Preface Adaptive constant force control of the cutting process through online adjustment of feed rate is an effective way to significantly improve the productivity of CNC milling machines, and it is also a topic that many scholars have been studying for many years. However, in the various adaptive constant force control algorithms proposed by predecessors [1-6], the parameter tuning of the controller usually only depends on the past and current dynamic behavior of the controlled system, without considering the influence of control input and system output prospects, and without imposing reasonable constraints on the controller. Therefore, when the milling force changes abruptly due to the abrupt change in depth of cut or width of cut, it usually leads to overshoot of the controlled system output or excessive control input. To overcome the above problems, this paper studies the method of constructing a constrained generalized predictive control law based on the characteristics of the CNC milling process, and proposes an analytical algorithm for controller parameter tuning accordingly. Simulation and experimental results show that the method has the advantages of strong engineering applicability, good robustness, and the ability to meet real-time control requirements. 1 Design of Unconstrained Generalized Predictive Control Law As shown in Figure 1, the constant force control system of the CNC milling process consists of a controller and a milling process. Among them, the servo feed and the milling machining are connected in series to form the milling process. In the figure, vf is the feed rate, F is the actual milling force, and Fr is the reference milling force. Figure 1. The constant force control system for CNC milling comprehensively considers the dynamic characteristics of the servo system and the tool deformation, and the transient milling process can be simplified to a first- or second-order linear system [9]. A(z-1)f(t)=B(z-1)vf(t-1) Where A and B are polynomials of the backward propagation operator z-1, and A(z-1)=1+a1z-1+a2z-2 B(z-1)=b0+b1z-1 The coefficients a1, a2, b0 and b1 can be estimated by the recursive least squares method. 2 Constraints Engineering practice shows that the constant force control of the CNC milling process needs to consider the following constraints: the feed rate should be within the design range of the machine tool and the milling force should be less than the limit load of the machine tool-tool-workpiece system. The feed rate increment should be less than the acceleration/deceleration limits of each coordinate, and the upper limit of the increment should be limited during tool idle cutting to prevent tool breakage due to excessive milling force during entry. When the actual milling force exceeds the set value, the feed rate should be reduced as soon as possible. Milling force overshoot should be effectively controlled to avoid the impact of tool deformation on surface quality. In summary, constant force control in CNC milling requires constraints on the feed rate and its increment, the rise time of the actual milling force, and overshoot. 3 Design of Constrained Generalized Predictive Control Law Analytical Algorithm: From the equation, the performance index function is generally a hypersurface in the optimization space (ΔvT, J) ∈ RNv+1. Note that the control foreground is Nv=2, so the QP problem can be solved in three-dimensional space. Since the components of Δv are Δvf(t) and Δvf(t+1), according to equation (14), the i-th constraint can be expressed as d1iΔv(t) + d2iΔv(t+1) ≤ ci. In practical control problems, since Δvf(t) and Δvf(t+1) are only in the first quadrant (F(t)) of the plane spanned by the two, Therefore, we can define two types of constraints: Definition: If a constraint forms a closed domain with the coordinate axes or a band-shaped region with the coordinate axes, it is called a first-type constraint; otherwise, it is called a second-type constraint. Let the number of the two types of constraints be k′ and kk′, respectively. Then, according to the above definition and Figure 2, there are three cases for the location of the unconstrained minimum point: ① The minimum point is within the feasible region, and the unconstrained solution is the same as the constrained solution. ② The first-type constraint pi (i=1,…,k′) is not satisfied. ③ The second-type constraint p′j (j=k′+1,…,k) is not satisfied. Based on this, we can construct an analytical solution for the constrained generalized predictive control law. Figure 2. Location of the unconstrained minimum point. If the unconstrained minimum point of the performance index function is outside the feasible region, note that plane Γ passes through the unconstrained minimum point (line) and intersects the feasible region, and the intersection line JΓ of plane Γ and performance index function J is monotonically increasing on either side of the minimum point. Therefore, the constrained minimum point must be the intersection point of Γ and the boundary of the feasible region. Based on this, the constraint boundary can be obtained by intersection operation and comparison of intersection point coordinates, and then the analytical solution of the feed rate can be obtained on the constraint boundary using analytical method. Computer simulation shows that compared with the unconstrained control strategy, the performance of the closed-loop system can be significantly improved, and the algorithm can meet the real-time control requirements [9]. 4. Cutting test. The cutting test equipment is a Cincinatti H1000 horizontal machining center, and the controller is a DSPmaster-C50 signal processing board. During the test, the photoelectric encoder installed at the end of the spindle provides 1024 pulses per revolution as the external clock of the DSP board to keep the sampling period synchronized with the spindle speed. The cutting force signal picked up by the KISTLER force gauge is amplified and filtered by charge amplification, then synthesized by the DSP board according to the effective value, and the control command is calculated according to the aforementioned algorithm. The control command is input to the built-in PLC of the CNC system via an optically isolated power amplifier, and the feed rate is adjusted in real time by rewriting the feed rate register (see Figure 3). The advantage of this interface scheme is that only the PLC program needs to be changed without changing the CNC hardware. After two effective force signals are measured, the estimator and controller start working. Figure 3 Schematic diagram of the test device The test conditions are as follows: Tool: 24 mm diameter three-tooth high-speed steel spiral bar milling cutter, helix angle 30°. Workpiece: Q235 high-quality carbon steel (geometric dimensions are shown in Figure 4). Figure 4 Workpiece shape Cutting conditions: Spindle speed 300 r/min, radial depth of cut 4 mm, axial depth of cut 15 mm, 20 mm, 25 mm and 30 mm respectively, cutting length 50 mm, 40 mm, 30 mm and 20 mm respectively, conventional milling, oil cooling. Controller parameter settings: Milling force setpoint 400 N, CNC programming speed 40 mm/min, minimum and maximum feed rates set according to the original CNC system values, 0 and 120% respectively, with each increment interval of 4%. Control commands are rounded using the decimal point discarding method to ensure the steady-state milling force is less than the setpoint. The controller's preset parameters are Nv=2, N=5, and ρ=80, and the control effect is shown in Figure 5. Figure 5: Constant Force Milling. As can be seen from the experimental results, during the air cutting phase, since the milling force is zero, the feed rate quickly rises to the set upper limit, i.e., 40 × 120% = 48 mm/min. Once cutting begins, because the milling force is greater than the setpoint, the controller adjusts the feed rate, allowing the milling force to quickly track the setpoint. Although the milling force is not exactly equal to the setpoint due to the feed rate increments, the steady-state milling force remains less than the setpoint throughout the entire cutting process, achieving the expected control effect. When the axial depth of cut is 30 mm, the feed rate reaches the set lower limit, i.e., 40×36%=14.4 mm/min. Figure 6 shows the milling force results when the cutting conditions remain unchanged and constant force control is not implemented, and the CNC programmed feed rate is 40 mm/min with a feed ratio of 36%. As can be seen from the figure, the maximum milling force is about 400 N, which is equal to the milling force set value in constant force milling. However, if the empty cutting stage is not considered, ordinary milling takes about 620 s, while constant force milling takes only 380 s, which shows that the improvement in milling efficiency is very significant. Figure 6 Ordinary milling feed rate 40×36%=14.4 mm/min 5 Conclusions This paper studies the constrained generalized predictive control method of CNC milling process and obtains the following conclusions: (1) Generalized predictive control has excellent stability and output performance because it takes into account the influence of the input and output prospects of the controlled system, and the algorithm structure is suitable for including constraints. (2) The four constraints proposed for the characteristics of CNC milling can significantly improve the output performance of the controlled system. (3) The proposed constrained generalized predictive control law analytical algorithm can meet the real-time requirements of constant force control in CNC milling. About the author: Huang Tian, ​​male, born in 1953, PhD, professor, doctoral supervisor, director of the Institute of Advanced Manufacturing Technology, Tianjin University, and expert of Tianjin Machinery Manufacturing. Research interests: parallel machine tool design and manufacturing, mechanical dynamics, intelligent control. He has published more than 60 papers and won two Ministry of Education Science and Technology Progress Awards. Author affiliation: School of Mechanical Engineering, Tianjin University, Tianjin 300072 Reference 1 Tomizuka M, Oh JH, Dornfeld D A. Model reference adaptive control of the milling process. InHardtD E, Book WJ eds. Control of Manufacturing Process and Robotic Systems, ASME, New York, 198337～44 2 Damehmend LK, Pak H A. Model reference adaptive control of cutting force in milling. InDonath Med. Dynamic Systems Modeling and Control, ASME, New York, 198543～50 3 Lauderbaugh LK, Ulsoy A G. Model reference adaptive control in milling. ASME J. Engng. Ind., (1).198913～21 4 Altintas Y. Direct adaptive control of end milling process. Int.J.Mach. Tools Manufact.,34(4),1994.461～474 5 Hsu PL, Hsieh M Y. Application of self-tuning control on industrial CNC machines. Int. J. Mach. ToolsManufact., 1994, 34(6): 859-877. 6 Huang SJ, Chiou K C. The application of neural networks in self-tuning constant force control. Int. J. Mach. ToolsManufact. 1996, 36(1): 17-31. 7 Clarke DW, Mohtadi C, Tuffs P S. Generalized predictive control—Part and Part. Automatica, 1987, 23(2): 137-160. 8 Astrom KJ, Wittenmark B. Adaptive control. New York Addison Wesley, 1989. 9 Li Weimin. Constrained generalized predictive control of CNC milling process based on multi-sensor information [Doctoral dissertation]. Tianjin University, 1998.