Multi-agent formation control for target tracking and circumnavigation missions

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INOM

EXAMENSARBETE ELECTRICAL ENGINEERING, AVANCERAD NIVÅ, 30 HP

STOCKHOLM SVERIGE 2017,

Multi-agent formation control for target tracking and

circumnavigation missions

ANTONIO BOCCIA

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Multi-agent formation control for target tracking and circumnavigation missions

ANTONIO BOCCIA

M.Sc. Thesis Stockholm, Sweden 2017

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TRITA-EE 2017:030 ISSN 1653-5146

ISRN KTH/xxx/xx--yy/nn--SE ISBN x-xxxx-xxx-x

KTH School of Electrical Engineering SE-100 44 Stockholm SWEDEN Akademisk avhandling som med tillst˚and av Kungl Tekniska högskolan framlägges till of- fentlig granskning för avläggande av examensarbete i reglerteknik i .

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Abstract

In this thesis, we study a problem of target tracking and circumnavigation with a network of autonomous agents. We propose a distributed algorithm to estimate the position of the target and to drive the agents to rotate around the target while forming a regular polygon and keeping a desired distance from it. We formally show that the algorithm attains exponential convergence of the agents to the desired polygon if the target is stationary, and bounded convergence if the target is moving with bounded speed. Numerical simulations in Matlab-Simulink and ROS corroborate the theoretical results and demonstrate the resilience of the network to addition and removal of agents.

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Sammanfattning

I denna avhandling studeras ett problem av m˚alföljning och omsegling med en nätverk av självstyrande agenter. Vi föresl˚ar en distribuerad algoritm för att upp- skatta positionen av m˚alet och för att driva agenterna att rotera runt m˚alet, medan de bildar en regelbunden polygon och h˚aller ett önskat avst˚and fr˚an m˚alet. Vi formellt visar att algoritmen uppn˚ar exponentiell konvergens av agenterna till den önskade polygonen om m˚alet är stillast˚aende, och avgränsad konvergens om m˚alet rör sig med avgränsad hastighet. Numeriska simuleringar i Matlab/Simulink och ROS bekräftar den teoretiska resultat och demonstrerar nätverkets spänstighet till addition och avfly- ttning av agenter.

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Acknowledgments

I thank my advisor Mario di Bernardo, for having introduced me to the world of nonlinear control and for having given me the opportunity to join the interesting Erasmus experience. I am grateful to my examiner Karl Henrik Johansson for having welcomed me at KTH and for having provided me interesting advices for my thesis work. I thank my supervisor Antonio Adaldo, for his huge availability and for the attention and professionality in revisioning my work.

I am grateful to my marvellous parents, Giuseppe and Maria Carmela, to have always supported me and for the exemplary teachings in every aspect of my life: this thesis is dedicated to you.

A special thank goes to my soul sister Simona, for having proved me how can someone be near even if she is thousands kilometers away: I am glad to have finally met you.

I thank my nearest neighbors Salvatore and Francesco, for the funny moments spent in the last twenty years, and I am grateful to the loyal and always present friend of a life Andrea.

I thank my friends and colleagues Marco and Vincenzo F., for the perfect synchronization in finishing our academic path together, as we started together six years ago.

I am grateful to my friends and colleagues that are living an academic experience abroad Agostino, Vincenzo S., Dario, Carmela and Alberto, for the personal growth derived from the exchange of views with them.

A friendly thank goes to Massimiliano, Matteo, Umar, Paul and Pedro for having shared good moments in Stockholm in the last six months.

Last, but not least, I am grateful to Armando, for his perfect timing.

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Chapter 1

Introduction

In this thesis, we present a multi-agent control for a network of agents in order to perform a circumnavigation task around a mobile target.

In the most general scenario, every agent in the network needs an estimator to localize the target, and a controller to approach it and to start circling around it; the multi-agent behavior achieved by the agents is the equidistance of them on a desired circle centred at the target position, thus the formation of a regular polygon.

In this work, we propose an estimate algorithm and a control algorithm to drive the agents of a directed network to achieve the collective behavior described above. The behavior of the closed-loop system is analyzed in three different scenarios. In the first scenario, the position of a stationary target is known to the network; this scenario allows us to separate the study of the formation on the circle from the study of the exponential convergence of the estimate. In the second scenario, an estimator is introduced to localize a stationary target;

the results achieved in the previous scenario on the convergence and the formation on the desired circle are extended. In the third scenario, the mobility of the target is added, and both analytical proofs and simulations in Matlab-Simulink show that the circumnavigation task and the polygonal formation are reached only through a bounded convergence of the system variables.

In the last part of this work, we present a modified version of the distributed control algorithm, that is adopted in order to simulate switching networks, where the addition and the removal of agents are considered.

1.1 Motivation

The problem of target tracking and circumnavigation finds applications in numerous fields.

A first motivation for driving a network of agents to circle around a mobile target is given by surveillance missions, where the agents (e.g., a team of UAVs, like the drone in Figure 1.1) are controlled to circumscribe an object (e.g., a building or an infrastructure) that they have to monitor. An interesting application is the escorting and patrolling mission by a network of unicycles, or more generally by a group of autonomous vehicles. A new challenging application is the tracking of an underwater target by a network of AUVs (see Figure 1.2), where the vehicles are employed to study, and possibly reproduce, the behavior of small sea animals.

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1.2. Literature review

Figure 1.1: Unmanned aerial vehicle.

Figure 1.2: Autonomous underwater vehicle.

1.2 Literature review

The design of a control algorithm that drives an agent to approach a target and to follow a circle trajectory around it has been studied in [1–5]. The solutions proposed for the single agent [5], have been extended to multi-agent systems, where a great attention has been given to the formation of a regular polygon inscribed in a desired circle, centered at the target position [6–9]. This type of formation is optimal to solve triangulation problems and it is a good solution to control agents that cannot stop moving, such as UAVs. For example, an application of the circumnavigation to escorting and patrolling missions is analyzed in [10], assuming that a unicycle vehicle can measure both the bearing angles formed with its neighbor and the distance from a target object.

In a large number of applications, the position of the target is unknown to the agents, so that a localization procedure is required to achieve the tracking. In [11] a peer-to-peer collaborative localization is studied for a network of sensors, while in [12] a stereo-vision- type estimation is realized by the leading agent, sending its visual measurements of the target to its followers.

The problem of using the information obtained from an identification process of unknown characteristics of a system to update a control algorithm is known in the literature as the

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1.3. Statement of contribution

dual problem [13,14]. In [15] and [5], the circumnavigation of a target with unknown position is formally modeled as a dual problem. In particular, in [15] the dual problem is solved using distance measurements, while in [5] it is solved using bearing measurements. In [16]

the robot and landmark localization problem is solved using an association of data from bearing-only measurements.

1.3 Statement of contribution

Similarly to the solution adopted in [5] and [9], we propose a distributed control algorithm based on bearing measurements, and an estimator to localize a mobile target. Specifically, in this thesis we propose a different control strategy where every agent has a tangential motion depending on its estimated distance from the target. Moreover, in our algorithm every agent updates its control signal on the basis of information it received by other agents within a defined communication radius, irrespectively of the distance from the target. Adopting this control strategy leads to some major improvements. Firstly, the angular velocity of the agents about the target does not grow unbounded when the desired distance from the target is small. Secondly, we formally prove the exponential convergence of the agents to a regular polygon.

The control algorithm is simulated in ROS [17], where each simulated agent is im- plemented as a separate ROS node. The simulations also demonstrate the resiliency of the algorithm to addition and removal of some agents, showing that the agents rearrange themselves to form a different polygon when one agent enters or leaves the network. Each agent starts in a monitoring position, and when it receives the first bearing measurement if effectively enters the network and approaches the desired circle.

1.4 Thesis outline

The rest of the thesis is organized as follows.

Chapter 2: Technical preliminaries

In Chapter 2, we introduce some technical definitions and results that are used in this thesis.

Chapter 3: Control algorithm

In Chapter 3, we present the estimator and the distributed control algorithm adopted in this thesis work, giving the objectives of the designed network control.

Chapter 4: Circumnavigation of a stationary target with known position

In Chapter 4, we analyze the simplest scenario for the circumnavigation, showing the asymptotic convergence of the network on the desired circle and the formation through simulations in Matlab-Simulink.

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1.4. Thesis outline

Chapter 5: Circumnavigation of a stationary target with estimated position

In Chapter 5, we introduce the estimate of the target position in the analysis of the algorithm for the circumnavigation, extending the results achieved for the formation in the previous chapter.

Chapter 6: Circumnavigation of a mobile target with estimated position In Chapter 6, we analyze the scenario of a network tracking a target moving with a slow drift, showing the bounded convergence of the system variables.

Chapter 7: Circumnavigation in switching networks

In Chapter 7, we introduce a modified version of the control algorithm and simulations performed in ROS, related to the scenario of switching networks.

Chapters 8: Conclusions and future developments

In Chapter 8, we present a summary of the results, and discuss directions for future research.

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Chapter 2

Technical preliminaries

The aim of this chapter is to provide the general notional and the technical concepts from the areas of algebraic graph theory and non linear control theory that are used to derive the main results presented in this thesis.

2.1 Notation

The set of the positive integers is denoted N, while N0= N ∪ {0}.

The metric spaces of real numbers are denoted Rⁿ, where n is the dimension of the space.

For n ∈ N, the vector made up of n unitary elements is denoted 1n, the vector made up of n null elements is denoted 0n, and the n-by-n identity matrix is denoted In.

The set of the symmetric matrices in R^n×nis denoted Sⁿ, the set of the positive semidefinite matrices in R^n×n is denoted Sⁿ≥0, and the set of positive definite matrices in R^n×n is denoted Sⁿ>0.

The set of the unit-norm vectos in R² is denoted S¹. The operator [·]i denotes the i-th element of a vector.

The operator [·]ij denotes the element at the i-th row and at the j-th column of a matrix.

The operator k·k denotes the Euclidean norm of a vector and the corresponding induced norm of a matrix.

The operators ∧ and ∨ denote the logical and and the logical or respectively.

2.2 Elements of graph theory

In this section, we review the main concepts of graph theory that are used to model multi- agent systems; the mentioned results can all be found in [18].

The multiagent system analyzed in this thesis can be described by a directed and un- weigthed graph, representing the topology of the interconnections among the agents in the network.

Definition 2.2.1 (Digraph). An unweighted directed graph or digraph is a tuple G = (V, E), where V = {1, · · · , n} with n ∈ N, E = {e1, · · · , em} ⊆ V × V. The elements of the set V are called the vertices of the graph, while the elements ek = (j, i) ∈ E, with j, i ∈ V, i 6= j and k ≤ m, are called the edges of the graph.

When an edge ek exists between the vertices i and j we call them adjacent, and the edge

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2.3. Agreement in directed networks

A directed path of length l in the graph G is a sequence of distinct vertices {v1, · · · , vl} such that vk and vk+1are adjacent for k = 1, · · · , l − 1.

Definition 2.2.2 (Strongly connected and weakly connected diagraph). A diagraph is strongly connected if, between every pair of vertices, there is a directed path.

We can draw a digraph with circles representing the vertices and arrows connecting the vertices, representing the edges: for the edge (j, i) ∈ E, j is said to be the tail (the vertex from the which the arrow starts), while i is said to be the head (the vertex where the arrows arrives).

The Laplacian matrix of a digraph is defined as L= ∆ − A,

where ∆ is the degree matrix and the A is the adiacency matrix. The adiacency matrix is defined as

[A]ij =(1 if (j, i) ∈ E, 0 otherwise, while the degree matrix is defined as

[∆]ii= d⁽ⁱ⁾in ∀i ∈ V,

where d⁽ⁱ⁾in is the in degree of the vertex i, defined as the number of incoming edges.

2.3 Agreement in directed networks

In this section we consider the problem of the agreement or consensus for directed networks, that are used in Chapter 4.

Consider a consensus protocol over a directed network of N agents (i.e., a network that can be represented with a digraph); denoting the r-dimensional state vector of every agent as xi(t) ∈ R^r, and collecting everyone of this vectors in the state vector of the network x(t) ∈ R^rN, the overall system can be described by

˙x = − L x(t).

We mention the following theorem, providing the conditions for the convergence of a directed network to the average consensus (i.e., the common value reached by the agreement protocol that is the average value of the initial nodes):

t→∞lim x(t) = 1

N1N r1^TN rx0.

Theorem 2.3.1 (Theorem 3.17 in [18]). The agreement protocol over a diagraph reaches the average consensus for every initial condition if and only if the diagraph is strongly- connected.

2.4 Elements of non linear control theory

Definition and conditions for the existence of limit cycles

In this section, we give the definition of limit cycle and we present a theorem that provides a sufficient condition to prove the existence of limit cycles; we present this concept in order to prove the exponential convergence of the agents to a desired circle in Chapter 3 .

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2.4. Elements of non linear control theory

In order to define a limit cycle, we need to give the definition of ω-limit set and α-limit set.

Definition 2.4.1(ω-limit set). Let φ(t; x0) be the trajectory of a n-dimensional non linear system rooted in x0∈ Rⁿ ; a point ω(x0) ∈ Rⁿis said to be an ω-limit point of the trajectory of the system φ(t; x0) if there exists a sequence of time instants t0, t₁, . . . , t_k such that

k→∞lim φ(tk; x0) = ω(x0).

We define the ω-limit set for the trajectory φ(t; x0) as the set of all the ω-limit points Ωφ(x0) = {ω(x0) ∈ Rⁿ}.

Definition 2.4.2 (α-limit set). Let φ(t; x0) the trajectory of a n-dimensional non linear system rooted in x0∈ Rⁿ ; a point α(x0) ∈ Rⁿis said to be an α-limit point of the trajectory of the system φ(t; x0) if there exists a sequence of time instants t0, t1, . . . , tk such that

k→∞lim φ(−tk; x0) = α(x0).

We define the α-limit set for the trajectory φ(t; x0) as the set of all the α-limit points A_φ(x0) = {α(x0) ∈ Rⁿ}.

Definition 2.4.3(Periodic orbit, Definition in [19]). Consider the autonomous system

˙x = f(x(t)), x ∈ Rⁿ, n ≥2.

A non-constant solution to this system, x(t), is periodic if there exists a T ≥ 0 such that x(t + T ) = x(t), ∀t.

The image of the periodicity interval [0, T ] under x in the state space Rⁿ is called the periodic orbit or cycle.

Definition 2.4.4 (Limit cycle, Definition in [19]). A limit cycle is defined as a periodic orbit Γ for which there exists at least a point x^?∈/ Γ such that

Γ = Ωφ(x^?) ∨ Γ = Aφ(x^?).

In non linear control theory it is important to estabilish whether a system exhibits a limit cycle or not; in [20] are presented methods to assest the existence of limit cycles, among the which the following theorem provides a sufficient condition for the existence.

Theorem 2.4.1 (Poincar´e-Bendixon Theorem, Theorem 7.3 in [20]). Suppose that R is a compact and invariant subset of the plane and consider the planar non linear system

˙x = f(x), where f is a continuosly differentiable vector field on an opet set containing R.

If R doesn’t contain any equilibria of the planar system, then R contains a periodic orbit.

Lyapunov stability theory

In this section, we recall two theorems to study the stability of a dynamic system with the Lyapunov theory.

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2.5. Properties of matrices defined on unit-norm vectors

Theorem 2.4.2 (Lyapunov Theorem for the global asymptotical stability, Theorem 4.2 in [21]). Consider an autonomous twodimensional system ˙x = f(x) and let x = 0 be an equilibrium point for this system. Let V : R → R² be a continuosly differentiable function

such that 





V(0) = 0, V (x) > 0, ∀x 6= 0, kx(t)k → ∞ =⇒ V (x) → ∞,

˙V (x) < 0, ∀x 6= 0.

Then x = 0 is globally asymptotically stable.

Definition 2.4.5(Positively invariant set, in [21]). Consider the twodimensional dynamic system ˙x = f(x) and let φ(t0; x0) be a trajectory of the system rooted at x0 at time t0: we define a set M ⊂ R² positively invariant with respect to the system if

φ(t0; x0) ∈ M =⇒ φ(t; x0) ∈ M ∀t ≥ t0

Theorem 2.4.3 (La Salle’s theorem, Theorem 4.4 in [21]). Consider the twodimensional dynamic system ˙x = f(x) and let Ω ⊂ D ⊂ R² be a positively invariant set for the system.

Let V : D → R a continuosly differentiable function such that ˙V (x) ≤ 0 in Ω. Let E be the set of points in Ω where ˙V (x) = 0. Let M the largest invariant set in E; then every solution starting in Ω approaches M as t → ∞.

2.5 Properties of matrices defined on unit-norm vectors

Both the control algorithm and the estimate presented in the next chapter are based on the updating of unit-norm vectors.

Let ϕ(t) ∈ S¹; defining ϑ(t) as the counterclockwise angle between the vector and the x-axis of a reference frame, ϕ(t) can be represented as

ϕ(t) =cos ϑ(t) sin ϑ(t)

.

In this section, some relevant properties of a matrix ϕ(t)ϕ^T(t) are presented.

Lemma 2.5.1. Let ϕ(t) ∈ S¹. The following properties hold:

• ϕ(t)ϕ^T(t) ∈ S²_≥0

• The eigenvalues of ϕ(t)ϕ^T(t) are 0 and 1

• Defining ¯ϕ(t) ∈ S¹ as the unit-norm vector obtaining rotating ϕ(t) clockwise, the following equality holds

I2− ϕ(t)ϕ^T(t) = ¯ϕ(t) ¯ϕ^T(t). (2.1) Proof. From the definition of the vector ϕ(t) we have

ϕ(t)ϕ^T(t) = cos²ϑ(t) sin ϑ(t) cos ϑ(t) sin ϑ(t) cos ϑ(t) sin²ϑ(t)

, (2.2)

hence the matrix is symmetric. Furthermore, we have that for every vector z ∈ R² the quadratic form with the considered matrix as hull is positive semidefinite

z^T(ϕ(t)ϕ^T(t))z = (z1sin ϑ(t) + z2cos ϑ(t))²≥0 ∀z ∈ R²,

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2.6. Persistency of excitation condition

hence the matrix is also positive semidefinite.

Since the the characteristic polynomial of the matrix is λ(λ − 1),

the eigenvalues of the matrix are 0 and 1.

Because of (2.2), we have that

I₂− ϕ(t)ϕ^T(t) = 1 − cos²ϑ(t) −sin ϑ(t) cos ϑ(t)

−sin ϑ(t) cos ϑ(t) 1 − sin²ϑ(t)

=

sin²ϑ(t) −sin ϑ(t) cos ϑ(t)

−sin ϑ(t) cos ϑ(t) cos²ϑ(t)

. Since the unit-norm vector ¯ϕ(t) ∈ S¹ is represented as

¯

ϕ(t) = sin ϑ(t)

−cos ϑ(t)

, we have that

I2− ϕ(t)ϕ^T(t) = sin ϑ(t) − cos ϑ(t) sin ϑ(t)

−cos ϑ(t)

= ¯ϕ(t) ¯ϕ^T(t), therefore the equality (2.1) holds.

2.6 Persistency of excitation condition

In this section, we recall the definition of the persistency of excitation condition from [22].

Definition 2.6.1 (Persistency of excitation condition). A vector ¯ϕ: R≥0 → R² is persis- tently exciting (p.e.) if there exist 1, ₂, T >0 such that

₁I₂≤ Z t0+T

t0

¯

ϕ(t) ¯ϕ^T(t)dt ≤ 2I₂, (2.3) for all t0≥0. Condition (2.3) is called persistency of excitation condition.

Condition (2.3) requires that the vector ¯ϕ rotates sufficiently in the plane that the integral of the semipositive definite matrix ¯ϕϕ¯^T is uniformly definite positive over any interval of some positive length T .

From [22] we have that the p.e. condition has another interpretation, and it can be expressed in an equivalent scalar form:

1≤ Z t₀+T

t0

(U^Tϕ¯(t))²dt ≤ 2, (2.4) for all t0≥0 and for every constant unit-norm vector U ∈ R²: (2.4) appears as a condition on the energy of ¯ϕin all the directions.

Given the previous definition, we recall the following convergence theorem:

Theorem 2.6.1(P.E. and Exponential Stability, Theorem 2.5.1 in [22]). Let ¯ϕ: R≥0→ R² be piecewise continuous; if ¯ϕis P.E., then the system

˙x(t) = −k ¯ϕ(t) ¯ϕ^T(t)x(t)

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2.7. Singular perturbations on the infinite interval

2.7 Singular perturbations on the infinite interval

In this section we recall the theorem from [23] that is used in Chapter 5.

Consider the following perturbed system

( ˙x = f(t, x, y, ), x(t0) = x0,

˙y = g(t, x, y, ), y(t0) = y0, (2.5) where x(t) ∈ R^k, y(t) ∈ R^j, f = (f1· · · , fk), g = (g1, · · · , gj) and is a small positive parameter.

To analize the behavior of (2.5) for → 0⁺ and for t0 ≤ t < ∞, we introduce the following systems.

The degenerate system (2.6) is obtained considering the perturbed system at = 0.

( ˙x = f(t, x, y, 0), x(t0) = x0,

0 = g(t, x, y, 0), y(t0) = y0. (2.6) In this section, we introduce two parameters γ and λ that have respectively the same role of parameters α and β in [23]. The following system is written by first making a stretching transformation of the independent variable s = ^t−γ , and then setting to 0.

(_dx

ds = 0,

dy

ds = g(γ, x, y, 0). (2.7)

Since the only solution of^dx_ds = 0 is x = λ and it is constant, we can rewrite the system (2.7) in the following form:

dy

ds = g(γ, λ, y, 0), y(0) = y0, (2.8)

where γ and λ are treated as parameters; the system (2.8) is called the boundary-layer system.

Let I = [0, ∞), SR = {(x, y) ∈ R^k+j: kxk + kyk ≤ R}, and let SR|x and SR|y represent the restrictions of SR to R^k and R^j.

Assume that f and g satisfy the following conditions.

I. The system (2.6) has x = 0, y = 0 as a solution for all t0 ≤ t < ∞. Therefore the system (2.6) can be written in a more convenient form as

˙x = f(t, x, 0j,0), x(t0) = x0. (2.9) II. f, g, fx, fy, gx, gy, gtare continuosly differentiable in I×SR×[0, 0], where [0, 0] is a set of values for . Here fxdenotes the matrix with components ^∂f_∂xⁱ_l, with i, l = 1, · · · , k, and similarly for gy,gx,fy.

III. It holds that

g(t, x, 0j,0) = 0 ∀(t, x) ∈ I × SR|x.

IV. The function f is continuous at y = 0, = 0, uniformly in (t, x) ∈ I × SR|x, and f(t, x, 0j,0) and fx(t, x, 0j,0) are bounded on I × SR|x.

V. The function g is continuous at = 0, uniformly in (t, x, y) ∈ I × SR, and g(t, x, y, 0) and its derivatives with respect to t and the components of x and y are bounded on I × S_R.

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2.7. Singular perturbations on the infinite interval

VI. We denote K the class of all continuous, strictly increasing, real-valued functions d(r), r ≥ 0, with d(0) = 0, and S the class of all continuous, stricly decreasing, nonnegative, real-valued functions σ(s), 0 ≤ s < ∞, for which σ(s) → 0 as s → ∞.

The zero solution of (2.9) is uniform-asymptotically stable. That is, if φ(t, t0, x₀) is the solution of (2.9), there exist d ∈ K and σ ∈ S such that

kφ(t, t0, x₀)k ≤ d(kx0k)σ(t − t0) for all kx0k ≤ Rand 0 ≤ t0≤ t < ∞. VII. The zero solution of (2.8) is uniform-asymptotically stable, uniformly in the parame-

ters (γ, λ) ∈ I × SR|x. That is, if y = ψ(s, y0, γ, λ) is the solution of (2.8), there exist l ∈K and ρ ∈ S such that

kψ(t, y0, γ, λ)k ≤ l(ky0k)ρ(s), for all ky0k ≤ R, 0 ≤ s < ∞ and (γ, λ) ∈ I × SR|x.

Theorem 2.7.1 (Theorem in [23]). Let conditions above be satisfied; then, for sufficiently small kx0k+ ky0k and , the solution of (2.5) exists and converges to the solution of (2.6) as → 0⁺ uniformly on any closed subset of t0< t < ∞.

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Chapter 3

Control algorithm

In this chapter, we present the distributed control algorithm used in this work to control a network of agents, in order to drive it to a desired configuration.

In Section 3.1, we present some relevant definitions for the network we analyze in this thesis; in Section 3.2 and in Section 3.3 we introduce the control algorithm and the estimate algorithm adopted in this work, for the most general scenario of mobile target with estimated position.

3.1 Dynamics of the network

We consider a directed network of N agents indexed as 1, · · · , N, with agent i at position yi(t) ∈ R² and a target at position x(t) ∈ R²; the network is represented by a digraph G= (V, E) and the agents are modelled as simple integrators

˙yi(t) = ui(t) i ∈ V, (3.1)

where ui(t) is the control action applied to the agent i.

Below we recall some definitions from [9].

Definition 3.1.1 (Counterclockwise angle). The counterclockwise angle βij(t) at time t is the angle subtended at x(t) by yi(t) and yj(t) (see Figure 3.1).

Definition 3.1.2(Counterclockwise neighbor). We define the agent j the counterclockwise neighbor of the agent i if βij(t) is the smallest counterclockwise angle among all βik(t) for k ∈ V \{i}.

From the previous definition it is convenient to define the counterclockwise neighborhood function νi(t) ∈ {1, ..., N} \ {i} returning the label of the agent that at time t is the counterclockwise neighbor of agent i, subtended at x(t).

It is important to notice that the definition of the neighborhood function allows the introduction and the removal of an agent in the network; this scenario is analyzed in Chap- ter 7. For this purpose, consider that at time t0 ν_i(t0) = j.

If at time t1the agent k is added to the network, and it is such that βik(t) < βij(t) then ν_i(t1) = k; on the other hand, if at time t2 there is a fault in the network and the agents j and k are removed, it holds that νi(t2) = r where r is the agent forming the third smallest counterclockwise angle with i in the original network.

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3.2. Collective bearing-only circumnavigation

Figure 3.1: Counterclockwise angles of the network.

Since the network considered in this thesis can be dynamic, and the counterclockwise neighbor is time-variant, it is convenient to denote the counterclockwise angle between agent iand its neighbor as βi(t) = βiν_i(t); from the previous notation, we have that

β_i(t) = βij(t) if j = νi(t).

3.2 Collective bearing-only circumnavigation

Definition 3.2.1(Bearing vector). The bearing vector for the agent i is defined as the unit- norm vector ϕi(t) ∈ S¹ in the direction from the agent to the target; defining the distance from the agent and the target as Di(t) = kx(t) − yi(t)k, we have

ϕi(t) = x(t) − yi(t)

Di(t) . (3.2)

In order to introduce the chosen control law, we consider the unit vector ¯ϕi(t) ∈ S¹, obtained by a π/2 clockwise rotation of ϕi(t), like shown in Figure 3.2. The control objective

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3.2. Collective bearing-only circumnavigation

Figure 3.2: Bearing vectors for the single agent: ϕicis the vector ¯ϕi(t).

is formally written as

t→∞lim Di(t) = D^∗, (3.3)

t→∞lim β_i(t) = 2π

N, (3.4)

for all i ∈ {1, · · · , N}. In order to achieve (3.3) and (3.4), we propose a distributed control algorithm, based on measurements of the bearing vectors and on the estimate of the distance between the agent and the target.

For the control law, we set:

u_i(t) = kd( ˆD_i(t) − D^∗)ϕi(t) + kϕDˆ_i(t)(α + βi(t)) ¯ϕ_i(t) (3.5) where ˆDi(t) = kˆxi(t) − yi(t)k is the estimated distance between the agent i and the target, and ˆxi(t) represents the estimate of the target position computed by the agent i at time t;

kd and kϕ are positive gains and D^∗ is the desired distance from the target.

Note that the control signal ui(t) is made up of two contributions: a radial term kd( ˆDi(t) − D^∗)ϕi(t) drives the agent towards the desired circle, and a tangential term kϕDˆi(t)(α + βi(t)) ¯ϕi(t) makes the agent circumnavigate the target while attaining the desired formation with the other agents. Differently from [5, 9], we let the tangential term depend on the estimated distance from the target, ˆDi(t), in order to avoid high angular velocities when the desired distance from the target is small. Another important property of control law (3.5) is that ui(t) is always nonzero. In fact, since ϕi(t) and ¯ϕ_i(t) are or- thogonal, and since α + βi(t) > 0, we have that ui(t) = 0 would require ˆD_i(t) − D^∗ = 0

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3.3. Estimate algorithm

Figure 3.3: Geometric illustration of the estimator.

and ˆDi(t) = 0, which is not possible since D^∗ > 0. This property also implies that the closed-loop system has no equilibria.

3.3 Estimate algorithm

The chosen algorithm is based on an estimate of the target position, according to the following dynamics

˙ˆxi(t) = −ke(I2− ϕ_i(t)ϕ^Ti(t))(ˆxi(t) − yi(t)), i ∈ V (3.6) where I2− ϕ_i(t)ϕ^Ti(t) is a projection matrix onto a plane perpendicular to the vector ϕi(t).

The goal of the estimator is to make the estimation error

˜xi(t) = ˆxi(t) − x(t)

converge to the null vector, so that the agent can localize the target. As shown in Figure 3.3, ˆxi(t) moves in a direction orthogonal to the bearing vector; in this way the term ˆxi(t)−yi(t) is rotated so that its direction converges to the direction of ϕi(t), with a rate dependent on the estimation gain ke.

Since, from Lemma (2.1),

I₂− ϕi(t)ϕ^Ti(t) = ¯ϕ_i(t) ¯ϕ^T_i(t), ∀i ∈ V

the dynamics of the estimator can be rewritten in terms of the estimation error in the following way

˙˜xi(t) = −keϕ¯_i(t) ¯ϕ^T_i(t)˜xi(t) − ˙x(t), ∀i ∈ V

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3.4. Summary

3.4 Summary

In this section, we have described the network of agents we will consider in this work, highlighting remarkable characteristics.

Subsequently, we have introduced the distributed control algorithm and the estimate algorithm we will analyze in this thesis; furthermore, we have both given a detailed motivation of the choise of the controller and a geometrical interpretation of the estimator.

In Chapter 7, we propose a modified version of (3.5), in order to achieve the control objectives in dynamic networks.

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Chapter 4

Circumnavigation of a stationary target with known position

In this chapter, we assume that every agent in the network knows the position of a stationary target; for this purpose there is no need of a localization algorithm.

Since in Chapter 5 we are extending the results achieved in this chapter, this scenario is useful because it allows to separate the study of the convergence and the formation on the desired circle from the analysis of the estimate convergence. In Section 4.1, we introduce the control algorithm adopted for this case and we give the main result for this scenario; in Sections 4.2, 4.3, 4.5, we give proofs that are used to prove the main result in Section 4.6; in the end, in Section 4.7, we show the results of simulations performed in Matlab-Simulink.

4.1 Problem statement and main result

In this scenario, every agent of the network has the following dynamics:

˙yi(t) = kd(Di(t) − D^∗)ϕi(t) + kϕDi(t)(α + βi(t)) ¯ϕi(t), (4.1) where kd, kϕ, αare positive constant.

For every agent we define the error on the distance between itself and the target ∆i(t) = Di(t) − D^∗ and the error on the counterclockwise angle ˜βi(t) = βi(t) −^2π_N.

The circle of desired radius D^∗ centered at the stationary target position x is denoted C(x, D^∗).

The main result of this chapter is formalized in the following theorem.

Theorem 4.1.1. Consider a network of N autonomous agents under control law (4.1);

then the agents converge to the desired circle C(x, D^∗) while forming a regular polygon; i.e., they achieve the control objective (3.3) and (3.4).

The proof of Theorem 4.1.1 derives from the following sections.

In Section 4.2, we prove that the circle C(x, D^∗) is an attractive limit cycle for the trajectories of the network; subsequently in Section 4.3, we present remarkable geometric relations for the network that are used to prove the asymptotic convergence of the counterclockwise angles in Section 4.5. Finally in Section 4.6, we use the results obtained in the previous sections to formalize the proof of Theorem 4.1.1.

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4.2. Proof of the convergence of the distance errors

4.2 Proof of the convergence of the distance errors

It is convenient to rewrite the system (4.1) using as polar coordinates Di(t) and ϕi(t).

Lemma 4.2.1. Consider a network of N agents under control law (4.1): the closed-loop system can be written using as polar coordinates Di(t) and ϕi(t) as follows:

( ˙Di(t) = −kd(Di(t) − D^∗),

˙ϕi(t) = −kϕ(α + βi(t)) ¯ϕ_i(t). (4.2) Proof. Since Di(t) = kx − yi(t)k and the target is stationary, differentiating Di(t) w.r.t.

time we have that

˙Di(t) = 1

Di(t)(x − yi(t))^T(− ˙yi(t)).

Expressing ˙yi(t) as (4.1) and remembering that x − yi(t) = Di(t)ϕi(t), we can rewrite the previous equation as

˙Di(t) = ϕ^Ti(t)[−kd(Di(t) − D^∗)ϕi(t) − kϕDi(t)(α + βi(t)) ¯ϕi(t)].

Exploiting the orthogonality between ϕi(t) and ¯ϕi(t), we obtain the following expression for the dynamics of Di(t):

˙Di(t) = −kd(Di(t) − D^∗).

Differentiating (3.2) w.r.t. time, we write the equation for the dynamics of the bearing vector as

˙ϕi(t) = d dt

 x − y_i(t) Di(t)

= − ˙yi(t)

Di(t)−(x − yi(t)) D²_i(t)

1

Di(t)(x − yi(t))^T(− ˙yi(t))

. Since x − yi(t) = Di(t)ϕi(t), we have

˙ϕi(t) = − ˙yi(t)

D_i(t)− ϕi(t)

D_i(t)(−ϕ^Ti (t) ˙yi(t)).

Expressing ˙yi(t) like in (4.1) and exploiting the orthogonality between ϕi(t) and ¯ϕi(t), the previous equation is rewritten as

˙ϕi(t) = −kϕ(α + βi(t)) ¯ϕi(t).

In order to apply the control law (4.1), we need that the bearing vector ϕi(t) is well defined for any t ≥ t0, or, in other words, that Di(t) > 0 for all t ≥ t0. Therefore, we will begin by showing that Di(t) > 0 is always guaranteed.

Lemma 4.2.2. Consider a network of N agents under control law (4.1): it holds that Di(t) > 0 ∀t ≥ t0, ∀i ∈ V .

Proof. From Lemma 4.2.1 we have that the dynamics of the distance is

˙Di(t) = −kdDi(t) + kdD^∗. Since the solution of the previous equation is

Di(t) = Di(t0)e^−k^d^(t−t⁰⁾+ D^∗(1 − e^−k^d^(t−t⁰⁾).

and D^∗ is a positive constant, Di(t) is always positive, for all the agents in the network.

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4.2. Proof of the convergence of the distance errors

Figure 4.1: Trapping region in the plane for the single agent.

Note that the tangential velocity kϕD_i(t)(α + βi(t)) is always not null, because both α and the counterclockwise angle βi(t) are positive and for Lemma 4.2.2 Di(t) is always positive; furthermore ϕi(t) ∈ S¹, hence it is never null. Therefore system (4.2) has no equilibria and consequently the agent approaches to the target and keeps on rotating around it, without stopping on the circle.

Since the dynamics of the distances Di(t) in (4.2) are decoupled, it is possible to analyze them independently; in this way we prove the existence of an attractive limit cycle for a single agent so that the proof could be automatically extended to the network.

In order to prove the existence of a limit cycle for the single agent we use Theorem 2.4.1:

it is sufficient to find an invariant region in R², that works like a trapping region for the trajectories of the single agent in the network.

The idea is to find the maximum radius Di,max∈ R and the minimum radius Di,min∈ R defining a region in the plane (see Figure 4.1), containing no equilibria, so that the twodimensional vector field of the system (4.1) points inside this region:

(Di,min∈ R : ˙Di(t) > 0, Di,max∈ R : ˙Di(t) < 0.

Using the equations of the closed-loop system (4.2), we have that (D_i(t) < D^∗=⇒ ˙Di(t) > 0,

Di(t) > D^∗=⇒ ˙Di(t) < 0.

Thus the closed orbit C(x, D^∗) is an invariant set for the trajectories of the agent i, and since there are no equilibria of (4.2) in R², it is an attractive limit cycle for the single agent.

Now consider the vector

D(t) = D1(t) . . . DN(t)^T ∈ R^N.

Since every element of the vector converge to the same closed orbit with radius D^∗ and centered at the target position, it holds that

lim (t) = D^∗1

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4.3. Geometric properties of the network

where the equality is element-wise.

Consequently, the invariant region C(x, D^∗) is an attractive limit cycle for the closed- loop system (4.1) and, because of Definition 2.4.4, the network converges asymptotically to this circle.

Furthermore, from the definition of ∆i(t) in Section 4.1, we write the dynamics of the distance error

˙∆i(t) = −kd∆i(t), (4.3)

from which it can be noticed that the convergence is also exponential, with a rate of convergence depending on the approach term gain.

Hence the agents of the network exponentially converge to the desired circle C(x, D^∗) as t → ∞.

4.3 Geometric properties of the network

In this section, we present remarkable relations between angles defined in the network, that are used to rewrite the dynamics of the counterclockwise angles as a consensus equation.

Recalling that βi(t) = βi,νi(t)(t) whenever νi(t) is defined, let us consider the generic coun- terclockwise angle βij(t) from ϕi(t) to ϕj(t). First we define ϑij(t) as the counterclockwise angle between the vectors ϕi(t) and ¯ϕj(t), for i, j ∈ V with j = νi(t), defined according to the following scalar product:

ϕ^T_i (t) ¯ϕj(t) = cos ϑij(t). (4.4) Similarly, the counterclockwise angle βij(t) is defined according to the following scalar product:

ϕ^T_i(t)ϕj(t) = cos βij(t). (4.5) Lemma 4.3.1. Consider the network described by (3.1); then

cos ϑij(t) = − cos ϑji(t), ∀i ∈ V, j = νi(t), ∀t ≥ t0. (4.6) Proof. In order to prove (4.6) we analyze all the four possible scenarios depending on the value of the angle βij(t).

If 0 ≤ βij(t) ≤^π₂, then, from Figure 4.2 we have that ((2π − ϑij(t)) + βij(t) = ^π₂,

(2π − ϑji(t)) = ^π₂+ βij(t), hence,

(2π − ϑij(t)) = π − (2π − ϑji(t)).

If ^π₂ ≤ βij(t) ≤ π, then, from Figure 4.3,

(ϑij(t) = βij(t) −^π₂, ϑji(t) = 2π − ^π₂− βij,(t) hence,

ϑ_ij(t) = π − ϑji(t).

If π ≤ βij(t) ≤ ^3π₂, then, from Figure 4.4, (ϑij(t) = βij(t) − ^π₂,

ϑ_ji(t) = 2π −^π₂ − βij(t),

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Figure 4.2: Relation between ϑij(t) and ϑji(t): 0 ≤ βij(t) ≤ ^π₂.

Figure 4.3: Relation between ϑij(t) and ϑji(t): ^π₂ ≤ βij(t) ≤ π.

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Figure 4.4: Relation between ϑij(t) and ϑji(t): π ≤ βij(t) ≤ ^3π₂ .

hence,

ϑ_ij(t) = π − ϑji(t).

If ^3π₂ ≤ βij(t) ≤ 2π, then, from Figure 4.5,

((2π − ϑij(t)) = (2π − βij(t)) +^π₂, (2π − ϑji(t)) + (2π − βij(t)) =^π₂, hence,

(2π − ϑij(t)) = π − (2π − ϑji(t)).

From the previous analysis, it holds that

cos ϑij(t) = − cos ϑji(t), ∀i ∈ V, j = νi(t), ∀t ≥ t0, and because of (4.4) we have

ϕ^T_i (t) ¯ϕj(t) = −ϕj^T(t) ¯ϕi(t), ∀i ∈ V, j = νi(t), ∀t ≥ t0.

Lemma 4.3.2. Consider the network described by (3.1); then

cos ϑij(t) = sin βij(t), ∀i ∈ V, j = νi(t), ∀t ≥ t0. (4.7)

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Figure 4.5: Relation between ϑij(t) and ϑji(t): ^3π₂ ≤ β_ij(t) ≤ 2π.

Proof. Similarly to what done in Lemma 4.3.1, in order to prove (4.6) we analyze all the four possible scenarios depending on the value of the angle βij(t).

If 0 ≤ βij(t) ≤^π₂, then, from Figure 4.6, β_ij(t) = π −π

2 −(2π − ϑij(t)), hence,

βij(t) = ϑij(t) −3π 2 . If ^π₂ ≤ βij(t) ≤ π, then, from Figure 4.7,

βij(t) = 2π − (π − ϑij(t)) −π 2, hence,

βij(t) = ϑij(t) +π 2. If π ≤ βij(t) ≤ ^3π₂, then, from Figure 4.8,

βij(t) = 2π − π

2 −(π − ϑij(t)), hence,

β_ij(t) = ϑij(t) +π 2. If ^3π₂ ≤ βij(t) ≤ 2π , then, from Figure 4.9,

β (t) = π +π

+ [π − (2π − ϑ (t))],

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Figure 4.6: Relation between ϑij(t) and βij(t): 0 ≤ βij(t) ≤ ^π₂.

Figure 4.7: Relation between ϑij(t) and βij(t): ^π₂ ≤ βij(t) ≤ π.

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Figure 4.8: Relation between ϑij(t) and βij(t): π ≤ βij(t) ≤^3π₂ .

Figure 4.9: Relation between ϑij(t) and βij(t): ^3π₂ ≤ βij(t) ≤ 2π.

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4.4. Proof of the convergence of the counterclockwise angles for a network of two agents

hence,

βij(t) = ϑij(t) +π 2. In all the scenarios it holds that

sin βij(t) = cos ϑij(t), ∀i ∈ V, j = νi(t), ∀t ≥ t0.

4.4 Proof of the convergence of the counterclockwise angles for a network of two agents

In this section, we use the Lyapunov stability theory to prove the convergence of the error

˜βij(t) to 0 for a network of N = 2 agents; starting from the results in Lemma 4.3.1 and Lemma 4.3.2, we analize the monotony of the derivative of a chosen Lyapunov function.

The goal is to prove that under control law (4.1), the counterclockwise angles βij(t) converge asymptotically to π; the idea is to introduce a vector Φ(t) defined as

Φ(t) =

2

X

i=1

ϕi(t).

If we manage to prove that limt→∞Φ(t) = 0, this means that the bearing vectors of the two agents are opposite and so that β12(t) = β21(t).

Lemma 4.4.1. Consider a direct network of 2 agents described by (3.1) under control law (4.1): defining the vector Φ(t) as

Φ(t) =

2

X

i=1

ϕ_i(t) = ϕ1(t) + ϕ2(t), it holds that it asymptotically converges to 0 as t → ∞.

Proof. The following Lyapunov function is chosen

V(Φ(t)) = 1

2kΦ(t)k²≥0. (4.8)

In order to study the monotony of the Lyapunov function, we compute its derivative

˙V (Φ(t)) = Φ^T(t) ˙Φ(t) = (ϕ^T1(t) + ϕ^T2(t))( ˙ϕ1(t) + ˙ϕ2(t)).

Exploiting the orthogonality of the vectors ϕi(t) and ¯ϕ_i(t) and using (4.2), the derivative of V (Φ(t)) can be written as

˙V (Φ(t)) = −kϕ(α + β21(t))ϕ^T1(t) ¯ϕ₂(t) − kϕ(α + β12(t))ϕ^T2(t) ¯ϕ₁(t).

Because of (4.4) and (4.6), the previous function is

˙V (Φ(t)) = k^ϕ(β12(t) − β21(t)) cos ϑ12(t).

For Theorem 2.4.2, a sufficient condition for the asymptotical convergence of βij(t) to π is the following

k_ϕ(β12(t) − β21(t)) cos ϑ12(t) < 0, (4.9)

Multi-agent formation control for target tracking and circumnavigation missions

Multi-agent formation control for target tracking and

circumnavigation missions

ANTONIO BOCCIA

Multi-agent formation control for target tracking and circumnavigation missions

Contents

Acknowledgments

Chapter 1

Introduction

1.1 Motivation

1.2 Literature review

1.3 Statement of contribution

1.4 Thesis outline

Chapter 2

Technical preliminaries

2.1 Notation

2.2 Elements of graph theory

2.3 Agreement in directed networks

2.4 Elements of non linear control theory

2.5 Properties of matrices defined on unit-norm vectors

2.6 Persistency of excitation condition

2.7 Singular perturbations on the infinite interval

Chapter 3

Control algorithm

3.1 Dynamics of the network

3.2 Collective bearing-only circumnavigation

3.3 Estimate algorithm

3.4 Summary

Chapter 4

Circumnavigation of a stationary target with known position

4.1 Problem statement and main result

4.2 Proof of the convergence of the distance errors

4.3 Geometric properties of the network

4.4 Proof of the convergence of the counterclockwise angles for a network of two agents