## 3 Robotic Deburring Tool Path Planning Method
### Robotic Deburring Tool Planning Method  
The proposed robotic deburring tool position planning method is discussed in two parts: surface deburring and curve deburring. In the first part (i.e., surface deburring), the tool contact path intersection line method [45] is used to generate the contact position of the deburring tool, i.e., the total locus of deburring tool contact points are derived from the intersection of a set of equidistant constraint planes and the target surface. Among them, the constraint plane spacing, the line spacing and step length of the tool path should be determined by some comprehensive factors such as deburring tool, deburring accuracy, deburring efficiency and deburring error. In the second part, i.e., curve deburring, the above comprehensive factors are also considered to directly generate the total locus of discrete deburring tool contact points. Upon obtaining the total locus of deburring tool contact points for surface deburring and curve deburring, considering the curvature variations of the surface and the curve, respectively, the total locus of deburring tool contact points is offset according to the layered \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \(L_{1}\) & \(L_{2}\) & \(L_{3}\) & \(L_{4}\) & \(L_{5}\) & \(L_{6}\) & \(e\) & \(L_{t}\) & \(L_{01}\) & \(s_{u}/s_{z}\) \\ \hline
420 & 50 & 450 & 160 & 210 & 95 & 0 & 297 & 465 & 10 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Robot manipulator structure parameters (mm). Figure 5: Reachable workspace of robot manipulator: (**a**) top view; (**b**) front view. deburring planning. After that, the deburring tool path of each layer is generated with some practical considerations of deburring tool length, blade edge and blade diameter, etc. In this article, the orientation of the deburring tool is aligned with its side edge, defined as follows. Deburring tool orientations, the movement direction of the tool and the end edge of the tool are expressed as vectors **n**, **o** and **a**, respectively. A target curve is used as an example to illustrate the robotic deburring tool orientation planning method. The target curve is shown in Figure 6, and unit tangent vectors \(\boldsymbol{\tau}\) and unit inner normal vectors **f** of discrete positions are represented by blue arrows and green arrows, respectively. The proposed robotic deburring tool orientation planning method is proposed for deburring manners with the tool end edge and the tool side edge. The assignment of orientation vectors for these two deburring manners is described as follows. For the tool end edge (i.e., the deburring manner with the tool end edge): (a) the deburring direction of tool is planned along the tool end edge vector **a**, which is chosen as the unit inner normal vectors **f** of each discrete position of the target curve; (b) the movement direction of tool (vector **o**) is planned along the unit tangent vectors \(\boldsymbol{\tau}\) of each discrete position of the target curve and points to the next discrete position to be deburred; and (c) the tool vector **n** is obtained by \(\mathbf{n}=\mathbf{o}\times\mathbf{a}\). For the tool side edge (i.e., the deburring manner with the tool side edge): (a) the deburring direction of tool is planned along the tool side edge vector **n**, which is chosen as the unit inner normal vectors **f** of each discrete position of the target curve; (b) the planned movement direction of tool (vector **o**) is the same as above deburring manner; and (c) the tool vector **a** is obtained by \(\mathbf{a}=\mathbf{n}\times\mathbf{o}\). In the above robotic deburring tool orientation planning, the feeding direction of the robot manipulator deburring corresponds to the deburring tool direction of the end edge or the side edge, that is, the feeding directions of the robot manipulator deburring along the unit inner normal vectors **f** for discrete positions of the target curve.