## 1 Introduction  
Flutes, as the major structure of end-mills, play an important role in the cutting performance [1; 2; 3; 4]. A flute can be defined by the following three parameters: core radius, flute angle and rake angle [5; 6; 7]. The rake angle influences the cutting force, while the core radius and the helix angle determine the stiffness and chip-removal capacity of the cutters. The performance of those flute parameters quietly depends on the manufacturing accuracy. Generally, the flute is manufactured by the CNC grinding machine through the determination of location and orientation for the grinding wheel path [8; 9; 10]. In the flute-grinding operations, the grinding wheel will move with a helix path to generate the grooves. In recent years, much attention has been paid to developing an advanced wheel path determination model and optimized algorithms for the CNC flute grinding to minimize the manufacture errors and improve calculation efficiency. The kinematics of flute-grinding operations for CNC grinders has been developed by many researchers. For instance, [PERSON] et al. (2011) developed a simulation method with Boolean operations to construct the helix motion and developed an iterative process to compute the wheel geometry and location data. Although this method could be used for virtual cutting tests in the CAM system, it is time-consuming to achieve a high machining accuracy. To improve the precision of generated flutes, [PERSON] (2012) established a novel algorithm to calculate the numerical data of flutes based on the enveloping theory, in which the flute profile was interpolated with appropriate discrete points using the cubic polynomial expression. However, this method might be invalid for grinding with a bevel-type wheel. In order to calculate the location and orientation of wheels with complex shapes (e.g., 1B1, 1E1, 1F1, and 4Y1 wheels), [PERSON] et al. (2013) used virtual grinding curves to formulate the grinding processes. Based on the virtual curves, they calculated the grinding error with the worn wheels and compensated the wheel path. Generally, the kinematic of flute-grinding was represented by several transcendental equations, which were supposed to be solved to get the generated helical flute. However, it is very complicated to give the analytical solution, and thus numerical analysis was generally used to describe the flute profile in current studies, which suffer from long computation time. For free-form grinding wheels, [PERSON] et al. (2014) presented a novel method for five-axis CNC grinding through the optimization of the grinding wheel geometry, which was constructed with line segments and circular arcs. Although this method could economically produce or dress the grinding wheel for accurately grinding the end-mill cutters, it ignored the optimization of the wheels' path. On the basis of the flute-grinding model, the following problem is how to achieve the desired flute parameters by setting the trajectory of the grinding wheel. In industry, the conventional way is to grind end-mill flutes by trial and error, which is costly and time-consuming (2015). Mathematically, the desired flute profile can be viewed as an optimization problem with regards to the wheel's shape and configuration. To solve this problem, [PERSON] et al. (2016) proposed an iteration algorithm to determine the wheel location and orientation. For each loop, the generated flute parameters were compared with the desired values until they converged within the target range. The iteration method has a high calculating speed and precision, but it required a proper initial value, which cannot be easily determined without experience. In [PERSON]'s research (2017), particle swarm optimization (PSO) was investigated to search the wheel location for a given helical flute and grinding wheel profile. Recently, [PERSON] et al. (2018) extended this work with a novel graphic analysis method and niche particle swarm optimization (NPSO) algorithm to solve the problem for multi-objective of machining accuracy.