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% 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 the everyday objects, up 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 on these objects with the hand, 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 are essential to form the haptic experience of interacting with the environment using the hand \cite{lederman2009haptic}.
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 across the body. They can be divided into two main modalities: \emph{cutaneous} and \emph{kinesthetic}.
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 Sensitivity}
\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 receptors that respond to mechanical deformation of the skin.
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}{Schema of cutaneous mechanoreceptors in a section of the skin \cite{blausen2014medical}.}
\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 determine the temporal and spatial resolution of these \emph{mechanoreceptors}, as summarized in \tabref{cutaneous_receptors}.
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 (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.
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 (named Type I) and are sensitive to fine skin deformations, while Ruffini and Pacinian receptors have a larger detection area (named Type II).
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.
\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}.
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. 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}.
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} & \textbf{Adaptation Rate} & \textbf{Receptive Size} & \textbf{Sensitivities} \\
\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}) \\
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 Sensitivity}
\subsubsection{Kinesthetic Modality}
\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.
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 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.
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 are essential for this perception because any movement of the body or contact with the environment necessarily deforms the skin \cite{johansson2009coding}.
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 thus composed of a rich and complex set of various cutaneous and kinesthetic receptors under the skin, in the muscles and in the joints.
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}, \Citeauthor{jones2006human} propose to delineate four categories of hand function on this continuum:
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, \eg in \textcite{gunther2022smooth}.
\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.
\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}.
\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.
Icons are from \href{https://thenounproject.com/creator/leremy/}{Gan Khoon Lay} / \href{https://creativecommons.org/licenses/by/3.0/}{CC BY}.
\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 augmentations (\partref{perception}) and grasping of \VOs (\partref{manipulation}) in the context of \AR and wearable haptics.
In this thesis, we are interested in exploring visuo-haptic texture augmentations (\partref{perception}) and grasping of \VOs (\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.
@@ -112,7 +122,7 @@ Thus the thumb has 5 DoFs, each of the other four fingers has 4 DoFs and the wri
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 Diagram of the hand skeleton. Adapted from \textcite{blausen2014medical}.
\item Kinematic model of the hand with 27 \DoFs \cite{erol2007visionbased}.
]
\subfigsheight{58mm}
@@ -131,7 +141,7 @@ For the other procedures, the whole hand is used: for example, approaching or po
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}.}
\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.
@@ -164,12 +174,13 @@ 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.
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}.
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}
@@ -183,13 +194,13 @@ But when running the finger over the surface with a lateral movement (\secref{ex
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}.
%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 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}.
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
@@ -197,7 +208,7 @@ As shown in \figref{lawrence2007haptic_2}, there is a quadratic relationship bet
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 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}
@@ -221,15 +232,21 @@ 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" 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 $v$ 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:
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{v}{f_p}
\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.
\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}.}
\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}.
@@ -247,30 +264,32 @@ When the finger presses on an object (\figref{exploratory_procedures}), its surf
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}.
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}{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:
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}
Y = \frac{F / A}{D / l}
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 by a force $F$ on an area $A$ by a distance $D$.
\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 the role of these two physical properties in the perception of hardness.
%\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}).
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}.
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}
@@ -363,8 +382,8 @@ In addition, an object with low stiffness but high Young's modulus can be percei
\subsection{Conclusion}
\label{haptic_sense_conclusion}
Haptic perception and manipulation of objects with the hand involves several simultaneous mechanisms with complex interactions.
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.
It is often the case that one perceptual cue is particularly important in the perception of a property, but perceptual constancy is possible by compensating for its absence with others.
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.