%&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.
        These robots can be sorted in four different configurations
        Each configuration explained in the following sections.
        From the possible configurations, the optimal configuration that fits the specifications 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 a 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 good, 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 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 vertical acceleration becomes.
            \autoref{fig:cable_angle} illustrates the vertical 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.
            A bigger challenge is the construction of the system, as the vertical slider has to stay vertical during operation.
            Especially the length of this setup makes twisting of the vertical slider more likely.
            \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 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 some disadvantages.
            The range of the robot is defined by the length of the prismatic joint.
            However, if the prismatic joint is fully retracted, the joint does not get shorter.
            In that case the arm still 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 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.
            However, the arm can be configurated to not protrude outside of the board.
            If the situation requires the arm to protrude, it is still significantly less than the polar arm (\autoref{fig:polar_protrude}).
            Furthermore, depending on the configuration the of the arm the area where it protrudes can be significantly smaller.
            However, the additional joint and extra arm length does add to the moment of inertia and gravitational torque similar to the polar robot.
            The SCARA is therefore not a robot that is convenient with large working areas.
            However, it can be really quick and precise in relative small areas.
            \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, dimension, speed, scaling and the interesting dynamics.
            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.
            The dimension looks at the number of states that require control and is for all systems defined as 2.5D.
            The half dimension is the binary state for the marker on or off the board.
            Except for the cable bot, all configurations score sufficient on speed.
            The cable bot can be quick, but is limited in acceleration, and depends on the type of cable configuration.
            The last one, how interesting or challenging are the dynamics. 
            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.

            \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|}{Dimension}                                                      & 2.5      & 2.5       & 2.5   & 2.5   & 4.5      \\ \hline
                    \multicolumn{1}{|l|}{Speed}                                                          & -        & +         & +     & + +   & +        \\ \hline
                    \multicolumn{1}{|l|}{\begin{tabular}[c]{@{}l@{}}Interesting\\ dynamics\end{tabular}} & +        & - -       & +     & +     & + +      \\ \hline
                \end{tabular}
            \end{table}           
            Based on the dimension, all configurations fail to meet the required four state minimum.
            By combining two configurations, it is possible to meet the minimum of four states.
            To get the best system, I decided to combine a 'speed' and a 'range' configuration.
            This results in a system that has both properties.
            Combining anything with the cartesian configurations, creates just a moving base for the other configurations.
            Together with the trivial dynamics, this option is discarded.
            Suspending the SCARA of the polar configuration with cables creates very interesting dynamics, as moving the end-effector also influences the cables.
            From both options, the SCARA is quicker and scales better with range than the polar.
            Therefore, the SCARA is chosen above the polar configuration to be combined with the cable bot.
            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 in blue. 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 a initial design that can be used in the next steps.
            However, I noticed that none of the previous steps gave some implementation threshold.
            For the problem description and the specifications steps this was a minimum implementation level.
            This step was a optimal implementation level, the minimum was reached rather quick.
            But at what level of implementation needs this step to be concluded?
            A related question: Would a simple dynamic model of the initial design be a useful insight or a waste of time?
