Conventional trajectory  optimization  strategies  use  model data of the machine tool and the cutting process. Model-based trajectory optimization focuses on free form surfaces that have been traditionally defined using large numbers of small line seg- ments. The optimization algorithms fit splines through the line segments that conform to desired constraints like chord error, velocity, acceleration, and jerk. This is a challenging fitting and optimization problem that is solved using quintic splines [2,3] or parametric curves [4–7]. A growing number of machine tool manufacturers have added non-uniform rational Basis Spline (NURBS) as a proprietary G-code extension to their CNC machines since they can represent freeform trajectories with smaller G-code files. When the tool paths are defined by NURBS, the optimization task reduces to selecting the optimum feedrate along the spline. For constraining velocity  and  acceleration in two or three axes Cartesian machine tools, analytical feedrate optimization strategies have been proposed [8,9]. One efficient algorithm to constrain acceleration employs an iterative two-pass strategy for first limiting acceleration in the forward pass, and Journal of Manufacturing Science and Engineering JANUARY 2017, Vol. 139 / 011012-1Copyright VC   2017 by ASME Measurements of position error were used in Refs. [18] and [19] to add this additional feedrate constraint, f servo  Fig. 1 CNC architectures based on modeled and measured process data decelerations in the backward pass [10,11]. Other researchers have added maximum jerk values to the constraint equations at the expense of computing efficiency, since numerical search techni- ques need to be applied for optimizing feedrate in the presence of jerk constraints [12–15]. Finally, trajectory optimization strategies for five axis non Cartesian machines have also been presented [16]. The model-based trajectory optimization techniques described above rely on machine tool data and cutting process data, which is often not available with a high degree of accuracy. To reduce the risk of tool or part breakage, measurements of path error and/or cutting forces can be used to monitor the machining process and

adjust feedrates in real-time [12,18–24] as shown in Fig. 1.

The real-time process  monitoring  and  control  systems in Refs. [20] and [21] avoid tool breakage constraints by continu- ously adjusting feedrate. Both in turning [20] and milling [21], tool edge breakage is avoided by constraining the equivalent chip thickness, he. For tool shank breakage in milling [21], the in-plane resultant force, Fr  is constrained.

In the work described in Ref. [21], the ratio of the measured equivalent chip thickness hemeasured over the allowable chip thick- ness is multiplied by the current feedrate override in order to arrive at an updated feedrate override, f edge  Measured process data-based feedrate optimization has been shown to significantly reduce contouring error during sharp cor- ners and circles [18,19]. However, the resulting acceleration pro- files during the corner transients are parabolic (zero  acceleration at zero velocity), which indicates that they are conservative and do not take full advantage of the machine’s acceleration capabil- ities. In addition, this architecture works best (lowest contouring errors) for symmetrical reversals, but can deteriorate for other corner scenarios.

The aim of this paper is to add a model-based feedrate override, fmodel, as a baseline to the measurement-based process control system shown in Refs. [18–21]f ¼ minðfRT; fmodelÞ (5)

3 Real-Time Feedrate Override

Conventional CNC machines allow the operator to manually adjust feedrate override, which effectively scales the feed velocity used by the machine tool. In this study, a dynamic feedrate over- ride is implemented that automatically adjusts the feedrate while the machine tool traverses along the programmed path. The pro- posed feedrate optimization strategy combines both model-based and measurement-based feedrate override using the relationship in Eq. (5).