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Tian Cheng (Riccardo) Xia
2025-04-26 18:08:17 +02:00
parent 5484e66406
commit 2ad67a3625
5 changed files with 23 additions and 21 deletions
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9
src/year2/distributed-autonomous-systems/sections/_cooperative_robotics.tex
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@ -6,7 +6,7 @@
Problem where $N$ agents want to optimize their positions $\z_i \in \mathbb{R}^2$ to perform multi-robot surveillance in an environment with:
\begin{itemize}
\item A static target to protect $\r_0 \in \mathbb{R}^2$.
\item Static intruders/opponents $\r_i \in \mathbb{R}^2$, each assigned to an agent $i$.
\item Static intruders/opponents $\r_i \in \mathbb{R}^2$, each assigned to the respective agent $i$.
\end{itemize}
The average position of the agents define the barycenter:
@ -16,7 +16,7 @@
\[
l_i(\z_i, \sigma(\z)) =
\gamma_i \underbrace{\Vert \z_i - \r_i \Vert^2}_{\text{close to opponent}} +
\underbrace{\Vert \sigma(\z) - \r_0 \Vert^2}_{\text{barycenter close to protectee}}
\underbrace{\Vert \sigma(\z) - \r_0 \Vert^2}_{\text{barycenter close to target}}
\]
Note that the opponent component only depends on local variables while the target component needs global information.
@ -69,8 +69,9 @@
&\frac{\partial}{\partial z_i} \left.\left( \sum_{j=1}^{N} l_j(z_j, \sigma(z_1, \dots, z_N)) \right) \right|_{z_j=z_j^k} \\
&=
\left.\frac{\partial}{\partial z_i} l_i(z_i, \sigma) \right|_{\substack{z_i = z_i^k,\\\sigma = \sigma(\z^k)}} +
\left.\left(\sum_{j=1}^{N} \frac{\partial}{\partial \sigma} l_j(z_j, \sigma) \right)\right|_{\substack{z_j = z_j^k,\\\sigma = \sigma(\z^k)}} \cdot
\left.\frac{\partial}{\partial z_i} \sigma(z_1, \dots, z_N)\right|_{\substack{z_j=z_j^k}}
\sum_{j=1}^{N} \left( \left. \left( \frac{\partial}{\partial \sigma} l_j(z_j, \sigma) \right)\right|_{\substack{z_j = z_j^k,\\\sigma = \sigma(\z^k)}}
\cdot
\left.\frac{\partial}{\partial z_i} \sigma(z_1, \dots, z_N)\right|_{\substack{z_j=z_j^k}} \right)
\end{split}
\]