Improve vhar_system equation and diagram

3-perception/vhar-system/2-method.tex

@@ -18,9 +18,9 @@ The visuo-haptic texture rendering system is based on:
 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. }[
+  \setstretch{1.2}
   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 $\poseFrame{c}$ are estimated, then filtered with an adaptive low-pass filter.
-  %These poses are transformed to the \AR/\VR headset frame $\poseFrame{h}$ 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 $\poseFrame{t}$.
@@ -74,7 +74,7 @@ The amplifier is connected to the audio output of a computer that generates the
 
 The represented haptic texture is a 1D series of parallels virtual grooves and ridges, similar to the real linear 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 $r$, sampled at \qty{48}{\kilo\hertz}, with a texture 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 $x_f = \poseX{s}{|\hat{\dot{X}}|}{f}$, and the texture period $\lambda$ \cite{friesen2024perceived}.
+Its frequency is a ratio of the absolute finger filtered (scalar) velocity $\dot{x} = \poseX{s}{|\hat{\dot{X}}|}{f}$, 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}).
 
@@ -82,14 +82,14 @@ This velocity modulation strategy is necessary as the finger position is estimat
 %
 %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 $x_f\,(t_j)$ is estimated at time $t_j$, the phase $\phi\,(t_j)$ of the signal $r$ needs also to be adjusted to ensure a continuity in the signal.
+When a new finger velocity $\dot{x}\,(t_j)$ is estimated at time $t_j$, the phase $\phi\,(t_j)$ of the signal $r$ 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 $r(x_f, t_j, t_k)$ of the audio signal at sampling time $t_k$, with $t_k >= t_j$, is thus given by:
+A sample $r(t_j, t_k)$ 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}
-    r(x_f, t_j, t_k) & = A\, \text{sgn} ( \sin (2 \pi \frac{x_f\,(t_j)}{\lambda} t_k + \phi(t_j) ) ) & \label{eq:signal_speed} \\
-    \phi(t_j)        & = \phi(t_{j-1}) + 2 \pi \frac{x_f\,(t_j) - x_f\,(t_j - 1)}{\lambda} t_k                 & \label{eq:signal_phase}
+    r(t_j, t_k) & = A\, \text{sgn} ( \sin (2 \pi \frac{\dot{x}\,(t_j)}{\lambda} t_k + \phi_j ) ) & \label{eq:signal_speed} \\
+    \phi_j        & = \phi_{j-1} + 2 \pi \frac{\dot{x}\,(t_j) - \dot{x}\,(t_j - 1)}{\lambda} t_k                 & \label{eq:signal_phase}
   \end{align}
 \end{subequations}