# Path Planning under Force Control in Robotic Polishing of the Complex Curved Surfaces
## 2 The Proposed Method
### Workstation Building and Force Control
#### 2.1.2 Force Control in Surface Polishing  
During the grinding and polishing process, the interacting force between the tool and surface plays a key role in achieving the surface finish. Generally, this force can be divided into two components, which are tangential force component (\(F_{T}\)) and normal force component (\(F_{N}\)) [21; 22; 23] as shown in Figure 4a. The direction of the \(F_{T}\) is parallel to the direction of the feed and affects the power consumption and tool life, whereas the direction of \(F_{N}\) is perpendicular to the surface plane and it has more influence on surface roughness and deformation of the workpiece [21]. A threefold strategy is presented in Figure 4b, which controls all three factors (wheel rotation speed, workpiece speed and direction, and normal contact force), which affect \(F_{T}\) and \(F_{N}\). Tangential Force Component Control  
Specific grinding energy (\(u\)) is an important process parameter, which can be defined as the energy consumed per unit volume of material removed [21; 22], it can relate \(F_{T}\) and other parameters as  
\[u=\frac{F_{T}\ v_{s}}{v_{w}\ a_{p}\ b} \tag{3}\]  
where \(b\) is the width of the wheel and \(a_{p}\) is the depth of the cutting (for polishing less material removal is always desired so it will be very shallow). The other two parameters are wheel speed (\(v_{s}\)) and workpiece speed (\(v_{w}\)), which are used to control the \(F_{T}\) in this work. Wheel speed or surface feed per minute: the polishing wheel speed against the normal polishing force (pressure applied on the workpiece) is very important for effective polishing. It is measured by surface feet per minute (\(SFPM\)), which dependents on the wheel diameter and the wheel rotation speed:  
\[SFPM=\pi\left(\frac{1\ (ft)}{12\ (in)}\ D\right)\mathrm{rpm}, \tag{4}\]  
where \(D\) is the diameter of the wheel in inches (\(in\)) and rpm is the speed of the wheel (revolutions per minute). From the \(SFPM\) formula, it can be seen that the required \(SFPM\) can be achieved with the various combination of the wheel diameter and the wheel rotation speed. Depending on the \(SFPM\)  
Figure 4: Grinding/polishing force: (**a**) workpiece against polishing wheel and (**b**) tree diagram of the force components control. the wheel creates a frictional force for the polishing process that can be further tuned by the applied pressure from the normal contact force. Workpiece speed and direction: the workpiece speed and direction against wheel rotation are other factors that contribute to polishing force. The direction of workpiece motion plays a significant role in handling the applied pressure. There are two motions for the polishing as shown in Figure 5. The first one is moving the workpiece against the wheel rotation with high pressure. It is usually used for smoothening and can generate the semi-brightness surface. The second motion is moving the workpiece in the direction of the wheel with comparatively medium or low pressure. It produces brightness and a shiny surface. Normal Force Component Control  
In order to achieve the full control of the polishing force, merely aligning up the workpiece and the speed regulation of the tool and workpiece is not enough, particularly when the workpiece has a freeform surface. Therefore, a force feedback control system active contact flange (ACF) manufactured by Ferrobotics under model ACF110-01 is used to keep the contact force constant. It shall be mentioned that many force feedback systems in robot applications use only one extensive setup to control the force and position because robot movement is associated with it [24; 25; 26]. It may cause a delay in the response of the force regulation. Conversely, ACF uses a separate setup independent of the robot movement, which provides quick response time. Therefore, the presented system has two different setups for robot force and motion control. The first setup has a robot and its controller for the robot movement and collision control.