Anti‐shake positioning algorithm of bridge crane based on phase plane analysis

The Journal of EngineeringVolume 2019, Issue 22 p. 8370-8373 Jiangsu Annual Conference on Automation (JACA 2019)Open Access Anti-shake positioning algorithm of bridge crane based on phase plane analysis Yuxuan Zhu, Yuxuan Zhu Key Laboratory of Measurement and Control of CSE, Ministry of Education, Southeast University, Nanjing, People's Republic of ChinaSearch for more papers by this authorDan Niu, Corresponding Author Dan Niu danniu1@163.com Key Laboratory of Measurement and Control of CSE, Ministry of Education, Southeast University, Nanjing, People's Republic of ChinaSearch for more papers by this authorQi Li, Qi Li Key Laboratory of Measurement and Control of CSE, Ministry of Education, Southeast University, Nanjing, People's Republic of ChinaSearch for more papers by this authorYoucheng Chen, Youcheng Chen Key Laboratory of Measurement and Control of CSE, Ministry of Education, Southeast University, Nanjing, People's Republic of ChinaSearch for more papers by this authorShuang Wei, Shuang Wei Key Laboratory of Measurement and Control of CSE, Ministry of Education, Southeast University, Nanjing, People's Republic of ChinaSearch for more papers by this authorJinbo Liu, Jinbo Liu Nanjing Sciyon Automation Group Co., Ltd, Nanjing, 211102 People's Republic of ChinaSearch for more papers by this author Yuxuan Zhu, Yuxuan Zhu Key Laboratory of Measurement and Control of CSE, Ministry of Education, Southeast University, Nanjing, People's Republic of ChinaSearch for more papers by this authorDan Niu, Corresponding Author Dan Niu danniu1@163.com Key Laboratory of Measurement and Control of CSE, Ministry of Education, Southeast University, Nanjing, People's Republic of ChinaSearch for more papers by this authorQi Li, Qi Li Key Laboratory of Measurement and Control of CSE, Ministry of Education, Southeast University, Nanjing, People's Republic of ChinaSearch for more papers by this authorYoucheng Chen, Youcheng Chen Key Laboratory of Measurement and Control of CSE, Ministry of Education, Southeast University, Nanjing, People's Republic of ChinaSearch for more papers by this authorShuang Wei, Shuang Wei Key Laboratory of Measurement and Control of CSE, Ministry of Education, Southeast University, Nanjing, People's Republic of ChinaSearch for more papers by this authorJinbo Liu, Jinbo Liu Nanjing Sciyon Automation Group Co., Ltd, Nanjing, 211102 People's Republic of ChinaSearch for more papers by this author First published: 25 November 2019 https://doi.org/10.1049/joe.2019.1083Citations: 4AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Abstract In this study, the dynamic model of bridge crane system is established based on Lagrange equation. The transfer function of crane running system is derived. A new crane anti-swaying scheme different from traditional mechanical anti-shake strategy is proposed and it is based on phase plane analysis algorithm. By the adaptive speed planning method, the industrial grade bridge crane can calculate different motion trajectories online without any off-line optimisation calculation under the given acceleration and maximum speed limit conditions. At the same time, the industrial grade bridge crane realises the purpose of anti-sway and positioning of the crane. The field experiment results show that the swing of the crane is obviously suppressed and the positioning accuracy fully meets the industrial requirements when the load of the crane reaches the target position. 1 Introduction With the advent of the domestic industry 4.0 era, industrial automation has become an inevitable development trend. The development of industry is accompanied by the expansion of production scale and the advancement of production technology [1]. The bridge crane is an important heavy-duty handling equipment, so the role of the bridge crane in the industrial field is increasing. However, the crane will certainly cause the goods to sway during the lifting of the cargo, and the swing of the lifting cargo will accelerate. The wear of mechanical equipment can even lead to safety accidents. Considering the factors such as industrial site safety and production efficiency, it is usually necessary for professional crane operators to control the movement of the crane by hand-operating operators to minimise the swing of the crane. The requirements of personnel are higher, so the electric anti-shake strategy is more popular in the industry [2]. The anti-shake positioning control can automatically eliminate the swing generated by the hanging object during the running process [3], what is more, an automated industrial bridge crane with positioning function can complete the transport of the hanging object more quickly. The anti-sway system can make the operation of the bridge crane more efficient and safer [4]. In this paper, an adaptive speed planning anti-shake positioning method based on phase plane geometry analysis is proposed. It can effectively reduce the swing of the bridge crane during travel and the residual oscillation after the lifting stop, and achieve the high positioning accuracy. The traditional mechanical anti-rolling is mainly achieved by installing a guide post under the small frame. It has the advantages of stability and reliability. However, it needs complicated mechanical mechanism of the car and high machining cost. Moreover, the entire anti-shake process achieves the anti-shake effect through the rigid structure of the bridge machine itself. Long-term use is prone to problems such as short mechanical life and increased clearance of the guide bush [5-7]. Compared with the traditional anti-shake, a multi-pulse adaptive speed planning anti-shake positioning algorithm has the following advantages and innovations as an electronic anti-shake strategy: (i) The maintenance and maintenance workload of anti-shake equipment and reduce unnecessary road modifications are reduced; (ii) The control strategy is relatively simple and easy to implement in the implementation process; (iii) The speed trajectory of the driving is adaptively programmed according to different moving distances. 2 Proposed methodology 2.1 Kinematics equation of bridge crane The bridge crane trolley is a complex under-actuated system [8]. The number of independent control variables of the system is less than the number of degrees of system freedom. After the simplified processing, the mass of the bridge crane is in force. Under the action of moving along the axis, the weight of the weight block is hung on the overhead crane trolley to make an approximate single pendulum motion through the wire rope [9]. The schematic diagram of the crane trolley is shown in Fig. 1. Fig. 1Open in figure viewerPowerPoint Schematic diagram of the crane crane trolley The system of motion differential equations is obtained by modelling the bridge crane by the Euler–Lagrange method [10, 11]: (1) Assuming that the weight of the wire rope and the air resistance are not counted, the stiffness of the wire rope is sufficiently large and its length variation is negligible [12, 13]. Due to the small load swing in the industrial field environment, the following equation can be obtained by substituting (1): (2) It can be seen from the mathematical model that there is only one input variable in the system. One is the angle and the other is the displacement. Therefore, the system is a non-linear second-order under-actuated system. 2.2 State equation and motion law analysis According to the differential equation obtained in (2), the initial conditions are assumed: , , , , The differential equation can be obtained: (3) where represents that the equation of state for the available system: (4) Then it can be obtained as follows: (5) According to (5), the model satisfies the elliptic curve equation. If the initial state satisfies , . θ (t), satisfy the following relationship at any time θ (t) from (4): (6) According to (6), the elimination of t can be obtained: (7) Assuming that the initial state θ (t) = 0, ω (t) = 0, the swinging law of the load appears as a phase plan as shown in Fig. 2. At this time, the relationship between the load swing law and the constant acceleration a of the crane can be divided into the following three cases: when a > 0, the trajectory of the state [θ (t), ω (t)] of the under-actuated crane system moves around the ellipse in the clockwise direction in the second and third quadrants; when a = 0, (7) evolves to , and the conclusion is that θ (t) ≡ 0, ω (t) ≡ 0. In other words, the crane and the load are obtained and the system is in a relatively static state. When a < 0, the trajectory of the state [θ (t), ω (t)] of the under-actuated crane system moves around the ellipse in the clockwise direction in one or four quadrants. Fig. 2Open in figure viewerPowerPoint Schematic diagram of the phase plane of the under-actuated system Therefore, the motion of the crane can be planned by using the law of the phase plane as shown in Figs. 3 and 4. At the initial state of the crane is θ (t) = 0, ω (t) = 0. The crane starts moving at a certain acceleration during the 0−t 1 time period. Also the load is from the initial position. The state point of load moves from point o to point a. During the period of t 1–t 2, the crane moves at a constant speed. In addition, the state point of load moves from point a to point b. The load has the same angle at point b and point a. What is more the angular velocity is the same in the opposite direction. During the t 2–t 3 period, the crane continues to move at the same acceleration, and the state point of load moves from point b to point o. At this time, since the load is moving in the class of pendulum, the process of swinging the load from point o to point a and the process of swinging from point b to point o are completely symmetrical. The acceleration time at both ends is equal. During the period of t 3–t 4, the crane moves at a constant speed, and the load has no residual oscillation during the uniform motion. During the t 4–t 7 time period, the trolley begins to enter the deceleration phase. The swing trajectory of the load is completely symmetrical with the acceleration phase. The trajectory of the load in the deceleration phase is o → c, c → d, d → o. According to the principle of symmetry, in the ideal case, the load is decelerated to 0 m/s and the load does not swing. Fig. 3Open in figure viewerPowerPoint Planned load phase plane trajectory Fig. 4Open in figure viewerPowerPoint Speed curve of the planned crane 2.3 Optimal control analysis By modelling and analysing the under-actuated system of the crane and phase plane analysis, an acceleration controller based on phase plane analysis is designed to select the mode of the acceleration controller through optimal control analysis. By analysing the state equations of known controlled systems is the model, where , , and the control vector u (t) are constrained by In order to seek the optimal control , the performance index of the system from the known initial state to the termination state is extremely small. The optimal solution at this time enables the crane under-driving system to reach the control target index in the shortest time, and the Hamilton function can be constructed: (8) According to the principle of minimum value, the necessary conditions for optimal control are: (9) The regular equation is and the boundary conditions satisfy , . From (9), the extreme condition is . Let , then . Each control component is set independent of each other, and the extreme condition is derived. In summary, the optimal control under constraint is (10) Therefore, considering the optimal control in the shortest time, the acceleration controller can be designed such that each component is a piecewise constant value function of time and the acceleration occurs at a set switching time from one constant value to another constant value. The mode, that is, the anti-sway control of the under-actuated system of the crane can be realised by editing the multi-pulse of the acceleration. 2.4 Proposed multi-pulse adaptive speed planning algorithm According to the above analysis, a multi-pulse adaptive speed planning anti-shake positioning algorithm is proposed in this paper. The crane anti-shake is realised by online editing the acceleration multi-pulse input model of the crane. Moreover, the crane positioning is realised by online planning the speed curve. The algorithm model is shown in Fig. 5. The target distance that the crane needs to travel is calculated from the input current position and the target position. In the safety consideration of the industrial site, the crane speed needs to be gradually increased. Therefore, the concept of multi-stage planning for the crane speed is introduced. The maximum speed of the crane is planned according to the target distance. The motor parameters of the crane, the maximum speed and acceleration, are combined online. The number of pulses in the acceleration process of the crane is accelerated. After the acceleration curve is planned, the displacement of the crane in the acceleration phase is obtained. According to the analysis in Section 2, the displacement of the crane in the acceleration and deceleration phase is the same, and the uniform speed of the crane after the acceleration is completed can be adjusted. The movement time is to adjust the uniform moving distance of the crane to achieve the purpose of precise positioning. Fig. 5Open in figure viewerPowerPoint Adaptive speed planning algorithm model 3 Simulation and experimental results The input speed curve obtained after the speed planning is input into the crane system model, and the swing angle after adding the anti-shake positioning algorithm based on the phase plane is obtained in the MATLAB simulation. The simulation results of the crane system with and without the anti-shake algorithm are compared, as shown in Fig. 6. From Fig. 6, the swing angle of the load is significantly suppressed. Fig. 6Open in figure viewerPowerPoint Comparison of simulation angle curves before and after adding anti-shake algorithm The algorithm is applied to the management system of a steel plant reservoir area. For the crane with a travel distance of 10 m, the maximum speed is 0.64 m/s, the acceleration is 0.4 m/s2, and the acceleration is three times. The actual speed curve is obtained. The angle curve is shown in Fig. 7. Through several field experiments, it is concluded that the load angle swing can be controlled within 0.5° after the crane stops. Fig. 7Open in figure viewerPowerPoint Actual speed and angle graph 4 Conclusion With the increase in productivity, the number of goods in industrial production is getting more and more. As a result, bridge cranes are applied to many aspects of production. In this paper, the theoretical modelling and method analysis of the crane running system are carried out. The mathematical model of the crane anti-sway system is established by using the Lagrange equation. The transfer function of the system is strictly derived. The anti-shake positioning algorithm of bridge crane based on phase plane analysis is proposed. Also the algorithm is applied to the actual steel plant reservoir area system. The field experiment results show that the load angle is effectively suppressed after the crane stops. The position accuracy is within 2 cm, which is satisfactory. In order to prove the correctness of the analysis and the effectiveness of the design controller, a large number of simulations and field experiments were carried out. The results show that the controller designed according to the idea of this paper is more in line with the characteristics of the bridge crane system and can suppress the transportation process very well. The proposed method has effectively improved the efficiency of bridge cranes. 5 Acknowledgments This work was supported by National key R&D Program of China (Grant no. 2018YFC1506900), Zhishan Youth Scholar Program of SEU, National NSF of China (Grant no. 61504027), the Key R&D Program of Jiangsu Province (Grant no. BE2017076), the Key R&D industrialization Program of Suzhou (Grant nos. 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