Broadway Brew #3 features Waitress, Sixpoint Brewing and the Adrienne Shelly Foundation!
Press is rolling with the latest news from the Broadway Brews Project - as of this week, we are HALF WAY to the Broadway Six-Pack!
Release of "It's for Adrienne" is Sunday 5/21!
NEW YORK – May 21, 2017 – The smash hit Broadway musical, WAITRESS is the latest Broadway show to create a craft beer for charity and partner with The Broadway Brews Project. A portion of the proceeds benefits The Adrienne Shelly Foundation, a NY-based charity that supports women in film.
The Broadway Brews Project is an initiative put together by The Happy Hour Guys (Mark Aldrich and Jimmy Ludwig) to pair hit Broadway shows with local craft brewers, to brew a special one-off beer, themed in relation to their show for a charity of the show's choosing. The Broadway Brews Project launched its first collaboration called Rise Up Rye, partnering with Gun Hill Brewing Company and the cast of the smash Broadway musical Hamilton, followed by School of Rock The Musical which launched School of Hops.
A lively cream ale bursting with the essence of cherries, Its for Adrienne Cream Ale will only be available on tap at select bars & venues throughout NYC. Those include: Beer Culture (both locations), Glass House Tavern, Rattle and Hum (both locations), Beer Authority, Arts & Crafts Beer Parlor (both locations), Alewife, The Jeffery, Fools Gold, L'Amico, 5 Napkin Burger (Midtown), Bronx Ale House, and Astoria Craft.
Plans for future Broadway Brew collaborations are already underway.
While some historical records indicate Sixpoint Brewery was founded in 2004, the real birth of Sixpoint begins at the dawn of civilization. That is when the earliest societies began cultivating cereal grains to make fermented beverages, and the desire for excellence in the craft of brewing was forged. One symbol has transcended and survived throughout the ages - the Sixpoint Brewers' star. The Sixpoint Brew Crew not only resurrected the Sixpoint Brewers' Star, but also breathed life into a patchwork of brewing equipment within an 800 square foot garage in a then-dilapidated neighborhood of Brooklyn, NYC called Red Hook. Out of this modest maritime enclave hundreds of delicious craft brews were concocted and disseminated, and the star had been reborn. Within years, a team of Sixpoint Brewers and Staff had gathered underneath the Sixpoint star, not knowing how they arrived there, but knowing they must keep the Sixpoint tradition alive.
WAITRESS opened April 24, 2016 at Broadway's Brooks Atkinson Theatre (256 West 47th Street). Based upon the 2007 motion picture written by Adrienne Shelly, WAITRESS is the first Broadway musical in history to have four women in the four top creative team spots, with a book by Jessie Nelson, a score by six-time Grammy Award-nominated singer-songwriter Sara Bareilles, choreography by Lorin Latarro and direction by Tony Award-winner Diane Paulus.
Waitress tells the story of Jenna, an expert pie maker in a small town, who dreams of a way out of her loveless marriage. A baking contest in a nearby county and the town's new doctor may offer her a chance at a new life, while her fellow waitresses offer their own recipes to happiness. But Jenna must find the courage and strength within herself to rebuild her life. This new American musical celebrates friendship, motherhood, and the magic of a well-made pie.
The Adrienne Shelly Foundation was founded by her husband Andy and created after her death in 2006. With partners including Sundance, AFI, IFP, Women in Film, Columbia University and Boston University (Adrienne's Alma Mater), and an advisory board that includes Paul Rudd, Michelle Williams and Keri Russell, ASF's mission is Supporting Women Filmmakers. Since its inception it has awarded 60 production grants to very talented women all over the world. Please visit www.adrienneshellyfoundation.org to honor Adrienne's memory and support ASF's critical artistic mission.


Hi i am currently writing exams and struggling to study for 5 papers which i write over a 3 day period , i am currently using the active memory product and im still not seeing results , can i increase my dosage from 2 per day to 3 per day ?
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All the recent talk of the hurricane hitting the East coast and the preparations families made to weather it out and/or evacuate was a good reminder for me to take a look at our family's emergency supplies and make sure things are updated.
Our first priority for emergency preparedness has always been our 72-hour kits (also often called "Go Bags"). I assembled my first when I was living on my own and have added more for my husband and kids as they've come along. The goal of a 72-hour kit is to supply each member of the family with enough food, clothing, toiletries, and emergency supplies to get through 3 days. It should be easy for each family member to grab quickly and carry independently.
We store each person's individual supplies in a backpack or diaper bag and have one duffel for shared family supplies. Containers with back or shoulder straps are ideal, because they lighten the somewhat heavy loads. I've marked the bags with each person's name both to help us know instantly whose is whose and to help emergency personnel identify us if necessary.
Old diaper bag. I like keeping all our bags the same color so we can find each other in a crowd if need be.
We purchased a pre-assembled first aid kit to ensure we didn't forget anything important, and because it was so compact.
Check out all the stuff this one little kit stores!
We rotate our kits twice a year to keep food fresh and clothes in current sizes.
Food should ideally have an expiration date at least 1 year from the date you're storing it, to give you time to use it in your pantry after replacing it in the kit.
Avoid pop-top lids, as they've been known to explode and make a mess in storage.
Pack foods that are not only nutritious, but that your family enjoys eating. Comfort foods are great in a stressful situation. Hard candy and gum are good for keeping spirits up and mouths busy.
Avoid foods that require water to prepare.
Pack wet cosmetics/toiletries in zip-top bags in case of leaks.
Pack tissues, toilet paper, paperwork, etc in zip-top bags to protect from moisture.
Consider your climate when packing clothes. We pack layers so we don't have to rotate clothes according to season.
Store enough cash (in a zip-top bag to protect it) in case you don't have access to an ATM or bank.
Remember to pack things to keep the family entertained. Playing cards, books, and notepads/pens all store compactly and provide a variety of ways to entertain both kids and adults.
Wet and dry things separately bagged, then stored in one large zip-top bag for easy location in the family bag.
Every year we use our tax refund to build our emergency supply storage (not just the 72-hour kits, but food/household goods storage, and our emergency saving's account). This year, we hope to buy a generator and a second water barrel.
Wow! I'm impressed and convinced that I will be in trouble in an emergency. I should make my own emergency bags, especially for power outages.
This is SUCH an important reminder and a great post. I'll be referring it to it as I get our kits ready this fall.


text:
\section*{Appendix}
Proofs are elaborated in this section to complete our discussion in Sec.~\ref{reallocation_approach}.
\begin{proposition-1*}
The global specification will be fulfilled if a path is found by Algorithm 1 on the local PA $\mathcal{P}^{(r)}$ and all agents successfully accomplish their tasks.
\end{proposition-1*}

\begin{proof}
According to Algorithm 1 and the definition of $S_{c,\mathcal{P}}^{(r)}$, the newly found path
\end{proof}

\begin{proposition-2*}
The global specification will be fulfilled if a path is found by Algorithm 2 on the revised local PA $\mathcal{P}^{(r)}(t)$ and all agents successfully accomplish their tasks.
\end{proposition-2*}

\begin{proof}
Algorithm 2 considers two
\end{proof}

\begin{proposition-3*}
The synchronized team automaton preserves the properties of the original team automaton.
\end{proposition-3*}

\begin{proof}

\end{proof}
\section{Introduction}

Mobile robots have been extensively investigated and deployed in various service applications such as assembly \cite{halperin2000general}, surveillance, \cite{ulusoy2013optimality} and search and rescue \cite{jennings1997cooperative}. In recent years, quadrupedal robots have been popularized for their superior traversability over \textcolor{black}{unstructured terrains} \cite{biswal2020development}. Nevertheless, even with exceptional locomotion capabilities, legged systems are often unstable, fragile, and less suitable for performing prolonged tasks compared to wheeled robots. However, distinct types of robots can form a heterogeneous team to compensate for their individual disadvantages.

Recent works on multi-robot systems have been focusing on mission planning problems with the assistance of formal languages such as Linear Temporal Logic (LTL) \cite{shoukry2017linear}. Originally proposed for model checking \cite{pnueli1977temporal}, LTL is a powerful tool used in the robotics community with a preponderance of research primarily conducted on wheeled robots \cite{kress2009temporal, decastro2018collision} and legged robots \cite{warnke2020towards, zhao2022reactive, kulgod2020temporal} for task and motion planning. There have also been works  \cite{kress2009temporal,caoleveraging,guo2015multi,tumova2016multi,kantaros2020stylus} on multi-agent systems. However, objectives are explicitly assigned to individual robots rather than having one global specification. This can be challenging for a large team of robots\cite{banks2020multi,kolling2015human}, where in most cases, a global task is simpler to define. Therefore, a simultaneous task allocation and planning (STAP) problem given a global LTL specification attracts more attention.

\begin{figure}[t]
    \centering
    \includegraphics[width=\linewidth]{figure1.pdf}
    \caption{A conceptual illustration of a three robots (Unitree A1, UBTECH DR, UBTECH Wassi) performing tasks in a simulated hospital setting. An illustration of team automaton is shown in the \emph{top left}. The role of this team automaton is described in detail in Fig.~\ref{fig:system_diagram}.}
    \label{fig:vision}
    \vspace{-0.2in}
\end{figure}

A line of research exists where STAP problems have been solved with global LTL specifications. In \cite{chen2011formal,ulusoy2013optimality} a product model is constructed with an exponential complexity, while \cite{schillinger2018decomposition} proposed a team model automaton with a linear complexity, assuming that each robot conducts its task independently. Other works also focused on advanced search algorithms \cite{banks2020multi,schillinger2017multi}, concrete time constraints \cite{leahy2020fast}, and collaborative tasks \cite{luo2021temporal}. However, failure recovery during real-world deployment is rarely studied, \textcolor{black}{especially considering unstructured environments such as rough terrain}. Single and multi-robot scenarios have been demonstrated with disturbances caused by a change in the environment \cite{guo2013revising,guo2013reconfiguration,livingston2012backtracking,li2021reactive} or a failure to perform an action \cite{tumova2014maximally}, but all have been limited to local LTL specifications. The work of \cite{faruq2018simultaneous} is conceptually similar to our goal, where each robot reallocates its tasks during execution failures, based on a Team Markov Decision Process. However, this work manually designs a set of LTL specifications in advance without identifying decomposable parts of a global task specification for optimal task sequences.
\textcolor{black}{Therefore, task reallocation capabilities during the online execution is desirable for STAP problems given global LTL specifications, which is addressed in this study.}

In this paper, multiple reallocation approaches to a rich class of online disturbances are proposed to replan on the updated team model given one global LTL specification.
\textcolor{black}{A local reallocation efficiently outputs a new action sequence only for the problematic robot, while a global reallocation searches on the entire team model and generates optimal action sequences for all robots.}
Meanwhile, the connection between the high-level reallocation and the low-level disturbance detection is interfaced by a mid-level Behavior Tree (BT). BT is a commonly used graphical and mathematical controller for a robot that enables fault-tolerant task executions \cite{schillinger2018simultaneous,colledanchise2019towards,abiyev2016robot,kim2018architecture,berenz2018playful,coronado2019robots}. In our work, we structure the BT to accept an action sequence assigned by the LTL planner similar to \cite{lan2019autonomous,li2021reactive}. \textcolor{black}{From an implementation perspective, our BT explicitly delegates all strategies while maintaining a simple structure that is extendable to other types of disturbances.} In addition, disturbance detection rates at different frequencies are employed by leveraging the BT's modularity property.

Our contributions are summarized as follows:
\begin{itemize}[leftmargin=*]

    \item \textcolor{black}{To handle potential disturbances from locomotion failures and environmental changes, we propose local and global task reallocation approaches given a global LTL specification. This approach eliminates the need to reconstruct the entire team model or resynthesize a new task.}

    \item \textcolor{black}{To select among reallocation strategies at run-time, we present a BT-based execution layer interfacing with low-level feedback. Our BT structure allows different types of disturbances to be detected at different rates.}

    \item We evaluate our pipeline in a simulated hospital environment with a heterogeneous robot team consisting of both \textcolor{black}{quadrupedal} and mobile robots \textcolor{black}{as shown in Fig.~\ref{fig:vision}}. \textcolor{black}{An open-source software package\footnote{\href{https://github.com/GTLIDAR/ltl\_multi\_agent}{https://github.com/GTLIDAR/ltl\_multi\_agent}.} is provided for the proposed reactive multi-robot task allocation and planning framework.}

\end{itemize}

\section{Preliminaries}
\label{sec:Prelim}
\subsection{LTL basics}
Linear temporal logic (LTL) has been widely used to encode temporal task specifications and automatically synthesize the system's transition behaviors. A specification $\phi$ is constructed from atomic propositions $\pi \in \Pi$, which is evaluated to be ${\rm True}$ or ${\rm False}$ and follows the syntax $\phi \: := \pi \:|\: \neg\: \phi \:|\: \phi_1 \:\wedge\: \phi_2 \:|\: \ocircle\phi \:|\: \phi_1 \:\mathcal{U}\: \phi_2 \:|\: \phi_1 \:\mathcal{R}\: \phi_2$. Boolean operators $\neg$ ``not'' and $\wedge$ ``and'' in addition to a set of temporal operators $\ocircle$ ``next'', $\mathcal{U}$ ``until'', and $\mathcal{R}$ ``release'', are denoted. To be concise, we omit the derivations of other boolean operators such as $\vee$ ``or'', $\rightarrow$ ``implies'', and $\leftrightarrow$ ``if and only if'', as well as temporal operators $\Diamond\,\phi$ ``eventually $\phi$'' and $\square\,\phi$ ``always $\phi$''.

A common usage of LTL is for constructing an automaton. A non-deterministic automaton (NFA) is defined as a tuple $\mathcal{Q} = (S_{\mathcal{Q}}, S_{0,\mathcal{Q}}, \Sigma, \delta_{\mathcal{Q}}, F)$ such that $S_{\mathcal{Q}}$ is a set of states ($s_{\mathcal{Q}} \in S_{\mathcal{Q}}$), $S_{0,\mathcal{Q}} \subseteq S_{\mathcal{Q}}$ is a set of initial state, $\Sigma$ is the input alphabet, $\delta_{\mathcal{Q}}$ is a set of transition relations such that $\delta_{\mathcal{Q}} : S_\mathcal{Q} \times \Sigma \rightarrow 2^{S_{\mathcal{Q}}}$, and $F$ is a set of accepting final states. In addition, LTL formulas are evaluated over a sequence $\sigma: \:\mathbb{N} \:\rightarrow\: 2^\Pi$ where $\sigma(t) \:\subset\: \Pi$ represents all true propositions at time $t$ \cite{schillinger2018simultaneous}.

For this framework, a transition system (TS) is created by combining data from the topological map and the robots' operating states. A TS is defined as a tuple $\mathcal{T} = (S_{\mathcal{T}}, s_{0,\mathcal{T}}, A_{\mathcal{T}}, \Pi_\mathcal{T}, \mathcal{L})$ such that $S_{\mathcal{T}}$ is a set of system states ($s_{\mathcal{T}} \in \mathcal{S}_{\mathcal{T}}$), $s_{0,\mathcal{T}} \in \mathcal{S}_{\mathcal{T}}$ is the initial system state, $A_{\mathcal{T}}$ is a set of available system actions, $\Pi_{\mathcal{T}}$ is the set of system propositions, and $\mathcal{L}$ : $S_{\mathcal{T}} \:\rightarrow\: 2^{\Pi_{\mathcal{T}}}$ is a labeling function that assigns atomic propositions to states\cite{banks2020multi}. We use $\texttt{Succ}(s_{\mathcal{T}})=\{s_{\mathcal{T}}^{\prime} \in S_{\mathcal{T}}|(s_{\mathcal{T}},s_{\mathcal{T}}^{\prime}) \in A_{\mathcal{T}}\}$ to denote the successors of $s_{\mathcal{T}}$ and $\texttt{Pred}(s_{\mathcal{T}})=\{s_{\mathcal{T}}^{*} \in S_{\mathcal{T}}|(s_{\mathcal{T}}^{*},s_{\mathcal{T}}) \in A_{\mathcal{T}}\}$ as the predecessors of $s_{\mathcal{T}}$. By combining a TS with an NFA, a product automaton (PA) $\mathcal{P}$ composed of system states and mission specifications is generated.
It is represented by $\mathcal{P} = \mathcal{Q} \otimes \mathcal{T} = (S_{\mathcal{P}}, S_{0,\mathcal{P}}, A_{\mathcal{P}})$ such that $S_{\mathcal{P}}=S_{\mathcal{Q}} \times S_{\mathcal{T}}$ is the set of states ($s_{\mathcal{P}} \in S_{\mathcal{P}}$), $S_{0,\mathcal{P}}=S_{0,\mathcal{Q}} \times \{s_{0,\mathcal{T}}\}$ is the set of initial states, and $A_{\mathcal{P}}=\{ ((s_{\mathcal{Q}},s_{\mathcal{T}}),(s_{\mathcal{Q}}^{\prime},s_{\mathcal{T}}^{\prime})) \in S_{\mathcal{P}} \times S_{\mathcal{P}}:(s_{\mathcal{T}},s_{\mathcal{T}}^{\prime}) \in A_{\mathcal{T}} \wedge s_{\mathcal{Q}}^{\prime} \in \delta_{\mathcal{Q}}(s_{\mathcal{Q}},\mathcal{L}(s_{\mathcal{T}}))\}$.

\begin{figure*}
    \centering
    \includegraphics[width=7.0in]{system_diagram.pdf}
    \caption{An overview of the hierarchical planning framework. The global LTL planner portrays the high-level LTL task planner. Once the product automaton is created, it is sent to the team automaton, where a sequence of actions are assigned to individual robots. During execution, the local task planner, the behavior tree, and the hardware on the robot will react to disturbances and handle accordingly.}
    \label{fig:system_diagram}
    \vspace{-0.2in}
\end{figure*}

\subsection{Offline task allocation}\label{subsec:offline_STAP}
We introduce the baseline task allocation method and terminologies used throughout this paper. Readers are referred to \cite{schillinger2018decomposition} for more details. Given the LTL semantics above, a global task $\phi$ along with its corresponding NFA $\mathcal{Q}$ can be specified for a whole team of $N$ agents, each of which has its own TS $\mathcal{T}^{(r)}$ and a corresponding PA $\mathcal{P}^{(r)} = \mathcal{Q} \otimes \mathcal{T}^{(r)}, r = \{1, \ldots, N\}$. Next, we will introduce a criterion that identifies the decomposed parts of $\phi$ based on the assumption that each agent executes its sub-task independently.
\begin{definition}[Finite Decomposition \cite{schillinger2018decomposition}]
Let $\mathcal{J}_r$ with $r \in \{1,\dotsc,N\}$ be a set of finite LTL task specifications and $\sigma_i$ is any sequence s.t. $\sigma_r \models \mathcal{J}_r$. These tasks are called decomposition of the global finite LTL specification $\phi$, iff:
\begin{equation}
    \sigma_{j_1}\dotsc\sigma_{j_r}\dotsc\sigma_{j_N} \models \phi
\end{equation}
for all permutations of $j_r \in \{1,\dotsc,N\}$ and all respective sequences $\sigma_r$.
\end{definition}
Based on this criterion, a decomposition set $\mathcal{D} \subseteq S_\mathcal{Q}$ can be derived from $\mathcal{Q}$, which includes all NFA states that allocate tasks. Then the action-state sequence allocated to each agent is computed by first constructing a team automaton.
\begin{definition}[Team automaton \cite{schillinger2018decomposition}]\label{def:team_automaton}
The team automaton $\mathcal{G}$ is a union of N local PA $\mathcal{P}^{(r)}$ with $r \in \{1,\dotsc,N\}$ and defined by $\mathcal{G} := (S_{\mathcal{G}},S_{0,\mathcal{G}},F_{\mathcal{G}},A_{\mathcal{G}})$, where:
\begin{itemize}
    \item $S_{\mathcal{G}}=\{(r, s_{\mathcal{Q}}, s_{\mathcal{T}}):r \in \{1,\dotsc,N\}, (s_{\mathcal{Q}},s_{\mathcal{T}}) \in S_{\mathcal{P}}^{(r)} \}$ is the set of states;
    \item $S_{0,\mathcal{G}}=\{(r, s_{\mathcal{Q}}, s_{\mathcal{T}}):r=1, (s_{\mathcal{Q}},s_{\mathcal{T}}) \in S_{0,\mathcal{P}}^{(1)} \}$ is the set of initial states;
    \item $F_{\mathcal{G}}=\{(r, s_{\mathcal{Q}}, s_{\mathcal{T}}) \in S_{\mathcal{G}}:s_{\mathcal{Q}} \in F\}$ is the set of final accepting states;
    \item $A_{\mathcal{G}}=\bigcup_r A_{\mathcal{P}}^{(r)} \cup \zeta$ is the set of actions including switch transitions $\zeta$.
\end{itemize}
\end{definition}
We use $\delta_{\mathcal{G}}:S_{\mathcal{G}} \rightarrow S_{\mathcal{G}}$ to denote the transitions corresponding to $A_{\mathcal{G}}$. The set $\zeta \subset S_{\mathcal{G}} \times S_{\mathcal{G}}$ denotes the \textit{switch transitions}, each of which $\varsigma=((i,s_{\mathcal{Q}},s_{\mathcal{T}}),(j,s_{\mathcal{Q}}^{\prime},s_{\mathcal{T}}^{\prime}))$ is defined as a transition between two states in $\mathcal{G}$ iff: 1) $j=i+1$: connects to the next agent; 2) $s_{\mathcal{Q}}=s_{\mathcal{Q}}^{\prime}$: the NFA state remains unchanged; 3) $s_{\mathcal{T}}^{\prime}=s_{0,\mathcal{T}}^{(j)}$: points to an initial agent state; 4) $s_{\mathcal{Q}} \in \mathcal{D}$: the NFA state is inside the decomposition set.

The team automaton is a combination of every agent's PA $\mathcal{P}^{(r)}$ \footnote{For simplicity, we abuse $\mathcal{P}^{(r)}$ to denote a sub-graph in $\mathcal{G}$ which contains the same states $S_{\mathcal{P}}^{(r)}$ and transitions except, the robot index $r$ is appended.} with additional switch transitions which can reassign an agent's remaining task to another agent. \textcolor{black}{As illustrated in Fig.~\ref{fig:cartoon_reallocation}(a)}, if \textcolor{black}{a global action sequence $\beta$} on the team automaton is found, we can state that task allocation and planning have been accomplished simultaneously (namely STAP \cite{schillinger2018decomposition}). By projecting $\beta$ onto the PA of each agent, tasks can be executed in parallel. This process of finding an initial set of action-state sequence is called the \textit{offline allocation}, \textcolor{black}{whereas the state space complexity scales linearly with the number of agents. The rest of this paper will focus on addressing external disturbances during real-world deployment.}

\section{Problem Formulation}\label{problem_formulation}

\subsection{Disturbance characterization}
In this section, we categorize disturbances into four classes in order to pair a reactive strategy that can efficiently resolve the problem. Unless specified, the following disturbances apply to both legged and wheeled robots.
\begin{itemize}[leftmargin=*]
  \item \textbf{Loss of balance} refers to a scenario where a \emph{legged robot} falls due to an unstable gait or erratic controller output.
  \item \textbf{Critical failure} refers to an irrecoverable hardware or software malfunction such as a damaged motor or a software glitch.
  \item \textbf{Unexpected robot state change} refers to a situation where a robot detects a sudden shift in the robot's state.
  \item \textbf{Environmental change} refers to an environmental event preventing the robot from continuing its current task.

\end{itemize}

\subsection{STAP reallocation}
Given the baseline STAP approach (Sec.~\ref{subsec:offline_STAP}), we seek to find an efficient reallocation strategy that is specific to each category of disturbance. For this problem, two aspects need to be investigated: 1) a formal guarantee to complete the global task; 2) a set of completed tasks by the whole team. To this end, we define a STAP reallocation problem as the following:

\begin{problem*}
Given an initial task assignment and the current TS of every agent, one finds a set of new action-state sequences for the agents to accomplish the global task without restarting the whole mission or re-synthesizing a new mission.
\end{problem*}

However, the task reallocation strategies are only compatible with high-level information such as changes in the NFA state or the TS state. Therefore, we propose a BT-based mid-level to organize different disturbance types into comprehensible inputs for the high-level planning framework.

Fig.~\ref{fig:system_diagram} shows the planning architecture consisting of 1) a high-level task planner that performs offline allocation and online reallocation; 2) a mid-level BT interface to execute the assigned action plan for each agent; 3) low-level controllers that power the actuators on legged and wheeled robots.

\section{Reallocation Approach}\label{reallocation_approach}
To solve the STAP reallocation problem, we propose two approaches: a local and global approach, both of which are designed at the high level. \textcolor{black}{Fig.~\ref{fig:cartoon_reallocation}(b) and \ref{fig:cartoon_reallocation}(c) show the workflow and conceptual examples of both local and global reallocation.}

\subsection{Local reallocation addressing unexpected state changes}\label{approach:state_change}
The offline task allocation process generates an action-state sequence for each agent, which assumes every action is performed successfully. During  execution unexpected interventions could occur, which would undermine the original plan. For instance, if a human removes a load carried by the robot before the robot reaches its destination, an unforeseen robot state change occurs.
To resolve this intervention, we introduce the local task reallocation approach. Suppose $\texttt{Path}^{(r)}$ is a planned state sequence for robot $r$, generated from the original team automaton. Let $\texttt{Path}^{(r)}({\rm acc})$ denote the agent's local accepting state and $s_{\mathcal{T}}^{\triangle}$ be the current state after the intervention. By comparing the state sequence execution history and $\texttt{Path}^{(r)}$, the last matched state is identified as $\texttt{Path}^{(r)}(m)$, whose NFA and TS are written as $\texttt{Path}^{(r)}_{\mathcal{Q}}(m)$ and $\texttt{Path}^{(r)}_{\mathcal{T}}(m)$. Thus, $\texttt{Path}^{(r)}_{\mathcal{T}}(m+1)=s_{\mathcal{T}}^{\triangle}$. Now, the problem is reformulated to find a path on the local PA, starting from an up-to-date initial set defined as
 $S_{c,\mathcal{P}}^{(r)}=\{(r,s_{\mathcal{Q}},s_{\mathcal{T}}) \in S_{\mathcal{P}}^{(r)} | s_{\mathcal{T}}=s_{\mathcal{T}}^{\triangle}, \forall s_{\mathcal{Q}} \in \delta_{\mathcal{Q}}(\texttt{Path}_{\mathcal{Q}}^{(r)}(m),\mathcal{L}(\texttt{Path}^{(r)}_{\mathcal{T}}(m)))  \}$. The $\texttt{find-path}$ method throughout this work is performed by Dijkstra's algorithm. Note that the original path can be reused if the current state happens to be on the agent's original path. This local task reallocation approach is summarized in Algorithm \ref{code:local-1}. $\texttt{Path}^{(r)}(i:)$ denotes a sub-path starting from the $i^{\rm th}$ element.

 \begin{theorem}
In the presence of one robot experiencing unexpected state changes and all other agents not interfered (i.e., can successfully accomplish their tasks), the global specification $\phi$ will be fulfilled if a path is found by the local reallocation method in Algorithm \ref{code:local-1} on the local PA $\mathcal{P}^{(r)}$.

\end{theorem}

\begin{algorithm}
    \small
  \caption{Local reallocation: unexpected state change}
  \textbf{Input:} $\mathcal{P}^{(r)},\texttt{Path}^{(r)}$\\
  \textbf{Output:} A new path $\texttt{Path}^{(r)+}$
  \label{code:local-1}
  \begin{algorithmic}
    \STATE{$\texttt{Path}^{(r)+} \gets \texttt{empty path}$}
    \STATE{$\texttt{Path-set}^{(r)} \gets \texttt{empty set}$}
    \FOR{$s \ \textbf{in} \ S_{c,\mathcal{P}}^{(r)}$}
        \IF{$s \ \textbf{in} \ \texttt{Path}^{(r)}$}
            \STATE $i \gets \texttt{getIndex}(\texttt{Path}^{(r)}(s))$
            \STATE $\texttt{Path-set}^{(r)}.\text{append}(\texttt{Path}^{(r)}(i:))$
            \STATE \textbf{continue}
        \ELSE
            \STATE $\texttt{Path-set}^{(r)}.\text{append}(\texttt{find-path}(s, \texttt{Path}^{(r)}({\rm acc})))$
        \ENDIF
    \ENDFOR
    \STATE $\texttt{Path}^{(r)+} \gets \texttt{find-best}(\texttt{Path-set}^{(r)})$
  \end{algorithmic}
\end{algorithm}

\begin{proof}
\textcolor{blue}{According to Algorithm \ref{code:local-1} and the definition of $S_{c,\mathcal{P}}^{(r)}$, the new state sequence $\texttt{Path}^{(r)+}$ (i) connects to agent $r$'s executed state sequence through a valid NFA transition since $s_{\mathcal{Q}} \in \delta_{\mathcal{Q}}(\texttt{Path}_{\mathcal{Q}}^{(r)}(m),\mathcal{L}(\texttt{Path}^{(r)}_{\mathcal{T}}(m))$; and (ii) ends with the original local accepting state $\texttt{Path}^{(r)}({\rm acc})$. In other words, the originally assigned sub-task for agent $r$ is fulfilled again. Given the assumption that the rest of the agents are not interfered by any disturbances, agent $r$'s newly concatenated sequence, along with the other agents' planned sequences, consist a global action sequence $\beta$ on the team automaton $\mathcal{G}$ again. Then according to the correctness property in \cite{schillinger2018simultaneous}, the projected global path onto the NFA satisfies the mission specification $\phi$.}
\end{proof}

\vspace{-0.1in}
\subsection{Local reallocation addressing environmental changes}
In the previous section, the disturbance shifts the robot's state but does not modify the environment, which will be addressed in this section. Such a disturbance will directly impact the TS. For instance, if the floor is occupied by an impassable object, the mobile robot would encounter a navigation failure and would not be able to transition to its next expected state. In this case, the robot will receive the changes to be made on TS called $Info(t)^{(r)}$. Each update contains three types of information: 1) $(s_{\mathcal{T}}, s_{\mathcal{T}}^{\prime}) \in Add(t)$ if $s_{\mathcal{T}}$ is allowed to transit to $s_{\mathcal{T}}^{\prime}$; 2) $(s_{\mathcal{T}}, s_{\mathcal{T}}^{\prime}) \in Delete(t)$ if $s_{\mathcal{T}}$ is not allowed to transit to $s_{\mathcal{T}}^{\prime}$; 3) $(b, s_{\mathcal{T}}) \in Relabel(t)$ if the labeling function of state $s_{\mathcal{T}}$ is updated to $b \subseteq 2^{AP}$.

\begin{figure*}
    \centering
    \includegraphics[width=\linewidth]{cartoon_reallocation.pdf}
    \caption{\textcolor{black}{Conceptual illustrations of local and global reallocations: (a) offline allocation given the team automaton. A global sequence (the red lines with arrows) is found during the offline phase as an initial task allocation. Assuming A1 undergoes a disturbance such as \textit{unexpected state change} and \textit{environmental change} as shown in subfigure (b). In correspondence to Algorithm \ref{code:local-1}, Case 1.1 refers to a scenario where the new initial state after the state jump is on the original offline-generated global sequence, while Case 1.2 corresponds to a completely new initial state and requires a replanning. Similarly in Algorithm \ref{code:local-2}, Case 2.1 demonstrates a scenario where the transition change doesn't affect the execution of the original global path, while a replanning is required in Case 2.2 due to a deleted edge on the original global path (represented by the cross marker). Subfigure (c) reveals a block diagram on the essential steps for a global reallocation. The synchronization step notifies the planner of each agent's current state (denoted by the yellow circle) and execution history to proceed with the remaining steps. More details are explained in Sec.~\ref{subsec:global_reallocation}.}}
    \label{fig:cartoon_reallocation}
    \vspace{-0.2in}
\end{figure*}

The TS change can be reflected by directly modifying the team automaton using the PA revision strategy in \cite{guo2013revising}. When a single agent $r$ receives an update, the latest team automaton is revised by only updating corresponding $\mathcal{P}^{(r)}(t)$\footnote{We use $\cdot (t)$ to denote an updated automaton at time $t$ given the transition relation is changed.}. All deleted transitions, i.e., edges, are added into a set $R(t)$.

\begin{definition}[Updating rules]
\textcolor{black}{
$\mathcal{G}(t)$ (more specifically, only $\mathcal{P}^{(r)}(t)$) is updated given the $Info(t)$ from agent $r$ following the rules:}
\begin{itemize}
    \item \textcolor{black}{If $(s_{\mathcal{T}}, s_{\mathcal{T}}^{\prime}) \in Add(t)$, $(r,s_{\mathcal{Q}}^n,s_{\mathcal{T}}^{\prime})$ is in $\delta_{\mathcal{G}}((r,s_{\mathcal{Q}}^m,s_{\mathcal{T}}))$ for $\forall s_{\mathcal{Q}}^n,s_{\mathcal{Q}}^m$ satisfying $s_{\mathcal{Q}}^n \in \delta_{\mathcal{Q}}(s_{\mathcal{Q}}^m,\mathcal{L}(s_{\mathcal{T}}))$;}

    \item \textcolor{black}{If $(s_{\mathcal{T}}, s_{\mathcal{T}}^{\prime}) \in Delete(t)$, $(r,s_{\mathcal{Q}}^n,s_{\mathcal{T}}^{\prime})$ is deleted from $\delta_{\mathcal{G}}((r,s_{\mathcal{Q}}^m,s_{\mathcal{T}}))$ for $\forall s_{\mathcal{Q}}^n,s_{\mathcal{Q}}^m \in S_\mathcal{Q}^{(r)}$;}

    \item \textcolor{black}{If $(b, s_{\mathcal{T}}) \in Relabel(t)$, then $\forall s_{\mathcal{T}}^{*} \in \texttt{Pred}(s_{\mathcal{T}})$: $(r,s_{\mathcal{Q}}^n,s_{\mathcal{T}}^{*})$ is added to $\delta_{\mathcal{G}}((r,s_{\mathcal{Q}}^m,s_{\mathcal{T}}))$ for $\forall s_{\mathcal{Q}}^n \in \delta_{\mathcal{Q}}(s_{\mathcal{Q}}^m,b)$; $(r,s_{\mathcal{Q}}^n,s_{\mathcal{T}}^{*})$ is deleted from $\delta_{\mathcal{G}}((r,s_{\mathcal{Q}}^m,s_{\mathcal{T}}))$ for $\forall s_{\mathcal{Q}}^n \notin \delta_{\mathcal{Q}}(s_{\mathcal{Q}}^m,b)$}
\end{itemize}
\end{definition}
If a disturbance was detected on an agent's TS, a new type of task reallocation algorithm is necessary. Note that no unexpected robot state is assumed in this case and the last matched state is equivalent to the current state, i.e. $\texttt{Path}^{(r)}_{\mathcal{T}}(m)=s_{\mathcal{T}}^{\triangle}$. Given the revised local PA $\mathcal{P}^{(r)}(t)$, we propose a different replanning approach in Algorithm \ref{code:local-2}, compared to the one in Sec.~\ref{approach:state_change}.

\begin{theorem}
In the presence of environmental change and all other agents that are not interfered (i.e., can successfully accomplish their tasks), the global specification $\phi$ will be fulfilled if a path is found by the local reallocation method in Algorithm \ref{code:local-2} on the revised local PA $\mathcal{P}^{(r)}(t)$.

\end{theorem}

\begin{proof}
\textcolor{blue}{According to Algorithm \ref{code:local-2}, the new state sequence $\texttt{Path}^{(r)+}$ starts from the last state $\texttt{Path}^{(r)}(m)$ in agent $r$'s execution history and reaches the same local accepting state $\texttt{Path}^{(r)}({\rm acc})$. Since the PA updating rules preserve valid NFA transitions, the originally assigned sub-task for agent $r$ is still fulfilled by the newly found path on $\mathcal{P}^{(r)}(t)$. Same as Proposition 1, a global action sequence $\beta$ is formed assuming the rest agents are not interrupted by any disturbances. Consequently, the global specification $\phi$ is satisfied again.}
\end{proof}

\begin{algorithm}[h]
\small
  \caption{Local reallocation: environmental change}
  \textbf{Input:} $\mathcal{P}^{(r)}(t),\texttt{Path}^{(r)},Info(t)$\\
  \textbf{Output:} A new path $\texttt{Path}^{(r)+}$
  \label{code:local-2}
  \begin{algorithmic}
    \STATE{$\texttt{Path}^{(r)+} \gets \texttt{empty path}$}
    \STATE $\mathcal{P}^{(r)}(t),R(t) \gets \texttt{UpdatePA}(\mathcal{P}^{(r)}(t),Info(t))$
    \IF{$R(t) \cap \texttt{edge}(\texttt{Path}^{(r)}) \neq \varnothing$}
    \STATE $\texttt{Path}^{(r)+} \gets \texttt{find-path}(\texttt{Path}^{(r)}(m),\texttt{Path}^{(r)}({\rm acc}))$
    \ELSE
    \STATE{$\texttt{Path}^{(r)+} \gets \texttt{Path}^{(r)}(m:)$}
    \ENDIF
  \end{algorithmic}
\end{algorithm}

\vspace{-0.1in}
\subsection{Global reallocation}\label{subsec:global_reallocation}
The two aforementioned local task reallocation approaches do not consider replanning for the whole team, which results in a sub-optimal strategy. Furthermore, if the local task reallocation fails to find a new plan for the agent, a succeeding global task reallocation over the entire team is activated. First, a synchronization step will be executed where the task planner requests for each agent's current TS state and sets it to be the latest initial TS state $s_{0,\mathcal{T}}^{(r)}(t)$. Then the initial PA set $S_{0,\mathcal{P}}^{(r)}(t)$ is updated accordingly by keeping $S_{0,\mathcal{Q}}^{(r)}$ the same. Since each $\mathcal{P}^{(r)}(t)$ has been updated during local reallocation if needed, the team automaton is only modified by updating the initial set of states and switch transitions, in addition to appending the \textit{synchronized transitions}.

\begin{definition}[Synchronized team automaton]
The synchronized team model $\mathcal{R} := \mathcal{G}(t)$ is a union of $N$ product automata $\mathcal{P}^{(r)}(t)$ given the updated product states after synchronization, where $r \in \{1,\dotsc,N\}$ and $\mathcal{R}:=(S_{\mathcal{R}},S_{0,\mathcal{R}},F_{\mathcal{R}},A_{\mathcal{R}})$ consists of:
\begin{itemize}
    \item $S_{\mathcal{R}}=\{(r, s_{\mathcal{Q}}, s_{\mathcal{T}}):r \in \{1,\dotsc,N\}, (s_{\mathcal{Q}},s_{\mathcal{T}}) \in S_{\mathcal{P}}^{(r)}(t) \}$ is the set of states;
    \item $S_{0,\mathcal{R}}=\{(r, s_{\mathcal{Q}}, s_{\mathcal{T}}):r=1, (s_{\mathcal{Q}},s_{\mathcal{T}}) \in S_{0,\mathcal{P}}^{(1)}(t) \}$ is the set of initial states;
    \item $F_{\mathcal{R}}=\{(r, s_{\mathcal{Q}}, s_{\mathcal{T}}) \in S_{\mathcal{R}}:q \in F\}$ is the set of final accepting states, which remains unchanged since the NFA accepting states are fixed;
    \item $A_{\mathcal{R}}=\bigcup_r A_{\mathcal{P}}^{(r)}(t) \cup \zeta(t) \cup \xi(t)$  is the set of actions that include the updated switch transitions $\zeta(t)$ and the newly proposed synchronized transitions $\xi(t)$.
\end{itemize}
\end{definition}
Suppose $\texttt{ExePath}^{(r)}$ is the executed state sequence acquired from each agent $r$ \textcolor{black}{and $\texttt{ExePath}_{\mathcal{Q}}^{(r)}$ is the projected NFA state sequence}. The definition of a synchronized transition is as follows:
\begin{definition}[Synchronized transition]\label{def:synchronized_transition}
The set $\xi \subset S_{\mathcal{R}} \times S_{\mathcal{R}}$ denotes synchronized transitions. Each element $\varepsilon=((i,s_{\mathcal{Q}},s_{\mathcal{T}}),(j,s_{\mathcal{Q}}^{\prime},s_{\mathcal{T}}^{\prime}))$ satisfies:
\begin{itemize}
    \item $i=j$: connects the same agent;
    \item $s_{\mathcal{Q}}=\texttt{ExePath}_{\mathcal{Q}}^{(r)}({\rm init}),s_{\mathcal{Q}}^{\prime}=\texttt{ExePath}_{\mathcal{Q}}^{(r)}({\rm final}),$ $r \in \{1,\dotsc,N\}$ starts from the initial NFA state and points to the most recent NFA nodes upon request for synchronization of each agent;
    \item $s_{\mathcal{T}}=s_{\mathcal{T}}^{\prime}$: TS state is preserved.
\end{itemize}
\end{definition}
This synchronized transition allows a new transition between two
NFA states inside each agent's $\mathcal{P}^{(r)}(t)$. Once this is complete, each agent will be aware of the task completion status of the whole team \textcolor{black}{and avoid performing redundant tasks}.
\textcolor{black}{
In the original team automaton \cite{schillinger2018simultaneous}, the four properties including correctness, independence, completeness, and ordered sequence are proposed to justify the rationale of finding a global path on the team automaton for a task allocation. Here we claim that our synchronized team automaton preserves these properties, so that a new global action sequence $\beta$ can be found by applying the same search algorithm performed during the offline phase (as presented in Sec.~\ref{sec:Prelim}). This process leads to a global task reallocation that assigns new sub-tasks to all agents.}

\begin{theorem}
The synchronized team automaton preserves the properties of the original team automaton, i.e., correctness, independence, completeness and ordered sequence.
\end{theorem}

\begin{proof}
\textcolor{black}{The synchronized team automaton is distinguished from the original team automaton in four folds: 1) the set of initial states $S_{0,\mathcal{R}}$ is different since the initial PA set $S_{0,\mathcal{P}}^{(1)}(t)$ is updated; 2) the set of actions $A_{\mathcal{P}}^{(r)}(t)$ from each agent $r$ is updated according to the latest $\mathcal{P}^{(r)}(t)$; 3) the switch transitions $\zeta(t)$ are removed and then reconstructed after setting the current TS state as the latest initial state $s_{0,\mathcal{T}}^{(r)}(t)$ for each agent $r$; 4) the synchronized transitions are added according to the execution history. The first three changes won't affect the properties of the team automaton in that every state $s_{\mathcal{R}} \in S_{\mathcal{R}}$ still has an NFA component $s_{\mathcal{Q}}$ constructed from $\phi$, and the switch transitions $\zeta(t)$ are reconstructed under the same definition. As for the fourth change, according to the second condition of Definition \ref{def:synchronized_transition}, new transitions between NFA states are added by connecting the initial and final NFA states from each agent $r$'s execution history $\texttt{ExePath}_{\mathcal{Q}}^{(r)}$. Although these transitions are not directly provided by $\phi$, since the states in $\texttt{ExePath}^{(r)}$ have already been traversed by agent $r$, there always exists a sequence of valid NFA transitions between $\texttt{ExePath}_{\mathcal{Q}}^{(r)}({\rm init})$ and $\texttt{ExePath}_{\mathcal{Q}}^{(r)}({\rm final})$. Therefore, synchronized transitions do not change the validness of the original NFA transitions, but only skip the executed transitions. Consequently, all of the original properties, including correctness, independence, completeness, and ordered sequence, are preserved.}
\end{proof}

\section{BT Execution Layer}\label{bt_interface}
As described in Sec.~\ref{reallocation_approach}, both local and global task reallocation approaches enable robot recovery from various disturbances and failures. We leverage the reactivity property of BT at the middle level to rapidly select appropriate reactive strategies.

\begin{figure}[t]
    \centering
    \includegraphics[width=3.45in]{BT_icra2022.pdf}
    \caption{A behavior tree for three levels of reactive strategies is shown. Basic types of BT nodes are thoroughly described in \cite{DBLP:journals/corr/abs-1709-00084}. Yellow oval nodes are condition nodes, and green rectangular nodes are action nodes. R1 is the recovery stand for legged robots, R2 is the reallocation for critical failure, R3 is the reallocation for unexpected robot state change, and R4 is the reallocation for environmental change. Force Failure custom decorator node terminates the BT to execute an appropriate reactive strategy for the specific condition node.}
    \label{fig:bt}
    \vspace{-0.2in}
\end{figure}

\subsection{Reactive strategies}
The decision to execute one reactive strategy over another is determined by the categoried disturbances in Sec.~\ref{problem_formulation}. Four corresponding types of reactive strategies are specified below.

\begin{itemize}[leftmargin=*]
    \item \textbf{Recovery stand}. This scenario responds to a loss of balance which is specific to legged robots. In this case, the low-level controller on the robot attempts to recover from the fallen state without triggering the high-level LTL planner.
    \item \textbf{Task reallocation: critical failure}. Under this scenario, the robot is deemed incapable of working and therefore, a local task reallocation will be impractical. To resolve this issue, the robot will directly request a global task reallocation assuming a disability to transit to any TS state, to allow the remaining agents to take over its task.
    \item \textbf{Task reallocation: unexpected robot state change}. If this type of disturbance is detected, the planner will locally reallocate its task sequence without the assistance of other robots. If no local plans are feasible, the planner will perform a global task reallocation.
    \item \textbf{Task reallocation: environmental change}. If a robot encounters an environmental change, it will perform a local task reallocation. However, if a solution is not found, a global task reallocation will be performed.

\end{itemize}
\subsection{Behavior tree structure}
The BT structure is constructed in the form of precondition $\rightarrow$ action $\rightarrow$ effect, which is similar to the works of\cite{lan2019autonomous,li2021reactive}. This structure allows us to encode all of the reactive strategies into separate nodes within a single BT. \textcolor{black}{Two special types of nodes, Repeat and Reactive Sequence, are utilized in our BT. The former node allows a simpler BT construction instead of concatenating all actions into one big tree, while the latter node enables condition nodes to check for disturbances at varying frequencies (i.e., detection rates).} For example, as shown in Fig.~\ref{fig:bt}, checking locomotion failure, reactive state change, and environmental change would be executed continuously under the Reactive Sequence node, while checking LTL precondition would only occur prior to receiving a new task. The BT for a wheeled robot is similar but omitted due to space limits.

\section{Evaluation and Discussion}\label{evaluation}
\subsection{Experiment set-up for simulation and hardware}

To evaluate the feasibility and robustness of the proposed multi-robot task allocation and planning framework, we first establish a simulation of a hospital environment in Gazebo\cite{IgnitionFuel-OpenRobotics-Hospital} and create a topological map for defining the TS, as shown in Fig.~\ref{fig:map_hospital}. The simulation architecture is composed of a high-level LTL planning layer based on a ROS package from \cite{baran2021rosltl}, a mid-level execution interface using BehaviorTree.CPP\cite{BehaviorTreeCPP}, and a low-level navigation and controller layer using ROS navigation stack and appropriate controllers for each robot model. A convex model predictive controller from MIT Mini Cheetah \cite{di2018dynamic,kim2019highly} is used to control a Unitree A1 quadruped over rough terrain, while a conventional holonomic drive model is used on the UBTECH DR and Wassi robot. For global path planning, the robot navigates between regions using the A* algorithm \cite{hart1968formal} and performs collision avoidance with the dynamic window approach \cite{fox1997dynamic}.

\begin{figure}[t]
    \centering
    \includegraphics[width=3.5in]{system_definition-compressed.pdf}
    \caption{Operating state diagram for legged robot capable of both delivery and training task. Each node refers to robot operating states, whereas each edge refer to an action performed by the robot. Unitree A1 quadruped \emph{(top left)} and UBTECH DR \emph{(bottom left)} are shown as delivery robots, and A1 and UBTECH Wassi \emph{(top right)} are shown as training robots.}
    \label{fig:robot_definition}
    \vspace{-0.2in}
\end{figure}

\subsection{Case study}

We evaluate our framework on a heterogeneous team of robots consisting of a delivery robot DR, a walk training robot Wassi, and a quadrupedal robot A1 with both capabilities. As shown in Fig.~\ref{fig:robot_definition}, a delivery robot consists of two simple operating states (\emph{Loaded, Standby}), where the \emph{Standby} state is equivalent to \emph{Unloaded State}. Likewise, a training robot consists of 4 operating states (\emph{Standby, Camera On, User Located, Training}), where the robot visually locates and helps seniors who need walking training assistance. These operating states are encoded into TS for each type of robot.

We conduct a series of case studies in a hospital environment simulation to evaluate the reactive strategies proposed in Sec.~\ref{bt_interface}. The global mission is defined as:
\begin{figure}[t]
    \centering
    \includegraphics[width=\linewidth]{map-compressed.pdf}
    \caption{Topological map of a hospital environment consisting of locations that are explicitly defined in each robot's transition system. The execution history for each robot is depicted for achieving global task $\phi$ under a scenario where an unexpected garbage heap is detected. The timeline for task execution and reallocation is illustrated at the bottom.}
    \label{fig:map_hospital}
    \vspace{-0.15in}
\end{figure}

\begin{scenario}
``Deliver medicines to locations p3 and p6; meet a patient at c1; complete a walk training along the corridor between c1 and c6, and then send the patient back to p4."
\begin{subequations}
\begin{align*}
    \phi_1 = & \Diamond(p3 \wedge Standby) \wedge \square\,((\neg\,p3 \wedge \ocircle \,p3) \rightarrow Loaded) \\
    & \wedge \Diamond(p6 \wedge Standby) \wedge \square\,((\neg\,p6 \wedge \ocircle \,p6) \rightarrow Loaded) \\
    & \wedge \Diamond(p4 \wedge Standby) \wedge \square\,((\neg\,p4 \wedge \ocircle \,p4) \rightarrow Training)\\
    & \wedge \Diamond(c1 \wedge Training \wedge \Diamond\,(c7 \wedge Training \wedge \Diamond\,(c6 \wedge Tr \\
    &\text{-}aining \wedge \Diamond\,(c7 \wedge Training \wedge \Diamond\,(c1 \wedge Training)))))
\end{align*}
\end{subequations}
\end{scenario}

\begin{table}[t]
\caption{\centering Triggered times and averaged computation time for four different
recovery strategies R1 -- R4. }
\label{table_10_trials}
\begin{center}
\begin{tabular}{|c||c||c||c||c|}
    \hline
     & R1 & R2 & R3 & R4\\
    \hline
    Triggered times & 18 & 13 & 10 & 16\\
    \hline
    Local reallocation time (s) & - & - & 0.023 & 0.018\\
    \hline
    Global reallocation time (s) & - & 3.31 & 2.93 & 3.42\\

    \hline
\end{tabular}
\end{center}
\vspace{-0.15in}
\end{table}

Three robots, A1, DR and Wassi, are placed at various locations in the hospital before the start of the simulation. Next, the task planner decomposes the specification and assigns sub-tasks to each robot offline, \textcolor{black}{which takes 21 seconds in total. 92\% of the computation time is taken by the generation of PA for each robot, which only needs to be performed once.}

During the simulation, each robot will encounter a disturbance. Since the integration with exteroceptive systems is out of this paper's scope, all the disturbances except the locomotion failure are triggered by manually setting the checkers in BT to be false. A diagram of this simulation is displayed in Fig.~\ref{fig:map_hospital}. More details can be found in the video\footnote{\href{https://youtu.be/AO7Ms3GKzYQ}{https://youtu.be/AO7Ms3GKzYQ}}.

\emph{1)} External force is applied to A1 and induces a loss of balance. A handcrafted whole-body recovery stand trajectory is tracked by a PD controller to assist A1 to resume its task.

\emph{2)} A critical failure is induced for Wassi when performing the walk training task from $c7$ to $c6$. A global reallocation strategy assigns A1 to complete its delivery task first, and then proceeds to finish Wassi's incomplete walk training task.

\emph{3)} The \emph{Loaded} state of DR suddenly becomes a \emph{Standby} state. This simulates a situation in which the robot unexpectedly loses its cargo. A local reallocation succeeds in instructing the robot to return to $s1$ and pick up another cargo. Then DR is instructed to complete the original delivery task.

\emph{4)} A garbage heap is placed in front of $p3$ to simulate an environmental change. This obstruction can only be traversed by the legged robot A1. The routes taken by each robot and the timeline for obstacle detection and task allocation are portrayed in Fig.~\ref{fig:map_hospital}. While performing its task, DR encounters the obstruction and fails to find an alternate plan via local reallocation. While A1 is returning home after completing the delivery task to $p6$, it is assigned to take over DR's incomplete delivery task at $c4$. As a result, A1 goes to $s1$ to retrieve the object for delivery, and completes the task by traveling to $p3$.

\textcolor{black}{In addition to applying individual disturbances, we also evaluate the the same scenario ten times with multiple manually triggered adversarial disturbances. Furthermore, the initial configuration for each robot is modified to prompt a different offline allocation result, for evaluating the generalization of our approach. At run-time, signals indicating robot state and environmental changes are sent to each robot's BT and prompt their reactive behaviors.
As shown in Table~\ref{table_10_trials}, we report the number of times that the reactivity strategy is triggered and the average computation time it takes for each trial. The LTL planner is not responsible for locomotion failure (R1) and the critical failure (R2) can only be handled at global level.
Although a global task reallocation in principle could result in an optimal task sequence according to Proposition 3, it is observed to be more computational expensive compared with the local reallocation. This is caused by a communication among all agents and extra computation for constructing the synchronized team automaton. Our framework demonstrates an 80\% success rate of completing the global mission $\phi$. The 20\% failed cases are caused by mission-level failures when more than one robot undergoes a critical failure, which causes the remaining task to be  unachievable.
}

\begin{figure}[h]
    \centering
    \includegraphics[width=0.6\linewidth]{scalability.png}
    \caption{The recorded planning time for offline allocation, local and global reallocation given different numbers of robots.}
    \label{fig:scalability}
\end{figure}

\textcolor{black}{To evaluate the scalability of our proposed framework, we deploy 3 (described above), 9, 15, and 30 robots in the same aforementioned scenario. Due to the high computation demand from the quadruped controller, we don't run the simulation but directly trigger the LTL planner to perform reallocation strategies. In all scalability tests, the delivery robot is assumed to encounter an abrupt state change that triggers local reallocation and a critical failure that triggers global reallocation. Without significant code optimization, Fig.~\ref{fig:scalability} reveals a linear complexity for both offline allocation and global reallocation, while the time required for local reallocation maintains at the same level. An alternative but \textit{ad hoc} way for reallocation is to reconstruct the entire team model with an updated specification, which is similar to repeating the expensive offline process and requires human designer knowledge in the loop.
}

\section{Conclusion and Discussions}
In this work, we present a heterogeneous, multi-robot task allocation and planning framework equipped with a hierarchically reactive mechanism from extensive disturbances. A local and global task reallocation is performed at the high level where an LTL-based team automaton is generated to follow a formal guarantee. At the middle level, a BT framework is incorporated to promptly select different replanning strategies which can be executed at different rates. Lastly, all the work mentioned is showcased in a dynamic simulation of a hospital scenario involving quadrupeds and wheeled robots.

\textcolor{black}{In certain scenarios, a global task reallocation could result in a more optimal task sequence than the one generated from a local reallocation. For instance, if a robot encounters a blocked path, a local strategy will first attempt to find a detour. However, if the task could be transferred over to a different robot closer to the destination, the global task reallocation will outperform the local one.
Our current framework does not consider optimality over computational efficiency. For our future work, we will incorporate a method to take into account optimality to determine whether replanning for the whole team over a single robot is favourable.}

\bibliographystyle{IEEEtran}

redpajama_set_name: RedPajamaArXiv


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Research Patient Care Press Releases
Skin Disease Researchers Win Renewed Grants
National leaders in research to cure debilitating skin diseases based at the Gates Center for Regenerative Medicine on the University of Colorado Anschutz Medical Campus have won a second NIH grant to further investigations of innovative treatments.
AVITA Medical Teams With Gates Center to Advance Therapeutic Skin Restoration
AVITA Medical (ASX: AVH, NASDAQ: RCEL), a regenerative medicine company with a technology platform positioned to address unmet medical needs in therapeutic skin restoration, and scientists at the Gates Center for Regenerative Medicine at the University of Colorado School of Medicine have announced a preclinical research collaboration to establish proof-of-concept and explore further development of a spray-on treatment of genetically modified cells for patients with epidermolysis bullosa (EB), with potential applicability to other genetic skin disorders.
Patient Care Education Community
CU Anschutz Grant Will Train Students to Prescribe Drugs That Fight Opioid Addiction
As the nation struggles with a persistent epidemic of opioid addiction, the University of Colorado Anschutz Medical Campus is taking a critical step that will allow its graduates to better treat patients and save lives.
David Kelly | November 21, 2019
Researchers Create Model to Predict Children Likely to Go Into Septic Shock
Researchers at the University of Colorado Anschutz Medical Campus have developed a unique model allowing them to predict which children arriving in emergency departments are most likely to go into septic shock, a life-threatening condition.
Heroes Clinic a 'Godsend' For Veterans Seeking Dental Care
John Berdeaux came up with a simple solution to his complex dental problem: chew a Chiclet and put it where his front tooth should be.

redpajama_set_name: RedPajamaCommonCrawl


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Curren$y Previews The Alchemist Collab Following The Game & Hit-Boy Studio Session
Gunna Reacts To Edging Out The Weeknd For Billboard 200 Supremacy: 'This Is The Big One!'
Kanye West & Big Sean Reunite Following 'Drink Champs' Spat
Ja Rule & Ludacris Dust Off 2000s Hits At Dwyane Wade's 40th Birthday Party
Key Glock Mourns Young Dolph On New Song 'Proud': 'I'm Really Lost Inside'
Logic Begins Work On New Album 'Vinyl Days' - With JAY-Z's Help
Kanye West's 'Eazy' Threat Prompts Pete Davidson To Beef Up His Security
Eazy-E's Daughter Co-Signs Kanye West & The Game's 'Eazy': 'It Don't Get More Gangsta'
Moneybagg Yo Looking To Follow In 50 Cent's $100M Footsteps With His Own Water Brand
Most Discussed News
10 Gunna Emerges Victorious Over The Weeknd After 'DS4EVER' Debuts At No. 1 On Billboard 200
9 Glasses Malone Insists 'Kanye Should've Never Married (That B*tch)'
7 Kanye West's 'Eazy' Threat Prompts Pete Davidson To Beef Up His Security
7 Houston Rapper Sad Frosty Dead At 24
6 Ice Cube To Perform At Upcoming NASCAR Race: 'I'm Excited To Be Part Of An Incredible Day'
6 Eazy-E's Daughter Co-Signs Kanye West & The Game's 'Eazy': 'It Don't Get More Gangsta'
6 Treach Talks Resurfaced Rapper Report Cards After Scoring Higher Than JAY-Z, Nas, Redman & Snoop Dogg
5 Cordae's New Album Gets Australian Open Love From Tennis Star Girlfriend Naomi Osaka
5 Big Daddy Kane Reveals Why He Shoved ASL Interpreter During Concert — & The Part People Didn't Post
5 Pharrell Challenges Corporate America To 'Do More' During MLK Event
53 Eminem Uses Meek Mill As Fuel For 6ix9ine Diss On Cordae 'Parables' Collab
46 T.I. Declares He's Never Been 'Dusted On A Record' While Naming JAY-Z, Nas & Kanye West
46 Jim Jones Demands Jadakiss Verzuz Following LOX & Dipset Battle: 'I Don't Like How He Did That'
40 A Drunk Wendy Williams 'Stripped Naked + Masturbated' In Front Of Manager Before Hospitalization
32 Kanye West Threatens To 'Beat Pete Davidson's Ass' Over Kim Kardashian On The Game 'Eazy' Single
27 NBA YoungBoy Disses King Von - & Gets Response From Lil Durk
21 JAY-Z Sets The Record Straight On 'In My Lifetime, Vol. 1' With The Help Of Questlove
19 Drake, Kendrick Lamar, J. Cole & Nicki Minaj Immortalized With Mt. Rushmore Murals
19 Nas Facing Potential Pete Rock Lawsuit Over 'Illmatic' Royalties
18 Kanye West Airs Out Kim Kardashian Over Daughter's Birthday Party Ban - Only To Show Up Anyway
110 The Best Hip Hop Albums of 2022 ...(so far)
104 Jaden Smith Admits He Gained 10lbs After Family Staged Intervention
62 The Best Rap Albums of 2022 ...(so far)
54 Rapper Drakeo The Ruler Confirmed Dead At 28 After Being Stabbed At L.A. Festival
49 Royce Da 5'9 Dubs Lil Xan Hip Hop's 1st Male 'Karen'
49 Corey Holcomb Claims Rick Ross Ditched 85 South Interview After Realizing 'Ain't None Of These N-ggas Gay'
49 Killer Mike 'Angry & Upset' After Atlanta Barbershop Vandalized By Conspiracy Theorist
49 Nas Calls J. Cole, Drake & Kendrick Lamar 3-Headed GOAT Successors For Biggie, JAY-Z & Himself
Kevin Mazur/WireImage
Alicia Keys Says JAY-Z Had To Tell Her About Lil Mama's Classic MTV Stage Crash
Published on: Dec 8, 2021, 3:50 PM
by Kyle Eustice
The MTV Video Music Awards typically provide at least one unforgettable moment — whether it's Kanye West boldly interrupting Taylor Swift's acceptance speech in 2009 or Madonna kissing Britney Spears on the mouth in 2003. Now, Alicia Keys is opening up about a night she'll never forget at the 2009 VMAs when Lil Mama infamously crashed her performance of "Empire State of Mind" with JAY-Z.
During a recent interview with Drink Champs, the Grammy Award-winning singer recalled completely missing Lil Mama's burst of excitement altogether; she was too focused on their performance. But Hov? Hov was annoyed and showed it all over his face.
"The truth of all the truths — all I know, we was on that stage," she said. "Jay was here, I was here. In my head, I was such a gorilla at the time and I was like, 'This is what's happening. I'm going to destroy this record tonight.' … Somehow, however she got over here, I didn't even bear witness to.
"The whole show I was iced — so focused on making it amazing. I went backstage and Jay was like, 'So, you ain't see that?' And I was like, 'What?' And he was like, 'Nah, you ain't just see what just happened?' And I was like, 'Yeah, we killed it. That's what happened.'"
A post shared by Drink Champs (@drinkchamps)
In October, Alicia Keys told the Morning Hustle she has long forgiven Lil Mama who was just a wide-eyed teenager at the time.
"Listen, to this day, I truly never knew she was even on the stage — that's the craziest part of the whole thing for me," she said. "I was on that stage, Jay was in the middle, she was on the other side of Jay, and I never saw her. Never. So umm, that was wild to me. But I absolutely — it's not even a statement.
Alicia Keys Reveals 'Empire State Of Mind' With JAY-Z Almost Didn't Happen
"That was then. You know, she's from New York, too. You know, I think she really felt the energy of the record and she loved it and she couldn't hold herself back. Look, this is what my music does to people. What can we say?"
Still, the stage crash resulted in a now-classic photo that shows Lil Mama posted up to JAY-Z in a b-boy pose with Alicia Keys on the other side.
Revisit the clip below and keep scrolling to see Beyoncé try to stop Lil Mama as well.


Exercise recovers the tone and class of muscle tissue and encourages the method of digestions, absorption, metabolism, and elimination.
The key to any sort of exercise may be a robust can and a sincere want to boost one's wholeness by weight loss. The most effective thanks to lose weight is doing activities or exercises that you just get pleasure from and might commemorate with.
An ideal weight loss program might embody many alternative sorts of the subsequent exercises. Calisthenics encompass lightweight exercises as well as sit-ups, push-ups, jumping jacks, that promote stress of building skeletal muscles. Dancing or rhythmic exercise is usually pleasant thanks to exercise and lose weight, by toning the body. Isometrics involves the pressure of a muscle or cluster of muscles against one another or an immoveable object. Stretching may be a natural exercise that ought to be practiced on an everyday basis, and may be a smart habit to develop. It will recover energy and endurance, encourage circulation, and improve stiffness. Walking is one amongst the most effective overall weight loss exercises that facilitate the complete system operate higher. Weight Lifting and exercise strengthen muscle tone, that is suggested with any weight loss management arrange.
Above all, don't forget the recreational exercises as well; this may facilitate to stay you on task along with your weight loss goals.
The next step in which will assist you is to feature correct nutrition to your weight loss management arrange. You wish to know what your body wants and the way to consume the right amounts of Carbohydrates, Fats and Proteins, that are the first sources of energy to the body. They provide the fuel necessary for body heat and work. This can be essential to quick weight loss. Therefore, foods that are high in energy worth are high in calories, whereas foods that are low in energy worth are low calories. Fats yield regarding nine calories per gram; carbohydrates and protein yield regarding four calories per gram.
By following the present food pyramid guide, you'll be able to scale back calories and with combined exercise (minimal), you are taking lose weight quick and simple. Keep in mind to stay your body hydrated by drinking lots of water and avoid soda, fattening foods like chips, and cookies, etc.
1- Eat six mini meals a day- this may permit your body to eat additional frequently curbing your appetite for snacking.
2- Drink a minimum of sixty four oz of water every day; however the quantity you drink ought to be primarily based on your weight and size.
3- Select healthy foods from every food cluster.
4- Embody some weight lifting in your weight loss management arrange, this may permit you to stay correct muscle tone whereas losing weight.
Hello I am Lisa Rose for New York. I have been spending most of my free time in writing, something that I hardly did before and I am enjoying it!


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Census Tract 2589, Eastpointe, Michigan
Census Tract 2589 is a Low-Income CommunityOpportunity Zone located in Eastpointe, Michigan.
This 0.4 square milecensus tract has a population of approximately 2,800and is one of 17 Opportunity Zones in Macomb County.
The adjacent map shows the location of this Opportunity Zone in Michigan.
Opportunity Zone Demographics
Michigan OZ Funds
List of Similar Opportunity Zones
Census Tract 2589 Demographics
The charts below illustrate how this Opportunity Zone compares to the broader city, county, and state on various socioeconomic indicators:
This Opportunity Zone has a median household incomeof approximately $46,000, which is 23% lowerthan the median household income for the state of Michigan of $60,000.
The percentage of households below the poverty line in this Opportunity Zone is 9.6%, which is 3% lowerthan the rate for the state of Michigan of 13%.
This Opportunity Zone has a median home valueof approximately $70,000, which is 59% lowerthan the median home value for the state of Michigan of $170,000.
This Opportunity Zone has a percentage of population with high school diplomaof approximately 85%, which is 6% lowerthan the rate for the state of Michigan of 91%.
This Opportunity Zone has a median ageof approximately 32, which is 20% lowerthan the median age for the state of Michigan of 40.
There are 6QOZ Fundsin the OpportunityDb database with an investment objective that specifically identify Michiganas a target market. Additional OZ funds may also invest in Michigan; see a complete list of Opportunity Zone Funds here.
Accredited Capital Real Estate Multi-Family Housing, Residential, Single-Family Housing $25M
Beacon Rose Opportunity Zone Fund Real Estate Industrial, Mixed-Use, Multi-Family Housing, Residential, Student Housing $100M
Greatwater Opportunity Capital Detroit Opportunity Fund Real Estate Commercial, Mixed-Use, Multi-Family Housing, Residential $100M
Urban Array Opportunity Fund Business, Real Estate Commercial, Office, Residential $5M
Wefunder Detroit Opportunity Fund Business $100K
There are no Opportunity Zone Funds in the OpportunityDb databasewith a focus specifically on Michigan properties.
Similar Opportunity Zones
See all OZs in Michigan
See all OZs in Macomb County
See all OZs in Eastpointe


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Q: Java Can l import a class to another class? l want to use variables from one class to another. How can l achieve this? I dont want to inherit from the other class. Is there any way to import the class / use package maybe? Haven't used packages yet so l dont know how they work.
(Using InteliJ Idea)
l am talking about classes made by us not already existing. Example - l make class Person and class Commands and l want to use fields from class Person in class Commands. I can't extend it because the user shouldn't be able to getID() in the Commands class. How can l get the fields in the Person class without inheriting?

A: I would suggest creating an instance variable in the class that you want to use this instance variable, Although, the class must be declared public to be available to all other packages, and make an import of the package where the class resides

redpajama_set_name: RedPajamaStackExchange


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Google parent Alphabet to acquire Fitbit for $2.1 billion
Daniel Howley
·Technology Editor
Google's parent company Alphabet (GOOG, GOOGL) is acquiring health tracking company Fitbit (FIT) for $2.1 billion. The deal will pay out $7.35 per Fitbit share, which was priced at $6.16 in pre-market trading Friday. The news follows an earlier report that the two firms were set for a tie up.
Fitbit CEO James Park unveils Fitbit's second smartwatch, Fitbit Versa, and first-ever device for kids, Fitbit Ace, along with the Fitbit family account and female health tracking at its launch event in New York, Monday, March 12, 2018. The newest devices and features from Fitbit support the company's vision of making the world healthier, while reaching more people in unique ways to continue to help them achieve their health and fitness goals. (Diane Bondareff/AP Images for Fitbit)
Fitbit was up 16% in pre-market trading on the news. Alphabet was up 0.3%.
The move gives Alphabet, which has struggled to gain traction in the growing wearables market, a much needed leg up. Fitbit would, naturally, benefit from the additional resources at Alphabet's disposal.
"Over the years, Google has made progress with partners in this space with Wear OS and Google Fit, but we see an opportunity to invest even more in Wear OS as well as introduce Made by Google wearable devices into the market," said Rick Osterloh, Google's senior vice president of devices and services.
"Fitbit has been a true pioneer in the industry and has created engaging products, experiences and a vibrant community of users. By working closely with Fitbit's team of experts, and bringing together the best AI, software and hardware, we can help spur innovation in wearables and build products to benefit even more people around the world," he added.
Google lags in wearables
Google was one of the first big-name players to jump into the smartwatch space with its Android Wear operating system. The company has since rebranded its wearable operating system to WearOS, but its market share is negligible at best.
The Fossil Group, which is the company that uses WearOS the most, saw its global share of the smartwatch market fall from 3.2% in Q1 2018 to 2.5% Q1 2019, according to Counterpoint Research. Fitbit, on the other hand, saw its share rise from 3.7% in Q1 2018 to 5.5% in Q1 2019.
Canalys, meanwhile, pegs Fossil Group's share of the North American wearables market at 4.1% as of Q2 2019, while Fitbit topped out at 24.1%. No matter how you slice it, Google is just a bit player in the wearables space.
The Google Wear OS watches are on display at CES International, Tuesday, Jan. 8, 2019, in Las Vegas. (AP Photo/Ross D. Franklin)
But Fitbit has had issues of its own, namely Apple. The Apple Watch maker dominates both the domestic and global smartwatch market. And while Fitbit was once able to undercut Apple's pricing, the Cupertino-based company's decision to lower the price of its Apple Watch Series 3 to $199 puts it on the same pricing level as Fitbit's new Versa 2.
For Google, adding Fitbit could be a boon to its fitness tracking efforts.
"Google WearOS and Android Wear has been one of the first smartwatch platforms around, however, its health and fitness components have lagged behind," IDC research director Ramon Llamas said during an interview after news of Google's interest in Fitbit broke earlier this week.
"And despite some updates and injection of talent, the health and fitness part hasn't moved significantly."
Adding Fitbit's fitness tracking and hardware capabilities to Google, however, could reverse the search giant's fortunes in the wearables space.
Fitbit has more than hardware
For most consumers, Fitbit is best known for its fitness trackers and smartwatches, but the company has been steadily pushing into the software market. It has inked deals with insurance providers to help subscribers improve their health, and recently announced a partnership with Bristol-Myers Squibb and Pfizer to develop technologies to detect atrial fibrillation within users.
In August, Fitbit debuted its Fitbit Premium subscription service, which provides subscribers with customized fitness and health guides based on their goals. Available for $9.99 per month or $79.99 for a full year, Fitbit Premium is being positioned. as a key differentiator for the company compared to Apple, which doesn't offer such a custom service.
Google, on the other hand, may be able to leverage Fitbit's efforts to build on its own health initiatives including its Google Fit service.
More from Dan:
Apple TV+ is now live — Here's how it compares to Disney+, HBO Max, Netflix, and others
Amazon's Echo Buds put Alexa in your ears — when they fit
Apple set to report Q4 earnings as iPhone 11 sales hopes build
The best features of the AirPods Pro
Got a tip? Email Daniel Howley at dhowley@yahoofinance.com, and follow him on Twitter at @DanielHowley.
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WRAPUP 1-Biden White House, McCarthy dig in ahead of debt meeting
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Yahoo Finance Live anchors discuss fourth-quarter earnings for Pfizer.


Established over 250 years Villeroy & Boch offer a complete bathroom solution with stunning sanitaryware, furniture solutions, baths and tiles to suit all tastes and budgets. Zest Bathrooms in Chester has a dedicated Villeroy & Boch studio within their showroom showcasing the latest ranges.
Visit our showroom now to view the Villeroy & Boch range.


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Visit Aquileia
Aquileia UNESCO Heritage
Opening times and entrance fees
Facilitations
Information kit
Arte.it guide
Aquileia 3D
Pictures of Aquileia
River Port
Early Christian Museum
Fondo CAL
Fondo Pasqualis Markets
Domus and Bishop's Palace
Südhalle
Decumanus of Aratria Galla
Wounded Archaeology
The Basilica of Aquileia
The Basilica of Aquileia makes an extraordinary architectural complex. The church today is essentially the result of the architectural works promoted by bishop (patriarch) Poppo (1019- 1042), who consecrated it in 1031. However, its origins go back to the second decade of the 4th century. By the end of the same century, under the rule of bishop Chromatius, the edifice had grown into the current dimensions, except for the transept and apse with the underground crypt.
A remarkable construction step comes in the early 9th century, under Patriarch Maxentius, who added a porch (originally on two floors) to connect the basilica with the Pagans' Church, as well as the two jutting portions of the transept, and the apse.
The reorganization promoted by Poppo in the 11th century included the refurbishing of the façade with a central mullioned window, and the fresco decoration of the apse. The two sets of 10 columns were provided with new capitals and a 73 m-high campanile was erected. Further works were carried out after an earthquake in 1348.
In 1909 the floor of Poppo's times was excavated up to the sanctuary portion to bring to light a mosaic floor belonging to the church erected during the rule of emperor Constantine, which was lying about one metre below.
• Mosaic floor: the mosaic floor of the basilica, discovered between the late 19th century and the first decades of the 20th century, offers a most stunning evidence of the original place of worship built under the initiative of bishop Theodore immediately after the Edict of Toleration of 313 AD. The church used to include two parallel halls (the second one can be seen in the Crypt of the Excavations), a connecting hall, an elliptic baptistery (no more visible today) and several rooms, probably related to the bishop's residence.
• Seven hundred and sixty square metres wide, the mosaic of Theodore's southern hall makes the largest mosaic floor of the Roman West. Divided in four sections and bearing an inscription with Theodore's dedication, the mosaic offers, with its allegories and images not always easily interpreted, an ideal itinerary towards Salvation. In particular, the eastern section is entirely covered with the representation of the biblical story of Jonah.
• Crypt of the frescoes: the crypt lying below the main altar of the basilica dates back to the time of patriarch Maxentius (9th century) and was created as a shrine to hold the relics of martyrs Hermagoras and Fortunatus from Aquileia. The walls are decorated with 12th century frescoes depicting the legend of Saint Mark the Evangelist.
• Crypt of the excavations: excavations here have made visible the foundations of Poppo's campanile standing on the mosaics of Theodore's northern hall (early 4th century).
The basilica complex also includes the campanile erected by patriarch Poppo in the 11th century, a model for other similar constructions in Friuli and Istria alike; the baptistery built under bishop Cromatius in the 4th century, with a hexagonal basin inside an octagonal structure; the so-called Südhalle, i.e. the southern hall of the early Christian baptistery recently turned into a museum, with over three hundred square metres of mosaic floor dated between the late 4th and early 5th centuries, and the wonderful mosaic "of the Peacock" coming from the narthex area of the basilica.
Visit the website dedicated to the Basilica of Aquileia http://basilicadiaquileia.it/it/basilica
3D Reconstructions
You'll receive updated information on exhibitions and events in Aquileia, news from the world of archaeology and a lot more.
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