## 2 Design of the Laser Scalpel
### Schematic Diagram for a Modified Stewart Platform
This optical setup is planned to be employed in further research in order to validate the compensation capabilities, with subsequent modifications (including miniaturization), as intended for its application in surgical laser systems. The collimator has an image focal length \(f^{\prime}\) equal to the imposed distance of 50 mm from the laser scalpel tip to the target plane. Considering only the vertical tremor of the hand, with a maximum amplitude of 150 \(\upmu\)m peak-to-peak along the X axis, the spot displacement on the target plane has the same amplitude as the hand tremor. In order to compensate for this unwanted vibration, the corresponding rotation angle \(\theta\) of the mobile platform that carries the laser subsystem can be calculated as:  
\[\theta=\tan^{-1}\frac{s}{f^{\prime}}=\tan^{-1}\frac{0.075}{50}\cong 0.085^{ \circ}, \tag{6}\]  
where \(f\) is the focal length imposed by the design and chosen to have the same length as a surgical needle, and \(s\) is half of the laser spot amplitude on the target plane (that is in turn equal to half of the vertical hand tremor amplitude). The rotation angle \(\psi\) can be determined similarly to Equation (6). It is equal to the value of \(\theta\). In the case of a small (i.e., \(<\)1\({}^{\circ}\)) angular displacement, the simplifying hypotheses of sin(x) \(\cong\) x, cos(x) \(\cong\) 1 can be applied. Therefore, the product xy can be neglected because of its small values. Considering this approach, the mathematical model of the system becomes:  
\[\text{AA}^{\prime 2}=-2\text{R}\delta\theta+\delta^{2} \tag{7}\]  
\[\text{BB}^{\prime 2}=\text{R}\delta\left(-\sqrt{3}\cdot\psi+\theta\right)+\delta^ {2} \tag{8}\]  
\[\text{CC}^{\prime 2}=\text{R}\delta\left(\sqrt{3}\cdot\psi+\theta\right)+\delta^ {2}. \tag{9}\]  
Regarding the mathematical model of the mechanism, one can see that trigonometric equations are involved, containing different position variables. Given the small displacements involved, a simplification method was proposed, leading to Equations (7)-(9). However, in our case, for small displacements, the distances AA\({}^{\prime}\), BB\({}^{\prime}\), and CC\({}^{\prime}\) were developed into a Taylor series with respect to the parameters \(\delta\), \(\theta\), and \(\psi\), around the initial point that was characterized by the position defined by \(\delta=\text{H}_{0}\) and \(\theta=\psi=0\). This leads to the following functions utilized for Taylor series:  
\[f_{1}(\delta,\theta,\psi)=\text{AA}^{\prime}=H_{0}+\Delta\text{AA}^{\prime}= \sqrt{-2\text{R}\delta\theta+\delta^{2}} \tag{10}\]  
\[f_{2}(\delta,\theta,\psi)=\text{BB}^{\prime}=H_{0}+\Delta\text{BB}^{\prime}= \sqrt{\text{R}\delta\left(-\sqrt{3}\cdot\psi+\theta\right)+\delta^{2}} \tag{11}\]  
\[\mathrm{BB}^{\prime 2}=2\mathrm{R}^{2}\Bigg{(}1+\frac{\sqrt{3}}{4}\mathrm{ sin}\,\mathrm{sin}\,\mathrm{\psi}-\frac{1}{4}\mathrm{cos}\,\mathrm{\theta}- \frac{3}{4}\mathrm{cos}\,\mathrm{\psi}\Bigg{)}+2\mathrm{R}\delta\Bigg{(}- \frac{\sqrt{3}}{2}\mathrm{cos}\,\mathrm{\theta}\mathrm{sin}\,\mathrm{\psi}+ \frac{1}{2}\mathrm{sin}\,\mathrm{\theta}\Bigg{)}+\delta^{2}. \tag{4}\]0 Schematic diagram of the proposed laser scalpel (side view). \[\mathrm{BB}^{\prime 2}=2\mathrm{R}^{2}\Bigg{(}1+\frac{\sqrt{3}}{4}\mathrm{ sin}\,\mathrm{sin}\,\mathrm{\psi}-\frac{1}{4}\mathrm{cos}\,\mathrm{\theta}- \frac{3}{4}\mathrm{cos}\,\mathrm{\psi}\Bigg{)}+2\mathrm{R}\delta\Bigg{(}- \frac{\sqrt{3}}{2}\mathrm{cos}\,\mathrm{\theta}\mathrm{sin}\,\mathrm{\psi}+ \frac{1}{2}\mathrm{sin}\,\mathrm{\theta}\Bigg{)}+\delta^{2}. \tag{4}\]1  
By developing the functions \(f_{1},f_{2}\), and \(f_{3}\) in a first-order Taylor series around the initial point \(P(H_{0},0,0)\), the following matrix equation is obtained:  
\[ \begin{bmatrix}\Delta AA^{\prime}\\ \Delta BB^{\prime}\\ \Delta CC^{\prime}\end{bmatrix} = \begin{bmatrix}1&-R&0\\ 1&\frac{R}{2}&-\frac{\sqrt{3}}{2}\\ 1&\frac{R}{2}&\frac{\sqrt{3}}{2}\end{bmatrix}\begin{bmatrix}\delta\\ \theta\\ \psi\end{bmatrix}