%&tex
    \subsection{Initial Design}
    \label{sec:initialdesign}
        The initial design started with a design space exploration.
        The goal was to collect possible solutions and ideas for the implementation.
        The exploration resulted in a lot of whiteboard writing robots ideas.
        These robots are sorted in four different configurations.
        Each configuration explained in the following sections.
        From the possible configurations, the one that fits the requirements best, is made into an initial design.
        
        \subsubsection{Cable-Driven}
            The cable-driven robot is suspended with multiple cables.
            The end-effector that contains the marker is moved along a board by changing the length of the cables.
            The cable-based positioning system results in an end-effector with a large range and high velocities.
            A basic setup is shown in \autoref{fig:cablebotdrawing}.
            This given setup contains two cables that are motorized.
            The big advantage of this system is that it scales well, as the cables can have almost any length.
            \begin{figure}
                \centering
                \includegraphics[width=10.8cm]{graphics/cablebot.pdf}
                \caption{Planar view of cable-driven robot. This setup contains two motorized pulleys in both top corners. From these two cables a mass is suspended at position $x,y$.
                By changing the length of the cables, the mass is moved over along the whole board.}
                \label{fig:cablebotdrawing}
            \end{figure}
            \begin{marginfigure}
                \centering
                \includegraphics[width=3.74cm]{graphics/cable_angle.pdf}
                \caption{Illustrating the limit for pure horizontal acceleration $a$, for different angles to compensate for gravitational acceleration $g$.
                The red arrow represents the acceleration as a result of the pulling force of the cable, which is vectorized in a vertical acceleration that compensates $g$ and a vertical acceleration $a$.}
                \label{fig:cable_angle}
            \end{marginfigure}

            Although it is possible to achieve high velocities, this system is limited by the gravitational acceleration.
            In case of vertical acceleration, the maximum downward acceleration or upward deceleration is limited by \SI{9.81}{\meter\per\second\squared}.
            The horizontal acceleration depends on the relative angle of the suspending cable.
            The closer the end-effector is below the cable pulley, the lower the pure horizontal acceleration becomes.
            \autoref{fig:cable_angle} illustrates the horizontal acceleration for different angles.
            
            A possible solution to this is to add one or two additional wires to the system.
            These can pull on the system to 'assist' the gravitational force.
            Depending on the implementation, the extra cables make the system over-constrained.
            Nevertheless, the extra cables allow for higher acceleration limits in vertical and horizontal direction.

        \subsubsection{Cartesian-coordinate robot}
            This configuration is a very common design for plotters and shown in \autoref{fig:plotter}.
            This setup is also known as a gantry robot or linear robot.
            It normally consists of two sliders, which behave as a prismatic joint. 
            Because each slider covers a single X or Y axis, the control and dynamics of this system are rather simple.
            The biggest challenge is in the construction of the system, especially when the size of the system is increased.
            The larger system requires longer sliders, which are expensive. 
            Another difficulty is the actuation of both horizontal sliders, if these sliders do not operate synchronous the vertical slider would slant and likely jam.
            \begin{figure}
                \centering
                \includegraphics[width=8.74cm]{graphics/plotter.pdf}
                \caption{This Cartesian plotter consists of two horizontal sliders to provide the $x$-movement and one vertical slider to provide the $y$-movement.}
                \label{fig:plotter}
            \end{figure}
            
        \subsubsection{Polar-coordinate robot}
            This robot is a combination of a prismatic and a revolute joint.
            Where the revolute joint can rotate the prismatic joint as shown in \autoref{fig:polar}.
            With this it can reach any point within a radius from the rotational joint.
            This is a little more complex design than the Cartesian robot.
            \begin{figure}
                \centering
                \includegraphics[width=8.74cm]{graphics/polar.pdf}
                \caption{A combination of a revolute joint and a prismatic joint, creating a polar-coordinate robot.}
                \label{fig:polar}
            \end{figure}
            \begin{marginfigure}
                \centering
                \includegraphics[width=3.74cm]{graphics/polar_protrude.pdf}
                \caption{The diagonal lined section shows the part of the protruding area that is used by the arm.}
                \label{fig:polar_protrude}
            \end{marginfigure}

            This robot has multiple disadvantages.
            The range of the robot is defined by the length of the prismatic joint.
            Thus when the operating range is doubled, the robot size has to be doubled or even more than that.
            Furthermore, when the arm of the robot is retracted, it protrudes on the other side.
            Therefore, the complete radius around the revolute joint cannot have any obstacles.
            \autoref{fig:polar_protrude} gives an impression of the required area.
            Even with this area, the arm cannot reach the complete board.
            This makes the required space of the setup very inefficient.
            Another disadvantage is that a long arm increases the moment of inertia and the gravitational torque on the joint quadratically.
            Furthermore, the long arm introduces stiffness problems and it amplifies any inaccuracy in the joint.
        
        \subsubsection{SCARA}
            The \ac{scara} robot is a configuration with two linkages that are connected via rotational joints.
            It compares to a human arm drawing on a table as shown in \autoref{fig:scara}.
            Similar to the polar robot it can reach all points within a radius from the base of the robot.
            But the \ac{scara} does not protrude like the polar arm (\autoref{fig:polar_protrude}).
            Depending on the configuration of the arm, it is possible to keep the arm completely within the area of operation.
            A downside is that the mass of the additional joint and extra arm length increase the moment of inertia and gravitational torque similar to the polar robot.
            This makes the \ac{scara} configuration convenient for small working areas as that keeps the forces manageable.
            Additionally, as the arms of the \ac{scara} have a fixed length, it is possible to create a counter balance.
            This can be used to remove any gravitational torque from the system. It would however increase the moment of inertia even further.
            For current requirements, the working area is too large for any practical application of the \ac{scara}.
            \begin{figure}
                \centering
                \includegraphics[width=8.74cm]{graphics/scara.pdf}
                \caption{Schematic example of a \ac{scara}, consisting of two rotation linkages. This setup can be compared to a human arm, where the gray base above the whiteboard represents the shoulder and the connections between both linkages the elbow.}
                \label{fig:scara}
            \end{figure}

        \subsubsection{Combining}
            A fifth option is to combine two of the discussed configurations, wherein the best properties of two configurations are used.
            The most interesting combination is the cable bot together with the \ac{scara}.
            In this combination, the \ac{scara} is small, only able to write a couple of characters.
            The smaller size of the \ac{scara} makes it quick.
            To write full sentences the \ac{scara} is placed on a carriage that is suspended by the cable bot.
            An example of this \ac{cdc} with the mounted \ac{scara} is shown in \autoref{fig:combined}.
            
            \begin{figure}[h]
                \centering
                \includegraphics[width=10.8cm]{graphics/combined.pdf}
                \caption{Combined system that integrates the cable bot together with the \ac{scara}. The \ac{scara} in red is mounted on the \ac{cdc}.}
                \label{fig:combined}
            \end{figure}

            This increases the complexity of the dynamics of the system, by having four degrees of freedom.
            Furthermore, the movement of the \ac{scara} also causes movement of the \ac{cdc}.
            Shrinking the \ac{scara} also decreases the challenges regarding construction, as long and unstable arms are out of the picture.

        \subsubsection{Choice of system}
            The previous sections have shown four different configurations.
            These configurations are compared in \autoref{tab:initial_design}.
            Each of the systems are scored on range, speed, cost, obstruction, effective area, and the interesting dynamics:
            \begin{description}
                \item{\emph{Range}}\\
                    The range scores the system on the practical dimension of the system, larger is better.
                    The cable, cartesian, and combined configuration scale very well, the cables or slider rails can be made longer without real difficulty.
                    The \ac{scara} or polar configuration run into problems with the arm lengths, as forces scale quadratically with their length.
                \item{\emph{Speed}}\\
                    Except for the cable bot, all configurations score sufficient on speed.
                    The cable bot can reach high velocities, but the acceleration is limited, depending on the configuration, to the gravitational acceleration.
                \item{\emph{Cost}}\\
                    For the cost, all systems fit within the €200 budget, except for the Cartesian setup.
                    All systems require DC or stepper motors, but the cartesian setup also requires linear sliders which are expensive, especially for longer distances.
                \item{\emph{Obstruction}}\\
                    The obstruction score depends on the capability of the system to move away from the text on the board, such that the system does not obstruct the written tweet.
                    All systems except for the cable and combined configuration can move themself outside of the working area.
                    It is possible that the wires of the cable or combined configuration obstruct the view.
                    However, the wires are expected to be thin enough to not block any text.
                \item{\emph{Scalability}}\\
                    For the scalability, the cable bot and the combined system score high. 
                    The cables make it possible to easily change the operating range of the system, only requiring reconfiguration.
                    The cartesian system scales poor because the length of the sliders is fixed, and longer sliders are expensive.
                    For the polar system and \ac{scara}, the forces on the joints scale quadratically with the length of the arms.
                    However, the \ac{scara} can be build with counter balance making it scale less worse than the Polar system.
                \item{\emph{Effective Area}}\\
                    With the effective area, the system is scored on the area it requires to operated versus the writable area.
                    The polar configuration has a low score due to the protruding arm.
                \item{\emph{Interesting Dynamics}}\\
                    The last metric, scores the system on the complexity of the dynamics.
                    This is a more subjective metric, but also a very important one.
                    In the problem description, the complexity of the dynamics was determined as one of the core requirements.
                    The cartesian configuration is trivial, both sliders operate completely separate from each other and the position coordinates can be mapped one to one with the sliders.
                    The combined configuration excels for this metric, as it has 4 degrees of freedom and the \ac{scara} movement can cause the carriage to swing.
            \end{description}

            \begin{table}[h]
                \caption{Table with comparison of the four proposed configurations and a combined configuration of the cable bot and the \ac{scara}.}
                \label{tab:initial_design}
                \rowcolors{2}{lightgray}{white!100}
                \begin{tabular}{l c c c c c }
                    \toprule
                    & Cable bot & Cartesian & Polar & \ac{scara} & Combined \\
                    \midrule
                    \multicolumn{1}{l}{Range}                                                          & $+ +$      & $+  $       & $- -$   & $-  $   & $+ +$      \\ 
                    \multicolumn{1}{l}{Speed}                                                          & $-  $      & $+  $       & $+  $   & $+ +$   & $+  $      \\ 
                    \multicolumn{1}{l}{Cost}                                                           & $+ +$      & $- -$       & $+  $   & $+  $   & $+  $      \\ 
                    \multicolumn{1}{l}{Obstruction}                                                    & $-  $      & $+  $       & $+  $   & $+  $   & $-  $      \\ 
                    \multicolumn{1}{l}{Scalability}                                                    & $+ +$      & $-  $       & $- -$   & $-  $   & $+  $      \\ 
                    \multicolumn{1}{l}{\begin{tabular}[c]{@{}l@{}}Effective\\ area\end{tabular}}       & $+ +$      & $+  $       & $- -$   & $+  $   & $+ +$      \\ 
                    \multicolumn{1}{l}{\begin{tabular}[c]{@{}l@{}}Interesting\\ dynamics\end{tabular}} & $-  $      & $- -$       & $-  $   & $+  $   & $+ +$      \\ 
                    \midrule
                    \hiderowcolors
                    \multicolumn{1}{l}{Total} & \multicolumn{1}{r}{$ +5$} & \multicolumn{1}{r}{$ -1$} & \multicolumn{1}{r}{$ -4$} & \multicolumn{1}{r}{$ +4$} & \multicolumn{1}{r}{$ +8$} \\
                    \bottomrule
                \end{tabular}
            \end{table}           

            The comparison in \autoref{tab:initial_design} shows that the combined configuration as preferred.
            Which is not surprising as it combines the advantages of both the cable bot and \ac{scara}.
            Although those systems have a good score of their own, they have disadvantages.
            The cable bot has low acceleration and no challenging dynamics.
            The main difficulty for the \ac{scara} is being able to build it large enough.

            The combined configurations, complement each other.
            The range of the \ac{cdc} allows for a small \ac{scara}.
            The small size of the \ac{scara} makes it quick.
            This compensates for the low acceleration of the cable bot and removes the need for a \ac{scara} with long arms.
            Therefore, the choice of configuration is the combined system of the \ac{scara} and \ac{cdc}.
            
        \subsubsection{Evaluation}
            This was the first step that felt really productive in the design process.
            It created a enormous amount of information and insight of the design.
            In hind sight, it would have been useful to have this information during the requirements step.
            However, as the requirements step are mainly on the "what" to solve, and specifically not on "how" to solve it, this information was avoided on purpose during the requirements step.

            This step did result in an initial design that is used in the next steps.
            However, I noticed that none of the previous steps have a clear start or end.
            For the problem description and the requirements steps the question is when all required information is collected.
            In the initial design it is always possible continue researching design options to come up with an even better design.
            Especially with complex system, it is unrealistic to create complete requirements before making design decisions.
            Resulting in the question: at what point do we have enough information and must we move to the next design step?
            This is also known as the \emph{requirement versus design paradox} \autocite{fitzgerald_collaborative_2014}.
