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Add DAS safety controllers

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Tian Cheng (Riccardo) Xia
2025-04-29 20:03:45 +02:00
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src/year2/distributed-autonomous-systems/das.tex
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@ -43,6 +43,7 @@
\def\v{{\vec{v}}}
\def\r{{\vec{r}}}
\def\s{{\vec{s}}}
\def\u{{\vec{u}}}
\begin{document}
@ -54,5 +55,6 @@
\include{./sections/_optimization.tex}
\include{./sections/_formation_control.tex}
\include{./sections/_cooperative_robotics.tex}
\include{./sections/_safety_controllers.tex}
\end{document}

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src/year2/distributed-autonomous-systems/sections/_safety_controllers.tex Normal file
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\chapter{Multi-robot safety controllers}
\begin{description}
\item[Control-affine non-linear dynamical system] \marginnote{Control-affine non-linear dynamical system}
System whose dynamics follows:
\[
\dot{\x}(t) = f(\x(t)) + g(\x(t)) \u(t) \quad \x(0) = \x_0
\]
with $\x(t) \in \mathbb{R}^n$, $\u(t) \in U \subseteq \mathbb{R}^m$, $f(\x(t)) \in \mathbb{R}^n$, and $g(\x(t)) \in \mathbb{R}^{n \times m}$.
$f(\x(t))$ can be seen as the drift of the system and $\u(t)$ a coefficient that controls how much $g(\x(t))$ is injected into $f(\x(t))$.
The overall system can be interpreted as composed of:
\begin{itemize}
\item A high-level controller that produces the direction $\u^\text{ref}(\x)$ towards the target position.
\item A safety layer that modifies $\u^\text{ref}(\x)$ into $\u(t) = \kappa(\x)$ to account for obstacles.
\end{itemize}
\item[Safety control] \marginnote{Safety control}
Given a (sufficiently regular) function $V^s: X \subseteq \mathbb{R}^n \rightarrow \mathbb{R}$, it is possible to define a safe state set as:
\[
X^s = \{ \x \in X \subseteq \mathbb{R}^n \mid V^s(\x) \geq 0 \}
\]
The goal is to design a feedback control law $\kappa^s: X \rightarrow \mathbb{R}^m$ for a control-affine non-linear dynamical system such that the set $X^s$ is forward invariant (i.e., any trajectory starting in $X^s$ remains in $X^s$).
\begin{remark}
The time derivative of $V^s(\x(t))$ along the system trajectories is given by:
\[
\begin{split}
\frac{d}{dt} V^s(\x(t))
&= \nabla V^s(\x(t))^T \frac{d}{dt} \x(t) \\
&= \nabla V^s(\x(t))^T \Big( f(\x(t)) + g(\x(t)) \u(t) \Big) \\
&= \nabla V^s(\x(t))^T f(\x(t)) + \sum_{h=1}^{m} \Big( \nabla V^s(\x(t))^T g_h(\x(t)) \u_h(t) \Big)\\
&= L_f V^s(\x(t)) + L_g V^s(\x(t)) \u(t) \\
\end{split}
\]
where $L_h V^s(\x(t)) = \nabla V^s(\x(t))^T h(\x(t))$ is the lie derivative.
\end{remark}
\item[Control barrier function (CBF)] \marginnote{Control barrier function (CBF)}
A function $V^s$ is a control barrier function if there exists a continuous strictly increasing function $\gamma: \mathbb{R} \rightarrow \mathbb{R}$ with $\gamma(0) = 0$ such that the following inequality (control barrier certificate) holds:
\[
\sup_{\u \in U} \{ L_fV^s(\x) + L_gV^s(\x)\u + \gamma(V^s(\x)) \} \geq 0 \quad \forall \x \in X
\]
$\gamma$ can be interpreted as a degree of movement freedom since, as long as it holds that $V^s(\x(t)) > 0$, it is allowed that $\frac{d}{dt} V^s(\x(t)) < 0$ (i.e., the agent can move closer to the border between safe and unsafe region).
\begin{remark}
In principle, the negative part of $\gamma$ is not necessary (the agent should start in a safe area). However, as it is strictly increasing, it allows to move out the unsafe region if the agent ever ends up there.
\end{remark}
\begin{example}
A simple choice for $\gamma$ is a linear function $\gamma(r) = \gamma r$ with $\gamma > 0$.
\end{example}
\item[Set of admissible safe controllers] \marginnote{Set of admissible safe controllers}
The set of inputs that satisfy the control barrier certificate for a given state $\x$ is:
\[
U^s(\x) = \{ \u \in U \mid L_f V^s(\x) + L_g V^s(\x) \u + \gamma(V^s(\x)) \geq 0 \}
\]
\end{description}
\section{Safety filter via control barrier certificate}
\begin{description}
\item[Safety filter via control barrier certificate] \marginnote{Safety filter via control barrier certificate}
Given a possibly unsafe reference input (from the high-level controller) $\u^\text{ref}(\x) \in \mathbb{R}^m$, the safety controller (i.e., rectifying controller) based on the control barrier certificate is designed to be minimally invasive (i.e., alter the reference as little as possible).
The policy $\u = \kappa^s(\x)$ can be defined as:
\[
\begin{gathered}
\kappa^s(\x) = \arg\min_{\u \in U} \Vert \u - \u^\text{ref}(\x) \Vert^2 \\
\text{subject to } -L_fV^s(\x) - L_gV^s(\x)\u - \gamma(V^s(\x)) \leq 0
\end{gathered}
\]
\begin{remark}
In the general case, this problem should be solved at each $t \geq 0$.
\end{remark}
\item[Single integrator model] \marginnote{Single integrator model}
Control-affine non-linear dynamical system where $f(\x(t)) = 0$ and $g(\x(t)) = \matr{I}$. The dynamics is:
\[
\begin{split}
\dot{\x}
&= 0 + \matr{I}\u \\
&= \u
\end{split}
\]
with $\x \in \mathbb{R}^d$ and $\u \in \mathbb{R}^d$.
\begin{remark}
In the case of single integrators, we have that:
\begin{itemize}
\item $L_f V^s(\x) = \nabla V^s(\x)^T 0 = 0$,
\item $L_g V^s(\x) = \nabla V^s(\x)^T \matr{I} = \nabla V^s(\x)^T$.
\end{itemize}
Therefore:
\[
\begin{split}
\frac{d}{dt} V^s(\x(t))
&= L_f V^s(\x(t)) + L_g V^s(\x(t)) \u(t) \\
&= \nabla V^s(\x(t))^T \u(t)
\end{split}
\]
\end{remark}
\end{description}
\subsection{Single-robot obstacle avoidance with single integrator models}
\begin{description}
\item[Single-robot obstacle avoidance] \marginnote{Single-robot obstacle avoidance}
Task where the goal is to keep an agent to a safety distance $\Delta > 0$ from an obstacle.
A control barrier function to solve the task can be:
\[
V^s(\x) = \Vert \x - \x_\text{obs} \Vert^2 - \Delta^2
\qquad
\nabla V^s(\x) = 2(\x - \x_\text{obs})
\]
The CBF-based safety policy $\kappa^s(\x)$ can be obtained by solving:
\[
\begin{gathered}
\arg\min_{\u \in U} \Vert \u - \u^\text{ref}(\x) \Vert^2 \\
\text{subject to } -2(\x-\x_\text{obs})^T \u - \gamma(\Vert \x-\x_\text{obs} \Vert^2 - \Delta^2) \leq 0
\end{gathered}
\]
As there are two constants in the constraint $a = -2(\x-\x_\text{obs})^T$ and $b = \gamma(\Vert \x-\x_\text{obs} \Vert^2 - \Delta^2)$, the problem can be reformulated as:
% \[
% \arg\min_{\u \in U} \u^T\u - 2\u^T\u^\text{ref} \quad \text{subject to } a^T \u + b \leq 0
% \]
\[
\arg\min_{\u \in U} \Vert \u - \u^\text{ref}(\x) \Vert^2 \quad \text{subject to } a^T \u + b \leq 0
\]
\begin{remark}
If $U$ is a polytope (or unconstrained: $U = \mathbb{R}^d$), the problem becomes a quadratic program.
\end{remark}
\end{description}
\subsection{Multi-robot collision avoidance with single integrator models}
\begin{description}
\item[Multi-robot collision avoidance] \marginnote{Multi-robot collision avoidance}
Task with $N$ single integrator agents that want to keep a safety distance $\Delta > 0$ among them.
The local control barrier function to solve the task can be defined as:
\[
V^s_{i,j}(\x_i, \x_j) = \Vert \x_i - \x_j \Vert^2 - \Delta^2
\qquad
\begin{aligned}
\nabla_{[\x_i]} V_{i,j}^s(\x_i, \x_j) &= 2(\x_i - \x_j) \\
\nabla_{[\x_j]} V_{i,j}^s(\x_i, \x_j) &= 2(\x_j - \x_i)
\end{aligned}
\]
The safe region $X_i$ for agent $i$ can be defined as:
\[
X_i = \{ \x \in \mathbb{R}^d \mid \forall j \in \mathcal{N}_i: V_{i,j}^s(\x) \geq 0 \}
\]
\end{description}