Fig. 1. Schematic diagram of removing function positioning error

Fig. 2. (a) The original surface error of the optical element; (b) The surface error obtained after the removal function processing; (c) The surface error obtained after the removal function was deflected 1° along the X-axis; (d) The surface error obtained after the removal function was deflected 2° along the X-axis; (e) The surface error obtained after the removal function was deflected 1° along the Y-axis; (f) The surface error obtained after the removal function was deflected 2° along the Y-axis

Fig. 3. Removal function of MRF finishing

Fig. 4. Process of (a) magnetorheological finishing and (b) surface shape detection

Fig. 5. (a) Initial surface error and (b) initial mid-spatial error of #A

Fig. 6. (a) Surface error and (b) mid-spatial error after simulation processing of #A

Fig. 7. Actual machining surface of #A. (a) After the first magnetorheological finishing; (b) After the second magnetorheological finishing; (c) After the smooth processing; (d) After the third magnetorheological finishing; (e) After the fourth magnetorheological finishing; (f) Mid-spatial error after the fourth magnetorheological finishing

Fig. 8. Process flow chart under uncertainty error

Fig. 9. (a) Initial surface and (b) initial mid-spatial error of #B

Fig. 10. (a) Surface error and (b) mid-spatial error after simulation processing of #B

Fig. 11. Actual machining surface of #B. (a) After the first magnetorheological finishing; (b) After the second magnetorheological finishing; (c) After the first smooth processing; (d) After the third magnetorheological finishing; (e) After the second smooth processing; (f) After the fourth magnetorheological finishing; (g) After the fifth magnetorheological finishing; (h) Mid-spatial error after the fifth magnetorheological finishing

Fig. 12. Actual machining mid-spatial error of #B. (a) Mid-spatial error in Fig.11(d); (b) Mid-spatial error in Fig.11(e)