Typo

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

@@ -46,9 +46,9 @@ To reduce the noise in the pose estimation while maintaining good responsiveness
 The filtered pose is denoted as $\pose{c}{\hat{T}}{i}$.
 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 finger marker, denoted as $\pose{c}{\dot{T}}{f}$, is estimated using the discrete derivative of the position.
-It is then filtered with another 1€ filter with the same parameters, and denoted as $\pose{c}{\hat{\dot{T}}}{f}$.
-Finally, this filtered finger velocity is transformed into the augmented surface frame $\poseFrame{s}$ to be used in the vibrotactile signal generation, such as $\pose{s}{\hat{\dot{T}}}{f} = \pose{c}{T}{s} \, \pose{c}{\hat{\dot{T}}}{f}$.
+The velocity (without angular velocity) of the finger marker, denoted as $\pose{c}{\dot{X}}{f}$, is estimated using the discrete derivative of the position.
+It is then filtered with another 1€ filter with the same parameters, and denoted as $\pose{c}{\hat{\dot{X}}}{f}$.
+Finally, this filtered finger velocity is transformed into the augmented surface frame $\poseFrame{s}$ to be used in the vibrotactile signal generation, such as $\pose{s}{\hat{\dot{X}}}{f} = \pose{s}{T}{c} \, \pose{c}{\hat{\dot{X}}}{f}$.
 
 \subsection{Virtual Environment Alignment}
 \label{virtual_real_alignment}
@@ -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 $x_f = \pose{s}{|\hat{\dot{T}}|}{f}$, and the texture period $\lambda$ \cite{friesen2024perceived}.
+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}.
 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}).
 
@@ -82,7 +82,7 @@ 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 $\phi$ of the signal $r$ needs also to be adjusted to ensure a continuity in the signal.
+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.
 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:
 \begin{subequations}