BayesianBased Domain Search Using Multiple ... - Wiley Online Library

Asian Journal of Control, Vol. 16, No. 1, pp. 20–29, January 2014 Published online 19 September 2012 in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/asjc.608

BAYESIAN-BASED DOMAIN SEARCH USING MULTIPLE AUTONOMOUS VEHICLES WITH INTERMITTENT INFORMATION SHARING Yue Wang and Islam I. Hussein ABSTRACT This paper focuses on the development of Bayesian-based domain search strategies using distributed multiple autonomous vehicles with intermittent communications. Multi-sensor fusion based on observations from neighboring vehicles is implemented via binary Bayesian filtering. We prove theoretically that, under appropriate sensor models, the belief of whether there exists an object at every discretized cell in the domain will converge to the true state. An uncertainty map representing the entropies associated with these probabilities is constructed to guide the vehicles’ motion for domain exploration. Under the proposed strategies, all objects in the search domain will be detected. Different motion control schemes are numerically tested to confirm the effectiveness of the proposed search strategies. Key Words: Bayesian filtering, domain search, intermittent information sharing, multiple autonomous vehicles, sensor fusion.

I. INTRODUCTION Sensor management and motion control for multiple autonomous vehicles is a challenging field of research with wide applications [1–4]. In this paper, we consider a stochastic domain search problem using distributed multiple autonomous vehicles with intermittent communications. The objective is to find each object within the domain and fix its position in space. We assume that the number and positions of objects of interest within the search domain are unknown beforehand. Each vehicle is only capable of taking observations within its own sensory range. This is consistent with the sensor models used in [5–7] and applicable to large-scale domain search problems where the sensor range is comparatively limited. It is also assumed that each vehicle can only communicate with neighboring vehicles within its limited communication range. The intermittent information sharing among vehicles reduces duplicated efforts by sharing measurements between neighboring vehicles. In practice, erroneous observations (false or missed detections of objects) are inevitable due to noise and/or sensor failure and hence the system performance is indeterministic [8]. A Bayesian-based probabilistic domain search strategy is adopted in this paper to capture the uncertainty in sensor perception. Furthermore, in order to reduce this Manuscript received December 1, 2011; revised May 5, 2012; accepted June 26, 2012. Y. Wang (corresponding author) is with Department of Mechanical Engineering at Clemson University, South Carolina, SC 29634, USA (e-mail: [email protected]). I. I. Hussein is with Department of Mechanical Engineering at University of New Mexico, Albuquerque, NM 87131, USA (e-mail: [email protected]).

uncertainty and maximize the probability of finding an object of interest, all the observations available to a vehicle (i.e., taken by both the vehicle itself and its neighboring vehicles) should be fused together and utilized as a combined observation sequence. We will prove that given sensors with a detection probability greater than 0.5, the search uncertainty will converge to a small neighborhood of zero, i.e., all unknown objects of interest will be found with near 100% confidence. Although this may seem intuitive, there lacks a rigorous mathematical proof in the literature due to the probabilistic nature of the problem. This, then, motivates the current work. We now review some related literature. In [9] a probabilistic domain search and path planning problem is proposed for multiple uninhabited aerial vehicles (UAVs) under global communications. In [10] the authors present an agent-based negotiation scheme for a multi-UAV search operation with limited sensory and communication ranges. In [11] a distributed fusion scheme is proposed for multi-sensor surveillance. In [12] the authors use the Modified Bayes Factor to model the level of confidence of target existence for an UAV search task in an uncertain environment. The multi-vehicle multitarget Bayesian search approach proposed in [13] maximizes the weighted sum of the “cumulative” probability of detection for each target, which requires the number of total targets to be known. In [14] the problem of searching an area containing both regions of opportunity and hazard with multiple cooperative UAVs is considered. In [15] the optimal domain coverage problem is studied using mobile sensors with limited ranges. An alternative approach related to search in an uncertain environment is simultaneous localization and

© 2012 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society

Y. Wang and I. I. Hussein: Bayesian-Based Domain Search using Multiple Autonomous Vehicles

Table I. Notation. Vi Ni(t) qi(t) ui(t) r* c q Oj pj X (c ) Wi(t) Yi (c ) β ij Hi Ji(t)

Vehicle i Set of Vi’s neighbors including itself Position of Vi Control input Communication range A cell in D Centroid of cell c Object j Position of Oj Binary state random variable Sensory domain Binary observation random variable of Vi at c Detection probability of Vi given X (c ) = j Information entropy of Vehicle i Search cost

mapping (SLAM) [16]. In [8], the occupancy grid mapping algorithm is addressed, which generates maps from noisy observations given foreknown robot pose. It is often used after solving a SLAM problem to generate robot navigation path from the raw sensor endpoints. The work presented in this paper is analogous to the binary Bayesian filtering and the occupancy grid mapping algorithm [8], which are very popular mapping techniques to deal with observations with sensor uncertainties in robotics and artificial intelligence. Nevertheless, we seek to (i) find the conditions that guarantee satisfactory detection results using multiple autonomous vehicles with limited-range sensors and intermittent communications, and (ii) provide a rigorous mathematical proof for these conditions which are also consistent with intuition. This is a nontrivial problem given limited theoretical results existing in the literature and it is significance for effective sensor management and vehicle motion control, especially when the sensing and communication resources are limited. The organization of the paper is as follows. In Section II, we introduce some basic notation and the sensor model. In Section III, the binary Bayesian filtering is derived based on the fused observations from multiple vehicle sensors. In Section IV we prove that the expected probability of object existence will converge to the true state under appropriate sensing assumptions. In Section V, an information uncertainty map is constructed to guide the vehicles’ motion and a search metric is defined to evaluate the search task. Section VI provides a simulation-based demonstration. We conclude the paper in Section VII. While the major results of the work have been summarized in [17], here we provide more detailed mathematical derivation, illustrative explanations and discussions on the simulation section. For the sake of clarity, Table I summarizes the main notation used in this paper.

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II. PROBLEM FORMULATION 2.1 Problem setup Let D ∈ R2 be a domain in which objects are located. Assume there are Nv autonomous vehicles Vi, i = 1,2, . . . , Nv, searching for an unknown number of objects of interest within D. Denote the position of vehicle Vi as qi(t). Let Q ∈ R2 be the configuration space of all the vehicles. Each vehicle Vi satisfies the first order discrete-time equation of motion qi(t+1) = qi(t) + ui(t), where ui ∈ R2 is the control input. Assume that a vehicle pair can only share information whenever they are within the communication range r*. The set Ni(t) = {Vj|||qi(t) - qj(t)|| ⱕ r*} defines vehicle Vi’s neighbors at time t including Vi itself. This model is consistent with the intermittent communication structure in [18, 19]. (Same communication range is assumed in this work for the sake of simplicity. The extension to heterogeneous communication ranges can be considered without any difficulty.) We discretize the domain D into Ntot unit cells, let c be an arbitrary cell in D and q be the centroid of cell c . Let 1 ⱕ No ⱕ Ntot be the total number of objects, which are i.i.d. over D† (By choosing a fine enough grid in the discretization, a cell is guaranteed to contain at most one object.). Denote the position of the static object Oj, j ∈ {1,2, . . . , No} as pj. Both No and pj are unknown beforehand. Define P as the set of all cells containing an object. Let X (c ) be a binary state random variable, where 0 corresponds to object absent, and 1 corresponds to object present. Note that the realization of X (c ) depends on the position of the observed cell c , that is, X (c ) = 1 if c ∈P and 0 otherwise. Since P is unknown and random, X (c ) is a random variable with respect to every c ∈D. Because we assume that the objects are immobile, X (c ) is invariant with respect to time.

2.2 Sensor model For each vehicle Vi, we assume that the onboard sensor follows a Bernoulli distribution within its limited sensory domain Wi(t), and gives binary outputs: object “absent” or “present” ∀c ∈ Wi (t ) . To be consistent with the binary state, we define a binary observation variable Yi (c ) for each Vi at cell c ∈D , where 0 indicates object absent, and 1 indicates object present. Given a state X (c ) = j , the conditional probability mass function f of the Bernoulli observation distribution of vehicle Vi is given by

fYi ( c ) (Yi (c ) = k |X (c ) = j ) ⎧β ij =⎨ i ⎩1 − β j


if k = j , j , k = 0, 1 if k ≠ j

(1)

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Asian Journal of Control, Vol. 16, No. 1, pp. 20–29, January 2014

Pi ( X (c ) = 1|Yt i (c ); t + 1) Π j ∈N i ( t ) (β j ) y j ,t ( c ) (1 − β j )(1− y j ,t ( c )) Pi ( X (c ) = 1; t ) Π j ∈N i ( t ) (β j ) y j ,t ( c ) (1 − β j )(1− y j ,t ( c )) Pi ( X (c ) = 1; t ) + Π j ∈N i ( t ) (1 − β j ) y j ,t ( c ) (β j )(1− y j ,t ( c )) Pi ( X (c ) = 0; t ) Pi ( X (c ) = 1; t ) . = 2 y j ,t ( c ) −1 ⎛⎛ 1 ⎞ ⎞ Pi ( X (c ) = 1; t ) + Π j ∈N i ( t ) ⎜ ⎜ j − 1⎟ ⎟ (1 − Pi ( X (c ) = 1; t )) ⎠ ⎝⎝ β ⎠ =

where Yi (c ) corresponds to the binary observation random variable taken by Vi and β ij is the detection probability of Vi given the actual state X (c ) = j . Because X (c ) are spatially i.i.d., the observations Yi (c ) taken by Vi at every cell c within D are spatially i.i.d. and hence the probability distribution for every c follows the same structure. The following matrix gives the general conditional probability matrix associated with Vi at c :

⎡ Prob(Yi = 0|X = 0; c ) Prob(Yi = 0|X = 1; c ) ⎤ Bi = ⎢ ⎥ ⎣ Prob(Yi = 1|X = 0; c ) Prob(Yi = 1|X = 1; c ) ⎦

(3)

as a normalizing function which ensures that the posterior probabilities Pi ( X (c ) = j|Yt i (c ); t + 1), j = 0, 1 sum to one. According to the law of total probability, the posterior probability of object presence at c updated according to all the observations available to vehicle Vi is given by (3), where y j ,t (c ) is the dummy variable for the random variable Y j ,t (c ) .

IV. CONVERGENCE ANALYSIS (2)

where Prob(Yi = k |X = j; c ), j , k = 0, 1, is the probability of vehicle Vi having observation k given that the actual state at cell c is X = j and is given by the Bernoulli observation distribution (1). For the sake of simplicity, we assume that β ij = β i for both states X = j, j = 0, 1. Let t be time index, conditioned on the state X (c ), the observations Yi ,t (c ) taken by vehicle Vi along time are temporally i.i.d.

III. BAYES PROBABILITY UPDATES In this section, we employ the binary Bayesian filtering to update the probability of object existence at c of vehicle Vi. Define Yt i (c ) = {Y j ,t (c ), Vj ∈ N i (t )} as the observation sequence taken by all the vehicles in Ni(t). Given Yt i (c ), Bayes’ rule gives, for each Vi,

Pi ( X (c ) = 1|Yt i (c ); t + 1) = α i Pi (Yt i (c )|X (c ) = 1) Pi ( X (c ) = 1; t ), where Pi ( X (c ) = 1|Yt i (c ); t + 1) is the posterior probability of object presence at cell c updated by Vi based on Yt i (c ) up to t. Pi (Yt i (c )|X (c ) = 1) is the probability of the particular observation sequence Yt i being taken given X (c ) = 1. Because the observations taken by different vehicles are i.i.d., we have Pi (Yt i (c ) | X (c ) = 1) = Π j ∈N i ( t ) Prob(Y j ,t (c )|X = 1; c ) , where Prob(Y j ,t (c )|X = 1; c ) is given by (1). Pi ( X (c ) = 1; t ) is the prior probability of object presence at time t, and ai serves

In this section, we discuss the convergence conditions for the sequence {Pi ( X (c ) = 1|Yt i (c ); t + 1} when bi is a deterministic parameter within [0,1]. For the sake of simplicity, denote Pi ( X (c ) = 1|Yt i (c ); t + 1) as Pt+1, Pi ( X (c ) = 1; t ) as Pt, and 2 y j ,t ( c ) −1 ⎞ ⎛⎛ 1 ⎞ Π j ∈N i ( t ) ⎜ ⎜ j − 1⎟ ⎟ ⎠ ⎠ ⎝⎝ β

as St, (3) simplifies to the

following non-autonomous nonlinear discrete-time system

Pt +1 =

Pt . Pt + St (1 − Pt )

(4)

Note that St is a random variable dependent on Yt i (c ). Let |Ni(t)| be the cardinality of the set Ni(t). The binary observation sequence Yt i (c ) has 2 N i ( t ) possible combinations at each time Ni ( t ) be the realizations of St step t for cell c . Let st1, st2, , st2 corresponding to each of the 2 N i ( t ) different observation sequences. The probability of having each particular observation sequence Yt i (c ) = {Y j ,t (c ) = y j ,t (c ), Vj ∈ N i (t )} given object present ( X (c ) = 1) is: Π j ∈N i ( t ) (β j ) y j ,t ( c ) (1 − β j )(1− y j ,t ( c )) . Consider the following conditional expectation

⎡ S (1 − Pt ) ⎤ E[1 − Pt +1|Pt ] = E ⎢ t |Pt = pt ⎥ P + S ( − P ) 1 ⎣ t ⎦ t t ⎡ St (1 − pt ) ⎤ |Pt = pt ⎥ =E⎢ ⎣ pt + St (1 − pt ) ⎦ =

2 Ni ( t )

∑

m =1

(5)

stm (1 − pt ) Prob( St = stm |Pt = pt ), pt + stm (1 − pt )


23


where pt is the dummy variable for Pt. We used the substitution law and the law of total probability to derive the second and third equations. Because the observation taken at each time step is a property of the sensor, it is not affected by the probability of object presence at the previous time step, i.e., St is independent of Pt. Therefore, (5) can be reduced to

E[1 − Pt +1|Pt ] =

2 Ni ( t )

∑

m =1

stm (1 − pt ) Prob( St = stm ). pt + stm (1 − pt )

(6)

⎛ ⎜ Π j ∈N i ( t ) (1 − β j ) E[1 − Pt +1|Pt ] = ⎜ ⎜ 1 − ε + Π j ∈N ( t ) ⎛ 1 − 1⎞ ε i ⎜⎝ ⎜⎝ β j ⎟⎠ +

Ni (t )

∑

⎞ ⎞ ⎛ 1 ⎛ 1 ⎜⎝ β k − 1⎟⎠ (1 − ε ) + Π j ∈N i ( t ), ≠ k ⎜⎝ β j − 1⎟⎠ ε

k =1

N (t ) C2 i

∑ k =1

Π j ∈N i ( t ) (1 − β j )

+…+

(7) Π j ∈N i ( t ) (1 − β j ) +… ⎞ ⎞⎛ 1 ⎞ ⎛ 1 ⎛ 1 ⎜⎝ β q − 1⎟⎠ ⎜⎝ β r − 1⎟⎠ (1 − ε ) + Π j ∈N i ( t ), ≠ q , r ⎜⎝ β j − 1⎟⎠ ε

Let us investigate the value of stm and the corresponding Prob( St = stm ) from m = 1 to 2 N i ( t ) .

•

Let m = 1 correspond to the observation sequence ⎛ 1 ⎞ {1, 1, . . . , 1}. We have st1 = Π j ∈N i ( t ) ⎜ j − 1⎟ , ⎝β ⎠ 1 j Prob( St = st ) = Π j ∈N i ( t )β Let m = k+1, where k = 1, . . . , |Ni(t)| correspond to the observation sequence where only the Kth vehicle in vehicle Vi’s neighborhood observes a 0. Define Ckn as the number of combinations that one can choose k objects from a set of size n. Because there are C1N i ( t ) such observation sequences with different orders out of the totally 2 N i ( t ) combinations, the value of k is in the set [1, |Ni(t)|]. Hence, we

have

⎛ ⎛ 1 ⎞⎞ ⎛ βk ⎞ stk +1 = ⎜ Π j ∈N i ( t ), ≠ k ⎜ j − 1⎟ ⎟ ⎜ ⎝ ⎝β ⎠ ⎠ ⎝ 1 − β k ⎟⎠

and

Prob( St = stk +1 ) = (Π j ∈N i ( t ), j ≠ k β j ) (1 − β k )

•

⎛ ⎛ 1 ⎞⎞ ⎛ βq ⎞ ⎛ βr ⎞ = ⎜ Π j ∈N i ( t ), ≠ q , r ⎜ j − 1⎟ ⎟ ⎜ , ⎝ ⎝β ⎠ ⎠ ⎝ 1 − β q ⎟⎠ ⎜⎝ 1 − β r ⎟⎠

Prob( St = stk +1+ N i ( t ) ) = (Π j ∈N i ( t ), j ≠ q , r β j ) (1 − β q )(1 − β r )

• •

And so on for other values of m Let m = 2 N i ( t ) correspond to the observation sequence ⎛ βj ⎞ and {0, 0, . . . , 0}. We have stm = Π j ∈N i ( t ) ⎜ ⎝ 1 − β j ⎟⎠ Prob( St = stm ) = Π j ∈N i ( t ) (1 − β j )

1 Suppose pt = 1-e, where ε ∈ ⎡0, ⎞⎟ is some positive ⎢⎣ 2 ⎠ constant, (6) can be rewritten as (7) if not all sensing parameters b j = 1, and E[1 - Pt+1|Pt] = 0 when all b j = 1, j ∈ Ni(t). Consider the following sensing condition:

⎞ ⎟ Π j ∈N i ( t ) (1 − β j ) + ⎟ ε. ⎞ ⎛ 1 Π j ∈N i ( t ) ⎜ j − 1⎟ (1 − ε ) + ε ⎟⎟ ⎠ ⎠ ⎝β

•

stk +1+ N i ( t )

Let m = k + 1 + N i (t ) , k = 1, , C2N i ( t ) correspond to the the observation sequences where two of the vehicles, e.g., the qth and rth vehicle, observe a 0. Because there are C2N i ( t ) such observation sequences in this case, k is within [1, C2N i ( t ) ] . Therefore, we have

1 Sensing Condition 1. β i ∈⎛⎜ ,1⎤⎥ , i = 1, 2, , N v . ⎝2 ⎦ This condition requires that all vehicle sensors are more 1 likely ( > chance) to take correct measurements. 2 Under Sensing Condition 1, it follows that ⎛ 1 ⎞ Π j ∈N i ( t ) ⎜ j − 1⎟ ∈[ 0, 1) . Now assume that e is a small ⎝β ⎠ ⎛ 1 ⎞ number in the neighborhood of zero, then Π j ∈N i ( t ) ⎜ j − 1⎟ ε ⎝β ⎠ is also a small number close to zero. Hence, (7) can be approximated as E[1 − Pt +1|Pt ] ≈ (Π j ∈N i ( t ) (1 − β j ) +

Ni (t )

∑

Π j ∈N i ( t ), ≠ k (1 − β j )β k

k =1

+…+

N (t ) C2 i

∑

(8)

Π j ∈N i ( t ), ≠ q , r (1 − β )β β j

q

r

k =1

+ … + Π j ∈N i ( t )β j )ε . The expression within the bracket in (8) gives the total probability of all possible observation sequences taken by the vehicles in Ni(t) given that there is an object within c , and is therefore equal to 1. If b j = b, the expression gives the total probability of a binomial distribution with parameter b and |Ni(t)|. Hence, E[1 - Pt+1|Pt = 1 - e] ª e and we have the following lemma. Lemma IV.1. Under Sensing Condition 1, if an object is present, given that the prior probability of object presence Pi ( X (c ) = 1; t ) of vehicle Vi is within a small neighborhood of radius e from 1 at time step t, then the conditional expectation of the posterior probability Pi ( X (c ) = 1|Yt i (c ); t + 1) will remain in this neighborhood at the next time step. If all the


24


(a)

(b)

(c)

(d)

1 1 Fig. 1. The term g(b, e, |Ni(t)|) as a function of ε ∈ ⎡0, ⎞⎟ and β ∈ ⎛⎜ , 1⎞⎟ for (a) |Ni(t)| = 1, (b) |Ni(t)| = 20, (c) |Ni(t)| = 50, and ⎢⎣ 2 ⎠ ⎝2 ⎠ (d) |Ni(t)| = 100.

sensors are “perfect” with zero probability of observation error, i.e., b j = b = 1, then the conditional expectation is 1.

g ( β , ε , N i (t ) ) = Ni (t )

∑

Following similar derivation, we come up with a lemma for the posterior probability of object absence. Lemma IV.2. Under Sensing Condition 1, if an object is absent, given that the prior probability of object absence Pi ( X (c ) = 0; t ) of vehicle Vi is within a small neighborhood of radius e from 1 at time step t, then the conditional expectation of the posterior probability Pi ( X (c ) = 0|Yt i (c ); t + 1) will remain in this neighborhood at the next time step. If all the sensors have zero observation error probability, i.e., b j = b = 1, then the conditional expectation is 1. The following theorem summarizes the above results.

1 Theorem IV.1. For β i ∈⎛⎜ , 1⎤ , i = 1, 2, , N v , if the initial ⎝ 2 ⎥⎦ probability of object existence is within a small neighborhood of the true state at time step t, the conditional expectation of the posterior probability will remain in this neighborhood at the next time step. If b i = b = 1, then the conditional expectation is 1. This theorem gives a weak result because it implies that only if the initial prior probability is close to the true state, given “good” sensors with detection probabilities greater than 0.5, the belief of object existence will remain near the true state. Next, we derive a stronger result for the case of homogeneous sensor properties across the network.

1 Sensing Condition 2. β i = β ∈⎛⎜ , 1⎤⎥ , i = 1, 2, , N v . ⎝2 ⎦ This condition implies that all the vehicles have identi1 cal sensors with the same detection probability β ∈⎛⎜ , 1⎤ . ⎝ 2 ⎥⎦ Under Sensing Condition 2, the term within the bracket in (7) is equivalent to

k =0

CkN i ( t ) (1 − β ) N i ( t ) k

⎛1 ⎞ ⎛1 ⎞ ⎜⎝ β − 1⎟⎠ (1 − ε ) + ⎜⎝ β − 1⎟⎠

, β ≠ 1.

Ni (t ) − k

(9)

ε

Lemma IV.3. The function g(b, e, |Ni(t)|) is less than 1 for

1 1 β ∈⎛⎜ , 1⎞⎟ , ε ∈⎛⎜ 0, ⎞⎟ , and |Ni(t)| ⱖ 1. Moreover, g(b, e, ⎝2 ⎠ ⎝ 2⎠ 1 |Ni(t)|) = 1 for e = 0, β ∈⎛⎜ , 1⎞⎟ , |Ni(t)| ⱖ 1. ⎝2 ⎠ Before we provide a rigorous proof, let us look at Fig. 1. It shows g(b, e, |Ni(t)|) as a function of b and e for |Ni(t)| = 1, |Ni(t)| = 20, |Ni(t)| = 50, and |Ni(t)| = 100. It shows that g ⱕ 1

1 1 for ε ∈ ⎡0, ⎞⎟ and β ∈⎛⎜ , 1⎞⎟ . ⎢⎣ 2 ⎠ ⎝2 ⎠ The following gives the proof for Lemma IV.3. Proof. For brevity, let n = |Ni(t)|. Proving that g(b, e, |Ni(t)|) is less than 1 is equivalent to proving that

Ckn (1 − β ) n

n

∑ k =0

n

k

⎛1 ⎞ ⎛1 ⎞ ⎜⎝ β − 1⎟⎠ (1 − ε ) + ⎜⎝ β − 1⎟⎠

n− k

ε

1 , or,, +1 n k =0

0 . If ⎝β ⎠ ⎢⎣⎝ β ⎠ ⎥⎦ n− k n ⎡⎛ 1 ⎞ k ⎤ ⎛1 ⎞ n n − + − ( n + 1 ) C ( 1 − β ) − − 1 ( 1 ε ) 1 ε ⎥ < 0, ⎢⎜ k ∑ ⎜ ⎟ ⎟ ⎝β ⎠ ⎢⎣⎝ β ⎠ ⎥⎦ k =0



25

this is true, first when m = 1, the numerator equals

then g(b,e,n) is less than 1. Note that

3

n

∑ (n + 1)C

n k

⎛1 ⎞ ( 4 − 6β ) + ⎜ − 1⎟ ( 2 − 6β ) < 0 . Next, we take the derivative ⎝β ⎠ of the numerator with respect to m, which gives

(1 − β ) n = ( n + 1)( 2 − 2β ) n,

k =0

k

n ⎛1 ⎞ ⎛1 ⎞ ∑ ⎜⎝ β − 1⎟⎠ = ∑ ⎜⎝ β − 1⎟⎠ k =0 k =0 n

n− k

⎛1 ⎞ 1 − ⎜ − 1⎟ ⎝β ⎠ = 1 2− β

n +1

⎛1 ⎞ (1 − 2β ) + ( m + 2) ⎜ − 1⎟ ⎝β ⎠

.

⎛1 ⎞ + (1 − 4β )] + ⎜ − 1⎟ ⎝β ⎠

Therefore, to prove the lemma, we only need to prove

⎛1 ⎞ 1 − ⎜ − 1⎟ ⎝β ⎠ ( n + 1)( 2 − 2β ) n − 1 2− β

n +1

< 0.

(10)

n

−( 2β − 1) and is hence less than 0. β Assume that for n = m,

[(1 − 2β )m

m+ 2

(1 − 2β ) < 0.

Therefore, the numerator is a monotonically decreasing function for m ⱖ 1 with a negative value at m = 1. When e = 0, g(b, e, n) reduces to

∑

Next, we use the principle of mathematical induction to prove the inequality (10). When n = 1, the left hand side of (10) is given by

m +1

Ckn (1 − β ) n

k =0

⎛1 ⎞ ⎜⎝ β − 1⎟⎠

k

n

= ∑ Cknβ k (1 − β ) n − k = 1. k =0

This completes the proof.

䊐

2

⎛1 ⎞ 1 − ⎜ − 1⎟ ⎝β ⎠ ( m + 1)( 2 − 2β ) m − 1 2− β

From Lemma IV.3, the expectation E[1 - Pt+1|Pt = 1 - e] is always less than e. Hence, we have the following lemmas and theorem.

m +1

0, or equivalently, the attained accuracy of the domain search task is 1-d. Furthermore, 100% certainty can be obtained if Sensing Condition 2 is satisfied according to Theorem IV.2. Therefore, under any vehicle motion control scheme that covers all the cells within the entire mission domain D, the cost function Ji→d, i.e., all the objects of interest will be guaranteed to be found with desired uncertainty. In this section, we present two different vehicle motion control strategies and compare their performance in simulation. Let us briefly introduce the mathematical model for bi used in these control schemes. The key feature of this kind of model is that the sensing capability is limited (see [18, 5, 6, 7, 19] for more details). Other sensor models can be used and the following control schemes still hold. Define si = qi (t ) − q as the relative distance between the vehicle’s position qi(t) and the centroid q of the observed cell c . Let bi be given by the following fourth-order polynomial function within the sensor range ri and bn = 0.5 otherwise

⎧ Mi 2 2 2 ⎪ ( si − ri ) + bn if si ≤ ri β i ( si ) = ⎨ ri 4 , ⎪⎩ if si > ri bn

(16)

where Mi + bn gives the peak value of bi if the cell c being observed is located at the vehicle’s location. ri is the sensor range. The sensing capability decreases with range and becomes 0.5 outside of the limited sensory domain Wi, i.e., the sensor returns an equally likely observation of “absent” or “present” regardless of the true state.

5.2 Search metric We now associate each vehicle Vi with the following search cost function to evaluate the progress of the search task

∑ J (t ) = i

H i ( Pi (c , t )) c ∈D H max AD

.

(15)

The cost Ji(t) is proportional to the sum of uncertainty over D. We divide the sum of the uncertainty over all cells by the area of the domain AD multiplied by Hmax in order to normalize Ji(t). Therefore, 0 ⱕ Ji(t) ⱕ 1. Initially,

H ( P (c , t = 0)) J i (t = 0 ) = i i ≤ 1 . If H i ( Pi (c , t s )) = 0 at some H max t = ts for all c ∈D , then Ji(ts) = 0 and the entire domain has been satisfactorily covered and we know with 100% certainty that there are no more objects yet to be found.

6.2 Memoryless motion control scheme We first consider a motion control scheme based on only the uncertainty map at current time step, that is, the control scheme is memoryless. For the sake of simplicity, we assume that there is no speed limit on the vehicles, i.e., a vehicle is able to move to any cell within D from its current location. Consider the set

QHi (t ) = {c ∈ D : argmax c H i ( Pi (c , t ))}, which is the set of cells with highest search uncertainty of vehicle Vi within D at time t. Next, let q ic (t ) be the centroid of the cell that Vi is currently located at and define the subset Qdi (t ) ⊆ QHi (t ) as



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Condition C1. H i ( Pi (c , t )) ≤ H iu, ∀c ∈Wi (t ), where H iu = −ε ln ε − (1 − ε ) ln(1 − ε ) > 0 is a preset threshold of some small value. For every vehicle Vi, the motion control scheme with memory is given as follows:

(a)

(b)

Fig. 2. (a) Position of objects and initial deployment of vehicles. (b) Probability of object presence of vehicle V1 at t = 1200.

⎧ui (t ) if C1 does not hold u*i (t ) = ⎨ ⎩ui (t ) if C1 holds

(17)

where

u i ( t ) = ki

∑

([( 2 Pi ( X (c ) = 1; t ) − 1) 2 − 1]2

c ∈Wi ( t ) t

⋅ ∑ (β i (τ + 1) − β i (τ ))), τ =0

is the nominal control law inspired by its deterministic continuous counterparts ([18, 5, 19]), where both the current probability of object presence Pi ( X (c ) = 1; t ) and the sensing capability b i up to the current time step are used, and (a)

(b)

Fig. 3. Fleet motion (green dots represent for vehicles’ initial positions and red dots for final positions) under (a) motion control scheme without memory, and (b) motion control scheme with memory.

ui (t ) = − ki (qi (t ) − q *i ) is the perturbation control law. The centroid q *i of cell c *i is chosen from the set Qi (t ) = {c ∈ D : H i ( Pi (c , t )) > H iu } based on the uncertainty information at the current time step and only available to vehicle Vi itself. 6.4 Simulation-based performance comparison

Qdi (t ) = {c ∈QHi (t ) : argmin c q ic (t ) − q } , where q is the centroid of c . The set Qdi (t ) contains the cells which have both the shortest distance from the current cell and the highest uncertainty. At every time step, vehicle Vi takes observations at all the cells within its sensory range. In general, b i ⫽ b j, j ∈ Ni(t). If Vi and its neighbor Vj have the same distance to the centroid of a certain cell c , we may have b i = b j, i ⫽ j. The posterior probabilities at these cells are updated according to (3) based on all the fused observations. The uncertainty map is then updated. The vehicle chooses the next cell to go to from Qdi (t ) based on the updated uncertainty map. Note that Qdi (t ) may have more than one cell. Let NHd be the number of cells in Qdi (t ), the sensor will randomly pick a cell 1 from Qdi (t ) with equal probability . This process is N Hd repeated until Hi is within a small neighborhood of zero with radius e for every cell c ∈D . 6.3 Motion control scheme with memory We now develop a motion control scheme that takes into account both the current probability information, uncertainty map and the sensing history. Let us consider the following condition.

We now provide numerical simulations to illustrate and compare the performance of both motion control schemes. We assume a square domain D with size 50 ¥ 50, and discretize it into 2500 unit cells. We set Mi = 0.4, ri = 10, and r = 8. The desired uncertainty level is H iu = 0.02, corresponding to e = 2 ¥ 10-4. There are 10 objects with a randomly selected deployment as indicated by the magenta dots in Fig. 2a. The position, sensory radius, and communication range for each of the six vehicle sensors is shown by the black dot, black circle, and green circle. Fig. 2b shows the probability of object presence of vehicle V1 at t = 1200 under both control schemes. All the peaks represent the positions of the objects with probability 1. The probability of object presence of other vehicles is similar to that shown in Fig. 2b. It indicates that all the 10 unknown objects have been found. Figs 3a and 3b show the trajectories of all the vehicles during the entire mission under the motion control scheme without and with memory, respectively. Fig. 4a shows the search cost function Ji(t) for vehicles V1 to V6 under the memoryless motion control scheme. Fig. 4b shows Ji(t) under the motion control scheme with memory, where the control law (17) is used with gains ki = 1, ki = 0.025. In both cases, every Ji(t) converges to zero at t = 1200, which is consistent with the result shown in Fig. 2b.


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1

0

200 400 600 800 10001200 t

0.5

200 400 600 800 10001200 t

0

200 400 600 800 10001200 t

(a)

200 400 600 800 10001200 t

Vehicle 2

1

0.5 0

0.5 0

200 400 600 800 10001200 t

1

0.5

200 400 600 800 10001200 t

1

0.5 0

200 400 600 800 10001200 t

1 Vehicle 6

Vehicle 5

1

0

0.5

0.5 0

200 400 600 800 10001200 t

1 Vehicle 3

0.5 0

0

200 400 600 800 10001200 t

1 Vehicle 4

Vehicle 3

1

0.5

Vehicle 4

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200 400 600 800 10001200 t

Vehicle 5

0

0.5

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Vehicle 6

0.5

1 Vehicle 1

Vehicle 2

Vehicle 1

1

200 400 600 800 10001200 t

0.5 0

200 400 600 800 10001200 t

(b)

Fig. 4. Cost function Ji(t) for V1-V6 under (a) memoryless motion control scheme, and (b) motion control scheme with memory.

Comparing the simulation results, there is more redundancy in vehicle trajectories under the memoryless motion control scheme. This is because the controller is only dependent on the current uncertainty map and does not take into account the history of the paths that the vehicles traveled before. However, the reduction of uncertainty is faster under the memoryless control scheme because it is a global controller that always seeks the cell with highest uncertainty within the entire search domain. On the other hand, under the motion control scheme with memory, the nominal controller is a local controller which drives the vehicle towards the cell with higher uncertainty within the sensory domain, and a perturbation controller is used whenever the vehicle is trapped in a local minimum. If fuel efficiency is a priority, one may want to avoid using a memoryless motion controller that spreads all over the domain. On the contrary, if time is a limited resource, one may prefer a memoryless motion controller in order to achieve the desired detection certainty faster.

VII. CONCLUSION In this paper, we developed a Bayesian-based domain search strategy using distributed multiple autonomous vehicles. Each vehicle is assumed to have limited sensory range and intermittent communications between neighboring vehicles. The expected probability of object existence is guaranteed to be within a desired uncertainty level of the true state under appropriate sensor condition. It is further proved that the belief of object existence converges to the true state given “good” homogenous sensors across the sensor network. Two vehicle motion control strategies utilizing the uncertainty

map are investigated and compared. Both show the effectiveness of the proposed search strategy.

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communications.” Amer. Control Conf., Seattle, Washington, pp. 4370–4375 (2008). 19. Wang, Y. and I. I. Hussein. “Awareness coverage control over large scale domains with intermittent communications.” IEEE Trans. Autom. Control (TAC), Vol. 55, No. 8, pp. 1850–1859 (2010). 20. Cover, T. M. and J. A. Thomas. “Elements Inform. theory.” Wiley, 2nd edition, New York, NY (2006).

Yue Wang received her Ph.D. (2011) and M.S. (2008) in Mechanical Engineering from Worcester Polytechnic Institute and B.S. (2005) in Mechanical Engineering and Automation from Shanghai University, China. She is Assistant Professor in Mechanical Engineering Department at Clemson University. From 2011 to 2012, she was a postdoc in Electrical Engineering Department at University of Notre Dame. Her research interests include multi-agent systems, cyber-physical systems, and networked control systems.

Islam Hussein recieved his Ph.D. (2005) in Aerospace Engineering and M.Sc. (2002) in Aerospace Engineering and Applied Mathematics from University of Michigan and B.Sc. (2000) in Mechanical Engineering from American University in Cairo. From 2005 to 2006, he was a postdoc at Coordinated Science Laboratory at University of Illinois at Urbana-Champaign. He is currently Research Associate Professor of Mechanical Engineering Department at University of New Mexico and National Academy of Sciences Senior Faculty Fellow with Air Force Research Laboratory in Albuquerque, NM. His research interests include cooperative coverage control of multi-sensor network, dynamics and control of virtual tether satellite systems, and nonlinear constrained dynamics and control.
