Overhead cranes fuzzy control design with deadzone.pdf

ORIGINAL ARTICLE Overhead cranes fuzzy control design with deadzone compensation Cheng-Yuan Chang Tung-Chien Chiang Received: 29 September 2007/Accepted: 24 March 2009/Published online: 9 April 2009 ? Springer-Verlag London Limited 2009 AbstractThis article proposes a simple but effective way to control 3D overhead crane. The proposed method uses PID control at the start for rapid transportation and fuzzy control with deadzone compensation when the crane is close to the goal for precise positioning and moving the load smoothly. Only the remaining distance and projection of swing angle are applied to design the fuzzy controller. No plant information of crane is necessary in this approach. Therefore, the proposed method greatly reduces the com- putational efforts. Several experiments illustrated the encouraging effectiveness of the proposed method through a scaled 3D crane model. The nonlinear disturbance, such as abrupt collision, is also taken into account to check the robustness of the proposed method. KeywordsOverhead crane ? Projection ? Swing angle ? Fuzzy controller ? Deadzone compensation 1 Introduction The 3D crane consists of a trolley, driving motors, and the fl exible wire. It is often used in the factories and harbors for moving heavy cargoes. The motors drive the trolley and the fl exible wire ties the load. Fast, smooth, and precise moving to the goal is the main objective of crane control. Usually, the experienced crane operator moved the load to the destination slowly, drove the trolley back and forth to make the load stationary, and tried to stop the trolley at the destination precisely and smoothly. However, due to the nonlinear load swing motion, smooth transportation and precise load positioning by the crane operator are not easy. In addition, transporting the load rapidly but without swing is an even more diffi cult objective. Hence, only the visual feedback of the crane operator is not enough to control the crane well during transportation. Theobjective ofthecranecontrol istotransferthe loadto the destination as fast as possible; meanwhile, to minimize the swing during the transportation and to stop the trolley preciselyatthedestination.However,the accelerationofthe cranealwaysaccompanieswithnonlinearloadswing,which might cause the load damage and even accidents. Some investigations have developed the anti-swing methods for the effi cient control of the overhead crane. Some articles are investigating the existing problems of the crane control system. Among these researches, Auernig and Troger 1 used minimal time control to minimize the load swing. Corrigaet al.2appliedanimplicitgain-schedulingmethod tocontrolthecrane.Someresearchersalsousedthedynamic model of the crane to evaluate an optimal speed or path reference that minimized the load swing 37. However, since the load swing depends on the movement and accel- eration of the trolley, minimizing the cycle time and load swing are partially confl icting requirements. Some resear- ches also applied nonlinear control theory to analyze the properties of the crane system 810. Besides, Karkoub and Zribi 11 also developed the modeling and energy-based nonlinear control of the crane lifter. These methods are too complex to implement for the industry use; meanwhile, they took much time to transfer the load smoothly and caused severe swing at the beginning of transportation. In addition, Yoshida and Kawabe 12 proposed the real-time saturating control strategy for cranes. Matsuo et al. 13 used the C.-Y. Chang ( meanwhile, the acceleration of trolley will also cause the additional swing of load. Hence, transferring the load fastly and smoothly in the meantime is not easy. Since one of the objectives of the crane control is to transfer the load as fast as possible; therefore, we utilize the PID control in the previous 95% distance for rapid transportation and then switch to the fuzzy projection method to restrain the load swing. The block diagram of the fuzzy crane control system is shown in Fig. 3. The present position of the trolley on the XY-plane is P,px;py? 2 R2and the destination is D,dx;dy? 2 R2. The PID control is Up upx;upy?, where UpKP KI s KD? s ? E1 The vector of the remaining distance E is E D ? P dx? px;dy? py?,ex; ey? 2 R22 where exand eyare the components of the vector E in the directions of X-axis and Y-axis, respectively. The constants Fig. 1 The physical apparatus of the 3D crane control system Fig. 2 The swing diagram of 3D crane 750Neural Comput kpy?, KI kix;kiy? and KD kdx;kdy? in the PID controller are designed to transfer the load rapidly under the assumed safety constraints: jhxzj15?andjhyzj15?3 where hxz; hyzrepresents the swing angle of load along the XZ-plane and YZ-plane, depicted in Fig. 2. The PID control helps to transfer the load rapidly under the assumed safety constraints. After the trolley has arrived at the 95% distance, the PID control is switched to fuzzy projection control. The pro- posed method utilizes the projection of the 3D swing angle to be another input variable of the fuzzy controller, shown in Fig. 4. The factor of swing angle is taken into consid- eration in this stage. One denotes the swing angle of the load in 3D space to be W hxz; hyz?4 We use the projection of 3D swing angle on a 2D plane to present the level of load swing. The severer swing will lead to longer projection, and vice versa. Hence, the pro- jection of swing angle can be represented by u 2 R2: u /x;/y?;5 where /xand /yare the projections of swing with respect to the X-axis and Y-axis. Since the projection and the remaining distance are all in length, it is more reasonable to apply both the factors to design the fuzzy output at the same time. The length of projections can be represented by /x L ? sinhxz6 /y L ? sinhyz7 L is the fi xed length of the fl exible wire hanging the load. These projection vectors are depicted in Fig. 4. In order to achieve the addressed control objective, fast traveling during the journey, stopping precisely and swinging smoothly at the end, the trolley should be driven with the following criteria. First of all, the trolley should be driven along the direction of E to reach the destination as fast as possible. Second, the trolley should be driven along the direction of u to eliminate the swing angle. However, the directions of E and u may not be the same, and driving the trolley along the directions of E and u in the meantime might be impossible. Therefore, the author applies the fuzzy method to control the trolley along the directions of E and u. Two motors for X- and Y-axes are used to drive the crane. One uses exand /xto be the antecedents of fuzzy controller to derive the fuzzy control for X-axis motor, and the other applies eyand /yto derive the control for Y-axis motor. Suppose that the output of the fuzzy controller is UF, where the output acts as the fuzzy function of input variables, UF ufx;ufy fxex;/x; fyey;/y;8 where ex, ey, /x, and /yare the input variables, fxand fy represent for the functions of fuzzy controller, and ufxand ufyare the fuzzy output power to drive the X- and Y-axes motors. In this case, we use fi ve linguistic states for each input variable e*and /*to fuzzify the input variables and use 11 states for output variables uf?. The notation * means x or y, and the linguistic states are shown in Fig. 5ac. The dynamic ranges are -0.1 m 0.1 m, -0.05 m 0.05 m, and -1 1 for e*, /*and uf?. One names the fi ve linguistic states of input variables e*and /* as Negative Big (NB), Negative Small (NS), Zero (ZO), Positive Small (PS), and Positive Big (PB). Besides, the author uses 11 linguistic variables to describe the output variables uf?including Negative Big_Big (NBB), Negative Big (NB), Negative Middle (NM), Negative Small (NS), Fig. 3 The block diagram of fuzzy crane control system E y e y x Load Trolley x e Destination Fig. 4 The concept of the projective method Neural Comput ucy? is used to be the consequent part, where jDEj jEt ? Et ? 1j jDexj;jDeyj?9 and Dex ext ? ext ? 1andDey eyt ? eyt ? 1: 10 The dynamic range of |DE| is partitioned into three lin- guistic terms: Zero (Z), Small (S), and Big (B). After fuzzifi cation, inference, and defuzzifi cation procedures, the output power Ucis obtained. Figure 8a and b shows the corresponding membership functions, and the fuzzy rules are shown in Table 2. Step 2 While the distance to the goal is still far enough, the fuzzy controller will generate the suffi cient power to drive the trolley crane. However, the power will reduce when the trolley is close to the destination. The speed of the trolley is hence slowed down. If the variation of the trolley position |DE| is less than a, then the trolley crane could gradually stop for the deadzone. The compensating method activates in this moment to provide the additional power, U0 c u0cx;u0cy?, to increase the control. This helps to avoid the crane stop in front of the destination. The com- pensating principles are shown in the following equations. ifjDEj?au0c?t 1 uc?t u0c?t11 otherwise;u0c?t 1 0;? 2 fx;yg12 The constant a is 5 9 10-4m and u0c?0 is set to 0 in the beginning. Step 3 In order to reduce the computation effort, the compensating fuzzy controller uses |DE| instead of DE as Fig. 7 The block diagram along with the compensating algorithm Fig. 8 The membership functions of a |DE|, b Uc Table 2 Rule table of the compensating fuzzy controller |DE|ZSB UcBSZ Neural Comput u 00 cy?: ifuf?0;u00 c? u0c?13 otherwise;u00 c? ?u0c? 2 fx;yg14 Hence, the additional voltage, U00 c, is activated to increase the driving power of the trolley when the variation is less than 5 9 10-4m. The actual power U to control the crane will be modifi ed: U UF U00 c 15 A fl exible wire is tied to the load. Therefore, the nonlinearity of the crane system is hence increased to examine the ability of the compensating control algorithm. The author sets the stop criterion of the crane control system with the following qualifi cations: the distance to the goal is less than 0.001 m; meanwhile, the swing angle of load is less than 0.5?. While the crane could stop due to the control deadzone, the compensating algorithm will activate to provide the extra power until the stop criterion is satisfi ed. 4 Experimental results A scaled crane model was built in the laboratory to illus- trate the effectiveness of the proposed method. Two DC motors for X- and Y-axes are applied to drive the overhead crane system. Four 12-bit encoders send the information of the present position (including the coordinates of X- and Y- axes) of the trolley and the swing angles of the load, hxzand hyz, to the controller. The weight of the load is 0.7 Kg and the length of the hanging fl exible wire is 1 m. Suppose that the destination of the load is set to be (1.5 m, 1.5 m), while the initial position of the load is at the location of (0 m, 0 m). The corresponding constants in the experiments are KP= 10, 9.85, KI= 0, 0.002, and KD= 9.6, 8.65. Figure 9 shows the experimental results with only PID controller. Figure 9a shows the remaining distance to the goal and Fig. 9b shows the swing angles hxzand hyz. One can fi nd that the trolley is driven fast but with severe swing. The fi nal position of the trolley is (1.47156 m, 1.50002 m) and the remaining swing amplitude is about 12? for hxzand 8? for hyz. It took only about 8 s to reach the goal, but the swing was unable to restrain well. Figure 10ab shows the experimental results with the proposed method. One can fi nd that the trolley took about 5 s to the destination; in the meanwhile, the swing angles of the projective method performed very well. The remaining swing amplitude is about 0.09? for hxzand 0.02? for hyz . However, it is diffi cult for the trolley to stop pre- cisely at the goal; the fi nal position of the trolley is (1.48274 m, 1.49956 m). Hence, the steady-state error of the trolley was 28.44 and 0.02 mm for the X- and Y-axes, respectively. This problem is caused by the friction of the X- and Y-axes track of trolley. Therefore, if the trolley is very close to the destination, the fuzzy controller will provide small power to reach the goal. When the power is not enough to overcome the control deadzone, the trolley would stop on the wrong place, making the performance worse. Besides, the positioning error of X-axis is worse than that of Y-axis. This result matches the control deadzone problem of X-motor is severer than Y-motor, shown in Fig. 6. Fig. 9 The experimental results with only PID controller a remaining distance, dashed: X-axis, solid: Y-axis, b swing angle, dashed hxz, solid hyz 754Neural Comput meanwhile, the trolley stopped precisely at the goal after the compensating algorithm was acti- vated. The fi nal position of the trolley is (1.49967 m, 1.49991 m). The positioning error is only 0.33 mm for X-axis and 0.09 mm for Y-axis. Besides, the remain- ing swing amplitude is about 0.04? for hxzand 0.02? for hyz. One sets two indexes to compare the experimental results, where the position index is posT 1 2 exT eyT16 and swing index is: swingT 1 2 amplitude of hxzT amplitude of hyzT 17 where T is the fi nal time to control. Comparisons between the presented methods are depicted in Fig. 12a and b. One can fi nd that the deadzone compensation greatly improves the control performance of the 3D overhead crane system. However, the overhead crane system is often used in the outdoors. The disturbance such as the abrupt collision of the load might affect the control performance. In the last Fig. 10 Theexperimentalresultswiththeprojectivemethod a remaining distance, dashed: X-axis, solid: Y-axis, b swing angle, dashed: hxz, solid: hyz Fig. 11 The experimental results with compensating algorithm a remaining distance, dashed: X-axis, solid: Y-axis, b swing angle, dashed: hxz, solid: hyz Neural Comput meanwhile, stop the trolley at the destination perfectly. 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