From 0ba933560b057beda54487ab94d5d520047d1c17 Mon Sep 17 00:00:00 2001
From: Sylvain CALINON <sylvain.calinon@idiap.ch>
Date: Thu, 21 Nov 2024 20:10:35 +0100
Subject: [PATCH] images added

---
 .../ergodicControl-fromCrudetoPrecise01.jpg   | Bin 0 -> 229601 bytes
 doc/images/f_ee01.png                         | Bin 194810 -> 174062 bytes
 doc/images/iLQR_objectBoundaries01.png        | Bin 0 -> 595819 bytes
 doc/images/transformations01.jpg              | Bin 260417 -> 0 bytes
 doc/images/transformations01.png              | Bin 0 -> 276095 bytes
 doc/rcfs.pdf                                  | Bin 13350581 -> 13585180 bytes
 doc/rcfs.tex                                  |  13 ++++++++-----
 7 files changed, 8 insertions(+), 5 deletions(-)
 create mode 100644 doc/images/ergodicControl-fromCrudetoPrecise01.jpg
 create mode 100755 doc/images/iLQR_objectBoundaries01.png
 delete mode 100644 doc/images/transformations01.jpg
 create mode 100755 doc/images/transformations01.png

diff --git a/doc/images/ergodicControl-fromCrudetoPrecise01.jpg b/doc/images/ergodicControl-fromCrudetoPrecise01.jpg
new file mode 100644
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diff --git a/doc/rcfs.tex b/doc/rcfs.tex
index 1664acc..8ad68ae 100644
--- a/doc/rcfs.tex
+++ b/doc/rcfs.tex
@@ -573,7 +573,7 @@ f = np.array([L @ np.diag(l) @ np.cos(L @ x), L @ np.diag(l) @ np.sin(L @ x)]) #
 
 \begin{figure}
 \centering
-\includegraphics[width=.5\columnwidth]{images/transformations01.jpg}
+\includegraphics[width=.5\columnwidth]{images/transformations01.png}
 \caption{\footnotesize
 Typical transformations involved in a manipulation task involving a robot, a vision system, a visual marker on the object, and a desired grasping location on the object.  
 }
@@ -601,12 +601,12 @@ where $\bm{J}\in\mathbb{R}^{R\times D}$ is the Jacobian matrix of $\bm{f}\in\mat
 
 For the orientation part of the data (if considered), the residual vector $\bm{f}(\bm{x}) = \bm{f}^\tp{ee}(\bm{x}) - \bm{\mu}$ is replaced by a geodesic residual computed with the logarithmic map $\bm{f}(\bm{x}) = \mathrm{Log}_{\bm{\mu}}\!\big(\bm{f}^\tp{ee}(\bm{x})\big)$, see \cite{Calinon20RAM} for details.
 
-The approach can also be extended to target objects/landmarks with positions $\bm{\mu}$ and rotation matrices $\bm{A}$, as depicted in Fig.~\ref{fig:transformations}. %, whose columns are basis vectors forming a coordinate system
+The approach can also be extended to target objects/landmarks with positions $\bm{\mu}$ and rotation matrices $\bm{U}$, as depicted in Fig.~\ref{fig:transformations}. %, whose columns are basis vectors forming a coordinate system
 We can then define an error between the robot endeffector and an object/landmark expressed in the object/landmark coordinate system as 
 \begin{equation}
 \begin{aligned}
-	\bm{f}(\bm{x}) &= \bm{A}^\trsp \big(\bm{f}^\tp{ee}(\bm{x}) - \bm{\mu}\big), \\
-	\bm{J}(\bm{x}) &= \bm{A}^\trsp \bm{J}^\tp{ee}(\bm{x}). 
+	\bm{f}(\bm{x}) &= \bm{U}^\trsp \big(\bm{f}^\tp{ee}(\bm{x}) - \bm{\mu}\big), \\
+	\bm{J}(\bm{x}) &= \bm{U}^\trsp \bm{J}^\tp{ee}(\bm{x}). 
 \end{aligned}
 \label{eq:fJU}
 \end{equation}
@@ -2483,7 +2483,7 @@ Based on the above definitions, $\bm{f}(\bm{x})$ and $\bm{J}(\bm{x})$ are in thi
 
 \begin{SCfigure}
 \centering
-\includegraphics[width=.6\textwidth]{images/iLQR_objectBoundaries01.jpg}
+\includegraphics[width=.6\textwidth]{images/iLQR_objectBoundaries01.png}
 \caption{\footnotesize
 Example of a viapoints task in which a planar robot with 3 joints needs to sequentially reach 2 objects, with object boundaries defining the allowed reaching points on the objects surfaces. \emph{Left:} Reaching task with two viapoints at $t=25$ and $t=50$. \emph{Right:} Corresponding values of the cost function for the endeffector space at $t=25$ and $t=50$.
 }
@@ -3226,6 +3226,9 @@ Search and exploration problems can be formulated in various ways. Ergodic contr
 }
 \end{figure}
 
+
+%ergodicControl-fromCrudetoPrecise01.jpg
+
 %\begin{figure}
 %\centering{
 
-- 
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