## 2 Materials and Methods
### Geometric Analysis of the DTF Method for 5-Axis Robotic Machining  
In the following, we provide a geometric analysis of the DTF method under the assumption that we consider the robotic motion problem to be a 5-axis robotic machining process in which the tool tip is required to follow the path embedded on the workpiece surface while maintaining normal orientation to the surface with zero tool rotation about its own axis. In this case, the angular velocity of the robot end-effector can be determined as:  
\[\omega=\hat{n}\times\hat{h}, \tag{7}\]  
where \(\hat{n}\) is the unit normal vector on the surface at a path point, \(\hat{h}\) is its time derivative, and \(\times\) stands for vector cross product. In addition, we can introduce the Darboux frame along the curve on the surface [43], which is associated with the path point and constructed on the surface as a moving frame along the path curve such that it consists of triple-unit orthogonal vectors \(\left\{\hat{t},\hat{b},\hat{n}\right\}\), where \(\hat{t}\) stands for the unit curve tangent vector and \(\hat{b}=\hat{n}\times\hat{t}\) can be called a unit bitangent vector; vectors \(\hat{t}\) and \(\hat{b}\) span the tangent plane that is perpendicular to the normal vector \(\hat{n}\) and is attached to the surface at the path point. The triple \(\left\{\hat{t},\hat{b},\hat{n}\right\}\) evolves along the path curve such that its differential can be described by the following formula:  
\[\mathrm{d} \begin{bmatrix}\hat{t}\\ \hat{b}\\ \hat{n}\end{bmatrix} = \begin{bmatrix}0&\kappa_{g}ds&\kappa_{n}ds\\ -\kappa_{g}ds&0&\tau_{g}ds\\ -\kappa_{n}ds&-\tau_{g}ds&0\end{bmatrix}\begin{bmatrix}\hat{t}\\ \hat{b}\\ \hat{n}\end{bmatrix} , \tag{8}\]  
where \(\kappa_{n},\kappa_{g}\), and \(\tau_{g}\) are the point-specific curve-on-surface characteristic properties, i.e., the normal curvature, the geodesic curvature, and the geodesic torsion of the path curve at that point, respectively; whereas, \(\mathrm{d}(. )\) denotes a differential and \(\mathrm{d}s\) stands for the differential path length. These functions determine important properties for surface curves. The normal curvature \(\kappa_{n}\) is related to the measure of the curve curvature that is due to the curvature of the underlying surface--it is projected onto the plane, which is spanned by the vectors, and can be used to tell us the extent to which the surface itself is curving along the curve. The geodesic curvature \(\kappa_{g}\) is the curvature of the curve projected onto the surface tangent plane spanned by vectors \(\hat{t}\) and \(\hat{b}\). And the geodesic torsion \(\tau_{g}\) is the relative torsion that measures the rate of change in the surface normal around the curve's tangent--it is related to the rate of rotation of the tangent plane around the tangent of the curve. From (8) one can deduce the time derivative of the unit normal vector as:  
\[\hat{h}=-\kappa_{n}\hat{s}\cdot\hat{t}-\tau_{g}\hat{s}\cdot\hat{b}, \tag{9}\]  
where \(\hat{s}\) represents the linear speed of a virtual particle, which travels along the path curve. Through the combination of (7) and (9) we can now derive the angular velocity vector as:  
\[\omega=\hat{n}\times\hat{h}=-(\kappa_{n}\hat{b}-\tau_{g}\hat{t})\hat{s}. \tag{10}\]  
The latter result shows that the angular velocity depends on the surface and path curve characteristics since it is embedded on the workpiece surface. If we consider \(\hat{b}\bot\hat{t}\) and \(\upsilon_{a}=\hat{s}\) then the angular speed can be further derived as:  
\[\omega_{a}=\upsilon_{a}\sqrt{\kappa_{n}^{2}+\tau_{g}^{2}}. \tag{11}\]  
Finally, the task-dependent parameter \(h\) can be expressed from (11) as:  
\[h=\frac{\upsilon_{a}}{\omega_{a}}=\frac{1}{\sqrt{\kappa_{n}^{2}+\tau_{g}^{2}}}. \tag{12}\]It is clearly shown that parameter \(h\) is determined only by the curve-on-surface geometry characteristics expressed via the normal curvature and the geodesic torsion.