%&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 can be sorted in four different configurations
        Each configuration explained in the following sections.
        From the possible configurations, the one that fits the specifications 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 systems result in an end-effector with a large range and high velocities.
            A basic setup can be seen 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 can be 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 bigger length sliders, which are expensive. 
            Another difficulty is the actuation of both horizontal sliders, if these sliders do not operate synchronous, the vertical slider rotates.
            However, the construction of the slider is not able to rotate, resulting in damage to the system.
            \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 seen 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 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 SCARA robot is a configuration with two linkages that are connected via rotational joints.
            It can be compared to a human arm drawing on a table as seen 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 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 SCARA configuration convenient for small working areas as that keeps the forces managable.
            Additionally, as the arms of the 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 specifications, the working area is too large for any practical application of the SCARA.
            \begin{figure}
                \centering
                \includegraphics[width=8.74cm]{graphics/scara.pdf}
                \caption{Schematic example of a 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{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 and cartesian configuration scale very well, the cables or slider rails can be made longer without real difficulty.
                    The 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 bot can move themself outside of the working area.
                    It is possible that the cables of the cable bot obstruct the view.
                    However, the wires are expected to be thin enough to not block any text.
                \item{\emph{Scalability}}\\
                    For the scalability, only the cable bot scores 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 SCARA, the forces on the joints scale quadratically with the length of the arms.
                    However, the 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.
                \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.
                    For the other configuration, some inverse kinematics are required to get from desired position to the control angles of the system.
            \end{description}

            \begin{table}[]
                \caption{Table with comparison of the four proposed configurations and a combined configuration of the cable bot and the SCARA.}
                \label{tab:initial_design}
                \begin{tabular}{l|l|l|l|l|l|}
                    \cline{2-6}
                    & Cable bot & Cartesian & Polar & SCARA & Combined \\ \hline
                    \multicolumn{1}{|l|}{Range}                                                          & + +      & +         & - -   & -     & + +      \\ \hline
                    \multicolumn{1}{|l|}{Speed}                                                          & -        & +         & +     & + +   & +        \\ \hline
                    \multicolumn{1}{|l|}{Cost}                                                           & + +      & - -       & +     & +     & +        \\ \hline
                    \multicolumn{1}{|l|}{Obstruction}                                                    & -        & +         & +     & +     & -        \\ \hline
                    \multicolumn{1}{|l|}{Scalability}                                                    & + +      & -         & - -   & -     & +        \\ \hline
                    \multicolumn{1}{|l|}{\begin{tabular}[c]{@{}l@{}}Effective\\ area\end{tabular}}       & + +      & +         & - -   & +     & + +      \\ \hline
                    \multicolumn{1}{|l|}{\begin{tabular}[c]{@{}l@{}}Interesting\\ dynamics\end{tabular}} & -        & - -       & -     & +     & + +      \\ \hline
                \end{tabular}
            \end{table}           
            
            Based on this comparison, I disqualified the cartesian and polar system.
            The cartesian has no interesting dynamics and is expensive to build at the current scale.
            The polar system is just not feasible, the arm length required to cover the writing area results forces that are too large.
            Making a rotational joint that delivers the torque and velocity required for such an arm, is just out of the scope of this case study.
            The two remaining configurations come with serious downsides as well.
            The cable bot is slow, and the arm length for the SCARA is also likely to cause problems.
            However, by combining both, it is possible to get a system that fits the requirements very well.
            By building a small SCARA that is the suspended by the cable bot, it combines the best of both worlds.
            The small SCARA is quick and accurate, while the cable bot gives the system an enormous range.
            Resulting in a system that scores high on all criteria except obstruction.
            The grading for the combined system is shown in the most right column in \autoref{tab:initial_design}.

            \begin{figure}
                \centering
                \includegraphics[width=10.8cm]{graphics/combined.pdf}
                \caption{Combined system that integrates the cable bot together with the SCARA. The SCARA in red is mounted on the carriage. This carriage is then suspended by cables.}
                \label{fig:combined}
            \end{figure}
            In the combined system, the SCARA will only be large enough to write a small number of characters at the time.
            This will alternate with the cable bot moving the base of the SCARA to the next position, so that it can write the next set of characters on the whiteboard.
            \autoref{fig:combined} shows a simple view of the system.

        \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 specifications step.
            However, as the specifications step are mainly on the "what" to solve, and specifically not on "how" to solve it, this information was avoided on purpose during the specifications step.

            This step did result in an initial design that can be 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 specification 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 specifications before making design decissions.
            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}.
