APPLI LICATION ON DOM OMAIN: N: SENS NSOR OR NE NETWOR ORKS KS
SENSOR NETWORK AS A CONTROL SYSTEM Know everything - must deploy resources Know nothing - must deploy resources Data fusion, build prob. map of to maximize benefit from interacting with (how many? where?) target locations (static) or data sources (targets): track, get data - Cooperate but operate autonomously trajectories (dynamic) - Manage Communication , Energy - Manage Communication , Energy Position/Move to optimize Localize DATA TARGET detection prob. targets COVERAGE COLLECTION DETECTION Position/Move Update to optimize event density data collection quality information Model for optimization COVERAGE: DATA COLLECTION: persistently look for optimize data quality TRADEOFF: new targets ⇒ congregate nodes Control node location ⇒ spread nodes out around known targets to optimize and search COVERAGE + DATA COLLECTION Christos G. Cassandras CODES Lab. - Boston University �
COVERAGE CONTROL Deploy sensors to maximize “ event ” detection probability – unknown event locations – event sources may be mobile – sensors may be mobile R ( x ) ( Hz / m 2 ) • Meguerdichian et al , IEEE INFOCOM, 2001 50 • Cortes et al , IEEE Trans. on Robotics and Automation, 2004 40 • Cassandras and Li , Eur. J. of Control, 2005 30 • Ganguli et al , American Control Conf., 2006 20 • Hussein and Stipanovic , American Control ? 10 ? ? ? ? Conf., 2007 ? ? ? Ω � 10 0 ? • Hokayem et al , American Control Conf., 2007 10 8 5 6 4 2 0 0 Perceived event density (data sources) over given region (mission space) Christos G. Cassandras CODES Lab. - Boston University �
COVERAGE: PROBLEM FORMULATION R ( x ) § N mobile sensors, each located at s i ∈ R 2 ( Hz / 50 m 2 ) 40 30 § Data source at x emits signal with energy E 20 ? ? 10 ? ? ? ? ? Ω � 10 ? 0 ? 10 8 5 6 4 § Signal observed by sensor node i (at s i ) 2 0 0 § SENSING MODEL: p ( x , s ) P [ Detected by i | A ( x ), s ] ≡ i i i ( A ( x ) = data source emits at x ) § Sensing attenuation: p i ( x, s i ) monotonically decreasing in d i ( x ) ≡ || x - s i || Christos G. Cassandras CODES Lab. - Boston University
COVERAGE: PROBLEM FORMULATION § Joint detection prob. assuming sensor independence ( s = [ s 1 ,…, s N ] : node locations) Event sensing probability N [ ] P ( x , s ) 1 1 p ( x , s ) ∏ = − − i i i 1 = § OBJECTIVE: Determine locations s = [ s 1 ,…, s N ] to maximize total Detection Probability : max R ( x ) P ( x , s ) dx ∫ s Ω Perceived event density Christos G. Cassandras CODES Lab. - Boston University
DISTRIBUTED COOPERATIVE SCHEME CONTINUED § Set N ⎧ ⎫ [ ] dx H ( s , … , s ) R ( x ) 1 1 p ( x ) ∏ = ∫ − − ⎨ ⎬ 1 N i ⎩ ⎭ i 1 = Ω § Maximize H ( s 1 , … , s N ) by forcing nodes to move using gradient information: N H p ( x ) s x ∂ ∂ − [ ] k k R ( x ) 1 p ( x ) dx ∏ = ∫ − i s d ( x ) d ( x ) ∂ ∂ i 1 , i k k k k = ≠ Ω H ∂ k 1 k s s + = + β Desired displacement = V · Δ t i i k k s ∂ i Christos G. Cassandras CODES Lab. - Boston University
DISTRIBUTED COOPERATIVE SCHEME CONTINUED CONTINUED N H p ( x ) s x ∂ ∂ − [ ] k k R ( x ) 1 p ( x ) dx ∏ = ∫ − i s d ( x ) d ( x ) ∂ ∂ i 1 , i k k k k = ≠ Ω … has to be autonomously evaluated by each node so as to determine how to move to next position: H ∂ k 1 k s s + = + β i i k k s ∂ i Ø Use truncated p i ( x ) ⇒ Ω replaced by node neighborhood Ø Discretize p i ( x ) using a local grid Cassandras and Li, EJC, 2005 Christos G. Cassandras CODES Lab. - Boston University
EXTENSION: POLYGONAL OBSTACLES CONTINUED • Constrain the navigation of mobile nodes p x s ( , ) if is visible from x s ⎧ i i i • Interfere with sensing: ˆ ( , ) p x s = ⎨ i i 0 otherwise ⎩ Visibility Region Christos G. Cassandras CODES Lab. - Boston University
GRADIENT CALCULATION WITH OBSTACLES CONTINUED Q s ( ) ˆ ( , ) N p x s s x H ∂ − i ∂ ˆ i i i R x ( ) [ 1 p x s ( , ) ] dx A ∑ ∏ = ∫ − + k k j s d x ( ) d x ( ) ∂ ∂ j 1 k 1, k i i i i = V s = ≠ ( ) i p visible Q ( s i ): # of occluding corner points ⎧ i ˆ p = ⎨ New term captures change in visibility region of s i � i 0 invisible ⎩ Mathematically: use extension of Leibnitz rule for differentiating integral where both integrand and integration domain are functions of the control variable � Christos G. Cassandras CODES Lab. - Boston University
OPTIMAL COVERAGE WITH OBSTACLES http://codescolor.bu.edu/coverage Zhong and Cassandras, 2008 Christos G. Cassandras CODES Lab. - Boston University
DEMO: OPTIMAL DISTRIBUTED DEPLOYMENT WITH OBSTACLES – SIMULATED AND REAL Christos G. Cassandras CODES Lab. - Boston University
COVERAGE + DATA COLLECTION Recall tradeoff: TRADEOFF: COVERAGE: DATA COLLECTION: persistently look for optimize data quality Control node location new targets ⇒ congregate nodes to optimize ⇒ spread nodes out around known targets COVERAGE + DATA COLLECTION MODIFIED DISTRIBUTED OPTIMIZATION OBJECTIVE: collect info from detected data sources (targets) while maintaining a good coverage to detect future events S ( u ) : data source value H s , t R x P x , s dx u D t S u F u , s F ( u, s ) : joint data collection D t : set of data sources, quality at u estimated based on sensor observations (e.g., covariance) Christos G. Cassandras CODES Lab. - Boston University �
DEMO: REACTING TO EVENT DETECTION Important to note: There is no external control causing this behavior. Algorithm includes tracking functionality automatically Christos G. Cassandras CODES Lab. - Boston University �
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