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\title[] % (optional, use only with long paper titles)
{Radar and the \\Speed \& Distance Unit (SDU) Library}
% based on Flyer "Radar and Odometry Dec. 2017"  [war in den eckigen Klammern]

{} % (optional)

\author[Wulf Kolbe] % (optional, use only with lots of authors)
{Wulf Kolbe, \\Radar Expert and Innovation Manager}
%% \\Blake Kozol\inst2 (CEO Deuta America)
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%   affiliation.

\institute[Universities of Somewhere and Elsewhere] % (optional, but mostly needed)
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%  Deuta Werke GmbH\\
%  Bergisch Gladbach Germany
%  \and
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%  Deuta America Corp\\
%  Richmond VA
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% - Keep it simple, no one is interested in your street address.
% \date[Date / Occasion]{based on Flyer "Radar and Odometry Dec. 2017"}

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\section{Radar and Speed \& Distance Unit (SDU) Library Dec. 2017}


% \transglitter
% \section{Questions related to Doppler Radar Applications}
\begin{frame}{What is the principle advantage of a Doppler Radar?}
\textbf{The Doppler Radar measures true speed over ground.}

\begin{alertblock}{Deuta Doppler radar gives\ldots}
    \item	speed measurement independent of the wheel slip \& slide
    \item	speed measurement from 0.2 km/h up to 600km/h
    \item	very accurate distance measurement

\begin{frame}{Why is a Doppler Radar used in ETCS odometry?}
 \begin{alertblock}{The principle considerations to use the Doppler Radar are\ldots}
	\item	Diversity of the principle for speed and distance measuement 
     \note[item]{Due to the diversity it can be assumed, 
                 that an common cause error of wheel sensor and
                 Doppler radar is highly unlikely.}
      \item	Speed and distance independent from slip \& slide 
      \note[item]{Wheel sensors on high-speed trains are influenced by the constant slip introduced by the                    traction control system to achieve optimal adhesion. 
                 The value is in a range between 1\%-5\%.}
	\item	Speed independent from wheel diameter degradation 
     \note[item]{Wheel diameter reduces due to wear and tear during operation, the speed is then
                 overestimated by several percent.}
	\item	Large measuring range of speed 0.2 km/h to 600km/h
      \note[item]{the speed measurement range can be in principle even higher, 
                  actual upper level depends on the maximal frequency of the DPS board}

\begin{frame}{What are typical Doppler Radar "Uses Cases"?}
 \begin{block}{Typical use Cases are \ldots}
    \item Speed / distance for ETCS (1-2 Dopplers per cab) 
     \note[item]{Typical ETCS systems have two Dopplers per ETCS onboard system. 
                 Most main line trains have two ETCS onbaord systems.}
	\item Speed / distance for CBTC (1-2 Dopplers per train) 
     \note[item]{Typical CBTC systems have two Dopplers per onboard system.
                 The system supports two drivers desks. 
                 In some cases one Doppler at each drivers cab.}
    \item Speed / distance for traction control on locomotives 
     \note[item]{Traction control has typical uses one Doppler near to the onboard system.}
	\item Speed / distance for Maglev and Mono-Rail trains 
     \note[item]{Maglev train has different mounting position very near to the surface of the track.
                 Typical only one or two Dopplers are used for Maglev.}
\item Speed for special machines (need no access to the bogie) 
     \note[item]{Machines for track laying, track maintenance and measuring work need
                 independent speed with accurate distance for localization. 
                 These small-volume applications take advantage from simple logistics and flexible mounting.}

\begin{frame}{What are the Deuta specific features\\to choose a DRS05/1S1 Doppler Radar?}
 \begin{exampleblock} {Doppler radar sensor DRS05/1Sa are chosen because \ldots}
    \item	High accuracy due to special two-channel algorithms 
      \note[item]{Our patented algorithm combines the signal from two microwave transceivers
                  in special way, which improves the accuracy while track ground
                  conditions are changing rapidly.}
	\item	Redundancy due to two transceivers
     \note[item]{If a microwave transceivers fails or the antenna receives
                 low reflection signal the firmware switches to a special 
                 single mode algorithm, which is able to deal with the situation.}
    \item	Reliability due to stone impact protection 
     \note[item]{DRS05/1S1 housing form and material thickness is designed to withstand stone 
                 impact even at highest speeds occurs on ballast tracks.}
    \item	Fexible mounting conditions 
    \item	Wide range power supply 24V-110V
    \item	Wide variety of output protocols and CRCs

\begin{frame}{What are the logistic reasons to rely on DRS05/1S1a Radar?}
 \begin{alertblock}{Most ETCS and CBTC system supplier worldwide selected Deuta Radar because \ldots}
	\item	the response time and support during project integration 
	\item	long-term availability (pro-active obsolescence process)
	\item	long term price by frame contracts
     \note[item]{Frame contracts optimize the purchase process and 
                 include also agreements about repair and RAMS reporting}
%% diese Zeile als Head-Line verwenden
%% \item	the large basis of installed devices \\more than 25,000+ units in operation

\begin{frame}{How is Deuta supporting the customer \\using the sensor products?}
 \begin{exampleblock} {there are several possibilities for support \ldots}
    \item	Answer to questions during integration phase 
      \note[item]{then it is often too late, architecture is already decided.}
     \item	Adapt Doppler Radar firmware to architecture of customer 
      \note[item]{only some changes are possible}
     \item	Encapsulate the part where sensor knowledge is needed
      \note[item]{then things will be easy for the customer, and he can focus on his part}
Deuta has developed an sensor fusion library called Speed \& Distance Unit (SDU) Library  based on a state of the art algorithm. This includes generalization of Kalman filter called Bayesian filtering derived from general probability theory.

\begin{frame}{Who needs such a library?}
 \begin{exampleblock} {Sensor fusion of wheel sensor, Doppler radar and other sensors is needed by\ldots}
    \item	ETCS system supplier who want to optimize their system 
    \item	ETCS newcomers who need short time to market 
    \item	CBTC system suppliers with high accuracy requirements 
  The encapsulation of all sensor specific know-how may also reduce the cost for the assessment because a large number of test cases related to sensor states is covered already by the SIL4 development documents.

\begin{frame}{What are the USPs of the SDU library?}
 \begin{exampleblock} {The unique selling proposition (USP) or unique selling points are related to the fact that the algorithm is able to deal with contradicting sensor information and can follow multiple hypotheses, this is achieved by \ldots}
    \item	Kalman filter for each sensor input
    \item	System models for most sensors types\\(RIG, Doppler, IMU, GPS) 
    \item	Bayesian filtering with the capability of\\following multiple hypotheses in parallel
    \item	Algorithm validated by experts (DFKI and Uni Bremen) 
    \item	Development process according to SIL4 (TÜV Nord)
    \item	Large number of test cases (16,000)

\begin{frame}{What are the functions of the SDU library?}
 \begin{exampleblock} {The SDU library contains several functions \ldots}
    \item	Read each sensor input and update Kalman filter for each sensor
    \item	Perform fusion step for all sensor information
    \item	Do the propagation in time
    \item	Prepare the output packet\\(results nearly identical to ETCS Odo Kernel)
 These functions are called periodically by the customer's software and the results can be used according to the ETCS Subset-035 Sect.12 and Subset-058 Sect.7.3..

\begin{frame}{Can you explain Bayesian filtering?}
  \begin{exampleblock}{here is a simple example for normal safe calculation with 2 sensors \ldots}
    \begin{equation} |v_{1}-v_{2}| < \Delta v \end{equation}
    \begin{equation} v = min(v_{1},v_{2}) \end{equation} or
    \begin{equation} v = max(v_{1},v_{2}) \end{equation}
    If the plausibility check fails, both values are ignored (2oo2).\\
    Selection of $max$ or $min$ depends on safety arguments.\\ 
    Transfer to probabilities for a logical "and" leads to a multiplication of probabilities.
    Values are calculated as expectation / mean value by integration over v. 

\begin{frame}{We map a sensor speed to a probability distribution}
  \begin{exampleblock}{to follow example system with 2 sensors \ldots}
    The probabilities are derived from a Likelihood functions $\mathcal{L}(v,\ldots)$. 
    The result of a function call $\mathcal{L}(v,\ldots)$ with for example speed $v_{1}$ as parameter is a probability distribution around $v_{1}$.
    This function $\mathcal{L}(v,\ldots)$ can model prior knowledge about the sensor or the system.
    \begin{equation} v_{1} =  \int_{0}^{vmax}  v' \cdot \mathcal{L}(v',v_{1}, \ldots)  \cdot dv' \end{equation}

\begin{frame}{Example for mapping to probabilities}
 \begin{exampleblock}{$ |v_{1}-v_{2}| < \Delta v $ is equivalent to rectangular distributions\ldots}

\begin{frame}{Example for mapping to probabilities cont.}
 \begin{exampleblock}{Gaussian distribution as example \ldots}

\begin{frame}{For safe speed we must consider evidence \ldots}
  \begin{exampleblock}{again simple example system with 2 sensors \ldots}
     The integral give the expectation value for the "and" operation by using the product of the distributions derived from $\mathcal{L}(v,\ldots) $ Likelihood functions.
    v= \frac{1}{T}\cdot \int_{0}^{vmax}  v' \cdot \mathcal{L}(v',v_{1}) \cdot \mathcal{L}(v',v_{2}) \cdot dv' \qquad (T>0)!
    Value of T is derived from evidence 
    % $P(0)=\frac{1}{vmax}$.
    T=  \int_{0}^{vmax}   \mathcal{L}(v',v_{1}) \cdot \mathcal{L}(v',v_{2}) \cdot dv'
    Again: Knowledge about typical sensor behaviour can be incorporated in the $\mathcal(L)(v',v,\ldots) $ Likelihood functions.

\begin{frame}{Bayesian Filter is generally known and accepted\ldots}
  \begin{exampleblock}{similar to a Kalman filter but general applicable \ldots}
     typical procedure :
     \item update system state by sensor information via $\mathcal{L}(v,\ldots) $ Likelihood functions.
        posterior(v,t) \propto prior(v,t) \cdot \mathcal{L}(v,v_{1},\ldots)
     \item propagate the system state based on speed and acceleration stored in probability distributions
        prior(v,t+\Delta t) \propto \\
        \int dynamic(v',\Delta t, \ldots) \cdot posterior(v',t) \cdot \mathcal{L}(v',v_{1},\ldots) \cdot dv'
        and a $dynamic(v',\Delta t, \ldots)$ function describing the transition probabilities.

\begin{frame}{How can we map a dynamic change on probabilities?}
\begin{exampleblock}{for example with Gaussian distributions we can have \ldots}
    \begin{equation} dynamic(v,v_1,v_2, \ldots) \propto exp(2v (v_2-v_1)))  \end{equation}
     \note[item]{the dynamic function for this example can be directly calculated 
     from the exponents of Gaussian function, below is shown only the simplified calculation with $\sigma = 1$. 
     \begin{gather*} prior(v,v_2) \propto \\
        dynamic(v,v_1,v_2, \ldots) \cdot posterior(v,v_1)
    \begin{gather*} (v-v_2)^2 = (v-v_1)^2-v_1^2+v_2^2-2 v (v_2-v_1)  
    The term $v_2^2-v_1^2$ is a constant scale factor,
    the important part is an exponential function $ exp(-2 v (v_2-v_1)) $ proportional v and the difference between old and new speed (acceleration). 

\begin{frame}{Summary for the sensor fusion principle \ldots}
 \begin{exampleblock}{product of two Gaussian distribution\ldots}

\begin{frame}{16,000 statistical test runs \ldots}
 \begin{exampleblock}{just one result with id 08763\ldots}
%%%%%%%%%%%%%%%%%%%%%%% todo %%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% new slide: 
%%% time step $\Delta t$ distribtuion will be broader after each time step due to the fact that
%%% $\sigma gets broader by the time step.
%%% to broaden from sigma = u to v, exponent (x-a)^2(u-v)/u*v should be used
%%% please check with Wolfram Alpa
%%% 2 change rates, one for all states unchanged, one for state changes  
%%% Each sensor update reduces the $\sigma$ again to the old values
%%% Evidence Integral and values of $\sigma$  $ E=\sqrt{\pi \sigma} $ 
%%% is it safe to be sure where the measurment is?
%%% new slide: 
%%% Basic slide about safety definitions
%%% new slide: 
%%% introduction of scaled values
%%% no function called without check of the units
%%% the scale to norm neads to be reordered

\begin{frame}{How is the library delivered\ldots}
 \begin{exampleblock} {The Speed and Distance library (SDU) is delivered as\ldots}
    \item   Compiled with the customer's compiler flags
    \item   Binary lib tested on final target with subset of test cases
    \item	Certificate SIL4 (TÜV Nord)
    \item	Detailed release notes
    \item	Simple license model\\(paid with DRS05 price)

\begin{frame}{What are the experiences of SDU customers?}
 \begin{exampleblock}{Customers reports about \ldots}
	\item	optimal speed information
	\item	reduced handling of exceptions
	\item	optimal values even in degraded modes
	\item	reduced time during assessment 

% [allowframebreaks] wenn es länger wird
\begin{frame}{Further Reading related to Deuta SDU Lib}
  \setbeamertemplate{bibliography item}[article]
  \bibitem{1} Dr.~Rainald~Koch, Deuta-Werke GmbH
    \newblock {\em Elements of a train odometer based on unsafe sensor signals}.
    \newblock {word document with performance graphs}
  \bibitem{2} Dr. Rainald Koch, Deuta-Werke GmbH
    \newblock {\em Deuta speed algo primer}
    \newblock {Deuta speed algo primer.doc}
  \bibitem{3} Dr. Rainald Koch, Deuta-Werke GmbH
    \newblock {\em SDU\_Performance}
    \newblock {VEK2268.SW.REP.SDU\_Performance.doc}
% g:\Downloads\Deuta speed algo primer.doc
% based on\\ \url{../2015-07/VEK2268.SW.REP.SDU_Performance.doc}
% \bibitem{D} ItemD

% ETCS Subset-035 Sect. 12 and Subset-058 Sect. 7.3.
% Vielleicht noch eoine Seite mit dem Protokol??
%T_ODO	32	 ms	Timestamp
%V_MAX	16	cm/s	Upper bound of the measured speed
%V_EST	16	cm/s	Estimated speed value
%V_MIN	16	cm/s	Lower bound of the measured speed
%D_MAX	32	 cm	Positive direction side of the confidence interval
%D_EST	32	 cm	Estimated value of distance
%D_MIN	32	 cm	Negative direction side of the confidence interval
%D_RES	  8	 cm	Resolution of distance measurement

% The SDU SW consists of four functions:
%-	sdu_init()
%-	sdu_eat_input()
%-	sdu_update()
%-	sdu_propagate(deltaT)
%A 5th function just bundles three of them:
%-	void sdu_run(){ sdu_eat_input(); sdu_update_RBF(); sdu_propagate_RBF(T_ODOCYCLE); } 

\begin{frame}{Probabilities can be used as general concept?}
 \begin{exampleblock}{if you want to calculate the average value \ldots}
% \vspace{10pt}
    \begin{equation} v=\frac{(v_1+v_2)}{2} \end{equation}
 seem optimal, but if one sensor fails to  $v_1=0.0 $ or $v_2=0.0$ the speed will be wrong. 
    If a weight factor is introduced, which can be changed due to sensor state, 
    then a sensor failure can be mitigated by $w_1=0$ or $w_2=0$ and then a failure has no influence on the final speed. 
    % Undetected error have influence on the result.
\begin{equation}  v=\frac{w_1 }{(w_1+w_2)}\cdot v_1+\frac{w_2 }{(w_1+w_2)} \cdot v_2 \end{equation}
then these values
        \begin{equation} \alpha_{i}=\frac{w_i}{(w_1+w_2)}  \qquad (i=1,2) \end{equation}
        $\alpha_{i} $ looks like probabilities because their sum is 1.

\begin{frame}{Again mapping to probabilities cont.}
 \begin{exampleblock}{Gaussian distribution as example and sum \ldots}

\begin{frame}{What are Deuta's unique selling points?}
 \begin{alertblock}{Deuta Doppler radar has\ldots}
   	\item	Experience - 25,000+ units in operation in extreme environments around the world\\
    (we have actually more then 750 Mio operating hours)
 	\item	Accuracy -  Special two-channel algorithms to reduce the Calibration-shift effects
	\item	Redundancy - system continues to operate with single sensor failure
	\item	Reliability - Spectral evaluation (winter conditions)