Antonio Loría

Emmanuel Nu
email: emmanuel.nuno@cucei.udg.mx

Elena Panteley

Observerless Output-feedback Consensus-based Formation Control of 2nd-order Nonholonomic Systems

Keywords: Formation control, persistency of excitation, nonholonomic systems, consensus, output feedback

We present a solution to the problem of formation consensus control of second-order nonholonomic systems via output feedback. We contribute with a distributed consensus controller for force-controlled nonholonomic systems under the assumption that the forward and angular velocities are not measurable. Our main statement establishes uniform global asymptotic stability for the closed-loop system; this guarantees robustness with respect to bounded disturbances, in the sense of Malkin's total stability, also known as local input-to-state stability.

I. INTRODUCTION

Formation control consists in making a group of multiple mobile robots acquire positions (and possibly orientations) in a desired geometric pattern, in order to maneuver as a whole [START_REF] Zhao | Affine formation maneuver control of multiagent systems[END_REF]. As such, it consists in two distinct problems: that of consensus-based formation, (also known as rendezvous [START_REF] Dimarogonas | On the rendezvous problem for multiple nonholonomic agents[END_REF], [START_REF] Roza | A Smooth Distributed Feedback for Global Rendezvous of Unicycles[END_REF]) and formation-tracking, in which case a trajectory imposed by a (possibly fictitious) leader is to be followed [START_REF] Consolini | On a class of hierarchical formations of unicycles and their internal dynamics[END_REF]- [START_REF] Marshall | Formations of vehicles in cyclic pursuit[END_REF]. The distinction is important because, in contrast to the case of holonomic systems, for systems with nonholonomic constraints stabilization is not a particular case of trajectory tracking. Indeed controllers that solve one problem generally cannot solve the other [START_REF] Lizárraga | Obstructions to the existence of universal stabilizers for smooth control systems[END_REF]. Let alone for multi-agent systems.

The consensus-based formation problem fundamentally differs from that of formation-tracking in that in the former there is no pre-specified reference. That is, consensus-based formation is a pure consensus-seeking problem in which the stabilized manifold of equilibria depends on the systems' initial conditions, the graph topology, and the systems' dynamics [START_REF] Ren | Distributed consensus in multivehicle cooperative control[END_REF]. Now, although the understanding of the consensus paradigm is well established for robots modeled as simple first-and second-order integrators, even under consideration of network aspects as proximity constraints [START_REF] Poonawala | Preserving strong connectivity in directed proximity graphs[END_REF] or communication delays [START_REF] Abdessameud | Synchronization of Lagrangian systems with irregular communication delays[END_REF], none of these approaches apply to consensus of nonholonomic systems. Indeed, this problem inherits the difficulties found both in cooperative control of networked systems and those related to setpoint stabilization under non-integrable velocity constraints [START_REF] Lin | Necessary and sufficient graphical conditions for formation control of unicycles[END_REF], so the literature on consensus of nonholonomic systems is more scarce.

For instance, in [START_REF] Dong | Cooperative control of multiple nonholonomic mobile agents[END_REF] and [START_REF] Defoort | Fixedtime stabilization and consensus of nonholonomic systems[END_REF] the problem for systems in so-called chain form is addressed and in [START_REF] Dimarogonas | On the rendezvous problem for multiple nonholonomic agents[END_REF] a discontinuous controller that applies even under proximity constraints is proposed; see also [START_REF] Poonawala | Collisionfree formation control with decentralized connectivity preservation for nonholonomic-wheeled mobile robots[END_REF] in which, in addition, the collision avoidance problem is addressed. In [START_REF] El-Hawwary | Distributed circular formation stabilization for dynamic unicycles[END_REF] the authors use the elegant reduction theorems to address a problem of consensus stabilization on circular formations. Nevertheless, with the exception of [START_REF] Cheng | Robust finite-time consensus formation control for multiple nonholonomic wheeled mobile robots via output feedback[END_REF] and [START_REF] El-Hawwary | Distributed circular formation stabilization for dynamic unicycles[END_REF], in the latter papers, as well as in several works on formation-tracking control, including [START_REF] Consolini | On a class of hierarchical formations of unicycles and their internal dynamics[END_REF] and [START_REF] Marshall | Formations of vehicles in cyclic pursuit[END_REF], nonholonomic systems are modeled as first-order systems involving only the kinematics equations that encode the non-integrable velocity constraints. Nonholonomic systems, however, are best modeled as second-order systems in which a force-balance equation, most typically expressed in Lagrangian form, is also considered -see, e.g., [START_REF] Do | A global output-feedback controller for simultaneous tracking and stabilization of unicycle-type mobile robots[END_REF]- [START_REF] Dhaouadi | Dynamic modelling of differentialdrive mobile robots using Lagrange and Newton-Euler methodologies: A unified framework[END_REF].

In this Technical Note we address the consensus-based formation control problem for 2nd-order nonholonomic systems -cf. [START_REF] Cheng | Robust finite-time consensus formation control for multiple nonholonomic wheeled mobile robots via output feedback[END_REF], [START_REF] El-Hawwary | Distributed circular formation stabilization for dynamic unicycles[END_REF]. The originality of the controller proposed here resides in that it relies only on position and orientation feedback, but the velocities are not measured. Indeed, output feedback control of nonholonomic systems has been addressed in the context of trajectory-tracking for a single nonholonomic robot, e.g., in [START_REF] Do | A global output-feedback controller for simultaneous tracking and stabilization of unicycle-type mobile robots[END_REF], [START_REF] Huang | Adaptive output feedback tracking control of a nonholonomic mobile robot[END_REF], [START_REF] Yousuf | Robust output-feedback formation control design for nonholonomic mobile robot[END_REF], and [START_REF] Wu | Output feedback control for nonholonomic systems with non-vanishing disturbances[END_REF], but it has been scarcely studied in a multi-agent-systems setting. For instance, in [START_REF] Cheng | Robust finite-time consensus formation control for multiple nonholonomic wheeled mobile robots via output feedback[END_REF] a discontinuous leader-follower formation-tracking controller is designed to follow a predetermined reference trajectory. The controller that we propose here is smooth and time-varying. It contains correcting terms proportional to consensus errors and their dirty derivatives, as well as a persistently-exciting term that excites all modes in the system to overcome the stabilization obstacles imposed by the nonholonomic constraints -cf. [START_REF] Nuño | Distributed consensus-formation of force-controlled nonholonomic robots with time-varying delays[END_REF]. A downside of such controllers, however, is the difficulty of tuning the control gains to avoid excessive oscillatory behavior [START_REF] Kim | Controllers for unicycle-type wheeled robots: Theoretical results and experimental validation[END_REF].

In contrast to some of the above-cited references, important aspects related to the network, such as preservation of connectivity, timedelays or, in robotics applications, obstacle-collision-avoidance, are not broached. Yet, a formal analysis is made to establish uniform global asymptotic stability for the closed-loop system. This property goes well beyond the mere (non)-uniform convergence property more often encountered in the literature. Indeed, for time-varying systems, only uniform global asymptotic stability (UGAS) guarantees robustness with respect to bounded disturbances, in the sense of Malkin's total stability [START_REF] Malkin | Theory of Stability of Motion[END_REF].

The rest of this Technical Note is organized as follows. The problem formulation is presented in Section II, followed by the rationale of our control approach, in Section III. The main result is stated in Section IV and numerical simulations are provided in Section V. Concluding remarks are given in Section VI.

II. MODEL AND PROBLEM FORMULATION

Consider a group of N differential-wheel mobile robots moving on a plane. Denoting by z i := [x i y i ] ∈ R 2 , with i ∈ N := {1, ..., N }, the Cartesian position of one robot's center of mass and by θ i ∈ R the vehicle's orientation, the velocity kinematics for the i-th robot is given by the equations

ẋi = v i cos(θ i ) (1a) ẏi = v i sin(θ i ) (1b) θi = ω i , i ∈ N (1c)
where v i , ω i ∈ R are the linear and angular velocities of the center of mass. Such model is often considered in the literature on cooperative control of nonholonomic systems -see, e.g., [START_REF] Dimarogonas | On the rendezvous problem for multiple nonholonomic agents[END_REF], [START_REF] Consolini | On a class of hierarchical formations of unicycles and their internal dynamics[END_REF], [START_REF] Marshall | Formations of vehicles in cyclic pursuit[END_REF] and [START_REF] Lin | Necessary and sufficient graphical conditions for formation control of unicycles[END_REF]. In this Technical Note, however, we adopt a more appropriate 2nd-order model which includes the velocity dynamics equation

m i 0 0 I i vi ωi = 1 r i 1 1 2R i -2R i τ i (2)
where m i is the robot's mass; I i is the moment of inertia; R i is the distance between the point z i and the wheels; r i is the radius of the wheels; and τ i is the control input torque of the left and right wheels, i.e., τ i = [τ il , τ ir ] . For analysis purposes, it is assumed that the center of mass is placed on the axis joining the two wheels' centers. Otherwise, Coriolis and centrifugal forces terms should be included in (2) -cf. [START_REF] Do | A global output-feedback controller for simultaneous tracking and stabilization of unicycle-type mobile robots[END_REF]- [START_REF] Dhaouadi | Dynamic modelling of differentialdrive mobile robots using Lagrange and Newton-Euler methodologies: A unified framework[END_REF]. These are considered in Section V.

Let zc := [xc yc] denote a point on the Cartesian plane that is not specified a priori and is unknown to any robot. Let this point be the center of a formation pattern constructed by defining constant vectors originating in zc, denoted δ i = [δ ix δ iy ] . For each i ∈ {1 . . . N }, the vector δ i is defined so as to determine a desired position for the i-th agent with respect to zc, thereby defining a formation centered at zc. Then, defining zi := z i -δ i the consensus-based formation control goal consists in ensuring that

lim t→∞ (z i (t), θ i (t), v i (t), ω i (t)) = (zc, θc, 0, 0), ∀i ∈ N . (3)
It is assumed that each robot possesses sensors to measure its Cartesian position, z i , and its orientation, θ i , but the forward and angular velocities, v i and ω i , are not measurable.

Regarding the network topology we assume that the robots communicate their own measurements with a set of neighbor robots located in sufficient proximity for the network interconnection to be established, but not necessarily close enough for relative-distance sensors to be effective [START_REF] Lin | Necessary and sufficient graphical conditions for formation control of unicycles[END_REF]. Note that it is natural to assume that if a pair of robots establish an interconnection, the flow of information is bidirectional hence, the robots' network topology may be described using an undirected weighted graph. This is commonly defined via a Laplacian matrix, L ∈ R N ×N , whose entries are defined as

ij = k∈N i w ik if i = j, ∀ i, j ∈ N -w ij otherwise, (4)
where N i is the set of agents communicating with the i-th robot. The interconnection weight, w ij , is positive if the pair of nodes (i, j) is connected; otherwise, w ij = 0. It is also assumed that the graph is connected, so L is symmetric positive semi-definite and possesses a unique null eigenvalue. Remark 1: Consensus of nonholonomic systems under less stringent graph conditions are considered, e.g., in [START_REF] Dimarogonas | On the rendezvous problem for multiple nonholonomic agents[END_REF], but in a statefeedback control context.

•

III. CONSENSUS-BASED FORMATION CONTROL

For the purpose of control design and stability analysis, we derive now a dynamical model for which the control problem may be formulated as one of stabilization of the origin. To that end, we define ϕ i : R → R 2 and Φ : R N → R 2N ×N , as

ϕ i (θ i ) := cos(θ i ) sin(θ i ) Φ(θ) := blockdiag ϕ i (θ i ) (5)
where

θ := [θ 1 • • • θ N ]
. Then, we use (1) and zi = z i -δ i to derive the equations

żi = ϕ i (θ i )v i , (6a) θi = ω i . (6b)
Next, we perform a change of variable to "normalize" the control inputs. That is, let

τ i = r i 2 m i I i /2R i m i -I i /2R i u vi u ωi , (7)
so the velocity-dynamics equations become

vi = u vi (8a) ωi = u ωi . (8b)
From a control viewpoint, beyond the weak convergence property expressed by [START_REF] Roza | A Smooth Distributed Feedback for Global Rendezvous of Unicycles[END_REF], it is desired to design decentralized dynamic outputfeedback controllers with state ζc and control laws (t, zi , θ i , ζc) → u vi (t, zi , θ i , ζc) and (t, zi , θ i , ζc) → u ωi (t, zi , θ i , ζc) which depend on the measurable positions and orientations, z i and θ i , as well as on the controller state variable, ζc, such that the origin for the closedloop system is uniformly globally asymptotically stable. To that end, we define the consensus errors

e i = j∈N i w ij zi -zj , (9)
e θi = j∈N i w ij θ i -θ j ( 10
)
and we introduce the smooth control laws

u vi := -k dvi ϕ i (θ i ) ϑ vi -k pvi ϕ i (θ i ) e i , (11)
u ωi := -k dωi ϑ ωi -k pωi e θi + α i (t, θ i , e i ) , (12a)
α i := k αi ψ i (t)ϕ i (θ i ) ⊥ e i , (12b)
where

k dvi , k pvi , k dωi , k pωi , k αi > 0, ϕ i (θ i ) ⊥ = [-sin(θ i ) cos(θ i )]
is the left annihilator of ϕ i (θ i ), i.e.,

ϕ i (θ i ) ⊥ ϕ i (θ i ) = ϕ i (θ i ) [ϕ i (θ i ) ⊥ ] = 0,
= -a vi (q vi + b vi zi ) , (13a)
ϑ vi = q vi + b vi zi , a vi , b vi > 0 (13b) and qωi = -a ωi (q ωi + b ωi θ i ) , (14a)
ϑ ωi = q ωi + b ωi θ i , a ωi , b ωi > 0. ( 14b
)
The dynamic controller defined by Eqs. ( 11)-( 14) is, essentially, of the proportional-derivative type, with the added time-varying term α i -cf. [START_REF] Nuño | Distributed consensus-formation of force-controlled nonholonomic robots with time-varying delays[END_REF]. In particular, the terms -k dvi ϕ i (θ i ) ϑ vi and -k dωi ϑ ωi are responsible for injecting appropriate damping to achieve asymptotic stabilization. However, in place of velocity measurements żi and ωi the variables ϑ vi and ϑ ωi are used. These may be regarded as the outputs of approximate-differentiation (low-pass) filters which, in the frequency domain, correspond to

ϑ vi = b vi s + a vi żi , ϑ ωi = b ωi s + a ωi ω i . (15)
Such ad hoc replacement for the velocities is often used, e.g., in control of robot manipulators, since the seminal paper [START_REF] Kelly | A simple set-point robot controller by using only position measurements[END_REF]. Now, the system (6) in closed-loop with the controller defined by Eqs. ( 11)-( 14) is given by

Σ ωi    θi = ω i ωi = -k dωi ϑ ωi -k pωi e θi + α i (t, θ i , e i ) θωi = -a ωi ϑ ωi + b ωi ω i . (16) Σ vi    żi = ϕ i (θ i )v i vi = -k dvi ϕ i (θ i ) ϑ vi -k pvi ϕ i (θ i ) e i θvi = -a vi ϑ vi + b vi ϕ i (θ i )v i . (17)
Several structural properties of the closed-loop system above are worth remarking. On one hand, note that both Σv i and Σ ωi are reminiscent of second-order systems coupled with first-order low pass filters. To better see this, consider Σ ωi with α i ≡ 0 and ϑω i = ω i . Then, we recover a linear second-order system for which the consensus manifold {ω i = 0, e θ i = 0} is asymptotically stable [START_REF] Ren | Distributed consensus in multivehicle cooperative control[END_REF]. What is more, replacing e θi with θ i in the first two equations in [START_REF] Do | A global output-feedback controller for simultaneous tracking and stabilization of unicycle-type mobile robots[END_REF], we observe that these equations correspond to an ordinary second-order linear system and the control gains k pωi and k dωi determine the fundamental frequency and damping coefficient respectively. Thus, with α i ≡ 0, the equations ( 16) correspond to those of two stable feedback-interconnected systems. Furthermore, letting aside the nonholonomy expressed via ϕ i in [START_REF] Tzafestas | Introduction to mobile robot control[END_REF], Σ vi has similar structural properties. On the other hand, note that if in [START_REF] Tzafestas | Introduction to mobile robot control[END_REF] we replace the state variable θ i with the corresponding trajectories θ i (t), which are (part of the) solutions of ( 16), then Σ vi may be regarded as a decoupled linear time-varying system -cf. [26, p. 657]. Correspondingly, Σ ωi may be considered as a system that is "perturbed" by α i , which is a function of e i so it is both an output of Σ vi and an "input" to Σ ωi . Thus, the closed-loop system Σ ωi -Σ vi may be considered as if interconnected in cascade 1 .

These observations are useful in understanding the stabilization mechanism in play. For cascaded systems uniform global asymptotic stability of the origin follows if three basic conditions are met: 1) the origin for the perturbed system without input (i.e., with α i ≡ 0), Σ • ωi , is UGAS; 2) the origin for the perturbing system, Σ vi , is UGAS, uniformly in the trajectories θ i (t); and 3) the solutions of Σ ωi are uniformly globally bounded. Now, UGAS of the origin for Σ • ωi may be established as per the rationale above. UGAS of the origin for Σ vi comes after similar considerations as for Σ ωi , in addition to the fact that ψ i is persistently exciting. Indeed, this property guarantees that α i ≡ 0 unless e i ≡ 0 -see (12b). Therefore, the "perturbation" α i persistently prevents Σω i to stabilize at unwanted equilibria while the consensus errors e i perdure. Uniform global boundedness of the solutions of Σ ωi with α i ≡ 0 follows by proving, in addition, that the trajectories e i (t), and hence the "perturbation" α i (t, e i (t), θ i (t)), are bounded. Our main result and its proof constitute a formal statement of these intuitive arguments.

IV. MAIN RESULT

We start by rewriting the closed-loop equations for the N robots in compact form. To that end, let e

:= [e 1 • • • e N ] , z := [z 1 • • • z N ] , v := [v 1 • • • v N ] , e θ := [e θ1 • • • e θN ] , θ := [θ 1 • • • θ N ] , ω := [ω 1 • • • ω N ] , ϑv := [ϑ v1 • • • ϑ vN ] , and ϑω := [ϑ ω1 • • • ϑ ωN ] , as well as α := [α 1 • • • α N ] and the control gains K dv := diag[k dvi ], K dω := diag[k dωi ], Kpv := diag[k pvi ], Kpω := diag[k pωi ], Av := diag[a vi ] ⊗ I 2 , Aω := diag[a ωi ], Bv := diag[b vi ] ⊗ I 2 , Bω := diag[b ωi ],
where the matrix I 2 corresponds to the 2 × 2-identity-matrix and ⊗ denotes the Kronecker product. Then, the closed-loop may be rewritten as

Σω    θ = ω ω = -K dω ϑω -Kpωe θ + α(t, θ, e) θω = -Aωϑω + Bωω (18)
1 See [START_REF] Loría | From feedback to cascade-interconnected systems: Breaking the loop[END_REF] for a more detailed explanation of how feedback-interconnected systems may be considered as if interconnected in cascade.

Σv

   ż = Φ(θ)v v = -K dv Φ(θ) ϑv -KpvΦ(θ) e θv = -Avϑv + BvΦ(θ)v (19)
where, in addition, we stress that e = Lz, with L := L ⊗ I 2 .

Proposition 1: (Main result) Consider the system ( 18)- [START_REF] Dhaouadi | Dynamic modelling of differentialdrive mobile robots using Lagrange and Newton-Euler methodologies: A unified framework[END_REF]. Assume that for each i ∈ N , there exist c ψi , µ i and T i > 0, such that

max |ψ i |∞, | ψi |∞ ≤ c ψi ( 20
)
t+T i t ψ i (s) 2 ds ≥ µ i ∀ t ≥ 0 (21)
where |ψ i |∞ := sup t≥0 |ψ i (t)|. In addition, assume that all the control gains Av, Aω, Bv, Bω, K dv , K dω , Kpv, Kpω, and Kα are positive definite and Bω is sufficiently large, such that

Bω ≥ 2L, (22)
i.e., (Bω -2L) is positive semi-definite. Then, the origin, (e, e θ , v, ω, ϑv, ϑω) = (0, 0, 0, 0, 0, 0) is uniformly globally asymptotically stable. In particular, under the action of the controller defined by Eqs. ( 11)-( 14), the system (6) achieves full consensus, i.e., (3) holds.

Remark 2: The condition ( 22) imposes a lower bound on each b wi depending only on a bound on the interconnection strength of the links involving the i-th robot and its neighbors. Also, the factor '2' is used to simplify some computations in the proof, but ( 22) may be relaxed to imposing that (Bω -L) be positive definite.

• Proof of Proposition 1. According with the rationale given previously, we follow three logical steps and we invoke a cascades argument, specifically, [28, Lemma 2].

Step 1: UGAS of Σ • w .-Consider the system ( 18) with e = 0, hence, with α ≡ 0. The proof follows invoking the Generalized Matrosov's Theorem [START_REF] Loría | A nested Matrosov theorem and persistency of excitation for uniform convergence in stable non-autonomous systems[END_REF]. Let Kω := K -1 pω K dω B -1 ω and consider the functions

W 1 (θ, ω, ϑω) := 1 2 ω K -1 pω ω + ϑ ω Kωϑω + θ Lθ , (23a)
W 2 (ω, ϑω) := -ω ϑω, (23b)
W 3 (θ, ω) := e θ ω, (23c)
where e θ = Lθ. The function W 1 is positive definite in the space of (e θ , ω, ϑω) since there exist c 1 and c 2 > 0 such that

c 1 |e θ | 2 ≤ θ Lθ ≤ c 2 |e θ | 2 . ( 24
)
-see [30, Lemma 1]. Next, we evaluate the derivatives of W 1 , W 2 , and W 3 , along the trajectories of ( 18) with α ≡ 0. For W 1 we obtain

Ẇ1 = -ϑ ω KωAωϑω =: Y w1 (e θ , ω, ϑω) ≤ 0. ( 25
)
It follows from the latter that the origin, {xω = 0}, where xω := [e , v , ϑ ω ] , is uniformly globally stable for Σ • ω . Now, the total derivatives of W 2 and W 3 yield

Ẇ2 = -ω Bωω + ϑ ω Aωω + Kpωe θ + K dω ϑω (26)
=: Y w2 (e θ , ω, ϑω) Ẇ3 = -e θ Kpωe θ -ϑ ω K dω e θ + ω Lω [START_REF] Loría | From feedback to cascade-interconnected systems: Breaking the loop[END_REF] =: Y w3 (e θ , ω, ϑω)

We see that Ẇ2 is negative semi-definite on the set {Y w1 = 0}, Ẇ3 is negative definite on the set {Y w1 = 0} ∩ {Y w2 = 0} and all Y wi , with i ∈ {1, 2, 3}, are zero simultaneously if and only if (e θ , ω, ϑω) = (0, 0, 0). UGAS of the origin for Σ • ω follows from [29, Theorem 1].

Step 2: Uniform boundedness of the solutions of Σω.-Let ε 1 , ε 2 > 0 and consider the function

W : R ≥0 × R 3N → R ≥0 , W (t, θ, ω, ϑω) := W 1 (θ, ω, ϑω) + ε 1 W 2 (ω, ϑω) + ε 2 W 3 (θ, ω).
This function is positive definite in (e θ , ω, ϑω) for sufficiently small ε 1 and ε 2 > 0. Its total derivative, using [START_REF] Huang | Adaptive output feedback tracking control of a nonholonomic mobile robot[END_REF], yields

Ẇ = Y w1 + ε 1 Y w2 + ε 2 Y w3 + α(t, θ, e) [K -1 pω ω -ε 1 ϑω + ε 2 e θ ].
Now, define k m * and k M * as the smallest and the largest elements of a diagonal matrix K * > 0. After some direct computations, using ( 26) and ( 27), we see that

Ẇ ≤ - 1 2 k m ω a m ω |ϑω| 2 + ε 1 2 b m ω |ω| 2 + ε 2 k m pω |e θ | 2 + |α| 1 k m pω |ω| + ε 1 |ϑω| + ε 2 |e θ | -ω ε 1 2 Bω -ε 2 L ω - ε 1 4 b m ω -2λa M ω |ω| 2 - 1 2 k m ω a m ω - ε 1 λ a M ω + k M pω + 2λk M dω - ε 2 λ k M dω |ϑω| 2 - 1 2 ε 2 k m pω -λ ε 1 k M pω + ε 2 k M dω |e θ | 2 (28)
where ε 1 , ε 2 , and λ ∈ (0, 1). On the right-hand side of the previous expression, the third term is non-positive, e.g., if ε 1 = ε 2 and in view of ( 22), the fourth and last terms are non-positive for sufficiently small values of λ, while the fifth term is non-positive for sufficiently small values of ε 1 and ε 2 . Hence, let

ε 1 = ε 2 =: ε and b m ω := 2cλa M ω , with c ≥ 1. Then, the last term is non-positive if λ ≤ k m pω k M pω + k M dω (29)
because

k m ω = k m dω b M ω k M pω = b m ω k m dω 2cλa M ω b M ω k M pω , while the before-last term is non-positive if ε ≤ b m ω k m dω a m ω 2c a M ω b M ω k M pω a M ω + k M pω + (2λ + 1)k M dω . (30)
In turn, [START_REF] Loría | A nested Matrosov theorem and persistency of excitation for uniform convergence in stable non-autonomous systems[END_REF]

and b m ω = 2cλa M ω impose that c ≥ k M pω + k M dω k m pω b m ω a M ω .
Remark 3: If necessary, the value of ε computed to satisfy (30) for an arbitrary c ≥ 1 may be redefined.

• Thus, Ẇ is bounded from above by the first two terms on the right-hand side of (28) so, for all |xω| 1, any cα > 0 and all α ≤ cα, we have Ẇ ≤ 0. That is, the solutions t → xω(t) are uniformly globally bounded provided that so is α(t, θ, e(t)) which, in view of [START_REF] Yousuf | Robust output-feedback formation control design for nonholonomic mobile robot[END_REF], holds if e(t) is uniformly bounded. This is proved next.

Consider, the system Σv with θ = θ(t), the latter corresponding to (part of) the solution to Eqs. [START_REF] Huang | Adaptive output feedback tracking control of a nonholonomic mobile robot[END_REF], and the Lyapunov function candidate

V 1 (z, v, ϑv) := 1 2 v K -1 pv v+ϑ v K -1 pv K dv B -1 v ϑv+z Lz . ( 31
)
This function is positive definite in (e, v, ϑ), where e = Lz. Indeed, there exist c 1 and c 2 > 0 such that

c 1 |e| 2 ≤ z Lz ≤ c 2 |e| 2 . (32)
-see [START_REF] Maghenem | Consensus of multi-agent systems with nonholonomic restrictions via Lyapunov's direct method[END_REF]Lemma 1]. Furthermore, using the first equation in [START_REF] Dhaouadi | Dynamic modelling of differentialdrive mobile robots using Lagrange and Newton-Euler methodologies: A unified framework[END_REF] we find that the total derivative of V 1 along the trajectories of Σv

yields V1 = -ϑ v Kvϑv, Kv := K -1 pv K dv B -1 v Av (33) for all (z, v, ϑv) ∈ R 5N . Since V1 ≤ 0, it also follows that V 1 (z(t), v(t), ϑv(t)) ≤ V 1 (z(t•), v(t•), ϑv(t•)) for all t ≥ t• ≥ 0.
In view of [START_REF] Kim | Controllers for unicycle-type wheeled robots: Theoretical results and experimental validation[END_REF], it also follows that, defining,

xv := [e , v , ϑ v ] , (34)
there exists cv > 0 such that

|xv|∞ ≤ cv|xv(t•)|, (35)
which implies uniform global stability for the origin of Σv, hence uniform global boundedness of xv(t). Remark 4: Technically, the equations ( 19) are defined only on the maximal interval of solutions of t → θ(t), [t•, tmax) with tmax ≤ +∞, so the bound (35) holds only for all t ∈ [t•, tmax). Forward completeness of ( 18)-( 19), however, follows from the fact that (33) and ( 28) hold along the system's solutions on any maximal interval of existence, [t•, tmax) for any tmax ≥ t•. By continuity of the solutions in the initial conditions this interval may be extended up to tmax = +∞.

• Step 3: UGAS of the origin for Σv with θ = θ(t).-We proceed as in Step 1. First, in view of (33) and the uniform global boundedness of the solutions, the origin for Σv with θ = θ(t) is globally stable, uniformly in the initial conditions and in θ(t). Next, consider the functions

Φ(•) := Φ(θ(•)), V 2 (t, v, ϑv) := -v Φ(t) ϑv, (36)
V 3 (t, e, v, ) := v Φ(t) e, (37)
and

V 4 (t, e) := -e Φ(t) ⊥ t+T t Ψ(τ ) Ψ(τ )dτ Φ(t) ⊥ e,
where Ψ(t) := diag[ψ i (t)], i ∈ N . These functions are defined on [t•, ∞) for any t• ≥ 0, are smooth, and uniformly bounded in t. Now, the total derivative of V 2 (t, v, ϑv) along the trajectories of Σv with θ = θ(t) yields

V2 (t, v, ϑv) = -v Bvv + [Ω(t) ⊗ I 2 ] Φ(t) ⊥ + Av Φ(t) v , + ϑ v Φ(t) K dv Φ(t) ϑv + Kpv Φ(t) e
which is negative semi-definite on the set { V1 = 0}. In the computation of V2 we used

Φ(θ) = [Ω ⊗ I 2 ]Φ(θ) ⊥ where Ω := diag[ω i ].
Next, define ξ 1 (t, e) := Φ(t) e; then, the total derivative of V 3 (t, e, v) yields

V3 (t, e, v) = -ξ 1 (t, e) Kpvξ 1 (t, e) -ϑ v Φ(t)K dv ξ 1 (t, e) + v Φ(t) ⊥ [Ω(t) ⊗ I 2 ]e + Φ(t) L Φ(t)v ,
which is negative semi-definite on the set { V1 = 0} ∩ { V2 = 0}. Finally, we evaluate the total derivative of V 4 . To that end, define ξ 2 (t, e) := Φ(t) ⊥ e then, V4 (t, e) = V 4 (t, e) -2ξ 2 (t, e) The proposed controller may be modified to address the problem in which the robots are required to converge to a given desired constant orientation θ d , as opposed to a common, not preimposed one, θc. This may be achieved by replacing the orientation error [START_REF] Abdessameud | Synchronization of Lagrangian systems with irregular communication delays[END_REF] with

e θi = j∈N i w ij θ i -θ j + j∈N i b i θi ,
where θi := θ i -θ d , b i > 0 if the ith-robot has access to the desired orientation θ d and b i = 0 otherwise. The result follows provided that at least one robot has access to θ d .

•

V. SIMULATIONS

Numerical simulations in Matlab TM Simulink TM under two different scenarii were performed to illustrate the controller's performance. The simulation setup consists in six robots required to meet at an unknown rendezvous point while forming a hexagonal pattern. The latter is defined by setting the offsets (δ xi , δ yi ) to the values showed in Table I below. The robots' initial conditions are also given in Table I and the robots' parameters were taken as in [START_REF] Shojaei | Adaptive trajectory tracking control of a differential drive wheeled mobile robot[END_REF]: m i = 10.4kg, I i = 3kgm 2 , R i = 0.3m, and r i = 0.05m.

θ i (0) 0 -π/4 -π/2 π/4 π/2 π/4 δx i 2 1 -1 -2 -1 1
δy i 0 2 2 0 -2 -2
It is assumed that the vehicles are interconnected according to a cyclic graph topology as illustrated in Fig. 1 below, with unitary interconnecting weights. The linear-and angular-velocity dynamics, Σv i and Σω i corresponding, essentially, to second-order systems coupled with low-pass filters, the controller gains may be selected following a rule of thumb based on the tuning of an ordinary second-order system. Hence, they were set to k pi = 300, k di = 600, k pωi = 30, k dωi = 60, k αi = 15 in order to damp transient oscillations. Regarding the gains of the filters, as expressed in [START_REF] El-Hawwary | Distributed circular formation stabilization for dynamic unicycles[END_REF], they were set so as to have a unitary DC gain and filter out "high" frequencies. Hence, they were set to a vi = b vi = a ωi = b ωi = 10. Finally, for ψ i to satisfy (21) we used the periodic (hence persistently exciting) function ψ i (t) := 2.5 + (4/π) sin(0.5t) for all agents, but it may be chosen differently for each of them.

The paths followed by the robots on the plane are illustrated in Figure 2. We observe that all the robots converge to the desired hexagonal formation. The center of the latter is at the point (xc, yc) = For the purpose of illustrating the robustness of the controller relative to parametric uncertainty, neglected dynamics, and measurement noise, a second simulation was performed in which there is a discrepancy between the actual values of the mass, the inertia, and the location of the center of mass and those considered for the control design. Also, white noise of zero mean and standard deviation equal to 2% was added to the position measurements, to account for sensor defects [START_REF] Kavanagh | Shaft encoder characterization via theoretical model of differentiator with both differential and integral nonlinearities[END_REF]. It is assumed that the value of the mass and moment of inertia are inaccurate by a 10% of their value and the center of mass is located off the axis connecting the two wheels. The misplacement of the center of mass entails neglected quadratic Coriolis terms, r i 3 ω 2 i on the left-hand side of equation (8a) and -r i m i 3I i ω i v i on the lefthand side of equation (8b) -cf. [START_REF] Tzafestas | Introduction to mobile robot control[END_REF]. In Figure 3 are showed the Cartesian positions and orientations converging to, and remaining within a neighborhood of, consensual values. The relatively small steady-state error is also appreciated in Figure 4 -notice the deformed gray hexagon.

The transient behavior, which in this simulation is clearly unfitting from a robotics viewpoint, may be improved by re-tuning the control gains. For the sake of fair comparison, however, the control gains and the initial conditions are deliberately set as in the first scenario (in which the Coriolis terms, the measurement noise, and the parametric uncertainty are neglected). Indeed, the purpose of the second simulation is to illustrate the robustness provided by uniform global asymptotic stability, that is, in the sense of total stability. In that regard, it is also worth recalling that only uniform asymptotic stability, and not mere non-uniform convergence (as it is more often encountered in the literature), guarantees total stability.

The paths on the plane are illustrated in Figure 4. Even though the intersections do not necessarily happen simultaneously, it shows the importance of improving our controller with collision-avoidance strategies.

VI. CONCLUSIONS

The rendezvous control problem for 2nd-order nonholonomic systems, without velocity measurements, may be addressed via smooth control laws of proportional-derivative type, by replacing the unavailable measurements with an ad hoc dirty-derivative filter. This result may serve as basis for further work on output-feedback formation control problems involving the relaxation of the assumptions made here on the network's topology and the nature of the interconnections. Also, it appears important to extend this work to models of nonholonomic systems involving Lagrangian dynamics.

Ψ

  (τ ) Ψ(τ )dτ × -Φ(t) [Ω ⊗ I 2 ]e + Φ(t) ⊥ L Φ(t)v , where we used Φ(θ) ⊥ = -Φ(θ) [Ω ⊗ I 2 ]. Now, let µ := min{µ i } and T := max{T i } where µ i and T i are as in[START_REF] Wu | Output feedback control for nonholonomic systems with non-vanishing disturbances[END_REF] and observe thatV 4 ≤ -µe -T |ξ 2 (t, e)| 2 . We see that V4 is negative semi-definite on the set { V1 = 0} ∩ { V2 = 0} ∩ { V3 = 0}. Moreover, Vi , with i ≤ 4are equivalently equal to zero if and only if v = ϑv = ξ 1 = ξ 2 = 0, while ξ 1 = ξ 2 = 0 if and only if e = 0. Thus, Vi = 0 for all i ≤ 4 only at the origin, {xv = 0}. UGAS follows invoking the Generalized Matrosov's Theorem[START_REF] Loría | A nested Matrosov theorem and persistency of excitation for uniform convergence in stable non-autonomous systems[END_REF] Theorem 1].Remark 5:

Fig. 1 .

 1 Fig. 1. Graph representation of the interconnection topology

Fig. 2 .

 2 Fig. 2. Path followed by the robots on the plane

Fig. 3 .

 3 Fig. 3. Cartesian positions and orientations converge to consensual values despite parametric uncertainty and unmodeled dynamics

Fig. 4 .

 4 Fig. 4. The robots on the plane converge to a hexagonal formation and consensual orientation despite parametric uncertainty and unmodeled dynamics, albeit an oscillatory transient

TABLE I INITIAL

 I CONDITIONS, RELATIVE DESIRED POSITIONS, AND DESIRED ORIENTATIONS.

E. Nu ño's work is partially supported by the Government of Mexico via the Basic Scientific Research grant CB-282807, sponsored by CONA-CyT. The work of A. Loría and E. Panteley is supported by the French National Research Agency (ANR) via the project "HANDY" -contract number: ANR-18-CE40-0010, and by CEFIPRA under the grant number 6001-A. The work of E. Panteley is also the supported Government of the Russian Federation under grant 074-U01.