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