\section{Perception and Interaction with the Hand}
\label{haptic_hand}

% describe how the hand senses and acts on its environment to perceive the haptic properties of real everyday objects.

The haptic sense has specific characteristics that make it unique in regard to other senses.
It enables us to perceive a large diversity of properties in the surrounding objects, through to a complex combination of sensations produced by numerous sensory receptors distributed throughout the body, but particularly in the hand.
It also allows us to act with the hand on these objects, to come into contact with them, to grasp them, to actively explore them, and to manipulate them.
This implies that the haptic perception is localized at the points of contact between the hand and the environment, \ie we cannot haptically perceive an object without actively touching it.
These two mechanisms, \emph{action} and \emph{perception}, are therefore closely associated and both essential to form the haptic experience of interacting with the environment using the hand~\cite{lederman2009haptic}.


\subsection{The Haptic Sense}
\label{haptic_sense}

Perceiving the properties of an object involves numerous sensory receptors embedded in the skin, but also in the muscles and joints of the hand, and distributed across the body. They are divided into two main modalities: \emph{cutaneous} and \emph{kinesthetic}.

\subsubsection{Cutaneous Sensitivity}
\label{cutaneous_sensitivity}

Cutaneous haptic receptors are specialized nerve endings implanted in the skin that respond differently to the various stimuli applied to the skin. \figref{blausen2014medical_skin} shows the location of the four main cutaneous receptors that respond to mechanical deformation of the skin.

\fig[0.6]{blausen2014medical_skin}{Schema of cutaneous mechanoreceptors in a section of the skin~\cite{blausen2014medical}.}

Adaptation rate and receptor size are the two key characteristics that respectively determine the temporal and spatial resolution of these \emph{mechanoreceptors}, as summarized in \tabref{cutaneous_receptors}.
The \emph{adaptation rate} is the speed and duration of the response to a stimulus.
Meissner and Pacinian receptors, known as fast-adapting (FA), respond rapidly to a stimulus but stop quickly even though the stimulus is still present, allowing the detection of high-frequency changes.
In contrast, Merkel and Ruffini receptors, known as slow-adapting (SA), have a slower but continuous response to a static, prolonged stimulus.
The \emph{size of the receptor} determines the area of skin that can be sensed by a single nerve ending.
Meissner and Merkel receptors have a small detection area (named Type I) and are sensitive to fine skin deformations, while Ruffini and Pacinian receptors have a larger detection area (named Type II).

\begin{tab}{cutaneous_receptors}{Characteristics of the cutaneous mechanoreceptors.}[
        Adaptation rate is the speed and duration of the receptor's response to a stimulus. Receptive size is the area of skin detectable by a single receptor. Sensitivities are the stimuli detected by the receptor. Adapted from \textcite{mcglone2007discriminative} and \textcite{johansson2009coding}.
    ]
    \begin{tabularx}{\linewidth}{p{1.7cm} p{2cm} p{2cm} X}
        \toprule
        \textbf{Receptor} & \textbf{Adaptation Rate} & \textbf{Receptive Size} & \textbf{Sensitivities}                                                          \\
        \midrule
        Meissner          & Fast                     & Small                   & Discontinuities (\eg edges), medium-frequency vibration (\qtyrange{5}{50}{\Hz}) \\
        Merkel            & Slow                     & Small                   & Pressure, low-frequency vibration (\qtyrange{0}{5}{\Hz})                        \\
        Pacinian          & Fast                     & Large                   & High-frequency vibration (\qtyrange{40}{400}{\Hz})                              \\
        Ruffini           & Slow                     & Large                   & Skin stretch                                                                    \\
        \bottomrule
    \end{tabularx}
\end{tab}

The density of mechanoreceptors varies according to skin type and body region.
\emph{Glabrous skin}, especially on the face, feet, hands, and more importantly, the fingers, is particularly rich in cutaneous receptors, giving these regions great tactile sensitivity.
The density of the Meissner and Merkel receptors, which are the most sensitive, is notably high in the fingertips~\cite{johansson2009coding}.
Conversely, \emph{hairy skin} is less sensitive and does not contain Meissner receptors, but has additional receptors at the base of the hairs, as well as receptors known as C-tactile, which are involved in pleasantness and affective touch~\cite{ackerley2014touch}.

There are also two types of thermal receptors implanted in the skin, which respond to increases or decreases in skin temperature, respectively, providing sensations of warmth or cold~\cite{lederman2009haptic}.
Finally, free nerve endings (without specialized receptors) provide information about pain~\cite{mcglone2007discriminative}.

\subsubsection{Kinesthetic Sensitivity}
\label{kinesthetic_sensitivity}

Kinesthetic receptors are also mechanoreceptors but are located in the muscles, tendons and joints~\cite{jones2006human}.
The muscle spindles respond to the length and the rate of stretch/contraction of the muscles.
Golgi tendon organs, located at the junction of muscles and tendons, respond to the force developed by the muscles.
Ruffini and Pacini receptors are found in the joints and respond to joint movement.
Together, these receptors provide sensory feedback about the movement, speed and strength of the muscles and the rotation of the joints during a movement.
They can also sense external forces and torques applied to the body.

Kinesthetic receptors are therefore closely linked to the motor control of the body.
By providing sensory feedback in response to the position and movement of our limbs, they enable us to perceive our body in space, a perception called \emph{proprioception}.
This allows us to plan and execute precise movements to touch or grasp a target, even with our eyes closed.
Cutaneous mechanoreceptors are essential for this perception because any movement of the body or contact with the environment necessarily deforms the skin~\cite{johansson2009coding}.


\subsection{Hand-Object Interactions}
\label{hand_object_interactions}

The sense of touch is thus composed of a rich and complex set of various cutaneous and kinesthetic receptors under the skin, in the muscles and in the joints.
These receptors give the hand its great tactile sensitivity and great dexterity in its movements.

\subsubsection{Sensorimotor Continuum of the Hand}
\label{sensorimotor_continuum}

\textcite{jones2006human} have proposed a sensorimotor continuum of hand functions, from mainly sensory activities to activities with a more important motor component.
As illustrated in the \figref{sensorimotor_continuum}, \Citeauthor{jones2006human} propose to delineate four categories of hand function on this continuum:
\begin{itemize}
    \item \emph{Passive touch}, or tactile sensing, is the ability to perceive an object through cutaneous sensations with a static hand contact. The object may be moving, but the hand remains static. It allows for relatively good surface perception, \eg in \textcite{gunther2022smooth}.
    \item \emph{Exploration}, or active haptic sensing, is the manual and voluntary exploration of an object with the hand, involving all cutaneous and kinesthetic sensations. It enables a more precise perception than passive touch~\cite{lederman2009haptic}.
    \item \emph{Prehension} is the action of grasping and holding an object with the hand. It involves fine coordination between hand and finger movements and the haptic sensations produced.
    \item \emph{Gestures}, or non-prehensible skilled movements, are motor activities without constant contact with an object. Examples include pointing at a target, typing on a keyboard, accompanying speech with gestures, or signing in sign language, \eg in \textcite{yoon2020evaluating}.
\end{itemize}

\fig[0.65]{sensorimotor_continuum}{
    The sensorimotor continuum of the hand function proposed by and adapted from \textcite{jones2006human}.
}[
    Functions of the hand are classified into four categories based on the relative importance of sensory and motor components.
    Icons are from \href{https://thenounproject.com/creator/leremy/}{Gan Khoon Lay} / \href{https://creativecommons.org/licenses/by/3.0/}{CC BY}.
]

This classification has been further refined by \textcite{bullock2013handcentric} into 15 categories of possible hand interactions with an object.
In this thesis, we are interested in exploring \vh augmentations (see \partref{perception}) and grasping of \VOs (see \partref{manipulation}) in the context of \AR and \WHs.

\subsubsection{Hand Anatomy and Motion}
\label{hand_anatomy}

Before we describe how the hand is used to explore and grasp objects, we need to look at its anatomy.
Underneath the skin, muscles and tendons can actually move because they are anchored to the bones.

As shown in the \figref{blausen2014medical_hand}, the skeleton of the hand is formed of 27 articulated bones.
The wrist, comprising 8 carpal bones, connects the hand to the arm and is the base for the 5 metacarpal bones of the palm, one for each finger.
Each finger is formed by a chain of 3 phalanges, proximal, middle and distal, except for the thumb which has only two phalanges, proximal and distal.
The joints at the base of each phalanx allow flexion and extension, \ie folding and unfolding movements relative to the preceding bone.
The proximal phalanges can also adduct and abduct, \ie move the fingers towards and away from each other.
Finally, the metacarpal of the thumb is capable of flexion/extension and adduction/abduction, which allows the thumb to oppose the other fingers.
These axes of movement are called DoFs and can be represented by a \emph{kinematic model} of the hand with 27 DoFs as shown in the \figref{blausen2014medical_hand}.
Thus the thumb has 5 DoFs, each of the other four fingers has 4 DoFs and the wrist has 6 DoFs and can take any position (3 DoFs) or orientation (3 DoFs) in space~\cite{erol2007visionbased}.

This complex structure enables the hand to perform a wide range of movements and gestures. However, the way we explore and grasp objects follows simpler patterns, depending on the object being touched and the aim of the interaction.

\begin{subfigs}{hand}{Anatomy and motion of the hand. }[
        \item Schema of the hand skeleton. Adapted from \textcite{blausen2014medical}.
        \item Kinematic model of the hand with 27 \DoFs~\cite{erol2007visionbased}.
    ]
    \subfigsheight{58mm}
    \subfig{blausen2014medical_hand}
    \subfig{kinematic_hand_model}
\end{subfigs}

\subsubsection{Exploratory Procedures}
\label{exploratory_procedures}

The exploration of an object by the hand follows patterns of movement, called exploratory procedures~\cite{lederman1987hand}.
As illustrated in the \figref{exploratory_procedures}, a specific and optimal movement of the hand is performed for a given property of the object being explored to acquire the most relevant sensory information for that property.
For example, a \emph{lateral movement} of the fingers on the surface to identify its texture, a \emph{pressure} with the finger to perceive its hardness, or a \emph{contour following} of the object to infer its shape.
These three procedures involve only the fingertips and in particular the index finger~\cite{gonzalez2014analysis}.
For the other procedures, the whole hand is used: for example, approaching or posing the palm to feel the temperature (\emph{static contact}), holding the object in the hand to estimate its weight (\emph{unsupported holding}).
The \emph{enclosure} with the hand makes it possible to judge the general shape and size of the object.
It takes only \qtyrange{2}{3}{\s} to perform these procedures, except for contour following, which can take about ten seconds~\cite{jones2006human}.

\fig{exploratory_procedures}{Exploratory procedures and their associated object properties (in parentheses). Adapted from \textcite{lederman2009haptic}.}

%Le sens haptique seul (sans la vision) nous permet ainsi de reconnaitre les objets et matériaux avec une grande précision.
%La reconnaissance des propriété matérielles, \ie la surface et sa texture, rigidité et température est meilleure qu'avec le sens visuel seul.
%Mais la reconnaissance des propriétés spatiales, la forme et la taille de l'objet, est moins bonne avec l'haptique qu'avec la vision~\cite{lederman2009haptic}.
%Quelques secondes (\qtyrange{2}{3}{\s}) suffisent pour effectuer ces procédures, à l'exception du suivi de contour qui peut prendre une dizaine de secondes~\cite{jones2006human}.

\subsubsection{Grasp Types}
\label{grasp_types}

Thanks to the degrees of freedom of its skeleton, the hand can take many postures to grasp an object (see \secref{hand_anatomy}).
By placing the thumb or palm against the other fingers (pad or palm grasps respectively), or by placing the fingers against each other as if holding a cigarette (side grasp), the hand can hold the object securely.
Grasping adapts to the shape of the object and the task to be performed, \eg grasping a pen with the fingertips then holding it to write, or taking a mug by the body to fill it and by the handle to drink it~\cite{cutkosky1986modeling}.
Three types of grasp are differentiated according to their degree of strength and precision.
In \emph{power grasps}, the object is held firmly and follows the movements of the hand rigidly.
In \emph{precision grasps}, the fingers can move the object within the hand but without moving the arm.
\emph{Intermediate grasps} combine strength and precision in equal proportions~\cite{feix2016grasp}.

For all possible objects and tasks, the number of grasp types can be reduced to 34 and classified as the taxonomy on \figref{gonzalez2014analysis}~\cite{gonzalez2014analysis}.\footnote{An updated taxonomy was then proposed by \textcite{feix2016grasp}: it is more complete but harder to present.}
For everyday objects, this number is even smaller, with between 5 and 10 grasp types depending on the activity~\cite{bullock2013grasp}.
Furthermore, the fingertips are the most involved areas of the hand, both in terms of frequency of use and time spent in contact: In particular, the thumb is almost always used, as well as the index and middle fingers, but the other fingers are used less frequently~\cite{gonzalez2014analysis}.
This can be explained by the sensitivity of the fingertips (see \secref{haptic_sense}) and the ease with which the thumb can be opposed to the index and middle fingers compared to the other fingers.

\fig{gonzalez2014analysis}{Taxonomy of grasp types of~\textcite{gonzalez2014analysis}}[, classified according to their type (power, precision or intermediate) and the shape of the grasped object. Each grasp shows the area of the palm and fingers in contact with the object and the grasp with an example of object.]


\subsection{Haptic Perception of Object Properties}
\label{object_properties}

The active exploration of an object with the hand is performed as a sensorimotor loop: The exploratory movements guide the search for and adapt to sensory information, allowing to construct a haptic perception of the object's properties.
There are two main types of \emph{perceptual properties}.
The \emph{material properties} are the perception of the roughness, hardness, temperature and friction of the surface of the object~\cite{bergmanntiest2010tactual}.
The \emph{spatial properties} are the perception of the weight, shape and size of the object~\cite{lederman2009haptic}.

Each of these properties is closely related to a physical property of the object, which is defined and measurable, but perception is a subjective experience and often differs from this physical measurement.
Perception also depends on many other factors, such as the movements made and the exploration time, but also on the person, their sensitivity~\cite{hollins2000individual} or age~\cite{jones2006human}, and the context of the interaction~\cite{kahrimanovic2009context,kappers2013haptic}.
These properties are described and rated\footnotemark using scales opposing two adjectives such as \enquote{rough/smooth} or \enquote{hot/cold}~\cite{okamoto2013psychophysical}.
\footnotetext{All the haptic perception measurements described in this chapter were performed by blindfolded participants, to control for the influence of vision.}


\subsubsection{Roughness}
\label{roughness}

Roughness (or smoothness) is the perception of the \emph{micro-geometry} of a surface, \ie asperities with differences in height on the order of millimeters to micrometers~\cite{bergmanntiest2010tactual}.
It is, for example, the perception of the fibers of fabric or wood and the texture of sandpaper or paint.
Roughness is what essentially characterises the perception of the \emph{texture} of the surface~\cite{hollins1993perceptual,baumgartner2013visual}.

When touching a surface in static touch, the asperities deform the skin and cause pressure sensations that allow a good perception of coarse roughness.
But when running the finger over the surface with a lateral movement (see \secref{exploratory_procedures}), vibrations are alos caused which give a better discrimination range and precision of roughness~\cite{bensmaia2005pacinian}.
In particular, when the asperities are smaller than \qty{0.1}{mm}, such as paper fibers, the pressure cues are no longer captured and only the movement, \ie the vibrations, can be used to detect the roughness~\cite{hollins2000evidence}.
This limit distinguishes \emph{macro-roughness} from \emph{micro-roughness}.

The physical properties of the surface determine the haptic perception of roughness.
The most important characteristic is the density of the surface elements, \ie the spacing between them: The perceived (subjective) intensity of roughness increases with spacing, for macro-roughness~\cite{klatzky2003feeling,lawrence2007haptic} and micro-roughness~\cite{bensmaia2003vibrations}.
For macro-textures, the size of the elements, the force applied and the speed of exploration have limited effects on the intensity perceived~\cite{klatzky2010multisensory}: macro-roughness is a \emph{spatial perception}.
This allows us to read Braille~\cite{lederman2009haptic}.
However, the speed of exploration affects the perceived intensity of micro-roughness~\cite{bensmaia2003vibrations}.

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}
    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.

\begin{subfigs}{lawrence2007hapti}{Estimation of haptic roughness of a linear grating surface by active exploration~\cite{lawrence2007haptic}. }[
        \item Schema of a linear grating surface, composed of ridges and grooves.
        \item Perceived intensity of roughness (vertical axis) of the surface as a function of the size of the grooves (horizontal axis, interval of \qtyrange{0.125}{4.5}{mm}), the size of the ridges (RW, circles and squares) and the mode of exploration (with the finger in white and via a rigid probe held in hand in black).
    ]
    \subfigsheight{56mm}
    \subfig{lawrence2007haptic_1}
    \subfig{lawrence2007haptic_2}
\end{subfigs}

It is also possible to perceive the roughness of a surface by \emph{indirect touch}, with a tool held in the hand, for example by writing with a pen on paper~\cite{klatzky2003feeling}.
The skin is no longer deformed and only the vibrations of the tool are transmitted.
But this information is sufficient to feel the roughness, which perceived intensity follows the same quadratic law.
The intensity peak varies with the size of the contact surface of the tool, \eg a small tool allows to perceive finer spaces between the elements than with the finger (see \figref{klatzky2003feeling_2}).
However, as the speed of exploration changes the transmitted vibrations, a faster speed shifts the perceived intensity peak slightly to the right, \ie decreasing perceived roughness for fine spacings and increasing it for large spacings~\cite{klatzky2003feeling}.

\begin{subfigs}{klatzky2003feeling}{Estimation of haptic roughness of a surface of conical micro-elements by active exploration~\cite{klatzky2003feeling}. }[
        \item Electron micrograph of conical micro-elements on the surface.
        \item Perceived intensity of roughness (vertical axis) of the surface as a function of the average spacing of the elements (horizontal axis, interval of \qtyrange{0.8}{4.5}{mm}) and the mode of exploration (with the finger in black and via a rigid probe held in hand in white).
    ]
    \subfig[.25]{klatzky2003feeling_1}
    \subfig[.5]{klatzky2003feeling_2}
\end{subfigs}

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}
    \lambda \sim \frac{v}{f_p}
\end{equation}

The vibrations generated by exploring natural textures are also very specific to each texture and similar between individuals, making them identifiable by vibration alone~\cite{greenspon2020effect}.
This shows the importance of vibration cues even for macro textures and the possibility of generating virtual texture sensations with vibrotactile rendering.

\fig[0.55]{delhaye2012textureinduced}{Speed of finger exploration (horizontal axis) on grating textures with different periods $\lambda$ of spacing (in color) and frequency of the vibration intensity peak $f_p$ propagated in the wrist (vertical axis)~\cite{delhaye2012textureinduced}.}

The everyday "natural" textures are more complex to study because they are composed of multiple elements of different sizes and spacings.
In addition, the perceptions of micro and macro roughness overlap and are difficult to distinguish~\cite{okamoto2013psychophysical}.
Thus, individuals have a subjective definition of roughness, with some paying more attention to larger elements and others to smaller ones~\cite{bergmanntiest2007haptic}, or even including other perceptual properties such as hardness or friction~\cite{bergmanntiest2010tactual}.


\subsubsection{Hardness}
\label{hardness}

Hardness (or softness) is the perception of the \emph{resistance to deformation} of an object when pressed or tapped~\cite{bergmanntiest2010tactual}.
The perceived softness of a fruit allows us to judge its ripeness, while ceramic is perceived as hard.
By tapping on a surface, metal will be perceived as harder than wood.
If the surface returns to its original shape after being deformed, the object is elastic (like a spring), otherwise it is plastic (like clay).

When the finger presses on an object (see \figref{exploratory_procedures}), its surface will move and deform with some resistance, and the contact area of the skin will also expand, changing the pressure distribution.
When the surface is touched or tapped, vibrations are also transmitted to the skin.
Passive touch (without voluntary hand movements) and tapping allow a perception of hardness as good as active touch~\cite{friedman2008magnitude}.

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} $\Delta l$ of the surface:
\begin{equation}
    \label{eq:stiffness}
    k = \frac{F}{\Delta l}
\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 $\Delta l / l$ (the relative displacement) of the object:
\begin{equation}
    \label{eq:young_modulus}
    Y = \frac{F / A}{\Delta l / l}
\end{equation}

\begin{subfigs}{stiffness_young}{Perceived hardness of an object by finger pressure. }[
        \item Diagram of an object with a stiffness coefficient $k$ and a length $l$ compressed by a force $F$ on an area $A$ by a distance $\Delta l$.
        \item Identical perceived hardness intensity between Young's modulus (horizontal axis) and stiffness (vertical axis). The dashed and dotted lines indicate the objects tested, the arrows the correspondences made between these objects, and the grey lines the predictions of the quadratic relationship~\cite{bergmanntiest2009cues}.
    ]
    \subfig[.3]{hardness}
    \subfig[.45]{bergmanntiest2009cues}
\end{subfigs}

\textcite{bergmanntiest2009cues} showed the role of these two physical properties in the perception of hardness.
With finger pressure, a relative difference (the \emph{Weber fraction}) of \percent{\sim 15} is required to discriminate between two objects of different stiffness or elasticity.
However, in the absence of pressure sensations (by placing a thin disc between the finger and the object), the necessary relative difference becomes much larger (Weber fraction of \percent{\sim 50}).
Thus, the perception of hardness relies on \percent{90} on surface deformation cues and \percent{10} on displacement cues.
In addition, an object with low stiffness but high Young's modulus can be perceived as hard, and vice versa, as shown in \figref{bergmanntiest2009cues}.

%En pressant du doigt, l'intensité perçue (subjective) de dureté suit avec la raideur une relation selon une loi de puissance avec un exposant de \num{0.8}~\cite{harper1964subjective}, \ie quand la raideur double, la dureté perçue augmente de \num{1.7}.
%\textcite{bergmanntiest2009cues} ont ainsi observé une relation quadratique d'égale intensité perçue de dureté, comme illustré sur la \figref{bergmanntiest2009cues}.


\subsubsection{Friction}
\label{friction}

Friction (or slipperiness) is the perception of \emph{resistance to movement} on a surface~\cite{bergmanntiest2010tactual}.
Sandpaper is typically perceived as sticky because it has a strong resistance to sliding on its surface, while glass is perceived as more slippery.
This perceptual property is closely related to the perception of roughness~\cite{hollins1993perceptual,baumgartner2013visual}.

When running the finger on a surface with a lateral movement (see \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 down by the friction before continuing its movement, both on 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 friction coefficient $\mu$:
\begin{equation}
    \label{eq:friction}
    F_s = \mu \, F_n
\end{equation}
The perceived intensity of friction is thus that of the friction coefficient $\mu$~\cite{smith1996subjective}.

\begin{subfigs}{smith1996subjective}{Perception des frottements haptique de plusieurs matériaux par exploration active avec le doigt~\cite{smith1996subjective}. }[
        \item Mesures des forces normales $F_n$ et tangentielles $F_t$ lors de l'exploration de deux surfaces : une lisse (le verre) et une rugueuse (le nyloprint). Les fluctuations de la force tangentielle sont dues au phénomène de stick-slip. Le coefficient de frottement $\mu$ peut être estimé par la pente de la relation entre les forces normales et tangentielle.
        \item Intensité perçue des frottements (axe vertical) en fonction du coefficient de frottement $\mu$ estimé de l'exploration (axe horizontal) pour quatre matériaux (formes et couleurs).
    ]
    \subfigsheight{55mm}
    \subfig{smith1996subjective_1}
    \subfig{smith1996subjective_2}
\end{subfigs}

Mais la perception de la friction est complexe car ce n'est pas seulement une propriété haptique de la surface.
Elle est, en effet, déterminée par les interactions à l'échelle micro entre la surface et la peau, et dépend donc de la force normale appliquée, de la vitesse du mouvement, de l'aire du contact et de l'humidité de la peau et de la surface~\cite{adams2013finger,messaoud2016relation}.
En ce sens, la perception de cette propriété est encore mal comprise~\cite{okamoto2013psychophysical}.

C'est pourtant une perception fondamentale pour la saisie et la manipulation d'objets : les forces de frottements permettent de tenir fermement l'objet en main pour éviter qu'il ne glisse, et la perception de la friction permet également d'ajuster automatiquement et très rapidement la force à appliquer à l'objet pour le saisir~\cite{johansson1984roles}.
Si le doigt est anesthésié, l'absence de sensations cutanées empêche d'ajuster efficacement la force de préhension: Les forces de l'objet sur le doigt ne sont plus correctement perçues et les doigts appuient alors plus fermement sur l'objet en compensation mais sans réaliser une bonne opposition des doigts~\cite{witney2004cutaneous}.


\subsubsection{Temperature}
\label{temperature}

La température (ou coldness/warmness) est la perception du \emph{transfert de chaleur} entre la surface touchée et la peau~\cite{bergmanntiest2010tactual}:
Si de la chaleur est extraite de (apportée à) la peau, la surface est perçue comme froide (chaude).
Le métal sera perçu comme plus froid que du bois avec la même température de la pièce ; c'est une propriété importante pour discriminer les matériaux~\cite{ho2006contribution}.
Cette perception est donc distincte de la température physique du matériau, et dépend de la conductance thermique et de la capacité thermique du matériau, du volume de l'objet, de la différence de température initiale entre la surface et la peau, et de l'aire du contact~\cite{kappers2013haptic}.
Par exemple, un objet plus volumineux ou une surface plus lisse, augmentant l'aire de contact, augmente la circulation thermique et rend une sensation de température plus intense~\cite{bergmanntiest2008thermosensory}.

%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}
%    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.
%Dans des conditions de la vie de tous les jours, avec une température de la pièce de \qty{20}{\celsius}, une différence relative du taux de transfert de chaleur de \percent{43} ou un écart de \qty{2}{\celsius} est nécessaire pour percevoir une différence de température~\cite{bergmanntiest2009tactile}.


\subsubsection{Spatial Properties}
\label{spatial_properties}

Le poids, la taille et la forme d'un objet sont des propriétés haptiques dites spatiales qui sont indépendantes des propriétés matérielles décrites précédemment.

Le poids (ou heaviness/lightness) est la perception de la \emph{masse} de l'objet~\cite{bergmanntiest2010haptic}.
Elle est typiquement estimé en tenant l'objet dans la paume de la main de façon statique, an \enquote{unsupported holding} (voir \figref{exploratory_procedures}).
Une différence relative de poids de \percent{8} est alors nécessaire être perceptible~\cite{brodie1985jiggling}.
En soulevant l'objet, il est possible de sentir en plus la force d'inertie de l'objet, \ie la résistance à la vitesse. Cela donne un indice perceptuel supplémentaire de sa masse et permet d'améliorer légèrement la discrimination de poids~\cite{brodie1985jiggling}.
Dans les deux cas, les indices kinesthésiques de force sont beaucoup plus importants que les indices cutanés de pression~\cite{bergmanntiest2012investigating}.
%Le lien entre le poids physique et l'intensité perçue est variable selon les individus~\cite{kappers2013haptic}.

La taille peut être la perception de la \emph{longueur} de l'objet (sur une dimension) ou de son \emph{volume} (en trois dimensions)~\cite{kappers2013haptic}.
Dans les deux cas et si l'objet est assez petit, un grip de précision (voir \figref{gonzalez2014analysis}) entre le pouce et l'index permet de discriminer des tailles avec une précision de \qty{1}{\mm} mais une sur-estimation de la longueur (loi de puissance avec un exposant \qty{1.3}).
Sinon, il est nécessaire de suivre les contours de l'objet avec les doigts pour estimer sa longueur (voir \figref{exploratory_procedures}), mais avec une précision dix fois moins bonne et une sous-estimation de la longueur (loi de puissance avec un exposant \qty{0.9})~\cite{bergmanntiest2011cutaneous}.
La perception du volume d'un objet qui n'est pas petit se fait typiquement par une enclosure de la main, mais l'estimation est fortement influencée par la taille, la forme et la masse de l'objet, pour un volume identique~\cite{kahrimanovic2010haptic}.

La forme de l'objet peut-être définie comme la perception de sa \emph{géométrie globale}, \ie de sa forme et de ses contours.
C'est par exemple quand on cherche une clé dans une poche.
L'exploration de contours et l'enclosure sont alors employés, comme pour l'estimation des longueur et du volume.
Si l'objet est non connu a priori, son identification objet est alors plutôt lente prends plusieurs secondes~\cite{norman2004visual}.
C'est pourquoi, l'exploration d'autres propriétés est alors privilégié pour reconnaitre l'objet plus rapidement, en particulier des bords marqués~\cite{klatzky1987there}, \eg une vis parmi des clous (voir \figref{plaisier2009salient_2}), ou des propriétés matérielles particulières~\cite{lakatos1999haptic,plaisier2009salient}, \eg un objet en métal parmi des objets en plastique.

\begin{subfigs}{plaisier2009salient}{Identification d'une sphère parmi des cubes~\cite{plaisier2009salient}. }[
        \item La forme a un effet important sur la perception du volume d'un objet, \eg une sphère est perçue plus petite qu'un cube de même volume.
        \item L'absence de bord marqué sur la sphère la rend facile à identifier parmi des cubes.
    ]
    \subfigsheight{40mm}
    \subfig{plaisier2009salient_1}
    \subfig{plaisier2009salient_2}
\end{subfigs}


\subsection{Conclusion}
\label{haptic_sense_conclusion}

La perception haptique et la manipulation des objets avec la main implique donc plusieurs mécanismes simultanés aux interactions complexes.
Ainsi, des mouvements exploratoires de la main sont effectués au contact de l'objet afin obtenir de multiples informations sensorielles venant de plusieurs capteurs cutanés et kinesthésiques.
Ces sensations traduisent des paramètres physiques en indices perceptuels qui sont alors intégrés pour former une perception de la propriété explorée.
Il est fréquent qu'un indice perceptuel soit particulièrement important dans la perception d'une propriété, mais une certaine constance perceptuelle est possible en compensant son absence par d'autres.
Ces perceptions permettent en retour de guider la préhension et la manipulation de l'objet en adoptant avec la main des préhensions et forces adaptées à la forme de l'objet et à la tâche à réaliser.
