Fuzzy logic controller for overhead cranes Mohammed T. Hayajneh Industrial Engineering Department, Jordan University of Science and Technology, Irbid, Jordan, and Saleh M. Radaideh and Issam A. Smadi Electrical Engineering Department, Jordan University of Science and Technology, Irbid, Jordan Abstract Purpose – To propose a new method for controlling the overhead crane systems based on the theory of fuzzy logic with a reduced number of rules than has appeared before in the literature. The proposed fuzzy logic controller (FLC) can be implemented to move the overhead crane along a desired path while ensuring that the payload is swing free at the end of the motion. Design/methodology/approach – In this study, a FLC that includes two rule bases, one for displacement control, the other for swing control, was designed and successfully implemented to move the overhead crane along a desired path while ensuring that the payload is swing free at the end of the motion. Findings – Control simulation results demonstrate that by using the proposed FLC, the overhead traveling crane smoothly moves to the destination in short time with small swing angle and almost no overshoot. Originality/value – This paper offers practical help to whom are working in controlling the transporting the payloads to the required position as fast and as accurately as possible without collision with other equipments. The proposed FLC can be implemented to move the overhead crane along a desired path while ensuring that the payload is swing free at the end of the motion. Keywords Modelling, Fuzzy logic, Cranes Paper type Research paper 1. Introduction Overhead cranes are widely used in industry for load transfer. In most of the applications, it is desired for the overhead crane to transport the payloads to the required position as fast and as accurately as possible without collision with other equipments. Such fast motion would induce undesirable payload swinging, which may cause load damage and other types of hazards, and hence reduces the operation efficiency. Therefore, these oscillations have to be damped before other payload manipulation is done. Performance of any motion control system is limited by variations of system parameters and disturbances such as payload change. This is specially applied for overhead crane. As known, overhead cranes are used widely where large and heavy payloads are handled such as shipyards and power stations (Manson, 1982). Expert systems (skill of operator) play a central role in safe payload transport and accurate positioning. Swinging of the suspended payload usually is due to accelerating and decelerating of the overhead crane, which reduces the operational efficiency and increases the risk of accidents. The current issue and full text archive of this journal is available at www.emeraldinsight.com/0264-4401.htm EC 23,1 84 Received August 2004 Revised May 2005 Accepted May 2005 Engineering Computations: International Journal for Computer-Aided Engineering and Software Vol. 23 No. 1, 2006 pp. 84-98 q Emerald Group Publishing Limited 0264-4401 DOI 10.1108/02644400610638989 Many works have been done in controlling the movement of the overhead cranes. Starr (1985) used an open loop control strategy applied to simple model with a bang-bang type acceleration profile. Also, Noakes and Jansen (1992) used an open loop path controlled manipulator to achieve swing free motion during the entire transport. On the other hand, Vaha and Martinen (1989) developed an open loop control algorithm based on the natural frequency of the suspended object, which use a specific case involving shaping of input to dampen vibration. A nonlinear model of an overhead crane was derived in Moustafa and Ebeid (1988) and based on it; a linearized state space was obtained by perturbing the system about its equilibriums state. Ridout (1989a) used a linear state feedback speed control law on a specific crane model, and Ridout (1989b) used a variable damping linear feedback control combined with contour mapping for a reduced order model obtained by perturbation. Park et al. (2000) and Singhose et al. (2000) adopted input shaping control method, but the input shaping must be pre-calculated accurately according to the system model. These approaches lacked robustness to external disturbances and could not damp residual swing well. Lee (1998) and Giua et al. (1999) proposed feedback control methods. These methods were greatly affected by system linearization and system parameters uncertainty. Moreover, they need accurate system model and onerous matrix computation. To accommodate crane system nonlinearity and parameter variations, researchers have recently directed their attention to fuzzy logic control. Lee and Cho (2001) used fuzzy logic only in anti-swing control and applied position servo control for positioning and swing damping. Hua (2000) only studied anti-swing control with fuzzy logic and did not take positioning control into consideration. Nalley and Trabia (2000) adopted fuzzy logic to both positioning control and swing damping. However, because of the large number of fuzzy rules for the complex overhead crane system, it was difficult to set both rules and parameters of the controller only according to experiences. The majority of the reviewed papers are based on the assumption that payload is a lumped load mass. However, in many cases the payload may have a considerable length and because of the complexity of overhead crane model, the fuzzy logic controlling seems to be the good choice for the task. Fuzzy set theory provides a systematic calculus to deal with incomplete information linguistically, and it performs numerical computation by using linguistic labels stipulated by membership function. Moreover, a selection of fuzzy if-then rules forms the key component of a fuzzy inference system (FIS) that can effectively model human expertise in a specific application (Zumberge and Passino, 1998). The basic principle of fuzzy logic controller (FLC) is to express the human knowledge in the form of linguistic IF-THEN rules. Every rule has two parts, condition part, which is the IF-PART of the rule, and conclusion part, which is the THEN-PART of the rule. The collection of such rules is called rule base. The basic structure of a FLC consists of three conceptual components: a rule, which contains a selection of fuzzy rules, a database, which defines the membership functions used in the fuzzy rules, and a reasoning mechanism, which performs the interface procedure upon the rules and given facts to derive conclusion. Finally, defuzzification is used to have a crisp value from the FLC. A FLC with crisp output is shown in Figure 1. With crisp input and output, a FIS implements a nonlinear mapping from its input space to output space (Ying, 2000). A typical process in developing the FLC incorporates the following steps: specify the problem and define Fuzzy logic controller for overhead cranes 85 linguistic variable, determine fuzzy set, encode the fuzzy sets, fuzzy rules and procedure to perform FIS, evaluate and tune the controller. The objective of this work is to design a FLC for an overhead crane capable of moving the overhead crane that carries considerably long and massive payload along a desired path while ensuring that the payload is swing free at the end of the motion. The proposed FLC for controlling the overhead crane systems based on the theory of fuzzy logic with a reduced number of rules than has appeared before in the literature. 2. Mathematical modeling of overhead crane system As shown in Figure 2, the overhead traveling crane consists of cross beam, trolley, an inextensible rope, and payload. The overhead crane is assumed rigid and the overhead trolley runs on friction-free rails, with the payload suspended on an inextensible rope. The payload mass is assumed to be uniformly distributed along it axis and does not twist during motion. The length of the rope is lr, the length of the payload is lp and mb and mt are the mass of the cross-beam and the mass of the trolley, respectively, r is linear density of the payload. The crane motion can be modeled using the following six degrees of freedom (Nalley and Trabia, 2000). q ¼ ðd1; d2;a3;b3;a4;b4ÞT ð1Þ where T refer for transpose. Figure 2 shows five coordinate frames which define the crane and payload, d1 is the cross-beam displacement with respect to frame 1, d2 is the trolley displacement with respect to frame 2, a3 is the rotation about frame 3 Z-axis by frame 4, b3 is the rotation about frame 3 Y-axis by frame 4. Finally, a4 is the rotation about frame 4 Z-axis by frame 5, b4 is the rotation about frame 4 Y-axis by frame 5. By using transformation matrices, the location of a point, rp, along the payload axis, X5, in terms of the fixed frame as: {rp} ¼ T12T23T34T45ð u 0 0 1 ÞT ð2Þ where Figure 1. Basic structure of FLC EC 23,1 86 T12 ¼ 1 0 0 0 0 1 0 0 0 0 1 d1 0 0 0 1 2 666664 3 777775 ð3Þ T23 ¼ 1 0 0 0 0 1 0 d2 0 0 1 0 0 0 0 1 2 666664 3 777775 ð4Þ T34 ¼ cosa3 cosb3 2sina3 cosa3 sinb3 2lrcosa3 cosb3 sina3 cosb3 cosa3 sina3 sinb3 2lrsina3 cosb3 2sinb3 0 cosb3 lrsinb3 0 0 0 1 2 666664 3 777775 ð5Þ Figure 2. Schematic diagram of an overhead crane carrying a load Fuzzy logic controller for overhead cranes 87 T45 ¼ cosa4 cosb4 2sina4 cosa4 sinb4 2u cosa4 cosb4 sina4 cosb4 cosa4 sina3 sinb4 2u sina3 cosb4 2sinb4 0 cosb4 u sinb4 0 0 0 1 2 666664 3 777775 ð6Þ Using Lagrangian dynamics, the equations of motion are derived as: d dt dKE dq0i � � 2 dKE dqi þ dU dqi ¼ f 1 f 2 0 0 0 0 � �T ð7Þ where KE and U are the kinetic and the potential energy, respectively, and f1 and f2 are the forces needed to move the cross-beam and trolley, respectively. KE ¼ 1 2 mb _d 2 1 þmt _d 2 1 þ _d 2 2 � � þ Z lp 0 r_r 2p � � du � � ð8Þ U ¼ Z lp 0 rgrpx � � du ð9Þ Using Taylor’s series, the resulting model are linearized around the operating point (a3 ¼ 0; b3 ¼ 0; a4 ¼ 0; b4 ¼ 0) since the payload swing angles are expected to remain relatively small. The mathematical model of overhead crane can be described as: Mq00 þ Kq ¼ C ð10Þ where C ¼ f 1 f 2 0 0 0 0 � �T ð11Þ K ¼ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 lrlpr g þ l 2 pr g 2 0 l2pr g 2 0 0 0 0 lrlpr g þ l 2 pr g 2 0 l2pr g 2 0 0 l2pr g 2 0 l2pr g 2 0 0 0 0 l2pr g 2 0 l2pr g 2 2 66666666666666664 3 77777777777777775 ð12Þ M ¼ M 11 M 12 M 21 M 22 " # ð13Þ where EC 23,1 88 M 11 ¼ mb þmt þ lpr 0 0 0 mt þ lpr lpr ð2lr þ lpÞ2 0 lpr ð2lr þ lpÞ 2 lpr 3l 2 r3lrlp þ l2p � � 2 2 666664 3 777775 ð14Þ M 12 ¼ 2lpr ð2lr þ lpÞ 2 0 2 l2pr 2 0 l2pr 2 0 0 l2pr ð3lr þ 2lpÞ 6 0 2 66666664 3 77777775 ð15Þ M 21 ¼ 2lpr ð2lr þ lpÞ 2 0 0 0 l2pr 2 l2pr ð3lr þ 2lpÞ 6 2 l2pr 2 0 0 2 6666664 3 7777775 ð16Þ M 22 ¼ lpr 3l 2 r þ 3lrlp þ l2p � � 3 0 l2pr ð3lr þ 2lpÞ 6 0 l3pr 3 0 l2pr ð3lr þ 2lpÞ 6 0 l3pr 3 2 666666664 3 777777775 ð17Þ 3. Fuzzy logic controller design As already mentioned, several controllers have been designed in the past, ranging from simple PID controllers to LQG controllers. In this research study, and due to uncertainties surrounding the system, a fuzzy logic-based controller is designed. Furthermore, recent emphasis in fuzzy logic control has focused on formulating rule bases with a reasonable number of fuzzy rules (Mahfouf and Linkens, 1998). Two good reasons for using the fuzzy control: first, there is no mathematical model available for the process or the model is complicated, and second, to satisfy a control strategy that can be developed empirically, without complicated mathematics. There are two primary criteria to consider when designing a fuzzy controller for an overhead crane. The first is related to motion profiles (the acceleration patterns and magnitude) of the crane cross-beam and trolley because the payload swing depends on it. Fuzzy logic controller for overhead cranes 89 Therefore, the controller should be designed to minimize payload swing. The second is to design a feedback control law to drive the overhead to its destination while controlling the swing of the payload. Therefore, the objective of the proposed controller is to control the overhead crane motion with minimum payload swing based on feedback control. An acceleration profile, Figure 3, is used for both the cross-beam and trolley to move them along a certain desired path, which consists of periods of constant acceleration, velocity and deceleration. It can be shown mathematically that a simply suspended object with no initial swing will end its path transport with no swing when subjected to this acceleration profile. The length of the deceleration and acceleration periods is dependent on the swing period of an equivalent simple pendulum and is equal to: ta ¼ 2p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi J rlp lr þ lp2 � � g vuut ð18Þ where ta is the acceleration period and g is the gravitational acceleration. The motion profile will be effective in providing swing-free transport if the swing period is precisely known and the payload is behaves as a simple pendulum. However, if the payload and rope parameters are not precisely known and external disturbances are present then the motion profile loses its effective. Therefore, feedback control is effective here especially considering that other external disturbance and unknowns is considered. FLC is a rule base control strategy, where rule is based on the observation of the system behavior. The input to the FLC is the difference between the actual and desired (error) and the output will be the force required to correct this error. The proposed controller consists of two inputs and single output as shown in Figure 4. When the inputs of the FLC are the error in displacement and the velocity of the cross-beam or the trolley with respect to desired path, the controller will refer to as displacement fuzzy controller. On the other hand, if the inputs are the error in angular displacement and velocity of the payload with respect to the stationary at-rest position it will be called swing fuzzy controller. Therefore, both types are used in the proposed fuzzy control system, which consists of four separate controllers. Two controllers are deal with the motion of the cross-beam Figure 3. Swing-damping acceleration profiles EC 23,1 90 and the trolley while the other two controllers handle swing of the payload in both planes, as shown in Figure 5, where T.S.FLC, C.B.S.FLC, C.B.D.FLC and T.D.FLC refer to trolley swing, cross-beam swing, cross-beam displacement and trolley displacement FLCs, respectively. Angles a4 and b4 are not incorporated in any controller because large magnitude of a4 and b4 are only excited when large accelerations are used. Three membership sets are used to describe the all variables. The same rules are used for both displacement controllers. The two swing controllers also share a common set of rules. Nine rules are specified for processing the fuzzy data of each controller. Table I lists the rules of the displacement controllers. From Table I, it can be seen that “IF the error of displacement and velocity are positive THEN a big negative force is required”. On the other hand, “IF the error of displacement and velocity are negative THEN a big positive force is required”. Table II, listed Figure 5. The proposed FLC for an overhead crane Figure 4. FLC of an overhead crane Fuzzy logic controller for overhead cranes 91 the rules for the swing controller. These rules are based on the idea that control forces are needed only when the payload is moving away from the vertical position. P, Z and N are linguistic variable describe positive, zero and negative, respectively. 4. Simulation results and discussion To validate the performance of the designed FLCs above, we simulated the overhead crane system given by Nalley and Trabia (2000), where the crane parameters were selected as follows: mt ¼ 300 kg; mb ¼ 1; 150 kg; r ¼ 1; 153 kg=m; lr ¼ 3 m; the payload is a cylinder with a length lp ¼ 5:7 m: The cross-beam moves 60 meters while the trolley moves 20 m both in 20 s. To simplify the design of the controller, normalized values between21 and 1 are defined to describe the input and output fuzzy sets of the FLC as shown in Table III (Nalley and Trabia, 2000). These fuzzy sets are shown in Figure 6 using triangular and trapezoidal symmetry membership functions. The force membership is also shown in Figure 7, which consists of three-singleton membership functions. In order to compare the effectiveness of the proposed FLC, it is compared with a PD controller. Only d1, d2, a3 and b3 are used in the feedback to make the result comparable to those of the proposed FLC. By linear quadratic regulator, the parameters of the PD controller are tuned. These parameters are (Nalley and Trabia, 2000): Displacement error Velocity error P Z N P N N Z Z N Z P N Z P P Table I. Rules for the displacement fuzzy controller Angular displacement error Angular velocity error P Z N P P P Z Z Z Z Z N Z N N Table II. Rules for the swing fuzzy controller Displacement error (m) Velocity error (m/s) Displacement force (N) Angular displacement error (rad) Angular velocity error (rad/s) Swing force (N) Cross-beam ^15 ^2.0 ^20,000 ^0.05 ^1.0 ^10,000 Trolley ^12.0 ^2.85 ^13,000 ^0.05 ^1.0 ^5,000 Table III. Ranges of fuzzy controller variables EC 23,1 92 Kp ¼ 21; 000 21; 000 22; 862 22; 517 0 0 � �T Kd ¼ 24; 062 23; 748 214; 934 215; 934 0 0 � �T ð19Þ Figures 8 and 9 show the displacement of the cross-beam and trolley, respectively, using PD and the proposed FLCs. In terms of percent peak overshoot, settling time Figure 6. Membership sets for the fuzzy controller input variable Figure 7. Membership function for the fuzzy controller output force Fuzzy logic controller for overhead cranes 93 and tracking, the proposed FLC has better performance than the traditional PD controller does. For example, the overshoot in the cross beam position is about 1.5 m comparing to about 4 m obtained using the PD controller. Figures 10 and 11 show the swing angles b3 and b4 using proposed fuzzy logic and PD controllers, respectively. Both controllers produce swinging of the same order for b4 while the proposed FLC produces less swinging of b3 and this result agrees with the Figure 8. Displacement of the cross-beam using FLC and PD controllers Figure 9. Displacement of the trolley using FLC and PD controllers EC 23,1 94 findings of Nalley and Trabia (2000). Figures 11 and 12 show the swing angles a3 and a4 using proposed fuzzy logic and PD controllers, respectively (Figure 13). Both controller produces swinging of the same order for a3 and a4. The required force to the cross-beam and trolley are shown in Figures 14 and 15, respectively, using proposed fuzzy logic and PD controllers. The input forces to both controllers are of the same order. Figure 10. Swing angles (b3 and b4) of the cross-beam and trolley using FLC controller Figure 11. Swing angles (b3 and b4) of the cross-beam and trolley using PD controller Fuzzy logic controller for overhead cranes 95 5. Conclusions In this study, a FLC is designed and successfully implemented to move the overhead crane along a desired path while ensuring that the payload is swing free at the end of the motion. The proposed FLC is divided into displacement and swing controllers. The inputs to these controllers are the displacement and velocity errors with respect to the desired motion profile. The outputs of the controllers are the forces needed to move the cross-beam and the trolley. Control simulation results demonstrate that by using Figure 12. Swing angles (a3 and a4) of the cross-beam and trolley using FLC controller Figure 13. Swing angles (a3 and a4) of the cross-beam and trolley using PD controller EC 23,1 96 the fuzzy controller, the overhead traveling crane smoothly moves to the destination in short time interval with small swing angle and almost no overshoot. The numerical simulation of the proposed FLC is compared with PD controller to demonstrate the effectiveness of the proposed FLC. Simulation results show that the proposed FLC improved the response of the system (e.g. reduced overshoot and faster settling time) over the PD controller. Figure 14. Input force to the cross-beam using FLC and PD controllers Figure 15. Input force to the trolley using FLC and PD controllers Fuzzy logic controller for overhead cranes 97 References Giua, A., Seatzu, C. and Usai, G. 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