 2008]

Objective

Given N mobile sensors si, i ∈ (1, · · · , N) each with unbounded energy, deployed in a sensing region R defined by field f and target contour value τ, the goal is to minimize latency.

Comparison with E

ACE E Minimizes L under Maximizes C and unbounded energy minimizes ℵ, L under bounded energy Periodic update Aperiodic update Latency sensitive to

• n demand

period Adaptive bias Energy aware adaptive bias factor bias factor Centralized Distributed

39 / 94

Bias factor under energy constraints

◮ When energy remaining in the sensor < the amount of

work to be done.

◮ A conservative estimate of amount of work remaining to be

done is Wi(t) = di(t) + Ppred(t) If eremi(t) < Wi(t), α = 0 ⇒ Perform surround to maximize coverage else, Perform approach or surround adaptively.

40 / 94

E Algorithm

41 / 94

Additional Optimizations

• 1. Should far lying sensors, outliers, participate in surround?

◮ Detect outliers. ◮ Set α = 1 for outliers and eliminate them fromtarget angle

computation and assignment.

• 2. Should nonconverged sensors be redirected to uncovered

sectors?

◮ Determine uncovered sectors. ◮ Assign uncovered sector to nonconverged sensors (explore

various schemes)

42 / 94

Evaluation Questions

◮ What is the effect of adaptation on the performance of E

when compared to the baseline algorithms?

◮ How does the latency of E and the baseline algorithms

compare to optimal latency whenever the latter can be computed?

◮ How does the performance of E compare wit other mobility

strategies in the literature?

◮ Is the performance sensitive to scaling of number of

sensors, information reduction due to decreasing communication range and sensing errors?

◮ What is the effect of the additional optimizations on the

performance of E?

43 / 94

Adaptation on performance metrics

In diffused depl., perf.

• f E ≥ G by (1-8%)

and E > S by (3-22%). In clustered depl.,

• perf. E ≥ G by

(0.4-14%) and E ≥ S by 1-14%. Performance of E ≥ S, G.

44 / 94

When is energy aware bias factor beneficial?

◮ In diffused deployment, CE > CEna by 1.5%, ℵE > ℵEna by 4.3% and

φE < φENA by 7.5%.

◮ In clustered deployment, CE > CEna by 3.5%, ℵE > ℵEna by 0.2% and

φE < φENA by 3%.

45 / 94

Comparison with optimal latency

LE > LOPT by 1.2, LG > LOPT by 1.3, LS > LOPT by 1.8

46 / 94

Comparison with near-optimal latency

◮ Under diffused deployments, LE > LMTSP by 1.2,

LS > LMTSP by 1.4 and LG > LMTSP by 1.25 (in the best case, LE = LG = LMTSP, LS > LMTSP by 1.3).

◮ LE > LMTSP by 1.3, LG > LMTSP by 1.4, , LS > LMTSP by 1.5

(in the worst case, LE > LMTSP by 1.2, LG > LMTSP by 2, LS > LMTSP by 1.6)

47 / 94

Sensitivity of E

Scaling

◮ Performance gain decreases (30%-0%) with scaling under

diffused deployment.

◮ Performance gain increases by 10% with scaling under

clustered deployment.

GO

Information reduction

◮ Performance gain improves (0-10%) with increasing rt. We

• bserve trade-off between precision and coverage.

(Validates claim - more information better performance).

GO

Sensing errors

◮ Coverage is unaffected. Precision is reduced up to 10% in

the presence of sensing errors.

◮ Latency is not sensitive to sensing errors.

GO 48 / 94

Effect of additional optimizations

Outlier Elimination

◮ Precision gain (8%) at the cost of loss in coverage (3%). ◮ Latency is unaffected by outlier elimination.

GO

Sensor redirection

◮ Sensor redirection does not always lead to improvement in

performance.

◮ Decision to redirect depends on size of the uncovered

sectors, distance to the uncovered sector and energy remaining in the sensors (Future Work).

GO 49 / 94

How does E compare to other mobility strategies?

Experiment Strategies Results Comparison with [Gupta and Ramanathan, 2007] LUM > LE by a factor of 15

• ther mobility

[A.Singh et al., 2006] LASA > LE by a factor of 15 strategies [Marthaler and Bertozzi, 2004] LSBE > LE by 12% [Ogren et al., 2004] MCD LMCD > LE by a factor of 4 [Srinivasan and Ramamritham, 2006] ACE LACE > LE by 5% [S.Srinivasan et al., 2008]

50 / 94

Conclusions

◮ Identified challenges and proposed an Energy aware,

Adaptive algorithm for CCP , E.

◮ Showed that the optimal path planning for CCP is NP

Complete and E is within 1.3 times the optimal latency whenever it can be determined.

◮ Showed by extensive performance evaluation that E has

significant “bang for the buck” over baseline algorithms and

• ther competing strategies.

◮ Demonstrated feasibility on a realistic testbed.

51 / 94

Future Work

◮ Covering discontinuous contours. ◮ Adaptive learning and spatial interpolation techniques to

improve contour estimate prediction.

◮ Tracking dynamic contours. ◮ Adaptively changing direction of movement in coverage

phase to improve coverage.

◮ Adaptively deciding whether to redirect or not based on

sizeof uncovered sectors, distance from them and energy remaining in the sensors.

52 / 94

Acknowledgements

◮ Krithi Ramamritham ◮ Puru Kulkarni ◮ Parmesh Ramanathan ◮ Folks at Embedded Systems Lab ◮ Folks at CSRE (GRAM++ team)

53 / 94

BACK UP SLIDES

54 / 94

Bang for the buck!!

Components Benefit/Cost Effort Outlier Gain in precision (8%) O(n) where n is elimination loss in coverage (3%) the number of nonconverged sensors Energy-aware Gain in coverage (6%) Same complexity bias factor Loss in precision (15%) as aperiodic update Aperiodic update reduction in latency factor of k, update by 7% period Adaptive bias Reduction in latency by Nelder Meaad scheme compared to fixed a factor of 4 complexity O(n) bias where n is number of simplex vertices (2) Graham scan for hull O(nlogn) where n is number of points in geometry Combination of Performance gain 5-20% Same complexity approach and Loss correlated to quality as surround surround

• f prediction model

Surround Performance good under clustered The Hungarian deployment, worse under diffused problem, bipartite deployment O(N4) Gradient Performance good under diffused O(k) Descent deployment and worse under clustered k , neighborhood deployment points 55 / 94

What caused the poor response?

◮ Delay in communication. ◮ A single helicopter with a dispersant in a bucket was used

to contain the spill as a first response to the disaster.

◮ For the first few days after the spill, most of the oil was in a

large concentrated patch near Bligh Island. On March 26, a storm with winds of over 70 mph weathered much of the

• il, and distributed it over a large area.

56 / 94

To explore or to exploit?

◮ Generalized motion planning algorithm is NP

Hard [LaValle, 2006]

◮ Search spaces of many realistic problems contain

information that can be exploited for intelligent motion planning.

◮ Contour covering is essentially a search problem with task

goals.

◮ Competing goals [Rickert et al., 2008] —

◮ exploration to maximize information gain and ◮ exploitation of information avaliable to attain task objectives.

◮ Balancing exploration and exploitation is a requisite for

efficient motion planning. Can strategies be designed which exploit information available for sensors and perform exploration only in the absence of such information for CCP?

57 / 94

Generalized Motion Planning Problem (LaValle, 2006)

The Piano Mover’s Problem

◮ A world W in which W = ℜ2 or W = ℜ3. ◮ A semi-algebraic obstacle region O ⊂ W in the world. ◮ A semi-algebraic robot is defined in W. It may be a rigid

robot A or a collection of mlinks A1, A2, . . . , Am.

58 / 94

Piano Mover’s problem is PSPACE-hard

◮ The configuration space C determined by specifying the set

• f all possible transformations that may be applied to the
• robot. From this Cobs and Cfree are derived.

◮ A configuratin qI ∈ Cfree designated as initial configuration. ◮ A configuration qG ∈ Cfree designated as final configuration. ◮ A complete algorithm must compute a (continuous) path,

τ : [0, 1] − → Cfree, such that τ(0) = qI and τ(1) = G, or correctly report that such a path does not exist. It was shown by J.H.Reif (Reif, 1979) that this problem is PSPACE-hard which implies NP-hard. The main problem is that the dimension of C is unbounded.

BACK 59 / 94

Novelty in Formulation

BACK

• FIG. 12: Optimization metrics in CCP

60 / 94

Motion Planning Algorithms I

◮ Gradient Descent Algorithm, G — direction of movement

using only the measurements available within the sensing range approaching the contour.

◮ Surround Algorithm, S — direction of movement using

measurements within the sensing range as well as an estimate of the centroid of the contour surrounding the contour

◮ Minimizing Centroid Distance Algorithm, MCD, a

combination of approach and surround with equal weightage for both, uses measurements within the sensing range as well as an estimate of the centroid of the contour [Srinivasan and Ramamritham, 2006]

61 / 94

Motion Planning Algorithms II

◮ Adaptive Contour Estimation Algorithm, ACE — an

adaptive choice between approach and surround uses measurements within sensing range, history and measurements from other sensors using a centralized periodic exchange of information [S.Srinivasan et al., 2008]

◮ Energy -aware, Distributed, Adaptive Contour Covering

Algorithm, E — an energy aware adaptive choice between approach and surround uses measurements within sensing range, history and measurements from other sensors using aperiodic exchange of information.(To be submitted)

◮ Additional optimizations

• 1. Effect of eliminating far way sensors (also known as
• utliers) from participating in surround, EOE
• 2. Schemes for redirecting nonconverged sensors to

uncovered sectors on the contour, ER

BACK 62 / 94

Notations

◮ Actual contour, Cact, is the convex hull bounding the points

in the level set {(x, y) ∈ R : f(x, y) = τ}.

◮ Estimated contour, Cest is the convex hull bounding the

points in the estimated level set, field value is equal to or closest to τ — {(x, y) ∈ R : f(x, y) ≃ τ}.

◮ Energy expended for movement is the most significant ∝ to

the distance the sensor has moved [Bartolini et al., 2008].

◮ If pt — the distance travelled by si up to time t , estep —

energy expended per sensor per unit distance travelled, then the total energy expended by the sensor until t is eexpi(t) = pt × estep. The energy remaining at time t for the ith sensor is given by eremi(t) = einiti − eexpi(t).

BACK 63 / 94

Coverage, C

Ratio of the area of intersection of estimated and actual contour to the area of the actual contour — how much of actual contour was covered by the estimated contour. C = Area of (Cest ∩ Cact) Area of Cact (9)

BACK 64 / 94

Coverage Error, ℵ

Ratio of the area of intersection of the nonrelevant region, Creg − Cact and the estimated contour, to the area of nonrelevant region Creg − Cact — how much of nonrelevant region covered by estimated contour ℵ = Area of ([Creg − Cact] ∩ Cest) Area of [Creg − Cact] (10)

BACK 65 / 94

Precision, φ

Ratio of the area of intersection of the estimated and actual contour to the area of estimated contour — how much of the estimated contour is the actual contour. Φ = Area of (Cest ∩ Cact) Area of Cest (11)

BACK 66 / 94

F-measures, Fβ

Weighted harmonic mean of precision and coverage — helps study precision/coverage trade-off when coverage weighs more than precision and vice-versa. Fβ = (1 + β2) × Φ × C β2 × Φ + C (12)

BACK 67 / 94

Latency, L

Maximum distance travelled by sensors to cover the contour. If Ti is the total distance travelled by the ith sensor from the beginning to termination then, L = max(T1, T2, . . . , Tm) (13)

BACK 68 / 94

Evaluation metrics comparison

BACK

• FIG. 13: Coverage, coverage error and precision metrics for different

types of overlap between the actual and the estimated contours at termination of contour covering.

69 / 94

Detection of Condition II

If eremi > 0, a converged sensor si terminates when,

◮ Approach 1: it encounters a point on the contour already

covered by another sensor. When all the converged sensors terminate A terminates. This minimizes latency but may not maximize coverage when sensors have low energy.

◮ Approach 2: it completes tracing the contour and returns

to its starting point on the contour. When atleast one converged sensor terminates, A terminates. This guarantees maximizing coverage but latency is not minimized since paths on the contour overlap.

◮ Approach 3: it encounters a point on the contour already

covered by another sensor and eremi is greater than the energy remaining in its clockwise neighboring sensor to cover the contour. If the converged sensors can predict the amount of work done to cover the portions of the contour accurately, this approach can maximize coverage and minimize latency.

BACK 70 / 94

Optimality and Terminating states of an algorithm A

Optimality Condition

An algorithm O is said to be optimal if it achieves 100% coverage and 0% coverage error, with lowest possible latency defined as LO upon termination.

Terminating States

For an algorithm A, CA, ℵA and LA represent coverage, coverage error and latency upon termination.

Energy CA = 100% CA < 100% remaining ℵA = 0 ℵA > 0 ℵA = 0 ℵA > 0 Erem > 0 LA − LO NA NA NA Erem = 0 LA − LO ℵA CA ℵA, CA

71 / 94

Measure of goodness at end states of an algorithm A solving CCP

Energy CA = 100% CA < 100% remaining ℵA = 0 ℵA > 0 ℵA = 0 ℵA > 0 Erem > 0 LA − LO NA NA NA Erem = 0 LA − LO ℵA CA ℵA, CA

Table:

72 / 94

Are measurable gradients present in realistic scenarios?

◮ Question 1: Does the sensing field exhibit measurable field

changes within rs?

◮ Question 2: Can these gradients be measured by sensors

available today?

◮ The spatial resolution of gradients in a pollutant spill

depends upon - time since the spill has occurred, distance from the spill, tonnage of the spill, type of pollutant, prevalent wind conditions and inherent current dynamics to name a few.

◮ Detection of gradients depends upon where the sensor is

located and sensitivity or the lowest detection ability of the sensor that has been deployed

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Numerical model of a spill

BACK

• FIG. 14: Emulsified Concentration Vs. Distance

◮ Modeling of a 28,000T marine fuel oil (979 kg/m3)

spill Tkalich.P [2000] in a two dimensional open sea channel [500km × 60m] (length and depth) is assumed.

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Numerical model of oil spill contd.

◮ The emulsified oil concentration in the water column is

measured and plotted against the distance travelled by the spill.

◮ The spill moves up to 400 kms from the spill site over a

time span of 4 days (indicated as 4.0d.

◮ If the robots are deployed in a timely fashion, near the

region of the spill (75kms of its traversal) and soon after the spill occurs (within half a day), they can measure a steep oil concentration gradient of (0-60 ppb, where 1 ppb = 10−6g/m3 and 1km = 1000m) within a distance of 1m.

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Variation of gradient with distance

BACK

• FIG. 15: Emulsified Concentration of Oil Gradient Vs. Distance

◮ We compute the gradient in ppb/meter units when the

maximum of the spill is at 75kms, 110kms, 180kms, 230kms, 285kms, 350kms from the initial spill site.

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Examples of laser fluoroscence sensors

BACK

• FIG. 16: SlickSLeuth installed on a sump

◮ The SlickSleuth C.R.Chase et al. [2005] sensor uses

fluorescence to detect presence of oil with as low concentration as 10ppb from a height of 1.5 - 3m over a target area spanning several meters.

◮ The laser fluorosensor Karpicz et al. [2005] based on laser

fluorometry principle has a maximum rsense = 50m with capability of detecting oil with submicrons thickness from a height of 3m

Summary

If sensors can be deployed in a timely manner in the vicinity of the spill, one can find gradients that can be measured by laser

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Detection of outliers

Three fundamental approaches Hodge and Austin [2004], where outliers are determined based on

◮ Type - I: No prior knowledge of data. This approach

considers data as a static distribution and names the most remote points as outliers.

◮ Type - II: Supervised classification. This requires

pre-labelled data tagged as normal and ubnormal.

◮ Type - III: Semisupervised leraning. This requires tagging

normal data and abnormality is learnt. In the case of CCP , since there is no prior knowledge of what is close to the contour and what is far, we use Type I based approach.

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Outlier elimination EOE

Detect outliers using Box Whisker plot [Tukey, 1977]. Input: Locations of other sensors ζ(t), Contour Estimate Cpred(t) Output: Compute the set of outliers Ω(t) foreach nonconverged sensor si do Compute Q3 and Q1; Compute IQR; if di(t) > k1 × IQR then Add si Ω(t); Set bias factor si to 1; end end Exclude outlier sensors from target angle computation;

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Schemes for determining and assigning uncovered sectors.

◮ Number of uncovered sectors = number of nonconverged

sensors, assign using Hungarian algorithm.

◮ Number of uncovered sectors < number of nonconverged

sensors, split uncovered sectors until they are equal.

◮ Number of uncovered sectors > number of nonconverged

sensors, sort the uncovered sectors based on size and assign to nonconverged sensors largest sector first (LM) or closest sector first (CM).

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Prediction Error

◮ Quality of prediction is measured by

PE(t) = Area of (Cpred(t)∆Cact) Area of Cact (14)

◮ Average prediction error (APE) is the mean of the values of

PE over all time steps in a single simulation run.

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Is prediction quality correlated to performance of E?

For pollutant contours, where the CE < CS up to 5% , LE > LS up to 10%

◮ A strong negative correlation, -0.72 between CE − CS

(when CS > CE) and the prediction error

◮ A strong positive correlation, 0.9, between LE and the

prediction error .

• FIG. 17: Correlation of prediction error with coverage difference for

MP and latency for SP contours.

Performance loss of E when compared to S under clustered deployment is correlated to quality of prediction.

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How does performance of E change with scaling of sensors?

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How does performance of E change with information reduction?

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How does performance E change with sensing errors?

Latency is unaffected!! Sensing errors make performance worse when there are already prediction errors.

• FIG. 18: Coverage under Diffused and Clustered deployments
• FIG. 19: Coverage Error Diffused and Clustered deployments

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How does performance E change with sensing errors?

Latency is unaffected!! Sensing errors make performance worse when there are already prediction errors.

• FIG. 20: Precision under Diffused and Clustered deployment
• FIG. 21: Latency under Diffused and Clustered deployments

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Effect of outlier elimination

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F-measures and latency with and without outliers

Take home: Elimination of

• utleirs from surround leads

to higher precision at the cost

• f lowering coverage.

Latency is unaffected.

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Effect of redirection

Does redirecting nonconverged sensors to the parts of the contour which are not covered help in improving performance? Case when LM coverage is better than CM coverage.

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Case when CM is better than LM

Take home

The decision regarding whether to redirect or not should be adaptive to distance to uncovered sector and energy remaining in the sensors (Future work).

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