## 3 Robotic Deburring Tool Path Planning Method
### Robotic Layered Deburring Planning Method  
The feature points on the object are selected based on the geometric characteristics of the deburring target, and the machining allowances are measured by the distance between each selected feature point and the target curve and/or surface. The detailed description of the proposed robotic layered deburring planning method for the layered deburring curve and/or surface is presented in the following steps:  
Step 1: calculate the distance between each selected feature point and the target curve and/or surface, and find out the maximum one (which is denoted as \(D_{\text{max}}\)) as the maximum machining allowance for the layered deburring curve and/or surface. Figure 6: Example of unit tangent and normal vectors for discrete positions of target curve. Step 2: calculate the number of deburring layers \(n\) (i.e., the times of layered deburring) by the formula \(n=\left\langle D_{\max}/d_{p}+1\right\rangle\), where \(d_{p}\) and symbolic form \(\left\langle\bullet\right\rangle\) represent the thickness of each selected deburring layer (i.e., cutting depth) and the rounding calculation, respectively. In order to improve the deburring quality, the deburring count is calculated based on different allowance \(d_{l}\) in the last layer: (a) set it to one when \(d_{l}<d_{p}/2\) and the deburring allowance sets \(d_{l}\), and (b) set it to two when \(d_{l}\geq d_{p}/2\), here, the first deburring allowance sets \(d_{p}/2\), and the second one sets the remains (i.e., \(d_{l}-d_{p}/2\)). Step 3: plan the deburring tool path on each layer (i.e., the total locus of deburring tool contact points between the deburring tool and the object) based on the index of each layer \(n_{j}\) and its corresponding thickness \(d_{p,j}\), that is, calculate the position offset relationship of each layer for points of the deburring tool path along the outward normal vector \(\mathbf{n}_{l}=\left[ \begin{array}{cc}n_{lx}&n_{ly}&n_{lz}\end{array} \right]^ {T}\) of each discrete point derived from the target curve and/or surface (i.e., the deburring tool contact positions on the target curve and/or surface are obtained by the tool contact path intersection line method [45]), as follows:  
\[\left[ \begin{array}{c}x\\ y\\ z\end{array} \right]_{n_{j}}=\left[ \begin{array}{c}x\\ y\\ z\end{array} \right]_{n_{j-1}}+\left[ \begin{array}{c}\Delta x\\ \Delta y\\ \Delta z\end{array} \right]_{n_{j}}\qquad j=1,2,\cdots,n, \tag{1}\]  
where, when \(j=1\), \(n_{0}\) and \(\left[ \begin{array}{cc}x&y&z\end{array} \right]_{n_{0}}^{T}\) represent the number of located layer of the target curve and/or surface and positions of each discrete point of the target curve and/or surface, respectively. And the incremental position calculation is related by  
\[\left[ \begin{array}{c}\Delta x\\ \Delta y\\ \Delta z\end{array} \right]_{n_{j}}=d_{p-j}\cdot\left[ \begin{array}{c}n_{lx }\\ n_{ly}\\ n_{lz}\end{array} \right]. \tag{2}\]  
Step 4: conduct the semi-finishing and finishing based on actual and special requirements according to Step 2 and Step 3. Step 5: introduce the orthogonal overlapping and reciprocating manner, and the shape of the basic body surface that is connected with the target curve to conduct planning the deburring tool path for surface deburring and curve deburring at each layer. In the above-mentioned deburring tool path planning for surface deburring and curve deburring, if the feature points of the object are unknown beforehand, the maximum machining allowance \(D_{\max}\) can be estimated by the robot manipulator teaching according to the actual workpiece to be deburred, and the remaining planning steps are the same as above. Usually, the loci of deburring tool contact points, which are obtained by only using the offset computed from the tool contact path intersection line method [45], may cause a large machining error due to the equal-distance offset.