# Path Planning under Force Control in Robotic Polishing of the Complex Curved Surfaces
## 1 Introduction  
Polishing is a tedious but important finishing process, which demands a considerable amount of manual work in many manufacturing sectors. It is equally essential in the restoration of the old or damaged parts. This time consuming, error-prone, and costly multistage process is facing a shortage of skilled workers due to the various problems, such as government regulation, deadly respiratory diseases, and increasing labor cost [1; 2; 3; 4]. Robotic polishing is the solution to these problems, which can improve the working condition of the labor, eliminate the manufacturing defects, and ensure the product quality giving an appealing glossiness to the product. As a serious contender for the robotic polishing, the industrial robot is an automated, programmable, and multipurpose robotic system, which can work in unhealthy and accident-prone environments with high accuracy. However, it has some mechanical constraints, which need to be addressed before using it into the surface polishing of the complex parts. For instance, even for thesimplest movement of the end effector between two points within the working envelope requires proper consideration of collision avoidance, work proficiency, best obtainable trajectory path, and other kinematic constraints such as singularity and joints limits. Therefore, path planning and trajectory planning are very significant for robot movement. Moreover, cumulative demand for high speed and less production time from the industrial robots challenge high precision and repeatability. These extreme demands place the path and trajectory planning algorithms in the limelight. Path planning utilizing various scanning patterns and space-filling curves have been discussed under automated polishing in several research studies [5; 6; 7; 8; 9]. [PERSON] et al. presented an investigation for surface roughness reduction by using a scanning path, which focuses the path evenness through pitch adaptation with surface slope changes and line spacing [5]. Fractal path generation was used in the robotic polishing of the metal molds [6]. It used a planer Peano curve with end effector orientation normal to the freeform surface. Contact mechanics was also used for contact area modeling and analysis to automate the polishing process in coverage based tool-path planning [7]. Similarly, the different force control methods and process control techniques have been used for better-controlled interaction during the polishing [10; 11; 12]. A fuzzy vector method was suggested for the grinding and polishing, which enables the controller to deal with force sensor signals caused by the working tool, to search the direction of the constraint surface of an unknown object [10]. Another study concentrated on the control issues that arise during the automation of die and mold surface polishing using an arm robot, particularly on the force controllers to maintain the required tool and workpiece interaction [12]. The major focus of this paper was on the active end effector force control because there are performance limitations in the force control loop of the arm robot. The third most adopted technique for the improvement of automated polishing is the implementation of specially designed polishing tools [13; 14; 15]. [PERSON] et al. presented an improved semirigid bonnet tool for the high-quality polishing on aspheric optics [13]. Based on the finite element analysis method, the tool influence function is modeled for the structural optimization to eliminate previous present defects. In paper [15], a collaborative tool for robotic polishing was presented to allow the simultaneous operation of the robot and human cooperation for polishing. The polishing tool was developed by using the minimum viable product approach. However, tool path planning plays a significant role in achieving the good surface quality during the polishing process in the complex structures and requires high reachability, but it offers some relaxation in terms of accuracy [11].