Nonlinear Cyclic Pursuit Based Cooperative Target Monitoring

Nonlinear Cyclic Pursuit based Cooperative Target Monitoring Sangeeta Daingade and Arpita Sinha Abstract This paper presents a nonlinear cyclic pursuit based target monitoring strategy for a group of autonomous vehicles. The vehicles are modeled as unicycles and are assumed to be heterogeneous. Each vehicle follows the next neighbor as well as the target. The detailed analysis is done for stationary target and the effectiveness of the proposed strategy against a moving target is shown through simulation. At equilibrium, the vehicles capture the target and move along concentric circles in a rigid polygonal formation around the target with equal angular speeds. A necessary condition for the existence of equilibrium formation is derived. Local stability analysis is carried out for homogeneous agents. Simulation results demonstrate the objective of the proposed method and verifies the derived results. 1 Introduction Cooperative control of multi-vehicle systems have attracted much attention, due to their various applications and advantages as compared to single vehicle missions. The various research problems in this area are consensus, rendezvous, formation control, motion coordination and cooperative target tracking. In this paper, we address the cooperative control problem for target monitoring with multiple autonomous vehicles. In various military and civil applications such as survellience, security systems, space and under water exploration, it is often required to track a target moving in a dynamic environment. In such situations, cooperative target tracking would be an attractive solution rather than employing a single, intelligent and sophisticated vehicle. In cooperative target monitoring, the objective is to coordinate the motion of vehicles in such a way that the vehicles reach the desired relative positions and oriSangeeta Daingade, Arpita Sinha Systems and Control Engineering, Indian Institute of Technology Bombay, India, e-mail: [email protected], [email protected] 1 2 Sangeeta Daingade and Arpita Sinha entations with respect to the target and keep following the target while maintaining the formation. The strategies discussed in [4] , [10] ,[7] assumes linear model for each vehicle. So these strategies does not take into account the kinematic constraints of the vehicles. In [14], the authors have studied the problem of vision based target tracking among a group of ground robots where it is assumed that the robots can measure the target’s position, velocity, and acceleration. They have developed control laws for both single-integrator and double-integrator type robot models. The unicycle model closely resembles the dynamics of mobile robots and unmanned aerial vehicles (UAVs). The target tracking strategies discussed in [16], [9], [8] , [5], [11], [13], [12] are based on unicycle model for the vehicles and assumes that all the agents are homogeneous. Switching control law is designed in [16] to track the center of mass of the agents. The agents follow piecewise linear trajectory. In [9], [8] the authors assume all to all communication topology and have presented analysis with only three agents. The strategy discussed in [5] considers a scenario where limited sensing capability of the agents is taken into account. The authors have shown that at equilibrium agents get distributed around the target in different platoons along the same circle, however the agents are not uniformally distributed. The splay state configuration introduced in [11] enables represention of the euilibrium state of the agents tracking a moving target. The control law proposed by the authors assumes ring topology and it is computationally intensive as it involves calculation of desired heading and it’s derivative. Our work is based on cyclic pursuit which is a simple strategy derived from the behavior of social insects. In cyclic pursuit, agent i follows agent i + 1 modulo n. This strategy can be used to obtain various behaviors like rendezvous, motion in formation, target capturing etc. Bruckstein et al. [3] modeled behaviors of ants, crickets and frogs with continuous and discrete cyclic pursuit laws. Stability and convergence of group of ants in linear pursuit are described in [2]. Marshall et al. [15] studied the formations of multivehicle system under linear as well as nonlinear cyclic pursuit. They have analyzed the equilibrium and stability of these formations in case of identical agents. In [18] and [19], Sinha and Ghosh studied generalization of the linear cyclic pursuit and nonlinear cyclic pursuit respectively. The authors have derived a necessary condition for equilibrium formation of heterogeneous agents. Cyclic pursuit based formation control strategies, discussed in [15] and [19], deal with formations about a point which cannot be specified a priori. In order to enable target enclosing, we should be able to achieve formations about a specific point (target). Rattan and Ghosh [17] have proposed Implicit Leader Cyclic Pursuit (ILCP) law for achieving formations about a given goal point and have used it for rendezvous of multiple vehicles. Cyclic pursuit based target tracking considering unicycle model of the agent has not been explored so far. We propose and analyze the target tracking strategy based on nonlinear cyclic pursuit strategy where each agent needs the position information from one of it’s neighbor. In this paper, we studied the case when the target is stationary. The extension of this work for moving target is underway and we present a simulation result to show that this strategy is equally applicable for moving target. The main contributions of the paper are: (1) Decentralized simple target Nonlinear Cyclic Pursuit based Cooperative Target Monitoring 3 tracking strategy for nonholonomic , heterogeneous agents; (2) Necessary conditions for achieving a formation about a stationary target; (3) Stability analysis of the equilibrium formations when the agents are identical. This paper is organized as follows. Analysis of the proposed strategy begins with modeling of the system in Section 2 followed by possible equilibrium formations in Section 3. In Section 4, we derive the necessary conditions for equilibrium. Then we discuss a special case in Section 5 where the agents are identical followed by stability analysis in Section 6. Simulation results are presented in Section 7 and conclusions and future research directions are discussed in Section 8. 2 Modeling of System Consider a group of n agents employed to track a target. The kinematics of each agent with a single nonholonomic constraint can be modeled as: ẋi = Vi cos θi ; ẏi = Vi sin θi ; θ̇i = ai Vi (1) where Pi ≡ [xi , yi ]T represents the position of agent i and θi represents the heading angle of the agent i with respect to a global reference frame. Equation (1) can represent a point mass model of a UAV flying at a fixed altitude or a point mass model of a wheeled robot on a plane. We use a generic term “agent ” to represent the aerial or ground vehicle. We assume that the agent i is moving with linear speed Vi which is constant over time. Therefore, the motion of the agent i is controlled using the lateral acceleration, ai . Vi+1 hi+1 Pi+1 Y (1 − ρ )r(i+1)g ′ Pi+1 ri zi ρ r(i+1)g φi ′ ri αi bi Vi hi Pi rig δ f i+1 fi P Fig. 1 Vehicle Formation geometry X Re f . 4 Sangeeta Daingade and Arpita Sinha Our objective is to enclose the target with n agents. It is assumed that each agent i has the information about the target position and i + 1th agent’s position. Consider the target to be located at point P as shown in Fig. 1. We modify the classical cyclic pursuit law for target enclosing problem such that agent i, positioned at Pi , follows not only i + 1th agent at Pi+1 but also the target at P. Let ρ be a constant, which decides the weightage agent i gives to the target position P, over the position of the ′ agent i + 1, Pi+1 . The agent i follows a virtual leader located at the point Pi+1 which ′ is a convex combination of P and Pi+1 . The point Pi+1 is calculated as: ′ Pi+1 = ρ Pi+1 + (1 − ρ ) P where 0 < ρ < 1. (2) Since we are considering a stationary target, let us assume a target centric reference frame and let us define the following variables (refer Fig. 1): rig – Distance between ith agent and the target; ri – Distance between ith agent and i + 1th agent; ′ ′ ri – Distance between ith agent and virtual leader at Pi+1 ; fi – angle made by the vector rig with respect to the Ref.;hi – heading angle of ith agent with respect to the Ref.;αi – angle between the heading and the line of sight (LOS) Pi Pi+1 of agent i; ′ φi – angle between the heading and modified LOS Pi Pi+1 of agent i. We define the control input to the ith agent, that is, the lateral acceleration ai , as ai = ki φi (3) where, ki > 0 is the controller gain. We assume 0 ≤ φi ≤ 2π for all time, t ≥ 0. This ensures that the agents always rotate in counter clockwise direction. Let ωi be angular speed of agent i and fii+1 = fi+1 − fi . The kinematics (1) can be written in V sin(h − f ) the target centric reference frame as, ṙig = Vi cos(hi − fi ), f˙ii+1 = i+1 r i+1 i+1 − (i+1)g i − fi ) and ḣi − f˙i = ωi − ḟi = kVi φi i − Vi sin(h . Let us define the states of the rig system as xi(1) = rig , xi(2) = fi+1 − fi and xi(3) = hi − fi . Then we can write state equations of the system as: Vi sin(hi − f i ) rig ẋi(1) = Vi cos(xi(3) ) (4a) ẋi(2) = Vi+1 sin(xi+1(3) ) Vi sin(xi(3) ) − xi+1(1) xi(1) (4b) ẋi(3) = ki φi Vi sin(xi(3) ) − . Vi xi(1) (4c) Equation (4) gives the kinematics of ith agent. In the subsequent sections, all the analysis are done based on this model. Note 1. In actual implementation, the agents will have a limit on the maximum lateral acceleration amax , that is, ai ≤ amax ∀i. We take into account this constraint by putting a bound on the value of ki as ki ≤ kmax where kmax = a2max π . Nonlinear Cyclic Pursuit based Cooperative Target Monitoring 5 3 Formation at equilibrium Theorem 1. Consider n agents with kinematics (4). At equilibrium, the agents move on concentric circles, with (i) the target at the center of concentric circles and (ii) equal angular velocities. Proof. At equilibrium ẋi( j) = 0 for i = 1, ..., n and j = 1, 2, 3, which implies, ṙig = 0, f˙ii+1 = 0 and ḣi − f˙i = 0. Then, from (4a) - (4c), at equilibrium xi(1) = rig = constant, xi(2) = fii+1 = constant and xi(3) = hi − fi = constant. So the distance between the target and agent i (for all i) remains constant at equilibrium. As ṙig = 0, using (4a) we can write, hi − fi = (2m + 1) π2 , where m = 0, ±1, ±2, · · ·. As ḣi − f˙i = 0, from i − fi ) (4c) we can write kVi φi i = Vi sin(h = ± rVigi . Since ki > 0, Vi > 0 , 0 ≤ φi ≤ 2π and rig rig ≥ 0, we get ki φi Vi = (5) Vi rig and therefore, m = 0, ±2, ±4, · · ·. Assuming hi ∈ [0, 2π ] and fi ∈ [0, 2π ], we get (hi − fi ) ∈ [−2π , 2π ]. Therefore m = 0 or m = −2. From geometry, m = 0 and m = −2 implies the same angle. Therefore hi − f i = π 2 (6) 2 From (3) and (5), ai = Vrigi . Since Vi and rig are constant, ai is constant for all i. Therefore all the agents move in a circular path with target at itś center and radius rig . This proves the first part of the theorem. The the angular velocity of agent i can be calculated as, ωi = Vaii = rVigi . As f˙ii+1 = 0, from equation (4b) and (6), we can write for all i, Vi Vi+1 = (7) rig r(i+1)g Using (7), we conclude that for all i, ωi = ωi+1 . Therefore, all the agents move around the target in concentric circles with equal angular speed.  Corollary 1. At equilibrium, the agents with kinematics (4), form a rigid polygon that rotates about the target. 4 Conditions for the existence of equilibrium Let the radius of the circle traversed by the first agent at equilibrium be r1g = R1 . Using (7), we can write, Vi (8) rig = R1 V1 6 Sangeeta Daingade and Arpita Sinha for all i. From (5) and (8), φi = ViV1 ki R 1 (9) ′ ′ Consider Fig. 1. Let ∠PPi Pi+1 = bi and ∠PPi+1Pi = zi . We assume that the angle is positive, if it is measured counter clockwise and negative if clockwise. Then, from Fig. 1, n ∑ ( fii+1 ) = 2π d, d = 0, ±1, ±2, · · · (10) φi + bi + (hi − fi ) = π bi + zi + fii+1 = π (11) i=1 (mod n) (12) ρ r(i+1)g rig ri = = sin(bi ) sin(zi ) sin( fii+1 ) (13) q 2 + ρ 2 r2 rig − 2rig ρ r(i+1)g cos( fii+1 ) (i+1)g (14) ′ ′ ri = From (6) and (11) φi + bi = π2 . Therefore from (13) and (7), sin(zi ) = From (9) and (12), we can write fii+1 = X , defined as π 2 Vi ρ Vi+1 cos(φi ). + VkiiRV11 − zi Now consider the set of agents X = {i : ρ < Vi } Vi+1 (15) Claim: The set X 6= 0. / Proof: We prove it by contradiction. Let us assume that X = 0. / Then ρ Vi+1 ≥ Vi for all i = 1, · · · , n ,which implies, ρ nV1 ≥ V1 . This can not be true as 0 < ρ < 1, which leads to a contradiction. So there exits at least one agent in the set X .  For the agents in set X , from (7), ρ r(i+1)g < rig . Then, the position of agent i ∈ X in the formation will be as shown in Fig. 2. The point P represents the target. At equilibrium, the agent i moves along the circle which has radius rig and center P. Let the agent i be at the point Pi . For a given ρ , the virtual leader of agent i ∈ X will lie on the circle of radius ρ r(i+1)g which is less than rig . Note that r(i+1)g can be greater or less than rig . Let Pi Q and Pi Q′ be the tangents drawn from point Pi to the circle of radius ρ r(i+1)g and φimin and φimax are the angles that the velocity vector of ith agent makes with Pi Q and Pi Q′ . Then, φimin + φimax = π . At equilibrium, φi must satisfy 0 < φimin ≤ φi ≤ φimax < π . (16) When φi = φimin , the virtual leader of i will be at Q and therefore fii+1 = φi . Similarly, ′ when φi = φimax , fii+1 = π + φi . When φi = φ which is within the bounds specified by (16), there are two possible locations where the virtual leader can lie (Q1 and Q2 ). So zi as defined in Fig. 1 can take two values. The two values of zi are zia = ∠PQ1 Pi Nonlinear Cyclic Pursuit based Cooperative Target Monitoring 7 Vi φ Q Q1 ′ Q0 φimin Q2 ρ r(i+1)g Pi P φimax rig Q′ Fig. 2 Formation of agents in set X and zib = ∠PQ2 Pi , which can be calculated as Vi cos(φi )) ρ Vi+1 Vi cos(φi )) zib = π − sin−1 ( ρ Vi+1 zia = sin−1 ( (17a) (17b) Similarly, ( fii+1 ) can take two values. At equilibrium, RHS of (17) should be real. So the condition for the existence of equilibrium can be stated as: Theorem 2. Consider n agent system with kinematics (4). The necessary condition for the equilibrium is (18) max ηi ≤ min µ j j∈X i∈X ηi = [cos−1 ( ρ Vi+1 ki )] Vi ViV1 and µj = [π − cos−1 ( ρ Vj+1 kj )] Vj Vj V1 (19) Proof. Equation (17) holds if, for all i, | cos(φi ) |≤ ρ Vi+1 Vi (20) Vi+1 ≥ Vi , equation (20) is always true. For i ∈ / X , i.e.  However, if i ∈ X , then  if ρ  pπ + cos−1 ρ Vi+1 Vi ≤ φi ≤ (p + 1)π − cos−1 ρ Vi+1 Vi where, p = 0, ±1, ±2, · · ·. From (16), p  can take Substituting the value of φi from (9)  only  onevalue, p = 0.   ρ Vi+1 ρ Vi+1 ViV1 −1 −1 we get cos ≤ ki R1 ≤ π − cos Let Vi Vi ℜi = {R1 : ηi ≤ 1/R1 ≤ µi , i ∈ X } (21) 8 Sangeeta Daingade and Arpita Sinha where ηi and µi are given by (19). For each i, (21) gives the range of values R1 can take. If \ ℜi 6= 0/ (22) i∈X then, there exists some R1 , such that (20) is satisfied for all i. If (22) has to be true, (18) must hold. Thus the necessary condition for equilibrium is given by (18).  Note 2. For the agents not in set X , that is, ρ Vi+1 ≥ Vi , from (7), ρ r(i+1)g ≥ rig . When ρ r(i+1)g = rig , the virtual leader lies on the circle of radius rig and φi can take value in [0, π ]. When ρ r(i+1)g > rig , the virtual leader lies outside the circle of radius rig and φi ∈ [0, 2π ]. In both the cases, given a φi , there exists an unique point where virtual leader can lie and hence the position of the next agent i + 1 is unique. In these cases, the value of zi lies in [0, π /2] and is given by (17a). Note 3. When 2 is satisfied, (11)-(13) will be always be true. However, (10) may not be satisfied. Therefore, to achieve a formation, one requires both 2 and (10) to be true simultaneously. 5 Formation in case of homogeneous agents Consider a special case when all the agents are identical, that is, Vi = V and ki = k for all i. Then at equilibrium, from (7) we have rig = R for all i. So the agents move on the same circle with the target at its center. From (9), φi = V2 = φeq kR (23) for all i. Thus, φimax = φmax and φimin = φmin for all i. Also from △PQPi (Fig. 2) ρ = cos(φmin ) As φi + bi = π 2, (24) from (13)and (14) ρ sin( fii+1 ) cos(φeq ) = q 1 + ρ 2 − 2ρ cos( fii+1 ) (25) for all i. We observe that ρ V < V and therefore all the agents belong to the set X as defined in (15). Then the necessary condition in 2 can be expressed as ρ ≥ ′ cos(φeq ) . Consider Fig. 2 and assume φeq = φ . Then ∠Q0 PPi = φeq . As long as (16) is satisfied, 2 is trivially satisfied. Let ∠Q0 PQ1 = ∠Q0 PQ2 = δ and fa = ∠Pi PQ1 = φeq + δ (26) fb = ∠Pi PQ2 = φeq − δ (27) Nonlinear Cyclic Pursuit based Cooperative Target Monitoring 9 The value of fii+1 can be either fa or fb . Note that, from (17), zia + δ = π /2 and zib − δ = π /2. Also from △Q1 PPi cos(φeq ) = ρ cos(δ ) (28) Let Ma = {i : fii+1 = fa } and Mb = {i : fii+1 = fb }. Then |Ma | + |Mb | = n. We have three possibilities: Case 1: When Ma is empty, that is, fii+1 = fb ∀i, from
(27) , we can
write cos( fb ) = cos(φeq − δ ). Using (25) and (28), ρ cos( fb ) − 1 ρ − cos( fb ) = 0. Since 0 < ρ < 1, ρ cos( fb ) 6= 1, so ρ = cos( fb ). From (24), fb = φmin = π − φmax . Then from Fig. 2, we observe that the agent i + 1 will be at Q or Q′ and thus we have either φeq = φmin or φeq = φmax . Case 2: When Mb is empty, that is, fii+1 = fa , ∀i, following similar procedure as in the previous case, cos( fa ) = cos(φeq + δ ) needs to
be true. Using
(25) and (28) and simplifying, we get ρ sin2 ( fa ) − 1 − ρ cos( fa ) ρ − cos( fa ) =
cos( fa ) 1 + ρ 2 − 2ρ cos( fa ) which is satisfied for all ρ . Case 3: When both Ma and Mb are non-empty, following similar procedure as in Case 1, it
is found that the equilibrium formation is possible if cos(δ ) = ρ cos 2π dn − mn δ is true for the values of δ satisfying (28), where m = |Ma | − |Mb |. For Cases 1 and 2, we have fii+1 = f¯ ∀i, where f¯ is some constant. However, in Case 3, fii+1 is not same for all i. A set of 50000 simulations were run with random initial conditions for team sizes, n = 3 to 11 and ρ = 0.1 to 0.9. It has been observed from the simulation that Case 3 never occured and so we are not analyzing Case 3 here. We will concentrate on Cases 1 and 2 only. In Case 1 and Case 2, φmin ≤ f¯ ≤ π + φmax (29) with Case 1 corresponding to equalities in (29). When fii+1 = f¯ ∀i, the agents will be uniformly distributed around the target. Then from (10) f¯ = 2π dn . From (23), the 2 distance between an agent and the target is, R = kVφeq . Then the distance between ith
2 and i + 1th agent will be r̄ = 2R sin 2f̄ = 2 kVφeq sin(π dn ). So the system equilibrium states are, V2 d π xieq (1) = xieq (3) = (30) ; xieq (2) = 2π ; kφeq n 2 Thus, at equilibrium, the agents arrange themselves in a regular formation around the target. This regular formation of n agents can be described by a regular polygon {n/d}, where d ∈ {1, 2, ..., n − 1}. This d is reflected in equilibrium state xieq (2) in (30). If Zi is the point on the circle and δ is clockwise rotation by an angle 2π dn along the circle from point Zi , then the regular polygon is defined by the set of vertices given by Zi+1 = δ Zi , and edges between Zi+1 and Zi . When d = 1, then polygon {n/d } is called an ordinary regular polygon, and its edges do not intersect. If d > 1 and if n and d are coprime, then the edges intersect, and the polygon is a star. If n and d have a common factor m > 1, then the polygon consists of m traversals of the same polygon with n/m vertices and n/m edges. 10 Sangeeta Daingade and Arpita Sinha ′ Note 4. When φeq = φ (as shown in Fig. 2), φmin ≤ f¯ ≤ π + φmax . Then, from (24), ρ = cos(φmin ) ≥ cos( f¯) = cos(2π dn ). Thus, ρ ≥ cos(2π dn ). Let us replace q = dn by a continuous variable q ∈ (0, 1). We can plot ρ = cos(2π q) as shown in Fig. 3. It can be seen that, for a given n, Region I corresponds to Case 2 (when Mb is empty) and the boundary of the Region I corresponds to Case 1 (when Ma is empty). 1 II 0.9 0.8 q −−−> 0.7 0.6 I 0.5 0.4 0.3 0.2 III 0.1 0 −1 0 1 ρ −−−> Fig. 3 q versus ρ plot when ρ = cos(2π q) 6 Stability Analysis for Homogeneous agents We study the stability of the equilibrium formation of homogeneous agents. The equilibrium points are given by (30). Since d can take (n − 1) values, there are (n − 1) formations possible for a given V , k and ρ . Linearizing (4) about the equilibrium point (30), we get x̂˙ i = Ax̂i + Bx̂i+1 where  A= a31 = 0 V R2 a31   0 −V 0 0  a32 −k V kρ sin(2π dn ) V R{1 + ρ 2 − 2ρ cos(2π dn )}  B= + V R2 and a32 = 0 −V R2 −kρ sin(2π dn ) V R{1+ρ 2 −2ρ cos(2π dn )}  00 0 0  00 kρ (ρ − cos(2π dn )) V{1 + ρ 2 − 2ρ cos(2π dn )} T So, the system of n vehicles, can be written as X̂˙ = ÂX̂, where X̂ =
[x̂1 , x̂2 , · · · , x̂n ] and  is a circulant matrix given by  = circ A B 03×3 · · · 03×3 . The stability of the formation depends on the eigen values of Â. Theorem 3. Consider n agents with kinematics (4), moving with unit linear velocity. For a given value of ρ , the equilibrium points given by (30) are locally asymptotically stable if d (31) ql < < qu where, n Nonlinear Cyclic Pursuit based Cooperative Target Monitoring ql = −0.17ρ + 0.254  0.17ρ + 0.75 qu = 2 −1 √ π cos ( 2ρ − 1) 11 for ρ ≤ 0.5305 for ρ > 0.5305 (32) Proof. The stability of a formation depends on the eigen values of the circulant matrix Â. We can find the eigen values of  by diagonalizing it using Fourier matrix Fn [6] as,  = (Fn ⊗ I3 )D(Fn ⊗ I3 )⋆ , where (⋆) indicates conjugate transpose. The diagonal matrix D is given by, D = diag(D1 , D2 , · · · , Dn ) = diag(P(1), P(ς ), · · · , P(ς n−1 )) √ 2π where ς = e j n , and j = −1. So we can write Di = A + ς i−1 B , i ∈ {1, 2, ....n}. The eigen values of  are same as eigenvalues of Di , i = 1, · · · n. So we can comment about the stability of n vehicle system, by observing the eigen values of each block Di . Assuming V = 1, each Di can be factorized as Di = 1k T D̃i T −1 , where T =diag[k, 1, 1] and   0 0 −1 D̃i =  φeq 2 (1 − ς i−1) 0 0  d˜31 d˜32 −1 d˜31 = φeq ρ sin(2π dn )(1 − ς i−1 ) 1 + ρ 2 − 2ρ cos(2π dn ) + φeq 2 and d̃32 = ρ (ρ − cos(2π dn )) 1 + ρ 2 − 2ρ cos(2π dn ) The spectrum σ (·) of Di and D̃i are related as σ (Di ) = 1k σ (D̃i ). Since k > 0, stability of Di can be determined from the stability of D̃i . D̃i does not have a term containing gain k. So we can conclude that the stability of {n/d} formation is independent of k, as long as k > 0. i−1 D̃i is a complex matrix, as ς i−1 = e2π j n is a complex quantity. We can write i−1 ς i−1 = βi + jξi ∈ C, where βi = cos(2π i−1 n ) and ξi = sin(2π n ). Then the charac3 teristic polynomial of D̃i can be written as PD̃i (λ ) = λ + c1 λ 2 + c2 λ + c3 , where c1 = 1, c2 = a2 + jb2 , c3 = a3 + jb3 . Let us define the Hermitian matrix H for the polynomial PD̃i (λ ) as,  c1 + c̄1 c2 − c̄2 c3 + c̄3 H =  −c2 + c̄2 c2 + c̄2 − c3 − c̄3 c3 − c̄3  c3 + c̄3 −c3 + c̄3 c2 c̄3 + c̄2 c3  Then the polynomial PD̃i (λ ) is asymptotically stable if and only if the principal minors h1 ,h2 and h3 of H are positive [1]. In this case h1 = 2, h2 = 4(a2 − a3 − b22 ) and h3 = 8(a2 2 a3 − a2 b2 b3 − 2a2 a3 2 − 3a3 b2 b3 − b3 2 − a3 b2 2 a2 − b2 3 b3 + a3 3 ). Here, h2 and h3 depends on dn and βi which takes discrete values. To find the range of values of dn and βi for which h2 > 0 and h3 > 0, we replace dn by a continuous variable q ∈ (0, 1) (as stated in 4). Also we replace βi by a continuous variable β̃ ∈ [−1, 1]. For a given β̃ , we can plot q versus ρ when h2 = 0 and h3 = 0. Figure 4 shows these plots for different values of β̃ . Let us define S2 = {(ρ , q, β̃ ) : h2 > 0, ρ ∈ (0, 1), q ∈ (0, 1), β̃ ∈ [−1, 1]} and S3 = {(ρ , q, β̃ ) : h3 > 0, ρ ∈ (0, 1), q ∈ (0, 1), β̃ ∈ [−1, 1]}. Then S = S2 ∩ S3, defines the stability region where both h2 and h3 are positive. In Fig. 4, Region I corresponds to S. Region I can be numerically approximated with 12 Sangeeta Daingade and Arpita Sinha 1 0.9 III 0.8 q=d/n 0.7 0.6 0.5 I 0.4 IV 0.3 0.2 II 0.1 0 0 0.2 0.4 ρ 0.6 0.8 1 Fig. 4 Stability region for continuous values of q, ρ and β̃ conservative bounds as q + 0.17ρ − 0.254 = 0 q − 0.17ρ − 0.75 = 0 πq cos2 ( ) − 2ρ + 1 = 0 for ρ ≥ 0.5 2 (33) (34) (35) Equations (33) and (34) are the linear approximations of ρ = cos(2π q). The Region I in Fig. 4 is a subset of the Region I of Fig. 3. So for a given ρ , the bounds on q can be defined by (31). Thus we can conclude that for a given ρ and n, if dn satisfies (31), then the eigen values of D̃i and hence that of  will have negative real part. So the formation will be asymptotically stable.  7 Simulation Results First we consider five heterogeneous agents which are randomly placed initially and the target is located at the origin (0, 0). Case 1: We consider five agents moving with speed V = [25 20 15 10 5] , controller T gain k = 5and ρ = 0.8. In this case X = {1, 2, 3, 4} and ℜi as defined by (21) is an empty set. So the system does not have an equilibrium as the condition in 2 is violated. Simulation results in Fig. 5-(a) confirm the same. Case 2: Now we modify the speed of the vehicles to V = [22 19 15 20 18] with controller gain and ρ same as in Case 1. Now we have X = {1, 2, 4, 5}. In this case 2 is satisfied and there exist a range of values of R1 = [41.4894.23] . However, as stated in 2 of Section 4, Equation (10) also needs to be satisfied. There are two values of R1 = {45.45, 64.52}, for which equation (10) is satisfied. Fig. 5-(b) shows that the system of 5 agents settles to R1 = 64.52 with d = 2. Now we consider seven homogeneous agents, all moving with unit speed and having controller gain k = 0.1. Nonlinear Cyclic Pursuit based Cooperative Target Monitoring 13 60 150 40 20 Y−axis Y−axis 100 50 0 Target −20 −40 0 Target −60 −50 −80 −200 −150 −100 −50 X−axis 0 50 100 −100 (a) −50 0 X−axis 50 100 (b) Fig. 5 Formation of heterogeneous agents (• :- initial position and △ :- final position) 70 8 4 60 6 6 50 4 40 2 Y−−−−> Y−axis 2 1 0 Target 30 20 10 −2 7 0 −4 −10 −6 3 −20 −8 −10 5 −8 −6 −4 −2 0 X−axis 2 4 6 8 10 0 (a) 20 40 X−−−> 60 80 100 (b) Fig. 6 Stable formation of seven homogeneous agents (• :- initial position and △ :- final position) Case 3: Here we assume that the target is stationary . The agents are initially in {7/3} formation with ρ = 0.4, which corresponds to Region I of Fig. 4. At equilibrium the agents stays in {7/3} formation as shown in Fig. 6-(a). The analytical values of r and R matches with the simulation values. Case 4: In this case we consider a moving target following a sinusoidal path, with a velocity 0.1 unit/sec. The agents start from random initial positions with ρ = 0.1. Fig. 6-(b) shows that they get into a {7/3} formation about the target and continue to enclose the target maintaining the formation. This illustrates the potential of the proposed strategy for cooperatively tracking a moving target. 8 Conclusion The paper has addressed cyclic pursuit based target monitoring using a group of heterogeneous agents modeled as planar kinematic unicycle model moving with constant speed. Mathematical formulation and the analysis are carried out for a stationary target. At equilibrium, the agents move with a polygonal formation around the target with equal angular speed. Necessary condition for the existence of the equilibrium is derived. This leaves us with the future work of deriving sufficient condition for equilibrium formation of heterogeneous agents. 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