## 1 Introduction  
With the rapid development of astronomy, space exploration, and advanced optical instruments, optical elements are being increasingly widely used. The application demands for high-quality and high-efficiency elements present distinct higher requirements for the process technology used in such elements [1]. Computer-controlled optical surfacing (CCOS) has been successfully applied in industrial production, as it can precisely correct the surface error by converting the dwell time of the polishing tool into the feed-rate along the polishing trajectory. Therefore, a trajectory planning method is the key factor affecting high-quality, high-efficiency polishing. Despite the extent of research on path planning in some fields [2; 3; 4], investigation of optical polishing has been rather limited. Although the problem of discontinuous surfaces between the adjacent mm-sized short line segments has attracted wide concern when using parametric curve theory [5; 6; 7], frequent acceleration and deceleration result in poor realisation precision of dwell time, i.e., the runtime of trajectory (hereafter referred to as runtime), which further influences the polishing quality of elements and the convergence rate of surfaces [8]. The main focus of this paper is to investigate an interpolation scheme to overcome the above problem, which has two main progressive aspects: fitting and interpolation of parametric curves. The fitting of parametric curves is a process of converting discrete short line segments into parametric curves. At present, many scholars have carried out research based on the dominant points using a non-uniform rational B-spline (NURBS). [PERSON] [9; 10] proposed a method for determining the dominant points according to the discrete curvature. [PERSON] [11] and [PERSON] [12] improved this method by taking concave-convex turning points and extreme points on the curvature curve as dominant points. To improve the fitting precision, [PERSON] [13] proposed curve fitting taking squared distance minimisation (SDM) as the evaluation index. Although the dominant points based method is easy with regard to calculation and interpolation, it may result in the loss of runtime at non-dominant points. Thus, only the global fitting method is suitable to the optical polishing. To do so, [PERSON] [14] proposed an optimisation algorithm by establishing the evaluation function for deviation of the fitted distance. [PERSON] [15] and [PERSON] [16] classified the trajectory into the different forms of NURBS and then employed a piecewise fitting method for real-time implementation. Based on Gaussian elimination and the continuous short block (CSB) look-ahead algorithm, [PERSON] [17] and [PERSON] [18] realised the on-line transformation from short line segments to NURBS: however, for the global fitting method, the simplified strategy that setting all the weight factors as 1 eliminates the regulating effect on the fairness of curves and easily causes curvature saltation on the trajectory. Therefore, generating a fair trajectory based on NURBS is the first problem facing the polishing trajectory planning technique. The interpolation of parametric curves is a process that discretises the NURBS into numerical control (NC) commands. Speed planning, as the critical step in the interpolation process, has been an area of research for numerous scholars: this can be classified into the time-optimal approach and the non-time-optimal approach. The time-optimal approach deals with the planning problem by taking the minimisation of the motion time as the objective to promote manufacturing efficiency [19; 20]. For example, [PERSON] [21] proposed a speed planning scheme for NC interpolation by the use of the optimal control theory. On this basis, [PERSON] [22] and [PERSON] [23] considered the driving capacity constraint and trajectory precision constraint in the interpolation, respectively. The non-time-optimal approach usually deals with the planning problem by taking the minimisation of speed fluctuations as the objective [24].