Cowlagi Cooper dddas 2016
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. Fair Use Disclaimer: This document may contain copyrighted material, such as photographs and diagrams, the use of which may not always have been specifically authorized by the copyright owner. The use of copyrighted material in this document is in accordance with the “fair use doctrine” as incorporated in Title 17 USC §107 of the United States Copyright Act of 1976.
Introduction
Cowlagi & Cooper (WPI)
Dynamic Sensor-Actor Interactions
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Motivation
Wildfire mapping and prediction to assist mitigation. c
2016 Matthew Keys. All rights reserved. http://feed.matthewkeys.net/firemap/
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/
Cowlagi & Cooper (WPI)
Dynamic Sensor-Actor Interactions
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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.
Cowlagi & Cooper (WPI)
Dynamic Sensor-Actor Interactions
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Problem Formulation: Actor Point-to-point route-planning in 2D with minimum exposure to a spatial threat field. 2 grid points in N rows and N columns, on a closed Grid-world: NG G G square 2D domain W ⊂ R2 . 2 Grid points labeled 1, . . . , NG ; 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 2 }. goal grid point ig with minimum threat exposure; is , ig ∈ {1, . . . , NG
Cowlagi & Cooper (WPI)
Dynamic Sensor-Actor Interactions
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Problem Formulation: Sensor 2 of A “small” number NS < NG 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?
Cowlagi & Cooper (WPI)
Dynamic Sensor-Actor Interactions
5 / 16
Problem Formulation Threat field parametrization: c(x) =
NP X θn φn (x) = Φ(x)Θ. n=1
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, σk2 ); denote R = diag(σ12 , . . . , σN2 S ) Denote z := [z1 . . . zNS ]T ; H := [Φ(xj1 ) . . . Φ(xjNS )]T .
Threat field parameter estimate: ˆ = H L z, Mean: Θ −1
Error covariance: P = (H T R −1 H)
.
Grid-world graph: G = (V , E ); vertices in V = {v1 , . . . , vN 2 } G uniquely associated with grid points . (vi , vj ) ∈ E ⇔ |i − j| = 1 or |i − j| = NG . Cowlagi & Cooper (WPI)
Dynamic Sensor-Actor Interactions
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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.
ˆ Expected edge transition cost: g ((vi , vj )) = cˆ(xj ) = Φ(xj )Θ. Incurred edge transition cost: g¯ ((vi , vj )) = c(xj ) = Φ(xj )Θ.
Cowlagi & Cooper (WPI)
Dynamic Sensor-Actor Interactions
7 / 16
Sensor Placement
vig
*
vis
2 = 400, N = 25. φn : 2D Gaussian functions; NP = 25, NG S Placement methods:
Uniformly distributed over W. Clustered near is . Placed at NS arbitrarily selected grid points. Placed to maximize determinant of Fisher information matrix.* Cowlagi & Cooper (WPI)
Dynamic Sensor-Actor Interactions
8 / 16
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. Cowlagi & Cooper (WPI)
ˆ with uniform placement. Field reconstruction with Θ
Dynamic Sensor-Actor Interactions
9 / 16
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.
Cowlagi & Cooper (WPI)
Dynamic Sensor-Actor Interactions
10 / 16
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.
Cowlagi & Cooper (WPI)
Dynamic Sensor-Actor Interactions
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Actor-Relevant Sensor Placement True
Iter. 1
Iter. 2
Iter. 3
True cost: 104.4.
Incurred cost: 131.5.
Incurred cost: 110.7.
Incurred cost: 107.6.
Iter. 4
Iter. 5
Iter. 6
Iter. 7
Incurred cost: 106.9.
Incurred cost: 105.4.
Incurred cost: 105.3.
Incurred cost: 104.8.
Iter. 8
Iter. 9
Iter. 10
Incurred cost: 104.9.
Incurred cost: 105.2.
Incurred cost: 104.9.
Cowlagi & Cooper (WPI)
Dynamic Sensor-Actor Interactions
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Actor-Relevant Sensor Placement Arbitrary 1
Arbitrary 2
Arbitrary 3
Clustered
Uniform
True Cost
140
135
Incurred Cost
130
125
120
115
110
105
100
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Iteration #
Cowlagi & Cooper (WPI)
Dynamic Sensor-Actor Interactions
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Actor-Relevant Sensor Placement (NS = 20) Arbitrary 1
Arbitrary 2
Arbitrary 3
Clustered
Uniform
True Cost
140
135
Incurred Cost
130
125
120
115
110
105
100
1
2
3
4
5
6
7
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Iteration #
Cowlagi & Cooper (WPI)
Dynamic Sensor-Actor Interactions
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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.
Cowlagi & Cooper (WPI)
Dynamic Sensor-Actor Interactions
15 / 16
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]
Cowlagi & Cooper (WPI)
Dynamic Sensor-Actor Interactions
wpi.edu/∼rvcowlagi.
16 / 16
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. Fair Use Disclaimer: This document may contain copyrighted material, such as photographs and diagrams, the use of which may not always have been specifically authorized by the copyright owner. The use of copyrighted material in this document is in accordance with the “fair use doctrine” as incorporated in Title 17 USC §107 of the United States Copyright Act of 1976.
Introduction
Cowlagi & Cooper (WPI)
Dynamic Sensor-Actor Interactions
1 / 16
Motivation
Wildfire mapping and prediction to assist mitigation. c
2016 Matthew Keys. All rights reserved. http://feed.matthewkeys.net/firemap/
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/
Cowlagi & Cooper (WPI)
Dynamic Sensor-Actor Interactions
2 / 16
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.
Cowlagi & Cooper (WPI)
Dynamic Sensor-Actor Interactions
3 / 16
Problem Formulation: Actor Point-to-point route-planning in 2D with minimum exposure to a spatial threat field. 2 grid points in N rows and N columns, on a closed Grid-world: NG G G square 2D domain W ⊂ R2 . 2 Grid points labeled 1, . . . , NG ; 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 2 }. goal grid point ig with minimum threat exposure; is , ig ∈ {1, . . . , NG
Cowlagi & Cooper (WPI)
Dynamic Sensor-Actor Interactions
4 / 16
Problem Formulation: Sensor 2 of A “small” number NS < NG 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?
Cowlagi & Cooper (WPI)
Dynamic Sensor-Actor Interactions
5 / 16
Problem Formulation Threat field parametrization: c(x) =
NP X θn φn (x) = Φ(x)Θ. n=1
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, σk2 ); denote R = diag(σ12 , . . . , σN2 S ) Denote z := [z1 . . . zNS ]T ; H := [Φ(xj1 ) . . . Φ(xjNS )]T .
Threat field parameter estimate: ˆ = H L z, Mean: Θ −1
Error covariance: P = (H T R −1 H)
.
Grid-world graph: G = (V , E ); vertices in V = {v1 , . . . , vN 2 } G uniquely associated with grid points . (vi , vj ) ∈ E ⇔ |i − j| = 1 or |i − j| = NG . Cowlagi & Cooper (WPI)
Dynamic Sensor-Actor Interactions
6 / 16
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.
ˆ Expected edge transition cost: g ((vi , vj )) = cˆ(xj ) = Φ(xj )Θ. Incurred edge transition cost: g¯ ((vi , vj )) = c(xj ) = Φ(xj )Θ.
Cowlagi & Cooper (WPI)
Dynamic Sensor-Actor Interactions
7 / 16
Sensor Placement
vig
*
vis
2 = 400, N = 25. φn : 2D Gaussian functions; NP = 25, NG S Placement methods:
Uniformly distributed over W. Clustered near is . Placed at NS arbitrarily selected grid points. Placed to maximize determinant of Fisher information matrix.* Cowlagi & Cooper (WPI)
Dynamic Sensor-Actor Interactions
8 / 16
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. Cowlagi & Cooper (WPI)
ˆ with uniform placement. Field reconstruction with Θ
Dynamic Sensor-Actor Interactions
9 / 16
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.
Cowlagi & Cooper (WPI)
Dynamic Sensor-Actor Interactions
10 / 16
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.
Cowlagi & Cooper (WPI)
Dynamic Sensor-Actor Interactions
11 / 16
Actor-Relevant Sensor Placement True
Iter. 1
Iter. 2
Iter. 3
True cost: 104.4.
Incurred cost: 131.5.
Incurred cost: 110.7.
Incurred cost: 107.6.
Iter. 4
Iter. 5
Iter. 6
Iter. 7
Incurred cost: 106.9.
Incurred cost: 105.4.
Incurred cost: 105.3.
Incurred cost: 104.8.
Iter. 8
Iter. 9
Iter. 10
Incurred cost: 104.9.
Incurred cost: 105.2.
Incurred cost: 104.9.
Cowlagi & Cooper (WPI)
Dynamic Sensor-Actor Interactions
12 / 16
Actor-Relevant Sensor Placement Arbitrary 1
Arbitrary 2
Arbitrary 3
Clustered
Uniform
True Cost
140
135
Incurred Cost
130
125
120
115
110
105
100
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
Iteration #
Cowlagi & Cooper (WPI)
Dynamic Sensor-Actor Interactions
13 / 16
Actor-Relevant Sensor Placement (NS = 20) Arbitrary 1
Arbitrary 2
Arbitrary 3
Clustered
Uniform
True Cost
140
135
Incurred Cost
130
125
120
115
110
105
100
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
Iteration #
Cowlagi & Cooper (WPI)
Dynamic Sensor-Actor Interactions
14 / 16
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.
Cowlagi & Cooper (WPI)
Dynamic Sensor-Actor Interactions
15 / 16
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]
Cowlagi & Cooper (WPI)
Dynamic Sensor-Actor Interactions
wpi.edu/∼rvcowlagi.
16 / 16
