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, bscooper@wpi.edu 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.

img/upload/images/information-overload.jpg

Boss asks you to talkto people about cars.

Cooper & Cowlagi (WPI) Interactive Planning and Sensing 1 / 20

Introduction

business-conversation-design_1133-88.jpg

Boss asks you to talk to peopleabout why they bought their car.

Introduction

business-conversation-design_1133-88.jpg

Boss asks you to talk to peopleabout why they bought their car.

Motivation

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/-gridlocks-city-closes-i93-google-maps/

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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.

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

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?

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

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

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.

Problem Formulation

NP∑n=1

1 , . . . , σ2NS

Mean: Θ = HL~z ,

(vi , vj) ∈ E ⇔ |i − j | = 1 or |i − j | = NG.

Problem Formulation

NP∑n=1

1 , . . . , σ2NS

Mean: Θ = HL~z ,

(vi , vj) ∈ E ⇔ |i − j | = 1 or |i − j | = NG.

Problem Formulation

NP∑n=1

1 , . . . , σ2NS

Mean: Θ = HL~z ,

(vi , vj) ∈ E ⇔ |i − j | = 1 or |i − j | = NG.

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.

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.

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.

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.

Current optimal path.

Identify basis functions of highcorrelation with path.

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.

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.

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

Iteration Example

True Iter. 1 Iter. 2

Iter. 3

StopCondition

0 = unoccupied.

Iteration Example

True Iter. 1 Iter. 2 Iter. 3

StopCondition

0 = unoccupied.

Iteration Example

True Iter. 1 Iter. 2 Iter. 3

StopCondition

0 = unoccupied.

General IPAS Results

Performance under Rank-Deficient Observations

Case When MI Outperforms Task-Driven

Parameter-Sensor Study Results

For most cases comparison is neutral

For many cases, NS < NP , Task-Driven outperforms

Parameter-Sensor Study Results

For most cases comparison is neutral

For many cases, NS < NP , Task-Driven outperforms

Computational Efficiency

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,

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, bscooper@wpi.edu wpi.edu/∼rvcowlagi.

Interactive Planning and Sensing in Uncertain …...Interactive Planning and Sensing in Uncertain...

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