Add DAS average consensus proof

src/year2/distributed-autonomous-systems/das.tex

@@ -45,6 +45,9 @@
 \def\s{{\vec{s}}}
 \def\u{{\vec{u}}}
 \def\D{\ensuremath{\mathcal{D}}}
+\def\A{{\matr{A}}}
+\def\w{{\vec{w}}}
+\def\R{\ensuremath{\mathbb{R}}}
 
 
 \begin{document}

src/year2/distributed-autonomous-systems/sections/_averaging_systems.tex

@@ -91,6 +91,171 @@
 
 \subsection{Consensus}
 
+% \begin{remark}
+%     The distributed consensus algorithm is a positive system (i.e., $\matr{A}$ is positive).
+% \end{remark}
+
+\begin{description}
+    \item[Positive matrix characterization]
+        Given $\A \in \mathbb{R}^{N \times N}$, it can be:
+        \begin{description}
+            \item[Non-negative] \marginnote{Non-negative matrix}
+                $\A \geq 0$.
+            \item[Irreducible] \marginnote{Irreducible matrix}
+                $\sum_{h=0}^{N-1} \A^h > 0$.
+            \item[Primitive] \marginnote{Primitive matrix}
+                $\exists h \in \{ 1, \dots, N \}: A^h > 0$.
+            % \begin{remark}
+            %     A graph with a primitive adjacency matrix is connected.
+            % \end{remark}
+            \item[Positive] \marginnote{Positive matrix}
+                $\A > 0$.
+        \end{description}
+\end{description}
+
+
+% Equilibrium:
+% \[
+%     \x^{k+1} = \matr{A}\x^k
+% \]
+% \[
+%     \begin{split}
+%         \x_\text{eq} = \matr{A} \x_\text{eq} \\
+%         \iff (\matr{I} - \matr{A}) \x_\text{eq} = 0
+%     \end{split}
+% \]
+% We are interested in the null space of $(\matr{I} - \matr{A})$ (at least one eigenvector).
+
+
+
+\begin{theorem} \label{th:positive_matrix_digraph_connected}
+    Given a weighted digraph $G$ with $N \geq 2$ nodes and adjacency matrix $\A$, it holds that:
+    \begin{itemize}
+        \item $\A$ is irreducible $\iff$ $G$ is strongly connected.
+        \item $\A$ is primitive $\iff$ $G$ is strongly connected and aperiodic.
+    \end{itemize}
+\end{theorem}
+
+
+
+\begin{theorem}[Gershgorin] \label{th:gershgorin} \marginnote{Gershgorin theorem}
+    For any square matrix $\A \in \mathbb{C}^{N \times N}$, it holds that the spectrum of $\A$ (i.e., set of eigenvalues) is contained in the Gershgorin disks:
+    \[
+        \text{spec}(\A) \subset \bigcup_{i=1}^{N} \left\{ s \in \mathbb{C} \,\,\bigg|\,\, |s - a_{ii}| \leq \sum_{j=1, j \neq i}^{N} |a_{ij}| \right\}
+    \]
+    In other words, it is the union of the disks with center $a_{ii}$ and radius $\sum_{j=1, j \neq i}^{N} |a_{ij}|$.
+
+    \indenttbox
+    \begin{remark}
+        This theorem provides an approximate location of the eigenvalues.
+    \end{remark}
+
+    \indenttbox
+    \begin{example}
+        Consider the matrix:
+        \[
+            \begin{bmatrix}
+                10 & 1 & 0 & 1 \\
+                0.2 & 8 & 0.2 & 0.2 \\
+                1 & 1 & 2 & 1 \\
+                -1 & -1 & -1 & -11
+            \end{bmatrix}
+        \]
+
+        Its eigenvalues are $\{ -10.870, 1.906, 7.918, 10.046 \}$.
+
+        The Gershgorin disks are:
+        \begin{figure}[H]
+            \centering
+            \includegraphics[width=0.4\linewidth]{./img/gershgorin.png}
+        \end{figure}
+    \end{example}
+\end{theorem}
+
+% \begin{lemma}
+%     If all the disks are within the unit disk, the eigenvalues are stable.
+%     \[
+%         \begin{bmatrix}
+%             \frac{1}{2} & \frac{1}{2} & 0 \\
+%             \frac{1}{3} & \frac{1}{3} & \frac{1}{3} \\
+%             0 & \frac{3}{4} & \frac{1}{4}
+%         \end{bmatrix}
+%     \]
+% \end{lemma}
+
+
+\begin{theorem}[Perron-Frobenius] \label{th:perron_frobenius} \marginnote{Perron-Frobenius theorem}
+    Let $\A \in \R^{N \times N}$ with $N \geq 2$ be a non-negative matrix. It holds that:
+    \begin{itemize}
+        \item There exists a real eigenvalue $\lambda \geq 0$ that is dominant for all the other eigenvalues $\mu \in \text{spec}(\A) \smallsetminus \{\lambda\}$ (i.e., $\lambda \geq |\mu|$),
+        \item The right eigenvector $\v \in \R^N$ and left eigenvector $\w \in \R^N$ associated to $\lambda$ can be chosen to be non-negative.
+    \end{itemize}
+    If $\A \in \R^{N \times N}$ is irreducible, then:
+    \begin{itemize}
+        \item The eigenvalue $\lambda$ is strictly positive ($\lambda > 0$) and simple.
+        \item The right and left eigenvalues $\v$ and $\w$ associated to $\lambda$ are unique and positive.
+    \end{itemize}
+    If $\A \in \R^{N \times N}$ is primitive, then:
+    \begin{itemize}
+        \item The eigenvalue $\lambda$ is strictly dominant for all $\mu \in \text{spec}(\A) \smallsetminus \{\lambda\}$ (i.e., $\lambda > |\mu|$).
+    \end{itemize}
+\end{theorem}
+
+\begin{lemma} \label{th:row_stochastic_unit_disk}
+    Given a row stochastic matrix $\A$, it holds that:
+    \begin{itemize}
+        \item $\lambda=1$ is an eigenvalue,
+        \item By \hyperref[th:gershgorin]{Gershgorin Theorem}, $\text{spec}(\A)$ is a subset of the unit disk (i.e., all Gershgorin disks lie inside the unit disk).
+    \end{itemize}
+
+    \begin{figure}[H]
+        \centering
+        \includegraphics[width=0.2\linewidth]{./img/gershgorin_unit.png}
+    \end{figure}
+
+    \indenttbox
+    \begin{corollary}
+        The eigenvalue $\lambda=1 \geq |\mu|$ is dominant.
+    \end{corollary}
+\end{lemma}
+
+\begin{lemma}
+    Given a row stochastic and primitive matrix $\A$, by \Cref{th:row_stochastic_unit_disk} and \hyperref[th:perron_frobenius]{Perron-Frobenius Theorem} it holds that $\lambda = 1$ is simple and strictly dominant.
+
+    \indenttbox
+    \begin{corollary}
+        The consensus averaging system is marginally stable (i.e., converges but not necessarily to the origin) as the largest distinct eigenvalue is $\lambda = 1$.
+    \end{corollary}
+\end{lemma}
+
+
+% \begin{lemma}
+%     \[
+%         \x_\text{eq} = ker(\matr{I} - \A) = \{ \vec{1}\beta \mid \beta \in \R \}
+%     \]
+
+%     \[
+%         \w^T \x^{k+1} = \w^T(\A \x^{k}) = \w^T \x^k
+%     \]
+%     i.e., $\w$ is left eigenvector of $\A$ with $\lambda = 1$.
+
+%     Therefore, the above must be true for:
+%     \[
+%         \begin{split}
+%             \w^T \x_\text{eq} \\
+%             \w^T \x^{0} \\
+%         \end{split}
+%     \]
+%     \[
+%         \w^T \vec{1}\beta \Rightarrow \beta = \frac{\w^T\x^{0}}{\w^T\vec{1}}
+%     \]
+% \end{lemma}
+
+
+
+
+
+
 \begin{theorem}[Discrete-time consensus] \marginnote{Discrete-time consensus}
     Consider a discrete-time averaging system with digraph $G$ and weighted adjacency matrix $\matr{A}$. Assume $G$ strongly connected and aperiodic, and $\matr{A}$ row stochastic. 
     
@@ -103,41 +268,157 @@
         \]
         where $\tilde{w}_i = \frac{w_i}{\sum_{i=j}^N w_j}$ are all normalized and sum to 1 (i.e., they produce a convex combination).
 
-    Moreover, if $\matr{A}$ is doubly stochastic, then it holds that the consensus is the average:
+    Moreover, if $\matr{A}$ is doubly stochastic, then it holds that the consensus is the average as $\vec{w} = 1$:
     \[ 
         \lim_{k \rightarrow \infty} \vec{x}^k = \vec{1} \frac{1}{N} \sum_{i=1}^N x_i^0
     \]
 
-    \begin{proof}[Sketch of proof]
-        Let $\matr{T} = \begin{bmatrix} \vec{1} & \vec{v}^2 & \cdots & \vec{v}^N \end{bmatrix}$ be a change in coordinates that transforms an adjacency matrix into its Jordan form $\matr{J}$:
-        \[ \matr{J} = \matr{T}^{-1} \matr{A} \matr{T} \]
-        As $\lambda=1$ is a simple eigenvalue (\Cref{th:strongly_connected_eigenvalues}), it holds that:
+    % \begin{proof}[Sketch of proof]
+    %     Let $\matr{T} = \begin{bmatrix} \vec{1} & \vec{v}^2 & \cdots & \vec{v}^N \end{bmatrix}$ be a change in coordinates that transforms an adjacency matrix into its Jordan form $\matr{J}$:
+    %     \[ \matr{J} = \matr{T}^{-1} \matr{A} \matr{T} \]
+    %     As $\lambda=1$ is a simple eigenvalue (\Cref{th:strongly_connected_eigenvalues}), it holds that:
+    %     \[
+    %         \matr{J} = \begin{bmatrix}
+    %             1 & 0 & \cdots & 0 \\
+    %             0 & & &  \\
+    %             \vdots & & \matr{J}_2 &  \\
+    %             0 & & &  \\
+    %         \end{bmatrix}
+    %     \]
+    %     where the eigenvalues of $\matr{J}_2 \in \mathbb{R}^{(N-1) \times (N-1)}$ lie inside the open unit disk.
+
+    %     Let $\vec{x}^k = \matr{T}\bar{\vec{x}}^k$, then we have that:
+    %     \[
+    %         \begin{split}
+    %             &\vec{x}^{k+1} = \matr{A} \vec{x}^{k} \\
+    %             &\iff \matr{T} \bar{\vec{x}}^{k+1} = \matr{A} (\matr{T} \bar{\vec{x}}^k) \\
+    %             &\iff \bar{\vec{x}}^{k+1} = \matr{T}^{-1} \matr{A} (\matr{T} \bar{\vec{x}}^k) = \matr{J}\bar{\vec{x}}^k 
+    %         \end{split}
+    %     \]
+    %     Therefore:
+    %     \[
+    %         \begin{gathered}
+    %             \lim_{k \rightarrow \infty} \bar{\vec{x}}^k = \bar{x}_1^0 \begin{bmatrix} 1 \\ 0 \\ \vdots \\ 0 \end{bmatrix} \\
+    %             \bar{x}_1^{k+1} = \bar{x}_1^k \quad \forall k \geq 0 \\
+    %             \lim_{k \rightarrow \infty} \bar{x}_i^{k} = 0 \quad \forall i = 2, \dots, N \\
+    %         \end{gathered}
+    %     \]
+    % \end{proof}
+
+    \begin{proof}[Proof (Jordan-form approach)]
+        As is $G$ strongly connected and aperiodic, and $\A$ is row stochastic, it holds that:
+        \begin{itemize}
+            \item By \Cref{th:positive_matrix_digraph_connected}, $\A$ is primitive.
+            \item By \hyperref[th:perron_frobenius]{Perron-Frobenius Theorem} and \Cref{th:row_stochastic_unit_disk}, the eigenvalue $\lambda=1$ is strictly dominant and it is associated to the right eigenvector $\vec{1}$ (row stochasticity) and left eigenvector $\w$.
+        \end{itemize}
+    
+        Consider the non-singular matrix $\matr{T} \in \R^{N \times N}$ defined as:
         \[
-            \matr{J} = \begin{bmatrix}
-                1 & 0 & \cdots & 0 \\
-                0 & & &  \\
-                \vdots & & \matr{J}_2 &  \\
-                0 & & &  \\
+            \matr{T} = \begin{bmatrix}
+                \vert & \vert & & \vert \\
+                \vec{1} & \v^2 & \dots & \v^N \\
+                \vert & \vert & & \vert \\
+            \end{bmatrix} = \begin{bmatrix}
+                \vec{1} & \matr{W}_R
+            \end{bmatrix}
+            \qquad
+            \matr{T}^{-1} = \begin{bmatrix}
+                - & (\w)^T & - \\
+                - & (\w^2)^T & - \\
+                - & \vdots & - \\
+                - & (\w^N)^T & - \\
+            \end{bmatrix} = \begin{bmatrix}
+                \w^T \\ \matr{W}_L
             \end{bmatrix}
         \]
-        where the eigenvalues of $\matr{J}_2 \in \mathbb{R}^{(N-1) \times (N-1)}$ lie inside the open unit disk.
-
-        Let $\vec{x}^k = \matr{T}\bar{\vec{x}}^k$, then we have that:
+    
+        A change in coordinates defined as:
+        \[
+            \x \mapsto \tilde{\x} = \matr{T}^{-1} \x
+        \]
+        allows to obtain the Jordan form $\matr{T}^{-1}\A\matr{T}$:
+        \[
+            \matr{T}^{-1}\A\matr{T} = \begin{bmatrix}
+                1 & 0 & \dots \\
+                0 & & \\
+                \vdots & & \matr{J}_2 \\
+            \end{bmatrix}
+        \]
+        with $\matr{J}_2 \in \mathbb{R}^{(N-1) \times (N-1)}$ Schur (i.e., $\text{spec}(\matr{J}_2)$ inside the open unit disk).
+    
+        The dynamics $\x^{k+1} = \A \x^k$ in the new coordinate system is:
         \[
             \begin{split}
-                &\vec{x}^{k+1} = \matr{A} \vec{x}^{k} \\
-                &\iff \matr{T} \bar{\vec{x}}^{k+1} = \matr{A} (\matr{T} \bar{\vec{x}}^k) \\
-                &\iff \bar{\vec{x}}^{k+1} = \matr{T}^{-1} \matr{A} (\matr{T} \bar{\vec{x}}^k) = \matr{J}\bar{\vec{x}}^k 
+                \tilde{\x}^{k+1} &= \matr{T}^{-1} \x^{k+1} = \matr{T}^{-1} \A \matr{T} \tilde{\x}^k \\
+                &= \begin{bmatrix}
+                    1 & 0 & \dots \\
+                    0 & & \\
+                    \vdots & & \matr{J}_2 \\
+                \end{bmatrix} \tilde{\x}^k
+                = \begin{bmatrix}
+                    1 & 0 & \dots \\
+                    0 & & \\
+                    \vdots & & \matr{J}_2 \\
+                \end{bmatrix}^{k+1} \tilde{\x}^0
             \end{split}
         \]
-        Therefore:
+        Let's denote:
         \[
-            \begin{gathered}
-                \lim_{k \rightarrow \infty} \bar{\vec{x}}^k = \bar{x}_1^0 \begin{bmatrix} 1 \\ 0 \\ \vdots \\ 0 \end{bmatrix} \\
-                \bar{x}_1^{k+1} = \bar{x}_1^k \quad \forall k \geq 0 \\
-                \lim_{k \rightarrow \infty} \bar{x}_i^{k} = 0 \quad \forall i = 2, \dots, N \\
-            \end{gathered}
+            \tilde{\x}^k = \matr{T}^{-1}\x^k = \begin{bmatrix}
+                \w^T\x^k \\ \matr{W}_L\x^k
+            \end{bmatrix}
+            = \begin{bmatrix}
+                \tilde{\x}^k_{m} \\ \tilde{\x}^k_{\bot}
+            \end{bmatrix}
         \]
+        We have that:
+        \[
+            \begin{split}
+                \lim_{k \rightarrow \infty} \tilde{\x}^k
+                &= \lim_{k \rightarrow \infty} \begin{bmatrix}
+                    1 & 0 & \dots \\
+                    0 & & \\
+                    \vdots & & \matr{J}_2 \\
+                \end{bmatrix}^k \tilde{\x}^0 \\
+                &= \lim_{k \rightarrow \infty} \begin{bmatrix}
+                    1 & 0 & \dots \\
+                    0 & & \\
+                    \vdots & & (\matr{J}_2)^k \\
+                \end{bmatrix} \begin{bmatrix}
+                    \tilde{\x}^0_{m} \\ \tilde{\x}^0_{\bot}
+                \end{bmatrix} \\
+                &= \begin{bmatrix}
+                    1 \cdot \tilde{\x}^0_{m} \\ 
+                    \lim_{k \rightarrow \infty} (\matr{J}_2)^k \tilde{\x}^0_{\bot}
+                \end{bmatrix} \\
+                &= \begin{bmatrix}
+                    \w^T \x^0 \\ 
+                    0
+                \end{bmatrix} \\
+            \end{split}
+        \]
+        Note that $\lim_{k \rightarrow \infty} \matr{J}_2^k = 0$ as it is stable (i.e., all eigenvalues are in the open unit disk $|\mu| < 1$).
+    
+        In the original coordinate system, the limit is:
+        \[
+            \begin{split}
+                \lim_{k \rightarrow \infty} \x^k 
+                &= \lim_{k \rightarrow \infty} \matr{T} \tilde{\x}^k \\
+                &= \matr{T} \lim_{k \rightarrow \infty} \tilde{\x}^k \\
+                &= \begin{bmatrix}
+                    \vec{1} & \matr{W}_R
+                \end{bmatrix} \begin{bmatrix}
+                    \w^T \x^0 \\ 
+                    0
+                \end{bmatrix} 
+                = \vec{1} (\w^T \x^0)
+            \end{split}
+        \]
+        
+        \indenttbox
+        \begin{remark}
+            It is assumed that $\Vert \w \Vert = 1$ (i.e., no normalization term).
+        \end{remark}
     \end{proof}
 \end{theorem}
 
@@ -154,6 +435,45 @@
 \end{example}
 
 
+% \begin{proof}[Lyapunov approach]
+%     $\A - \vec{1}\w^T$ is rank-1. This is to change one specific eigenvalue (move 1 to 0).
+
+%     Dissensus vector represents error:
+%     \[
+%         \begin{split}
+%             delta^{k+1} 
+%             = \x^{k+1} - \vec{1}\w^T \x^0 \\
+%             = \x^{k+1} - \vec{1}\w^T \x^{k+1} \\
+%             = (\matr{I} - \vec{1}\w^T) \x^{k+1} \\
+%             = (\matr{I} - \vec{1}\w^T) \A\x^{k} \\
+%             = (\A - \vec{1}\w^T) \x^{k} \\
+%             = (\A - \vec{1}\w^T) \delta^{k} \\
+%         \end{split}
+%     \]
+
+%     Study:
+%     \[
+%         \delta^{k+1} = (\A - \vec{1}\w^T) \delta{k}
+%     \]
+%     If $\delta^k \rightarrow 0$, then $\x^k \rightarrow\vec{1}\w^T\x^0$.
+%     Note $(\A - \vec{1}\w^T)$ is Schur.
+
+%     Lyapunov equation for discrete time systems:
+%     \[
+%         \bar{\A}^T \matr{P} \bar{\A} = - \matr{P} = - \matr{Q}
+%     \]
+%     where $\bar{\A}$ is the Jordan-form of $(\A - \vec{1}\w^T)$
+
+%     Select $Q_2$ to be block-diagonal and $p_1$
+
+
+%     \[
+%         V(\delta) = \delta^T (\matr{T}^{-1})^T \matr{P} \matr{T}^{-1} \delta
+%     \]
+% \end{proof}
+
+
+
 
 \section{Discrete-time averaging algorithm over time-varying graphs}
 
@@ -228,7 +548,7 @@
         with states $x_i \in \mathbb{R}$, inputs $u_i \in \mathbb{R}$, and communication following a digraph $G$.
 
     \item[Laplacian dynamics system] \marginnote{Laplacian dynamics system}
-        Consider a network of dynamic systems where $u_i$ is defined as a proportional controller (i.e., only communicating $(i, j)$ have a non-zero weight $a_{ij}$):
+        Consider a network of dynamic systems where $u_i$ is defined as a proportional controller (i.e., only communicating $(i, j)$ have a non-zero weight):
         \[ 
             \begin{split}
                 u_i(t) 

src/year2/distributed-autonomous-systems/sections/_safety_controllers.tex

@@ -1,4 +1,4 @@
-\chapter{Multi-robot safety controllers}
+\chapter{Safety controllers}
 
 
 \begin{description}
@@ -126,7 +126,7 @@
             \includegraphics[width=0.35\linewidth]{./img/safety_control_single.png}
         \end{figure}
 
-        A control barrier function to solve the task can be:
+        A control barrier function to solve the task (i.e., rectify the trajectory of the high level controller) can be:
         \[
             V^s(\x) = \Vert \x - \x_\text{obs} \Vert^2 - \Delta^2
             \qquad
@@ -180,4 +180,151 @@
         \[
             X_i = \{ \x \in \mathbb{R}^d \mid \forall j \in \mathcal{N}_i: V_{i,j}^s(\x) \geq 0 \}
         \]
+
+        The set of admissible controllers is:
+        \[
+            \begin{aligned}
+                \begin{aligned}
+                    U^s(\x) = \Big\{ \u \in \mathbb{R}^{dN} \mid 
+                    -\nabla_{[\x_i]} V_{ij}^s(\x_i, \x_j)^T \u_i 
+                    - \nabla_{[\x_i]} V_{ji}^s(\x_j, \x_i)^T \u_j 
+                    - &\gamma(V_{ij}^{s}(\x_i, \x_j)) \leq 0 \\
+                    &\forall j \in \mathcal{N}_i, \forall i \in \{1, \dots, N\} \Big\}
+                \end{aligned} \\
+                = \Big\{ \u \in \mathbb{R}^{dN} \mid -2(\x_i, \x_j)^T \u_i - 2(\x_j-\x_i)^T \u_j - \gamma(V_{ij}^s(\x_i, \x_j)) \leq 0 \,\,\forall j \in \mathcal{N}_i, \forall i \in \{1, \dots, N\} \Big\}
+            \end{aligned}
+        \]
+
+        % \[
+        %     L_g V_{ij}^s(\x) = \nabla_{[\x_i]} V^s(\x_i, \x_j)^T \u_i + \nabla_{[\x_j]} V^s(\x_i, \x_j)^T \u_j
+        % \]
+\end{description}
+
+\begin{description}
+    \item[Centralized safety controller] \marginnote{Centralized safety controller}
+        The CBF-based policy can be obtained by solving:
+        \[
+            \begin{gathered}
+                \arg\min_{\u \in \mathbb{R}^N} \sum_{i=1}^{N} \Vert \u_i - \u_i^\text{ref} \Vert^2 \\
+                \begin{aligned}
+                    \text{subject to } 
+                    &-2(\x_i, \x_j)^T \u_i - 2(\x_j-\x_i)^T \u_j - \gamma(V_{ij}^s(\x_i, \x_j)) \leq 0 \\
+                    & \Vert \u_i \Vert \leq \u_i^\text{max} \\
+                    & \forall j \in \mathcal{N}_i, \forall i \in \{ 1, \dots, N \}
+                \end{aligned}
+            \end{gathered}
+        \]
+        where $\u_i^\text{ref}$ is the reference input of the high level controller and $\u_i^\text{max}$ is the bound.
+
+        \begin{remark}
+            The policy should be computed continuously for each $x_i(t)$.
+        \end{remark}
+
+    \item[Decentralized safety controller] \marginnote{Decentralized safety controller}
+        The CBF-based policy can be obtained by solving a more constrained problem compared to the centralized formulation:
+        \[
+            \begin{gathered}
+                \arg\min_{\u_i \mathbb{R}^d} \Vert \u_i - \u_i^\text{ref} \Vert^2 \\
+                \begin{aligned}
+                    \text{subject to } &- \nabla_{[\x_i]} V_{ij}^s(\x_i, \x_j)^T \u_i - \frac{1}{2} \gamma (V_{ij}^s(\x_i, \x_j)) \leq 0 \\
+                    & \Vert \u_i \Vert \leq \u_i^\text{max} \\
+                    & \forall j \in \mathcal{N}_i
+                \end{aligned}
+            \end{gathered}
+        \]
+
+        \begin{remark}
+            If $\forall i \in \{1, \dots, N\}: \nabla_{[\x_i]} V_{ij}^s(\x_i, \x_j)^T \u_i \geq \frac{1}{2} \gamma (V_{ij}^s(\x_i, \x_j))$, then it holds that:
+            \[
+                \begin{split}
+                    \nabla_{[\x_i]} V_{ij}^s(\x_i, \x_j)^T \u_i + \nabla_{[\x_i]} V_{ji}^s(\x_j, \x_i)^T \u_j 
+                    &\geq -\frac{1}{2} \gamma\left( V_{ij}^s(\x_i, \x_j) \right) - \frac{1}{2} \gamma\left( V_{ji}^s(\x_j, \x_i) \right) \\
+                    &\geq - \gamma\left( V_{ij}^s(\x_i, \x_j) \right)
+                \end{split}
+            \]
+        \end{remark}
+\end{description}
+
+
+\subsection{Multi-robot collision avoidance with unicycle control}
+
+\begin{description}
+    \item[Unicycle model with non-holonomic constraints] 
+        Model that captures the constraints given by wheels. Its dynamics is:
+        \[
+            \begin{split}
+                \dot{\vec{p}}_x &= v \cos(\theta) \\
+                \dot{\vec{p}}_y &= v \sin(\theta) \\
+                \theta &= \omega \\
+            \end{split}
+        \]
+        where:
+        \begin{itemize}
+            \item $(\vec{p}_x, \vec{p}_y)$ is the position of the center of mass,
+            \item $\theta$ is the orientation,
+            \item $v$ is the linear velocity,
+            \item $\omega$ is the angular velocity.
+        \end{itemize}
+
+        \begin{figure}[H]
+            \centering
+            \includegraphics[width=0.25\linewidth]{./img/unicycle_model.png}
+        \end{figure}
+
+        \begin{remark}
+            It is assumed that the robot does not drift sideways ($v_{\bot} = 0$).
+        \end{remark}
+
+    \item[Single integrator to unicycle control mapping] \marginnote{Single integrator to unicycle control mapping}
+        Consider a point $\x^\text{int}$ longitudinal to $v$ that is not the barycenter:
+        \[
+            \x^\text{int} = \begin{bmatrix}
+                \vec{p}_x \\ \vec{p}_y
+            \end{bmatrix}
+            +
+            \rho \begin{bmatrix}
+                \cos(\theta) \\ \sin(\theta)
+            \end{bmatrix}
+        \]
+        where $\rho > 0$ is the distance to the barycenter.
+
+        By differentiating w.r.t. time, the dynamics is:
+        \[
+            \dot{\x}^\text{int} = \begin{bmatrix}
+                \dot{\vec{p}}_x \\ \dot{\vec{p}}_y
+            \end{bmatrix}
+            +
+            \rho \dot{\theta} \begin{bmatrix}
+                - \sin(\theta) \\ \cos(\theta)
+            \end{bmatrix}
+        \]
+
+        \begin{figure}[H]
+            \centering
+            \includegraphics[width=0.2\linewidth]{./img/single_unicycle_map.png}
+        \end{figure}
+
+        By using the unicycle model dynamics, it becomes:
+        \[
+            \dot{\x}^\text{int} = \begin{bmatrix}
+                \cos(\theta) & -\rho\sin(\theta) \\
+                \sin(\theta) & \rho\cos(\theta) \\
+            \end{bmatrix}
+            \begin{bmatrix}
+                v \\ \omega
+            \end{bmatrix}
+        \]
+
+        By formulating $v$ and $\omega$ as a state-feedback control with input $\u^\text{int} \in \mathbb{R}^2$ as:
+        \[
+            \begin{bmatrix}
+                v \\ \omega
+            \end{bmatrix}
+            =
+            \begin{bmatrix}
+                \cos(\theta) & \sin(\theta) \\
+                -\frac{1}{\rho} \sin(\theta) & \frac{1}{\rho} \cos(\theta)
+            \end{bmatrix} \u^\text{int}
+        \]
+        The result is a single-integrator $\dot{\x}^\text{int} = \u^\text{int}$.
 \end{description}