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\input{pst-node}
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\input{pogbase_new_par}
\input{BG_base.tex}

\input{mathbase}
\input{VPPC09_POG_Automotive_Modelling_Base}

 \newrgbcolor{mygreen}{0 0.6 0}
 \newrgbcolor{vgreen}{0.1 0.5 0.1}
 \newrgbcolor{vred}{0.5 0.1 0.1}
 %\newrgbcolor{green}{0 0 0}
 %\newrgbcolor{red}{0 0 0}

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\begin{document}

\title{The POG Technique for Modeling Planetary Gears and Hybrid Automotive Systems
 %\\[-2mm] {\large (Draft paper)}
 }

\author{\IEEEauthorblockN{Roberto Zanasi}
\IEEEauthorblockA{Information Engineering Department\\
University of Modena e Reggio Emilia\\
Via Vignolese 905\\ 41100
Modena, Italy\\
roberto.zanasi@unimore.it} \and
\IEEEauthorblockN{Federica Grossi}
\IEEEauthorblockA{Information Engineering Department\\
University of Modena e Reggio Emilia\\
Via Vignolese 905\\ 41100 Modena, Italy\\
federica.grossi@unimore.it} }


\maketitle


\begin{abstract}
In this paper the Power-Oriented Graphs (POG) technique is used for
modeling planetary gears and hybrid automotive systems. Some basic
properties of the POG technique are firstly given. An extended
dynamic model of a planetary gear with internal elasticity is
presented. Then  a POG congruent state space transformation is used
to transform and reduce the system when the elasticities or the
inertias go to zero. The obtained reduced model is used as central
element of an hybrid automotive power structure (endothermic engine,
multi-phase synchronous motor and vehicle dynamics). Simulation
results of the modeled hybrid system ends the paper.
\end{abstract}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% no \IEEEPARstart

Nowadays, the planetary gears are key elements for the design of new
hybrid power structures in the automotive area. In this paper, some
detailed dynamic models of a planetary gear and  an  hybrid
automotive system are given using the Power-Oriented Graphs (POG)
technique. This  technique allows to graphically describes the
dynamic model of any type of physical system putting in evidence the
powers which flow within the modeled systems. The POG schemes are
easy to use, easy to understand and can be directly implemented in
Simulink. The paper is organized as follows:
Sec.~\ref{POGproperties} describes the basic properties of the POG
modeling technique. Sec.~\ref{Planetary_Gears} and
Sec.~\ref{Hybrid_Structure} show, respectively, the POG dynamic
models of full and reduced  planetary gears, and of an hybrid
automotive power structure composed by an endothermic engine, a
multi-phase synchronous motor and the vehicle dynamics. Finally, in
Sec.~\ref{Simulations} some simulation results are reported.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Power-Oriented Graphs basic principles}\label{POGproperties}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The Power-Oriented Graphs technique, see \cite{Zana1} and
\cite{bodypap93}, is suitable for modeling physical systems. The POG
block schemes are normal block diagrams combined with a particular
modular structure essentially based on the use of the two blocks
shown in Fig.~\ref{figflow}.a and Fig.~\ref{figflow}.b: the {\em
elaboration block} (e.b.) stores and/or dissipates energy (i.e.
springs, masses, dampers, capacities, inductances, resistances,
etc.); the {\em connection block} (c.b.) redistributes the power
within the system without storing nor dissipating energy (i.e. any
type of gear reduction, transformers, etc.).
 % The c.b. transforms the power variables with the constraint $\x_{1}\tras\y_{1}=\x_{2}\tras\y_{2}$.
The e.b.~and the c.b.~are suitable for representing both scalar and
vectorial systems. In the vectorial case, $\G(s)$ and $\K$ are
matrices: $\G(s)$ is always a square matrix composed by positive
real transfer functions; matrix $\K$ can also be rectangular. The
circle present in the e.b. is a summation element and the black spot
represents a minus sign that multiplies the entering variable. The
main feature of the Power-Oriented Graphs is to keep a direct
correspondence between the dashed sections of the graphs and real
power sections of the modeled systems: the scalar product
$\x\tras\y$ of the two {\em power vectors} $\x$ and $\y$ involved in
each dashed line of a power-oriented graph, see Fig.~\ref{figflow},
has the physical meaning of {\em the power flowing through that
particular section}. The Bond Graphs technique, see \cite{Paynter}
and \cite{karnopp}, is based on the same idea, but it uses a
different and specific graphical representation.

 \begin{figure}[t]
 \centering
 \schemaPOG{(-2.0,-1.0)(40,12)}{2.1mm}{
 \thicklines
 \bloin{\x_{1}}{\y}
 \rput(3,-2.9){\mbox{\footnotesize a) {\em e. b.}}}
 \blovigiu{\ds \G(s)}{}{\dst}
 \bloout{\x_{2}}{\y}
 \addspace{6}
 \bloin{\x_{1}}{\y_{1}}
 \rput(2.25,-2.9){\mbox{\footnotesize b) {\em c. b.}}}
 \blokxyor{2.5}{2.5}{$\K$}{$\K$\tras}{}{}
 \bloout{\x_{2}}{\y_{2}}
 %
 \rput[l](5.5,-2.9){\mbox{\footnotesize c) {\em Across and through variables}}}
 \rput[l](4.5,5){\scriptsize
 \begin{array}{|@{\;}l@{\;}|@{\;}l@{\;}|@{\;}l@{\;}|}\hline
 & \multicolumn{2}{c|}{\mbox{\bf POG Variables}}  \\ \hline
   \mbox{\bf Domains:} & \mbox{\em Across: }     \vgreen v_e
                       &   \mbox{\em Through: }  \vred v_f
 \\ \hline
 \mbox{Electrical}&  \mbox{Voltage } \vgreen V  &  \mbox{Current } \vred I \\ \hline
 \mbox{Mec. Tra.} & \mbox{Velocity }   \vgreen v & \mbox{Force } \vred  F \\ \hline
 \mbox{Mec. Rot.} & \mbox{Ang. Vel. }  \vgreen \omega   & \mbox{Torque } \vred \tau \\ \hline
 \mbox{Hydraulic} & \mbox{Pressure }  \vgreen P  & \mbox{Flow rate }  \vred    Q  \\  \hline
\end{array}
 } }
 %%
%
% %%%%%%%%%%%%%%%%
\caption{\label{figflow} POG basic blocks and variables:  a) {\em
elaboration block}; b) {\em connection block}; c) {\em across and
through variables}.}
 \end{figure}
 %
\begin{figure}[t]
 \begin{center}
 %
 %
 \schemaPOG{(-3.0,-3)(9.0,22)}{2.2mm}{
 \put(0,19){
 \setlength{\unitlength}{1.6mm}
 \psset{unit=1.0\unitlength}
 \pscircle[fillstyle=solid,fillcolor=lightgray,linecolor=lightgray](0,0){0.5}
 \pscircle[fillstyle=solid,fillcolor=lightgray,linecolor=lightgray](8,0){0.5}
 \rput(0,0){\punto}
 \rput(8,0){\punto}
 \psline[linewidth=1.2pt]{-}(0,0)(2,0)
 \psline[linewidth=1.2pt]{-}(6,0)(8,0)
 \psline[linewidth=0.4pt]{-}(0,-8)(8,-8)
 \rput[l](2,0){\framebox(4,3){PE}}
 \rput(0,-8){\punto}
 \psline{->}(0,-7.5)(0,-0.5)
 \rput[r](-0.5,-4){$\vgreen v_{e1}$}
 \rput[r](-0.5,-8){\footnotesize $0$}
 \psline{->}(-3,0)(-0.5,0)
 \rput[b](-2,0.5){$\vred v_{f}$} %%%
 \rput(8,-8){\punto}
 \psline{->}(8,-7.5)(8,-0.5)
 \rput[l](8.5,-4){$\vgreen v_{e2}$}
 \rput[l](8.5,-8){\footnotesize $0$}
 \psline{->}(8.5,0)(11,0)
 \rput[b](10,0.5){$\vred v_{f}$}
 %
 \rput[b](0,2.25){\footnotesize $1$}
 \rput[b](8,2.25){\footnotesize $2$}
 \psline[linewidth=0.4pt,linestyle=dashed]{-}(0,-7.5)(0,1.75)
 \psline[linewidth=0.4pt,linestyle=dashed]{-}(8,-7.5)(8,1.75)
  }
 \thicklines %
%   \psellipse[linestyle=dashed,linecolor=mygreen,linewidth=0.3pt](3,10)(2.2,1.2)
%   \psellipticarc[linestyle=dashed,linecolor=mygreen,linewidth=0.3pt]{->}(-9.75,4.0)(2.1,2.1){-70}{250}
%   \psline[linestyle=dashed,linecolor=mygreen,linewidth=0.3pt]{<-}(-8.0,5.0)(1,9)
%   \rput(-9.75,4.0){\tiny \mbox{\shortstack{Kirchhoff's \\voltage law}}}
 \bloin{\vgreen v_{e1}}{\vred v_{f}}
 \rput[t](3,-2.5){\mbox{\footnotesize a) Element in series}.}
 \POGvigiu{\scr f_e(\!{\vgreen v_{e}}\!)}{\vgreen v_{e}}{\vred v_{f}}{\dst}
 \bloout{\vgreen v_{e2}}{\vred v_{f}}
 }
 %
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\hspace{10mm}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
 \schemaPOG{(-3.0,-3)(9.0,22)}{2.2mm}{
 \put(0,19){
 \setlength{\unitlength}{1.6mm}
 \psset{unit=1.0\unitlength}
  \pscircle[fillstyle=solid,fillcolor=lightgray,linecolor=lightgray](4,0){0.5}
 \pscircle[fillstyle=solid,fillcolor=lightgray,linecolor=lightgray](4,-8){0.5}
 \rput(0,0){\punto}
 \psline[linewidth=1.2pt]{-}(0,0)(4,0)(4,-2)
 \rput[t](4,-2){\framebox(3,4){PE}}
 \psline[linewidth=1.2pt]{-}(4,-6)(4,-8)(0,-8)
 \rput(0,-8){\punto}
 \psline{->}(0,-7.5)(0,-0.5)
 \rput[r](-0.5,-4){$\vgreen v_{e}$}
 \psline{->}(-3,0)(-0.5,0)
 \rput[b](-2,0.5){$\vred v_{f1}$} %
 %
 \rput(4,0){\punto}
 \rput(4,-8){\punto}
 \rput(8,0){\punto}
 \rput(8,-8){\punto}
 \psline[linewidth=1.2pt]{-}(4,0)(8,0)
 \psline[linewidth=1.2pt]{-}(4,-8)(8,-8)
 \psline{->}(8,-7.5)(8,-0.5)
 \rput[l](8.5,-4){$\vgreen v_{e}$}
 \psline{->}(8.5,0)(11,0)
 \rput[b](10.0,0.5){$\vred v_{f2}$}
 %
 \rput[b](4,1.0){\footnotesize$1$}
 \rput[t](4,-9){\footnotesize$2$}
 \psline[linewidth=0.4pt,linestyle=dashed]{-}(0,-7.5)(0,1.75)
 \psline[linewidth=0.4pt,linestyle=dashed]{-}(8,-7.5)(8,1.75)
% \rput[t](4,-11.5){\footnotesize a) Elemento parallelo}
  }
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \thicklines %
 %   \psellipse[linestyle=dashed,linecolor=mygreen,linewidth=0.3pt](3,10)(2.2,1.2)
 %   \psellipticarc[linestyle=dashed,linecolor=mygreen,linewidth=0.3pt]{->}(-9.75,4.0)(2.1,2.1){-70}{250}
 %   \psline[linestyle=dashed,linecolor=mygreen,linewidth=0.3pt]{<-}(-8.0,5.0)(1,9)
 %   \rput(-9.75,4.0){\tiny \mbox{\shortstack{Kirchhoff's \\voltage law}}}
 \bloin{\vgreen v_{e}}{\vred v_{f1}}
 \rput[t](3,-2.5){\mbox{\footnotesize b) Element in parallel}.}
 \POGvisu{\scr f_f(\!{\vred v_{f}}\!)}{\vgreen v_{e}}{\vred v_{f}}{\dst}
 \bloout{\vgreen v_{e}}{\vred v_{f2}}
 }
 %
 \end{center}
\caption{POG representations of  Physical Elements (PE): a)
connected in series (inputs  {\vgreen $v_{e1}$}, {\vgreen
$v_{e2}$}); b) connected in parallel (inputs  {\vred $v_{f1}$},
{\vred $v_{f2}$}).}\label{PE_connected_in_series}
\end{figure}
 %
The main energetic domains encountered in modeling physical systems
are the electrical, the mechanical (translational and rotational)
and the hydraulic, see Fig.~\ref{figflow}.c. Each energetic domain
is characterized by two {\em power variables}: an {\em
across-variable} ${\vgreen v_e}$ defined between two points (i.e.
the voltage $V$, the velocity $\dot x$, etc.), and a {\em
through-variable} ${\vred v_f}$ defined in each point of the space
(i.e. the current $I$, the force $F$, etc.). Each Physical Element
(PE) interacts with the external world through the power sections
associated to its terminals. A Physical Element is connected {\em in
series} when its terminals share the same through-variable ${\vred
v_f}$: see the physical element and the corresponding POG scheme in
Fig.~\ref{PE_connected_in_series}.a. A Physical Element is connected
{\em in parallel} when its terminals share the same across-variable
${\vgreen v_e}$: see the physical element and the POG scheme in
Fig.~\ref{PE_connected_in_series}.b. An example of POG modeling is
shown in Fig.~\ref{Esempio_CP_RS} where a C-parallel element is
connected with an R-series element.
 % The $C$ and $R$ physical elements are described by two POG elaboration blocks.
There is a direct correspondence between physical power sections and
dashed sections in the POG model. Note:  the summation elements
present in the elaboration blocks are a mathematical description of
the current and voltage Kirchhoff's laws applied to the considered
electrical system.
%
%
% \begin{figure}[t]
% \centering
% \schemaPOG{(-2.0,-1.0)(20,13)}{2.8mm}{
% \thicklines
% \bloin{\x_{1}}{\y}
% \rput(3,12.5){\footnotesize \mbox{a) {\em elaboration block}}}
% \blovigiu{\ds \G(s)}{}{\dst}
% \bloout{\x_{2}}{\y}
% \addspace{7}
% \bloin{\x_{1}}{\y_{1}}
% \rput(2.25,12.5){\footnotesize \mbox{b) {\em connection block}}}
% \blokxyor{2.5}{2.5}{$\K$}{$\K$\tras}{}{}
% \bloout{\x_{2}}{\y_{2}}
% }
% %%
%\caption{\label{figflow} POG basic blocks:  a) {\em elaboration
%block}; b) {\em connection block}.}
% \end{figure}
% %
% %
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %
 \begin{figure}[t]
 \centering
\begin{tabular}{@{}cc}
  % \hline  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \psfrag{V1}[r][r]{$\scr V_{1}$}
  \psfrag{V2}[l][l]{$\scr V_{2}$}
  \psfrag{V3}[r][r]{$\scr $}
  \psfrag{V4}[t][t]{$\scr V_{4}$}
  \psfrag{I1}[b][b]{$\scr I_{1}$}
  \psfrag{I2}[b][b]{$\scr I_{2}$}
  \psfrag{I3}[r][r]{$\scr I_{3}$}
  \psfrag{I4}[t][t]{$\scr I_{2}$}
  \psfrag{C}[l][l]{$\scr C$}
  \psfrag{R}[b][b]{$\scr R$}
  \includegraphics[width=3.8cm]{eC_0_R_1.eps}
   &
 \schemaPOG{(-2,-0.5)(13,13)}{2.1mm}{
 \thicklines
   \rput(-20,0){\nrsezgiu{$\scr 1$}{5.25}}
   \rput(-14.5,0){\nrsezgiu{$\scr 2$}{5.25}}
   \rput(-14.5,0){\psline[linestyle=dashed,linewidth=0.2pt](0,-0.75)(0,8.25)}
   \rput(-7.5,0){\nrsezgiu{$\scr 3$}{5.25}}
 \bloin{\scr I_1}{\scr V_1}
   \nrsezgiu{$\scr 1$}{-0.0}
   \rput[l](3.5,8.5){\scr I_{3}}
   \psellipse[linestyle=dashed,linecolor=red,linewidth=0.3pt](3,10)(2.2,1.2)
   \psellipse[linestyle=dashed,linecolor=red,linewidth=0.3pt](-16.5,10)(2.2,1.2)
   \pscurve[linestyle=dashed,linecolor=red,linewidth=0.3pt]{<-}(-15,11)(-11,12.5)(-2,12.5)(2,11)
   \rput[b](-16.75,11.5){\tiny \mbox{\shortstack{Kirchhoff's \\current law}}}
 \blovigiu{\ds \frac{1}{C\,s}}{\scr }{\dst}
   \nrsezgiu{$\scr 2$}{-0.0}
   \rput[l](3.5,1.5){\scr V_{4}}
   \psellipse[linestyle=dashed,linecolor=vgreen,linewidth=0.3pt](3,0)(2.2,1.2)
   \psellipticarc[linestyle=dashed,linecolor=vgreen,linewidth=0.3pt]{->}(-17,3.5)(3,3){-70}{250}
   \pscurve[linestyle=dashed,linecolor=vgreen,linewidth=0.3pt]{<-}(-14.5,1.5)(-10,-0.25)(-2,-1.25)(1,-1)
   \rput[b](-17.0,2.5){\tiny \mbox{\shortstack{Kirchhoff's \\voltage law}}}
 \blovisu{\ds \frac{1}{R}}{\scr }{\dst}
   \nrsezgiu{$\scr 3$}{-0.0}
 \bloout{\scr I_{2}}{\scr V_2}
 }
  \\
   &\\
  % \hline  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{tabular}
%
 %%
\caption{\label{Esempio_CP_RS} POG modeling of an electrical RC
circuit.}
 \end{figure}
 %

Another important property of the POG technique is the direct
correspondence between the POG schemes and the corresponding state
space dynamic equations. For example, the POG scheme shown in
Fig.~\ref{basesysgraph} can be represented by the state space
equations (\ref{generic}) where the {\em energy matrix} $\L$ is
symmetric and positive definite: $\L=\L\tras>0$.
 % For the POG systems described in form (\ref{generic})
 It can be easily shown that when $\D=0$
it follows that  $\C=\B\tras$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \begin{figure}[t]
 \centering
 \schemaPOG{(-1.5,0)(34,12)}{2.6mm}{
 \thicklines
  \bloin{\u}{\y}
  \bloivgiuor{\D}{}{}
  \bloko{\B}{\C}{}{}
  \bloivgiuintm{\L\muno}{\x}{\dst}
  \bloseiretaor{\A}{}
  \rput(4,5){
  \begin{minipage}{4.25cm}
 \small
 \begin{equation}
  \left\{
 \begin{array}{@{}r@{\;}c@{\;}l@{}}
 \L\,\dot{\x}&=&-\A\, \x + \B\, \u \\
 \y &=& \C\,\x+ \D\, \u
 \end{array}
 \right.
 \label{generic}
 \end{equation}
 \vspace{-1mm}
 \[
 \Downarrow
 (\x=\T\,\z)
 \]
 \vspace{-1mm}
 \begin{equation}
 \left\{
 \begin{array}{@{\,}r@{\;}c@{\;}l@{\,}}
 \ooL\,\dot{\z}&=&-\ooA \,\z\! +\! \ooB\, \u \\[1mm]
 \y &=& \ooC\,\z+\! \ooD\, \u
 \end{array}
 \right.
 \label{genericTrans}
 \end{equation}
  \end{minipage}}
 }
 \caption{\label{basesysgraph} POG block scheme of a generic
dynamic system.}
 \end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \begin{figure}[t]
 \centering
 \schemaPOG{(-1.5,0)(34,12)}{2.6mm}{
 \thicklines
  \bloin{\u}{\y}
  \bloivsu{\D\muno}{}{\dst}
  \bloko{\B}{\C}{}{}
  \bloivgiuintm{\L\muno}{\x}{\dst}
  \bloseiretaor{\A}{}
  \rput(4.25,5){
  \begin{minipage}{4cm}
 \small
 \begin{equation}
 \left\{
 \begin{array}{@{\,}r@{\;}c@{\;}l@{\,}}
 \tL\,\dot{\x}&=&-\tA \,\x\! +\! \tB\, \y \\[1mm]
 \u &=& \tC\,\x+\! \tD\, \y
 \end{array}
 \right.
 \label{genericTrans_qua}
 \end{equation}
% \vspace{-2mm}
\[
\mbox{where}
\]
% \vspace{-2mm}
\[
 \tL\!=\!\L,
 \hspace{3mm}
 \tB\!=\!\B\D\muno,
\]
\[
 \tA\!=\!\A\!+\!\B\D\muno\C,
\]
\[
 \tC\!=\!-\!\D\muno\C,
 \hspace{3mm}
 \tD\!=\!\D\muno.
\]
  \end{minipage}}
 }
 \caption{\label{basesysgraphinverted} POG block scheme of the input-output inverted system.}
 \end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%For such a system, the stored energy $E_{s}$ and the dissipating
%power $P_{d}$ can be expressed as follows:
% $E_{s}=\frac{1}{2}\x\tras\L\,\x$,
%   $P_{d}   =\x\tras\A\,\x$.
% where $\A_{s}=(\A+\A\tras)/2$ is the symmetric part of the {\em
% power matrix} $\A$.
% The skew-symmetric part
%$\A_{w}=(\A-\A\tras)/2$ of matrix  $\A$ represents power
%redistribution within the system.
 When an eigenvalue of matrix $\L$ tends to zero (or to infinity),
system (\ref{generic}) degenerates towards a lower dimension dynamic
system.  In this case, the dynamic model of the ``reduced'' system,
see (\ref{genericTrans}), can be directly obtained from
(\ref{generic}) using a simple ``congruent'' transformation
$\x=\T\,\z$ where  (if $\T$ is constant)  $\ooL=\T\tras\L\T$,
$\ooA=\T\tras\A\T$, $\ooB=\T\tras\B$, $\ooC=\C\T$  and $\ooD=\D$.
% If matrix $\T$ is time-varying, an additional term $\T\tras\L\dot{\T}\z$ appears in
% the transformed system. When matrix $\T$ is rectangular, the system
% is transformed and reduced at the same time.
 %.
The POG scheme of Fig.~\ref{basesysgraph} can also be easily
input-output inverted, both graphically and mathematically, as shown
in Fig.~\ref{basesysgraphinverted}. In this case $\tL\!=\!\L$,
$\tA\!=\!\A\!+\!\B\D\muno\C$, $\tB\!=\!\B\D\muno$,
$\tC\!=\!-\!\D\muno\C$ and $\tD\!=\!\D\muno$ are the matrices of the
inverted system (\ref{genericTrans_qua}).

 %  More details on Power-Oriented Graphs are reported in \cite{lille},
 % \cite{mitmemo}, \cite{mim93}.



\section{POG modeling of a planetary gear} \label{Planetary_Gears}

Let us consider the planetary gear shown in Fig.~\ref{epicicloida}.
The main parameters of the system are: $r_s$ and $r_{p}$ are the sun
and planet radii;  $J_{s}$, $b_{s}$, $J_{c}$, $b_{c}$, $J_{r}$ and
$b_{r}$ are the inertia and linear friction coefficients of the sun,
carrier and ring, respectively; $K_{sc}$, $d_{sc}$, $K_{cr}$ and
$d_{cr}$ are the stiffness and friction coefficients of the
sun-carrier and carrier-ring elastic elements, respectively.
 %
\begin{figure} [t]
 \centering
 \includegraphics[angle=0,width=0.8\columnwidth]{{epicicloida3.eps}}
 \caption{Planetary gear and related parameters.}
 \label{epicicloida}
\end{figure}
%
 \begin{figure*}[t]
  \centering
  \schemaPOG{(-1.5,-5)(66,13)}{2.9mm}{
 \thicklines
 \bloin{\tau_{s}}{\omega_{s}}
 \flowrightsu{}
 \nrsezxsu{0.65}{\scriptsize $2$}{1}
 \scrivigiu{9.5}{3}{\footnotesize Sun}
 \blovigiu{\ds \frac{1}{J_{s}s\!+\!b_{s}}}{\omega_{s}}{\dst}
 \flowrightsu{}
 \blokxyaor{1.5}{1.5}{r_{s}}{r_{s}}{}{}
 \flowrightsu{}
 \scrivigiu{8}{3}{\footnotesize Elastic element}
 \put(1,8.3){\makebox(0,0){$F_{sc}$}}
 \blosuattdue{}{K_{sc}}{d_{sc}}{}{}{\dst}
 \flowrightsu{}
 \scrivigiu{24}{3}{\footnotesize Planets and Carrier: two-dimensional dynamics}
 \blokxyor{4.0}{4.0}{\scr\mat{@{}c@{}}{-r_{p} \\ r_{s}} }{\scr\mat{@{}c@{\,}c@{}}{-r_{p} & r_{s}}}{}{}
 \flowrightsu{}
 \put(3,12.5){\vector(0,-1){2}}
 \put(3,12){$\mat{@{}c@{}}{ \tau_{p} \\ \tau_{c}}$}
 \blovigiumxm{\scr\mat{@{}c@{\;}c@{}}{\frac{1}{J_{p}} & 0 \\ 0 & \frac{1}{J_{c}}}}{\mat{@{}c@{}}{\\[-5mm]\omega_{p} \\[-1mm] \omega_{c}\\[-0.3mm]} }{\dst
 }{\frac{1}{s}}{}
 \flowrightsu{}
 \blovisuaor{\mat{@{}c@{\;}c@{}}{b_{p} & 0 \\ 0 & b_{c}}}{}{}
 \flowrightsu{}
 \blokxyaor{4.0}{4.0}{\mat{@{}c@{}}{r_{p} \\ r_{r}}}{\mat{@{}c@{\;}c@{}}{r_{p} & r_{r}}}{}{}
 \flowrightsu{}
 \scrivigiu{8}{3}{\footnotesize Elastic element}
 \put(1,8.3){\makebox(0,0){$F_{cr}$}}
 \blosuattdue{}{K_{cr}}{d_{cr}}{}{}{\dst}
 \flowrightsu{}
 \scrivigiu{10}{3}{\footnotesize Ring}
 \blokxyor{2}{2}{r_{r}}{r_{r}}{}{}
 \flowrightsu{}
 \blovigiu{\ds \frac{1}{J_{r}s\!+\!b_r}}{\omega_r}{}
 \bloout{\tau_{r}}{\omega_{r}}
 \scrivigiu{10}{3}{}
 \nrsezxsu{0.65}{\scriptsize  $3$}{1}
 \flowleftsu{}
 }
  \centering
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{equation}\label{epicicloidale_completo_SS}
 \footnotesize
 \underbrace{\mat{@{}c@{}c@{\;}c@{}c@{\,}c@{\,}c@{}}{
 J_{s} &  0 &  0 &  0 &  0 &  0 \\
 0 &  \frac{1}{K_{sc}} &  0 &  0 &  0 &  0 \\
 0 &  0 &  J_{p} &  0 &  0 &  0 \\
 0 &  0 &  0 &  J_{c} &  0 &  0 \\
 0 &  0 &  0 &  0 &  \frac{1}{K_{cr}} &  0 \\
 0 &  0 &  0 &  0 & 0 &  J_{r}}}_{\ds \ooL}\!
 \underbrace{\mat{@{}c@{}}{\dot\omega_{s}\\ \dot F_{sc} \\ \dot \omega_{p}\\ \dot \omega_{c}\\ \dot F_{cr} \\ \dot \omega_{r}}}_{\ds \dot\oox}\!
 =
 \underbrace{\mat{@{}c@{\;\;}c@{\;\;}c@{\;\;}c@{\;\;}c@{\;\;}c@{}}{
 -\!b_{s}\!-\!r_{s}^2d_{sc} & -\!r_{s} & -\!r_{s}d_{sc}r_{p} & r_{s}^2 d_{sc} & 0 & 0\\
 r_{s}         &   0  &    r_{p}       &   -\!r_{s}     & 0 & 0\\
 -\!r_{s}d_{sc}r_{p}   & -\!r_{p} & -\!b_{p}\!-\!d_{sc}r_{p}^2\!-\!d_{cr}r_{p}^2 & r_{s}d_{sc}r_{p}\!-\!d_{cr}r_{p}r_{r} & -\!r_{p} & d_{cr}r_{p}r_{r}\\
 r_{s}^2d_{sc} & r_{s} & r_{s}d_{sc}r_{p}\!-\!d_{cr}r_{p}r_{r} & -\!b_{c}\!-\!r_{s}^2d_{sc}\!-\!d_{cr}r_{r}^2 & -\!r_{r} & d_{cr}r_{r}^2\\
 0 & 0 & r_{p} & r_{r} & 0 & -\!r_{r}\\
 0 & 0 & d_{cr}r_{p}r_{r} & d_{cr}r_{r}^2 & r_{r} & -\!b_r\!-\!d_{cr}r_{r}^2
 }}_{\ds-\ooA}\!
 \underbrace{\mat{@{}c@{}}{\omega_{s}\\ F_{sc} \\ \omega_{p}\\ \omega_{c}\\  F_{cr} \\ \omega_{r}}}_{\ds\oox}\!
 +
 \underbrace{\mat{@{}c@{\,}c@{\,}c@{}}{
 1 &  \,0 &  \,0 \\
 0 &  \,0 &  \,0 \\
 0 &  \,0 &  \,0 \\
 0 &  \,1 &  \,0 \\
 0 &  \,0 &  \,0 \\
 0 &  \,0 &  \,1 }}_{\ds\ooB}\!
  \underbrace{\mat{@{}c@{}}{\tau_{s}\\ \tau_{c}\\ \tau_{r}}}_{\ds\u}
\end{equation}
\caption{The POG  block diagram and the POG state space equations of
the considered planetary gear: the sun, the planets-carrier and the
ring interact through elastic elements. The considered  inputs are:
$\tau_{s}$, $\tau_{c}$ and $\tau_{r}$.}
  \label{POG_epicicloidale}
\end{figure*}
 %
The extended POG dynamic model of the considered planetary gear is
shown in  the upper part of Fig.~\ref{POG_epicicloidale}: the
carrier, the planets and the ring interact each other through the
two elastic elements $K_{cr}$ and $K_{sc}$. The corresponding state
space dynamic equations are shown in lower part of
Fig.~\ref{POG_epicicloidale}:
\begin{equation}
\label{expanded_POG_model}
 \ooL\,\dot\oox
 =
 -\ooA\,\oox+\ooB\,\u,
 \hspace{12mm}
 \y=\ooB\tras\oox
\end{equation}
The POG linear systems described in form (\ref{expanded_POG_model})
always satisfy the following properties:
 \\
 1) the energy $E_{s}$ stored in the system and the dissipating power $P_{d}$ are
quadratic functions of matrices $\ooL$ and $\ooA_{s}$, respectively:
 \[
 E_{s}=\frac{1}{2}\,\oox\tras\,\ooL\,\oox,
 \hspace{15mm}
 P_{d}   =\oox\tras\,\ooA_{s}\,\oox
 \]
where $\ooA_{s}=(\ooA+\ooA\tras)/2$ is the symmetric part of the
{\em power matrix} $\ooA$.  The skew-symmetric part
$\ooA_{w}=(\ooA-\ooA\tras)/2$ of matrix $\ooA$ represents the power
redistribution within the system. One can easily verify that all the
dissipating parameters of the system appear only in matrix
$\ooA_{s}$, while matrix $\ooA_{w}$ is completely characterized by
all the connection parameters;
 \\
2)  all the loops of the POG schemes {\em always contain  an ``odd''
number of signs  ``$-$''} (i.e. the black spots) in the summation
blocks of the loop;
 \\
 3) the direction of the power flowing through a section is
positive if {\em an ``even'' number of signs ``$-$'' is present
along all the paths}  which link the input and the output of the
section. Note, for example, that in Fig.~\ref{POG_epicicloidale} the
power is entering the system in both sections \nrsec{2} and
\nrsec{3}.

\subsection{Reduced inertial model}

For certain applications the POG model of
Fig.~\ref{POG_epicicloidale} can be too much detailed. In these
cases it can be of interest to find the reduced model when, for
example, the stiffness coefficients $K_{cr}$ and $K_{sc}$ tend to
infinity. Applying to system (\ref{expanded_POG_model}) the
following congruent transformation:
\[
 \oox=\T_{1}\,\x,
 \hspace{8mm}
 \mbox{where}
 \hspace{8mm}
 \T_{1}=\mat{@{\;}c@{\;}c@{\;}c@{\;}c@{\;}c@{\;}c@{\;}}{
 1 &  0 &  0 &  0 &  0 &  0 \\
 0 &  0 &  0 &  0 &  0 &  1 \\
 0 &  0 &  1 &  0 &  0 &  0 \\
 0 &  1 &  0 &  0 &  0 &  0 \\
 0 &  0 &  0 &  0 &  1 &  0 \\
 0 &  0 &  0 &  1 &  0 &  0
 }.
\]
 one obtains the following transformed system:
\begin{equation}\label{tran_sys_pog_t}
 \underbrace{\mat{@{}c@{}c@{}c@{}}{
 \J_{1} &  0 &  0 \\
 0 & \J_{2} &   0 \\
 0 & 0 &    0 }}_{\ds \L}\!
 \underbrace{\mat{@{}c@{}}{\dot\x_{1}\\ \dot\x_{2} \\ \dot\x_{3}}}_{\ds \dot\x}\!
 =
\underbrace{\mat{@{}c@{\;}c@{\;}c@{}}{
 \A_{11} & \A_{12} & \A_{13} \\
 \A_{21} & \A_{22} & \A_{23} \\
 \A_{31} & \A_{32} & 0
  }}_{\ds-\A}\!
 \underbrace{\mat{@{}c@{}}{\x_{1}\\ \x_{2} \\ \x_{3}}}_{\ds \x}\!
 +
 \underbrace{\mat{@{}c@{}}{
 \B_{1} \\
 \B_{2} \\
 0
 }}_{\ds\B}\!
 \u
\end{equation}
 %
where $\x=\T_{1}\tras\,\oox$, $\L=\T_{1}\tras\ooL\,\T_{1}$,
$\B=\T_{1}\tras\,\ooB$, $\A=\T_{1}\tras\ooA\,\T_{1}$,
$\y=\B\tras\,\x$ and:
\[
\footnotesize
 \x_{1}\!=\!\!\mat{@{}c@{}}{\omega_{s}\\ \omega_{c}}\!\!,
 \hspace{1mm}
 \x_{2}\!=\!\!\mat{@{}c@{}}{\omega_{p}\\ \omega_{r}}\!\!,
 \hspace{1mm}
 \x_{3}\!=\!\!\mat{@{}c@{}}{F_{cr} \\ F_{sc}}\!\!,
 \hspace{1mm}
 \J_{1}\!=\!\mat{@{}c@{}c@{}}{
 J_{s} &  0 \\
 0 &  J_{c} }\!\!,
 \hspace{1mm}
 \J_{2}\!=\!\mat{@{}c@{}c@{}}{
 J_{p}  &  0 \\
0 &  J_{r} }\!\!,
\]
\[
\footnotesize
\B_{1}\!=\!\!\mat{@{}c@{\,}c@{\,}c@{}}{
 1 &  0 &  0 \\
 0 &  1 &  0 }\!,\!
 \hspace{1mm}
\B_{2}\!=\!\!\mat{@{}c@{\,}c@{\,}c@{}}{
 0 &  0 &  0 \\
 0 &  0 &  1 }\!,\!
 \hspace{1mm}
 \A_{11}\!=\!\mat{@{}c@{}c@{}}{
 -\!b_{s}\!-\!r_{s}^2d_{sc} & r_{s}^2d_{sc}                          \\
 r_{s}^2d_{sc}    & -\!b_{c}\!-\!r_{s}^2d_{sc}\!-\!d_{cr}r_{r}^2 }\!\!,
 \]
 \[
 \footnotesize
 \!\A_{12}\!\!=\!\!\mat{@{}c@{\;}c@{}}{
 -\!r_{s}d_{sc}r_{p}                          & 0                      \\
 r_{s}d_{sc}r_{p}\!-\!d_{cr}r_{p}r_{r} & d_{cr}r_{r}^2    }\!\!,
 \hspace{0mm}
 \A_{13}\!\!=\!\!\mat{@{}c@{}c@{}}{
 0            & -\!r_{s}     \\
 -\!r_{r}       & r_{s}}\!\!,
 \hspace{0mm}
 \A_{31}\!\!=\!\!\mat{@{}c@{\,}c@{}}{
 0                   & r_{r} \\
 r_{s}            & -\!r_{s}         }\!\!,
\]
\[
 \footnotesize
\!\A_{21}\!=\!\mat{@{}c@{\;}c@{}}{
-\!r_{s}d_{sc}r_{p}        & r_{s}d_{sc}r_{p}\!-\!d_{cr}r_{p}r_{r}  \\
0                   & d_{cr}r_{r}^2 }\!\!,
 \hspace{8mm}
 \A_{23}\!=\!\mat{@{}c@{}c@{}}{
 -\!r_{p}          & -\!r_{p}  \\
 r_{r}  & 0    }\!\!,
\]
 \[
 \footnotesize
 \A_{22}\!=\!\mat{@{}c@{\;}c@{}}{
  -\!b_{p}\!-\!d_{sc}r_{p}^2\!-\!d_{cr}r_{p}^2   & d_{cr}r_{p}r_{r}   \\
  d_{cr}r_{p}r_{r}                  & -\!b_r\!-\!d_{cr}r_{r}^2 }\!\!,
 \hspace{6mm} \scriptsize
 \A_{32}\!=\!\mat{@{}c@{\;}c@{}}{
 r_{p}                              & -\!r_{r}  \\
 r_{p}                              & 0     }\!\!.
\]
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
The last equation of system (\ref{tran_sys_pog_t}) shows an
algebraic relation between the state variables:
$\A_{31}\,\x_{1}+\A_{32}\,\x_{2}=0$. Since matrix $\A_{32}$ is
invertible, vector $\x_{2}$ can be expressed as a function of vector
$\x_{1}$:
\[
 \x_{2}
 \!=\!\mat{@{}c@{}}{\omega_{p}\\ \omega_{r}}
 \!=\!
 -\A_{32}\muno\,\A_{31}\,\x_{1}
 \!=\!
 \mat{@{}c@{\;}c@{}}{
  -\frac{r_{s}}{r_{p}} &  \frac{r_{s}}{r_{p}} \\
  -\frac{r_{s}}{r_{r}} &  1\!+\!\frac{r_{s}}{r_{r}}  }
  \mat{@{}c@{}}{\omega_{s}\\ \omega_{c}}
%  \!=\!
%  \mat{@{}c@{}}{
%  \frac{r_{s}}{r_{p}}(\omega_{c}\!-\!\omega_s)\\
%  \omega_{c}\!+\!\frac{r_{s}}{r_{r}}(\omega_{c}\!-\!\omega_s)  }
\]
Applying to system (\ref{tran_sys_pog_t}) the following
``rectangular'' and ``congruent'' state space transformation:
\[
\x=\T_{2}\,\x_{1},
 \hspace{4mm}
 \mbox{where}
 \hspace{4mm}
 \T_{2}
 \!=\!
 \mat{@{}c@{}}{
 \I_{2} \\[1mm]
-\A_{32}\muno\,\A_{31}\\[1mm]
 0 }
 \!=\!
 \mat{@{}c@{\;}c@{}}{
  1 & 0  \\
  0 & 1 \\
  -\frac{r_{s}}{r_{p}} &  \frac{r_{s}}{r_{p}} \\
  -\frac{r_{s}}{r_{r}} &  1\!+\!\frac{r_{s}}{r_{r}} \\
  0 & 0 \\
  0 & 0 }
\]
one obtains the following second order transformed and reduced
system:
\begin{equation}
\label{modello_epiciclo_ridotto}
\L_{r}\,\dot\x_{1}=-\A_{r}\,\x_{1}+\B_{r}\,\u,
 \hspace{10mm}
 \y=\B_{r}\tras\x_{1},
\end{equation}
where $\L_{r}=\T_{2}\tras\L\,\T_{2}$, $\A_{r}=\T_{2}\tras\A\,\T_{2}$
and $\B_{r}=\T_{2}\tras\,\B$ are:
\[
 \L_{r}
 \!=\!
 \mat{@{}c@{\;\,}c@{}}{
 J_{s}\!+\!\frac{r_{s}^2 }{r_{p}^2} J_{p}\!+\!\frac{r_{s}^2 }{r_{r}^2} J_r &
 \!-\!\frac{r_{s}^2 }{r_{p}^2 } J_{p}  \!-\!\frac{r_{s}}{r_{r}} \!\left(1\!+\!  \frac{r_{s} }{ r_{r}}\right)\! J_r \\[1mm]
 \!-\frac{r_{s}^2 }{r_{p}^2 } J_{p}  \!-\!\frac{r_{s}}{r_{r}} \!\left(1\!+\!  \frac{r_{s} }{ r_{r}}\right)\! J_r &
 J_{c}\!+\!\frac{r_{s}^2}{r_{p}^2} J_{p} \!+\! \left(1\!+\!\frac{r_{s} }{ r_{r}}\right)^2 \!\!J_r}
\]
\[
 \A_{r}
\!=\!
 \mat{@{}c@{\;\,}c@{}}{
 b_{s}\!+\!\frac{r_{s}^2 }{r_{p}^2} b_{p}\!+\!\frac{r_{s}^2 }{r_{r}^2} b_r &
 \!-\!\frac{r_{s}^2 }{r_{p}^2 } b_{p}  \!-\!\frac{r_{s}}{r_{r}} \!\left(1\!+\!  \frac{r_{s} }{ r_{r}}\right)\! b_r \\[1mm]
 \!-\frac{r_{s}^2 }{r_{p}^2 } b_{p}  \!-\!\frac{r_{s}}{r_{r}} \!\left(1\!+\!  \frac{r_{s} }{ r_{r}}\right)\! b_r &
 b_{c}\!+\!\frac{r_{s}^2}{r_{p}^2} b_{p} \!+\! \left(1\!+\!\frac{r_{s} }{ r_{r}}\right)^2 \!\!b_r}
%\mat{@{}c@{\;\;}c@{}}{
% b_{s}\!+\!\frac{r_{s}^2 }{r_{p}^2 } b_{p}\!+\! \frac{r_{s}^2 }{r_{r}^2} b_r &
% \!\!\!\!-\!\left(\!1\!+\!\!\frac{r_{s} }{ r_{r}}\!\right)\!\!\frac{r_{s}}{r_{r}} b_r\!-\!\frac{r_{s}^2 }{r_{p}^2 } b_{p} \\[1mm]
% -\!\left(\!1\!+\!\frac{r_{s} }{ r_{r}}\!\right)\!\frac{r_{s}}{r_{r}} b_r\!-\!\frac{r_{s}^2 }{r_{p}^2 } b_{p} &
% b_{c} \!+\!\frac{r_{s}^2 }{r_{p}^2 } b_{p} \!+\!\left(\!1\!+\!  \frac{r_{s} }{ r_{r}}\!\right)^2 \!\!b_r }
\]
\[
 \B_{r}
 \!=\! \mat{@{\,}c@{\;\;}c@{\;\;}c@{\,}}{
1 & 0 & \!-\frac{r_{s}}{r_{r}}\\[1mm]
0 & 1 & 1\!+\!\frac{r_{s}}{r_{r}}
 },
 \hspace{4mm}
\x_{1}\!=\!\mat{@{}c@{}}{\omega_{s}\\\omega_{c}},
 \hspace{4mm}
 \y
 \!=\!\mat{@{}c@{}}{\omega_{s}\\ \omega_{c}\\ \omega_{r}}.
% \!=\!\mat{@{}c@{}}{\omega_{s}\\ \omega_{c}\\  }
\]
Note: the possibility of using a ``rectangular'' matrix $\T_{2}$ for
transforming and reducing a dynamical system is a ``specific
characteristics'' of the POG technique and it holds only for system
described in the POG form (\ref{generic}). A simulative comparison
between the extended model (\ref{epicicloidale_completo_SS}) of the
planetary gear and the reduced model
(\ref{modello_epiciclo_ridotto}) is shown in
Fig.~\ref{Simulative_comparison}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
 \centering
 \psfrag{Velocita del solare}[c][tc]{\footnotesize Angular velocity $\omega_{s}$ of the sun}
 \psfrag{Wc [rad/s]}[c][c]{\footnotesize $\omega_s$ [rad/s]}
 % \includegraphics[width=8cm,height=4cm]{VPPC09_Sun_Figure.eps}
 \psfrag{Velocità del solare}[c][tc]{\footnotesize Angular velocity $\omega_{s}$ of the sun}
 \psfrag{Wc [rad/s]}[c][c]{\footnotesize $\omega_s$ [rad/s]}
 \psfrag{modello esteso}[c][c]{\tiny extended model}
 \psfrag{modello ridotto}[c][c]{\tiny reduced model}
 \psfrag{Time [s]}[ct][c]{\footnotesize Time [s]}
 \includegraphics[width=9cm]{epicicloidale_da_solo.eps}
 \caption{Simulative comparison between the extended (blue solid line) and the reduced (magenta dashed line) model of the planetary gear.}
 \label{Simulative_comparison}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The parameters used in simulation are (SI units):
 sun parameters $[J_s,\, b_s,\, r_s]=[0.049,\,    4.946,\, 0.102]$;
 ring parameters $[J_r,\, b_r,\, r_r]=[2.180,\, 218.02,\, 0.248]$;
% carrier parameters $[J_p,\, J_c,\, b_p,\, b_c,\, r_p]=[0.0812,\, 0.9290,\, 8.1228,\, 92.8975,\, 0.0732]$;
 carrier parameters $[J_c,\, b_c]=[0.929,\, 92.89]$;
 planet parameters $[J_p,\, b_p,\, r_p]=[0.081,\, 8.123,\, 0.073]$;
 stiffness parameters $[K_{sc},\, d_{sc}]=[K_{cr},\, d_{cr}]=[10^7,\, 10]$;
 initial conditions $[\omega_s(0),\, \omega_c(0),\, \omega_r(0)]=[0.1,\, -0.2,\, -0.323]$ rad/s;
 constant external torques $[\tau_s,\,\tau_c,\,\tau_r]=[0,\,4,\,0]$ Nm.
 %

With different choices of matrix $\T_{1}$ it is possible to obtain
different but equivalent reduced systems, similar to
(\ref{modello_epiciclo_ridotto}), with different state vectors
$\x_{1}$, for example
$\x_{1}=\mat{@{\,}c@{\;}c@{\,}}{\omega_{s},&\omega_{r}}\tras$,
$\x_{1}=\mat{@{\,}c@{\;}c@{\,}}{\omega_{r}, & \omega_{c}}\tras$,
etc.

\subsection{Reduced elastic model}

Using the POG reduction technique it is also possible to obtain from
(\ref{epicicloidale_completo_SS}) the reduced elastic model when
inertias $J_{s}$, $J_{c}$ and $J_{r}$ go to zero. Applying to system
(\ref{expanded_POG_model}) the following congruent transformation:
\[
 \oox=\T_{3}\,\z,
 \hspace{8mm}
 \mbox{where}
 \hspace{8mm}
 \T_{3}=\mat{@{\;}c@{\;}c@{\;}c@{\;}c@{\;}c@{\;}c@{\;}}{
 0 &  0 &  0 &  1 &  0 &  0 \\
 1 &  0 &  0 &  0 &  0 &  0 \\
 0 &  1 &  0 &  0 &  0 &  0 \\
 0 &  0 &  0 &  0 &  1 &  0 \\
 0 &  0 &  1 &  0 &  0 &  0 \\
 0 &  0 &  0 &  0 &  0 &  1
 }.
\]
and using the constraint $J_{s}=J_{c}=J_{r}=0$, one obtains the
following transformed system:
\begin{equation}\label{tran_sys_pog_t_elastic}
 \underbrace{\mat{@{}c@{\;}c@{}}{
 \L_{1} &  0 \\
 0 & 0 }}_{\ds \L}\!
 \underbrace{\mat{@{}c@{}}{\dot\z_{1}\\ \dot\z_{2} }}_{\ds \dot\z}\!
 =
 \underbrace{\mat{@{}c@{\;}c@{}}{
 \A_{11} & \A_{12}  \\
 \A_{21} & \A_{22}  }}_{\ds-\A}\!
 \underbrace{\mat{@{}c@{}}{\z_{1}\\ \z_{2} }}_{\ds \z}\!
 +
 \underbrace{\mat{@{}c@{}}{
 \B_{1} \\
 \B_{2}
 }}_{\ds\B}\!
 \u,
 \hspace{4mm}
 \y=\C\,\z+\D\,\u,
\end{equation}
 %
where $\z=\T_{1}\tras\,\oox$, $\L=\T_{3}\tras\ooL\,\T_{3}$,
$\A=\T_{3}\tras\ooA\,\T_{3}$, $\B=\T_{3}\tras\,\ooB$,
$\C=[\C_1,\;\C_2]=\B\tras=[\B_1\tras,\;\B_2\tras]$, $\D=0$ and
where:
\[
\footnotesize
 \z_{1}\!=\!\!\mat{@{}c@{}}{F_{sc}\\ \omega_{p}\\ F_{cr}}\!\!,
 \hspace{7mm}
 \z_{2}\!=\!\!\mat{@{}c@{}}{\omega_{s}\\ \omega_{c}\\ \omega_{r}}\!\!,
 \hspace{7mm}
 \L_{1}\!=\!\mat{@{}c@{}c@{}c@{}}{
 \frac{1}{K_{sc}} &  0  &  0 \\
 0 &  J_p  &  0 \\
 0 & 0 &  \frac{1}{K_{cr}} }\!\!,
\]
\[
\footnotesize
 \A_{11}\!=\!\mat{@{}c@{\;\;}c@{\;\;}c@{}}{
  0 & r_p &  0\\
  -\!r_p & -\!d_{sc} r_p^2\!-\!r_p^2 d_{cr}\!-\!b_p &  -\!r_p\\
  0 & r_p &  0 }\!\!,
 \hspace{5mm}
\B_{1}\!=\!\!\mat{@{}c@{\,}c@{\,}c@{}}{
 0 &  0 &  0 \\
 0 &  0 &  0 \\
 0 &  0 &  0 }\!,\!
 \]
 \[
 \footnotesize
 \!\A_{12}\!\!=\!\!\mat{@{}c@{\;\;}c@{\;\;}c@{}}{
 r_s & -\!r_s & 0 \\
 -\!r_s d_{sc} r_p & r_s d_{sc} r_p\!-\!r_p r_r d_{cr} & r_p r_r d_{cr} \\
 0 & r_r &-\!r_r    }\!\!,
 \hspace{1mm}
\B_{2}\!=\!\!\mat{@{}c@{\,}c@{\,}c@{}}{
 1 &  0 &  0 \\
 0 &  1 &  0 \\
 0 &  0 &  1 }\!,\!
\]
\[
 \footnotesize
\!\A_{21}\!=\!\mat{@{}c@{\;}c@{\;}c@{}}{
 -r_s & -r_s d_{sc} r_p & 0 \\
 r_s & r_s d_{sc} r_p-r_p r_r d_{cr} & -r_r \\
 0 & r_p r_r d_{cr} & r_r }\!\!,
 \hspace{5mm}
\D\!=\!\!\mat{@{}c@{\,}c@{\,}c@{}}{
 0 &  0 &  0 \\
 0 &  0 &  0 \\
 0 &  0 &  0 }\!,\!
\]
 \[
 \footnotesize
 \A_{22}\!=\!\mat{@{}c@{\;}c@{\;}c@{}}{
 -b_s-r_s^2 d_{sc} & r_s^2 d_{sc} & 0 \\
 r_s^2 d_{sc} & -b_c-r_s^2 d_{sc}-r_r^2 d_{cr} & r_r^2 d_{cr} \\
 0 & r_r^2 d_{cr} & -b_r-r_r^2 d_{cr}  }\!\!.
\]
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \begin{figure*}[t]
  \centering
  \schemaPOG{(-2,-12)(58,18)}{2.9mm}{
  \thicklines
  \bloin{\tau_{s}}{\omega_{s}}
  \flowrightsu{}
  \scrivigiu{8}{3}{\footnotesize Sun}
  \bloivsuaor{b_{s}}{}{}
  \flowrightsu{}
  \blokxyaor{2}{2}{r_{s}}{r_{s}}{}{}
  \flowrightsu{}
  \scrivigiu{8}{3}{\footnotesize Elastic element}
  \put(1,8.3){\makebox(0,0){$F_{sc}$}}
  \blosuattdue{}{K_{sc}}{d_{sc}}{}{}{\dst}
  \flowrightsu{}
  \scrivigiu{0}{3}{\footnotesize }
%
  \bloivparingiu{
     \flowrightsu{}
     \blokxyor{2}{2}{{\scr-}r_{p}}{{\scr-}r_{p}}{}{}
     \flowrightsu{}
     \bloivgiumxm{\frac{1}{J_{p}}}{\omega_{p}}{\dst}{\frac{1}{s}}{}
     \flowrightsu{}
     \bloivsuaor{b_{p}}{}{}
     \flowrightsu{}
     \blokxyaor{2}{2}{r_{p}}{r_{p}}{}{}
     \flowrightsu{}
 }{
    \scrivisu{16}{3}{\footnotesize Planets and Carrier}
    \flowrightsu{}
    \blokxyor{2}{2}{r_{s} }{ r_{s}}{}{}
    \flowrightsu{}
    \addsu{\omega_{c}}{\snt}{\tau_{c}}{b_{c}}
    \flowrightsu{}
    \blokxyaor{2}{2}{r_{r}}{r_{r}}{}{}
    \scrivisu{10}{3}{}
    \flowrightsu{}
 }{}{}{}{}{}
  %
  \addspace{16}
  \bloivparoutgiu{}{}{}{}{}
  \scrivigiu{8}{3}{\footnotesize Elastic element}
  \put(1,8.3){\makebox(0,0){$F_{cr}$}}
  \flowrightsu{}
  \blosuattdue{}{K_{cr}}{d_{cr}}{}{}{\dst}
  \scrivigiu{8}{3}{\footnotesize Ring}
  \flowrightsu{}
  \blokxyor{2}{2}{r_{r}}{r_{r}}{}{}
  \flowrightsu{}
  \bloivsuor{b_r}{}{\snt}
  \scrivigiu{10}{3}{}
  \flowleftsu{}
  \bloout{\tau_{r}}{\omega_{r}}
 }
  \centering
\begin{equation}\label{sistema_elastico_ridotto_invertito}
\begin{array}{@{}r@{\;}c@{\;}l@{}}
 \underbrace{\mat{@{}c@{}c@{}c@{}}{
  \frac{1}{K_{sc}} &  0 &  0 \\
  0 &  J_{p} &  0 \\
  0 &  0 &  \frac{1}{K_{cr}}}}_{\ds \tL_e}\!
 \underbrace{\mat{@{}c@{}}{\dot F_{sc} \\ \dot \omega_{p}\\ \dot F_{cr} }}_{\ds \dot\tx}\!
 &=&
 \underbrace{\mat{@{}c@{\;\,}c@{\;\,}c@{}}{
  0 &  r_{p} &  0 \\
  -r_{p} &  -r_{p}^{2} d_{sc} \!-\!b_{p}\!-\!r_{p}^{2} d_{cr} &  -r_{p} \\
  0 &  r_{p} &  0}}_{\ds-\tA_e}\!
 \underbrace{\mat{@{}c@{}}{F_{sc} \\ \omega_{p}\\  F_{cr} }}_{\ds\tx}\!
 +
 \underbrace{\mat{@{}ccc@{}}{
  r_{s} &  -r_{s} &  0 \\
  -r_{s}d_{sc}r_{p} &  r_{s}d_{sc}r_{p}\!-\!r_{r}d_{cr}r_{p}  &  r_{r}d_{cr}r_{p} \\
  0 &  r_{r} &  -r_{r}}}_{\ds\tB_e}\!
  \underbrace{\mat{@{}c@{}}{\omega_{s}\\ \omega_{c}\\ \omega_{r}}}_{\ds\tu}
 \\[13mm]
 \underbrace{\mat{@{}c@{}}{\tau_{s} \\ \tau_{c}\\ \tau_{r} }}_{\ds \ty}\!
  &=&
 \underbrace{\mat{@{}c@{\;\;}c@{\;\;}c@{}}{
  r_{s} &  r_{s}d_{sc}r_{p} &  0 \\
  -r_{s} &  -r_{s}d_{sc}r_{p}\!+\!r_{r}d_{cr}r_{p} &  r_{r} \\
  0 &  -r_{r}d_{cr}r_{p} &  -r_{r}}}_{\ds\tC_e}\!
 \underbrace{\mat{@{}c@{}}{F_{sc} \\ \omega_{p}\\  F_{cr} }}_{\ds\tx}\!
 +
 \underbrace{\mat{@{}c@{\;}c@{\;}c@{}}{
  b_{s}+r_{s}^2 d_{sc} &  -r_{s}^2 d_{sc} &  0 \\
  -r_{s}^2 d_{sc} &  b_{c}+r_{s}^2 d_{sc}+r_{r}^2 d_{cr}  &  -r_{r}^2 d_{cr} \\
  0 &  -r_{r}^2 d_{cr} &  b_{r}+r_{r}^2 d_{cr}}}_{\ds\tD_e}\!
  \underbrace{\mat{@{}c@{}}{\omega_{s}\\ \omega_{c}\\ \omega_{r}}}_{\ds\tu}
\end{array}
\end{equation}
  \caption{POG block diagram and POG state space equations of the reduced planetary gear when $J_s=J_c=J_r=0$ and the velocities are the inputs.}
  \label{POG_epicicloidale_elastico}
\end{figure*}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The last equation of system (\ref{tran_sys_pog_t_elastic}) shows the
following algebraic relation between the state and  input variables:
\[
  \A_{21}\,\z_{1}+\A_{22}\,\z_{2}+\B_2\,\u=0
\]
 Since
matrix $\A_{22}$ is invertible, vector $\z_{2}$ can be expressed as:
\[
 \z_{2}=-\A_{22}\muno \A_{21}\,\z_{1}-\A_{22}\muno\B_2\,\u
\]
Applying  the  following rectangular transformation:
 % $ \z = \T_{z}\,\z_{1}+ \T_{u}\,\u$:
\[
 \z
 =
 \underbrace{ \mat{@{}c@{}}{
 \I_{2} \\[1mm]
-\A_{22}\muno\,\A_{21}}
  }_{\ds \T_{z}} \,\z_{1}
 +
 \underbrace{
 \mat{@{}c@{}}{
 0 \\[1mm]
-\A_{22}\muno\,\B_{2}}
 }_{\ds \T_{u}}\,\u
\]
to system (\ref{tran_sys_pog_t_elastic}) (i.e.
$\L\,\dot\z=-\A\,\z+\B\,\u$ and $\y=\C\,\z+\D\,\u$), one obtains the
following reduced system (note that $\L\,\T_{u}=0$):
%%\[
%%\left\{
%%\begin{array}{@{}r@{\;}c@{\;}l@{}}
%% \T_{z}\tras\L\,\T_{z}\,\dot\z_{1}
%% &=&-\T_{z}\tras\A\,(\T_{z}\,\z_{1}+ \T_{u}\,\u)+\T_{z}\tras\B\,\u
%%  \\[1mm]
%%  \y&=&\C(\T_{z}\,\z_{1}+ \T_{u}\,\u)+\D\,\u
%%\end{array}
%%\right.
%%\]
\[
\left\{
\begin{array}{@{}r@{\;}c@{\;}l@{}}
 \T_{z}\tras\L\,\T_{z}\,\dot\z_{1}
 &=&-\T_{z}\tras\A\,\T_{z}\,\z_{1}+\T_{z}\tras(\B-\A\,\T_{u})\,\u
  \\[1mm]
  \y&=&\C\,\T_{z}\,\z_{1}+ (\D+\C\,\T_{u})\,\u
\end{array}
\right.
\]
that  in compact form is:
\begin{equation}
\label{modello_epiciclo_ridotto_e}
\L_{e}\,\dot\z_{1}=-\A_{e}\,\z_{1}+\B_{e}\,\u,
 \hspace{6mm}
 \y=\C_{e}\tras\z_{1}+\D_{e}\tras\u
\end{equation}
where matrices $\L_{e}$, $\A_{e}$, $\B_{e}$, $\C_{e}$ and $\D_{e}$
have the following structure:
\[
\begin{array}{@{}r@{\;}c@{\;}l@{}}
 \L_{e}
 &=&
  \T_{z}\tras\L\,\T_{z}=\L_{1},
 \\[2mm]
-\A_{e}
 &=&
 -\T_{z}\tras\A\,\T_{z} =
%% \\ &=&
%%  \mat{@{}c@{\;}c@{}}{
%% \I_{2} &
%%-\A_{21}\tras\A_{22}\mtras}
%% \mat{@{}c@{\;}c@{}}{
%% \A_{11} & \A_{12}  \\
%% \A_{21} & \A_{22}  }
%%\mat{@{}c@{}}{
%% \I_{2} \\[1mm]
%%-\A_{22}\muno\,\A_{21}}
%% \\ &=&
%%\mat{@{}c@{\;}c@{}}{
%% \I_{2} &
%%-\A_{21}\tras\A_{22}\mtras}
%% \mat{@{}c@{}}{
%% \A_{11} - \A_{12} \A_{22}\muno\,\A_{21} \\
%% 0 }
%% \\ &=&
\A_{11} - \A_{12} \,\A_{22}\muno\,\A_{21},
 \\[2mm]
%%\end{array}
%%\]
%%\[
%%\begin{array}{@{}r@{\;}c@{\;}l@{}}
\B_{e}
 &=&
 \T_{z}\tras(\B-\A\,\T_{u}) =
%% \\ &=&
%% \T_{z}\tras\left(
%% \mat{@{}c@{}}{ \B_{1} \\\B_{2}}
%% +
%% \mat{@{}c@{\;}c@{}}{
%% \A_{11} & \A_{12}  \\
%% \A_{21} & \A_{22}  }\,
%% \mat{@{}c@{}}{ 0 \\[1mm] -\A_{22}\muno\,\B_{2}}
%% \right)
%% \\ &=&
%% \T_{z}\tras\left(
%% \mat{@{}c@{}}{ \B_{1} \\ \B_{2}}
%% +
%% \mat{@{}c@{}}{
%%  -\A_{12}\,\A_{22}\muno\,\B_{2}  \\
%%  -\A_{22}\,\A_{22}\muno\,\B_{2}  }
%% \right)
%% \\ &=&
%%\mat{@{}c@{\;}c@{}}{
%% \I_{2} &
%%-\A_{21}\tras\A_{22}\mtras}
%% \mat{@{}c@{}}{ \B_{1} -\A_{12}\,\A_{22}\muno\,\B_{2}\\ 0}
%% \\ &=&
\B_{1} -\A_{12}\,\A_{22}\muno\,\B_{2},
 \\[2mm]
\C_{e}
 &=&
 \C\,\T_{z} =
%% \\ &=&
%% \mat{@{}c@{\;}c@{}}{ \C_{1} &\C_{2}}
%%\mat{@{}c@{}}{
%%\I_{2} \\[1mm]
%%-\A_{22}\muno\,\A_{21}}
%% \\ &=&
 \C_{1} - \C_{2}\,\A_{22}\muno\,\A_{21},
 \\[2mm]
\D_{e}
 &=&
 \D+\C\,\T_{u} =
%% \\ &=&
%% \mat{@{}c@{\;}c@{}}{ \\C_{1} &\C_{2}}
%%\mat{@{}c@{}}{
%%0 \\[1mm]
%%-\A_{22}\muno\,\B_{2}}
%% \\ &=&
 \D - \C_{2}\,\A_{22}\muno\,\B_{2}.
\end{array}
\]
Since matrix $\D_e$ is invertible,  system
(\ref{modello_epiciclo_ridotto_e}) can be input-output inverted
using the relations shown in Fig.~\ref{basesysgraphinverted}. The
state space equations $\tL_e\,\dot\tx=-\tA_e\,\tx+\tB_e\,\tu$ and
$\ty=\tC_e\,\tx+\tD_e\,\tu$ of the inverted-reduced system are shown
in (\ref{sistema_elastico_ridotto_invertito}) in the lower part of
Fig.~\ref{POG_epicicloidale_elastico} where:
\[
 \tL_e\!=\!\L_e,
 \hspace{2mm}
 \tA_e\!=\!\A_e+\B_e\D_e\muno\C_e,
 \hspace{2mm}
 \tB_e\!=\!\B_e\D_e\muno,
 \hspace{2mm}
 \tC_e\!=\!-\D_e\muno\C_e,
\]
\[
 \tx\!=\!\z_1,
 \hspace{2mm}
 \tu\!=\!\y\!=\![\omega_s, \omega_c, \omega_r]\tras,
 \hspace{2mm}
 \ty\!=\!\u\!=\![\tau_s, \tau_c, \tau_r]\tras,
 \hspace{2mm}
 \tD_e\!=\!\D_e\muno.
\]
A POG graphical representation of this inverted-reduced system is
shown in the upper part of Fig.~\ref{POG_epicicloidale_elastico}.
%
 %


\subsection{Reduced dissipative models}

A first dissipative static model of the planetary gear can be
obtained from the inertial model (\ref{modello_epiciclo_ridotto})
when $J_s=J_c=J_p=J_r=0$, that is when $\L_{r}=0$:
\begin{equation}
\label{modello_statico_ridotto_massivo}
 \x_{1}=\A_{r}\muno\,\B_{r}\,\u,
 \hspace{4mm}
 \rightarrow
 \hspace{4mm}
  \y
   =\D_{s1}\,\u
   =\B_{r}\tras\,\A_{r}\muno\,\B_{r}\,\u.
\end{equation}
Matrix $\D_{s1}$ is singular and therefore all the torque vectors
$\u=[\tau_s,\,\tau_c,\,\tau_r]\tras$ which belong to the kernel of
matrix $\B_{r}$, i.e. which are parallel to vector
$\k_1=[r_s,\,-r_r\!-\!r_s,\,r_r]\tras$,  do not influence the output
velocities $\y=[\omega_s,\,\omega_c,\,\omega_r]\tras$. Moreover, all
the velocities $\y$ obtained from
(\ref{modello_statico_ridotto_massivo}) are perpendicular to vector
$\k_1$, i.e. the vector $\y$ satisfies the relation:
\begin{equation}
 \label{k1perpendicoare}
 \k_1\tras\,\y=0
 \hspace{4mm}
 \leftrightarrow
 \hspace{4mm}
r_s\,\omega_s-(r_r\!+\!r_s)\,\omega_c+r_r\,\omega_r=0.
\end{equation}
A second dissipative static model can be obtained from the elastic
model (\ref{modello_epiciclo_ridotto_e}) when $K_{sc}=K_{sc}=\infty$
and $J_p=0$, that is when $\tL_{e}=0$:
\begin{equation}
\label{modello_statico_ridotto_elastico}
 \tA_{e}\,\tx=\,\tB_{e}\,\tu,
 \hspace{6mm}
% \rightarrow
 \hspace{6mm}
  \ty
%   =\D_{s2}\,\tu
   =\tC_{e}\,\tx+\tD_{e}\,\tu.
\end{equation}
In this case a solution in closed form does not exists because
matrix $\tA_{e}$ is singular.
 % $\tB_{e}$ and $\tC_{e}$ aresingular: $\ker(\tA_{e})=[-1,\,0,\,1]\tras$, $\ker(\tB_{e})=[1,\,1,\,1]\tras$ and $\ker(\tB_{e})=[r_p d_{sc},\,-1,\,r_p d_{sc}]\tras$.
One can easily verify that system
(\ref{modello_statico_ridotto_elastico}) has a solution only when
the input vector $\tu=[\omega_s,\,\omega_c,\,\omega_r]\tras$ is
perpendicular to vector $\k_1$, i.e. when
$\k_1\tras\,\tu=\k_1\tras\,\y=0$ as shown in
(\ref{k1perpendicoare}). So, the static dissipative model
(\ref{modello_statico_ridotto_elastico}) cannot be used in practice
because  the input vector $\tu$ is ``constrained''.




\section{POG modeling of an hybrid automotive system}
\label{Hybrid_Structure}

It is well known, see for example \cite{miller}, that the planetary
gear is a key element for modeling hybrid automotive power
structures.
\begin{figure}[t]
 \centering
 \psfrag{r}[c]{R}
 \psfrag{c}[c]{C}
 \psfrag{s}[c]{S}
 \psfrag{i}[l]{ICE}
 \psfrag{p}[c]{PMSM}
 \psfrag{k}[b]{$K_{2}$}
 \psfrag{h}[b]{$K_{1}$}
 \psfrag{a}[b]{\nrsec{5}}
 \psfrag{g}[b]{\nrsec{1}}
 \psfrag{d}[b]{\nrsec{2}}
 %\psfrag{b}[b]{\nrsec{3}}
 \psfrag{e}[c]{\scriptsize Control}
 \psfrag{f}[b]{\nrsec{3}}
 \includegraphics[width=7.5cm, angle=0]{epicicloidale_senza_K.eps}
 \caption{Scheme of the considered power structure of the vehicle}
 \label{schema_epicicloidale}
\end{figure}
 %
The hybrid structure shown in Fig.~\ref{schema_epicicloidale}
includes an internal combustion engine (ICE), a multi-phase
Permanent Magnet Synchronous Machine (PMSM) and the vehicle. The
planetary gear is the element connecting the two motors and the
driving wheels. The ICE is rigidly connected to the Carrier (C), the
PMSM is connected to the Sun (S) and the vehicle driving axle is
connected to the Ring (R). This hybrid system can be dynamically
described by the  ``high level'' POG block scheme shown in
Fig.~\ref{POG_hybrid}.
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \begin{figure}[t]
 \schemaPOG{(2,-3)(50,12)}{2.15mm}{
 \blosuretor{\mbox{\footnotesize Control}}{\scr \V_{e}}{\scr\I_{e}}
 \nrsezxgiu{0.65}{$\scr 1$}{-1.75}
 \scrivigiu{9}{2}{\scriptsize Synchronous machine}
 \bloxboxgiu{6}{\mbox{PMSM}}{}{}
 \put(-0.6,10.5){\makebox(0,0)[b]{$\scr \tau_{s}$}}
 \nrsezxgiu{0.65}{$\scr 2$}{-1.75}
 \scrivigiu{9}{2}{\scriptsize Planetary Gear}
 \put(2,10.5){\vector(0,1){1.5}} \put(6,12){\vector(0,-1){1.5}}
 \psline[linestyle=dashed, linewidth=0.2pt](1,11.25)(7,11.25)
 \put(2,12.5){\makebox(0,0)[c]{$\scr \tau_{c}$}}
 \put(6,12.5){\makebox(0,0)[c]{$\scr \omega_{c}$}}
 \bloxboxsu{6}{\begin{minipage}[l]{0.7cm} Plan. \\ Gear \end{minipage}}{\scr \omega_{s}}{\scr \omega_{r}}
 \nrsezxgiu{0.65}{$\scr 3$}{-1.75}
 \put(0.6,10.5){\makebox(0,0)[b]{$\scr \tau_{r}$}}
 \put(-0.9,0.25){\nrsezxsu{0.65}{$\scr 5$}{-0.2}}
 \scrivigiu{10}{2}{\scriptsize Vehicle}
 \bloxboxgiu{7}{\includegraphics[width=1.25cm]{icon_car.eps} }{}{}
 \nrsezxgiu{0.65}{$\scr 4$}{-1.75}
 \scrivigiu{7}{2}{\scriptsize Air Resistance}
 \blosuretao{\small \begin{minipage}[l]{0.7cm} Air \\ Res. \end{minipage}}{\scr F_{r}}{\scr \dot{x}_v}
 }
 \caption{\label{POG_hybrid} POG graphical representation of the considered hybrid vehicle.}
 \end{figure}
%
Note that the power sections \nrsec{1}-\nrsec{5} shown in
Fig.~\ref{POG_hybrid} correspond to the physical power sections
indicated  in Fig.~\ref{schema_epicicloidale}.
 The POG block ``Control'' of Fig.~\ref{POG_hybrid} represents the electric control
of the PMSM. The POG model of the synchronous machine present
between sections \nrsec{1} and \nrsec{2} is  given in
Fig.~\ref{POG_PMSM}, see \cite{ElectrIMACS_2008_Multi_phase} and
\cite{EPE}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\begin{figure}[t]
   \begin{center}
 \schemaPOG{(0,-4)(25,12)}{2.8mm}{
 \thicklines
 \bloin{\V_{e}}{\I_{e}}
 \nrsezgiu{$\scr 1$}{-1.75}
 \scrivigiu{10}{1.8}{ \footnotesize Electrical part}
 \blovigiumxm{\ds\L\muno}{}{\dst}{\frac{1}{s}}{ \tiny \boldsymbol{\Phi}}
 \bloivsuaor{\R}{}{}
 %\nrsezgiu{$\scr 2$}{-1.75}
 \scrivigiu{6}{1.8}{\footnotesize Energy \\ \footnotesize conversion}
 \blokxyaor{4}{3}{\ds \K_{\tau}(\theta)}
                 {\ds \K_{\tau}\tras(\theta)}{\E}{\tau_{m}}
 %\nrsezgiu{$\scr 3$}{-1.75}
 \scrivigiu{10}{1.8}{ \footnotesize Mechanical part}
 \blovisumxm{\ds\frac{1}{J_m}}{\omega_{m}}{\dst}{\frac{1}{s}}{}
 \bloivgiuor{b_m}{}{}
 \nrsezgiu{$\scr 2$}{-1.75}
 \bloout{}{\tau_{s}}
 \scrivigiu{}{1.8}{}
 }
  \end{center}
  \caption{POG scheme of the multi-phase synchronous motor (PMSM).}\label{POG_PMSM}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
The POG model of the Planetary Gear present between sections
\nrsec{2} and \nrsec{3} is the elastic model given in
Fig.~\ref{POG_epicicloidale_elastico}.
% The corresponding POG reduced model proposed in Sec.~\ref{Planetary_Gears} can also be used.
In \nrsec{3} the planetary gear is connected to  the driving shaft
of the vehicle. The dynamics of the vehicle is described by the POG
model present between sections \nrsec{3} and \nrsec{4}: the details
of this POG model can be found in \cite{2009_ELECTROMOTION}.
Finally,  the last POG block describes the resistance of the air
$F_{r}$ as a function of the vehicle velocity $\dot{x}_v$.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Simulation of the hybrid power structure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{Simulations}

The hybrid automotive system shown in
Fig.~\ref{schema_epicicloidale}  and Fig.~\ref{POG_hybrid} has been
implemented in Matlab/Simulink. The obtained Simulink block scheme
is  shown in Fig.~\ref{architetture_epi}.
 %
\begin{figure}[t]
 \centering
% \includegraphics[angle=-90,width=9cm]{architetture_HEV_disegno_riduttore_mdl.eps}
 \includegraphics[angle=-90,width=9cm]{architetture_HEV_disegno_senza_riduttore_mdl.eps}
 \caption{Simulink block scheme of the considered hybrid power structure.}
 \label{architetture_epi}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
All subsystems have been modeled with the POG technique and in
particular the vehicle in the right part of the scheme is a bicycle
model of a car that includes the tire-road elastic interaction, see
\cite{2009_ELECTROMOTION}.


%The main electrical and mechanical parameters used for the
%simulations presented in this Section are the following: 1)
%
%The simulation results presented in this Section have been obtained
%using the following electrical and mechanical parameters: $m=9$,
%$p=1$, $R=3\,\Omega$, $L_0=0.1$ H, $M_0=0.08$ H, $N_c=30$,
%$\varphi_{r}=0.02\mbox{ W}$, $J_r=0.5\mbox{ kg m}^2$, $b_r=1.8\mbox{
%N\,m\,s/rad}$ and  $\tau_{e}=0$ Nm. The odd harmonics
%$\{1,\,3,\,5,\,7\}$ of the rotor flux $\bar\phi(\theta)$ when $m=9$
%and $p=1$ are shown in Fig.~\ref{EPE_Figura_Optimal_Flux_Shape}. The
%motor phases are supposed to be star connected. The input vector
%$^{\omega}\V$ is given by control law (\ref{controllaw}) where
%$^{\omega}\I_{d}$ has been calculated using relation
%(\ref{tauyomega}) considering the desired torque $\tau_{d}=10 \mbox{
%Nm}$ for $t\in[0,\,1.5]$ s and $\tau_{d}=5 \mbox{ Nm}$ for
%$t\in[1.5,\,3]$ s. The elements $K_{s_{i}}$ of matrix $\K_{s}$, see
%eq.~(\ref{Ks}), are chosen in order to have the time constants
%$\tau_{i}= \{0.33 \mbox{s},\,0.25 \mbox{s},\,0.17 \mbox{s},\,0.09
%\mbox{s}\}$ for $i\in\{1,\,3,\,5,\,7\}$.

The ``start-and-stop'' simulation results reported in
Fig.~\ref{startstop_V_veicolo}$\div$\ref{startstop_Potenze} have
been obtained with the ICE switched off, i.e. $\omega_{c}=0$, and
controlling the multi-phase electric motor in order to force to the
vehicle the start-and-stop longitudinal movement described by the
trapezoidal velocity shape $\dot{x}_{d}$ shown  in
Fig.~\ref{startstop_V_veicolo} (the red dashed line). The velocity
$\dot{x}_{v}$ of the vehicle follows the desired velocity
$\dot{x}_{d}$ with a small delay because of the elastic horizontal
slipping of the tires on the ground. Main parameters of the vehicle:
$M_{v}=1272$ kg is the  mass of the vehicle; $J_w=3.459$ kg\,m$^2$
is the inertia of the wheels; $R_{w}=32.55$ cm is the radius of the
wheels; $K_t=360000$ N/m is the longitudinal stiffness of the tires.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
 \centering
 \psfrag{Velocita veicolo}[b][b][0.8]{Velocity of the vehicle $\dot x_v$}
 \psfrag{V [Km/ora]}[b][b][0.8]{$\dot x_v$ [Km/h]}
 \psfrag{Time [s]}[t][t][0.8]{Time [s]}
 \includegraphics[width=8.5cm]{architetture_HEV_startstop_V_veicolo.eps}
 \caption{Velocity of the vehicle $\dot x_v$ (blue, solid) compared with the desired velocity  $\dot x_d$ (red, dashed).}
 \label{startstop_V_veicolo}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[p]
 \centering
 \psfrag{Velocita epicicploidale}[b][b][0.8]{Velocities of the planetary gear: sun $\omega_s$,  carrier $\omega_c$ and ring  $\omega_r$}
 \psfrag{W [rpm]}[b][b][0.8]{$\omega_s$, $\omega_c$, $\omega_r$ [rpm]}
 \psfrag{Time [s]}[t][t][0.8]{Time [s]}
 \includegraphics[width=8.5cm]{architetture_HEV_startstop_Ws_Wc_Wr.eps}
 \caption{Velocities of the planetary gear: sun $\omega_s$ (red, solid),  carrier $\omega_c$ (blue, dashed) and ring  $\omega_r$ (black, dash-dotted).}
 \label{startstop_Ws_Wc_Wr}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
The angular velocities of the planetary gear are shown in
Fig.~\ref{startstop_Ws_Wc_Wr}: sun $\omega_s$ (red, solid), carrier
$\omega_c$ (blue, dashed) and ring $\omega_r$ (black, dash-dotted).
Parameters of the planetary gear: $r_{s}=10.2$ cm and $r_{r}=24.8$
cm are the radii; $K_{sc}=K_{cr}=10^7$ N/m and $d_{sc}=d_{cr}=3003$
N\,s/m are the stiffness and friction coefficients; $J_{p}=0.0812$
kg\,m$^2$ is the planet inertia;
$[b_{p},\,b_{s},\,b_{c},\,b_{r}]=[0.243,\,0.148,\,2.787,\,6.541]$
N\,m/rad are the linear friction coefficients.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[p]
 \centering
 \psfrag{Coppie epicicloidali}[b][b][0.8]{Torques of the planetary gear: sun $\tau_s$,  carrier $\tau_c$ and ring  $\tau_r$}
 \psfrag{Tau [Nm]}[b][b][0.8]{$\tau_s$, $\tau_c$, $\tau_r$ [Nm]}
 \psfrag{Time [s]}[t][t][0.8]{Time [s]}
 \includegraphics[width=8.5cm]{architetture_HEV_startstop_Ts_Tc_Tr.eps}
 \caption{Torques within the planetary gear: sun $\tau_s$ (red, solid),  carrier $\tau_c$ (blue, dashed) and ring  $\tau_r$ (black, dash-dotted).}
 \label{startstop_Ts_Tc_Tr}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
The torques acting on the  wheels of the planetary gear are shown in
Fig.~\ref{startstop_Ts_Tc_Tr}: sun $\tau_s$ (red, solid), carrier
$\tau_c$ (blue, dashed) and ring $\tau_r$ (black, dash-dotted).
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[p]
 \centering
 \psfrag{Tensioni di alimentazione}[b][b][0.8]{Voltages $\V_e$ of the multi-phase electric motor}
 \psfrag{Vt [V]}[b][b][0.7]{$\V_e$ [V]}
 %\psfrag{Time [s]}[t][t][0.8]{Time [s]}
 \psfrag{Time [s]}[t][t][0.8]{}
 \includegraphics[width=8.5cm]{architetture_HEV_startstop_Vt.eps}
 \psfrag{Correnti di alimentazione}[b][b][0.8]{Currents $\I_e$ of the multi-phase electric motor}
 \psfrag{It [A]}[b][b][0.7]{$\I_e$ [A]}
 \psfrag{Time [s]}[t][t][0.8]{Time [s]}
 \includegraphics[width=8.5cm]{architetture_HEV_startstop_It.eps}
% \caption{Phase voltages $\V_e$ of the multi-phase electric motor.}
 \caption{Phase voltages $\V_e$ and phase currents $\I_e$ of the multi-phase electric motor.}
 \label{startstop_Vt}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{figure}[t]
% \centering
% \psfrag{Correnti di alimentazione}[b][b][0.8]{Currents $\I_e$ of the multi-phase electric motor}
% \psfrag{It [A]}[b][b][0.7]{$\I_e$ [A]}
% \psfrag{Time [s]}[t][t][0.8]{Time [s]}
% \includegraphics[width=8.5cm]{architetture_HEV_startstop_It.eps}
% \caption{Phase currents  $\I_e$ of the multi-phase electric motor.}
% \label{startstop_It}
%\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
The voltages $\V_{e}$ and the currents $\I_{e}$ of the $m$-phases
electrical motor ($m=5$) are shown in Fig.~\ref{startstop_Vt}.
 % and Fig.~\ref{startstop_It}, respectively.
Parameters of the multi-phase electric motor: the shape of the rotor
flux is sinusoidal; the $m$ phases are star connected;
$R_{s}=1\;\Omega$ and $L_{s}=0.01$ H are the resistance and self
inductance coefficient of the stator phases; $M_{s0}=0.008$ H is the
maximum value of mutual inductance between stator phases;
$\varphi_{c}=6$ W is the maximum value of the rotor flux chained
with the stator phases; $J_m=0.529$ kg\,m$^2$ and  $b_m=0.08$
N\,m\,s/rad are the inertia and the linear friction coefficient of
the rotor.
%The electric motor is connected to the sun of the
%planetary gear by means of a reduction gear with $R_{id}=0.5$, see
%Fig.~\ref{architetture_epi}.
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[p]
 \centering
 \psfrag{Potenze}[b][b][0.8]{Power flows : electric  $P_1$, planetary  $P_2$, vehicle  $P_3$.}
 \psfrag{P [W]}[b][b][0.7]{$P_1$, $P_2$, $P_3$ [W]}
 \psfrag{Time [s]}[t][t][0.8]{Time [s]}
 \psfrag{Mot}[c][c][0.7]{Motor}
 \psfrag{Gen}[c][c][0.7]{Generator}
 \includegraphics[width=8.5cm]{architetture_HEV_startstop_Potenze.eps}
 \caption{Powers  entering the subsystems:  $P_1$ electric motor (red, plain), $P_2$ planetary gear (blue, dashed), $P_3$ vehicle  (black, dash-dotted).}
 \label{startstop_Potenze}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
The main powers flowing through the sections of the system are shown
in Fig.~\ref{startstop_Potenze}: $P_1$ (red, plain) is the power in
section \nrsec{1} flowing from the control unit towards the electric
motor; $P_2$ (blue, dashed) is the mechanical power in section
\nrsec{2} flowing from the electrical motor towards the planetary
gear; $P_3$ (black, dash-dotted) is the mechanical power in section
\nrsec{3} which flows towards the vehicle. When  $t\in[0,\,2]$ s the
vehicle accelerates and part of the power flowing towards the
vehicle dissipates within the electric motor (i.e. the difference
between the solid and dashed lines) and within the planetary gear
(i.e. the difference between the dashed and dash-dotted lines). When
$t\in[4,\,7]$ s the vehicle decelerates and a consistent amount of
mechanical power flows from the vehicle towards the electric motor:
in this phase the motor acts as a generator, see
Fig.~\ref{startstop_Potenze}.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusions}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In this paper a planetary gear has been modeled with different level
of details using the Power-Oriented Graphs (POG) technique. POG
congruent state space transformations have been used to obtain the
reduced models when the elasticities and/or the inertias of the
system go to zero. A full hybrid automotive power system
(endothermic engine, multi-phase synchronous motor, planetary gear
and vehicle dynamics) has been modeled using POG. The final
simulations results
% of the control an hybrid automotive system
clearly show that the POG technique is also suitable to easily,
clearly and precisely model complex dynamic systems.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\end{thebibliography}



\end{document}
