Attack-resilient observer pruning for path-tracking control of Wheeled Mobile Robot
PProceedings of the ASME 2020Dynamic Systems and Control ConferenceDSCC2020October 4-7, 2020, Pittsburgh, PA, USA
DSCC2020-3139
ATTACK-RESILIENT OBSERVER PRUNING FOR PATH-TRACKING CONTROL OFWHEELED MOBILE ROBOT
Yu Zheng, Olugbenga Moses Anubi
FAMU-FSU College of EngineeringTallahassee, FL 32310, USAEmail: [email protected], [email protected]
ABSTRACT
Path-tracking control of wheeled mobile robot (WMR) hasgained a lot of research attention, primarily because of its wideapplicability – for example intelligent wheelchairs, exploration-assistant remote WMR. Recent increase in remote and au-tonomous operations/requirements for WMR has led to more andmore use of IoT devices within the control loop. Consequently,providing interfaces for malicious interactions through falsedata injection attacks (FDIA). Moreover, optimization-basedFDIAs have been shown to cause catastrophic consequences infeedback control systems while by-passing any residual-basedmonitoring system. Since these attacks target system measure-ment process, this paper focuses on the problem of improvingthe resiliency of dynamical observers against FDIA. Specifically,we propose an attack-resilient pruning algorithm which attemptsto exclude compromised channels from being processed by theobserver. The proposed pruning algorithm improves attack-localization precision to with high probability, which cor-respondingly improves the resiliency of the underlying UKF toFDIA. The improvements due to the developed resilient pruning-based observer is validated through a numerical simulation ofa two-layer path-tracking control platform of differential-drivenwheeled mobile robot (DDWMR) under FDIA.
NOMENCLATURE
The following notations and definitions are used through-out the whole paper: R , R n , R n × m denote the space of real num-bers, real vectors of length n and real matrices of n rows and m columns respectively. R + denotes positive real numbers.Normal-face lower-case letters ( e . g . x ∈ R ) are used to representreal scalar, bold-face lower-case letter ( e . g . x ∈ R n ) representsvectors, while normal-face upper case ( e . g . X ∈ R n × m ) repre-sents matrices. represents all-ones vector. Let T ⊆ { , . . . , n } ,then for a matrix X ∈ R n × m , X T ∈ R | T |× m is the sub-matrix ob-tained by extracting the rows of X corresponding to the indicesin T . T c denotes the complement of a set T and the universal seton which it is defined will be clear from the context. The sym-bol ◦ denotes element-wise multiplication of two vectors and isdefined as z = x ◦ y , where z i = x i · y i . The symbol ∗ denotesthe convolution operator for vectors. supp ( x ) donotes the sup-port of the vector x given by the set T = supp ( x ) = { i | x i (cid:54) = } . argsort ↓ ( x ) denotes a function that returns the sorted indices ofvector x in descending order. The space of all square integrablesignals is denoted by L . The space of all point-wise boundedsignals is denoted by L ∞ . Nonholonomic wheeled mobile robots (WMRs) have at-tracted much attention in the past two decades due to its greatmobility and the broad range of applications [1]. Quite a lot ofresearchers have developed path-tracking controllers for wheeledmobile robots considering nonlinearities [2, 3, 4], robustnessagainst model uncertainties [5,6], robustness against noise [7,8].The control strategies depend on the measurements of the robots’velocities and/or location coordinates. However, due to the in-creasing dependence on IoT devices and wireless communica-1 Copyright © 2020 by ASME a r X i v : . [ m a t h . O C ] S e p ion, the resulting tight coupling of computation, communicationand physical components enables malicious agents to inject at-tacks via the sensors and actuators [9]. Consequently, controllerwould make decision based on attacked measurements or the ve-hicle would receive malicious control signals. One type of at-tacks, false data injection attacks (FDIAs), has been shown to becapable of fooling bad data detection (BDD) scheme to compro-mise the integrity of the state estimator, even with very sparsemeasurements corruption. This results in false operations of thewhole system without any alarm [10, 11]. Therefore, it is neces-sary to develop an attack-resilient observer-based control schemeto mitigate the effect of those attacks.Many authors [9, 12, 13] have proposed an (cid:96) -based resilientstate estimators with different modifications or under differentscenarios. These estimators have been validated using cruisecontrol of autonomous ground vehicle, electrical power systems,industrial control systems. However, with the exception of [13],none of the estimators were validated against large percentageFDIA. Also, in [14], a robot intrusion detection system (RIDS)is designed by leveraging physical dynamics of mobile robots.However, the detection engine is a residual-based Chi-squarescheme, which is known to be vulnerable to coordinated FDIAsconsidered in this paper.Inspired by recent developments in estimation and compres-sive sensing, we propose a pruning algorithm to mitigate the ef-fect of FDIA on UKF. Consider a linear measurement model un-der attack: y = H x + e , where, H ∈ R m × n is the linear measurement operator, x ∈ R n isthe state vector, y ∈ R m is the attacked measurement corrupted bya sparse attack vector e ∈ R m . Consequently, attack-resilient es-timation is often formulated as a classical error correction prob-lem [12, 15, 13, 9]: Minimize : (cid:107) e (cid:107) (cid:96) Subject to: y = F e , where F ∈ R n × m is a coding matrix with n (cid:28) m and FH = . Itis known [16, 17] that if the number of attacked nodes is smallenough, exact state estimation can be guaranteed by solving theabove problem. However, it is shown [18] that exact recoveryis unattainable by solving the problem above if more than 50%of the sensor nodes are attacked. Moreover, the (cid:96) optimiza-tion problem above is NP-hard and is often relaxed by solvinga convex problem if coding matrix satisfies Restricted IsometryProperty (RIP) [16, 19].Suppose there is an oracle which gives the exact supp ( e ) apriori, then the resilient state estimation problem becomes trivialsince any decent regression algorithm will be able to recover the states exactly from the non-attacked set. The challenge, how-ever, is that no such oracle exists. Although, there is a host oflocalization algorithms [20] designed to serve this purpose, theyare always not exact with significant false positive and false neg-ative rates . This observation is the central motivation for de-veloping the pruning algorithm. Therefore, the pruning problemto increase the signal-to-attack-ratio of the measure-ment system using any pre-designed inexact attack localizationscheme (subsequently referred to as the oracle ). Then the ex-isting least-square based robust estimation algorithms can be im-plemented for the pruned measurements sets to create a resilientestimator. This process requires a certain amount of redundancyin the measurement system. Otherwise, the estimation prob-lem will be rendered under-determined by the pruning process.Quantifying the required redundancy level for a given oracle isbeyond the scope of this present work and will be addressed infuture work.Although, there is a lot of work in the literature on re-silient Kalman filtering, typical least-square based robust estima-tor, mitigating sensors failures, distortion, delay, strong noise in-terference and more reasons for corrupt signals [21,22,23]. How-ever, the specific characteristics of attack, unbounded but sparse,make those resilient filters be hard to perform attack-resiliently.To the best of the authors’ knowledge, this paper represents oneof the earliest approach to prune measurement channels in real-time in order to improve the resiliency of an underlying observeragainst FDIA.The rest of paper is organized as follows. In Section 2,a two-layer controller is designed, with UKF, to track a refer-ence trajectory with noisy measurement system. In Section 3,an optimization-based FDIA algorithm designed to bypass themonitor is also implemented. In Section 4, the channel pruningalgorithm is developed and combined with traditional UKF tocreate a resilient observer. In Section 5, simulation results arepresented to validate the proposed pruning-based resilient ob-server. In Section 6, concluding remarks and future directionsare given. In this section, we present a basic two-layer observer-basedpath tracking controller for a differential-driven wheeled mobilerobot (DDWMR). This will be the platform where subsequentpruning algorithm and FDIA are implemented. Figure. 1 showsthe schematic of the DDWMR considered in this paper.The dynamic and kinematic models of DDWMR are given2 Copyright © 2020 by ASME igure 1. Schematic Diagram of the Considered DDWMR Showing Rel-evant Kinematic and Geometric QuantitiesFigure 2. Schematic Diagram of the two-layer observer based controlsystem and the attack injection by [24]: ˙ q = M − ( − D q + B τ ) + w (cid:44) g ( x , u ) + w ˙ θ · · · ˙ z = · · · C ( θ ) q (cid:44) ¯ C ( θ ) q , (1)where, q = [ v ω ] (cid:62) is the generalized body velocities vector, u (cid:44) τ = [ τ R τ L ] (cid:62) is a vector of the wheels torques, and z = [ x y ] (cid:62) isthe task-space position vector, x = [ θ v ω ] (cid:62) is defined as a statevector, w ∼ N ( , R ) is the process noise in dynamics.The kinematic and dynamical parameters are given by: M = (cid:20) m md + J (cid:21) , D = (cid:20) − md ω md ω (cid:21) B = r (cid:20) L − L (cid:21) , C ( θ ) = (cid:20) cos ( θ ) − d sin ( θ ) sin ( θ ) d cos ( θ ) (cid:21) . Given a reference task-space trajectory [ θ d ( t ) z d ( t ) (cid:62) ] (cid:62) ,where z d ( t ) ∈ R is the corresponding planar Cartesian coordi-nates of the desired trajectory. We assume that z d ( t ) is contin-uously differentiable with bounded derivatives, and that all itsderivative up to the 2nd order are known. Next, consider thetracking error given by (cid:101) e = (cid:20) θ − θ d z − z d (cid:21) = (cid:20) e θ e z (cid:21) . (2)Then, the control law is then designed as: τ = B − ( M u + D q ) , (3)where, u = − k q ( q − q d ) + ˙ q d − ¯ C ( θ ) (cid:62) (cid:101) e with q d = C − ( θ )( ˙ z d − k e e z ) ˙ q d = − k e ( ˙ C − ( θ ) e z + q ) + C − ( θ )[ ¨ z d + ( k e + C ( θ ) ˙ C − ( θ )) ˙ z d ] and k q , k e are positive scalar control gains. Proposition 1.
Consider the control law given in (3), if con-trol gains k q > and k e > , then the tracking errors in (2) con-verges to zero asymptotically. Moreover, the generalized veloc-ities tracking error (cid:101) q = q − q d converges to zero asymptoticallywith ˙ z d = C ( θ ) q d satisfied in the limit.Proof. Consider the candidate Lyapunov function: V = (cid:107) (cid:101) q (cid:107) + (cid:107) (cid:101) e (cid:107) (4)taking the first time derivative and substituting (1), (2), (3)yields˙ V = (cid:101) q (cid:62) (cid:18) − k q (cid:101) q − (cid:20) e θ (cid:21) − C ( θ ) e z (cid:19) + e (cid:62) θ ˙ e θ + e (cid:62) z (cid:0) C ( θ ) q d − ˙ z d (cid:1) = − k q (cid:107) (cid:101) q (cid:107) − ( ω − ω d ) e θ − e (cid:62) z C ( θ )( q − q d ) + e (cid:62) θ ˙ e θ + e (cid:62) z ( C ( θ ) q − ˙ z d )= − k q (cid:107) (cid:101) q (cid:107) + e (cid:62) z ( C ( θ ) q d − ˙ z d )= − k q (cid:107) (cid:101) q (cid:107) − k e (cid:107) e z (cid:107) (5)3 Copyright © 2020 by ASMEhis implies that ˙ V is negative semi-definite, and since V ispositive, it follows that V ∈ L ∞ . From (4), it follows that (cid:101) q , (cid:101) e ∈ L ∞ , which also implies that e θ ∈ L ∞ .Integrating (5) yields V − V ( ) ≤ − (cid:90) t (cid:0) k q (cid:107) (cid:101) q ( τ ) (cid:107) + k e (cid:107) e z ( τ ) (cid:107) (cid:1) d τ from which it follows that (cid:101) q , e z ∈ L . Also, ˙ (cid:101) q = − k q (cid:101) q − ¯ C ( θ ) (cid:62) (cid:101) e ∈ L ∞ and ˙ (cid:101) e = ¯ C ( θ ) (cid:101) q − k e (cid:20) e z (cid:21) ∈ L ∞ , which implies that (cid:101) e and (cid:101) q are uniformly continuous. Thus, by Barbalat’s Lemma[25], it follows that (cid:101) e ( t ) → , (cid:101) q ( t ) → An attacker can inject false data computed based on a partialor complete knowledge of system model, in order to covertly andaccurately change the physical behavior of the plant [26]. Thissection gives the notion of a monitor used in this paper. Based onthe monitor, we give a design of FDIA algorithm while assumingan attacker has complete knowledge of system.For the DDWMR described in previous section, we considera redundant measurement system of the form: y = / r L / r / r − L / r cos ( θ ) − d sin ( θ ) sin ( θ ) d cos ( θ ) · q + v (cid:44) f ( x ) + v (6)consisting of both linear and nonlinear components, where x =[ θ v ω ] (cid:62) is defined as a state vector, and v denotes measurementnoises. Definition 1 (Residual-based Monitor of Horizon T ). Based on the closed-loop system in Figure. 2, a monitor schemeis any mapping of the form: Ψ T : { Y T , U T } (cid:55)→ { Ψ , Ψ } where, Y T = ∈ R m × T , U T ∈ R l × T are historical measurementsand controlled inputs for T horizon respectively, Ψ = { ( sa f e ) , ( unsa f e ) } is the first output argument indicatingwhether or not the data contains attacks, Ψ = { , , ··· , m } is thesecond output argument indicating the support of attacks’ loca-tion. The monitor outputs Ψ = { } for any measurement vectorhistory Y T = [ y k , y k − , · · · , y k − T + ] and corresponding controlhistory U T = [ τ k − τ k − · · · τ k − T ] if there exists estimate historyˆ X T = [ ˆ q k , ˆ q k − , · · · , ˆ q k − T ] such that (cid:107) ˆ q j + − g ( ˆ q j , τ j ) (cid:107) ≤ ε w , j = k − T , · · · , k − (cid:107) y j − f ( ˆ q j ) (cid:107) ≤ ε v , j = k − T + , · · · , k where ε w and ε v are any real numbers related to process noiseand measurement noise.Otherwise, the monitor outputs Ψ = { } and the supportof the sparsest attack vector history E T = { e k , e k − , · · · , e k − T + } such that (cid:107) ˆ q j + − g ( ˆ q j , τ j ) (cid:107) ≤ ε w , j = k − T , · · · , k − (cid:107) y j − f ( ˆ q j ) − e j (cid:107) ≤ ε v , j = k − T + , · · · , k After linearizing (6) about the operating point x =[ θ v ω ] (cid:62) , we discretize it using Euler’s approximation witha sampling time T s , and iterate forward T f samples, one obtains: Y f = H x k + G u f + e (7)where, Y f = (cid:104) y k y k + · · · y k + T f (cid:105) (cid:62) , H = C d C d A m C d A m ... C d A T f m , G = T s · · · C d B m · · · C d A m B m C d B m · · · C d A T f − m B m C d A T f − m B m · · · C d B m with A m = I + T s · d ω − md ω md + J − mdv md + J , B m = T s (cid:20) M − B (cid:21) , C d = / r L / r / r − L / r − v sin ( θ ) − d ω cos ( θ ) cos ( θ ) − d sin ( θ ) v cos ( θ ) + d ω sin ( θ ) sin ( θ ) − d cos ( θ ) H admits the singular value decomposition: H = [ U U ] (cid:20) ∑ (cid:21) V , where, U ∈ R m × n , U ∈ R m × ( m − n ) , ∑ = diag ( σ , σ , · · · , σ n ) ,and V ∈ R n × n , it is obvious that, the FDIA would pass the mon-itor if the attack vector e is defined such that the attack mea-surement y a is in the range space of the observation matrix H ( = Range ( U ) ). Consequently, the FDIA is generated by solv-ing the optimization problem: Maximize (cid:107) U T (cid:62) . y A (cid:107) Subject to (cid:107) U T (cid:62) . y A (cid:107) ≤ α (8)for a given support T of attack locations under upper bound ofpercentage of attack injection, and α is a threshold value relatedto observation matrix H and monitor’s threshold ε v . Data-driven attack localization algorithms [27, 28] are ef-fective ways of achieving resiliency under FDIA. However, itis challenging to correctly locate all attacked nodes due to thefundamental inexactness associated with data-driven algorithms.In this section, we propose a pruning algorithm to improve theaccuracy of localization algorithms. The underlying philosophyis that if the measurement set is sufficiently redundant, a subsetwith reduced attacked percentage can be obtained by systemat-ically pruning the measurement set. If the attack percentage isreduced to 0, the pruned measurement set is then used with UKFto produce an improved resilient state estimation under FDIA.Let the unknown actual support of safe measurements be T c = supp ( − e ) with an indicator vector q given, element-wise, as: q i = (cid:26) i ∈ T c T c with ˆ q . Then, the disagreement between the oracle and the actualsupport can be modeled as: q i = ε i ˆ q i + ( − ε i )( − ˆ q i ) , (10)where ε i depicts the agreement between the estimated and actualsupport as follows: ε i = (cid:26) q i = q i q i = − q i (11) It is assumed that ε i ∼ B ( , p i ) , where p i is given by the truepositive rate from the oracle ROC statistics. Moreover, one cansee that ∑ mi = ε i is Poisson-Binomially distributed with probabil-ity mass function given by: Pr (cid:18) m ∑ i = ε i = k − (cid:19) = r ( k ) , k = , · · · , m + r = m ∏ i = P i · (cid:20) − P P (cid:21) ∗ (cid:20) − P P (cid:21) ∗ · · · ∗ (cid:20) − P m P m (cid:21) , r ∈ R m + .Thus, given a reliability level η ∈ ( , ) , we define the max-imum integer l η ≤ m for which oracle will correctly localize atleast l η nodes with a probability of at least η : l η = max (cid:26) k (cid:12)(cid:12)(cid:12)(cid:12) Pr (cid:18) m ∑ i = ε i ≥ k (cid:19) ≥ η (cid:27) = max (cid:26) k (cid:12)(cid:12)(cid:12)(cid:12) − k + ∑ i = r i ≥ η (cid:27) = max (cid:26) k (cid:12)(cid:12)(cid:12)(cid:12) k + ∑ i = r i ≤ − η (cid:27) (13)Next, we retain the oracle output for the first l η most trustednodes. Let s ∈ [ , ] m be a vector of confidence values for the or-acle output for each node, then a robust support can be estimatedas: ˆ T c η = ˆ T c ∩ (cid:8) argsort ↓ ( p ◦ s ) (cid:9) l η . (14) Remark 1. (13) and (14) constitute a pruning scheme for whichthe resulting ˆ T c η excludes all attacked channel with a probabilitylarger than η , Pr { ˆ T c η ∩ T = /0 } ≥ η . Following the pruning operation, the safe measurementmodel used for a UKF is: y ˆ T c η = f ˆ T c η ( x ) + v ˆ T c η . (15)Following standard unscented transformation [30], we use2 n + n -dimensional normallydistributed state x with assumed mean ¯ x and covariance P x asfollows: χ = x χ i = x + ( (cid:112) ( λ + n ) P x ) i , i = , · · · , n χ i + n = x + ( (cid:112) ( λ + n ) P x ) i − n , i = n + , · · · , n W m = λ / ( n + λ ) , W c = W m + ( − α + β ) W i = / ( L + λ ) where, λ = α ( n + κ ) − n represents how far the sigma points areaway from the state, κ ≥ , α ∈ ( , ] , and β = x k − ∼ N ( ¯ x k − , P x , k − ) , sigma points updatethrough time in sequence with the pruning measurement modelin (15). Moreover, according to the corresponding weight, wecan predict the new time step state and calculate the new errorcovariances between the sigma points and the predicted state asfollow: X (cid:63) k = g ( X k − , L ˆ T c η ( ˆ x k )) ˆ x − k = n ∑ i = W i X (cid:63) k , i ˆ P x , k = n ∑ i = W i ( X (cid:63) k , i − ˆ x k )( X (cid:63) k , i − ˆ x k ) T + R Y k , ˆ T c η = f ˆ T c η ( X k ) Next, the measurements and Kalman gains updates are givenby: ˆ y k , ˆ T c η = n ∑ i = W i Y ( k , i ) , ˆ T c η ˆ P y , k = n ∑ i = W i ( Y ( k , i ) , ˆ T c η − ˆ y k , ˆ T c η )( Y ( k , i ) , ˆ T c η − ˆ y k , ˆ T c η ) T + Q ˆ P xy = n ∑ i = W i ( X (cid:63) k , i − ˆ x k )( Y ( k , i ) , ˆ T c η − ˆ y k , ˆ T c η ) T K k = ˆ P xy ˆ P − y , k x k = ˆ x k + K k ( y k , ˆ T c η − ˆ y k , ˆ T c η ) , P x , k = ˆ P x , k − K k ˆ P y , k K Tk where, Q and R are the measurement and process noise covari-ance matrices respectively.In order to numerically verify that the robust support gen-erated by (14) can achieve 100% localization with a probabilityof at least η , we implemented the pruning localization algorithmin a numerical simulation with time-varying FDIAs. The resultsFigure. 3 shows that the algorithm achieves 100% localizationeven for reliability setting η = .
5! When the reliability is setto just 0 .
1, this algorithm misses only two attacked measurementnodes.
Figure 3. Numerical Simulation of pruning algorithm with time-varyingFDIAs with η = . , η = . and η = . ( cross → pruning, circle → oracle, dots at 0 → attacked nodes, the perfect result is all dots at are covered) In this section, numerical simulation is carried out forDDWMR using three observer strategies under FDIA and theresulting path tracking performance and estimated inner statesare compared. The observers compared are: (1) Only UKF,(2) UKF combine directly with the oracle and (3) the proposedpruning-based UKF. For the path-tracking control system, thecontrol gains are set as k = k =
10. The nominal performanceof the control system with UKF in an attack-free setting is shownin Figure 4. It is seen that the control system, together with UKF,performs well when measurement contains no attack. Next, aFDIA is implemented and the generated attack vector is added tothe system measurements. The oracle is simulated based on theuncertainty model in (10) with defined true positive rate r p = . s = .
5. Localizationresults were then generated to match the specified ROC statistics.The pruning algorithm is implemented with η = .
8. The codesfor simulation can be found in https://github.com/ZYblend/Resilient-Pruning-Observer-against-False-Data-Injection-Attacks.Figures 5, 6 and 7 show the comparison of the performanceof three observer strategies under FDIA: " only UKF ", "
UKF withmachine learning " and " pruning observer . The results show thatrobot cannot track the trajectory under FDIA without any local-ization and pruning operation, and the estimated dynamic stateshas very large deviation from the true states. With the oracle, dueto the uncertainty, the tracking path is very oscillatory althoughnot as bad as with UKF alone. However, with the proposed ob-server, the robot was able to track the reference path very closelyand smoothly.6 Copyright © 2020 by ASME igure 4. Path-tracking and state estimation results of the proposed con-trol system without attacksFigure 5. A comparison of path tracking results. The proposed pruningobserver-based control scheme is able to focus robot to track better, whileUKF cannot handle the attacks and machine learning cannot smooth thetrajectory. (dot line: reference trajectory, solid line: actual path)
In this paper, an attack-resilient path tracking controlscheme for wheeled mobile robot under an optimization-basedFDIA was designed. The main contributions include: (1) Sta-ble path-tracking control system for DDWMR, (2) Optimization-based FDIA for DDWMR, and (3) The pruning-based observerdesign using UKF as the underlying observer. It was shown thatthe proposed pruning-based observer significantly improves thesignal-to-attack ratio such that the UKF is able to resiliently es-timate the state of the DDWMR even when portion of the sen-
Figure 6. A comparison of estimated forward velocity. The proposedpruning observer gives more stable and accurate estimation.Figure 7. A comparison of estimated angular velocity. The proposedpruning observer gives more stable and accurate estimation. sor measurements were subject to an FDIA. Although this papershows how promising the resiliency boosting through pruning al-gorithm is, the results presented only represent the initial stagesof this development. Hence there are several open problems thatneed to be addressed. We name a few:1. As with other resilient observers, the pruning-based resilientobserver relies heavily on the inherent redundancy in themeasurement system [31]. However, there is no systematicway to quantify the level of redundancy required given anyoracle. With (cid:96) -based methods, the RIP property partly pro-vide answers to this question. What would be interestingto see is how much of a relaxation do we get on the RIP7 Copyright © 2020 by ASME igure 8. FDIAs are localized wrongly by residual-based monitor requirements by including pruning? Partial answer to thisquestion can be found in [32]. We plan to expand on theresults as it applies to this problem.2. It would be beneficial to see some results on the potentialgain by combining pruning and (cid:96) -based methods.3. There are indications from this paper that it is possible tocombine pruning directly with the update laws of Kalmanfiltering algorithms. In future, we will develop a systematicway to achieve this.4. We plan to generalize and identify the salient properties fora class of oracles that would combine well with a given un-derlying estimator. ACKNOWLEDGMENT
Thanks to Florida State University and the Center for Ad-vanced Power Systems for remote working support on this pa-per during the COVID-19 outbreak. The authors wish everyonesafety in these difficult times.
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