June 27, 2018 — American Control Conference, Milwaukee

This research was funded by the Laboratory Directed Research and Development (LDRD) program at SandiaNational Laboratories and National Science Foundation grants — NSF CMMI-1254990 (CAREER, Oishi) and NSFCNS-1329878. Sandia National Laboratories is a multi-mission laboratory managed and operated by NationalTechnology and Engineering Solutions of Sandia, LLC., a wholly owned subsidiary of Honeywell International, Inc.,for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525.

Multiple Pursuer-Based Intercept via ForwardStochastic Reachability

Abraham Vinod†, Baisravan HomChaudhuri†, Christoph Hintz†,Anup Parikh∗, Stephen Buerger∗, Meeko M. K. Oishi†, Greg

Brunson†, Shakeeb Ahmad†, and Rafael Fierro†

†Electrical and Computer Engineering,University of New Mexico

∗Sandia National Laboratories

Abraham P. Vinod Multiple Pursuer-Based Intercept via Forward Stochastic Reachability 1 / 12

Motivation: Aerial Suppression of Airborne Platform

I Protect assets from an intruder UAVI Modeled as a (non-reactive) stochastically moving goal/threat

I Pursue the intruder with UAV teamsI Stochastic reachability-based pursuit

Maximize capture probability via a team of pursuer UAVs

Abraham P. Vinod Multiple Pursuer-Based Intercept via Forward Stochastic Reachability 2 / 12

Related work

I Multi-agent differential game-based pursuitI Vidal, Shakernia, Kim, Shim, & Sastry (2002); Tomlin,

Mitchell, Bayen, & Oishi(2003); Mitchell, Bayen, & Tomlin(2005); C. F. Chung, T. Furukawa, & A. H. Goktogan (2006);C. F. Chung, & A. H. Goktogan (2008); Huang, Zhang, Ding,Stipanovıc, & Tomlin (2011);

I Stochastic reachabilityI Abate, Prandini, Lygeros, & Sastry (2008); G. Hollinger, S.

Singh, J. Djugash, & A. Kehagias (2009); Summers, & Lygeros(2011); Kariotoglou, Raimondo, Summers, & Lygeros (2011);Lesser, Oishi, & Erwin (2013); Vinod, HomChaudhuri, & Oishi(2017); N. Malone, K. Lesser, M. Oishi, & L. Tapia (2017);

Abraham P. Vinod Multiple Pursuer-Based Intercept via Forward Stochastic Reachability 3 / 12

System description

I Linearized UAV dynamics (n = 12)I Based on Asctec Hummingbird

I Goal UAV GxG [t + 1] = AxG [t] + BK (xG [t]− xa) + Bw [t]

I LQR to an asset location xaI Model uncertainty via w ∼ N (µw ,Σw )I Uncontrolled stochastic dynamics

I N pursuers UAV PixPi [t + 1] = AxPi [t] + BuPi [t]

I Bounded control authority uPi ∈ UPI Controlled deterministic dynamicsI Capture set for each Pi

I Known initial states for goal and pursuer UAVs

Mahony, Kumar, & Corke, IEEE Rob. Auto. Mag., 2012

Abraham P. Vinod Multiple Pursuer-Based Intercept via Forward Stochastic Reachability 4 / 12

Capture of stochastic target via convex optimization

Pursuer Pi Goal G

Single Pursuer Problem :

maximizeτ, xPi [τ ]

CatchPrxPi(τ , xPi [τ ]; x0)

subject to{

τ ∈ Z[1,N]xPi [τ ] ∈ ReachPi (τ ; xPi [0])

Vinod, HomChaudhuri, and Oishi, HSCC 2017

Abraham P. Vinod Multiple Pursuer-Based Intercept via Forward Stochastic Reachability 5 / 12

Capture of stochastic target via convex optimization

Pursuer Pi Goal G

Pi ’s τ -timereach set

G’s statePDF at τ

Single Pursuer Problem :

maximizeτ, xPi [τ ]

CatchPrxPi(τ , xPi [τ ]; x0)

subject to{

τ ∈ Z[1,N]xPi [τ ] ∈ ReachPi (τ ; xPi [0])

Vinod, HomChaudhuri, and Oishi, HSCC 2017

Abraham P. Vinod Multiple Pursuer-Based Intercept via Forward Stochastic Reachability 5 / 12

Capture of stochastic target via convex optimization

Pursuer Pi Goal G

Optimal capturelocation at τ1

Single Pursuer Problem :

maximizeτ, xPi [τ ]

CatchPrxPi(τ , xPi [τ ]; x0)

subject to{

τ ∈ Z[1,N]xPi [τ ] ∈ ReachPi (τ ; xPi [0])

Vinod, HomChaudhuri, and Oishi, HSCC 2017

Abraham P. Vinod Multiple Pursuer-Based Intercept via Forward Stochastic Reachability 5 / 12

Forward reachability for goal and pursuer UAVs

ψW ΨW Ψw

ψwΨx(·; τ, xG [0])ψx(·; τ, , xG [0])

IID noise ΨW =∏

Ψw

Fourier transformx = Aτ xG [0] + Cτ W

Fourier transform

Inverse Fourier transform

Ψw (α) = Ew[exp

(jα>w

)]=∫Rp

ejα>zψw (z)dz

ψw (z) =( 1

2π

)p ∫Rp

e−jα>zΨw (α)dα

I Catch probability

CatchPrxPi(τ, xPi [τ ]; xG [0]) =

∫CatchSet(xPi [τ ])

ψxG (y ; τ, xG [0])dy

Vinod, HomChaudhuri, and Oishi, HSCC 2017

Abraham P. Vinod Multiple Pursuer-Based Intercept via Forward Stochastic Reachability 6 / 12

Forward reachability for goal and pursuer UAVs

I Log-concave (over xPi [τ ]) τ -time catch probability

CatchPrxPi(τ, xPi [τ ]; xG [0]) =

∫CatchSet(xPi [τ ])

ψxG (y ; τ, xG [0])dy

I Goal UAV G’s state is Gaussian (x0 = xG [0])

xG [τ ; x0] ∼ N (µ[τ ],Σ[τ ])

I Deterministic forward reach set for pursuer Pi

ReachPi (τ ; xPi [0]) ={y ∈ X |∃πτ ∈ Uτ

P s.t. xPi [τ ] = y}

= {Aτ xPi [0]} ⊕ CτUτP

Single Pursuer Problem-τ :

minimizexPi [τ ]

− log(

CatchPrxPi(τ, xPi [τ ]; x0)

)subject to xPi [τ ] ∈ ReachPi (τ ; xPi [0])

Single Pursuer Problem-τ is convex! Vinod, HomChaudhuri, and Oishi, HSCC 2017

Abraham P. Vinod Multiple Pursuer-Based Intercept via Forward Stochastic Reachability 6 / 12

Problem statements

Q1 Maximize the team capture probability over a finite timehorizon

Q2 Perform experimental validation of forward stochasticreachability-based interception

Abraham P. Vinod Multiple Pursuer-Based Intercept via Forward Stochastic Reachability 7 / 12

Extension to multiple pursuers w/ convexity preserved

I For XP [τ ] =[(xP1 [τ ])>, (xP2 [τ ])>, . . . , (xPN [τ ])>

]>,

TeamCatchPr(τ,XP ; xG [0]) = maxPi

[CatchPrxPi(τ, xPi [τ ]; xG [0])]

Multiple Pursuer Problem :

maximizeτ, XP [τ ]

TeamCatchPr(τ ,X P [τ ]; x0)

subject to{

τ ∈ Z[1,N]X P [τ ] from dynamics

Single Pursuer Problem-i :

maximizeτ, xPi [τ ]

CatchPrxPi(τ , xPi [τ ]; x0)

subject to{

τ ∈ Z[1,N]xPi [τ ] ∈ ReachPi (τ ; xPi [0])

Single Pursuer Problem-(τ, i) :

minimizexPi [τ ]

− log(

CatchPrxPi(τ, xPi [τ ]; x0)

)subject to xPi [τ ] ∈ ReachPi (τ ; xPi [0])

Single Pursuer Problem-(τ, i) is convex!I Enforce coordination via other definitions for TeamCatchPr

Abraham P. Vinod Multiple Pursuer-Based Intercept via Forward Stochastic Reachability 8 / 12

Experimental Setup

Multi-Agent, Robotics, Heterogeneous Systems (MARHES) Lab(http://marhes.unm.edu/)

Abraham P. Vinod Multiple Pursuer-Based Intercept via Forward Stochastic Reachability 9 / 12

Robustness to model uncertainty

I Mismatch in dynamics (human-controlled)I Robustness due to receding horizon control

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Robustness to model uncertainty

https://www.youtube.com/watch?v=eFGg7U7gEQw

Abraham P. Vinod Multiple Pursuer-Based Intercept via Forward Stochastic Reachability 10 / 12

Robustness to model uncertainty

Abraham P. Vinod Multiple Pursuer-Based Intercept via Forward Stochastic Reachability 10 / 12

Open-loop vs receding horizon control

I Threat UAV follows the LQR-based stochastic dynamicsI Receding horizon control > open-loop control

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Open-loop vs receding horizon control

https://www.youtube.com/watch?v=H0BZrk9Goxg

Abraham P. Vinod Multiple Pursuer-Based Intercept via Forward Stochastic Reachability 11 / 12

Open-loop vs receding horizon control

Abraham P. Vinod Multiple Pursuer-Based Intercept via Forward Stochastic Reachability 11 / 12

Summary, future work, and acknowledgmentsSummaryI Stochastic reachability-based pursuit w/ multiple pursuersI Experimental validation of forward stochastic reachability

Future workI Experiments with multiple pursuersI Enforcing more coordinationI Enforcing keep-out regions

AcknowledgmentsThis work was supported byI Laboratory Directed Research and Development (LDRD)

program at Sandia National LaboratoriesI NSF CMMI-1254990 (CAREER, Oishi)I NSF CNS-1329878

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