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Fix xr-perception equation labels

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Erwan Normand
2024-06-26 18:56:33 +02:00
parent 935e7b094b
commit 74f2f271bc
1 changed files with 5 additions and 5 deletions
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10
2-perception/xr-perception/3-method.tex
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@@ -36,7 +36,7 @@ The visuo-haptic texture rendering system is based on
\item and a modulation of the signal frequency by the estimated finger speed with a phase matching. \item and a modulation of the signal frequency by the estimated finger speed with a phase matching.
\end{enumerate*} \end{enumerate*}
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\figref{method/diagram} shows the diagram of the interaction loop and \eqref{signal} the definition of the vibrotactile signal. \figref{method/diagram} shows the diagram of the interaction loop and \eqref{xr_perception:signal} the definition of the vibrotactile signal.
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The system is composed of three main components: the pose estimation of the tracked real elements, the visual rendering of the virtual environment, and the vibrotactile signal generation and rendering. The system is composed of three main components: the pose estimation of the tracked real elements, the visual rendering of the virtual environment, and the vibrotactile signal generation and rendering.
@@ -116,10 +116,10 @@ It is generated as a square wave audio signal, sampled at \qty{48}{\kilo\hertz},
A sample $s_k$ of the audio signal at sampling time $t_k$ is given by: A sample $s_k$ of the audio signal at sampling time $t_k$ is given by:
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\begin{subequations} \begin{subequations}
\label{eq:signal} \label{eq:\labelprefix:signal}
\begin{align} \begin{align}
s(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} \\ s(x_{f,j}, t_k) & = A \text{\,sgn} ( \sin (2 \pi \frac{\dot{x}_{f,j}}{\lambda} t_k + \phi_j) ) & \label{eq:\labelprefix:signal_speed} \\
\phi_j & = \phi_{j-1} + 2 \pi \frac{x_{f,j} - x_{f,{j-1}}}{\lambda} t_k & \label{eq:signal_phase} \phi_j & = \phi_{j-1} + 2 \pi \frac{x_{f,j} - x_{f,{j-1}}}{\lambda} t_k & \label{eq:\labelprefix:signal_phase}
\end{align} \end{align}
\end{subequations} \end{subequations}
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@@ -133,7 +133,7 @@ This is important because it preserves the sensation of a constant spatial frequ
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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. 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.
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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}. 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{xr_perception:signal_phase}.
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This approach avoids sudden changes in the actuator movement thus affecting the texture perception in an uncontrolled way (see \figref{method/phase_adjustment}) and, contrary to previous work~\cite{asano2015vibrotactile,friesen2024perceived}, it enables no constraints a free exploration of the texture by the user with no constraints on the finger speed. This approach avoids sudden changes in the actuator movement thus affecting the texture perception in an uncontrolled way (see \figref{method/phase_adjustment}) and, contrary to previous work~\cite{asano2015vibrotactile,friesen2024perceived}, it enables no constraints a free exploration of the texture by the user with no constraints on the finger speed.
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