\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 of everyday objects, up to a complex combination of sensations produced by numerous sensory receptors distributed throughout the body, but especially in the hand \cite{johansson2009coding}.
It also allows us to act on these objects with the hand, to come into contact with them, to grasp them and 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 closely associated and both are essential to form a complete 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 throughout the body. They can be divided into two main modalities: \emph{cutaneous} and \emph{kinesthetic} \cite{lederman2009haptic}.

\subsubsection{Cutaneous Modality}
\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 \emph{mechanoreceptors} that respond to mechanical deformation of the skin.

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

Adaptation rate and receptor size are the two key characteristics that respectively define \emph{temporal and spatial sensitivity} of mechanoreceptors, \ie their ability to detect changes in stimuli over time and space \cite{johansson2009coding}.
They are 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, 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, 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 and are sensitive to fine skin deformations.
Conversely, Ruffini and Pacinian receptors have a larger detection area, which means they can detect more distant events on the skin, but at the cost of poor spatial sensitivity.

The density of mechanoreceptors varies according to skin type and body region \cite{johansson2009coding}.
\emph{Glabrous skin}, is particularly rich in cutaneous receptors, especially on the face, feet, hands and fingers.
The density of Meissner and Merkel receptors, which are the most sensitive, is particularly high in the fingertips.
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}.

\begin{tab}{cutaneous_receptors}{Characteristics of the cutaneous mechanoreceptors.}[
    Adaptation rate is the speed and duration of the receptor's response to a stimulus, either fast or slow adapting.
    Receptor size is the area of skin detectable by a single receptor, either small or large.
    Sensitivity describes the spatial and temporal variation of 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 Name} & \textbf{Adaptation Rate} & \textbf{Receptor Size} & \textbf{Sensitivity}                                                          \\
    \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}) from distant events                             \\
    Ruffini           & Slow                     & Large                   & Skin stretch                                                                    \\
    \bottomrule
  \end{tabularx}
\end{tab}

\subsubsection{Kinesthetic Modality}
\label{kinesthetic_sensitivity}

Kinesthetic receptors are also mechanoreceptors, but are located in muscles, tendons and joints \cite{jones2006human}.
Muscle spindles respond to the length and rate of stretch/contraction of 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 located 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 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 (\secref{cutaneous_sensitivity}) are also involved in proprioception \cite{johansson2009coding}.

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

The sense of touch is 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 \figref{sensorimotor_continuum}, \textcite{jones2006human} 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 \cite{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 \cite{feix2016grasp}.
  \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 \cite{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.
  \protect\footnotemark
]

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 visuo-haptic texture augmentations (\partref{perception}) and grasping of virtual objects (\partref{manipulation}) using immersive \AR and wearable haptics.

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

\footnotetext{%
  All icons are from \href{https://thenounproject.com/creator/leremy/}{Gan Khoon Lay} / \href{https://creativecommons.org/licenses/by/3.0/}{CC BY}.
}

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 \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 \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 Diagram 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 \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 brackets). 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 (\secref{hand_anatomy}).
By placing the thumb or palm against the other fingers (pad or palm opposition respectively), or by placing the fingers against each other as if holding a cigarette (side opposition), 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 in \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 (\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 Roughness and Hardness}
\label{object_properties}

The active exploration of an object with the hand is performed as a sensorimotor loop: The exploratory movements (\secref{exploratory_procedures}) guide the search for and adapt to sensory information (\secref{haptic_sense}), 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 \cite{bergmanntiest2010tactual}.
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.}

The most salient and fundamental perceived material properties are the roughness and hardness of the object \cite{hollins1993perceptual,baumgartner2013visual}, which are also the most studied and best understood \cite{bergmanntiest2010tactual}.
To be able to render virtual haptic sensations that reproduce these properties, as we will detail in \secref{tactile_rendering}, it is important to understand how they are perceived when interacting with real objects \cite{klatzky2013haptic}.

\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 characterizes 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 (\secref{exploratory_procedures}), vibrations are also 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 perception of roughness can be characterized by the density of the surface elements: the perceived (subjective) intensity of roughness increases with the spacing between the elements. %, 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 elements' 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}{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 Diagram 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 perceiving finer spaces between the elements than with the finger (\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" everyday 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 patterned textures, as illustrated in \figref{delhaye2012textureinduced}, the ratio of the finger speed $\dot{x}$ 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}{grating_vibrations}
  \lambda \sim \frac{\dot{x}}{f_p}
\end{equation}

The vibrations generated by exploring everyday textures are also specific to each texture and similar between individuals, making them identifiable by vibration alone \cite{manfredi2014natural,greenspon2020effect}.
This shows the importance of vibration cues even for macro textures and the possibility of generating virtual texture sensations with vibrotactile rendering.

\begin{subfigs}{delhaye2012textureinduced}{Scanning grating textures with different periods $\lambda$ of spacing with the fingertip, and measure of the propagated vibrations in the wrist \cite{delhaye2012textureinduced}. }[][
  \item Experimental setup.
  \item Speed of finger exploration (horizontal axis) for texture spacings (in color) and frequency of the vibration intensity peak $f_p$ propagated in the wrist (vertical axis).
  ]
  \subfig[.43]{delhaye2012textureinduced_1}
  \subfig[.53]{delhaye2012textureinduced_2}
\end{subfigs}

The everyday 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 (\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 \cite{higashi2019hardness}.
Passive touch (without voluntary hand movements) and tapping allow a perception of hardness as good as active touch \cite{friedman2008magnitude}.

Perceived hardness is related to \emph{physical elasticity} of the material and the structure of the object \cite{bergmanntiest2009cues}.
The physical elasticity of a material can be expressed by its Young's modulus $E$ (in \unit{Pa}), which is the ratio between stress $\sigma$ and strain $\varepsilon$ (\figref{hardness}):
\begin{equation}{young_modulus}
  E = \frac{\sigma}{\varepsilon} = \frac{F / A}{\Delta L / L}
\end{equation}
Stress is the force per unit area $F / A$ (in \unit{N/m^2}) applied to the object and strain is the relative displacement $\Delta L / L$ (in \unit{m/m}) of the object.

The physical resistance to deformation of an object can also be expressed as its \emph{stiffness} $k$ (in \unit{N/mm}), which is the ratio between the applied force $F$ and the resulting displacement $\Delta L$ along one axis of deformation (\figref{hardness}):
\begin{equation}{stiffness}
  k = \frac{F}{\Delta L}
\end{equation}
Stiffness depends on the structure of the object: a thick object can be more compressed than a thin object, with the same force applied and the same physical elasticity.

\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 on an axis 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 how  of these two physical measures in the perception of hardness.
An object with low stiffness, but high Young's modulus can be perceived as hard, and vice versa, as shown in \figref{bergmanntiest2009cues}.
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}).
That is, \textcite{bergmanntiest2009cues} showed the perception of hardness relies on \percent{90} on surface deformation cues and \percent{10} on displacement cues.
%Finally, when pressing with the finger, the perceived hardness intensity $h$ follows a power law with the stiffness $k$ \cite{harper1964subjective}:
%\begin{equation}{hardness_intensity}
%    h = k^{0.8}
%\end{equation}

%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 (\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}{friction}
%    F_s = \mu \, F_n
%\end{equation}
%The perceived intensity of friction is thus roughly related to the friction coefficient $\mu$ \cite{smith1996subjective}.
%However, it is a complex perception because it is more determined by the micro-scale interactions between the surface and the skin: It depends on many factors such as the normal force applied, the speed of movement, the contact area and the moisture of the skin and the surface \cite{adams2013finger,messaoud2016relation}.
%In this sense, the perception of friction is still poorly understood \cite{okamoto2013psychophysical}.
%
%\begin{subfigs}{smith1996subjective}{Perceived intensity of friction of different materials by active exploration with the finger \cite{smith1996subjective}. }[
%        \item Measurements of normal $F_n$ and tangential $F_t$ forces when exploring two surfaces: one smooth (glass) and one rough (nyloprint). The fluctuations in the tangential force are due to the stick-slip phenomenon. The coefficient of friction $\mu$ can be estimated as the slope of the relationship between the normal and tangential forces.
%        \item Perceived friction intensity (vertical axis) as a function of the estimated friction coefficient $\mu$ of the exploration (horizontal axis) for four materials (shapes and colors).
%    ]
%    \subfigsheight{55mm}
%    \subfig{smith1996subjective_1}
%    \subfig{smith1996subjective_2}
%\end{subfigs}
%
%Yet, it is a fundamental perception for grasping and manipulating objects.
%The forces of friction make it indeed possible to hold the object firmly in the hand and prevent it from slipping
%The perception of friction also allows us to automatically and very quickly adjust the force we apply to the object in order to grasp it \cite{johansson1984roles}.
%If the finger is anaesthetized, the lack of cutaneous sensation prevents effective adjustment of the gripping force: the forces of the object on the finger are no longer correctly perceived, and the fingers then press harder on the object in compensation, but without achieving good opposition of the fingers \cite{witney2004cutaneous}.

%\subsubsection{Temperature}
%\label{temperature}
%
%Temperature (or coldness/warmness) is the perception of the \emph{transfer of heat} between the touched surface and the skin \cite{bergmanntiest2010tactual}:
%When heat is removed from (added to) the skin, the surface is perceived as cold (hot).
%Metal will be perceived as colder than wood at the same room temperature: This perception is different from the physical temperature of the material and is therefore an important property for distinguishing between materials \cite{ho2006contribution}.
%This perception depends on the thermal conductivity and heat capacity of the material, the volume of the object, the initial temperature difference and the area of contact between the surface and the skin \cite{kappers2013haptic}.
%For example, a larger object or a smoother surface, which increases the contact area, causes more heat circulation and a more intense temperature sensation (hot or cold) \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}{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}

%Weight, size and shape are haptic spatial properties that are independent of the material properties described above.

%Weight (or heaviness/lightness) is the perceived \emph{mass} of the object \cite{bergmanntiest2010haptic}.
%It is typically estimated by holding the object statically in the palm of the hand to feel the gravitational force (\secref{exploratory_procedures}).
%A relative weight difference of \percent{8} is then required to be perceptible \cite{brodie1985jiggling}.
%By lifting the object, it is also possible to feel the object's force of inertia, \ie its resistance to velocity.
%This provides an additional perceptual cue to its mass and slightly improves weight discrimination.
%For both gravity and inertia, kinesthetic cues to force are much more important than cutaneous cues to pressure \cite{bergmanntiest2012investigating}.
%Le lien entre le poids physique et l'intensité perçue est variable selon les individus \cite{kappers2013haptic}.

%Size can be perceived as the object's \emph{length} (in one dimension) or its \emph{volume} (in three dimensions) \cite{kappers2013haptic}.
%In both cases, and if the object is small enough, a precision grip (\figref{gonzalez2014analysis}) between the thumb and index finger can discriminate between sizes with an accuracy of \qty{1}{\mm}, but with an overestimation of length (power law with exponent \qty{1.3}).
%Alternatively, it is necessary to follow the contours of the object with the fingers to estimate its length (\secref{exploratory_procedures}), but with ten times less accuracy and an underestimation of length (power law with an exponent of \qty{0.9}) \cite{bergmanntiest2011cutaneous}.
%The perception of the volume of an object that is not small is typically done by hand enclosure, but the estimate is strongly influenced by the size, shape and mass of the object, for an identical volume \cite{kahrimanovic2010haptic}.

%The shape of an object can be defined as the perception of its \emph{global geometry}, \ie its shape and contours.
%This is the case, for example, when looking for a key in a pocket.
%The exploration of contours and enclosure are then employed, as for the estimation of length and volume.
%If the object is not known in advance, object identification is rather slow, taking several seconds \cite{norman2004visual}.
%Therefore, the exploration of other properties is favoured to recognize the object more quickly, in particular marked edges \cite{klatzky1987there}, \eg a screw among nails (\figref{plaisier2009salient_2}), or certain material properties \cite{lakatos1999haptic,plaisier2009salient}, \eg a metal object among plastic objects.

%\begin{subfigs}{plaisier2009salient}{Identifcation of a sphere among cubes \cite{plaisier2009salient}. }[
%        \item The shape has a significant effect on the perception of the volume of an object, \eg a sphere is perceived smaller than a cube of the same volume.
%        \item The absence of a marked edge on the sphere makes it easy to identify among cubes.
%    ]
%    \subfigsheight{40mm}
%    \subfig{plaisier2009salient_1}
%    \subfig{plaisier2009salient_2}
%\end{subfigs}

\subsection{Conclusion}
\label{haptic_sense_conclusion}

Haptic perception and manipulation of objects with the hand involve several simultaneous mechanisms with complex interactions.
Exploratory movements of the hand are performed on contact with the object to obtain multiple sensory information from several cutaneous and kinaesthetic receptors.
These sensations express physical parameters in the form of perceptual cues, which are then integrated to form a perception of the property being explored.
For the perception of roughness (texture) or hardness, one perceptual cue is particularly important, but perceptual constancy is possible by compensating for its absence with others.
In turn, these perceptions help to guide the grasping and manipulation of the object by adapting the grasp type and the forces applied to the shape of the object and the task to be performed.
