# Workspace Analysis of a Mobile Manipulator with Obstacle Avoidance in 3D Printing Tasks
## 1 Introduction
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{1}{c}{\begin{tabular}{c} **Industrial** \\ **Prolotypes** \\ \end{tabular} } & \multicolumn{2}{c}{**Optimal Workspace Analysis**} & **References** \\ \hline \multirow{4}{*}{\begin{tabular}{c} Parallel Robot \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} Delta 3 DOF \\ and \end{tabular} } & The algorithm of Cylindrical Algebraic Decomposition is studied to obtain the optimal & \multirow{2}{*}{[26]} \\  & workspace of the industrial manipulator. The SIROPA library represents the complexity & & \\  & and singularity of the workspace. & & \\ \hline \multirow{2}{*}{\
\begin{tabular}{c} Parallel Robot \\ \end{tabular} } & Delta 5 DOF & Genetic algorithms are analyzed to find the optimal workspace. & [27] \\ \cline{2-4}  & Delta 10 DOF & The Monte Carlo method is used along with a Gaussian distribution to evaluate the limits & \multirow{2}{*}{[28]} \\  & of the optimal workspace. & & \\ \hline \multirow{2}{*}{
\begin{tabular}{c} FANUC-Delta \\ \end{tabular} } & The SolidWorks software determines the limits of the robot arm joints. The dynamic & \multirow{2}{*}{[29]} \\  & simulation function accurately describes the motion of the robot platform in an & & \\ \multicolumn{2}{c}{optimal workspace.} & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: \(Cont.\)  
Figure 1: Dynamic environment: construction areas; (**a**) formwork system, (**b**) distribution of a formwork system, (**c,d**) common cracks in construction environments. The images are courtesy of the Department of Design and Theory of Architecture of the University of Bio Bio. context, the parameters of the mobile manipulator workspace study its shape, scale and position within a construction site with obstacles. The printing workspace of the mobile manipulator for 3D printing profiles with variable heights was studied, considering fixed obstacles and avoiding possible collisions between the mobile platform with the robotic arm and the end-effector with the printing profiles. For the remainder of this work, we will refer to the printing profile as building elements. This work focuses on studying the printing workspace of a mobile manipulator in construction sites, emphasising the effect produced by detecting obstacles in the robot's workspace during printing tasks. To that end, we implement a linear algebra controller to guarantee 3D printing and use an obstacle avoidance method widely studied in robotics [33; 34]. Furthermore, we aim to provide an analysis strategy from the viewpoint of robotic technology and suggest a suitable methodology to study the printing workspace of a robotic platform in building construction. This paper is organized as follows: Section 2 shows the mathematical derivation of the mobile manipulator model, the strategy followed to ensure 3D printing of building elements and the implemented method for obstacle avoidance. Section 3 presents the results of the workspace analysis of a 6 DOF mobile manipulator. Section 4 shows a discussion on the three-dimensional interpretation of the mobile manipulator's workspace in construction sites. Finally, conclusions are shown in Section 5.