# Path Planning under Force Control in Robotic Polishing of the Complex Curved Surfaces
## 2 The Proposed Method
### Workstation Building and Force Control  
#### 2.1.1 Polishing Tool and End Effector Selection  
There are large amounts of polishing tools available in the market. To select an appropriate tool would need to consider various factors such as tool cost, required roughness, polishing time, tool changing time, tool compliance, material removal rate, etc. [16]. For polishing plain and/or simple curved surfaces, it is relatively easy to find a suitable tool that can reach every point of the workpiece. However, for complex surfaces, it is sometimes difficult to find a perfect tool. A sensible solution is to test several polishing tools manually on the workpiece. In our application (polishing eyeglass frame), based on the know-how of experienced workers, we choose the 150 mm \(\times\) 15 mm loose cotton wheel (LCW) for polishing (Manufacturer: Bolpol, Model: HL150/1) with dry polishing wax. It is a flexible polishing tool that can reach the difficult corners of the eyeglass frame. The target is to achieve the arithmetic average roughness (\(R_{a}\)) of 0.02 \(\upmu\)m.  
For robot polishing, the end effector of the robot can be either the polishing tool or the workpiece. The decision depends on the application [17; 18; 19; 20]. Figure 3a shows the polishing tool used as the end effector. It is effective when the workpiece is large and/or heavy. Additionally, when multiple polishing tools are needed, it is convenient to use an automatic tool changer (ATC) for tool change. In other configurations, workpiece is used as an end effector by attaching to the robot through a specifically designed fixture as shown in Figure 3b. The advantage of this setup is that the orientation of the end effector can be easily controlled to retain a constant orientation between the tool and the workpiece. Additionally, the polishing force control unit can be separated from the robot and there is no delay in polishing for the various tools change. Thus, in this study, the workpiece is used as an end effector.  
Note that when using a workpiece as the end effector, both the payload and the gripping force of the robot must be considered. The payload is the maximum mass that can be attached as the end effector of the robot. It is a key parameter when selecting the robot, whereas, the gripping force is the maximum force of the gripper to hold the workpiece as  
\[F_{g}=\frac{m_{g}(g+a)}{\mu} \tag{1}\]  
where \(F_{g}\) is the minimal gripping force to hold the workpiece, \(\mu\) is the friction coefficient, \(g\) is the gravitational constant, \(m_{g}\) is the mass of the gripped body and \(a\) is the acceleration. In the polishing operation, the tool exerts force on the workpiece to smoothen the surface. This additional polishing  
Figure 3: Two different workstation building: (**a**) use tool as the end effector and (**b**) use workpiece as the end effector.  
force \(\left(F_{p}\right)\) must be added in the gripping force to firmly hold the workpiece during the polishing process. Hence, the total gripping force (\(F_{t}\)) can be represented as  
\[F_{t}=F_{g}+F_{p}=\frac{m_{g}\left(g+a\right)}{\mu}+F_{p} \tag{2}\]