From 65453e06051e7a438036f6bc7ad482d69e11dbe2 Mon Sep 17 00:00:00 2001 From: Erwan Normand Date: Fri, 20 Sep 2024 09:03:53 +0200 Subject: [PATCH] Fix equation references --- 1-introduction/related-work/1-haptic-hand.tex | 18 ++++++------------ .../related-work/4-visuo-haptic-ar.tex | 7 ++----- config/thesis_commands.tex | 10 +++++++++- 3 files changed, 17 insertions(+), 18 deletions(-) diff --git a/1-introduction/related-work/1-haptic-hand.tex b/1-introduction/related-work/1-haptic-hand.tex index d4c4314..6676b87 100644 --- a/1-introduction/related-work/1-haptic-hand.tex +++ b/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. diff --git a/1-introduction/related-work/4-visuo-haptic-ar.tex b/1-introduction/related-work/4-visuo-haptic-ar.tex index 714fa78..a7f86b9 100644 --- a/1-introduction/related-work/4-visuo-haptic-ar.tex +++ b/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}. diff --git a/config/thesis_commands.tex b/config/thesis_commands.tex index 2b165d6..6238525 100644 --- a/config/thesis_commands.tex +++ b/config/thesis_commands.tex @@ -30,12 +30,20 @@ \renewcommand{\secref}[2][\labelprefix]{Section~\ref{#1:#2}} \renewcommand{\tabref}[2][\labelprefix]{Table~\ref{#1:tab:#2}} +\NewEnvironmentCopy{oldequation}{equation} +\RenewDocumentEnvironment{equation}{m}{% + \begin{oldequation}% + \label{\labelprefix:eq:#1}% + }{% + \end{oldequation}% +} + % Images % example: \fig[1]{universe}{The Universe}[Additional caption text, not shown in the list of figures] % reference later with: \figref{universe} % 1 = \linewidth = 150 mm \RenewDocumentCommand{\fig}{O{1} O{htbp} m m O{}}{% #1 = width, #2 = position, #3 = filename, #4 = caption, #5 = additional caption - \begin{figure}[#2] + \begin{figure}[#2]% \centering% \includegraphics[width=#1\linewidth]{#3}% \caption[#4]{#4#5}