Pre-compensation of Friction for CNC Machine Tools through Constructing a Nonlinear Model Predictive Scheme

In the feed drive system of CNC machine tools, friction, which is from the ball-screw and the linear motion guide, is one of the most crucial factors that restrict the positioning accuracy of machine tools [1]. At the velocity reversal points, the reversals of friction cause large tracking errors. To improve the positioning accuracy of the feed drives, various friction compensation techniques have been proposed.

The friction compensation methods can be classified into two groups, i.e., the model-free and the model-based friction compensation methods [2]. The model-free friction compensation method usually treats the friction as a disturbance, and develops various advanced controllers to suppress the adverse effect of friction [3]. Papageorgiou et al. [4, 5] implemented and experimentally compared several friction-resilient controllers. Ren et al. [6] estimated the friction with a reduced-order extended state observer, and designed a super-twisting sliding mode controller to control the three-wheeled omnidirectional mobile robot. Tian et al. [7] treated the friction as a rapidly changing disturbance during velocity reversal, and proposed an adaptive switching-gain sliding-mode-assisted disturbance observer to suppress the friction-induced tracking error. Su et al. [8] proposed a robust output feedback nonlinear proportional-derivative (PD) controller for the positioning of uncertain motion systems subject to the unknown friction with consideration of the actuator constraint.

The model-based friction compensation, which can be realized in the feedback or feedforward manner [9], cancels the friction torque by applying an additional drive torque according to the friction model. Armstrong-H\(\acute{\text{e}}\)louvry et al. [10] made a survey on the friction modeling and compensation, and proposed the well-known Stribeck curve to describe the friction characteristic of the lubricated metallic surfaces. The friction is divided into four regimes, i.e., the static friction, the boundary lubrication, the partial fluid lubrication and the full fluid lubrication. However, because the friction predicted by the Stribeck friction model is not continuous around zero velocity [11], the Stribeck friction model cannot be directly used in the friction compensation of precision motion control.

To realize the continuous and precise friction compensation, various friction models and compensation methods have been developed. The existing friction compensation methods can be classified into the static friction model-based methods and the dynamic friction model-based methods [12]. In terms of the static friction model-based methods, Makkar et al. [13] proposed a continuously differentiable friction model by expressing friction as the sum of three tanh functions and a linear function with respect to velocity. Based on this model, researchers developed various controllers, e.g., adaptive prescribed performance motion controller [14], robust adaptive tracking controller [15] and rise-based controller [16]. Xi et al. [17] proposed a two-stage tracking error-based static friction compensation method through expressing the compensation signal as a function of tracking error and velocity. Feng et al. [18] used a trapezoidal compensation pulse to compensate the friction-induced errors, and designed a generalized regression neural network algorithm to generate the optimal pulse amplitude function. Verbert et al. [19] expressed the friction force as a time-varying coefficient multiplied by the sign of velocity, and designed an online updating law to estimate this coefficient based on the position and velocity errors. Yang et al. [20] proposed a method to distinguish the boundary between the pre-sliding and sliding regimes, and proposed a two stage friction model that uniformly expresses the friction corresponding to the presliding and sliding regimes as functions of velocity. Huang et al. [21] deduced an analytical formulation to distinguish the breakaway point between the pre-sliding and sliding stages, and proposed a triple-stage friction compensation method to cancel the effect of static friction.

In terms of the dynamic friction model-based methods, Dahl [22] presented a friction model that describes the presliding friction as an elastic deflection of surface asperities. However, this model did not incorporate the Stribeck effect. Canudas de Wit et al. [23] proposed the well-known LuGre model, which captures most of the friction behaviours including the Stribeck effect, hysteresis, spring-like characteristics of friction, and varying break-away force. Yao et al. [24] proposed a continuously differentiable version of the LuGre model. Based on the LuGre model and its modifications, researchers developed parameter estimation and compensation method [25], neural network-based adaptive funnel sliding mode controller [26], adaptive barrier controller [27], adaptive compensator [28, 29], observer-based friction compensator [30], distributed friction compensator [31], and adaptive load friction compensator [32]. Dupont et al. [33] proposed the elastoplastic friction model for the control applications involving small displacements and velocities. Based on this model, Keck et al. [34] developed a friction parameter identification and compensation method. Al-Bender et al. [35] proposed the generalized Maxwell-slip (GMS) model. Based on the GMS model, various modifications [36, 37] and controllers [38] were developed. Bui et al. [39] proposed a new friction model, which combines the conventional Coulomb-viscous friction model and a nonlinear sinusoidal component, to better describe the friction in case of high speed motion or insufficient lubration. Guo et al. [40] proposed a neural network-based friction model, which takes the position and velocity information as input to predict the friction.

The above methods can realize precise prediction and compensation of friction. However, in order to obtain continuous compensation signal at the velocity reversal points, both the static and dynamic model-based methods usually introduce presliding-related parameters besides the Stribeck model parameters [13‐40]. For example, the boundary between presliding and sliding is required for the static model-based methods [17, 20, 21]. The stiffness and damping coefficients of the bristles [23‐28, 30‐32] are required for the LuGre model-based methods. The attraction parameters are required for the GMS model-based methods [35‐38]. Identifications of these parameters depend on the position measurements by rotary encoders or linear scales [21]. If the resolutions of rotary encoders or linear scales are high and the measurement noise is small [11], those presliding-related parameters can be precisely identified. However, if the resolutions of rotary encoders or linear scales are low or the measurement noise is relatively large, those parameters are difficult to be precisely identified whether by offline [20, 41‐44] methods or online methods [19, 27, 28, 30‐32]. This fact limits the scope of application of the existing methods.

To avoid the identification of the fine parameters related to pre-sliding, this paper proposes a non-linear model predictive scheme for friction pre-compensation of CNC machine tools based on Stribeck friction model. Through taking the friction-induced tracking error as the optimization objective, the optimal compensation signal is computed with the help of non-linear model predictive scheme. The derivative of the compensation signal is constrained to avoid the vibration of the machine tool. The advantages of the proposed method are three-folds. The contributions of this paper are three-folds. First, a precise prediction model of the nonlinear friction-induced tracking error of the P-PI controller with velocity feedforward is established, as described in Section 2. Second, the nonlinear model predictive friction pre-compensation method is developed with the constraining of derivative of compensation signal, as described in Section 3. Third, both air cutting and real cutting experiments are conducted to validate the proposed method, as presented in Section 4. Section 5 gives the conclusions.

In this section, the prediction model of the friction-induced tracking error is established, which includes the following three steps. First, the model of the actual feed drive system with non-linear friction and compensation signal is developed. Second, the model of the ideal feed drive system without friction is developed. Third, through calculating the difference between the predicted position of the actual feed drive system and that of the ideal feed drive system, the friction-induced tracking error is obtained.

Since the proposed method is aimed to compensate the friction, the friction-induced tracking error is adopted as the optimization target. If the aim is to directly reduce the tracking error, the tracking error can be directly taken as the optimization target.

Figure 3 shows the block diagram of the proposed nonlinear model predictive friction pre-compensation method.

The compensation signals generated by the NMPFP at time step k can be denoted bywhere \(n_\text{c}\) is the control horizon. According to the compensation signals, the friction-induced tracking error \(\mathcal {E}_k\) can be predicted by Eq. (10):where \(n_\text{p}\ge n_\text{c}\) is the prediction horizon. The compensation signals beyond the control horizon and within the prediction horizon are chosen to be the same as \(u_\text{c}(k+n_\text{c}-1)\), i.e.,The following objective function is chosen to minimize the friction-induced tracking error:If this objective function is directly applied without constraints, the optimal solution of \(\mathcal {U}_k\) will be a sharp step signal, as shown in Figure 4. This signal will impact the ball screw and cause the vibration of the machine tool.

Therefore, the derivative of the compensation signal needs to be constrained. However, the derivative of the compensation signal is not convenient to be constrained for the system model given in Eq. (6). Therefore, the system model in Eq. (6) is updated by the following incremental model.where \(\Delta u_\text{c}(k)=u_\text{c}(k+1)-u_\text{c}(k)\) is the incremental compensation signal of friction. Correspondingly, the compensation signals and the predicted tracking errors are updated byTo avoid sharp change of the compensation signal, the incremental compensation signal \(\Delta u_\text{c}(k)\) needs to be constrained:where \(\Delta \overline{u_\text{c}}\) is the upper bound of the incremental compensation signal. The nonlinear optimization problem at time step k can be summarized as follows:The optimization problem in Eq. (18) can be solved by the interior-point method [46] or trust-region method [47].

To evaluate the performance of the proposed method, simulations, air cutting experiments and real-cutting experiments are conducted. The experiments are conducted on an in-house developed open-architecture CNC machine tool, as shown in Figure 5.

The machine tool is controlled by a real-time control board, where various controllers can be implemented. The sampling frequency is set as 1000 Hz. The servo model parameters and friction model parameters of each axis are identified utilizing the method reported in Ref. [11]. The results are listed in Table 1.

The P-PI controller with velocity feedforward shown in Figure 1 is implemented on the real-time control board. The parameters of the P-PI controllers of each axis are tuned to minimize the tracking errors without causing vibration, and the parameters are listed in Table 1.

Three common trajectories, i.e., the 3D circular trajectory, the fan-shaped trajectory, and the butterfly trajectory, are adopted to validate the controlling performance in both ideal and complex cases. All the trajectories are planned under a maximum feedrate of 20 mm/s, a maximum acceleration of 200 mm/s\(^2\) and a maximum jerk of 2000 mm/s\(^3\) by using the S-shaped feedrate planning method reported in Ref. [45].

This paper proposes a nonlinear model predictive scheme to pre-compensate the friction for CNC machine tools. The developed method aims at improving the practical applicability through avoiding fine identification of the parameters in relation to pre-sliding, since these parameters are usually difficult to be accurately identified because of the small displacement of pre-sliding, the quantization error and noise of position measurement. Besides the model parameters of the Stribeck friction, the proposed approach only introduces a single parameter, i.e., the boundary of the derivative of compensation signal. This makes the proposed method easy to be parameterized.

The developed method is experimentally validated by evaluating its tracking and contouring performances, the robustness in terms of the uncertainties of model parameters, and the performance in the real cutting processes. Experiments show that compared with the P-PI controller without friction compensation, the proposed method can reduce the maximum tracking error by 56% to 81%, and reduce the maximum contour error by 50% to 69% for different trajectories. It is also validated that the proposed method has good robustness in terms of the uncertainties of servo model parameters and friction model parameters. For fluctuations of ± 10% in the servo model parameters J and B, the tracking errors are only 1-2 \(\mathrm{\upmu }\)m larger. For fluctuations of ± 10% in the friction model parameters, the tracking errors can still be reduced by more than 52%. Furthermore, experiments demonstrate that the proposed method shows similar tracking and contouring error reductions in real cutting as those in air cutting.

Since the nonlinear model predictive control method takes a relatively long time to calculate and is difficult to calculate online, the friction compensation signals are pre-calculated offline. However, this method lacks adaptivity to model parameter uncertainties and unknown disturbances. In the future research, the calculation speed of the algorithm can be improved to realize online compensation.