# Design of a Planar Cable-Driven Parallel Robot for Non-Contact Tasks
## 4 Experimental Demonstration: Laser Engraving  
A prototype of a 3-DoF 4-cable \(CDPR\) with a _crossed layout_ was built at IRMA L@B of the University of Bologna (Figure 13a). It employs 4 of the actuation systems described in Section 3.1, connected to the corners of an aluminum frame, and 4 guidance units, whose distal swivel pulleys are bolted to the robot platform. The \(CDPR\) has rectangular base (0.875 m \(\times\) 0.700 m) and mobile platform (0.080 m \(\times\) 0.100 m). The inertial frame \(Oxy\) is located in the center of the base and the moving frame \(Px^{\prime}y^{\prime}\) is located at the center of the \(EE\), coinciding with its center of mass. Its wrench-feasible translational workspace [50] is represented in Figure 13b with a regular discrete grid of \(100\times 100\) points, considering the platform mass equal to \(m=2.5\) kg, and \(\tau_{min}=10\) N and \(\tau_{max}=80\) N as cable tension limits. The \(CDPR\) is controlled by a Real-Time Linux PC, which runs its control algorithm at 1 kHz rate, and feeds motor commands through Ethereat protocol [51] to low-level servo drives. The low-level controller is also developed in-house and run on a ST Nucleo-H743 ZI development board at 10 kHz: in addition to controlling motor angles, this allows to directly feed a cable-tension command to the drive. This latter feature simplifies the adoption of the hybrid-input control strategy for the \(CDPR\) described in [50]: once an  
Figure 13: Laser-engraving prototype at IRMA L@B. (**a**) Picture of the prototype. (**b**) Wrench-feasible translational workspace of the prototype. \(EE\)-pose set-point is assigned in the Cartesian space, 3 cables are length controlled (i.e., motor angles are commanded to the drives), and one cable is tension controlled (i.e., cable tension is commanded to the drive, and the error with respect to the readings of the load cell is compensated). A 2.5 W laser diode is mounted on the robot \(EE\) so that its focal axis is perpendicular to the working plane (Figure 14), thus allowing the engraving of a paper sheet (for additional details, see Supplementary Video Material). The name of our laboratory, IRMA L@B, was stylized in a way that every letter was a union of a finite number of linear segments (Figure 15a): accordingly, a quantitative comparison between the digital model and the engraved result could be performed, by comparing each segment length. A series of consecutive points \(P_{1},\ P_{2},\ldots,\ P_{n}\), with \(n=15\), were extracted from the digital model (Figure 15b), and the robot reference point commanded to follow the path between them with a trapezoidal velocity profile in the task space, while keeping the orientation identically zero. If the trajectory is expressed as:  
\[\mathbf{p}(t)=\mathbf{p}_{i}+(\mathbf{p}_{i+1}-\mathbf{p}_{i})u(t),\qquad u(0) =0,\,u(T)=1,\qquad i=1,\ldots,n-1 \tag{3}\]  
the motion law \(u(t)\)[52] takes the form:  
\[u(t)= \begin{cases}\frac{1-\alpha}{2\pi}(yt)^{2},&\text{for }0\leq t<\alpha T\\ -\frac{\alpha}{2(1-\alpha)}+vt,&\text{for }\alpha T\leq t<(1-\alpha)T,\\ -\frac{2\alpha^{2}-2\alpha+1}{2\alpha(1-\alpha)}+\frac{1}{\alpha}vt-\frac{1- \alpha}{2\alpha}(vt)^{2},&\text{for }(1-\alpha)T\leq t\leq T\end{cases} \qquad T= \frac{1}{v(1-\alpha)} \tag{4}\]  
Since the set-points \(P_{i}\) of Equation (3) are set by the task, the only parameters to be determined are \(v\) and \(\alpha\) of Equation (4). They represent, respectively, the motion law maximum speed, and the ratio between the accelerating phase duration and the total time; they are set considering the requirements of the technological operation to be performed. Since the diode laser can only be on or off, the accelerating phase duration should be as limited as possible, to avoid burning the paper when the \(EE\) is moving too slow; on the other hand, motors have limited power, and too quick accelerations may result in their stall: by trial and error, \(\alpha=0.1\) was established as a good compromise.