Stochastic Target Interception in Non-convex Domain Using MILP

Transcription

1 Stochastic Target Interception in Non-convex Domain Using MILP Apoorva Shende, Matthew J. Bays, and Daniel J. Stilwell Virginia Polytechnic Institute and State University Blacksburg, VA {apoorva, mjb222, Abstract In this paper we present a planning approach for the stochastic target interception problem, in which, a team of mobile sensor agents is tasked with intercepting multiple targets. We extend our previous work on stochastic target interception to non-convex domains and propose a cost that addresses minimum time requirement for probabilistically intercepting all the targets if possible over a finite horizon. Indeed, our optimization problem for the stochastic case has similar computational costs as the optimization program for the corresponding deterministic case. Our solution presumes that the system can be approximated by linear dynamics and Gaussian noise, with Gaussian localization uncertainty. I. INTRODUCTION We present an algorithm for multi-agent motion planning in the presence of Gaussian uncertainties. The objective of the desired multi-agent motion is to intercept a set of moving targets as quickly as possible while conforming to a nonconvex operational domain. We address this problem in a mixed integer linear programming (MILP) framework for a finite planning horizon. MILP has been effectively used in planning problems, where event detections in the forward planning need to be accounted for in computing the agent control actions. Examples of such planning problems addressed using MILP can be found in [1], [2], [3], [4], [5] etc. In [4] and [5], MILP has been used for efficient target interception. [4] addresses the case of deterministic target interception for the area protection problem. To the best of our knowledge, our previous work [5], is the first effort in integrating uncertainty in localization and dynamics in the target interception problem in the MILP framework. The work described herein extends our previous work [5] on incorporating Gaussian uncertainties in target interception, in 2 ways. While [5] requires the sensor operational domains to be convex, nonconvex domains often arise in realistic applications. Hence we first address the issue of non-convex operational domains using chance constraints in MILP. Secondly, we present an alternative to planning cost in [5], which proposed an additive cost over the planning horizon. The novel cost that we present here, explicitly deals with the time required to intercept all the targets in the finite planning horizon case, where the interception is evaluated through the chance constraints. The issue of agent maneuvering in non-convex domain under Gaussian uncertainty has been addressed in [6], [7] and [8] for goal-reaching problems using disjunctive linear programming (DLP). As is evident in these works, DLP can be efficiently utilized to integrate Gaussian uncertainty in non-convex operational domains for agent maneuvering. For planning problems in which the efficiency of trajectories depends on the occurrence of certain discrete events, it is possible to use DLP by representing the discrete events in the forward planning through disjunctions. However the number of disjunctions in the constraint set of the resulting DLP will be exponential in the number of possible events in the forward planning trajectories, making the disjunctive notation very lengthy and cumbersome. MILP provides a notationally compact alternative to DLP in formulating discrete event trajectory planning problems where the events are represented using binary variables instead of disjunctions. The target interception problem is an example of such a trajectory planning problem where the events of intercepting targets largely determine the cost associated with a multi-agent trajectory. Due to its notational compactness we use MILP to formulate our target interception problem over DLP, which is consistent with our work in [5]. In [1] and [2], non-convex domain constraints have been enforced on agent motion using MILP. However [1] and [2] do not incorporate uncertainty in the constraints. Hence we specifically address the conformance of the agents to the nonconvex operational domain in the MILP framework under Gaussian uncertainty in localization and dynamics. In many recent works, [5], [6], [7], [8], [9], [10] etc., it has been shown that incorporation of Gaussian uncertainty through closed form chance constraints, leads to computationally efficient and accurate solutions. Sampling-based approaches for chance constraints, are also available in the literature [11], [12], [13] etc. These approaches accommodate a more general form of uncertainty than assumed herein, but are more computationally intensive. This paper is organized as follows. In Section II, we describe the linear Gaussian discrete time dynamical model of the system required for the proposed algorithm. In Section III, we describe 2 discrete events that are dealt with in this paper, namely, non-convex domain conformance by the sensor agents and target interception. In Section IV, we propose our MILP formulation of the target interception planning problem. Simulation results for a novel coordinated autonomous riverine rescue team are described in Section V and the paper is concluded in Section VI.

2 II. SYSTEM DYNAMICS In this section we state the principal assumptions concerning the system dynamics that we use in our problem, and we briefly outline the stochastic RHC problem. Most of this section is reproduced from our work [5], as we have used same system dynamics here. We are reproducing it here for the sake of completeness. We denote the number of un-intercepted targets at time t by N(t). As new targets appear, N(t) increases, and it decreases when the un-intercepted targets get intercepted. We denote the state of target i for i {1,..., N(t)} at time t by x i (t) R n and the state of the sensor j for j {1,..., M} at time t by s j (t) R m. We assume that the target and sensor dynamics are described by discrete-time, time-invariant, linear state space equations, x i (t + 1) = A i x i (t) + B i ν i (t) (1) s j (t + 1) = Ājs j (t) + C j u j (t) + B j ν j (t). (2) In the target state equation (1), ν i (t) R nν is zero mean Gaussian noise distributed as N (0, Q i (t)). The sensor control signal is u j (t) R mu and ν j (t) R mν denotes the zero mean Gaussian noise vector distributed as N (0, Q j (t)). The 2-norm of the control vector u j (t) is bounded by a constant u max, i.e., u j (t) u max. (3) If m u = 2, then u j (t) R 2, and the control constraint (3) represents the interior of a circle. As (3) is a nonlinear constraint, we conservatively approximate it using linear constraints, that represent a regular polygon that inscribes the circle. These constraints can be incorporated in the MILP formulation. The constraints corresponding to an N U sided inscribing polygon are given by, ( ) π < u j (t), v r > u max cos for all r {1,..., N U }, N U (4) where, [ <.,. > is] used to denote the inner product in R 2 and, sin(θr ) v r =, for θ r = 2πr N cos(θ r ) U. An important aspect of our modeling is that the noise terms ν i (t) and ν j (t) are independently distributed with respect to each other and the target and sensor states x i (t) and s j (t) for all i {1,..., N(t)} and j {1,..., M}. As a consequence for a given set of control actions the target and sensor trajectories are independent. We assume the target positions χ i (t) R 2 and the sensor positions ς j (t) R 2 are linear functions of the states x i (t) and s j (t) respectively. These relations are stated, χ i (t) = Lx i (t), (5) ς j (t) = Λs j (t), (6) where L and Λ are matrix operators. We require the posterior PDFs of target and sensor states to be Gaussian or well approximated by Gaussian. If the observation equations are linear, then, for linear dynamics, a Kalman filter can be used to compute the mean and covariance of the state distributions. If the state dynamics or observation equations are not linear, then we presume that an adequate approximation of the state distribution can be computed using an extended Kalman filter, unscented filter, etc. Forward predictions are computed for the interval [t, t + T ]. Since the system is time-invariant, and to simplify notation, we denote time in the planning interval by the variable τ [0, T ]. A fixed number N RH = N(t) targets are considered during the planning interval. The initial target and sensor state PDFs at τ = 0 of the planning interval will correspond to the posterior PDFs at time t in the real or simulation environment. At a time instant τ in the planning interval, we denote the target and sensor state means by ˆx i (τ) and ŝ j (τ) respectively and, covariances by P i (τ) and Π j (τ) respectively. These would be updated by the following open-loop predictor equations as in and [6], ˆx i (τ + 1) = A iˆx i (τ) (7) ŝ j (τ + 1) = Ājŝ j (τ) + C j u j (τ) (8) P i (τ + 1) = A i P i (τ)a T i + B i Q i (τ)b T i (9) Π j (τ + 1) = ĀjΠ j (τ)āt j + B j Qj (τ)āt j (10) As the target and sensor positions at all times in the planning interval depend linearly through (6) on the states they also have a Gaussian distribution. We denote the target and sensor position means by ˆχ i (τ) and ˆς j (τ) respectively and, covariances by P i (τ) and Π j (τ) respectively at time τ in the planning interval. From (6) these are related to the state estimates and covariances as follows, ˆχ i (τ) = Lˆx i (τ) (11) ˆς j (τ) = Λŝ j (τ), (12) P i (τ) = LP i (τ)l T (13) Π j (τ) = ΛΠ j (τ)λ T. (14) III. DISCRETE EVENTS AND BINARY VARIABLES A. Non-convex Sensor Domain In most practical applications, the positions of the mobile sensor agents ς j (τ), need to be constrained within a domain Ω R 2. The domain Ω, which we refer to as the operational domain, is typically non-convex. We consider those nonconvex domains Ω R 2 that are connected and can be represented or conservatively approximated as a union of convex polygons Ω k, Ω N D k=1 Ω k, (15) where N D is the number of convex domains Ω k. Approximation of a part of a riverine domain, where such autonomous sensor agents could operate, as a union of disjoint convex polygon is shown in Figure 1. Figure 1 is a part of Peek Creek, located in Pulaski county, VA. While it is possible to use DLP based notation [6] to represent the non-convex polynomial domain constraint, due to its notational compactness for the target interception problem (as was noted in the introduction),

3 Due to Gaussianity of the sensor position we can adopt an approach similar to Subsection III-B to get the following constraint on the sensor mean, < ˆς j (τ), u kl > ˆQ jkl (τ) for all l {1,..., L k }, (23) where, ˆQ jkl (τ) = Q kl Φ 1 (β k ) u T Π kl j (τ)u kl. (24) We use binary indicator variables d jk (τ) to denote that the event ς j (t) Ω k occurs probabilistically, i.e. (21) and hence (23) is satisfied. Thus, Fig. 1. Riverine domain approximated by a union of disjoint polygons. we choose to use MILP based notation to model the nonconvex domains instead. As the sets Ω k for k {1,..., N D } are convex polygons, we can define them by a conjunction of L k linear constraints, < ς, u kl > Q kl for all l {1,..., L k }, (16) where, ς R 2 is a variable point in Ω k and, u kl R 2 and Q kl are constants corresponding to the l th constraint of Ω k. From (15), the constraint on sensor position j can be written as a union of events, ς j (τ) Ω N D k=1 ς j (τ) Ω k. (17) As the sensor positions ς j (τ) are random vectors we cannot assert with certainty that the event ς j (τ) Ω will occur in the planning horizon at every time instant. Thus we define the probabilistic occurrence of the event ς j (τ) Ω by specifying a lower bound β Ω on its probability, P (ς j (t) Ω) β Ω. (18) However due to (17), and the property of union of events, we have, P (ς j (τ) Ω) P (ς j (τ) Ω k ), for all k = 1,..., N D. (19) Thus by imposing the chance constraint, k {1,..., N D } s.t. P (ς j (t) Ω k ) β Ω, (20) the chance constraint (18) is guaranteed to be satisfied. From (16), ς j (t) Ω k is a conjunction of events < ς j (τ), u kl > Q kl for all l {1,..., L k }. Hence, using Boole s inequality, we can achieve the lower bound β Ω in (20), by imposing a lower bound on the probability of satisfaction of the events, P (< ς j (τ), u kl > Q kl ) β k for all l {1,..., L k }, (21) where, β k = 1 1 β Ω L k. (22) d jk (τ) < ˆς j (τ), u kl > ˆQ jkl (τ) for all l {1,..., L k }. (25) The logical relation (25) can be algebraically represented using the big-m notation [14] as follows, < ˆς j (τ), u kl > ˆQ jkl (τ) + M big (1 d jk (τ)) for all l {1,..., L k }, (26) where M big is a large positive number greater than the maximum value that the right hand side can take. As the sensor j has to be in at least one of the convex subdomains Ω k, we additionally require that, N D d jk (τ) 1. (27) k=1 Thus using constraints (26) and (27), we can guarantee that the chance constraint (18), which corresponds to the nonconvex agent position constraint ς j (τ) Ω, is satisfied. B. Target Interception In this section we propose an approach to evaluate target interception in the planning interval. The constraints that define the target-sensor interception proximity are reposed as chance constraints. This ensures their deterministic evaluation in the MILP framework. We require that a target needs to be intercepted by any sensor only once to be considered intercepted for the rest of the planning interval. Most of the derivations in this section are reproduced from our work [5]. We are presenting it here for the sake of completeness. We define a target i to be intercepted by a sensor j at time τ if it is within a distance of R from sensor j at τ. Thus for this interception to occur we require, χ i (τ) ς j (τ)) R, (28) where is the Eucledian norm in R 2. However in order to formulate a MILP we require linear constraints. Thus we conservatively approximate the interception constraint (28), which represents the interior of a circle in R 2 with the interior of a sided inscribing regular polygon through the following set of linear inequalities, ( ) π < (χ i (τ) ς j (τ)), v q > R cos (29) for all q {1,..., },

4 [ ] sin(θq ) where v q =, for θ q = 2πq N cos(θ q ) I. We denote the discrete event of target i being intercepted by sensor j at time τ by I ij (τ). Thus occurrence of I ij (τ) implies joint satisfaction of the linear constraints in (29). Since χ i (τ) and ς j (τ) are random vectors, the satisfaction of the constraints (29) cannot be deterministically guaranteed. We reformulate the constraints in probabilistic form and require a lower bound α I such that, P (I ij (τ)) α I (30) implies that the target i has been probabilistically intercepted by sensor j at time τ. It should be noted that the interception event I ij (τ) is a conjunction of the events corresponding to the satisfaction of the linear constraints in (29) by ς j (t). Thus using Boole s inequality we can achieve the lower bound α I in (30) by imposing a lower bound α on the probability of satisfaction of each individual constraint in (29), ( ( )) π P < (χ i (τ) ς j (τ)), v q > R cos α (31) for all q {1,..., }, such that α I = (α 1) + 1, which can be rewritten as, α = 1 1 α I. (32) Thus (31) is the probabilistic reformulation of the target interception constraints (29) wherein we use (32) to obtain α that gives us the prescribed lower bound α I in (30). Thus satisfaction of (31) implies probabilistic interception (30). Since χ i (τ) and ς j (τ) are independent and Gaussian, < (χ i (τ) ς j (τ)), v q > is a Gaussian random variable with distribution N (µ q ij (τ), σq ij (τ)), where µ q ij (τ) =< (ˆχ i(τ) ˆς j (τ)), v q > (33) σ q ij (τ) = v T q ( Π j (τ) + P i (τ))v q (34) The constraints (31) thus can be rewritten as, ( ) Φ R cos π µ q ij (τ) σ q ij (τ) α for all q {1,..., }. (35) Taking Φ 1 (.) on both sides of (35) and rearranging we get, < (ˆχ i (τ) ˆς j (τ)), v q > ˆR q ij (τ) cos ( π ) where, for all q {1,..., }, (36) ˆR q ij (τ) = R Φ 1 (α)σ q ij ( ) (τ). (37) π cos Thus joint satisfaction of the constraints (36) is equivalent to joint satisfaction of the constraints (31) implying probabilistic interception (30). Again due to Gaussianity, the structure of the probabilistically reformulated interception constraints (36) with respect to ˆς j (τ) is the same as the original interception constraints (29) with respect to ς j (τ). As all quantities in (36) are deterministic, its satisfaction can be evaluated in the MILP framework, which is not the case with (29). We use variables b ij (τ) {0, 1} for i {1,...N RH } and j {1,..., M}, to indicate probabilistic interception (30). If b ij (τ) = 1, then target i will be probabilistically intercepted by mobile sensor j at time τ (or (30) will be satisfied). As satisfaction of (36) implies probabilistic interception (30), we require b ij (τ) = 1 to imply satisfaction of (36). Thus, for i {1,...N RH } and j {1,..., M} we have, b ij (τ) = 1 < (ˆχ i (τ) ˆς j (τ)), v q > ˆR q ij (τ) cos ( π ) for all q {1,..., }. (38) The logical relation (38) can be equivalently represented in terms of the following linear inequalities [14] for i {1,...N RH } and j {1,..., M}, < (ˆχ i (τ) ˆς j (τ)), v q > ˆR q ij (τ) cos ( π ) M big (1 b ij (τ)) for all q {1,..., }, (39) where M big is the big-m constant. M big should be greater than the largest possible value of the left-hand side of (39). We use the variables δ i (τ) {0, 1} as indicator variables for the target i {1,..., N RH } being probabilistically intercepted at a time τ τ. Thus, 1 If target i probabilistically intercepted at τ τ. δ i (τ) = 0 If target i not probabilistically intercepted for any τ τ. (40) From (38) and (40) we get, τ τ =0 j=1 M b ij (τ ) 1 δ i (τ) = 1 (41) As illustrated in [14], the logical relation (41) can be represented in terms of the linear inequalities, 1 1 τ τ =0 j=1 τ τ =0 j=1 M b ij (τ ) (1 δ i (τ)) M b ij (τ ) ɛ + (1 (τ + 1)M ɛ)δ i (τ), (42) where ɛ > 0 is a small positive number, typically machine precision for practical implementation. IV. MILP PROBLEM In this section we formulate a MILP problem for target interception that incorporates the non-convex agent domain constraints, (26) and (27) formulated in Subsection III-A, and target interception constraints (39) and (42) formulated

5 in Subsection III-B. In addition we have (8), which is the linear dynamic equation for the propagation of the sensor state mean and (12), which linearly relates the sensor state mean to the sensor position mean, as constraints. The stochastic optimization problem consists of sensor control action variables u j (τ) R mu, variables that depend on the sensor control actions, constraints that relate the variables, and the cost function. Thus in addition to u j (τ), the sensor state means ŝ j (τ), the sensor position means ˆς j (τ), the interception indicators, b ij (τ) and δ i (τ), and the sensor domain indicators d jk (τ) are also the variables of the optimization problem that has the sensor motion, domain, and target interception constraints. The initial state means, however, are given parameters, ŝ j (0) = ŝ 0 j for j {1,..., M}. (43) The target state and position means, ˆx i (τ) and ˆχ i (τ), and the sensor and target covariances evolve independently of the control variables and hence are not optimization variables. These are obtained using (7), (11), (9), (10),(13) and (14) and are supplied to the optimization problem as given parameters. A key aspect of our formulation is the linearity of the constraints and the cost leading to a MILP-based framework. In order to efficiently intercept the targets, an additive cost over all targets and all planning time instants, J = RH N RH + T 1 E[d i (x i (τ))](1 δ i (τ)) τ=0 E[D i (x i (T ))](1 δ i (T )). (44) that was proposed in [5], is an reasonable option, where E[d i (x i (τ))] and E[D i (x i (T ))] are the planning and terminal costs for target i. However, in this section we formulate another cost that cannot be represented in the same form as (44) and that addresses the issue of intercepting all targets in a minimum time for finite horizon planning. In order to formulate such a cost, first note from the definition of δ i (τ) (40), that the time to probabilistically intercept a target i is given by, τ i = T τ=0 (1 δ i(τ)), if the target is probabilistically intercepted at or before the terminal planning instant T. Thus in order to define a cost that would minimize the time required to intercept all the targets, we define the variable T max as an upper bound on all the target interception times, T (1 δ i (τ)) T max. (45) τ=0 We define the planning cost of the receding horizon problem, J plan = T max. (46) Due to the finiteness of the planning horizon T, it is possible that some of the targets are not probabilistically interceptable at or before T in a planing iteration. Hence a cost J = J plan = T max for the receding horizon problem would be useless, as it would always be T +1 if there do not exist feasible trajectories that can intercept all the targets within T. In such a case we would like the planning iteration to result in as many target interceptions as possible. Hence we add a terminal cost, J term = (1 δ i (T )), (47) that signifies the number of un-intercepted targets at T. Thus the receding horizon cost that minimizes the time to probabilistically intercept all the targets, if there exists feasible trajectories that can do so or minimizes the number of unintercepted targets at the end of the planning horizon is given by, J = J plan + J term = T max + (1 δ i (T )). (48) This follows by observing that if there exist sensor trajectories that can intercept all the targets in the finite horizon T then minimizing the cost J in (48) will require that all the targets are intercepted. Note that if at least one target is not intercepted then minimizing J in (48), will require, T max = T + 1 (from (45)) and N (1 δ i(t )) 1. Thus J T +2 in (48) for the scenario that at least one target is not intercepted. Compared to this if all the targets are intercepted in T then as a result of minimizing J in (48), T max = max( T τ=0 (1 δ i(τ))) T (from (45)) and N (1 δ i(t )) = 0, resulting in J = max( T τ=0 (1 δ i(τ))) T in (48). Thus interception of all the targets whenever possible will result in consistently lower cost J in (48) as compared to the scenario where at least one of the targets is not intercepted. For the case that all the targets are intercepted in the finite planning horizon, the time required to intercept all the target is given by max( T τ=0 (1 δ i (τ))), which also corresponds to the cost J for this case. Thus minimizing J in (48) will result in minimizing the time to intercept all the targets whenever possible in the finite horizon T. The corresponding planning optimization problem is a MILP given by, Minimize J = J plan + J term = T max + w.r.t.: Subject to: (1 δ i (T )) T max, u j (τ), ŝ j (τ), δ i (τ), b ij (τ) and d jk (τ) for all τ {0,..., T }, j {1,..., M} k {1,..., N D } and i {1,..., N RH } Sensor Motion Constraints:(3), (8) and (12) Sensor Domain Constraints: (26) and (27) Target Interception Constraints: (39) and (42) Interception Time Upper Bound Constraints: (45) (49) We use the same argument as in [5] to justify that the MILP (49) has same computational cost as the corresponding deterministic problem. If the initial covariances satisfy

6 P i (0) = Π j (0) = 0 and state noise covariances satisfy Q i (τ) = Q j (τ) = 0, then from (9) and (10) the predicted covariances for τ [0, T ] in the planning interval are zero, i.e. P i (τ) = Π j (τ) = 0. Thus for this case the MILP in (49) is equivalent to the MILP for the deterministic case. This implies the number and form of the optimization variables and constraints will be the same in the MILP for the Gaussian case (49) and the corresponding deterministic case. This results in the solution of (49) having similar computational costs as the MILP for deterministic case. However the deterministic MILP problem can be formulated directly in terms of the sensor state without requiring the use of chance constraints. Thus there is a slight computational overhead involved in reformulating the constraints to incorporate the probabilistic information for the stochastic Gaussian case over the deterministic case. V. SIMULATION RESULTS In this section we present the simulation results for the proposed MILP-based stochastic RHC. Our application is a target interception problem, that would occur in a potential novel system of autonomous multi-agent system for riverine rescue. The objective of such an autonomous multi-agent system would be to intercept the objects that are drifting away due to riverine flow. We use MATLAB for simulations and use the optimization software GUROBI through a MATLAB MEX interface to solve the MILP on a Windows machine with 2.53 GHz Intel Core 2 Duo Processor and 4GB memory. We assume a double integrator model for the target motion and a single integrator model for the sensor motion. The target state x i R 4 has 2 position components (x and y) and 2 velocity components (x and y). The sensor state s j R 2 has 2 position components (x and y). The sensor velocity u j R 2 is assumed to be a control variable. Thus we have, 1 0 T T A i = , B i = I 4 (50) Ā j = I 2, B j = I 2, C j = T I 2, (51) where, T is the time between two consecutive discrete time instants. We use T = 5s in our simulation. We use the following covariances for the zero mean Gaussian process noise for the targets and sensors, Q i = E [ ν i (t)ν i (t) T] = diag [ ] 10 3 Q j = E [ ν j (t) ν j (t) T] = diag [ 2 2 ] 10 2 (52) We also use Q i and Q j for the Gaussian initial states. We set the lower bound on the probability of joint satisfaction of the sensor domain constraints, β Ω = This results in the lower bound on the probability of satisfaction of individual linear domain constraints to be β = We use R = 12 as the target-sensor interception distance and use a = 10 sided regular polygon to approximate the target-sensor interception proximity. For the target to be probabilistically intercepted by a sensor we use a lower bound α I = 0.9 on the joint satisfaction of the linear constraints (sides of the regular polygon) defining interception proximity. This translates to a lower bound α = 0.99 on the satisfaction of each of the linear constraints defining interception. The upper and lower bound vectors on the sensor [ control ] (velocity) vector are set 3 to (u j ) ub = (u j ) lb = m/s. The velocities of the 3 targets were randomly generated between 1 2.2m/s for the sumultion. Figure 2 shows a simulation consisting of 2 riverine rescue sensor agents and 4 targets, and a 12-step time horizon for the riverine rescue problem. The circles represent sensor interception region at each time instant, corresponding to the sensor position indicated by the dot of the same color. Red lines indicate target path. Initial target locations highlighted by red circles. The target motion stops when it comes in an interception radius R of the rescue agent, signifying interception. As can be seen in the figure, the 2 sensors intercept all Fig. 2. Riverine rescue simulation with 4 targets and 2 rescue agents. 4 targets. An average of 1.02 s and a maximum of 1.84 s were required to set up and solve the MILP at every iteration. However average and maximum of just the solve times for the MILP at each iteration were 0.24 s and 0.42 s respectively. VI. CONCLUSIONS The work presented in this paper extends our previous work on stochastic target interception [5]. While [5] required the sensor domain to be convex, we now have a MILP framework for target interception that is flexible enough to accommodate non-convex domains in the presence of Gaussian uncertainty. As most realistic scenarios scenarios for the operation of sensor agent teams will involve non-convex domains, the reasons for such an extension are quite compelling. In addition we have come up with a cost function that directly addresses the issue of intercepting all the target in a minimum time. We simulate the proposed algorithm on a non-convex polygonal

7 approximation of a riverine domain for a potential riverine rescue team of autonomous sensor agents. The computational times required for the 2 sensor agent and 4 target scenario are encouraging. VII. ACKNOLEDGEMENT This work was funded by the Office of Naval Research via grant N , and the DoD SMART fellowship program. REFERENCES [1] E. Feron T. Schouwenaars, B. DeMoor and J. How. Mixed integer programming for safe multi-vehicle cooperative path planning. In ECC, [2] A. Richards and J.P. How. Aircraft trajectory planning with collision avoidance using mixed integer linear programming. In American Control Conference., [3] N.K. Yilmaz, C. Evangelinos, P. Lermusiaux, and N.M. Patrikalakis. Path planning of autonomous underwater vehicles for adaptive sampling using mixed integer linear programming. Oceanic Engineering, IEEE Journal of, 33(4): , [4] M.G. Earl and R. D Andrea. Modeling and control of a multi-agent system using mixed integer linear programming. In Decision and Control, 2002, Proceedings of the 41st IEEE Conference on, volume 1, pages vol.1, [5] A. Shende, M. Bays, and D. Stilwell. Multiple agent coordination for stochastic target interception using milp. In IEEE/RSJ International Conference on Intelligent Robots and Systems, to appear, [6] L. Blackmore, Hui Li, and B. Williams. A probabilistic approach to optimal robust path planning with obstacles. In American Control Conference, 2006, page 7 pp., [7] M. Ono, L. Blackmore, and B.C. Williams. Chance constrained finite horizon optimal control with nonconvex constraints. In American Control Conference (ACC), 2010, pages , july [8] L. Blackmore, M. Ono, and B. C. Williams. Chance-constrained optimal path planning with obstacles. Robotics, IEEE Transactions on, 27(6), [9] N. E. Du Toit and J. W. Burdick. Probabilistic collision checking with chance constraints. Robotics, IEEE Transactions on, 27(4): , aug [10] Jun Yan and Robert R. Bitmead. Incorporating state estimation into model predictive control and its application to network traffic control. Automatica, 41(4): , [11] L. Blackmore. A probabilistic particle control approach to optimal, robust predictive control. In AIAA Guidance, Navigation and Control Conference, 2006, [12] A. Ono M. Williams B. C. Blackmore, L. Bektassov. Robust, optimal predictive control of jump markov linear systems using particles. LEC- TURE NOTES IN COMPUTER SCIENCE, (4416): , [13] Lars Blackmore, Masahiro Ono, Askar Bektassov, and Brian C. Williams. A probabilistic particle-control approximation of chanceconstrained stochastic predictive control. IEEE Transactions on Robotics, 26: , [14] Alberto Bemporad and Manfred Morari. Control of systems integrating logic, dynamics, and constraints. Automatica, 35(3): , 1999.