Merge pull request #49 from HanaeRateau/pr_algosimagestokatex

Adds algorithms md files with katex and deletes images

Author: EulalieCoevoet

data/algorithms/algorithm1.md

@@ -0,0 +1,23 @@
+$$
+\begin{array}{ll}
+\textbf{Algorithm 1 } \text{Statics of the continuum legs} \\
+\hline
+1:\ \text{Initialize } \mathbf{q}^0 \\
+2:\ \text{Define a tolerance } \epsilon \\
+3:\ i \leftarrow 0 \\
+4:\ \textbf{while } \|\mathbf{q}^i - \mathbf{q}^{i-1}\| > \epsilon \textbf{ do} \\
+5:\ \quad \text{Find the residual:} \\
+    \qquad\qquad \mathbf{F}(\mathbf{q}^{i-1}) + \mathbf{Mg} + \mathbf{F}_{ext} = \mathbf{b} & (3)\\
+6:\ \quad \text{Approximate } \mathbf{F}(\mathbf{q}^i) \text{ by a linearization around } \mathbf{q}^{i-1}\text{:} \\
+  \qquad\qquad \mathbf{F}(\mathbf{q}^i) \approx \mathbf{F}(\mathbf{q}^{i-1}) + \underbrace{\dfrac{\partial \mathbf{F}(\mathbf{q}^{i-1})}{\partial \mathbf{q}}}_{-\mathbf{A}} d\mathbf{q} & (4)\\
+7:\ \quad \text{Solve the following system:} \\
+    \qquad\qquad \mathbf{A}\, d\mathbf{q} = \mathbf{b} & (5)\\
+8:\ \quad \text{Update } \mathbf{q}^i \text{ (with } 0 < \alpha < 1\text{):} \\
+    \qquad\qquad \mathbf{q}^i = \mathbf{q}^{i-1} + \alpha\, d\mathbf{q} & (6)\\
+9:\ \quad i \leftarrow i + 1 \\
+10:\ \textbf{end while} \\
+11:\ \textbf{Convergence reached: } \mathbf{q}^i \\
+\hline
+\end{array}
+$$
+{width=65%, .center}

data/algorithms/algorithm2.md

@@ -0,0 +1,31 @@
+$$
+\begin{array}{ll}
+\textbf{Algorithm 2 } \text{Kinematics of the parallel continuum robot computation } f(\textcolor{red}{\mathbf{u}_a}) = \textcolor{green}{\mathbf{y}_e} \\[0.5em]
+\hline \\[-0.5em]
+1:\ \text{Initialize } \mathbf{q}^0 = [\mathbf{q}_1^0;\ \mathbf{q}_2^0;\ \mathbf{q}_3^0;\ \mathbf{q}_4^0] \\
+2:\ \text{Step 2 and 3 of Algorithm 1} \\
+3:\ \textbf{while } \|\mathbf{q}^i - \mathbf{q}^{i-1}\| > \epsilon \textbf{ do} \\
+4:\ \quad \text{Find the residual:} \\[0.5em]
+\qquad\qquad \left\{
+\begin{array}{l}
+\mathbf{F}_1(\mathbf{q}_1^{i-1}) + \mathbf{M}_1\mathbf{g} + \mathbf{F}_{ext} = \mathbf{b}_1 \\
+\vdots \\
+\mathbf{F}_4(\mathbf{q}_4^{i-1}) + \mathbf{M}_4\mathbf{g} + \mathbf{F}_{ext} = \mathbf{b}_4
+\end{array}
+\right. & (14) \\[1em]
+5:\ \quad \text{Compute } \mathbf{A}_1(\mathbf{q}_1^{i-1}),\ \ldots,\ \mathbf{A}_4(\mathbf{q}_4^{i-1}) \text{ by linearizing } \mathbf{F}_j\ (j \in [1,4]) \\
+\qquad\ \quad \text{and use the coupling method to obtain } \mathbf{A} \text{ and } \mathbf{b}. \text{ Then solve:} \\[0.5em]
+\qquad\qquad \left\{
+\begin{array}{l}
+\mathbf{A}\, d\mathbf{q} = \mathbf{b} + \mathbf{H}_\mathrm{a}^T \boldsymbol{\lambda}_\mathrm{a} \\
+\text{subject to}\\
+\boldsymbol{\delta}_\mathrm{a}(\mathbf{q}^{i-1}) + \mathbf{H}_\mathrm{a}\, d\mathbf{q} = \textcolor{red}{\mathbf{u}_a}
+\end{array}
+\right. & (15) \\[1em]
+6:\ \quad \text{Step 8 and 9 of Algorithm 1} \\
+7:\ \textbf{end while} \\
+8:\ \textbf{return: } \textcolor{green}{\mathbf{y}_e} = \boldsymbol{\delta}_\mathrm{e}(\mathbf{q}^{i-1}) + \mathbf{H}_\mathrm{e}\, d\mathbf{q}\\
+\hline 
+\end{array}
+$$ 
+{width=55%, .center}

data/algorithms/algorithm3.md

@@ -0,0 +1,17 @@
+$$
+\begin{array}{ll}
+\textbf{Algorithm 3 } \text{Inverse Kinematics obtained by optimization } \textcolor{red}{\mathbf{u}_a} = f^{-1}(\textcolor{green}{\mathbf{y}_e}) \\
+\hline
+1:\ \text{Initialize } \mathbf{q}^0 = [\mathbf{q}_1^0; \mathbf{q}_2^0] \\
+2:\ \text{Step 2 and 3 of Algorithm 1} \\
+3:\ \textbf{while } \|\mathbf{q}^i - \mathbf{q}^{i-1}\| > \epsilon \textbf{ do} \\
+4:\ \quad \text{Find the residual (like Algorithme 2):} \\
+   \qquad\qquad \mathbf{F}(\mathbf{q}^{i-1}) + \mathbf{Mg} + \mathbf{F}_{ext} = \mathbf{b} & (19)\\
+5:\ \quad \text{Compute } \mathbf{A} \text{ and } \mathbf{b} \text{ and solve} \\
+   \qquad\qquad \left\{ \begin{array}{l} \mathbf{A}\,d\mathbf{q} = \mathbf{b} + \mathbf{H}_a^T \boldsymbol{\lambda}_a \\ \text{subject to} \\ \displaystyle\min_{\boldsymbol{\lambda}_a} \tfrac{1}{2}(\boldsymbol{\delta}_e(\mathbf{q}_{i-1}) + \mathbf{H}_e d\mathbf{q}) - \textcolor{green}{\mathbf{y}_e})^2 \end{array} \right. & (20) \\
+6:\ \quad \text{Step 8 and 9 of Algorithm 1} \\
+7:\ \textbf{end while} \\
+8:\ \textbf{return: } \textcolor{red}{\mathbf{u}_a} = \boldsymbol{\delta}_a(\mathbf{q}^{i-1}) + \mathbf{H}_a\,d\mathbf{q} \\
+\hline
+\end{array}
+$$

data/images/lab1-algorithm1.png

@@ -18,7 +18,16 @@ Lagrange multiplier $\boldsymbol{\lambda}_{\mathrm{a}}$. Furthermore, the motor
 $\boldsymbol{\delta}_{\mathrm{a}}$, is a function of the robot's position $\mathbf{q}$ 
 (which is a concatenation of the leg positions: $\mathbf{q}_1,  ..., \mathbf{q}_4$).
 
-![](assets/data/images/lab2-algorithm2.png){width=65%, .center}
+
+#include(assets/data/algorithms/algorithm2.md)
+
+::: spoiler Algorithm 1: Statics of the continuum legs
+
+#include(assets/data/algorithms/algorithm1.md)
+
+:::
+
+<br/>
 
 To compute Equation 15 (in the algorithm above), it is more efficient to proceed with an indirect solution. 
 We will decompose the movement at each time step by separating the contributions from the force $\mathbf{b}$, which is 

data/images/lab2-algorithm2.png

@@ -10,7 +10,15 @@ There are several challenges in solving this inverse problem:
 
 To handle these challenges, we typically employ QP optimization techniques. 
 
-![](assets/data/images/lab2-algorithm3.png){width=65%, .center}
+#include(assets/data/algorithms/algorithm3.md){width=65%, .center}
+
+::: spoiler Algorithm 1: Statics of the continuum legs
+
+#include(assets/data/algorithms/algorithm1.md)
+
+:::
+
+<br/>
 
 With the indirect solving, the optimization presented in equation 20 (in the algorithm above) can be rewriten:
 

data/images/lab2-algorithm3.png

@@ -30,7 +30,7 @@ $$
 
 In practice $\mathbf{F}(\mathbf{q})$ being non-linear, the computation of the equilibrium position is obtained by iterative resolution (see algorithm 1 below).
 
-![](assets/data/images/lab1-algorithm1.png){width=65%, .center}
+#include(assets/data/algorithms/algorithm1.md)
 
 To model the various legs and their deformations, we propose three types of modeling of these internal forces:
 

labs/lab_inversekinematics/sections/2_kinematics.md

@@ -1,25 +1 @@
-{
-    "labs": [
-        {
-            "name": "introduction"
-        },    
-        {
-            "name": "lab_models"
-        },
-        {
-            "name": "lab_inversekinematics"
-        },
-        {
-            "name": "project_pickandplace"
-        },
-        {
-            "name": "lab_design"
-        },
-        {
-            "name": "lab_closedloop"
-        },
-        {
-            "name": "sandbox"
-        }
-    ]
-}
+{"labs":[{"name":"introduction"},{"name":"lab_models"},{"name":"lab_inversekinematics"},{"name":"project_pickandplace"},{"name":"lab_design"},{"name":"lab_closedloop"},{"name":"sandbox"}]}