## 1 Introduction  
Magnetroheological finishing (MRF) is a deterministic sub-aperture polishing process that causes no sub-surface damage [1]. The deterministic removal of material is realized primarily by the dwell time method in numerical control polishing [2; 3; 4], meaning that the amount of material removal is controlled by the residence time of the polishing tool. The longer the polishing tool is in contact with the workpiece on a specific surface, the more material is removed from that area. Hence the magnetorheological removal function (i.e., the distribution function of the removal amount of the processed material in unit time) and its stability are critical in the polishing process. Recently MRF technology has a relatively mature application and development on computer-numerical-control (CNC) machines, and magnetorheological technology and equipment have been engineered and commercialized [5]. Nonetheless, conventional MRF machine tools have drawbacks such as excessive weight and volume, restricted processing range, and high cost. Since the birth of industrial robots in the 1960s, the robot has been widely and effectively applied with the advantages of high flexibility and low cost in various fields, including welding, assembly, medical treatment, and life services. In the field of polishing, different from high-precision CNC machines, the robot is a rigid-flexible coupling body composed of multi-joints and multi-links. Thereby, the robot has poor rigidity and it is difficult to meet the requirements of positioning precision during the polishing process. To realize the application of robots in precision occasions, researchers have proposed a variety of solutions, classified into two categories of compensation: direct compensation of robotic errors and indirect compensation using work actuators. The direct correction of robotic error is also categorized into model-based error compensation and sensor-based error compensation [6]. [PERSON] et al. [7] used the finite element model to simulate and predict the robotic workload, combined with the robot stiffness model to realize the compensation of the robotic milling trajectory error and keep the overall trajectory error within 0.3 mm. [PERSON] et al. [8] combine offline and online compensation by incorporating the theoretical milling force into the robotic stiffness model. This method pre-corrects the robotic work trajectory in the control program and re-corrects the work trajectory in the actual operation based on milling force feedback from the sensor, improving the precision of the robotic processing trajectory. [PERSON] et al. [9] measured the robot end tool position based on the laser tracker, developing a closed-loop feedback system for robotic error compensation. The absolute positioning accuracy was increased to 0.087 mm, and the attitude accuracy of the robotic end reached 0.01\({}^{\circ}\). Nevertheless, since commercial industrial robots have closed architecture systems, it is difficult to modify the closed control system of those robots using direct correction. To meet the demands of robot polishing, the method of indirect compensation using work actuators mainly includes active compliance and passive compliance. Generally speaking, an active compliance actuator has higher accuracy and better adaptability, achieving the positive effects of grinding technology [10; 11]. [PERSON] et al. [12] developed a pneumatic actuator for robotic gadget polishing and realized the force control adjustment of the robot surface polishing process with 0.5N resolution, greatly improving the robot polishing quality. In terms of the combination of magnetorheological finishing technology and robot technology, [PERSON] et al. [13] integrated the exiting magnetorheological finishing tool on a six-axis industrial robot. A 1000 mm \(\times\) 470 mm off-axis aspherical mirror was processed and achieved positive results. Nonetheless, the compensation of robotic normal positioning error is not disclosed.