- Email: [email protected]
A Distributed Control Law for Acyclic Formations
Proceedings of the 18th World Congress The International Federation of Automatic Control Milano (Italy) August 28 - September 2, 2011
A Distributed Control Law for Acyclic Formations ⋆ S. Mou ∗ M. Cao ∗∗ A. S. Morse ∗ ∗
Department of Electrical Engineering, Yale University, New Haven, CT, USA (e-mail: {shaoshuai.mou,as.morse}@ yale.edu). ∗∗ Faculty of Mathematics and Natural Sciences, ITM, University of Groningen, The Netherlands (e-mail: [email protected])
Abstract: New distributed control laws using relative positions are proposed to control a formation of mobile autonomous agents in the plane. When the formation is generated by Henneberg sequence operations, control laws proposed here are capable of driving it to its desired shape even when the triangular sub-formations are initially formed by some collinear neighboring agents or in the wrong orientations. We prove that the conclusion holds globally and further validate it by simulations. Keywords: autonomous agents, exponential convergence, formation control, global stability 1. INTRODUCTION Inspired by the cooperative behavior in nature, formations of mobile autonomous agents have been studied over the past decade, and have found extensive applications into surveillance, search and rescue, and exploration in unknown environment (Lin et al. (2004); Marshall et al. (2004); Anderson et al. (2007)). One research problem that is especially important for the application of multiagent formations is the formation control problem (Cao et al. (2011b); Dimarogonas and Johansson (2009, 2010); Dimarogonas and Kyriakopoulosx (2008); Basiri et al. (2010)), the goal of which is to stabilize a formation to a given desired shape. In Anderson et al. (2008), this problem is solved by maintaining desired distances between chosen pairs of agents provided that the graph of the formation is rigid. When each distance is maintained by both associated agents, the formation is said to be undirected; otherwise, when some distances in the formation are maintained by only one of the two associated agents, the formation is directed. For convenience, an agent is called a follower in the sequel if it has to maintain its distances to some other leading agents and correspondingly we call these leading agents leaders. There is an increasing interest in the study of formation control for directed formations (Cao et al. (2008a,b, 2011a); Yu et al. (2009)). Using linearization techniques and center manifold theory, it has been shown in Krick et al. (2009) that gradient-like control laws guarantee local asymptotic stability for infinitesimally rigid formations. However, such results only hold locally; in other words, a formation under gradient-like control laws is guaranteed to be stabilized only when its shape is close enough to the ⋆ The research of Mou and Morse was supported in part by the US Army Research Office, the US Air Force Office of Scientific, and the National Science Foundation. The research of Cao was supported in part by grants from the Dutch Organization for Scientific Research (NWO) and the Dutch Technology Foundation (STW).
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desired one. Global stability analysis has also been carried out for gradient-like controllers, but only for classes of special formations, such as triangles in Cao et al. (2007) and Cao et al. (2008a). One reason for the difficulty in the global analysis for gradient-like control laws is their inherent tendency to converge to local minima of the associated potential function. What this implies in terms of the evolution of the shape dynamics of a formation is that there are initial agent positions that may lead to degenerate formation shapes. One example is that when guided by gradient-like control laws, a formation of more than two agents starting from collinear positions will always remain collinear no matter which pairwise distances we set the agents to maintain. Another example is that gradient-like control laws fail to drive the formation of more than three agents to converge to the desired one when the agents start with wrong orientation. Moreover, the global stability result under gradient-like control laws for acyclic triangles in Cao et al. (2008a) is based on an assumption that the leader strictly keeps still. This assumption limits the application of this result since the result may not hold even when there is small disturbance to the leader. Hence, there is a need to look at other types of control strategies to complement the existing gradientlike control approach. It is the aim of this paper to develop globally stabilizable formation control strategies, which also work for the challenging situations just described. New control laws using agents’ relative position information are proposed for the one-leader case in Section 2 and for the two-leader case in Section 3. It is shown in both cases that the proposed control laws drive the follower to converge to its desired position exponentially fast. Based on these results, we apply the proposed control laws to formations that are generated by Henneberg sequence operations in Section 4. It is shown these control laws work even when the agents start from collinear positions or with wrong orientations.
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The advantages of the control laws are finally verified by simulation in Section 5. 2. ONE-LEADER CASE We consider a two-agent formation in the plane, where a follower is required to maintain desired distance away from one leader. Let x, y ∈ R2 denote the positions of the follower and the leader respectively in a common coordinate system. Let d denote the desired distance between them. Let · denote the Euclidean norm. We propose to use the following control law for the follower x˙ = −( y − x 2 −d2 )(x − y). (1) To analyze the behavior of the follower, we find it convenient to study the system (2) z˙ = (d2 − z 2 )z − v, where z = x − y and v = y. ˙ Suppose v is at least piecewise continuous. Since system (2) is locally Lipschitz continuous with z as the state, for each z0 ∈ R2 , there exists a maximal interval I = [0, T ) and a unique continuous solution to (2) on [0, T ) which starts at z0 at t = 0. It will be shown later that this maximal interval of existence is [0, ∞). Note that if v = 0, system (2) has two equilibrium sets, namely the point z = 0 and the set of z for which z = d. Moreover, the former is repulsive and the latter is attractive. Thus, if v = 0, all trajectories starting outside of the set z = 0 remain bounded away from z = 0. Perturbing (2) with nonzero v will change everything. However, if v is “small” in some sense one might still expect trajectories starting outside of z = 0 to be bounded away from z = 0 for all t. In the sequel we only consider the situation when z(0) is nonzero for this one leader case. Lemma 1. Suppose z(0) is nonzero, v converges to 0 ∞ exponentially fast as t goes to infinity and 0 v dt < min{ z(0) , d}. Then I = [0, ∞); z(t) ≥ min{ z(0) , d} −
∞
v dτ ;
if the desired orientation of leader 1, leader 2 and the follower is clockwise; otherwise, s − δ 2 − s2 R(s) = . δ 2 − s2 s The control law we use for the follower is x˙ = −δ 2 (x − y1 ) + d1 R(s)(y2 − y1 )
In this two-leader case, gradient-like control laws work well as long as the follower and the two leaders are not in collinear positions. In practice, since neither of the leaders knows the existence of the follower, even if the three agents are not initially collinear, the motions of the leaders may still cause the three agents’ positions become collinear. In addition, the new control law employs a rotation matrix R(s) in the plane, which rotates a vector according to the orientation of the follower and the two leaders. By considering orientation here, we have distinguished the desired position of the follower from its reflection with respect to the line passing the two leaders. When the follower starts from a position with the wrong orientation with respect to its two leaders, the gradient-like control laws may guide the three agents to converge to their final positions with the wrong orientation. However, the control law (4) does not have such deficiencies as stated in the following lemma. Lemma 2. Given positive constants d1 , d2 with d1 ≥ d2 and constant vectors y1∗ , y2∗ ∈ R2 satisfying d1 −d2 < y1∗ − y2∗ < d1 + d2 . Suppose there exists a finite time T such that after T , d1 − d2 < y1 − y2 < d1 + d2 and y1 , y2 converge exponentially fast to y1∗ , y2∗ respectively. Then after T , x converges exponentially fast to a constant vector x∗ ∈ E = {x ∈ R2 : x − y1 = d1 , x − y2 = d2 } where x∗ , y1∗ and y2∗ are in the desired orientation for the threeagent formation. We give the proof for this Lemma in the Appendix. In the next section, we apply Lemmas 1 and 2 for formations that are generated by Henneberg sequence operations.
(3)
4. APPLICATION TO A CLASS OF ACYCLIC DIRECTED FORMATIONS
0 ∗
x converges exponentially fast to a constant vector x ∈ E = {x ∈ R2 : x − y = d}. We provide the proof for this lemma in the Appendix. 3. TWO-LEADER CASE Now we consider a three-agent formation in which a follower follows two leaders, labeled by 1 and 2, by trying to maintain the given desired distances d1 and d2 to leaders 1 and 2 respectively. Without loss of generality, assume d1 ≥ d2 > 0. Let x, y1 , y2 ∈ R2 denote the positions of the follower, leader 1 and leader 2 respectively in a common coordinate system. Let δ = y2 − y1 and δ 2 + d21 − d22 , d1 − d 2 < δ < d1 + d 2 ; s= δ, 2d1 otherwise.
Let
s R(s) = − δ 2 − s2
δ 2 − s2 s
(4)
Now we apply the control laws discussed in the previous sections to a class of acyclic directed formations that are generated by a standard procedure for obtaining minimally rigid graphs called Henneberg sequence operations (Anderson et al. (2008)). When n = 2, the corresponding formation consists of a leader agent 1 and a follower agent 2; when n > 2, the agents in the formation can be labeled in such a way that agent 2 follows agent 1 and agent i, i > 2, follows two other agents j and k with 1 ≤ j, k < i. In the sequel, let xi denote the position of agent i. The main result that we want to prove is as follows. Theorem 1. For an n-agent, n > 2, acyclic directed formation generated by Henneberg sequence operations, suppose x1∞(0) = x2 (0), x˙ 1 converges exponentially fast to 0 and x˙ 1 dt < min{ x1 (0) − x2 (0) , d}, where d is the 0 desired distance for agent 2 to maintain with respect to agent 1. If control law (1) is applied to agent 2 and control law (4) to agent i, i > 2, then there exists a finite time T after which the n-agent formation converges exponentially fast to its desired shape.
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Proof of Theorem 1: We prove this theorem by induc¯ dˆ denote tion. We first consider the case when n = 3. Let d, the desired distances for agent 3 to maintain with respect to agents 1 and 2 respctively. Without loss of generality, ˆ Since x˙ 1 converges to 0 exponentially fast, assume d¯ ≥ d. we know agent 1 converges to some fixed position in the plane exponentially fast and we denote this position by x∗1 . From Lemma 1, we know that agent 2 also converges exponentially fast to some fixed position, denoted by x∗2 , at which the desired distance between agents 1 and 2 is obtained. Since x1 and x2 converge exponentially fast to x∗1 and x∗2 respectively, we know there exists a finite time ˆ T ≥ 0 such that for all t ≥ T , d¯ − dˆ < x2 − x1 < d¯ + d. In view of Lemma 2, it must be true that x3 converges exponentially fast to its desired position after T . Now we assume that the conclusion holds for k-agent formations of interest with k ≥ 3 and study the stability of the formation consisting of k + 1 agents. We denote the position of the (k + 1)th agent and the positions of its two leaders by x, y1 and y2 respectively and denote the associated two desired distances by d¯ and dˆ respectively. Since the sub-formation of the first k agents converges after some fixed time T exponentially fast to the desired shape, we know the two leaders of agent k + 1 converges after T exponentially fast to some fixed positions, denoted ˆ by y1∗ and y2∗ , such that d¯ − dˆ < y1∗ − y2∗ < d¯ + d. Then there exists a finite time T¯ ≥ T such that after T¯, ˆ From Lemma 2, we know then d¯ − dˆ < y1 − y2 < d¯ + d. xk+1 converges exponentially fast to its desired position after T¯. Hence, we have also proved that the conclusion holds for the (k + 1)-agent formation. In view of the discussion above, we conclude by induction that the conclusion holds for all n-agent formations with n ≥ 3. Now we use some examples to demonstrate how the proposed control laws can be applied.
3
3
d7
3 d1 1
d2
2 1
5 d6
d3
1 2
2 d5
d4 4
1
2
Remark 1. In Cao et al. (2008a), the leader in such a twoagent formation is assumed to be stationary. In comparison, we have shown here that the follower still converges exponentially fast to its desired position under control law (1) even when agent 1 moves under certain constraints. As shown in Fig. 1, a 3-agent formation is obtained by adding agent 3 to the previous two-agent formation with d2 and d3 being the given desired distances. We apply control law (4) to agent 3. Let E2 = {x3 ∈ R2 : x3 − x1 = d2 , x3 −x2 = d3 }. Since x1 converges to x∗1 and x2 converges to x∗2 exponentially fast, where d3 − d2 < x∗1 − x∗2 < d2 + d3 , then there exists a finite time T such that d3 − d2 < x1 − x2 < d2 + d3 after T . By Lemma 2, we know x3 converges to x∗3 exponentially fast after T , where x∗3 ∈ E2 and x∗2 , x∗1 , x∗3 are in a clockwise orientation. Remark 2. When agent 3 starts from an initial position collinear with the positions of agents 1 and 2, gradientlike control laws in Cao et al. (2008a) fail to drive this triangular formation to its desired formation. However, in comparison, control law (4) can guarantee that agents 1, 2 and 3 converge to the desired positions with the correct orientation. This is especially advantageous for larger formations with more than 3 agents. Now we add agent 4 to the previous acyclic triangular formation with d4 and d5 being the desired distances to get the 4-agent formation shown in Fig. 1. The control law (4) is applied to agent 4. From theorem 1, we know x4 converges to x∗4 exponentially fast after some finite time T¯, where x∗4 ∈ E3 = {x4 ∈ R2 : x4 − x1 = d4 , x4 − x2 = d5 } such that x∗1 , x∗2 , x∗4 are in a clockwise orientation. Remark 3. The proposed control law takes care of the orientation of the triangular sub-formations while gradientlike control laws may fail to do so. For the desired fouragent formation in Fig. 1, if the agents start from a formation in which the two triangular sub-formations are twisted as shown in Fig. 2, gradient-like control laws are not able to drive this four-agent formation to its desired shape. However, from Theorem 1, the proposed control law can still guide the 4-agent formation to converge to a shape satisfying all the distance constraints and the relative orientation constraints. This guarantees that the final formation is as shown in Fig. 1 instead of the deformed one shown in Fig. 2.
1
4
Fig. 1. Minimally rigid formations generated by Henneberg sequence operations. First consider the two-agent formation in Fig. 1, where agent 2 follows agent 1 with the given desired distance d1 . Denote agent 1’s velocity by v and assume that v satisfies the conditions in Lemma 1, then x1 converges to some constant x∗1 ∈ R2 exponentially fast. Applying control law (1) to agent 2 and assume x2 (0) satisfies the conditions of Lemma 1, then from the lemma we know that x2 − x1 is bounded away from 0 and agent 2 converges to some desired position x∗2 ∈ E1 = {x2 ∈ R2 : x2 − x1 = d1 } exponentially fast.
3
2
4
Fig. 2. A four-agent formation with the wrong shape.
5. SIMULATION Example 1: Consider the acyclic triangular formation in Fig. 1 with −t d1 = 3, d2 = 4, d3 = 5. Let agent 1’s velocity e . The three agents are initially positioned at be v = 0
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{x1 =
2 3 0 }. , x3 = , x2 = 0 0 0
d6 = d7 = 5. Let v =
Note that these initial positions are collinear. We present the simulation results in Fig. 3, and it is clear that the 4.5
e−t . These five agents are initially e−t
0 2 {x1 = , x2 = x3 = x4 = x5 = }. 0 0
As shown in Fig. 5, even in this challenging situation when agents 2, 3, 4 and 5 are initially coincident, the five-agent
3
4
positioned at
3.5
6
3
3
5
2.5
4
2
3
1.5
2
1
1
0.5
5 1
0
1
2
0
2
−1 −2
−0.5 −0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
−3
Fig. 3. Convergence of the acyclic triangular formation
4
−4 −5 −2
triangular formation converges to its desired shape under the proposed control law.
0
2
4
6
8
Fig. 5. Convergence of the five-agent formation Example 2: Consider the four-agent formation in Fig. 1 with the same d1 , d2 , d3 as in Example 1, d4 = 5 and 0 d5 = 4. Let v = −t . These four agents start from e
0 3 10 0 {x1 = }. , x4 = , x3 = , x2 = 4 4 0 0
Note that agents 1, 2 and 4 are required to be clockwise while they are initially positioned counterclockwise. As shown in Fig. 4, the four-agent formation still converges 6
3
5 4 3
6. CONCLUDING REMARKS In this paper, new control laws have been proposed to control the shape of formations of autonomous agents. These proposed control laws do not require that the leader of the formation has to be held still. It can also deal with the challenging situation where a follower’s position is collinear with its leaders. In addition, the control law takes into account the orientation of the triangular sub-formations of agents. When these control laws are applied to acyclic directed formations generated by Henneberg sequence operations, it has been shown that such formations converge after a finite time exponentially fast to their desired shapes. It is still not clear how to obtain similar results for arbitrary directed formations with cycles, even for a four-agent formation consisting of two cyclic triangles. In future, we would like to deign control laws for directed formations containing cycles.
2
1
1
formation still converges to its desired shape.
2
0 −1 −2
REFERENCES
−3
4 −4 0
2
4
6
8
10
Fig. 4. Convergence of the four-agent formation to its desired shape. Example 3: Consider the five-agent formation in Fig. 1 with the same d1 , d2 , d3 , d4 , d5 as in Example 2 and
Anderson, B.D.O., Yu, C., Dasgupta, S., and Morse, A.S. (2007). Control of a three coleader persistent formation in the plane. Systems and Control Letters, 573–578. Anderson, B.D.O., Yu, C., Fidan, B., and Hendrickx, J.M. (2008). Rigid graph control architectures for autonomous formations. IEEE Contr. Syst. Mag., 28(6), 48–63. Basiri, M., Bishop, A.N., and Jensfelt, P. (2010). Distributed control of triangular formations with angle-only constraints. Systems and Control Letters, 59(2), 147– 154.
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Cao, M., Anderson, B.D.O., Morse, A.S., and Yu, C. (2008a). Control of acyclic formations of mobile autonomous agents. Proc. of the 47th Conf. on Decision and Contr., 1187–1192. Cao, M., Morse, A.S., Yu, C., Anderson, B.D.O., and Dasgupta, S. (2007). Controlling a triangular formation of mobile autonomous agents. Proc. of the 46th Conf. on Decision and Contr., 3603–3608. Cao, M., Morse, A.S., Yu, C., Anderson, B.D.O., and Dasgupta, S. (2011a). Maintaining a directed, triangular formation of mobile autonomous agents. Communications in Information and Systems, 11, 1–16. Cao, M., Yu, C., and Anderson, B.D.O. (2011b). Formation control using range-only measurements. Automatica, 47, 776–781. Cao, M., Yu, C., Morse, A.S., Anderson, B.D.O., and Dasgupta, S. (2008b). Generalized controller for directed triangle formations. Proc. of the 17th IFAC Congress, 6590–6595. Dimarogonas, D.V. and Johansson, K.H. (2009). Further results on the stability of distance-based multi-robot formations. Proc. of the 2009 American Control Conference. Dimarogonas, D.V. and Johansson, K.H. (2010). Stability analysis for multi-agent systems using the incidence matrix: quantized communication and formation control. Automatica, 46(4), 695–700. Dimarogonas, D.V. and Kyriakopoulosx, K.J. (2008). A connection between formation infeasibility and velocity alignment in kinematic multi-agent systems. Automatica, 44(10), 2648–2654. Krick, L., Broucke, M., and Francis, B. (2009). Stabilization of infinitesimally rigid formations of multi-robot networks. Int. J. of Control, 82(3), 423–439. Lin, Z., Francis, B., and Maggiore, M. (2004). Local control strategies for groups of mobile autonomous agents. IEEE Trans. Automatic Control, 49, 622–629. Marshall, J., Broucke, M.E., and Francis, B.A. (2004). Formation of vehicles in cyclic pursuit. IEEE Trans. Automatic Control, 9, 1963–1974. Yu, C., Anderson, B.D.O., Dasgupta, S., and Fidan, B. (2009). Control of minimally persistent formations in the plane. SIAM J. Control Optim., 206–233.
ez 2 ≤ e2 (e + d2 ) e2 + e + d2 2 ) ≤( 2 3 1 1 ≤ ( e2 + + d2 )2 , 4 4 2 it is true that 3 2 1 1 2 e + + d 4 4 2 In view of (6) and (7), ez ≤
t ∈ [0, T ).
(7)
V˙ ≤ 3 v V + (1 + 2d2 ) v t ∈ [0, T ), which implies t v(s) ds V (t) ≤ V (0) + (1 + 2d2 ) 0 t +3 v(s) V (s)ds. 0
From the Bellman-Gronwall Lemma, one thus has
V (t) ≤ (V (0) + (1 + 2d2 )a)e3a t ∈ [0, T ), which means V is bounded on [0, T ). So do e and z. Since the maximal solution z(t) is bounded on [0, T ) and continuous, the limit z ∗ = lim z(t) t→T
exists. Now if T were finite there would have to be some interval [T, T¯) of positive length on which there were a continuous solution z¯(t) to (2) which passed through z ∗ at t = T. This in turn would imply that the concatanated function θ defined as θ(t) = z(t), t ∈ [0, T ) and θ(t) = z¯(t), t ∈ [T, T¯) would be a continuous function to (2) passing through z0 at t = 0. But this can not be so because T¯ ≥ T and [0, T ) is the maximal interval of existence. Therefore, the maximal interval of existence is [0, ∞). Since system (2) is locally Lipschitz continuous with z as the state, z is bounded on [0, ∞). Now we prove (3) by contradiction. Assume that (3) is false. Then there exists a finite t2 such that ∞ v dt (8) z(t2 ) < a − 0
7. APPENDIX Proof of Lemma 1: We first show there exists a unique solution to (2) on [0, ∞). It suffices to show that z is bounded on [0, T ), which as defined before is the maximal interval of existence of solution to (2). Let a = min{ z(0) , d} and e = z 2 −d2 . Introduce the function V = e2 . Along the solution to (2), V˙ = −4 z 2 V − 4ez ′ v
Since
≤ −4 z 2 V + 4 ez v
(5)
≤ 4 ez v .
(6)
In addition since z(0) > 0, we can choose t2 in such a way that z(t) > 0, t ∈ [0, t2 ]. Since z(0) ≥ a and z(t2 ) < a, then there must exist a time t1 ∈ [0, t2 ) such that z(t1 ) = a and z(t) ≤ a for t ∈ [t1 , t2 ]. From (2), one has z d z = (d2 − z 2 ) z −v ′ , t ∈ [t1 , t2 ]. dt z z But from the Cauchy Schwartz inequality v ′ z ≤ v , so d z ≥ (d2 − z 2 ) z − v , t ∈ [t1 , t2 ]. dt Since z ≤ a for t ∈ [t1 , t2 ] and a ≤ d, then d z ≥ − v , t ∈ [t1 , t2 ]. dt Hence,
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z(t2 ) ≥ z(t1 ) − Therefore, z(t2 ) ≥ a −
t2
v dt. t1
∞
v dt,
0
which contradicts (8). Thus, (3) is true. This also implies z is bounded away from 0. From z is bounded and also bounded away from 0, v converges to 0 exponentially fast as t goes to infinity and (5), one has V converges to 0 and y converges to a constant exponentially fast. Then x converges to a constant x∗ ∈ E exponentially fast.
y Proof of Lemma 2: Let y = 1 and define y2 τ (y) =
d1 R(s)(y2 − y1 ) + y1 δ2
(9)
when δ > 0 and τ (y) = y1
(10)
when δ = 0. Since d1 − d2 < y1 − y2 < d1 + d2 for all t ≥ T and in view of the definitions of s, R(s) and τ (y), one has τ (y) is continuously differentiable for t ≥ T with respect to y and thus Lipschitz continuous. Then there exists a ∗ ∗ positive constant l such that
τ (y) − τ (y ) ≤ l y − y ∗ y for t ≥ T , where y ∗ = 1∗ . Therefore τ (y) converges to y2 τ (y ∗ ) exponentially fast after T since y converges to y ∗ exponentially fast for t ≥ T , Note that (4) can be written as x˙ = −δ 2 (x − τ (y)), (11) where δ > 0 and is bounded away from 0 for t ≥ T. Therefore, x converges to x∗ = τ (y ∗ ) exponentially fast after T because of the exponential convergence of the trajectory of system (11) to its equilibrium point τ (y ∗ ), where τ (y ∗ ) ǫE and is in the desired orientation with respect to y1∗ , y2∗ .
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A Distributed Control Law for Acyclic Formations ⋆ S. Mou ∗ M. Cao ∗∗ A. S. Morse ∗ ∗
Department of Electrical Engineering, Yale University, New Haven, CT, USA (e-mail: {shaoshuai.mou,as.morse}@ yale.edu). ∗∗ Faculty of Mathematics and Natural Sciences, ITM, University of Groningen, The Netherlands (e-mail: [email protected])
Abstract: New distributed control laws using relative positions are proposed to control a formation of mobile autonomous agents in the plane. When the formation is generated by Henneberg sequence operations, control laws proposed here are capable of driving it to its desired shape even when the triangular sub-formations are initially formed by some collinear neighboring agents or in the wrong orientations. We prove that the conclusion holds globally and further validate it by simulations. Keywords: autonomous agents, exponential convergence, formation control, global stability 1. INTRODUCTION Inspired by the cooperative behavior in nature, formations of mobile autonomous agents have been studied over the past decade, and have found extensive applications into surveillance, search and rescue, and exploration in unknown environment (Lin et al. (2004); Marshall et al. (2004); Anderson et al. (2007)). One research problem that is especially important for the application of multiagent formations is the formation control problem (Cao et al. (2011b); Dimarogonas and Johansson (2009, 2010); Dimarogonas and Kyriakopoulosx (2008); Basiri et al. (2010)), the goal of which is to stabilize a formation to a given desired shape. In Anderson et al. (2008), this problem is solved by maintaining desired distances between chosen pairs of agents provided that the graph of the formation is rigid. When each distance is maintained by both associated agents, the formation is said to be undirected; otherwise, when some distances in the formation are maintained by only one of the two associated agents, the formation is directed. For convenience, an agent is called a follower in the sequel if it has to maintain its distances to some other leading agents and correspondingly we call these leading agents leaders. There is an increasing interest in the study of formation control for directed formations (Cao et al. (2008a,b, 2011a); Yu et al. (2009)). Using linearization techniques and center manifold theory, it has been shown in Krick et al. (2009) that gradient-like control laws guarantee local asymptotic stability for infinitesimally rigid formations. However, such results only hold locally; in other words, a formation under gradient-like control laws is guaranteed to be stabilized only when its shape is close enough to the ⋆ The research of Mou and Morse was supported in part by the US Army Research Office, the US Air Force Office of Scientific, and the National Science Foundation. The research of Cao was supported in part by grants from the Dutch Organization for Scientific Research (NWO) and the Dutch Technology Foundation (STW).
978-3-902661-93-7/11/$20.00 © 2011 IFAC
desired one. Global stability analysis has also been carried out for gradient-like controllers, but only for classes of special formations, such as triangles in Cao et al. (2007) and Cao et al. (2008a). One reason for the difficulty in the global analysis for gradient-like control laws is their inherent tendency to converge to local minima of the associated potential function. What this implies in terms of the evolution of the shape dynamics of a formation is that there are initial agent positions that may lead to degenerate formation shapes. One example is that when guided by gradient-like control laws, a formation of more than two agents starting from collinear positions will always remain collinear no matter which pairwise distances we set the agents to maintain. Another example is that gradient-like control laws fail to drive the formation of more than three agents to converge to the desired one when the agents start with wrong orientation. Moreover, the global stability result under gradient-like control laws for acyclic triangles in Cao et al. (2008a) is based on an assumption that the leader strictly keeps still. This assumption limits the application of this result since the result may not hold even when there is small disturbance to the leader. Hence, there is a need to look at other types of control strategies to complement the existing gradientlike control approach. It is the aim of this paper to develop globally stabilizable formation control strategies, which also work for the challenging situations just described. New control laws using agents’ relative position information are proposed for the one-leader case in Section 2 and for the two-leader case in Section 3. It is shown in both cases that the proposed control laws drive the follower to converge to its desired position exponentially fast. Based on these results, we apply the proposed control laws to formations that are generated by Henneberg sequence operations in Section 4. It is shown these control laws work even when the agents start from collinear positions or with wrong orientations.
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
The advantages of the control laws are finally verified by simulation in Section 5. 2. ONE-LEADER CASE We consider a two-agent formation in the plane, where a follower is required to maintain desired distance away from one leader. Let x, y ∈ R2 denote the positions of the follower and the leader respectively in a common coordinate system. Let d denote the desired distance between them. Let · denote the Euclidean norm. We propose to use the following control law for the follower x˙ = −( y − x 2 −d2 )(x − y). (1) To analyze the behavior of the follower, we find it convenient to study the system (2) z˙ = (d2 − z 2 )z − v, where z = x − y and v = y. ˙ Suppose v is at least piecewise continuous. Since system (2) is locally Lipschitz continuous with z as the state, for each z0 ∈ R2 , there exists a maximal interval I = [0, T ) and a unique continuous solution to (2) on [0, T ) which starts at z0 at t = 0. It will be shown later that this maximal interval of existence is [0, ∞). Note that if v = 0, system (2) has two equilibrium sets, namely the point z = 0 and the set of z for which z = d. Moreover, the former is repulsive and the latter is attractive. Thus, if v = 0, all trajectories starting outside of the set z = 0 remain bounded away from z = 0. Perturbing (2) with nonzero v will change everything. However, if v is “small” in some sense one might still expect trajectories starting outside of z = 0 to be bounded away from z = 0 for all t. In the sequel we only consider the situation when z(0) is nonzero for this one leader case. Lemma 1. Suppose z(0) is nonzero, v converges to 0 ∞ exponentially fast as t goes to infinity and 0 v dt < min{ z(0) , d}. Then I = [0, ∞); z(t) ≥ min{ z(0) , d} −
∞
v dτ ;
if the desired orientation of leader 1, leader 2 and the follower is clockwise; otherwise, s − δ 2 − s2 R(s) = . δ 2 − s2 s The control law we use for the follower is x˙ = −δ 2 (x − y1 ) + d1 R(s)(y2 − y1 )
In this two-leader case, gradient-like control laws work well as long as the follower and the two leaders are not in collinear positions. In practice, since neither of the leaders knows the existence of the follower, even if the three agents are not initially collinear, the motions of the leaders may still cause the three agents’ positions become collinear. In addition, the new control law employs a rotation matrix R(s) in the plane, which rotates a vector according to the orientation of the follower and the two leaders. By considering orientation here, we have distinguished the desired position of the follower from its reflection with respect to the line passing the two leaders. When the follower starts from a position with the wrong orientation with respect to its two leaders, the gradient-like control laws may guide the three agents to converge to their final positions with the wrong orientation. However, the control law (4) does not have such deficiencies as stated in the following lemma. Lemma 2. Given positive constants d1 , d2 with d1 ≥ d2 and constant vectors y1∗ , y2∗ ∈ R2 satisfying d1 −d2 < y1∗ − y2∗ < d1 + d2 . Suppose there exists a finite time T such that after T , d1 − d2 < y1 − y2 < d1 + d2 and y1 , y2 converge exponentially fast to y1∗ , y2∗ respectively. Then after T , x converges exponentially fast to a constant vector x∗ ∈ E = {x ∈ R2 : x − y1 = d1 , x − y2 = d2 } where x∗ , y1∗ and y2∗ are in the desired orientation for the threeagent formation. We give the proof for this Lemma in the Appendix. In the next section, we apply Lemmas 1 and 2 for formations that are generated by Henneberg sequence operations.
(3)
4. APPLICATION TO A CLASS OF ACYCLIC DIRECTED FORMATIONS
0 ∗
x converges exponentially fast to a constant vector x ∈ E = {x ∈ R2 : x − y = d}. We provide the proof for this lemma in the Appendix. 3. TWO-LEADER CASE Now we consider a three-agent formation in which a follower follows two leaders, labeled by 1 and 2, by trying to maintain the given desired distances d1 and d2 to leaders 1 and 2 respectively. Without loss of generality, assume d1 ≥ d2 > 0. Let x, y1 , y2 ∈ R2 denote the positions of the follower, leader 1 and leader 2 respectively in a common coordinate system. Let δ = y2 − y1 and δ 2 + d21 − d22 , d1 − d 2 < δ < d1 + d 2 ; s= δ, 2d1 otherwise.
Let
s R(s) = − δ 2 − s2
δ 2 − s2 s
(4)
Now we apply the control laws discussed in the previous sections to a class of acyclic directed formations that are generated by a standard procedure for obtaining minimally rigid graphs called Henneberg sequence operations (Anderson et al. (2008)). When n = 2, the corresponding formation consists of a leader agent 1 and a follower agent 2; when n > 2, the agents in the formation can be labeled in such a way that agent 2 follows agent 1 and agent i, i > 2, follows two other agents j and k with 1 ≤ j, k < i. In the sequel, let xi denote the position of agent i. The main result that we want to prove is as follows. Theorem 1. For an n-agent, n > 2, acyclic directed formation generated by Henneberg sequence operations, suppose x1∞(0) = x2 (0), x˙ 1 converges exponentially fast to 0 and x˙ 1 dt < min{ x1 (0) − x2 (0) , d}, where d is the 0 desired distance for agent 2 to maintain with respect to agent 1. If control law (1) is applied to agent 2 and control law (4) to agent i, i > 2, then there exists a finite time T after which the n-agent formation converges exponentially fast to its desired shape.
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
Proof of Theorem 1: We prove this theorem by induc¯ dˆ denote tion. We first consider the case when n = 3. Let d, the desired distances for agent 3 to maintain with respect to agents 1 and 2 respctively. Without loss of generality, ˆ Since x˙ 1 converges to 0 exponentially fast, assume d¯ ≥ d. we know agent 1 converges to some fixed position in the plane exponentially fast and we denote this position by x∗1 . From Lemma 1, we know that agent 2 also converges exponentially fast to some fixed position, denoted by x∗2 , at which the desired distance between agents 1 and 2 is obtained. Since x1 and x2 converge exponentially fast to x∗1 and x∗2 respectively, we know there exists a finite time ˆ T ≥ 0 such that for all t ≥ T , d¯ − dˆ < x2 − x1 < d¯ + d. In view of Lemma 2, it must be true that x3 converges exponentially fast to its desired position after T . Now we assume that the conclusion holds for k-agent formations of interest with k ≥ 3 and study the stability of the formation consisting of k + 1 agents. We denote the position of the (k + 1)th agent and the positions of its two leaders by x, y1 and y2 respectively and denote the associated two desired distances by d¯ and dˆ respectively. Since the sub-formation of the first k agents converges after some fixed time T exponentially fast to the desired shape, we know the two leaders of agent k + 1 converges after T exponentially fast to some fixed positions, denoted ˆ by y1∗ and y2∗ , such that d¯ − dˆ < y1∗ − y2∗ < d¯ + d. Then there exists a finite time T¯ ≥ T such that after T¯, ˆ From Lemma 2, we know then d¯ − dˆ < y1 − y2 < d¯ + d. xk+1 converges exponentially fast to its desired position after T¯. Hence, we have also proved that the conclusion holds for the (k + 1)-agent formation. In view of the discussion above, we conclude by induction that the conclusion holds for all n-agent formations with n ≥ 3. Now we use some examples to demonstrate how the proposed control laws can be applied.
3
3
d7
3 d1 1
d2
2 1
5 d6
d3
1 2
2 d5
d4 4
1
2
Remark 1. In Cao et al. (2008a), the leader in such a twoagent formation is assumed to be stationary. In comparison, we have shown here that the follower still converges exponentially fast to its desired position under control law (1) even when agent 1 moves under certain constraints. As shown in Fig. 1, a 3-agent formation is obtained by adding agent 3 to the previous two-agent formation with d2 and d3 being the given desired distances. We apply control law (4) to agent 3. Let E2 = {x3 ∈ R2 : x3 − x1 = d2 , x3 −x2 = d3 }. Since x1 converges to x∗1 and x2 converges to x∗2 exponentially fast, where d3 − d2 < x∗1 − x∗2 < d2 + d3 , then there exists a finite time T such that d3 − d2 < x1 − x2 < d2 + d3 after T . By Lemma 2, we know x3 converges to x∗3 exponentially fast after T , where x∗3 ∈ E2 and x∗2 , x∗1 , x∗3 are in a clockwise orientation. Remark 2. When agent 3 starts from an initial position collinear with the positions of agents 1 and 2, gradientlike control laws in Cao et al. (2008a) fail to drive this triangular formation to its desired formation. However, in comparison, control law (4) can guarantee that agents 1, 2 and 3 converge to the desired positions with the correct orientation. This is especially advantageous for larger formations with more than 3 agents. Now we add agent 4 to the previous acyclic triangular formation with d4 and d5 being the desired distances to get the 4-agent formation shown in Fig. 1. The control law (4) is applied to agent 4. From theorem 1, we know x4 converges to x∗4 exponentially fast after some finite time T¯, where x∗4 ∈ E3 = {x4 ∈ R2 : x4 − x1 = d4 , x4 − x2 = d5 } such that x∗1 , x∗2 , x∗4 are in a clockwise orientation. Remark 3. The proposed control law takes care of the orientation of the triangular sub-formations while gradientlike control laws may fail to do so. For the desired fouragent formation in Fig. 1, if the agents start from a formation in which the two triangular sub-formations are twisted as shown in Fig. 2, gradient-like control laws are not able to drive this four-agent formation to its desired shape. However, from Theorem 1, the proposed control law can still guide the 4-agent formation to converge to a shape satisfying all the distance constraints and the relative orientation constraints. This guarantees that the final formation is as shown in Fig. 1 instead of the deformed one shown in Fig. 2.
1
4
Fig. 1. Minimally rigid formations generated by Henneberg sequence operations. First consider the two-agent formation in Fig. 1, where agent 2 follows agent 1 with the given desired distance d1 . Denote agent 1’s velocity by v and assume that v satisfies the conditions in Lemma 1, then x1 converges to some constant x∗1 ∈ R2 exponentially fast. Applying control law (1) to agent 2 and assume x2 (0) satisfies the conditions of Lemma 1, then from the lemma we know that x2 − x1 is bounded away from 0 and agent 2 converges to some desired position x∗2 ∈ E1 = {x2 ∈ R2 : x2 − x1 = d1 } exponentially fast.
3
2
4
Fig. 2. A four-agent formation with the wrong shape.
5. SIMULATION Example 1: Consider the acyclic triangular formation in Fig. 1 with −t d1 = 3, d2 = 4, d3 = 5. Let agent 1’s velocity e . The three agents are initially positioned at be v = 0
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
{x1 =
2 3 0 }. , x3 = , x2 = 0 0 0
d6 = d7 = 5. Let v =
Note that these initial positions are collinear. We present the simulation results in Fig. 3, and it is clear that the 4.5
e−t . These five agents are initially e−t
0 2 {x1 = , x2 = x3 = x4 = x5 = }. 0 0
As shown in Fig. 5, even in this challenging situation when agents 2, 3, 4 and 5 are initially coincident, the five-agent
3
4
positioned at
3.5
6
3
3
5
2.5
4
2
3
1.5
2
1
1
0.5
5 1
0
1
2
0
2
−1 −2
−0.5 −0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
−3
Fig. 3. Convergence of the acyclic triangular formation
4
−4 −5 −2
triangular formation converges to its desired shape under the proposed control law.
0
2
4
6
8
Fig. 5. Convergence of the five-agent formation Example 2: Consider the four-agent formation in Fig. 1 with the same d1 , d2 , d3 as in Example 1, d4 = 5 and 0 d5 = 4. Let v = −t . These four agents start from e
0 3 10 0 {x1 = }. , x4 = , x3 = , x2 = 4 4 0 0
Note that agents 1, 2 and 4 are required to be clockwise while they are initially positioned counterclockwise. As shown in Fig. 4, the four-agent formation still converges 6
3
5 4 3
6. CONCLUDING REMARKS In this paper, new control laws have been proposed to control the shape of formations of autonomous agents. These proposed control laws do not require that the leader of the formation has to be held still. It can also deal with the challenging situation where a follower’s position is collinear with its leaders. In addition, the control law takes into account the orientation of the triangular sub-formations of agents. When these control laws are applied to acyclic directed formations generated by Henneberg sequence operations, it has been shown that such formations converge after a finite time exponentially fast to their desired shapes. It is still not clear how to obtain similar results for arbitrary directed formations with cycles, even for a four-agent formation consisting of two cyclic triangles. In future, we would like to deign control laws for directed formations containing cycles.
2
1
1
formation still converges to its desired shape.
2
0 −1 −2
REFERENCES
−3
4 −4 0
2
4
6
8
10
Fig. 4. Convergence of the four-agent formation to its desired shape. Example 3: Consider the five-agent formation in Fig. 1 with the same d1 , d2 , d3 , d4 , d5 as in Example 2 and
Anderson, B.D.O., Yu, C., Dasgupta, S., and Morse, A.S. (2007). Control of a three coleader persistent formation in the plane. Systems and Control Letters, 573–578. Anderson, B.D.O., Yu, C., Fidan, B., and Hendrickx, J.M. (2008). Rigid graph control architectures for autonomous formations. IEEE Contr. Syst. Mag., 28(6), 48–63. Basiri, M., Bishop, A.N., and Jensfelt, P. (2010). Distributed control of triangular formations with angle-only constraints. Systems and Control Letters, 59(2), 147– 154.
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Cao, M., Anderson, B.D.O., Morse, A.S., and Yu, C. (2008a). Control of acyclic formations of mobile autonomous agents. Proc. of the 47th Conf. on Decision and Contr., 1187–1192. Cao, M., Morse, A.S., Yu, C., Anderson, B.D.O., and Dasgupta, S. (2007). Controlling a triangular formation of mobile autonomous agents. Proc. of the 46th Conf. on Decision and Contr., 3603–3608. Cao, M., Morse, A.S., Yu, C., Anderson, B.D.O., and Dasgupta, S. (2011a). Maintaining a directed, triangular formation of mobile autonomous agents. Communications in Information and Systems, 11, 1–16. Cao, M., Yu, C., and Anderson, B.D.O. (2011b). Formation control using range-only measurements. Automatica, 47, 776–781. Cao, M., Yu, C., Morse, A.S., Anderson, B.D.O., and Dasgupta, S. (2008b). Generalized controller for directed triangle formations. Proc. of the 17th IFAC Congress, 6590–6595. Dimarogonas, D.V. and Johansson, K.H. (2009). Further results on the stability of distance-based multi-robot formations. Proc. of the 2009 American Control Conference. Dimarogonas, D.V. and Johansson, K.H. (2010). Stability analysis for multi-agent systems using the incidence matrix: quantized communication and formation control. Automatica, 46(4), 695–700. Dimarogonas, D.V. and Kyriakopoulosx, K.J. (2008). A connection between formation infeasibility and velocity alignment in kinematic multi-agent systems. Automatica, 44(10), 2648–2654. Krick, L., Broucke, M., and Francis, B. (2009). Stabilization of infinitesimally rigid formations of multi-robot networks. Int. J. of Control, 82(3), 423–439. Lin, Z., Francis, B., and Maggiore, M. (2004). Local control strategies for groups of mobile autonomous agents. IEEE Trans. Automatic Control, 49, 622–629. Marshall, J., Broucke, M.E., and Francis, B.A. (2004). Formation of vehicles in cyclic pursuit. IEEE Trans. Automatic Control, 9, 1963–1974. Yu, C., Anderson, B.D.O., Dasgupta, S., and Fidan, B. (2009). Control of minimally persistent formations in the plane. SIAM J. Control Optim., 206–233.
ez 2 ≤ e2 (e + d2 ) e2 + e + d2 2 ) ≤( 2 3 1 1 ≤ ( e2 + + d2 )2 , 4 4 2 it is true that 3 2 1 1 2 e + + d 4 4 2 In view of (6) and (7), ez ≤
t ∈ [0, T ).
(7)
V˙ ≤ 3 v V + (1 + 2d2 ) v t ∈ [0, T ), which implies t v(s) ds V (t) ≤ V (0) + (1 + 2d2 ) 0 t +3 v(s) V (s)ds. 0
From the Bellman-Gronwall Lemma, one thus has
V (t) ≤ (V (0) + (1 + 2d2 )a)e3a t ∈ [0, T ), which means V is bounded on [0, T ). So do e and z. Since the maximal solution z(t) is bounded on [0, T ) and continuous, the limit z ∗ = lim z(t) t→T
exists. Now if T were finite there would have to be some interval [T, T¯) of positive length on which there were a continuous solution z¯(t) to (2) which passed through z ∗ at t = T. This in turn would imply that the concatanated function θ defined as θ(t) = z(t), t ∈ [0, T ) and θ(t) = z¯(t), t ∈ [T, T¯) would be a continuous function to (2) passing through z0 at t = 0. But this can not be so because T¯ ≥ T and [0, T ) is the maximal interval of existence. Therefore, the maximal interval of existence is [0, ∞). Since system (2) is locally Lipschitz continuous with z as the state, z is bounded on [0, ∞). Now we prove (3) by contradiction. Assume that (3) is false. Then there exists a finite t2 such that ∞ v dt (8) z(t2 ) < a − 0
7. APPENDIX Proof of Lemma 1: We first show there exists a unique solution to (2) on [0, ∞). It suffices to show that z is bounded on [0, T ), which as defined before is the maximal interval of existence of solution to (2). Let a = min{ z(0) , d} and e = z 2 −d2 . Introduce the function V = e2 . Along the solution to (2), V˙ = −4 z 2 V − 4ez ′ v
Since
≤ −4 z 2 V + 4 ez v
(5)
≤ 4 ez v .
(6)
In addition since z(0) > 0, we can choose t2 in such a way that z(t) > 0, t ∈ [0, t2 ]. Since z(0) ≥ a and z(t2 ) < a, then there must exist a time t1 ∈ [0, t2 ) such that z(t1 ) = a and z(t) ≤ a for t ∈ [t1 , t2 ]. From (2), one has z d z = (d2 − z 2 ) z −v ′ , t ∈ [t1 , t2 ]. dt z z But from the Cauchy Schwartz inequality v ′ z ≤ v , so d z ≥ (d2 − z 2 ) z − v , t ∈ [t1 , t2 ]. dt Since z ≤ a for t ∈ [t1 , t2 ] and a ≤ d, then d z ≥ − v , t ∈ [t1 , t2 ]. dt Hence,
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
z(t2 ) ≥ z(t1 ) − Therefore, z(t2 ) ≥ a −
t2
v dt. t1
∞
v dt,
0
which contradicts (8). Thus, (3) is true. This also implies z is bounded away from 0. From z is bounded and also bounded away from 0, v converges to 0 exponentially fast as t goes to infinity and (5), one has V converges to 0 and y converges to a constant exponentially fast. Then x converges to a constant x∗ ∈ E exponentially fast.
y Proof of Lemma 2: Let y = 1 and define y2 τ (y) =
d1 R(s)(y2 − y1 ) + y1 δ2
(9)
when δ > 0 and τ (y) = y1
(10)
when δ = 0. Since d1 − d2 < y1 − y2 < d1 + d2 for all t ≥ T and in view of the definitions of s, R(s) and τ (y), one has τ (y) is continuously differentiable for t ≥ T with respect to y and thus Lipschitz continuous. Then there exists a ∗ ∗ positive constant l such that
τ (y) − τ (y ) ≤ l y − y ∗ y for t ≥ T , where y ∗ = 1∗ . Therefore τ (y) converges to y2 τ (y ∗ ) exponentially fast after T since y converges to y ∗ exponentially fast for t ≥ T , Note that (4) can be written as x˙ = −δ 2 (x − τ (y)), (11) where δ > 0 and is bounded away from 0 for t ≥ T. Therefore, x converges to x∗ = τ (y ∗ ) exponentially fast after T because of the exponential convergence of the trajectory of system (11) to its equilibrium point τ (y ∗ ), where τ (y ∗ ) ǫE and is in the desired orientation with respect to y1∗ , y2∗ .
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