Hector Garcia De Marina

Zhiyong Sun

Ming Cao

Brian D O Anderson

Controlling a triangular flexible formation of autonomous agents. ⋆

Keywords: Formation control, Distributed control, Multi-agent system

In formation control, triangular formations consisting of three autonomous agents serve as a class of benchmarks that can be used to test and compare the performances of different controllers. We present an algorithm that combines the advantages of both positionand distance-based gradient descent control laws. For example, only two pairs of neighboring agents need to be controlled, agents can work in their own local frame of coordinates and the orientation of the formation with respect to a global frame of coordinates is not prescribed. We first present a novel technique based on adding artificial biases to neighboring agents' range sensors such that their eventual positions correspond to a collinear configuration. Right after, a small modification in the bias terms by introducing a prescribed rotation matrix will allow the control of the bearing of the neighboring agents.

INTRODUCTION

The theory of rigidity and in particular the concepts of infinitesimal and minimal rigidity have been proven to be very useful in formation control with the goal to define and achieve prescribed shapes by just controlling the distances between neighboring agents (see [START_REF] Anderson | Rigid graph control architectures for autonomous formations[END_REF] and [START_REF] Krick | Stabilization of infinitesimally rigid formations of multi-robot networks[END_REF]). The popular distancebased gradient descent algorithm for rigid formations has appealing properties. For example, the agents can work in their own local frame of coordinates, the system can be made robust against biased sensors, and since the orientation of the desired shape is not prescribed, one can induce rotational motions to the formation as presented in Garcia de Marina et al. (2015[START_REF] Garcia De Marina | Distributed rotational and translational maneuvering of rigid formations and their applications[END_REF]. In comparison, position-based formation control needs a common frame of coordinates and the steady-state orientation of the shape is restricted, which implies that a free rotational motion is not allowed; however, the same formation shape can be achieved by fewer pairs of neighboring agents in positionbased than in distance-based control. This prompts a ⋆ The work of Hector Garcia de Marina was supported by Mistrale project, http://mistrale.eu. The work of Zhiyong Sun and Brian Anderson was supported by the Australian Research Council Grants DP160104500, DP130103610 and the Prime Ministers Australia Asia Incoming Endeavour Postgraduate Award. The work of Cao was supported in part by the European Research and the Netherlands Organization for Scientific Research (NWO-vidi-14134). E-mails: hgdemarina@ieee.org; {zhiyong.sun,brian.anderson}@anu.edu.au; m.cao@rug.nl.

search for compromise among the advantages of both formation control techniques.

The minimum total number of pairs of neighboring agents in a distance-based rigid formation control system is given by the necessary conditions for infinitesimal and minimal rigidity, which is equal to 2n -3 for 2D scenarios, where n is the total number of agents. This is directly related to the necessary number of inter-agent distances to be controlled such that the desired shape is (at least locally) uniquely defined. For example, the triangle is the simplest rigid shape in the 2D case and the necessary number of desired distances to define such a shape is three. On the other hand, one can achieve a desired triangular shape by just controlling (the relative positions rather than distances of) two neighboring agent pairs in the position-based setup. We illustrate these basic concepts in Figure 1. This paper focuses on triangular formations consisting of three agents. This apparently simple setup has been considered as a benchmark in formation control (see [START_REF] Cao | Controlling a triangular formation of mobile autonomous agents[END_REF]; [START_REF] Anderson | Control of a three-coleader formation in the plane[END_REF]; [START_REF] Cao | Generalized controller for directed triangle formations[END_REF]; [START_REF] Liu | Controlling triangular formations of autonomous agents in finite time using coarse measurements[END_REF]; [START_REF] Mou | Undirected rigid formations are problematic[END_REF]), since it allows detailed rigorous analysis for novel techniques as we aim at in this work. This provides a starting point in order to achieve more general formations. The goal of this paper is to propose an algorithm that combines the advantages of both distanceand position-based control, i.e., agents employ their own local frames of coordinates, no prescribed orientation for the desired shape (so rotational motions are allowed) and smaller number of controlled pairs of neighboring agents than in distance-based minimally rigid formations.

The novelty of this algorithm lies in exploiting the sensingerror-induced collective motion, e.g., those caused by biased range sensors among neighboring agents. For example, it has been reported in [START_REF] Mou | Undirected rigid formations are problematic[END_REF] that small constant biases in range sensors for a rigid formation cause the whole team of agents to converge to a distorted version of the desired shape and eventually it will exhibit some steady-state motion. On the other hand, it has been shown in Garcia de Marina et al. ( 2016) that one can introduce artificially such biases in order to steer the desired rigid formation in a controlled way.

This paper will first study the effect of these biases in a flexible (or non-rigid) distance-based formation of three agents, i.e., we control only two distances instead of three for a rigid triangular formation. We prove that if the range measurements between agents are not perfectly accurate then the three agents converge to collinear positions. This is somewhat counter-intuitive since for a non-rigid formation the steady-state shape is, depending on the initial conditions, arbitrary within a constraint set. Furthermore, we will show that depending on the value of these biases, the eventual collinear formation will move with a constant velocity or remain stationary. In our first step, we take advantage of this effect in order to align the three agents with the desired distances between them. Then, we will show how a small modification in this biased control law can achieve the control of the angle between the two relative vectors of the flexible formation, leading to the control of a stationary desired triangular shape with no restriction on its orientation.

In the following section we introduce some notations, briefly review the (exponential) stability of formation systems under the flexible shape and introduce the biases in the range measurements. We continue in Section 3 analyzing the consequences of such biases in the control of a non-rigid formation. We introduce a technique motivated by [START_REF] Mou | Undirected rigid formations are problematic[END_REF], where the authors study nonminimally but infinitesimally rigid formations by reducing the problem to minimally rigid ones. In this paper since we are dealing with fewer pairs of neighboring agents than in minimally rigid setups, we instead augment our formation to a minimally rigid one in order to study the stability of a biased non-rigid formation. In Section 4 we introduce and analyze a modified version of the biased algorithm in order to control triangular shapes. We finish the paper with simulations in Section 5 and some conclusions in Section 6.

NON-RIGID FORMATION OF THREE AGENTS

Gradient descent distance-based formation control

We consider a team of three agents governed by the firstorder kinematic model ṗi = u i , (1) where p i ∈ R 2 is the position of the agent i = {1, 2, 3}, and u i ∈ R 2 is the control action over agent i.

Let us define the two vectors

z 1 ∆ = p 1 -p 2 , z 2 ∆ = p 2 -p 3 ,
(2) which are the relative positions corresponding to the two links available to the agents. For each link one can construct a potential function V k , k ∈ {1, 2} with its minimum at the desired distance d k , so that the gradient of such functions can be used to control inter-agent distances distributively. We consider the following shape potential function

V k (z k ) = 1 2 (||z k || -d k ) 2 , (3)
with the following gradient along

z k ∇ z k V k (z k ) = ẑk (||z k || -d k ), (4)
where ẑk

∆ = z k ||z k || .
We then apply to each agent i in (1) the following gradient descent control

u i = -∇ pi 2 k=1 V k (z k ), (5)
and by denoting the distance error for the kth link by (7) System (7) has some interesting properties. For example, the agents can employ their own local systems of coordinates, i.e., a global or common frame of coordinates is not necessary, and collision avoidance is guaranteed among pairs of neighboring agents [START_REF] Oh | A survey of multi-agent formation control[END_REF]). For tree graph topology, one can prove the (almost global) exponential convergence of the error system, i.e., the signals e 1 (t) and e 2 (t) both converge to zero exponentially fast if the agents do not start at the same positions [START_REF] Dimarogonas | On the stability of distance-based formation control[END_REF]; [START_REF] Sun | Exponential stability for formation control systems with generalized controllers: A unified approach[END_REF]).

e k = ||z k || -d k , (
It is clear that although the error signals e 1 (t) and e 2 (t) converge to zero, the final relative positions may not guarantee any prescribed shapes as one would like to have for minimally rigid formations. In particular, agents p 1 and p 3 will lie somewhere on a circumference with the center at p 2 and the radius as d 1 and d 2 respectively. More precisely, the relative positions converge to the set

Z ∆ = {z : ||z k || = d k , ∀k ∈ {1, 2}}.
(8) Furthermore, the exponential convergence of e(t) and z(t) to the origin and Z, respectively, implies that ṗ1 (t) and ṗ2 (t) converge to zero exponentially fast, therefore the agents converge to only one stationary shape among all the ones defined by Z, i.e., all the agents eventually stop.

Biases in distance-based formation control

It is clear from (5) with V k as in (3) that neighboring agents i and j share the same potential V k in the implementation of the gradient formation control. Let us focus on agent i whose control is

u i = - 2 k=1 u k i ,
where u k i is the corresponding gradient ∇ pi V k . More precisely, for the link k where i and j are neighbors we have that

u k i = ∇ pi V k = ẑk (||z k || -d k ) (9) u k j = ∇ pj V k = -ẑ k (||z k || -d k ).
(10) However, when the range sensors of the two agents are biased with respect to each other by a constant µ k , without loss of generality, the pair ( 9)-( 10) should be modified as

u k i = ẑk (||z k || -d k + µ k ) (11) u k j = -ẑ k (||z k || -d k ). ( 12
)
The effects of these biases on undirected rigid formations have been studied in [START_REF] Mou | Undirected rigid formations are problematic[END_REF] and how to remove or take advantage of them have been shown in Garcia de Marina et al. (2015[START_REF] Garcia De Marina | Distributed rotational and translational maneuvering of rigid formations and their applications[END_REF]. Flexible formations have not received much attention for formation control problems as the rigid ones since the sets like (8) do not define (locally) uniquely any shape. Nevertheless, we will show that the intentional introduction of biases (following the spirit of Garcia de Marina et al. ( 2016)) can make these apparently flexible formations appealing for formation control.

BIASED NON-RIGID FORMATION OF THREE AGENTS: CONVERGING TO A COLLINEAR STEADY-STATE

Let us consider biases in the range sensors of the agents in system (7). In particular, as it has been shown in ( 11)-( 12), we can always write these biases from the point of view of agent 2, namely ṗ2 = ẑ1 (e 1 + µ 1 ) -ẑ2 (e 2µ 2 ), (13) which leads to the following biased system

   ṗ1 = -ẑ 1 e 1 ṗ2 = ẑ1 e 1 -ẑ2 e 2 + µ 1 ẑ1 + µ 2 ẑ2 ṗ3 = ẑ2 e 2 . (14)
We shall now study the effect of imposing a special condition on the biases. This condition will be achievable if we are free to introduce biases into the controls, namely

µ 1 = -µ 2 = c, (15
) for some constant c ∈ R\{0}. These biases can be considered as a parametric disturbance to the following closedloop system involving the dynamics of z and e derived from ( 14):

       ż1 = -2ẑ 1 e 1 + ẑ2 e 2 -cẑ 1 + cẑ 2 ż2 = -2ẑ 2 e 2 + ẑ1 e 1 + cẑ 1 -cẑ 2 ė1 = -2e 1 + ẑT 1 ẑ2 e 2 -c + cẑ T 1 ẑ2 ė2 = -2e 2 + ẑT 2 ẑ1 e 1 -c + cẑ T 2 ẑ1 . (16)
As we have discussed, one can derive the exponential stability of the closed-loop system ( 16) with c = 0 (Dimarogonas and Johansson ( 2008)). Therefore the stability of ( 16) for small values of c is not compromised but at the cost of (probably) shifting its equilibrium from the one described in (8).

One realizes by a quick inspection in ( 14) together with (15)1 that e 1 = e 2 = 0 with ẑ1 = ẑ2 = ẑ * is an equilibrium for the system ( 16), implying that all the agents are stationary. Note that the agents are collinear with agent 2 in the middle for this configuration. We define this equilibrium for ( 16) by

U d ∆ = {z, e : (e 1 = e 2 = 0) ∧ (ẑ 1 = ẑ2 )}.
(17) However, a more careful inspection of ( 16) reveals that there is another equilibrium corresponding to e 1 = e 2 = -2c

3 with ẑ1 = -ẑ 2 = ẑ * , i.e., agents 1 and 3 converge to the same side of agent 2. This implies by inspecting ( 14) that the whole formation is travelling with velocity ṗi = 2c 3 ẑ * , ∀i = {1, 2, 3}. We define this equilibrium for ( 16) by

U u ∆ = {z, e : (e 1 = e 2 = - 2c 3 ) ∧ (ẑ 1 = -ẑ 2 )}. ( 18
)
Proposition 1. Consider the closed-loop system (16) derived from the biased flexible formation control ( 14). If µ 1 = -µ 2 = c with c ∈ R\{0}, then the only two equilibrium sets for ( 16) are U d and U u .

Proof. By using the last two equations of ( 16) one has that the following condition holds at the equilibrium (2 + ẑT 1 ẑ2 )(e 2e 1 ) = 0, (19) from which one can derive that regardless of ẑ1 and ẑ2 , one has that e 1 = e 2 = e * at an equilibrium. Note that this condition implies that the norms of z 1 and z 2 at the equilibrium are constant. We are going to check that such a situation can only occur at the equilibrium. For example, it might be possible that both vectors could be rotating while keeping their norms constant. Let us write the first two equations of system ( 16) and require that the error signals are constant d dt

z 1 z 2 =       - 2e * -c ||z 1 || * e * + c ||z 2 || * e * + c ||z 1 || * - 2e * -c ||z 2 || *    ⊗ I 2    z 1 z 2 , (20)
where I 2 is the 2×2 identity matrix, the symbol ⊗ denotes the Kronecker product, and we have that ||z 1 || * and ||z 2 || * are constant since e 1 = e 2 = e * . Note that we have split ẑ{1,2} =

1 ||z {1,2} || * z {1,2}
. It is clear that the matrix in (20) cannot be skew-symmetric since the off-diagonal terms always have the same sign, therefore we discard the possibility of rotations for z 1 and z 2 when the error signals are constant. Therefore ė1 = ė2 = 0 implies that the time derivatives ż1 = ż2 = 0 as well. By using the first two equations in (16) we can derive the following condition at an equilibrium of ( 16) It is clear from (21) that we only have two possible equilibrium sets. Let us check the case ẑ * 1 = ẑ * 2 . By checking the equilibrium of the third equation in ( 16) one can quickly derive that e * = 0. So we have checked the equilibrium U d . For the second case e * = -2c 3 , one can check in the first equation in ( 16) that for its equilibrium the relation ẑ * 1 = -ẑ * 2 must be satisfied. So we have checked the equilibrium U u .

(ẑ * 2 -ẑ * 1 )(3e * + 2c) = 0, ( 21
One may wonder about the stability of U d and U u . In fact, if one is interested in having the three agents in a collinear fashion while they are stationary it would be desirable to know that at least U d is locally stable. We will see this fact in more detail in the following subsection.

Stability analysis of the error system

By inspecting the system (16) one realizes that the error system for the flexible formation is not autonomous. This is due to the fact that the term

ẑT 1 ẑ2 = 1 2||z 1 ||||z 2 || (||p 3 -p 1 || 2 -||z 1 || 2 -||z 2 || 2 ), ( 22
)
depends on ||p 3p 1 || 2 and it cannot be written as a function of e 1 and e 2 . It would be desirable to handle a system involving only scalar variables than a mixture of scalar and vectorial ones as in ( 16). The key idea for starting the stability analysis of ( 16) is by including the following states

e 3 ∆ = ||z 3 || -d 3 , z 3 ∆ = p 3 -p 1 ,
where one can choose the value for d 3 depending on the equilibrium to be studied. For example, for the equilibrium U d we set

d 3 = d 1 + d 2 .
The inclusion of these states will make the augmented system for the error vector e ∆ = [e 1 e 2 e 3 ]

T autonomous and therefore easier to analyze.

Let us first write the dynamics of e derived from ( 14)

   ė1 = -2e 1 + ẑT 1 ẑ2 e 2 -c + cẑ T 1 ẑ2 ė2 = -2e 2 + ẑT 2 ẑ1 e 1 -c + cẑ T 2 ẑ1 ė3 = ẑT 3 ẑ1 e 1 + ẑT 3 ẑ2 e 2 . ( 23
)
By employing the relation ( 22) and similarly for ẑT 3 ẑ2 and ẑT 3 ẑ1 one can derive the following self-contained dynamics for the error system

                                 ė1 = -2e 1 -c + (e 2 + c) (e 3 + d 3 ) 2 -(e 1 + d 1 ) 2 -(e 2 + d 2 ) 2 2(e 1 + d 1 )(e 2 + d 2 ) ė2 = -2e 2 -c + (e 1 + c) (e 3 + d 3 ) 2 -(e 1 + d 1 ) 2 -(e 2 + d 2 ) 2 2(e 1 + d 1 )(e 2 + d 2 ) ė3 = e 1 (e 2 + d 2 ) 2 -(e 1 + d 1 ) 2 -(e 3 + d 3 ) 2 2(e 1 + d 1 )(e 3 + d 3 ) + e 2 (e 1 + d 1 ) 2 -(e 2 + d 2 ) 2 -(e 3 + d 3 ) 2 2(e 2 + d 2 )(e 3 + d 3 ) .
(24) Now we linearize the system (24) at the equilibrium U d or equivalently e = 0 for d 3 = d 1 + d 2 . We first compute the partial derivatives of

f (e) = (e 3 + d 3 ) 2 -(e 1 + d 1 ) 2 -(e 2 + d 2 ) 2 2(e 1 + d 1 )(e 2 + d 2 ) (25) ∂f (e) ∂e 1 | e=0 = - (e 3 + d 3 ) 2 -(e 1 + d 1 ) 2 -(e 2 + d 2 ) 2 2(e 1 + d 1 ) 2 (e 2 + d 2 ) | e=0 - 2(e 1 + d 1 ) 2(e 1 + d 1 )(e 2 + d 2 ) | e=0 = - d 2 3 -d 2 1 -d 2 2 2d 2 1 d 2 - 1 d 2 = - d 1 + d 2 d 1 d 2 ∂f (e) ∂e 2 | e=0 = - d 2 3 -d 2 1 -d 2 2 2d 1 d 2 2 - 1 d 1 = - d 1 + d 2 d 1 d 2 ∂f (e) ∂e 3 | e=0 = d 3 d 1 d 2 = d 1 + d 2 d 1 d 2 ,
that in combination with

f (0) = d 2 3 -d 2 1 -d 2 2 2d 1 d 2 = 1, (26)
helps us to arrive at the Jacobian matrix defining the linearization of the autonomous system ( 24) at e = 0 as below

J e = -2 -ca 1 -ca ca 1 -ca -2 -ca ca -1 -1 0 , ( 27
)
where a = d1+d2 d1d2 is always positive. Note that for the particular case c = 0 we have that in ( 27) the error signals e 1 and e 2 do not depend on e 3 and the points e 1 = e 2 = 0 are stable regardless of e 3 . In fact, in such a case we have a flexible shape where the steady state of e 3 will depend on the initial conditions p(0), i.e., the agents converge to the set (8). Remark 1. Note that by just adding a bias in a range sensor, no matter how small it is, we are linking the dynamics of the error signals e 1 and e 2 with e 3 . This has an important implication since in practice it is not common (or arguably possible) to have perfect measurements in the agents' sensors. Therefore, to have a truly flexible formation in practice would be quite challenging without any further action in (7). A technique employing estimators in order to remove the biases has been introduced in Garcia de Marina et al. (2015).

In order to check the stability of the origin of system (24) we compute the characteristic polynomial of ( 27)

P (λ) = -λ(2 + ca + λ) 2 -2ca(1 -ca) -2ca(2 + ca + λ) + λ(1 -ca) 2 = (λ + 1)(λ + 3)(λ + 2ca), ( 28
) therefore we can conclude that for any positive (negative) c the equilibrium U d is locally exponentially stable (unstable). The same analysis can be done for checking the stability of U u which is unstable (stable) for positive (negative) values of c. We summarize these findings with the following proposition. Proposition 2. Consider system ( 14) for positive values of c = µ 1 = -µ 2 , then the equilibria U d and U u for system ( 16) are (exponentially) stable and unstable respectively. The converse holds for negative values of c. Remark 2. For negative values of c we have that the agents not only converge to a collinear formation but by inspecting ( 14) we have that ṗi (t) → z * 3c 2 , as t → ∞, ∀i ∈ {1, 2, 3}, i.e., the agents will travel with a constant velocity. Remark 3. Checking the eigenvalues given by P (λ) show that one cannot increase the convergence speed by increasing c. Conversely it can be made arbitrarily slow by choosing a small positive c. In fact, the effect of a small bias in the range sensors of system (7) will be practically noticed after a sufficiently long time.

FLEXIBLE TRIANGULAR SHAPE

In this section we are going to show that the previous setup for collinear formations is just a particular case of a general setup where we can design controllers to achieve a prescribed angle between z 1 and z 2 .

Consider the following system

   ṗ1 = -ẑ 1 e 1 ṗ2 = ẑ1 e 1 -ẑ2 e 2 + W (θ)ẑ 1 -ẑ2 ṗ3 = ẑ2 e 2 . , (29)
where

W (θ) = cos θ -sin θ sin θ cos θ , (30)
is a rotation matrix. One can check that the two cases µ 1 = -µ 2 = 1 and µ 1 = µ 2 = -1 in ( 14) are in fact particular cases of (29) with θ = 0 and ±π radians respectively. In fact, the angle θ defines the prescribed angle between ẑ1 and ẑ2 in a target formation and note that the agents are not required to measure the actual angle between z 1 and z 2 . We proceed to show that the agents indeed can employ their own local frames. Lemma 3. The agents in system (29) can employ their own local frames of coordinates.

Proof. Consider y ij for the position of agent j with respect to a fixed local frame of coordinates for agent i with j, i ∈ {1, 2, 3}. Therefore, there exists a rotation matrix R i and a translation vector τ i such that y ij = R i p j + τ i . Note that the vectors τ i with i ∈ {1, 2, 3} vanish for the calculation of z 1 and z 2 and the rotation matrices do not affect the value of e 1 and e 2 . Hence, agent 2 measures the signals R 2 z 1 , R 2 z 2 and we have that W (θ)R 2 z 2 = R 2 W (θ)z 2 since rotation matrices commute in 2D. Thus, for agent 2 in global coordinates we have that

ṗ2 = R T 2 (R 2 ẑ1 e 1 -R 2 ẑ2 e 2 + R 2 W (θ)ẑ 1 -R 2 ẑ2 ) = ẑ1 e 1 -ẑ2 e 2 + W (θ)ẑ 1 -ẑ2 . (31)
Therefore, one can conclude that the control action of agent 2 does not need any global information, e.g., a common frame of coordinates. The same reasoning can be applied to agents 1 and 3. .

For the sake of convenience for the stability analysis, let us reformulate (29) as

       ṗ1 = -ẑ 1 e 1 ṗ2 = ẑ1 e 1 -ẑ2 e 2 + W ( θ 2 )ẑ 1 -W ( θ 2 ) T ẑ2 ṗ3 = ẑ2 e 2 , . (32)
Note that we are still defining the same desired steadystate angle between ẑ1 and ẑ2 . The corresponding augmented error system derived from (32), with

2 d 2 3 = d 2 1 + d 2 2 + 2d 1 d 2 cos θ, is given by          ė1 = -2e 1 + ẑT 1 ẑ2 e 2 -ẑT 1 W ( θ 2 )ẑ 1 + ẑT 1 W ( θ 2 ) T ẑ2 ė2 = -2e 2 + ẑT 2 ẑ1 e 1 -ẑT 2 W ( θ 2 ) T ẑ2 + ẑT 2 W ( θ 2 )ẑ 1 ė3 = ẑT 3 ẑ2 e 2 + ẑT 3 ẑ1 e 1 . (33)
In order to analyze the stability of the linearization of ( 33) around e = 0 we need to work out some technical results first.

Lemma 4. Consider the matrix

M = p 1 p 2 a p 2 p 1 a b c 0 ,
where p 1 , p 2 , a, b, c ∈ R with p 1 > 0, p 1 > p 2 and p2 1 > p 2 2 , so the second principal minor of M is positive definite. Consider a small perturbation a < 0. If (b + c) > 0, then -M is Hurwitz.

Proof. First we note that if a = 0 the three eigenvalues of M are defined by the two positive eigenvalues λ 1 and λ 2 of its 2x2 leading principal submatrix and λ 3 = 0. Therefore a small perturbation a will not change the sign of the positive eigenvalues of M but we do not know about the sensitivity of the zero eigenvalue λ 3 with respect to the disturbance a. In order to check such sensitivity we derive the characteristic polynomial of M P

(λ) = -λ(p 1 -λ) 2 + abp 2 + acp 2 -ab(p 1 -λ) + λp 2 2 -ac(p 1 -λ) = λ p 2 2 -(p 1 -λ) 2 + a(b + c) + a(b + c)(p 2 -p 1 ).
(34) Note that a can be chosen arbitrarily small. Since λ 3 is a continuous function of a, it can also be made arbitrarily small, so

|p 1 | ≫ |λ 3 | and |(p 2 2 -p 2 1 )| ≫ |a(b + c)|.
Then, without affecting the analysis of the sign of the perturbed λ3 , we can approximate the first bracket in (34) by (p 2 2p 2 1 ). Therefore we can look at

P ( λ3 ) ≈ λ3 (p 2 2 -p 2 1 ) + a(b + c)(p 2 -p 1 ) = 0, ( 35
)
in order to check the sign of the perturbed λ3 . We note that under the assumptions in the statement of the lemma one concludes that for a small negative a the eigenvalue λ3 becomes positive. Hence -M is Hurwitz.

Remark 4. The Jacobian in (27) fits the description of the matrix M in Lemma 4. Note that from Proposition 2 the sign of λ3 from (27) only depends on the sign of a and not on its magnitude.

We now work out a series of calculations that will be needed for the calculation of the Jacobian matrix of ( 33) at e = 0. Consider the expression

ẑT 2 W (α)ẑ 1 = cos(γ -α), ( 36
)
where γ is the actual angle between z 1 and z 2 and α ∈ R is a constant angle.We also have that = sin(θα)

∂ ẑT 2 W (α)ẑ 1 ∂e = ∂ cos(γ(e) -α) ∂e , (37)
1 √ 1 -cos 2 θ - 2d 1 2d 1 d 2 - d 2 3 -d 2 1 -d 2 2 d 2 1 d 2 = - sin(θ -α) sin(θ) d 1 + d 2 cos θ d 1 d 2 =: a 1 (39) ∂ cos(γ(e) -α) ∂e 2 | e=0 = - sin(θ -α) sin(θ) d 2 + d 1 cos θ d 1 d 2 =: a 2 (40) ∂ cos(γ(e) -α) ∂e 3 | e=0 = sin(θ -α) sin θ d 3 d 1 d 2 =: a 3 . (41)
Remark 5. We highlight again that α is a constant parameter that can be chosen arbitrarily. For example, the computation of a 3 for θ = 0, i.e., d 3 = d 1 + d 2 , and α = 0 becomes

lim θ→0 a 3 (θ) = lim θ→0 sin(θ -0) sin θ d 1 + d 2 d 1 d 2 = d 1 + d 2 d 1 d 2 , (42)
which is consistent with the values obtained for the Jacobian in (27).

Similarly as in ( 14) with (15), we also introduce a gain c > 0 in (32) multiplying the terms W ( θ 2 )ẑ 1 and W ( θ 2 ) T ẑ2 . Now we are ready to present our main result. Theorem 5. For the system

       ṗ1 = -ẑ 1 e 1 ṗ2 = ẑ1 e 1 -ẑ2 e 2 + c W ( θ 2 )ẑ 1 -W ( θ 2 ) T ẑ2 ṗ3 = ẑ2 e 2 , (43)
its equilibrium defined by the set π,π) is locally exponentially stable for a sufficiently small c > 0.

U ∆ = {p 1 , p 2 , p 3 : W (θ)ẑ 1 -ẑ2 = 0, ||z 1 || = d 1 , ||z 2 || = d 2 }, for every θ ∈ (-
Proof. We first note that the study of the stability of the equilibrium U d corresponds to checking the eigenvalues of the Jacobian of the autonomous augmented error system (33), with the introduced gain c, at e = 0. The Jacobian is given by

J e =    -2 + ca 1 cos(θ) + ca 2 ca 3 cos(θ) + ca 1 -2 + ca 2 ca 3 - d 1 + d 2 cos θ d 3 - d 2 + d 1 cos θ d 3 0    , (44)
where a 1 , a 2 and a 3 are as in ( 39)-( 41) respectively. We notice that the following expression

-( d 1 + d 2 cos θ d 3 + d 2 + d 1 cos θ d 3 ) = - d 1 (1 + cos θ) + d 2 (1 + cos θ) d 3 , (45)
is always negative for θ ∈ (-π, π). Since α = θ 2 one can also check that ( 41) is always positive. Indeed, the second principal minor is not symmetric (negative definite) anymore because of the added small perturbations ca 1 and ca 2 , but its eigenvalues are still with real negative part and bounded from zero for all θ. Furthermore, the diagonal elements and off-diagonal elements of the second principal minor are very similar respectively for a very small c > 0. Therefore from the eigenvalue sensitivity analysis in Lemma 4 one can derive that indeed ( 44) is Hurwitz for a very small gain c > 0. Note that the local exponential convergence of the augmented e(t) to the origin implies that the agents' velocities in (32) converge to zero as well, so the agents converge to fixed positions. Remark 6. Note that in system (43), the set of collinear positions for all the agents is not invariant. Therefore, it outperforms in that sense the distance-based control of rigid formations, where if the agents start collinear, they remain collinear.

SIMULATIONS

In this section we validate the results in Proposition 2 and Theorem 5. We first show in Figure 3 that for µ 1 = -µ 2 = 1 in system ( 14) the equilibrium U d is stable, where all the agents converge to collinear positions with ẑ1 = ẑ2 . The desired distances are d 1 = 30 and d 2 = 10, and the agents 1, 2 and 3 are marked with red, green and blue colors respectively. We then show in Figure 4 that the equilibrium U u is stable for µ 2 = -µ 1 = 1. Note that the agents do not stop moving and the distance errors converge both to 2 3 . We move on to control a triangular shape by employing the results from Theorem 5. The desired angle betwen ẑ1 and ẑ2 has been set to 60 degrees. The evolution of the agents can be seen in Figure 5, where the left inner plot shows the evolution of the inter-angle and the right inner plot the evolution of the distance errors.

CONCLUSIONS

We have presented a formation control algorithm for achieving triangular shapes. The algorithm is based on a new technique derived from the biased distance-based gradient descent control of two links. The algorithm enjoys the advantages of both position-based and distance-based formation controls. Current research is aimed at providing a systematic approach for achieving stable arbitrary shapes consisting of more than three agents in non-rigid formation. We have already made some progress for chain topologies Garcia de [START_REF] Garcia De Marina | Distributed algorithm for controlling scale-free polygonal formations[END_REF]. Fig. 3. Simulation result from Proposition 2. If µ 1 = -µ 2 = 1, then the equilibrium set U d is stable and all the agents stop eventually once the distance errors converge to zero (shown in the inner plot). In this equilibrium, we recall that ẑ1 = ẑ2 , therefore we have that the agent 2 marked in green will finish between the other two agents. Note that ẑ1 = -ẑ 2 , therefore the agent 2 marked in green is leading the motion. Both distance errors e 1 and e 2 (in the inner plot) converge to 2 3 , regardless of the prescribed distances.

Fig. 1 .

 1 Fig. 1. The same triangular shape depicted by the three agents (dots) can be achieved by controlling the two vectors z 1 and z 2 (position-based control) or the three distances d 1 , d 2 and d 3 (distance-based control). Note that in a) the orientation of the triangle is prescribed while it is not in b).

  ẑ 1 e 1 ṗ2 = ẑ1 e 1 -ẑ2 e 2 ṗ3 = ẑ2 e 2 .

  ) where z * 1 and z * 2 are two fixed vectors for z 1 and z 2 such that e * 1 = e * 2 = e * .

Fig. 2 .

 2 Fig. 2. The triangular shape can be defined by setting the norms ||z 1 || = d 1 , ||z 2 || = d 2 and the angle θ between z 1 and z 2 .

  with γ(e) = arccos (e 3 + d 3 ) 2 -(e 1 + d 1 ) 2 -(e 2 + d 2 ) 2 2(e 1 + d 1 )(e 2 + d 2 ) , (38) and note that γ(0) = θ. Therefore by applying the chain rule ∂ cos(γ(e)α) ∂e 1 | e=0 = -sin(γ(e)α) ∂γ(e) ∂e 1 | e=0

Fig. 4 .

 4 Fig. 4. Simulation result from Proposition 2. If µ 2 =-µ 1 = 1, then the equilibrium set U u is stable and the formation describes a steady-state collective motion (denoted by the black arrow) in the direction of ẑ2 .Note that ẑ1 = -ẑ 2 , therefore the agent 2 marked in green is leading the motion. Both distance errors e 1 and e 2 (in the inner plot) converge to 2 3 , regardless of the prescribed distances.

For arbitrary values of µ 1 and µ

the results of this paper apply as well but with different steady-state values for e 1 and e 2 . For the sake of simplicity and clarity we only consider the case where (15) holds. For more details, we refer to[START_REF] Mou | Undirected rigid formations are problematic[END_REF] and Garcia de[START_REF] Garcia De Marina | Distributed rotational and translational maneuvering of rigid formations and their applications[END_REF].

Note z 1 points at agent 2 and z 2 starts at agent 2, therefore we will have cos(πθ) = -cos(θ) in the law of cosines for Figure2.

Conference on, 1200-1205. IEEE. Garcia de Marina, H., Cao, M., and Jayawardhana, B. (2015). Controlling rigid formations of mobile agents un- Fig. 5. Simulation result from Theorem 5. In addition to the distance control between agents 1, 2 and 2, 3 (right inner plot), the agent 2 marked in green color is controlling the angle between ẑ1 and ẑ2 (left inner plot). Note that the final orientation of the shape is not controlled and depends on the initial conditions.