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Rendezvous of multiple mobile agents with preserved network connectivity
Systems & Control Letters 59 (2010) 313–322
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Rendezvous of multiple mobile agents with preserved network connectivityI Housheng Su a,b,c,∗ , Xiaofan Wang b , Guanrong Chen c a
Department of Control Science and Engineering, Huazhong University of Science and Technology, Wuhan 430074, China
b
Department of Automation, Shanghai Jiao Tong University, Shanghai 200240, China
c
Department of Electronic Engineering, City University of Hong Kong, Hong Kong, China
article
info
Article history: Received 16 April 2008 Received in revised form 30 July 2009 Accepted 15 March 2010 Available online 24 April 2010 Keywords: Distributed control Nonlinear system Rendezvous Multi-agent Virtual leader
abstract In coordinative control of a network of multi-agent systems, to guarantee the stability of the coordinated motion, a basic assumption typically is that the underlying topology of the network can maintain its connectivity frequently enough during the motion evolution. However, for a given set of initial conditions, this assumption is very difficult to satisfy and verify. In particular, the connectivity of the initial network generally cannot guarantee the connectivity of the network throughout the evolution. In this paper, we propose a rendezvous protocol with double-integrator dynamics, which combines the functions of motion control and connectivity preservation. This protocol can enable the group of mobile agents to converge to the same position and move with the same velocity while preserving the connectivity of the whole network during the evolution if the initial network is connected. We find that there is a trade-off between the maximum overshoot and the settling time of the velocity convergence. Furthermore, we investigate the rendezvous protocol with a virtual leader and show that all agents can asymptotically attain a desired velocity even if only one agent in the team has information about the virtual leader. We finally show some numerical simulations to verify and illustrate the theoretical results. © 2010 Elsevier B.V. All rights reserved.
1. Introduction In recent years, multi-agent coordinated control problems have attracted much attention among researchers studying biology, physics, computer science and control engineering [1–3]. Many versions of distributed coordinated control protocols have been proposed, including consensus [3–11], swarming [12–17], flocking [18–23], and rendezvous [24–29]. In particular, rendezvous is the motion in which a large number of agents, using only limited environmental information and simple rules, arrive at a location simultaneously, which are characterized by distributed control, local interactions and self-organization. Understanding the mechanisms of rendezvous behaviors can help comprehending some natural behaviors and developing many artificial autonomous systems such as schooling of fish for avoiding predators and finding food, and mobile robots for rescue, surveillance, fire fighting and disaster control [30].
I This work was partly supported by the NSF of PR China under Grant Nos. 60674045 and 60731160629, and the Major State Basic Research Development Program of China (973 Program) (No. 2010CB731400), as well as the NSFC-HKRGC Joint Research Scheme under Grant N_CityU 107/07. ∗ Corresponding author at: Department of Control Science and Engineering, Huazhong University of Science and Technology, Wuhan 430074, China. Tel.: +86 27 87540210; fax: +86 27 87540210. E-mail addresses: [email protected] (H. Su), [email protected] (X. Wang), [email protected] (G. Chen).
0167-6911/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.sysconle.2010.03.006
Subjected to limited sensing and communication capabilities of agents, the interaction topology among agents may change over time. A basic assumption made in most previous works on stability analysis of collective dynamics is that the underlying topology can remain connected frequently enough during the motion evolution. However, for a given set of initial states and parameters, it is very difficult or even impossible to satisfy and verify this assumption in practice. In fact, the connectivity of the initial network generally cannot guarantee the connectivity of the whole network throughout a long process of motion. For example, the consensus algorithm proposed in [3], the flocking algorithms in [18,19], and the rendezvous algorithm in [29], have been known to possibly result in fragmentation even if the initial network is connected. Recently, some efforts had been made to preserve the connectivity of the networks. For example, Cortes et al. [31] proposed a circumcenter algorithm to avoid the loss of existing connections. Degennaro and Jadbabaie [32] suggested an iterative decentralized algorithm to maximize the smallest nonzero eigenvalue of the Laplacian matrix for the connectivity control. Spanos and Murray [33] used a measure of local connectivity to ensure the global connectivity of a network. Dimarogonas and Kyriakopoulos [34,35] used potential functions to maintain the network connectivity for rendezvous and swarming. Ji and Egerstedt [36] proposed a hysteresis in adding new links and a special potential function to achieve rendezvous while preserving the network connectivity. However, all these works are concerned with agents described
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H. Su et al. / Systems & Control Letters 59 (2010) 313–322
by single-integrator dynamics. Recently, Zavlanos et al. [37] used the hysteresis and potential function method to study a flocking problem combining with the network connectivity for doubleintegrator dynamics to achieve velocity alignment. In this work, we investigate the rendezvous problem with connectivity preservation for double-integrator dynamics, also employing hysteresis in adding new links and using the potential function method. It should be noted, however, that in [36,37], two specific potential functions are constructed for single-integrator dynamics and double-integrator dynamics, respectively, which tend to infinity as the distance between two already connected agents tends to the sensing radius, which is rather impractical. In this work, we construct a general class of bounded potential functions that can preserve the network connectivity for doubleintegrator dynamics and the aforementioned divergence problem can be avoided. We also investigate the rendezvous protocol with a virtual leader and show that all agents can asymptotically attain a desired velocity even if only one agent in the team has communication with, hence has information about the virtual leader. The remainder of the paper is organized as follows. Section 2 describes the problem to be investigated. A new rendezvous algorithm and its stability analysis are established in Section 3. Section 4 discusses the case with a virtual leader. Section 5 presents some illustrative simulation results. Finally, Section 6 draws some conclusions. 2. Problem statement Consider N agents moving in an n-dimensional Euclidean space. The motion of each agent is described by a double-integrator equation of the form q˙ i = pi , p˙ i = ui ,
i = 1, . . . , N ,
(1)
where qi ∈ Rn is the position vector of agent i, pi ∈ Rn is its velocity vector, and ui ∈ Rn is the (force) control input acting on agent i, i = 1, . . . , N. Suppose that all agents have the same influencing/sensing radius r. Let ε ∈ (0, r ] and ε0 ∈ (0, r ) be the given constants and ε0 ≤ ε . Call G(t ) = (V , E (t )) a undirected dynamic graph consisting of a set of vertices V = {1, 2, . . . , N } indexed by the set of agents and a time-varying set of links E (t ) = {(i, j)|i, j ∈ V } such that (i) initial links are generated by E (0) = { (i, j)| kqi (0) − qj (0)k < r − ε0 , i , j ∈ V } ; (ii) if (i, j) 6∈ E (t − ) and kqi (t ) − qj (t )k < r − ε , then (i, j) is a new link being added to E (t ); (iii) if kqi (t ) − qj (t )k ≥ r, then (i, j) 6∈ E (t ). Throughout this paper, k·k denote the Euclidean norm. One can use a symmetric indicator function σ (i, j) ∈ {0, 1} to describe whether there is a link between agent i and agent j at time t, which is defined as
0, if ((σ (i, j)[t − ] = 0) ∩(kqi (t ) − qj (t )k ≥ r − ε)) ∪((σ (i, j)[t − ] = 1) ∩ (kqi (t ) − qj (t )k ≥ r )), σ (i, j)[t ] = 1, if ((σ (i, j)[t − ] = 1) ∩(kqi (t ) − −qj (t )k < r )) ∪((σ (i, j)[t ] = 0) ∩ (kqi (t ) − qj (t )k < r − ε)). One can see from Fig. 1 that there is a hysteresis in the indicator function for adding new links to the graph [36,37]. When ε = r, no new links can be added to the graph. Moreover, one will see that the selection of the parameter ε (ε0 ) is a trade-off between the maximum overshoot and the settling time of velocity convergence (the feasible energy and the radius of the initial network):
2 1.8 1.6 1.4 1.2 σ(i,j)
1 0.8 0.6 0.4 0.2 0
0
1
2
3 r–ε
||qij||
4 r
5
6
Fig. 1. Indicator function σ (i, j).
the smaller the value of ε (ε0 ), the larger the maximum overshoot and the smaller the settling time (the larger the magnitude of the potential function and the radius of the initial network). Our objective is to make all agents converge to the same position and move with the same velocity under the assumption that the initial network is connected. Specifically, we want pi (t ) − pj (t ) → 0 and qi (t ) − qj (t ) → 0 for all i, j ∈ V . To achieve this goal, we choose the control law ui for agent i in (1) to be ui = αi + βi ,
(2)
where αi is the gradient-based term, which is designed to enforce the position of each agent to a common value and also to maintain the connectivity of the network, and βi is the consensus term, which regulates the velocity of each agent to a common value. In some cases, the regulation of agents is for certain purposes, e.g., achieving a desired common velocity or arriving at a desired destination. In the situation where there is a virtual leader, we modify the control law (2) to ui = αi + βi + γi ,
(3)
where γi is the navigational feedback term, which drives agent i to track the virtual leader. 3. Fundamental rendezvous algorithm 3.1. Algorithm design and main results We now specify an explicit control input in Eq. (2) as ui = −
X
∇qi ψ(kqij k) −
j∈Ni (t )
|
X
wij (pi − pj ) .
(4)
j∈Ni (t )
{z αi
}
|
{z βi
}
Here, qij = qi − qj , the constant wij = wji > 0 is bounded for all i, j ∈ V , Ni (t ) is the neighborhood region of agent i at time t defined as Ni (t ) = {j|σ (i, j)[t ] = 1, j 6= i, j = 1, . . . , N }.
(5)
The nonnegative potential ψ(kqij k) is defined to be a function of the distance kqij k between agent i and agent j, differentiable with respect to kqij k ∈ [0, r ] such that ∂ψ(kqij k) ∂kqij k
> 0 for kqij k ∈ (0, r ); ∂ψ(kqij k) 1 (ii) limkqij k→0 is nonnegative and bounded; · ∂kqij k kqij k PN T (iii) ψ(r ) = Qˆ ∈ [Qmax , +∞), where Qmax , 12 i=1 pi (0)pi (0) + N (N −1) ψ(k r − ε k) . 0 2 (i)
Condition (i) illustrates that the potential between the two agents is an increasing function of their distance, which makes two agents
H. Su et al. / Systems & Control Letters 59 (2010) 313–322
of L(t ) as λ1 (L(t )) ≤ · · · ≤ λN (L(t )). Then, λ1 (L(t )) = 0 and 1 = [1, 1, . . . , 1]T ∈ RN is its corresponding eigenvector. Moreover, if G(t ) is a connected graph, then λ2 (L(t )) > 0 [38]. The _ corresponding n-dimensional graph Laplacian is defined as L (t ) = L(t ) ⊗ In , where In is the identity matrix of order n and ⊗ stands for the Kronecker product. This multi-dimensional Laplacian satisfies the following sum-of-squares property [19]:
200 180 160 140 120 ψij 100 80
1 X
_
z T L (t )z =
60
20 0
0.5
1
1.5 ||qij||
2
2.5
3
3.5
where z = [z1T , z2T , . . . , zNT ]T ∈ RNn and zi ∈ Rn for all i. Denote the position and velocity of the center of mass (COM) of all agents in the group by
4 r
N P
Fig. 2. The potential function (6).
attract each other; Condition (ii) requires that the magnitude of gradient between two agents is the same or a higher-order term of their distance when the two agents converge to the same position; Condition (iii) states that the potential between two agents will be sufficiently large when the distance between the two agents reaches the sensing radius, which guarantees all existing edges not be lost. One example of such a potential function is the following (see Fig. 2):
ψ(kqij k) =
kqij k2 2
r − kqij k + rˆ Q
,
kqij k ∈ [0, r ].
(6)
ψ(kqij k) =
kqij k
r − kqij k
,
kqij k ∈ [0, r ),
(7)
and in [37],
ψ(kqij k) =
1
kqij k2
+
1 r 2 − kqij k2
,
kqij k ∈ (0, r ).
Q =
2 i=1
j∈Ni (t )
ψ(kqij k) +
pTi pi
,
(9)
with initial energy Q0 = Q (q(0), p(0)). Clearly, Q is a positive semi-definite function. The adjacent matrix A(t ) = (aij (t )) of the above graph G(t ) is defined as aij (t ) =
wij , 0,
pi
i=1
, p¯ = . N N The main results of this section are given in the following theorem: q¯ =
Theorem 1. Consider a system of N mobile agents with dynamics (1), each steered by protocol (4). Suppose that the initial network G(0) is connected and the initial energy Q0 is finite. Then, the following hold: (i) G(t ) is connected for all t ≥ 0; (ii) All agents asymptotically converge to the same position and move with the same velocity; (iii) The velocity p¯ (t ) of the COM is invariant for all t ≥ 0.
3.2.1. Proof of part (i) Assume that G(t ) switches at time tk (k = 1, 2, . . .), otherwise G(t ) is a fixed graph in each time interval [tk−1 , tk ). Note that Q0 is finite and the time derivative of Q (t ) in [t0 , t1 ) is
(8)
! X
i=1
Q˙ (t ) =
Potential function (7) satisfies conditions (i) and (ii) but does not satisfy condition (iii) of our general definition of the potential function, and potential function (8) does not satisfy any of the three conditions. In particular, potential functions (7) and (8) tend to infinity when the distance between two agents i and j tends to r, which may not be practical since it will require extremely large control input. If the hysteresis parameter ε = 0, from simple calculation, Qˆ = +∞ in the potential function (6). Therefore, the potential function (6) is the same as the potential function (7). Note that, here as well as in [36,37], without the hysteresis mechanism (see Fig. 1), no new links will be added to the network. This is because the potential force is so big as to prevent the addition of new links when the distance between two agents i and j tends to r. The sum of the total artificial potential energy and the total kinetic energy is defined as follows: N 1X
N P
qi
3.2. Proof of the theorem
Note that in [36,37], only two specific potential functions are proposed: in [36], 2
aij (t )kzj − zi k2 ,
2 (i,j)∈E
40
0
315
(i, j) ∈ E (t ), otherwise.
The corresponding Laplacian is L(t ) = ∆(A(t )) − A(t ), where the degree matrix ∆(A(t )) is a diagonal matrix with the ith PN diagonal element equal to j=1, j6=i aij (t ). Denote the eigenvalues
=
N 1 XX
2 i=1 j∈N i N X
pTi
X
N X
pTi p˙ i
i=1
∇qi ψ(kqij k)
j∈Ni
i =1
+
˙ qij k) + ψ(k
N X
! pTi
i =1
−
X
∇qi ψ(kqij k) −
X
wij (pi − pj )
j∈Ni
j∈Ni
= −pT [L(t ) ⊗ In ]p. Since L(t ) is positive semi-definite [39], Q˙ (t ) ≤ 0 in [t0 , t1 ), which implies that Q (t ) ≤ Q0 < Qmax
for t ∈ [t0 , t1 ).
From the definition of the potential function, one has ψ(r ) > Qmax > Q0 . Therefore, no edge-distances will tend to r for t ∈ [t0 , t1 ), which implies that no existing edges will be lost at time t1 . Hence, new edges must be added in the interaction network at the switching time t1 . Without loss of generality, assume that there are m1 new links being added to the interaction network (N −1)(N −2) at time t1 . Clearly, 0 < m1 ≤ M and M , , thus, 2 Q (t1 ) = Q0 + m1 ψ(kr − εk) ≤ Qmax . Similar to the aforementioned analysis, the time derivative of Q (t ) in [tk−1 , tk ) is Q˙ (t ) = −pT [L(t ) ⊗ In ]p ≤ 0,
(10)
which implies that Q (t ) ≤ Q (tk−1 ) < Qmax
for t ∈ [tk−1 , tk ), k = 1, 2, . . . .
(11)
Therefore, no edge-distances will tend to r for t ∈ [tk−1 , tk ), which implies that no edges will be lost at time tk and Q (tk ) ≤ Qmax . Since G(0) is connected and no edges in E (0) will be lost, G(t ) will be connected for all t ≥ 0.
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H. Su et al. / Systems & Control Letters 59 (2010) 313–322
3.2.2. Proof of parts (ii) and (iii) Assume that there are mk new links being added to the interaction network at time tk . Clearly, 0 < mk ≤ M. From (9) and (11), one has Q (tk ) ≤ Q0 + (m1 + · · · + mk )ψ(kr − εk) ≤ Qmax . Since there are at most M new links that can be added to G(t ), one has k ≤ M and Q (t ) ≤ Qmax for all t ≥ 0. Therefore, the number of switching times k of the system (1) is finite, which implies the interaction network G(t ) eventually becomes fixed. Thus, the rest discussions can be restricted on the time interval (tk , ∞). Note that the distance of edge is not longer than ψ −1 (Qmax ). Hence, the set
Ω = q_ ∈ Dg , p ∈ RNn |Q ( q_, p) ≤ Qmax , _
(12) N2n
| kqij k ∈ [0, is positively invariant, where Dg = { q ∈ R ψ −1 (Qmax )], ∀ (i, j) ∈ E (t )}, q_ = [qT11 , . . . , qT1N , . . . , qTN1 , . . . , qTNN ]T and p = [pT1 , pT2 , . . . , pTN ]T . Since G(t ) is connected for all t ≥ 0, one has kqij k < (N − 1)r for all i and j. Since Q (t ) ≤ Qmax , one has pTi pi ≤ 2Qmax , and √ thus kpi k ≤ 2Qmax . Therefore, the set Ω satisfying Q (t ) ≤ Qmax is closed and bounded, hence compact. Note that system (1) with control input (4) is an autonomous system, at least on the concerned time interval (tk , ∞). Therefore, the LaSalle’s invariance principle [40] can be applied to infer that if the initial conditions of the system lies in Ω , its trajectories will converge to the largest invariant set inside the region S = { q_ ∈ Dg , p ∈ RNn |Q˙ = 0}. From (10), one has Q˙ = −pT (L(t ) ⊗ In )p =
1 X 2 (i,j)∈E
=−
ui
N
! X
N i =1
j∈Ni (t )
∇qi ψ(kqij k) +
X
X
wij (pi − pj ) = 0, (14)
j∈Ni (t )
i
i = 1, . . . , N .
(15)
Rewrite (15) in a matrix form as
where qT = [qT1 , . . . , qTN ] and Lˆ (t ) = [ˆlij ] is a matrix with
ˆlij = ∂ψ(kqij k) · 1 , ∂kqij k kqij k
for i 6= j.
∂ψ(kqij k) ∂kqij k
·
1
kqij k
In this subsection, we present some qualitative analysis of the convergence rate of the system. From (10), the decay rate of the total energy Q is influenced by the topology of the interaction network. From (13), Q˙ = 0 if and only if p1 = · · · = pN . Since the interaction network is connected all the time, from the matrix theory [39] and (10), one has
− λN (L(t ))pT p ≤ Q˙ (t ) ≤ −λ2 (L(t ))pT p.
(16)
Thus, λ2 (L(t )) characterizes the convergence rate of the system. Therefore, we conclude that the convergence rate of the system relies on the smallest nonzero eigenvalue of the Laplacian of the interaction network. For an undirected graph, adding edges increases the smallest nonzero eigenvalue λ2 (L(t )), and λ2 (L(t )) attains its maximum value N when the topology of the network is fully connected [41]. Thus, λ2 (L(0)) ≤ λ2 (L(t )) ≤ N, showing that λ2 (L(0)) characterizes the bound of the convergence rate of the system.
In nature, few individuals have some pertinent information, such as knowledge of the location of a food source or of a migration route [42]. Here, we assume that only one agent is informed with information about the virtual leader. Certainly, the result can be generalized to the case with multiple informed agents, which may lead to faster convergence. Suppose that velocity of the virtual leader can be received by the informed agent through some designed mechanism, such as broadcasting. The control input to agent i is designed as ui = −
X
∇qi ψ(kqij k) −
j∈Ni (t )
X
wij (pi − pj )
j∈Ni (t )
|
{z
}
αi
− hi c1 (pi − pγ ), | {z }
|
{z
}
βi
c1 > 0,
(17)
where pγ is a desired constant velocity of the virtual leader. If agent i is the informed agent, then hi = 1; otherwise, hi = 0. Without loss of generality, assume that the first agent is the informed agent, that is, hi = 1 for i = 1 and hi = 0 for all the others. Define the sum of the total artificial potential energy and the total relative kinetic energy between all agents and the virtual leader as follows:
−(Lˆ (t ) ⊗ In )q = 0,
and
γi
j∈Ni (t )
N X ∂ψ(kqij k) 1 · ∂kqij k kqij k j=1, j6=i
(Qmax )] and limkqij k→0
3.3. Convergence rate analysis
∇qi ψ(kqij k)
X ∂ψ(kqij k) 1 =− · (qi − qj ) = 0, ∂kqij k kqij k j∈N (t )
ˆlii = −
is positive and uniformly bou-
is nonnegative and bounded. Since the dynamic network is connected all the time, from the property of the Laplacian matrix Lˆ (t ) ⊗ In [38], q converges asymptotically to span{1}, which implies that q1 = q2 = · · · = qN .
which implies that the velocity of the COM is invariant for all t ≥ 0. In steady state, p1 = · · · = pN = p¯ , so that p˙ i = p˙¯ = 0, ∀i ∈ V . From (4), one has ui = p˙ i = −
1
kqij k
4.1. Main results
i =1
N 1 X
nded for kqij k ∈ (0, ψ
−1
·
(13)
all agents asymptotically move with the same velocity. Since ψ(kqij k) = ψ(kqji k) and wij = wji , from the control input (4), one has
u¯ = p˙¯ =
∂ψ(kqij k) ∂kqij k
4. Rendezvous with a virtual leader
wij kpj − pi k2 .
Therefore, Q˙ = 0 if and only if p1 = · · · = pN , which implies that
N P
From the definition,
U =
!
N 1X
X
2 i=1
j∈Ni (t )
ψ(kqij k) + (pi − pγ ) (pi − pγ ) . T
(18)
Clearly, U is a positive semi-definite function. Here, the potential function ψ should satisfy conditions (i)–(iii) defined in the last section with Qmax in condition (iii) being replaced by Umax , PN N (N −1) 1 T ψ(kr − ε0 k). i=1 (pi (0) − pγ (0)) (pi (0) − pγ (0)) + 2 2 Our main results on tracking the virtual leader can be stated in the following theorem.
H. Su et al. / Systems & Control Letters 59 (2010) 313–322
Theorem 2. Consider a system of N mobile agents with dynamics (1), each steered by protocol (17). Suppose that the initial network G(0) is connected and the initial energy U0 is finite. Then, the following hold: (i) G(t ) is connected for all t ≥ 0. (ii) All agents asymptotically converge to the same position and move with the desired velocity pγ .
4.3. Extensions and discussions In general, all agents asymptotically converge to the desired velocity pγ , which implies that the velocity p¯ (t ) of the COM asymptotically converge to the desired velocity pγ . From the control input (17), N P
4.2. Proof of the theorem We first prove part (i) of Theorem 2. Denote the position difference vector and the velocity difference vector between agent i and the virtual leader by q˜ i = qi − pγ t and p˜ i = pi − pγ , respectively. Then, q˙˜ i = p˜ i , p˙˜ i = ui ,
(19)
i = 1, . . . , N .
By the definition of ψ(kqij k), where qij = qi − qj , we have, for q˜ ij = q˜ i − q˜ j ,
ψ(kqij k) = ψ(k˜qij k).
(20)
Thus, the control input (17) for agent i can be rewritten as
X
ui = −
∇q˜ i ψ(k˜qij k) −
j∈Ni (t )
X
wij (˜pi − p˜ j ) − hi c1 p˜ i ,
(21)
j∈Ni (t )
and the positive semi-definite energy function (18) can be rewritten as N 1X
(ψ(k˜qij k) + p˜ Ti p˜ i ). (22) 2 i=1 Similar to the proof of part (i) of Theorem 1, we can show that the time derivative of U (t ) in [tk−1 , tk ) is U =
U˙ (t ) =
N 1X X
2 i=1 j∈N (t ) i
˙ qij k) + ψ(k˜
N X
(25)
2
is compact, where Dg = { q˜ ∈ RN n |k˜qij k ∈ [0, ψ −1 (Umax )], _
∀ (i, j) ∈ E (t )}, q˜ = [˜qT11 , . . . , q˜ T1N , . . . , q˜ TN1 , . . . , q˜ TNN ]T and p˜ = [˜pT1 , p˜ T2 , . . . , p˜ TN ]T . From LaSalle’s invariance principle [40], we have U˙ = −˜pT [(L(t ) + c1 H ) ⊗ In ]˜p = −˜pT (L(t ) ⊗ In )˜p − c1 p˜ T (H ⊗ In )˜p = 0, (26) which implies that p˜ 1 = · · · = p˜ N and p˜ 1 = 0, i.e. p1 = · · · = pN = pγ . This also implies that, in steady state, ui = p˙ i = p˙ γ = 0. Thus, from (17),
∇qi ψ(kqij k)
Therefore, we conclude that the convergence rate of the system relies on the smallest nonzero eigenvalue of the Laplacian of the interaction network. Thus, λ2 (L(0)) ≤ λ2 (L(t )) ≤ N. When all agents have reached the common desired velocity, the velocity of COM must reach the desired velocity. Then, the convergence rate of the system is not faster than the convergence rate of COM. Therefore, λ2 (L(0)) characterizes the bound of the convergence rate of COM. In the situation where each agent is an informed agent and the velocity pγ (t ) of the virtual leader varies with time, we propose the following protocol: ui = −
∇qi ψ(kqij k) −
X
wij (pi − pj )
j∈Ni (t )
{z αi
}
|
{z βi
+ p˙ γ − c1 (pi − pγ ) . | {z }
} (30)
γi
In practical implementation, the derivative of the velocity of the virtual leader can be calculated by numerical differentiation [9]. The results on tracking the virtual leader with a varying velocity can then be stated in the following theorem. Theorem 3. Consider a system of N mobile agents with dynamics (1), each steered by protocol (30). Suppose that the initial network G(0) is connected and the initial energy U0 is finite. Then, the following hold: (i) G(t ) is connected for all t ≥ 0. (ii) All agents asymptotically converge to the same position and move with the desired velocity pγ . (iii) If the initial velocity of the COM, p¯ (0), is equal to the initial velocity of virtual leader, pγ (0), then p¯ (t ) is equal to the desired velocity pγ (t ) for all t ≥ 0; otherwise p¯ (t ) will exponentially converge to the desired velocity pγ (t ) with a time constant, c1 s. Proof. Denote the position difference vector and the velocity difference vector between agent i and the virtual leader as q˜ i = qi − pγ t and p˜ i = pi − pγ , respectively. Then, q˙˜ i = p˜ i , i = 1, . . . , N .
(31)
Similar to the proof of part (i) of Theorem 2, the time derivative of U (t ) in [tk−1 , tk ) is
X ∂ψ(kqij k) 1 · (qi − qj ) = 0, ∂k q k k q ij ij k j∈N (t ) i
i = 1, . . . , N .
X
p˙˜ i = ui − p˙ γ ,
j∈Ni (t )
=−
(29) − (λN (L(t )) + c1 )˜pT p˜ ≤ U˙ (t ) ≤ −λ2 (L(t ))˜pT p˜ . Thus, λ2 (L(t )) characterizes the convergence rate of the system.
|
We now prove part (ii) of Theorem 2. Similar to the proof of part (ii) of Theorem 1, the set
ui = p˙ i = −
N 1 X
j∈Ni (t )
= −˜pT [(L(t ) + c1 H ) ⊗ In ]˜p ≤ 0. (23) Since H = diag (h1 , h2 , . . . , hN ) is positive semi-definite, U˙ (t ) ≤ 0 in [tk−1 , tk ), which implies that U (t ) ≤ U (tk−1 ) < Umax for t ∈ [tk−1 , tk ), k = 1, 2, . . . . (24) Therefore, no edge distance will tend to r for t ∈ [tk−1 , tk ), which implies that no edge will lost at time tk , and U (tk ) ≤ Umax . Since G(0) is connected and no edges in E (0) will be lost, G(t ) will remain to be connected for all t ≥ 0.
X
= −c1
N
i =1
_
ui
i =1
hi (pi − pγ ) = 0. (28) N i=1 Even if only one agent in the group is not the informed agent, it is difficult to explicitly estimate the convergence rate of the COM. However, one can get a bound for the convergence rate of the system. From (23), the decay rate of the total energy U is influenced by the topology of the interaction network and the coefficient c1 . From (26), U˙ = 0 if and only if p1 = · · · = pN = pγ . Similar to (16), one has p˙¯ =
p˜ Ti p˙˜ i
_ n_ o Ω = q˜ ∈ Dg , p˜ ∈ RNn U ( q˜ , p˜ ) ≤ Umax ,
317
(27)
Similar to the proof of part (ii) of Theorem 1, we have q1 = q2 = · · · = qN .
U˙ (t ) =
N 1X X
2 i=1 j∈N (t ) i
˙ qij k) + ψ(k˜
N X
p˜ Ti p˙˜ i
i=1
= −˜p [(L(t ) + c1 IN ) ⊗ In ]˜p ≤ 0, T
(32)
318
H. Su et al. / Systems & Control Letters 59 (2010) 313–322
which implies that U (t ) ≤ U (tk−1 ) < Umax
for t ∈ [tk−1 , tk ), k = 1, 2, . . . .
(33)
Therefore, no edge-distances will tend to r for t ∈ [tk−1 , tk ), which implies that no edges will be lost at time tk , and U (tk ) ≤ Umax . Since G(0) is connected and no edges in E (0) will be lost, G(t ) will be connected for all t ≥ 0. We now prove part (ii) of Theorem 3. Similar to the proof of part (ii) of Theorem 2, the set
_ o n_ Ω = q˜ ∈ Dg , p˜ ∈ RNn U ( q˜ , p˜ ) ≤ Umax
(34)
is compact. From LaSalle’s invariance principle [40], U˙ = −˜pT [(L(t ) + c1 IN ) ⊗ In ]˜p
= −˜pT (L(t ) ⊗ In )˜p − c1 p˜ T p˜ = 0,
(35)
which implies that p˜ 1 = · · · = p˜ N = 0, i.e. p1 = · · · = pN = pγ . This also implies that, in steady state, ui = p˙ i = p˙ γ . From (30),
X
ui − p˙ γ = −
∇qi ψ(kqij k)
j∈Ni (t )
=−
X ∂ψ(kqij k) 1 · (qi − qj ) = 0, ∂kqij k kqij k j∈N (t ) i
i = 1, . . . , N .
(36)
Similar to the proof of part (ii) of Theorem 2, q1 = q2 = · · · = qN . Finally, we prove part (iii) of Theorem 3. Since ψ(kqij k) = ψ(kqji k) and wij = wji , from the control protocol (30), p˙¯ − p˙ γ = −
N 1 X X
N i =1
∇qi ψ(kqij k) −
j∈Ni (t )
X
wij (pi − pj )
j∈Ni (t )
− c1 (pi − pγ ) = −c1 p¯ + c1 pγ .
(37)
By solving (37), p¯ = pγ + (¯p(0) − pγ (0))e−c1 t .
(38)
Thus, it follows that if p¯ (0) = pγ (0) then p¯ (t ) = pγ for all t ≥ 0; otherwise, p¯ (t ) will exponentially converge to the desired velocity with a time constant, c1 s. 5. Simulation study 5.1. Rendezvous without a virtual leader We consider 10 agents moving in a 2-dimensional plane under the influence of the control protocol (4). The influencing/sensing radius is chosen as r = 4. Initial positions and initial velocities of the 10 agents are chosen randomly from the boxes [0, 8] × [0, 8] and [0, 4] × [0, 4], respectively. Here, the initial interaction network is set to be connected. Potential function (6) is selected for the protocol (4). If we choose ε0 = ε , then Qmax =
≤ =
N 1X
pTi (0)pi (0) +
2 i=1 N (N − 1) 2
2
ψ(kr − εk) +
N (N − 1) (r − ε)2 2
N (N − 1)
ε+
r2 Qmax
+
N 2
N 2
ψ(kr − εk)
kqij k2 r − kqij k +
1 4500
,
kqij k ∈ [0, 4].
(40)
In Fig. 3, potential function (40) was selected for the algorithm (4) and ε0 = 0.01. Fig. 3(a) shows the initial states of the agents, where the solid lines represent the neighboring relations, and the solid lines with arrows represent the directions of velocities; Fig. 3(b) depicts the motion trajectories of all agents from t = 0 to 7 s; Fig. 3(c–d) and (e–f) show the convergence of positions for ε = 0.01 and ε = 1.0, respectively, from which one can see that all agents eventually achieve the same position; Fig. 3(g–h) and (i–j) show the convergence of velocities for ε = 0.01 and ε = 1.0, respectively. If we choose the percentage set for the settling time as cts = 0.04, then the settling times in Fig. 3(g–j) are 2.7 s, 3.2 s, 5.6 s and 6.6 s, respectively. We find that the smaller the value of the constant ε , the larger the maximum overshoot and the smaller the settling time of the velocity. Therefore, there exists a fundamental trade-off between maximum overshoot and settling time for choosing the parameter ε . This is due to the fact that a smaller ε corresponds to a larger potential force, but also leads to earlier adding of edges. In the following, we present simulation results for the situation without adding new edges to the network (ε = 4). In the simulation, potential function (40) was selected for the algorithm (4), and the initial states of the agents and all parameters were chosen the same as those in Fig. 3. Fig. 4(a) and (b) show the convergence of positions over the x-axis and the y-axis, respectively; Fig. 4(c) and (d) show the convergence of velocities over the x-axis and the yaxis, respectively. One can see that all agents eventually reach the same position and velocity. This is because that initial network is connected and no existing edges are lost. Compared to both cases in Fig. 3, the maximum overshoot becomes smaller but the settling time becomes larger. This is due to the fact that no new edges are added to the network. 5.2. Rendezvous with a virtual leader Simulations on protocol (17) and on protocol (30) are performed with 50 agents moving in a 2-dimensional plane. In both simulations, r = 4 and ε = ε0 = 0.5. Initial positions and velocities of the 50 agents are chosen randomly from the boxes [0, 15] × [0, 15] and [0, 4] × [0, 4], respectively. Once again, the initial interaction network is set to be connected. The desired velocity of the virtual leader is chosen as pγ = [3, 3], and potential function (6) is selected for both protocol (17) and protocol (30). We have N (N − 1) ψ(kr − εk) p˜ Ti (0)˜pi (0) + 2 i=1 2 N (N − 1) N ≤ ψ(kr − εk) + max p˜ Ti (0)˜pi (0) 2 2
Umax =
N 1X
N (N − 1) (r − ε)2 2
ε+
r2
+
Umax
N 2
max p˜ Ti (0)˜pi (0) .
(41)
From (41), Umax ≤ 30205.7. If we choose Uˆ = 32000, then the potential function (6) can be explicitly described as
ψ(kqij k) =
=
max pTi (0)pi (0)
max pTi (0)pi (0) .
and velocities, the simulations are performed with ε = 0.01 and ε = 1.0, respectively. From (39), we have Qmax ≤ 70122.2 for ε = 0.01 and Qmax ≤ 471.7 for ε = 1.0. If we choose Qˆ = 72000, then the potential function (6) can be explicitly described as
(39)
Note that we can evaluate Qmax by using an estimate of the bound of the initial velocities of all agents. Under the same initial positions
ψ(kqij k) =
kqij k2 r − kqij k +
1 2000
,
kqij k ∈ [0, 4].
(42)
In Fig. 5, the informed agent is chosen randomly from the group and marked with a star; Fig. 5(a) shows the initial states of
H. Su et al. / Systems & Control Letters 59 (2010) 313–322
a
b
Initial state
8
319
c
Paths and final states 20 ε=0.01 ε=1.0
18
7
16
6
Position Convergence for ε=0.01 (x axis)
25
20
14 12
15 qx
4
y
y
5
10 8
3
10
6
2
4 1 0
d
5
2 1
2
3
4
5 x
6
7
8
0
5
10
15
20
0
25
e
2
3
5
6
7
Position Convergence for ε=1.0 (y axis)
f
Position Convergence for ε=1.0 (x axis)
25
4
20 18
16
16
20
14
14
12
15
12 qy
10
qx
qy
1
t
18
8
10
10
8
6
6
4
5
4
2 0
0
x
Position Convergence for ε=0.01 (y axis)
20
0
9
2 0
1
2
3
4
5
6
0
7
0
1
2
3
4
g
5
0
7
6
Velocity Convergence for ε=0.01 (x axis)
h
60
1
2
3
60 40
2.4
–80
2.2
–100
2
–120
py
px
2.6
–60
0
0.5
6
7
20
3 2.8
–40
5
Velocity Convergence for ε=0.01 (y axis)
80
20
–20
4 t
40
0
0
2.8
–20
2.6 2.4
–40
2.2 –60 1
2
1
3 1.5
4
5
2
6 2.5
2
7 –80
3
0
0.5
1
1
2 1.5
3
4 2
5 6 2.5
7 3
t
t
i
0
t
t
Velocity Convergence for ε=1.0 (x axis)
j
6
Velocity Convergence for ε=1.0 (y axis)
8 7
5
6 4
5 4 py
px
3 2
3 2
1
1
0
0 –1 –2
0
–1 1
2
3
4
5
6
7
–2
0
1
2
3
4
5
6
7
t
t
Fig. 3. Rendezvous of 10 agents under algorithm (4).
the agents; Fig. 5(b) depicts the motion trajectories of all agents from t = 0 to 35 s; Fig. 5(c) and (d) show the convergence of positions over the x-axis and the y-axis, respectively; Fig. 5(e) and (f) show the convergence of velocities over the x-axis and the y-axis, respectively. One can see that all agents eventually reach
the same position and velocity. The final common velocity is equal to the desired velocity pγ . Fig. 6(a) shows the initial states of the agents; Fig. 6(b) depicts the motion trajectories of all agents from t = 0 to 5 s; Fig. 6(c) and (d) show the convergence of positions over the x-axis and
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H. Su et al. / Systems & Control Letters 59 (2010) 313–322
a
Position Convergence(x axis)
b
45 40
35
35
30
30
25 qy
25 qx
Position Convergence(y axis) 40
20
20 15
15
10
10
5
5 0
0
5
10
0
15
0
5
10
c 4.5
15
t
t Velocity Convergence(x axis)
d
4
Velocity Convergence(y axis) 7 6
3.5
5
3
4
px
py
2.5 3
2 2
1.5
1
1 0.5
0
0
–1
0
5
10
t
15
0
5
10
15
t
Fig. 4. Rendezvous of 10 agents without adding new edges under algorithm (4).
a
b 100
Initial state
16 14 12
c
Paths and final states
90
80
80
70
70
60
60
50
50
qx
8
y
y
10
40
6
2 0
20
20
10
10
0 0
5
10
15
0
10
20
30
40
e
Position Convergence(y axis) 100
50
60
70
80
0
90
f
Velocity Convergence(x axis) 15
90
prx=3
10
15
20
t
25
30
35
Velocity Convergence(y axis) 20 15
pry=3
10
70
px
50
py
5
60 qy
5
10
80
3 0
40 30
5 3 0 –5
20
–5 –10
10 0
0
x
x
d
40 30
30 4
Position Convergence(x axis) 90
0
5
10
15
20
25
30
35
–10
0
5
10
t
15
20
25
30
35
t
–15
0
5
10
15
20
25
30
35
t
Fig. 5. Rendezvous of 50 agents under algorithm (17).
the y-axis, respectively; Fig. 6(e) and (f) show the convergence of velocities over the x-axis and the y-axis, respectively. One can see that all agents eventually reach the same position and velocity. The
final common velocity is equal to the desired velocity pγ . One can clearly see that the group using control protocol (17) needs a longer convergent time than that using control protocol (30).
H. Su et al. / Systems & Control Letters 59 (2010) 313–322
a 16
Initial state
b
c
Paths and final states
25
321 Position Convergence(x axis) 25
14 12
8
20
15
15 qx
y
y
10
20
6
10
10
5
5
4 2 0
0 0
5
10
15
0
5
10
d
15
20
0
25
0
1
2
x
x
e 15
Position Convergence(y axis) 25
f
Velocity Convergence(x axis)
px
3 py
qy
pry=3
10
5
0
10
5
15 prx=3
15
4
Velocity Convergence(y axis)
20
10
20
3 t
–5
5 3 0 –5
5 –10 0
0
1
2
t
3
4
5
–15
–10
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
t
–15
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
t
Fig. 6. Rendezvous of 50 agents under algorithm (30).
6. Conclusions In this paper, we have investigated the rendezvous problem with connectivity preservation for double-integrator dynamics employing hysteresis for adding new links and using the potential function method. The hysteresis idea is adopted from [36,37]; yet, we found that there is a trade-off between the maximum overshoot and the settling time of velocity convergence for choosing the hysteresis parameter. We have constructed a class of bounded potential functions to guarantee the existing links not to be lost, which is in some sense more practical than the two specific unbounded potential functions used in [36,37]. Moreover, we have investigated the coordinated control with a virtual leader and show that all agents can asymptotically attain the desired velocity even if only one agent in the team has information about the virtual leader. Future work will consider the situation where parts of agents are informed and the velocity of the virtual leader varies with time, and the effects of time delay and disturbance on these new algorithms. References [1] T. Vicsek, A. Cziro’ok, E. Ben-Jacob, O. Cohen, I. Shochet, Novel type of phase transition in a system of self-deriven particles, Physical Review Letters 75 (1995) 1226–1229. [2] C.W. Reynolds, Flocks, herds, and schools: a distributed behavioral model, in: ACM SIGGRAPH 87 Conference Proceedings, Computer Graphics 21 (1987) 25–34. [3] A. Jadbabaie, J. Lin, A.S. Morse, Coordination of groups of mobile agents using nearest neighbor rules, IEEE Transactions on Automatic Control 48 (2003) 988–1001. [4] R. Olfati-Saber, R.M. Murray, Consensus problems in networks of agents with switching topology and time-delays, IEEE Transactions on Automatic Control 49 (2004) 1520–1533. [5] W. Ren, R.W. Beard, Consensus seeking in multi-agent systems under dynamically changing interaction topologies, IEEE Transactions on Automatic Control 50 (2005) 655–661.
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Contents lists available at ScienceDirect
Systems & Control Letters journal homepage: www.elsevier.com/locate/sysconle
Rendezvous of multiple mobile agents with preserved network connectivityI Housheng Su a,b,c,∗ , Xiaofan Wang b , Guanrong Chen c a
Department of Control Science and Engineering, Huazhong University of Science and Technology, Wuhan 430074, China
b
Department of Automation, Shanghai Jiao Tong University, Shanghai 200240, China
c
Department of Electronic Engineering, City University of Hong Kong, Hong Kong, China
article
info
Article history: Received 16 April 2008 Received in revised form 30 July 2009 Accepted 15 March 2010 Available online 24 April 2010 Keywords: Distributed control Nonlinear system Rendezvous Multi-agent Virtual leader
abstract In coordinative control of a network of multi-agent systems, to guarantee the stability of the coordinated motion, a basic assumption typically is that the underlying topology of the network can maintain its connectivity frequently enough during the motion evolution. However, for a given set of initial conditions, this assumption is very difficult to satisfy and verify. In particular, the connectivity of the initial network generally cannot guarantee the connectivity of the network throughout the evolution. In this paper, we propose a rendezvous protocol with double-integrator dynamics, which combines the functions of motion control and connectivity preservation. This protocol can enable the group of mobile agents to converge to the same position and move with the same velocity while preserving the connectivity of the whole network during the evolution if the initial network is connected. We find that there is a trade-off between the maximum overshoot and the settling time of the velocity convergence. Furthermore, we investigate the rendezvous protocol with a virtual leader and show that all agents can asymptotically attain a desired velocity even if only one agent in the team has information about the virtual leader. We finally show some numerical simulations to verify and illustrate the theoretical results. © 2010 Elsevier B.V. All rights reserved.
1. Introduction In recent years, multi-agent coordinated control problems have attracted much attention among researchers studying biology, physics, computer science and control engineering [1–3]. Many versions of distributed coordinated control protocols have been proposed, including consensus [3–11], swarming [12–17], flocking [18–23], and rendezvous [24–29]. In particular, rendezvous is the motion in which a large number of agents, using only limited environmental information and simple rules, arrive at a location simultaneously, which are characterized by distributed control, local interactions and self-organization. Understanding the mechanisms of rendezvous behaviors can help comprehending some natural behaviors and developing many artificial autonomous systems such as schooling of fish for avoiding predators and finding food, and mobile robots for rescue, surveillance, fire fighting and disaster control [30].
I This work was partly supported by the NSF of PR China under Grant Nos. 60674045 and 60731160629, and the Major State Basic Research Development Program of China (973 Program) (No. 2010CB731400), as well as the NSFC-HKRGC Joint Research Scheme under Grant N_CityU 107/07. ∗ Corresponding author at: Department of Control Science and Engineering, Huazhong University of Science and Technology, Wuhan 430074, China. Tel.: +86 27 87540210; fax: +86 27 87540210. E-mail addresses: [email protected] (H. Su), [email protected] (X. Wang), [email protected] (G. Chen).
0167-6911/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.sysconle.2010.03.006
Subjected to limited sensing and communication capabilities of agents, the interaction topology among agents may change over time. A basic assumption made in most previous works on stability analysis of collective dynamics is that the underlying topology can remain connected frequently enough during the motion evolution. However, for a given set of initial states and parameters, it is very difficult or even impossible to satisfy and verify this assumption in practice. In fact, the connectivity of the initial network generally cannot guarantee the connectivity of the whole network throughout a long process of motion. For example, the consensus algorithm proposed in [3], the flocking algorithms in [18,19], and the rendezvous algorithm in [29], have been known to possibly result in fragmentation even if the initial network is connected. Recently, some efforts had been made to preserve the connectivity of the networks. For example, Cortes et al. [31] proposed a circumcenter algorithm to avoid the loss of existing connections. Degennaro and Jadbabaie [32] suggested an iterative decentralized algorithm to maximize the smallest nonzero eigenvalue of the Laplacian matrix for the connectivity control. Spanos and Murray [33] used a measure of local connectivity to ensure the global connectivity of a network. Dimarogonas and Kyriakopoulos [34,35] used potential functions to maintain the network connectivity for rendezvous and swarming. Ji and Egerstedt [36] proposed a hysteresis in adding new links and a special potential function to achieve rendezvous while preserving the network connectivity. However, all these works are concerned with agents described
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by single-integrator dynamics. Recently, Zavlanos et al. [37] used the hysteresis and potential function method to study a flocking problem combining with the network connectivity for doubleintegrator dynamics to achieve velocity alignment. In this work, we investigate the rendezvous problem with connectivity preservation for double-integrator dynamics, also employing hysteresis in adding new links and using the potential function method. It should be noted, however, that in [36,37], two specific potential functions are constructed for single-integrator dynamics and double-integrator dynamics, respectively, which tend to infinity as the distance between two already connected agents tends to the sensing radius, which is rather impractical. In this work, we construct a general class of bounded potential functions that can preserve the network connectivity for doubleintegrator dynamics and the aforementioned divergence problem can be avoided. We also investigate the rendezvous protocol with a virtual leader and show that all agents can asymptotically attain a desired velocity even if only one agent in the team has communication with, hence has information about the virtual leader. The remainder of the paper is organized as follows. Section 2 describes the problem to be investigated. A new rendezvous algorithm and its stability analysis are established in Section 3. Section 4 discusses the case with a virtual leader. Section 5 presents some illustrative simulation results. Finally, Section 6 draws some conclusions. 2. Problem statement Consider N agents moving in an n-dimensional Euclidean space. The motion of each agent is described by a double-integrator equation of the form q˙ i = pi , p˙ i = ui ,
i = 1, . . . , N ,
(1)
where qi ∈ Rn is the position vector of agent i, pi ∈ Rn is its velocity vector, and ui ∈ Rn is the (force) control input acting on agent i, i = 1, . . . , N. Suppose that all agents have the same influencing/sensing radius r. Let ε ∈ (0, r ] and ε0 ∈ (0, r ) be the given constants and ε0 ≤ ε . Call G(t ) = (V , E (t )) a undirected dynamic graph consisting of a set of vertices V = {1, 2, . . . , N } indexed by the set of agents and a time-varying set of links E (t ) = {(i, j)|i, j ∈ V } such that (i) initial links are generated by E (0) = { (i, j)| kqi (0) − qj (0)k < r − ε0 , i , j ∈ V } ; (ii) if (i, j) 6∈ E (t − ) and kqi (t ) − qj (t )k < r − ε , then (i, j) is a new link being added to E (t ); (iii) if kqi (t ) − qj (t )k ≥ r, then (i, j) 6∈ E (t ). Throughout this paper, k·k denote the Euclidean norm. One can use a symmetric indicator function σ (i, j) ∈ {0, 1} to describe whether there is a link between agent i and agent j at time t, which is defined as
0, if ((σ (i, j)[t − ] = 0) ∩(kqi (t ) − qj (t )k ≥ r − ε)) ∪((σ (i, j)[t − ] = 1) ∩ (kqi (t ) − qj (t )k ≥ r )), σ (i, j)[t ] = 1, if ((σ (i, j)[t − ] = 1) ∩(kqi (t ) − −qj (t )k < r )) ∪((σ (i, j)[t ] = 0) ∩ (kqi (t ) − qj (t )k < r − ε)). One can see from Fig. 1 that there is a hysteresis in the indicator function for adding new links to the graph [36,37]. When ε = r, no new links can be added to the graph. Moreover, one will see that the selection of the parameter ε (ε0 ) is a trade-off between the maximum overshoot and the settling time of velocity convergence (the feasible energy and the radius of the initial network):
2 1.8 1.6 1.4 1.2 σ(i,j)
1 0.8 0.6 0.4 0.2 0
0
1
2
3 r–ε
||qij||
4 r
5
6
Fig. 1. Indicator function σ (i, j).
the smaller the value of ε (ε0 ), the larger the maximum overshoot and the smaller the settling time (the larger the magnitude of the potential function and the radius of the initial network). Our objective is to make all agents converge to the same position and move with the same velocity under the assumption that the initial network is connected. Specifically, we want pi (t ) − pj (t ) → 0 and qi (t ) − qj (t ) → 0 for all i, j ∈ V . To achieve this goal, we choose the control law ui for agent i in (1) to be ui = αi + βi ,
(2)
where αi is the gradient-based term, which is designed to enforce the position of each agent to a common value and also to maintain the connectivity of the network, and βi is the consensus term, which regulates the velocity of each agent to a common value. In some cases, the regulation of agents is for certain purposes, e.g., achieving a desired common velocity or arriving at a desired destination. In the situation where there is a virtual leader, we modify the control law (2) to ui = αi + βi + γi ,
(3)
where γi is the navigational feedback term, which drives agent i to track the virtual leader. 3. Fundamental rendezvous algorithm 3.1. Algorithm design and main results We now specify an explicit control input in Eq. (2) as ui = −
X
∇qi ψ(kqij k) −
j∈Ni (t )
|
X
wij (pi − pj ) .
(4)
j∈Ni (t )
{z αi
}
|
{z βi
}
Here, qij = qi − qj , the constant wij = wji > 0 is bounded for all i, j ∈ V , Ni (t ) is the neighborhood region of agent i at time t defined as Ni (t ) = {j|σ (i, j)[t ] = 1, j 6= i, j = 1, . . . , N }.
(5)
The nonnegative potential ψ(kqij k) is defined to be a function of the distance kqij k between agent i and agent j, differentiable with respect to kqij k ∈ [0, r ] such that ∂ψ(kqij k) ∂kqij k
> 0 for kqij k ∈ (0, r ); ∂ψ(kqij k) 1 (ii) limkqij k→0 is nonnegative and bounded; · ∂kqij k kqij k PN T (iii) ψ(r ) = Qˆ ∈ [Qmax , +∞), where Qmax , 12 i=1 pi (0)pi (0) + N (N −1) ψ(k r − ε k) . 0 2 (i)
Condition (i) illustrates that the potential between the two agents is an increasing function of their distance, which makes two agents
H. Su et al. / Systems & Control Letters 59 (2010) 313–322
of L(t ) as λ1 (L(t )) ≤ · · · ≤ λN (L(t )). Then, λ1 (L(t )) = 0 and 1 = [1, 1, . . . , 1]T ∈ RN is its corresponding eigenvector. Moreover, if G(t ) is a connected graph, then λ2 (L(t )) > 0 [38]. The _ corresponding n-dimensional graph Laplacian is defined as L (t ) = L(t ) ⊗ In , where In is the identity matrix of order n and ⊗ stands for the Kronecker product. This multi-dimensional Laplacian satisfies the following sum-of-squares property [19]:
200 180 160 140 120 ψij 100 80
1 X
_
z T L (t )z =
60
20 0
0.5
1
1.5 ||qij||
2
2.5
3
3.5
where z = [z1T , z2T , . . . , zNT ]T ∈ RNn and zi ∈ Rn for all i. Denote the position and velocity of the center of mass (COM) of all agents in the group by
4 r
N P
Fig. 2. The potential function (6).
attract each other; Condition (ii) requires that the magnitude of gradient between two agents is the same or a higher-order term of their distance when the two agents converge to the same position; Condition (iii) states that the potential between two agents will be sufficiently large when the distance between the two agents reaches the sensing radius, which guarantees all existing edges not be lost. One example of such a potential function is the following (see Fig. 2):
ψ(kqij k) =
kqij k2 2
r − kqij k + rˆ Q
,
kqij k ∈ [0, r ].
(6)
ψ(kqij k) =
kqij k
r − kqij k
,
kqij k ∈ [0, r ),
(7)
and in [37],
ψ(kqij k) =
1
kqij k2
+
1 r 2 − kqij k2
,
kqij k ∈ (0, r ).
Q =
2 i=1
j∈Ni (t )
ψ(kqij k) +
pTi pi
,
(9)
with initial energy Q0 = Q (q(0), p(0)). Clearly, Q is a positive semi-definite function. The adjacent matrix A(t ) = (aij (t )) of the above graph G(t ) is defined as aij (t ) =
wij , 0,
pi
i=1
, p¯ = . N N The main results of this section are given in the following theorem: q¯ =
Theorem 1. Consider a system of N mobile agents with dynamics (1), each steered by protocol (4). Suppose that the initial network G(0) is connected and the initial energy Q0 is finite. Then, the following hold: (i) G(t ) is connected for all t ≥ 0; (ii) All agents asymptotically converge to the same position and move with the same velocity; (iii) The velocity p¯ (t ) of the COM is invariant for all t ≥ 0.
3.2.1. Proof of part (i) Assume that G(t ) switches at time tk (k = 1, 2, . . .), otherwise G(t ) is a fixed graph in each time interval [tk−1 , tk ). Note that Q0 is finite and the time derivative of Q (t ) in [t0 , t1 ) is
(8)
! X
i=1
Q˙ (t ) =
Potential function (7) satisfies conditions (i) and (ii) but does not satisfy condition (iii) of our general definition of the potential function, and potential function (8) does not satisfy any of the three conditions. In particular, potential functions (7) and (8) tend to infinity when the distance between two agents i and j tends to r, which may not be practical since it will require extremely large control input. If the hysteresis parameter ε = 0, from simple calculation, Qˆ = +∞ in the potential function (6). Therefore, the potential function (6) is the same as the potential function (7). Note that, here as well as in [36,37], without the hysteresis mechanism (see Fig. 1), no new links will be added to the network. This is because the potential force is so big as to prevent the addition of new links when the distance between two agents i and j tends to r. The sum of the total artificial potential energy and the total kinetic energy is defined as follows: N 1X
N P
qi
3.2. Proof of the theorem
Note that in [36,37], only two specific potential functions are proposed: in [36], 2
aij (t )kzj − zi k2 ,
2 (i,j)∈E
40
0
315
(i, j) ∈ E (t ), otherwise.
The corresponding Laplacian is L(t ) = ∆(A(t )) − A(t ), where the degree matrix ∆(A(t )) is a diagonal matrix with the ith PN diagonal element equal to j=1, j6=i aij (t ). Denote the eigenvalues
=
N 1 XX
2 i=1 j∈N i N X
pTi
X
N X
pTi p˙ i
i=1
∇qi ψ(kqij k)
j∈Ni
i =1
+
˙ qij k) + ψ(k
N X
! pTi
i =1
−
X
∇qi ψ(kqij k) −
X
wij (pi − pj )
j∈Ni
j∈Ni
= −pT [L(t ) ⊗ In ]p. Since L(t ) is positive semi-definite [39], Q˙ (t ) ≤ 0 in [t0 , t1 ), which implies that Q (t ) ≤ Q0 < Qmax
for t ∈ [t0 , t1 ).
From the definition of the potential function, one has ψ(r ) > Qmax > Q0 . Therefore, no edge-distances will tend to r for t ∈ [t0 , t1 ), which implies that no existing edges will be lost at time t1 . Hence, new edges must be added in the interaction network at the switching time t1 . Without loss of generality, assume that there are m1 new links being added to the interaction network (N −1)(N −2) at time t1 . Clearly, 0 < m1 ≤ M and M , , thus, 2 Q (t1 ) = Q0 + m1 ψ(kr − εk) ≤ Qmax . Similar to the aforementioned analysis, the time derivative of Q (t ) in [tk−1 , tk ) is Q˙ (t ) = −pT [L(t ) ⊗ In ]p ≤ 0,
(10)
which implies that Q (t ) ≤ Q (tk−1 ) < Qmax
for t ∈ [tk−1 , tk ), k = 1, 2, . . . .
(11)
Therefore, no edge-distances will tend to r for t ∈ [tk−1 , tk ), which implies that no edges will be lost at time tk and Q (tk ) ≤ Qmax . Since G(0) is connected and no edges in E (0) will be lost, G(t ) will be connected for all t ≥ 0.
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H. Su et al. / Systems & Control Letters 59 (2010) 313–322
3.2.2. Proof of parts (ii) and (iii) Assume that there are mk new links being added to the interaction network at time tk . Clearly, 0 < mk ≤ M. From (9) and (11), one has Q (tk ) ≤ Q0 + (m1 + · · · + mk )ψ(kr − εk) ≤ Qmax . Since there are at most M new links that can be added to G(t ), one has k ≤ M and Q (t ) ≤ Qmax for all t ≥ 0. Therefore, the number of switching times k of the system (1) is finite, which implies the interaction network G(t ) eventually becomes fixed. Thus, the rest discussions can be restricted on the time interval (tk , ∞). Note that the distance of edge is not longer than ψ −1 (Qmax ). Hence, the set
Ω = q_ ∈ Dg , p ∈ RNn |Q ( q_, p) ≤ Qmax , _
(12) N2n
| kqij k ∈ [0, is positively invariant, where Dg = { q ∈ R ψ −1 (Qmax )], ∀ (i, j) ∈ E (t )}, q_ = [qT11 , . . . , qT1N , . . . , qTN1 , . . . , qTNN ]T and p = [pT1 , pT2 , . . . , pTN ]T . Since G(t ) is connected for all t ≥ 0, one has kqij k < (N − 1)r for all i and j. Since Q (t ) ≤ Qmax , one has pTi pi ≤ 2Qmax , and √ thus kpi k ≤ 2Qmax . Therefore, the set Ω satisfying Q (t ) ≤ Qmax is closed and bounded, hence compact. Note that system (1) with control input (4) is an autonomous system, at least on the concerned time interval (tk , ∞). Therefore, the LaSalle’s invariance principle [40] can be applied to infer that if the initial conditions of the system lies in Ω , its trajectories will converge to the largest invariant set inside the region S = { q_ ∈ Dg , p ∈ RNn |Q˙ = 0}. From (10), one has Q˙ = −pT (L(t ) ⊗ In )p =
1 X 2 (i,j)∈E
=−
ui
N
! X
N i =1
j∈Ni (t )
∇qi ψ(kqij k) +
X
X
wij (pi − pj ) = 0, (14)
j∈Ni (t )
i
i = 1, . . . , N .
(15)
Rewrite (15) in a matrix form as
where qT = [qT1 , . . . , qTN ] and Lˆ (t ) = [ˆlij ] is a matrix with
ˆlij = ∂ψ(kqij k) · 1 , ∂kqij k kqij k
for i 6= j.
∂ψ(kqij k) ∂kqij k
·
1
kqij k
In this subsection, we present some qualitative analysis of the convergence rate of the system. From (10), the decay rate of the total energy Q is influenced by the topology of the interaction network. From (13), Q˙ = 0 if and only if p1 = · · · = pN . Since the interaction network is connected all the time, from the matrix theory [39] and (10), one has
− λN (L(t ))pT p ≤ Q˙ (t ) ≤ −λ2 (L(t ))pT p.
(16)
Thus, λ2 (L(t )) characterizes the convergence rate of the system. Therefore, we conclude that the convergence rate of the system relies on the smallest nonzero eigenvalue of the Laplacian of the interaction network. For an undirected graph, adding edges increases the smallest nonzero eigenvalue λ2 (L(t )), and λ2 (L(t )) attains its maximum value N when the topology of the network is fully connected [41]. Thus, λ2 (L(0)) ≤ λ2 (L(t )) ≤ N, showing that λ2 (L(0)) characterizes the bound of the convergence rate of the system.
In nature, few individuals have some pertinent information, such as knowledge of the location of a food source or of a migration route [42]. Here, we assume that only one agent is informed with information about the virtual leader. Certainly, the result can be generalized to the case with multiple informed agents, which may lead to faster convergence. Suppose that velocity of the virtual leader can be received by the informed agent through some designed mechanism, such as broadcasting. The control input to agent i is designed as ui = −
X
∇qi ψ(kqij k) −
j∈Ni (t )
X
wij (pi − pj )
j∈Ni (t )
|
{z
}
αi
− hi c1 (pi − pγ ), | {z }
|
{z
}
βi
c1 > 0,
(17)
where pγ is a desired constant velocity of the virtual leader. If agent i is the informed agent, then hi = 1; otherwise, hi = 0. Without loss of generality, assume that the first agent is the informed agent, that is, hi = 1 for i = 1 and hi = 0 for all the others. Define the sum of the total artificial potential energy and the total relative kinetic energy between all agents and the virtual leader as follows:
−(Lˆ (t ) ⊗ In )q = 0,
and
γi
j∈Ni (t )
N X ∂ψ(kqij k) 1 · ∂kqij k kqij k j=1, j6=i
(Qmax )] and limkqij k→0
3.3. Convergence rate analysis
∇qi ψ(kqij k)
X ∂ψ(kqij k) 1 =− · (qi − qj ) = 0, ∂kqij k kqij k j∈N (t )
ˆlii = −
is positive and uniformly bou-
is nonnegative and bounded. Since the dynamic network is connected all the time, from the property of the Laplacian matrix Lˆ (t ) ⊗ In [38], q converges asymptotically to span{1}, which implies that q1 = q2 = · · · = qN .
which implies that the velocity of the COM is invariant for all t ≥ 0. In steady state, p1 = · · · = pN = p¯ , so that p˙ i = p˙¯ = 0, ∀i ∈ V . From (4), one has ui = p˙ i = −
1
kqij k
4.1. Main results
i =1
N 1 X
nded for kqij k ∈ (0, ψ
−1
·
(13)
all agents asymptotically move with the same velocity. Since ψ(kqij k) = ψ(kqji k) and wij = wji , from the control input (4), one has
u¯ = p˙¯ =
∂ψ(kqij k) ∂kqij k
4. Rendezvous with a virtual leader
wij kpj − pi k2 .
Therefore, Q˙ = 0 if and only if p1 = · · · = pN , which implies that
N P
From the definition,
U =
!
N 1X
X
2 i=1
j∈Ni (t )
ψ(kqij k) + (pi − pγ ) (pi − pγ ) . T
(18)
Clearly, U is a positive semi-definite function. Here, the potential function ψ should satisfy conditions (i)–(iii) defined in the last section with Qmax in condition (iii) being replaced by Umax , PN N (N −1) 1 T ψ(kr − ε0 k). i=1 (pi (0) − pγ (0)) (pi (0) − pγ (0)) + 2 2 Our main results on tracking the virtual leader can be stated in the following theorem.
H. Su et al. / Systems & Control Letters 59 (2010) 313–322
Theorem 2. Consider a system of N mobile agents with dynamics (1), each steered by protocol (17). Suppose that the initial network G(0) is connected and the initial energy U0 is finite. Then, the following hold: (i) G(t ) is connected for all t ≥ 0. (ii) All agents asymptotically converge to the same position and move with the desired velocity pγ .
4.3. Extensions and discussions In general, all agents asymptotically converge to the desired velocity pγ , which implies that the velocity p¯ (t ) of the COM asymptotically converge to the desired velocity pγ . From the control input (17), N P
4.2. Proof of the theorem We first prove part (i) of Theorem 2. Denote the position difference vector and the velocity difference vector between agent i and the virtual leader by q˜ i = qi − pγ t and p˜ i = pi − pγ , respectively. Then, q˙˜ i = p˜ i , p˙˜ i = ui ,
(19)
i = 1, . . . , N .
By the definition of ψ(kqij k), where qij = qi − qj , we have, for q˜ ij = q˜ i − q˜ j ,
ψ(kqij k) = ψ(k˜qij k).
(20)
Thus, the control input (17) for agent i can be rewritten as
X
ui = −
∇q˜ i ψ(k˜qij k) −
j∈Ni (t )
X
wij (˜pi − p˜ j ) − hi c1 p˜ i ,
(21)
j∈Ni (t )
and the positive semi-definite energy function (18) can be rewritten as N 1X
(ψ(k˜qij k) + p˜ Ti p˜ i ). (22) 2 i=1 Similar to the proof of part (i) of Theorem 1, we can show that the time derivative of U (t ) in [tk−1 , tk ) is U =
U˙ (t ) =
N 1X X
2 i=1 j∈N (t ) i
˙ qij k) + ψ(k˜
N X
(25)
2
is compact, where Dg = { q˜ ∈ RN n |k˜qij k ∈ [0, ψ −1 (Umax )], _
∀ (i, j) ∈ E (t )}, q˜ = [˜qT11 , . . . , q˜ T1N , . . . , q˜ TN1 , . . . , q˜ TNN ]T and p˜ = [˜pT1 , p˜ T2 , . . . , p˜ TN ]T . From LaSalle’s invariance principle [40], we have U˙ = −˜pT [(L(t ) + c1 H ) ⊗ In ]˜p = −˜pT (L(t ) ⊗ In )˜p − c1 p˜ T (H ⊗ In )˜p = 0, (26) which implies that p˜ 1 = · · · = p˜ N and p˜ 1 = 0, i.e. p1 = · · · = pN = pγ . This also implies that, in steady state, ui = p˙ i = p˙ γ = 0. Thus, from (17),
∇qi ψ(kqij k)
Therefore, we conclude that the convergence rate of the system relies on the smallest nonzero eigenvalue of the Laplacian of the interaction network. Thus, λ2 (L(0)) ≤ λ2 (L(t )) ≤ N. When all agents have reached the common desired velocity, the velocity of COM must reach the desired velocity. Then, the convergence rate of the system is not faster than the convergence rate of COM. Therefore, λ2 (L(0)) characterizes the bound of the convergence rate of COM. In the situation where each agent is an informed agent and the velocity pγ (t ) of the virtual leader varies with time, we propose the following protocol: ui = −
∇qi ψ(kqij k) −
X
wij (pi − pj )
j∈Ni (t )
{z αi
}
|
{z βi
+ p˙ γ − c1 (pi − pγ ) . | {z }
} (30)
γi
In practical implementation, the derivative of the velocity of the virtual leader can be calculated by numerical differentiation [9]. The results on tracking the virtual leader with a varying velocity can then be stated in the following theorem. Theorem 3. Consider a system of N mobile agents with dynamics (1), each steered by protocol (30). Suppose that the initial network G(0) is connected and the initial energy U0 is finite. Then, the following hold: (i) G(t ) is connected for all t ≥ 0. (ii) All agents asymptotically converge to the same position and move with the desired velocity pγ . (iii) If the initial velocity of the COM, p¯ (0), is equal to the initial velocity of virtual leader, pγ (0), then p¯ (t ) is equal to the desired velocity pγ (t ) for all t ≥ 0; otherwise p¯ (t ) will exponentially converge to the desired velocity pγ (t ) with a time constant, c1 s. Proof. Denote the position difference vector and the velocity difference vector between agent i and the virtual leader as q˜ i = qi − pγ t and p˜ i = pi − pγ , respectively. Then, q˙˜ i = p˜ i , i = 1, . . . , N .
(31)
Similar to the proof of part (i) of Theorem 2, the time derivative of U (t ) in [tk−1 , tk ) is
X ∂ψ(kqij k) 1 · (qi − qj ) = 0, ∂k q k k q ij ij k j∈N (t ) i
i = 1, . . . , N .
X
p˙˜ i = ui − p˙ γ ,
j∈Ni (t )
=−
(29) − (λN (L(t )) + c1 )˜pT p˜ ≤ U˙ (t ) ≤ −λ2 (L(t ))˜pT p˜ . Thus, λ2 (L(t )) characterizes the convergence rate of the system.
|
We now prove part (ii) of Theorem 2. Similar to the proof of part (ii) of Theorem 1, the set
ui = p˙ i = −
N 1 X
j∈Ni (t )
= −˜pT [(L(t ) + c1 H ) ⊗ In ]˜p ≤ 0. (23) Since H = diag (h1 , h2 , . . . , hN ) is positive semi-definite, U˙ (t ) ≤ 0 in [tk−1 , tk ), which implies that U (t ) ≤ U (tk−1 ) < Umax for t ∈ [tk−1 , tk ), k = 1, 2, . . . . (24) Therefore, no edge distance will tend to r for t ∈ [tk−1 , tk ), which implies that no edge will lost at time tk , and U (tk ) ≤ Umax . Since G(0) is connected and no edges in E (0) will be lost, G(t ) will remain to be connected for all t ≥ 0.
X
= −c1
N
i =1
_
ui
i =1
hi (pi − pγ ) = 0. (28) N i=1 Even if only one agent in the group is not the informed agent, it is difficult to explicitly estimate the convergence rate of the COM. However, one can get a bound for the convergence rate of the system. From (23), the decay rate of the total energy U is influenced by the topology of the interaction network and the coefficient c1 . From (26), U˙ = 0 if and only if p1 = · · · = pN = pγ . Similar to (16), one has p˙¯ =
p˜ Ti p˙˜ i
_ n_ o Ω = q˜ ∈ Dg , p˜ ∈ RNn U ( q˜ , p˜ ) ≤ Umax ,
317
(27)
Similar to the proof of part (ii) of Theorem 1, we have q1 = q2 = · · · = qN .
U˙ (t ) =
N 1X X
2 i=1 j∈N (t ) i
˙ qij k) + ψ(k˜
N X
p˜ Ti p˙˜ i
i=1
= −˜p [(L(t ) + c1 IN ) ⊗ In ]˜p ≤ 0, T
(32)
318
H. Su et al. / Systems & Control Letters 59 (2010) 313–322
which implies that U (t ) ≤ U (tk−1 ) < Umax
for t ∈ [tk−1 , tk ), k = 1, 2, . . . .
(33)
Therefore, no edge-distances will tend to r for t ∈ [tk−1 , tk ), which implies that no edges will be lost at time tk , and U (tk ) ≤ Umax . Since G(0) is connected and no edges in E (0) will be lost, G(t ) will be connected for all t ≥ 0. We now prove part (ii) of Theorem 3. Similar to the proof of part (ii) of Theorem 2, the set
_ o n_ Ω = q˜ ∈ Dg , p˜ ∈ RNn U ( q˜ , p˜ ) ≤ Umax
(34)
is compact. From LaSalle’s invariance principle [40], U˙ = −˜pT [(L(t ) + c1 IN ) ⊗ In ]˜p
= −˜pT (L(t ) ⊗ In )˜p − c1 p˜ T p˜ = 0,
(35)
which implies that p˜ 1 = · · · = p˜ N = 0, i.e. p1 = · · · = pN = pγ . This also implies that, in steady state, ui = p˙ i = p˙ γ . From (30),
X
ui − p˙ γ = −
∇qi ψ(kqij k)
j∈Ni (t )
=−
X ∂ψ(kqij k) 1 · (qi − qj ) = 0, ∂kqij k kqij k j∈N (t ) i
i = 1, . . . , N .
(36)
Similar to the proof of part (ii) of Theorem 2, q1 = q2 = · · · = qN . Finally, we prove part (iii) of Theorem 3. Since ψ(kqij k) = ψ(kqji k) and wij = wji , from the control protocol (30), p˙¯ − p˙ γ = −
N 1 X X
N i =1
∇qi ψ(kqij k) −
j∈Ni (t )
X
wij (pi − pj )
j∈Ni (t )
− c1 (pi − pγ ) = −c1 p¯ + c1 pγ .
(37)
By solving (37), p¯ = pγ + (¯p(0) − pγ (0))e−c1 t .
(38)
Thus, it follows that if p¯ (0) = pγ (0) then p¯ (t ) = pγ for all t ≥ 0; otherwise, p¯ (t ) will exponentially converge to the desired velocity with a time constant, c1 s. 5. Simulation study 5.1. Rendezvous without a virtual leader We consider 10 agents moving in a 2-dimensional plane under the influence of the control protocol (4). The influencing/sensing radius is chosen as r = 4. Initial positions and initial velocities of the 10 agents are chosen randomly from the boxes [0, 8] × [0, 8] and [0, 4] × [0, 4], respectively. Here, the initial interaction network is set to be connected. Potential function (6) is selected for the protocol (4). If we choose ε0 = ε , then Qmax =
≤ =
N 1X
pTi (0)pi (0) +
2 i=1 N (N − 1) 2
2
ψ(kr − εk) +
N (N − 1) (r − ε)2 2
N (N − 1)
ε+
r2 Qmax
+
N 2
N 2
ψ(kr − εk)
kqij k2 r − kqij k +
1 4500
,
kqij k ∈ [0, 4].
(40)
In Fig. 3, potential function (40) was selected for the algorithm (4) and ε0 = 0.01. Fig. 3(a) shows the initial states of the agents, where the solid lines represent the neighboring relations, and the solid lines with arrows represent the directions of velocities; Fig. 3(b) depicts the motion trajectories of all agents from t = 0 to 7 s; Fig. 3(c–d) and (e–f) show the convergence of positions for ε = 0.01 and ε = 1.0, respectively, from which one can see that all agents eventually achieve the same position; Fig. 3(g–h) and (i–j) show the convergence of velocities for ε = 0.01 and ε = 1.0, respectively. If we choose the percentage set for the settling time as cts = 0.04, then the settling times in Fig. 3(g–j) are 2.7 s, 3.2 s, 5.6 s and 6.6 s, respectively. We find that the smaller the value of the constant ε , the larger the maximum overshoot and the smaller the settling time of the velocity. Therefore, there exists a fundamental trade-off between maximum overshoot and settling time for choosing the parameter ε . This is due to the fact that a smaller ε corresponds to a larger potential force, but also leads to earlier adding of edges. In the following, we present simulation results for the situation without adding new edges to the network (ε = 4). In the simulation, potential function (40) was selected for the algorithm (4), and the initial states of the agents and all parameters were chosen the same as those in Fig. 3. Fig. 4(a) and (b) show the convergence of positions over the x-axis and the y-axis, respectively; Fig. 4(c) and (d) show the convergence of velocities over the x-axis and the yaxis, respectively. One can see that all agents eventually reach the same position and velocity. This is because that initial network is connected and no existing edges are lost. Compared to both cases in Fig. 3, the maximum overshoot becomes smaller but the settling time becomes larger. This is due to the fact that no new edges are added to the network. 5.2. Rendezvous with a virtual leader Simulations on protocol (17) and on protocol (30) are performed with 50 agents moving in a 2-dimensional plane. In both simulations, r = 4 and ε = ε0 = 0.5. Initial positions and velocities of the 50 agents are chosen randomly from the boxes [0, 15] × [0, 15] and [0, 4] × [0, 4], respectively. Once again, the initial interaction network is set to be connected. The desired velocity of the virtual leader is chosen as pγ = [3, 3], and potential function (6) is selected for both protocol (17) and protocol (30). We have N (N − 1) ψ(kr − εk) p˜ Ti (0)˜pi (0) + 2 i=1 2 N (N − 1) N ≤ ψ(kr − εk) + max p˜ Ti (0)˜pi (0) 2 2
Umax =
N 1X
N (N − 1) (r − ε)2 2
ε+
r2
+
Umax
N 2
max p˜ Ti (0)˜pi (0) .
(41)
From (41), Umax ≤ 30205.7. If we choose Uˆ = 32000, then the potential function (6) can be explicitly described as
ψ(kqij k) =
=
max pTi (0)pi (0)
max pTi (0)pi (0) .
and velocities, the simulations are performed with ε = 0.01 and ε = 1.0, respectively. From (39), we have Qmax ≤ 70122.2 for ε = 0.01 and Qmax ≤ 471.7 for ε = 1.0. If we choose Qˆ = 72000, then the potential function (6) can be explicitly described as
(39)
Note that we can evaluate Qmax by using an estimate of the bound of the initial velocities of all agents. Under the same initial positions
ψ(kqij k) =
kqij k2 r − kqij k +
1 2000
,
kqij k ∈ [0, 4].
(42)
In Fig. 5, the informed agent is chosen randomly from the group and marked with a star; Fig. 5(a) shows the initial states of
H. Su et al. / Systems & Control Letters 59 (2010) 313–322
a
b
Initial state
8
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c
Paths and final states 20 ε=0.01 ε=1.0
18
7
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6
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y
y
5
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d
5
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e
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Position Convergence for ε=1.0 (y axis)
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Position Convergence for ε=1.0 (x axis)
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4
20 18
16
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20
14
14
12
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12 qy
10
qx
qy
1
t
18
8
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6
4
5
4
2 0
0
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Position Convergence for ε=0.01 (y axis)
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9
2 0
1
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3
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5
6
0
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0
1
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3
4
g
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0
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h
60
1
2
3
60 40
2.4
–80
2.2
–100
2
–120
py
px
2.6
–60
0
0.5
6
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20
3 2.8
–40
5
Velocity Convergence for ε=0.01 (y axis)
80
20
–20
4 t
40
0
0
2.8
–20
2.6 2.4
–40
2.2 –60 1
2
1
3 1.5
4
5
2
6 2.5
2
7 –80
3
0
0.5
1
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2 1.5
3
4 2
5 6 2.5
7 3
t
t
i
0
t
t
Velocity Convergence for ε=1.0 (x axis)
j
6
Velocity Convergence for ε=1.0 (y axis)
8 7
5
6 4
5 4 py
px
3 2
3 2
1
1
0
0 –1 –2
0
–1 1
2
3
4
5
6
7
–2
0
1
2
3
4
5
6
7
t
t
Fig. 3. Rendezvous of 10 agents under algorithm (4).
the agents; Fig. 5(b) depicts the motion trajectories of all agents from t = 0 to 35 s; Fig. 5(c) and (d) show the convergence of positions over the x-axis and the y-axis, respectively; Fig. 5(e) and (f) show the convergence of velocities over the x-axis and the y-axis, respectively. One can see that all agents eventually reach
the same position and velocity. The final common velocity is equal to the desired velocity pγ . Fig. 6(a) shows the initial states of the agents; Fig. 6(b) depicts the motion trajectories of all agents from t = 0 to 5 s; Fig. 6(c) and (d) show the convergence of positions over the x-axis and
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Position Convergence(x axis)
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45 40
35
35
30
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25 qy
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20
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5
5 0
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c 4.5
15
t
t Velocity Convergence(x axis)
d
4
Velocity Convergence(y axis) 7 6
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px
py
2.5 3
2 2
1.5
1
1 0.5
0
0
–1
0
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10
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15
t
Fig. 4. Rendezvous of 10 agents without adding new edges under algorithm (4).
a
b 100
Initial state
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Paths and final states
90
80
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50
qx
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y
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90
f
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90
prx=3
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20
t
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Velocity Convergence(y axis) 20 15
pry=3
10
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px
50
py
5
60 qy
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10
80
3 0
40 30
5 3 0 –5
20
–5 –10
10 0
0
x
x
d
40 30
30 4
Position Convergence(x axis) 90
0
5
10
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25
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35
–10
0
5
10
t
15
20
25
30
35
t
–15
0
5
10
15
20
25
30
35
t
Fig. 5. Rendezvous of 50 agents under algorithm (17).
the y-axis, respectively; Fig. 6(e) and (f) show the convergence of velocities over the x-axis and the y-axis, respectively. One can see that all agents eventually reach the same position and velocity. The
final common velocity is equal to the desired velocity pγ . One can clearly see that the group using control protocol (17) needs a longer convergent time than that using control protocol (30).
H. Su et al. / Systems & Control Letters 59 (2010) 313–322
a 16
Initial state
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c
Paths and final states
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321 Position Convergence(x axis) 25
14 12
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2
x
x
e 15
Position Convergence(y axis) 25
f
Velocity Convergence(x axis)
px
3 py
qy
pry=3
10
5
0
10
5
15 prx=3
15
4
Velocity Convergence(y axis)
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10
20
3 t
–5
5 3 0 –5
5 –10 0
0
1
2
t
3
4
5
–15
–10
0
0.5
1
1.5
2
2.5
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3.5
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4.5
5
t
–15
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
t
Fig. 6. Rendezvous of 50 agents under algorithm (30).
6. Conclusions In this paper, we have investigated the rendezvous problem with connectivity preservation for double-integrator dynamics employing hysteresis for adding new links and using the potential function method. The hysteresis idea is adopted from [36,37]; yet, we found that there is a trade-off between the maximum overshoot and the settling time of velocity convergence for choosing the hysteresis parameter. We have constructed a class of bounded potential functions to guarantee the existing links not to be lost, which is in some sense more practical than the two specific unbounded potential functions used in [36,37]. Moreover, we have investigated the coordinated control with a virtual leader and show that all agents can asymptotically attain the desired velocity even if only one agent in the team has information about the virtual leader. Future work will consider the situation where parts of agents are informed and the velocity of the virtual leader varies with time, and the effects of time delay and disturbance on these new algorithms. References [1] T. Vicsek, A. Cziro’ok, E. Ben-Jacob, O. Cohen, I. Shochet, Novel type of phase transition in a system of self-deriven particles, Physical Review Letters 75 (1995) 1226–1229. [2] C.W. Reynolds, Flocks, herds, and schools: a distributed behavioral model, in: ACM SIGGRAPH 87 Conference Proceedings, Computer Graphics 21 (1987) 25–34. [3] A. Jadbabaie, J. Lin, A.S. Morse, Coordination of groups of mobile agents using nearest neighbor rules, IEEE Transactions on Automatic Control 48 (2003) 988–1001. [4] R. Olfati-Saber, R.M. Murray, Consensus problems in networks of agents with switching topology and time-delays, IEEE Transactions on Automatic Control 49 (2004) 1520–1533. [5] W. Ren, R.W. Beard, Consensus seeking in multi-agent systems under dynamically changing interaction topologies, IEEE Transactions on Automatic Control 50 (2005) 655–661.
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