Proceedings of the 2006 American Control Conference Minneapolis, Minnesota, USA, June 14-16, 2006
ThC02.6
Theoretical Foundations for Multiple Rendezvous of Glowworm-inspired Mobile Agents with Variable Local-decision Domains K.N. Krishnanand and D. Ghose Abstract— We present the theoretical foundations for the multiple rendezvous problem involving design of local control strategies that enable groups of visibility-limited mobile agents to split into subgroups, exhibit simultaneous taxis behavior towards, and eventually rendezvous at, multiple unknown locations of interest. The theoretical results are proved under certain restricted set of assumptions. The algorithm used to solve the above problem is based on a glowworm swarm optimization (GSO) technique, developed earlier, that finds multiple optima of multimodal objective functions. The significant difference between our work and most earlier approaches to agreement problems is the use of a virtual local-decision domain by the agents in order to compute their movements. The range of the virtual domain is adaptive in nature and is bounded above by the maximum sensor/visibility range of the agent. We introduce a new decision domain update rule that enhances the rate of convergence by a factor of approximately two. We use some illustrative simulations to support the algorithmic correctness and theoretical findings of the paper.
I. INTRODUCTION Consensus problems in multi-agent networks appear in different forms [1], [2], [3], where a collection of agents transits from an initially random state to a final steady state such that all the members of the group eventually agree upon their individual state values. The state could represent physical quantities such as heading angle, frequency of oscillation, position, and so on. Vicsek et al. [1] analyze alignment of heading angles of multiple particles using the approach of statistical mechanics. In synchronization of coupled oscillators, a consensus is reached regarding the frequency of oscillation of all agents [2]. The multi-agent rendezvous problem, posed by Ando et al. [3], involves devising local control laws that enable all the members to steer towards and eventually rendezvous at a single unspecified location. Localization of multiple emission sources using networked mobile robots has received some attention recently [4], [5] in the collective robotics community. The goal of the above problem is to drive groups of robots to multiple sources of a general nutrient profile that is distributed spatially on a two dimensional workspace. This problem is representative of a wide variety of applications that include detection of multiple radiating sources such as nuclear/hazardous chemical spills and origins of fire calamities. In this paper, we provide theoretical foundations for the multiple rendezvous problem involving design of local conThis work is partially supported by a project grant from the Ministry of Human Resources Development, India. K.N. Krishnanand is a Ph.D student and D. Ghose is a professor at the Guidance, Control, and Decision Systems Laboratory in the Department of Aerospace Engineering, Indian Institute of Science, Bangalore, India
[email protected]
1-4244-0210-7/06/$20.00 ©2006 IEEE
trol strategies that enable groups of visibility-limited mobile agents to split into subgroups, exhibit simultaneous taxis behavior towards, and eventually rendezvous at multiple unknown locations of interest. Solutions to the above problem can be used to perform the class of multiple source localization tasks discussed above. The algorithm used to solve the above problem is based on a glowworm swarm optimization (GSO) algorithm [5], [6] that finds multiple optima of multimodal objective functions. The GSO method is inspired from group intelligence − a result of actions performed by a large number of relatively simple individuals that are solely based on neighbor-interactions and local information from the environment − exhibited by biological swarms (e.g., ants, termites, bees, wasps, and bacteria) [9]. Usually, the emergent behavior is utilized in an appropriate manner in order to perform a desired complex task. The significant difference between our work and most earlier approaches to agreement problems is the use of a virtual local-decision domain by the agents in order to compute their movements. We introduce a new decision domain update rule that enhances the rate of convergence by a factor of approximately two. We present some preliminary theoretical results to prove convergence under a restricted set of assumptions. II. T HE GSO A LGORITHM A. Overview The GSO algorithm is in the same spirit as the ant colony optimization (ACO) and particle swarm optimization (PSO) techniques, but with several significant differences [7]. In this algorithm, the agents are initially deployed randomly in the parameter-space of the given objective function. The agents in the GSO algorithm carry a luminescence quantity called luciferin along with them. Agents are thought of as glowworms that emit a light whose intensity of luminescence is proportional to the associated luciferin. Each glowworm uses the luciferin to (indirectly) communicate the functionprofile information at its current location to the neighbors. The number of peaks captured is a strong function of the radial sensor range. For instance, if the sensor range of each agent covers the entire workspace, all the agents move to the global optimum point and the local optima remain undetected. Since we assume that a priori information about the objective function (e.g., number of maxima and minima) is not available, in order to detect multiple peaks, the sensor range must be made a varying parameter. For this purpose, each glowworm i depends on a variable localdecision domain rdi (Figure 1(a)), which is bounded above by a circular sensor range rsi (0 < rdi ≤ rsi ), to compute
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local-decision domains
3
local-decision range 1
10
glowworm
1
a
rsj
i
j
d
k
Y
8
b
rk
4
−1 4
c
−2
3 5
radial sensor range of agent j radial sensor range of agent k
(a)
−3 −3
rdj
rsk
e
6
rsj . Agent i is in the sensor
its movements. Each glowworm selects a neighbor that has a luciferin value more than its own, using a probabilistic mechanism, and moves towards it. B. Algorithm description The algorithm starts by placing the glowworms randomly in the workspace so that they are well dispersed. Initially, they contain equal quantity of luciferin. Each iteration consists of a luciferin update phase followed by a movement phase based on a transition rule. During the movement phase, each glowworm decides, using a probabilistic mechanism, to move towards a neighbor that has a luciferin value more than its own. Figure 1(b) shows the emergence of a directed graph among a set of six glowworms based on their relative luciferin levels and availability of only local information. For each glowworm i, the probability of moving towards a neighbor j at iteration t is given by: =
−1
0 X
1
2
3
0
0
200
400
600
800
1000
(t)
Fig. 2. Update rule (2) is used. a) Emergence of solution b) Luciferin histories of all the glowworms.
Fig. 1. < d(i, k) = d(i, j) < < < range of both j and k. But, they have different decision-domains. Hence, only j uses the information of i. b) Emergence of a directed graph based on the relative luciferin level of each agent and availability of only local information. Agents are ranked according to the increasing order of their luciferin values. For instance, the agent a whose luciferin value is highest is ranked ‘1’.
pj (t)
2
−2
f
(b)
a) rdk
0
6
2
d
k
rs
12
(li)
rdj
14
2
(j (t) − i (t)) k∈Ni (t) (k (t) − i (t))
(1)
where, j Li (t), Li (t) = {j : d(i, j) < rdi and i (t) < j (t)}, t is the time index, d(i, j) is the euclidian distance between glowworms i and j, and j (t) is the luciferin level associated with glowworm j at time t. Let the glowworm i select a glowworm j ∈ Li (t) with pj (t) given by (1). The local-decision domain update rule given in [5] results in an oscillatory behavior of rdi (t). To smoothen the response, we propose a new update rule where an explicit threshold parameter nt is used to control the number of neighbors at each iteration. We notice that there is a substantial enhancement in performance by using this rule: rdi (t + 1) = min{rs , max{0, rdi (t) + β(nt − |Ni (t)|)}} (2) where, β is a constant parameter and nt is used as a threshold parameter to control the number of neighbors. The luciferin update rule is given by: j (t + 1) = (1 − ρ)j (t) + γJj (t + 1)
(3)
where, ρ is the luciferin decay constant (0 < ρ < 1) and γ is a proportionality constant for enhancing the luciferin level as a function of Jj (t) which represents the value of the objective function at agent j’s location at time t. III. S IMULATIONS Simulation results demonstrating the capability of the glowworm algorithm to capture multiple peaks of a number of complex multimodal functions have been reported in [5], [6] for constant and variable local-decision range cases. We have shown that when constant decision range is used, the number of peaks captured decreases with increase in the value of decision range. Interestingly, when the decisionrange is made adaptive, even though rdi (0) is chosen to be greater than the maximum distance between the peaks, all the peaks are captured. In this work, we apply the new update rule (2) and compare its performance against the results obtained with the update rule [5] by using number of iterations for convergence as a performance metric. We consider the following function to model the multimodal nature of the sources in the environment: J1 (x, y) = 3(1 − x)2 exp(−(x2 ) − (y + 1)2 ) − 10(x/5 − x3 − y 5 ) exp(−x2 − y 2 ) − (1/3) exp(−(x + 1)2 − y 2 ) (4) The function J1 (x, y) consists of a set of three peaks and three valleys. A set of 50 glowworms are randomly deployed in a two-dimensional workspace of size 6X6 square units. Figures 2 shows the emergence of the solution when the local-decision domain range is made to vary according to (2). A value of rdi (0) = 3 is chosen. The luciferin histories of all the glowworms is shown in Figure 2. Using the earlier update rule, it takes 900 iterations for all the glowworms co-locate at sources [8]. However, in the modified case, all glowworms, excepting two, get co-located within 360 iterations. IV. THEORETICAL FOUNDATIONS In this work, we provide some basic convergence results constituting the theoretical foundations to the glowworm algorithm used to solve the multiple rendezvous problem. In the glowworm algorithm, the luciferin values vary with time according to (3). However, we initially assume that glowworms are associated with constant and distinct luciferin values. We also assume that the range of local-decision domain rdi is kept constant and made equal to the maximum
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sensor range rs . Note that the above assumptions lead to a preliminary analysis based on a static framework and does not completely represent the dynamics of the GSO algorithm. A. Notations We use the following notations: I = {1, 2, · · · , n} is the index set used to identify the n glowworms; x = (x1 , ..., xn ) is the state vector containing the states (positions) of all members of the group; dij = xi − xj is the euclidian distance between locations of agents i and j; rs is the radial sensor range of each agent; i is the luciferin-value associated with agent i; Ni = {j : j = i, j ∈ I, dij ≤ rs } is the set of distance-based neighbors of agent i; Hi = {j : j ∈ I and i < j } is the set of luciferin-based neighbors of agent i; Li = Ni Hi is the leader-set of agent i; tk is the kth discrete time instant, G(V, E) is a directed graph with the set of nodes (glowworms) V = {v1 , v2 , · · · vn }; the edge set ∈ I E ⊆ V ×V (an edge (i, j) exists iff j ∈ Li ); L = {i : i and Li (tk ) = φ, ∀ tk } is the leader-set of G. (G = L F where L F = φ); = {0, 1, · · · , k, k + 1, · · ·} is the discrete time-index set. The terms glowworm, node, and agent are used interchangeably. Some of the questions that motivate our analysis are: 1) Do the nodes become stationary as t → ∞? 2) Do the nodes become stationary at a finite time tf ? 3) Does a time T () exist for every small > 0 such that a glowworm i reaches a leader with probability 1 − within T () steps? B. Assumptions 1) Each agent is represented by a point. 2) Agents move instantaneously with a maximum stepsize of δ (0 < δ < rs ) at each instant tk . 3) Collisions between the agents are ignored. 4) The value of j remains constant, ∀j ∈ I. 5) i = j , ∀ i, j ∈ I, i = j; that is agents have distinct values of . 6) The local-decision domain range is kept constant and made equal to the maximum sensor range, i.e., rdi = rs . C. Glowworm Dynamics The continuous-time model of the glowworm dynamics can be stated as: x˙i
=
ui
=
ui
(5) 0, if Li = φ aij (xj − xi ), otherwise
(6)
where j ∈ Li and i selects to move towards j with a probability pj given by (1). In particular, the discrete-time model of the agent dynamics is given by: xj (tk ) − xi (tk ) (7) xi (tk+1 ) = xi (tk ) + s xj (tk ) − xi (tk ) where
s =
δ, if dij (tk ) ≥ δ dij (tk ), otherwise
(8)
Definition 1: An agent i is said to be stationary at time tk , if xi (tk+1 ) = xi (tk ). Definition 2: An agent i is said to be stationary for all time after tj , if xi (tk+1 ) = xi (tk ), ∀ k ≥ j. Definition 3: G(V, E) is said to be stationary at time tk , if all the agents are stationary at time tk , i.e., xi (tk+1 ) = xi (tk ), ∀ i I. Definition 4: G(V, E) is said to be stationary for all time after tj , if all the agents are stationary for all time after tj , i.e., xi (tk+1 ) = xi (tk ), ∀ k ≥ j and ∀ i ∈ I. Definition 5: Two nodes i and j are co-located at time tk if xi (tk ) = xj (tk ) or dij (tk ) = 0. Definition 6: A leader li is said to be isolated, if li − lj > 3rs ∀ j ∈ L, j = i. Proposition 1: At least one agent remains stationary always. Proof: From assumption (4), ∃ i∗ ∈ I such that i∗ (tk ) ≥ i (tk ), ∀ i ∈ I and i = i∗ and ∀ tk . Therefore, Hi∗ (tk ) = φ, ∀ tk . This implies that Li∗ (tk ) = Ni∗ (tk ) ∩ Hi∗ (tk ) = φ, ∀ tk . This gives ui∗ (tk ) = 0, ∀ tk . From (7), xi∗ (tk+1 ) = xi∗ (tk ), ∀ tk . Therefore, the agent i∗ remains stationary for [] all time tk . Note: Assumption (5) is not required to prove the above. Proposition 2: If G is stationary at time tq , then it is stationary ∀ tk , k > q. Proof: We prove the result using mathematical induction. Let G be stationary at time tq . From definition 1, xi (tq+1 ) = xi (tq , ) ∀ i ∈ I. Therefore, the result is true for k = q + 1. Now, let xi (tk+1 ) = xi (tk ), ∀ i ∈ I and for some k > q. From (6), (7), and (8), this is true, if and only if for all i, either (i) Li (tk ) = φ or (ii) Li (tk ) = φ and dij (tk ) = 0, ∀ j ∈ Li (tk ). Suppose (i) is true. Since there are no agent movements at tk , the leader-sets continue to be empty at tk+1 , i.e., Li (tk+1 ) = φ, ∀ i ∈ I. From (5) and (6), we get xi (tk+2 ) = xi (tk+1 ), ∀ i ∈ I. Suppose (ii) is true. Since there are no agent movements, Li (tk+1 ) = Li (tk ) = φ ⇒ dij (tk+1 ) = dij (tk ) = 0, ∀j ∈ Li (tk+1 ). Using (6), (7), and (8) again, we get xi (tk+2 ) = xi (tk+1 ). We proved that if G is stationary at time tk , k > q, then it is stationary at tk+1 . Thus, the result follows by induction. [] Proposition 3: For any agent i, if Li (tk ) = φ at some time tk , then Li (tk ) = φ, ∀ q ≥ k. Proof: Given Li (tk ) = φ. Let j ∈ Li (tk ) such that i selects to move towards it. This implies that dij (tk ) ≤ rs . Agent i moves a distance s1 units on the line joining xi (tk ) and xj (tk ). During the same time, let j move s2 units in a direction that makes an angle θ with the vector xj (tk ) − xi (tk ) (see Figure 3). Case 1: Suppose s1 = δ and s2 ≤ δ Using the triangle inequality, we have dij (tk+1 )
≤ (dij (tk ) − s1 ) + s2 = (dij (tk )) − (s1 − s2 ) ≤ dij (tk ) ≤ rs (9)
Case 2: Suppose 0 < s1 < δ and s2 = δ. This implies, i reaches j’s position at tk in one step i.e., xi (tk+1 ) =
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X j (t k+1 ) d
ij
(t
s2
θ
x j (t k )
s1
xi (t k )
k+1
)
X i (t k+1 )
) d ij (t k Fig. 3.
Movements of agents i and j at iteration tk
xj (tk ) while j moves a step distance of δ units. Therefore, dij (tk+1 ) = δ. From assumption 2 and (9), we get Li (tk+1 ) = φ. In a similar way, we can show that, if Li (tq ) = φ for some q > k, then Li (tq+1 ) = φ. Thus, we get the desired result by induction. [] Remark: Proposition 3 implies that once an agent acquires at least one leader, it continues to have a leader. Proposition 4: Suppose G becomes stationary at some time tk . If nodes i and j are not co-located and i = j , then xi (tk ) − xj (tk ) > rs . Proof: Since the values of are distinct, we can let i < j ⇒ j ∈ Hi (tk ). Now, xi (tk+1 ) = xi (tk ) (since G becomes stationary at time tk ) and dij (tk ) = 0 ⇒ j ∈ / Li (tk ) (otherwise i can movetowards j). This is true only if j ∈ / Ni (tk ) (since Li = Ni Hi and j ∈ Hi ). [] Proposition 5: If for a node i, Li (tq ) = φ at any time tq , then when node i becomes stationary at tk , there exists at least one j co-located with i such that j > i . Proof: Li (tq ) = φ ⇒ Li (tk ) = φ (from proposition 3). Suppose dij (tk ) = 0, ∀j ∈ Li (tk ). From (6), xi (tk+1 ) = xi (tk ), which is a contradiction. [] ˆ k ) = {i : Li (tk ) = φ}, Proposition 6: At any time tk , let L(t ˆ k ). then L ⊆ L(t Proof: L is the leader-set of G, i.e., if i ∈ L then Li (tk ) = ˆ k ), ∀ i ∈ L ⇒ L ⊆ L(t ˆ k ). [] φ, ∀ tk ⇒ i ∈ L(t ˆ be defined as in Proposition 6. Then, Proposition 7: Let L ˆ k ). ˆ k+1 ) ⊆ L(t L(t ˆ k ), Li (tk ) = φ ⇒ Li (tk+1 ) = Proof: For i ∈ I and i ∈ / L(t ˆ k+1 ). This implies that φ (from Proposition 2) ⇒ i ∈ / L(t ˆ ˆ [] i ∈ L(tk+1 ) ⇒ i ∈ L(tk ). Hence the result. ˆ k) Remark: Proposition 7 implies that a non-member of L(t ˆ k+1 ). However, a member cannot become a member of L(t ˆ k+1 ). For instance, ˆ k ) may lose its membership in L(t i ∈ L(t this happens when an agent, say j, with i < j enters within the rs range of i at tk+1 . ˆ k) = Proposition 8: There exists a time tq so that L(t ˆ k+1 ), ∀ k ≥ q. L(t ˆ k ) is true only if ∃ at ˆ k+1 ) ⊂ L(t Proof: The relation L(t ˆ k ), but i ∈ ˆ k+1 ) (this occurs least one i such that i ∈ L(t / L(t ˆ k) ˆ k+1 ) ⊆ L(t when i acquires a leader). Since L ⊆ L(t (from Propositions 6 and 7), members can continue to lose ˆ q ) = L. Therefore, ˆ only until tq when L(t membership in L
ˆ k ) = L(t ˆ k+1 ). for all tk , k ≥ q, L(t [] Theorem 1: If a leader l is isolated (Definition 5), then all the with l, ∀ tk , k ≥ K(q), members of Nl (t0 ) are co-located i where, q = |F |, K(i) = j=1 dfj l (tK(j−1) , K(0) = 0, fj ∈ F, j = 1, · · · , q, and f1 > f2 > ... > fq . Proof: Consider an isolated leader l. Since l > fj ∀ fj ∈ Nl (t0 ), all followers are influenced by the leader. Since the values of are distinct, we can sort the followers as f1 , f2 , ..., fq according to the hierarchy with respect to their associated values. Therefore, we have l1 > f1 > f2 > ... > fq . Consider the movements of f1 . Since Lf1 (tk ) = {l}, ∀ tk , f1 makes deterministic moves towards l and reaches it in tK(1) iterations, where K(1) = df1 l (0). Therefore, ∀ tk , k ≥ K(1), f1 is co-located with l. Consider the movements of f2 . Since Lf2 (tk ) = {l, f1 }, ∀ tk , at each iteration it moves either towards l or f1 . Based on its distance to l at t0 and its moves, it may or may not reach l within tK(1) iterations. However, at tK(1) (assuming that f2 is not co-located with l at that time), since l and f1 are co-located, f2 has only one direction to move. Let df2 l (tK(1) ) be the distance between f2 and l at tK(1) . Then f2 converges to l in df2 l (tK(1) ) steps after tK(1) . Therefore, ∀ tk , k ≥ K(2), f2 is co-located with l, where K(1) + df2 l (tK(1) ), if df2 l (tK(1) ) = 0 (10) K(2) = K(1), Otherwise Note: However, this is a conservative bound because f2 could get co-located with l much before tK(2) . Now, suppose fi is co-located with l, ∀ tk , k ≥ K(i). Let dfi+1 l (tK(i) ) be the distance between fi+1 and l at time tK(i) . fi+1 takes dfi+1 l (tK(i) ) steps to reach l. This i implies, K(i + 1) = K(i) + dfi+1 l (tK(i) ) = j=1 dfj l (tK(j−1) + dfi+1 l (tK(i) ) = i+1 j=1 dfj l (tK(j−1) . Therefore, by induction, we can show that all the members of Nl (t0 ) will be co-located with l, ∀ tk , k ≥ K(q). [] Theorem 2: Let K be defined as in Theorem 1 and q = |F |. Suppose two leaders l1 and l2 are not isolated but their neighborhoods are non-overlapping, i.e., 2rs < l1 − l2 ≤ 3rs , then all the followers get co-located with either one of the leaders, ∀ tk , k ≥ K(q). Proof: Consider followers f1 , f2 , ..., fq such that l1 > f1 > ...fg > l2 > fg+1 > ... > fq . Consider the movements of f1 . Obviously, f1 ∈ Nl1 Nl2 . Otherwise, there will be no attracting influence on f1 and hence it remains stationary, which then contradicts the characteristics of a follower. / Nl2 because the relationship f1 > l2 Moreover, f1 ∈ contradicts the fact that a leader must have maximum value in its neighborhood. Therefore, f1 ∈ Nl1 (tk ), ∀ k. This implies that f1 gets co-located with l1 , ∀ tk , k ≥ K(1). / Nl2 . Also, Consider themovements of f2 . Clearly, f2 ∈ f2 ∈ / Nl1∗ Nl2∗ is possible for tk such that k ≤ K(1). However, during this time Lf2 = {f1 }. Otherwise, f2 becomes a leader which is a contradiction. But once f1 is co-located with l1 , f2 should be within rs -distance of l1 . Therefore, ∀ tk , k ≥ K(2), f2 is co-located with l1 . Similar analysis can be carried out till follower fg and we can show
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p
p
1 2
1
l1
l2
l1
xa
xb
l2
p
1
1
3
xa
p
p
f g+1 case 1
Fig. 4.
Fig. 5.
f g+1
p
2
2
n
n-1
xb
2
The n-state finite state Markov chain
case 2
The two different cases that occur when fg+1 ∈ Nl1 ∩ Nl2
that f1 , f2 , ..., fg will be co-located with l1 ∀ tk , k ≥ K(g). Consider the movements of fg+1 . If fg+1 ∈ Nl1 (t0 ), it remains in Nl1 and gets co-located for all tk , k ≥ K(g+1). If fg+1 ∈ Nl2 (t0 ), there is a possibility that it leaves Nl2 and enters Nl1 before f1 , ..., fg reach l1 . However, once itenters Nl1 , it cannot leave Nl1 . After tK(g) , fg+1 ∈ Nl1 Nl2 . Therefore, if fg+1 ∈ Nl1 (or fg+1 ∈ Nl2 ), it gets colocated with l1 (or l2 ), ∀ tk , k ≥ K(g + 1). Note that while evaluating the expression dfj l (tK(j−1) , l = l1 (l2 ) if fj ∈ Nl1 (tK(j−1) ) (fj ∈ Nl2 (tK(j−1) )). In general, if fi ∈ {fg+1 , ..., fq } reaches either one of the leaders in time tK(i) , then fi+1 reaches either one of them in time tK(i+1) . Therefore, all the followers f1 , ..., fg are colocated at l1 , ∀ tk , k ≥ K(g) and the rest of the followers fg+1 , ..., fq are co-located at either one of the leaders [] ∀ tk , k ≥ K(q). Theorem 3: Let K be defined as in Theorem 1 and q = |F |. If two leaders l1 and l2 have overlapping neighborhoods, i.e., rs < l1 − l2 ≤ 2rs , then all followers fj such that j ∈ F and l2 < fj < l1 converge to one of the leaders in finite time and the remaining followers located in the overlap region of the neighborhoods asymptotically converge to one of the leaders. Proof: Let the followers f1 , f2 , ..., fq satisfy l1 > f1 > ...fg > l2 > fg+1 > ... > fq . Consider the movements of the members of F1 = {f1 , ..., fg }. The set F1 has the property that fi ∈ Nl1 \ Nl1 ∩ Nl2 , ∀ fi ∈ F1 . Using Theorem 1, we can show that all members of F1 are co-located with l1 , ∀ tk , k ≥ K(g). Consider the movements of fg+1 after tK(g) . If fg+1 ∈ Nl1 \ Nl1 ∩ Nl2 , then fg+1 reaches l1 in time tK (g + 1), with l = l1 . If fg+1 ∈ Nl2 \ Nl1 ∩ Nl2 , then fg+1 reaches l2 in time tK (g + 1), with l = l2 . Suppose fg+1 ∈ Nl1∗ ∩ Nl2∗ . Let P = {x : x ∈ (αxa + (1 − α)xb ), 0 ≤ α ≤ 1}, where xa and xb are the points of intersection of the line joining l1 and l2 with circles centered at xl1 and xl2 , respectively. Case 1: fg+1 ∈ P . Let p1 and p2 (p1 + p2 = 1) be the probabilities that fg+1 makes a step movement towards l1 and l2 , respectively, at each time tk . Since the agent moves in discrete steps, the convergence problem can be formulated as a finite state Markov chain with n = 2rs − dl1 l2 , where the agent’s position at any iteration tk is represented by one of the states. For instance, in the Figure 5, xa coincides with state 1 and xb coincides with state n. Note that we need not consider
the movements of the follower on the line segments l1 xa and l2 xb because, once the agent reaches xa (xb ) it will be influenced by l1 (l2 ) only, thus reaching it in finite number of steps. The one-step probabilities can be expressed in the form of a transition probability matrix given by: ⎞ ⎛ 1 0 . . . 0 ⎜ p1 0 p2 0 . 0 ⎟ ⎟ ⎜ ⎜ 0 p1 0 p2 . 0 ⎟ ⎟ (11) P = [pij ]nxn = ⎜ ⎜ . . . . . . ⎟ ⎟ ⎜ ⎝ 0 . . p 1 0 p2 ⎠ 0 0 0 0 0 1 Let P n = [pnij ] be the n-state transition matrix where pnij represents the probability that an agent initially in state i ends up in state j at the nth step. The steady state probability state vector equation is then given by: P ∞P
P∞
=
where P ∞ = limn→∞ [pnij ] Let ⎛ 1 0 ⎜ p21 p 22 ⎜ ⎜ . . P∞ = ⎜ ⎜ . . ⎜ ⎝ p(n−1)1 p(n−1)2 0 0
. . . . . .
. . . . . .
(12)
. 0 . p2n . . . . . p(n−1)n . 1
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
(13)
Consider Pi∞ P = Pi∞ , where Pi∞ is the ith row of P ∞ . Then, p1 pi2 + pi1 = p1 pi4 + p2 pi2 = .. . p1 pi(n−1) + p2 i(n − 3) = p2 pi(n−2) = pi(n−1) ;
pi1 ; p1 pi3 = pi2 pi3 pi(n−2) pin + p2 pi(n−1) = pin(14)
From (14), we get pi2 = p1 pi(j+2) + p2 pij = p2 pi(n−2) = pi(n−1) ;
0; pi3 = 0 pi(j+1) , j = 2, 3, · · · , (n − 3) pin + p2 pi(n−1) = pin (15)
Solving the set of equations in (15), we get
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pij
=
0 for j = 2, 3, · · · , (n − 1)
pij
=
1, ∀ i = 1, 2, · · · , n
⇒ pi1 + pin
=
1, ∀ i = 1, 2, · · · , n
n j=1
(16)
Xf
df
ε
Xl
Fig. 6.
g+1 l 1
df θ2
θ1
δ
g+1 l 2
h(t0)
δ dl l 1 2
φ
1
(t0)
g+1
Xl
2
Follower movements when fg+1 ∈ Nl1
Nl2 \ P .
The result in (16) shows that the agent starting in any one of the n states, converges to either xa or xb asymptotically with probability one. Once fg+1 reaches xa (xb ), it reaches l1 (l2 ) in dl1 l2 − rs time steps. As an illustration, the explicit probabilities of reaching either l1 or l2 from any intermediate state when n = 5 is obtained below: ⎞ ⎛ 1 0 0 0 0 ⎟ ⎜ p1 (1−p1 p2 ) p32 0 0 0 ⎟ ⎜ 1−2p1 p2 1−2p1 p2 ⎟ ⎜ p21 p22 ∞ ⎟(17) ⎜ = ⎜ 1−2p p P5 0 0 0 ⎟ 1−2p1 p2 1 2 ⎜ p31 p2 (1−p1 p2 ) ⎟ ⎠ ⎝ 1−2p p 0 0 0 1−2p1 p2 1 2 0 0 0 0 1 Case 2: fg+1 ∈ Nl1 ∩ Nl2 \ P . At each iteration, fg+1 makes a step movement δ towards either l1 or l2 . Let dl1 l2 be the distance between l1 and l2 . Let h(t) and h(t + 1) be the perpendicular distances from xfg+1 (t) and xfg+1 (t + 1), respectively, to the line joining xl1 and xl2 . Clearly, h(t + 1) < h(t). Therefore, h(t) is a monotonically decreasing function. Now, we show that there exists a time tq such that dl1 l2 < dfg+1 l1 (tk ) + dfg+1 l2 (tk ) < dl1 l2 + δ, ∀ tk , k ≥ q (18) Referring to Figure 6, the angles θ1 and θ2 are given by: 2 dfg+1 l1 + d2l1 l2 − d2fg+1 l2 (19) θ1 = arccos 2dfg+1 l1 dl1 l2 2 dfg+1 l2 + d2l1 l2 − d2fg+1 l1 (20) θ2 = arccos 2dfg+1 l2 dl1 l2 A lower bound on the perpendicular distance to the line joining l1 and l2 such that dfg+1 l1 (tk ) + dfg+1 l2 (tk ) = dl1 l2 + δ can be calculated as: =
δ(δ + 2dl1 l2 ) 2(dl1 l2 + δ)
(21)
Using (21), the minimum perpendicular distance dp moved by fg+1 at each step is given by: dp
=
φ =
δ cos φ
90o − arctan
tan θ d tan θ −
and θ = min{θ1 , θ2 }
From Figure 6, the initial perpendicular distance h(t0 ) is given by: h(t0 ) =
dl1 l2 tan θ1 tan θ2 tan θ1 tan θ2
(22)
Therefore, after tq , q = h(td0p)− , the perpendicular distance to the line segment l1 l2 is within and hence (18) is satisfied. The consequence of (18) is that dfg+1 l1 (tk )+dfg+1 l2 (tk ) = dl1 l2 , ∀ k ≥ q. Hence a similar analysis as in Case 1 can be used to prove asymptotic convergence to one of the leaders. Moreover, since the movement of fg+1 is always on the line joining its own position and that of l1 or l2 , it cannot converge to a position between the leader positions and gets co-located with either l1 or l2 . A similar analysis can be done to show asymptotic convergence of the remaining followers [] (fg+2 , · · · , fq ) to either one of the leaders. Remarks: We can infer from the above theoretical results that, even though agent movements are probabilistic in nature, agents get co-located with leaders in finite time if the leader-neighborhoods are non-overlapping. Otherwise, the agents get co-located with leaders asymptotically. V. C ONCLUSIONS We presented the theoretical foundations for the multiple rendezvous of glowworm-inspired mobile agents that use a variable local-decision domain to distinguish between source locations of interest. The main convergence results are proved for the constant luciferin and constant decision domain case. The significance of a variable decision domain is described using an illustrative simulation experiment. Future work involves proving convergence results in a more general setting where the luciferin values and decision domain ranges vary according to their respective update rules. R EFERENCES [1] T. Vicsek, A. Cziro´ok, E. Ben-Jacob, O. Cohen, and I. Shochet, “Novel type of phase transition in a system of self-driven particles,” Phys. Rev. Lett., Vol. 75, No. 6, August 1995, pp. 1226- 1229. [2] S.H. Strogatz, “‘Exploring complex networks,” Nature, Vol. 410, 2001, pp. 268-276. [3] H. Ando, Y. Oasa, I. Suzuki, and M. Yamashita, “Distributed memoryless point convergence algorithm for mobile robots with limited visibility,” IEEE Trans. on Robotics and Automation, Vol. 15, No. 5, October 1999, pp. 818-828. [4] X. Cui, C. T. Hardin, R. K. Ragade, and A. S . Elmaghraby, “A swarm approach for emission sources localization,” Proceedings of the 16th International Conference on Tools with Artificial Intelligence (ICTAI 2004), Boca Raton, Florida, November 2004, pp. 424-430. [5] K.N. Krishnanand and D. Ghose, “Detection of multiple source locations using a glowworm metaphor with applications to collective robotics,” Proceedings of IEEE Swarm Intelligence Symposium, Pasadena, California, June 2005, pp. 84-91. [6] K.N. Krishnanand and D. Ghose, “Multimodal function optimization using a glowworm metaphor with applications to aollective robotics,” 2nd Indian International Conference on Artificial Intelligence, Pune, India (to appear). [7] K.N. Krishnanand and D. Ghose,“Glowworm-inspired Swarms with Adaptive Local-decision Domains for Multimodal Function Optimization,” Proceedings of IEEE Swarm Intelligence Symposium, Indianapolis, May 2006 (to appear). [8] K.N. Krishnanand, P. Amruth, M.H. Guruprasad, Sharschchandra V. Bidargaddi, and D. Ghose.“Glowworm-inspired robot swarm for simultaneous taxis towards multiple radiation sources,” Proceedings of IEEE International Conference on Robotics and Automation, Orlando, Florida, May 2006 (to appear). [9] E. Bonabeau, M. Dorigo, G. Theraulaz, Swarm Intelligence: From Natural to Artificial Systems, Oxford University Press 1999 pp. 183203. [10] R Olfati-Saber and R.M. Murray, “Consensus problems in networks of agents with switching topology and time-delays,” IEEE Trans. on Automatic Control, Vol. 48, No. 9, September 2004, pp. 1520-1533.
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