# Remote Servo Tuning System for Multi-Axis CNC Machine Tools Using a Virtual Machine Tool Approach
## 2 Preliminaries
### Feed Drive Systems  
Machine tool feed drive servo control systems are designed to accomplish a task, that is, to control the positions and velocities of machine tool axes in follow the commons generated by CNC controller. In general, the feed drive system contains two major parts. The first one is the electronic control system or servo control system. The second one is the mechanical system such as the lead screw, bearing, motor, and worktable, etc. The servo system consists of plant, actuator, controller, and sensor. The control system with feedback is a closed-loop system, and without feedback is an open-loop system. The most common structure of servo control is closed loop, which belonging to passive control, and it has multiple loop characteristics and each loop has close relationships.  
Many feed drive models have been proposed for system identification and controller design [2; 3; 6; 16; 31; 32]. Literature [31] proposes a virtual simulation model, the model contains the influence of the cutting force, friction and backlash nonlinear phenomenon. However, it assumes that the lead-screw is a rigid body, and the table mass and damping ratio were not considered. These reduce the accuracy and response of feed drive system. Figure 1 shows the dynamic model of the single-axis feed drive system with mechanical vibration, where \(J_{m}\): rotary inertia (kg\(\cdot\)m\({}^{2}\)), \(B_{m}\): damping coefficient of the rotary motion (\(N\)s/m), \(M_{t}\): table mass (kg), \(C_{t}\): damping coefficient of the lead-screw (\(N\)s/m), \(T\): motor torque(\(N\)m), \(K\): stiffness coefficient of the lead-screw (\(N\)/m), \(\theta_{m}\): motor rotation angle (rad), \(x_{t}\): table position (m), \(R\): transform ratio from motor rotation angle to table position (mm/rad) (\(R=p/2\pi\)), \(p\) is the lead-screw pitch (mm). The dynamic equation of this model can be expressed as:  
\[\left[\begin{array}{cc}J_{m}&0\\ 0&M_{t}\end{array}\right]\left[\begin{array}{c}\ddot{\theta}_{m}\\ \ddot{x}_{t}\end{array}\right]+\left[\begin{array}{cc}B_{m}&0\\ 0&M_{t}\end{array}\right]\left[\begin{array}{c}\dot{\theta}_{m}\\ \dot{x}_{t}\end{array}\right]+\left[\begin{array  
resistance (\(\Omega\)), \(L_{a}\) is motor inductance (H), \(e_{b}\) is total voltage (V) induced in the armature, and \(\omega\) is motor velocity (rad/s). Therefore, we have  
\[\frac{\omega(s)}{v_{a}(s)}=\frac{K_{t}}{(J_{m}s+B_{m})(L_{a}s+R_{a})+K_{e}K_{t}} \tag{2}\]  
The corresponding block diagram of the feed drive system is shown in Figure 2; Figure 2a-c show the servo motor system, feed mechanical dynamic systems, and the overall closed-loop systems, respectively. The proportional (P) or proportional-integral (PI) or proportional-integral-derivative (PID) controller are typically used to control the servo systems of machine tools. In general, current and velocity loops are usually part of the PI controller and the position loop is in the P controller [13; 22; 31].  
Figure 1: Dynamic model of single-axis feed drive system; (**a**) mechanical dynamics, (**b**) motor dynamics.  
Figure 2: \(Cont\).  
From Figure 2c, the closed-loop consists of a current loop, velocity loop and position loop. The current loop has a high-frequency response and the bandwidth is higher than processing. The typical values of bandwidths of the current, velocity, and position loops are 100-3000 Hz, 10-100 Hz, and 1-50 Hz, respectively. Since the control target of the milling application is the position control and the current loop gain is at unity in the low frequency band, therefore, the current loop can be bypassed in the applications of milling. Thus, the transfer function is:  
\[G_{p}(s)=\frac{x_{t}(s)}{x_{m}(s)}=\frac{\big{(}K_{p}K_{t}K_{vp}K\big{)}s+K_{p} K_{t}K_{vi}K}{s_{5}5^{5}+a_{4}s^{4}+a_{3}s^{3}+a_{2}s^{2}+a_{1}as+a_{0}} \tag{3}\]  
where \(a_{5}=J_{m}M_{t}\), \(a_{4}=C_{t}J_{m}+B_{m}M_{t}+K_{t}K_{vp}M_{t}\), \(a_{3}=J_{m}K+B_{m}C_{t}+C_{t}K_{vp}+K_{t}K_{vi}M_{t}\), \(a_{2}=B_{m}K+C_{t}K_{vi}K_{vi}\), \(a_{1}=K_{p}K_{t}K_{vp}K\), and \(a_{0}=K_{p}K_{t}K_{vi}K\).