Multi-Objective Motion Control Optimization for the Bridge Crane System

Renxin Xiao; Zelin Wang; Ningyuan Guo; Yitao Wu; Jiangwei Shen; Zheng Chen

doi:10.3390/app8030473

Xiao, Renxin;Wang, Zelin;Guo, Ningyuan;Wu, Yitao;Shen, Jiangwei;Chen, Zheng

2018-03-20 00:00:00

applied sciences Article Multi-Objective Motion Control Optimization for the Bridge Crane System ID ID Renxin Xiao, Zelin Wang , Ningyuan Guo, Yitao Wu, Jiangwei Shen and Zheng Chen * Faculty of Transportation Engineering, Kunming University of Science and Technology, Kunming 650500, China; xrx1127@foxmail.com (R.X.); zelinwang0@foxmail.com (Z.W.); gnywin@163.com (N.G.); yitaowumail@gmail.com (Y.W.); shenjiangwei6@163.com (J.S.) * Correspondence: chen@kmust.edu.cn; Tel.: +86-186-6908-3001 Received: 19 February 2018; Accepted: 19 March 2018; Published: 20 March 2018 Featured Application: The speciﬁc application of the research aims to the port transportation, the working efﬁciency can be improved considering ﬁndings of this article. Abstract: A novel control algorithm combining the linear quadratic regulator (LQR) control and trajectory planning (TP) is proposed for the control of an underactuated crane system, targeting position adjustment and swing suppression. The TP is employed to control the swing angle within certain constraints, and the LQR is applied to achieve anti-disturbance. In order to improve the accuracy of the position control, a differential-integral control loop is applied. The weighted LQR matrices representing priorities of the state variables for the bridge crane motion are searched by the multi-objective genetic algorithm (MOGA). The stability proof is provided in order to validate the effectiveness of the proposed algorithm. Numerous simulation and experimental validations justify the feasibility of the proposed method. Keywords: anti-disturbance; bridge crane system; linear quadratic regulator (LQR); multi-objective genetic optimization (MOGA); trajectory planning 1. Introduction Nowadays, the development of port transportation brings with it increasing demands of cargo movement. The bridge crane system is widely applied in order to move the cargo with less cost compared to other transit systems [1]. For the crane system, it is imperative to design an effective controller that can achieve fast and safe cargo movement. For the cargo movement control, in addition to satisfying the target position, some constraints and disturbances need to be properly dealt with, including maximum moving speed and distance, maximum swing angle, etc. Focusing on these practical challenges, the designed controller should satisfy the demands, including fast response, high robustness, and stability, as well as strong anti-disturbance. Among them, the most important task is to regulate the crane to the desired position, and meanwhile to suppress the payload swing. In terms of this, the traditional control methods can be divided into two categories, i.e., open loop and closed loop methods. For open loop control methods, the control commands are determined in advance according to the requirements before operation. The popular methods include input shaping control [2,3], trajectory planning (TP) control [4,5], etc. The input shaping control is imposed in order to produce a series of pulses according to the ratio between the system frequency and the damping; then, the convolution between the pulse and the reference trajectory is conducted in order to generate the control command. This method can guarantee the control performance for the crane position when the cable length changes, whereas the residual swing of the payload cannot be eliminated. By TP, Appl. Sci. 2018, 8, 473; doi:10.3390/app8030473 www.mdpi.com/journal/applsci Appl. Sci. 2018, 8, 473 2 of 19 the crane trajectory can be designed based on the nonlinear coupling relationship between the crane position and the swing angle, thereby ensuring the cargo’s safety when moving and suppressing the residual oscillation. Both methods can satisfy the controlling target. However, when the system is under external disturbance, such as wind and human touch, the payload oscillation can be potentially triggered, and yet cannot be eliminated effectively based on the open loop methods. In order to overcome the inﬂuence of the external disturbance, various closed loop methods are proposed, which can be divided into adaptive control [6–8], fuzzy logic-based control [9,10], genetic algorithms (GA) [11,12], feedback linearization control [13,14], linear quadratic regulator (LQR) control [15–17], proportional–integral–differential (PID) control [18–20], etc. In [21], Zhang and He considered the crane velocity, the position error, and the swing angle totally in a linearized manner, and an adaptive sliding mode controller is designed. However, the convergence and boundedness proof of the position error and swing angle are not given. Based on linearization, the PID control algorithm, regulated by the fuzzy logic algorithm, is employed in order to improve the crane’s transient performance and robustness. Since the GA is capable of ﬁnding the optimal solutions, a nonlinear control algorithm is proposed based on a real-time GA in order to solve the difﬁculty of regulating the control gain [12,22–24]. In this manner, the whole control performance of the crane system is improved. Actually, it is difﬁcult for the GA algorithm to ensure the stability strictly at the equilibrium point theoretically, and the controller design is relatively complex, bringing inconvenience for practical application. For sake of reducing the controller complexity and easily applying it in actual operation, the feedback linearization control algorithm is introduced based on the simpliﬁed crane model, which divides the crane motion and the payload swing into two subsystems [25–28]. A nonlinear coupling control method is designed to ensure that the crane position and the payload swing can ﬁnally converge to speciﬁed ranges. Based on the model linearization, the LQR method is harnessed to achieve the control using the feedback gain of the linear systems [29]. Although the LQR method is simple and highly efﬁcient, the system states can possibly deviate from the equilibrium points when the system is disturbed by external signals, and thus, it is difﬁcult to guarantee the stability of the control performance. To solve this problem, the constraint of the payload swing and the capability of anti-disturbances need to be taken into account, and some combined algorithms of open loop and closed loop methods are introduced [30–32]. In [30], Blajer and Kołodziejczyk employed the feedforward control and the feedback PID control together, according to the inverse dynamics analysis. However, this method lacks the stability analysis, and it is difﬁcult to satisfy the demand of the payload angle and crane position simultaneously with the PID control. Aiming at these considerations, a combined open loop and closed loop method is herein proposed to satisfy the constraint of the payload swing, and precisely realize the position control of the crane. First, the motion path of the crane is designed by TP, which keeps the payload swing states in the vicinity of the equilibrium point due to the underactuated characteristics. By this manner, the constraints of the swing angle can be satisﬁed, and the residual swing can be eliminated. Then, an updated LQR algorithm is introduced to improve the precision of the crane position and the payload swing control, which are immune from external disturbance. Here, for the sake of the multi-freedom control for the crane position, a differential and integral control loop is added to track the crane position with fast speed. In order to gain preferable control effects, a multi-objective GA (MOGA) algorithm is employed to achieve the different motion state optimization by searching the weighed matrices of the LQR [33]. The MOGA is characterized by the weight of the multi-objective functions being assigned randomly; thus, its search direction is not ﬁxed. A set of Pareto optimal solutions can be retained and uniformly distributed during implementation [34,35]. By this manner, the MOGA exhibits the advantage of the diversity of solutions, and meanwhile, the computational complexity is lower compared with the traditional GA [36,37]. Finally, the quasioptimal control of the crane is reached by simulation and experiment validation. Appl. Sci. 2018, 8, 473 3 of 19 Appl. Sci. 2018, 8, x FOR PEER REVIEW 3 of 19 The remainder of this paper is structured as follows. The detailed modeling and associated constraints are introduced in Section 2. The controller design and its stability proof are provided in Section 3. The feasibility of the proposed method is verified by the simulation and experiments in Section 3. The feasibility of the proposed method is veriﬁed by the simulation and experiments in Sections 4 and 5, respectively. Finally, conclusions are drawn in Section 6. Sections 4 and 5, respectively. Finally, conclusions are drawn in Section 6. 2. Dynamic Modeling of the Crane System 2. Dynamic Modeling of the Crane System In order to design an algorithm with the target of precise position control and anti-disturbance, In order to design an algorithm with the target of precise position control and anti-disturbance, the physical model of the two-dimensional (2D) underactuated system is established, as shown in the physical model of the two-dimensional (2D) underactuated system is established, as shown in Figure 1. In order to simplify the modeling, some assumptions are made. Figure 1. In order to simplify the modeling, some assumptions are made. Yy / x 0 Xx / Figure 1. Crane physical model. Figure 1. Crane physical model. Assumption 1. The crane is a rigid driving system, and the payload is assumed to be a rigid body. Additionally, Assumption 1. The crane is a rigid driving system, and the payload is assumed to be a rigid body. Additionally, the air friction is neglected. the air friction is neglected. Assumption 2. The cable length is unchanged. Assumption 2. The cable length is unchanged. Assumption 3. The payload swing angle q satisﬁes p/2 < q(t) < p/2. Assumption 3. The payload swing angle θ satisfies −π / 2 < < π / 2 . θ () t With these assumptions, dynamic equations can be formulated [32]: With these assumptions, dynamic equations can be formulated [32]: .. . .. (m + m )x + m Lq cos q m Lq sin q = F (1) (1) T P T P P    (m ++ m ) x mLθθ cos − mLθ sinθ = F TP TP P .. .. x cos q + Lq + g sin q = 0 (2) (2)   x cosθθ ++ Lg sinθ = 0 where x is the horizontal position of the payload, x is the crane horizontal position, L is the cable P T length, m and m are the mass of the crane and payload, respectively, g presents the gravity factor, T P x x where is the horizontal position of the payload, is the crane horizontal position, L is the P T and F is the driving force. Through the geometric operation and analysis, the payload position x can m m cable length, and are the mass of the crane and payload, respectively, g presents the T P be obtained: gravity factor, and F is the driving force. Through the geometric operation and analysis, the x = x + l sin q (3) P T payload position can be obtained: It can be found that due to the underactuated characteristics, the payload swing cannot be designed directly, and instead it can be planned by the coupling relationship between the crane x xl+ sinθ (3) P T and the payload. From Equation (2), we can provide the theoretical analysis for the subsequent contrIol t ca methods. n be found that due to the underactuated characteristics, the payload swing cannot be designed directly, and instead it can be planned by the coupling relationship between the crane and During the process of the actual transportation, the payload swing should be imposed within the th small e paangle, yload.i.e.,: From Equation (2), we can provide the theoretical analysis for the subsequent control methods. > sin q q During the process of the actual transpor cos tatiqon, t 1he payload swing should be imposed within (4) . .. the small angle, i.e.,: L = L 0 sinθθ ≈ cosθ ≈ 1  (4)   LL≈ 0 According to Equations (2) and (4), the dynamic equations can be simplified as: = Appl. Sci. 2018, 8, 473 4 of 19 Appl. Sci. 2018, 8, 473 4 of 19 (5) x  L  g  0 According to Equations (2) and (4), the dynamic equations can be simpliﬁed as: .. In order to fully consider the Stribec ..k effect, the friction, and the crane mass, is selected as x + Lq + gq = 0 (5) the control command. Now, the matrix equation of model can be established according to Equation .. (5): In order to fully consider the Stribeck effect, the friction, and the crane mass, x is selected as the control command. Now, the matrix equation of model can be established according to Equation (5): 0 1 0 0 0         2 3 2 3 0 0 0 0 1 0  1 0 0   0 X X u 6 7 6 7 0 0 0 1  0  (6) 0 0 0 0 1 6 7 6 7     X = X + u 6 7 6 7 0 0 g / L 0 1 / L 4 0 0 0 1 5 4 0 5     (6) 0 0YX g/1111 L 0 1/L   h i Y = X 1 1 1 1 X  [x , x , , ] where is the state matrix of the bridge crane, T states the matrix transposition, and TT u() t  x denotes the crane acceleration. where X = [x , x , q, q] is the state matrix of the bridge crane, T states the matrix transposition, T T .. Aiming to satisfy the real-world requirement, three constraints are considered. and u(t) = x denotes the crane acceleration. Aiming to satisfy the real-world requirement, three constraints are considered. xR  t Constraint 1. The crane is required to reach the target position within a limited time : d total Constraint 1. The crane is required to reach the target position x 2 R within a limited time t : d total x (t) x , t  t (7) T d total x (t) = x , 8t t (7) d total Constraint 2. The velocity and acceleration of the crane should be restricted due to the actual limitation of Constraint 2. The velocity and acceleration of the crane should be restricted due to the actual limitation actuators: of actuators: . .. x (t) v , x (t) a x (t) v , a ((8) 8) x (t) TTT limlim T lim lim Constraint 3. The maximum swing angle of the payload should be bounded: Constraint 3. The maximum swing angle of the payload should be bounded:  () t  (9) jq(t)j q (9) lim lim where v ,a R are defined as the maximum velocity and maximum acceleration, respectively, lim lim where v , a 2 R are deﬁned as the maximum velocity and maximum acceleration, respectively, lim lim   R and andq 2 R is is th the e limited limited maximum maximum swing swing a angle. ngle. lim lim Next, the corresponding controller design is detailed in the following section. Next, the corresponding controller design is detailed in the following section. 3. Control Design 3. Control Design In this paper, a LQR algorithm based on the TP is proposed for the 2D underactuated system In this paper, a LQR algorithm based on the TP is proposed for the 2D underactuated system with with the target of minimizing the regulating time and guaranteeing the swing angle under the the target of minimizing the regulating time and guaranteeing the swing angle under the constraints. constraints. The flowchart of the overall controller design is shown in Figure 2. The ﬂowchart of the overall controller design is shown in Figure 2.   J  t e () t dt T  Multi-objective k k k k   QR 1 2 3 4   Genetic Algorithm J  t e () t dt TT x x   T T LQR T Controller ut () Trajectory Differential and Planning  integral control Crane Model KK  ID Figure 2. Flowchart of the controller design. Figure 2. Flowchart of the controller design. Appl. Sci. 2018, 8, 473 5 of 19 As can be seen in Figure 2, the whole control framework consists of four parts, namely: the Appl. Sci. 2018, 8, 473 5 of 19 trajectory planning, the LQR controller, the crane model, and the MOGA algorithm. First, according to the desired position, the planning trajectory of the crane motion is planned in advance. Then, the modified LQR algorithm, which combines the differential and integral controllers, is implemented to As can be seen in Figure 2, the whole control framework consists of four parts, namely: control the bridge crane system with the help of the designed trajectory. The weighted matrixes of the trajectory planning, the LQR controller, the crane model, and the MOGA algorithm. First, according the improved LQR controller are determined based on the MOGA, with the target of minimizing the to the desired position, the planning trajectory of the crane motion is planned in advance. Then, integration of the time-weighted absolute value of the errors (ITAEs) for the payload swing and the the modiﬁed LQR algorithm, which combines the differential and integral controllers, is implemented crane position. By this manner, the control merits of the TP and LQR can be combined to better reach to control the bridge crane system with the help of the designed trajectory. The weighted matrixes of the control target. the improved LQR controller are determined based on the MOGA, with the target of minimizing the integration of the time-weighted absolute value of the errors (ITAEs) for the payload swing and the 3.1. Trajectory Planning Control crane position. By this manner, the control merits of the TP and LQR can be combined to better reach the control target. Based on the geometrical analysis, an efficient trajectory imposed by Constraints 1 to 3 can be acquired. The second-order ordinary differential equation and its derivative equation are solved 3.1. Trajectory Planning Control based on Equation (5): Based on the geometrical analysis, an efﬁcient trajectory imposed by Constraints 1 to 3 can be () t a  (t )  (t ) cos t sin t (1 cos t ) acquired. The second-order ordinary differential equation and its derivative equation are solved based (10) 0 n n n  g on Equation (5): q(t ) a  a q(t) = q(t ) cos w t + sin w t (1 cos w t) (10) n n n n  (t) (t ) sin t (t ) cos t sin t w g (11) 00 n n n n n . . w a q(t) = q(t )w sin w t + q(t ) cos w t sin w t (11) 0 n n 0 n n ax  () t g   gL / () t where , , and denote the initial swing and the angle velocity of the n T 0 0 .. payload, respectively. where w = g/L, a = x , q(t ) and q(t ) denote the initial swing and the angle velocity of the T 0 0 () t Since the payload does not oscillate at the initial motion point, and () t equal zero. payload, respectively. 0 0 Based Since on Ethe quapayload tions (10does ) andnot (11oscillate ), we can at get the : initial motion point, q(t ) and q(t ) equal zero. Based on 0 0 Equations (10) and (11), we can get:  aa    () t  (t) (12)  2   2 ag g a     q(t) + + F (t) = (12) g g  (tt )  ( ) / where . . . where F(t) = q(t)/w . () t Now, the phase plane is employed in terms of Equation (12) for further analysis, of which . Now, the phase plane is employed in terms of Equation (12) for further analysis, of which F(t) () t and refer to the longitudinal coordinate and the horizontal coordinate, respectively, as shown and q(t) refer to the longitudinal coordinate and the horizontal coordinate, respectively, as shown in in Figure 3. To implement the intuitive illustration, the following three cases are discussed according Figure 3. To implement the intuitive illustration, the following three cases are discussed according to to different values of the acceleration. different values of the acceleration. () t Case3 A A 1 2 () t Case1 Case2 Figure 3. Phase plane of the payload. Figure 3. Phase plane of the payload. Case 1. When a  0 , the vector in the phase plane moves in the clockwise direction at a constant angular Case 1. When a > 0, the vector in the phase plane moves in the clockwise direction at a constant angular velocity ; velocity w ; Case 2. When a < 0, the vector moves in the counterclockwise direction at the same velocity; Case 2. When a  0 , the vector moves in the counterclockwise direction at the same velocity; Appl. Sci. 2018, 8, 473 6 of 19 Case 3. When a = 0, the vector stays in the original point; namely, there does not exist any relative motion between the crane and the payload. Here, an appropriate trajectory is determined considering the geometric properties. Based on the above analysis, the primary expression can be furnished as: a , 0 t t < max acc .. x = (13) a , t + t t 2t + t max acc cnst acc cnst 0, else where t is the duration of the acceleration, and t is the duration for the constant speed stage. acc cnst As shown in Figure 3, t , t and a can be determined according to the geometric properties. acc cnst max At the beginning of acceleration, the vector starts to move from the original point towards O with the angular velocity w . After reaching A , the maximum payload swing is achieved, and the relationship with respect to q and a can be determined, i.e.,: max max gjq j max ja j = (14) max and they should be satisﬁed with the following constraints: jq j q ,ja j a (15) max max lim lim The payload swing reaches zero after one complete acceleration cycle. At T, the crane stops accelerating and keeps its speed unchanged, while the load’s swing angle maintains zero, as stated in Case 2. In addition, during the deceleration process presented in Case 3, the similar phenomenon can also be observed: that one cycle is required to be undergone in order to reach the target position through point A , during which the residual angle can be eliminated. Based on the previous analysis, the duration in terms of acceleration/deceleration stages requires the same time cost, i.e., T. Given the chosen target position, the motion states can be determined after an integral calculation: a t, 0 t t max acc . a t , t t t + t max acc acc acc cnst x(t) = (16) a (t T ), t + t t t max acc cnst > total total 0, else 1 2 > a t , 0 t t max acc a t t, t t t + t max acc acc acc cnst x(t) = (17) > a ( t T t), t + t t t max total acc cnst total 0, else The displacement of the entire operation period can be calculated as: total S = x(t)dt = a t (t + t ) (18) max acc acc cnst total where T = 2t + t . Moreover, the duration during the acceleration and constant velocity acc cnst total period can be calculated: t = T = 2p L/g acc (19) t = (S /v ) 2p L/g cnst max total Appl. Sci. 2018, 8, 473 7 of 19 where v = a t is the maximum velocity of the crane. Note that a pivotal condition versus v max max acc max should be considered here due to t 0, which can be furnished as: cnst total v v (20) max lim 2p L/g Combining with Equations (9), (14), and (19), the constraint can be set: 2a max q = q (21) max lim Now, based on Equations (8) and (21), the upper limit of the maximum acceleration can be yielded: gq lim a min a , (22) max lim where a indicates the boundary value of the acceleration, and ﬁnally, v = a t . The velocity lim max max acc constraint can be calculated considering Equation (20) in order to improve the efﬁciency, and thus, the maximum velocity can be deﬁned as: total v = min v , p , 2p a L/g (23) max lim lim 2p L/g Meanwhile, the maximum acceleration can be determined: max a = (24) max 2p L/g Now, all of the parameters are determined considering the designed requirements. In the next step, the corresponding controller and its parameters are designed with care. 3.2. LQR Control Based on Trajectory Planning The TP algorithm is easy to apply, which can take the moving efﬁciency and maximum payload swing into account. However, it is difﬁcult to adaptively adjust the swing for the payload in practical conditions, and the oscillation cannot be suppressed when external disturbance exists. Hence, the TP algorithm combined with the closed-loop control can effectively solve these problems. For the underactuated system, there exists multiple motion states, and thus, it is necessary to attain the feedback control. According to Equation (6), the LQR control law from the feedback input u = KX yields: LQR X = ( A BK)X (25) where K can be calculated by minimizing the energy function of LQR E , as: LQR T T E = X QX + u Ru dt (26) LQR LQR LQR where Q is a semi-deﬁnite matrix of 4 4, and R is a positive deﬁne matrix constant of 1 1. Then, K can be calculated as: 1 T K = R B P (27) where P is a positive deﬁnite symmetric matrix, which is evaluated from the Algebraic Riccati equation (ARE) as: T 1 T A P + PA + Q PBR B P = 0 (28) Appl. Sci. 2018, 8, 473 8 of 19 By this manner, the gain of LQR feedback K=[k , k , k , k ] can be estimated, and the crane 1 2 3 4 acceleration u can thus be determined. In order to accomplish the position control accurately, the differential and integral control is introduced for the crane’s position adjustment. Hence, the improved controller can be expressed as: t . . . u(t) = K e dt + K e k x k x k q k q (29) I T D T 1 T 2 T 3 4 where u(t) is the control input; and k , k , k , and k , belonging to K, are deﬁned as feedback gains for 1 2 3 4 the crane position and velocity, the payload swing, and the angular velocity, respectively. e = x x T d T is the error trajectory of the crane system, and K can be determined based on Q and R by engineering experience. However, it is difﬁcult to ﬁnd the optimal solution in this manner. K and K are integral I D and differential gains, which are designed by the response optimization, and the constraint of the response optimization can be expressed as: o ptimal(K , K ) = min max (x x ) I D Tsim Tbnd 0t10 (30) 0 x 0.320 0 t 1.2 s.t. 0.285 x 0.302 1.2 t 10 where x is the simulated response of the crane position, and x is the piecewise linear bound of Tsim Tbnd the crane position. 3.3. Multi-Objective Optimization It is critical to determine the control parameters of the designed algorithm, i.e., k , k , k , and k , 1 2 3 4 and thus, the weighted matrixes Q and R need to be optimized. Here, integration of time-weighted absolute value of the error (ITAE) is introduced in order to evaluate the convergence and oscillation of the system, as: J = tje (t)jdt q q (31) J = tje (t)jdt T T where J and J denote the ITAEs of the payload swing and the crane position, respectively. Here, q T the MOGA is applied to ﬁnd optimal solutions among multiple objective functions. According to the optimal method, a series of points reﬂecting the control effects can be presented based on the ITAEs of the crane and payload. In addition, the optimization process should be subject to the following constraints: f = min J , J ( ) Pareto q T > S total v = min v , , 2p a L/g > max lim lim 2p L/g (32) s.t p/2 < q(t) < p/2 gq : lim a min a , max lim Here, we employed the MOGA algorithm to calculate the optimization parameters with the criteria of minimizing the ITAE of the crane position and the payload swing. As can be seen in the Figure 4, the whole process will ﬁrst choose the initial weighted matrixes Q and R randomly. Then, the control commands can be generated based on the constraints and the updated ITAE of the crane position, and the payload swing can be calculated based on the ﬁtness function. In the next step, the MOGA is applied to search the optimal parameters through a series of the selection, crossover, and mutation, wherein the roulette wheel selection and the two-point crossover are adopted. For the MOGA algorithm, ﬁrst, some existing populations are selected to generate the next generation, and this process is known as selection. Some existing populations are considered as elitists, and are directly selected as next generations without any change. The selection criteria is based on the ﬁtness function Appl. Sci. 2018, 8, 473 9 of 19 f  min J , J  Pareto  T  total v  min v , , 2 a L / g  max lim lim  2/  Lg   (32) s.t  / 2 (t ) / 2  g lim a  min a , max  lim  2  Here, we employed the MOGA algorithm to calculate the optimization parameters with the criteria of minimizing the ITAE of the crane position and the payload swing. As can be seen in the Figure 4, the whole process will first choose the initial weighted matrixes and R randomly. Then, the control commands can be generated based on the constraints and the updated ITAE of the crane position, and the payload swing can be calculated based on the fitness function. In the next step, the MOGA is applied to search the optimal parameters through a series of the selection, crossover, and mutation, wherein the roulette wheel selection and the two-point crossover are adopted. For the MOGA algorithm, first, some existing populations are selected to generate the next generation, and this process is known as selection. Some existing populations are considered as elitists, and are directly selected as next generations without any change. The selection criteria is ba Appl. sed Sci. on 2018 th,e 8,f473 itness function and their corresponding constraints. During the crossover process 9, of th 19 e parent chromosomes are hybridized to generate new offsprings, and meanwhile, some bits of chromosomes are uniformly or randomly changed with a certain possibility. This is the so-called and their corresponding constraints. During the crossover process, the parent chromosomes are mutation, of which the main function is to avoid falling into a local optimum. By means of these hybridized to generate new offsprings, and meanwhile, some bits of chromosomes are uniformly actions, a new chromosome group is generated, which is different from the previous version. The or randomly changed with a certain possibility. This is the so-called mutation, of which the main whole action is operated iteratively until the terminal condition is reached, which usually includes function is to avoid falling into a local optimum. By means of these actions, a new chromosome group exceeding the budget time, reaching the maximum allowable amount, etc. Finally, the Pareto fronts is generated, which is different from the previous version. The whole action is operated iteratively and the optimized parameters can be found. In this paper, the population size, the crossover rate, until the terminal condition is reached, which usually includes exceeding the budget time, reaching the and the mutation rate of the MOGA algorithm are set to 40, 0.8, and 0.05, respectively, and the number maximum allowable amount, etc. Finally, the Pareto fronts and the optimized parameters can be of iterations is set to 50 after repetitive tunning. Furthermore, the constraint of the payload lim found. In this paper, the population size, the crossover rate, and the mutation rate of the MOGA swing is set to be less than 4°, the maximum acceleration is defined as 5 m/s , the maximum crane algorithm are set to 40, 0.8, and 0.05, respectively, and the number of iterations is set to 50 after x  0.4 repetitive tunning. Furthermore, the constraint q of the payload swing is set to be less than 4 , velocity is 1.5 m/s, the target position is m, and the error of two objective functions is limited lim 4 the maximum acceleration is deﬁned as 5 m/s , the maximum crane velocity is 1.5 m/s, the target within . 1  10 position is x = 0.4 m, and the error of two objective functions is limited within 1 10 . Next, the stability analysis is conducted, and the stability proof is given. Pareto Front Yes Initialize Population No min J , J Matrixes Q and R    T Fitness Functions Decoding Evaluation Individuals Selection Rank Fitness Functions Encoding Selection Crossover Mutation Population Evaluation Offspring Elitism Fig Figure ure 4 4. . F Flowchart lowchart o of f m multi-objective ulti-objective g genetic enetic a algorithm. lgorithm. Next, the stability analysis is conducted, and the stability proof is given. 3.4. Stability Analysis 3.4. Stability Analysis Theorem. Independently from the stochastic initial payload swing, the proposed algorithm given by Equation (29) can control the crane to the desired position and suppress the payload swing, i.e.,: Theorem. Independently from the stochastic initial payload swing, the proposed algorithm given by Equation (29) can control the crane to the desired position and suppress the payload swing, i.e.,: h i h i . .. . .. lim = x 0 0 0 0 0 (33) x x x q q q T T T t!¥ Proof. In order to meet the demand of the theorem, a non-negative function is selected as: V(t) = Lq + g(1 cos q) 0 (34) Differentiating Equation (34) and combining Equation (2) with Equation (29), we can get: . .. V(t) = q(Lq + g sin q) . . (35) . . = q cos q(K e dt + K e k x k x k q k q) I T D T 1 T 2 T 3 4 According to the principle of the arithmetic mean–geometric mean (AM–GM) inequality, Equation (35) can be rewritten as: . . . 2 R 2 . . 1 1 V(t) q cos q + K e dt + K e k x k x k q k q I T D T 1 T 2 T 3 4 2 2 0 (36) . . 2 R 2 2 t 2 1 5 2 2 2 2 2 2 2 q cos q + K e dt + (K + k ) x + k x + k q + k q T D 2 3 4 I T T 2 2 0 1 Appl. Sci. 2018, 8, 473 10 of 19 By integrating Equation (34), we can further get: R 2 R R 2 t t t 1 5 2 2 V(t) V(0) + q cos qdt + K e dt dt 2 0 2 0 0 (37) R R R R 2 2 t t t t 5 5 2 2 5 2 2 5 2 + (K + k ) x dt + k x dt + k q dt + k q dt D 2 T 3 4 2 2 1 T 2 2 0 0 0 0 In Equation (37), we can ﬁnd that all of the integration items are bounded. Based on Equations (4) and (34), it can prove that: t . q cos qdt 2 L (38) Therefore, Equation (37) can be utilized to show that V(t) 2 L , Based on this fact, the following conclusion can be drawn from Equation (2). It can be proven that: .. .. q(t), x (t) 2 L (39) T ¥ Then, Barbalat lemma [38] can then be directly utilized to show that: .. .. lim q = 0, lim x = 0 (40) t!¥ t!¥ And Equation (37) can be rewritten as: R R 2 R R R R . t t t 2 t t t 5 2 2 2 2 2 2 K e dt dt + (K + k ) x dt + k x dt + k q dt + k q dt T D 2 3 T 4 2 I 0 0 0 1 0 T 0 0 (41) R 2 V(0) V(t) + q cos qdt 2 0 Based on V(t) 2 L and Equation (38), Equation (41) can then be employed to conclude that: q(t), q(t), x (t), x (t) 2 L (42) T T ¥ According to Equations (37) and (42), it is easy to show that: lim q = 0, lim x = 0 (43) t!¥ t!¥ Based on Equations (2), (40), and (43), we can go through a similar analysis to show that: lim sin q = 0 (44) t!¥ Thus, Assumption 3 and Equation (4) can be employed to conclude that: lim q = 0 (45) t!¥ From Equations (29), (41) and (45), it is clear that: lim x = x (46) T d t!¥ Now, the designed controller is proved to be asymptotically stable. Next, numerical simulation and experimental validation are performed in order to validate the proposed algorithm. 4. Numerical Simulation In this section, the simulation validation and control performance validation are conducted. All simulations are carried out based on Matlab/Simulink, and the MOGA is implemented with its built-in standard code. The crane trajectory is designed with the constraints provided by the TP. Appl. Sci. 2018, 8, 473 11 of 19 The anti-disturbance of the control is solved combining the LQR. The model parameters are shown in Table 1. Table 1. Model parameters. Parameter Note Value L Payload length 0.122 m g Gravity 9.81 m/s Based on the MOGA, a series of Pareto fronts can be achieved in light of the different requirements after iterations. Since the security of the cargo has a higher priority, the swing optimal solution is considered chieﬂy in this section, and the optimal control solutions of the Pareto front are shown in Figure 5. The horizontal coordinate and the vertical coordinate denote the ITAEs of the swing angle and the crane’s position, respectively. According to the actual application, three solutions, including the time optimization, swing angle minimization, and the trade-off in between, are selected for further analysis, as marked in Figure 6. The weighted matrixes of three solutions, Q and R, are obtained by the MOGA; then, the feedback gain K is achieved with these weighted matrixes. The gain of the integration control and differential control, i.e., K and K , are acquired by the response optimization. I D The weighted matrixes and the parameters of these three solutions are displayed in Table 2, and the performances of them are shown in Table 3. The main tasks of Solutions 1 and 3 are to minimize the swing angle and the regulation time, respectively. The main destination of Solution 2 is the trade-off Appl. Sci. 2018, 8, x FOR PEER REVIEW 12 of 19 Appl. Sci. 2018, 8, x FOR PEER REVIEW 12 of 19 between solutions 1 and 3. 1000 Pareto Front Pareto Front Solution1 Solution1 Solution2 Solution2 Solution3 800 Solution3 2400 2600 2800 3000 3200 3400 3600 3800 4000 2400 2600 2800 3000 3200 3400 3600 3800 4000 ITAE of Swing ITAE of Swing Figure 5. Pareto optimal front. Figure 5. Pareto optimal front. Figure 5. Pareto optimal front. 0.4 0.4 0.3 0.3 0.2 Solution1 0.2 Solution1 Solution2 0.1 Solution2 0.1 Solution3 0 Solution3 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -2 -2 -4 -4 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Time (seconds) Time (seconds) Figure 6. The response of three kinds of solutions. Figure 6. The response of three kinds of solutions. Figure 6. The response of three kinds of solutions. Table 3. Performance of three solutions. ITAE: integration of the time-weighted absolute Table 3. Performance of three solutions. ITAE: integration of the time-weighted absolute value of the errors. value of the errors. Performance Solution 1 Solution 2 Solution 3 Performance Solution 1 Solution 2 Solution 3 Settling time (s) 4.21 2.68 1.86 Settling time (s) 4.21 2.68 1.86 Maximum payload swing (deg) 1.00 2.00 4.00 Maximum payload swing (deg) 1.00 2.00 4.00 ITAE of swing 2469.21 2993.82 3999.16 ITAE of swing 2469.21 2993.82 3999.16 ITAE of positon 910.92 328.69 154.58 ITAE of positon 910.92 328.69 154.58 It can be observed that Solution 1 is able to achieve the minimal payload swing. The maximum It can be observed that Solution 1 is able to achieve the minimal payload swing. The maximum payload swing is 1°, and the ITAE of the swing is 2469.21, whereas it takes a longer time, i.e., 4.21 s, payload swing is 1°, and the ITAE of the swing is 2469.21, whereas it takes a longer time, i.e., 4.21 s, to reach control command. Solution 3 achieves the optimized time regulation, in which the minimum to reach control command. Solution 3 achieves the optimized time regulation, in which the minimum settling time is required to sacrifice the regulation swing. The settling time is 2.35 s, and the ITAE of settling time is required to sacrifice the regulation swing. The settling time is 2.35 s, and the ITAE of the position control declines by 83.03%. However, the maximum payload swing increases by 3° the position control declines by 83.03%. However, the maximum payload swing increases by 3° compared with Solution 1. Solution 2 is the trade-off regulation, which means that the time is not at compared with Solution 1. Solution 2 is the trade-off regulation, which means that the time is not at its minimum, and the swing is not yet smallest when compared with solutions 1 and 3. This solution its minimum, and the swing is not yet smallest when compared with solutions 1 and 3. This solution is the most reasonable choice, since that its settling time is reduced by 1.53 s compared with Solution is the most reasonable choice, since that its settling time is reduced by 1.53 s compared with Solution 1, and the payload swing decreases by 2° compared with Solution 3. Thus, Solution 2 can be selected 1, and the payload swing decreases by 2° compared with Solution 3. Thus, Solution 2 can be selected as the control algorithm for the following study, since it can not only ensure the load swing angle as the control algorithm for the following study, since it can not only ensure the load swing angle within a small range, it can also control the load to reach the target position with fast speed. within a small range, it can also control the load to reach the target position with fast speed. By simulation, the parameters based on the LQR and the proposed algorithm are shown in Table By simulation, the parameters based on the LQR and the proposed algorithm are shown in Table 4, respectively. There exists obvious difference among the algorithm parameters, since the crane 4, respectively. There exists obvious difference among the algorithm parameters, since the crane trajectory is predetermined by the proposed algorithm. From Figure 7, although the TP and LQR can trajectory is predetermined by the proposed algorithm. From Figure 7, although the TP and LQR can achieve the effective control, the proposed controller can improve the efficiency that the TP algorithm achieve the effective control, the proposed controller can improve the efficiency that the TP algorithm cannot handle. As displayed in Figure 8, the performance indicators of the proposed algorithm are cannot handle. As displayed in Figure 8, the performance indicators of the proposed algorithm are preferably superior to those of the other two control methods under the same swing constraint. preferably superior to those of the other two control methods under the same swing constraint. Compared with the TP, the settling time shortens to be 1.35 s, the ITAE of the swing angle decreases Compared with the TP, the settling time shortens to be 1.35 s, the ITAE of the swing angle decreases ITAE of Position ITAE of Position Position (m) Theta (deg) Position (m) Theta (deg) Appl. Sci. 2018, 8, 473 12 of 19 Table 2. Performance of three solutions. Parameter Solution 1 Solution 2 Solution 3 2 32 32 3 470.19 0 0 0 498.57 0 0 0 474.60 0 0 0 6 76 76 7 0 219.74 0 0 0 230.79 0 0 0 230.78 0 0 6 76 76 7 4 54 54 5 0 0 465.33 0 0 0 503.24 0 0 0 450.99 0 0 0 0 2.01 0 0 0 2.01 0 0 0 2.06 R 2.21 2.05 2.42 [k , k , k , k ] [14.60, 13.13,14.66,0.93] [15.89, 13.93,15.89,0.95] [13.99, 12.75,13.75,0.90] 1 2 3 4 K 0.02 0.02 0.04 K 166.82 169.34 175.86 Table 3. Performance of three solutions. ITAE: integration of the time-weighted absolute value of the errors. Performance Solution 1 Solution 2 Solution 3 Settling time (s) 4.21 2.68 1.86 Maximum payload swing (deg) 1.00 2.00 4.00 ITAE of swing 2469.21 2993.82 3999.16 ITAE of positon 910.92 328.69 154.58 It can be observed that Solution 1 is able to achieve the minimal payload swing. The maximum payload swing is 1 , and the ITAE of the swing is 2469.21, whereas it takes a longer time, i.e., 4.21 s, to reach control command. Solution 3 achieves the optimized time regulation, in which the minimum settling time is required to sacriﬁce the regulation swing. The settling time is 2.35 s, and the ITAE of the position control declines by 83.03%. However, the maximum payload swing increases by 3 compared with Solution 1. Solution 2 is the trade-off regulation, which means that the time is not at its minimum, and the swing is not yet smallest when compared with solutions 1 and 3. This solution is the most reasonable choice, since that its settling time is reduced by 1.53 s compared with Solution 1, and the payload swing decreases by 2 compared with Solution 3. Thus, Solution 2 can be selected as the control algorithm for the following study, since it can not only ensure the load swing angle within a small range, it can also control the load to reach the target position with fast speed. By simulation, the parameters based on the LQR and the proposed algorithm are shown in Table 4, respectively. There exists obvious difference among the algorithm parameters, since the crane trajectory is predetermined by the proposed algorithm. From Figure 7, although the TP and LQR can achieve the effective control, the proposed controller can improve the efﬁciency that the TP algorithm cannot handle. As displayed in Figure 8, the performance indicators of the proposed algorithm are preferably superior to those of the other two control methods under the same swing constraint. Compared with the TP, the settling time shortens to be 1.35 s, the ITAE of the swing angle decreases by 3.24%, and the ITAE of the position decreases by 98.05%. Compared with LQR, the settling time reduced by 1.11 s, the ITAE of swing decreased by 12.82%, and the ITAE of position fell by 68.13%. Based on the above comparative analysis, we can conclude that the proposed method can achieve the position control rapidly, and meanwhile satisfy the swing constraint requirement. Table 4. Parameters of the linear quadratic regulator (LQR), and the proposed method. Parameter LQR Proposed Method K NA 0.02 K NA 169.34 [k , k , k , k ] [1.46, 2.56,3.13,1.35] [15.89, 13.93,15.89,0.95] 1 2 3 4 Appl. Sci. 2018, 8, x FOR PEER REVIEW 13 of 19 Appl. Sci. 2018, 8, x FOR PEER REVIEW 13 of 19 by 3.24%, and the ITAE of the position decreases by 98.05%. Compared with LQR, the settling time by 3.24%, and the ITAE of the position decreases by 98.05%. Compared with LQR, the settling time reduced by 1.11 s, the ITAE of swing decreased by 12.82%, and the ITAE of position fell by 68.13%. reduced by 1.11 s, the ITAE of swing decreased by 12.82%, and the ITAE of position fell by 68.13%. Based on the above comparative analysis, we can conclude that the proposed method can achieve the Based on the above comparative analysis, we can conclude that the proposed method can achieve the position control rapidly, and meanwhile satisfy the swing constraint requirement. position control rapidly, and meanwhile satisfy the swing constraint requirement. Table 4. Parameters of the linear quadratic regulator (LQR), and the proposed method. Table 4. Parameters of the linear quadratic regulator (LQR), and the proposed method. Parameter LQR Proposed Method Parameter LQR Proposed Method NA 0.02 K NA 0.02 K NA 169.34 NA 169.34 [1.46, 2.56,−− 3.13, 1.35] [15.89,13.93,−− 15.89, 0.95] [, k k , k , k ] 12 3 4 [1.46, 2.56,−− 3.13, 1.35] Appl. Sci. 2018, 8, 473 [15.89,13.93,−− 15.89, 0.95] 13 of 19 [, k k , k , k ] 12 3 4 0.4 0.4 Trajectory Planning 0.2 LQR Trajectory Planning 0.2 Proposed Method LQR 0 0.5 1 1.5 2 2.5 3 3.5 4 Propos 4.ed M 5 ethod 5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -2 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -2 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -1 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -1 Time (seconds) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Time (seconds) Figure 7. Control response of trajectory planning (TP), linear quadratic regulator (LQR), and the Figure 7. Control response of trajectory planning (TP), linear quadratic regulator (LQR), and the Figure 7. Control response of trajectory planning (TP), linear quadratic regulator (LQR), and the proposed method. proposed method. proposed method. Settling Time Payload Swing ITAE of Swing ITAE of Position Settling Time Payload Swing ITAE of Swing ITAE of Position 16867.56 4.5 18000 16867.56 4.5 18000 4 16000 4 16000 3.5 4.03 14000 3.78 3.5 4.03 14000 3 12000 3.78 3 12000 2.5 10000 2.68 2.5 10000 2 8000 2.68 2 8000 1.5 6000 3434.09 3094.15 2993.82 1.5 6000 1 4000 3434.09 3094.15 2993.82 1 4000 1031.32 0.5 2000 328.69 1031.32 0.5 328.69 0 0 0 0 Trajectory Planning LQR Proposed Method Trajectory Plannin C gontrol S LQ traR tegies Proposed Method Control Strategies Figure 8. Performance of three controllers. Figure 8. Performance of three controllers. Figure 8. Performance of three controllers. In order to verify the anti-disturbance performance, an acceleration excitation is given to the crane, In order to verify the anti-disturbance performance, an acceleration excitation is given to the In order to verify the anti-disturbance performance, an acceleration excitation is given to th 2e thereby leading to a certain swing angle for the payload. Here, the acceleration is set as 0.8 m/s , crane, thereby leading to a certain swing angle for the payload. Here, the acceleration is set as 0.8 crane, thereby leading to a certain swing angle for the payload. Here, the acceleration is set as 0.8 and its duration as 0.2 s. The responses of the TP, LQR, and proposed algorithm are compared m/s , and its duration as 0.2 s. The responses of the TP, LQR, and proposed algorithm are compared m/s , and its duration as 0.2 s. The responses of the TP, LQR, and proposed algorithm are compared in Figure 9. It can be observed that the maximum payload swing of TP is 7.3 when the external in Figure 9. It can be observed that the maximum payload swing of TP is 7.3° when the external in Figure 9. It can be observed that the maximum payload swing of TP is 7.3° when the external disturbance occurs; obviously, it cannot meet the control demands. In addition, the LQR algorithm disturbance occurs; obviously, it cannot meet the control demands. In addition, the LQR algorithm disturbance occurs; obviously, it cannot meet the control demands. In addition, the LQR algorithm can suppress the swing, and the maximum swing is 2.03 . However, the whole duration is still 1.66 s. can suppress the swing, and the maximum swing is 2.03°. However, the whole duration is still 1.66 can suppress the swing, and the maximum swing is 2.03°. However, the whole duration is still 1.66 The performance of the proposed algorithm is superior to that of other control methods, and the s. The performance of the proposed algorithm is superior to that of other control methods, and the s. The performance of the proposed algorithm is superior to that of other control methods, and the settling time is 1.44 s. Thus, it proves that the algorithm can achieve fast, stable ability and realize settling time is 1.44 s. Thus, it proves that the algorithm can achieve fast, stable ability and realize Appl. Sci. 2018, 8, 473 14 of 19 settling time is 1.44 s. Thus, it proves that the algorithm can achieve fast, stable ability and realize immune control of external disturbances. immune control of external disturbances. immune control of external disturbances. 0.6 0.4 Trajectory Planning 0.2 LQR Proposed Method 0 1 2 3 4 5 6 7 -5 0 1 2 3 4 5 6 7 Time (seconds) Figure 9. Response of different controllers with external disturbances. Figure 9. Response of different controllers with external disturbances. Next, experimental validation is performed to further justify the feasibility of the Next, experimental validation is performed to further justify the feasibility of the proposed proposed algorithm. algorithm. 5. Experimental Verification In order to ensure the safety of the cargo, the payload swing should be the prior control object, and Solution 2 is selected as the method of the experiment. The experimental validation is conducted based on a test platform of B&R Industrial Automation Ltd., which supplies an integrated solution for automation systems [39]. The experimental equipment is designed based on a downsized model according to the actual bridge crane [40], as shown in Figure 10. It employs a metal lever and a metal load to simulate the actual crane line and the payload, respectively. The system applies a motor to move the crane system in a horizontal direction, thereby simulating the actual cargo transportation. Compared with actual cranes, the downsized model can meet the requirements of control algorithm regulation, and yet, the weight and vertical swing of the cable and the friction of the crane during movement cannot be taken into account [41]. Program mabl e ACOPOS imicro Power S upply Logic Controller Hosting P C Servo Motor Crane & Payload Com pact Si mulation P anel Figure 10. Anti-swing control equipment. In the actual application, there are several factors that can possibly influence the control precision, e.g., the target position, the payload weight, the cable length, and the external disturbance. In order to justify the advance of the proposed algorithm, a series of experimental validations are carried out considering different constraints, different payloads, different cable lengths, and disturbances, respectively. 5.1. Constraints Condition In order to verify whether the constraints of the payload swing and the settling time can be both satisfied under different target positions, two groups of experiments are conducted. Here, the maximum swing angle is set to be 2°. The corresponding responses with respect to different moving ranges are shown in Figure 11 and Table 5, respectively. It can be found that the actual maximum Settling Time (seconds)/Maximum Settling Time (seconds)/Maximum payload swing (deg) payload swing (deg) Position (m) Control (m/s ) Theta (deg) Position (m) 2 Position (m) Theta (deg) Control (m/s ) Theta (deg) ITAE of Position(Swing) ITAE of Position(Swing) Appl. Sci. 2018, 8, x FOR PEER REVIEW 14 of 19 0.6 0.4 Trajectory Planning 0.2 LQR Proposed Method 0 1 2 3 4 5 6 7 -5 0 1 2 3 4 5 6 7 Time (seconds) Figure 9. Response of different controllers with external disturbances. Next, experimental validation is performed to further justify the feasibility of the proposed Appl. Sci. 2018, 8, 473 14 of 19 algorithm. 5. Experimental Veriﬁcation 5. Experimental Verification I In n or ord der er t to o e ensur nsure e t the he s safety afety o of f t the he c car arg go, o, t the he p payload ayload s swing wing s should hould b be e tthe he p prior rior con contr trol ol ob object, ject, and Solution 2 is selected as the method of the experiment. The experimental validation is conducted and Solution 2 is selected as the method of the experiment. The experimental validation is conducted b based ased o on n a a t test est p platform latform of of B B&R &R I Industrial ndustrial A Automation utomation L Ltd., td., w which hich s supplies upplies an an in int tegrated egrated s solution olution for automation systems [39]. The experimental equipment is designed based on a downsized model for automation systems [39]. The experimental equipment is designed based on a downsized model a accor ccord ding ing t to o t the he a actual ctual b bridge ridge crane crane [ [40 40], ], a as s s shown hown in in F Figur igure e 10 10.. I It t e employs mploys a a m metal etal lleve ever r a and nd a a m metal etal load to simulate the actual crane line and the payload, respectively. The system applies a motor to load to simulate the actual crane line and the payload, respectively. The system applies a motor to mo move ve t the he c crane rane s system ystem iin n a a hor horizontal izontal d dir ire ection, ction, tther here eby by s simulating imulating tthe he a actual ctual ca car rg go o ttransportation. ransportation. Co Compar mpared ed w with ith a actual ctual cr cranes, anes, t the he d downsized ownsized m model odel c can an m meet eet t th he e r re equir quire ements ments of of cont contr ro ol l a algorithm lgorithm r re egulation, gulation, a and nd y yet, et, t the he w weight eight a and nd v vertical ertical s swing wing of of t the he c cable able a and nd t the he f friction riction of of t the he cr crane ane during during m movement ovement ca cannot nnot b be e t taken aken into into account account [ [41 41]. ]. Program mabl e ACOPOS imicro Power S upply Logic Controller Hosting P C Servo Motor Crane & Payload Com pact Si mulation P anel Figure 10. Anti-swing control equipment. Figure 10. Anti-swing control equipment. In the actual application, there are several factors that can possibly influence the control In the actual application, there are several factors that can possibly inﬂuence the control precision, e.g., the target position, the payload weight, the cable length, and the external disturbance. precision, e.g., the target position, the payload weight, the cable length, and the external In order to justify the advance of the proposed algorithm, a series of experimental validations are disturbance. In order to justify the advance of the proposed algorithm, a series of experimental carried out considering different constraints, different payloads, different cable lengths, and validations are carried out considering different constraints, different payloads, different cable lengths, disturbances, respectively. and disturbances, respectively. 5.1. Constraints Condition 5.1. Constraints Condition In order to verify whether the constraints of the payload swing and the settling time can be In order to verify whether the constraints of the payload swing and the settling time can be both both satisﬁed under different target positions, two groups of experiments are conducted. Here, satisfied under different target positions, two groups of experiments are conducted. Here, the the maximum swing angle is set to be 2 . The corresponding responses with respect to different maximum swing angle is set to be 2°. The corresponding responses with respect to different moving moving ranges are shown in Figure 11 and Table 5, respectively. It can be found that the actual ranges are shown in Figure 11 and Table 5, respectively. It can be found that the actual maximum maximum swing angle is 1.92 , which satisﬁes the setting requirement. The regulation time is almost the same, and the responses are shown in Figure 12 and Table 6. The results show that all of the regulated time is less than 1.64 s. When the target position changes, the regulated time basically remains unchanged. To summarize, for different target positions, the proposed algorithm can achieve effective control. Table 5. Performance within angle constraint. Target Position Settling Time (s) Maximum Payload Swing (deg) 0.2 m 1.87 1.93 0.3 m 2.35 1.85 0.4 m 2.67 1.93 Position (m) Theta (deg) Appl. Sci. 2018, 8, x FOR PEER REVIEW 15 of 19 swing angle is 1.92°, which satisfies the setting requirement. The regulation time is almost the same, and the responses are shown in Figure 12 and Table 6. The results show that all of the regulated time is less than 1.64 s. When the target position changes, the regulated time basically remains unchanged. Appl. Sci. 2018, 8, 473 15 of 19 To summarize, for different target positions, the proposed algorithm can achieve effective control. Table 6. Performance within a time optimal solution. Appl. Sci. 2018, 8, x FOR PEER REVIEW 15 of 19 0.4 0.3 swing angle is T1 arget .92°,Position which satisfieSettling s the setti Time ng r (s) equireme Maximum nt. The re Payload gulation Swing time (Deg) is almost the same, 0.2 and the responses are shown in Figure 12 and Table 6. The results show that all of the regulated time Target Position=0.2m 0.2 m 1.63 2.0 0.1 Target Position=0.3m 0.3 m 1.64 Target P 3.2 osition=0.4m is less than 1.64 s. When the target position changes, the regulated time basically remains unchanged. 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0.4 m 1.66 4.0 To summarize, for different target positions, the proposed algorithm can achieve effective control. 0.4 -1 0.3 -2 0.2 Target Position=0.2m 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0.1 Target Position=0.3m Time (seconds) Target Position=0.4m 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Figure 11. The response within an angle constraint scheme. Table 5. Performance within angle constraint. Target Po -1sition Settling Time (s) Maximum Payload Swing (deg) -2 0.2 m 1.87 1.93 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Time (seconds) 0.3 m 2.35 1.85 0.4 m 2.67 1.93 Figure 11. The response within an angle constraint scheme. Figure 11. The response within an angle constraint scheme. Table 5. Performance within angle constraint. 0.4 0.3 Target Position Settling Time (s) Maximum Payload Swing (deg) 0.2 Position=0.2m 0.2 m 1.87 1.93 0.1 Position=0.3m Position=0.4m 0.3 m 2.35 1.85 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0.4 m 2.67 1.93 0.4 -2 0.3 -4 0.2 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Position=0.2m 0.1 Time (seconds) Position=0.3m Position=0.4m 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Figure 12. The time-optimization response within a time-optimal scheme. Figure 12. The time-optimization response within a time-optimal scheme. 5.2. Different Payload Condition Table 6. Performance within a time optimal solution. Some magnetic sheets are attached to the original load to simulate the weight variation, as shown Target Po-2 sition Settling Time (s) Maximum Payload Swing (Deg) in Figure 13. When the weight changes, the controller can still move the cargo to the desired position -4 0.2 m 1.63 2.0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 with almost the same maximum swing angle. TimThe e (secon maximum ds) difference is less than 0.06 . Thus, 0.3 m 1.64 3.2 the variation of the payload cannot affect the settling time and the maximum swing, and the robustness 0.4 m 1.66 4.0 Appl. Sci. 2018, 8, x FOR PEER REVIEW 16 of 19 Figure 12. The time-optimization response within a time-optimal scheme. of the proposed algorithm is proved to some extent. 5.2. Different Payload Condition Table 6. Performance within a time optimal solution. 0.4 0.3977 0.3977 Some magnetic shee 0.t 3s are attached to the original load to simulate the weight variation, as shown Target Position Settling Time (s) Maximum Payload Swing (Deg) 0.3977 Payload=0.50Kg 0.3977 0.2 Payload=0.51Kg in Figure 13. When the weight changes, the controller ca n still move the cargo to the desired position 0.2 m 1.63 2.0 2.4222 2.4222 2.4222 0.1 Payload=0.52Kg with almost the same maximum swing angle. The maximum difference is less than 0.06°. Thus, the Payload=0.53Kg 0.3 m 1.64 3.2 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 variation of the payload cannot affect the settling time and the maximum swing, and the robustness 0.4 m 1.66 4.0 of the proposed algorithm is proved to some extent. 1.7 1.6 1.5 5.2. Different Payload Condition 2.1 2.15 2.2 -1 Some magnetic sheets are attached to the original load to simulate the weight variation, as shown -2 in Figure 13. When the weight changes, the controller can still move the cargo to the desired position 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Time (seconds) with almost the same maximum swing angle. The maximum difference is less than 0.06°. Thus, the Figure 13. The response for different weights of payloads. variation of the payload cannot affect the settling time and the maximum swing, and the robustness Figure 13. The response for different weights of payloads. of the proposed algorithm is proved to some extent. 5.3. Different Cable Length Condition In addition, it is imperative to consider different cable lengths in order to verify the algorithm. The response is shown in Figure 14. The maximum payload swing increases by 0.2° when the cable length varies from 0.092 m to 0.122 m. Certainly, longer cable length brings larger swing angle. Nonetheless, it can still satisfy the constraint of the maximum angle, i.e., 2°. Thus, the controller can effectively control the crane to reach the target position, and meanwhile can satisfy the maximum angle constraint when the payload and cable length change. 0.4 0.3 0.371 Length=0.092m 0.371 0.2 Length=0.102m 0.371 Length=0.112m 0.1 2.0438 2.0438 2.0438 2.0438 Length=0.122m 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 1.8 1.6 1.4 2 2.1 2.2 2.3 -1 -2 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Time (seconds) Figure 14. The response in different cable lengths. 5.4. Disturbance Condition Moreover, the control performance needs to be validated when external disturbance exists. Here, the disturbance is maintained for 3 s to 5 s. Similar to that of the simulation, an external acceleration of 0.8 m/s is added, and its duration is 0.2 s. By comparing the responses of three different controllers in Figure 15, the proposed algorithm can realize the optimal control effect compared with the other methods. The proposed algorithm can reach the target position with the shortest time. It can also be observed that the TP algorithm cannot suppress the payload swing, and the LQR algorithm can induce the swing angle by 1.41°. In this manner, conclusions can be made that the proposed method can effectively suppress the external disturbance. Position (m) Position (m) Position (m) Position (m) Position (m) Theta (deg) Position (m) Theta (deg) Theta (deg) Theta (deg) Theta (deg) Theta (deg) Appl. Sci. 2018, 8, x FOR PEER REVIEW 16 of 19 0.4 0.3977 0.3977 0.3 0.3977 Payload=0.50Kg 0.2 0.3977 Payload=0.51Kg 2.4222 2.4222 2.4222 Payload=0.52Kg 0.1 Payload=0.53Kg 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 1.7 1.6 1.5 2.1 2.15 2.2 -1 -2 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Time (seconds) Appl. Sci. 2018, 8, 473 16 of 19 Figure 13. The response for different weights of payloads. 5.3. Different Cable Length Condition 5.3. Different Cable Length Condition In addition, it is imperative to consider different cable lengths in order to verify the algorithm. In addition, it is imperative to consider different cable lengths in order to verify the algorithm. The response is shown in Figure 14. The maximum payload swing increases by 0.2° when the cable The response is shown in Figure 14. The maximum payload swing increases by 0.2 when the cable length varies from 0.092 m to 0.122 m. Certainly, longer cable length brings larger swing angle. length varies from 0.092 m to 0.122 m. Certainly, longer cable length brings larger swing angle. Nonetheless, it can still satisfy the constraint of the maximum angle, i.e., 2°. Thus, the controller can Nonetheless, it can still satisfy the constraint of the maximum angle, i.e., 2 . Thus, the controller can effectively control the crane to reach the target position, and meanwhile can satisfy the maximum effectively control the crane to reach the target position, and meanwhile can satisfy the maximum angle constraint when the payload and cable length change. angle constraint when the payload and cable length change. 0.4 0.3 0.371 Length=0.092m 0.371 0.2 Length=0.102m 0.371 2.0438 2.0438 2.0438 2.0438 Length=0.112m 0.1 Length=0.122m 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 2 1.8 1.6 1.4 2 2.1 2.2 2.3 -1 -2 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Time (seconds) Figure 14. The response in different cable lengths. Figure 14. The response in different cable lengths. 5.4. Disturbance Condition 5.4. Disturbance Condition Moreover, the control performance needs to be validated when external disturbance exists. Here, Moreover, the control performance needs to be validated when external disturbance exists. Here, the disturbance is maintained for 3 s to 5 s. Similar to that of the simulation, an external acceleration of the disturbance is maintained for 3 s to 5 s. Similar to that of the simulation, an external acceleration 0.8 m/s is 2 added, and its duration is 0.2 s. By comparing the responses of three different controllers of 0.8 m/s is added, and its duration is 0.2 s. By comparing the responses of three different controllers in Figure 15, the proposed algorithm can realize the optimal control effect compared with the other in Figure 15, the proposed algorithm can realize the optimal control effect compared with the other methods. The proposed algorithm can reach the target position with the shortest time. It can also methods. The proposed algorithm can reach the target position with the shortest time. It can also be be observed that the TP algorithm cannot suppress the payload swing, and the LQR algorithm can observed that the TP algorithm cannot suppress the payload swing, and the LQR algorithm can induce the swing angle by 1.41 . In this manner, conclusions can be made that the proposed method induce the swing angle by 1.41°. In this manner, conclusions can be made that the proposed method Appl. Sci. 2018, 8, x FOR PEER REVIEW 17 of 19 can effectively suppress the external disturbance. can effectively suppress the external disturbance. 0.4 0.3 0.2 Trajectory Planning LQR 0.1 Proposed Method 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -0.5 -1 -1.5 4.6 4.8 -5 -10 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Time (seconds) Figure Figure 15. 15. The The rresponse esponse of of th thrree ee c contr ontrols olswith with d disturbance. isturbance. 6. Conclusions 6. Conclusions In this paper, a novel algorithm combining the TP and the LQR is employed to achieve control In this paper, a novel algorithm combining the TP and the LQR is employed to achieve control of the anti-swing crane system. The proposed algorithm takes the acceleration of the payload as the of the anti-swing crane system. The proposed algorithm takes the acceleration of the payload as the control variable, and the ITAEs of the position and the swing angle as the criterion of evaluation. control variable, and the ITAEs of the position and the swing angle as the criterion of evaluation. The The MOGA is applied to ﬁnd the optimal algorithm parameters. Compared with the TP algorithm and MOGA is applied to find the optimal algorithm parameters. Compared with the TP algorithm and the LQR algorithm, the proposed algorithm can reach the control settings. The regulation time can be the LQR algorithm, the proposed algorithm can reach the control settings. The regulation time can be shortened, and the maximum swing angle can be reduced. Simulation and experimental validation justify the feasibility of the proposed algorithm. Following works will focus on the real implementation of the proposed algorithm in port cargo transportation. Acknowledgments: The research is supported by National Science Foundation of China (Grant No. 51567012, 61763021) in part, the Innovation Foundation of Kunming University of Science and Technology (Grant No. 2015YB053) in part, the Innovation Team Program of Kunming University of Science and Technology (No. 14078368) in part, and the Scientific Research Start-up Funding of Kunming University of Science and Technology (Grant No. 14078337) in part. In addition, the authors would like to thank the B&R Automation LLC for their hardware and program training support. Moreover, the authors would also like to thank reviewers for their corrections and helpful suggestions. Author Contributions: Renxin Xiao and Zelin Wang conceived the paper, discussed the multi-objective genetic optimization and the trajectory planning. Zheng Chen designed the bridge crane model and revised the paper. Ningyuan Guo and Yitao Wu performed the experiments and conducted the figures drawing. Jiangwei Shen analyzed the data and provided some valuable suggestions. Conflicts of Interest: The authors declare no conflict of interest. References 1. Abdel-Rahman, E.M.; Nayfeh, A.H.; Masoud, Z.N. Dynamics and Control of Cranes: A Review. J. Vib. Control 2003, 9, 863–908. 2. Maghsoudi, M.J.; Mohamed, Z.; Sudin, S.; Buyamin, S.; Jaafar, H.; Ahmad, S. An improved input shaping design for an efficient sway control of a nonlinear 3D overhead crane with friction. Mech. Syst. Signal Process. 2017, 92, 364–378. 3. Abdullahi, A.M.; Mohamed, Z.; Abidin, M.Z.; Buyamin, S.; Bature, A.A. Output-based command shaping technique for an effective payload sway control of a 3D crane with hoisting. Trans. Inst. Meas. Control 2017, 39, 1443–1453. 4. Zhang, M.; Ma, X.; Gao, F.; Tian, X.; Li, Y. A motion planning method for underactuated 3D overhead crane systems. In Proceedings of the 2015 34th Chinese Control Conference (CCC), Hangzhou, China, 28–30 July 2015; pp. 4286–4291. 5. Sun, N.; Fang, Y.; Zhang, X.; Yuan, Y. Transportation task-oriented trajectory planning for underactuated overhead cranes using geometric analysis. IET Control Theory Appl. 2012, 6, 1410–1423. 6. Chiu, C.-H.; Lin, C.-H. Adaptive output recurrent neural network for overhead crane system. In Proceedings of the SICE Annual Conference 2010, Taipei, Taiwan, 18–21 August 2010; pp. 1082–1087. Theta (deg) Position (m) Position (m) Position (m) Theta (deg) Theta (deg) Appl. Sci. 2018, 8, 473 17 of 19 shortened, and the maximum swing angle can be reduced. Simulation and experimental validation justify the feasibility of the proposed algorithm. Following works will focus on the real implementation of the proposed algorithm in port cargo transportation. Acknowledgments: The research is supported by National Science Foundation of China (Grant No. 51567012, 61763021) in part, the Innovation Foundation of Kunming University of Science and Technology (Grant No. 2015YB053) in part, the Innovation Team Program of Kunming University of Science and Technology (No. 14078368) in part, and the Scientiﬁc Research Start-up Funding of Kunming University of Science and Technology (Grant No. 14078337) in part. In addition, the authors would like to thank the B&R Automation LLC for their hardware and program training support. Moreover, the authors would also like to thank reviewers for their corrections and helpful suggestions. Author Contributions: Renxin Xiao and Zelin Wang conceived the paper, discussed the multi-objective genetic optimization and the trajectory planning. Zheng Chen designed the bridge crane model and revised the paper. Ningyuan Guo and Yitao Wu performed the experiments and conducted the ﬁgures drawing. Jiangwei Shen analyzed the data and provided some valuable suggestions. Conﬂicts of Interest: The authors declare no conﬂict of interest. References 1. Abdel-Rahman, E.M.; Nayfeh, A.H.; Masoud, Z.N. Dynamics and Control of Cranes: A Review. J. Vib. Control 2003, 9, 863–908. [CrossRef] 2. Maghsoudi, M.J.; Mohamed, Z.; Sudin, S.; Buyamin, S.; Jaafar, H.; Ahmad, S. An improved input shaping design for an efﬁcient sway control of a nonlinear 3D overhead crane with friction. Mech. Syst. Signal Process. 2017, 92, 364–378. [CrossRef] 3. Abdullahi, A.M.; Mohamed, Z.; Abidin, M.Z.; Buyamin, S.; Bature, A.A. Output-based command shaping technique for an effective payload sway control of a 3D crane with hoisting. Trans. Inst. Meas. Control 2017, 39, 1443–1453. [CrossRef] 4. Zhang, M.; Ma, X.; Gao, F.; Tian, X.; Li, Y. A motion planning method for underactuated 3D overhead crane systems. In Proceedings of the 2015 34th Chinese Control Conference (CCC), Hangzhou, China, 28–30 July 2015; pp. 4286–4291. 5. Sun, N.; Fang, Y.; Zhang, X.; Yuan, Y. Transportation task-oriented trajectory planning for underactuated overhead cranes using geometric analysis. IET Control Theory Appl. 2012, 6, 1410–1423. [CrossRef] 6. Chiu, C.-H.; Lin, C.-H. Adaptive output recurrent neural network for overhead crane system. In Proceedings of the SICE Annual Conference 2010, Taipei, Taiwan, 18–21 August 2010; pp. 1082–1087. 7. He, W.; Zhang, S.; Ge, S.S. Adaptive Control of a Flexible Crane System with the Boundary Output Constraint. IEEE Trans. Ind. Electron. 2014, 61, 4126–4133. [CrossRef] 8. Fang, Y.; Ma, B.; Wang, P.; Zhang, X. A motion planning-based adaptive control method for an underactuated crane system. IEEE Trans. Control Syst. Technol. 2012, 20, 241–248. [CrossRef] 9. Rahmani, R.; Karimi, H.; Yusof, R.; Othman, M.F. A precise fuzzy controller developed for overhead crane. In Proceedings of the 2015 10th Asian Control Conference (ASCC), Sabah, Malaysia, 31 May–3 June 2015; pp. 1–5. 10. Li, C.; Lee, C.-Y. Fuzzy motion control of an auto-warehousing crane system. IEEE Trans. Ind. Electron. 2001, 48, 983–994. 11. Nakazono, K.; Ohnisihi, K.; Kinjo, H. Load swing suppression in jib crane systems using a genetic algorithm-trained neuro-controller. In Proceedings of the ICM2007 4th IEEE International Conference on Mechatronics, Tokyo, Japan, 8–10 May 2007; pp. 1–4. 12. Kimiaghalam, B.; Homaifar, A.; Bikdash, M.; Dozier, G. Genetic algorithms solution for unconstrained optimal crane control. In Proceedings of the 1999 Congress on Evolutionary Computation CEC ’99, Washington, DC, USA, 6–9 July 1999; pp. 2124–2130. 13. Le, T.A.; Kim, G.H.; Min, Y.K. Partial feedback linearization control of overhead cranes with varying cable lengths. Int. J. Precis. Eng. Manuf. 2012, 13, 501–507. [CrossRef] 14. Sun, N.; Fang, Y.; Zhang, X. Energy coupling output feedback control of 4-DOF underactuated cranes with saturated inputs. Automatica 2013, 49, 1318–1325. [CrossRef] Appl. Sci. 2018, 8, 473 18 of 19 15. Yang, B.; Xiong, B. Application of LQR techniques to the anti-sway controller of overhead crane. Adv. Mater. Res. 2010, 139, 1933–1936. [CrossRef] 16. Win, T.M.; Hesketh, T.; Eaton, R. SimMechanics Visualization of Experimental Model Overhead Crane, Its Linearization and Reference Tracking-LQR Control. AIRCC Int. J. Chaos Control Model. Simul. 2013, 2, 1–16. [CrossRef] 17. Abdullah, J.; Ruslee, R.; Jalani, J. Performance Comparison between LQR and FLC for Automatic 3 DOF Crane Systems. Int. J. Control Autom. 2011, 4, 163–178. 18. Choi, S.; Kim, J.; Lee, J.; Lee, Y.; Lee, K. A study on gantry crane control using neural network two degree of PID controller. In Proceedings of the ISIE 2001 IEEE International Symposium on Industrial Electronics Proceedings, Pusan, Korea, 12–16 June 2001; pp. 1896–1900. 19. Solihin, M.I.; Wahyudi; Legowo, A. Fuzzy-tuned PID anti-swing control of automatic gantry crane. J. Vib. Control 2010, 16, 127–145. [CrossRef] 20. Lee, C.H.; Lee, Y.H.; Teng, C.C. A novel robust PID controllers design by fuzzy neural network. In Proceedings of the American Control Conference, Anchorage, AK, USA, 8–10 May 2002; pp. 1561–1566. 21. Zhang, S.; He, X. Adaptive HJI sliding mode control of three dimensional overhead cranes. In Proceedings of the 2016 Chinese Control and Decision Conference (CCDC), Yinchuan, China, 28–30 May 2016; pp. 5820–5855. 22. Solihin, M.I.; Kamal, M.; Legowo, A. Objective function selection of GA-based PID control optimization for automatic gantry crane. In Proceedings of the 2008 International Conference on Computer and Communication Engineering (ICCCE), Kuala Lumpur, Malaysia, 13–15 May 2008; pp. 883–887. 23. Lin, J.; Zheng, Y. Vibration suppression control of smart piezoelectric rotating truss structure by parallel neuro-fuzzy control with genetic algorithm tuning. J. Sound Vib. 2012, 331, 3677–3694. [CrossRef] 24. Petrenko, Y.N.; Alavi, S.E. Fuzzy logic and genetic algorithm technique for non-liner system of overhead crane. In Proceedings of the 2010 IEEE Region 8 International Conference on Computational Technologies in Electrical and Electronics Engineering (SIBIRCON), Irkutsk, Russia, 11–15 July 2010; pp. 848–851. 25. Wang, X.J.; Chen, Z.M. Two-degree-of-freedom sliding mode anti-swing and positioning controller for overhead cranes. In Proceedings of the 2016 Chinese Control and Decision Conference (CCDC), Yinchuan, China, 28–30 May 2016; pp. 673–677. 26. Wang, X.; Wang, H.; Tian, Y.; Christov, N. Predictive Observer based Lyapunov antiswing control for overhead visual crane. In Proceedings of the 2015 Chinese Automation Congress (CAC), Wuhan, China, 27–29 November 2015; pp. 1670–1675. 27. Wu, X.; He, X.; Ou, X. A coupling control method applied to 2-D overhead cranes. In Proceedings of the 2016 35th Chinese Control Conference (CCC), Chengdu, China, 27–29 July 2016; pp. 1658–1662. 28. Chen, H.; Fang, Y.; Sun, N. A swing constraint guaranteed MPC algorithm for underactuated overhead cranes. IEEE/ASME Trans. Mechatron. 2016, 21, 2543–2555. [CrossRef] 29. Jafari, J.; Ghazal, M.; Nazemizadeh, M. A LQR Optimal Method to Control the Position of an Overhead Crane. IAES Int. J. Robot. Autom. 2014, 3, 252. 30. Blajer, W.; Kołodziejczyk, K. Control of underactuated mechanical systems with servo-constraints. Nonlinear Dyn. 2007, 50, 781–791. [CrossRef] 31. Wang, J.; Noda, Y.; Inomata, A. Straight transfer control system using PI control and trajectory planning in overhead traveling crane. In Proceedings of the 2015 IEEE/SICE International Symposium on System Integration (SII), Nagoya, Japan, 11–13 December 2015; pp. 522–527. 32. Chen, H.; Fang, Y.; Sun, N. Optimal trajectory planning and tracking control method for overhead cranes. IET Control Theory Appl. 2016, 10, 692–699. [CrossRef] 33. Summanwar, V.S.; Jayaraman, V.K.; Kulkarni, B.D.; Kusumakar, H.S.; Gupta, K.; Rajesh, J. Solution of constrained optimization problems by multi-objective genetic algorithm. Comput. Chem. Eng. 2002, 26, 1481–1492. [CrossRef] 34. Coello, C.A.C. Multi-objective Evolutionary Algorithms in Real-World Applications: Some Recent Results and Current Challenges; Springer International Publishing: Cham, Switzerland, 2015; pp. 3–18. 35. Konak, A.; Coit, D.W.; Smith, A.E. Multi-objective optimization using genetic algorithms: A tutorial. Reliab. Eng. Syst. Saf. 2006, 91, 992–1007. [CrossRef] 36. Cui, Y.; Geng, Z.; Zhu, Q.; Han, Y. Review: Multi-objective optimization methods and application in energy saving. Energy 2017, 125, 681–704. [CrossRef] Appl. Sci. 2018, 8, 473 19 of 19 37. Yeh, W.C.; Chuang, M.C. Using multi-objective genetic algorithm for partner selection in green supply chain problems. Expert Syst. Appl. 2011, 38, 4244–4253. [CrossRef] 38. Khalil, H.K. Nonlinear Systems, 3rd ed.; Prentice-Hall, Inc.: Upper Saddle River, NJ, USA, 2002. 39. Padula, F.; Adamini, R.; Finzi, G.; Visioli, A. Revealing the Hidden Technology by Means of an Overhead Crane. IFAC Papersonline 2017, 50, 9126–9131. [CrossRef] 40. 2016 B&R Scholastic Union Competition. Available online: http://www.br-education.com/activities/index. asp?ColumnId=45&Style_ID=3&ID=376 (accessed on 18 March 2018). 41. B&R Automation. Available online: https://www.br-automation.com/en-au/about-us/customer- magazine/2013/201303/swift-steady/ (accessed on 18 March 2018). © 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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Multi-Objective Motion Control Optimization for the Bridge Crane System

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Publisher: Multidisciplinary Digital Publishing Institute
ISSN: 2076-3417
DOI: 10.3390/app8030473
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Multi-Objective Motion Control Optimization for the Bridge Crane System

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Multi-Objective Motion Control Optimization for the Bridge Crane System

Multi-Objective Motion Control Optimization for the Bridge Crane System

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