diff --git a/tex/chapters/experiments.tex b/tex/chapters/experiments.tex
index 8bf1689..c348448 100644
--- a/tex/chapters/experiments.tex
+++ b/tex/chapters/experiments.tex
@@ -65,9 +65,8 @@
 		\input{gfx/eval/paths}
 		\caption{The four paths that were part of the evaluation.
 		Starting positions are marked with black circles.
-		For a better visualisation they were slightly shifted to avoid overlapping.
+		For a better visualisation they were slightly shifted to avoid overlapping.}
 		\commentByFrank{font war korrekt, aber die groesse war zu gross im vgl. zu den anderen}
-		}
 		\label{fig:paths}
 	\end{figure}
 	% error development over time while walking along a path
@@ -76,9 +75,8 @@
 		\caption{Development of the error while walking along Path 4 using the Motorola Nexus 6.
 		When leaving the suggested route (3), the error of \textbf{shortest} path \refeq{eq:transShortestPath}
 		and \textbf{multi}path \refeq{eq:transMultiPath} increases.
-		The same issues arise when facing multimodalities between two staircases just before the destination (9).
+		The same issues arise when facing multimodalities between two staircases just before the destination (9).}
 		\commentByFrank{hilft das bold vlt. schon um die legende zu verstehen?}
-		}
 		\label{fig:errorTimedNexus}
 	\end{figure}
 	% detailed analysis of path 4
@@ -139,8 +137,8 @@
 	\begin{figure}
 		\input{gfx/eval/error_dist_nexus}
 		\caption{Error distribution of all walks conducted with the Motorola Nexus 6 for distinct percentile values.
-		Our proposed methods clearly provide an enhancement for the overall localization process.
-		\commentByFrank{percentile erwaehnt}}
+		Our proposed methods clearly provide an enhancement for the overall localization process.}
+		\commentByFrank{percentile erwaehnt}
 		\label{fig:errorDistNexus}
 	\end{figure}
 	%\begin{figure}
diff --git a/tex/chapters/grid.tex b/tex/chapters/grid.tex
index 360f994..588822a 100644
--- a/tex/chapters/grid.tex
+++ b/tex/chapters/grid.tex
@@ -258,9 +258,11 @@
 			\newcommand{\pathRef}{v_\text{ref}}
 		
 		
-			Before every transition, the centre-position $\pathCentroid = (x,y,z)^T$ of the current
-			\commentByFrank{reicht das so schon?}
-			sample-set, representing the posterior distribution at time $t-1$, is calculated.
+			Before every transition, the centre-position $\pathCentroid = \fPos{\mStateVec_{t-1}^*}$ of the current sample-set, where 
+\begin{equation}
+\mStateVec_{t-1}^* = \underset{\mStateVec_{t-1}}{\argmax} \enspace p(\mStateVec_{t-1} | \mObsVec_{t-1}) 
+\end{equation}
+represents the most proper state of the posterior distribution at time $t-1$, is calculated.
 			%
 			%
 			%\commentByFrank{avg-state vom sample-set. frank d. meinte ja hier muessen wir aufpassen. bin noch unschluessig wie.}
@@ -345,8 +347,8 @@
 				\caption{Heat-Map showing the number of visits per vertex after $30.000$ walks using \refeq{eq:transMultiPath}.
 				Both possible paths are covered and slight deviations are possible.
 				Additionally shows the shortest-path calculation without (dashed) and with (solid) importance-factors
-				used for edge-weight-adjustment.
-				\commentByFrank{so besser?}}
+				used for edge-weight-adjustment.}
+				\commentByFrank{so besser?}
 				\label{fig:multiHeatMap}
 			\end{figure}
 
diff --git a/tex/chapters/sensors.tex b/tex/chapters/sensors.tex
index 3b718e4..a89f09c 100644
--- a/tex/chapters/sensors.tex
+++ b/tex/chapters/sensors.tex
@@ -18,8 +18,8 @@
 			\include{gfx/baro/baro_setup_issue}
 			\caption{Sometimes the smartphone's barometer (here: Motorola Nexus 6) provides erroneous pressure readings
 			during the first seconds. Those need to be omitted before $\sigma_\text{baro}$ and
-			$\overline{\mObsPressure}$ are estimated.
-			\commentByFrank{fixed}}
+			$\overline{\mObsPressure}$ are estimated.}
+			\commentByFrank{fixed}
 			\label{fig:baroSetupError}
 		\end{figure}
 		%	
@@ -84,17 +84,15 @@
 	
 	
 	\subsection{Step- \& Turn-Detection}
-
-		Step- and turn-detection use the smartphone's IMU and are implemented as described in \cite{Ebner-15}.
 		%
-		However, a big disadvantage of using the state transition as proposal distribution is the high possibility of sample
+		A big disadvantage of using the state transition as proposal distribution is the high possibility of sample
 		impoverishment due to a small measurement noise. This happens since accurate observations result in high peaks
 		of the evaluation density and therefore the proposal density is not able to sample outside that peak \cite{Isard98:CCD}. 
 		Additionally, erroneous or delayed measurements from absolute positioning sensors like \docWIFI{} may lead to misplaced turns. 
-		This causes a downvoting of all states $\mStateVec_t$ with increased heading deviation. 
-		\commentByFrank{so besser?: downvoting of states statt particles}
-		Therefore, we incorporate the turn-detection, as well as the related step-detection, directly into the state transition
+		This causes a downvoting of the posterior distribution in areas where the heading deviates.
+		Therefore, we incorporate the pedestrian's heading $\mObsHeading$, as well as the number of steps $\mObsSteps$, directly into the state transition
 		$p(\mStateVec_{t} \mid \mStateVec_{t-1}, \mObsVec_{t-1})$, which leads to a more directed sampling instead of a truly random one.
+		Steps and turns are detected using the smartphone's IMU and are implemented as described in \cite{Ebner-15}. 
 
 
 
diff --git a/tex/chapters/system.tex b/tex/chapters/system.tex
index 9d2df8c..4bcc696 100644
--- a/tex/chapters/system.tex
+++ b/tex/chapters/system.tex
@@ -24,7 +24,7 @@
 	where $x, y, z$ represent the position in 3D space, $\mStateHeading$ the user's heading and $\mStatePressure$ the
 	relative atmospheric pressure prediction in hectopascal (hPa). 
 	The recursive part of the density estimation contains all information up to time $t-1$.
-	Furthermore, the state transition models the pedestrian's movement as described in section \ref{sec:trans}. 
+	Furthermore, the state transition $p(\mStateVec_{t} \mid \mStateVec_{t-1}, \mObsVec_{t-1})$ models the pedestrian's movement as described in section \ref{sec:trans}. 
 	%It should be noted, that we also include the current observation $\mObsVec_{t}$ in it.
 	As \cite{Koeping14-PSA} has proven, we are able to include the observation $\mObsVec_{t-1}$ into the state transition. 
 
@@ -62,7 +62,7 @@
 	Therefore, numerical solutions like Gaussian filters or the broad class of Monte Carlo methods are deployed \cite{sarkka2013bayesian}. 
 	Since we assume indoor localisation to be a time-sequential, non-linear and non-Gaussian process,
 	a particle filter is chosen as approximation of the posterior distribution. 
-	Within this work the state transition $p(\mStateVec_{t} \mid \mStateVec_{t-1}, \mObsVec_{t})$ is used as proposal distribution,
+	Within this work the state transition $p(\mStateVec_{t} \mid \mStateVec_{t-1}, \mObsVec_{t-1})$ is used as proposal distribution,
 	what is also known as CONDENSATION algorithm \cite{Isard98:CCD}.