# Path Planning under Force Control in Robotic Polishing of the Complex Curved Surfaces
## 2 The Proposed Method
### Tool Path Planning
#### 2.2.2 Tool Path Generation for Individual Part Surfaces
\[q=J^{-1}\;v, \tag{7}\]  
where \(J\) is a Jacobian matrix of order \(6\times 6\), which is a function of the joint positions (\(q\)) and the robot geometry. This equation can calculate joint rates at every point on the path for the robot movement  
Figure 11: Effect of the tool center point (TCP) position on flexible polishing tool (LCW) against the plain and curve surface: (**a**) curve surface with TCP at \(R\); (**b**) curve surface with TCP at \(R\)-\(h\) and (**c**) plain surface with TCP at \(R\)-\(h\). with some velocity vector in the Cartesian space. However, the robots have some values of \(q\) for which Jacobians becomes singular. Those places are considered as singularities. While using the robot in Cartesian mode, trajectories that instruct the robot to move close to a singularity must be avoided. Normally, robots are programmed to stop the movement near singularity to avoid any harm to the robot. Some robots are designed to move slowly near a singularity but lose constant speed in line and force the tool to linger on at the same point, which is damaging for the polishing surface. Sometimes to avoid the most commonly occurred wrist singularity, a small angle is added to the end effector, which reduces the chance of two links alignment. Optimization of polishing parameters: optimized control of the polishing parameters can lead to the required surface quality (surface roughness \(R_{a}\)) in a short time. However, it is difficult to optimize the input parameters due to the complexity of the polishing process. Additionally, the random effects of the flexible polishing tool on the polishing surface make this process difficult to represent by simple equations. Therefore, a full factorial DOE is used to investigate the effect of input factors (_SFPM_, feed rate, and no. of the polishing cycle) on the surface roughness of the eyeglass frame in the polishing process. It can help to find the economical polishing parameters for the required surface finish. The full factorial design consists of all possible combinations of the input factors and their levels [33; 34]. The number of the experimental run for a full factorial design is the product of the levels of each factor (\(F_{n}\)):  
\[\text{{No.}}\;\text{{of}}\;\text{{experimental}}\;\text{{run}}=\;\text{Levels}\; \text{{in}}\;F_{1}\times\text{{Levels}}\;\text{{in}}\;F_{2}\times\ldots \times\text{Levels}\;\text{{in}}\;F_{n}. \tag{8}\]  
Using the above equation, thirty no. of experimental runs were designed with the various combinations of the process parameters. Besides the full factorial design, other methods under DOE can be used such as the fractional factorial design, Plackett-Burman design, and Taguchi method. However, depending on the required accuracy and process cost a full factorial design is used. \[\text{{PE}}= \frac{A\Delta R_{a}}{T} \tag{9}\]  
According to references [29; 30], the best parameters can be selected by using the polishing efficiency (_PE_) formula given in Equation (9). It is defined as the amount of roughness reduction (\(\Delta R_{a}\)) for the specified area (\(A\)) in time (\(T\)). A plot of the _PE_ against the \(R_{a}\) can help to extract the best possible polishing parameters based on the highest polishing efficiency achieve. After the surface division of the eyeglass frame, most of the part surfaces are strips. The maximum width (\(W_{PS}\)) of each part surface is less than the tool width (\(W_{Tool}\)). As a result, the polishing tool can cover the entire strip or part surface just in one pass starting from one point and ending at the opposite side. However, to achieve the required surface quality, multiple polishing passes are needed. The number of polishing passes, the starting, and ending points is dependent on the polishing parameters. Figure 12 shows the polishing trajectory of the four part surfaces with the tool and robot target frames \(\{R\}\). Based on the optimized polishing parameters achieved by full factorial DOE from the part surface one (\(S_{1}\)), two polishing cycles are suggested for each part surface.