# Using a Cable-Driven Parallel Robot with Applications in 3D Concrete Printing
## 3 Analysis of the Workspace of the CDPR  
The workspace of parallel cable-driven mechanisms depends on various factors, such as the dimension of the base frame and moving platform, payload, cable configuration, and cable characteristics; on the other hand, the condition of the cable tension forces is non-negative, causing many difficulties and challenges for defining the workspace. This problem has been the subject of several studies and, recently, efficient methods that allow determining the workspace of parallel cable-driven mechanisms have been proposed [10]. [PERSON] et al. [11] proposed an efficient way of determining analytically the Wrench Closure Workspace (WCW) of spatially redundant cable-driven parallel mechanisms (CDPMs); the method was based on the boundary surface equations obtained from the null space of the structure matrix of a CDPM. [PERSON] et al. [12] proposed a method to determine the cross of the WCW 6-degree-of-freedom (6-DOF) CDPR, corresponding to a specific orientation of the moving platform. This method was called the constant-orientation WCW (COWCW). Consequently, in case of 6-DOF mechanisms, only a partial workspace is obtained because three of the six posed variables must be fixed to compute the workspace. Another limitation of these methods lies in the difficulty taking into account the requirement of a minimum and a maximum tension in each cable (bounds on the allowed tensions), especially for 6-DOF mechanisms driven by more than six cables. Moreover, as a tool to design parallel cable-driven mechanisms, geometric methods seem to be limited to trial-and-error approaches. The kinematic problem of CDPRS is not enough to control the position or force of the moving platform of the CDPRs. The cables cannot provide pushing forces but can only provide pulling forces. For CDPMs, the cable tension greatly affects the determination workspace. Figure 3 shows that, to balance any external wrench (\(\mathbf{F}_{\text{p}}\), \(\mathbf{M}_{\text{p}}\)) applied on  
Figure 2: Kinematics of the moving platform of a cable robot. a moving platform, the cables must create tension forces, achieving equilibrium of the platform. The forces and torques acting on the CDPRs can be obtained from the following equilibrium equations:  
\[\sum_{i=1}^{m}\tau_{i}+\mathbf{F}_{p}=0 \tag{5}\]  
\[\sum_{i=1}^{m}\left(b_{i}\times\tau_{i}\right)+\mathbf{M}_{p}=0 \tag{6}\]  
where \(\mathbf{F}_{\text{P}}\) and \(\mathbf{M}_{\text{P}}\) are the vectors of the external forces and torques acting on the mobile platform, respectively; \(\tau_{i}\) is the force vector that each cable acts upon on the end-effector; and \(\mathbf{b_{i}}\) vectors are the positions of the distal attachment point \(\mathbf{B}_{i}\) on the mobile platform relative to the origin of Op, given in local coordinates of the frame OP. From the cable direction show in Figure 3, with the unit vectors \(\mathbf{u}_{i}\), the direction of the forces \(\tau_{i}\) can be obtained as below. \[\boldsymbol{\tau}_{i}=\tau_{i}\mathbf{u}_{i} \tag{7}\]  
Equation (7) represents the relationship between the kinematics, statics, and dynamics of the CDPRs; from (5) and (6) we can have  
\[\left[ \begin{array}{ccc}\mathbf{u}_{1}&\ldots&\mathbf{u}_{\text{m}}\\ \mathbf{b}_{\text{i}}\times\mathbf{u}_{1}&\ldots&\mathbf{b}_{\text{m}}\times \mathbf{u}_{\text{m}}\end{array} \right]\left[ \begin{array}{c}\tau_{1}\\ \vdots\\ \tau_{m}\end{array} \right]+\left[ \begin{array}{c}\mathbf{F}_{p}\\ \mathbf{M}_{p}\end{array} \right]=0 \tag{8}\]  
by defining  
\[\mathbf{A}=\left[ \begin{array}{ccc}\mathbf{u}_{1}&\ldots&\mathbf{u}_{\text{ m}}\\ \mathbf{b}_{\text{i}}\times\mathbf{u}_{1}&\ldots&\mathbf{b}_{\text{m}}\times \mathbf{u}_{\text{m}}\end{array} \right] \tag{9}\]  
\[\mathbf{w}_{\text{P}}=\left[ \begin{array}{c}\text{F}_{\text{P}}\\ \mathbf{M}_{\text{P}}\end{array} \right] \tag{10}\]  
where \(\boldsymbol{\tau}=\left[ \begin{array}{cc}\tau_{1}&\tau_{2}&\ldots&\tau_{ \text{m}}\end{array}