Dwell time algorithm based on optimization theory for magnetorheological finishing

Magnetorheological finishing (MRF) is a deterministic polishing technique capable of rapidly converging to the required surface figure. This process can deterministically control the amount of removed material by varying the time to dwell at each particular position on the workpiece surface. The dwell time algorithm is one of the most important key techniques of the MRF. A dwell time algorithm based on matrix equation and optimization theory was presented in this paper. The previous mathematical model of the dwell time was transferred to a matrix equation containing initial surface error and removal function. The required dwell time was just the solution to the large, sparse matrix equation. A new mathematical model of the dwell time based on the optimization theory was established, which aims to minimize the 2-norm or -norm of the residual error. The solution meets almost all the requirements of precise computer numerical control (CNC) without any need for extra data processing, because this optimization model has taken some polishing condition as the constraints. Practical approaches to find a minimal least-squares solution and a minimal maximum solution are also discussed in the paper. Simulations have shown that the proposed algorithm is numerically robust and reliable. With this algorithm an experiment has been performed on the MRF machine developed by ourselves. After 4.7 minutespolishing, the figure error of a flat workpiece with a 50mm diameter is improved by PV from 0.191 to 0.087 and RMS 0.041 to 0.010. This algorithm can be constructed to polish workpieces of all shapes including flats, spheres, aspheres and prisms.