%With a vibrotactile actuator attached to a hand-held device or directly on the finger, it is possible to simulate virtual haptic sensations as vibrations, such as texture, friction or contact vibrations \cite{culbertson2018haptics}.
%
%We describe a system for rendering vibrotactile roughness textures in real time, on any real surface, touched directly with the index fingertip, with no constraints on hand movement and using a simple camera to track the finger pose.
%
%We also describe how to pair this tactile rendering with an immersive \AR or \VR headset visual display to provide a coherent visuo-haptic augmentation of the \RE.

\section{Concept}
\label{principle}

The visuo-haptic texture rendering system is based on:
\begin{enumerate}[label=(\arabic*)]
  \item a real-time interaction loop between the finger movements and a coherent visuo-haptic feedback simulating the sensation of a touched texture,
  \item a precise alignment of the \VE with its real counterpart, and
  \item a modulation of the signal frequency by the estimated finger speed with a phase matching.
\end{enumerate}

\figref{diagram} shows the interaction loop diagram and \eqref{signal} the definition of the vibrotactile signal.
The system consists of three main components: the pose estimation of the tracked real elements, the visual rendering of the \VE, and the vibrotactile signal generation and rendering.

\figwide{diagram}{Diagram of the visuo-haptic texture rendering system. }[
  Fiducial markers attached to the voice-coil actuator and to augmented surfaces to track are captured by a camera.
  The positions and rotations (the poses) ${}^c\mathbf{T}_i$, $i=1..n$ of the $n$ defined markers in the camera frame $\mathcal{F}_c$ are estimated, then filtered with an adaptive low-pass filter.
  %These poses are transformed to the \AR/\VR headset frame $\mathcal{F}_h$ and applied to the virtual model replicas to display them superimposed and aligned with the \RE.
  These poses are used to move and display the virtual model replicas aligned with the \RE.
  A collision detection algorithm detects a contact of the virtual hand with the virtual textures.
  If so, the velocity of the finger marker ${}^c\dot{\mathbf{X}}_f$ is estimated using discrete derivative of position and adaptive low-pass filtering, then transformed onto the texture frame $\mathcal{F}_t$.
  The vibrotactile signal $s_k$ is generated by modulating the (scalar) finger velocity ${}^t\hat{\dot{X}}_f$ in the texture direction with the texture period $\lambda$ (\eqref{signal}).
  The signal is sampled at 48~kHz and sent to the voice-coil actuator via an audio amplifier.
  All computation steps except signal sampling are performed at 60~Hz and in separate threads to parallelize them.
]

\section{Description of the System Components}
\label{system_components}

\subsection{Pose Estimation}
\label{pose_estimation}

A \qty{2}{\cm} AprilTag fiducial marker \cite{wang2016apriltag} is glued to the top of the actuator (\figref{device}) to track the finger pose with a camera (StreamCam, Logitech) which is placed above the experimental setup and capturing \qtyproduct{1280 x 720}{px} images at \qty{60}{\hertz} (\figref{apparatus}).
Other markers are placed on the real surfaces to augment (\figref{setup}) to estimate the relative position of the finger with respect to the surfaces.
Contrary to similar work, using vision-based tracking allows both to free the hand movements and to augment any real surface.
A camera external to the \AR headset with a marker-based technique is employed to provide accurate and robust tracking with a constant view of the markers \cite{marchand2016pose}.
We denote ${}^c\mathbf{T}_i$, $i=1..n$ the homogenous transformation matrix that defines the position and rotation of the $i$-th marker out of the $n$ defined markers in the camera frame $\mathcal{F}_c$, \eg the finger pose ${}^c\mathbf{T}_f$ and the texture pose ${}^c\mathbf{T}_t$.

To reduce the noise in the pose estimation while maintaining good responsiveness, the 1€ filter \cite{casiez2012filter} is applied; a low-pass filter with an adaptive cut-off frequency, specifically designed for human motion tracking.
The filtered pose is denoted as ${}^c\hat{\mathbf{T}}_i$.
The optimal filter parameters were determined using the method of \textcite{casiez2012filter}, with a minimum cut-off frequency of \qty{10}{\hertz} and a slope of \num{0.01}.
The velocity (without angular velocity) of the marker, denoted as ${}^c\dot{\mathbf{X}}_i$, is estimated using the discrete derivative of the position and another 1€ filter with the same parameters.

\subsection{Virtual Environment Alignment}
\label{virtual_real_alignment}

%To be able to compare virtual and augmented realities, we then create a \VE that closely replicate the real one.
Before a user interacts with the system, it is necessary to design a \VE that will be registered with the \RE during the experiment.
Each real element tracked by a marker is modelled virtually, \eg the hand and the augmented surface (\figref{device}).
In addition, the pose and size of the virtual textures were defined on the virtual replicas.
During the experiment, the system uses marker pose estimates to align the virtual models with their real world counterparts. %, according to the condition being tested.
This allows to detect if a finger touches a virtual texture using a collision detection algorithm (Nvidia PhysX), and to show the virtual elements and textures in real-time, aligned with the \RE, using the considered \AR or \VR headset.

In our implementation, the \VE is designed with Unity and the Mixed Reality Toolkit (MRTK)\footnoteurl{https://learn.microsoft.com/windows/mixed-reality/mrtk-unity}.
The visual rendering is achieved using the Microsoft HoloLens~2, an \OST-\AR headset with a \qtyproduct{43 x 29}{\degree} \FoV, a \qty{60}{\Hz} refresh rate, and self-localisation capabilities.
A \VST-\AR or a \VR headset could have been used as well.

\subsection{Vibrotactile Signal Generation and Rendering}
\label{texture_generation}

A voice-coil actuator (HapCoil-One, Actronika) is used to display the vibrotactile signal, as it allows the frequency and amplitude of the signal to be controlled independently over time, covers a wide frequency range (\qtyrange{10}{1000}{\Hz}), and outputs the signal accurately with relatively low acceleration distortion\footnote{HapCoil-One specific characteristics are described in its data sheet: \url{https://tactilelabs.com/wp-content/uploads/2023/11/HapCoil_One_datasheet.pdf}}.
The voice-coil actuator is encased in a \ThreeD printed plastic shell and firmly attached to the middle phalanx of the user's index finger with a Velcro strap, to enable the fingertip to directly touch the environment (\figref{device}).
The actuator is driven by a class D audio amplifier (XY-502 / TPA3116D2, Texas Instrument). %, which has proven to be an effective type of amplifier for driving moving-coil \cite{mcmahan2014dynamic}.
The amplifier is connected to the audio output of a computer that generates the signal using the WASAPI driver in exclusive mode and the NAudio library\footnoteurl{https://github.com/naudio/NAudio}.

The represented haptic texture is a series of parallels virtual grooves and ridges, similar to real grating textures manufactured for psychophysical roughness perception studies \secref[related_work]{roughness}. %\cite{friesen2024perceived,klatzky2003feeling,unger2011roughness}.
It is generated as a square wave audio signal $s_k$, sampled at \qty{48}{\kilo\hertz}, with a period $\lambda$ and an amplitude $A$, similar to \eqref[related_work]{grating_rendering}.
Its frequency is a ratio of the absolute finger filtered (scalar) velocity ${}^t\hat{\dot{|X|}}_f$, transformed into the texture frame $\mathcal{F}_t$, and the texture period $\lambda$ \cite{friesen2024perceived}.
As the finger is moving horizontally on the texture, only the $x$ component of the velocity is used.
This velocity modulation strategy is necessary as the finger position is estimated at a far lower rate (\qty{60}{\hertz}) than the audio signal (unlike high-fidelity force-feedback devices \cite{unger2011roughness}).

%As the finger position is estimated at a far lower rate (\qty{60}{\hertz}), the filtered finger (scalar) position ${}^t\hat{X}_f$ in the texture frame $\mathcal{F}_t$ cannot be directly used. % to render the signal if the finger moves fast or if the texture period is small.
%
%The best strategy instead is to modulate the frequency of the signal as a ratio of the filtered finger velocity ${}^t\hat{\dot{\mathbf{X}}}_f$ and the texture period $\lambda$ \cite{friesen2024perceived}.
%
When a new finger velocity ${}^t\hat{\dot{X}}_{f,j}$ is estimated at time $t_j$, the phase $\phi_j$ of the signal $s$ needs also to be adjusted to ensure a continuity in the signal.
In other words, the sampling of the audio signal runs at \qty{48}{\kilo\hertz}, and its frequency and phase is updated at a far lower rate of \qty{60}{\hertz} when a new finger velocity is estimated.
A sample $s_k$ of the audio signal at sampling time $t_k$, with $t_k >= t_j$, is thus given by:
\begin{subequations}
  \label{eq:signal}
  \begin{align}
    s_k(x_{f,j}, t_k) & = A\, \text{sgn} ( \sin (2 \pi \frac{|\dot{X}_{f,j}|}{\lambda} t_k + \phi_j) ) & \label{eq:signal_speed} \\
    \phi_j            & = \phi_{j-1} + 2 \pi \frac{x_{f,j} - x_{f,{j-1}}}{\lambda} t_k                 & \label{eq:signal_phase}
  \end{align}
\end{subequations}

This rendering preserves the sensation of a constant spatial frequency of the virtual texture while the finger moves at various speeds, which is crucial for the perception of roughness \cite{klatzky2003feeling,unger2011roughness}.
%
%Note that the finger position and velocity are transformed from the camera frame $\mathcal{F}_c$ to the texture frame $\mathcal{F}_t$, with the $x$ axis aligned with the texture direction.
%
%However, when a new finger position is estimated at time $t_j$, the phase $\phi_j$ needs to be adjusted as well with the frequency to ensure a continuity in the signal as described in \eqref{signal_phase}.
%
The phase matching avoids sudden changes in the actuator movement thus affecting the texture perception in an uncontrolled way (\figref{phase_adjustment}) and, contrary to previous work \cite{asano2015vibrotactile,ujitoko2019modulating}, it enables a free exploration of the texture by the user with no constraints on the finger speed.
A square wave is chosen to get a rendering closer to a real grating texture with the sensation of crossing edges \cite{ujitoko2019modulating}, and because the roughness perception of sine wave textures has been shown not to reproduce the roughness perception of real grating textures \cite{unger2011roughness}.
A square wave also makes it possible to render low signal frequencies that occur when the finger moves slowly or the texture period is large, as the actuator cannot render a pure sine wave signal below \qty{\approx 20}{\Hz} with sufficient amplitude to be perceived.

The vibrotactile texture is described and rendered in this chapter as a 1D signal by integrating the relative finger movement to the texture on a single direction, but it is easily extended to a two-dimensional texture by simply generating a second signal for the orthogonal direction and summing the two signals in the rendering \cite{girard2016haptip}.

\fig[0.68]{phase_adjustment}{
  Change in frequency of a sinusoidal signal with and without phase matching.
}[
  Phase matching ensures a continuity and avoids glitches in the rendering of the signal.
  A sinusoidal signal is shown here for clarity, but a different waveform will give a similar effect.
]

\section{System Latency}
\label{latency}

As shown in \figref{diagram} and described above, the system includes various haptic and visual sensors and rendering devices linked by software processes for image processing, \ThreeD rendering and audio generation.
Because the chosen \AR headset is a standalone device (like most current \AR/\VR headsets) and cannot directly control the sound card and haptic actuator, the image capture, pose estimation and audio signal generation steps are performed on an external computer.
All computation steps run in a separate thread to parallelize them and reduce latency, and are synchronized with the headset via a local network and the ZeroMQ library\footnoteurl{https://zeromq.org/}.
This complex assembly inevitably introduces latency, which must be measured.

The rendering system provides a user with two interaction loops between the movements of their hand and the visual (loop 1) and haptic (loop 2) feedbacks.
Measures are shown as (mean $\pm$ standard deviation), when it is known.
The end-to-end latency from finger movement to feedback is measured at \qty{36 \pm 4}{\ms} in the haptic loop and \qty{43 \pm 9}{\ms} in the visual loop.
Both are the result of latency in image capture \qty{16 \pm 1}{\ms}, markers tracking \qty{2 \pm 1}{\ms} and network communication \qty{4 \pm 1}{\ms}.
The haptic loop also includes the voice-coil latency \qty{15}{\ms} (as specified by the manufacturer\footnotemark[1]), whereas the visual loop includes the latency in \ThreeD rendering \qty{16 \pm 5}{\ms} (60 frames per second) and display \qty{5}{\ms}.
The total haptic latency is below the \qty{60}{\ms} detection threshold in vibrotactile feedback \cite{okamoto2009detectability}.
The total visual latency can be considered slightly high, yet it is typical for an \AR rendering involving vision-based tracking \cite{knorlein2009influence}.

The two filters also introduce a constant lag between the finger movement and the estimated position and velocity, measured at \qty{160 \pm 30}{\ms}.
With respect to the real hand position, it causes a distance error in the displayed virtual hand position, and thus a delay in the triggering of the vibrotactile signal.
This is proportional to the speed of the finger, \eg distance error is \qty{12 \pm 2.3}{\mm} when the finger moves at \qty{75}{\mm\per\second}.
%and of the vibrotactile signal frequency with respect to the finger speed.%, that is proportional to the speed of the finger.