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authorTobias Klauser <tobias.klauser@uzh.ch>2013-04-20 18:16:14 +0200
committerTobias Klauser <tobias.klauser@uzh.ch>2013-04-20 18:16:14 +0200
commit772a929d30eb18130ab7338041037e91a0665edd (patch)
tree9b4e3e3a2a50e7ed616b9558a9c342890e3b3587 /doc
parent04d88711816f8ddd994b05e818919039402664c3 (diff)
Update report
Diffstat (limited to 'doc')
-rw-r--r--doc/.gitignore7
-rw-r--r--doc/bibliography_additional.bib23
-rw-r--r--doc/conclusion.tex4
-rw-r--r--doc/discussion.tex1
-rw-r--r--doc/images/screenshot_opensim_gait2354.pngbin123184 -> 1907170 bytes
-rw-r--r--doc/introduction.tex18
-rw-r--r--doc/masterproject_klauser.tex9
-rw-r--r--doc/methods.tex151
-rw-r--r--doc/results.tex5
-rw-r--r--doc/title.tex41
10 files changed, 207 insertions, 52 deletions
diff --git a/doc/.gitignore b/doc/.gitignore
index a55db0b..a1297dd 100644
--- a/doc/.gitignore
+++ b/doc/.gitignore
@@ -1,4 +1,11 @@
+.DS_Store
*.aux
+*.bbl
+*.blg
+*.lof
*.log
+*.lot
+*.out
+*.run.xml
*.synctex.gz
*.toc
diff --git a/doc/bibliography_additional.bib b/doc/bibliography_additional.bib
index e5b2e6d..5473e42 100644
--- a/doc/bibliography_additional.bib
+++ b/doc/bibliography_additional.bib
@@ -1,6 +1,21 @@
@book{Bernstein1967,
-author = {Bernstein, Nikolai A},
-title = {The co-ordination and regulation of movements},
-publisher = {Pergamon Press},
-year = {1967}
+ author = {Bernstein, Nikolai A.},
+ title = {The co-ordination and regulation of movements},
+ publisher = {Pergamon Press},
+ address = {Oxford},
+ year = {1967}
+}
+@book{Hill1970,
+ author = {Hill, A. V.},
+ title = {First and last experiments in muscle mechanics},
+ publisher = {Cambridge University Press},
+ address = {Cambridge},
+ year = {1970}
+}
+@inbook{Shadmehr2005,
+ author = {Shadmehr, R. and Wise, S. P.},
+ title = {A Simple Muscle Model},
+ booktitle = {Supplementary documents for ``Computational Neurobiology of Reaching and Pointing''.},
+ publisher = {MIT Press},
+ year = {2005}
} \ No newline at end of file
diff --git a/doc/conclusion.tex b/doc/conclusion.tex
index 9c43ca4..55d03d4 100644
--- a/doc/conclusion.tex
+++ b/doc/conclusion.tex
@@ -1,4 +1,5 @@
\chapter{Conclusion}
+\label{ch:conclusion}
\TODO{Conclusion about work done}
@@ -8,8 +9,9 @@ not feasible for the simulation of tasks such as those that were subject of this
\begin{itemize}
\item Precise EMG data from an actual subject (no averaged an processed patterns) would be needed in order to achieve realistic behavior in simulation with OpenSim.
\item Focus of OpenSim more on clinical applications where such data is available.
- \item A more abstract muscoloskeletal model of a human would need to be developed in OpenSim, for such simulations, but is outside of the focus of this project. The OpenSim API would provide a decent foundation for developing such a model from scratch.
+ \item A more abstract musculoskeletal model of a human would need to be developed in OpenSim, for such simulations, but is outside of the focus of this project. The OpenSim API would provide a decent foundation for developing such a model from scratch.
\item Ground contact forces would need to be incorporated (and provided by the study the project is based on).
+ \item Muscles were removed from the model ``arbitrarily'' (because no data is available for them), consequences for behavior of model unclear
\end{itemize}
More abstract models such as the ones used by Geyer and Herr \cite{Geyer2010}, Marques et al \cite{Marques2012a, Marques2012} might be more suitable. \ No newline at end of file
diff --git a/doc/discussion.tex b/doc/discussion.tex
index 149b5f0..6fb55ed 100644
--- a/doc/discussion.tex
+++ b/doc/discussion.tex
@@ -1,4 +1,5 @@
\chapter{Discussion}
+\label{ch:discussion}
Problems with data (missing ground contact forces), OpenSim more suited for analysis of precise data acquired in subject studies. Ground contact forces key to investigate walking.
diff --git a/doc/images/screenshot_opensim_gait2354.png b/doc/images/screenshot_opensim_gait2354.png
index a69a66d..ac08810 100644
--- a/doc/images/screenshot_opensim_gait2354.png
+++ b/doc/images/screenshot_opensim_gait2354.png
Binary files differ
diff --git a/doc/introduction.tex b/doc/introduction.tex
index bc78116..03dd594 100644
--- a/doc/introduction.tex
+++ b/doc/introduction.tex
@@ -1,9 +1,19 @@
\chapter{Introduction}
-Controlling and learning to control movements of a many degree of freedom system such as the human body is a non-trivial task. This problem is famously known as Bernstein's problem \cite{Bernstein1967}.
+Controlling and learning to control movements of a many degree of freedom system such as the human body is a non-trivial task. How does the central nervous system (CNS) choose from the infinitely many possibilities of how to achieve a certain movement? This problem is famously known as Bernstein's problem \cite{Bernstein1967}.
-Complex movements of humans (and other animals) are thought to develop based on a simple, rudimentary set of building blocks -- commonly called motor primitives -- during ontogenetic development \cite{Flash2005, Hart2010}. Contrary to the hitherto belief that these innate simple reflexes get completely replaced by more complex ones with age, a recent publication by Dominici et al. \cite{Dominici2011} shows that certain basic patterns of muscle activation are instead retained and are being built upon during development in order to achieve a variety of walking movement behaviours.
+Complex movements of humans (and other animals) are thought to develop based on a simple, rudimentary set of building blocks -- commonly called motor primitives -- during ontogenetic development \cite{Flash2005, Hart2010}. These primitives are defined by the pattern of simultaneous activation of different muscles and are then combined in order to achieve a certain movement.
-This project thesis attempts to investigate the role such locomotor primitives play in the development of human biped walking by building a muscoloskeletal simulation model in the OpenSim software \cite{Delp2007} and then applying the basic activation patterns at different stages of human development (neonate, toddlers, preschooler and adults) identified by Dominici et al. \cite{Dominici2011} to it.
+Contrary to the hitherto belief that these innate simple reflexes get completely replaced by more complex ones with age, a recent publication by Dominici et al. \cite{Dominici2011} shows that certain basic patterns of muscle activation are instead retained and are being built upon during development in order to achieve a variety of walking movement behaviors. They measured EMG activity of 24 muscles simultaneously in neonates\footnote{Even though neonates of course can't walk by themselves, stepping can be evoked when they're being held upright and by gently pushing the infant along the pathway. However this behavior typically disappears 4 to 6 weeks after birth.} ($\sim{}3$ days old), toddlers (11-14 months old), preschoolers (22-48 months old) and adults (25-40 years old). Additionally foot pressure was recorded as well as the limb kinematics.
-\TODO{Describe steps taken and goals of the individual steps.} \ No newline at end of file
+This project thesis attempts to investigate the role these locomotor primitives play in the development of human bipedal walking by building a musculoskeletal simulation model in the OpenSim software \cite{Delp2007} and then applying the basic activation patterns at different stages of human development (neonate, toddlers, preschooler and adults) identified by Dominici et al. \cite{Dominici2011} to it in order to achieve a walking behavior.
+
+The basic process of developing the necessary components and finally reproducing the walking behavior will consist of the following distinct steps:
+
+\begin{description}
+ \item[Single joint model] Implement a simple musculoskeletal model with one joint to get familiar with OpenSim. Actuate it using a reflex controller.
+ \item[Full two-legged model] Implement the full musculoskeletal model with two legs, fixed in space without ground forces. Actuate it using a simple reflex controller and pseudo-random data resembling the activation patterns from the study.
+ \item[Extract data and investigate walking] Extend the model to include the ground-contact forces accordingly. Extract the real activation pattern data from \cite{Dominici2011} and apply it to the model. Examine the walking behavior of the musculoskeletal model.
+\end{description}
+
+The remainder of this report is organized as follows: In chapter \ref{ch:methods}, the simulation software environment, the musculoskeletal model and the developed simulation controller are described. Furthermore the data preparation is explained. In chapter \ref{ch:results} the data produced and the simulation behaviors are presented. In chapter \ref{ch:discussion} the obtained results are discussed and the usability of the OpenSim environment for these kinds of experiments is evaluated. Finally in chapter \ref{ch:conclusion} some conclusions are drawn. \ No newline at end of file
diff --git a/doc/masterproject_klauser.tex b/doc/masterproject_klauser.tex
index 767e0cb..c69c952 100644
--- a/doc/masterproject_klauser.tex
+++ b/doc/masterproject_klauser.tex
@@ -9,8 +9,11 @@
\usepackage[latin1]{inputenc}
\usepackage[english]{babel}
+\usepackage{textcomp}
+\usepackage{longtable}
+
\usepackage{geometry}
-\geometry{top=40mm,bottom=50mm,textwidth=142mm}
+\geometry{top=45mm,bottom=50mm,textwidth=142mm}
\usepackage{hyperref}
\usepackage[pdftex]{graphicx}
@@ -62,8 +65,8 @@
%%%
\appendix
-\listoffigures
-\listoftables
+%\listoffigures
+%\listoftables
%\bibliographystyle{numeric-comp}
%\bibliography{bibliography}
\printbibliography
diff --git a/doc/methods.tex b/doc/methods.tex
index f732e9e..a08a982 100644
--- a/doc/methods.tex
+++ b/doc/methods.tex
@@ -1,48 +1,153 @@
\chapter{Methods}
+\label{ch:methods}
\section{Simulation Software}
-Simulation is performed using the OpenSim software platform \cite{Delp2007}. OpenSim is an open-source platform for modeling and analyzing neuromusculoskeletal structures and simulating their dynamic movement behavior. It's developed and maintained as part of the Simtk.org repository.
+Simulation is performed using the OpenSim\footnote{Not to be confused with the 3D virtual world software OpenSimulator, which is sometimes also abbreviated as OpenSim.} software \cite{Delp2007}. OpenSim is an open-source platform for modeling and analyzing neuromusculoskeletal structures and simulating their dynamic movement behavior. It's developed and maintained at the NIH Center for Biomedical Computation at Stanford University and is distributed as part of the Simtk.org repository.
-The software provides an extensive application programming interface (API) in order to develop custom simulation controllers. Additionally it contains a graphical user interface (GUI) used to visualize simulations, edit models and muscle excitation data, as well as to access predefined simulation controllers.
+\TODO{Describe working principle (controllers, models etc)}
-The provided controllers turned out to require a very specific data collection as input, which is typically generated during experiments using motion capturing systems and force plates.
+The software provides an extensive application programming interface (API) in order to develop custom simulation controllers. Additionally it contains a graphical user interface (GUI) used to visualize simulations, edit models and muscle excitation data, as well as to access predefined simulation controllers.
-In the context of this project, no such data is generated or used, but we rather solely rely on relatively sparse muscle activation data and have no ground force measurements available. Also we are interested in general principles of movement rather than the detailed analysis of the walking behaviour of an individual subject.
+The controllers provided with the OpenSim distribution turned out to require a very specific data collection as its input. These types of data are typically generated during experiments using motion capturing systems and force plates.
-Thus a custom controller making use of the provided OpenSIM API was developed. This allows to specifically tune the simulation behavior to the requirements of this project and the data available.
+In the context of this project, no such data was or is available or could easily be acquired. Instead we rather rely on relatively sparse muscle activation data and have no ground force measurements available. Also we are interested in general principles of movement rather than the detailed analysis of the walking behavior of an individual subject.
-For all tasks performed as part of this project, version 3.0 of OpenSim was used as provided on \url{http://www.simtk.org}.
+Thus a custom controller making use of the provided OpenSIM API was developed which should actuate the model based on the basic activation patterns. This approach also allows to specifically tune the simulation behavior to the requirements of this project and the data available.
-\section{Muscoloskeletal Model}
+For all tasks performed as part of this project, version 3.0 of OpenSim was used as provided on \url{http://www.simtk.org}. The source version was used and built accord
-The muscoloskeletal model used in the project was developed based on the Gait2354 model provided with OpenSim 3.0. This is a 23-degree-of-freedom model of the human muscoloskeletal system. By default it contains 54 musclotendon actuators and represents a subject with a height of 1.8 m and a mass of 75.16 kg\footnote{More information on the Gait2354 model and its kinematic and dynamic properties as well as references to the studies the model is based on can be found on \url{http://simtk-confluence.stanford.edu:8080/display/OpenSim/Gait+2392+and+2354+Models}}.
+\section{Musculoskeletal Model}
-Gait model provided with OpenSim (\TODO{Add reference to paper describing model}), muscles removes where no data was available from the paper. Muscles which were measured in combination in the study (hamstring, \dots) needed to be properly accounted for.
+\subsection{Simple Leg Model}
-\TODO{Add Screenshot}
+\TODO{Describe SimpleLeg}
-In order to verify the principal working of the OpenSim model, random data was generated and applied to it.
+\subsection{Two-legged Model}
-\section{Muscle Model}
+In the study by Dominici et al. \cite{Dominici2011} EMG activity of the following set of 14 muscles was recorded for each leg of the subject:
-For all muscles in the model, the Thelen muscle model \cite{Thelen2003} was used. This model is the default muscle model used OpenSim 3.0\footnote{\TODO{Add note about deprecated model of same name from previous versions of OpenSim}} and is employed in all muscloloskeletal models provided with OpenSim. The Thelen muscle model is based on the well-known and widely used Hill muscle-tendon model (\TODO{add reference}), but incorporates adjustments in order to better account for age-related changes in the muscle properties.
+\begin{itemize}
+ \item Tibialis Anterior (TA)
+ \item Gastrocnemius Lateralis (LG)
+ \item Gastrocnemius Medialis (MG)
+ \item Soleus (Sol)
+ \item Vastus Lateralis (VL)
+ \item Vastus Medialis (VM)
+ \item Rectus Femoris (RF)
+ \item Hamstring (HS) -- a group of four muscles, namely Semitendinosus, Semimembranosus, Biceps Femoris-Long Head and Biceps Femoris-Short Head
+ \item Adductor Longus (Add)
+ \item Tensor Fascia Latae (TFL)
+ \item Gluteus Maximus (GM)
+ \item Erector Spinae (ES)
+ \item External Oblique (OE)
+ \item Latissimus Dorsi (LD)
+\end{itemize}
-The behavior of the Thelen muscles can be tuned using various parameters, see table \ref{table:thelen2003_parameters}. These were left in their
+The musculoskeletal model to be used thus had to incorporate all these muscles, so the proper activation pattern could be directly applied. Several musculoskeletal models of human legs are already provided with OpenSim 3.0. For this project the Gait2354 model was used as a basis. This is a 23-degree-of-freedom model of the musculoskeletal system of the human leg. By default it contains 54 musculo-tendon actuators and represents a subject with a height of 1.8 m and a mass of 75.16 kg\footnote{More information on the Gait2354 model and its kinematic and dynamic properties as well as references to the studies the model is based on can be found on \url{http://simtk-confluence.stanford.edu:8080/display/OpenSim/Gait+2392+and+2354+Models}}. See figure \ref{fig:screenshot_opensim_gait2354} for a visualization of the model.
-\begin{table}[htb!]
-\label{table:thelen2003_parameters}
+\begin{figure}[htb!]
+\label{fig:screenshot_opensim_gait2354}
\centering
-\begin{tabular}{|c|c|c|c|}
-\hline
-\textbf{Parameter} & \textbf{Description} & \textbf{Value range} & \textbf{Chosen Value} \\
-\hline
-\end{tabular}
-\caption{Parameters of the Thelen muscle model \cite{Thelen2003}}
-\end{table}%
+\includegraphics[width=\textwidth]{images/screenshot_opensim_gait2354.png}
+\caption{Screenshot of the Gait2354 model visualized in the OpenSim 3.0 GUI}
+\end{figure}
+
+The model was then adapted in order to only in its final version incorporate only the 28 muscles for which data was available. Table \ref{table:muscles_correspondence} shows the muscles from the study and the Gait 2354 musculoskeletal model respectively and how their activation was determined from the study data in the final model. The table shows only the muscles for one leg, the muscles of second leg were handled correspondingly. The leg to which each muscle in the model belongs, is signified by it having a postfix \emph{\_l} for the left leg, or a postfix \emph{\_r} for the right leg.
+
+\begin{center}
+\begin{longtable}{llll}
+\label{table:muscles_correspondence}
+\\
+\textbf{Muscle} & \textbf{Name in model} & \textbf{Name in \cite{Dominici2011}} & \textbf{Activation Usage} \\
+[1ex] \hline \\ [-1.5ex]
+Adductor Longus & add\_long & Add & used directly \\
+[1ex] \hline \\ [-1.5ex]
+Erector Spinae & ercspn & ES & used directly \\
+[1ex] \hline \\ [-1.5ex]
+External Oblique & extobl & OE & used directly \\
+[1ex] \hline \\ [-1.5ex]
+Gastrocnemius Lateralis & lat\_gas & LG & used directly \\
+[1ex] \hline \\ [-1.5ex]
+Gastrocnemius Medialis & med\_gas & MG & used directly \\
+[1ex] \hline \\ [-1.5ex]
+Rectus Femoris & rect\_fem & RF & used directly \\
+[1ex] \hline \\ [-1.5ex]
+Soleus & soleus & Sol & used directly \\
+[1ex] \hline \\ [-1.5ex]
+Tensor Fasciae Latae & tfl & TFL & used directly \\
+[1ex] \hline \\ [-1.5ex]
+Tibialis Anterior & tib\_ant & TA & used directly \\
+[1ex] \hline \\ [-1.5ex]
+Vastus Lateralis & vas\_lat & VL & used directly \\
+[1ex] \hline \\ [-1.5ex]
+Vastus Medialis & vas\_med & VM & used directly \\
+[1ex] \hline \\ [-1.5ex]
+Biceps Femoris-Long Head & bifemlh\_r & part of HS & $\frac{1}{4}$ of HS activation \\
+[1ex] \hline \\ [-1.5ex]
+Biceps Femoris-Short Head & bifemsh\_r & part of HS & $\frac{1}{4}$ of HS activation \\
+[1ex] \hline \\ [-1.5ex]
+Semimembranosus & semimem & part of HS & $\frac{1}{4}$ of HS activation \\
+[1ex] \hline \\ [-1.5ex]
+Semitendinosus & semiten & part of HS & $\frac{1}{4}$ of HS activation \\
+[1ex] \hline \\ [-1.5ex]
+Gluteus Maximus 1 & glut\_max1 & part of GM & $\frac{1}{3}$ of GM activation \\
+[1ex] \hline \\ [-1.5ex]
+Gluteus Maximus 2 & glut\_max2 & part of GM & $\frac{1}{3}$ of GM activation \\
+[1ex] \hline \\ [-1.5ex]
+Gluteus Maximus 3 & glut\_max3 & part of GM & $\frac{1}{3}$ of GM activation \\
+[1ex] \hline \\ [-1.5ex]
+Adductor Magnus 2 & add\_mag2 & - & removed from model \\
+[1ex] \hline \\ [-1.5ex]
+Gluteus Medius 1 & glut\_med1 & - & removed from model \\
+[1ex] \hline \\ [-1.5ex]
+Gluteus Medius 2 & glut\_med2 & - & removed from model \\
+[1ex] \hline \\ [-1.5ex]
+Gluteus Medius 3 & glut\_med3 & - & removed from model \\
+[1ex] \hline \\ [-1.5ex]
+Gracilis & grac & - & removed from model \\
+[1ex] \hline \\ [-1.5ex]
+Iliacus & iliacus & - & removed from model \\
+[1ex] \hline \\ [-1.5ex]
+Inferio gemellus & gem & - & removed from model \\
+[1ex] \hline \\ [-1.5ex]
+Pectineus & pect & - & removed from model \\
+[1ex] \hline \\ [-1.5ex]
+Piriformis & peri & - & removed from model \\
+[1ex] \hline \\ [-1.5ex]
+Psoas Major & psoas & - & removed from model \\
+[1ex] \hline \\ [-1.5ex]
+Quadratus Femoris & quad\_fem & - & removed from model \\
+[1ex] \hline \\ [-1.5ex]
+Sartorius & sar & - & removed from model \\
+[1ex] \hline \\ [-1.5ex]
+Tibialis Posterior & tib\_post & - & removed from model \\
+[1ex] \hline \\ [-1.5ex]
+Vastus Intermedius & vas\_int & - & removed from model \\
+[1ex] \hline \\ [-1.5ex]
+Latissmus Dorsi & n/a & LD & no data available in \cite{Dominici2011} \\
+\caption{Muscles in the Gait2354 model and their correspondents in \cite{Dominici2011}}
+\end{longtable}
+\end{center}
+
+Gait model provided with OpenSim (\TODO{Add reference to paper describing model}), muscles removes where no data was available from the paper. Muscles which were measured in combination in the study (hamstring, \dots) needed to be properly accounted for.
+
+\section{Muscle Model}
+
+\subsection{Thelen Muscle Model}
+
+For all muscles in the model, the Thelen muscle model \cite{Thelen2003} was used. This model is the default muscle model used OpenSim 3.0\footnote{\TODO{Add note about deprecated model of same name from previous versions of OpenSim}} and is employed in all muscluloskeletal models provided with OpenSim. The Thelen muscle model is based on the well-known and widely used Hill-type muscle-tendon model \cite{Hill1922,Hill1970,Shadmehr2005}, but incorporates adjustments in order to better account for age-related changes in the muscle properties.
\section{Simulation Controller}
+\subsection{Simple Leg Controller}
+
+\TODO{Describe ideas behind it}
+
+\subsection{Locomotor Primitives Controller}
+
+\TODO{Describe ideas behind it}
+
\section{Data Preparation}
Since the original data from the publication by Dominici et al. \cite{Dominici2011} could not be used, the respective pattern curves had to be extracted from the paper. In order to gather the data from there, a graph digitizer software (GraphClick by Arizona Software) was used. The data was then preprocessed by applying a curve interpolation in order to allow arbitrary, linearly spaced sampling of data points later on in the OpenSim model controller. \ No newline at end of file
diff --git a/doc/results.tex b/doc/results.tex
index 6269112..191588c 100644
--- a/doc/results.tex
+++ b/doc/results.tex
@@ -1,11 +1,14 @@
\chapter{Results}
+\label{ch:results}
\section{Random Data}
-In order to validate the model and the developed controller, random data resembling muscle activation patterns was generated for all the 53 muscles of the model.
+In order to validate the principal working of the model and the developed controller, random data resembling muscle activation patterns was generated for all the 53 muscles of the model.
\TODO{Add example plot}
+Problems with random data: discontinuities, not fitted to muscle model and musculoskeletal model. Simulation time thus very high (or even unable to simulate). Smoothing of data needed, smaller activations. Works, but leads to unrealistic behavior.
+
\section{Locomotor Primitives Study Data}
\TODO{Add example plot}
diff --git a/doc/title.tex b/doc/title.tex
index cc556a5..966e7f9 100644
--- a/doc/title.tex
+++ b/doc/title.tex
@@ -44,27 +44,36 @@
% split the page into two columns
% left column
- \begin{minipage}[t]{0.48\textwidth}
- \begin{flushleft} \large
- \emph{Author:}\\
- Tobias Klauser\\
+% \begin{minipage}[t]{0.48\textwidth}
+% \begin{flushleft} \large
+% \emph{Author:}\\
+% Tobias Klauser\\
%Winterthur, Switzerland\\
- Registration Number: 05-535-927
- \end{flushleft}
- \end{minipage}
+% Registration Number: 05-535-927
+% \end{flushleft}
+% \end{minipage}
%
% right column
- \begin{minipage}[t]{0.48\textwidth}
- \begin{flushright} \large
- \emph{Supervised by:} \\
- Prof. Dr. Rolf Pfeifer\\
- Dr. Hugo Gravato Marques\\
+% \begin{minipage}[t]{0.48\textwidth}
+% \begin{flushright} \large
+% \emph{Supervised by:} \\
+% Prof. Dr. Rolf Pfeifer\\
+% Dr. Hugo Gravato Marques\\
% ensure vertical alignment with LHS
- \footnotesize{\ }
- \end{flushright}
- \end{minipage}
+% \footnotesize{\ }
+% \end{flushright}
+% \end{minipage}
- \vspace*{15ex}
+ Tobias Klauser \\
+ Registration Number: 05-535-927 \\
+
+ \vspace*{8ex}
+
+ \emph{Supervized by:} \\
+ Prof. Dr. Rolf Pfeifer \\
+ Dr. Hugo Gravato Marques \\
+
+ \vspace*{9ex}
%
\vfill % fill vertical space
%