Nonlinear Dynamics (2006) 45: 149–170 DOI: 10.1007/s11071-006-2425-3
c Springer 2006
Nonlinear Input-Shaping Controller for Quay-Side Container Cranes MOHAMMED F. DAQAQ∗ and ZIYAD N. MASOUD Department of Engineering Science and Mechanics, MC 0219, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, U.S.A.; ∗ Author for correspondence (e-mail:
[email protected]) (Received: 21 September 2004; accepted: 2 August 2005)
Abstract. Input-shaping is one of the most practical open-loop control strategies for gantry cranes, especially those having predefined paths and operating at constant cable lengths. However, when applied to quay-side container cranes, its performance is far from satisfactory. A major source of the poor performance can be linked to the significant difference between the gantry crane model and the quay-side container crane model. Gantry cranes are traditionally modeled as a simple pendulum. However, a quay-side container crane has a multi-cable hoisting mechanism. In this paper, a two-dimensional four-bar-mechanism model of a container crane is developed. For the purpose of controller design, the crane model is reduced to a double pendulum with two fixed-length links and a kinematic constraint. The method of multiple scales is used to develop a nonlinear approximation of the oscillation frequency of the simplified model. The resulting frequency approximation is used to determine the switching times for a bang-off-bang input-shaping controller. The performance of the controller is numerically simulated on the full model of the container crane, and is compared to the performance of similar controllers based on a nonlinear frequency approximation of a simple pendulum and a linear frequency approximation of a constraint double pendulum. Results demonstrate a superior performance of the controller based on the nonlinear frequency approximation of the constraint double pendulum. The effect of the oscillation frequency on the controller performance is investigated by varying the model’s frequency around the design value. Simulations revealed that the performance of the controller suffers serious degradation due to small changes in the model frequency. To alleviate the shortcomings of the input-shaping controller, a delayed-position feedback controller is successfully applied at the end of each transfer maneuver to eliminate residual oscillations without affecting the commands of the input-shaping controller. Key words: container cranes, delayed-position feedback control, input-shaping, nonlinear stability, sway control
1. Introduction In general, cranes play a very important role in transportation and construction. As a result there is an increasing demand on faster, bigger, and more efficient cranes to guarantee fast turn-around time, while meeting safety requirements. This steered recent research into developing more efficient crane controllers. Inertial forces on the payload due to crane commanded trajectories or operators commands can cause the payload to experience large sway oscillations. To avoid exciting these oscillations, crane operators resolve to slowing down the operations such that the oscillations do not cause safety concerns and possible damage of the payload. However, slowing down operations increases the cost of loading and unloading operations. Traditionally, a gantry crane is modeled as a simple pendulum with a rigid or flexible hoisting cable and a lumped mass at the end of the cable. However, in the case of quay-side container cranes the model is significantly different (Figure 1). The actual hoisting mechanism of a container crane consists typically of a set of four hoisting cable arrangement. The cables are hoisted from four different
150 M. F. Daqaq and Z. N. Masoud
Figure 1. Typical quay-side container crane.
points on a trolley and are attached on the payload side to four points on a spreader bar used to lift containers. The two most commonly used modeling approaches for gantry cranes are the lumped-mass and distributed-mass models. In the distributed-mass model, the hoisting cable is modeled as a distributed mass, and the hook and payload are lumped as a point mass and are applied as a boundary condition to this system. d’Andrea-Novel et al. [1, 2] and d’Andrea-Novel and Boustany [3] used the distributed-mass model. They ignored the inertia of the payload and modeled the cable as perfectly flexible, inextensible body using the wave equation. Others [4–6] extended the model to include the inertia of the payload by changing the boundary conditions at the payload end. However, the most widely used approach for crane modeling, is the lumped-mass approach. The hoisting line is treated as a massless rigid link, and the payload is lumped with the hook and modeled as a point mass. For gantry cranes, the level of control of payload oscillations varies according to the application at hand. In some applications, oscillations are acceptable while the payload is on its way to target, while the settling time and residual oscillations are kept very small to allow for accurate payload positioning. In other applications, such as nuclear reactors, or where the space around the crane is populated, the safety requirements are very strict. Thus, large oscillations are not acceptable during and at the end of a transfer maneuver. Input-shaping is one of the practical open-loop control strategies for gantry cranes. Controllers using various forms of input-shaping are incorporated into gantry cranes and currently used in ports [7]. This technique moves the crane a known distance along a set path, and depends significantly on system parameters such as cable length and system delays. Alsop et al. [8] were the first to propose a controller based on input-shaping. The controller accelerates the trolley in steps of constant acceleration then kills the acceleration when the payload reaches zero oscillation angle (after multiples of a full period). The trolley then coasts at constant speed along the path for a period of time. A replicate of the acceleration procedure is used in the deceleration stage. Alsop et al. used constant-amplitude acceleration/deceleration steps, and a linear frequency approximation of a simple pendulum model. Their results showed very little residual oscillations, while transient oscillation angles were of order of 10◦ during the acceleration/deceleration stages.
Nonlinear Input-Shaping Controller for Quay-Side Container Cranes 151 Jones and Petterson [9] extended the work of Alsop et al. [8] using a nonlinear approximation of the payload period to generate an analytical expression for the duration of the coasting stage as a function of the amplitude and duration of the constant acceleration steps. This analytical expression is then used to generate a two-step acceleration profile. Numerical simulations using various acceleration profiles show that this technique was not able to dampen out initial disturbances of the payload and could even amplify them. Starr [10] used a symmetric two-step acceleration/deceleration profile to transport a suspended object with minimal oscillations. The duration of the constant acceleration steps is assumed to be negligible compared to the period of the payload. A linear approximation of the period of the payload is used to generate analytically the acceleration profile. Strip [11] extended this work by employing a nonlinear approximation of the payload frequency to generate one-step and two-step symmetric acceleration profiles. Kress et al. [12] showed analytically that input-shaping is equivalent to a notch filter applied to a general input signal and centered around the natural frequency of the payload. Based on that, they proposed a Robust Notch Filter, a second-order notch filter, applied to the acceleration input. Numerical simulation and experimental verification of this strategy on an actual bidirectional crane, moving at an arbitrary step acceleration and changing cable length at a slow constant speed, showed that the strategy was able to suppress residual payload oscillation. Parker et al. [13] developed a command shaping notch filter to reduce payload oscillation on rotary cranes excited by the operator commands. They reported that in general, there was no guarantee that applying such filter to the operator’s speed commands would result in excitation terms having the desired frequency content, and that it only works for small speed and acceleration commands. Parker et al. [14] experimentally verified their simulation results. Input-shaping techniques are limited by the fact that they are sensitive to variations in the parameter values about the nominal values and changes in the initial conditions and external disturbances and that they require “highly accurate values of the system parameters” to achieve satisfactory system response [15–17]. While a good design can minimize the controller’s sensitivity to changes in the payload mass, it is much harder to alleviate the controller’s sensitivity to changes in the cable length. Singhose et al. [18] developed four input-shaping controllers. Simulations of their best controller produced a reduction of 73% in transient oscillations over the time-optimal rigid-body commands. However, they reported that “transient deflection with shaping increases with hoist distance, but not as severely as the residual oscillations”. Their simulations showed that “the percentage in reduction with shaping is dependent on system parameters”. As a result, their controllers suffered significant degradation in crane maneuvers that involved hoisting. Alzinger et al. [19] showed that a two-step acceleration/deceleration profile results in significant reductions in travel time over a one-step acceleration profile. Testing on an actual crane has shown that the two-step acceleration profile can deliver both faster travel and minimal payload oscillations at the target point. However, any deviation from the prescribed parameters, causes significant payload oscillations. Dadone and Vanlandingham [20] used the method of multiple scales to generate a nonlinear approximation of the oscillation period of a simple pendulum model. They produced numerical simulations and compared the response of the nonlinear controller to the linear controller. They found that the acceleration profile based on the nonlinear frequency approximation could damp the residual oscillations 2 orders of magnitude more than the profile based on the linear approximation. The enhanced performance of the nonlinear strategy was most pronounced for longer coasting distances and higher accelerations.
152 M. F. Daqaq and Z. N. Masoud In addition to feedforward controllers, researchers used linear and nonlinear feedback techniques for anti-sway control of suspended objects [21–23]. In an application to quay-side container cranes, Yong-Seok et al. [24] and Yong-Seok, Keum-Shik, and Seung-Ki [25] developed a state feedback controller to control oscillations on a rail-mounted quay-side crane. They proposed a novel technique to measure the sway angle. This technique is based on mounting an inclinometer on the top surface of the spreader bar, then using the geometry of the crane and the angle relations to recover the actual sway angle. Although, the sway angle measurement was based on the actual geometry of the crane, the authors used a simple pendulum model to express the dynamics of the payload, and verified the results experimentally using a single-cable rubber-tired gantry crane. Researchers [26, 27] have extensively studied the possibility of using time delay to control mechanical systems. It has been noticed that systems with time delays exhibit interesting complex responses. Time delay has the capability of stabilizing or destabilizing dynamic systems. For this reason, they have been used as simple switches to control the behavior of systems, either by damping out the oscillations or creating chaotic responses that are sometimes desirable to secure communication signals [28]. Cheng and Chen [29] were among the first to use time delay to control gantry cranes. They proposed a control strategy which employs time delay control and feedback linearization to move a crane along a predefined path and to eliminate residual oscillations. Their results showed that the strategy is capable of delivering the payload to its goal with minimal transient oscillations and almost no residual oscillations. Masoud and Nayfeh [30] introduced a two-dimensional four-bar-mechanism model of a container crane. They approximated the model with a constrained double pendulum to find a linear frequency approximation of the payload oscillation. The resulting frequency was then used to find the delay and gain of a nonlinear delayed-position feedback controller. The resulting controller was applied to the full crane model. Masoud et al. [31] verified the modeling approach and the controller performance experimentally on a 1/10 scale model of a container crane. Despite their practicality, the performance of open-loop input-shaping controllers designed for gantry cranes is far from satisfactory when applied to quay-side container cranes. One of the main reasons behind this poor performance, is the lack of a good model that resembles the dynamics and approximates the frequency of a real container crane. In this paper, we analyze the nonlinear dynamics of the traversing motion of the crane payload, then develop a one-step input-shaping controller based on a four-barmechanism model of a quay-side container crane. We use the method of multiple scales to develop a nonlinear frequency approximation of the payload oscillation. To eliminate residual oscillations, the open-loop input-shaping controller is augmented with a closed-loop delayed-position feedback controller applied at the end of each transfer maneuver.
2. Mathematical Models In this section, a four-bar-mechanism is developed to model the actual hoisting mechanism of the quayside container crane. This model is further simplified to a double pendulum model with a kinematic constraint between the angles of both links of the pendulum. The simplified model is used for the purpose of controller design, however, numerical simulations are performed on the full model of the crane. To study the performance of input-shaping controllers developed using a simple pendulum model, a nonlinear version of the traditional simple pendulum model is used.
Nonlinear Input-Shaping Controller for Quay-Side Container Cranes 153
Figure 2. A schematic model of a simple pendulum model of a container crane.
2.1. SIMPLE PENDULUM We start by deriving the nonlinear equation of motion of a constant length simple pendulum model (Figure 2). The position vector to the center of mass of the payload is r = ( f + L sin θ )i − L cos θj
(1)
where L is the length of the cable, and f is the position of the trolley. The equation of motion is derived using the Euler–Lagrangian equation d dt
∂L ∂ q˙
−
∂L =0 ∂q
(2)
where q = θ and L is the Lagrangian and is defined as L=T −V
(3)
T and V are the kinetic and potential energy of the payload, which can be expressed as 1 1 m r˙ r˙ T = m( ˙f 2 + L 2 θ˙ 2 + 2 ˙f θ˙ L cos θ ) 2 2 V = mgr y = −mgL cos θ T=
(4) (5)
where g is the gravitational acceleration. Substituting Equations (3)–(5) into Equation (2) yields the following nonlinear equation of motion: θ¨ + η cos θ + 2◦ sin θ = 0
(6)
where η = f¨/L is the normalized trolley acceleration, and 2◦ = g/L is the linear frequency of the pendulum.
154 M. F. Daqaq and Z. N. Masoud
Figure 3. A schematic model of a container crane.
2.2. FOUR-BAR-MECHANISM MODEL The two-dimensional model of a quay-side container crane is shown in Figure 3. The container is grabbed using a spreader bar. The spreader bar is then hoisted from the trolley by means of four cables, two of which are shown in the figure. The cables are spaced a distance d at the trolley and a distance w at the spreader bar. The hoisting cables in the model are considered as rigid massless links with constant lengths. The position vector to the combined center of mass of the payload and the spreader bar is r = xi − yj
(7)
The unconstrained equations of motion of the payload mechanism can be derived using the Euler– Lagrangian Equation (2), where q = [x, y, θ ]T . The kinetic energy and potential energy of the payload and spreader bar are given by 1 1 1 1 m r˙ r˙ T + J θ˙ 2 = m(x˙ 2 + y˙ 2 ) + J θ˙2 2 2 2 2 V = mgr y = −mgy T=
(8) (9)
where m is the combined mass of the payload and the spreader bar, and J is the combined moment of inertia of the payload and the spreader bar about point Q. Since this is a single-degree-of-freedom system, two constraints are used governing the distances AB and DC. The constraints are
2 1 + y − R cos θ − w sin θ − L 2 = 0 2 2 2 1 1 1 x + R sin θ + w cos θ − f − d + y − R cos θ + w sin θ − L 2 = 0 2 2 2 1 1 x + R sin θ − w cos θ − f + d 2 2
2
(10) (11)
Nonlinear Input-Shaping Controller for Quay-Side Container Cranes 155 We multiply these two constraint by the Lagrange multipliers λ1 and λ2 then append them to the Lagrangian L to form the augmented Lagrangian La La =
1 1 m(x˙ 2 + y˙ 2 ) + J θ˙2 + mgy 2 2 2 1 1 2 1 2 + λ1 x + R sin θ − w cos θ − f + d + y − R cos θ − w sin θ − L 2 2 2 2 2 1 1 1 + λ2 x + R sin θ + w cos θ − f − d + y − R cos θ + w sin θ − L 2 2 2 2
(12)
Substituting Equation (12) into (2), we get the following three nonlinear differential equations of motion: 1 1 m x¨ − 2λ1 x + R sin θ − w cos θ − f + d 2 2 1 1 + 2λ2 x + R sin θ + w cos θ − f − d =0 2 2 1 1 m y¨ − 2λ1 y − R cos θ − w sin θ + 2λ2 y − R cos θ + w sin θ − mg = 0 2 2 1 1 1 J θ¨ − 2λ1 R cos θ + w sin θ x − w cos θ − f + R sin θ + d 2 2 2 1 1 + R sin θ − w cos θ y − w sin θ − R cos θ 2 2 1 1 1 − 2λ2 R cos θ − w sin θ x + w cos θ − f + R sin θ − d 2 2 2 1 1 + R sin θ + w cos θ y − R cos θ + w sin θ =0 2 2
(13) (14)
(15)
In order to solve the equations of motion with the two constraint equations, we differentiate the constraints Equations (10) and (11) twice with respect to time to obtain α11 x¨ + α12 y¨ + α13 θ¨ + 2x˙ 2 + 2 y˙ 2 + α14 θ˙ 2 + α15 θ˙ x˙ + α16 θ˙ y˙ + α17 θ˙ ˙f + α18 f¨ − 4 f˙ x˙ + 2 f˙2 = 0 (16) α21 x¨ + α22 y¨ + α23 θ¨ + 2x˙ + 2 y˙ + α24 θ + α25 θ˙ x˙ + α26 θ˙ y˙ + α27 θ˙ ˙f + α28 f¨ − 4 ˙f x˙ + 2 f˙2 = 0 2
2
˙2
(17) where αk1 = −(−1)k d + (−1)k w cos θ − 2 f + 2R sin θ + 2x αk2 = −2R cos θ + (−1)k w sin θ + 2y 1 αk3 = −(−1)k d R cos θ − 2R f cos θ + dw sin θ 2 + (−1)k w f sin θ + 2Rx cos θ − (−1)k wx sin θ + (−1)k wy cos θ + 2Ry sin θ
156 M. F. Daqaq and Z. N. Masoud 1 αk4 = (−1)k d R sin θ + 2R f sin θ + dw cos θ 2 + (−1)k w f cos θ − 2Rx sin θ − (−1)k wx cos θ
(18)
− (−1)k wy sin θ + 2Ry cos θ αk5 = 4R cos θ − 2(−1)k w sin θ αk6 = 4R sin θ + 2(−1)k w cos θ αk7 = −4R cos θ + 2(−1)k w sin θ αk8 = −2R sin θ − 2x + 2 f + (−1)k d − (−1)k w cos θ Equations (13)–(17) are the full nonlinear equations of motion of the four-bar-mechanism of the container crane. 2.3. CONSTRAINED DOUBLE PENDULUM For the purpose of controller design, the four-bar-mechanism model is simplified to a double pendulum system with two fixed-length links, and a kinematic constraint relating the angles φ and θ as shown in Figure 4. In Figure 3, point O is the midpoint between points A and D, and point P is the midpoint between points B and C. The closing constraints of the loop ABPO are 1 1 l sin φ − w cos θ + d − L sin φ1 = 0 2 2 1 l cos φ − w sin θ − L cos φ1 = 0 2
(19) (20)
Similarly, the closing constraints of the loop ODCP are written as 1 1 l sin φ + w cos θ − d − L sin φ2 = 0 2 2 1 l cos φ + w sin θ − L cos φ2 = 0 2
Figure 4. A schematic model of a constrained double pendulum model of a container crane.
(21) (22)
Nonlinear Input-Shaping Controller for Quay-Side Container Cranes 157 Squaring and adding Equations (19) and (20), and squaring and adding Equations (21) and (22), we can eliminate L from the resulting equations and obtain the following relation between φ and θ: d sin φ = w sin(θ + φ)
(23)
Equation (23) represents the kinematic constraint between the angles φ and θ, which reduces the two-degrees-of-freedom double pendulum model to a single-degree-of-freedom model [32]. Using Equation (23), we simplify the closing constraints of the loop ABPO to obtain 1 l 2 = L 2 − (d 2 + w 2 − 2dw cos θ ) 4
(24)
The position vector to the center of mass of the payload of the constrained double pendulum is r = ( f + l sin φ − R sin θ )i − (l cos φ + R cos θ)j
(25)
Substituting the position vector into Equations (8) and (9), the kinetic and potential energies of the constrained double pendulum can be written as 1 2 ˙2 1 1 ml φ + (J + m R 2 )θ˙ 2 + m ˙f 2 − m Rl φ˙ θ˙ cos(φ + θ) + ml φ˙ ˙f cos φ − m R θ˙ ˙f cos θ (26) 2 2 2 V = −mg(l cos φ + R cos θ ) (27) T=
The equation of motion expressed in terms of the angle φ is obtained by substituting the constraint Equations (23) and (24) into Equations (26) and (27), then applying the Euler–Lagrange equation to the Lagrangian (3). Due to the lengthy expressions in this equation, we only show a Taylor series expansion of the resulting equations up to cubic terms. φ¨ + ω◦2 φ + c1 φ 2 φ¨ + c1 φ φ˙ 2 − c2 ξ φ 2 − c3 φ 3 + c4 ξ = 0
(28)
where
mg k32 + 2k42 + k12 k3 R ω◦ = k3 J k12 + m(k3 − k1 R)2 m 4k42 + (k1 (1 + k1 )2 − 6k2 )k33 R c1 = k32 J k12 + m(k3 − k1 R)2 2mk1 (3 + 2k1 )k3 k5 R − 2k32 (k4 − 3k1 k2 R 2 ) + k3 2 J k12 + m(k3 − k1 R)2 6J k1 k2 k3 2 k3 + m(k3 − k1 R)2 m 6k4 + k3 k3 − k13 R + 6k2 R c2 = 2 J k12 + m(k3 − k1 R)2 +
2
J k12
(29)
158 M. F. Daqaq and Z. N. Masoud gm 12k4 + k4 (k3 − 24k3 k5 + k1 k13 − 24k2 R c3 = − 6k3 J k12 + m(k3 − k1 R)2 mk3 (k3 − k1 R) c4 = J k12 + m(k3 − k1 R)2 f¨ ξ= k3 and d −w w d(d 2 − w 2 ) = 3 6w 1 = L 2 − (d − w)2 4 dw = 8 dw(d 2 − 4L 2 + dw + w 2 ) = 24((d − w)2 − 4L 2 )2
k1 = k2 k3 k4 k5
(30)
3. Nonlinear Analysis A nonlinear frequency approximation of the simple pendulum and constrained double pendulum models will be derived. These frequency approximations along with the linear frequency ω◦ of the constrained double pendulum model will be used to generate acceleration profiles for the same input-shaping controller. 3.1. CONSTRAINED DOUBLE PENDULUM To find a nonlinear frequency approximation we use the method of multiple scales [33]. First we scale Equation (28) by introducing a bookkeeping parameter which is set to one at the end of this analysis. ξ and θ are scaled at order , that is φ = φ
ξ = ξ
(31)
Substituting the scaled parameters into Equation (28) yields φ¨ + ω◦2 φ = −c4 ξ − 2 (c1 φ 2 φ¨ + c1 φ φ˙ 2 − c2 ξ φ 2 + c3 φ 3 )
(32)
The time dependence is expanded into multiple time scales as following t = T0 + T1 + 2 T2 + O( 3 ) d = D0 + D1 + 2 D2 + O( 3 ) dt d2 = D02 + 2 D0 D1 + 2 D12 + 2 2 D0 D2 + O( 3 ) dt2
(33)
Nonlinear Input-Shaping Controller for Quay-Side Container Cranes 159 where Dn =
∂ . ∂ Tn
We then seek a solution in the form
φ(T0 , T1 , T2 ) = φ0 (T0 , T1 , T2 ) + φ1 (T0 , T1 , T2 ) + 2 φ2 (T0 , T1 , T2 ) + O( 3 )
(34)
Substituting Equations (33) and (34) into Equation (32) and equating coefficients of like powers of we obtain O(1) :
D02 φ0 + ω◦2 φ0 = −c4 ξ
O( ) :
D02 φ1 + ω◦2 φ1 = −2D0 D1 φ0
O( 2 ) :
D02 φ2 + ω◦2 φ2 = −2D0 D1 φ1 − 2D0 D2 φ0
(35)
− D12 φ0 − c2 ξ φ02 − c1 φ0 (D0 φ0 )2 − c1 φ02 D02 φ0 − c3 φ03 The solution of the O(1) equation can be expressed as φ0 (T0 , T1 , T2 ) = A(T1 , T2 )eiω◦ T0 −
c4 ξ + cc ω02
(36)
where A is complex. Substituting Equation (36) into O( ) of Equation (35) and eliminating the terms that lead to secular terms, we obtain ∂A = 0 ⇒ A = A(T2 ) ∂ T1
(37)
Now, substituting Equations (36) and (37) into O( 2 ) of Equation (35) and eliminating the coefficients of eiω◦ T0 which leads to secular terms, we obtain the solvability condition ∂A ξ 2 c4 −2iω◦ + 2 ∂ T2 ω◦
c 3 c4 c1 c4 − 3 2 − 2c2 A + 2ω◦2 c1 − 3c3 A2 A¯ = 0 ω◦
(38)
Introducing the polar transformation A=
1 a(T2 )eiβ(T2 ) 2
(39)
into Equation (38) and separating real and imaginary parts, we obtain the following two modulation equations: ∂a =0 ∂ T2 ∂β a2 3c3 ξ2 3c3 c42 2 2c − − = c + − c c + c ω 2 4 1 1 ◦ 4 ∂ T2 2ω◦3 ω◦2 4 2ω◦
(40) (41)
Solving Equations (41) and (41) we get the following approximate solution of Equation (32): φ = a cos(ωt + β0 ) −
c4 ξ ω◦2
(42)
160 M. F. Daqaq and Z. N. Masoud 5 Numerical Solution Nonlinear Frequency Linear Frequency
Angle [deg]
0
-5
-10
-15
-20 0
2
4
6
8
10
12
14
16
18
20
Time [sec] Figure 5. Payload sway angle for the constrained double pendulum model of a container crane. The results are obtained for L = 17.5 m, R = 2.5 m, ξ = 0.1 s−2 , θ0 = 0, and θ˙0 = 0.
where 3c3 c1 ξ 2 −3c3 c42 2 ω = ω◦ 1 − a 2 + + 2c c − c c + 2 4 1 4 8ω◦2 4 2ω◦4 2ω◦2
(43)
is the nonlinear frequency approximation of the constrained double pendulum. We tested the resulting approximation against a numerical solution of the full nonlinear equations of motion Equations (13)–(17) for different values of ξ . The results showed that this approximation holds for values of ξ approximately less than 0.1 s−2 . This value sets a limit above which the nonlinear solution starts to deteriorate. However, container cranes normally operate at lower accelerations (0.015–0.03). Figure 5 shows that the numerical integration and the multiple scale approximate solution closely match, while the linear frequency approximate solution quickly drifts away from the numerical solution as a result of the inaccurate frequency approximation. 3.2. SIMPLE PENDULUM To compare controllers based on the constrained double pendulum model with those based on the simple pendulum model, we develop a nonlinear frequency approximation of the payload oscillations for the traditional simple pendulum model of the container crane. Using Equation (6) the procedure described in Section 3.1 is followed to obtain the following nonlinear frequency approximation: a2 η2 = ◦ 1 − + 16 44◦
(44)
where η = f¨ /L is the normalized acceleration, and ◦ is the linear frequency approximation of the payload oscillations of a simple pendulum. We tested the resulting approximation against a numerical
Nonlinear Input-Shaping Controller for Quay-Side Container Cranes 161 Numerical Simulation Nonlinear Frequency Linear Frequency
10
0
Angle [deg]
-10
-20
-30
-40
-50
0
2
4
6
8
10
12
14
16
18
20
Time [sec] Figure 6. Payload sway angle for a simple pendulum model of a container crane. The results are obtained for L = 20 m, η = 0.2 s−2 , θ0 = 0, and θ˙0 = 0.
solution of Equation (6) for different values of η. The results showed that this approximation holds for values of η approximately less than 0.2 s−2 . Figure 6 shows that the solution based on the multiple scales method has an excellent agreement with the numerical solution. 4. Controllers Design 4.1. INPUT-SHAPING CONTROLLER A bang-off-bang input-shaping controller is shown in Figure 7. The controller works by generating an acceleration profile designed to cancel only its own oscillations. The controller will be used to perform a transfer maneuver that produces zero residual oscillations while taking into consideration any delays in the system. Assuming that both the cable length and the system delays are known, the controller determines the magnitude of the constant acceleration and the switching times to reach the target point with zero residual oscillations. We will follow the procedure developed by Dadone et al. [12]. First the trolley accelerates for the period of a half swing cycle ta = Ta /2, where Ta is the period of the sway oscillation in the acceleration mode. The acceleration is then switched off for a period of time (necessary to accomplish the load transfer). This period is called the coast mode. To bring the load to complete stop a negative acceleration is applied taking into account the known system delays. For known system delays τs , the length of τs will determine the minimum time that the trolley must spend in the coast mode. By design the coast time must be an odd multiple of half the period of the sway oscillation Tc in the coast mode. The coast time tc depends also on the maximum magnitude of acceleration achievable by the trolley drives. Thus, we can define the coast time as
tc =
2n + 1 Tc , 2
n = 0, 1, 2, 3, . . .
(45)
162 M. F. Daqaq and Z. N. Masoud
Figure 7. A schematic drawing showing the bang-off-bang acceleration profile.
where n is known as the number of coasting cycles. To calculate the least number of cycles the trolley needs to spend in the coast mode, the constraint tc ≥ τs must be satisfied. Thus, we can write n as 1 2τs n≥ −1 2 Tc
(46)
here n is rounded up to the nearest integer including zero. The total distance travelled by the trolley in a full maneuver S derived from Figure 7 is S = f¨( ta )2 + f¨( ta )( tc )
(47)
Using the normalized acceleration ξ = f¨/k3 and ta = Ta /2, and substituting Equation (45) into (47), the normalized travel distance δ = S/k3 is δ=
1 2 2n + 1 ξT + ξ Ta Tc 4 a 4
(48)
To adapt the input-shaping controller to the constrained double pendulum, the nonlinear frequency approximation, Equation (43) is used. Assuming zero initial conditions we solve Equation (42) for a and then substitute back into Equation (43) to get Ta =
ξ2 2π 15c3 c42 3c1 c42 −1 1+ 2c c + − 2 4 ω◦ 2ω◦4 4ω◦2 2
(49)
The period in the coast mode can be obtained by setting ξ to zero and substituting a = −2c4 ξ/ω◦2 in Equation (43). a here is the amplitude of oscillation in the coast mode and is found by substituting t = π/ω in Equation (42) −1 4ξ 2 c42 c1 2π 3c3 Tc = 1− − ω◦ ω◦4 4 8ω◦2
(50)
Nonlinear Input-Shaping Controller for Quay-Side Container Cranes 163 Now we can summarize the controller design process as following: • Using the known values for τs and Tc , determine the minimum number of coast cycles n from Equation (46). • Solve Equation (48) for ξ and compare it with ξmax . ξmax is the minimum of two values; the first being the maximum acceleration that the trolley drives can provide and the second being the maximum acceleration above which the nonlinear solution starts to deteriorate, which in this case is equal to 0.1 s−2 . If ξ is greater than ξmax increment n up by one and recompute a new ξ . • Compute the switching times t1 , t2 , and t3 according to the following equations: t1 =
Ta , 2
t 2 = t1 +
2n + 1 Tc , 2
t 3 = t2 + t 1
(51)
The design of the input-shaping controller based on the simple pendulum model follows the same procedure. However, in Equation (48) δ = LS is used and η and ηmax are substituted for ξ and ξmax , respectively. Assuming zero initial conditions, Equation (44) is used to calculate the nonlinear approximation of the periods in the bang mode as follows η2 −1 2π Ta = 1+ ◦ 44◦
(52)
In the coast mode, the normalized acceleration η is set to zero and a to η/ 2◦ , which is the linear approximation of the amplitude of oscillation at the end of the bang mode obtained from Equation (6) η2 −1 2π Tc = 1− ◦ 164◦
(53)
4.2. DELAYED-POSITION FEEDBACK CONTROLLER The delayed-position feedback controller creates damping in the system by adding a delayed feedback component to the operator command to the trolley drives. This component is proportional to the swing angle of the hoist line. The controller takes the following general form f = f ◦ + kˆ sin φτd
(54)
where kˆ is the feedback gain, τd is the delay time, f ◦ is the operator input, and φτd = φ(t − τd ). To study the linear stability of the controller and to make a proper choice of the gain-delay combination, we substitute Equation (54) into (28) and add a linear damping term to account for the damping in the system. Assuming that the operator input is slowly varying and keeping only linear terms yield to kˆ ¨ φ¨ + ω◦2 φ + 2μφ˙ + φ τd + 2μφ˙ τd = 0 k3
(55)
where μ is a linear damping coefficient. Now we seek an exponentially damped solution of the equation in the form φ = ceσ t cos(ωt + ψ)
(56)
164 M. F. Daqaq and Z. N. Masoud where c, σ , ω, and ψ are real constants. Substituting Equation (56) into (55), and setting the coefficients of cos(ωt + ψ) and sin(ωt + ψ) equal to zero independently, we get the following two equations K (σ 2 + 2μσ − ω2 ) sin(ωτd ) − 2K ω(μ + σ ) cos(ωτd ) − 2ω(σ + μ)eσ τd = 0 K (σ + 2μσ − ω ) cos(ωτd ) + 2K ω(μ + σ ) sin(ωτd ) + (σ + 2μσ − ω + 2
2
2
2
ω◦2 )eσ τd
=0
(57) (58)
ˆ 3 = K . The stability of the system depends on the value of the parameter σ . The system is where k/k asymptotically stable for σ < 0 and unstable for σ > 0. To determine the boundaries of linear stability and instability we set σ equals to zero in Equations (57) and (58) to obtain K ω2 sin(ωτd ) + 2K μ cos(ωτd ) + 2μω = 0 2K ωμ sin(ωτd ) − ω (1 + K cos(ωτd )) + 2
ω◦2
=0
(59) (60)
Equations (59) and (60) can be normalized by dividing them by ω◦2 , and set the time delay τd proportional to the linear period of oscillation T = 2π/ω◦ . This yields to K λ2 sin(2πλγ ) + 2K νλ cos(2πλγ ) + 2νλ = 0
(61)
2K νλ sin(2πλγ ) − λ (1 + K cos(2πλγ )) + 1 = 0
(62)
2
where λ = ω/ω◦ , γ = τd /T , and ν = μ/ω◦ . Manipulating Equations (61) and (62), K and γ as functions of λ are
4ν 2 + λ2 (λ2 + 4ν 2 − 1)2 K (λ) = − (63) λ(λ2 + 4ν 2 ) 1 2ν γ (λ) = arctan + jπ j = 0, 1, 2, . . . (64) 2πλ λ(λ2 + 4ν 2 − 1) The stability boundaries are determined by varying λ in Equations (63) and (64) and solving for K and γ . Figure 8 shows the stable and the unstable regions as predicted by the linear theory. Any gain-delay combination that lies inside the shaded areas of Figure 8 leads to an asymptotically stable response. It is worth mentioning that there are infinite number of stability pockets, which decrease in size as the time delay of the controller τd increases. To determine the magnitude σ of the damping factor resulting from each gain-delay combination, τd and K are varied in Equation (57) and (58) and calculate σ . Figure 9 shows a contour plot of the value of σ in the first pocket of stability.
5. Simulations To verify the accuracy of our nonlinear approximation of the simple pendulum frequency, an inputshaping profile is generated using the linear and the nonlinear frequency approximations of the simple pendulum model. The following values are used for the model parameters in the simulation: R = 2.5 m, d = 2.825 m, w = 1.4125 m, and m = 50000 kg, and for the IS controller n = 1 and S = 50 are used. Figure 10 shows the acceleration profiles and the payload sway simulated on the full nonlinear model of the simple pendulum. The figure shows that the residual oscillations associated with the linear simple
Nonlinear Input-Shaping Controller for Quay-Side Container Cranes 165
Figure 8. A stability plot of the delayed-position feedback controller for a relative damping ν = 0.0033. The shaded areas represent the pockets of stability.
Figure 9. A contour plot of the damping as a function of the gain K and the delay τd , where τd is given in terms of the natural period of the uncontrolled system. The darker the areas are the higher the damping is.
pendulum (LSP) profile are about 8 orders of magnitude larger than those associated with the nonlinear simple pendulum model (NSP). Although the residual oscillations associated with both models are small and can be neglected for the given system parameters, thorough investigation shows that the residual oscillations associated with a LSP increase for shorter cables and longer tracks. Similarly, to verify the accuracy of the nonlinear frequency approximation of the constrained double pendulum, acceleration profiles based on the constrained double pendulum model are generated using both a linear and a nonlinear frequency approximations and then applied to the same constrained double pendulum model. Figure 11 shows the shaped acceleration profiles for the linear and the nonlinear cases. It can be easily seen that the residual oscillations associated with the linear double pendulum model (LDP) profile are orders of magnitude larger than those associated with the nonlinear double pendulum (NDP) profile.
166 M. F. Daqaq and Z. N. Masoud 1.5
4 LSP NSP
LSP NSP 3
1
1
Sway[m]
2
Acceleration[m/s ]
2 0.5
0
0
-1 -0.5 -2 -1 -3
-1.5 0
5
10
15
20
25
30
35
40
-4 0
5
10
15
Time [sec]
20
25
30
35
40
Time [sec]
LSP NSP
0.06
0.04
Sway [m]
0.02
0
-0.02
-0.04
-0.06 24
26
28
30
32
34
36
38
Time [sec]
Figure 10. Sway response of a simple pendulum model of a container crane to shaped operator commands. Results are obtained for L = 20 m.
The shaped acceleration profiles based on the NSP, the LDP, and the NDP frequency approximations are then applied to the full four-bar-mechanism model as shown in Figure 12. Results show that the NSP acceleration profile, will not reduce the residual oscillations. On the contrary, this shaped profile will amplify residual oscillations to magnitudes that are even larger than the transient oscillations. The LDP and NDP models did reduce the residual oscillations significantly. However, the figure shows that the NDP acceleration profile has a much better performance than the LDP profile. Figure 13 demonstrates the sensitivity of the input-shaping controller to changes in system parameters. The NDP acceleration profile shown in Figure 12(a) is applied to the full model while varying the length of the hoisting cables. Simulations show that a change of 1.0 m in the cables length causes significant degradation in the controller performance. To overcome the problem of the open-loop input-shaping controller sensitivity, and to eliminate the residual oscillations, we apply a delayed-position feedback controller at the end of the transfer
Nonlinear Input-Shaping Controller for Quay-Side Container Cranes 167 4
1.5
LDP NDP
LDP NDP 3 1
1
Sway[m]
Acceleration[m/s2]
2 0.5
0
0
-1 -0.5 -2 -1
-1.5 0
-3
5
10
15
20
25
30
35
-4 0
40
5
10
15
Time[sec]
20
25
30
35
40
Time [sec]
Figure 11. Sway response of a constrained double pendulum model of a container crane to shaped operator commands. Results are obtained for L = 17.5 m.
1.5
4
NSP LDP NDP
NSP LDP NDP
3
1
1
Sway[m]
Acceleration[m/s2]
2 0.5
0
0
-1
-0.5
-2 -1 -3 -1.5 0
5
10
15
20
Time [sec]
25
30
35
40
-4 0
5
10
15
20
25
30
35
40
Time [sec]
Figure 12. Sway response of the full model of the container crane to shaped operator commands. Results are obtained for L = 17.5 m.
maneuver. The choice of delayed-position feedback controller is based on its ability to handle systems with inherent time delays. The controller can incorporate these delays in its parametric delay. A number of factors were considered behind the choice of applying a feedback controller only at the end of the transfer maneuver. A significant factor is reducing the power required to perform a transfer maneuver with minimal residual oscillations. This stems from the fact that for large sway angles feedback control systems may require input accelerations that are beyond the normal operating accelerations, which may overload the trolley motors during the acceleration and deceleration stages. When applied at the end of the transfer maneuver performed using an input-shaping control system, feedback controllers require lower accelerations. This is due to the fact that a welldesigned input-shaping controller is expected to produce significant reduction in residual payload oscillations.
168 M. F. Daqaq and Z. N. Masoud 4 16.5 m 17.5 m 18.5 m
3
2
Sway[m]
1
0
-1
-2
-3
-4 0
5
10
15
20
25
30
35
40
Time [sec]
Figure 13. Sway response of the full model of the container crane to the shaped operator commands shown in Figure 12(a).
4
1.5
IS IS + FB
IS IS + FB
3
1
1
Sway[m]
2
Acceleration[m/s ]
2 0.5
0
0
-1 -0.5 -2 -1
-1.5 0
-3
5
10
15
20
Time [sec]
25
30
35
40
-4 0
5
10
15
20
25
30
35
40
Time [sec]
Figure 14. Sway response of the full model of the container crane with input-shaping (IS) and combination (IS+FB (feedback)) controllers. Results are obtained for L = 16.5 m.
To demonstrate the performance of the combination controller, an NDP acceleration profile based on a 17.5 m cable is applied to a 16.5 m cable model. Figure 14 shows that an input-shaping controller based on the NDP frequency approximation results in a residual sway of approximately 1.25 m, while a combination of input-shaping and delayed-position feedback controller suppresses the residual sway to a magnitude less than 0.05 m within 4.5 s of the end of the transfer maneuver. The delayed-position feedback controller parameters used are K = 0.4 and τd = 0.28 T.
Nonlinear Input-Shaping Controller for Quay-Side Container Cranes 169 6. Concluding Remarks A bang-off-bang input-shaping controller based on a simple pendulum model falls short of satisfying the goal of eliminating residual oscillations on quay-side container cranes. The residual oscillations are many orders of magnitude larger than those when the controller is applied to a simple pendulum model. To achieve satisfactory performance, input-shaping controllers should be based on accurate mathematical models. Our results show that a constrained double pendulum model based on a four-barmechanism model of a container crane is a considerably accurate model for quay-side container cranes. For predefined system parameters, the input-shaping acceleration profiles based on a nonlinear frequency approximation of this model are capable of reducing the residual oscillations to magnitudes less than 0.01 m. A delayed-position feedback controller is used to minimize the sensitivity of the input-shaping controller to variations in system parameters and to improve its robustness. The addition of such feedback controller will account for system delays, external disturbances, and will compensate for uncertainties in the system. The delayed-position feedback controller is shown to be capable of reducing the residual oscillations of the payload to less than 0.05 m within 5 s of the end of the transfer maneuver. Feedback controllers produce continuously changing acceleration profiles, which affects the crane operator performance and comfort. Therefore, applying an efficient feedback controller, such as the delayed-position feedback, at the end of the transfer maneuver for a very short period of time eliminates excessive trolley motion, thus maintaining comfortable working conditions for the crane operator.
Acknowledgement The authors are indebted to Dr. Ali H. Nayfeh for his guidance and continuous support.
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