text:
\section{Introduction}
The scientific community and industry anticipate distributed robot swarms assisting humans in challenges involving vast, hard accessible, and dangerous areas \cite{yang2018grand}. These challenges are related to specific missions in environmental monitoring, intensive agriculture, search \& rescue, and disaster management, among others, \cite{delmerico2019current}. In particular, the \emph{distributed} nature of these groups of robots (or agents in general) concentrates on the local interaction between the individuals to create collective behaviors. One of these global behaviors focuses on displaying geometrical patterns to assist the team in higher-level tasks \cite{Oh2015}. In this regard, formation control algorithms offer a repertoire of solutions depending on the sensing capabilities of the agents and the desired geometrical pattern. Far from being a solved problem, scientists are still on the development of reliable methods for the control and coordination of robot swarms, or multi-agent systems in general \cite{yang2018grand}. In robotics, it is common to demand from the swarm not only to display a shape but to move in a coordinated fashion.

In this paper, we focus on maneuvering formations of multi-agent systems based on the complex Laplacian matrix \cite{lin2014distributed}. In particular, we show that it is possible to achieve the (simultaneous) coordinated motions of translation, rotation, and scaling by only modifying a set of the original complex weights, designed only for static formations, in the Laplacian matrix. This fact enables us to preserve three interesting properties. Firstly, it is \emph{distributed}, i.e., an individual agent only needs local information.
Secondly, the agents do not need to share any common frame of coordinates. Thirdly, the agents control \emph{tensions}, e.g., they aim to have the zero weighted sum of their available relative positions for having an eventual static formation. We will show that a designed non-zero weighted sum of the relative positions will be responsible for the collective motion. Indeed, that is why we propose the modification of the original weights in \cite{lin2014distributed}.

The simplicity of our maneuvering technique on modifying the weights can be analyzed in detail by explicitly solving the resultant matrix differential equation involving the \emph{modified} complex Laplacian matrix. We prove how to \emph{relocate} one of the two Laplacian's original zero eigenvalues while we keep its associated eigenvector, which describes the \emph{appearance} of the desired shape. We move such a zero eigenvalue over the real and the imaginary axis to achieve scaling and rotational motions. For pure translations, we prove that our technique reduces from two to one the geometric multiplicity of the zero eigenvalue. The relocation of such zero eigenvalue of the original complex Laplacian matrix comes with an inconvenience, i.e., the rest of the non-zero eigenvalues are relocated as well. We provide an explicit condition that can be satisfied by design on the modified weights such that the original non-zero eigenvalues do not cross the imaginary axis, i.e., we can guarantee the global convergence to the desired steady-state motion and shape simultaneously.

We present three practical applications as the motivation for the proposed maneuvering technique, namely, the \emph{shaped consensus}, the \emph{enclosing of a target}, and \emph{the travelling formation with controlled speed and heading}. Shaped consensus consists in the rendezvous of all the agents while they display the desired shape, in contrast with the \emph{standard} consensus algorithm where the shape is not under control \cite{olfati2007consensus}. Also, this shaped consensus can be done while describing an inwards circular spiral trajectory. This spiral-shaped consensus can be done outwards as well, where all the agents spread out in the plane with a specific desired shape. Both collective motions (inwards and outwards) can be of interest in area coverage scenarios. The enclosing of a target maneuver chooses one agent and has the rest orbiting around it with a constant angular speed. For example, this collective motion is of interest in escorting missions. In contrast with other approaches \cite{lan2010distributed,Mar12}, among other works, we do not require \emph{all} the enclosing agents to track the target, or to follow the same circular path. Finally, we will show that the formation can travel with an arbitrary speed and heading controlled by only one agent, i.e., without the need of any \emph{leader-follower} architecture/estimators as it is common in the literature \cite{lin2013leader,zhao2018affine}.

This paper has been organized in the following way. Section \ref{sec: pre} introduces the required notation and the notion of \emph{desired shape}. In Section \ref{sec: dyn}, we debrief the complex-Laplacian-based formation control, and we introduce our strategy on modifying the Laplacian's weights to induce collective motions in Section \ref{sec: modL}. In Section \ref{sec: complex}, we derive the solutions of the resulting linear system explicitly after applying our maneuvering technique; in particular, we show how to satisfy a sufficient condition such that the global convergence to the desired collective motion is guaranteed. We continue in Section \ref{sec: simulations}, discussing the applications of our technique with illustrative simulations. Finally, we end the paper in Section \ref{sec: conc} with some conclusions and future work.

\section{Preliminaries}
\label{sec: pre}
\subsection{Notation and graph theory}
We consider the complex-Laplacian-based formation control of $n \in \mathbb{N} \geq 2$ mobile agents on the plane. We represent the complex unit by the symbol $\iota$. We denote by $||x||$ the Euclidean norm of the vector $x\in\mathbb{C}^p, p\in\mathbb{N}$. Given a set $\mathcal{X}$, we denote by $|\mathcal{X}|$ its cardinality. Finally, we denote by $\mathbf{1}_p\in\mathbb{C}^p, p\in\mathbb{N}$, the all-one column vector.

A \emph{graph} $\mathcal{G} = (\mathcal{V}, \mathcal{E})$ consists of two non-empty sets: the node set $\mathcal{V} = \{1,2,\dots,n\}$, and the edge set $\mathcal{E} \subseteq (\mathcal{V}\times\mathcal{V})$. In this paper we deal with the special case of \emph{undirected} graphs. In particular, undirected graphs are \emph{bidirectional} graphs where if the edge $(i,j)\in\mathcal{E}$, then the edge $(j,i)\in\mathcal{E}$ as well. The set $\mathcal{N}_i$ containing the neighbors of the node $i$ is defined by $\mathcal{N}_i:=\{j\in\mathcal{V}:(i,j)\in\mathcal{E}\}$. Let $w_{ij}\in\mathbb{C} \neq 0$ be a weight associated with the edge $(i,j)\in\mathcal{E}$, then the complex \emph{Laplacian} matrix $L\in\mathbb{C}^{n\times n}$ of $\mathcal{G}$ is defined as
\begin{equation}
	l_{ij} := \begin{cases}\sum_{k\in\mathcal{N}_i}w_{ik} & \text{if} \quad i = j \\
		-w_{ij} & \text{if} \quad i \neq j \wedge j\in\mathcal{N}_i \\
		0 & \text{if} \quad i \neq j \wedge j\notin\mathcal{N}_i.
	\end{cases}
	\label{eq: L}
\end{equation}
We note that if $\mathcal{G}$ is connected, then $L\mathbf{1}_n = 0$. For an undirected graph, we choose one of the two arbitrary directions for each pair of neighboring nodes to construct the ordered set of edges $\mathcal{Z}$. For an arbitrary edge $\mathcal{Z}_k = (\mathcal{Z}_k^{\text{head}},\mathcal{Z}_k^{\text{tail}})$, we call to its first and second element the \emph{tail} and the \emph{head} respectively. From such an ordered set, we construct the following \emph{incidence matrix} $B\in\mathbb{C}^{|\mathcal{V}|\times |\mathcal{Z}|}$
\begin{equation}
	b_{ik} := \begin{cases}+1 \quad \text{if} \quad i = {\mathcal{Z}}^{\text{tail}}_k \\
		-1 \quad \text{if} \quad i = \mathcal{Z}^\text{head}_k \\
		0 \quad \text{otherwise.}
	\end{cases}
	\label{eq: B}
\end{equation}
Note that $B^T\mathbf{1}_n = 0$.

\subsection{Frameworks and desired shape}
We codify the 2D position of each agent $i\in\mathcal{V}$ in $p_i\in\mathbb{C}$, where we take the real and imaginary parts respectively for the coordinates of the two dimensions of the Euclidean plane. We stack all the positions $p_i$ in a single vector $p\in\mathbb{C}^{n}$ and we call it \emph{configuration}. We define a \emph{framework} $\mathcal{F}$ as the pair $(\mathcal{G}, p)$, where we assign each agent's position $p_i$ to the node $i\in\mathcal{V}$, and the graph $\mathcal{G}$ establishes the set of neighbors $\mathcal{N}_i$ for each agent $i$.

We choose an arbitrary configuration of interest or \emph{reference shape $p^*$} for the team of agents, and we split it as
\begin{equation}
	p^* = p_{\text{c.m.}}\mathbf{1}_n  + p^*_c,
	\label{eq: pstar}
\end{equation}
where $p_{\text{c.m.}}\in\mathbb{C}$ is the position of the \emph{center of mass} of the configuration and $p_c^*\in\mathbb{C}$, starting from $p_{\text{c.m.}}$, gives the \emph{appearance} to the formation as in the example shown in Figure \ref{fig: pstar}. Without loss of generality, and for the sake of simplicity, we set $p_{\text{c.m.}} = 0$ in (\ref{eq: pstar}), i.e., $p^* = p^*_c$.
\begin{figure}
\centering
\begin{tikzpicture}[line join=round]
\filldraw(0,0) circle (2pt);
\filldraw(1.5,0) circle (2pt);
\filldraw(1.5,1.5) circle (2pt);
\filldraw(0,1.5) circle (2pt);
\draw[draw=black,arrows=->](-2,0.75)--(-1.5,0.75);
\draw[draw=black,arrows=->](-2,0.75)--(-2,1.25);
\draw[draw=black,arrows=->](-2,0.75)--(0.75,0.75);
\draw[draw=black,arrows=->](0.75,0.75)--(0+0.05,1.5-0.05);
\draw[draw=black,arrows=->](0.75,0.75)--(0+0.05,0+0.05);
\draw[draw=black,arrows=->](0.75,0.75)--(1.5-0.05,1.5-0.05);
\draw[draw=black,arrows=->](0.75,0.75)--(1.5-0.05,0+0.05);
\node at (-1.95,0.5) {\small $O_g$ \normalsize};
\node at (0-0.2,1.2) {\small $p_{c_1}^*$ \normalsize};
\node at (0-0.2,0.75-0.5) {\small $p_{c_4}^*$ \normalsize};
\node at (1.5+0.2,0.75+0.45) {\small $p_{c_2}^*$ \normalsize};
\node at (1.5+0.2,0.75-0.4) {\small $p_{c_3}^*$ \normalsize};
\node at (1.3,0.75) {\small $p_{\text{c.m.}}$ \normalsize};
\end{tikzpicture}
	\caption{The vector $p^* =  p_{\text{c.m.}}\mathbf{1}_4 + [p_{c_1}^* \, \dots \, p_{c_4}^*]^T$ is an example of \emph{reference shape} to construct a \emph{desired shape} $\mathcal{S}$.}
\label{fig: pstar}
\end{figure}

We now define the concept of \emph{desired shape} constructed from the reference shape $p^*$:
\begin{defn}
The framework or the formation is at the \emph{desired shape} when
	\begin{equation}
		p\in\mathcal{S}:=\{p = c_1 \mathbf{1}_{n} + c_2 p^* \, | \, c_1,c_2\in\mathbb{C}\},
	\label{eq: dshape}
\end{equation}
\end{defn}
Note that $c_1$ accounts for translations without restrictions in the space, while $c_2$ accounts for the scaling and rotation of all the elements of the reference $p^*$ equally.

\section{Agents' dynamics and shape stabilization}
\label{sec: dyn}
We consider that the position of each agent is modelled by the single-integrator dynamics
\begin{equation}
	\dot p_i = u_i, \quad \forall i\in\mathcal{V},
	\label{eq: dyn}
\end{equation}
where $u_i\in\mathbb{C}$ is the control action for the corresponding agent $i$.
Later, for the analysis of the whole system, let us consider the following compact form for all the agent's dynamics in (\ref{eq: dyn})
\begin{equation}
\dot p = u,
	\label{eq: pdyn}
\end{equation}
where $u\in\mathbb{C}^{n}$ is the stacked vector of control actions $u_i$.

We will consider only distributed control actions; therefore the agent $i$ only has access to relative information with respect to its neighbors in $\mathcal{N}_i$. This implies that the maneuvering technique to be introduced in this paper can only count on the same available information as well. In particular, such a local available information is the set of relative positions $z_{ij} = p_i - p_j, (i,j)\in\mathcal{E}$. For the later analysis of several compact forms, we will exploit the stacked vector of relative positions $z = \overline B^Tp$.

In order to drive $p(t)$ to $\mathcal{S}$, our technique will start from the distributed complex-Laplacian-based formation control analyzed in \cite{lin2014distributed}, i.e.,
\begin{equation}
	u_i = -k_i \sum_{j\in\mathcal{N}_i} \omega_{ij} (p_i - p_j) = -k_i\sum_{j\in\mathcal{N}_i} \omega_{ij} z_{ij},
	\label{eq: uLcomplex}
\end{equation}
where $\omega_{ij} \in \mathbb{C}$ is a complex weight to be designed later for the construction of the complex Laplacian matrix (\ref{eq: L}), and $k_i\in\mathbb{C} \setminus \{0\}$ is a non-zero gain to be designed to guarantee the convergence of the formation to the desired (static) shape. Considering all the control actions (\ref{eq: uLcomplex}), we can arrive at the following closed loop in compact form
\begin{equation}
\dot p = -KLp,
	\label{eq: comcom}
\end{equation}
where $K = \operatorname{diag}\{k_1, \dots, k_n\}$. We need the assistance of the matrix $K$ in (\ref{eq: comcom}) since some of the non-zero eigenvalues of the complex $L$ might be in the left-half plane. Note that $L$ is not positive semi-definite necessarily since $\omega_{ij}\neq\omega_{ji}$ in general. For regular polygonal formations with $(n-1)$ edges, it has been proved that $K=I_n$ is sufficient to guarantee the stability of (\ref{eq: comcom}) \cite{Marina2017}.

The authors in \cite{lin2014distributed} identified a (graph) requirement for the design of the weights in (\ref{eq: uLcomplex}). In particular, we can see that the desired shape must satisfy the following $n$ linear constraints
\begin{equation}
	\sum_{j\in\mathcal{N}_i} \omega_{ij} (p_{i}^* - p_{j}^*) = 0,\quad \forall i\in\mathcal{V}.
	\label{eq: wcomcon}
\end{equation}
These conditions can be satisfied if and only if the graph $\mathcal{G}$ is \emph{2-rooted}, i.e., if there exists a subset of two nodes, from which every other node is \emph{2-reachable} \cite{lin2014distributed}. A node $v\in\mathcal{V}$ is \emph{2-reachable} from a non-singleton set $\mathcal{U}$ of nodes if there exists a path from a node in $\mathcal{U}$ to $v$ after removing any one node except node $v$.

The complex Laplacian, with its weights satisfying the constrains (\ref{eq: wcomcon}), has two zero eigenvalues with eigenvectors $\mathbf{1}_n$ and $p^*$ respectively. Therefore, if the rest of eigenvalues of $KL$ are in the right half plane, then $p(t)$ in (\ref{eq: comcom}) converges to the kernel of the complex $L$, i.e., $p(t) \to \mathcal{S}$ as $t\to\infty$. More specifically, the configuration $p(t)$ converges to a point in $\mathcal{S}$, i.e., the formation will be stopped eventually, where $c_1$ and $c_2$ in (\ref{eq: dshape}) will be determined by the initial condition $p(0)$.

\section{Modified Laplacian matrix}
\label{sec: modL}
The maneuvering technique in this paper consists in modifying a (non-unique) subset of the weights $\omega_{ij}$ in (\ref{eq: wcomcon}) such that the formation converges to a steady-state motion within the desired shape $\mathcal{S}$. In particular, the modification of the weights $\omega_{ij}$ will be designed by exploiting the available (given by $\mathcal{G}$) relative positions between the agents in the reference shape $p^*$.

Let us consider the following weights to construct a modified Laplacian matrix
\begin{equation}
	\tilde\omega_{ij} = \omega_{ij} - \frac{\tilde\kappa}{k_i}\mu_{ij}, \quad (i,j)\in\mathcal{E}
\label{eq: wmod}
\end{equation}
where the \emph{motion parameters} $\mu_{ij}\in\mathbb{C}$ will be designed in Section \ref{sec: mpd} for the translation, rotation and scaling of the formation, and $\tilde\kappa\in\mathbb{R}_+$ will regulate the speed of such motions. We recall that $k_i$ comes from the gain in (\ref{eq: uLcomplex}), and we will need it to compensate for $K$ in the design of the steady-state motions. Since our maneuvering technique is also distributed, then if $j\notin\mathcal{N}_i$, then $\mu_{ij} = 0$. In general, we will have that $\mu_{ij} \neq \mu_{ji}$ and $\tilde\omega_{ij} \neq \tilde\omega_{ji}, (i,j)\in\mathcal{E}$, for the modified Laplacian matrix.

Similarly to the incidence matrix $B$ in (\ref{eq: B}), consider the ordered set of edges $\mathcal{Z}$, where we take one (arbitrary) edge per pair of neighboring agents, and let us define the components of the following matrix $M\in\mathbb{C}^{|\mathcal{V}|\times |\mathcal{Z}|}$
\begin{equation}
	m_{ik} := \begin{cases}\mu_{i\mathcal{Z}^\text{head}_k} \quad \text{if} \quad i = \mathcal{Z}^\text{tail}_k \\
		-\mu_{i\mathcal{Z}^\text{tail}_k} \quad \text{if} \quad i = \mathcal{Z}^\text{head}_k \\
		0 \quad \text{otherwise.}
	\end{cases}.
	\label{eq: M}
\end{equation}
The definition (\ref{eq: M}) enables us to write the modified Laplacian matrix from the modified weights (\ref{eq: wmod}) in compact form as
\begin{equation}
	\tilde L = L - \tilde\kappa K^{-1}MB^T.
	\label{eq: Ltilde}
\end{equation}

\begin{comment}
The steady-state velocity vectors will be designed from the reference shape $p^*$; in particular, as linear combinations of the relative positions $(p^*_i - p^*_j), (i,j)\in\mathcal{E}$. The steady-state velocity for agent $i$ is designed as
\begin{equation}
	v^*_i = \sum_{j\in\mathcal{N}_i} \mu_{ij} (p^*_i - p^*_j), \quad \forall i\in\mathcal{V}.
	\label{eq: vs}
\end{equation}
Note that the design of the velocities in (\ref{eq: vs}), that can be split into three for the combinations of translations, rotations and scalings of the formation, can be done with respect to a frame of coordinates $O_b$ at the centroid of the formation, i.e., it is independent of a global coordinate system. We illustrate this fact graphically in Figure \ref{fig: velcom}.

\end{comment}

\section{Shape maneuvering}
\label{sec: complex}
 We first show how to modify the complex weights of the complex Laplacian via the parameters $\mu_{ij}$ in (\ref{eq: wmod}) to achieve combined motions of translation, rotation and scaling for the formation. Then, we analyze the dynamics of the closed loop system with the modified complex Laplacian matrix.

\subsection{Motion parameters design}
\label{sec: mpd}

\begin{figure}
\centering
\begin{tikzpicture}[line join=round]
\filldraw(1,1) circle (1pt);
\filldraw(-1,1) circle (1pt);
\draw[dashed](1,1)--(-1,1);
\draw[draw=black,arrows=->](0,0)--(0.5,0);
\draw[draw=black,arrows=->](0,0)--(0,0.5);
\draw[draw=black,arrows=->](0,0)--(1,1);
\draw[draw=black,arrows=->](0,0)--(-1,1);
\draw[draw=black,color=green,arrows=->](1,1)--(1.5,1.5);
\draw[draw=black,color=red,arrows=->](1,1)--(0.5,1.5);
\draw[draw=black,color=blue,arrows=->](1,1)--(1.5,1);
\draw[draw=black,color=green,arrows=->](-1,1)--(-1.5,1.5);
\draw[draw=black,color=red,arrows=->](-1,1)--(-1.5,0.5);
\draw[draw=black,color=blue,arrows=->](-1,1)--(-0.5,1);
	\node at (-0.25,-0.25) {\small $O_b$ \normalsize};
	\node at (0.25,-0.25) {\small Re \normalsize};
	\node at (-0.25,0.65) {\small Im \normalsize};
\node at (0.8, 0.45) {\small $p_{i}^*$ \normalsize};
\node at (-1+0.2,0.45) {\small $p_{j}^*$ \normalsize};
\end{tikzpicture}
	\caption{Construction of the three velocities for neighboring agents $i$ and $j$ such that $p^*$ stays in $\mathcal{S}$. In red color, the velocities for the rotation around $O_b$ (the centroid of the formation). In green color, the scaling velocity must be parallel to the agent's position. Finally, in blue color, the translational velocity can be arbitrary as long as all the agents follow the same velocity. Note that each of these three velocities (represented as a complex number) can be constructed by multiplying the relative complex number $(p_{i}^* - p_{i}^*)$ (dashed relative position) by an appropriate complex number $\mu_{ij}$.}
\label{fig: velcom}
\end{figure}
Similarly as in ($\ref{eq: wcomcon}$), we will design the parameters $\mu_{ij}$ by satisfying the linear constraint
\begin{equation}
	{^b}v_i^* = \sum_{j\in\mathcal{N}_i}\mu_{ij}\, ({^b}p_i^* - {^b}p_j^*)  = \sum_{j\in\mathcal{N}_i}\mu_{ij}\,{^b}z_{ij}^*, \, \forall i\in\mathcal{V},
\label{eq: vbi}
\end{equation}
where ${^b}v_i^*\in\mathbb{C}$ is the desired velocity for each agent $i$ with respect to a frame of coordinates $O_b$ fixed at the centroid of $p^*$ as shown in Figure \ref{fig: velcom}. Since $\mu_{ij}$ \emph{scales and rotates} the complex number ${^b}z_{ij}^*$ (representing a relative position), only one non-zero $\mu_{ij}$ is enough per agent $i$ to construct an arbitrary desired ${^b}v_i^*$. For example, choose an arbitrary neighbor $\tilde j\in\mathcal{N}_i$, and calculate directly
\begin{equation}
	\mu_{ij} := \begin{cases}{^b}v_i^* \, / \, {^b}z_{ij}^* \quad \text{if} \quad j=\tilde j \\
		0 \quad \text{otherwise}
	\end{cases}, \quad \forall i\in\mathcal{N}.
	\label{eq: mus}
\end{equation}

We can stack (\ref{eq: vbi}) for all the agents and arrive at the following compact form
\begin{equation}
	{^b}v_f = MB^T\,{^b}p^*,
	\label{eq: vf}
\end{equation}
where ${^b}v_f \in \mathbb{C}^n$ is the stacked vector with all the desired agents' velocities. In fact, in order to keep a formation invariant in $\mathcal{S}$, we can split ${^b}v_f$ into the three terms as in Figure \ref{fig: velcom}, i.e.,
\begin{equation}
{^b}v_f = {^b}v^* \mathbf{1}_n + a\,{^b}p^* + \iota \omega \, {^b}p^*,
	\label{eq: dvf}
\end{equation}
where $v^*\in\mathbb{C}$ accounts for the common translational velocity in \emph{distance units}/sec, $a\in\mathbb{R}$ sets whether the formation grows or contracts in \emph{current size}/sec, and $\omega\in\mathbb{R}$ sets the angular speed in radians/sec. Likewise, we can split (\ref{eq: vf}) into three terms, namely
\begin{equation}
	MB^T\,{^b}p^* = (M_t + M_r + M_s)B^T\,{^b}p^*,
	\label{eq: ormo}
\end{equation}
where the matrices $M_t,M_r,M_s\in\mathbb{C}^{|\mathcal{V}|\times|\mathcal{Z}|}$ have their elements as in (\ref{eq: M}) designed only for the translation, rotation and scaling of the formation respectively.

\subsection{Modified complex Laplacian matrix for translational, rotational and scaling motion control}
Let us define
\begin{equation}
\tilde M := \kappa_t M_t + \kappa_r M_r + \kappa_s M_s, \, \kappa_t,\kappa_r,\kappa_s\in\mathbb{R}_+.
	\label{eq: mtilde}
\end{equation}
	Let us set $M = \tilde M$ in (\ref{eq: Ltilde}), then we can see $\kappa_{\{t,r,s\}}$ as coordinates for the different orthogonal motions\footnote{Note that $\kappa_tM_t$ defining translation can be split further into other two terms, i.e., \emph{horizontal} and \emph{vertical} motion.} to assist in the design of the desired behavior of the formation (see Section \ref{sec: simulations}), and $\tilde\kappa$ in (\ref{eq: Ltilde}) sets the \emph{global speed}. For the sake of clarity in the main results, we consider only positive $\kappa_{\{t,r,s\}}$ since the sign of the resultant velocities is encoded in $M_{\{t,r,s\}}$, e.g., in the sign of $\mu_{ij}$.

The term $K^{-1}\tilde MB^T$ in (\ref{eq: Ltilde}) modifies the properties of $L$ substantially. In particular, we modify the original two zero eigenvalues of $L$ in different ways while preserving the associated eigenvectors that define the desired shape $\mathcal{S}$. After analyzing such differences, we will show that this modification will impact the closed-loop dynamics by going from a static to an eventual moving configuration $p(t)$ in $\mathcal{S}$. Without loss of generality, consider $\tilde\kappa = 1$ in (\ref{eq: Ltilde}) for the following result.

\begin{lemma}
\label{lem: eigtr}
We consider the following two mutually exclusive cases:
	\begin{enumerate}
\item If at least one of the speeds $\omega$ or $a$ is different from zero for the design in (\ref{eq: dvf}), and $\kappa_{\{t,r,s\}}>0$ are sufficiently small, then the matrix $K\tilde L$ has one eigenvalue equals $0$ and one eigenvalue equals $-(k_sa + \iota\kappa_r\omega)$ whose corresponding eigenvectors are $\mathbf{1}_n$ and $(\frac{\kappa_tv^*}{\kappa_sa + \iota\kappa_r\omega}\mathbf{1}_n + p^*)$ respectively with $v^*$ being the designed translational velocity in (\ref{eq: dvf}). The rest of eigenvalues are in the right-half complex plane.
\item If $\omega = a = 0$ and $v^*\neq 0$ in (\ref{eq: dvf}). Then, the matrix $K\tilde L$ has $0$ as an eigenvalue with algebraic multiplicity $2$ but geometric multiplicity equals $1$. Furthermore, the chain of generalized (complex) eigenvectors associated with the eigenvalue $0$ is $\{-\kappa_t v^* \mathbf{1}_n, p^*\}$. In addition, if $\kappa_t > 0$ is sufficiently small, then the rest of eigenvalues are in the right-half complex plane.
	\end{enumerate}
\end{lemma}

\begin{pf}
The original complex Laplacian matrix $L$ has two zeros as eigenvalues with algebraic and geometric multiplicity equal $2$ whose two independent eigenvectors are $\mathbf{1}_n$ and $p^*$ respectively, i.e., according to the kernel $\mathcal{S}$ of $L$ we have that $L\mathbf{1}_n = 0$ and $Lp^* = 0$, and we note that the zero eigenvalues of $KL$ and their eigenvectors are the same as in $L$.

Let us analyze the first case. For $K\tilde L$ we have that
\begin{equation}
	K\tilde L \mathbf{1}_n = (KL - \kappa_t M_tB^T - \kappa_r M_rB^T - \kappa_s M_sB^T)\mathbf{1}_n = 0 \nonumber
\end{equation}
and
\begin{align}
	&(KL - \tilde M B^T) (\frac{\kappa_tv^*}{\kappa_sa + \iota\kappa_r\omega}\mathbf{1}_n + p^*) = \nonumber \\
	&= -\kappa_t M_tB^T p^* - \kappa_r M_sB^T p^* -\kappa_s M_rB^Tp^* \nonumber \\
	&= -\kappa_t v^*\mathbf{1}_n - (\kappa_sa + \iota\kappa_r\omega)p^* \nonumber \\
	&= - (\kappa_sa + \iota\kappa_r\omega)(\frac{\kappa_tv^*}{\kappa_sa + \iota\kappa_r\omega}\mathbf{1}_n + p^*).
\end{align}
Since by continuity the eigenvalues of $K\tilde L$ can get arbitrarily close to the eigenvalues of $KL$ for a sufficiently small $\kappa_{\{t,r,s\}}>0$, and that all the eigenvalues of $KL$ (excepting the two zero eigenvalues) can be placed arbitrarily far from the imaginary axis because of $K$, then we can conclude that $K\tilde L$ has one eigenvalue equals $0$ whose eigenvector is $\mathbf{1}_n$, another eigenvalue equals $- (\kappa_sa + \iota\kappa_r\omega)$ whose eigenvector is $(\frac{\kappa_tv^*}{\kappa_sa + \iota\kappa_r\omega}\mathbf{1}_n + p^*)$, and the rest of eigenvalues are on the right-half complex plane.

Now, let us analyze the second case. For $K\tilde L$ with $\omega = a = 0$, we have that
\begin{align}
	K\tilde Lp^* = (KL - \kappa_t M_tB^T)p^* = -\kappa_tv^* \mathbf{1}_n,
	\label{eq: genv}
\end{align}
and
\begin{align}
	&K\tilde L (-\kappa_t v^*\mathbf{1}_n) = -\kappa_t v^*(KL - \kappa_t M_tB^T)\mathbf{1}_n = 0,
	\label{eq: genv2}
\end{align}
	consequently, we have that $(K\tilde L)^2p^* = 0$ but $K\tilde Lp^* \neq 0$, and $K\tilde L \mathbf{1}_n = 0$. Now we check the algebraic multiplicity of the zero eigenvalue of $K\tilde L$. We have that $KL = K\tilde L$ if $\kappa_{\{t,r,s\}} = 0$, and by continuity the eigenvalues of $K\tilde L$ can get arbitrarily close to the eigenvalues of $KL$ by making $\kappa_t > 0$ sufficiently small. Therefore, the non-zero eigenvalues of $K\tilde L$ can stay \emph{far} from zero. For a sufficiently small $\kappa_t > 0$, we can deduce then that from (\ref{eq: genv}) and (\ref{eq: genv2}) the matrix $K\tilde L$ must have an eigenvalue $0$ with algebraic multiplicity $2$ (as $KL$) but geometric multiplicity $1$, whose chain of generalized (complex) eigenvectors is $\{-\kappa_t v^*\mathbf{1}_n, p^*\}$, and the rest of eigenvalues are far from zero on the right-half complex plane. \qed
\end{pf}

	We will show in Subsection \ref{sec: kappa} how small $\kappa_{\{t,r,s\}}$ (or $\tilde\kappa$ scaling down all of them at once in (\ref{eq: Ltilde})) should be. Nonetheless, we will see that we can choose any arbitrary value for these gains since the perturbation to the original complex-Laplacian $L$ in (\ref{eq: Ltilde}) can be also mitigated by choosing a higher gain $K$, e.g., by multiplying $K$ by a real constant $h > 1$ in (\ref{eq: Ltilde}). For the following main result, without loss of generality and for the sake of clarity, let us consider that $\kappa_t = \kappa_r = \kappa_s = \kappa$, and $\tilde\kappa = 1$. Therefore, we have that $\tilde L = L - \kappa K^{-1}(M_t+M_r+M_s)B^T$.

\begin{theorem}
\label{pro: comt}
Given a framework $\mathcal{F} = (\mathcal{G},p)$, whose graph is \textit{2-rooted} so that there exist weights $\omega_{ij}$ that satisfy the constrains (\ref{eq: wcomcon}) for a desired shape $\mathcal{S}$ as in (\ref{eq: dshape}) constructed from the reference shape $p^*$ as in (\ref{eq: pstar}). Consider the following control law for the dynamics (\ref{eq: dyn})
\begin{equation}
	u_i = -k_i \sum_{j\in\mathcal{N}_i} \tilde\omega_{ij}(p_i - p_j), \quad \forall i\in\mathcal{V},
	\label{eq: uimodcomt}
\end{equation}
	where $\tilde \omega_{ij}$ is as in (\ref{eq: wmod}) with $\kappa > 0$ sufficiently small as required in Lemma \ref{lem: eigtr}, and $k_i$ has been designed such that the matrix $KL$ does not have eigenvalues on the left-half complex plane. Consider the following two cases:
	\begin{enumerate}
\item If $a=\omega=0$ and ${^b}v^*\neq 0$  in (\ref{eq: dvf}), then $p(t)\to \mathcal{S}$, where $\dot p_i(t) \to -c_2\kappa \, v^*, \forall i\in\mathcal{V}$, as $t\to\infty$, and $c_2 \in \mathbb{C}$, which depends on the initial condition $p(0)$, determines the eventual fixed scale of the formation with respect to $p^*$, and the orientation of $O_b$ with respect to $O_g$.
\item If at least one of the desired speeds $\omega$ or $a$ in (\ref{eq: dvf}) is different from zero, then $p(t)\to \mathcal{S}$ as $t\to\infty$. This eventual configuration $p(t)\in\mathcal{S}$ will describe a motion compatible with $\mathcal{S}$, i.e., $p(t)\to c_1\mathbf{1}_n + c_2(\frac{v^*}{a + \iota\omega}\mathbf{1}_n + p^*)e^{\kappa(a + \iota\omega) t}$ as $t\to\infty$, where $c_{\{1,2\}}\in\mathbb{C}$ depend on the initial condition $p(0)$.
\end{enumerate}
\end{theorem}
\begin{pf}
Similarly to the closed loop (\ref{eq: comcom}), we can write the following compact form for the closed loop derived from (\ref{eq: uimodcomt}) and the dynamics (\ref{eq: dyn})
\begin{equation}
	\dot p = -K\tilde Lp.
	\label{eq: pklp}
\end{equation}
Let us consider the first case. According to Lemma \ref{lem: eigtr}, the non-zero eigenvalues of $-K\tilde L$ are in the half-left complex plane if $\kappa > 0$ is sufficiently small. Furthermore, the eigenvalue $0$ of $-K\tilde L$ has algebraic multiplicity $2$ but geometric multiplicity $1$, and its chain of generalized eigenvectors is $\{-\kappa v^*\mathbf{1}_n,p^*\}$. Therefore, from the exponential of the Jordan form of ($-K\tilde L$) we have that the solution of (\ref{eq: pklp}) is
\begin{equation}
	p(t) = -c_1 \kappa v^*\mathbf{1}_n - c_2 (\kappa v^*\mathbf{1}_n t - p^*) + \sum_{l=3}^n f_le^{\lambda_l t},
\label{eq: solptcom}
\end{equation}
where $c_1,c_2\in\mathbb{C}$ depend on the initial condition $p(0)$, for the rest of eigenvalues of $-K\tilde L$ we have that $\operatorname{Re}\{\lambda_l\} < 0,  l\geq 3$, and $f_l(t,c_l,\dots,c_n,w_l,\dots,w_n)$ are different functions corresponding to linear combinations like $c_lw_l + c_{l+1}(w_{l+1}+w_lt)+\dots$ depending on the (possibly generalized) eigenvectors $w_l$ of $-K\tilde L$ and constants $c_l$ given by the initial condition $p(0)$. Nevertheless, since these functions $f_l$ (polynomials on $t$) are multiplied by exponentials $e^{\lambda t}$, when we focus on the limit of (\ref{eq: solptcom}) for $t\to\infty$, we have that $p(t) \to -\kappa v^* \mathbf{1}_n(c_1 + c_2t) + c_2p^*$
	and $\dot p(t) \to -c_2 \kappa v^*\mathbf{1}_n$. Since ${^b}v^*$ is a designed complex constant, then we can conclude that $p(t) \to \mathcal{S}$ as $t\to\infty$. Furthermore, we can conclude that $c_2$ will determine the eventual direction and speed of the velocity of the formation in global coordinates with origin at $O_g$. In particular, the speed will be proportional to the scale of $p(t)\in\mathcal{S}$ with respect to $p^*$, which is given by $c_2$ as well. Consequently, the eventual configuration $^bp(t)$ will travel with the constant translational velocity $|c_2|\kappa \, ^bv^*\mathbf{1}_n$ whose \emph{heading} or orientation of $O_b$ with respect to $O_g$ will depend on $c_2$.

Now, let us consider the second case. Similarly as in the first case, if $\kappa > 0$ is sufficiently small, then according to Lemma \ref{lem: eigtr}, the solution of the closed loop (\ref{eq: pklp}) is given by
\begin{equation}
	p(t) = c_1 \mathbf{1}_n + c_2(\frac{v^*}{a + \iota\omega}\mathbf{1}_n + p^*)e^{\kappa(a + \iota\omega) t} + \sum_{l=3}^n f_le^{\lambda_l t}, \nonumber
\end{equation}
	where $c_1,c_2\in\mathbb{C}$ depend on the initial condition $p(0)$, the eigenvalues of $-K\tilde L$ satisfy $\operatorname{Re}\{\lambda_l\} < 0,  l\geq 3$, and $f_l$ are polynomial functions on $t$ as discussed in the first case. Then, we can deduce that $p(t)\to\mathcal{S}$ as $t\to\infty$, since $p(t)\to c_1\mathbf{1}_n +  c_2(\frac{v^*}{a + \iota\omega}\mathbf{1}_n + p^*)e^{\kappa(a + \iota\omega) t}$ as $t\to\infty$.\qed
\end{pf}

\subsection{How sufficiently small should be $\kappa_{\{r,t,s\}}$ or $\tilde\kappa$?}
\label{sec: kappa}
Our strategy to find an upper bound focuses on showing that $p(t) \to \operatorname{Ker}\{KL\}$, i.e., $p(t)\to\mathcal{S}$ as $t\to\infty$, via one Lyapunov analysis so that $\tilde\kappa$ or $\kappa_{\{r,t,s\}}$ can be taken into account explicitly in the stability analysis. Let us define the following complementary subspace $\mathcal{S}^\perp := (\operatorname{Ker}\{KL\})^\perp$ and let $P_\mathcal{S}, P_{\mathcal{S}^\perp}\in\mathbb{C}^{n\times n}$ be the projection matrices over the spaces $\mathcal{S}$ and $\mathcal{S}^\perp$ respectively. We then split
\begin{equation}
	p = P_\mathcal{S}\, p + P_{\mathcal{S}^\perp}\, p = p_\parallel + p_\perp.
\end{equation}
We are interested in showing that $p_\perp(t) \to 0$ as $t\to\infty$. Let us write the following dynamics derived from (\ref{eq: pklp})
\begin{align}
	\dot p_\perp &=  P_{\mathcal{S}^\perp} \dot p =  -P_{\mathcal{S}^\perp} KL(p_\parallel + p_\perp) + P_{\mathcal{S}^\perp}\tilde\kappa \tilde MB^T(p_\parallel + p_\perp).
	\label{eq: aux76}
\end{align}
We have shown in Lemma \ref{lem: eigtr} that $MB^Tp_\parallel\in\mathcal{S}$, i.e., $P_{\mathcal{S}^\perp} MB^Tp_\parallel = 0$, and together with $KL p_\parallel = 0$ and $P_{\mathcal{S}^\perp}KL p_\perp = KL p_\perp$, we can simplify (\ref{eq: aux76}) as
\begin{equation}
	\dot p_\perp = -KLp_\perp + \tilde\kappa P_{\mathcal{S}^\perp} \tilde MB^T p_\perp.
	\label{eq: aux77}
\end{equation}

Consider the Jordan form $J\in\mathbb{C}^{n\times n}$ of $KL$, i.e., $J = TKLT^{-1} = \left[\begin{smallmatrix}J_1 & 0 \\ 0 & J_2 \end{smallmatrix}\right]$ for some invertible matrix $T$, and $J_1\in\mathbb{C}^{2\times 2}$ and $J_2\in\mathbb{C}^{n-2\times n-2}$. In fact, let the first submatrix $J_1$ corresponds to the two zero eigenvalues of $KL$, i.e., the zero matrix since each zero eigenvalue has an independent eigenvector ($\mathbf{1}_n$ and $p^*$). Consequently, $-J_2$ is a Hurwitz matrix. Now consider the coordinate transformation $Tp = \begin{bmatrix}q_1^T & q_2^T\end{bmatrix}^T,$
with $q_1\in\mathbb{C}^2$ and $q_2\in\mathbb{C}^{n-2}$. Since $J_1$ corresponds to the zero eigenvalues of $KL$, then we have that $Tp_\parallel = \left[\begin{smallmatrix}q_1^T & 0\end{smallmatrix}\right]^T$ and $Tp_\perp = \left[\begin{smallmatrix}0 & q_2^T\end{smallmatrix}\right]^T$. Hence, by applying the same coordinate transformation to the dynamics (\ref{eq: aux77}) we have that
\begin{align}
	&\frac{\mathrm{d}}{\mathrm{dt}}\left[\begin{smallmatrix}0 \\ q_2\end{smallmatrix}\right] = -TKLT^{-1}\left[\begin{smallmatrix}0 \\ q_2\end{smallmatrix}\right] + \tilde\kappa TP_{\mathcal{S}^\perp}  \tilde MB^TT^{-1}\left[\begin{smallmatrix}0 \\ q_2\end{smallmatrix}\right] \nonumber \\
		\dot q_2 &= -J_2 q_2 + \tilde\kappa \left(TP_{\mathcal{S}^\perp} \tilde MB^TT^{-1}\right)_{\scriptscriptstyle (n-2 \times n-2)}q_2,
	\label{eq: aux79}
\end{align}
where the subindex for the matrix in the second term of (\ref{eq: aux79}) means that we have eliminated its first two rows and two columns. Then if $q_2(t)$ converges asymptotically to zero, then $p_\perp(t)$ will do it as well. Given a fixed designed $\kappa_{\{r,t,s\}}$ in $\tilde M$ for a desired formation's behavior, we can calculate a bound for $\tilde\kappa$, i.e., the \emph{global speed} of the formation, that guarantees the convergence with a standard Lyapunov analysis on the linear system (\ref{eq: aux79}) where its second term is considered as a perturbation.
\begin{proposition}
\label{pro: kappa}
	The system (\ref{eq: aux79}) is exponentially stable if $0 \leq \tilde\kappa < \Big(||Q\left(T\tilde MB^TT^{-1}\right)_{\scriptscriptstyle(n-2 \times n-2)}||_2\Big)^{-1}$, where $Q$ is a Hermitian positive definite matrix such that $QJ_2 + J_2^HQ \geq 2I_{(n-2)}$, where $X^H$ denotes the conjugate transpose of $X$.
\end{proposition}
\begin{pf}
Since $-J_2$ is Hurwitz matrix, then there exists a Hermitian positive definite matrix $Q$ such that $QJ_2 + J_2^HQ \geq 2I_{(n-2)}$. Now consider the Lyapunov candidate $V = q_2^HQq_2$, whose time derivative satisfies
{\small
	$
		\frac{\mathrm{d}V}{\mathrm{dt}} \leq -2||q_2||^2 +2\tilde\kappa \, ||Q\left(TP_{\mathcal{S}^\perp} \tilde MB^TT^{-1}\right)_{\scriptscriptstyle (n-2 \times n-2)}||_2 \, ||q_2||^2$,

}
	and noting that $||P_{\mathcal{S}^\perp}||_2 = 1$, then the system (\ref{eq: aux79}) is stable if $0 \leq \tilde\kappa < \Big(||Q\left(T\tilde MB^TT^{-1}\right)_{\scriptscriptstyle (n-2 \times n-2)}||_2\Big)^{-1}.$
\qed
\end{pf}
\begin{remark}
	Note that if we multiply the gain matrix $K$ by a real constant $h > 1$, then $Q$ in Proposition \ref{pro: kappa} can be substituted by $\tilde Q = \frac{Q}{h}$. Therefore, the upper bound in Proposition \ref{pro: kappa} can be made arbitrarily large by increasing the free parameter $h$. Indeed, if we make $K$ bigger, then we mitigate the perturbation in (\ref{eq: Ltilde}) of the original complex-Laplacian matrix $L$ as required by Lemma \ref{lem: eigtr}.
\end{remark}
The following algorithm summarizes the design process to guarantee the convergence of the formation to the desired eventual behavior.
\begin{alg}
\begin{enumerate}
\item Given a desired $p^*$, we calculate the weights $\omega_{ij}$ and the gains $k_i$ according to \cite{lin2014distributed} to construct the Laplacian $L$ and the gain matrix $K$ in (\ref{eq: comcom}).
\item We calculate the sets of motion parameters $\mu_{ij}$ in (\ref{eq: ormo}) for the translation, rotation, and scaling.
\item We design $\kappa_{\{r,t,s\}}$ in (\ref{eq: mtilde}) to define the eventual behavior of the formation. Then, we choose $\tilde\kappa$ to set the \emph{global speed}, i.e., we scale $\kappa_{\{r,t,s\}}$ simultaneously in (\ref{eq: Ltilde}) where $M = \tilde M$.
\item Finally, we check whether with the given $L$, $K$, $\tilde M$ and $\tilde\kappa$ the condition in Proposition \ref{pro: kappa} is satisfied. If not, then we multiply $K$ by a real constant $h > 1$ such that the condition is satisfied.
\end{enumerate}
\end{alg}

\section{Applications and simulations}
\label{sec: simulations}
We will show how to use Theorem \ref{pro: comt} in escorting scenarios via \emph{enclosing of a target}, area coverage via the \emph{shaped consensus}, and formation travelling via the \emph{translation with controlled speed and heading}.  We will compare the proposed technique with existing popular ones.

Let us start with considering Theorem \ref{pro: comt} when $a = 0$ (or $\kappa_s=0$), i.e., there is no scaling motion. In such a case, the eventual formation will orbit around an instantaneous center of rotation in the complex plane. Such a center of rotation can be set at the agent $q\in\mathcal{V}$; therefore, ${^b}v_i^*, \forall i\in\mathcal{V}$, must be designed to orbit around agent $q$ similarly as in Figure \ref{fig: velcom} the rotation velocity is around the centroid. We show this scenario in Figure \ref{fig: enc}. A similar technique based on distance-based formation control has been proposed in \cite{Marina16}, where instead of weights the authors manipulate desired distances. While distance-based can guarantee collision avoidance between neighboring agents and can be employed in 3D (or higher), only local-convergence is given due to nonlinearities.

If we consider $v^* = a = 0$ (or $\kappa_{\{t,s\}} = 0$) in Theorem \ref{pro: comt}. Then, the formation will describe an eventual spinning motion around its centroid. If we combine it with a scaling motion, then we can cover an area by describing circular spirals within $\mathcal{S}$ if $a>0$. Conversely, we have the consensus of the formation, i.e., $p(t)\to b\mathbf{1}_n, b\in\mathbb{C}$ as $t\to\infty$ if we set $a<0$. Our extension from the \emph{standard} consensus \cite{olfati2007consensus} is that in Theorem \ref{pro: comt} we set $p^*$ as the eigenvector associated to the algebraic connectivity $\lambda_2 = \kappa_s a + \iota\kappa_r \omega$ of the Laplacian matrix. Therefore, a \emph{shaped consensus} occurs, i.e, the formation will display the desired formation while rendezvous. We show both maneuvers in Figure \ref{fig: decacon}.

Let us finally consider $v^*\neq 0$ and $\omega = a = 0$ (or $\kappa_{\{r,s\}} = 0$). Then, according to Theorem \ref{pro: comt}, the formation will travel with constant velocity eventually. However, the translational velocity has been designed with respect to $O_b$ and not $O_g$ as it is shown in Figure \ref{fig: velcom}. We could choose one agent $i\in\mathcal{V}$ to control one relative position $z_{ij}, j\in\mathcal{N}_i$, in global coordinates, e.g., with adding the simple proportional controller $-(z_{ij} - z_{ij}^*)$ to its control action with $z_{ij}^*\in\mathbb{C}$ with respect to $O_g$ and compatible with $\mathcal{S}$. Since $z_{ij}^*$ has both direction and magnitude, then we are controlling both, the heading of the translational velocity in $O_g$ and its speed since it depends on the size of $p(t)\in\mathcal{S}$. This approach is free of any extra complexities/estimators as it is required in a \emph{leader-follower} approach \cite{lin2013leader,zhao2018affine}.

\begin{comment}
\section{Numerical experiments}
\label{sec: simulations}
We consider a regular decagon for the demonstration of our maneuvering technique with the complex-Laplacian-based formation control. We choose as a reference shape $p^*$ a decagon whose sides are equal to $1$. We follow the design shown in \cite{Marina2017}, i.e., the set of edges is given by $\mathcal{E} = \{(1,2),(2,3),\dots,(9,10),(10,1)\}$, and the weights for the agent $i\in\mathcal{V}$ are $\omega_{i\,\,\operatorname{mod}_{10}(i-1)} = e^{-\iota \frac{\pi(1-\frac{10-2}{10})}{2}}$ and $\omega_{i\,\,\operatorname{mod}_{10}(i+1)} = e^{\iota \frac{\pi(1-\frac{10-2}{10})}{2}}$ so that $K$ in (\ref{eq: comcom}) can be the identity matrix. Thanks to the symmetry of the decagon we can easily modify the Laplacian weights $\omega_{ij},(i,j)\in\mathcal{E}$ for the design of the rotational motion by setting $\mu_{{ij}_r} = \mu_{{ji}_r} = 1 + \iota 0, (i,j)\in\mathcal{E}$ to construct $M_r$. Similarly, for the scaling motion we set $\mu_{{ij}_s} = -\mu_{{ji}_s} = 1 + \iota 0, (i,j)\in\mathcal{E}$. Prior to the design of the scaling and rotational speeds, we note that the interior angle of a regular polygon is given by $\psi = \frac{1}{n}(n-2)\pi$ and that $r = ||p_i^*|| = (2\sin{(\pi/n)})^{-1},i\in\mathcal{V}$. If we consider $\kappa_r \omega = r(2\sin\frac{\psi}{2})^{-1}$ and $\kappa_s a = r(2\cos\frac{\psi}{2})^{-1}$, with $\omega = a = \pm1$ then $k_r$ and $k_s$ sets the desired angular and scaling speeds respectively.

In the numerical experiments in Figure \ref{fig: decacon} we have set $a = \mp 1$ \emph{current size}/sec respectively with a common $k_s = 0.025$, $\omega = 1$ rads/sec and $k_r = 0.025$. The agents describe circular spirals towards and away from the centroid of the eventual formed decagon respectively. We show now the numerically calculated ten eigenvalues of $-I_{10}\tilde L$ for $\omega = -a = 1$ and $k_r = k_s = 0.025$:
\begin{equation}
\scriptsize
\begin{smallmatrix}
       \boldsymbol{2.68264624e-16+1.09742323e-16j} & \boldsymbol{-2.50000000e-02+2.50000000e-02j} \\
	   -7.51542528e-01-2.50000000e-02j& -8.16993378e-01+4.04508497e-02j \\
	   -1.99256388e+00-4.04508497e-02j& -2.07346558e+00+4.04508497e-02j \\
	   -4.06602946e+00+4.58025544e-16j& -4.04102946e+00-2.50000000e-02j \\
	   -3.31448694e+00+2.50000000e-02j& -3.24903609e+00-4.04508497e-02j
\end{smallmatrix}, \nonumber
\end{equation}
that verifies the predictions about \emph{relocating} one of the zero eigenvalues to the desired value so that we induce the scaling plus rotational motions of the decagon. Note that the rest of eigenvalues have negative real part, and they are one order of magnitude bigger than $a$ in absolute value so that the eventual shape during the \emph{shaped consensus} is a decagon. We provide the Python code (employing only open-source libraries) of the numerical experiments in Figures \ref{fig: decacon} at \url{https://github.com/noether/multiagents/blob/master/complex_laplacian_regular_polygon.py}.
\end{comment}

\begin{figure}
\centering
\includegraphics[width=1\columnwidth]{./images/enclosing.eps}
	\caption{The formation converges to a square and cooperates to enclose and orbit around one of its agents (in black color). The desired rotational and translational velocities of the formation have been designed such that the enclosed agent is the instantaneous center of rotation. The crosses and the dots denote for the initial and the $t = 250$ secs positions. The dashed lines denote the edges of the graph.}
\label{fig: enc}
\end{figure}

\begin{figure}
\centering
\includegraphics[width=0.48\columnwidth]{./images/deca_conse.eps}
\includegraphics[width=0.48\columnwidth]{./images/deca_grow.eps}
	\caption{On the left-hand side, the formation eventually forms a regular decagon that describes a circular spiral towards its centroid, i.e., a \emph{shaped} consensus. On the right-hand side the formation describes a circular spiral while the eventual displayed decagon grows exponentially fast. The crosses and the dots denote for the initial and the $t = 250$ secs positions. The dashed lines denote the edges of the graph.}
\label{fig: decacon}
\end{figure}

\begin{figure}
\centering
\includegraphics[width=1\columnwidth]{./images/vel_heading4.eps}
	\caption{The formation converges to a square with only translational velocity w.r.t. $O_b$. The eventual formation's velocity (blue arrow) is designed to be parallel to the side $z_{12}$. The red agent controls the global orientation of such a side, e.g., with the proportional controller ($z_{12} - z_{12}^*$). Every $50$ seconds, the desired $z_{12}^*$ is rotated by $\frac{\pi}{4}$ radians. In the last rotation, $z_{12}^*$ is also four times bigger than at the beginning, hence the square not only grows, but the velocity's speed is increased accordingly. The crosses denote for the initial positions, and the dashed lines denote the edges of the graph.}
\label{fig: trav}
\end{figure}

\section{Conclusions}
\label{sec: conc}
We have presented a maneuvering technique to induce collective motions with global convergence in \emph{complex-Laplacian-based} formation control. This technique can be exploited in the problems of \emph{shape consensus}, \emph{enclosing of a target}, and \emph{travelling formation with controlled speed and heading}.

redpajama_set_name: RedPajamaArXiv


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2.	In 1980, a first addendum (A/AC.183/L.2/Add.1) was issued, covering the years 1976-1979. Since 1980, seven addenda (A/AC.183/L.2/Add.2-8) have been prepared by the Secretariat.
3.	The present document, covering 1988, is intended to bring this chronological compilation up to date. This conmpilation also includes General Assembly resolution 43/233 adopted by the General Assembly at its resumed Forty-third session on 20 April 1989.


1.1 The mission of the North Orange County Community College District is to serve and enrich our diverse communities by providing a comprehensive program of educational opportunities that are accessible, relevant and academically excellent. We are unequivocally committed to student success and lifelong learning.
1.2 Cypress College and Fullerton College will offer associate degrees, vocational certificates, and transfer education, as well as developmental instruction and a broad array of specialized training. The School of Continuing Education will offer noncollege credit programs including high school diploma completion, basic skills, vocational certificates, and self-development courses. Specific activities in both the colleges and School of Continuing Education will be directed toward economic development within the community.
1.3 The mission is evaluated and revised every three years.
2.0 Our diverse learners will be prepared to shape their future and become engaged in our democracy.
and training services. We promote engaged and ethical participation in global, civil society.
3.2 High Standards: We uphold high standards in academics and promote joy in teaching, learning, and work.
3.3 Adaptability: We respond to the changing needs of our communities in a global environment through continuous improvement and creative innovation.
3.4 Professionalism: We hold high standards for our faculty and staff and create an environment in which staff development and other continuing education are valued and promoted.
3.5 Stewardship: We uphold the responsibility for public trust of our mission and resources.
3.6 Respect: We cultivate an atmosphere of courtesy, civility, and transparency with all students and employees in the District by promoting a willingness to collaborate and a responsibility for all to be engaged as collegial partners in carrying out the District's mission.
3.7 Inclusiveness: We welcome and respect the diverse backgrounds and beliefs of our students, faculty, and staff, and the many communities we serve.


Why is social media really nothing new? Why is Twitter king and LinkedIn boring? How do you deal with trolls on social media? What's vaguebooking?
To get some answers, I speak to Susie Boniface, journalist and author of The Bluffer's Guide to Social Media. Have a listen to the interview on iTunes and Soundcloud or keep reading for a special summary. And don't forget to subscribe to the Employer Branding Podcast.
Is social media really something new?
It's simply communication. It's not difficult. We've communicated by cave paintings, we've communicated by quill pen and ink, and we now communicate via Facebook and Instagram, and Snapchat. The thing to remember with all social media, whether you're intimidated by it, trying to understand it or trying to make money out of it, is that it's fundamentally human interaction, and that's it. There's no difference between someone painting a picture of the bison they killed on the cave wall and someone tweeting a picture of their lunch. It's exactly the same thing.
Is Twitter the king of social networks?
Twitter is… but not in terms of making money or amount of people that are involved in it. It is, probably at the moment, king in terms of the level of interaction and socialisation, and the fact that news tends to break on Twitter, and that's where people do tend to get a lot of their immediate information about the world around them.
But Facebook is the one that makes all the money. But it's arguably the one that's got the most adverts and it's got slightly more conservative with the small c, say, reactions to things. For example, the difference between Twitter and Facebook is that on Twitter, you only see what people say, you don't see all the responses from everybody else.
What about the professional network, LinkedIn?
LinkedIn, right, yes. LinkedIn's difficult for me to talk about on the basis that I, long ago, spammed all the invitations to join it. Basically it's an online Rolodex. It's not interesting enough to be banned in China like most of the other social media forms are. It's basically just a place to put your online CVs, and for marketing and recruitment professionals to go and find people and say that, "We will find you jobs. Upload your details to us". There's a huge amount of users on it. It's inexplicably popular in terms of just the sheer number of people and it does make some money – it's very successful, but it's incredibly dull. You're not going to be on there very much really just sort of chat or interact with people on a normal human basis. You might be looking for work or trying to find an employee, that kind of thing. Or if you do it very wrong, as was the case a little while ago with a female barrister and a male lawyer, you're trying to chat someone up or give them a compliment on LinkedIn, it tends to backfire.
How do you deal with trolls on social media?
There are general rules for anybody on social media, what I call the bus-stop rules. This applies whether someone is lovely, whether someone is asking on date, whether someone's a troll, whether anything else, normal human interaction, what would you do at the bus stop if this happened to you?
Just because it happens on your computer immediately and in front of your face, and you can't quite see the body language, and you can't quite see whether someone is being sarcastic or not, or trying to make a bad joke, if this was said to you at a bus stop, would you call the police? Would you get this upset? Would you decide that this is the person you want to marry? And if you would do that at the bus stop, that's the way you should behave. And if you wouldn't, then, most cases, people like trolls, for example, if someone comes up to you at a bus stop and says, "I'm going to come in your house and blow you up at 10pm tonight," would you actually call the police, or would you just move further away from the bus stop, or get the next one as it came along?
Tell us about citizen journalism on social?
Well, there's no such thing. There are citizens and then there are journalists. Journalists have some training, normally. And part of that, when you are at a news event, is to see what everyone else has to say about it, try to get their photographs and their names spelt correctly, and it's got nothing to do with you, as a journalist. You don't put yourself in that story. Citizens, however, when they are at an event, they blog it or tweet it or Facebook it, or take pictures of it, video it and they go, "Look at me, I'm here, I'm in the middle of everything," and that's not journalism.
The best example of why citizen journalism isn't journalism, is coverage of the Sydney siege a few years ago. People were tweeting and blogging pictures of themselves, selfies, outside a siege where a gunman had people held at gunpoint. Now that is not what journalists would do.
FTW: For the win. This means that you are promoting one particular thing as being the likely to win. So you might say, "Trump for the win, FTW," or you might be using a more sarcastic fashion and saying, "Eighteen slices of chocolate gateau for the win." Depends on the context.
H/T: That's a hat tip. So particularly on Twitter if you are trying to share what someone else has tweeted, and you want to attribute what you're tweeting to somebody else to say, "this isn't me saying this but I've seen this somewhere else," then you just, "hat tip to," and include the other person's handle.
Vaguebooking: Vaguebooking is just updating your Facebook status in a vague way, perhaps in a way to try to encourage other people to ask you if you are all right, or, "what are you talking about?" kind of thing.
Follow Susie on Twitter @FleetStreetFox and subscribe to the Employer Branding Podcast.


Exhibition "Anniversary" is open in the Institute of Contemporary Art, Ukraine's NAA. The exhibition is dedicated to the tenth anniversary of the foundation of this art and research institution. The works presented at the exposition show mosaic of authors' strategies and tactics, many individual gestures, and artistic events. The curators of the exhibition – Iryna Yatsyk, Hlib Vysheslavsky, and Oleh Sydor-Hibelynda tried to gather as many artists of different generations as possible, those artists, who, in their opinion, were the most active in the sphere of modern art. The spirit of the new millennium "hangs in the air" – metaphorical red human figure created by the director of the institute artist Viktor Sydorenko. White installation by Oleksii Lytvynenko and Petro Bevza which takes up a whole room reminds what a brave step it was to put "wild" natural materials in the exhibition hall. The wooden object by Roman Minin resembles a sarcophagus, in which all the problems of Donbas miners are buried. Somehow it doesn't even feel right to use the word "academic" referring to this artistic mischief, however, modern art is worth to be treated very seriously. Institute workers study all the manifestations of the new way of thinking and sensuality that appear on our art arena. It was really impressive how many publications were presented on display stands.
In the same exhibition space we can see bright examples of "classics and contemporary artists," "old guard" and "vigorous young generation." There are also presented rough and living exhibits, made using the traditional technique, as well as virtual art documentation, which took place "once somewhere," but, in fact, "recently and somewhere close." Sculptures and graphics are more and more often replaced with photograph, land art with installations, and video art, while painting is going through a rebirth, as well as performance, which seemed to fade away in our country, after it just began to bloom in the early 1990s. Entirely new genres such as graffiti are emerging and only now are being slowly acknowledged seriously. We see a variety of names, talents, and trends. We also see artists, who began their artistic career in the spirit of Kharkiv school and later came up on the international level (Viktor Sydorenko); those who grew from the new wave of Ukrainian postmodern (Hlib Vysheslavsky) and found their individual stylistics (Oleksandr Klymenko); those who are still searching for the "alternative way" like representatives of the plastics trend (Petro Bevza and Oleksii Lytvynenko), as well as "new archaic" trend (Oleksandr Babak); the ones who feel confident with new technologies (Oksana Chepelyk); and, finally, those who entered the artistic life in the past decade responding more sensitively then others to its bizarre pulsations (Andrii Sydorenko, Hamlet Zinkivsky, Roman Minin, Yurii Kruchak, and Dmytro Korniienko). The exposition is completed with the media project by Hlib Vysheslavsky, Virtual Museum.
Objective look at all the participants of the process of modern art creation can be compared favorably with the one, which strictly filters artists leaving out of a thousand only dozens of those looking promising in the eyes of art business dealers. Commercial approach to those "chosen ones" exhausts them and doesn't make them any better. Looking closely at the variety of works from different parts of Ukraine you understand how many interesting and talented individuals didn't get into those top rankings, always presented in art advertising.
Photos of land art compositions in midst of beautiful scenery of the Carpathians and Vinnytsia (report by Oleksii Malykh, Oleksandr Nikitiuk, Hanna Naduda, Hanna Sydorenko, Serhii Yakunin, Kostiantyn Alieninsky, and Volodymyr Nois) strike with their beauty. Photo documentation of Hurzuf Seasons by curator Olesia Avramenko reproduces the interesting experiments of different artists in urban environment. Along with retrospective projects by well-known artists the new ones are presented. Thus, fierce abstractionist Oleksandr Klymenko presented pieces of paper printed on a computer on which he expressed a new philosophical view on modern time. Photo installation by Olena Holub Rising of the Wooden Sun, that was presented at art festival in Hungary, is exhibited for the first time in Ukraine. Both fascinating and ironic mood of these works has also a culturological connotation referring to the famous painting by Claude Monet, Impression. Sunrise, stressing the fact that times change.


The International DOI Foundation has adopted a Patent Policy covering the rights and obligations of DOI Registration Agencies, following completion of its Strategic Review and approval at the 2005 members' meeting. This policy is now binding on all current and future IDF Registration Agencies.
The Patent Policy, and accompanying DOI Core Specification, was approved by the Board of the International DOI Foundation as a formalization of past policies and practices. The DOI Core Specification, which is also at the heart of the proposed ISO TC46 SC9 specification, represents the current state of the DOI System and completion of the first phase of developments. The Patent Policy aims for a balance between an open, core system (the system developed collectively) and value-added services created by RAs.
Support the existing commitment that DOI be an open standard and system available to all who want to use it on equal terms (see "IDF re-affirms DOI as an open specification", October 2004).
Preserve and protect the collective investment in the DOI system and standard.
Encourage Registration Agencies to develop added-value services and features on top of the Core DOI System.
To be simple, practical and easy to implement.
Royalty-free licensing with respect to patents for which the RA fails to satisfy the notice requirement.
The IDF believes this offers the best way forward in encouraging the widest possible distribution and use of DOIs and the DOI System, ensuring both that the DOI remains an open standard and system available to all who wish to adopt it and that a core common infrastructure is maintained.
For further information about the policy, please contact n.paskin@doi.org.


If you're considering getting Sony's virtual-reality PlayStation VR accessory, included free is a PS VR Demo Disc chocked full of games naturally!
The demo disc will also be available for download at PlayStation Store for free after PS VR launches.
Please note, content included in the demo disc differs across countries and regions. The demo disc with the above content will be included in retail PS VR units in the US and Canada.
It's now Apr 18, 2019 at 6:24 PM.


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Q: Should I say "taking the assignment" or "doing the assignment"? Let's say I am working on my assignments and my friend calls me up. What should I tell him?

I am doing my math's assignment (so I can't come).

or

I am taking my math's assignment. (similar to taking the exam)

Which of these two is more appropriate here? Or should I say something else?

A: Math vs. Maths
It is either math (American English) or maths (British English), but never math's. See this for more about that.
Doing vs. Taking
In regards to a math(s) assignment, "doing" is a better word choice. As used in your question, "working on" is also appropriate. You could also simply say, "I need to finish my homework."

A: We often use common verbs like have and take with nouns. They are called delexical verbs because the important part of the meaning is taken out of the verb and put into the noun. We often put adjectives in front of the noun as well
in your example "doing" is right because taking implies that you are taking it away with you which isn't case here

redpajama_set_name: RedPajamaStackExchange


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Home Politics Volker Bouffier: This is the man who wants to rule in Hesse,...
Volker Bouffier: This is the man who wants to rule in Hesse, Germany more
State election in Hesse: All the info in the News Ticker of FOCUS Online
Sharp sounds or provocative theses you hear from Volker Bouffier, meanwhile, rarely. As the inventor of the black-green coalition in Hesse is a Motto of the 66-Year-old: listen to, also changes the Position of the other, forming alliances.
When it comes to the AfD, and especially to the Federal chief Alexander Gauland, with Bouffier, many years ago, still in the CDU's policy in the state of Hesse has made, but again the hard-liners of earlier days. Then, the already rough voice of the government is still rough, and the vocabulary is sharp and uncompromising.
Bouffier is considered to be loyal supporters of Chancellor Angela Merkel
The is pour the root end of a lawyer, as a faithful supporter of German Chancellor Angela Merkel, and is a man of clear language. The popular festivals and in the halls of the CDU-Federal Vice-moves, as the Berlin or the international parquet. Small Talk, shake hands, shoulder, tapping and pull strings, for the father of three adult children, of course, and no annoying torment.
Self-Bouffiers political opponents as his SPD adversary Thorsten Schäfer-Gümbel trusted that the CDU politician uses his years of experience of Hesse and the interests of the people. The Left keeps Bouffier, however, his role as the then Minister of the interior in the investigation of the NSU murder series. Agree friend and enemy are the fact that the father of the country has lied to a slope, some of them strenuous, long mono.
Bouffier 2010 succeeded by Roland Koch in Hesse
Bouffier, who had to give up after a serious car accident as a young man, the dream of professional Basketball career, was under the CDU government chief Walter Wallmann of justice, the Secretary of state. In 1999, he was Minister of the interior in Hessen. After the withdrawal of Prime Minister Roland Koch of politics, he took up in 2010, its successor as the head of a black-yellow coalition.
The state government with the Green Bouffier leads since the beginning of 2014. It was the first Alliance of this type in a German area of the country and a Gamble. With Deputy Prime Minister Tarek Al-Wazir, from the Greens, the CDU's Deputy by You and by now a well-oiled Team. Very gladly he would, therefore, also the smooth cooperation with the eco-in the state of Hesse for a further five years.
even Though he is in December, 67 years of age and often because of his accident back pain plagued, want to ride Bouffier in the case of a re-election to the full term as father of the country. To a question, he replied: "you will Remain relaxed, with me, you must expect for a long time."
In the FOCUS Online/Wochit the CDU and the SPD in poll-Deep – Green in the height of flight jmü/dpa

redpajama_set_name: RedPajamaCommonCrawl


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Q: How can I tell if a weapon is one or two handed? In Skyrim is there a way to tell if a sword is one or two handed without picking it up. I'm not a fan of two handed weapons and I don't want to pick them up just to drop them when I realize it's not one handed. Specially, I want to know about swords but the answer may apply to other weapons as well. Is there some icon I'm missing? Or is it just a matter of weight?

A: It's generally a matter of weight and weapon type.
One-handed weapons include daggers, swords (not greatswords), war axes (not battle axes), and maces (not warhammers). These will generally max out at 20 weight (with maces being the heaviest)
Two-handed weapons include greatswords (not regular swords), battle axes, and warhammers. Most have over 20 weight, but greatswords can weigh a little less (~15 weight).
Jesse Cox of OMFGcata also notes you can tell by the orientation of the weapon in the preview window:

*

*Two-handed weapons are tilted to the side

*One-handed weapons are shown vertically straight

A: You can see whether a weapon requires two hands if you equip it - if it's a one handed weapon it will say (L) or (R) (depending on which hand you put it in) and on two handed weapons it will show (LR) - just like with some spells that need two hands to cast (because it's too powerful for one hand).


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Tom Holland wants to make a live-action Jak and Daxter movie
Uncharted and Spider-Man star, Tom Holland has shared that he wants to make and star in a live-action version of the videogame series Jak and Daxter
Emma-Jane Betts
Tom Holland, star of the upcoming action movie Uncharted, already has another big role in mind. When asked what his dream video game adaptation might be during an interview with GameSpot, the Spider-Man: No Way Home actor revealed that he'd love to be on board a live-action version of the Jak and Daxter game series.
With its first entry released in 2001, Jak and Daxter was one of the earliest games to hit the PlayStation 2 console, and has since gathered a huge fanbase. Set in a fictional cyberpunk inspired world, the series follows Jak and Daxter as they embark on adventures, and try to uncover the mysteries left behind by an ancient race – the Precursors. The last main instalment in the franchise, Jak 3, came about in 2004, and since then there have been a couple of spin-offs, but no significant developments or additions to the IP have come about.
However, all that may now change considering Holland's excitement about the 2000s story. "I would like to make a Jak and Daxter movie, and I would play Jak," Holland said. "But I would make it at A24, so it was really weird and like dark. I would do like a really weird, live-action version of Jak and Daxter."
Despite Holland's answer, there hasn't been any official announcement or hint that a Jax and Daxter movie is in the works. But considering that the franchise is owned by Sony Interactive Entertainment, and that Holland and the company have a long-running professional relationship, it isn't unfeasible – the star may still get his Jak and Daxter dream yet.
Hey, @A24 and @Naughty_Dog. Tom Holland has a great video game adaptation pitch for you guys. pic.twitter.com/QKKbJYWedf
— GameSpot (@GameSpot) February 4, 2022
While we wait on updates, fans can see Holland in another videogame adaptation, Uncharted, which Venom director Ruben Fleischer helms. In the thriller movie, Holland stars as the leading protagonist Nathan Drake, alongside other well known Hollywood actors such as Mark Wahlberg and Antonio Banderas.
Uncharted hits theatres in the UK on February 11, and US on February 18. For more action, here is our list of the best adventure movies of all time.
Uncharted director "working on" Jak and Daxter movie
Staff writer, horror movie and The Lord of the Rings expert, Emma-Jane moved from South Africa to the UK after breaking her arm playing 'duck, duck, goose'. She has worked on several independent films and wrote for The Upcoming before joining The Digital Fix.


In the 50's, Rene Thom solved the problem of determining when a closed smooth manifold bounds a compact manifold. Subsequent work of Milnor and Wall solved the analogous oriented problem. These works comprise an important early example of the fundamental method of geometric topology via reduction to algebraic topology. The basic question (posed by Gromov) is: given a complexity measure for manifolds, how complicated must a nullcobordism of a given manifold be? I won't solve this problem, but I will explain (based on joint work with Steve Ferry, and with Greg Chambers, Dominic Dotterer, and Fedor Manin) some recent estimates.


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Q: Manual (or examples) of r.ros, r.spreadpath and r.spread in GRASS you can report me references or applications for use of the following modules (GRASS):
- r.ros,
- r.spreadpath,
- r.spread.

A: The Spread distance is an estimate of the distance field from one or more target cells, such that the distances are routed through the grid lattice (see below). It's calculated iterative, in a way that minimizes the distance field, i.e. shorter routes are sought in each iteration until no shorter route between each grid cell and their nearest target cell can be found.
Before Shih and Wu (2003) published their extremely efficient and accurate 4-pass Euclidean distance transform method, Spread was the dominant method for estimating distances on a raster in GIS. These days, Spread distance isn't used for estimating Euclidean distance (including raster buffering operations) in GIS so much. However, because its iterative approach is conducive to an anisotropic distance calculation, the Spread algorithm is still widely used for performing least-cost pathway analysis. Shih and Wu's method, while faster and more accurate, is useless for estimating a directionally dependent or cost-factor dependent distance. The Spread algorithm also provides the opportunity to calculate the 'back-link' for each cell, essentially equivalent to the D8 flow-direction grid in flow path analysis. In this way, it is possible to identify the 'distance watersheds' that connect each cell in the grid to it's nearest target and to trace individual least-cost pathways for any location to a target. Since least-cost analysis tends to find itself applied in all sorts of spatial optimization problems, the Spread tool is still very relevant today.

If you're looking for an academic reference for the Spread algorithm then I should think Chase (1984) and Rothermel (1983) would be good starting places. If you're just looking for a user reference then the GRASS user manual (http://grass.osgeo.org/grass64/manuals/r.spread.html) is the place to start.


Trump warns Mexico on guns drawn on US troops: "Better not happen again!"
749c, the first Earth-sized planet found by NASA's Transit Exoplanets Survey Satellite (TESS), as well as its companion, HD 21749b, and hot sub-Neptune-sized world.
An Earth-sized world has been discovered in a star system 53 light years from our solar system just days after a study found that life could exist on four close exoplanets .
The first Earth-sized planet discovered using NASA's Transit Exoplanets Survey Satellite (TESS) was found orbiting a star along with a much larger, hot planet smaller than Neptune. a milestone for TESS, "said Diana Dragomir, an exoplanetologist at MIT Kavli Institute and lead author of " TESS Shares Its First Earth-sized Planet and a Hot Sub-Neptune" published in Astrophysical Journal Letters. sets the path for finding smaller plan ets around even smaller stars, and those planets may potentially be habitable. "& nbsp; This is the 10th confirmed planet discovered by TESS.
What did the astronomers find?
Orbiting the K dwarf star HD 21749, the astronomers found HD 21749c, which takes about eight days to orbit the host star and is similar in size to Earth at 89% its diameter. A likely rocky world, it's thought to have surface temperatures as high as 800 & deg; F /427°C. They also found HD 21749b, a hot sub-Neptune sized world about 23 times Earth's mass and a radius of about 2.7 times Earth's, which orbits its host every 36 days. That one was a bit of a surprise since TESS is not designed to detect planets that take longer than 10 days to orbit their star.
Launched atop a SpaceX Falcon 9 rocket on April 18, 2018, TESS is a planet-hunting spacecraft designed to search for Earth-like planets. TESS will observe 400,000 stars across the whole sky to catch a glimpse of a planet transiting across the face of its star. TESS will enable astronomers to measure masses, atmospheric compositions, and other properties of many smaller exoplanets for the first time.
Why is TESS a "game changer?"
Which ground-based telescope was used?
The Transiting Satellite Survey (TESS) is NASA's newest planet hunter.
How do you measure the mass of an exoplanet?
Wobbly orbits. The mass of a planet is important to help astronomers find out more about its density and chemical composition. The PFS tool uses the "radial velocity" technique, which detects not only the tiny wobbles in the orbit of a planet, caused by the gravity of the host star, but also how the planet's gravity also affects the star's orbit. "The PTA is one of the only instruments in the Southern Hemisphere that can do these types of measurements," said Teske added. "So, it will be a very important part or further characterizing the planets found by the TESS mission."
What will the astronomers do next?
Potentially the first mass measurement of an Earth size planet found by TESS. "Measuring the exact mass and composition of such a small plane will be challenging, but important for comparing HD 21749c to Earth," said Wang. "Carnegie's PFS team is continuing to collect data on this object with this goal in mind. "
Will TESS help find more Earth-sized planets?
Expected to last for at least two years, and as long as a decade, TESS is expected to find over 20,000 exoplanets, hopefully, many Earth-sized. "For stars that are very close and very bright, we expected to find a couple of boxes of Earth-sized planets," said Dragomir. "And this is our first one, and it's a milestone for TESS."
Artist's conception of HD 21749c, the first Earth-sized planet found by NASA's Transit Exoplanets Survey Satellite ( TESS), as well as its band, HD 21749b, in a hot sub-Neptune-sized world.
An Earth-sized world has been discovered in a star system 53 light years from our solar system just days after a study found that life could exist on four close exoplanets.
The first Earth-sized planet discovered using NASA's Transit Exoplanets Survey Satellite (TESS) was found orbiting a star along with a much larger, hot planet narrower than Neptune "It's a milestone for TESS," said Diana Dragomir, an exoplanetologist at MIT Kavli Institute and lead author of "TESS Delivers Its First Earth-sized Planet and a Hot Sub-Neptune", published in Astrophysical Journal Letters. "The plan for finding smaller planets around even smaller stars, and those planets may potentially be habitable." This is the 10th planet discovered by TESS.
Launched atop a SpaceX Falcon 9 rocket on April 18, 2018, TESS is a planet-hunting spacecraft designed to search for Earth-like planets. TESS will observe 400,000 stars across the whole sky to catch a glimpse of a planet transiting across the face of its star. TESS will enable astronomers to measure the masses, atmospheric compositions, and other properties of many smaller exoplanets for the first time.
TESS is designed to make a shortlist of targets for large space and ground-based telescopes to investigate further. In this case, Magellan II, a 6.5-meter telescope situated at Carnegie's Las Campanas Observatory in Chile, was used to follow up data from TESS. It hosts the Planet Finder Spectrograph (PFS), a tool built by Shectman and Crane using a method pioneered by Butler and his collaborators, which helped confirm that the signal from TESS were indeed planets. It was also used to measure the mass of the Neptune-like HD 21749b, which was possible to do for the much smaller HD 21749c.
Potentially the first mass measurement of an Earth size planet found by TESS. "Measuring the exact mass and composition of such a small plane will be challenging, but important for comparing HD 21749c to Earth," said Wang. "Carnegie's PFS team is continuing to collect data on this object with this goal in mind."
Expected to last for at least two years, and as long as a decade, TESS is expected to find over 20,000 exoplanets, hopefully, many Earth-sized. "For stars that are very close and very bright, we expected to find a couple of box-sized planets," said Dragomir. "And here we are – this would be our first one, and it's a milestone for TESS."


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Inflation in supergravity with a single chiral superfield
Sergei V. Ketov, Takahiro Terada
Исследовательская школа физики высокоэнергетических процессов
Результат исследований: Материалы для журнала › Статья › рецензирование
We propose new supergravity models describing chaotic Linde- and Starobinsky-like inflation in terms of a single chiral superfield. The key ideas to obtain a positive vacuum energy during large field inflation are (i) stabilization of the real or imaginary partner of the inflaton by modifying a Kähler potential, and (ii) use of the crossing terms in the scalar potential originating from a polynomial superpotential. Our inflationary models are constructed by starting from the minimal Kähler potential with a shift symmetry, and are extended to the no-scale case. Our methods can be applied to more general inflationary models in supergravity with only one chiral superfield.
Опубликовано - 7 сен 2014
http://www.scopus.com/inward/record.url?scp=84907401373&partnerID=8YFLogxK
http://www.scopus.com/inward/citedby.url?scp=84907401373&partnerID=8YFLogxK
Fingerprint Подробные сведения о темах исследования «Inflation in supergravity with a single chiral superfield». Вместе они формируют уникальный семантический отпечаток (fingerprint).
supergravity Physics & Astronomy
polynomials Physics & Astronomy
stabilization Physics & Astronomy
scalars Physics & Astronomy
vacuum Physics & Astronomy
symmetry Physics & Astronomy
energy Physics & Astronomy
Ketov, S. V., & Terada, T. (2014). Inflation in supergravity with a single chiral superfield. Physics Letters, Section B: Nuclear, Elementary Particle and High-Energy Physics, 736, 272-277. https://doi.org/10.1016/j.physletb.2014.07.036
Inflation in supergravity with a single chiral superfield. / Ketov, Sergei V.; Terada, Takahiro.
В: Physics Letters, Section B: Nuclear, Elementary Particle and High-Energy Physics, Том 736, 07.09.2014, стр. 272-277.
Ketov, SV & Terada, T 2014, 'Inflation in supergravity with a single chiral superfield', Physics Letters, Section B: Nuclear, Elementary Particle and High-Energy Physics, том. 736, стр. 272-277. https://doi.org/10.1016/j.physletb.2014.07.036
Ketov SV, Terada T. Inflation in supergravity with a single chiral superfield. Physics Letters, Section B: Nuclear, Elementary Particle and High-Energy Physics. 2014 Сент. 7;736:272-277. https://doi.org/10.1016/j.physletb.2014.07.036
Ketov, Sergei V. ; Terada, Takahiro. / Inflation in supergravity with a single chiral superfield. В: Physics Letters, Section B: Nuclear, Elementary Particle and High-Energy Physics. 2014 ; Том 736. стр. 272-277.
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N2 - We propose new supergravity models describing chaotic Linde- and Starobinsky-like inflation in terms of a single chiral superfield. The key ideas to obtain a positive vacuum energy during large field inflation are (i) stabilization of the real or imaginary partner of the inflaton by modifying a Kähler potential, and (ii) use of the crossing terms in the scalar potential originating from a polynomial superpotential. Our inflationary models are constructed by starting from the minimal Kähler potential with a shift symmetry, and are extended to the no-scale case. Our methods can be applied to more general inflationary models in supergravity with only one chiral superfield.
AB - We propose new supergravity models describing chaotic Linde- and Starobinsky-like inflation in terms of a single chiral superfield. The key ideas to obtain a positive vacuum energy during large field inflation are (i) stabilization of the real or imaginary partner of the inflaton by modifying a Kähler potential, and (ii) use of the crossing terms in the scalar potential originating from a polynomial superpotential. Our inflationary models are constructed by starting from the minimal Kähler potential with a shift symmetry, and are extended to the no-scale case. Our methods can be applied to more general inflationary models in supergravity with only one chiral superfield.


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2021 Human Rights Watch Film Festival Announces Lineup
Festivals, News
hrwff
The second virtual edition of the Human Rights Watch Film Festival is gearing up to kick off, and the full lineup of titles has been revealed.
Ten socially important feature films will comprise this year's festival, which will be streaming online from May 19 – 27.
Tickets are available for purchase now, so check out details about each film below and click over to hrwfilmfestivalstream.org for more.
Ameen Nayfeh, Palestine/Jordan/Qatar/Italy/Sweden, 2020, 96 minutes, Arabic, Hebrew, English. This film is fully subtitled in English.
Live online Q&A with filmmaker and guests on Sunday, May 23, 2pm (EDT)
U.S. Digital Festival Premiere
Mustafa and his wife Salwa are from two Palestinian West Bank villages only 200 meters apart but split by Israel's separation wall. Mustafa lives on the Palestinian-controlled side of the wall, and Salwa and their children on the Israeli side. One day Mustafa gets the call every parent dreads: his son has been in an accident and is in the hospital. Mustafa will do anything to reach him, and after being denied access through the checkpoint on a technicality, he embarks upon a journey to cross the border illegally. The 200-meter distance soon becomes a 200-kilometer odyssey. Palestinian director Ameen Nayfeh's debut drama shows the life-threatening struggles of daily life under occupation in an urgent story of resistance, dignity, family and hope.
"200 Meters illustrates how sweeping Israeli movement restrictions separate families, disrupt daily life and otherwise violate the basic rights of Palestinians in the occupied territory." — Khulood Badawi, Israel and Palestine consultant, Human Rights Watch
Winner, Audience Award, Giornate degli Autori, Venice International Film Festival 2020
A Once and Future Peace
Eric Daniel Metzgar, USA, 2021, 95 minutes, English, Spanish. Closed-captioning available in English.
Live online Q&A with filmmaker and guests on Sunday May 23, 8:30pm (EDT)
World Digital Festival Premiere
In Seattle, communities are working to break the cycle of incarceration. A promising new restorative justice program based on Indigenous peace-making circles aims to bring healing to families and communities while reforming the justice system. Using beautifully crafted animation, the film follows "Andy," a teenager facing felony charges, and his family as they work through the program shepherded by Saroeum, a former gang leader. As they look at the status of the broken justice system – prosecutors, judges, and those running the program ask: how much is our society willing to invest to truly change the trajectory of our communities for the better?
"When a young person comes into court, their attorney rightfully says 'Don't talk, I'll do all the talking for you.' When you come into a peace-making circle and this talking stick is in the hands of a 16-year-old boy, and all of the adults and everyone else in the room is leaning in to hear what he has to say, that's a powerful moment." — Daniel T. Satterberg, film participant, A Once and Future Peace
Jennifer Redfearn, USA, 2020, 85 minutes, English. Closed-captioning available in English.
Live online Q&A with filmmaker and guests on Thursday, May 20, 8:30pm (EDT)
In a midwestern U.S. state caught between the opioid epidemic and rising incarceration of women, three unforgettable mothers — Tomika, Lydia, and Amanda — prepare to rejoin their families after years of separation. With the number of women in the US prison system increasing at double the rate of men, mothers are now the fastest-growing population in the criminal justice system. With the help of a unique re-entry program run by Malika, an advocate formerly incarcerated in the same prison, the women lean on each other to find the tools and resiliency needed to rebuild their lives and relationships with their children. In Apart, viewers bear witness to how familial love and courage combat the intergenerational trauma caused by the war on drugs.
"Breaking that cycle of addiction is what I think about every single day. I want Tyler to know that he doesn't have to live how I lived, and my mom lived, and all of his aunts and uncles live. I want him to know that there is a world outside of that." — Amanda, film participant, Apart
Bajo Fuego (Under Siege)
Sjoerd Van Grootheest, Irene Vélez-Torres, Colombia, 2020, 85 minutes, Spanish. Closed-captioning available in English.
Live online Q&A with filmmaker and guests on Saturday, May 22, 8:30pm (EDT)
In November 2016, the Revolutionary Armed Forces of Colombia (FARC) and the office of President Juan Manuel Santos signed the Colombian peace deal. Many hoped this would mark an end to 52 years of armed conflict. For farmers in the coca-growing region of Cauca however, this "peace" has proven to be short-lived. Bajo Fuego follows "cocaleros" as they mobilize to protect their livelihoods after the government instructs them to destroy their crops as part of the "war on drugs." As new armed groups arise, the promised peace turns out to be an illusion for these farmers whose lives are threatened and who are displaced from their homes. Bajo Fuego exposes the lived reality behind the politics that has left many Colombians in a continued state of war.
"What was signed were just agreements, they did not sign peace. We still have to build peace." — Farmer, film participant, Bajo Fuego
Official Selection, HotDocs 2021
Daughter Of A Lost Bird — Closing Night
Brooke Pepion Swaney, USA, 2021, 66 minutes, English. Closed-captioning available in English.
Live online Q&A with filmmaker and guests on Wednesday, May 26, 8:30pm (EDT)
Kendra Mylnechuk Potter was adopted into a white family and raised with no knowledge of her Native parentage. Serving as both investigator and witness, this beautifully personal film documents Kendra on her journey as a new mother to discover her Native identity. Upon finding her birth mother April, who is also a Native adoptee, Kendra returns to her Lummi homelands in Washington State, and uncovers a wealth of emotional and spiritual beauty and pain. The film also serves an entry point into a more complicated national issue — intentional government actions to erase an entire culture, including the 1958 Indian Adoption Project, which removed Native children from their families and placed them in white homes in an effort to "kill the Indian and save the man." This poignant story provides living proof that history is not only the past, but the present too.
"I identified as white. This strange confusion of white guilt, and native anger. Where does it sit in me? And how do I sit with both of those things?" — Kendra Mylnechuk Potter, film participant, Daughter of a Lost Bird
"This story we have been telling for seven years can't be wrapped up in a neat bow, because it's such a complex experience to be Native in this country. And sometimes painful, but also beautiful, and powerful, and a million other things." — Brooke Pepion Swaney, director, Daughter of a Lost Bird
Forget Me Not — Opening Night
Olivier Bernier, USA, 2021, 100 minutes, English. Closed-captioning available in English.
As 3-year-old Emilio is ready to start school, his family finds itself cornered in the United States' most segregated education system — New York City public schools. Fighting for their son's right to an inclusive education — where Emilio and other children with disabilities would be taught alongside their classmates without disabilities — film director Olivier and his wife Hilda investigate the personal stories of students and their parents in the US. With children with disabilities worldwide less likely to attend school, these experiences expose just a handful of the widespread injustices currently taking place in the educational system and beyond for children with disabilities. Forget Me Not reveals a path to a more inclusive society that starts with welcoming diversity in the classroom.
"As a director, this is not only a story I will be telling — this is a story I am living. When I was a child, I didn't go to an inclusive school. I was never exposed to anyone with intellectual disabilities, and I was ill-prepared for my own son's arrival. I want to use this opportunity to make sure this never happens to anyone again." — Olivier Bernier, director, Forget Me Not
"What we're talking about here is the society we want to have in the future in which people with disabilities are welcome. And inclusion is the beginning of that." — Thomas Hehir, film participant, Forget Me Not
Nanfu Wang, USA/China, 2021, 95 minutes, English, Mandarin. Closed-captioning available in English.
Live online Q&A with filmmaker and guests on Monday, May 24, 8:30pm (EDT)
In The Same Breath, directed by Nanfu Wang (One Child Nation), explores the parallel campaigns of misinformation waged by the Chinese and U.S. leadership and their devastating impact on millions of lives since the beginning of the Covid-19 pandemic. In a deeply personal approach, Wang, who was born in China and now lives in the United States, shares emotional first-hand accounts and startling, on-the-ground footage captured by camerapersons in China and the U.S. as she weaves a revelatory picture of mass cover-ups and dishonesty while also highlighting the strength and resilience of the healthcare workers, activists, and family members who risked everything to communicate the truth. An HBO Documentary Film
"We all think of ourselves as capable of separating truth from falsehood. But how can we make that distinction, when misinformation comes from the people we are supposed to trust?" — Nanfu Wang, director, In The Same Breath
"It's really about the 'same breath' drawn by people across China and the U.S.—breath that isn't valued by political leaders, who lie to them and abandon them, the failures of democracy and the costs of authoritarianism." — Sophie Richardson, China director, Asia Division, Human Rights Watch
Winner, Audience Award, Festival Favorites, SXSW 2021
Official Selection, Sundance Film Festival 2021
The Return: Life After ISIS
Alba Sotorra Clua, Spain/UK, 2021, 90 minutes, English. Fully subtitled in English.
Live online Q&A with filmmaker and guests on Thursday, May 20, 5pm (EDT)
Hoda Muthana and Shamima Begum made world headlines after leaving their homes in the U.S. and UK as teenagers to join the extremist group Islamic State, also known as ISIS. The Return: Life After ISIS is a unique portrait of a group of Western women who pledged their lives to ISIS, but now want to return home to restart their lives. While facing hostile journalists and governments who have left them de-facto stateless, the women confront their truths and try to heal from their trauma in a locked camp in northeast Syria, with the help of Kurdish women's rights activists. With its rare access to Roj camp, this film is a sensitive portrayal of just a few of the 63,000 women and children held, in dire conditions with no due process, as ISIS suspects and family members in northeast Syria by a Kurdish-led armed group.
"When you are brainwashed, you don't realize it until you snap out of it. I took everything too fast, and too deep." — Hoda Muthana, film participant, The Return: Life After ISIS
"This film cuts through the stereotypes about Western women who joined ISIS and makes a compelling case for bringing them home. In the process, it reveals an unusual sisterhood between these women and a Kurdish social worker who helps them even though ISIS sought to destroy her community." — Letta Tayler, associate director, Crisis and Conflict Division, Human Rights Watch
Nominee, Grand Jury Award, SXSW 2021
Tacheles – The Heart of the Matter
Jana Matthes & Andrea Schramm, Germany, 2020, 104 minutes, German, Hebrew, English
Live online Q&A with filmmaker and guests on Saturday, May 22, 2pm (EDT)
North American Digital Festival Premiere
Three generations removed from the Holocaust, Yaar is a young Jewish Berliner desperate to leave the past behind him. Along with his friend Marcel, they develop a computer game set in 1940s Germany featuring a young Jewish girl based on Yaar's grandmother Rina, and an SS officer inspired by Marcel's ancestor, in which Jews can defend themselves and Nazis can act humanely. Yaar's father is shocked, and the work opens up old family wounds. Thus begins a painful confrontation with history that will forever change Yaar's relationships with his father and his friend: the grandchildren of victims and perpetrators. As the game ceases to be a game, Tacheles — The Heart of the Matter explores with growing self-awareness how trauma of survivors is inherited, and asks the burning question from the perspective of a 21-year-old: what does the Holocaust have to do with me?
"Tacheles is a very important film. It shows how crimes and injustice can ruin the lives not merely of the survivors but also of the following generations decades after they took place." — Wenzel Michalski, Germany director, Human Rights Watch
Ashley O'Shay, USA, 2020, 86 minutes, English. Closed-captioning available in English.
Live online Q&A with filmmaker and guests on Friday, May 21, 8:30pm (EDT)
This is a profound and necessary story ripe for a countrywide, and indeed a worldwide, reckoning with racial injustice. After two Black Chicagoans, Rekia Boyd and Laquan McDonald, are killed by police, the Movement for Black Lives demands justice and organizes to challenge an administration complicit in violence against its residents. Unapologetic introduces Janaé and Bella, two fierce activist leaders whose upbringing and experiences have shaped their view of what liberation could and should look like, as they urge an expansive view of public safety that does not depend on police. This invigorating documentary illuminates the love underpinning the anger and frustration that comes with being Black, queer women in the United States, and elevates those who are most often leading the way while being denied the spotlight.
"If Black, queer, feminist people are not free, nobody else is going to be free." — Kush, film participant, Unapologetic


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