Tuyển tập công trình Hội nghị Cơ điện tử toàn quốc lần thứ 6 795 Mã bài: 168 Mô hình Maxwell-Slip để bù ma sát trong điều khiển khớp mềm khí nén một bậc tự do: một bộ bù Lead-Lag phi tuyến tương đương Maxwell-Slip Model for Hysteresis Compensation in Controlling an 1-DOF Pneumatic Soft Joint: An Equivalently Nonlinear Lead-Lag Compensator Võ Minh Trí Trường Đại học Cần Thơ e-Mail: vmtri@ctu.edu.vn Tóm tắt Một trong những ứng dụng điển hình của cơ nhân tạo khí nén là khớp đôi, được tạo thành từ một cặp cơ. Điều khiển khớp này vẫn còn còn nhiều vấn đề cần nghiên cứu, bởi (i) tính phi tuyến của hệ thống khí nén, (ii) trễ phi tuyến xảy ra trong lớp vỏ của cơ bắp, và (iii) tính biến đổi theo thời gian. Trong hầu hết các nghiên cứu gần đây, mô hình khớp này thường được tính gần đúng, tính phi tuyến và tính bất định (nếu có) của mô hình đã được khắc phục bằng hệ thống điều khiển đi kèm. Trong nghiên cứu này, mô hình khớp đã được phát triển bao gồm cả phần trễ của áp suất theo vị trí, theo đó phần này được mô tả chính xác bằng cách sử dụng mô hình Maxwell-Slip (MS). Mô hình này rất thích hợp để nhúng vào các chương trình điều khiển để bù sự trễ. Hệ thống điều khiển đi kèm do đó đơn giản hóa và chỉ cần áp dụng một bộ điều khiển PID thông thường. Hoạt đồng của hệ thống điều khiển vị trí của khớp quay với đáp ứng bước rất hiệu quả khi bộ bù trễ hoạt động. Phân tích FRF cũng cho thấy rằng việc bù trực tuyến tạo ra một kiểu giống với bộ bù phi tuyến lead-lag. Abstract One of the typical applications of pneumatic artificial muscles (PAMs) is the antagonistic manipulator, which is made up of a pair of PAMs. Control of this manipulator is still an on-going issue, facing with (i) the nonlinearity of a pneumatic system, (ii) the nonlinear hysteresis occurring in the PAM sheath, and (iii) the time-varying problem due to the PAM creep. In most recent studies, the manipulator model is a rough approximation, and the nonlinearities and/or the model uncertainties were left to be overcome by the associated control system. In this paper, the novel joint model was developed including the extracted pressure difference-joint angle hysteresis part, which can be accurately described by using the Maxwell-Slip (MS) model. This hysteresis model is suitable for embedding in the control scheme in terms of hysteresis compensation. The associated control system is therefore simplified to the application of a conventional PID controller. The position control performance of the manipulator with step response showed effectively improved when the hysteresis compensation is introduced. The FRF analysis also showed that the online compensation generated a kind of equivalently nonlinear lead-lag compensator. Nomenclature DOF Degree of Freedom PAM Pneumatic Artificial Muscle MS PID FRF HC Maxwell-Slip Proportional Integral Derivative Frequency Response Function Hysteresis Compensation 1. Introduction One of the typical applications of a pneumatic artificial muscle (PAM) is the antagonistic manipulator, which is made up of a pair of antagonistically arranged PAMs, simply called muscle. This one degree-of-freedom pneumatic muscle manipulator can operate as a soft joint owing to its inherent compliance with a well- known advantage, i. e. its angular position and its stiffness can be controlled independently. This setup is considered as a common case-study for most researchers before they go to more complex and multi-degree-of-freedom robot manipulators [1, 2, 3, 4, 5, 6], or for those who want to investigate the control problems involved [7]. The antagonistic system manifests the advantages of the individual muscles, such as large power/volume or power/weight ratios, compliance, direct power, cleanliness, low cost, etc. However, the existing drawbacks of the individual muscles come also to exist in the coupled system, i.e. poor performance in position control due to the load variations affecting on the position [2], and the 796 Võ Minh Trí VCM2012 inherent hysteresis and muscle creep [8] which leads to nonlinearities and uncertainties in the system. The occurrence of hysteresis, muscle creep and model uncertainties appear to be the most challenging for controlling the pneumatic muscle manipulator. As difficult as finding an accurate model for an individual PAM, the investigators usually approximate the manipulator dynamics with a rough model. Also muscle creep due to the environmental effects on the muscle sheath, results in extra degrees of model uncertainties. In addition, the hystereses of the two PAMs do not cancel each other when integrated in the antagonistic system, leading to nonlinearities in this coupled system. These nonlinearities and uncertainties are a source of difficulty for developing a position control which is still an on- going issue challenging researchers. Literature shows that most investigators choose for developing advanced control algorithms to control the joint position. Some advanced techniques were proposed to overcome the nonlinearities and uncertainties such as pole placement methods [1], neural networks control [8,14], variable structure control or sliding mode control [15,16], or using a combination of more advanced techniques such as nonlinear PID control [6], adaptive fuzzy logic sliding mode control [17]. Most of these studies investigate on tracking position problems. A few authors [11, 15] consider the load effects. In this work, all possibilities leading to the nonlinearities, uncertainties, and time-varying parameters are carefully analyzed and synthesized in an empirical model. In this approach, the constraint model of the static torque of the antagonistic manipulator joint, incorporated with the muscle creep, is determined [18]. The separated hysteresis using the constraint model makes it nullified to the creep effect. This hysteresis is in the form of the extracted pressure difference/joint angle hysteresis which is found suitable to be implemented in the control scheme in terms of hysteresis compensation as shown in Fig.1. The paper is organized as follows. Section 2 describes the experiment apparatus and defines the system variables. Section 3 shows how to model the pressure difference/angle hysteresis. Section 4 presents the experimental results supported by an extensive discussion. Section 5 closes with the conclusions and future work. 2. Experimental setup The test setup consists of a pair of FESTO fluidic muscles (type MAS-20-200N) configured into an antagonistic configuration. A pair of FESTO directional proportional valves (type MYPE-5- M5-010B) was used to exactly control the pressure in each muscle. Two similar pressure sensors (SENSORTECHNICS type of PTE5151D1A) were mounted between the valves’ outlet and the muscle inlet to provide feedback for the pressure control loop. In order to create a rotary joint, one muscle end was connected to the other via a timing belt that spanned a timing gear, which was mounted on an axis with two supporting bearings. Two coupling adaptors were designed to connect the other ends of the two muscles to the load cells (type DBBP-200 from BONGSHIN), which were fixed to the support. The joint axis and the load cell support were fixed to a rigid frame. The joint rotation was measured by an incremental rotary encoder (PANASONIC type of E6C-CWZ1C-M). An arm or a lever, fixed to the joint axis, was used to exert an external torque to the joint. All I/O information from/to the plant setup was processed by a 16-bit data acquisition card DAQmx NI-6229 from National Instruments, embedded in a real- time desktop PC. The control and measurement algorithms were developed based on LabView Professional Development System for Windows with the add-on LabView Real-Time Module. In the two branches of the setup (e.g. agonistic and antagonistic) the components and control elements (Fig. 2) were selected identically such that the behaviour of the antagonistic system is symmetric with respect to the centre (initial) position. The initial position is also called the zero position since with our design the joint angle starts at 0 degrees at this position. PI Muscle volume - + d r r P d P Mechanical system Hys. Compensation + + hys P H1. Proposed control scheme of antagonistic setup with hysteresis compensation Tuyển tập công trình Hội nghị Cơ điện tử toàn quốc lần thứ 6 797 Mã bài: 168 H. 2 The antagonistic setup The joint angle starting from the centre position is positive, measured clockwise and negative, measured counterclockwise. The pressure difference, 1 2 p p p , plays a role as input to activate the joint motion, which is considered as the output of the control system. 1 2 ( ) T r F F is the joint torque Subscript 1, 2 indicate the right-hand side muscle and the left-hand side muscle The constraint model takes the form: ( ) ( ) const T a p b (1) where ( ) a is the constraint model’s slope and ( ) b the constraint model’s intercept. They are functions of the joint angle [18]. The model of Eq. 1 allows us to calculate the pressure difference needed to act against the torque exerted at a certain joint angle as follows: ( ) ( ) T b p a (2) 3. Modeling 3.1 Pressure difference/joint angle hysteresis extraction In the isotonic condition test [13], the pressure differences needed to move the external payload forward and backward, are different. This is not due to the gravity torque acting on the joint, which, depends only on the joint position, not on the direction of motion. Meanwhile, the force generated by each muscle depends on the motion direction due to hysteresis effects [12]. If the required torque generated by the coupled muscles to counter the gravity torque which is regardless of the motion direction, the pressure of each muscle has to adapt automatically to the requested force that is contributing to the joint torque. As a result, the pressure differences needed at the same equilibrium position of the pendulum arm will be different depending on which direction the joint is moving towards. The pressure difference hysteresis loops linked to the joint motion, are shown in Fig. 3. -40 -30 -20 -10 0 10 20 30 40 -5 -4 -3 -2 -1 0 1 2 3 4 5 Joint angle (degree) Pressure difference (bar) load 1kg load 2.75kg load 4.5kg simulated at 1kg H. 3 The isotonic measurements at different loads compared to simulated pressure difference reconstructed from the constraint model. This hysteresis is called the pressure difference/joint angle hysteresis. It behaves similarly to the torque/joint angle hysteresis exposing nonlocal memory and quasi rate independency. This type of hysteresis can be modeled by using the Maxwell-slip model [18]. From this figure, it can be observed that the heavier the load, the shorter the interval motion of the pendulum arm and the higher the slope of the hysteresis loop will be. If each pressure difference/joint angle hysteresis at an isotonic condition would be modeled separately. This would lead to a number of models which can be avoided as follows. As discussed earlier, the constraint model was formed in the condition without hysteresis. From this model, the pressure difference needed for a certain exerted torque can be predicted exactly using (Eq.2). This pressure difference is also called the constraint pressure difference. However the pressure difference/joint angle hysteresis is inevitable and makes the actual pressure difference needed a bit higher than the constraint pressure. The actual pressure difference needed is referred to as the isotonic pressure difference and is governed by Eq 3. If a comparison is made between the measured (loop 2 c U 2 P s P 2 CV s P 1 CV 1 P 1 c U 0 P: muscle pressure θ: joint angle P s : supply pressure r: timing gear radius CV: control valve F: muscle force U c : valve control signal Δ P=P 1 - P 2 : pressure difference 2 F 1 F r T T 1 Muscle 2 Muscle arm payloa d 798 Võ Minh Trí VCM2012 type) and the simulated (single line) at the same load (for example a 1 kg load as shown in Fig. 3), the extracted hysteresis loop of a heavier load is always enclosed by the extracted hysteresis loop of a lighter load, as shown in Fig. 4. From this figure, it may be visibly concluded that all of them match the shape of hysteresis with nonlocal memory [12]. This allows us to select the biggest loop to derive the model. This loop is referred to as a global loop since it can represent all other smaller local loops corresponding to the smaller loads. isot cons hys p p p (3) hys isot cons p p p (4) where isot p is the actual pressure difference measured from the isotonic tests, cons p is the pressure difference reconstructed from the constraint model with respect to the measured torque and joint angle in the isotonic tests, and hys p is the extracted hysteresis resulting from differentiating between the actual pressure difference and the reconstructed pressure difference. 3.2 Modeling process The procedure to model hysteresis using the MS model has been discussed in previous papers [12, 13, 18] and is applied in this paper. The boundary of the global hysteresis loop is geometrically defined as the dotted curve also shown in Fig. 4. Notice that the intuitive selection of the asymptotes for identifying the MS elements is usually made by visual inspection of the of the hysteresis loop, in which the trade-off between the model accuracy and the nonlinear behavior of the two muscle interaction close to the outmost pendulum positions is considered. Four steps of hysteresis identification are recalled hereby: Obtain the hysteresis loop experimentally. Shrink the upper (or lower) half of hysteresis loop to get the virgin curve. Pick up intuitively the segments which are asymptotes to the virgin curve. Calculate the stiffness and saturation level for each element by establishing and solving a system of equations that is formulated based on the coordinates of the end points of the selected segments on the plot. -40 -30 -20 -10 0 10 20 30 40 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 Joint angle (degree) Extracted hysteresis (bar) load 1kg load 2.75kg load 4.5kg H. 4 The extracted hysteresis loops: the bigger loop corresponding to the lighter load encloses the smaller loop of the heavier load. The shape of the loops matches that of hysteresis with nonlocal memory. 4. Results and discussion 4.1 Pressure difference/joint angle hysteresis capture -40 -30 -20 -10 0 10 20 30 40 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 Joint angle (degree) Extracted hysteresis (bar) measured modelled H. 5 Extracted pressure difference, captured for arbitrary joint trajectories The boundary of the global loop shown in Fig. 4 is constructed by the double-stretched virgin curve (upper half loop) and flipped-double-stretched virgin curve (lower half loop). Based on this boundary, the virgin curve is withdrawn, and the parameters for four MS elements are calculated and shown in Table I, where ‘k’ and ‘w’ are slopes and saturation values respectively of each Maxwell-element 1 to 4. These parameters are used for online simulation and to compare the measurement to the extracted hysteresis loop. Fig. 5 illustrates that the model captures very well the experimental hysteresis loops. The mismatch of the model could be foreseen as a tradeoff between the model accuracy and the nonlinear regions had to be made beforehand when picking up the MS elements. It means that we accept larger model errors in the outmost pendulum positions while reducing the errors at rest of the joint motion interval, as shown in Fig. 6. This compromise is Tuyển tập công trình Hội nghị Cơ điện tử toàn quốc lần thứ 6 799 Mã bài: 168 based on the fact that close to the outermost positions, the pressure loop has a higher damping ratio owing to a friction flow (Fig. 7, at 40-degree amplitude of excitation), while in the middle range we try to keep a better fit between the model and the measurements, for serving hysteresis compensation later on. Table 1. Four Maxwell-slip elements’ parameters 1 2 3 4 k 0.0557 0.0170 0.0021 0.0033 w 0.1680 0.1383 0.0342 0.1335 -40 -30 -20 -10 0 10 20 30 40 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 Joint angle (degree) Model error (bar) H. 6 Model errors as a function of the joint angle, increasing in the outermost positions. 4.2 Performance of the manipulator joint through hysteresis compensation. Figure 7 demonstrates the position control performance of the system with and without hysteresis compensation. 0 500 1000 -10 0 10 Amplitude 10 degree 0 500 1000 -40 -20 0 20 40 Amplitude 30 degree 0 500 1000 -20 0 20 Amplitude 20 degree time (x20ms) 0 500 1000 -50 0 50 Amplitude 40 degree time (x20ms) H. 7 Position response of the antagonistic system with 1kg payload at different excitation amplitudes with and without hysteresis compensation. The system was excited under a square wave of 0.1 Hz, using 4 different amplitudes of 10, 20, 30, and 40 degrees. In each subplot, two cycles are illustrated; the first one is the system response without hysteresis compensation while the second one is the system response with hysteresis compensation. Notice that the PI controller parameters were tuned by trial-and-error, and kept unchanged during the test performance. Based on this result, we recognize that: 10 0 -60 -40 -20 0 Amplitude 3deg [ r / d ] (dB) 10 0 -60 -40 -20 0 Amplitude 6deg 10 0 -60 -40 -20 0 Amplitude 9deg 10 0 -60 -40 -20 0 Amplitude 12deg 10 0 -300 -200 -100 0 Hz Phase (deg) 10 0 -300 -200 -100 0 10 0 -300 -200 -100 0 10 0 -300 -200 -100 0 10 0 -60 -40 -20 0 Amplitude 3deg [ p r / d ] (dB) 10 0 -60 -40 -20 0 Amplitude 6deg 10 0 -60 -40 -20 0 Amplitude 9deg 10 0 -60 -40 -20 0 Amplitude 12deg 10 0 -300 -200 -100 0 Hz Phase (deg) 10 0 -300 -200 -100 0 10 0 -300 -200 -100 0 10 0 -300 -200 -100 0 H. 8 FRF measurement of the pressure loop responses to the different amplitudes of position excitation: (a) without hysteresis compensation, (b) with hysteresis compensation. In general, before and after hysteresis compensation is activated, the occurrence of the oscillation is nonlinear with respect to the amplitude of excitation. The system is more oscillating at 20 degrees than at 10 degrees, but is less oscillating at 30 and 40 degrees than at 20 degrees. This phenomenon could be a result of increased nonlinear flow effects at higher pressure buildup as discussed in [13]. The compensation action provides a certain damping ratio to the system. The effectiveness a) b) 800 Võ Minh Trí VCM2012 of the compensation indicates that it is appropriate to reduce the oscillation around the desired position. 4.3 FRF analysis for pressure loop To perform the FRF measurements, we excite the system with a multisine of 10Hz of the interest band, and 4 different amplitudes, i.e. 3, 6, 9, and 12 degrees, to reveal the nonlinearities of the system. To guarantee the symmetric behaviour of the system, it was only tested around its centre position. The pressure block reflects the interaction between the response position and the pressure loop since there the response position is fed back. Both cases, with and without hysteresis compensation, are compared. -60 -50 -40 -30 -20 -10 0 Magnitude[ r / d ] (dB) 10 -1 10 0 10 1 -270 -180 -90 0 Phase (deg) Bode Diagram Frequency (Hz) w ithout HC w ith HC H. 9 FRF measurement of the pressure loop responses to the 12-degree amplitude of the position excitation: (a) with/without hysteresis compensation, (b) with hysterersis compensation. -20 -10 0 10 20 Magnitude (dB) 10 -1 10 0 10 1 -135 -90 -45 0 45 90 Phase (deg) Bode Diagram Frequency (Hz) H. 10 Nonlinear ‘lead-lag’ like compensation The pressure loop is highly nonlinear with respect to the position amplitude. The nonlinearity of the pressure loop is due to the nonlinear process of the pressure buildup as well as due to the effect of the position on the pressure loop. Fig. 8a shows that (i) there is a delay in the pressure loop, and (ii) complex interactions in the pressure loop decrease the damping of this loop. The system delay in connection with the phase lag potentially pushes the system into the unstable region. When the HC is in function, the pressure loop performs quite stable with an improvement of the damping and an additional phase lead. The pressure loop response with HC especially shows the effectiveness of the compensation regardless of the different amplitudes (Fig. 8b). This could benefit the hysteresis compensation as it is a kind of position- based compensation. Fig.9 is withdrawn from Fig. 8 with a view to look closer to a typical case where the HC plays very well its role. This comparison shows that the hysteresis compensation dramatically improves control performance in adding a phase lead to the system as well as providing damping to the pressure loop. The action of the hysteresis compensator is somehow similar to lead-lag compensation (Fig. 10). However, this is a kind of nonlinear ‘lead-lag’ compensator with varying parameter. 5. Conclusion The MS model of the extracted hysteresis is simple and well-suited to build in the control scheme, in which the PID feedback linearizes the system, whereas the hysteresis compensation functions effectively as a feedback compensator providing suitable damping and phase lead for the pressure loop at any positions. The nonlinear effects of hysteresis lead to a complex interaction between the pressure loop and the joint position. An FRF analysis does not provide an explicit explanation for the hysteresis effects in the system, but helps to see the effectiveness of the compensation, so-called nonlinear lead-lag compensator. Reference [1] Medrano-Cerda, G.; Bowler, C. & Caldwell, D. Adaptive position control of antagonistic pneumatic muscle actuators IEEE/RSJ International Conference on Intelligent Robots and Systems, 1995, 1, 378-383 [2] Caldwell, D.; Medrano-Cerda, G. & Goodwin, M. Control of pneumatic muscle actuators IEEE Control Systems Magazine, 1995, 15, 40- 48 [3] Tondu, B. & Lopez, P. Modeling and control of McKibben artificial muscle robot actuators IEEE Control Systems Magazine, 2000, 20, 15- 38 [4] Tondu, B.; Ippolito, S.; Guiochet, J. & Daidie, A., A seven-degrees-of-freedom robot-arm driven by pneumatic artificial muscles for humanoid robots The International Journal of Robotics Research, Multimedia Archives, 2005, 24, 257 [5] Ahn, K. K., Thanh, T.,& Ahn Y. K. Intelligent Switching Control of Pneumatic Artificial Tuyển tập công trình Hội nghị Cơ điện tử toàn quốc lần thứ 6 801 Mã bài: 168 Muscle Manipulator JSME International Journal Series C, J-STAGE, 2005, 48, 657-667 [6] Thanh, T. & Ahn, K. Nonlinear PID control to improve the control performance of 2 axes pneumatic artificial muscle manipulator using neural network Mechatronics, Elsevier, 2006, 16, 577-587 [7] Cai, D. & Yamaura, H. A robust controller for manipulator driven by artificial muscleactuator Control Applications, 1996., Proceedings of the 1996 IEEE International Conference on, 1996, 540-545 [8] Van der Smagt, P.; Groen, F. & Schulten, K. Analysis and control of a rubbertuator arm Biological Cybernetics, Springer, 1996, 75, 433-440 [9] Daerden, F.; Lefeber, D.; Verrelst, B. & Van Ham, R. Pleated pneumatic artificial muscles: actuators for automation and robotics IEEE/ASME International Conference on Advanced Intellegent Mechatronics, 2001, 738- 743 [10] Caldwell, D. & Tsagarakis, N. Biomimetic actuators in prosthetic and rehabilitation applications Technology and Health Care, IOS Press, 2002, 10, 107-120 [11] Verrelst, B.; Ham, R.; Vanderborght, B.; Daerden, F.; Lefeber, D. & Vermeulen, J. The pneumatic biped “Lucy” actuated with pleated pneumatic artificial muscles Autonomous Robots, Springer, 2005, 18, 201-213 [12] Tri, V.; Bram, K.; Tjahjowidodo, T.; Ramon, H. & Van Brussel, H. Modeling Torque-Angle Hysteresis in A Pneumatic Muscle Manipulator IEEE/ASME Int. Conf. on Advanced Intelligent Mechatronics, IEEE, 2010, 0, - [13] Tri, V.; Tjahjowidodo, T.; Ramon, H. & Van Brussel, H. Cascade position control of a single pneumatic artificial muscle-mass system with hysteresis compensation, Mechatronics, Elsevier, 2010,20(3),402-414 [14] Kyoung Kwan Ahn , Huynh Thai Chau Nguyen, Intelligent switching control of a pneumatic muscle robot arm using learning vector quantization neural network, Mechatronics, Vol. 17, 2007, pp. 255-262. [15] M. Hamerlain, Anthropomorphic robot arm driven by artificial muscles using a variable structure control, Proc. IEEE Int. Conf. Intelligent Robots Systems, Pittsburgh, PA, Aug. 1995, pp. 550–555. [16] J. H. Lilly, L. Yang, Sliding mode tracking for pneumatic muscles actuators in opposing pair configuration, IEEE Transactions on Control Systems Technology, Vol. 13, No. 4, 2005, pp. 550–558. [17] J. Schröder, D. Erol, K. Kawamura, and R. Dillmann, Dynamic pneumatic actuator model for a model-based torque controller, IEEE Int. Symp. On Computational Intelligence in Robotics and Automation, 2003, pp. 342-347. [18] Tri, V.; Bram, K.; Ramon, H. & Van Brussel, H. Modeling and control of a pneumatic artificial muscle manipulator. Part I: Modeling of a pneumatic artificial muscle manipulator with accounting for creep effect, Mechatronics, Elsevier, 2012,22(7),923-933 Vo Minh Tri, born in Bentre (1970), obtained his Ph.D from Katholieke Universiteit Leuven, Belgium in 2010. Formerly, graduated as Agricultural Mechanical Engineer at Cantho University, and received M.Sc. degree in Mechanical Engineering at National University of Ho Chi Minh, Vietnam in 1993 and 1998 respectively. Currently, he is senior lecturer at Cantho University, Department of Automation Technology where he is involved in teaching Mechatronics Design, Measurement systems, Systems Engineering. His main research interest concerns the implementation of mechatronic into agricultural production. . lần thứ 6 795 Mã bài: 168 Mô hình Maxwell- Slip để bù ma sát trong điều khiển khớp mềm khí nén một bậc tự do: một bộ bù Lead-Lag phi tuyến tương đương Maxwell- Slip Model for Hysteresis Compensation. này, mô hình khớp đã được phát triển bao gồm cả phần trễ của áp suất theo vị trí, theo đó phần này được mô tả chính xác bằng cách sử dụng mô hình Maxwell- Slip (MS). Mô hình này rất thích hợp để. chương trình điều khiển để bù sự trễ. Hệ thống điều khiển đi kèm do đó đơn giản hóa và chỉ cần áp dụng một bộ điều khiển PID thông thường. Hoạt đồng của hệ thống điều khiển vị trí của khớp quay