## 2 Error of Removal Function  
Equation (1) gives a classical version of [PERSON]'s [17] ideas for the material removal rate. \[R=KPV=K\frac{F_{n}}{A_{c}}V, \tag{1}\]where \(K\) is [PERSON]'s coefficient determined by the physical and chemical properties of the polishing fluid (e.g., PH value, slurry, abrasive type, friction force) and the material of the machined workpiece, \(P\) is the normal pressure applied (i.e., normal force \(F_{n}\), the contact area \(A_{c}\) between the polishing tool and the substrate being polished), and \(V\) is the relative velocity between the workpiece surface and the magnetorheological fluid. [PERSON] [18] found that the material removal of MRF finishing is a process dominated by sheer force and supplemented by normal force. To address MRF, [PERSON] proposed a modified [PERSON]'s model, as is shown in Equation (2):  
\[R_{s}=C\mathit{t}_{p,MRF(F_{n})}\frac{\mu F_{n}}{A_{s}}V, \tag{2}\]  
where \(C\mathit{t}_{p,MRF(F_{n})}\) is the modified [PERSON]'s coefficient, instead of \(A_{c}\) in Equation (1), \(A_{s}\) is the projected spot area where polishing occurs, and the coefficient of friction is \(\mu\) correlating with the sheer force. The experiment indicates that \(\mu\) lies between 0.4 and 0.6, within the typical range for the coefficient of friction reported for most materials in the sliding friction model [19; 20]. Equation (2) reveals that the normal force \(F_{n}\) between the MRF tool and the workpiece is one of the key factors affecting the removal function. The normal force \(F_{n}\) of polishing can be affected by changing immersion depths (the depth at which the ribbon is immersed into the workpiece surface). The change of immersion depth results in the change of ribbon hardness and abrasive adhesion degree at the contact layer of the ribbon. Hence, the immersion depth is an important factor affecting the polishing force [21; 22; 23]. Under the existing experimental conditions, the sensitivity of the removal function to the normal positioning error is investigated by taking spots under different immersion depths on a high-precision MRF machine with a positioning precision of 3 \(\upmu\)m on the Z-axis, as illustrated in Figure 1.  
Figure 1: Taking spots under different immersion depths on a high-precision MRF machine. (**a**) MRF tool on a high-precision machine; (**b**) MRF ribbon when taking a spot; (**c**) The figure map after taking spots; (**d**) The distribution of the removal amount of MRF spot taking. Eight groups with different immersion depths from 0.2 mm to 0.55 mm are taken, with two spots in each group. A standard MR fluid is used in this experiment, consisting of Nano-diamond abrasives, iron powder, deionized water, and stabilizers. The Magnetorheological flow rate is 1800 mL/min, the wheel speed is 200 rpm with the diameter of the polishing wheel at 300 mm, the magnetorheological fluid moisture is 14.5%, and the ribbon height is adjusted to 1.3 mm. The volumetric removal rate (VRR) is obtained from MRF spots. As illustrated in Figure 2, the result indicates a positive linear dependence on immersion depth under the conditions used in this experiment. The normal error of 0.05 mm results in a 0.000935 mm\({}^{3}\)/s error in volumetric removal rate. Since the immersion depth of 0.3-0.4 mm commonly is used in polishing, the relative error of the volumetric removal rate reaches 19-30% for the normal error 0.05 mm. Overall, these results indicate that the stability of the MRF process is sensitive to normal positioning error. In the stable polishing process, the normal distance between the polishing tool and the workpiece surface should keep constant, guaranteeing the stability of the normal polishing force \(F_{n}\) and the projected spot area \(A_{s}\). That ensures the sheer force affecting the material removal efficiency is stable. The normal positioning error between the polishing tool and the workpiece leads to uneven polishing force.