on the Tight Formation for Multi-Agent Dynamical Systems

On the Tight Formationfor Multi-agent Dynamical Systems

Ionela Prodan1,2, Sorin Olaru1, Cristina Stoica1, and Silviu-Iulian Niculescu2

1 SUPELEC Systems Sciences (E3S) - Automatic Control Department,Gif sur Yvette, France

{ionela.prodan,sorin.olaru,cristina.stoica}@supelec.fr2 SUPELEC CNRS - Laboratory of Signal and Systems, Gif sur Yvette, France

{ionela.prodan,silviu.niculescu}@lss.supelec.fr

Abstract. This paper addresses the real-time control of multiple agentsin the presence of disturbances and non-convex collision avoidance con-straints. The goal is to guarantee the convergence towards a tight forma-tion. A single optimal control problem is solved based on a prediction ofthe future evolution of the system and the resulting controller is imple-mented in a centralized way. At the supervision level, it is shown thatthe decision about which agents should take on what role in the desiredtight formation is equivalent with a classical pairing (or task assignment)problem. Furthermore, the pairing is re-evaluated at each iteration. Theproposed method exhibits eective performance validated through someillustrative examples.

Keywords: Multi-Agent Systems, invariant sets, non-convex constraints,constrained MPC.

1 Introduction

The formation of multiple agents is important in many applications involvingthe control of cooperative systems [3], as for example, in adversary games appli-cations where the sensor assets are limited. Agent formations allow each memberof the team to concentrate its sensors across a specied region of the workspace,while their neighboring members cover the rest of the workspace. Furthermore,in applications like search and rescue, coverage tasks and security patrols, theagents direct their visual and radar search responsibilities depending on theirpositions in the formation [1]. Other applications include coordinated ocean plat-form control for a mobile oshore base [2]. The homogeneous modules formingthe oshore base must be able to perform long-term station keeping at sea, inthe presence of waves, winds and currents. Therefore, the independent moduleshave to be controlled in order to be maintained aligned (i.e., they converge to atight formation). The research of Ionela Prodan is nancially supported by the EADS Corporate

Foundation (091-AO09-1006).

G. Jezic et al. (Eds.): KES-AMSTA 2012, LNAI 7327, pp. 554565, 2012.c Springer-Verlag Berlin Heidelberg 2012

On the Tight Formation for Multi-agent Dynamical Systems 555

Furthermore, the problem of maintaining a formation becomes even morechallenging if one needs to ensure that all the agents avoid collisions inside thegroup [14].

There is a large literature dedicated to the formation control for a collection ofvehicles using potential eld approach [16], or approaches based on graph theory[7]. The authors of [5] and [16] investigate the motions of vehicles modeled asdouble integrators. Their objective is for the vehicles to achieve a common veloc-ity while avoiding collisions with obstacles and/or agents assumed to be points.The derived control laws involve graph Laplacians for an associated undirectedgraph and also nonlinear terms resulting from articial potential functions.

The main goal of this paper is to control a set of agents having independentdynamics while achieving a global objective, such as a tight formation withdesired specications and collision free behavior. For reducing the computationtime we use the nominal behavior of the agents and consider safety regionsaround them to compensate for the eects of the disturbances aecting the realsystems. Further, these regions are dened within the theory of invariant sets inorder to avoid recomputations during the real-time functioning. The formationcontrol problem is decomposed in two separate problems: The o-line denition of the ideal conguration. A minimal conguration isdetermined with respect to a given cost function under the constraints imposedby the safety regions. In real-time, a receding horizon optimization combined with task assignmentrelative to the minimal conguration will be employed.

The real-time control is designed based on the following two-stage procedure:1. Determine "who-goes-where" in the formation. This is equivalent with solv-

ing a standard assignment problem, which is a special case of the so-calledHitchcock Transportation Problem (TP) [11].

2. Solve a mixed-integer optimization problem according to the target geometryof the formation and the associated safety regions.

Finally, this two separate problems are embedded within a Model PredictiveControl (MPC) problem (see, for instance, [9] for basic notions in MPC), lead-ing to an optimization problem for driving the group of agents to a speciedformation with associated target locations.

In the present paper, we revise the preliminary results obtained in [12] andintroduce enhancements in the control design method which enables the stabi-lization of the multi-agent formation. We show that for the convergence to thepredened formation an additional xed point constraint (i.e., the target posi-tions are also equilibrium points for the considered dynamics) must be takeninto account. Moreover other contributions of the paper are, on the one hand,the reduction in the computational cost and, on the other hand, the ecienthandling of an increased number of constraints.

The rest of the paper is organized as follows. Section 2 describes the individualagents model and the set invariance concept. Furthermore, Section 3 presents theconguration of the desired multi-agent formation. Section 4 states themixed-integer optimal control problem embedded within MPC. Finally, several

556 I. Prodan et al.

concluding remarks are drawn in Section 6 and the illustrative examples are pre-sented in Section 5.

The following notations will be used throughout the paper. Minkowskis ad-dition of two sets X and Y is dened as X Y = {A + B : A X , B Y}. Letxk+1|k denote the value of x at time instant k+1, predicted upon the informationavailable at time k N.

2 Preliminaries and Prerequisites

2.1 System DescriptionConsider a set of Na linear systems (vehicles, pedestrians or agents in a generalform) which model the behavior of individual agents. The ith system is describedby the following discrete LTI dynamics aected by additive disturbances:

xik+1d = Aixikd

+ Biuikd + wik, i = 1, . . . , Na, (1)

where xikd Rn are the state variables, uikd Rm is the control input andwik Rn represents a bounded disturbance for the agent i. Henceforth we assumethe following:1. The pair (Ai, Bi) is stabilizable, with Ai Rnn, Bi Rnm.2. The disturbance wi is bounded, i.e. wi W i, where W i is a convex and

compact set containing the origin.Theoretically, formulation (1) suces for solving any typical multi-agent controlproblem (e.g., formation stability, trajectory tracking and so forth). However,the presence of additive noises makes the numerical computation dicult andseverely limits the practical implementability. This is particularly true for cen-tralized schemes where the computations are to be made into an extended space.

The solution followed here is based on the ideas in [10]. As a rst step, weconsider the nominal systems associated to (1):

xik+1 = Aixik + Biuik, i = 1, . . . , Na. (2)

By linking the control laws associated to dynamics (1) and (2), respectively,through the relation

uikd = uik + Ki(xikd xik), (3)

we observe that the tracking error of the ith system, dened as zik xikd xik,is given by:

zik+1 = (Ai + BiKi)zik + wik. (4)Assuming that Ki makes the closed-loop state matrix Ai + BiKi to be Schur1,it follows that an RPI (see Denition 1) set Si can be determined and the real1 The stabilizability hypothesis on the pair (Ai, Bi) implies the existence of an optimal

control law for each agent i, Ki Rnm such that the matrices Ai + BiKi arestable, where the controller Ki, i = 1, . . . , Na is constructed either by a LinearQuadratic (LQ) design using the solution of the discrete algebraic Riccati equationor alternatively by pole placement technique.


trajectory generated by (1) will reside in a tube centered along the nominaltrajectory generated by (2):

xikd xik Si xikd {xik} Si (5)as long as2 zi0 Si for any k 0.

This permits to consider the nominal system in the subsequent optimizationproblems and thus minimize the necessary numerical computations.

Denition 1. A set Rn is robustly positive invariant (RPI) for the discrete-time system xk+1 = xk + with the perturbation Rn, i .A set is minimal robustly invariant (mRPI) for some given dynamics i itis a RPI set in Rn contained in every RPI set for the given dynamics.

There are various algorithms able to oer arbitrarily close RPI approxima-tion for a mRPI set (as for example, the approaches proposed by [13]). It isworth mentioning that these algorithms ignore the exponentially increase in thecomplexity of representation.

2.2 Collision Avoidance Formulation

A typical multi-agent problem is the minimization of some cost problem withconstraints. As stated before, the original formulation, with dynamics (1) isnot optimal since it requires to take into account the bounded disturbancesaecting the dynamics. Hereafter we will use the nominal dynamics (2) andwe will analyze how conditions on the real dynamics (1) are transposed to thenominal dynamics case.

A classical issue in multi-agent formations is the collision avoidance. Usingthe notation of (1), the condition that any two agents do not collide translatesinto:

{xid} {xjd} = , i, j = 1, . . . , Na, i = j. (6)Using the notation from (2) and assuming that the conditions validating relation(5) are veried, we reach the equivalent formulation:

{xi Si} {xj Sj} = , i, j = 1, . . . , Na, i = j. (7)This condition takes explicitly (through the use of the sets Si, Sj) into accountthe uncertainties introduced by the bounded perturbation in (1).

Let us recall that for any two convex sets A,B the following equivalence istrue:

A B = 0 / A {B} . (8)Using (8), we obtain the equivalent formulation for (7):

xi xj / Sj (Si) , i, j = 1, . . . , Na, i = j. (9)2 The assumption that the tracking error starts inside the set is made for simplication

reasons. As long as the set is contractive, after a nite number of steps, any trajectorystarting outside will enter inside the set.


Remark 1. The obstacle avoidance problem can be treated in a similar way: xi /Ol, where {Ol}l=1,...,M denotes the collection of xed obstacles (if a particularobstacle is non-convex, it may be seen as a union of convex sets). Then theconditions that have to be veried is:

xi / Ol (Si) , i = 1, . . . , Na, l = 1, . . . , M. (10)Remark 2. Note that, a solution using parametrized polyhedra (see, for instance,[8]) to describe the safety regions of the agents is presented in [12]. For guarantee-ing that two (or more) agents do not superpose, the parametrized intersections ofthe invariant sets are considered and then, the domain for which the intersectionsare void is described. However, we note that this approach is computationallydemanding.Remark 3. The main technical diculty encountered in this paper is the factthat, often, the feasible regions are non-convex. This problem rises naturallyfrom separation conditions (see condition (9) and Remark 1). The solution is touse the mixed-integer programming techniques [6]. This allows us to express theoriginal non-convex feasible region as a convex set in an extended space. Suchan approach leads to a signicant number of binary variables in the problemformulation, thus leading to unrealistic computational times (in the worst-casescenarios, an exponential increase dependent of the number of binary variables).A method for reducing the computational time is detailed in [15], where wepropose a technique for making the time of computation P-hard in the numberof Linear/Quadratic Programming (LP/QP) subproblems that have to be solved.

3 The Configuration of Multi-agent Formations

3.1 Minimal Conguration of the Multi-agent Formation O-Line

The goal of clustering the agents as close as possible to the origin is realizedthrough a minimal conguration for the group of agents (2). We pose the problemas an optimization problem where the cost function is the sum of the squaredistances of each agent from the origin and the constraints are the ones imposingcollision avoidance (9):

min(xi,ui), i=1,...,Na

Na

i=1xi22, subject to:

{xi xj / Sj {Si} ,xi = Aixi + Biui, i, j = 1, . . . , Na, i = j.

(11)Solving the mixed-integer optimization problem (11), a set of target positionsand the associated control laws are obtained:

T ={(x1f , u1f), (x2f , u2f ), . . . , (x

Naf , u

Naf )

}, (12)

where every pair (xif , uif) is a xed state/input of the ith agent.


Remark 4. Note that the second constraint in (11) is a xed point condition.That is, the optimization problem will nd only pairs (xif , uif), i = 1, . . . , Nawhich are also a xed point for the considered dynamics, (2). Geometrically,this means that the points xif will nd themselves on the associated subspacesspanned by (In Ai)1Bi. In particular, if the agents have the same dynamics(i.e., homogeneous agents), they will have a common subspace over which toselect the xed points xif .

3.2 Task Assignment Formulation On-LineIn the particular case of homogeneous3 agents (understood here as agents withthe same safety regions) we can intercalate an additional step in the controlmechanism. Since the agents have the same safety regions, we can change whogoes where in the minimal conguration computed in the previous subsection.This is equivalent with nding the best permutation over the set of the nal posi-tions in the target formation, T from (12). This is in fact the optimal assignmentproblem encountered in the eld of combinatorial optimization [4].

If one associates a cost with the assignment of agent j to target xif as cij , theproblem of nding the best assignment is dened as:

minij , i,j=1,...,Na

Na

i=1

Na

j=1cijij , subject to:

Na

i=1ij = 1,

Na

j=1ij = 1,

ij {0, 1},

(13)

where ij are the decision variables: 1 if target xif is assigned to agent j and0 otherwise. These binary variables ensure that each agent is assigned to oneunique target position.

The problem is dened by the choice of the cost weights cij , the simplestway is to choose it as the distance between the actual position of agent j andthe desired target position in the formation. Hence, the problem would be todetermine the minimal distance that an agent has to travel to establish theoptimal assignment in the specied formation. A more insightful way is to usethe unconstrained dynamics (2) of the agents to describe the cost of reachingfrom the initial position to the desired position. Then, cij can be described by aweighted norm:

cij = (xj xif )T P (xj xif ), i, j = 1, . . . , Na (14)with the matrix P = P T 0 given by the Lyapunov function or the innite timecost-to-go, as long as the agents follow the unconstrained optimum through thecontrol action:

uj = Kj(xj xif ) + uj , i, j = 1, . . . , Na, i = j, (15)3 In the heterogeneous case the reassignment of the nal destination points is no longer

feasible since the swapping of the safety regions will result in collisions of the agents.


where uj is chosen such that xjf = Ajxif + Bj uj , with uj = B1j (I Aj)xif , if

the matrix Bj is invertible (or the alternative pseudo-inverse which allows thedenition of a xed point for the nominal trajectory). This optimization problemcan be reduced to a simple LP problem, hence it can be eciently computed.

4 Receding Horizon Optimization Problem

The goal is to drive the agents to a minimal conguration (12) (if possible,applying optimization (13)) while in the same time avoiding collisions alongtheir evolution towards the formation.

To this end, we will consider the set of Na constrained systems as a globalsystem dened as:

xk+1 = Agxk + Bguk, (16)

with the corresponding vectors which collects the states and the inputs ofeach individual nominal system (2) at time k, i.e., xk = [x1k

T | |xNakT ]T ,

u = [u1kT | |uNak

T ]T and the matrices which describe the model: Ag =diag[A1, . . . , ANa ], Bg = diag[B1, . . . , BNa ].

We consider an optimal control problem for the global system where thecost function and the constraints couple the dynamic behavior of the individualagents. Also, perfect knowledge of each agent dynamics described by equation (2)is available to all the other agents. Consequently, the global model will be usedin a predictive control context which permits the use of non-convex constraintsfor collision avoidance behavior.

A nite receding horizon implementation is typically based on the solutionof an open-loop optimization problem. An optimal control action u is obtainedfrom the control sequence u

{uk|k, uk+1|k, . . . , uk+N1|k

}as a result of the

optimization problem:

u = argminu

VN (xk+1|k, . . . , xk+N |k, uk|k, . . . , uk+N1|k),

subject to:{

xk+l|k = Agxk+l1|k + Bguk+l1|k, l = 1, . . . , NHij xk+l|k / Sj {Si} , i, j = 1, . . . , Na, i = j,

(17)

where Hij [0 . . . Ii

. . . Ij

. . . 0] is a projection matrix which permits to

rewrite the collision avoidance between two agents i and j in the notation of thecentralized system (16)4.

In order to assure that the target positions (12) we require a cost functionwhich is minimized in the destination points (and not the origin):

4 Using the elements provided in Remark 3, the computational complexity of (17) canbe assessed to a polynomial number of QP problems.


VN (xk|k, uk|k) = (xk+N|kxi(k)f )T P (k+N|kxi(k)f )+

N1

l=1

(xk+N|kxi(k)f )T Q(xk+N|kxi(k)f )+

+N1

l=0

(uTk+l|k ui(k)f )R(uk+l|k ui(k)f

), (18)

with (xi(k)f , ui(k)f ) represents the optimal target positions and the associated

control laws at current time k, i = 1, . . . , Na. Here Q = QT 0, R > 0 arethe weighting matrices with appropriate dimensions, P = P T 0 denes theterminal cost and N denotes the length of the prediction horizon.

First let us summarize in the following algorithm the receding horizon strategytogether with task assignment mechanism:

Algorithm 1. Centralized scheme strategy for a group of agentsInput: initial positions of the agents

1 compute the safety regions associated to each agent;2 compute the minimal conguration as in (11);3 compute the matrices dening the cost function (18);4 for k = 1 : kmax do5 if the safety regions are identical then6 execute task assignment xikref xif as in (13);7 end8 nd the optimal control action u as in (17);9 compute the next value of the state: xk+1 = Agxk + Bguk

10 end

Due to the fact that we use invariant sets, steps 1, 2 and 3 can be executed inan o-line procedure. In the on-line part of the algorithm, we apply a nite hori-zon trajectory optimization: in step 6 we execute a task assignment if possible(only if the safety regions are identical) and then proceed with the actual com-putation of the receding horizon control (step 8). Finally, the rst component ofthe resulting control sequence is eectively applied to the global system (step 9)and the optimization procedure is reiterated using the available measurementsbased on the receding horizon principle [9].Remark 5. Although functional, this scheme will not scale favorably with anincreased number of agents or a large prediction horizon due to the numericaldiculties. In particular, the mixed programming algorithms are very sensitiveto the number of binary auxiliary variables. In this case a decentralized approachis to be envisaged in order to minimize the numerical computations.Remark 6. Note that although desirable, an increase in the length of the pre-diction horizon is not always practical, especially when using mixed-integer pro-gramming. We observed that a two-stage MPC, where in the rst stage a taskassignment procedure is carried and in the second, the usual optimization prob-lem is solved oers good performances with a reduced computational eort.


5 Illustrative Example

For the illustrative example we consider that each of the agent is described bythe following dynamics and disturbances:

Ai =

0 0 1 00 0 0 10 0 imi 00 0 0 imi

, Bi =

0 00 01

mi0

0 1mi

, W i =

wi : |wi|

0.50.30.50.2

, (19)

where [xi yi vix viy]T , [uix uiy]T are the state and the input of each system. Thecomponents of the state are the position (xi, yi) and the velocity (vix, viy) of theith agent, i = 1, . . . , Na. The parameters mi, i are the mass and the dampingfactor, respectively.

10 8 6 4 2 0 2 4 6 8 1010

8

6

4

2

0

2

4

6

8

10

x1

x2

(a) Projection of the RPI set on the positionsubspace

20 15 10 5 0 5 10 1520

15

10

5

0

5

10

15

20

x1

x2

(b) The trajectories of the real and nomi-nal system

10 8 6 4 2 0 2 4 6 8 1020

15

10

5

0

5

10

15

20

x1

x2

(c) The minimal configuration of fourhomogeneous agents

50 40 30 20 10 0 10 20 30 40 5050

40

30

20

10

0

10

20

30

40

50

step 1

step 10

step 20

x1

x2

(d) The evolution of 4 homogeneousagents with task assignment

50 40 30 20 10 0 10 20 30 40 5050

40

30

20

10

0

10

20

30

40

50

step 1

step 10step 20

x1

x2

(e) The evolution of 4 homogeneousagents withought task assignment

Fig. 1. The tight formation of 4 homogeneous agents

First, let us consider Na = 4 homogeneous agents with mi = 45kg,i = 15Ns/m, i = 1, . . . , 4. For the sake of illustration, we construct the RPIsets (i.e.,the safety regions) for the homogeneous agents aected by distur-bances. Using pole placement methods we derive the feedback gain matrices


Ki, i = 1, . . . , 4, which placed the poles of the closed-loop system in the 0.6 to0.9 interval. The RPI sets Si are obtained as detailed in [13] and the projectionon the position subspace of one set is depicted in Figure 1 (a). For this system anominal trajectory (2) (in blue) is constructed and one observes in Figure 1 (b)that any trajectory of system (1) aected by disturbance will verify relation (5)(i.e., resides in a tube described by the RPI set Si ). Furthermore, by solving themixed-integer optimization problem (11) we obtain the set of target positionsT =

{(xif , uif)

}, i = 1, . . . , 4 as in (12), which satisfy the anti-collision conditions

(9) (see, Figure 1 (c)). As it can be seen in Figure 1 (c), for the case of agentswith the same dynamics (as detailed also in Remark 4) the equilibrium pointsof (12) stay on the same subspace, which in this particular case is a line passingthrough the origin.

We next apply the receding horizon scheme (17) for the global system with aprediction horizon N = 2 and the tunning parameters P = 5I4, Q = I4, R = I2.The optimal trajectories for the agents are obtained such that the set of targetpoints is reached through task optimization and under state constraints. Wesummarized the details in Algorithm 1. The eectiveness of the present algorithmis conrmed by the simulation depicted in Figure 1 (d), where the evolution of theagents is represented at three dierent time instances. The agents successfullyreach their target positions in the predened formation without violating theconstraints and with a minimum cost.

For comparison purposes we execute Algorithm 1 with and without the taskassignment stage. As it can be seen in Figure 1 (e) for a prediction horizon of N =2 and without the task assignment procedure, the agents do not converge to thedesired conguration (two agents, depicted in blue and red, respectively switchplaces). Note that if the prediction horizon is long enough (in this particular caseN = 8) the desired conguration is achieved but the computational complexityof the mixed-integer optimization problem (11) increases signicantly.

6 4 2 0 2 4 6 810

5

0

5

10

15

x1

x2

(a) The target configuration of 4 heterogeneousagents

50 40 30 20 10 0 10 20 30 40 5050

40

30

20

10

0

10

20

30

40

50

step 1

step 20

x1

x2

(b) The evolution of 4 heterogeneous agents

Fig. 2. The tight formation of 4 heterogeneous agents


Second, let us consider Na = 4 heterogeneous agents with dierent valuesof the parameters mi and i, i = 1, . . . , 4. The prediction horizon is N = 7and the tunning parameters are P = 50I4, Q = I4, R = I2. Following thesame procedure we depict in Figure 2 (a) the agents with the associated safetyregions in a minimal conguration. Furthermore, in Figure 2 (b) we illustratethe evolution of the agents at two dierent time instances.

6 Conclusions

In this paper, we rst present several tools in order to provide a systematic o-line procedure for the control of a group of agents towards a minimal congura-tion. Second, in real-time a two stage receding horizon control design is adoptedfor driving the agents to the predened formation. Also, we provide several re-marks, leading to computational improvements of the mixed-integer techniquesused to assure a collision free behavior along the evolution of the agents. Theresults are presented through some illustrative simulations of several examples.The current research is to develop software-in-the-loop simulations and subse-quent ight tests for the control of small Unmanned Aerial Vehicles (UAVs). Thevehicle dynamics is simulated by a Piccolo software and then, the control algo-rithm is transmitted in real ight simulations through a communication routinerunning on a PC on the ground.

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On the Tight Formation for Multi-agent Dynamical SystemsIntroductionPreliminaries and PrerequisitesSystem DescriptionCollision Avoidance Formulation

The Configuration of Multi-agent FormationsMinimal Configuration of the Multi-agent Formation Off-LineTask Assignment Formulation On-Line

Receding Horizon Optimization ProblemIllustrative ExampleConclusionsReferences

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on the Tight Formation for Multi-Agent Dynamical Systems