# 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  
Tool path pattern: the prime objective of path planning is the achievement of the required level of the surface finish on the entire area of the workpiece in the shortest time. Several scanning patterns and area filling curves have been implemented on the different shapes of the workpieces for the better path consistency, stock removal, surface texture, and quality [5; 6; 28; 29]. Figure 10 presents some commonly used tool path patterns to cover the plain or simple curved surfaces. Each pattern has its advantages, disadvantages, and parameters that control its performances. According to the literature, the scanning path is one of the most adopted tool path patterns in polishing [5; 8; 9; 30; 31]. In this study, the scanning path pattern was selected for the eyeglass frame because mostly part surfaces were long stripes. Scanning path consists of a starting point, ending point, and the middle body. It can be classified into two categories, \"S\" pattern and \"Z\" pattern. Path density of the middle body is responsible for the quality and the productivity of the polishing process that can  
Figure 10: Tool path patterns for plain and simple curved surfaces. Figure 9: Surface divisions of the eyeglass frame: (**a**) division into symmetric parts and faces and (**b**) complete division into part surfaces. be controlled by the distance between two consecutive lines. In S scan, tool moves in an alternative direction, however, Z scan moves in the same direction all over the scans. In tangential polishing, tool feed direction had a significant impact on the resultant surface as discussed in Section 2.1.2. The implementation of the scanning path patterns for the polishing curve surface utilizing adopted LCW can be seen in Figure 8. High curvature surfaces demand high path densities because these provide less surface contact. Depending on the tool effective width (\(w_{e}\)), line spacing can vary from 0 to \(W_{Tool}\). Tool compliance: the flexibility of the polishing tool is another important factor that offers smooth interaction, additional tool contact, and help to reach the difficult places of the workpiece. Examples of flexible tools include polishing belts, wool pad, and cloth wheels. Figure 11 shows the impact of using different tool center point (TCP) at flexible LCW in polishing the plain and simply curved surface. TCP defines the zero position of the tool coordinate system. All robots have predefined tool coordinates system and new tool coordinates are defined as an offset from this predefined location. TCP with \(R\)-\(h\) gives large surface contact compare to TCP at \(R\). However, it offers different effects on plain and curved surfaces (\(a_{2}>a_{1}\)) as shown in Figure 11b,c. Smart surface division can convert the high curvature surfaces into nearly a plain or less curved surface that can provide approximately the same contact area as plain surface (i.e., \(a_{2}\approx a_{1}\)). Equation (6) affirms that \(w_{e}\) and \(s\) controls the surface contact area (\(A_{C}\)) between the tool and plain surface. In our study, TCP is taken at \(R\)-\(h\) with a small value of \(s\) because the compliance force of the tool can easily deform the frame for a larger value. \[A_{C}=2\;w_{e}\;(R\;-\;h)\;tan\;(\frac{\theta_{1}}{2})\;\text{and}\;\theta_{1} =\cos^{-1}\left[\frac{(R-h)}{R}\right]. \tag{6}\]  
TCP or end effector velocity in polishing: the composite robot applications like painting, welding, and polishing require continuous path control with explicit orientation and uniform velocity. Within the working space of a robot, there are some spaces where the position and orientation of the end effector of the robot cannot be asserted. These locations are referred to as singularities in which robot displacements and velocities are volatile [32]. For the six-axis robot, singularity can be mathematically described by the relation between the vector of joint velocities (\(i\)) and the Cartesian velocity vector (\(v\)) of the end-effector [27].