Fix equation references

1-introduction/related-work/1-haptic-hand.tex

@@ -195,8 +195,7 @@ However, the speed of exploration affects the perceived intensity of micro-rough
 
 To establish the relationship between spacing and intensity for macro-roughness, patterned textured surfaces were manufactured: as a linear grating (on one axis) composed of ridges and grooves, \eg in \figref{lawrence2007haptic_1}~\cite{lederman1972fingertip,lawrence2007haptic}, or as a surface composed of micro conical elements on two axes, \eg in \figref{klatzky2003feeling_1}~\cite{klatzky2003feeling}.
 As shown in \figref{lawrence2007haptic_2}, there is a quadratic relationship between the logarithm of the perceived roughness intensity $R$ and the logarithm of the space between the elements $s$ ($a$, $b$ and $c$ are empirical parameters to be estimated)~\cite{klatzky2003feeling}:
-\begin{equation}
-    \label{eq:roughness_intensity}
+\begin{equation}{roughness_intensity}
     log(R) \sim a \, log(s)^2 + b \, s + c
 \end{equation}
 A larger spacing between elements increases the perceived roughness, but reaches a plateau from \qty{\sim 5}{\mm} for the linear grating~\cite{lawrence2007haptic}, while the roughness decreases from \qty{\sim 2.5}{\mm}~\cite{klatzky2003feeling} for the conical elements.
@@ -227,8 +226,7 @@ However, as the speed of exploration changes the transmitted vibrations, a faste
 Even when the fingertips are deafferented (absence of cutaneous sensations), the perception of roughness is maintained~\cite{libouton2012tactile}, thanks to the propagation of vibrations in the finger, hand and wrist, for both pattern and natural textures~\cite{delhaye2012textureinduced}.
 The spectrum of vibrations shifts to higher frequencies as the exploration speed increases, but the brain integrates this change with proprioception to keep the \emph{perception constant} of the texture.
 For grid textures, as illustrated in \figref{delhaye2012textureinduced}, the ratio of the finger speed $v$ to the frequency of the vibration intensity peak $f_p$ is measured most of the time equal to the period $\lambda$ of the spacing of the elements:
-\begin{equation}
-    \label{eq:grating_vibrations}
+\begin{equation}{grating_vibrations}
     \lambda \sim \frac{v}{f_p}
 \end{equation}
 
@@ -256,14 +254,12 @@ Passive touch (without voluntary hand movements) and tapping allow a perception
 
 Two physical properties determine the haptic perception of hardness: its stiffness and elasticity, as shown in \figref{hardness}~\cite{bergmanntiest2010tactual}.
 The \emph{stiffness} $k$ of an object is the ratio between the applied force $F$ and the resulting \emph{displacement} $D$ of the surface:
-\begin{equation}
-    \label{eq:stiffness}
+\begin{equation}{stiffness}
     k = \frac{F}{D}
 \end{equation}
 
 The \emph{elasticity} of an object is expressed by its Young's modulus $Y$, which is the ratio between the applied pressure (the force $F$ per unit area $A$) and the resulting deformation $D / l$ (the relative displacement) of the object:
-\begin{equation}
-    \label{eq:young_modulus}
+\begin{equation}{young_modulus}
     Y = \frac{F / A}{D / l}
 \end{equation}
 
@@ -295,8 +291,7 @@ In addition, an object with low stiffness but high Young's modulus can be percei
 %When running the finger on a surface with a lateral movement (\secref{exploratory_procedures}), the skin-surface contacts generate frictional forces in the opposite direction to the finger movement, giving kinesthetic cues, and also stretch the skin, giving cutaneous cues.
 %As illustrated in \figref{smith1996subjective_1}, a stick-slip phenomenon can also occur, where the finger is intermittently slowed by friction before continuing to move, on both rough and smooth surfaces~\cite{derler2013stick}.
 %The amplitude of the frictional force $F_s$ is proportional to the normal force of the finger $F_n$, \ie the force perpendicular to the surface, according to a coefficient of friction $\mu$:
-%\begin{equation}
-%    \label{eq:friction}
+%\begin{equation}{friction}
 %    F_s = \mu \, F_n
 %\end{equation}
 %The perceived intensity of friction is thus roughly related to the friction coefficient $\mu$~\cite{smith1996subjective}.
@@ -328,8 +323,7 @@ In addition, an object with low stiffness but high Young's modulus can be percei
 
 %Parce qu'elle est basée sur la circulation de la chaleur, la perception de la température est plus lente que les autres propriétés matérielles et demande un toucher statique (voir \figref{exploratory_procedures}) de plusieurs secondes pour que la température de la peau s'équilibre avec celle de l'objet.
 %La température $T(t)$ du doigt à l'instant $t$ et au contact avec une surface suit une loi décroissante exponentielle, où $T_s$ est la température initiale de la peau, $T_e$ est la température de la surface, $t$ est le temps et $\tau$ est la constante de temps:
-%\begin{equation}
-%    \label{eq:temperature}
+%\begin{equation}{temperature}
 %    T(t) = (T_s - T_e) \, e^{-t / \tau} + T_e
 %\end{equation}
 %Le taux de transfert de chaleur, décrit par $\tau$, et l'écart de température $T_s - T_e$, sont les deux indices essentiels pour la perception de la température.

1-introduction/related-work/4-visuo-haptic-ar.tex

@@ -87,11 +87,8 @@ Adding a visual delay increased the perceived stiffness of the reference piston,
 \end{subfigs}
 
 %explained how these delays affected the integration of the visual and haptic perceptual cues of stiffness.
-The stiffness $k$ of the piston is indeed estimated by both sight and proprioception as the ratio of the exerted force $F$ and the displacement $D$ of the piston, following \eqref{stiffness}.
-But a delay $\Delta t$ modify the equation to:
-\begin{equation}
-    \label{eq:stiffness_delay}
-    k = \frac{F(t_A)}{D (t_B)}
+\begin{equation}{stiffness_delay}
+    k = \frac{F(t_H)}{D(t_V)} \quad \text{with} \quad t_H = t_V + \Delta t
 \end{equation}
 where $t_B = t_A + \Delta t$.
 Therefore, a haptic delay (positive $\Delta t$) increases the perceived stiffness $k$, while a visual delay in displacement (negative $\Delta t$) decreases perceived $k$~\cite{diluca2011effects}.

config/thesis_commands.tex

@@ -30,12 +30,20 @@
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+}
+
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