Dynamic Sensor-Actor Interactions for Path-Planning in a ... · Actor:planning + acting....
Transcript of 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.
Fair Use Disclaimer: This document may contain copyrighted material, such as photographs and diagrams, the use of which maynot always have been specifically authorized by the copyright owner. The use of copyrighted material in this document is inaccordance 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
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/
Wildfire mapping and predictionto assist mitigation.c©2016 Matthew Keys. All rights reserved.
http://feed.matthewkeys.net/firemap/
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.
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}.
Cowlagi & Cooper (WPI) Dynamic Sensor-Actor Interactions 4 / 16
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?
Cowlagi & Cooper (WPI) Dynamic Sensor-Actor Interactions 5 / 16
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.
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.
Expected edge transition cost: g((vi , vj)) = c(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
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.*
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.Field reconstruction with Θ with uniform placement.
Cowlagi & Cooper (WPI) 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 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.
Cowlagi & Cooper (WPI) Dynamic Sensor-Actor Interactions 12 / 16
Actor-Relevant Sensor Placement
100
105
110
115
120
125
130
135
140
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
Incu
rre
d C
ost
Iteration #
Arbitrary 1 Arbitrary 2 Arbitrary 3 Clustered Uniform True Cost
Cowlagi & Cooper (WPI) Dynamic Sensor-Actor Interactions 13 / 16
Actor-Relevant Sensor Placement (NS = 20)
100
105
110
115
120
125
130
135
140
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
Incu
rre
d C
ost
Iteration #
Arbitrary 1 Arbitrary 2 Arbitrary 3 Clustered Uniform True Cost
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] wpi.edu/∼rvcowlagi.
Cowlagi & Cooper (WPI) Dynamic Sensor-Actor Interactions 16 / 16