text:
\section{Introduction}
In practice, positioning and tracking problems are of special interest to many important applications which involve the trajectory/path following of autonomous vehicles \cite{aguiar2007trajectory,BWang2021leader},  formation/containment control of multiple autonomous vehicles \cite{WANG201826,liu2018event} and so on.
It is noted that there exist two important scales in positioning and tracking: space and time. Existing researches usually focus on when to reach a designated location, or what kind of trajectory in space, but few considerations involve both of them at the same time, which inspires this work.

First of all, a concept named 'synchronization of space and time' is proposed.

\emph{Synchronization of space and time:}
 We summarize the following three characteristics for the synchronization of space and time:
\begin{enumerate}[(i)]
\item  Time and space are separated and can be controlled independently;
\item  In the time control, the specified position can be reached at the predetermined instant;
\item  In the space control, the trajectory in space is independent of time, that is, the trajectory in space is fixed regardless of the terminal time.
\end{enumerate}

In the following, two aspects are discussed.

\emph{Prescribed-time/distance control:} Finite/fixed /prescribed-time control has drawn  increasing attention in recent years in view of the fact that many practical applications require severe time response constraints.  Homogeneity property gives a useful analysis for HOSM (high-order sliding mode) control, and if an asymptotically stable system is homogeneous of a negative degree, then it is finite-time stable to a sliding manifold (see \cite{Davila13,Levant05c,Bhat05}).
However, the settling time in these finite-time methods grows unbounded when initial conditions tend to infinity, and the initial conditions are hard
to acquire in some practical situations \cite{Hu2019concurrent}.
Fixed-time stability requires a controller (observer) to provide some desired control (observation) precision at a given time, independent of the initial conditions, and  the settling time is subject to an upper bound but varies due to uncertainties and nonlinearities
 \cite{Tian17,Polyakov15}.
 The reviews upon the fixed-time stability and its applications can be found in \cite{Zuo2018overview}.
 However, the fixed-time settling time can not be preassigned arbitrarily since its upper bound is subject to certain restrictions \cite{YWang2018}.
Moreover, prescribed-time control employs a scaling of state by a function of time that grows unbounded to the terminal time and yields the regulation in prescribed finite time(see \cite{song2017time,wang2018leader}).
Furthermore, regarding the prescribed-distance control problem, the system regulation can be achieved before the prescribed distance is reached.
The conversion between the prescribed-distance control and the existing prescribed-time control only requires the distance-driven transformation for the differential dynamic system model, which will be answered in this paper.

\emph{Simultaneity in time: }The finite-time distributed consensus has been a popular topic in recent years, which is known as one of the most important issues in cooperative control\cite{cao2014finite,zhao2017adaptive,wang2018leader}. However, for some classes of practical systems, this finite- time convergence is not enough operating, for example, for a service robot hand, its joints are usually required to reach the desired angle at the same time; the proximity operations require that the pursuer perform the translational and rotational maneuvers with respect to the target simultaneously \cite{shao2020adaptive}; for the multi-missile interception active defense \cite{zhou2016distributed,CHEN2019454}, multiple missiles are expected to strike the target simultaneously. In this regard, simultaneity in time for multi-agent systems are of great importance.
The works in \cite{DLi21Ontime,DLi21Fixedtime} proposed the
time-synchronized
consensus control, focusing on how to design a controller which drives
all the elements of all the agent states to consensus at the
same time; and fixed-time-synchronized consensus, where the
upper bound of the synchronized settling time is independent of the initial states of multi-agent systems.

\emph{Contributions of this work:} The main contributions are stated as following
\begin{enumerate}[(i)]
\item  {In the framework of polar coordinates, given a reference relative trajectory, the developed distance-driven controller enables the prescribed-distance stability and achieves the separation between space and time. In comparison, few of the existing works consider the distance-driven transformation about the tracking problem. Moreover, the work in \cite{CHEN2019454} employs a different distance-driven transformation and obtains the travelling range in advance, while  the velocity value is set fixed and the target is assumed stationary;}
\item Through correlating and decoupling the reference/actual kinematics between the self vehicle and target, the control variables related to velocity and inclination angle are controlled respectively to form synchronization between the arrival time and path trajectory. {This kind of correlating and decoupling operations give the freedom to the velocity control in polar coordinates, which is proposed for the first time, to the best of the authors' knowledge};
\item  A cascaded prescribed-time control theorem is proposed which enables the prescribed-time stabilization for the general strict-feedback systems. Compared with the prescribed-time control \cite{song2017time,wang2018leader},  the proposed theorem eliminates restrictions on the system model and allows the uncertainties and the remaining-time-related interconnections in each subsystem; the cascaded prescribed-time tracking problem can be directly solved through the separately prescribed-time control design of each subsystem;
\item The synchronization of space and time control enables the simultaneous arrival of multiple vehicles which reach the designated position or achieve formation simultaneously.
{Compared with the existing simultaneous control work, the target in the simultaneous tracking/formation problem does not have to be set stationary as in \cite{zhou2016distributed,CHEN2019454}, and the simultaneous property can be directly derived from the proposed space-and-time-synchronized control.}
\end{enumerate}

\emph{Notation}:
Throughout this paper, ${\mathcal{R}}$ denotes the set of real numbers, ${\mathcal{R}}^{n\times m}$ represents the set of $n \times m$ real matrix, $\mathcal{N}_+$ is the set of positive integers.

$( \cdot )^T$ is the transpose of one matrix,
diag$( \cdot )$ represents the diagonal matrix. $\left|  \cdot  \right|$ is the absolute value of a scalar and $\left\|  \cdot  \right\|$ is defined as the Euclidean norm of a vector. Given a matrix $M$, $M>0$ (or $M<0$) means that $M$ is a positive definite (or negative definite) matrix, $\lambda_\text{min}(M)$ and $\lambda_\text{max}(M)$ denote the the minimum and maximum eigenvalue of matrix $M$, respectively.

\section{Formulation of space-and-time-synchronized problem}
As the geometry of planer scenario depicted in Fig. \ref{fig:Fig_geometry}, the subscripts $s$ and $t$ denote the self vehicle and target, respectively, the subscripts $0$ and $f$ represent the initial time and the predetermined time, respectively.  $R \in \mathcal{R}$ represents their relative distance along the line of sight (LOS) angle $q \in \mathcal{R}$. ${{V}_{s}}\in \mathcal{R}$ and ${{V}_{t}} \in \mathcal{R}$ denote the velocities of the vehicle and target, respectively, ${{\theta }_{s}} \in \mathcal{R}$ and ${{\theta }_{t}} \in \mathcal{R}$ are their corresponding path angles, ${{\eta }_{s}} \in \mathcal{R}$ and ${{\eta }_{t}} \in \mathcal{R}$ represent the heading angles. The tracking kinematics between the vehicle and target in the framework of polar coordinates are modeled as
\begin{equation}\label{Eq_2_1}
\begin{aligned}
  & \dot{R}=-{{V}_{s}}\cos {{\eta }_{s}}+{{V}_{t}}\cos {{\eta }_{t}}, \\
 & R\dot{q}={{V}_{s}}\sin {{\eta }_{s}}-{{V}_{t}}\sin {{\eta }_{t}}, \\
 & {{\eta }_{s}}=q-{{\theta }_{s}},\ {{\eta }_{t}}=q-{{\theta }_{t}}
\end{aligned}
\end{equation}
with the initial condition ${{R}}(0)=R_0$,  ${{q}}(0)=q_0$, ${{\eta}}_s(0)=\eta_{s0}$ and  ${{V}}_s(0)=V_{s0}$. When it comes to a special case $V_t=0$ which means a stationary target point, the kinematics model (\ref{Eq_2_1})  is reduced to a positioning problem.

\begin{figure}[!t]
\centerline{\includegraphics[width=0.4\textwidth]{geometry.pdf}}
\caption{Geometry description}
\label{fig:Fig_geometry}
\end{figure}

Define the pair $(t,R, q)$: at time $t$, the distance and LOS angle between the vehicle and target are $R$ and $q$, respectively. We can now formulate the problem as follows:

\begin{problem} \label{Problem_2_1}
(Synchronization of space and time) For a vehicle tracking problem (\ref{Eq_2_1}), how to ensure the vehicle starting from $(t_0,R_0, q_0)$ eventually realizes $(t_f,R_f,q_f)$ and gets maintained in $(t,R_f,q_f)$ for $t>t_f$?
\end{problem}

\begin{algorithm}
For the sake of generality, the solution with respect to Problem \ref{Problem_2_1} can be summarized as
\begin{enumerate}[(i)]
\item   In the framework of polar coordinates, introduce the reference trajectory $(R_d,q_d)$ for the vehicle-target relative motion, which achieves the separation of time and space, see Section II.A;
\item   Correlate and decouple the reference/actual relative kinematics  to obtain the separated control variables related to velocity $V_s$ and inclination angle $\eta_s$, see Section II.B;
\item   The travelling  time and path trajectory are controlled respectively to form synchronization between space and time.
\end{enumerate}
\end{algorithm}
\subsection{Reference relative tracking trajectory }
The motion equations for the reference tracking problem are given by
\begin{equation}\label{Eq_2_2}
\begin{aligned}
  & {{{\dot{R}}}_{d}}=-{{V}_{d}}\cos {{\eta }_{d}}, \\
 & {{R}_{d}}{{{\dot{q}}}_{d}}={{V}_{d}}\sin {{\eta }_{d}} \\
\end{aligned}
\end{equation}
where $R_d$ and $q_d$ denote the distance and LOS angle of the reference trajectory, respectively, $V_d$ and $\eta_d$ are the corresponding velocity and heading angle, respectively. Their initial conditions are set ${{R}_{d}}(0)=R_0$,  ${{q}_{d}}(0)=q_0$, ${{\eta}_{d}}(0)=\eta_{d0}$ and ${{V}_{d}}(0)=V_{d0}$.

Define a new distance-scaled variable ${{{r}}_{d}}$ and a prescribed value ${{{r}}_{f}}$
\begin{align} \label{Eq_2_3}
&{{{r}}_{d}}=
\ln (R_0-{R}_{f1})-\ln ({{R}_{d}}(t)-{R}_{f1}), \ \ R_d \in \left( R_f,R_0  \right]    ,  \notag \\
&{{{r}}_{f}}=\ln (R_0-{R}_{f1})
\end{align}
where ${R}_{f1}=R_f-1$. It is obvious that $r_d=r_f$ when $R_d=R_f$.
The relationship between the distance-scaled variable ${r}_d$ and the actual time $t$ is subject to

\begin{equation}\label{Eq_2_4}
{{{{\dot{r}}}_{d}}}=
{{{V}_{d}}\cos {{\eta }_{d}}}/({{{R}_{d}}-{R}_{f1}}), \ \ R_d \in \left( R_f,R_0  \right].
\end{equation}

Taking derivative with respect to the distance-scaled variable $r_d$, rather than the time variable $t$, yields the rewritten dynamics: {when $r_d \in \left[0, r_f \right)$,}
\begin{align} \label{Eq_2_5}
  & \frac{\text{d}{{q}_{d}}}{\text{d}{{{r}}_{d}}}=\frac{{{R}_{d}}-{R}_{f1}}{{{R}_{d}}}\tan {{\eta }_{d}}, \\
 & \frac{{{\text{d}}^{2}}{{q}_{d}}}{\text{d}r_{d}^{2}}=\frac{R_d-{R}_{f1}}{{{V}_{d}}{{\cos }}{{\eta }_{d}}}\left[\frac{R_d-{R}_{f1}}{R_d \cos^2{\eta_d}}\frac{\text{d}{{\eta }_{d}}}{\text{d}t}
- \frac{{R}_{f1} V_d \sin{\eta_d}}{R_d^2}
 \right]. \notag
\end{align}

\begin{theorem} \label{Theorem_1}
Consider the system in form of (\ref{Eq_2_2}).

(i)  When $R_d \in  [0, r_f  )$,  the control law of $\eta_d$ is designed according to (\ref{Eq_2_6}). The system state pair $(q_d,\eta_d)$  asymptotically converges to the desired value $(q_f,0)$ before ${{r}_{d}}$ approaching the prescribed value ${{r}_{f}}$.
\begin{align} \label{Eq_2_6}
{\dot{\eta }_{d}}=&-\frac{R_d{{V}_{d}}{{\cos }^{3}}{{\eta }_{d}}}{({{R}_{d}}-{R}_{f1})^2}\left[ \frac{k_{d2}}{{{{r}}_{f}}-{{{r}}_{d}}}\frac{\text{d}{{q}_{d}}}{\text{d}{{{r}}_{d}}}+\frac{k_{d1}({{q}_{d}}-{{q}_{f}})}{{{({{{r}}_{f}}-{{{r}}_{d}})}^{2}}}  \right]  \notag \\
& + \frac{{R}_{f1} V_d \cos^2{\eta_d}\sin{\eta_d}}{(R_d-{R}_{f1})R_d}.
\end{align}
When letting
${{k}_{d1}}=N_1N_2, \  {{k}_{d2}}=N_1+N_2-1$, $N_1<N_2$ and
$N_1, N_2 \in \mathcal{N}_+$, the travel range can be calculated as
\begin{equation}
\begin{aligned}
S_d=\int_{R_f}^{{{R}_{0}}}{\sqrt{1+{{(\tan {{\eta }_{d}})}^{2}}}}\text{d}R_d .
\end{aligned}
\end{equation}
In this sequel, $S_d$ is irrespective of $V_{d}$, the travel time $t_{f}$ is only related to the velocity $V_{d}$.
For any constant velocity ${{V}_{d}}$, the travel time is ${{t}_{f}} ={S_d}/{V_d}$.

(ii) When $R_d$ approaches $R_f$, through $\dot{V}_{d}=0$ and  $\dot{\eta}_d=0$, the system state pair $(R_d,q_d,\eta_d)$  gets maintained in the desired value $(R_f,q_f,0)$.
\end{theorem}

\begin{Proof}
See Appendix A.
\end{Proof}

\begin{remark}
Here comes the motivation of introducing the proposed distance-scaled variable $r$ and doing  the $r$-related dynamics transformation:
\begin{enumerate} [(i)]
\item The system dynamics (\ref{Eq_2_5}) with respect to the introduced distance-scaled variable $r_d$ is independent of time and directly linked to the terminal distance $R_f$. As a result, this proposed distance-related transformation is more intuitive to the distance tracking problem like the prescribed-distance control;
\item This kind of transformation (\ref{Eq_2_3}) together with the controller (\ref{Eq_2_6}) enables the achievement of regulation objective before $r_d=r_f$. The resultant irrelevance between $S_d$ and $V_{d}$ creates a separation between space and time, which provides the possibility for the space-and-time-synchronized control;
\item Such a state transformation brings $\eta_d$ into a state-related form, i.e. $\tan \eta_d=\frac{\text{d}{{q}_{d}}}{\text{d}{{{r}}_{d}}} \frac{R_d}{R_d-R_{f1}}$, such that the specific expression $\eta_d$ in space can be directly obtained in (\ref{Eq_2_11}). Some heading-angle-constrained problems can be handled through appropriate parameter selection when considering the field-of-view limits.
\end{enumerate}
\end{remark}

\begin{figure*}[htbp]
\normalsize
\begin{equation*}
\begin{aligned}
& M=\left[ \begin{matrix}
   -\cos {{\eta }_{d}} & {{V}_{s}}\left( \cos {{\eta }_{d}}\int_{0}^{1}{\sin (\tau {{\eta }_{e}})\text{d}\tau }+\sin {{\eta }_{d}}\int_{0}^{1}{\cos (\tau {{\eta }_{e}})\text{d}\tau } \right)  \\
   \sin {{\eta }_{d}} & {{V}_{s}}\left( \cos {{\eta }_{d}}\int_{0}^{1}{\cos (\tau {{\eta }_{e}})\text{d}\tau }-\sin {{\eta }_{d}}\int_{0}^{1}{\sin (\tau {{\eta }_{e}})\text{d}\tau } \right)  \\
\end{matrix} \right], \ \ G=\left[ \begin{matrix}
   {{V}_{t}}\cos {{\eta }_{t}}  \\
   -{{V}_{t}}\sin {{\eta }_{t}}-{{V}_{d}}\sin {{\eta }_{d}} ({{{R}_{e}}}/{R_d} ) \\
\end{matrix} \right].
\end{aligned}
\end{equation*}
\hrulefill
\end{figure*}

\subsection{Kinematic modelling}
To correlate the reference/actual relative kinematics, we define the following error variables, i.e., distance error ${{R}_{e}}=R-{{R}_{d}}$, LOS angle error ${{q}_{e}}=q-{{q}_{d}}$, angle ${{\eta }_{e}}={{\eta }_{s}}-{{\eta }_{d}}$, velocity error ${{V}_{e}}={{V}_{s}}-{{V}_{d}}$.

Taking time derivative of ${{R}_{e}}$ along the solutions of (\ref{Eq_2_1}) and (\ref{Eq_2_2}) leads to

\begin{align}\label{Eq_2_16}
 {{{\dot{R}}}_{e}} & =-{{V}_{s}}\cos {{\eta }_{s}}+{{V}_{s}}\cos {{\eta }_{d}}-{{V}_{s}}\cos {{\eta }_{d}}\nonumber \\
 & \ \ \ +{{V}_{d}}\cos {{\eta }_{d}}+{{V}_{t}}\cos {{\eta }_{t}} \nonumber\\
 & =-{{V}_{s}}(\cos ({{\eta }_{d}}+{{\eta }_{e}})-\cos {{\eta }_{d}})-{{V}_{e}}\cos {{\eta }_{d}}+{{V}_{t}}\cos {{\eta }_{t}} \nonumber\\
 & =-{{V}_{e}}\cos {{\eta }_{d}}+{{V}_{t}}\cos {{\eta }_{t}} \nonumber \\
 & \ \ \ +{{V}_{s}}\left[ \cos {{\eta }_{d}}\frac{1-\cos {{\eta }_{e}}}{{{\eta }_{e}}}+\sin {{\eta }_{d}}\frac{\sin {{\eta }_{e}}}{{{\eta }_{e}}} \right]{{\eta }_{e}}.
\end{align}

A similar operation for ${{q}_{e}}$ brings
\begin{align}\label{Eq_2_17}
  R{{{\dot{q}}}_{e}} =&{{{V}_{s}}\sin {{\eta }_{s}}}-{{{V}_{s}}\sin {{\eta }_{d}}}+{{{V}_{s}}\sin {{\eta }_{d}}}  -{{{V}_{d}}\sin {{\eta }_{d}}} \nonumber\\
  &  -{{{V}_{d}}\sin {{\eta }_{d}}}({R_e}/{R_d})-{{{V}_{t}}\sin {{\eta }_{t}}} \nonumber\\
   =&{{{V}_{e}}\sin {{\eta }_{d}}}+
{V_s} \left[ \sin {{\eta }_{d}}\frac{\cos {{\eta }_{e}}-1}{\eta_e}+\cos {{\eta }_{d}} \frac{\sin {{\eta }_{e}}}{\eta_e} \right]{\eta_e}
 \nonumber\\
 & -{{{V}_{d}}\sin {{\eta }_{d}}}({R_e}/{R_d})-{{{V}_{t}}\sin {{\eta }_{t}}} .
\end{align}

Invoking the fact
\begin{equation} \label{Eq_2_19a}
\begin{aligned}
& \frac{1-\cos {{\eta }_{e}}}{{\eta }_{e}}=\int_{0}^{1}{\sin (\tau {{\eta }_{e}})\text{d}\tau },  \ \ \frac{\sin {{\eta }_{e}}}{{\eta }_{e}}=\int_{0}^{1}{\cos (\tau {{\eta }_{e}})\text{d}\tau }
\end{aligned}
\end{equation}
and combing the above dynamics of ${{R}_{e}}$ and ${{q}_{e}}$, we readily obtain the kinematic model of vehicle tracking problem
\begin{equation}\label{Eq_2_19}
\begin{aligned}
\left[ \begin{matrix}
   {{{\dot{R}}}_{e}}  \\
   R{{{\dot{q}}}_{e}}  \\
\end{matrix} \right]= &M({{\eta }_{s}},{{\eta }_{e}},V_s)\left[ \begin{matrix}
   {{V}_{e}}  \\
   {{\eta }_{e}}  \\
\end{matrix} \right] +G(R_e,R_d,{{V}_{t}},{{\eta }_{t}},{{V}_{d}},{{\eta }_{d}})
\end{aligned}
\end{equation}
where the detailed expressions of $M({{\eta }_{s}},{{\eta }_{e}},V_s)$ and $G(R_e,R_d,{{V}_{t}},{{\eta }_{t}},{{V}_{d}},{{\eta }_{d}})$ are at the top of next page.

\begin{remark}
With regard to (\ref{Eq_2_19a}), a special case is when $\eta_e=0$, { through the L'Hospital rule and limit of indeterminate form, one obtains $(1-\cos \eta_e)/\eta_e=0$ and $\sin \eta_e /\eta_e=1$.}
\end{remark}

\begin{remark}
As for the parameter matrix $M({{\eta }_{s}},{{\eta }_{e}},V_s)$, its determinant is {
$\left| M({{\eta }_{s}},{{\eta }_{e}},V_s) \right|= -{{V}_{s}} (\sin \eta_e /\eta_e)<0
$},
which contributes to the full rank of $M({{\eta }_{s}},{{\eta }_{e}},V_s)$ and enables the later matrix inversion operation.
\end{remark}

\subsection{Dynamic modelling}
Define the control variable $u={{\left[ \begin{matrix}
   {{u}_{V}} & {{u}_{\theta }}  \\
\end{matrix} \right]}^{T}}$  where ${{u}_{V}}$ is the tangential acceleration and ${{u}_{\theta }}$ is the lateral acceleration.
 The vehicle's dynamic model is expressed by
\begin{equation}\label{Eq_2_21}
\begin{aligned}
  & {{{\dot{V}}}_{s}}={{u}_{V}}+d_V ,\\
 & {{{\dot{\theta }}}_{s}}={u_\theta}/{{{V}_{s}}} +d_\theta
\end{aligned}
\end{equation}
where $d_V$ and $d_\theta$ are the external disturbances.

As such, the compact system model combing (\ref{Eq_2_19}) and (\ref{Eq_2_21}) is written as
\begin{subequations}
\begin{align}
&   \left[ \begin{matrix}
   {\dot{R}_{e}}  \\
   {\dot{q}_{e}}  \\
\end{matrix} \right]=  \bar{M} \left[ \begin{matrix}
   V_s \\
    {\eta}_s   \\
\end{matrix} \right]- \bar{M} \left[ \begin{matrix}
   V_d \\
   \eta_d    \\
\end{matrix} \right]+\bar{G},   \label{Eq_3_2} \\
&
 \left[ \begin{matrix}
   {{{\dot{V}}}_{s}}  \\
   {{{\dot{{\eta }}}}_{s}}  \\
\end{matrix} \right]= {B}\left[ \begin{matrix}
   {{u}_{V}}  \\
   {{u}_{\theta }}  \\
\end{matrix} \right]+ H+ {F}  \label{Eq_3_3}
\end{align}
\end{subequations}
where
$ \bar{M}(R,{{\eta }_{s}},{{\eta }_{e}},V_s)=\text{diag}(1,1/R)M$,
$ \bar{G}(R,R_e,R_d,{{V}_{t}},{{\eta }_{t}},{{V}_{d}},{{\eta }_{d}})=\text{diag}(1,1/R)G$ ,
$ B(V_s)=\text{diag}(1,-1/V_s)$ ,
$  H(\dot{q},\dot{R},R,\eta_s)=\left[ \begin{matrix}
   0   &
   \dot{q}
\end{matrix} \right]^T,  {F}(d_V,d_\theta)= \left[ \begin{matrix}
   d_V   &
   -d_\theta
\end{matrix} \right]^T$ .

\section{Prescribed-time Control Design}
For the prescribed-time control design, we reach the following consensus:
as long as $R_e$ and $q_e$ converge to 0 before the final time $t_f$ (before $R$ converges to ${R}_{f}$), the vehicle $(R,q)$ reaches the specified state $(R_f,q_f)$ at the time $t_f$.

\begin{problem} \label{Problem_2}
(Prescribed-time stabilization) For some kind of strict feedback systems like system (\ref{Eq_3_2})(\ref{Eq_3_3}), how to ensure the system stability within the prescribed time $t \in \left[ 0, t_f \right)$ ?
\end{problem}
\begin{algorithm}
The corresponding solution with respect to Problem \ref{Problem_2} is summarized as
 \begin{enumerate}[(i)]
\item   The prescribed-time stability of a single first-order system can be achieved by employing a scaling of the state by a function of $t_f-t$;
\item   During the backstepping design procedure, the above first-order prescribed-time control design is repeated for each subsystem. In this sequel, a cascaded system consisting of two prescribed-time stable subsystems is obtained;
\item   The prescribed-time stability of the cascaded system is addressed and extended to cascade cases of multiple subsystems.
\end{enumerate}
\end{algorithm}
\subsection{Prescribed-time stabilization of cascaded systems}
Before processing the prescribed-time control, a cascaded prescribed-time theorem is presented as a preliminary.
Cascades-based control essentially uses the designed control law to make the closed-loop system in a cascaded structure, which usually has the advantage of reducing the complexity of the controller and the difficulty of stability analysis.

Consider a kind of cascaded system described by
\begin{equation}  \label{EQ_Cascaded_N}
\begin{aligned}
 & {{\Sigma }_{i}}:\dot{x}_i(t)={{f}_{i}}(t,x_i,{{\Delta }_{x,i}})+g_i(t,x_i,x_{i+1})x_{i+1}, \\
  & {{\Sigma }_{n}}:\dot{x}_n(t)={{f}_{n}}(t,x_n,{{\Delta }_{x,n}}),  i=1,\cdots,n-1
\end{aligned}
\end{equation}
where $x_i(t) \in {\mathcal{R}^{n_i }}$ is the system state of subsystem ${{\Sigma }_{i}}$. ${{f}_{i}}(t,x_i,{{\Delta }_{x,i}})$, $g(t,x_i,x_{i+1})$ are continuous in their arguments, and locally Lipschitz in $x_i$, $(x_i,x_{i+1})$ respectively. For a simplified presentation, we view the cascaded system as the subsystem ${{\hat{\Sigma }}_{i}}$ perturbed by the output of subsystem ${{\Sigma }_{i+1}}$.
\begin{equation}\label{Eq_3_5}
{{\hat{\Sigma }}_{i}}:\dot{x}_i(t)={{f}_{i}}(t,x_1,{{\Delta }_{x,i}}).
\end{equation}

\begin{definition}
\cite{Panteley01} A continuous function $\alpha :\left[ 0,a \right)\to \left[ 0,\infty  \right)$ belongs to class $\mathcal{K}$ if it is increasing and $\alpha (0)=0$, {if moreover $\alpha (a)\to \infty$ as $a\to \infty $, it belongs to class $\mathcal{K}_\infty$.} A continuous function $\beta :\left[ 0,a \right)\to \left[ 0,\infty  \right)$ belongs to class $\mathcal{L}$ if it is decreasing and $\beta (s)\to 0$ as $s\to \infty $.
Again, a continuous function  $\gamma :\left[ 0,a \right)\times \left[ 0,a \right)\to \left[ 0,\infty  \right)$ belongs to class $\mathcal{KL}$ if the mapping $\gamma (r,s)$ belongs to class $\mathcal{K}$ with respect to $r$ when fixed $s$, and the mapping $\gamma (r,s)$ decreases along $s$, i.e., $\gamma (r,s)\to 0$ as $s\to \infty $, when fixed $r$.
\end{definition}

\begin{definition} \label{definition_2}
Consider the system $\dot{x}=f(t,x)$, where $f(t,x)$ is piecewise continuous in $t$ and locally Lipschitz in $x$.  The system is globally uniformly prescribed-time stable (GUPTS), if there exist a class $\mathcal{KL}$ function $\gamma (\cdot ,\cdot )$ and a settling-time $t_f$ such that for any initial values $x({{t}_{0}})$:
\begin{equation}
\left\| x(t) \right\|\le \gamma (\left\| x({{t}_{0}}) \right\|,\tau(t,t_f) ),\ \ \ {{t}_{0}}\le t< {{t}_{f}}
\end{equation}
where $\tau(t,t_f) \to \infty$ as $t \to t_f$, from which one obtains the strictly prescribed-time convergence of $x$: $x(t)\to 0$ as $t\to {{t}_{f}}$.
\end{definition}

\begin{theorem} \label{Theorem_Cascade_N}
Consider the cascaded system   (\ref{EQ_Cascaded_N}) with $n$ subsystems.
The globally uniformly prescribed-time stability (GUPTS) of the subsystem $\Sigma_i$, $i=1,\cdots,n$ can be obtained, if the following assumptions hold:

(i) Assumption on subsystem  ${{\hat{\Sigma }}_{i}}$, $i=1,\cdots, n$: the subsystem ${{\hat{\Sigma }}_{i}}$ is GUPTS, and there exists a continuously differentiable Lyapunov function ${{V}_{i}}(t,x_i)$ such that
\begin{align} \label{assumption_cascaded}
  & {{\alpha }_{i1}}(\tau){{\left\| x_i \right\|}^{2}}\le {{V}_{i}}(t,x_i)\le {{\alpha }_{i2}}(\tau){{\left\| x_i \right\|}^{2}},  \notag \\
 & {{{\dot{V}}}_{i}}={\partial {{V}_{i}}}/{\partial t}\;+({\partial {{V}_{i}}}/{\partial x_i}\;){{f}_{i}}(t,x_i,{{\Delta }_{x,i}}) \notag \\
 & \ \ \ \ \le -{{\alpha }_{i3}}(\tau){{\left\| x_i \right\|}^{2}}+{{b}_{i}}({{\alpha }^2_{i4}}(\tau)/\alpha_{i3}(\tau)){{\left\| {{\Delta }_{x,i}} \right\|}^{2}},\notag  \\
 & \left\| {\partial {{V}_{i}}}/{\partial x_i}\; \right\|\le {{\alpha }_{i4}}(\tau)\left\| x_i \right\|,\notag \\
 & {{\alpha }_{i3}}(\tau)={{\alpha }_{i4}}(\tau)\times {{\alpha }_{i5}}(\tau)
\end{align}
where ${\alpha}_{ij}(\cdot)$, $j=1,\ldots,5$ are {class $\mathcal{K}_\infty$ functions}, $b_i>0$.

(ii) Assumption on interconnection $g_i(t,x_{i},x_{i+1})$, $i=1,\cdots, n-1$:   there exist a class $\mathcal{K}_\infty $ function  $\alpha_{i6}(\cdot)$ and  a positive constant $c_{g,i}$ satisfying
\begin{equation} \label{assumption_g}
\begin{aligned}
&\left\| ({\partial {{V}_{i}}}/{\partial x_i}\; ) g_i(t,x_{i},x_{i+1}) \right\| \le {{\alpha }_{i6}}(\tau)c_{g,i}\left\| x_i \right\|, \\
& {{\lim }_{t\to {{t}_{f}}}} {{{\alpha }_{i6}} (\tau)/ (\alpha_{i3}(\tau) \alpha_{i+1,5}(\tau))}=0.
\end{aligned}
\end{equation}
\end{theorem}

\begin{Proof}
See Appendix B.
\end{Proof}

Here we give an example which has been introduced in \cite{song2017time} for justification of assumption (\ref{assumption_cascaded}).
Consider the first-order system governed by
$\dot{x}=b(x,t)u+f(x,t)$
where $x,u \in \mathcal{R}$ and $0<\underline{b}\le \left| b(x,t) \right|$, $\left| f(x,t) \right|\le d(t)\psi (x)$. Employing the residual-time-based scaling function $\mu (t-{{t}_{0}})=\frac{{{T}^{1+m}}}{{{(T+{{t}_{0}}-t)}^{1+m}}},t\in [{{t}_{0}},{{t}_{0}}+T]$ with positive integer $m$, the controller was derived
$u=-\frac{1}{{\underline{b}}}(k+\lambda \psi {{(x)}^{2}}+\frac{1+m}{T})\mu (t-{{t}_{0}})x$ with $k, \lambda>0$.
Following the Lyapunov function candidate $V={{{\omega }^{2}}}/{2}\;$, $\omega =\mu (t-{{t}_{0}})x$ and the presentation in \cite{song2017time}, upon applying the control law, one obtains
$\dot{V}\le -2k\mu V+\frac{\mu }{4\lambda }{{d}^{2}}$.
In accordance with the proposed assumption, it is obvious
${{\alpha }_{i1}}(\tau )={{a}_{1}}{{\mu }^{2}}$$(0<{{a}_{1}}<0.5)$, ${{\alpha }_{i2}}(\tau )={{a}_{2}}{{\mu }^{2}}$$({{a}_{2}}>0.5)$,
${{\alpha }_{i3}}(\tau )=k{{\mu }^{3}}$, ${{\alpha }_{i4}}(\tau )=\mu^2 $, ${{\alpha }_{i5}}(\tau )=k{{\mu }}$, ${{\alpha }_{i6}}(\tau )=\mu$, ${{b}_{i}}={1}/{(4\lambda }\;)$,
all of which satisfy the condition in the applied assumption.

\begin{remark} \label{Remark_strict}
A strict feedback form of the cascaded system (\ref{EQ_Cascaded_N})  can be obtained through the backstepping procedure
\begin{equation}
\begin{aligned}
 & {{\Sigma }_{i}}:\dot{x}_i(t)=-k_i(\tau)x_i+g_i(t,x_i,x_{i+1})x_{i+1}+{{\Delta }_{x,i}}, \\
  & {{\Sigma }_{n}}:\dot{x}_n(t)=-k_n(\tau)x_n+{{\Delta }_{x,n}},  i=1,\cdots,n-1
\end{aligned}
\end{equation}
where ${k}_{i}(\cdot)$, $i=1,\ldots,n$ are class $\mathcal{K}_\infty$ functions.
{ In principle, each subsystem can achieve the prescribed-time convergence due to the high-gain nature of $k_i(\tau)$ at the prescribed time $t_f$. Under the premise of (\ref{assumption_g}), since the convergence of $\Sigma_i$ is influenced by $\alpha_{i4} \Vert x_{i}\Vert \Vert x_{i+1} \Vert$ or $\alpha_{i4} \Vert x_{i}\Vert ^2 \Vert x_{i+1} \Vert$, an ideal manner would be for $x_{i+1}$ to converge to zero before $\alpha_{i4} $ approaches infinity, which inspires the design $k_{i+1}(\tau)>k_{i}(\tau)$.}
\end{remark}

\begin{remark}
In the high-order prescribed-time control proposed
in \cite{song2017time}, the system model is in a standard integral form
which limits that no uncertainties occur in the  $x_i$'s dynamics. The reason lies in the repeated differentiation of the time-scaling transformation $\mu(t)x_1$ where $\mu(t)$ is a time-varying function which goes to infinity when $t$ approaches $t_f$, as the form.
 However, the proposed cascaded prescribed-time control scheme in Theorem \ref{Theorem_Cascade_N} has the distinguished features:
\begin{enumerate} [(i)]
\item  Eliminate restrictions on the system model, and allow the uncertainties in $x_i$' dynamics;
\item {Allow the remaining-time ($t_f-t$)-related interconnection $g_i(t,x_{i},x_{i+1})$ appear in $x_i$' dynamics; }
\item Each subsystem can also be individually designed for its prescribed-time stability performance.
\end{enumerate}
\end{remark}
\subsection{Prescribed-time Controller design}
For the sake of clarity, the following prescribed-time control design borrows from the techniques of backstepping and the overall dynamics are organized into a cascaded form.
Invoking the definition of $\tau(t,t_f)$ in Definition \ref{definition_2}, let a user-defined $\tau(t,t_f)$ satisfy $\tau(0,t_f)=1$.
Introduce now a time-varying scaling function of $t_f-t$ with
${{h}}>0$
{
\begin{equation}\label{Eq_3_9}
{{\mu }}(t,h)=\left\{ \begin{matrix}
  \tau^h(t,t_f), \ \ \ t\in \left[ 0,{{t}_{f}} \right)   \\
   1, \ \ \ \ \ \ \ \ \ \ \ \ t\in \left[ {{t}_{f}},\infty  \right)    \\
\end{matrix} \right.
\end{equation}
whose derivative with respect to $t$ yields
\begin{equation}
\begin{aligned}
& {\dot{\mu}}(t,h) =\left\{ \begin{matrix}
 h{{\mu }}(t,h)(\dot{\tau}/\tau), \ \ t\in \left[ 0,{{t}_{f}} \right)   \\
   0,   \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ t\in \left[ {{t}_{f}},\infty  \right)    \\
\end{matrix} \right. \\
& \dot{\tau}(t,t_f)=0, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  t\in \left[ {{t}_{f}},\infty  \right)
\end{aligned}
\end{equation}
For simplicity, define $\mu_1(t)=\mu(t,h_1)$, $\mu_2(t)=\mu(t,h_2)$ with $h_1, h_2>0$.}

{
\begin{theorem} \label{Theorem_Kinematics}
Consider the kinematic model (\ref{Eq_3_2}) (\ref{Eq_3_3}). Assume there exists a positive $\sigma $ satisfying ${{ {F}}^{T}} {F} \le  \sigma^2$, $\sigma>0$ due to the boundedness of $d_V$ and $d_{\theta}$. Choose the controller
\begin{subequations}
\begin{align}
& \left[ \begin{matrix}
   \alpha_{V}^{*}  \\
   \alpha_{\eta}^{*}  \\
\end{matrix} \right]=  -({{k}_{1}} \mu_1 + \frac{h_1 \dot{\tau}} {\tau}) \left[ \begin{matrix}
   {{R}_{e}}  \\
   R{{q}_{e}}  \\
\end{matrix} \right]-{G}  +{M} \left[ \begin{matrix}
   {{V}_{d}}  \\
   {{\eta}_{d} }  \\
\end{matrix} \right],  \\
& \left[ \begin{matrix}
   {V}_{s}^{*}  \\
   {\eta }_{s}^{*}  \\
\end{matrix} \right]=\left\{ \begin{matrix}
{{M}^{-1}} \left[ \begin{matrix}
   \alpha_{V}^{*}  &
   \alpha_{\eta}^{*}  \\
\end{matrix} \right]^T, \ \ \ t  \in \left[ 0,t_f  \right)   \\
f_{M} (\left[ \begin{matrix}
   \alpha_{V}^{*}  &
   \alpha_{\eta}^{*}  \\
\end{matrix} \right]), \ \ \ \ \ \ t \in \left[ t_f, \infty \right)    \\
\end{matrix} \right. \label{Eq_3_10}
\end{align}
\begin{align}
& \left[ \begin{matrix}
   {{u}_{V}}  \\
   {{u}_{\theta }}  \\
\end{matrix} \right]={{{B}}^{-1}} \left\{ -(({{k}_{2}}+{{\lambda }_{2}}){{\mu }_{2}}+\frac{h_2\dot{\tau}}{\tau})\left[ \begin{matrix}
   {{e}_{V}}  \\
   {{e}_{\eta }}  \\
\end{matrix} \right]-\left[ \begin{matrix}
   \dot{V}_{s}^{*}  \\
   \dot{{\eta }}_{s}^{*}  \\
\end{matrix} \right]-H\right\} \label{Eq_3_16}
\end{align}
\end{subequations}
where ${{k}_{1}}>0$, ${{k}_{2}}>0,{{\lambda }_{2}}>0$ and the function
\[ f_{M} (\left[ \begin{matrix}
   \alpha_{V}^{*}  &
   \alpha_{\eta}^{*}  \\
\end{matrix} \right]) =\left[ \begin{matrix}
\Vert \left[ \begin{matrix}
   \alpha_{V}^{*}  &
   \alpha_{\eta}^{*}  \\
\end{matrix} \right]\Vert    \\
 \text{atan}(-\alpha_{V}^{*}/\alpha_{\eta}^{*})    \\
\end{matrix} \right]. \]
 Define the tracking errors  ${{e}_{V}}={{{V}}_{s}}- {V}_{s}^{*}$, ${{e}_{\eta }}={{{\eta }}_{s}}-{\eta }_{s}^{*}$. Consequently, the closed-loop system  is prescribed-time stable, which means, $({{R}_{e}},{{q}_{e}},e_{V},e_{\eta})$ converges to the origin before $t=t_f$. Furthermore, one obtains
\begin{enumerate}[(i)]
\item when $R_f=0$, the choices of $\mu_1(t)$ and $\mu_2(t)$ should satisfy ${{\lim }_{t\to {{t}_{f}}}}  (t_f-t){ \mu_1 \mu_2}=\infty$;
\item when $R_f>0$,  the formation gets maintained in $(R_f,q_f)$ through $V_{df}=0$, $\dot{\eta}_{d}=0$ and $\dot{V}_{d}=0$ for $t \ge t_f$.
\end{enumerate}
\end{theorem}}

\begin{Proof}
The proof is divided into four step:

\emph{Step 1.} Prescribed-time convergences of $R_e$ and $q_e$ under $V_s^*$ and $\eta_s^*$ in (\ref{Eq_3_10}) before $t=t_f$.

{
With the controller (\ref{Eq_3_10}), the error dynamics is obtained as
\begin{equation}\label{Eq_3_11}
 \left[ \begin{matrix}
   {\dot{R}_{e}}  \\
   {\dot{q}_{e}}  \\
\end{matrix} \right]=-({{k}_{1}} \mu_1 + \frac{h_1 \dot{\tau}} {\tau}) ) \left[ \begin{matrix}
   {{R}_{e}}  \\
   {{q}_{e}}  \\
\end{matrix} \right].
\end{equation}
Consider the Lyapunov candidate ${{V}_{1}}(r)=x_{1}^{T}{{x}_{1}}$ and ${{W}_{1}}(r)=\mu _{1}^{2}x_{1}^{T}{{x}_{1}}=\omega _{1}^{T}{{\omega }_{1}}$ with ${{x}_{1}}={{\left[ \begin{matrix}
   {{R}_{e}} & {{q}_{e}}  \\
\end{matrix} \right]}^{T}}$ and ${{\omega }_{1}}={{\mu }_{1}}{{\left[ \begin{matrix}
   {{R}_{e}} & {{q}_{e}}  \\
\end{matrix} \right]}^{T}}$. The derivative of ${{W}_{1}}(r)$ along (\ref{Eq_3_11}) is
\begin{equation}\label{Eq_3_12}
\begin{aligned}
 {\dot{W}_{1}}(t) & =2\omega _{1}^{T}\left[   {\dot{\mu }_{1}}  {{x}_{1}}-({{k}_{1}} \mu_1 + {\dot{\tau}}/{\tau}){{\mu }_{1}}{{\omega }_{1}} \right] \\
 & =2\omega _{1}^{T}\left[  h_1(\dot{\tau}/\tau)  {{\omega}_{1}}-({{k}_{1}} \mu_1 + h_1 {\dot{\tau}}/{\tau}) {{\omega }_{1}} \right] \\
 &\le -2{{k}_{1}}{{\mu }_{1}}{{W}_{1}}.
 \end{aligned}
\end{equation}}
Solving this differential inequality  and invoking the fact $\int_0^t{\mu_1(\nu)\text{d}\nu}=-\phi_1(t)$ give
$ \mu _{1}^{2}(t){{V}_{1}}(t)<{{\exp }^{2{{k}_{1}}{{\phi }_{1}}}}{{W}_{1}}(0)$
which yields
\begin{align} \label{Eq_3_10a}
& \left\| \left[ \begin{matrix}
   {{R}_{e}}  \\ {{q}_{e}}  \\
\end{matrix} \right] \right\|\le \frac{1}{{{\mu }_{1}}(t)}{{\exp }^{{{k}_{1}}{{\phi }_{1}}}}(R_{e}^{2}(0)+q_{e}^{2}(0)).
\end{align}
where ${{\phi }_{1}}(t)=\frac{{{{t}}_{f}}}{{{h}_{1}}-1}(1-\mu _{1}^{{({{h}_{1}}-1)}/{{{h}_{1}}}\;})$. Thus, the origin of system (\ref{Eq_3_11}) is globally prescribed-time stable within the prescribed time ${{t}_{f}}$.

\emph{Step 2.} Prescribed-time convergences of $ {V}_s$ to $ {V}_{s}^{*}$ and $ {\eta}_s$ to ${\eta}_{s}^{*}$ under the proposed $u_V$ and $u_\theta$ in (\ref{Eq_3_16}) before $t=t_f$.

{
With the controller (\ref{Eq_3_16}), the dynamic model of ${{e}_{V}}$, ${{e}_{\eta }}$ is obtained as
\begin{equation}\label{Eq_3_18}
 \left[ \begin{matrix}
   {\dot{e}_{V}}  \\
   {\dot{e}_{\eta }}  \\
\end{matrix} \right]=-(({{k}_{2}}+{{\lambda }_{2}}){{\mu }_{2}}+ \frac{h_2 \dot{\tau}}{\tau}) \left[ \begin{matrix}
   {{e}_{V}}  \\
   {{e}_{\eta }}  \\
\end{matrix} \right]+ {F}.
\end{equation}
Consider the Lyapunov candidate ${{V}_{2}}(t)=x_{2}^{T}{{x}_{2}}$ and ${{W}_{2}}(t)=\mu _{2}^{2}x_{2}^{T}{{x}_{2}}=\omega _{2}^{T}{{\omega }_{2}}$ with ${{x}_{2}}={{\left[ \begin{matrix}
   {{e}_{V}} & {{e}_{\eta }}  \\
\end{matrix} \right]}^{T}}$ and ${{\omega }_{2}}={{\mu }_{2}}{{\left[ \begin{matrix}
   {{e}_{V}} & {{e}_{\eta }}  \\
\end{matrix} \right]}^{T}}$.}
By applying Young's inequality with ${{\lambda }_{2}}>0$, one obtains
$\omega _{2}^{T}{{\mu }_{2}}  {F}\le {{\lambda }_{2}}{{\mu }_{2}}\omega _{2}^{T}{{\omega }_{2}}+ {{{\mu }_{2}}{{{  {F}}}^{T}} {F}}/({4{{\lambda }_{2}}})$.
In this way, the derivative of ${{W}_{2}}(t)$ with respect to the variable $t$ yields
\begin{align} \label{Eq_3_20}
{\dot{W}_{2}}(t)&=2\omega _{2}^{T}\left[
\dot{\mu}_2{{x}_{2}}-(({{k}_{2}}+{{\lambda }_{2}}){{\mu }_{2}}+h_2\dot{\tau}/\tau){{\omega }_{2}}+{{\mu }_{2}}  {F} \right] \notag \\
 & \le -2{{k}_{2}}{{\mu }_{2}}{{W}_{2}}+ {{{\mu }_{2}}{{{  {F}}}^{T}} {F}}/({2{{\lambda }_{2}}}).
\end{align}

In view of ${{ {F}}^{T}} {F} \le  \sigma^2$ and invoking Lemma 1 in \cite{song2017time}, it is direct to deduce
$\mu_2^2(t){{V}_{2}}(t)< \left({{\exp }^{2{{k}_{2}}{{\phi }_{2}}}}{{W}_{2}}(0)+ {\sigma^2}/({4k_2 \lambda_2})\right)
$
which yields
\begin{align}
\left\| \left[ \begin{matrix}
   {{e}_{V}} \\  {{e}_{\eta}}  \\
\end{matrix} \right] \right\|\le \frac{1}{{{\mu }_{2}}}\left( {{\exp }^{{{k}_{2}}{{\phi }_{2}}}}(e_{V}^{2}(0)+e_{\theta}^{2}(0))+ \frac{\sigma}{2\sqrt{k_2 \lambda_2 }}\right) \label{Eq_3_17}
\end{align}
where ${{\phi }_{2}}(t)=\frac{{{{t}}_{f}}}{{{h}_{2}}-1}(1-\mu _{2}^{{({{h}_{2}}-1)}/{{{h}_{2}}}\;})$. Thus, it follows the prescribed-time stability of $(e_V,e_{\eta})$.

\emph{Step 3:} Prescribed-time convergence of the cascaded system.

{Combining the two system models into a cascaded form, we obtain
\begin{subequations}\label{Eq_3_26}
\begin{align}
  &  \left[ \begin{matrix}
   {\dot{R}_{e}}  \\
   {\dot{q}_{e}}  \\
\end{matrix} \right]=-({{k}_{1}} \mu_1 + \frac{h_1 \dot{\tau}} {\tau}) ) \left[ \begin{matrix}
   {{R}_{e}}  \\
   {{q}_{e}}  \\
\end{matrix} \right]+\bar{M}\left[ \begin{matrix}
   {{e}_{V}}  \\
   {{e}_{\eta }}  \\
\end{matrix} \right], \\
 &  \left[ \begin{matrix}
   {\dot{e}_{V}}  \\
   {\dot{e}_{\eta }}  \\
\end{matrix} \right]=-(({{k}_{2}}+{{\lambda }_{2}}){{\mu }_{2}}+ \frac{h_2 \dot{\tau}}{\tau}) \left[ \begin{matrix}
   {{e}_{V}}  \\
   {{e}_{\eta }}  \\
\end{matrix} \right]+ {F}.
\end{align}
\end{subequations}
\begin{enumerate}[(i)]
\item Under the case $R_f>0$,  it is obvious that $\Vert \bar{M} \Vert$ is bounded. In addition, $\alpha_{13}(\tau)=2k_1\mu_1^3$, $\alpha_{14}(\tau)=2\mu_1^2$,  $\alpha_{15}(\tau)=k_1 \mu_1$; $\alpha_{23}(\tau)=2k_2 \mu_2^3$, $\alpha_{24}(\tau)=2\mu^2_2$, $\alpha_{25}(\tau)=k_2 \mu_2$.
Invoking the proposed Theorem \ref{Theorem_Cascade_N}, the cascaded system is globally prescribed-time stable in the sense that the reference trajectory gets tracked before $t_f$;
\item Under the case $R_f=0$,  it is obvious that $\Vert \bar{M} \Vert$ is  related to $1/(t_f-t)$ such that $\alpha_{16}(\tau)=\mu^2_1 /(t_f-t)$.
Invoking the proposed Theorem \ref{Theorem_Cascade_N}, the cascaded system is globally prescribed-time stable if only  ${{\lim }_{t\to {{t}_{f}}}} \frac{ 1/(t_f-t)}{ \mu_1 \mu_2}=0$.
\end{enumerate}}

{\emph{Step 4:} Formation keeping over $[t_f, \infty)$ when $R_f>0$.}

{
When $t \in [t_f, \infty)$, $\mu_1=\mu_2=1$, $\dot{\tau}=0$ such that the cascaded system is rewritten into
\begin{subequations}
\begin{align}
  &  \left[ \begin{matrix}
   {\dot{R}_{e}}  \\
   {\dot{q}_{e}}  \\
\end{matrix} \right]=-{{k}_{1}} \left[ \begin{matrix}
   {{R}_{e}}  \\
   {{q}_{e}}  \\
\end{matrix} \right]+\bar{M}_m \left[ \begin{matrix}
   {{e}_{V}}  \\
   {{e}_{\eta }}  \\
\end{matrix} \right], \\
 &  \left[ \begin{matrix}
   {\dot{e}_{V}}  \\
   {\dot{e}_{\eta }}  \\
\end{matrix} \right]=-({{k}_{2}}+{{\lambda }_{2}}) \left[ \begin{matrix}
   {{e}_{V}}  \\
   {{e}_{\eta }}  \\
\end{matrix} \right]+ {F}
\end{align}
\end{subequations}
where $V_d(t)=0$, $\eta_d(t)=0$, $t \ge t_f$ and
$ \bar{M}_m=\text{diag}(1,1/R)  {M}(R,{{\eta }_{s}},e_\eta,V_s)$.
Define the Lyapunov candidate $V=x_1^T x_1+ x_2^T x_2$ whose time derivative yields
\begin{equation}
\dot{V} \le -(2k_1 -\Vert \bar{M}_m \Vert)x_1^T x_1 -(2k_2 -\Vert \bar{M}_m \Vert) x_2^T x_2
\end{equation}
where $k_1, k_2 >0.5 \Vert \bar{M}_m \Vert$.
The corresponding stabilization is obvious established according to the above proof.}
\end{Proof}

\begin{remark}
It is obvious that the larger $k_i$, $h_i$ ($i=1,2$) are, the faster $R_e$, $q_e$ converge. Invoking Remark \ref{Remark_strict},  it is better to design $k_2 \mu_2 >k_1 \mu_1$  to make the states of different subsystems converge in a reasonable order. { In addition,
the high-order scaling of state by a function of time in \cite{song2017time} requires the parameter $h$ to be positive integers greater than 1. Our results do not require such limitations.}
\end{remark}

\section{Extensions to simultaneous tracking/formation of multi-vehicles}
For the multi-agent tracking/formation scenario, a distinguishing requirement is the simultaneous arrival. Suppose that $n$ vehicles, $M_1, \cdots, M_n$, participates in a tracking/formation with respect to a single target. In the following, the variable $X_i, 1 \le i \le n$ stands for the corresponding $X$ of the $i$-th vehicle. With the above design and analysis, each vehicle realizes the prescribed-time stabilization, that is, it realizes the tracking of $R_i$ to $R_{d,i}$ and $q_i$ to $q_{d,i}$ before the terminal time $t_{f,i}$.

According to the proposed reference relative tracking trajectory theorem in Section II.A, for the simultaneous tracking/formation with $t_{f,i}=T_d$, the sufficient condition is
$\int_{0}^{T_d}{{{V}_{d,i}}(t)\text{d}t}=S_{d,i}$, $i=1,\cdots,n$.

Assume a centralized coordination strategy exists between vehicles and the desired terminal time can be determined at the initial time. Thus, it is a direct solution to allocate the velocity reasonably to obtain simultaneous arrival. Below is an adjustment strategy which can achieve the smooth arrival to the terminal situation.

Construct a $m$-order polynomial function to describe the law of ${{V}_{d,i}}(t)$
\begin{equation}\label{Eq_4_4}
{{V}_{d,i}}(t)= I_1 t^m+I_2 t^{m-1}+\cdots+I_{m} t+I_{m+1}
\end{equation}
which satisfies $m+1$ equality constraints, including the following 3 necessary conditions
\[{{V}_{d,i}}(0)=V_{d0,i} ,  {{V}_{d,i}}(T_d)=V_{df,i}, \int_0^{T_d}{{{V}_{d,i}}}(\tau)\text{d}\tau =S_{d,i}. \]

\begin{figure}[t]
\centerline{\includegraphics[width=0.47\textwidth]{whole_movingtarget.pdf}}
\caption{Trajectories of four vehicles}
\label{fig:Fig_whole_movingtarget}
\end{figure}

\begin{figure}[tbp]
\centerline{\includegraphics[width=0.5\textwidth]{vehicle_movingtarget.pdf}}
\caption{States of four vehicles}
\label{fig:Fig_vehicle_movingtarget}
\end{figure}

\section{Numerical examples}

\begin{table}[bp] \footnotesize
 \newcommand{\tabincell}[2]{\begin{tabular}{@{}#1@{}}#2\end{tabular}}
 \caption{Scenarios for simultaneous formation.}
 \label{Table_5_1}
 \centering
 \setlength{\tabcolsep}{0.2mm}{
 \begin{tabular}{ccccccc}%
  \toprule

  \multirow{3}{*}{}
  &  \multicolumn{3}{c@{}@{}@{}}{Initial condition} & \multirow{3}{*}{}
  & \multicolumn{2}{@{}c@{}}{Terminal condition} \\[2.5pt]
  \cmidrule{2-4} \cmidrule{6-7}

  & Distance $R_0$ & LOS $q_0$ &Path angle $\theta_0$ & & Distance $R_f$  & LOS $q_f$  \\
  \midrule

  Vehicle 1 & 7000 m  &  220 deg &  190 deg  &   & 1000 m & 230 deg \\
  Vehicle 2 & 6000 m  &  -60 deg &  -10 deg  &   & 1000 m & -40 deg \\
  Vehicle 3 & 5000 m  &  30 deg  &  0   deg  &   & 1000 m & 50 deg \\
  Vehicle 4 & 5000 m  &  150 deg &  180 deg  &   & 1000 m & 140 deg \\
  \bottomrule

 \end{tabular}}
\end{table}

Suppose that four vehicles track a single target with the initial and terminal condition shown in Table \ref{Table_5_1}. Each vehicle has the same initial velocity, 50 m/s.

\begin{figure}[t]
\centerline{\includegraphics[width=0.5\textwidth]{error_movingtarget.pdf}}
\caption{Tracking errors of four vehicles}
\label{fig:Fig_error_movingtarget}
\end{figure}

\begin{figure}[t]
\centerline{\includegraphics[width=0.5\textwidth]{control_movingtarget.pdf}}
\caption{Control input of four vehicles}
\label{fig:Fig_control_movingtarget}
\end{figure}

For a target moving at a speed of $V_t=100$m/s in a 45-degree direction, the scenario at $T_d=t_{f,i}=20s$ expects the four vehicles to be at a 90-degree angle to each other and a distance of 1000m from the target, meanwhile, it requires  $V_{df,i}=0$m/s for the gentle transition to the phase of formation keeping $t>T_d$.

In accordance with (\ref{Eq_4_4}), use a third-order polynomial for velocity planning. { The function $\tau(t,t_f)$ is chosen in form of $\tau(t,t_f)=1+\text{ln}(\frac{t_f}{t_f-t})$.} The control parameters are as follows:
$N_1=6$, $N_2=10$, $k_1=1$, $k_2=2$, $h_1=1$, $h_2=2$ and $\lambda_2=1$.
The corresponding results for Case 2 can be found in Fig. \ref{fig:Fig_whole_movingtarget}-\ref{fig:Fig_control_movingtarget}. {From the curves of $R_i$ and $q_i$ in Fig. \ref{fig:Fig_whole_movingtarget},
the simultaneous formation and formation keeping are obvious achieved at $T_d$, and the transmission process to formation keeping is
smooth due to the small value of the desired relative velocity $V_{df,i}$.}
The prescribed-time convergence of tracking errors can be obtained in Fig. \ref{fig:Fig_error_movingtarget}.
It is observed from Fig. \ref{fig:Fig_vehicle_movingtarget} that the changes of both velocities and angles are slow towards the prescribed time $T_d$,  which promotes the limited amount of control input near the time $T_d$ as shown in Fig. \ref{fig:Fig_control_movingtarget}.
After the prescribed time $T_d$, the task becomes formation keeping relative to the target.
The controller switching brings the controller mutation at time $T_d$ in Fig. \ref{fig:Fig_control_movingtarget}. As shown in the short period after $T_d$ in Fig. \ref{fig:Fig_error_movingtarget},
there is a regulation stage for the angle $\eta_{s,i}$ such that the speed of each vehicle gets to be parallel to that of the target.

It is worth emphasizing that the realization of simultaneous tracking and formation is completely dependent on the proposed space-and-time-synchronized strategy.

\section{Conclusion}
In this paper, we presented a space-and-time-synchronized control method for simultaneous tracking/ formation. The resultant control is able to achieve the predetermined state at the prescribed terminal time with a fixed relative space trajectory independent of time. As a result, simultaneous tracking/formation of multiple vehicles can be directly implemented. Extending this method to the simultaneous tracking/formation with no predetermined terminal time under communication topologies is an interesting topic for future research.

\appendices
\section{Proof of Theorem 1}
\emph{Step 1.} Prescribed-distance stability before $r_d=r_f$.

{Similar to the prescribed-time design philosophy in \cite{PWang2021prescribed}, we choose the controller (\ref{Eq_2_6}) as a prescribed-distance controller by treating $r_f-r_d$ as a measure of the remaining distance to go.} Substitution of (\ref{Eq_2_6}) into (\ref{Eq_2_5}) leads to second-order differential equation
\begin{equation}\label{Eq_2_7}
\frac{{\text{d}^{2}}{{q}_{d}}}{\text{d}r_{d}^{2}}+\frac{{{k}_{d2}}}{{{{r}}_{f}}-{{{r}}_{d}}}\frac{\text{d}{{q}_{d}}}{\text{d}{{{r}}_{d}}}+\frac{{{k}_{d1}}}{{{({{{r}}_{f}}-{{{r}}_{d}})}^{2}}}({{q}_{d}}-{{q}_{f}})=0.
\end{equation}
This is a kind of second-order Cauchy equation. Here, we introduce a scaling transformation $\zeta =\ln ({{r}_{f}}-{{r}_{d}})$
with which the differential equation is transformed into
\begin{equation}\label{Eq_2_8}
\frac{{\text{d}^{2}}{{q}_{d}}}{\text{d}{{\zeta }^{2}}}-({{k}_{d2}}+1)\frac{\text{d}{{q}_{d}}}{\text{d}\zeta }+{{k}_{d1}}({{q}_{d}}-{{q}_{f}})=0.
\end{equation}
The corresponding characteristic equation is obtained by letting ${{q}_{d}}-{{q}_{f}}={{e}^{\lambda \zeta }}$ as
$
{{\lambda }^{2}}-({{k}_{d2}}+1)\lambda +{{k}_{d1}}=0 $
whose discriminant is
$\Delta ={{({{k}_{d2}}+1)}^{2}}-4{{k}_{d1}}$.
In order to obtain an elegant description of the solution, we make ${{\lambda }_{1}}=N_1 \in \mathcal{N}_+$, ${{\lambda }_{2}}=N_2 \in \mathcal{N}_+$, $N_1<N_2$ which brings about ${{k}_{d1}}=N_1N_2, \  {{k}_{d2}}=N_1+N_2-1$, and the discriminant $\Delta >0$. Hence, with the initial condition ${{q}_{d}}(0)={{q}_{0}}$ and $\tan ({{\eta }_{d}}(0))=\tan {{\eta }_{d0}}$, the solution of this Cauchy equation can be obtained
\begin{equation}\label{Eq_2_10}
{{q}_{d}}({{r}_{d}})={{C}_{1}}({{r}_{f}}-{{r}_{d}})^{N_1}+{{C}_{2}}{{({{r}_{f}}-{{r}_{d}})}^{N_2}}+{{q}_{f}}
\end{equation}
where
\begin{align}
&{{C}_{1}}=\frac{N_2({{q}_{0}}-{{q}_{f}})+{{{r}}_{f}}\tan {{\eta }_{d0}}(R_0-{R}_{f1})/R_0}{(N_2-	N_1){{{r}}^{N_1}_{f}}}, \notag \\
& {{C}_{2}}=\frac{N_1({{q}_{f}}-{{q}_{0}})-{{{r}}_{f}}\tan {{\eta }_{d0}}(R_0-{R}_{f1})/R_0}{(N_2-N_1)r_{f}^{N_2}}. \notag
\end{align}
Differentiating ${{q}_{d}}(r_d)$ with respect to $r_d$ yields
\begin{equation}\label{Eq_2_11}
\begin{aligned}
\tan {{\eta }_{d}}= -\frac{R_d}{R_{d}-{R}_{f1}} &\left [  N_1{{C}_{1}}{{({{r}_{f}}-{{r}_{d}})}^{N_1-1}} \right. \\
&  \ \ \ \left. +N_2{{C}_{2}}{{({{r}_{f}}-{{r}_{d}})}^{N_2-1}} \right].
\end{aligned}
\end{equation}
It is thus clear that, given the positive integers ${{N}_{1}}$, ${{N}_{2}}$,  as $r_d$ approaching $r_f$, ${{q}_{d}}({{r}_{d}}) $ and  ${{\eta }_{d}}({{r}_{d}})$converges to $q_f$ and 0, respectively.

\emph{Step 2.} Position maintaining after $R_d$ approaching $R_f$.

Since $\eta_d(r_f)=0$ in (\ref{Eq_2_11}), $\eta_d(r_d) \equiv 0$  through $\dot{\eta}_d=0$ for $r_d \ge r_f$. When $ {V}_{df}=0$ and $\dot{V}_d=0$ for $r_d \ge r_f$, it is apparent $R_d \equiv R_f$ after $R_d$ approaching $R_f$. Invoking $ {{{\text{d}}^{2}}{{q}_{d}}}/{\text{d}r_{d}^{2}}=0$  in  (\ref{Eq_2_5}), one obtains  $q_d(r_d) \equiv 0$.

\section{Proof of Theorem 2}
We start from the simplified cascaded system with $n=2$ subsystems.
Evidently the prescribed-time stability of $x_2(t)$ in system ${{\Sigma }_{2}}$ can be obtained by taking time derivative of ${{V}_{2}}(t,x_2)$
\begin{equation}\label{Eq_3_6}
\begin{aligned}
   {{{\dot{V}}}_{2}}
 & \le -{{\alpha }_{23}}(\tau){{\left\| x_2 \right\|}^{2}}+{{b}_{2}}({{\alpha }^2_{24}}(\tau)/\alpha_{23}(\tau)){{\left\| {{\Delta }_{x,2}} \right\|}^{2}} \\
 & =-{{\delta }_{2}}{{\alpha }_{23}}{{\left\| x_2 \right\|}^{2}} \\
 & \ \ \ -{{\alpha }_{23}}\left[ (1-{{\delta }_{2}}){{\left\| x_2 \right\|}^{2}}-{{b}_{2}}{{{\left\| {{\Delta }_{x,2}} \right\|}^{2}}}/{{{\alpha }^2_{25}}}\; \right]
\end{aligned}
\end{equation}
with ${{\delta }_{2}}\in (0,1)$. Define the region ${{\Omega }_{x_2}}(r)=\left\{ x_2|{{\alpha }^2_{25}}{{\left\| x_2 \right\|}^{2}}\le {{b}_{2}}{{\left\| {{\Delta }_{x,2}} \right\|}^{2}}/(1-{{\delta }_{2}}) \right\}$ and we get ${{\Omega }_{{x_2}}}({{t}_{f}})={{\lim }_{t\to {{t}_{f}}}}{{\Omega }_{{x_2}}}(t)=0$, following from ${{\lim }_{t\to {{t}_{f}}}}{{\alpha }_{25}}(\tau)=\infty $.  When $x_2$ is outside of ${{\Omega }_{x_2}}(r)$, the increasing property of ${{\alpha }_{23}}(\tau)$ will force $x_2$ to ${{\Omega }_{x_2}}(t)$;  once $x_2$ is inside of ${{\Omega }_{x_2}}(t)$,  $x_2$ will never escape  ${{\Omega }_{x_2}}(t)$ and converge to 0 as  $t$ approaching ${{t}_{f}}$.

   Then, the time derivative of ${{V}_{1}}(t,x_1)$ along ${{\Sigma }_{1}}$ becomes
\begin{align}
  {{{\dot{V}}}_{\text{1}}}
 & \le -{{\alpha }_{13}}{{\left\| x_1 \right\|}^{2}}+{{b}_{1}}({{\alpha }^2_{14}}/\alpha_{13}){{\left\| {{\Delta }_{x,1}} \right\|}^{2}} +{{\alpha }_{16}} {{c}_{g,1 }} \left\| x_1 \right\|  \left\| x_2 \right\| \notag \\
 & =-0.5{{\delta }_{1}}{{\alpha }_{13}}{{\left\| x_1 \right\|}^{2}}-{{\alpha }_{13}}\left[ (1-{{\delta }_{1}}){{\left\| x_1 \right\|}^{2}} \right. \notag \\
 & \ \ \ \ \left. -\gamma \alpha^2_{16} {{{\left\| x_2 \right\|}^{2}}}/ {\alpha^2_{13}}  -{{b}_{1}}{{{\left\| {{\Delta }_{x,1}} \right\|}^{2}}}/{{{\alpha }^2_{15}}}\; \right]
\end{align}
where $\gamma ={0.5c_{\theta }^{2}}/{{{\delta }_{1}}}\;$. In view of the relationship among $\alpha_{13}$, $\alpha_{16}$ and $\alpha_{25}$, one obtains
\begin{align}
 {{\lim }_{t\to {{t}_{f}}}}{\alpha^2_{16} {{{\left\| x_2 \right\|}^{2}}}/{\alpha^2_{13}}}&= \alpha^2_{16} /(\alpha^2_{13}\alpha^2_{25})  \left\| {{\Delta }_{x,2}} \right\|^2    {{{b}_{2}}/(1-{{\delta }_{2}})}   \notag \\
 &=0.
 \end{align}
Invoking the boundedness of ${{\left\| {{\Delta }_{x,1}} \right\|}^{2}}$, the prescribed-time convergence of $x_1$ in system ${{\Sigma }_{1}}$ can be proved.

The prescribed-time convergence of cascaded system with $n=2$ is thus concluded.
From the prescribed-time stability of subsystems $\Sigma_n$ and $\hat{\Sigma}_{n-1}$, the prescribed-time stability of ${\Sigma}_{n-1}$ system is obtained; from the prescribed-time stability of subsystems ${\Sigma}_{n-1}$ and $\hat{\Sigma}_{n-2}$, the prescribed-time stability of system ${\Sigma}_{n-2}$ is obtained; and so on, the prescribed-time stability of ${\Sigma}_{1}$ system can be obtained.
The proof is thus completed.

\textbf{ }

\small
\bibliographystyle{elsarticle-num-names}

redpajama_set_name: RedPajamaArXiv


London's oldest bus route is commonly cited as Route 24 which runs over seven miles from Hampstead to Pimlico.
The route was first launched in 1910 but initially stopped at Victoria Station. It was extended to Pimlico just two years later in 1912 and has largely unchanged ever since (apparently with the exception of some minor adjustments due to one-way traffic schemes).
The route, which operates 24 hours a day, does take in some key landmarks of London – among them Trafalgar Square, Horse Guards Parade and Parliament Square. In 2013, Transport for London, said some 28,000 people used the route each day.
In 1965, the double-decker buses on the route – which have always been powered by motors rather than horses – became the first to have front entry. In 1988, it became the first route through central London to be privatised when purchased by Grey-Green (the line is now operated by Metroline).
Mostly recently, in 2013, it became the first route to fully implement the curvaceous new 'Routemasters' (while they've commonly been called that, the new buses are actually just called the 'new bus for London').


The IT services provider CHAMP Cargosystems transmits IATA C/XML messages for Kuehne + Nagel to the U.S. Customs and Border Protection and PSN response messages back to Kuehne + Nagel. The logistics company has thus been able to further enhance the quality of its air cargo services.
Luxembourg, May 2013 – The global logistics service provider Kuehne + Nagel (K+N) is pro-actively filing all air cargo shipments bound for the USA according to ACAS dual filing requirements as part of its quality airfreight services. On receipt of a shipment order the airfreight department transmits the House Air Waybill data to the U.S. Customs and Border Protection (CBP) to launch enhanced screening prior loading. When K+N receives the ACAS response (PSN) the company proceeds with consolidation and delivery to the air carrier. If there are problems with the shipment the logistics services provider is able to solve these pro-actively before tendering the shipment to the carrier.
K+N has opted for a CHAMP application to facilitate the message exchange with the US CBP. The solution transmits IATA C/XML messages (XFZB) directly from K+N's enterprise system to US CBP without manual intervention thereby optimizing data quality. It also transmits the CBP response messages back to K+N.
"Kuehne + Nagel participates in the ACAS dual filing project to detect high risk shipments and customs clearance problems as early as possible in the supply chain process. We feel it is our duty to solve any problems that might arise from a product, supplier or recipient pro-actively for our customers as quickly as possible after receiving their orders", explains Michael Weber, Vice President Interfaces and Processes, Air Logistics, Kuehne + Nagel Management AG.
"We are proud that we have been able to expand our cooperation with K+N, one of the leading global logistics players", says Rene E. Kircil, Senior Area Manager Sales at CHAMP Cargosystems. "CHAMP Cargosystems is well known for its premium messaging services for airlines and for its e-customs solutions. However, it is also offering similar applications (meeting IATA Cargo-IMP and XML standards) that are tailored to the needs of freight forwarders. K+N has opted for IATA Cargo-XML as it better facilitates the implementation and adoption of e-cargo initiatives such as the electronic Air Waybill (e-AWB) or e-freight. It also aids compliance to customs and security requirements by giving more flexibility in providing complete data elements." "Our Air AMS and ACAS messaging services enable logistics services providers to give a better service to their customers, to improve supply chain visibility and control over their data, to speed up transport processes and avoid shipments being held up on arrival in the USA", adds James Fernandez, VP Global Commercial Operations at CHAMP Cargosystems.
· Two-Part Carrier Filing – Air carrier transmits advance shipment data for security filing while subsequently transmitting post-departure Automated Manifest System (AMS) data in accordance with Trade Act requirements.
· Forwarder and Carrier Dual Filing – Freight forwarder may transmit House Air Waybill data directly to CBP to meet pre-loading targeting and manifest requirements; forwarder and carrier receive ACAS responses prior to consolidation and delivery to air carriers. Air carriers accept forwarder cargo for transport after confirming successful ACAS transmission, and perform required screening.
· Single Filing of ACAS and AMS data – Air carriers transmit the AMS manifest to CBP prior to loading of cargo, and perform all required screening. The pre-loading AMS manifest transmission would be used by CBP to satisfy both ACAS and AMS requirements.
CHAMP Cargosystems provides the most comprehensive range of integrated, IT solutions and distribution services for the air cargo transport chain. The portfolio spans core management systems, messaging services, and eCargo solutions. These include applications to meet customs and security requirements, quality optimization, as well as e-freight and mobility needs. The products and services are renowned under the Cargospot and Traxon brands.
The company serves over 100 airlines and GSAs, and links these with some 3,000 forwarders and GHAs worldwide. CHAMP's solutions help its customers, and their clients, adapt to critical and continuous changes in air transport logistics to meet global trade demands. CHAMP Cargosystems is headquartered in Luxembourg and operates offices in London, Zurich, Frankfurt / Main and Manila.
For further information see: www.champ.aero.


Jalna has positioned its pot-set yoghurt as 'a little pot of purity' in its first campaign created by Spinach.
Spinach won the creative account for Jalna Dairy Foods following a pitch process and has a media spend of around $2m for the campaign.
The 30-second TV ad features young school children visiting Jalna's farm in Victoria to see how each pot of yoghurt is made.
Frank Morabito, executive creative director at Spinach, said in a statement: "It was such a delight working with the Jalna management, marketing and dairy farm teams to bring the long-standing theme, 'A little pot of purity', to life in a new and engaging way.


We ship all orders USPS with delivery in 3-7 days of your purchase. We offer free shipping* in the USA on all orders over $49. Orders under $49 pay $6.99.
At this time we don't ship outside the USA. Sorry!
**Please note that to qualify for free shipping your order total must be $49 or more after a discount code has been applied.
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This glittery party hat for dogs is available in blue and pink and will be perfect for that birthday photo you want to post on Instagram.
The adjustable dog birthday hat is one size fits all.


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Home » Risk-reward proposition for Bunge stock seen as favorable
Risk-reward proposition for Bunge stock seen as favorable
Credit Suisse lowers Bunge's 2016 e.p.s. but still remains upbeat about long-term outlook.
NEW YORK — Slow farmer selling, particularly in Brazil and Argentina, is expected to weigh on third-quarter results at Bunge Ltd., according to a Sept. 21 research report from Credit Suisse. Despite the forecast for challenges in the third quarter, Credit Suisse believes the business is poised for a quick turnaround in the fourth quarter.
Robert Moskow, research analyst with Credit Suisse
"We are lowering our 2016 e.p.s. to account for a weak 3Q and modestly lowering 2017, but we believe the risk-reward proposition for the stock is highly favorable," Robert Moskow, research analyst with Credit Suisse, wrote in the report. "In the near-term, Brazilian and Argentinian farmers are holding back on selling their crops because they are unhappy with the falling prices and because the Brazilian currency strengthened. Falling soybean meal prices have hurt soybean crush margins and have caused customers to buy on the spot markets rather than buy forward. In addition, China remains challenging, Brazilian edible oils remain under pressure, and the mark-to-market benefit Bunge enjoyed in 2Q through crush margin hedges will reverse in 3Q."
Mr. Moskow said the ratings agency now expects e.p.s. for 2016 of $5.16, down from the previous estimate of $5.41. For fiscal 2017 Bunge's e.p.s. was lowered to $6.20 from $6.38. The company will release its third-quarter financials at the end of October.
Despite lowering its forecasts for e.p.s., Credit Suisse is positive on Bunge's longer-term outlook, Mr. Moskow said, noting that the White Plains, N.Y.-based company remains "poised for a quick turnaround in 4Q '16 and set up for strong 2017."
"Crush margins are likely to rebound as the U.S. and Brazil produce strong crops, U.S. exports increase, and the processing industry maintains capacity discipline," he said. "The glut of crop supplies and the probability of tighter credit conditions should spur more farmer selling as well. On top of that, the business is poised to grow EBIT by at least $200 million in 2017 and $6.20 of e.p.s. through acquisitions, sugar market improvements (which Bunge can hedge), and cost savings."
Another area that is expected to benefit Bunge's long-term growth is the company's emphasis on expanding its presence in specialty oils. Soren Schroder, chief executive officer of Bunge, has expressed a desire to significantly expand the company's business-to-business capabilities in specialty oils to meet the needs of food processors that are reformulating products, and the company already has a sizeable position in the United States with high-oleic oils and is a leading producer of softseed oil in Europe, Mr. Moskow said.
But to get into the laboratories of food processors the company will need to acquire and develop stronger technological capabilities and applications, Mr. Moskow said. He said the company will need to acquire more companies like Walter Rau Neusser, a German supplier of mid-specialty oils and fats that was acquired earlier this spring, if it hopes to achieve its long-term objective of making its value-added businesses a 35% share of its portfolio, up from 20% currently.
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China skeptics hammer corn, wheat, soybean prices
F.D.A. to propose glucomannan meets fiber definition
Nielsen forecasts big decade for C.B.D.
Slideshow: New products from Kellogg, General Mills, Clif Bar
Companies combine their brands in new pastries, cereals, cookie dough and more.
The big brand mashups of 2019
Coca-Cola, Mars, General Mills and more debuted co-branded products in 2019.

redpajama_set_name: RedPajamaCommonCrawl


I watch Top Chef, so I consider myself an expert in American cuisine and am willing to judge people solely on the way their food looks on my TV. But I'm not super excited about a documentary about the comeback of some old chef who changed the game but then disappeared. I imagine only a few foodies even know this guy's name or care if he succeeds in his return to the culinary world.


Today is International Human Rights Day. Check out some of the issues still plaguing our world, thanks to DC Progressive.
The President accepted his Nobel Peace Prize in Oslo this morning.
Prof Rah has a good little Christmas shopping list.
Are the naughties the best decade ever? Discuss.


Mother [M] and Father [F] were Ventura County, California residents. They had two children in the custody of the Ventura County Human Services Agency [HSA] that had been permanently removed from the parents. Both M and F had numerous problems with the law and drug problems. Both of them had been convicted numerous times on charges of child endangerment, domestic abuse and drug abuse.
The judge in Clark County contacted the Ventura County Court system, speaking with Judge Cody numerous times to determine which court would better serve the needs of the children and parents in this case based on the UCCJEA. The Clark County Judge determined it would be in the best interests of everyone to allow Ventura County to hear the case.
In July of 2017, the HSA filed juvenile dependency petitions regarding the twins under California's Welfare and Institutions Codes of child endangerment. The HSA also informed the court that the twins were currently in a confidential foster home in Nevada. A hearing was scheduled for August 1, 2017.
At the hearing the attorney for M argued that since the twins were in Nevada, Nevada should retain jurisdiction. Judge Cody disagreed and granted custody of the children to HSA. (Under the UCCJEA, Nevada turned over custody to HSA.) At a later hearing, HSA argued that both M and F's parental rights regarding the twins should be terminated for the same reasons as their parental rights had been terminated regarding their other children.
M appealed arguing that the UCCJEA states that any jurisdictional issues should take place in the child's home state – in this case Nevada.
Here both parents lived in Ventura County. Most of their families lived in Ventura County. The Ventura County HSA was very familiar with M and F's problems. And, further, M just went to Nevada to give birth and prevent Ventura County from obtaining custody of the children.


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GreenPanthera is a brand new website where you can earn cash for surveys, offers, shopping and lots more. By registering with them today you will get a free £3 ($5) registration bonus added to your account to get you started. By visiting the 'earn' tab within your account you can earn money by taking the poll of the day, surveys and other tasks. Once you have £19 ($30) on your balance you can cash out and your money is sent to your Paypal account within 4 - 10 days.
The Survey Rewardz survey panel is a brand new and exclusive site where you can earn money via different methods by taking surveys. The site is different to others as their system is much more efficient at matching surveys with your profile information. This means that you will not be screened out as often and can earn more in the time you have available. As soon as you have registered you can take numerous surveys straight away by logging into your account. Once you have earned just $1 (64p) you can cash out by Paypal or for a minimum of $5 (£3.20) you can cash out by cheque, virtual Visa card or Amazon voucher. Due to these low minimum cash out amounts and the large amount of surveys available here you could claim your first payment within the first couple of days and get lots more cash thereafter.
1) Sign up for a new e-mail address so you don't miss any survey invitations in your existing e-mail account.
2) Register an account with as many decent research sites as you can find.
3) Fill in any profile information that is required.
4) Check your inbox regularly and login to any new accounts you have created.
5) Complete surveys and redeem your points for cash or gift vouchers - It really is as as easy as that!
Each questionnaire you complete will be easy to fill out and will cover a wide range of topics. Some of the most popular topics include Movies, TV Programmes, Finance, Sports, Politics, Hobbies, Health, Motoring, Technology, Gaming, in fact pretty much any subject you can think of could be covered.
Participating in these research activities is a perfect part time jobs for university and college students, teenagers, kids, professionals, stay at home parents or anyone with a little extra time, an Internet connect and the desire to earn some extra money online.
After registering with survey panels you should then look for other research opportunities such as mystery shopping jobs, telephone and sms surveys, focus groups and product testing. These extra opportunities will offer higher rewards than standard online surveys so should always be opted into when possible.
Above all have fun, the more you are enjoying the time you spend taking research surveys the more money you are likely to earn.


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text:
Collars and ties mask the real white monopoly capitalism
Patrick Bond
The conversation on white monopoly capital is being extended to include corruption
Last week a conceptual barrier was broken at the University of the Witwatersrand's Great Hall: two leading critics of KPMG's enabling role in the "Zupta" perversion of governance, Pravin Gordhan and Iraj Abedian, made an unusual case against white monopoly capital (though they obviously couldn't name the beast as such).
Gordhan served from 2009 to 2017 as a white monopoly capital-compliant finance minister. But given how deeply rooted he now describes the culture of corruption, the solution isn't self-motivated internal reform.
Instead, he concluded, mass action is required against big capital and parastatals, perhaps to the bitter end: the fate Bell Pottinger just suffered.
Protest against corporate injustice remains one of South Africa's greatest strengths, the World Economic Forum indirectly acknowledged in last week's Global Competitiveness Report. Since 2012, our workers have been considered the world's most confrontational in the Swiss-based forum's survey of 14 000 executives.
On the other side of the class struggle, recall that PwC regularly names the Sandton elite as the world's most prone to corruption — especially procurement fraud, money-laundering, asset misappropriation and bribery— and that 80% of our corporate managers "do crime".
Before leaving his top treasury job after death threats earlier this year, Kenneth Brown estimated that more than a third of the government's R600‑billion annual procurement budget is lost to corporate theft.
As a result, the International Monetary Fund reports, South African firms now claim average profits among the top five of all major economies, and the World Bank acknowledges that the highest-income 1% doubled their consumption of our national pie from 10% to 12% in the 1990-1994 period to 18% to 20% since 2008.
So, are corruption and inequality hard-wired into South African capitalism?
Gordhan explained: "There's a particular culture in the dominant part of business in South Africa, or some sections of that business, that we've inherited from our past, in the sanctions-busting era. Elements of that DNA are still persisting, 23 years later."
Abedian, who recently quit insurance company Munich Re's local board because of its KPMG contracts, confirmed: "There is an embedded culture where national resources are for the benefit of the rich … What we should be doing first is looking at the actions of those who sit on boards of insurance companies' banks and investment companies."
In contrast to the no-holds-barred truth-telling by Gordhan and Abedian, others retain an either/or Gupta/white monopoly capital bias.
As social critic Jonny Steinberg complained in Business Day last week, debates he is having with (pro-Zuma) interviewees are frustrating: "It seems that we believe what we believe; any new evidence simply fills the contours of the story we are already telling. Mine is that an unholy alliance of politicians and bureaucrats in hock to a rich family has hijacked public institutions. Theirs is that global corporations have stolen South Africa."
Both positions are correct. South Africans from all sorts of backgrounds are expressing hostility against several firms linking the Guptas and white monopoly capital: spin-doctors Bell Pottinger, accountants KPMG, consultants McKinsey and two German software firms.
Such healthy hatred is by no means new. Since the origins of white-settler profiteering began with the Dutch East India Company invasion in 1652, later amplified by the likes of Cecil Rhodes and Ernest Oppenheimer's Anglo American Corporation, resistance arose from grass-roots, labour and nationalist (both Boer and black) activists. Mostly, the targets of these protests have represented malevolent greed and often outright theft. They were:
• Hundreds of Western multinational corporations and banks that ignored anti-apartheid sanctions called initially by Albert Luthuli;
• Big Pharma profiteers that denied access to Aids medicines, until the Treatment Action Campaign removed their monopoly patents (raising average life expectancy from 52 in 2004 to 64 today);
• Post-apartheid's public-private plunderers, including municipal water firms (Suez, Biwater and Veolia) and Gauteng's highway e-toll managers (Kapsch TrafficCom), which were pushed back by unions, township activists and the Organisation Undoing Tax Abuse;
• The Zurich Fifa mafia, whose 2010 World Cup ran into local protests;
• Collusive construction, bread and cellphone companies, and bankers fiddling the currency — all under fire from the Competition Commission, in turn fuelled by social outrage;
• Lonmin's extreme labour exploitation and illicit financial outflows, fought against valiantly by mining union Amcu;
• The World Bank in lucrative roles — as apartheid lender (Jubilee 2000 and Khulumani demanded reparations), Lonmin investor (taken to task by Marikana grass-roots feminists and the Wits Centre for Applied Legal Studies), primary creditor for Eskom's corruption-riddled Medupi power plant (hauled over the coals by Lephalale critics and Earthlife Africa) and lead owner of Net1-CPS, the social-grant disburser that illegitimately debit-ordered millions of poor people (until Black Sash forced its chief executive out this year);
• The "big three" credit rating agencies (Standard & Poor's, Fitch and Moody's) which, since 1994, have state-captured the treasury, thus compelling cutbacks in social infrastructure and higher-education spending in the world's most unequal country — they were fiercely contested (albeit indirectly) by myriad service delivery protests and #FeesMustFall; and
• Three Gupta brothers who, along with their allies in ethically challenged British, United States and German corporations, are now forever despised and brand-degraded.
Self-correcting mechanisms appear broken. The World Bank's inspection panel is toothless and, if South Africa's Independent Regulatory Board for Auditors belatedly finds KPMG guilty the maximum penalty is R250 000, small change given the fees it has raked in from its "consultancy services".
Instead, calls for the corporate death sentence ring out. Daily Maverick columnist Richard Poplak advises: "We need to go after the likes of KPMG and Bell Pottinger, and bury them — not just because they're complicit in Zuma's state capture project, but because they're shitty institutions that do shitty work and they deserve to die."
Wits School of Governance director David Everatt warned these firms on The Conversation: "South Africans will hold them accountable or, if necessary, break them."
After applauding last Wednesday's anti-corruption marches by labour and communists, Gordhan told the Wits audience: "While debates like these are important, in our political culture it's mass action that eventually counts. It's the involvement of people who are willing to put some effort into bringing about changes that actually makes a difference."
Patrick Bond teaches political economy at the University of the Witwatersrand
Patrick Bond teaches political economy at the Wits School of Governance. He is the co-editor of Brics: An Anti-Capitalist Critique (Jacana Media, 2015).
Bell Pottinger
Iraj Abedian
Zondo commission testimony: Billions wasted at Eskom
High court sets aside public protector report on Gordhan and Sars 'rogue unit'
Eusebius McKaiser: Mpofu, Gordhan caught in the crosshairs
War of words at Zondo commission: 'Grow up Mr Gordhan, don't be cheeky'
Gordhan tells Zondo how Moyane wanted to advance the objectives of state capture
Khaya Sithole: The decline and fall of the South African auditing profession


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Pictured: From left, Mentee Mary Lear and Mentor Chris Ynzunza.
Christine (Chris) Ynzunza, Human Resources Manager, Department of Natural Resources and Parks, ended up being the perfect match for Mary. Prior to making the mentor-mentee relationship official, Mary and Chris took part in the Mentorship Orientation. The purpose of the training was to allow both parties to learn more about the program, learn more about each other, and to form the foundation needed for a successful mentoring relationship.
Whether a mentor or mentee, mentoring is a beneficial and rewarding partnership that encourages professional and personal development in powerful ways. Mentors get a chance to improve their coaching skills, help engage and empower others, and give back to the success of King County. Mentees get to build relationship with a trusted advisor, assess their strengths and challenges, and have someone to help them navigate complex situations.
The program launched in September 2017. Mary and Chris began meeting near the start of the program in November 2017, and since then have continued to meet monthly.
There are currently around 200 employees taking part in the King County Mentorship Program. As King County continues to nourish a learning culture, mentoring is set to play an important role in the "Investing in You" strategy.
To sign up and learn more about the program employees should register for the King County Mentoring Program lunch and learn. During the sessions, participants will learn about the program, including the application process and how to select a mentor.