Exact Formation Control with Very Coarse Information
2013 American Control Conference (ACC) Washington, DC, USA, June 17-19, 2013
Exact formation control with very coarse information Matin Jafarian and Claudio De Persis
Abstract— This paper investigates a formation control problem for agents modeled as double integrators when very coarse information is exchanged. We assume that neighboring agents only know whether their relative position is larger or smaller than the prescribed one. The use of this binary information results in very simple control inputs that direct the agents closer or away from each other and take values in finite sets. We also show that the task of keeping a formation and tracking a reference velocity which is only known to the formation’s leader is still achievable under the very coarse information scenario that we consider. In contrast with the other results of practical convergence with coarse or quantized information, here the control task is achieved exactly.
I. INTRODUCTION Distributed formation keeping control of a group of mobile agents is a specific case of motion coordination which aims at reaching a desired geometrical shape for the positions of the agents while maintaining or following a desired velocity. This problem has been addressed with different approaches [1], [3], [5], [18], [22], [23], [24]. In formation keeping control, an important component, besides the dynamics of the agents and the graph topology, is the flow of information among the agents. In fact, the usual assumption in the literature on cooperative control is that a continuous flow of perfect information is exchanged among the agents. Due to the coarseness of sensors and/or to communication constraints, the latter might be a restrictive requirement. To cope with the problem in the case of continuous-time agents’ dynamics, the use of distributed quantized feedback control has been proposed in the literature [7], [8], [9], [14]. In fact, in formation control by quantized feedback control, information is transmitted among the agents whenever measurements cross the thresholds of the quantizer. At these times a value taken from a discrete set and corresponding to the quantization level is transmitted. This allows dealing naturally both with the continuous-time nature of the agents’ dynamics and with the discrete nature of the transmission information process without the need to rely on sampled-data models [7], [10], [13]. Literature review. Coordination in the presence of quantized or coarse information has attracted a great deal of interest, mainly for discrete-time systems ([6], [15], [17], [20] to name a few). Motivated by the problem of reaching consensus in finite-time, [9] adopted binary control laws and M. Jafarian and C. De Persis are with ITM, Faculty of Mathematics and Natural Sciences, University of Groningen, the Netherlands, Tel: +31 50 363 3080. {m.jafarian,c.de.persis}@rug.nl. This work is partially supported by the Dutch Organization for Scientific Research (NWO) under the auspices of the project QUantized Information Control for formation Keeping (QUICK).
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cast the problem in the context of discontinuous control systems. The paper [8] proposed a consensus control algorithm in which the information collected from the neighbors is in binary form. The paper [14] has investigated a similar problem but in the presence of quantized measurements, while [7] has rigorously cast the problem in the framework of non-smooth control systems. The latter has also introduced a new class of hybrid quantizers to deal with possible chattering phenomena. Results on the formation control problem for continuous-time models of the agents in the presence of coarse information have also appeared. The authors of [25] studied a rendezvous problem for Dubins cars based on a ternary feedback which depends on whether or not the neighbor’s position is within the range of observation of the agent’s sensor. Deployment on the line of a group of kinematic agents was shown to be achievable by distributed feedback which uses binary information control [11]. In [10] and [13] formation control problems for groups of agents with second-order non linear dynamics have been studied in the presence of quantized measurements. Main contribution. In the current paper, we study the problem of distributed position-based formation keeping of a group of agents with second-order dynamics which exchange binary information. The binary information models a sensing scenario in which each agent detects whether or not the components of its current distance vector from a neighbor are above or below the prescribed distance and applies a force (in which each component takes a binary value) to reduce or increase the actual distance. Here we show that, remarkably, despite such a coarse information and control action, the control law guarantees exact achievement of the desired formation. This is an interesting result, since statically quantized control inputs typically generate practical convergence, namely the achievement of an approximate formation in which the distance from the actual desired formation depends on the quantizer resolution [10]. Here the use of binary information allows us to conclude asymptotic convergence without the need to dynamically update the quantizer resolution. Moreover, our approach results in control laws which are implemented by very simple directional commands (such as “move north”, “move north-east”, etc. or “stay still”). Furthermore, it is quite straightforward to include the effect of actuator saturation [21]. We also show that the presence of coarse information does not affect the ability of the proposed controllers to achieve the formation in a leader-follower setting in which the prescribed reference velocity is only known to a leader. The paper generalizes to higher-dimensional systems the results of [11]. It adopts a similar setting as in [10] and
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[13], but controllers and analysis are different. Compared with [25], where coarse information was also used for rendezvous problem, the results in our contribution apply to a different class of systems and to a different cooperative control problem. In this paper, we focus on agents that are double integrators. However, our results can be extended to the class of strictly passive systems [1] (cf. Remark 2 below). This paper is organized as follows: Section II introduces the problem statement along with notation and motivations. Analysis of the formation keeping problem with coarse data is studied in Section III. Also, this section presents the analysis of the system in the presence of saturation on the actuators. Section IV investigates the problem of formation keeping with coarse data when the desired reference velocity is just known to one of the agents. Related simulations and results are presented in Section V. The paper is summarized in Section VI. The proofs are omitted due to space constraints1 . Notation. Given two sets S1 , S2 , the symbol S1 × S2 denotes the Cartesian product of two sets. This can be iterated. The m symbol ×k=1 Sk denotes S1 × S2 × . . . × Sm . For a set S, card(S) denotes the cardinality of the set S. Given a matrix M of real numbers, we denote by R(M) and N (M) the range and the null space, respectively. The symbols 1, 0 denotes vectors or matrices of all 1 and 0 respectively. I p is the p × p identity matrix. Sometimes the size of the matrix is explicitly given. Thus, 1n is the n-dimensional vector of all 1. II. P ROBLEM F ORMULATION We consider n agents evolving in R p and modeled as double integrators of the form r˙i v˙i
= =
vi ui , i = 1, 2, . . . , n
(1)
where ri ∈ R p is the position and vi ∈ R p is the velocity of agent i while ui ∈ R p is the control input. The way in which the agents exchange information is modeled with a connected undirected graph G = (V, E), where the set of nodes V coincides with the set of agents (and hence |V | = n) and an edge (i, j) ∈ E ⊂ V ×V models the fact that agents i and j can exchange information. Label one end of each edge in E with a positive sign and the other end with a negative sign. We define the relative position zk between two agents i and j as follows ri − r j if node i is the positive end of the edge k zk = r j − ri if node j is the positive end of the edge k. Let m = |E| be the number of edges. We define B as the (n×m) incidence matrix associated to the graph G as follows +1 if node i is the positive end of the edge k −1 if node i is the negative end of the edge k bik = 0 otherwise.
Define z = (BT ⊗ I p )r as the vector of distances between agents connected by an edge, where the symbol ⊗ denotes 1 A full version of the paper that includes proofs can be found at www.rug.nl/staff/c.de.persis/mj-cdp-acc13.pdf.
the Kronecker product. We define the concatenated vectors r , [r1T . . . rnT ]T r ∈ Rnp ri ∈ R p , z , [zT1 . . . zTm ]T z ∈ Rmp zk ∈ R p , z∗ , [z∗1 T . . . z∗m T ]T z∗ ∈ Rmp z∗k ∈ R p . z∗ is the desired relative position vector. A. Model and Motivation We are interested in control inputs which can guarantee the achievement and maintenance of a certain formation. This formation is specified by a constant vector z∗ ∈ Rmp . Without loss of generality, we assume that z∗ ∈ R(BT ⊗ I p ). It is well known (see e.g. [1]) that the control law u = −Kv − (B ⊗ I p )(z − z∗ ) with K a diagonal and positive definite matrix, guarantees the convergence of the velocity vector to the origin, v → 0, and achievement of the formation, namely z → z∗ . Observe that, apart from the velocity damping −ki vi , which is supposed to be available to the agent i, each agent requires only relative position information from its neighbors, namely m
ui = −ki vi − ∑ bik (zk − z∗k ). k=1
The aim of this paper is to investigate the possibility of achieving the same control objective when only a very approximate knowledge of zk − z∗k is available to the agents. Consider the component ℓ of zk − z∗k , and suppose that the agent is only able to sense whether the actual inter-agent distance zkℓ is above or below the desired distance z∗kℓ and apply a control law to reduce or increase the actual distance. In mathematical terms, this control action has the form −bik sign(zk − z∗k ), where sign y represents the vector (sign y1 . . . sign y p )T , and where the function sign : R → {−1, 1} is defined as sign(y) = +1 if y ≥ 0 and sign(y) = −1 if y < 0. To visualize such a sensing scenario we consider the agents in the plane. Represent the position of the agents with respect to an inertial frame in R2 . Assign a local Cartesian coordinate system with an orientation identical to the inertial frame to each agent such that the current position of the agent is the origin of its local coordinate system. Therefore, each agent partitions the space R2 into four quadrants. Formally, the partition induced by the agent i, whose position with respect to the inertial frame is given by ri , is represented by R2 =
4 [
Qi , where
i=1
Q1 Q2 Q3 Q4
= = = =
{r ∈ R2 : r1 ≥ ri1 , r2 ≥ ri2 } {r ∈ R2 : r1 < ri1 , r2 ≥ ri2 } {r ∈ R2 : r1 < ri1 , r2 < ri2 } {r ∈ R2 : r1 ≥ ri1 , r2 < ri2 }.
Our control scenario assumes that each agent i only detects the quadrant Qi in which the endpoint of the vector −bik (zk − z∗k ) lies. Then, the controller of the agent i exerts the corre√ sponding force along the bisector of Qi with magnitude 2. Thus, if the endpoint of −bik (zk − z∗k ) lies in Q1 (see Fig. 1), then the control action is given by uik = −bik sign(zk − z∗k ) (in the figure, it is denoted as uk for simplicity). We stress
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alizes to higher-dimensional systems and to complex coordination tasks the results of [11], [8]. (vi) The proposed control law paves the way towards selftriggered coordination control ([12]). In the next section we are going to investigate the stability properties of the closed-loop system
Fui
ZiÛ Zi
ui
r˙i
= vi
v˙i
= −kvi vi − kr
m
∑ bik sign(zk − z∗k ),
i = 1, . . . , n
(2)
k=1
where kr , kvi are positive gains.
Zi F ZiÛ
III. A NALYSIS Fig. 1. Agent i (bottom-left), a neighbor j (right) and the desired position of agent i with respect to agent j (top-left, in gray). Agent i detects the quadrant where zk − z∗k and applies the control action −bik sign(zk − z∗k ).
that any other vector whose endpoint lies in Q1 would return the same control value as the one due to −bik (zk − z∗k ). Such a control scheme has several advantages, which we discuss below for the case in which p = 2. (i) Given any quadrant of the partition, the proposed control law returns the same control value for any vector −bik (zk − z∗k ), corresponding to one of the neighbors of agent i, which lies in that quadrant. This shows that the formation is achieved even with large inaccuracies in the measurements of zk − z∗k . In fact, the error |(zk − z∗k ) − sign(zk − z∗k )| ranges between 0 and +∞. (ii) The agent applies a new control action whenever it detects that −bik (zk − z∗k ) is crossing the boundary between two quadrants.2 In our analysis we show that the formation will converge to the desired shape, even if the control action at the boundary of two quadrants takes any value in the convex hull of the two control inputs associated to the two quadrants. (iii) Neglecting the term −ki vi , the control ui takes on values in the discrete set {−di , . . . , −1, 0, +1, . . . , +di } with di the degree of agent i. This means that if one replaces the damping term −ki vi with a bounded function, then the proposed control law implicitly includes saturation effects. (iv) The proposed control scenario explains how a complex task such as achieving a desired formation is guaranteed by simple control directives. To be specific, the force that each agent applies as a result of the interaction with its neighbors takes values in the following finite set i { s.t. i, j ∈ {−1, 0, 1}}. j (v) It contributes to the literature on quantized control systems showing that exact (and not practical as in other statically quantized control systems) convergence is achieved with extremely coarse information. In this respect it gener2 In that regard, the control law is reminiscent of the controllers of [25] that changes when a neighbor enters or leaves the field-of-view sector of the agent.
As a first step in the analysis we express the system (2) in the coordinates (˜z, v), with z˜ = z − z∗ . Thus z˙˜ = (BT ⊗ I p )v v˙ = −Kv v − kr (B ⊗ I p )sign z˜
(3)
where Kv = diag(kv1 , . . . , kvn ) ⊗ I p , and kvi , kr > 0. As a second step we define an appropriate notion of solutions. In fact, due to the discontinuous map sign z˜, the right-hand side of (3) is discontinuous. In this paper, the solutions to the system above are intended in the Krasowskii sense. For lack of space we omit a number of results from non-smooth control theory which are used throughout the paper and for which we refer the interested reader to e.g. [9]. Let X = (˜z, v) and let F(X) be the set valued map T 0 (B ⊗ I p )v F(X) = K sign z˜ (4) − kr (B ⊗ I p ) −Kv v m
p
where K sign z˜ = ×k=1 ×i=1 K sign z˜kℓ and {sign z˜kℓ } i f z˜kℓ 6= 0 K sign z˜kℓ = [−1, 1] i f z˜kℓ = 0. We define X(t) = (˜z(t), v(t)) a Krasowkii solution to (3) on the interval [0,t1 ] if it is an absolutely continuous function which satisfies the differential inclusion ˙ ∈ F(X(t)) X(t)
(5)
for almost every t ∈ [0,t1 ], with F defined as in (4). Krasowskii solutions to the differential inclusion above always exist for sufficiently small values of t. In the following result we state the desired convergence property: Proposition 1 Any Krasowskii solution to (5) exists for all t ≥ 0 and converges to the origin. Remark 1 (Finite-time convergence) In the case of single integrators, the use of signed controllers ([9]) or control with signed measurements ([8]) leads to finite-time convergence. It is not hard to find examples of system (3) for which finitetime convergence does not occur. Remark 2 (Strictly passive nonlinear systems) The result can be extended to systems that have attracted considerable interest in the literature ([1], [4], [10], [19]) and in which the velocity equation in (1) has the form
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v˙i
=
fi (vi ) + gi (vi )ui
(6)
provided that fi , gi satisfy suitable assumptions, namely fi (0) = 0, gi is a full-column rank for all vi and there exists a positive definite and radially unbounded storage function Si (vi ) such that ∇Si (vi )( fi (vi ) + gi (vi )ui ) ≤ −αi (vi ) + uTi yi , for some positive definite function αi and some output function yi = hi (vi ). A. Saturated input A natural modification of the control input proposed above, which could take into account a possible control saturation, is the following: u = −Kv sign v − kr (B ⊗ I p )sign z˜. As a matter of fact, a similar analysis as for the proposition above yields that every Krasowskii solution converges to the largest weakly invariant set Z such that v = 0. Differently from the previous case, however, such a largest weakly invariant set strictly contains the origin. Proposition 2 Let p = 1.3 The largest weakly invariant set Z with respect to the system z˜˙ = (BT ⊗ I p )v v˙ = −Kv sign v − kr (B ⊗ I p )sign z˜
(7)
such that v = 0 contains all the points (˜z, 0) for which two components z˜k , z˜l exist that share one node and are such that z˜k 6= 0, z˜ℓ = 0. The latter result allows to conclude that the introduction of the discontinuous damping term −Kv sign v creates new undesired (Krasowskii) equilibria and should be avoided. To limit the velocity damping term, an intuitive alternative is to resort to a standard saturation function in the control input. In this case the control input becomes m
ui = −kvi sat vi − kr
∑ k=1
where sat : R → R is defined as r if sat r = 1 if
Consider now the scenario in which the formation should converge to the desired shape and evolve with a given reference velocity v∗ . Define the velocity error v˜ = v − 1n ⊗ v∗ . It satisfies v˙˜ = u − 1n ⊗ v˙∗ . Assuming that both v∗ and v˙∗ are available to each agent, one can set u = 1n ⊗ v˙∗ − Kv v˜ − kr (B ⊗ I p )sign z˜ which results in the closed-loop system z˙˜ = (BT ⊗ I p )v˜ v˙˜ = −Kv v˜ − kr (B ⊗ I p )sign z˜. Notice that in the first equation we are exploiting the property N (BT ) = R(1n ). The system above is the same as (3) and we can similarly conclude that all the Krasowskii solutions to the system converge to the set of points where z˜ = 0 and v˜ = 0, that is the agents converge to the prescribed inter-agent vector z∗ and the velocities vi converge to v∗ . Assuming the knowledge of both v∗ and v˙∗ for all the agents is very restrictive and one wonders whether it is possible to relax such assumption despite the very coarse information scenario we are confining ourselves. The positive answer to this question is given in the analysis below. The problem of the reference velocity unknown to the agents is tackled here in the scenario in which at least one of the agents is aware of v∗ . We refer to this agent as the leader ([18], [22]). As in [3], we further assume that the class of reference velocities that the formation can achieve are those generated by an autonomous system that we refer to as the exosystem. Given two matrices Φ and Γv , whose properties will be made precise later on, the exosystem obeys the following equations w˙ = Φw,
η˙ i vˆi
r≤1 r > 1.
In the case r is a vector, the symbol sat r has to be intended component-wise. The closed-loop system becomes (8)
and in the associated differential inclusion (5) the set-valued map F is T 0 (B ⊗ I p )satv F(X) = − K sign z˜. kr (B ⊗ I p ) −Kv sat v (9) Then the following can be proven: Corollary 1 Any Krasowskii solution to (5) with F as in (9) exists for all t ≥ 0 and converges to the origin. 3 The
VELOCITY
v∗ = Γv w.
(10)
Taking inspiration from the theory of output regulation (see e.g. [16]), an internal-model-based controller can be adopted for each agent i = 2, 3, . . . , n, namely
bik sign(zk − z∗k )
z˙˜ = (BT ⊗ I p )v v˙ = −Kv satv − kr (B ⊗ I p )sign z˜
IV. F ORMATION CONTROL WITH UNKNOWN REFERENCE
= =
Φηi + Gu˘i Γv ηi ,
i = 2, 3, . . . , n.
(11)
When u˘i = 0 and the system is appropriately initialized, the latter system is able to generate any w solution to (10). Because v∗ is known to agent 1, there is no need to implement system (11) at agent 1. The input u˘i , as well as the input matrix G are to be designed later. We define the new velocity error v˜ as v˜ = v − v, ˆ where vˆ = ˙ˆ Since vˆ is available (v∗ T vˆT2 . . . vˆTn ). Then, it satisfies v˙˜ = u− v. ˜ then and v˙ˆi = Γv (Φηi + Gu˘i ), if we choose u = v˙ˆ − Kv v˜ + u, this is still a distributed controller provided that both u˘ and u˜ are distributed inputs. The control input above gives v˙˜ = −Kv v˜ + u. ˜ Further introduce the new variables η˜ i = ηi − w,
v˘i = vˆi − v∗ ,
i = 2, . . . , n,
along with the fictitious variables η˜ 1 = 0, v˘1 = 0. We obtain
case p > 1 does not change the analysis but complicates the notation.
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η˙˜ i v˘i
= =
Φη˜ i + Gu˘i Γv η˜ i ,
i = 2, 3, . . . , n,
(12)
0
v˘ = |
0 0 0 0 .. η˜ + .. . . Φ 0 } |
... ... .. . ...
0
{z
|
14
form
˜ Φ
0 0 .. .
0 Γv .. .
0
0
0 G .. .
... ... .. .
0
... {z G˜
12
0 u˘1 0 u˘2 .. . . . . u˘n G | {z } }
10 8 Position−y (m)
and in compact 0 0 0 Φ η˙˜ = . . .. ..
u˘
−2 −4 0
Γv
...
2
4
6 8 Position−x (m)
10
12
14
Fig. 2. The evolution of five agents in R2 tracking a constant reference velocity just known to the leader. The system reaches the desired formation and exhibits a translational motion with the desired constant velocity.
}
Γ˜ v
trajectory of agent 1 trajectory of agent 2 trajectory of agent 3 trajectory of agent 4 trajectory of agent 5
0
0 0 ˜ .. η. .
{z
4 2
... ... .. .
6
Finally, write the equation for z˜ as 2
z˙˜ = (BT ⊗ I p )v = (BT ⊗ I p )(v − vˆ + vˆ − 1n ⊗ v∗ )
z˜11
1
T
= (B ⊗ I p )(v˜ + v). ˘
The overall closed-loop system is (13)
3
4
5
6
7
8
9
10
0
1
2
3
4
5
6
7
8
9
10
0
1
2
3
4
5
6
7
8
9
10
u˜11
(15)
(16)
converge to the point z˜ = 0, v˜ = 0, η˜ = 0. V. S IMULATIONS In this section we present the simulation results for a group of five agents with unit mass and double integrator dynamics evolving in R2 . The agents are communicating over a connected graph. The associated incidence matrix is +1 0 0 0 −1 +1 0 0 . 0 −1 +1 0 B= 0 0 −1 +1 0 0 0 −1 The desired formation has a pentagon shape with edge length equal to two and is defined by the following inter-agent √ distance vectors: z∗1 = [0 − 2]T , z∗2 = [−1 − 1 − 3]T , √ z∗3 = [−1 1 + 3]T , z∗4 = [0 2]T . Note that the number of edges of the graph is four. The initial position of the agent is set to r(0) = [0.5 − 0.5 0.5 1 1 0.5 0.8 0 1.1 0]T . Gains kr and Kv are equal to one and I2 . The simulation refers to the case of a formation evolving with a constant reference velocity known only to the formation leader (agent
0
−2
(14)
Then all the Krasowskii solutions of (13) in closed-loop with the control input u˜ = u˘ = −kr (B ⊗ I p )sign z˜
2
2
(Γv , Φ) is an observable pair and let G = ΓvT .
1
0
−2
Proposition 3 Assume that Φ is skew-symmetric, namely ΦT + Φ = 0,
0
2
sign(˜z11 )
˜ z˜˙ = (BT ⊗ I p )(v˜ + Γ˜ v η) v˙˜ = −Kv v˜ + u˜ ˜ η˜ + G˜ u. η˙˜ = Φ ˘
0 −1
Time (s)
Fig. 3. The plot of the (horizontal) x-component of z˜1 , sign z˜1 and the corresponding control u˜1 .
1). The following are set as internal model parameters for all agents: Φ = 02×2 , Γv = I2 , G = I2 , w(0) = [0.6 0.6]T , and ηi (0) = [0 0]T . Figure 2 shows the evolution of the described system with desired constant velocity v∗ = [0.6 0.6]T . The other agents generate the same reference velocity using the control laws based on the internal model principle. Figure 3 shows the time behavior of the horizontal component of z˜1 , sign z˜1 and the corresponding control u˜1 . As time elapses, z˜1 converges to the origin implying convergence to the desired relative position. While z˜1 converges to zero, sign z˜1 and u˜1 converge to the discontinuity surface and oscillate between +1 and −1. Figure 4 shows the horizontal and vertical components of the estimated velocities vˆi for the other four agents. The leader generates a constant desired velocity and the remaining agents estimate the same reference velocity after some time. Another example concerns the case of a time-varying reference velocity. In this simulation, the following are set as the internalmodel parameters: 0 1 0.5 0.5 0 0 v Φ = I2 ⊗ , Γ = , −1 0 0 0 −0.5 0.5 G = ΓvT , w(0) = [1 1 1 1]T , and ηi (0) = [0 0 0 0]T . Figure 5 shows the horizontal and vertical components of the estimated velocities vˆi for the other four agents. The leader generates a sinusoidal desired velocity and the remaining agents asymptotically estimate the reference velocity.
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R EFERENCES
In this paper we presented the analysis of the distributed formation keeping problem with very coarse exchanged data with two scenarios. First, the desired reference velocity is known to all agents (or is zero), and second the desired velocity is just known to one of the agents (namely the leader). For the first scenario, we have also showed that the result holds considering saturated control laws. Although our results have been stated for agents modeled as double integrators, they can be extended to a general class of nonlinear strictly passive systems. Possible future avenues of research include the extension of the results to deal with disturbance rejection properties and time-varying topologies. Moreover, discontinuous control laws as those considered in this paper can be viewed as the outcome of a nonsmooth optimization problem associated with the original control problem ([4]). Finally, we observe that as the system converges to the prescribed formation, fast oscillation of the control inputs between +1 and −1 may occur. This could be overcome by the self-triggered approach of [12].
[1] M.Arcak. Passivity as a design tool for group coordination. IEEE Transactions on Automatic Control 52 (8): 1380-1390, 2007. [2] A. Bacciotti and F. Ceragioli. Stability and stabilization of discontinuous systems and nonsmooth Lyapunov functions. ESAIM Control, Optimisation and Calculus of Variations, (4):361–376, 1999. [3] H. Bai, M. Arcak, and J. Wen. Cooperative Control Design: A Systematic, Passivity-Based Approach. Communications and Control Engineering. Springer, New York, 2011. [4] M. B¨urger, D. Zelazo, and F. Allg¨ower. Network clustering: A dynamical systems and saddle-point perspective. In Proceedings of the 50th IEEE Conference on Decision and Control, Orlando, FL, 7825–7830, 2011. [5] F. Bullo, J. Cort´es and S. Mart´ınez. Distributed Control of Robotic Networks. Series in Applied Mathematics, Princeton University Press, 2009. [6] R. Carli, F. Bullo, and S. Zampieri. Quantized average consensus via dynamic coding/decoding schemes. International Journal of Robust and Nonlinear Control, 20(2):156–175, 2010. [7] F. Ceragioli, C. De Persis, and P. Frasca. Discontinuities and hysteresis in quantized average consensus. Automatica, 47:1916–1928, 2011. [8] G. Chen, F. L. Lewis, and L. Xie. Finite-time distributed consensus via binary control protocols. Automatica 47 (9): 1962-1968, 2011. [9] J. Cort´es. Finite-time convergent gradient flows with applications to network consensus. Automatica 42 (11): 1993–2000, 2006. [10] C. De Persis. On the passivity approach to quantized coordination problems. In Proceedings of the 2011 50th IEEE Conference on Decision and Control and European Control Conference, pages 1086– 1091, 2011. [11] C. De Persis, M. Cao, and F. Ceragioli. A note on the deployment of kinematic agents by binary information. Proceedings of the IEEE Conference on Decision and Control, Orlando, FL, pp. 2487-2492, 2011. [12] C. De Persis and P. Frasca. Self-triggered coordination with ternary controllers. In Proceedings of the 3rd IFAC Workshop on Distributed Estimation and Control in Networked Systems (NecSys’12), Santa Barbara, CA, 2012, pp. 43–48. Extended version available at http://arxiv.org/abs/1205.6917. [13] C. De Persis and B. Jayawardhana. Coordination of passive systems under quantized measurements. SIAM Journal on Control and Optimization, to appear. Available in ArXiv e-prints 1108.4216, http://arxiv.org/abs/1108.4216. [14] D. V. Dimarogonas and K. H. Johansson. Stability analysis for multiagent systems using the incidence matrix: Quantized communication and formation control. Automatica, 46(4):695–700, 2010. [15] P. Frasca, R. Carli, F. Fagnani, and S. Zampieri. Average consensus on networks with quantized communication. International Journal of Robust and Nonlinear Control, 19(16):1787–1816, 2009. [16] A. Isidori, L. Marconi, and A. Serrani. Robust Autonomous Guidance:An Internal Model Approach. London, U.K., Springer, 2003. [17] A. Kashyap, T. Basar, and R. Srikant. Quantized consensus. Automatica, 43(7):1192–1203, 2007. [18] M. Mesbahi and M. Egerstedt. Graph Theoretic Methods in Multiagent Networks. Princeton University Press, 2010. [19] U. M¨unz, A. Papachristodoulou and F. Allg¨ower. Robust consensus controller design for nonlinear relative degree two multi-agent systems with communication constraints. IEEE Transactions on Automatic Control, 56(1):145-151, 2011. [20] A. Nedic, A. Olshevsky, A. Ozdaglar, and J. N. Tsitsiklis. On distributed averaging algorithms and quantization effects. IEEE Transactions on Automatic Control, 54(11):2506–2517, 2009. [21] W. Ren. On consensus algorithms for double-integrator-dynamics. IEEE Transactions on Automatic Control, 53, 6, 1503–1509, 2008. [22] W. Ren, R. Beard. Distributed Consensus in Multi-vehicle Cooperative Control: Theory and Applications. Communications and Control Engineering. Springer, 2007. [23] W. Ren, Y. Cao. Distributed Coordination of Multi-agent Networks. Communications and Control Engineering Series, Springer-Verlag, London, 2011. [24] H. G. Tanner, A. Jadbabaie, and G. J. Pappas. Stable flocking of mobile agents, part I: Fixed topology. volume 2, pages 2010–2015, 2003. [25] J. Yu, S. La Valle, D. Liberzon. Rendezvous without coordinates. IEEE Transactions on Automatic Control 57 (2): 421-434, 2012.
vˆ in x-direction (m/s)
VI. C ONCLUSION
2
v∗
1.5
vˆ2
vˆ3
vˆ4
vˆ5
1 0.5 0 −0.5 −1
vˆ in y-direction (m/s)
−1.5
0
5
10
0
5
10
15
20
25
30
15
20
25
30
1.5 1 0.5 0 −0.5
Time (s)
vˆ in x-direction (m/s)
Fig. 4. The horizontal and vertical component of the estimated reference velocity vˆi for each agent i. The leader (agent 1) generates the desired velocity v∗ = [0.6 0.6]T (m/s).
vˆ2
vˆ3
vˆ4
vˆ5
1 0.5 0 −0.5 −1 −1.5
vˆ in y-direction (m/s)
v∗
1.5
0
5
10
15
20
0
5
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15
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25
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40
45
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50
1.5 1 0.5 0 −0.5 −1 −1.5
Time (s)
Fig. 5. The horizontal and vertical component of the estimated reference velocity vˆi for each agent i. The leader (agent 1) generates the desired velocity v∗ = [−sin(t) cos(t)]T .
3031
Exact formation control with very coarse information Matin Jafarian and Claudio De Persis
Abstract— This paper investigates a formation control problem for agents modeled as double integrators when very coarse information is exchanged. We assume that neighboring agents only know whether their relative position is larger or smaller than the prescribed one. The use of this binary information results in very simple control inputs that direct the agents closer or away from each other and take values in finite sets. We also show that the task of keeping a formation and tracking a reference velocity which is only known to the formation’s leader is still achievable under the very coarse information scenario that we consider. In contrast with the other results of practical convergence with coarse or quantized information, here the control task is achieved exactly.
I. INTRODUCTION Distributed formation keeping control of a group of mobile agents is a specific case of motion coordination which aims at reaching a desired geometrical shape for the positions of the agents while maintaining or following a desired velocity. This problem has been addressed with different approaches [1], [3], [5], [18], [22], [23], [24]. In formation keeping control, an important component, besides the dynamics of the agents and the graph topology, is the flow of information among the agents. In fact, the usual assumption in the literature on cooperative control is that a continuous flow of perfect information is exchanged among the agents. Due to the coarseness of sensors and/or to communication constraints, the latter might be a restrictive requirement. To cope with the problem in the case of continuous-time agents’ dynamics, the use of distributed quantized feedback control has been proposed in the literature [7], [8], [9], [14]. In fact, in formation control by quantized feedback control, information is transmitted among the agents whenever measurements cross the thresholds of the quantizer. At these times a value taken from a discrete set and corresponding to the quantization level is transmitted. This allows dealing naturally both with the continuous-time nature of the agents’ dynamics and with the discrete nature of the transmission information process without the need to rely on sampled-data models [7], [10], [13]. Literature review. Coordination in the presence of quantized or coarse information has attracted a great deal of interest, mainly for discrete-time systems ([6], [15], [17], [20] to name a few). Motivated by the problem of reaching consensus in finite-time, [9] adopted binary control laws and M. Jafarian and C. De Persis are with ITM, Faculty of Mathematics and Natural Sciences, University of Groningen, the Netherlands, Tel: +31 50 363 3080. {m.jafarian,c.de.persis}@rug.nl. This work is partially supported by the Dutch Organization for Scientific Research (NWO) under the auspices of the project QUantized Information Control for formation Keeping (QUICK).
978-1-4799-0178-4/$31.00 ©2013 AACC
cast the problem in the context of discontinuous control systems. The paper [8] proposed a consensus control algorithm in which the information collected from the neighbors is in binary form. The paper [14] has investigated a similar problem but in the presence of quantized measurements, while [7] has rigorously cast the problem in the framework of non-smooth control systems. The latter has also introduced a new class of hybrid quantizers to deal with possible chattering phenomena. Results on the formation control problem for continuous-time models of the agents in the presence of coarse information have also appeared. The authors of [25] studied a rendezvous problem for Dubins cars based on a ternary feedback which depends on whether or not the neighbor’s position is within the range of observation of the agent’s sensor. Deployment on the line of a group of kinematic agents was shown to be achievable by distributed feedback which uses binary information control [11]. In [10] and [13] formation control problems for groups of agents with second-order non linear dynamics have been studied in the presence of quantized measurements. Main contribution. In the current paper, we study the problem of distributed position-based formation keeping of a group of agents with second-order dynamics which exchange binary information. The binary information models a sensing scenario in which each agent detects whether or not the components of its current distance vector from a neighbor are above or below the prescribed distance and applies a force (in which each component takes a binary value) to reduce or increase the actual distance. Here we show that, remarkably, despite such a coarse information and control action, the control law guarantees exact achievement of the desired formation. This is an interesting result, since statically quantized control inputs typically generate practical convergence, namely the achievement of an approximate formation in which the distance from the actual desired formation depends on the quantizer resolution [10]. Here the use of binary information allows us to conclude asymptotic convergence without the need to dynamically update the quantizer resolution. Moreover, our approach results in control laws which are implemented by very simple directional commands (such as “move north”, “move north-east”, etc. or “stay still”). Furthermore, it is quite straightforward to include the effect of actuator saturation [21]. We also show that the presence of coarse information does not affect the ability of the proposed controllers to achieve the formation in a leader-follower setting in which the prescribed reference velocity is only known to a leader. The paper generalizes to higher-dimensional systems the results of [11]. It adopts a similar setting as in [10] and
3026
[13], but controllers and analysis are different. Compared with [25], where coarse information was also used for rendezvous problem, the results in our contribution apply to a different class of systems and to a different cooperative control problem. In this paper, we focus on agents that are double integrators. However, our results can be extended to the class of strictly passive systems [1] (cf. Remark 2 below). This paper is organized as follows: Section II introduces the problem statement along with notation and motivations. Analysis of the formation keeping problem with coarse data is studied in Section III. Also, this section presents the analysis of the system in the presence of saturation on the actuators. Section IV investigates the problem of formation keeping with coarse data when the desired reference velocity is just known to one of the agents. Related simulations and results are presented in Section V. The paper is summarized in Section VI. The proofs are omitted due to space constraints1 . Notation. Given two sets S1 , S2 , the symbol S1 × S2 denotes the Cartesian product of two sets. This can be iterated. The m symbol ×k=1 Sk denotes S1 × S2 × . . . × Sm . For a set S, card(S) denotes the cardinality of the set S. Given a matrix M of real numbers, we denote by R(M) and N (M) the range and the null space, respectively. The symbols 1, 0 denotes vectors or matrices of all 1 and 0 respectively. I p is the p × p identity matrix. Sometimes the size of the matrix is explicitly given. Thus, 1n is the n-dimensional vector of all 1. II. P ROBLEM F ORMULATION We consider n agents evolving in R p and modeled as double integrators of the form r˙i v˙i
= =
vi ui , i = 1, 2, . . . , n
(1)
where ri ∈ R p is the position and vi ∈ R p is the velocity of agent i while ui ∈ R p is the control input. The way in which the agents exchange information is modeled with a connected undirected graph G = (V, E), where the set of nodes V coincides with the set of agents (and hence |V | = n) and an edge (i, j) ∈ E ⊂ V ×V models the fact that agents i and j can exchange information. Label one end of each edge in E with a positive sign and the other end with a negative sign. We define the relative position zk between two agents i and j as follows ri − r j if node i is the positive end of the edge k zk = r j − ri if node j is the positive end of the edge k. Let m = |E| be the number of edges. We define B as the (n×m) incidence matrix associated to the graph G as follows +1 if node i is the positive end of the edge k −1 if node i is the negative end of the edge k bik = 0 otherwise.
Define z = (BT ⊗ I p )r as the vector of distances between agents connected by an edge, where the symbol ⊗ denotes 1 A full version of the paper that includes proofs can be found at www.rug.nl/staff/c.de.persis/mj-cdp-acc13.pdf.
the Kronecker product. We define the concatenated vectors r , [r1T . . . rnT ]T r ∈ Rnp ri ∈ R p , z , [zT1 . . . zTm ]T z ∈ Rmp zk ∈ R p , z∗ , [z∗1 T . . . z∗m T ]T z∗ ∈ Rmp z∗k ∈ R p . z∗ is the desired relative position vector. A. Model and Motivation We are interested in control inputs which can guarantee the achievement and maintenance of a certain formation. This formation is specified by a constant vector z∗ ∈ Rmp . Without loss of generality, we assume that z∗ ∈ R(BT ⊗ I p ). It is well known (see e.g. [1]) that the control law u = −Kv − (B ⊗ I p )(z − z∗ ) with K a diagonal and positive definite matrix, guarantees the convergence of the velocity vector to the origin, v → 0, and achievement of the formation, namely z → z∗ . Observe that, apart from the velocity damping −ki vi , which is supposed to be available to the agent i, each agent requires only relative position information from its neighbors, namely m
ui = −ki vi − ∑ bik (zk − z∗k ). k=1
The aim of this paper is to investigate the possibility of achieving the same control objective when only a very approximate knowledge of zk − z∗k is available to the agents. Consider the component ℓ of zk − z∗k , and suppose that the agent is only able to sense whether the actual inter-agent distance zkℓ is above or below the desired distance z∗kℓ and apply a control law to reduce or increase the actual distance. In mathematical terms, this control action has the form −bik sign(zk − z∗k ), where sign y represents the vector (sign y1 . . . sign y p )T , and where the function sign : R → {−1, 1} is defined as sign(y) = +1 if y ≥ 0 and sign(y) = −1 if y < 0. To visualize such a sensing scenario we consider the agents in the plane. Represent the position of the agents with respect to an inertial frame in R2 . Assign a local Cartesian coordinate system with an orientation identical to the inertial frame to each agent such that the current position of the agent is the origin of its local coordinate system. Therefore, each agent partitions the space R2 into four quadrants. Formally, the partition induced by the agent i, whose position with respect to the inertial frame is given by ri , is represented by R2 =
4 [
Qi , where
i=1
Q1 Q2 Q3 Q4
= = = =
{r ∈ R2 : r1 ≥ ri1 , r2 ≥ ri2 } {r ∈ R2 : r1 < ri1 , r2 ≥ ri2 } {r ∈ R2 : r1 < ri1 , r2 < ri2 } {r ∈ R2 : r1 ≥ ri1 , r2 < ri2 }.
Our control scenario assumes that each agent i only detects the quadrant Qi in which the endpoint of the vector −bik (zk − z∗k ) lies. Then, the controller of the agent i exerts the corre√ sponding force along the bisector of Qi with magnitude 2. Thus, if the endpoint of −bik (zk − z∗k ) lies in Q1 (see Fig. 1), then the control action is given by uik = −bik sign(zk − z∗k ) (in the figure, it is denoted as uk for simplicity). We stress
3027
alizes to higher-dimensional systems and to complex coordination tasks the results of [11], [8]. (vi) The proposed control law paves the way towards selftriggered coordination control ([12]). In the next section we are going to investigate the stability properties of the closed-loop system
Fui
ZiÛ Zi
ui
r˙i
= vi
v˙i
= −kvi vi − kr
m
∑ bik sign(zk − z∗k ),
i = 1, . . . , n
(2)
k=1
where kr , kvi are positive gains.
Zi F ZiÛ
III. A NALYSIS Fig. 1. Agent i (bottom-left), a neighbor j (right) and the desired position of agent i with respect to agent j (top-left, in gray). Agent i detects the quadrant where zk − z∗k and applies the control action −bik sign(zk − z∗k ).
that any other vector whose endpoint lies in Q1 would return the same control value as the one due to −bik (zk − z∗k ). Such a control scheme has several advantages, which we discuss below for the case in which p = 2. (i) Given any quadrant of the partition, the proposed control law returns the same control value for any vector −bik (zk − z∗k ), corresponding to one of the neighbors of agent i, which lies in that quadrant. This shows that the formation is achieved even with large inaccuracies in the measurements of zk − z∗k . In fact, the error |(zk − z∗k ) − sign(zk − z∗k )| ranges between 0 and +∞. (ii) The agent applies a new control action whenever it detects that −bik (zk − z∗k ) is crossing the boundary between two quadrants.2 In our analysis we show that the formation will converge to the desired shape, even if the control action at the boundary of two quadrants takes any value in the convex hull of the two control inputs associated to the two quadrants. (iii) Neglecting the term −ki vi , the control ui takes on values in the discrete set {−di , . . . , −1, 0, +1, . . . , +di } with di the degree of agent i. This means that if one replaces the damping term −ki vi with a bounded function, then the proposed control law implicitly includes saturation effects. (iv) The proposed control scenario explains how a complex task such as achieving a desired formation is guaranteed by simple control directives. To be specific, the force that each agent applies as a result of the interaction with its neighbors takes values in the following finite set i { s.t. i, j ∈ {−1, 0, 1}}. j (v) It contributes to the literature on quantized control systems showing that exact (and not practical as in other statically quantized control systems) convergence is achieved with extremely coarse information. In this respect it gener2 In that regard, the control law is reminiscent of the controllers of [25] that changes when a neighbor enters or leaves the field-of-view sector of the agent.
As a first step in the analysis we express the system (2) in the coordinates (˜z, v), with z˜ = z − z∗ . Thus z˙˜ = (BT ⊗ I p )v v˙ = −Kv v − kr (B ⊗ I p )sign z˜
(3)
where Kv = diag(kv1 , . . . , kvn ) ⊗ I p , and kvi , kr > 0. As a second step we define an appropriate notion of solutions. In fact, due to the discontinuous map sign z˜, the right-hand side of (3) is discontinuous. In this paper, the solutions to the system above are intended in the Krasowskii sense. For lack of space we omit a number of results from non-smooth control theory which are used throughout the paper and for which we refer the interested reader to e.g. [9]. Let X = (˜z, v) and let F(X) be the set valued map T 0 (B ⊗ I p )v F(X) = K sign z˜ (4) − kr (B ⊗ I p ) −Kv v m
p
where K sign z˜ = ×k=1 ×i=1 K sign z˜kℓ and {sign z˜kℓ } i f z˜kℓ 6= 0 K sign z˜kℓ = [−1, 1] i f z˜kℓ = 0. We define X(t) = (˜z(t), v(t)) a Krasowkii solution to (3) on the interval [0,t1 ] if it is an absolutely continuous function which satisfies the differential inclusion ˙ ∈ F(X(t)) X(t)
(5)
for almost every t ∈ [0,t1 ], with F defined as in (4). Krasowskii solutions to the differential inclusion above always exist for sufficiently small values of t. In the following result we state the desired convergence property: Proposition 1 Any Krasowskii solution to (5) exists for all t ≥ 0 and converges to the origin. Remark 1 (Finite-time convergence) In the case of single integrators, the use of signed controllers ([9]) or control with signed measurements ([8]) leads to finite-time convergence. It is not hard to find examples of system (3) for which finitetime convergence does not occur. Remark 2 (Strictly passive nonlinear systems) The result can be extended to systems that have attracted considerable interest in the literature ([1], [4], [10], [19]) and in which the velocity equation in (1) has the form
3028
v˙i
=
fi (vi ) + gi (vi )ui
(6)
provided that fi , gi satisfy suitable assumptions, namely fi (0) = 0, gi is a full-column rank for all vi and there exists a positive definite and radially unbounded storage function Si (vi ) such that ∇Si (vi )( fi (vi ) + gi (vi )ui ) ≤ −αi (vi ) + uTi yi , for some positive definite function αi and some output function yi = hi (vi ). A. Saturated input A natural modification of the control input proposed above, which could take into account a possible control saturation, is the following: u = −Kv sign v − kr (B ⊗ I p )sign z˜. As a matter of fact, a similar analysis as for the proposition above yields that every Krasowskii solution converges to the largest weakly invariant set Z such that v = 0. Differently from the previous case, however, such a largest weakly invariant set strictly contains the origin. Proposition 2 Let p = 1.3 The largest weakly invariant set Z with respect to the system z˜˙ = (BT ⊗ I p )v v˙ = −Kv sign v − kr (B ⊗ I p )sign z˜
(7)
such that v = 0 contains all the points (˜z, 0) for which two components z˜k , z˜l exist that share one node and are such that z˜k 6= 0, z˜ℓ = 0. The latter result allows to conclude that the introduction of the discontinuous damping term −Kv sign v creates new undesired (Krasowskii) equilibria and should be avoided. To limit the velocity damping term, an intuitive alternative is to resort to a standard saturation function in the control input. In this case the control input becomes m
ui = −kvi sat vi − kr
∑ k=1
where sat : R → R is defined as r if sat r = 1 if
Consider now the scenario in which the formation should converge to the desired shape and evolve with a given reference velocity v∗ . Define the velocity error v˜ = v − 1n ⊗ v∗ . It satisfies v˙˜ = u − 1n ⊗ v˙∗ . Assuming that both v∗ and v˙∗ are available to each agent, one can set u = 1n ⊗ v˙∗ − Kv v˜ − kr (B ⊗ I p )sign z˜ which results in the closed-loop system z˙˜ = (BT ⊗ I p )v˜ v˙˜ = −Kv v˜ − kr (B ⊗ I p )sign z˜. Notice that in the first equation we are exploiting the property N (BT ) = R(1n ). The system above is the same as (3) and we can similarly conclude that all the Krasowskii solutions to the system converge to the set of points where z˜ = 0 and v˜ = 0, that is the agents converge to the prescribed inter-agent vector z∗ and the velocities vi converge to v∗ . Assuming the knowledge of both v∗ and v˙∗ for all the agents is very restrictive and one wonders whether it is possible to relax such assumption despite the very coarse information scenario we are confining ourselves. The positive answer to this question is given in the analysis below. The problem of the reference velocity unknown to the agents is tackled here in the scenario in which at least one of the agents is aware of v∗ . We refer to this agent as the leader ([18], [22]). As in [3], we further assume that the class of reference velocities that the formation can achieve are those generated by an autonomous system that we refer to as the exosystem. Given two matrices Φ and Γv , whose properties will be made precise later on, the exosystem obeys the following equations w˙ = Φw,
η˙ i vˆi
r≤1 r > 1.
In the case r is a vector, the symbol sat r has to be intended component-wise. The closed-loop system becomes (8)
and in the associated differential inclusion (5) the set-valued map F is T 0 (B ⊗ I p )satv F(X) = − K sign z˜. kr (B ⊗ I p ) −Kv sat v (9) Then the following can be proven: Corollary 1 Any Krasowskii solution to (5) with F as in (9) exists for all t ≥ 0 and converges to the origin. 3 The
VELOCITY
v∗ = Γv w.
(10)
Taking inspiration from the theory of output regulation (see e.g. [16]), an internal-model-based controller can be adopted for each agent i = 2, 3, . . . , n, namely
bik sign(zk − z∗k )
z˙˜ = (BT ⊗ I p )v v˙ = −Kv satv − kr (B ⊗ I p )sign z˜
IV. F ORMATION CONTROL WITH UNKNOWN REFERENCE
= =
Φηi + Gu˘i Γv ηi ,
i = 2, 3, . . . , n.
(11)
When u˘i = 0 and the system is appropriately initialized, the latter system is able to generate any w solution to (10). Because v∗ is known to agent 1, there is no need to implement system (11) at agent 1. The input u˘i , as well as the input matrix G are to be designed later. We define the new velocity error v˜ as v˜ = v − v, ˆ where vˆ = ˙ˆ Since vˆ is available (v∗ T vˆT2 . . . vˆTn ). Then, it satisfies v˙˜ = u− v. ˜ then and v˙ˆi = Γv (Φηi + Gu˘i ), if we choose u = v˙ˆ − Kv v˜ + u, this is still a distributed controller provided that both u˘ and u˜ are distributed inputs. The control input above gives v˙˜ = −Kv v˜ + u. ˜ Further introduce the new variables η˜ i = ηi − w,
v˘i = vˆi − v∗ ,
i = 2, . . . , n,
along with the fictitious variables η˜ 1 = 0, v˘1 = 0. We obtain
case p > 1 does not change the analysis but complicates the notation.
3029
η˙˜ i v˘i
= =
Φη˜ i + Gu˘i Γv η˜ i ,
i = 2, 3, . . . , n,
(12)
0
v˘ = |
0 0 0 0 .. η˜ + .. . . Φ 0 } |
... ... .. . ...
0
{z
|
14
form
˜ Φ
0 0 .. .
0 Γv .. .
0
0
0 G .. .
... ... .. .
0
... {z G˜
12
0 u˘1 0 u˘2 .. . . . . u˘n G | {z } }
10 8 Position−y (m)
and in compact 0 0 0 Φ η˙˜ = . . .. ..
u˘
−2 −4 0
Γv
...
2
4
6 8 Position−x (m)
10
12
14
Fig. 2. The evolution of five agents in R2 tracking a constant reference velocity just known to the leader. The system reaches the desired formation and exhibits a translational motion with the desired constant velocity.
}
Γ˜ v
trajectory of agent 1 trajectory of agent 2 trajectory of agent 3 trajectory of agent 4 trajectory of agent 5
0
0 0 ˜ .. η. .
{z
4 2
... ... .. .
6
Finally, write the equation for z˜ as 2
z˙˜ = (BT ⊗ I p )v = (BT ⊗ I p )(v − vˆ + vˆ − 1n ⊗ v∗ )
z˜11
1
T
= (B ⊗ I p )(v˜ + v). ˘
The overall closed-loop system is (13)
3
4
5
6
7
8
9
10
0
1
2
3
4
5
6
7
8
9
10
0
1
2
3
4
5
6
7
8
9
10
u˜11
(15)
(16)
converge to the point z˜ = 0, v˜ = 0, η˜ = 0. V. S IMULATIONS In this section we present the simulation results for a group of five agents with unit mass and double integrator dynamics evolving in R2 . The agents are communicating over a connected graph. The associated incidence matrix is +1 0 0 0 −1 +1 0 0 . 0 −1 +1 0 B= 0 0 −1 +1 0 0 0 −1 The desired formation has a pentagon shape with edge length equal to two and is defined by the following inter-agent √ distance vectors: z∗1 = [0 − 2]T , z∗2 = [−1 − 1 − 3]T , √ z∗3 = [−1 1 + 3]T , z∗4 = [0 2]T . Note that the number of edges of the graph is four. The initial position of the agent is set to r(0) = [0.5 − 0.5 0.5 1 1 0.5 0.8 0 1.1 0]T . Gains kr and Kv are equal to one and I2 . The simulation refers to the case of a formation evolving with a constant reference velocity known only to the formation leader (agent
0
−2
(14)
Then all the Krasowskii solutions of (13) in closed-loop with the control input u˜ = u˘ = −kr (B ⊗ I p )sign z˜
2
2
(Γv , Φ) is an observable pair and let G = ΓvT .
1
0
−2
Proposition 3 Assume that Φ is skew-symmetric, namely ΦT + Φ = 0,
0
2
sign(˜z11 )
˜ z˜˙ = (BT ⊗ I p )(v˜ + Γ˜ v η) v˙˜ = −Kv v˜ + u˜ ˜ η˜ + G˜ u. η˙˜ = Φ ˘
0 −1
Time (s)
Fig. 3. The plot of the (horizontal) x-component of z˜1 , sign z˜1 and the corresponding control u˜1 .
1). The following are set as internal model parameters for all agents: Φ = 02×2 , Γv = I2 , G = I2 , w(0) = [0.6 0.6]T , and ηi (0) = [0 0]T . Figure 2 shows the evolution of the described system with desired constant velocity v∗ = [0.6 0.6]T . The other agents generate the same reference velocity using the control laws based on the internal model principle. Figure 3 shows the time behavior of the horizontal component of z˜1 , sign z˜1 and the corresponding control u˜1 . As time elapses, z˜1 converges to the origin implying convergence to the desired relative position. While z˜1 converges to zero, sign z˜1 and u˜1 converge to the discontinuity surface and oscillate between +1 and −1. Figure 4 shows the horizontal and vertical components of the estimated velocities vˆi for the other four agents. The leader generates a constant desired velocity and the remaining agents estimate the same reference velocity after some time. Another example concerns the case of a time-varying reference velocity. In this simulation, the following are set as the internalmodel parameters: 0 1 0.5 0.5 0 0 v Φ = I2 ⊗ , Γ = , −1 0 0 0 −0.5 0.5 G = ΓvT , w(0) = [1 1 1 1]T , and ηi (0) = [0 0 0 0]T . Figure 5 shows the horizontal and vertical components of the estimated velocities vˆi for the other four agents. The leader generates a sinusoidal desired velocity and the remaining agents asymptotically estimate the reference velocity.
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R EFERENCES
In this paper we presented the analysis of the distributed formation keeping problem with very coarse exchanged data with two scenarios. First, the desired reference velocity is known to all agents (or is zero), and second the desired velocity is just known to one of the agents (namely the leader). For the first scenario, we have also showed that the result holds considering saturated control laws. Although our results have been stated for agents modeled as double integrators, they can be extended to a general class of nonlinear strictly passive systems. Possible future avenues of research include the extension of the results to deal with disturbance rejection properties and time-varying topologies. Moreover, discontinuous control laws as those considered in this paper can be viewed as the outcome of a nonsmooth optimization problem associated with the original control problem ([4]). Finally, we observe that as the system converges to the prescribed formation, fast oscillation of the control inputs between +1 and −1 may occur. This could be overcome by the self-triggered approach of [12].
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vˆ in x-direction (m/s)
VI. C ONCLUSION
2
v∗
1.5
vˆ2
vˆ3
vˆ4
vˆ5
1 0.5 0 −0.5 −1
vˆ in y-direction (m/s)
−1.5
0
5
10
0
5
10
15
20
25
30
15
20
25
30
1.5 1 0.5 0 −0.5
Time (s)
vˆ in x-direction (m/s)
Fig. 4. The horizontal and vertical component of the estimated reference velocity vˆi for each agent i. The leader (agent 1) generates the desired velocity v∗ = [0.6 0.6]T (m/s).
vˆ2
vˆ3
vˆ4
vˆ5
1 0.5 0 −0.5 −1 −1.5
vˆ in y-direction (m/s)
v∗
1.5
0
5
10
15
20
0
5
10
15
20
25
30
35
40
45
50
25
30
35
40
45
50
1.5 1 0.5 0 −0.5 −1 −1.5
Time (s)
Fig. 5. The horizontal and vertical component of the estimated reference velocity vˆi for each agent i. The leader (agent 1) generates the desired velocity v∗ = [−sin(t) cos(t)]T .
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