Distributed algorithm for controlling scale-free polygonal formations

TL;DR

This paper introduces a distributed control algorithm for polygonal formations of mobile agents, enabling shape, scale, and movement control with fewer inter-agent links, supported by theoretical analysis and simulations.

Contribution

It proposes a novel distributed algorithm for polygonal formations that reduces the number of control links and allows scale and movement control, extending existing rigid formation methods.

Findings

The algorithm successfully controls polygonal formations with fewer links.

The formation's scale can be adjusted by a single agent pair.

Exponential stability ensures reliable formation control and steering.

Abstract

This paper presents a distributed algorithm for controlling the deployment of a team of mobile agents in formations whose shapes can be characterized by a broad class of polygons, including regular ones, where each agent occupies a corner of the polygon. The algorithm shares the appealing properties of the popular distance-based rigid formation control, but with the additional advantage of requiring the control of fewer pairs of neighboring agents. Furthermore, the scale of the polygon can be controlled by only one pair of neighboring agents. We also exploit the exponential stability of the controlled formation in order to steer the formation as a whole with translations and rotations in a prescribed way. We provide both theoretical analysis and illustrative simulations.

Figures

Equations

$$b_{ik}\stackrel{\Delta }{=}\{\begin{array}{l}+1\text{if}i={\mathcal{E}}_k^{\text{tail}}\\ -1\text{if}i={\mathcal{E}}_k^{\text{head}}\\ 0\text{otherwise}\end{array},$$

$$z={\overline{B}}^{T}p.$$

$$\dot{p}=u,$$

$$B=\left[\begin{array}{ccccc}1& 0& \dots & 0& 0\\ -1& 1& \dots & 0& 0\\ ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& \dots & -1& 1\\ 0& 0& \dots & 0& -1\end{array}\right],$$

$$V(z)=\frac{1}{4}(||z_1|{|}^2-d_1^2{)}^2+\frac{1}{4}(||z_2|{|}^2-d_2^2{)}^2,$$

$$\{\begin{array}{ll}{\dot{p}}_1& =-z_1{e}_1\\ {\dot{p}}_2& =z_1{e}_1-z_2{e}_2\\ {\dot{p}}_3& =z_2{e}_2,\end{array}$$

$$d_k^{2\text{tail}}=d_k^{2\text{head}}-{\mu }_k,$$

$$\{\begin{array}{ll}{\dot{p}}_1& =-z_1{e}_1\\ {\dot{p}}_2& =z_1{e}_1-z_2{e}_2+{\mu }_1{z}_1-{\mu }_2{z}_2\\ {\dot{p}}_3& =z_2{e}_2.\end{array}$$

$$\{\begin{array}{ll}{\dot{p}}_1& =0\\ {\dot{p}}_2& ={\mu }_1{z}_1-{\mu }_2{z}_2\\ {\dot{p}}_3& =0.\end{array}$$

$$e_{\theta }={\overline{B}}_{\theta }^{T}z,$$

$$\{\begin{array}{ll}{\dot{p}}_1& =0\\ {\dot{p}}_2& =c{e}_{{\theta }_1}\\ ⋮& \\ {\dot{p}}_{n-1}& =c{e}_{{\theta }_{n-2}}\\ {\dot{p}}_n& =0,\end{array}$$

$$\dot{z}={\overline{B}}^T\dot{p}=-c{\overline{B}}_{\theta }{\overline{B}}_{\theta }^{T}z=-c{\overline{B}}_{\theta }{e}_{\theta },$$

$${\dot{e}}_{\theta }=-c\overline{B_{\theta }^T{B}_{\theta }}{e}_{\theta }.$$

$$\frac{\mathrm{d}V}{\mathrm{dt}}=e_{\theta }^T{\dot{e}}_{\theta }^T=-c{e}_{\theta }^T\overline{B_{\theta }^T{B}_{\theta }}{e}_{\theta }.$$

$$\tilde{z}={\overline{D}}_{r}z,$$

$$e_{\theta }={\overline{B}}_{\theta }^T\tilde{z},$$

$${\dot{e}}_{\theta }=-c\overline{B_{\theta }^T{D}_r{B}_{\theta }}{e}_{\theta },$$

$$e_{{\theta }_k}=W\left(\frac{{\theta }_k}{2}\right)z_k-W{\left(\frac{{\theta }_k}{2}\right)}^T{z}_{k+1},\forall k\in \{1,\dots ,|\mathcal{V}|-2\},$$

$$W(\alpha )=\left[\begin{array}{cc}\cos (\alpha )& -\sin (\alpha )\\ \sin (\alpha )& \cos (\alpha )\end{array}\right].$$

$$e_{\theta }=B_W^{T}z,$$

$$B_W=\left[\begin{array}{ccccc}W\left(\frac{{\theta }_1}{2}\right)& 0& \dots & 0& 0\\ -W{\left(\frac{{\theta }_1}{2}\right)}^T& W\left(\frac{{\theta }_2}{2}\right)& \dots & 0& 0\\ ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& \dots & -W{\left(\frac{{\theta }_{n-1}}{2}\right)}^T& W\left(\frac{{\theta }_{n-2}}{2}\right)\\ 0& 0& \dots & 0& -W{\left(\frac{{\theta }_{n-2}}{2}\right)}^T\end{array}\right].$$

$$\dot{z}=-c{\overline{B}}_{\theta }{B}_W^{T}z,$$

$${\dot{e}}_{\theta }=-c{B}_W^T{\overline{B}}_{\theta }{e}_{\theta }=-A(\theta )e_{\theta },$$

$$A(\theta )=\left[\begin{array}{ccccccc}W(\frac{{\theta }_1}{2})+W(\frac{{\theta }_1}{2}{)}^T& -W(\frac{{\theta }_2}{2})& 0& \dots & 0& 0& 0\\ -W(\frac{{\theta }_2}{2}{)}^T& W(\frac{{\theta }_2}{2})+W(\frac{{\theta }_2}{2}{)}^T& -W(\frac{{\theta }_3}{2})& \dots & 0& 0& 0\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮& ⋮\\ 0& 0& 0& \dots & -W(\frac{{\theta }_{n-3}}{2}{)}^T& W(\frac{{\theta }_{n-3}}{2})+W(\frac{{\theta }_{n-3}}{2}{)}^T& -W(\frac{{\theta }_{n-2}}{2})\\ 0& 0& 0& \dots & 0& -W(\frac{{\theta }_{n-2}}{2}{)}^T& W(\frac{{\theta }_{n-2}}{2})+W(\frac{{\theta }_{n-2}}{2}{)}^T\end{array}\right].$$

$$C(\theta )=\left[\begin{array}{cccc}2\cos \left(\frac{{\theta }^{\ast }}{2}\right)& -w({\theta }^{\ast })& & \\ -w({\theta }^{\ast }{)}^{†}& \ddots & \ddots & \\ & \ddots & \ddots & -w({\theta }^{\ast })\\ & & -w({\theta }^{\ast }{)}^{†}& 2\cos \left(\frac{{\theta }^{\ast }}{2}\right)\end{array}\right],$$

$${\lambda }_k({\theta }^{\ast })=2\cos \frac{{\theta }^{\ast }}{2}+2\cos \left(\frac{k\pi }{n-1}\right),k\in \{1,\dots ,n-2\}$$

$$\{\begin{array}{ll}{\dot{p}}_1& =-(p_n-p_1)(||p_n-p_1|{|}^2-d^2)\\ {\dot{p}}_n& =(p_n-p_1)(||p_n-p_1|{|}^2-d^2)\end{array}.$$

$$\{\begin{array}{ll}{\dot{p}}_1& =-(p_n-p_1)(||p_n-p_1|{|}^2-d^2)\\ & +{\mu }_{1_1}(p_n-p_1)+{\mu }_{1_2}{z}_1\\ & ⋮\\ {\dot{p}}_i& =c{e}_{{\theta }_{i+1}}+{\mu }_{i_1}{z}_{i-1}+{\mu }_{i_2}{z}_i\\ & ⋮\\ p_n& =(p_n-p_1)(||p_n-p_1|{|}^2-d^2)\\ & +{\mu }_{n_1}{p}_n-p_1+{\mu }_{n_2}{z}_{n-1},\end{array}$$

Full text

††thanks: The work of Hector Garcia de Marina was supported by Mistrale project, http://mistrale.eu. The work of Cao was supported in part by the European Research Council (ERC-StG-307207) and the Netherlands Organization for Scientific Research (NWO-vidi-14134).

1 Introduction

Accomplishing the tasks of surveillance, exploration, and search & rescue, among others, often requires formation control of multi-agent systems (see for instance Oh et al. (2015)). In particular, an appealing formation control framework based on rigid frameworks for the above mentioned tasks has been discussed in Krick et al. (2009). In such setups the agents can form a desired shape by only controlling the distances between neighbors. It is worth mentioning as follows some of the properties of distance-based rigid formation control. Firstly, the agents do not need to share a common frame of coordinates. Secondly, the system is robust against biases when sensing the distances between of neighboring agents (see Garcia de Marina et al. (2015)). Thirdly, the motion of the formation can be de-coupled into rotational, translational and scaling movements (we refer to Garcia de Marina et al. (2016a, b)). Forthly, the stability of the closed-loop system is exponentially stable for agents modelled by first or second-order integrators Sun et al. (2016). On the other hand, the main drawback of this distance-based approach is that to control the formation, one needs to control at least $2n-3$ distances in 2D, in order to be able to achieve a desired shape. This is not required for other approaches such as the position-based control Oh et al. (2015), but with the possible price of loosing many of the above listed advantages.

This paper presents an algorithm for controlling formations in the shapes from a broad class of polygons, i.e., a plane figure that is bounded by a finite chain of straight line segments closing in a loop to form a closed chain, where each agent occupies a corner of the polygon. We will show that the algorithm has all the advantages from distance-based rigid formation control and at the same time it only needs to control a smaller number of agent pairs. In particular, the assignment of neighboring agents, e.g., the sensing topology of the team, is based on a daisy chain configuration, i.e., a configuration where the agents are connected in series. We will also show that by controlling only the distance between the first and last agent in the sensing topology, one can scale the size of the whole formation up and down freely.

The algorithm is motivated by the distance-based control of non-rigid formations as recently studied in Dimarogonas and Johansson (2008). In particular, we exploit the effect of having mismatches in the prescribed distances between neighboring agents. Although one cannot define a particular shape by controlling a non-rigid setup, it is reported in Garcia de Marina and Sun (2017) that biases in the range sensors of neighboring agents111In the cited paper, the mismatches have been addressed as a biases. Nevertheless, mathematically speaking in the cited paper both concepts are equivalent. can be used to lead the formation to converge to a collinear configuration for a daisy chain network consisting of three agents. In this work we will employ a slightly different approach than in Garcia de Marina and Sun (2017). In fact, we will strengthen the resultant mismatched control law by clearly identifying two terms. The first term is responsible for controlling distances and derived from the standard gradient descent technique for the chosen potential function. The second term involving the mismatches has a clear interpretation and is responsible for the steady-state collinear configuration. Furthermore, as hindsight, the first term is surprisingly in the same form as the control law presented in Kvinto and Parsegov (2012) and Proskurnikov and Parsegov (2016) for steering equally-spaced agents to a line. We will show that with the technique introduced in Garcia de Marina and Sun (2017), the second term responsible for the alignment of the formation can be modified in order to control a prescribed angle and a prescribed distance between two pairs of consecutive neighboring agents in the daisy chain. Furthermore, the scale of the whole formation can be set by one pair of neighboring agents. The proposed algorithm renders the controlled formation convergning to the prescribed shape exponentially stable. This property, combined with a non-fixed steady-state orientation, allows us to achieve translations and rotations of the desired shape by following the technique introduced in Garcia de Marina et al. (2016a).

This paper is organized as follows. We introduce some notation and the notion of framework in Section 2. Then in Section 3 we introduce the daisy chain topology for distance-based control. The addition of distance mismatches between neighboring agents in the control terms leads to an algorithm for deploying agents in a collinear and equally (or relatively) spaced configuration. We modify this algorithm by the addition of rotational matrices in Section 4 in order to control the relative angle between two consecutive relative positions in the framework. We prove the exponential stability under the new algorithm for a broad class of polygons, including regular ones. At the end of Section 4 we exploit this exponential stability in order to control the scale of the desired shape by only controlling the distance between the first and the last agent of the framework, and to induce rigid body motions, i.e., rotations and translations, to the polygon. We present a numerical simulation in Section 5 in order to validate the theoretical findings and we finish the paper with some conclusions in Section 6.

2 Notations and definitions

For a given matrix $A\in\mathbb{R}^{n\times p}$, define $\overline{A}\stackrel{{\scriptstyle\Delta}}{{=}}A\otimes I_{2}\in\mathbb{R}^{2n\times 2p}$, where the symbol $\otimes$ denotes the Kronecker product and $I_{2}$ is the $2\times 2$ identity matrix. We denote by $|\mathcal{X}|$ the cardinality of the set $\mathcal{X}$.

Consider a formation of $n\geq 3$ autonomous agents whose positions are denoted by $p_{i}\in\mathbb{R}^{2}$ with $i\in\{1,\dots,n\}$. The agents are able to sense the relative positions of their neighboring agents. The neighbor relationships are described by an undirected graph $\mathbb{G}=(\mathcal{V},\mathcal{E})$ with the vertex set $\mathcal{V}=\{1,\dots,n\}$ and the ordered edge set $\mathcal{E}\subseteq\mathcal{V}\times\mathcal{V}$. The set $\mathcal{N}_{i}$ of the neighbors of agent $i$ is defined by $\mathcal{N}_{i}\stackrel{{\scriptstyle\Delta}}{{=}}\{j\in\mathcal{V}:(i,j)\in\mathcal{E}\}$. We define the elements of the incidence matrix $B\in\mathbb{R}^{|\mathcal{V}|\times|\mathcal{E}|}$ for $\mathbb{G}$ by

[TABLE]

where $\mathcal{E}_{k}^{\text{tail}}$ and $\mathcal{E}_{k}^{\text{head}}$ denote the tail and head nodes, respectively, of the edge $\mathcal{E}_{k}$, i.e., $\mathcal{E}_{k}=(\mathcal{E}_{k}^{\text{tail}},\mathcal{E}_{k}^{\text{head}})$.

A framework is defined by the pair $(\mathbb{G},p)$, where $p$ is the stacked vector of the agents’ positions $p_{i}$ with $i\in\{1,\dots,n\}$. The stacked vector of the sensed relative positions by the agents can then be described by

[TABLE]

Note that each vector $z_{k}=p_{i}-p_{j}$ in $z$ corresponds to the relative position associated with the edge $\mathcal{E}_{k}=(i,j)$.

3 Distance-based daisy chain frameworks, mismatches and the uniform deployment on a line problem

Assume that the agent’s dynamics are described by the first-order model

[TABLE]

where $u$ is the stacked vector of control inputs $u_{i}\in\mathbb{R}^{2}$ for $i=\{1,\dots,n\}$.

Consider the following incidence matrix defining a daisy-chain topology

[TABLE]

where $B\in\mathbb{R}^{|\mathcal{V}|\times(|\mathcal{V}|-1)}$. The incidence matrix that will help us to control the angles defined by the vectors $z_{k}$ and $z_{k+1}$ also corresponds to a daisy chain topology but $B_{\theta}\in\mathbb{R}^{(|\mathcal{V}|-1)\times(|\mathcal{V}|-2)}$, i.e., the first column of $B_{\theta}$ is related to $\theta_{1}$ as the angle between $z_{1}$ and $z_{2}$ and so on.

3.1 Distance-based mismatched gradient-descent control

For illustration, let us consider the case when our daisy chain framework consists of three agents. We then choose for the distance-based control the following potential function

[TABLE]

where $d_{1}$ and $d_{2}$ are the desired distances between the corresponding neighboring agents. Taking the gradient-descent of (5) (as used in Garcia de Marina et al. (2016a)) we arrive at the following system

[TABLE]

where $e_{k}=||z_{k}||^{2}-d_{k}^{2},k\in\{1,2\}$ are the distance error signals. Inspired by Garcia de Marina et al. (2016a), let us now include a distance mismatch $\mu_{k}\in\mathbb{R}$ in the edge $\mathcal{E}_{k}=(i,j)$, namely

[TABLE]

where $d_{k}^{\text{tail}}$ and $d_{k}^{\text{head}}$ are the different desired distances that the agents $i$ and $j$ respectively in $\mathcal{E}_{k}=(i,j)$ want to maintain for the same edge. We consider that the mismatches are assigned to the second agent in (6) such that we can arrive at the following expression

[TABLE]

One can identify that system (8) can be derived from a potential function as it has been done for system (6) with the exception of the term $\mu_{1}z_{1}-\mu_{2}z_{2}$. In fact, the gradient-descent-derived terms are responsible for the distance control between neighboring agents. If one drops all the terms in (8) involving the control of the $d_{k}$’s, then one gets

[TABLE]

If one considers $\mu_{1}=\mu_{2}=c$ then system (9) follows precisely from the algorithm presented in Kvinto and Parsegov (2012) and in Proskurnikov and Parsegov (2016) for solving the problem of deployment on a line, i.e., two fixed points $p_{1}$ and $p_{n}$ defining a segment and the rest of agents will be deployed on such a segment at spots equally separated. In particular, as one will see in the following section, the algorithm is stable for $c\in\mathbb{R}^{+}$ and its compatibility with the distance-based gradient-descent control and its relation with biases in range sensors have been studied in Garcia de Marina and Sun (2017).

3.2 Deployment on a line problem

In this section we will prove the stability of the algorithm introduced in system (9) for $\mu_{1}=\mu_{2}=c$ but for a general daisy chain topology. The stability analysis in this section is different from the one presented in Kvinto and Parsegov (2012) and in Proskurnikov and Parsegov (2016). In particular, in this paper we analyze the derived error signals from the algorithm. This approach serves as a starting point for controlling polygons in the plane, not only collinear configurations. Let us define the following error vector

[TABLE]

where $e_{\theta}\in\mathbb{R}^{(|\mathcal{V}|-2)}$. Then the extension to $n$ agents from system (9) can be generalized as

[TABLE]

where $c\in\mathbb{R}^{+}$ is a constant gain. Let us write the dynamics of the signal $e_{\theta}(t)$. We first derive the dynamics of $z$ from system (11), namely

[TABLE]

and noting that $\dot{e}_{\theta}=\overline{B}^{T}_{\theta}\dot{z}$ we have that

[TABLE]

Proposition 1

The origin of system (13) is globally exponentially stable. That is, all the agents from system (11) will converge to a fixed point, namely $p(t)\to p^{*}$ as $t\to\infty$, where all the agents are equally spaced with respect to each other in a collinear fashion.

{pf}

Consider the following Lyapunov function $V=\frac{1}{2}||e_{\theta}||^{2}$, whose time derivative is given by

[TABLE]

We know that $B_{\theta}$ defines a daisy chain topology, i.e., it does not contain any cycles, therefore the matrix $B^{T}_{\theta}B_{\theta}$ is positive definite (Dimarogonas and Johansson (2008)). Hence the exponential stability of the origin of $e_{\theta}$ follows. Since the signal $e_{\theta}(t)$ converges exponentially fast to zero, we know $\overline{B}^{T}_{\theta}z(t)\to 0$ as $t\to\infty$, i.e., $z_{k}(t)-z_{k+1}(t)\to 0$ as $t\to\infty$. Thus, by observing system (11), we have that $\dot{p}(t)$ also converges exponentially fast to zero. So one can conclude that $p(t)$ converges to a fixed point $p^{*}$ where all the agents are equally spaced and collinearly positioned.$\blacksquare$

Remark 3.1

Note that for the case $p_{1}(0)=p_{n}(0)$, all the agents will converge to the same point, i.e., $z(t)\to 0$ as $t\to\infty$.

3.3 Controlling relative magnitudes between relative positions

The relative magnitude between two consecutive relative positions $z_{k}$ and $z_{k+1}$ can be trivially defined as $r_{k}z_{k}=r_{k+1}z_{k+1}$, where $r_{k},r_{k+1}\in\mathbb{R}^{+}$ are the scaling factors that determine how the magnitude of one relative position with respect to its next neighboring one. This case encompasses, as in (9), the particular case of having all the agents equally spaced in the steady state, e.g., $r_{k}=1,\forall k\{1,\dots,|\mathcal{E}|\}$. In particular, we have that

[TABLE]

where $D_{r}\stackrel{{\scriptstyle\Delta}}{{=}}\operatorname{diag}\left({\begin{bmatrix}r_{1}&\dots&r_{k}\end{bmatrix}}\right),$ with $k\in\{1,\dots,|\mathcal{E}|\}$. So by redefining

[TABLE]

we have that the error dynamics derived from (11), as we have done before in Proposition 1, are given by

[TABLE]

where the matrix $-B^{T}_{\theta}D_{r}B_{\theta}$ is Hurwitz since $D_{r}$ is a diagonal positive definite matrix. Therefore, the set defined by the vanishing of the signal (16) is globally exponentially stable for system (11).

4 Controlling polygonal formations in the plane

It is possible to extend the results of Proposition 1 in order to deploy the team of agents on the plane in a more general way. We are going to show that by following the technique introduced in Garcia de Marina and Sun (2017) one is able to control the relative angle $\theta_{k}$ between two consecutive vectors $z_{k}$ and $z_{k+1}$. For formations where all these consecutive angles are equal to $\theta^{*}$, we provide a bound to such an angle in order to assert the (exponential) stability of the system. In particular, we will show that such a bound applies to the particular case of controlling regular polygons in the plane.

We introduce the consecutive angles $\theta_{k}$ to be controlled in the redefinition of the error signal $e_{\theta}$ as follows

[TABLE]

where $\theta_{k}\in(-\pi,\pi]$ and $W(\alpha)$ is the rotational matrix

[TABLE]

Note that in (18) we are comparing the clockwise rotated $z_{k}$ with the counterclockwise rotated $z_{k+1}$.

We now write in a compact form the stacked vector of errors in (18) as

[TABLE]

where

[TABLE]

Note that trivially $B_{W}$ is equal to $\overline{B}_{\theta}$, as defined in the end of Section 2, if we set $\theta_{k}=0,\forall k\in\{1,\dots,|\mathcal{V}|-2\}$. Therefore to deploy on a line is a particular case of the problem considered in this section.

Let us now write the dynamics of $z$ derived from system (11) by employing the error signal (18)

[TABLE]

so it allows us to derive the new error (linear) system dynamics given by

[TABLE]

where $\theta\in\mathbb{R}^{(|\mathcal{V}|-2)}$ is the stacked vector of all $\theta_{k}$ and $A(\theta)$ is shown in (24) in the next page.

Note that now $A(\theta)$ is not positive definite in general. Therefore in order to check the stability of the origin of $e_{\theta}$ in (23), one has to do an eigenvalue analysis for $A(\theta)$.

Our numerical simulations have shown that not for all the values of $\theta$ the origin of (23) is stable. In fact, the team of agents might converge to a different shape at the same time when they perform a steady-state motion. This effect has been shown not only for rigid formations with distance mismatches (see Mou et al. (2016)), but also for flexible formations (as in Garcia de Marina and Sun (2017)) as the daisy chain setup described in this paper. Nevertheless, we can provide an analytical result for the stability of a broad class of polygons where $\theta_{k}=\theta^{*}$. In particular we provide a bound for $\theta^{*}$ such that the formation is stable. Fortunately, this bound also applies to the set of regular polygons, which can be of interest in the field of formation control Krick et al. (2009).

Theorem 2

Consider the $n\geq 3$ agent system (11) with $e_{\theta}$ defined as in (18) and $\theta_{k}=\theta^{*},\,\forall k\in\{1,\dots,n-2\}$ . Then, the origin of $e_{\theta}(t)$ in system (23) is exponentially stable if and only if $|\theta^{*}|\leq\frac{2\pi}{n-1}$.

{pf}

Since all the 2D rotational matrices $W(\alpha)$ as in (19) commute, then $A(\theta)$ is unitarily similar to $\operatorname{diag}\{C(\theta),C(\theta)^{\dagger}\}$, where

[TABLE]

with $C(\theta)\in\mathbb{R}^{(n-2)\times(n-2)},w(\theta)=e^{j\frac{\theta}{2}}$, $j$ is the imaginary unit, and the symbol $\dagger$ denotes for the complex conjugate transpose. The matrix $C$ is tridiagonal and Toeplitz, so its eigenvalues have the following analytical expression (Noschese et al. (2013)):

[TABLE]

hence $C$ is positive definite (so the eigenvalues of $A(\theta)$ are positive) if and only if $|\theta^{*}|\leq\frac{2\pi}{n-1}$. Therefore the origin of $e_{\theta}(t)$ in system (23) is exponentially stable. This fact implies the (exponential) convergence of system (11) to the set given by $e_{\theta}=0$ with $e_{\theta_{k}}$ as in (18). $\blacksquare$

Remark 4.1

For the particular case of $n=3$, we have that for all the values of $\theta_{1}\in(-\pi,\pi]$ the system is stable, i.e., we are defining a triangle where its scale is determined by the constant positions $p_{1}(0)$ and $p_{3}(0)$.

Remark 4.2

Note that for regular polygons we have that $\theta^{*}=\pi-\frac{\pi(n-2)}{n}$ which satisfies the bound in Theorem 2. The angle $\theta_{k}$ is not an inner angle of the polygon, but the angle between two consecutives $z_{k}$ and $z_{k+1}$.

Note that the algorithm presented in (11) with $e_{\theta}$ as in (20) is able to control 2D shapes employing ($n-2$) edges, which requires fewer than the ($2n-3$) edges as in the gradient-descent control of the distance-based rigid setup. This is also the case if one employs position-based control (Oh et al. (2015)). However, the algorithm presented in this paper has two important features that are not present in the position-based approach. First, the agents can work employing their own local frames of coordinates. Second, the steady-state orientation of the shape is not fixed, therefore allowing for rotational motion as we will show. These two advantages come from the fact that the presented algorithm is an extension of the mismatched distance-based formation control.

4.1 Controlling the scale of the prescribed shape

Consider the example where six agents want to form an hexagon, so the four agents in the middle of the chain would control the inter-angles $\theta_{1},\dots,\theta_{4}=\pi-\frac{2}{3}\pi$ and by looking at (11) we notice that agents $1$ and $6$ are stationary. The idea is to apply the distance-based control to these two agents at the ends of the daisy chain, and therefore closing the chain. For example, if we are controlling a regular polygon, then all the side-lengths are equal, i.e., $D_{r}$ is the identity matrix in (15). Therefore, if we control the distance $d$ between the first and the last agent, then the rest of distances between neighboring agents will also be equal to $d$.

For controlling the scale we assign the following control law, derived from a potential function like in (5), to agents $1$ and $n$

[TABLE]

We have already noted that the convergence to the desired distance between agents $1$ and $n$ in the nonlinear system (27) is exponential (Sun et al. (2016)). One can treat the terms (27) as a disturbance that vanishes exponentially fast when they are into the dynamics of (11), and hence the stability result in Theorem 2 is not compromised and the scale of the formation can be controlled by only two agents. In fact, only one agent is needed if we set $\dot{p}_{n}=0$ in (27).

In case that one also desires to control the steady-state orientation of the formation, then a position-based control (with an exponential equilibrium) can be applied to agents $1$ and $n$.

Remark 4.3

For the three agents example, the main difference of system (11) with respect to the one presented in Garcia de Marina and Sun (2017) is that in the latter the sensing of $p_{3}-p_{1}$ is not necessary for determining the scale of the triangular formation. In Garcia de Marina and Sun (2017), the agents are also controlling the size of $z_{k}$ as in system (8) where the gradient descent terms have not been dropped out.

4.2 Steering the prescribed shape in the plane

The exponential stability in Theorem 2 can be further exploited. For example, one can employ the technique in Garcia de Marina et al. (2016a) in order to steer the whole group with rotational and translational motions. It is obvious that a vector in the plane can be constructed as a linear combination of two non-parallel vectors. Therefore, agent $i$ can construct a velocity vector $\dot{p}_{i}^{*}$ by just combining two relative positions (available for the agent) from the formation. We note that this is indeed possible for agent $i$ from the system (11) in combination with (27). The main idea is to design a collection of steady-state velocities $\dot{p}_{i}^{*}$ by employing the relative positions such that $p\in\{z\,:\,(e_{\theta}=0)\land(||p_{1}-p_{n}||=d)\}$ such that the desired shape is not destroyed, i.e., rigid body motions. For example, the control law introducing such an idea is given by

[TABLE]

where $i\in\{2,\dots,(n-1)\}$ and $\mu_{n_{\{1,2\}}}$ are the motion parameters responsible for the design of the velocities $\dot{p}_{i}^{*}$. We illustrate the physical meaning of (28) in Figure 1.

An algorithm describing how to compute these motion parameters such that they define rigid motions for generic shapes can be found in Garcia de Marina et al. (2016a). In fact, these motion parameters can be considered as a parametric disturbances for system (11) considered in Theorem 2. In particular, it has been introduced in Garcia de Marina and Sun (2017), inspired by the work in Mou et al. (2016), that the error-distance system defined by a rigid framework, whose equilibrium is the desired shape described by $\theta$, is autonomous and exponentially stable. Therefore, the stability of the error-distance system will not be compromised for small $\mu_{n_{\{1,2\}}}$’s (Mou et al. (2016)), or for big control gains (Garcia de Marina et al. (2016a)). This fact can be employed for giving bounds to the parameters $\mu_{n_{\{1,2\}}}$’s and the gain $c$ in (28) in order to guarantee the exponential stability of the system for a set of desired velocities $\dot{p}_{i}^{*}$ (Garcia de Marina et al. (2016a)).

Remark 4.4

It is important to note that by the addition of the motion parameters in (28) we might add undesired equilibria in the system.

5 Simulations

In this section we are going to validate the result of Theorem 2 together with system (28). We consider a team of six agents for achieving a regular hexagon, so $D_{r}$ is the identity matrix in (15). Since the inner angles of the hexagon are $\frac{2\pi}{3}$, we then set $\theta^{*}=\pi-\frac{2\pi}{3}=\frac{\pi}{3}$ which is smaller than the bound $\frac{2\pi}{5}$ given in Theorem 2. We define the error distance to be controlled by the agents $1$ and $6$ as $e_{d}\stackrel{{\scriptstyle\Delta}}{{=}}||p_{1}-p_{6}||-d$ and we set $d=10$ in (28). After time $t=150$ we set $d=30$. In addition we want to induce a spinning motion of the hexagon around its centroid. Such a motion can be accomplished by setting all the motion parameters in (28) equal to $\mu=0.025$. This can be checked by simple geometrical arguments or by employing the algorithm given in Garcia de Marina et al. (2016a). The simulation results are described in Figure 2.

6 Conclusions

In this paper we have presented a distributed algorithm for controlling formations of a broad class of polygonal shapes, including regular ones, defined by a daisy chain topology. The first step is to construct an algorithm for deploying agents equally spaced on a line, which is derived from adding distance mismatches to a standard distance-based controller in the literature. Consequently, a number of properties, such as having agents working using their own local frames of coordinates and non-controlled steady-state orientation, are preserved. In the second step we show that with the addition of rotational matrices one can control the relative angle between two consecutive relative positions in the framework. It turns out that the desired shape is then exponentially stable for a broad class of polygons including regular ones. By exploiting this stability property, one can add a series of useful properties to the formation. Firstly, one can control the relative size of consecutive relative positions in the framework. Secondly, the scale of the whole shape can be achieved by only controlling the distance between the first and the last agent of the framework. Thirdly, motion parameters can be employed in order to steer the formation as a combination of translations and rotations. We are currently working to implement the proposed algorithm in a robotic testbed Garcia de Marina et al. (2017).