## 2 Design of the Laser Scalpel
### Schematic Diagram for a Modified Stewart Platform
the \(X_{2}Y_{2}Z_{2}\) coordinate system to the \(X_{3}Y_{3}Z_{3}\) coordinates system; Trans\({}_{6}\),7 (x, y, z) represents the translation matrix from the \(X_{6}Y_{6}Z_{6}\) coordinate system to the \(X_{7}Y_{7}Z_{7}\) coordinates system when performed with the x, y, and z distances; Rot\({}_{21}\)2 (\(\beta_{1}\)) represents the rotation matrix around the \(Z_{1}\) axis of the input coordinate system \(X_{1}Y_{1}Z_{1}\) to the final coordinate system \(X_{2}Y_{2}Z_{2}\), performed with the angle \(\beta_{1}\); Rot\({}_{23}\)4 (\(-\beta_{2}\)) represents the rotation matrix around the \(Z_{3}\) axis of the input coordinate system \(X_{3}Y_{3}Z_{3}\) to the final coordinate system \(X_{4}Y_{5}Z_{4}\), performed with the angle \(-\beta_{2}\); Rot\({}_{23}\)5 (\(\theta\)) represents the rotation matrix around the \(X_{4}\) axis of the input coordinate system \(X_{4}Y_{4}Z_{4}\) to the final coordinate system \(X_{5}Y_{5}Z_{5}\), performed with the angle \(\theta\); Rot\({}_{55}\)6 (\(\psi\)) represents the rotation matrix around the \(Y_{5}\) axis of the input coordinate system \(X_{5}Y_{5}Z_{5}\) to the final coordinate system \(X_{6}Y_{6}Z_{6}\), performed with the angle \(\psi\). The coordinates of the point A\({}^{\prime}\) in the \(X_{0}Y_{0}Z_{0}\) system of coordinates are given by  
\[ \begin{pmatrix}x_{A^{\prime}}\\ y_{A^{\prime}}\\ z_{A^{\prime}}\\ 1\end{pmatrix} =\mathrm{H}_{0}^{7*}(x_{A},y_{A},z_{A},1). \tag{2}\]  
Considering Equations (1) and (2), the AA\({}^{\prime}\), BB\({}^{\prime}\), and CC\({}^{\prime}\) distances can be obtained. They represent the displacement of the actuators under the action of the input parameters (i.e., the rotations \(\theta\) and \(\psi\), as well as translation \(\delta\)). The AA\({}^{\prime}\) vector in the \(X_{0}Y_{0}Z_{0}\) system of coordinates that was customized for AO\({}_{1}\) = A\({}^{\prime}\)O\({}_{2}\) = R and \(\beta_{1}\) = \(\beta_{2}\) = 0 is given by the relation:  
\[\mathrm{AA}^{\prime 2}=(\mathrm{R}-\mathrm{R}\cdot\mathrm{cos}\theta)^{2}+( \delta-\mathrm{R}\cdot\mathrm{sin}\theta)^{2}=2\mathrm{R}^{2}(1-\mathrm{cos} \theta)-2\mathrm{R}\delta\cdot\mathrm{sin}\theta+\delta^{2}. \tag{3}\]  
Similarly, the BB\({}^{\prime}\) vector in the \(X_{0}Y_{0}Z_{0}\) system of coordinates, which was particularized for AO\({}_{1}\) = A\({}^{\prime}\)O\({}_{2}\) = R and for \(\beta_{1}\) = \(\beta_{2}\) = 2\(\pi\)/3 is given by  
\[\mathrm{BB}^{\prime 2}=2\mathrm{R}^{2}\Bigg{(}1+\frac{\sqrt{3}}{4}\mathrm{ sin}\,\mathrm{sin}\,\mathrm{\psi}-\frac{1}{4}\mathrm{cos}\,\mathrm{\theta}- \frac{3}{4}\mathrm{cos}\,\mathrm{\psi}\Bigg{)}+2\mathrm{R}\delta\Bigg{(}- \frac{\sqrt{3}}{2}\mathrm{cos}\,\mathrm{\theta}\mathrm{sin}\,\mathrm{\psi}+ \frac{1}{2}\mathrm{sin}\,\mathrm{\theta}\Bigg{)}+\delta^{2}. \tag{4}\]  
Also, in the \(X_{0}Y_{0}Z_{0}\) system of coordinates, the CC\({}^{\prime}\) vector that was particularized for AO\({}_{1}\) = A\({}^{\prime}\)O\({}_{2}\) = R and for \(\beta_{1}\) = \(\beta_{2}\) = 4\(\pi\)/3 is given by the relation:  
\[\mathrm{CC}^{\prime 2}=2\mathrm{R}^{2}\Bigg{(}1+\frac{\sqrt{3}}{4}\mathrm{ sin}\,\mathrm{sin}\,\mathrm{\psi}-\frac{1}{4}\mathrm{cos}\,\mathrm{\theta}- \frac{3}{4}\mathrm{cos}\,\mathrm{\psi}\Bigg{)}+2\mathrm{R}\delta\Bigg{(}\frac{ \sqrt{3}}{2}\mathrm{cos}\,\mathrm{\theta}\mathrm{sin}\,\mathrm{\psi}+\frac{1} {2}\mathrm{sin}\,\mathrm{\theta}\Bigg{)}+\delta^{2}, \tag{5}\]  
where \(\delta\) represents the sum of the distance O\({}_{1}\)O\({}_{2}\) and of the translation of the upper platform. The side view of the real system is presented in Figure 4. The high-power fiber-coupled diode laser system FC-W-808 (Changchun New Industries Optoelectronics Technology Co., (CNI) Ltd., Changchun, China) has a central wavelength of 808 nm. It is connected by an optical fiber to the upper (in Figure 4, the right-side) platform, where it interfaces through an SMA905 connector with an FOC-02 collimator (CNI Ltd., Changchun, China) equipped with an anti-reflective coated aspherical lens.