dynamic sensor-actor interactions for path-planning in a ... · actor:planning + acting....

Dynamic Sensor-Actor Interactions for

Path-Planning in a Threat Field

Raghvendra V. Cowlagi∗ Benjamin S. Cooper∗

∗Aerospace Engineering Program,

Worcester Polytechnic Institute, Worcester, MA.

rvcowlagi, [email protected] wpi.edu/∼rvcowlagi

1st International Conference InfoSymbiotics/DDDAS.

August 10, 2016. Hartford, CT.

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Introduction

Cowlagi & Cooper (WPI) Dynamic Sensor-Actor Interactions 1 / 16

Motivation

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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.

Motion to execute a symbolic planning task.

Kinematic and dynamic vehicle models.

Discrete route: e.g., sequence of waypoints.

Acting: generating and tracking a reference trajectory.

Execute the planned route with a trajectory feasible w.r.t the vehicle’s

kinematic-, dynamic-, and input- constraints.


Problem Formulation: Actor

Point-to-point route-planning in 2D with

minimum 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 grid

point 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 prespecified

goal grid point ig with minimum threat exposure; is, ig ∈ {1, . . . ,N2G}.


Problem Formulation: Sensor

A “small” number NS < N2G of

sensors that take noisy

pointwise measurements of the

threat field.

Least-squares estimate of threat

field parameters.

Available to actor: threat field

estimate, not true field values.

What if the actor could decide where to place sensors? Where would it

place the sensors, and is there a benefit to this interaction?


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 = HLz,

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 (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 Placement

vis��*

vig�

φn : 2D Gaussian functions; NP = 25, N2G = 400, NS = 25.

Placement methods:

Uniformly distributed over W.

Clustered near is.

Placed at NS arbitrarily selected grid points.

Placed to maximize determinant of Fisher information matrix.*


Sensor Placement

Uniformly distributed over W :

trace(P) ≈ 257;

# diagonal elements of P less than 1: 3.

Incurred cost: ≈ 131 units, for comparison, incurred cost of true

optimal path is 104.4 units.

Clustered near is : trace(P)� 103 units; incurred cost: ≈ 133 units.

True field and optimal path; illustration of

clustered placement.Field reconstruction with Θ with uniform placement.


Sensor Placement (continued)

Arbitrary placement:

In 20 arbitrarily chosen placements, only 1 resulted in trace(P) < 103.

Median # diagonal elements of P less than 1: 2.

Incurred cost: avg. 126.7, min: 107.5, max: 144.2.


Actor-Relevant Sensor Placement

Intuitive argument:

”A flashlight can illuminate a narrow path, floodlights not needed.”

True optimal path consists of a small number of grid points; can be

covered by a subset of the basis function family.

Heuristic iterative algorithm:

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 basis

functions.

4 Place sensors arbitrarily in this subset of grid points. REPEAT.


Actor-Relevant Sensor PlacementTrue Iter. 1 Iter. 2 Iter. 3

Iter. 4 Iter. 5 Iter. 6 Iter. 7

Iter. 8 Iter. 9 Iter. 10

True cost: 104.4. Incurred cost: 131.5. Incurred cost: 110.7. Incurred cost: 107.6.

Incurred cost: 106.9. Incurred cost: 105.4. Incurred cost: 105.3. Incurred cost: 104.8.

Incurred cost: 104.9. Incurred cost: 105.2. Incurred cost: 104.9.


Actor-Relevant Sensor Placement

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Arbitrary 1 Arbitrary 2 Arbitrary 3 Clustered Uniform True Cost


Actor-Relevant Sensor Placement (NS = 20)

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

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Arbitrary 1 Arbitrary 2 Arbitrary 3 Clustered Uniform True Cost


Issues to be Resolved

Actor-relevant sensor placement strategy.

Optimal placement within domain of interest.

Convergence and performance guarantees.

Under what conditions, if any, do iterations converge? Bounds on

suboptimality of incurred cost?

Risk-sensitive utility for path-planning.

Exponential or HARA utility functions are used in stochastic optimal

control.

Basis functions.

Compact support, e.g., Daubechies wavelets.


Summary & Future Work

Traditionally, a “principle of separation” between planning and

sensing subsystems is assumed. However, the placement of sensors

can/should be influenced by the planning problem at hand.

(Very) preliminary results indicate that actor’s performance can

improve with task-relevant sensor placement.

Future work: more sophisticated planning formulations; vehicle

kinematic/dynamic models.

Acknowledgment: Funding from the AFOSR 2016 Young

Investigator Program.

rvcowlagi, [email protected] wpi.edu/∼rvcowlagi.


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