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Tian Cheng (Riccardo) Xia
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src/year2/distributed-autonomous-systems/sections/_formation_control.tex
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@ -9,40 +9,45 @@
\begin{figure}[H] \begin{figure}[H]
\centering \centering
\includegraphics[width=0.5\linewidth]{./img/mass_spring_system.png} \includegraphics[width=0.35\linewidth]{./img/mass_spring_system.png}
\end{figure} \end{figure}
The elastic force $F_{e,i}(x)$ at mass $i$ is given by: The elastic force $F_{e,i}(\x)$ at mass $i$ is given by:
\[ \[
F_{e,i}(x) = -a_{i,i-1}(x_i-x_{i-1}) - a_{i,i+1}(x_i - x_{i+1}) F_{e,i}(\x) = -a_{i,i-1}(x_i-x_{i-1}) - a_{i,i+1}(x_i - x_{i+1})
\] \]
Equivalently, it is possible to express the elastic force as the negative gradient of the elastic potential energy: Equivalently, it is possible to express the elastic force as the negative gradient of the elastic potential energy:
\[ \[
F_{e,i}(x) = -\frac{\partial}{\partial x_i}\left( \frac{1}{2} a_{i,i-1} \Vert x_i - x_{i-1} \Vert^2 + \frac{1}{2} a_{i,i+1} \Vert x_i - x_{i+1} \Vert^2 \right) F_{e,i}(\x) = -\frac{\partial}{\partial x_i}\left( \frac{1}{2} a_{i,i-1} \Vert x_i - x_{i-1} \Vert^2 + \frac{1}{2} a_{i,i+1} \Vert x_i - x_{i+1} \Vert^2 \right)
\] \]
\item[Mass-spring system with two springs] \marginnote{Mass-spring system with two springs} \item[Mass-spring system with two springs] \marginnote{Mass-spring system with two springs}
Assume that the springs of a mass-spring system can be split with halved elastic constants. Assume that the springs of a mass-spring system can be split with halved elastic constants.
\begin{figure}[H] \begin{figure}[H]
\centering \centering
\includegraphics[width=0.5\linewidth]{./img/mass_spring_system2.png} \includegraphics[width=0.35\linewidth]{./img/mass_spring_system2.png}
\end{figure} \end{figure}
Accordingly, the elastic force can be defined as: Accordingly, the elastic force can be defined as:
\[ \[
\small \small
\begin{split} \begin{split}
F_{e,i}(x) F_{e,i}(\x)
&= &=
- \frac{1}{2} a_{i,i-1}(x_i - x_{i-1}) - \frac{1}{2} a_{i,i-1}(x_i - x_{i-1})
- \frac{1}{2} a_{i-1,i}(x_i - x_{i-1}) - \frac{1}{2} a_{i-1,i}(x_i - x_{i-1})
- \frac{1}{2} a_{i,i+1}(x_i - x_{i+1}) - \frac{1}{2} a_{i,i+1}(x_i - x_{i+1})
- \frac{1}{2} a_{i+1,i}(x_i - x_{i+1}) \\ - \frac{1}{2} a_{i+1,i}(x_i - x_{i+1}) \\
&= &= -\frac{\partial}{\partial x_i} \left(
- \frac{1}{2} a_{i,i-1}(x_i - x_{i-1}) \frac{1}{2} \frac{a_{i,i-1}}{2} \Vert x_i - x_{i-1} \Vert^2 +
+ \frac{1}{2} a_{i-1,i}(x_{i-1} - x_i) \frac{1}{2} \frac{a_{i-1,i}}{2} \Vert x_{i} - x_{i-1} \Vert^2 +
- \frac{1}{2} a_{i,i+1}(x_i - x_{i+1}) \frac{1}{2} \frac{a_{i,i+1}}{2} \Vert x_i - x_{i+1} \Vert^2 +
+ \frac{1}{2} a_{i+1,i}(x_{i+1} - x_i) \\ \frac{1}{2} \frac{a_{i+1,i}}{2} \Vert x_{i} - x_{i+1} \Vert^2 \right) \\
% &=
% - \frac{1}{2} a_{i,i-1}(x_i - x_{i-1})
% + \frac{1}{2} a_{i-1,i}(x_{i-1} - x_i)
% - \frac{1}{2} a_{i,i+1}(x_i - x_{i+1})
% + \frac{1}{2} a_{i+1,i}(x_{i+1} - x_i) \\
&= -\frac{\partial}{\partial x_i} \left( &= -\frac{\partial}{\partial x_i} \left(
\frac{1}{2} \frac{a_{i,i-1}}{2} \Vert x_i - x_{i-1} \Vert^2 + \frac{1}{2} \frac{a_{i,i-1}}{2} \Vert x_i - x_{i-1} \Vert^2 +
\frac{1}{2} \frac{a_{i-1,i}}{2} \Vert x_{i-1} - x_{i} \Vert^2 + \frac{1}{2} \frac{a_{i-1,i}}{2} \Vert x_{i-1} - x_{i} \Vert^2 +
@ -53,7 +58,7 @@
The total potential energy (i.e., sum of the function in the derivative over all masses) can be compactly defined as: The total potential energy (i.e., sum of the function in the derivative over all masses) can be compactly defined as:
\[ \[
\begin{split} \begin{split}
V(x) &= \sum_{i=1}^{N} \sum_{j \in \mathcal{N}_i} \frac{1}{2} \frac{a_{i,j}}{2} \Vert x_i - x_j \Vert^2 \\ V(\x) &= \sum_{i=1}^{N} \sum_{j \in \mathcal{N}_i} \frac{1}{2} \frac{a_{i,j}}{2} \Vert x_i - x_j \Vert^2 \\
&= \sum_{i=1}^{N} \sum_{j \in \mathcal{N}_i} V_{ij}(x_i, x_j) &= \sum_{i=1}^{N} \sum_{j \in \mathcal{N}_i} V_{ij}(x_i, x_j)
\end{split} \end{split}
\quad \quad
@ -63,17 +68,17 @@
Then, the potential energy at mass $i$ can be written as: Then, the potential energy at mass $i$ can be written as:
\[ \[
V_i(x) = \sum_{j \in \mathcal{N}_i} ( V_{i,j}(x_i, x_j) + V_{j,i}(x_j, x_i) ) V_i(\x) = \sum_{j \in \mathcal{N}_i} ( V_{i,j}(x_i, x_j) + V_{j,i}(x_j, x_i) )
\] \]
Finally, the elastic force at mass $i$ can be reformulated as: Finally, the elastic force at mass $i$ can be reformulated as:
\[ \[
\begin{split} \begin{split}
F_{e,i}(x) F_{e,i}(\x)
&= - \frac{\partial}{\partial x_i} \Big( V_{i,i-1}(x_i, x_{i-1}) + V_{i-1,i}(x_{i-1}, x_i) + V_{i,i+1}(x_i, x_{i+1}) + V_{i+1,i}(x_{i+1}, x_i) \Big) \\ &= - \frac{\partial}{\partial x_i} \Big( V_{i,i-1}(x_i, x_{i-1}) + V_{i-1,i}(x_{i-1}, x_i) + V_{i,i+1}(x_i, x_{i+1}) + V_{i+1,i}(x_{i+1}, x_i) \Big) \\
&= - \frac{\partial}{\partial x_i} \left( \sum_{j \in \mathcal{N}_i} ( V_{i,j}(x_i, x_j) + V_{j,i}(x_j, x_i) ) \right) \\ &= - \frac{\partial}{\partial x_i} \left( \sum_{j \in \mathcal{N}_i} ( V_{i,j}(x_i, x_j) + V_{j,i}(x_j, x_i) ) \right) \\
&= - \frac{\partial}{\partial x_i} V_i(x) \\ &= - \frac{\partial}{\partial x_i} V_i(\x) \\
&= - \frac{\partial}{\partial x_i} V(x) &= - \frac{\partial}{\partial x_i} V(\x)
\end{split} \end{split}
\] \]
@ -90,7 +95,7 @@
\[ \[
\begin{split} \begin{split}
\dot{x}_i &= v_i \\ \dot{x}_i &= v_i \\
m_i \dot{v}_i &= - v_i - c\frac{\partial}{\partial x_i} V(x) = - v_i - \frac{\partial}{\partial x_i} V(x) m_i \dot{v}_i &= - v_i - c\frac{\partial}{\partial x_i} V(\x) = - v_i - \frac{\partial}{\partial x_i} V(\x)
\end{split} \end{split}
\] \]
where $m_i$ is the mass of the $i$-th mass. where $m_i$ is the mass of the $i$-th mass.
@ -98,8 +103,8 @@
By assuming small masses $m_i$, the following approximation can be made: By assuming small masses $m_i$, the following approximation can be made:
\[ \[
\begin{gathered} \begin{gathered}
\cancel{m_i \dot{v}_i} = - v_i - \frac{\partial}{\partial x_i} V(x) \Rightarrow v_i \approx -\frac{\partial}{\partial x_i} V(x) \\ \cancel{m_i \dot{v}_i} = - v_i - \frac{\partial}{\partial x_i} V(\x) \Rightarrow v_i \approx -\frac{\partial}{\partial x_i} V(\x) \\
\dot{x_i} = -\frac{\partial}{\partial x_i} V(x) = F_{e,i}(x) \dot{x_i} = -\frac{\partial}{\partial x_i} V(\x) = F_{e,i}(\x)
\end{gathered} \end{gathered}
\] \]
@ -117,20 +122,20 @@
Therefore, the overall system follows a Laplacian dynamics and can be equivalently formulated as the gradient flow of $V$: Therefore, the overall system follows a Laplacian dynamics and can be equivalently formulated as the gradient flow of $V$:
\[ \[
\begin{split} \begin{split}
\dot{\x} &= -\matr{L}\x = - \nabla V(x) \\ \dot{\x} &= -\matr{L}\x = - \nabla V(\x) \\
\begin{bmatrix} \begin{bmatrix}
\dot{x}_1 \\ \vdots \\ \dot{x}_N \dot{x}_1 \\ \vdots \\ \dot{x}_N
\end{bmatrix} \end{bmatrix}
&= &=
- \begin{bmatrix} - \begin{bmatrix}
\frac{\partial}{\partial x_1} V(x) \\ \frac{\partial}{\partial x_1} V(\x) \\
\vdots \vdots
\\ \\
\frac{\partial}{\partial x_N} V(x) \frac{\partial}{\partial x_N} V(\x)
\end{bmatrix} \end{bmatrix}
\end{split} \end{split}
\] \]
And consensus is reached at a stationary point of $V(x)$. And consensus is reached at a stationary point of $V(\x)$.
\end{description} \end{description}
@ -160,7 +165,7 @@
V_{ij}^\text{form}(\x_i, \x_j) = \frac{1}{8} \left( \Vert \x_i - \x_j \Vert^2 - d_{ij}^2 \right)^2 V_{ij}^\text{form}(\x_i, \x_j) = \frac{1}{8} \left( \Vert \x_i - \x_j \Vert^2 - d_{ij}^2 \right)^2
\end{gathered} \end{gathered}
\] \]
where $\frac{1}{8}$ is used to cancel out the fraction when deriving. where $\frac{1}{8}$ is used to cancel out the coefficients when deriving.
The gradient flow dynamics is then: The gradient flow dynamics is then:
\[ \[