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  \title{\huge Power Oriented Graphs for modeling planetary gear}
  \author{{\bf R.~Zanasi, F.~Grossi}\\[2mm]
   DII, University of Modena and Reggio Emilia, Modena, Italy, ({\small federica.grossi@unimore.it, roberto.zanasi@unimore.it})  
  \vspace{-1mm}}

\begin{document}

 \maketitle \thispagestyle{empty} \pagestyle{empty}

\begin{abstract}

In this paper a vehicle is described using two different energy-based
graphical techniques: Power-Oriented Graphs (POG) and Energetic
Macroscopic Representation (EMR). The main features these techniques are presented and an integration between them is proposed: they can be used together to
model different parts of the same system.
%The POG technique is
%used to model the interaction between tire and road in order to
%have a dynamic energetic model, while the EMR technique is used to
%represent the whole system and deduce the control structure
%through the inversion-based methodology.
Using both
techniques the whole vehicle can be represented and a structure to
control the vehicle velocity is given.

\end{abstract}

\section{Introduction}

Nowadays, a lot of modelling methods are available to describe
electromechanical systems such as Bond Graphs
\cite{Paynter}-\cite{karnopp}, Causal Ordering Graph \cite{ref5}
Power-Oriented Graphs \cite{p.o.g.-lille}-\cite{bodypap93} and
Energetic Macroscopic Representation \cite{EMR_Subway}. All these
techniques use power interaction between subsystems as the basic
concept for modelling. In this paper Power-Oriented Graphs (POG)
and Energetic Macroscopic Representation (EMR) are exploited and
merged: they can be considered as complementary tools and can be
exploited depending on the desired goal.

 The main target of the POG modelling
technique is the analysis of physical systems, it allows to write the system equations
in the state space form, it is suitable for education as it is
easily comprehensible. Its main drawbacks are the linear topology
structure of the scheme and the absence of a methodology to build
the control structure.

The EMR technique gives a methodology to build a control
architecture starting from the model of the system. The
representation is planar, it clearly shows the coupling among
elements and the energy flux through the system. The structure is
easy to be read, however it does not show mathematical details of
the model.

The aim of this paper is to give a model of a vehicle involving
the tire-road contact law and implement a velocity control by using an integration of both modeling techniques:
the POG technique is used to represent an energetic
model of the tire-road interaction with a detailed mathematical
description, while the EMR technique is used to represent the
whole system, including the traction and the control structure.
%Both techniques are merged as the EMR is a macroscopic
%description, while through POG a detailed mathematical model of
%the EMR blocks inner structure can be given.

%\noindent In section \ref{studied_system} the studied system is
%presented. In section \ref{section_POG} the basic principles of
%POG are explained and the POG representation of the system is
%given. The main features of EMR and the EMR of the considered
%system are shown in section \ref{section_EMR}. Section
%\ref{section_control} deals with the global control structure, in
%section \ref{simulations} some simulation results are given.
%Finally a comparison between POG and EMR techniques is given and
%some conclusions are drawn in sections \ref{section_comparison}
%and \ref{section_conclusion}.

% One unique
%tool cannot afford all aspects of modelling physical systems.
\section{The studied system}\label{studied_system}



\section{Representation using Power-Oriented Graphs}\label{section_POG}

\subsection{Power-Oriented Graphs basic
principles}\label{POGproperties}

The ``Power-Oriented Graphs'', are ``signal flow graphs'' combined
with a particular ``modular'' structure which essentially uses
only two blocks. The basic
characteristic of this modular structure is the direct
correspondence between pairs of system variables and real power
flows: the product of the two variables involved in each dashed
line of the graph has the physical meaning of ``power flowing
through the section''.
% \begin{figure}[htbp]
% \centering
% \schemaPOG{(0,-1)(16,11)}{3mm}{
% \thicklines
% \bloin{\x_{1}}{\y}
% \blovigiu{\ds \G(s)}{}{\dst}
% \bloout{\x_{2}}{\y}
% \hspace{2cm}
% \bloin{\x_{1}}{\y}
% % \bloko{$\K$}{$\K$\tras}{}{}
% \blokxyor{3}{3}{$\K$}{$\K$\tras}{}{}
% \bloout{\x_{2}}{\y}
% }
%\caption{\label{figflow} Basic blocks: elaboration block (e.b.)
%and connection block (c.b.).}
% \end{figure}
 The two basic blocks
 %shown in Fig.~\ref{figflow}
 are named
``elaboration block'' (e.b.) and ``connection block'' (c.b.).
 %The circle present in the e.b. is a summation element and the black
%spot represents a minus sign that multiplies the entering
%variable. There is no restriction on variables $\x$ and $\y$ other
%than the fact that their inner product
%$\langle\x,\y\rangle=\x\tras\y$ must have the physical meaning of
%a ``power''. In this way the POG schemes always show a direct
%correspondence between pairs of system variables and real power
%flows. 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 square, $\K$ can also be
%rectangular.
While the elaboration block can store and dissipate
energy (i.e. springs, masses and dampers), the connection block
can only ``transform'' the energy, that is, transform the system
variables from one type of energy-field to another (i.e. any type
of gear reduction).

\subsection{POG representation of the system}









\section{Simulation}\label{simulations}

In order to simulate the model the scheme of the vehicle
has been implemented in
Matlab-Simulink$^{TM}$.
%(Fig.~\ref{Auto_REM_SMC_figure_mdl}).
%%
%\begin{figure}[h!]
% \centering
% ~\hspace{-4mm} \includegraphics[angle=-90, width=9.0cm]{Auto_REM_SMC_figure_mdl.eps}
%  \caption{Matlab-Simulink$^{TM}$ model of the system with the control structure in the EMR formalism.}
%  \label{Auto_REM_SMC_figure_mdl}
%\end{figure}
%%

\section{Conclusion}\label{section_conclusion}
 A vehicle has been modelled considering the interaction
of the tire with the road using POG and EMR
 techniques.
 A Maximum Control Structure has been given in order to control the
 vehicle longitudinal speed. The whole model has been implemented
 in Matlab-Simulink$^{TM}$ and some simulation results have been
 provided. In particular a comparison between POG and EMR
techniques is given to underline the possibility of cooperation
between the two techniques in order to merge their different
features for the desired purposes.



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\end{thebibliography}





\end{document}
