Read and understand new problem statement formulation using UDQDG

[yzsshen]

\documentclass{article}
\usepackage{graphicx}
\usepackage{amsfonts}
\usepackage{amsmath}
\usepackage{amsthm}

\newtheorem{lemma}{Lemma}

\title{Damage-Responsive Reconfiguration}

\begin{document}

\maketitle

\section{Problem Formulation}

We represent the modular spacecraft at time $t$ as a
\emph{Unit Dual Quaternion Directed Graph} (UDQDG)
[
\mathcal{G}t = (V_t, E_t, m_t, Q_t, \hat q_t),
]
with:
\begin{itemize}
\item $V_t$ the set of modules (vertices).
\item $E_t \subset V_t \times V_t$ the set of directed edges representing physical connections.
\item $m_t: V_t \to {0,1}$ the \emph{activity map} (active $=1$, inactive $=0$).
\item $Q_t: V_t \to \mathbb{S}^3$ (optional) the \emph{absolute attitude} (unit quaternion) per vertex; $Q_t$ is an independent attribute and does not affect geometry below.
\item $\hat q_t : E_t \to \mathbb{DH}u$ the \emph{edge gains}, restricted to
\emph{pure translations}:
[
\hat q{uv} ;=; 1 + \frac{\varepsilon}{2},q^{\mathrm{tr}}{uv}, \qquad
q^{\mathrm{tr}}{uv} = (0,; \mathbf t{uv}), \ \ \mathbf t_{uv}\in\mathbb{R}^3,
]
where $\varepsilon^2=0$ and $q^{\mathrm{tr}}{uv}$ is a pure imaginary quaternion encoding the translation vector $\mathbf t{uv}$. Gains satisfy
[
\hat q_{vu} ;=; \hat q_{uv}^{-1} ;=; 1 - \frac{\varepsilon}{2},q^{\mathrm{tr}}{uv},
]
i.e., $\mathbf t{vu} = -,\mathbf t_{uv}$.
\end{itemize}

\paragraph{Active connectivity.}
The \emph{active subgraph} is
[
\mathcal{G}^A_t = (V^A_t, E^A_t),\qquad
V^A_t = {v \in V_t \mid m_t(v)=1},\quad
E^A_t = {(u,v)\in E_t \mid m_t(u)=m_t(v)=1}.
]
Only edges in $E^A_t$ participate in reconfiguration.

\paragraph{Geometric consistency (translation-only).}
Along any active path
$P : r=v_0 \to v_1 \to \cdots \to v_k=v$,
the cumulative transform is
[
\hat q_{rv} ;=; \prod_{i=0}^{k-1} \left(1 + \frac{\varepsilon}{2} q^{\mathrm{tr}}{v_i v{i+1}}\right)
;=; 1 + \frac{\varepsilon}{2}\sum_{i=0}^{k-1} q^{\mathrm{tr}}{v_i v{i+1}},
]
since $\varepsilon^2=0$. Equivalently, positions (up to a global translation) are obtained by summing translations:
[
\mathbf p_v ;=; \sum_{i=0}^{k-1} \mathbf t_{v_i v_{i+1}} \in \mathbb{R}^3.
]
Path-independence holds iff every directed cycle $C$ in $\mathcal{G}^A_t$ satisfies the closure condition
[
\sum_{(i\to j)\in C} \mathbf t_{ij} ;=; \mathbf 0
\quad\Longleftrightarrow\quad
\prod_{(i\to j)\in C} \hat q_{ij} ;=; 1.
]

\paragraph{Lattice constraint.}
To enforce a lattice, constrain each translation to be a unit step in a discrete direction set:
[
\mathcal{D} \subset \mathbb{S}^2, \qquad
|\mathbf t_{uv}| = 1, \ \ \frac{\mathbf t_{uv}}{|\mathbf t_{uv}|} \in \mathcal{D}, \ \ \mathbf t_{vu}=-\mathbf t_{uv}.
]
For a cubic lattice, $\mathcal{D} = {\pm \hat{\mathbf x},,\pm \hat{\mathbf y},,\pm \hat{\mathbf z}}$.

\subsection*{Damage-Responsive Reconfiguration Objective}
A \emph{damage event} at time $t$ changes activity of one or more vertices:
[
m_{t+1}(v) = 0 \quad \text{for some } v \text{ with } m_t(v)=1.
]
Inactive vertices remain physically connected but cannot pivot or serve as pivot axes.

The goal is to find a finite sequence of allowable \emph{pivot operations} that restores connectivity of the active graph in the fewest steps.

\subsection*{Pivotability Rules}
A vertex $u\in V_t$:
\begin{itemize}
\item \textbf{Can pivot} if $m_t(u)=1$ and all neighbors $v$ with $(u,v)\in E_t$ are also active.
\item \textbf{Can be pivoted on} if $m_t(u)=1$.
\end{itemize}
Inactive vertices cannot serve as pivot axes.

\subsection*{Pivot Operations in Dual-Quaternion (Translation) Space}
A \emph{pivot} updates translations without using lattice coordinates:
\begin{enumerate}
\item Choose an active pivot axis $p$ and an active neighbor $u$ with $(p,u)\in E^A_t$.
\item Detach $u$ from $p$ (remove $(p,u)$ and $(u,p)$) and reattach at a new unit direction $\tilde{\mathbf t}\in\mathcal D$ by inserting edges with gains
[
\hat q'{pu} = 1 + \frac{\varepsilon}{2}(0,\tilde{\mathbf t}), \qquad
\hat q'{up} = \left(\hat q'_{pu}\right)^{-1} = 1 - \frac{\varepsilon}{2}(0,\tilde{\mathbf t}).
]
\item Treat the connected component of $u$ (excluding $p$) as a rigid substructure in the \emph{relative} sense: internal edges keep their translations; only the cut edge at $p$ is changed.
\end{enumerate}
The transition set $T$ consists of all such updates that maintain validity.

\subsection*{Optimization Problem}

Given $\mathcal{G}t$ at time $t$ and a damage event producing the activity update $m{t+1}$, find an integer
[
t_f \ge t+1
]
and a sequence
[
{\mathcal{G}{t+1}, \mathcal{G}{t+2}, \dots, \mathcal{G}{t_f}}
]
such that:
\begin{align*}
\min{{\mathcal{G}{\tau}}{\tau=t+1}^{t_f}} \quad & t_f \
\text{s.t.}\quad
& (\mathcal{G}{\tau}, \mathcal{G}{\tau+1}) \in T
&& \forall \tau \in {t+1,\dots,t_f-1} \quad \text{(valid pivot transitions)}, \
& \mathcal{G}{\tau} \in S
&& \forall \tau \in {t+1,\dots,t_f} \quad \text{(all intermediate states allowable)}, \
& \mathcal{G}^A{t_f} \ \text{is connected}
&& \text{(final active graph connected)}, \
& \mathcal{G}_{t+1} \ \text{matches the damage at time } t
&& \text{(boundary condition)}.
\end{align*}

\noindent
Here, $S$ is the set of structurally valid UDQDG states under the translation-only restriction (unit dual quaternion edges with identity rotation; unit-length lattice-constrained translations; and cycle-closure where required), and $T$ is the set of allowable pivot transitions that respect pivotability and preserve validity.

\subsection*{Graph-Theoretic Pivoting Rules (In Progress)}
Each directed edge $(u,v)\in E_t$ carries a unit dual quaternion $\hat q_{uv}$ representing a pure translation, with
[
d = \mathrm{trans}(\hat q_{uv}) \in \mathcal{D}, \qquad
\hat q_{vu} = \hat q_{uv}^{-1}, \quad \mathrm{trans}(\hat q_{vu})=-d.
]
Here $\mathcal{D}={\pm \hat{\mathbf{x}}, \pm \hat{\mathbf{y}}, \pm \hat{\mathbf{z}}}$ is the set of allowable lattice directions.
Each vertex face may be occupied by at most one edge (\emph{port exclusivity}).
A vertex $u$ is \emph{pivotable} if $m(u)=1$ and all its neighbors are active.

\vspace{1ex}
\noindent\textbf{Corner Pivot.}
A \emph{corner pivot} moves a module $u$ around a corner of its neighbor $v$, changing which face of $v$ it attaches to:
\begin{itemize}
\item Current edge: $(u,v)$ with $d=\mathrm{trans}(\hat q_{uv})$.
\item Choose a new direction $d'\in\mathcal{D}$ such that:
\begin{enumerate}
\item $d\cdot d'=0$ (orthogonal directions).
\item $v$ has no other neighbor using face $-d'$.
\item $u$ has no other neighbor using face $d'$.
\item $m(u)=m(v)=1$ and all neighbors of $u$ are active.
\end{enumerate}
\item Update the edge:
[
\hat q_{uv} \leftarrow \text{UDQ}(d'), \qquad
\hat q_{vu} \leftarrow \hat q_{uv}^{-1}.
]
\end{itemize}

\vspace{1ex}
\noindent\textbf{Flat Pivot.}
A \emph{flat pivot} rolls a module $u$ off of its neighbor $v$ and onto a different neighbor $w$, while keeping the same face direction $d$:
\begin{itemize}
\item Current edge: $(u,v)$ with $d=\mathrm{trans}(\hat q_{uv})$.
\item Select another neighbor $w$ of $v$ such that:
\begin{enumerate}
\item $w$ has no other neighbor using face $-d$.
\item $u$ has no other neighbor using face $d$.
\item $m(u)=m(v)=m(w)=1$ and all neighbors of $u$ are active.
\end{enumerate}
\item Update the connection:
[
\text{remove }(u,v),\quad
\text{add }(u,w)\text{ with }\mathrm{trans}(\hat q'{uw})=d,\quad
\hat q'{wu} = (\hat q'_{uw})^{-1}.
]
\end{itemize}

\noindent
In both pivot types, only the specified edge(s) are updated; all other edges remain unchanged. These operations preserve the lattice constraints and respect the port exclusivity rules.

\subsection*{Connectivity Recovery Lemma}
\begin{lemma}[Connectivity-Preserving Reconfiguration]
Let $\mathcal{G}^A_t$ be disconnected immediately after a damage event at time $t$. Suppose there exist components $C_1,C_2\subset V^A_t$ and a vertex $u\in C_1$ such that:
\begin{itemize}
\item $u$ is pivotable,
\item there exists a pivot operation assigning a new pure-translation gain $\hat q'{pu}$ whose translation is $\tilde{\mathbf t}\in\mathcal D$ so that $u$ becomes adjacent (unit step) to some $v\in C_2$ with $m_t(v)=1$,
\end{itemize}
then there exists a valid transition $\mathcal{G}{t+1}$ such that $\mathcal{G}^A_{t+1}$ is connected.
\end{lemma}

\end{document}