# Consensus in networks of uncertain euler-lagrange agents using adaptive gravity compensation

Mostafa Almodarresi, Marzieh Kamali, Emmanuel Nuño, Romeo Ortega, Farid Sheikholeslam

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## To cite this version:

Mostafa Almodarresi, Marzieh Kamali, Emmanuel Nuño, Romeo Ortega, Farid Sheikholeslam. Consensus in networks of uncertain euler-lagrange agents using adaptive gravity compensation. European Journal of Control, 2021, 57, pp.194-204. 10.1016/j.ejcon.2020.05.010. hal-04177825

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Submitted on 8 Aug 2023

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# Consensus in Networks of Uncertain Euler-Lagrange Agents Using Adaptive Gravity Compensation

Seyed-Mostafa Almodarresi<sup>a,c</sup>, Marzieh Kamali<sup>a</sup>, Emmanuel Nuño<sup>b</sup>, Romeo Ortega<sup>c</sup>, and Farid Sheikholeslam<sup>a</sup>

<sup>a</sup>Department of Electrical Engineering, Isfahan University of Technology. Isfahan, Iran

<sup>b</sup>Department of Computer Science, University of Guadalajara. Guadalajara, Mexico

<sup>c</sup>Laboratoire des Signaux et Systèmes. SUPÉLEC. Gif-sur-Yvette, France

## Abstract

The leaderless and the leader-follower consensus are the most basic synchronization behaviors for multi-agent systems. For networks of Euler-Lagrange (EL) agents different controllers have been proposed to achieve consensus, requiring in all cases, either the cancellation or the estimation of the gravity forces. While, in the first case, it is shown that a simple Proportional plus damping (P+d) scheme with exact gravity cancellation can achieve consensus, in the latter case, it is necessary to estimate, not just the gravity forces, but the parameters of the whole dynamics. This requires the computation of a complicated regressor matrix, that grows in complexity as the degrees-of-freedom of the EL-agents increase. To simplify the controller implementation we propose in this paper a simple P+d scheme with only adaptive gravity compensation. In particular, two adaptive controllers that solve both consensus problems by only estimating the gravitational term of the agents and hence without requiring the complete regressor matrix are reported. The first controller is a simple P+d scheme that does not require to exchange velocity information between the agents but requires centralized information. The second controller is a Proportional-Derivative plus damping (PD+d) scheme that is fully decentralized but requires exchanges of speed information between the agents. Simulation results demonstrate the performance of the proposed controllers.

**Keywords:** Euler-Lagrange Dynamics; multi-agent Systems; Consensus; Distributed Control; Adaptive Gravity Compensation

## 1. Introduction

Synchronization and cooperative behaviors are ubiquitous in nature. Social groups of animals such as schools of fish, flocks of birds and fireflies show the power of these phenomena. Inspired from nature, synchronization and other cooperative control problems, in networks of dynamical systems, has attracted the attention of many scientists [7]. The objective in cooperative control is to design a distributed controller so that the aggregate system achieves specified behaviors, such as flocking [21, 13], synchronization [34, 1], coordination [42, 31], rendezvous and formation control. The fundamental cooperative behavior is *consensus*, where all agents in a network reach an agreement in some coordinates of interest. Consensus control can be split into two classes, namely, the leader-follower, where a network of agents agree at a given leader reference; and the leaderless, where in the absence of a leader, the agents converge to a certain common value [5, 33, 22, 3, 20].

The consensus control problem presents significant theoretical and practical challenges, specially when dealing with networks of nonlinear systems, as Euler-Lagrange (EL) agents [6, 33, 15]. The control of EL-agents is interesting because these systems describe the behavior of several physical systems—including mechanical, electrical and electromechanical systems [30]. The first results on consensus (synchronization) of a particular class of EL-agents has been reported in [10] and the case of general, nonidentical, EL-systems has been re-

ported in [26]. Since then, a plethora of different controllers have been proposed to solve both consensus problems. See [40] for a unified treatment of different consensus controllers for EL-systems and [15] for a survey of recent developments on this topic.

As is well-known [30] the Coriolis and centrifugal forces that appear in EL systems are work-less, therefore they don't play any role on the regulation of the position of the system—a feature that is encrypted in the well-known “Skew symmetry property” of EL systems. On the other hand, the presence of the gravity forces has to be taken into account when solving this task, because they affect the equilibrium point. For fully actuated EL systems it is possible to propose the exact cancellation of these forces and then solve the regulation task with a simple Proportional plus damping (P+d) controller [38]. Since these forces are usually uncertain it is necessary to appeal to an adaptive version of the controller. Within the context of robotics, this problem was first solved in [39] adding an ingenious gravity estimation feature.

In the context of multi-agent systems P+d schemes with gravity cancellation have been proposed in [32, 28, 25, 43]. Besides having the advantage of a simple implementation they do not require the agents to exchange their velocities with their neighbours. However, their disadvantage is that they are not robust to parameter uncertainty—refer to [27] where the uncertainty problem has been analyzed. To deal with parametric uncertainty adaptive schemes have been proposed in
[11, 26, 23, 8, 2, 9, 41]. Unlike the single EL-system case of [39]—see also [18]—the adaptive multi-agent schemes estimate, not just the gravity forces, but the parameters of the whole dynamics. There are two important drawbacks of estimating all dynamical parameters. First, to ensure a successful parameter search in a bigger dimensional space requires higher excitation levels and longer convergence times—both factors stymying the achievement of good transient performances. Second, it requires the computation of a complicated regressor matrix, that grows in complexity as the degrees-of-freedom of the EL-agents increase, rendering the controllers of limited practical interest.

In this paper, we propose two adaptive controllers that solve both consensus problems by only estimating the gravitational term of the agents, and hence without requiring the complete regressor matrix. To the best of our knowledge, this is the first controller for multi-agent systems with only adaptive gravity compensation. One controller is a simple adaptive P+d scheme that does not require velocity information exchange between the agents, however the setting of the controller gains requires the knowledge of the largest eigenvalue of the Laplacian interconnection matrix and the largest of the upper bounds of the inertia matrices—therefore, it is not truly decentralized. The second controller is an adaptive Proportional-Derivative plus damping (PD+d) scheme whose gain condition setting is fully distributed but, because of the addition of the derivative action, requires the exchange of the neighbours velocities. Since these schemes do not estimate the whole EL-dynamics their complexity is significantly smaller than all the previously reported adaptive controllers that require the computation of the regressor matrix for the full dynamics. The regressor matrix of the gravity term is fairly simple because one only needs to calculate the gradient of the potential function associated to gravity and such potential function is the simple sum of the height of the link times its mass times the acceleration of the gravity constant. We have just became aware of the recent work [16] that also deals with the problem at hand and it provides a Proportional-Integral-Derivative (PID) controller to solve the consensus problem.

The rest of the paper is organized as follows. Section 2 presents the dynamic description of the network of EL-agents. Section 3 reports the controllers that solve the leaderless consensus problem and Section 4 shows the proposed schemes for the leader-follower consensus. Section 5 depicts a simulation study and, finally, Section 6 draws the conclusions of this work and discusses future research.

## 2. Background

**Notation** $\mathbb{R} := (-\infty, \infty)$, $\mathbb{R}_{>0} := (0, \infty)$, $\mathbb{R}_{\ge 0} := [0, \infty)$ and $\mathbb{N} := \{1, 2, 3, ...\}$. $\mathbf{I}_n \in \mathbb{R}^{n \times n}$ is the $n \times n$ identity matrix, $\mathbf{1}_n \in \mathbb{R}^n$ is the vector of $n$ elements equal to one and $\mathbf{0}_n \in \mathbb{R}^n$ is the all-zeros vector. For any $\mathbf{x} \in \mathbb{R}^n$, $|\mathbf{x}|$ is its Euclidean norm and $\tanh(\mathbf{x}) := [\tanh(x_1), ..., \tanh(x_n)]^\top$, where $\tanh(\mathbf{x})$ is the standard hyperbolic tangent. $\lambda_m\{\mathbf{A}\}$ and $\lambda_M\{\mathbf{A}\}$ are the minimum and the maximum eigenvalues of the symmetric matrix $\mathbf{A} \in \mathbb{R}^{n \times n}$. $\bar{\mathbf{N}} := \{1, 2, ..., N\}$ for $N \in \mathbb{N}$.

## 2.1. Euler-Lagrange Agents

We consider a network composed of $N$ fully-actuated and conservative EL-agents, with $n$-Degrees-of-Freedom (DoF). The dynamics of the $i$-th-agent is given by

$$M_i(\mathbf{q}_i)\ddot{\mathbf{q}}_i + C_i(\mathbf{q}_i, \dot{\mathbf{q}}_i)\dot{\mathbf{q}}_i + g_i(\mathbf{q}_i) = \tau_i \quad (1)$$

where $\mathbf{q}_i, \dot{\mathbf{q}}_i, \ddot{\mathbf{q}}_i \in \mathbb{R}^n$ are the generalized position, velocity and acceleration, respectively; $M_i(\mathbf{q}_i) \in \mathbb{R}^{n \times n}$ is the generalized inertia matrix, which is symmetric positive definite and lower bounded for all $\mathbf{q}_i \in \mathbb{R}^n$; $C_i(\mathbf{q}_i, \dot{\mathbf{q}}_i) \in \mathbb{R}^{n \times n}$ is the Coriolis and centrifugal forces matrix, defined via the Christoffel symbols of the first kind; and $\tau_i \in \mathbb{R}^n$ is the control input.

We restrict to EL-agents (1) that satisfy the following assumption:

**Assumption A1:** There exists a known $m_{2i} \in \mathbb{R}_{>0}$ such that, for all $\mathbf{q}_i \in \mathbb{R}^n$, $M_i(\mathbf{q}_i) \le m_{2i}\mathbf{I}_n$. ◦

Model (1) has the following properties [19]:

**Property P1:** Matrix $\dot{M}_i(\mathbf{q}_i) - 2C_i(\mathbf{q}_i, \dot{\mathbf{q}}_i)$ is skew-symmetric. Further, $\dot{M}_i(\mathbf{q}_i) = C_i(\mathbf{q}_i, \dot{\mathbf{q}}_i) + C_i^\top(\mathbf{q}_i, \dot{\mathbf{q}}_i)$. ◇

**Property P2:** There exists $k_{ci} \in \mathbb{R}_{>0}$ such that, for all $\mathbf{q}_i \in \mathbb{R}^n$, $\|C_i(\mathbf{q}_i, \dot{\mathbf{q}}_i)\| \le k_{ci}\|\dot{\mathbf{q}}_i\|^2$. ◇

**Property P3:** The gravity vector $g_i(\mathbf{q}_i)$ is linearly parameterizable. Thus $g_i(\mathbf{q}_i) = \mathbf{Y}_i(\mathbf{q}_i)\theta_i$, where $\mathbf{Y}_i(\mathbf{q}_i) \in \mathbb{R}^{n \times m}$ is a matrix of known functions and $\theta_i \in \mathbb{R}^m$ is a constant vector of the manipulator physical parameters. ◇

**Remark 1.** It is a well-known property that the whole dynamic of an EL system can be written in a linearly parametrized form as [35, 36, 19],

$$M_i(\mathbf{q}_i)\ddot{\mathbf{q}}_i + C_i(\mathbf{q}_i, \dot{\mathbf{q}}_i)\dot{\mathbf{q}}_i + g_i(\mathbf{q}_i) = \bar{\mathbf{Y}}_i(\mathbf{q}_i, \dot{\mathbf{q}}_i, \ddot{\mathbf{q}}_i)\bar{\theta}_i. \quad (2)$$

Setting $\ddot{\mathbf{q}}_i \equiv 0$ and $\dot{\mathbf{q}}_i \equiv 0$ results,

$$g_i(\mathbf{q}_i) = \Phi_i(0, 0, \dot{\mathbf{q}}_i, \mathbf{q}_i)\bar{\theta}_i := \mathbf{Y}_i(\mathbf{q}_i)\theta_i. \quad (3)$$

It is clear from (2) that in the matrix $\bar{\mathbf{Y}}_i(\mathbf{q}_i, 0, 0)$ appear only the terms of the gravity force. Hence, the resulting vector to be estimated has a significantly smaller size.

## 2.2. Interconnection Topology

As it is customary, we use graphs to represent the interconnection topology among the $N$ EL-agents. In particular, we employ the graph Laplacian matrix $\mathbf{L} := \{L_{ij}\} \in \mathbb{R}^{N \times N}$ that is defined as $L_{ii} = \sum_{j \in N_i} a_{ij}$ and $L_{ij} = -a_{ij}$, where $a_{ij} > 0$ if $j \in N_i$ and $a_{ij} = 0$ otherwise. The set $N_i$ contains all the neighbours of the $i$-th-EL-agent. Note that, by construction, $\mathbf{L}$ has a zero row sum. Therefore $\mathbf{L}\mathbf{1}_N = \mathbf{0}_n$.

We assume that the agents exchange information according to the following assumption.

**Assumption A2.** The EL-agents interconnection graph is *undirected*, *static* and *connected*. ◦

From A2, the Laplacian $\mathbf{L}$ is symmetric; positive semi-definite; it has a single zero-eigenvalue, with the associated eigenvector $\mathbf{1}_N$, and all of the other eigenvalues are strictly positive; and $\mathrm{rank}(\mathbf{L}) = N - 1$. Further, $\mathrm{ker}(\mathbf{L}) = \alpha\mathbf{1}_N$, $\forall \alpha \in \mathbb{R}$.
In addition to A2, in order to ensure that there is at least one follower agent that receives the leader desired position, we make the following assumption.

**Assumption A3.** There is a nonempty set of follower agents that has direct access to the desired constant position $\mathbf{q}_{\ell}$. ◇

Next, let us define a diagonal matrix $\mathbf{A}_{\ell} := \text{diag}\{a_{i\ell}\} \in \mathbb{R}^{N \times N}$ to model the leader-follower interconnections, such that $a_{i\ell} > 0$ if node $i$ receives the leader desired position and $a_{i\ell} = 0$, otherwise. Define $N_{\ell}$ as the set of all follower agents that receive the leader desired position. The following lemma, which is a special case of Lemma 3 in [17] and Lemma 1.6 in [33], provides a fundamental property of the composed Laplacian matrix $\mathbf{L}_{\ell} := \mathbf{L} + \mathbf{A}_{\ell}$.

**Lemma 1.** Consider the matrix $\mathbf{A}_{\ell} := \text{diag}\{a_{i\ell}\} \ge 0 \in \mathbb{R}^{N \times N}$ and suppose that $N_{\ell}$ is non-empty. Assume that A2 holds, then the matrix $\mathbf{L}_{\ell}$ is symmetric, positive definite and of full rank. ◇

### 2.3. Leaderless Consensus

We say that leaderless consensus problem is solved if there exists $\mathbf{q}_c \in \mathbb{R}^n$ such that

$$ \lim_{t \to \infty} |\dot{\mathbf{q}}_i(t)| = 0, \quad \lim_{t \to \infty} \mathbf{q}_i(t) = \mathbf{q}_c \quad (4) $$

for all $i \in \tilde{\mathcal{N}}$.

In order to achieve (4), let us define the position error of the $i$-th-agent as

$$ \mathbf{e}_i := \sum_{j \in \mathcal{N}_i} a_{ij}(\mathbf{q}_i - \mathbf{q}_j). \quad (5) $$

Then (4) holds if $\lim_{t \to \infty} |\dot{\mathbf{q}}_i(t)| = 0$ and $\lim_{t \to \infty} |\mathbf{e}_i(t)| = 0$. Note that, using the Laplacian matrix and defining $\mathbf{q} := [\mathbf{q}_1^\top, \dots, \mathbf{q}_N^\top]^\top$, we can write $\mathbf{e} = (\mathbf{L} \otimes \mathbf{I}_n)\mathbf{q}$. Thus $\lim_{t \to \infty} |\mathbf{e}_i(t)| = 0 \Leftrightarrow \lim_{t \to \infty} |\mathbf{e}(t)| = 0$. Moreover, $(\mathbf{L} \otimes \mathbf{I}_n)\mathbf{q} = \mathbf{0} \Leftrightarrow \mathbf{q} = \mathbf{I}_N \otimes \mathbf{q}_c$.

### 2.4. Leader-Follower Consensus

In the leader-follower consensus we aim at showing that

$$ \lim_{t \to \infty} |\dot{\mathbf{q}}_i(t)| = 0, \quad \lim_{t \to \infty} \mathbf{q}_i(t) = \mathbf{q}_{\ell} \quad (6) $$

for all $i \in \tilde{\mathcal{N}}$, where $\mathbf{q}_{\ell} \in \mathbb{R}^n$ is the leader constant position that is available to a set of followers $N_{\ell}$ that we assume is non-empty.

In order to ensure (6), we redefine the position error (5) as

$$ \begin{aligned} \mathbf{e}_i &= a_{i\ell}(\mathbf{q}_i - \mathbf{q}_{\ell}) + \sum_{j \in \mathcal{N}_i} a_{ij}(\mathbf{q}_i - \mathbf{q}_j) \\ &= \sum_{j \in \mathcal{N}_i \cup N_{\ell}} a_{ij}(\mathbf{q}_i - \mathbf{q}_j). \end{aligned} \quad (7) $$

Therefore (6) is achieved if $\lim_{t \to \infty} |\dot{\mathbf{q}}_i(t)| = 0$ and $\lim_{t \to \infty} |\mathbf{e}_i(t)| = 0$. To show that this is indeed the case, let us define $\tilde{\mathbf{q}}_i := \mathbf{q}_i - \mathbf{q}_{\ell}$ and $\tilde{\mathbf{q}} := [\tilde{\mathbf{q}}_1^\top, \dots, \tilde{\mathbf{q}}_N^\top]^\top = \mathbf{q} - \mathbf{I}_N \otimes \mathbf{q}_{\ell}$. Hence we can write $\mathbf{e} = (\mathbf{A}_{\ell} \otimes \mathbf{I}_n)\tilde{\mathbf{q}} + (\mathbf{L} \otimes \mathbf{I}_n)\mathbf{q}$ and since $\mathbf{L}\mathbf{I}_N = \mathbf{0}$ then $\mathbf{e} = (\mathbf{A}_{\ell} \otimes \mathbf{I}_n)\tilde{\mathbf{q}} + (\mathbf{L} \otimes \mathbf{I}_n)\tilde{\mathbf{q}} = (\mathbf{L}_{\ell} \otimes \mathbf{I}_n)\mathbf{q}$. Assumption A3 and Lemma 1 ensure that $\mathbf{e} = (\mathbf{L}_{\ell} \otimes \mathbf{I}_n)\tilde{\mathbf{q}} = \mathbf{0} \Leftrightarrow \tilde{\mathbf{q}} = \mathbf{0}$ and thus $\mathbf{q}_i = \mathbf{q}_{\ell}$, for all $i \in \tilde{\mathcal{N}}$. Henceforth, the error definition $\mathbf{e}_i$ is given in (7).

In what follows we design two different adaptive controllers that ensure that velocities and position errors globally and asymptotically converge to zero. The first scheme does not employ the neighbours velocities —but requires the largest Laplacian eigenvalue and the largest of the upper bounds of the inertia matrices— and the second scheme does not require global information to set the control gains —but it requires the neighbours velocities. To have a uniform exposition of the leaderless and the leader-follower problems, in the sequel, we only consider the position error $\mathbf{e}_i$ defined in (7), and we set $a_{i\ell} = 0 \forall i \in \tilde{\mathcal{N}}$ for leaderless case.

## 3. Adaptive P+d Controller

The proposed adaptive P+d scheme is given by

$$ \tau_i = -d_i \dot{\mathbf{q}}_i - p_i \mathbf{e}_i + \mathbf{Y}_i(\mathbf{q}_i) \hat{\mathbf{e}}_i, \quad (8) $$

where $p_i, d_i \in \mathbb{R}_{>0}$ are the proportional and damping injection gains. The parameter estimation law is

$$ \dot{\hat{\mathbf{e}}}_i := -\Gamma_i \mathbf{Y}_i(\mathbf{q}_i)^{\top} \left( \frac{1}{p_i} \dot{\mathbf{q}}_i + \epsilon \tanh(\mathbf{e}_i) \right), \quad (9) $$

where $\epsilon > 0$, $\Gamma_i$ is a positive definite diagonal matrix and $\tanh(\mathbf{e}_i)$ is a vector defined with the standard hyperbolic tangent as $\tanh(\mathbf{e}_i) := [\tanh(e_{i1}), \dots, \tanh(e_{in})]^\top$.

The resulting closed-loop system is

$$ \ddot{\mathbf{q}}_i = -\mathbf{M}_i^{-1}(\mathbf{q}_i) [\mathbf{C}_i(\mathbf{q}_i, \dot{\mathbf{q}}_i)\dot{\mathbf{q}}_i + d_i\dot{\mathbf{q}}_i + p_i\mathbf{e}_i + \mathbf{Y}_i(\mathbf{q}_i)\tilde{\mathbf{e}}_i] \quad (10) $$

where $\tilde{\mathbf{e}}_i := \theta_i - \hat{\mathbf{e}}_i$.

**Proposition 1.** For each $i \in \tilde{\mathcal{N}}$, consider the system (1) in closed loop with (8) and (9). Then, under Assumptions A1, A2, and A3, (4) and (6) hold for the leaderless and leader-follower setups respectively, provided that $\epsilon$ is set satisfying

$$ \epsilon < \min_{i \in \tilde{\mathcal{N}}} \left\{ \sqrt{\frac{1}{2m_2 L_{i\ell} p_i}} ; \frac{d_i}{p_i(m_2^{\max} \lambda_M(\mathbf{L}_{\ell}) + \sqrt{n}k_{ci} + \frac{d_i}{2\mu_i})} \right\} \quad (11) $$

where $\mu_i < \frac{2p_i}{d_i}$ is an arbitrary constant, $m_2^{\max} := \max_{i \in \tilde{\mathcal{N}}} \{m_{2i}\}$, and $L_{i\ell} := L_{ii} + a_{i\ell}$. △

*Proof.* Consider the following Lyapunov candidate function

$$ \mathcal{W} = \sum_{i \in \tilde{\mathcal{N}}} \left( V_i + \frac{1}{4} a_{i\ell} |\mathbf{q}_i - \mathbf{q}_{\ell}|^2 + \epsilon U_i \right), \quad (12) $$

where $V_i$ is a positive definite function and $U_i$ is a cross term between position error and velocity. These functions are

$$ V_i = \frac{1}{2} \left[ \frac{1}{p_i} \dot{\mathbf{q}}_i^\top \mathbf{M}_i \dot{\mathbf{q}}_i + \frac{1}{2} \sum_{j \in \mathcal{N}_i} a_{ij} |\mathbf{q}_i - \mathbf{q}_j|^2 + \tilde{\mathbf{\theta}}_i^\top \Gamma_i^{-1} \tilde{\mathbf{\theta}}_i \right], \quad (13) $$

and

$$ U_i = \tanh^\top(\mathbf{e}_i) \mathbf{M}_i(\mathbf{q}_i) \dot{\mathbf{q}}_i, \quad (14) $$
respectively. Then, $\mathcal{W}$ can be written as

$$ \mathcal{W} = \frac{1}{2} \sum_{i \in \mathcal{N}} \left[ \frac{1}{p_i} (\dot{\mathbf{q}}_i + \epsilon p_i \tanh(\mathbf{e}_i))^\top \mathbf{M}_i (\dot{\mathbf{q}}_i + \epsilon p_i \tanh(\mathbf{e}_i)) \right. \\ \left. + \tilde{\mathbf{\theta}}_i^\top \mathbf{\Gamma}_i^{-1} \tilde{\mathbf{\theta}}_i - \epsilon^2 p_i \tanh^\top(\mathbf{e}_i) \mathbf{M}_i \tanh(\mathbf{e}_i) \right. \\ \left. + \frac{1}{2} \sum_{j \in \mathcal{N}_i \cup \mathcal{N}_\ell} a_{ij} |\mathbf{q}_i - \mathbf{q}_j|^2 \right]. $$

Further, using the fact that $|\tanh(\mathbf{e}_i)|^2 \le |\mathbf{e}_i|^2$, the term $\tanh^\top(\mathbf{e}_i)\mathbf{M}_i\tanh(\mathbf{e}_i)$ can be bounded as

$$ \tanh^\top(\mathbf{e}_i)\mathbf{M}_i\tanh(\mathbf{e}_i) \le m_{2j}|\mathbf{e}_i|^2 \le m_{2i}L_{i\ell} \sum_{j \in \mathcal{N}_i \cup \mathcal{N}_\ell} a_{ij}|\mathbf{q}_i - \mathbf{q}_j|^2. \quad (15) $$

Thus, $\mathcal{W}$ admits the following bound

$$ \mathcal{W} \ge \frac{1}{2} \sum_{i \in \mathcal{N}} \left[ \frac{1}{p_i} (\dot{\mathbf{q}}_i + \epsilon p_i \tanh^\top(\mathbf{e}_i) \mathbf{M}_i (\dot{\mathbf{q}}_i + \epsilon p_i \tanh(\mathbf{e}_i)) \right. \\ \left. + \left(\frac{1}{2} - \epsilon^2 m_{2i} L_{i\ell} p_i\right) \sum_{j \in \mathcal{N}_i \cup \mathcal{N}_\ell} a_{ij} |\mathbf{q}_i - \mathbf{q}_j|^2 \right. \\ \left. + \frac{1}{2} a_{i\ell} |\mathbf{q}_i - \mathbf{q}_\ell|^2 + \tilde{\mathbf{\theta}}_i^\top \mathbf{\Gamma}_i^{-1} \tilde{\mathbf{\theta}}_i \right], \quad (16) $$

Note that setting

$$ \epsilon < \min_{i \in \mathcal{N}} \left\{ \sqrt{\frac{1}{2m_{2i}L_{i\ell}p_i}} \right\} $$

ensures that $(\frac{1}{2} - \epsilon^2 m_{2i} L_{i\ell} p_i) > 0$ and hence $\mathcal{W}$ is positive definite and radially unbounded with regards to $\dot{\mathbf{q}}_i, \mathbf{q}_i - \mathbf{q}_j$ and $\tilde{\mathbf{\theta}}_i$.

Now evaluating $\dot{V}_i$ along (10), and using the fact that $\dot{\tilde{\mathbf{\theta}}}_i = -\dot{\tilde{\mathbf{\theta}}}_i$, yields

$$ \dot{V}_i = - \frac{d_i}{p_i} |\dot{\mathbf{q}}_i|^2 - \dot{\mathbf{q}}_i^\top \mathbf{e}_i - \tilde{\mathbf{\theta}}_i^\top \left[ \frac{1}{p_i} \mathbf{Y}_i^\top (\mathbf{q}_i) \dot{\mathbf{q}}_i + \mathbf{\Gamma}_i^{-1} \dot{\tilde{\mathbf{\theta}}}_i \right] \\ + \frac{1}{2} \sum_{j \in \mathcal{N}_i} a_{ij} (\dot{\mathbf{q}}_i - \dot{\mathbf{q}}_j)^\top (\mathbf{q}_i - \mathbf{q}_j), $$

and using (9) it becomes

$$ \dot{V}_i = - \frac{d_i}{p_i} |\dot{\mathbf{q}}_i|^2 - \dot{\mathbf{q}}_i^\top \mathbf{e}_i + \epsilon \tanh^\top(\mathbf{e}_i) \mathbf{Y}_i(\mathbf{q}_i) \tilde{\mathbf{\theta}}_i \\ + \frac{1}{2} \sum_{j \in \mathcal{N}_i} a_{ij} (\dot{\mathbf{q}}_i - \dot{\mathbf{q}}_j)^\top (\mathbf{q}_i - \mathbf{q}_j). \quad (17) $$

From the property that the Laplacian matrix is symmetric, we have that

$$ \sum_{i \in \mathcal{N}} \left[ \dot{\mathbf{q}}_i^\top \mathbf{e}_i - \frac{1}{2} \sum_{j \in \mathcal{N}_i} a_{ij} (\dot{\mathbf{q}}_i - \dot{\mathbf{q}}_j)^\top (\mathbf{q}_i - \mathbf{q}_j) \right] = 0. \quad (18) $$

Hence, defining $V = \sum_{i \in \mathcal{N}} V_i$ yields

$$ \dot{V} = - \sum_{i \in \mathcal{N}} \left[ \frac{d_i}{p_i} |\dot{\mathbf{q}}_i|^2 - \epsilon \tanh^\top(\mathbf{e}_i) \mathbf{Y}_i(\mathbf{q}_i) \tilde{\mathbf{\theta}}_i \right]. \quad (19) $$

The time-derivative of $U_i$ is

$$ \dot{U}_i = \left( \frac{d}{dt} \tanh^\top(\mathbf{e}_i) \right) \mathbf{M}_i \dot{\mathbf{q}}_i + \tanh^\top(\mathbf{e}_i) \left[ \dot{\mathbf{M}}_i \dot{\mathbf{q}}_i + \mathbf{M}_i \ddot{\mathbf{q}}_i \right]. $$

Using **P1** and (10) yields

$$ \dot{U}_i = \left( \frac{d}{dt} \tanh^\top(\mathbf{e}_i) \right) \mathbf{M}_i(\mathbf{q}_i) \dot{\mathbf{q}}_i + \tanh^\top(\mathbf{e}_i) \mathbf{C}_i^\top(\mathbf{q}_i, \dot{\mathbf{q}}_i) \dot{\mathbf{q}}_i \\ - \tanh^\top(\mathbf{e}_i) [d_i \dot{\mathbf{q}}_i + p_i \mathbf{e}_i + \mathbf{Y}_i(\mathbf{q}_i) \tilde{\mathbf{\theta}}_i]. \quad (20) $$

In what follows we find an upper bound of $\dot{U}_i$. For, note that the term $(\frac{d}{dt} \tanh^\top(\mathbf{e}_i)) \mathbf{M}_i(\mathbf{q}_i) \dot{\mathbf{q}}_i$ can be manipulated as

$$ \sum_{i \in \mathcal{N}} \left( \frac{d}{dt} \tanh^\top(\mathbf{e}_i) \right) \mathbf{M}_i(\mathbf{q}_i) \dot{\mathbf{q}}_i = \left( \frac{d}{dt} \tanh^\top(\mathbf{e}_i) \right) \mathbf{M}(\mathbf{q}) \dot{\mathbf{q}}_i, \quad (21) $$

where $\dot{\mathbf{q}}$ is defined pilling-up their $N$ elements and $\mathbf{M}(\mathbf{q}) := \text{blockdiag}\{\mathbf{M}_i(\mathbf{q}_i)\}$. Now the fact that $|\frac{d}{dt}\tanh(\mathbf{e}_i)| \le |\dot{\mathbf{e}}| = |(\mathbf{L} \otimes \mathbf{I}_n)\dot{\mathbf{q}}| \le \lambda_M\{\mathbf{L}\}|\dot{\mathbf{q}}|$ is employed to obtain the following inequality

$$ \left( \frac{d}{dt} \tanh^\top(\mathbf{e}_i) \right) \mathbf{M}(\mathbf{q}) \dot{\mathbf{q}}_i \le m_2^{\max} \lambda_M \{\mathbf{L}\} |\dot{\mathbf{q}}|^2 \quad (22) $$

Moreover, using **P2** and $|\tanh(\mathbf{e}_i)|^2 \le n$, yields

$$ \tanh^\top(\mathbf{e}_i) \mathbf{C}_i^\top(\mathbf{q}_i, \dot{\mathbf{q}}_i) \dot{\mathbf{q}}_i \le \sqrt{n} k_{ci} |\dot{\mathbf{q}}_i|^2. \quad (23) $$

Applying Young's inequality [14] to $-\tanh^\top(\mathbf{e}_i)\dot{\mathbf{q}}_i$, results

$$ -\tanh^\top(\mathbf{e}_i) \dot{\mathbf{q}}_i \le \frac{\mu_i}{2} |\tanh(\mathbf{e}_i)|^2 + \frac{1}{2\mu_i} |\dot{\mathbf{q}}_i|^2, \quad (24) $$

for any $\mu_i > 0$. Hence

$$ \sum_{i \in \mathcal{N}} \dot{U}_i \le \sum_{i \in \mathcal{N}} \left[ m_2^{\max} \lambda_M \{\mathbf{L}\} + \sqrt{n} k_{ci} + \frac{d_i}{2\mu_i} \right] |\dot{\mathbf{q}}_i|^2 \\ - \left( p_i - \frac{\mu_i d_i}{2} \right) |\tanh(\mathbf{e}_i)|^2 - \tanh^\top(\mathbf{e}_i) \mathbf{Y}_i(\mathbf{q}_i) \tilde{\mathbf{\theta}}_i, $$

where we have also employed the fact that

$$ \tanh^\top(\mathbf{e}_i) \mathbf{e}_i \ge |\tanh(\mathbf{e}_i)|^2. \quad (25) $$

This fact is proved in Appendix A. Thus $\dot{\mathcal{W}}$ can be bounded as

$$ \dot{\mathcal{W}} \le - \sum_{i \in \mathcal{N}} [c_{1i} |\dot{\mathbf{q}}_i|^2 + c_{2i} |\tanh(\mathbf{e}_i)|^2], \quad (26) $$

where

$$ c_{1i} := \frac{d_i}{p_i} - \epsilon \left( m_2^{\max} \lambda_M \{\mathbf{L}_\ell\} + \sqrt{n} k_{ci} + \frac{d_i}{2\mu_i} \right), \quad (27) $$

and

$$ c_{2i} := p_i - \frac{\mu_i d_i}{2}. \quad (28) $$

Clearly, setting $\epsilon$ satisfying (11) ensures that $\dot{\mathcal{W}}$ is negative definite with regards to $\dot{\mathbf{q}}_i$ and $\mathbf{e}_i$. Therefore we have a positive definite and radially unbounded Lyapunov function $\mathcal{W}$ with a negative definite time-derivative. Since $\dot{\mathbf{q}}_i = \mathbf{e}_i = \mathbf{0}$ is the only
possible trajectory that lives in the set {$\dot{\mathbf{W}} = 0$}, invoking the LaSalle-Krasovskii Invariance Theorem [24] we conclude that $\dot{\mathbf{q}}_i \to 0$ and $\mathbf{e}_i \to 0$ as $t \to \infty$. Using the properties of the Laplacian matrix, we finish the proof. ▲

The following remarks are in order:

**Remark 2.** While the gain condition setting (11) requires some global information, i.e., the maximum eigenvalue of the Laplacian matrix and the maximum upper bound of the inertia matrices $m_2^{\max}$, a distributed algorithm can be used to obtain these values. For example, the algorithm proposed in [12, 4] can be used to find the maximum eigenvalue of the Laplacian; and the scheme of [37] is able to calculate $m_2^{\max}$.

**Remark 3.** The adaptive P+d controller is inspired in the control law reported in [28], for networks of EL-agents, and in the adaptive scheme reported in [39, 18]. In fact, instead of the hyperbolic tangent function, other bounded functions of the error can be employed in the adaptation law, e.g., the function $\mathbf{h}_i(\mathbf{e}_i) := \frac{1}{1+|\mathbf{e}_i|^2} \mathbf{e}_i$ that is used in [39], or function $\mathbf{f}_i(\mathbf{e}_i) := \frac{1}{1+|\mathbf{e}_i|} \mathbf{e}_i$, that is applied in [18]. Refer to [30] for the generalization of these bounded functions.

**Remark 4.** Controller (8)-(9) does not require the neighbours velocities and the regressor matrix $\mathbf{Y}_i(\mathbf{q}_i)$ is only related to the gravity vector in (1).

## 4. Adaptive PD+d Controller

In this section we design a fully distributed controller that only requires local information available at each $i$th-agent but requires the exchange of the neighbours velocities.

The proposed adaptive PD+d scheme is

$$ \tau_i = -d_i \dot{\mathbf{q}}_i - p_i \mathbf{e}_i - k_i \dot{\mathbf{e}}_i + \mathbf{Y}_i(\mathbf{q}_i) \hat{\mathbf{e}}_i, \quad (29) $$

where $p_i, k_i, d_i \in \mathbb{R}_{\ge 0}$ are the proportional, the derivative and the damping injection control gains; and

$$ \dot{\mathbf{e}}_i := \sum_{j \in \mathcal{N}_i \cup \mathcal{N}_t} a_{ij}(\dot{\mathbf{q}}_i - \dot{\mathbf{q}}_j). \quad (30) $$

The parameter estimation law is now given by

$$ \dot{\hat{\mathbf{e}}}_i := -\Gamma_i \mathbf{Y}_i(\mathbf{q}_i)^\top \left( \frac{1}{p_i} \dot{\mathbf{q}}_i + \epsilon_i \tanh(\mathbf{e}_i) \right), \quad (31) $$

where $\epsilon_i > 0$, $\Gamma_i \in \mathbb{R}^{m \times m}$ is a strictly positive diagonal matrix and function $\tanh(\mathbf{e}_i)$ is the vector of hyperbolic functions of $\mathbf{e}_i$. The resulting closed-loop system is

$$ \ddot{\mathbf{q}}_i = -\mathbf{M}_i^{-1}(\mathbf{q}_i) [\mathbf{C}_i(\mathbf{q}_i, \dot{\mathbf{q}}_i)\dot{\mathbf{q}}_i + d_i\dot{\mathbf{q}}_i + p_i\mathbf{e}_i + k_i\dot{\mathbf{e}}_i] \\ - \mathbf{M}_i^{-1}(\mathbf{q}_i)\mathbf{Y}_i(\mathbf{q}_i)\hat{\mathbf{e}}_i. \quad (32) $$

**Proposition 2.** For each $i \in \bar{N}$, consider the system (1) in closed loop with (29) and (31). Then, under Assumptions A1, A2, and A3, (4) and (6) hold for the leaderless and leader-follower setups respectively, provided that $\epsilon$ is set satisfying

$$ \epsilon_i < \min \left\{ \frac{k_i}{p_i m_{2i}}, \frac{2d_i + k_i a_{i\ell}}{p_i(m_{2i}(L_{ii} + a_{i\ell}) + 2\sqrt{n}k_{ci} + \frac{d_i}{\mu_i})} \right\} \quad (33) $$

where $\mu_i < \frac{2p_i}{d_i}$ is an arbitrary constant. △

**Proof.** Consider the following Lyapunov function candidate

$$ \mathcal{E} = \sum_{i \in \bar{N}} \left( V_i + \frac{1}{2} a_{i\ell} |\mathbf{q}_i - \mathbf{q}_{\ell}|^2 + \epsilon_i U_i + \frac{1}{2} \epsilon_i k_i |\tanh(\mathbf{e}_i)|^2 \right), \quad (34) $$

where $V_i$ and $U_i$ have been defined in (13) and (14). Hence we have that

$$ \mathcal{E} \ge \frac{1}{2} \sum_{i \in \bar{N}} \left[ \frac{1}{p_i} (\dot{\mathbf{q}}_i + \epsilon_i p_i \tanh(\mathbf{e}_i))^\top \mathbf{M}_i (\dot{\mathbf{q}}_i + \epsilon_i p_i \tanh(\mathbf{e}_i)) \right. \\ \left. + \epsilon_i (k_i - \epsilon_i p_i m_{2i}) |\tanh(\mathbf{e}_i)|^2 + \frac{1}{2} a_{i\ell} |\mathbf{q}_i - \mathbf{q}_{\ell}|^2 \right. \\ \left. + \frac{1}{2} \sum_{j \in \mathcal{N}_i \cup \mathcal{N}_t} a_{ij} |\mathbf{q}_i - \mathbf{q}_j|^2 + \tilde{\theta}_i^\top \Gamma_i^{-1} \tilde{\theta}_i \right] $$

Therefore, setting $\epsilon_i < \frac{k_i}{p_i m_{2i}}$ ensures that $\mathcal{E}$ is positive definite and radially unbounded.

We compute next the total derivative of $\mathcal{E}$. To that end, $\dot{V}$ evaluated along (31) and (32) yields

$$ \dot{V} = - \sum_{i \in \bar{N}} \left[ \frac{d_i}{p_i} |\dot{\mathbf{q}}_i|^2 + a_{i\ell} \dot{\mathbf{q}}_i^\top (\mathbf{q}_i - \mathbf{q}_{\ell}) + \frac{k_i}{p_i} \dot{\mathbf{q}}_i^\top \dot{\mathbf{e}}_i - \epsilon_i \tanh^\top(\mathbf{e}_i) \mathbf{Y}_i(\mathbf{q}_i) \tilde{\theta}_i \right]. \quad (35) $$

Now, since

$$ \begin{aligned} \dot{\mathbf{q}}_i^\top \dot{\mathbf{e}}_i &= a_{i\ell} |\dot{\mathbf{q}}_i|^2 + \sum_{j \in \mathcal{N}_i} a_{ij} \dot{\mathbf{q}}_i^\top (\dot{\mathbf{q}}_i - \dot{\mathbf{q}}_j) \\ &= a_{i\ell} |\dot{\mathbf{q}}_i|^2 + \frac{1}{2} \sum_{j \in \mathcal{N}_i} a_{ij} (|\dot{\mathbf{q}}_i - \dot{\mathbf{q}}_j|^2 + |\dot{\mathbf{q}}_i|^2 - |\dot{\mathbf{q}}_j|^2), \end{aligned} \quad (36) $$

defining $\mathbf{Q} := [|\dot{\mathbf{q}}_1|^2, \dots, |\dot{\mathbf{q}}_N|^2]^\top$, we can write

$$ \begin{aligned} \sum_{i \in \bar{N}} \dot{\mathbf{q}}_i^\top \dot{\mathbf{e}}_i &= \frac{1}{2} \sum_{i \in \bar{N}} \left[ a_{i\ell} |\dot{\mathbf{q}}_i|^2 + \sum_{j \in \mathcal{N}_i \cup \mathcal{N}_t} a_{ij} |\dot{\mathbf{q}}_i - \dot{\mathbf{q}}_j|^2 \right] + \mathbf{1}_N^\top \mathbf{L} \mathbf{Q} \\ &= \frac{1}{2} \sum_{i \in \bar{N}} \left[ a_{i\ell} |\dot{\mathbf{q}}_i|^2 + \sum_{j \in \mathcal{N}_i \cup \mathcal{N}_t} a_{ij} |\dot{\mathbf{q}}_i - \dot{\mathbf{q}}_j|^2 \right], \end{aligned} \quad (37) $$

where we employed the fact that $\mathbf{1}_N^\top \mathbf{L} = \mathbf{0}$. Thus $\dot{V}$ becomes

$$ \dot{V} = - \sum_{i \in \bar{N}} \left[ \frac{1}{2p_i} (2d_i + k_i a_{i\ell}) |\dot{\mathbf{q}}_i|^2 + a_{i\ell} \dot{\mathbf{q}}_i^\top (\mathbf{q}_i - \mathbf{q}_{\ell}) \right. \\ \left. - \epsilon_i \tanh^\top(\mathbf{e}_i) \mathbf{Y}_i(\mathbf{q}_i) \tilde{\theta}_i + \frac{k_i}{2p_i} \sum_{j \in \mathcal{N}_i \cup \mathcal{N}_t} a_{ij} |\dot{\mathbf{q}}_i - \dot{\mathbf{q}}_j|^2 \right]. $$
<table><caption>Table 1: Physical Parameters of the Robot Manipulators</caption><thead><tr><th rowspan="2"></th><th colspan="2">Manipulators 1-3</th><th colspan="2">Manipulators 4-6</th><th colspan="2">Manipulators 7-10</th></tr><tr><th>Link 1</th><th>Link 2</th><th>Link 1</th><th>Link 2</th><th>Link 1</th><th>Link 2</th></tr></thead><tbody><tr><th>Link length l<sub>ki</sub> (m)</th><td>0.4</td><td>0.4</td><td>0.3</td><td>0.5</td><td>0.5</td><td>0.2</td></tr><tr><th>Link mass m<sub>ki</sub> (kg)</th><td>4</td><td>2</td><td>2.5</td><td>3</td><td>3</td><td>2.5</td></tr><tr><th>Link inertia I<sub>ki</sub> (kg.m<sup>2</sup>)</th><td>0.478</td><td>0.044</td><td>0.678</td><td>0.144</td><td>0.03</td><td>0.01</td></tr><tr><th>Center of mass l<sub>cki</sub> (m)</th><td>0.2</td><td>0.2</td><td>0.15</td><td>0.25</td><td>0.25</td><td>0.1</td></tr></tbody></table>

Moreover, $\dot{U}_i$ becomes

$$ \dot{U}_i = \left(\frac{d}{dt}\tanh^\top(\mathbf{e}_i)\mathbf{M}_i(\mathbf{q}_i)\dot{\mathbf{q}}_i + \tanh^\top(\mathbf{e}_i)\mathbf{C}_i^\top(\mathbf{q}_i, \dot{\mathbf{q}}_i)\dot{\mathbf{q}}_i - \tanh^\top(\mathbf{e}_i)\left[d_i\dot{\mathbf{q}}_i + p_i\mathbf{e}_i + k_i\dot{\mathbf{e}}_i + \mathbf{Y}_i(\mathbf{q}_i)\tilde{\mathbf{\theta}}_i\right]\right) $$

Now, for any $\alpha_i > 0$,

$$ \left(\frac{d}{dt}\tanh^\top(\mathbf{e}_i)\mathbf{M}_i(\mathbf{q}_i)\dot{\mathbf{q}}_i \le m_{2i}|\dot{\mathbf{e}}_i||\dot{\mathbf{q}}_i| \le \frac{\alpha_i m_{2i}}{2}|\dot{\mathbf{e}}_i|^2 + \frac{m_{2i}}{2\alpha_i}|\dot{\mathbf{q}}_i|^2\right), \quad (38) $$

where we used Young's inequality [14] to conclude the last inequality. Additionally we have,

$$ |\dot{\mathbf{e}}_i|^2 = \left| \sum_{j \in N_i \cup N_f} a_{ij}(\dot{\mathbf{q}}_i - \dot{\mathbf{q}}_j) \right|^2 \le (L_{ii} + a_{i\ell}) \sum_{j \in N_i \cup N_f} a_{ij}|\dot{\mathbf{q}}_i - \dot{\mathbf{q}}_j|^2. \quad (39) $$

Moreover, the term $\tanh^\top(\mathbf{e}_i)\dot{\mathbf{e}}_i$ satisfies

$$ \tanh^\top(\mathbf{e}_i)\dot{\mathbf{e}}_i \ge \tanh^\top(\mathbf{e}_i)\frac{d}{dt}\tanh(\mathbf{e}_i). \quad (40) $$

Hence, setting $\alpha_i = \frac{1}{L_{ii}+a_{i\ell}}$, $\dot{U}_i$ can be upper bounded as

$$ \begin{aligned} \dot{U}_i \le & \frac{1}{2} \left( m_{2i}(L_{ii} + a_{i\ell}) + 2\sqrt{n}k_{ci} + \frac{d_i}{\mu_i} \right) |\dot{\mathbf{q}}_i|^2 \\ & - k_i \tanh^\top(\mathbf{e}_i) \frac{d}{dt} \tanh(\mathbf{e}_i) - \tanh^\top(\mathbf{e}_i) \mathbf{Y}_i(\mathbf{q}_i) \tilde{\mathbf{\theta}}_i \\ & + \frac{m_{2i}}{2} \sum_{j \in N_i \cup N_f} a_{ij} |\dot{\mathbf{q}}_i - \dot{\mathbf{q}}_j|^2 - \left( p_i - \frac{\mu_i d_i}{2} \right) |\tanh(\mathbf{e}_i)|^2. \end{aligned} $$

Therefore, $\dot{\mathcal{E}}$ satisfies

$$ \dot{\mathcal{E}} \le - \sum_{i \in \mathcal{N}} \left[ c_{1i} |\dot{\mathbf{q}}_i|^2 + c_{2i} \epsilon_i |\tanh(\mathbf{e}_i)|^2 + c_{3i} \sum_{j \in \mathcal{N}_i \cup \mathcal{N}_f} a_{ij} |\dot{\mathbf{q}}_i - \dot{\mathbf{q}}_j|^2 \right], $$

where

$$ \begin{aligned} c_{1i} &:= \frac{1}{2p_i} (2d_i + k_i a_{i\ell}) - \frac{\epsilon_i}{2} \left( m_{2i}(L_{ii} + a_{i\ell}) + 2\sqrt{n}k_{ci} + \frac{d_i}{\mu_i} \right) \\ c_{3i} &:= \frac{1}{2} \left( \frac{k_i}{p_i} - \epsilon_i m_{2i} \right) \end{aligned} \quad (41) $$

and $c_{2i}$ is defined in (28).

Setting $\epsilon_i$ satisfying (33) ensures that $c_{1i}$, $c_{2i}$ and $c_{3i}$ are strictly positive and thus $\dot{\mathcal{E}}$ is negative definite with regards to

$\dot{\mathbf{q}}_i, \dot{\mathbf{q}}_i - \dot{\mathbf{q}}_j$ and $\mathbf{e}_i$. The proof is finished proceeding as in the final step of the proof of Proposition 2. ▼

The following remarks are in order:

**Remark 5.** The estimation gain $\epsilon$ in (9) is the same for all the agents. However, for (31) this gain is different for each agent.

**Remark 6.** Condition (33) requires only information available to the ith-agent.

## 5. Simulation

Consider a network of ten 2-DOF planar manipulators. Physical parameters of each manipulator is determined with a vector $\bar{P}_i = [m_{1i}, m_{2i}, l_{1i}, l_{c1i}, l_{2i}, l_{c2i}, I_{1i}, I_{2i}]$, where $m_{ki}$, $l_{ki}$, and $l_{cki}$ are the mass, the length and the distance to the center of mass of link $k$. Also, $I_{1i}$ and $I_{2i}$ are the diagonal elements of the inertia matrix. Three different groups of manipulators are considered. The physical parameters of these manipulators are given in Table 1. The graph Laplacian matrix is given by $L = 0.1L_g$, where

$$ L_g = \begin{bmatrix} 14 & 0 & -3 & 0 & 0 & 0 & 0 & -4 & 0 & -7 \\ 0 & 9 & 0 & -8 & 0 & 0 & 0 & 0 & 0 & -1 \\ -3 & 0 & 5 & 0 & 0 & 0 & 0 & -2 & 0 & 0 \\ 0 & -8 & 0 & 10 & 0 & 0 & 0 & 0 & -2 & 0 \\ 0 & 0 & 0 & 0 & 8 & 0 & -5 & 0 & -3 & 0 \\ 0 & 0 & 0 & 0 & 0 & 6 & 0 & -4 & -2 & 0 \\ 0 & 0 & 0 & 0 & -5 & 0 & 14 & 0 & 0 & -9 \\ -4 & 0 & -2 & 0 & 0 & -4 & 0 & 10 & 0 & 0 \\ 0 & 0 & 0 & -2 & -3 & -2 & 0 & 0 & 7 & 0 \\ -7 & -1 & 0 & 0 & 0 & 0 & -9 & 0 & 0 & 17 \end{bmatrix}. $$

The initial joint velocities are assumed to be zero while the initial joint positions are set as $\mathbf{q}(0) = [2, 6, -7, 3, 1, 8, 0, 1, -6, 9, -5, 0, -4, 5, -3, 4, -2, 7, -8, 1]^\top$.

The simulation scenario is the same as in [25] with the sole difference being that the distance to the center of mass is distinct from the length of the link. The initial parameter estimates are set as $\hat{\theta}(0) = 10\mathbf{I}_2$. The controller gains are chosen as $p_i = 4$, $d_i = 8$, $k_i = 2$, $\Gamma_i = 80\mathbf{I}_2$, $\epsilon = 0.25$.

To have a more realistic simulation, we consider additive Gaussian noises with variance 0.05 in the joint velocity measurements of all manipulators. Fig. 1 shows the level of the noise in the velocity measurements of the manipulator 1.
<figure><img src="image_3.png" /><figcaption>Figure 1: Joint velocities of the manipulator 1 (blue) versus their measured values (green)</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Figure 3: Joint velocities using the adaptive P+d controller.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Figure 2: Position consensus using the adaptive P+d controller.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Figure 4: Estimated parameters for the leaderless consensus with the adaptive P+d controller.</figcaption></figure>

## 5.1. Leaderless Consensus

### 5.1.1. Adaptive P+d Controller

Simulation results for the leaderless consensus controller without using the neighbours velocities are shown in Figs. 2-5. Fig. 2 depicts the joint positions agreement in a consensus point. Fig. 3 shows the joint velocities. Fig. 4 shows the evolution of the estimated parameters. Input torques of the robot manipulators are drawn in Fig. 5.

### 5.1.2. Adaptive PD+d Controller

The simulation results for the leaderless consensus controller without requiring global information are shown in Figs. 6-9. Fig. 6 depicts the joint positions agreement in a consensus point. Fig. 7 shows the joint velocities. Fig. 8 shows the time evolution of the estimated parameters. Input torques of the robot manipulators are drawn in Fig. 9.

<figure><img src="image_7.png" /><figcaption>Figure 5: Input torques for the leaderless consensus with the adaptive P+d controller.</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>Figure 6: Leaderless consensus using the adaptive PD+d controller.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Figure 7: Joint velocities using the adaptive PD+d controller.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Figure 8: Estimated parameters for the leaderless consensus with the adaptive PD+d controller.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Figure 9: Input torques for the leaderless consensus with the adaptive PD+d controller.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>Figure 10: Leader-follower synchronization of the positions without using the neighbours velocities.</figcaption></figure>

## 5.2. Leader-Follower Consensus

In this case we consider a constant leader position $q_l = [-2, 3]^T \text{ rad}$. We further assume that the leader position is only available for agents 8 and 9. Therefore the corresponding gains of these agents in the matrix $A_l$ are set to $a_{8l} = a_{9l} = 1$ and the other gains $a_{il}$ are set to zero.

### 5.2.1. Adaptive P+d Controller

In this section the performance of the leader-follower controller without using the neighbours velocities is investigated. The time evolution of the robot positions, joint velocities, estimated parameters and input torques are shown in Fig. 10, Fig. 11, Fig. 12 and Fig. 13, respectively.

### 5.2.2. Adaptive PD+d Controller

In this section the performance of the leader-follower controller without requiring global information is investigated. The time evolution of the robot positions, joint velocities, estimated
<figure><img src="image_13.png" /><figcaption>Figure 11: Joint velocities of the leader-follower controller without using the neighbours velocities.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>Figure 12: Estimation parameters of the leader-follower controller without using the neighbours velocities.</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>Figure 13: Input torques of the leader-follower controller without using the neighbours velocities.</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>Figure 14: Leader-follower synchronization of the positions without requiring global information.</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>Figure 15: Joint velocities of the leader-follower controller without requiring global information.</figcaption></figure>

parameters and input torques are shown in Fig. 14, Fig. 15, Fig. 16 and Fig. 17, respectively.

### 5.3. Controller Performance with Parameter Variations

In this section, the effect of a payload variation in the performance of the proposed adaptive controller (8) and the non-adaptive controller [28] is compared, for the leaderless case. Sensors are assumed to be noise-free. Suppose that at time 20sec, the payload of Manipulator 10 is increased by 2kg and the payload of Manipulator 9 is decreased by 1kg. As expected, this payload variation does not affect the consensus behavior of the proposed controller, while the controller of [28] is not capable to withstand the payload variation. In fact, the proposed controller adapts itself with the new payloads by modifying the estimated parameters. These conclusions are drawn from the simulation results shown in Fig. 18. Due to the space limitations, only the positions of Joint 1 are given.
<figure><img src="image_18.png" /><figcaption>Figure 16: Estimation parameters of the leader-follower controller without requiring global information.</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>Figure 17: Input torques of the leader-follower controller without requiring global information.</figcaption></figure>

<figure><img src="image_20.png" /><figcaption>Figure 18: Positions of joint 1 when a payload variation occur at t = 20s in manipulators 9 and 10.</figcaption></figure>

## 5.4. Comparison with a complete regressor adaptive controller

In this section, the performance of the proposed adaptive gravity compensation (8) and the complete regressor adaptive controller [26] is compared, for the leaderless scenario. Consider the following linearly parametrized model of each ith-EL-agent,

$$M_i(\mathbf{q}_i)\mathbf{x}_i + C_i(\mathbf{q}_i, \dot{\mathbf{q}}_i)\mathbf{y}_i + g_i(\mathbf{q}_i) = \Phi_i(\mathbf{x}_i, \mathbf{y}_i, \mathbf{q}_i, \dot{\mathbf{q}}_i)\theta_i. \quad (42)$$

The proposed scheme in [26] is given by

$$\begin{aligned} \tau_i &= \Phi_i(\dot{\mathbf{e}}_i, \mathbf{e}_i, \mathbf{q}_i, \dot{\mathbf{q}}_i)\hat{\theta}_i - D_i\epsilon_i, \\ \hat{\theta}_i &= -\Gamma_i\Phi_i^T\epsilon_i, \end{aligned} \quad (43)$$

where $\epsilon_i = \dot{\mathbf{q}}_i + \lambda\mathbf{e}_i$ and $\mathbf{e}_i$ is defined in (5). Clearly, the implementation of this scheme requires the complete regressor $\Phi_i$ computation plus velocity information exchange between the agents. To have similar gains for both schemes, the gains of the controller (43) are chosen as $D_i = 8$, $\lambda = 0.5$ and $\Gamma_i = 80I_2$. Figs. 19-21 show the performance of both schemes. Fig. 19 shows that the convergence speed of the proposed method (8) is faster than the controller of [26]. The evolution of the estimated parameters is shown in Fig. 20. For the complete regressor adaptive controller, we have estimated five parameters for each manipulator while for the adaptive gravity compensation method, the estimation of just two parameters is enough. As expected, the convergence time of the proposed gravity parameter estimator (9) is lower than that of the complete parameter estimator of [26]. The input torques of the robot manipulators are shown in Fig. 21. In comparison with the controller (8), the initial torques of the controller [26] are larger.
<figure><img src="image_21.png" /><figcaption>Figure 19: Position consensus of the controller [26] (first row) versus the proposed controller (8).</figcaption></figure>

<figure><img src="image_22.png" /><figcaption>Figure 21: Input torques of the controller [26] (above) versus the proposed controller (8).</figcaption></figure>

<figure><img src="image_23.png" /><figcaption>Figure 20: Estimated parameters of the controller [26] (above) versus the proposed estimator (9).</figcaption></figure>

## 6. Conclusions

In this paper we report two novel decentralized adaptive controllers to solve the leaderless and leader-follower consensus in networks of uncertain EL-systems. The controllers estimate only the gravitational term of the dynamics. Therefore, they do not require the computational burden of estimating the whole EL-dynamics. Moreover, one of the proposals is a simple Proportional plus damping scheme and it does not require to exchange velocity information between the agents. The other proposed scheme is a Proportional-Derivative plus damping controller with a fully distributed gain condition setting. Simulations show the performance of the proposed controllers using a network composed of ten EL-agents.

Future research goes along two different avenues: 1) including time-delays in the communications of the agents; and 2) extending the solution to directed graphs. None of these problems can be, however, trivially solved. For the delayed case one has to design a proper Lyapunov function such that its time-derivative is negative definite with regards to the delayed error, which is rather difficult —to our knowledge, the first strict Lyapunov function for the delayed case has been designed in [29] for the P+d with gravity cancellation, i.e., without parametric uncertainty. As for the directed case, we are analyzing the solution of [43]. However the Lyapunov function employed in [43] can only be employed to conclude a local stability result.

## Acknowledgements

This work has been partially supported by the Mexican CONACYT basic scientific research grant CB-282807.
# Appendix A. Proof that $\tanh^T(e_i)e_i \ge |\tanh(e_i)|^2$

To prove this fact, it is enough to show that

$$x \tanh(x) \ge \tanh^2(x) \quad (A.1)$$

holds for any scalar $x \in \mathbb{R}$. For, first define $A(x) = \tanh(x) (x - \tanh(x))$ and note that (A.1) is equivalent to $A(x) \ge 0$ for all $x \in \mathbb{R}$. Since $\tanh(x)$ has the same sign as $x$ and it vanishes only when $x = 0$, then we only require to show that $|x| \ge |\tanh(x)|$. To that end let us define $f_1(x) = |x|$ and $f_2(x) = |\tanh(x)|$. Therefore, $f_1(0) = f_2(0) = 0$. Additionally $|\dot{f}_1(x)| = 1$ and $|\dot{f}_2(x)| = \frac{1}{\cosh^2(x)}$. From the fact that $\cosh(x) \ge 1$, we conclude that $|\dot{f}_2(x)| \le |\dot{f}_1(x)|$ for all $x \in \mathbb{R}$. Using the comparison lemma, it follows that $f_2(x) \le f_1(x)$ for all $x \in \mathbb{R}$ as required.

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