Interactive Planning and Sensing in Uncertain …...Interactive Planning and Sensing in Uncertain...
Transcript of Interactive Planning and Sensing in Uncertain …...Interactive Planning and Sensing in Uncertain...
Interactive Planning and Sensing in UncertainEnvironments with Task-Driven Sensor Placement
Benjamin S. Cooper* Raghvendra V. Cowlagi*
∗Aerospace Engineering Program,Worcester Polytechnic Institute, Worcester, MA.
rvcowlagi, [email protected] wpi.edu/∼rvcowlagi
2nd International Conference on InfoSymbiotics/DDDAS.August 07, 2017. Cambridge, MA.
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Introduction
You are an employee at car manufacturing company.
Plan: Design next year’s car and marketing approach.
Act: Manufacture and market the car.
Sense: Obtain information about cars.
Information overload.c©2017 Savvysme. All rights reserved.https://www.savvysme.com.au/themes/savvy_bootstrap/
img/upload/images/information-overload.jpg
Boss asks you to talkto people about cars.
Cooper & Cowlagi (WPI) Interactive Planning and Sensing 1 / 20
Introduction
You are an employee at car manufacturing company.
Plan: Design next year’s car and marketing approach.
Act: Manufacture and market the car.
Sense: Obtain information about cars.
Information overload.c©2017 Savvysme. All rights reserved.https://www.savvysme.com.au/themes/savvy_bootstrap/
img/upload/images/information-overload.jpg
Boss asks you to talkto people about cars.
Cooper & Cowlagi (WPI) Interactive Planning and Sensing 1 / 20
Introduction
You are an employee at car manufacturing company.
Plan: Design next year’s car and marketing approach.
Act: Manufacture and market the car.
Sense: Obtain information about cars.
Information overload.c©2017 Savvysme. All rights reserved.https://www.savvysme.com.au/themes/savvy_bootstrap/
img/upload/images/information-overload.jpg
Boss asks you to talkto people about cars.
Cooper & Cowlagi (WPI) Interactive Planning and Sensing 1 / 20
Introduction
Meaningful Information.c©2017 Freepik. All rights reserved.https://image.freepik.com/free-vector/
business-conversation-design_1133-88.jpg
Boss asks you to talk to peopleabout why they bought their car.
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Introduction
Meaningful Information.c©2017 Freepik. All rights reserved.https://image.freepik.com/free-vector/
business-conversation-design_1133-88.jpg
Boss asks you to talk to peopleabout why they bought their car.
Cooper & Cowlagi (WPI) Interactive Planning and Sensing 2 / 20
Motivation
Turn gaze to check blind spot.c©2016 North West Crash Courses. All rights reserved.
http://www.northwestcrashcourses.co.uk/
latin-words-comined-handful-of-mode/
Snow removal gridlocks Boston.c©2016 Americaninno. All rights reserved.https://www.americaninno.com/boston/
boston-traffic-update-snow-removal/
/-gridlocks-city-closes-i93-google-maps/
Ambulance stuck in traffic.c©2016 Asiannet news. All rights reserved.http://newsable.asianetnews.tv/south/
ambulance-traffic-bengaluru-deaths
Cooper & Cowlagi (WPI) Interactive Planning and Sensing 3 / 20
Introduction: Terminology
Actor: planning + acting.
Planning: route-planning for a mobile vehicle; many possibilities:
Point-to-point motion.Motion to satisfy temporal logic specifications.Kinematic and dynamic vehicle models.
Acting: generating and tracking a reference trajectory.
Execute the planned route with a trajectory feasible w.r.t the vehicle’skinematic-, dynamic-, and input- constraints.
Interactive Planning and Sensing: (IPAS) overall strategy of planning,acting, and sensing based on a task-driven sensor placement approach.
Task-driven: sensor placement approach driven by the needs or taskof the actor.
Cooper & Cowlagi (WPI) Interactive Planning and Sensing 4 / 20
Problem Formulation: Actor
Point-to-point route-planning in 2D withminimum exposure to a spatial threat field.
Grid-world: N2G grid points in NG rows and NG columns, on a closed
square 2D domain W ⊂ R2.
Grid points labeled 1, . . . ,N2G; denote coordinates of i th point by xi
Strictly positive threat field c :W → R+.
No vehicle kinematic or dynamic model: particle jumps from one gridpoint to the next (4-connectivity, i.e., up, down, left, right).
No uncertainty in localization or grid-point transition.
Objective: Move from prespecified initial grid point is to prespecifiedgoal grid point ig with minimum threat exposure; is, ig ∈ {1, . . . ,N2
G}.
Cooper & Cowlagi (WPI) Interactive Planning and Sensing 5 / 20
Problem Formulation: Actor
Point-to-point route-planning in 2D withminimum exposure to a spatial threat field.
Grid-world: N2G grid points in NG rows and NG columns, on a closed
square 2D domain W ⊂ R2.
Grid points labeled 1, . . . ,N2G; denote coordinates of i th point by xi
Strictly positive threat field c :W → R+.
No vehicle kinematic or dynamic model: particle jumps from one gridpoint to the next (4-connectivity, i.e., up, down, left, right).
No uncertainty in localization or grid-point transition.
Objective: Move from prespecified initial grid point is to prespecifiedgoal grid point ig with minimum threat exposure; is, ig ∈ {1, . . . ,N2
G}.
Cooper & Cowlagi (WPI) Interactive Planning and Sensing 5 / 20
Problem Formulation: Actor
Point-to-point route-planning in 2D withminimum exposure to a spatial threat field.
Grid-world: N2G grid points in NG rows and NG columns, on a closed
square 2D domain W ⊂ R2.
Grid points labeled 1, . . . ,N2G; denote coordinates of i th point by xi
Strictly positive threat field c :W → R+.
No vehicle kinematic or dynamic model: particle jumps from one gridpoint to the next (4-connectivity, i.e., up, down, left, right).
No uncertainty in localization or grid-point transition.
Objective: Move from prespecified initial grid point is to prespecifiedgoal grid point ig with minimum threat exposure; is, ig ∈ {1, . . . ,N2
G}.
Cooper & Cowlagi (WPI) Interactive Planning and Sensing 5 / 20
Problem Formulation: Actor
Point-to-point route-planning in 2D withminimum exposure to a spatial threat field.
Grid-world: N2G grid points in NG rows and NG columns, on a closed
square 2D domain W ⊂ R2.
Grid points labeled 1, . . . ,N2G; denote coordinates of i th point by xi
Strictly positive threat field c :W → R+.
No vehicle kinematic or dynamic model: particle jumps from one gridpoint to the next (4-connectivity, i.e., up, down, left, right).
No uncertainty in localization or grid-point transition.
Objective: Move from prespecified initial grid point is to prespecifiedgoal grid point ig with minimum threat exposure; is, ig ∈ {1, . . . ,N2
G}.
Cooper & Cowlagi (WPI) Interactive Planning and Sensing 5 / 20
Problem Formulation: Actor
Point-to-point route-planning in 2D withminimum exposure to a spatial threat field.
Grid-world: N2G grid points in NG rows and NG columns, on a closed
square 2D domain W ⊂ R2.
Grid points labeled 1, . . . ,N2G; denote coordinates of i th point by xi
Strictly positive threat field c :W → R+.
No vehicle kinematic or dynamic model: particle jumps from one gridpoint to the next (4-connectivity, i.e., up, down, left, right).
No uncertainty in localization or grid-point transition.
Objective: Move from prespecified initial grid point is to prespecifiedgoal grid point ig with minimum threat exposure; is, ig ∈ {1, . . . ,N2
G}.
Cooper & Cowlagi (WPI) Interactive Planning and Sensing 5 / 20
Problem Formulation: Sensor
A “small” number NS < N2G of
sensors that take noisypointwise measurements of thethreat field.
Least-squares estimate of threatfield parameters available toactor.
Optimal sensor placement acombinatorial problem.
What if the actor could decide where to place sensors? In somemeaningful subdomain? How can the actor determine this subdomain,
and place sensors within?
Cooper & Cowlagi (WPI) Interactive Planning and Sensing 6 / 20
Problem Formulation: Sensor
A “small” number NS < N2G of
sensors that take noisypointwise measurements of thethreat field.
Least-squares estimate of threatfield parameters available toactor.
Optimal sensor placement acombinatorial problem.
What if the actor could decide where to place sensors? In somemeaningful subdomain? How can the actor determine this subdomain,
and place sensors within?
Cooper & Cowlagi (WPI) Interactive Planning and Sensing 6 / 20
Problem Formulation: Sensor
A “small” number NS < N2G of
sensors that take noisypointwise measurements of thethreat field.
Least-squares estimate of threatfield parameters available toactor.
Optimal sensor placement acombinatorial problem.
What if the actor could decide where to place sensors? In somemeaningful subdomain? How can the actor determine this subdomain,
and place sensors within?
Cooper & Cowlagi (WPI) Interactive Planning and Sensing 6 / 20
Problem Formulation
Threat field parametrization: c(x) =
NP∑n=1
θnφn(x) = Φ(x)Θ.
φn : spatial basis functions, Φ := [φ1 . . . φNP], Θ := [θ1 . . . θNP
]T.
Sensor grid point locations: j1, . . . , jNS;
measurements zk := c(xjk ) + ηk .
ηk ∼ N (0, σ2k); denote R = diag(σ2
1 , . . . , σ2NS
)
Denote ~z := [z1 . . . zNS]T; H := [Φ(xj1 ) . . . Φ(xjNS
)]T.
Threat field parameter estimate:
Mean: Θ = HL~z ,
Error covariance: P = (HTR−1H)−1.
Grid-world graph: G = (V ,E ); vertices in V = {v1, . . . , vN2G}
uniquely associated with grid points .
(vi , vj) ∈ E ⇔ |i − j | = 1 or |i − j | = NG.
Cooper & Cowlagi (WPI) Interactive Planning and Sensing 7 / 20
Problem Formulation
Threat field parametrization: c(x) =
NP∑n=1
θnφn(x) = Φ(x)Θ.
φn : spatial basis functions, Φ := [φ1 . . . φNP], Θ := [θ1 . . . θNP
]T.
Sensor grid point locations: j1, . . . , jNS;
measurements zk := c(xjk ) + ηk .
ηk ∼ N (0, σ2k); denote R = diag(σ2
1 , . . . , σ2NS
)
Denote ~z := [z1 . . . zNS]T; H := [Φ(xj1 ) . . . Φ(xjNS
)]T.
Threat field parameter estimate:
Mean: Θ = HL~z ,
Error covariance: P = (HTR−1H)−1.
Grid-world graph: G = (V ,E ); vertices in V = {v1, . . . , vN2G}
uniquely associated with grid points .
(vi , vj) ∈ E ⇔ |i − j | = 1 or |i − j | = NG.
Cooper & Cowlagi (WPI) Interactive Planning and Sensing 7 / 20
Problem Formulation
Threat field parametrization: c(x) =
NP∑n=1
θnφn(x) = Φ(x)Θ.
φn : spatial basis functions, Φ := [φ1 . . . φNP], Θ := [θ1 . . . θNP
]T.
Sensor grid point locations: j1, . . . , jNS;
measurements zk := c(xjk ) + ηk .
ηk ∼ N (0, σ2k); denote R = diag(σ2
1 , . . . , σ2NS
)
Denote ~z := [z1 . . . zNS]T; H := [Φ(xj1 ) . . . Φ(xjNS
)]T.
Threat field parameter estimate:
Mean: Θ = HL~z ,
Error covariance: P = (HTR−1H)−1.
Grid-world graph: G = (V ,E ); vertices in V = {v1, . . . , vN2G}
uniquely associated with grid points .
(vi , vj) ∈ E ⇔ |i − j | = 1 or |i − j | = NG.
Cooper & Cowlagi (WPI) Interactive Planning and Sensing 7 / 20
Problem Formulation
Threat field parametrization: c(x) =
NP∑n=1
θnφn(x) = Φ(x)Θ.
φn : spatial basis functions, Φ := [φ1 . . . φNP], Θ := [θ1 . . . θNP
]T.
Sensor grid point locations: j1, . . . , jNS;
measurements zk := c(xjk ) + ηk .
ηk ∼ N (0, σ2k); denote R = diag(σ2
1 , . . . , σ2NS
)
Denote ~z := [z1 . . . zNS]T; H := [Φ(xj1 ) . . . Φ(xjNS
)]T.
Threat field parameter estimate:
Mean: Θ = HL~z ,
Error covariance: P = (HTR−1H)−1.
Grid-world graph: G = (V ,E ); vertices in V = {v1, . . . , vN2G}
uniquely associated with grid points .
(vi , vj) ∈ E ⇔ |i − j | = 1 or |i − j | = NG.
Cooper & Cowlagi (WPI) Interactive Planning and Sensing 7 / 20
Problem Formulation (continued)
Actor’s problem: find a path in G from vis to vig with minimum cost.
Cost of path = sum of edge transition costs.
Expected edge transition cost: g((vi , vj)) = c(xj) = Φ(xj)Θ.
Incurred edge transition cost: g((vi , vj)) = c(xj) = Φ(xj)Θ.
Sensor reconfiguration: implied subproblem in which actor identifies asubdomain and directs sensors to new location.
Reconfiguration occurs until the norm of the sensor locations reachessome threshold.
Cooper & Cowlagi (WPI) Interactive Planning and Sensing 8 / 20
Problem Formulation (continued)
Actor’s problem: find a path in G from vis to vig with minimum cost.
Cost of path = sum of edge transition costs.
Expected edge transition cost: g((vi , vj)) = c(xj) = Φ(xj)Θ.
Incurred edge transition cost: g((vi , vj)) = c(xj) = Φ(xj)Θ.
Sensor reconfiguration: implied subproblem in which actor identifies asubdomain and directs sensors to new location.
Reconfiguration occurs until the norm of the sensor locations reachessome threshold.
Cooper & Cowlagi (WPI) Interactive Planning and Sensing 8 / 20
Problem Formulation (continued)
Actor’s problem: find a path in G from vis to vig with minimum cost.
Cost of path = sum of edge transition costs.
Expected edge transition cost: g((vi , vj)) = c(xj) = Φ(xj)Θ.
Incurred edge transition cost: g((vi , vj)) = c(xj) = Φ(xj)Θ.
Sensor reconfiguration: implied subproblem in which actor identifies asubdomain and directs sensors to new location.
Reconfiguration occurs until the norm of the sensor locations reachessome threshold.
Cooper & Cowlagi (WPI) Interactive Planning and Sensing 8 / 20
Task-Driven Sensor Placement
Sensor placement problems often focus on Information Maximization.
Fisher information, mutual information, entropy,...
The combinatorial problem can be dealt with in a few ways,
Convex optimization: relax boolean constraints.Greedy placement: if objective function submodular, near-optimal.Heuristics: reduce the search space.
Task-driven: Place sensors to minimize the path cost.
We take the current optimal path to identify a subdomain for sensorplacement.
Cooper & Cowlagi (WPI) Interactive Planning and Sensing 9 / 20
Identify Significant Basis with 2D Cross-Correlation
XMxN : A map with a proto-basis.
HPxQ : a proto-threat at path location.
C (k, l) =M−1∑m=0
N−1∑n=0
X (m, n)H(m − k , n − l),−(P − 1) 6 k 6 M − 1
−(Q − 1) 6 l 6 N − 1
Sum the correlation of each basis along the path.
Rank the basis by largest cumulative correlation.
Cooper & Cowlagi (WPI) Interactive Planning and Sensing 10 / 20
Task-Driven Sensor Placement Approach
Cross-correlation of path locations with basis functions
Place sensors near basis functions with high correlation to currentpath OR
Use mutual information to place sensors within subdomain ofcorrelated basis functions
Current optimal path. Identify basis functions of highcorrelation with path.
Define subdomain to place sensorswithin.
Cooper & Cowlagi (WPI) Interactive Planning and Sensing 11 / 20
Task-Driven Sensor Placement Approach
Cross-correlation of path locations with basis functions
Place sensors near basis functions with high correlation to currentpath OR
Use mutual information to place sensors within subdomain ofcorrelated basis functions
Current optimal path.
Identify basis functions of highcorrelation with path.
Define subdomain to place sensorswithin.
Cooper & Cowlagi (WPI) Interactive Planning and Sensing 11 / 20
Task-Driven Sensor Placement Approach
Cross-correlation of path locations with basis functions
Place sensors near basis functions with high correlation to currentpath OR
Use mutual information to place sensors within subdomain ofcorrelated basis functions
Current optimal path. Identify basis functions of highcorrelation with path.
Define subdomain to place sensorswithin.
Cooper & Cowlagi (WPI) Interactive Planning and Sensing 11 / 20
Task-Driven Sensor Placement Approach
Cross-correlation of path locations with basis functions
Place sensors near basis functions with high correlation to currentpath OR
Use mutual information to place sensors within subdomain ofcorrelated basis functions
Current optimal path. Identify basis functions of highcorrelation with path.
Define subdomain to place sensorswithin.
Cooper & Cowlagi (WPI) Interactive Planning and Sensing 11 / 20
Interactive Planning and Sensing
Intuitive argument:
True optimal path consists of a small number of grid points; can becovered by a subset of the basis function family.
Heuristic iterative algorithm: while !StopCondition
1 Find a path with minimum expected cost with current threat estimate.
2 Identify subset of basis functions that cover this path.
3 Identify subset of grid points within the support of these basisfunctions.
4 Place sensors near highly correlated basis functions OR using MI.
5 Evaluate StopCondition. REPEAT.
StopCondition: When “most” sensors stop moving. Norm of change insensor position less than some integer tolerance.
Cooper & Cowlagi (WPI) Interactive Planning and Sensing 12 / 20
Interactive Planning and Sensing
Intuitive argument:
True optimal path consists of a small number of grid points; can becovered by a subset of the basis function family.
Heuristic iterative algorithm: while !StopCondition
1 Find a path with minimum expected cost with current threat estimate.
2 Identify subset of basis functions that cover this path.
3 Identify subset of grid points within the support of these basisfunctions.
4 Place sensors near highly correlated basis functions OR using MI.
5 Evaluate StopCondition. REPEAT.
StopCondition: When “most” sensors stop moving. Norm of change insensor position less than some integer tolerance.
Cooper & Cowlagi (WPI) Interactive Planning and Sensing 12 / 20
Iteration Example
True Iter. 1
Iter. 2 Iter. 3
True cost: 176.4. Incurred cost: 181.5. Incurred cost: 180.5. Incurred cost: 180.5.
StopCondition
Binary Sensor Location vector vj :For each grid location,
1 = occupied by sensor,
0 = unoccupied.
Until ||vj − vj−1|| < Tol
Cooper & Cowlagi (WPI) Interactive Planning and Sensing 13 / 20
Iteration Example
True Iter. 1 Iter. 2
Iter. 3
True cost: 176.4. Incurred cost: 181.5. Incurred cost: 180.5. Incurred cost: 180.5.
StopCondition
Binary Sensor Location vector vj :For each grid location,
1 = occupied by sensor,
0 = unoccupied.
Until ||vj − vj−1|| < Tol
Cooper & Cowlagi (WPI) Interactive Planning and Sensing 13 / 20
Iteration Example
True Iter. 1 Iter. 2 Iter. 3
True cost: 176.4. Incurred cost: 181.5. Incurred cost: 180.5. Incurred cost: 180.5.
StopCondition
Binary Sensor Location vector vj :For each grid location,
1 = occupied by sensor,
0 = unoccupied.
Until ||vj − vj−1|| < Tol
Cooper & Cowlagi (WPI) Interactive Planning and Sensing 13 / 20
Iteration Example
True Iter. 1 Iter. 2 Iter. 3
True cost: 176.4. Incurred cost: 181.5. Incurred cost: 180.5. Incurred cost: 180.5.
StopCondition
Binary Sensor Location vector vj :For each grid location,
1 = occupied by sensor,
0 = unoccupied.
Until ||vj − vj−1|| < Tol
Cooper & Cowlagi (WPI) Interactive Planning and Sensing 13 / 20
General IPAS Results
Cooper & Cowlagi (WPI) Interactive Planning and Sensing 14 / 20
Performance under Rank-Deficient Observations
Cooper & Cowlagi (WPI) Interactive Planning and Sensing 15 / 20
Case When MI Outperforms Task-Driven
Cooper & Cowlagi (WPI) Interactive Planning and Sensing 16 / 20
Parameter-Sensor Study Results
For most cases comparison is neutral
For many cases, NS < NP , Task-Driven outperforms
Cooper & Cowlagi (WPI) Interactive Planning and Sensing 17 / 20
Parameter-Sensor Study Results
For most cases comparison is neutral
For many cases, NS < NP , Task-Driven outperforms
Cooper & Cowlagi (WPI) Interactive Planning and Sensing 17 / 20
Computational Efficiency
Cooper & Cowlagi (WPI) Interactive Planning and Sensing 18 / 20
Issues to be Resolved
Actor-relevant sensor reconfiguration.
Finding meaningful subdomain of basis functions for min-cost path.
Convergence and performance guarantees.
So far, empirical results on convergence, performance.Characterize performance bounds from information bounds?Cramer-Rao Lower Bound?
“Reconfiguration Cost”.
Sensor vehicle dynamics, communication network,
Cooper & Cowlagi (WPI) Interactive Planning and Sensing 19 / 20
Summary & Future Work
Traditionally, a “principle of separation” between planning andsensing subsystems is assumed. However, the placement of sensorscan/should be influenced by the planning problem at hand.
Task-driven sensor placement reduces search space for sensorplacement; demonstrates expected computational speed up, andstrong performance even under rank-deficient observations.
Future work: Convergence proofs, performance bounds;reconfiguration cost, vehicle kinematic models.
Acknowledgment: Funding from award #FA9550-17-1-0028.
rvcowlagi, [email protected] wpi.edu/∼rvcowlagi.
Cooper & Cowlagi (WPI) Interactive Planning and Sensing 20 / 20