- Email: [email protected]
Discrete-time command profile for simultaneous travel and hoist maneuvers of overhead cranes
Journal of Sound and Vibration 345 (2015) 47–57
Contents lists available at ScienceDirect
Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi
Discrete-time command profile for simultaneous travel and hoist maneuvers of overhead cranes Khaled A. Alghanim a, Khaled A. Alhazza a,n, Ziyad N. Masoud b a b
Department of Mechanical Engineering, Kuwait University, P.O.Box 5969, Safat 13060, Kuwait Department of Mechatronics Engineering, German Jordanian University, Amman 11180, Jordan
a r t i c l e i n f o
abstract
Article history: Received 31 March 2014 Received in revised form 12 January 2015 Accepted 20 January 2015 Handling Editor: W. Lacarbonara Available online 2 March 2015
A strategy for generating optimal shaped command-profile for the reduction of residual vibrations in rest-to-rest crane maneuvers is proposed. The technique used is based on generating discrete-time shaped acceleration profile. This profile accommodates simultaneous travel and hoist maneuvers in the presence of damping. The discrete-time acceleration profile is derived analytically using finite step segments. These segments are integrated into an input matrix coupled with a response matrix through a time varying equation of motion of the system. The input acceleration matrix is then optimized to satisfy specific rest-to-rest maneuver conditions. Several parameters are incorporated in the proposed strategy including maneuver time, time step, hoisting speed, damping, and maximum travel velocity and acceleration. Unlike traditional command shapers, the proposed strategy matches the discrete command signal steps to the machine minimum time step of the control hardware used. To achieve an optimum maneuvering time, the shaped command utilizes the full system acceleration and velocity capabilities. The resulting shaped command profile is capable of eliminating travel and residual oscillations. Successful performance of the proposed command shaping strategy is validated through numerical simulations and experiments on a scaled model of an overhead crane. & 2015 Elsevier Ltd. All rights reserved.
1. Introduction Digital control systems and associated hardware are becoming more common in large scale industrial applications. Crane industry is one application that is advancing rapidly towards automation. This trend commands a shift from continuoustime to discrete-time control design. The need for this shift becomes evident and crucial in open-loop control strategies such as command shaping. Timing is the key to successfully shaped commands. The mismatch between continuous commands and discrete commands compromises performance effectiveness and may even render a shaped command useless. Several examples that illustrate the effect of this mismatch on the performance of the shaped commands are presented in [1]. Examples demonstrate that such mismatch has a direct impact on the carefully timed acceleration profiles. The mismatch led to variations in terminal velocity and significant residual payload oscillations. Shaped commands are widely used for rest-to-rest maneuvers. Industrial cranes are one of the most important applications of such maneuvers. Dynamics and control of cranes have been the focus of vast theoretical and experimental
n
Corresponding author. E-mail addresses: [email protected] (K.A. Alghanim), [email protected] (K.A. Alhazza), [email protected] (Z.N. Masoud).
http://dx.doi.org/10.1016/j.jsv.2015.01.042 0022-460X/& 2015 Elsevier Ltd. All rights reserved.
48
K.A. Alghanim et al. / Journal of Sound and Vibration 345 (2015) 47–57
research [2,3]. The main objectives of research were reducing or eliminating residual vibrations, reducing maneuver time, and enhancing safety in operating conditions. Many control techniques were developed to achieve those goals. Feedback control [4–6] was used because of its robustness to modeling uncertainties. Feedback control systems may outperform open-loop control systems. However, this comes at a price of adding extra hardware components, including jib motion sensors and oscillations angle sensors. Payload sway sensing systems are particularly challenging and expensive. In many overhead crane systems, this price is too expensive for the required operation tasks. Open-loop control, on the other hand, does not require changes or additions to the existing crane hardware, which makes it more attractive. Command shaping is one open-loop approach that was developed to eliminate the extra requirements of feedback control. Command shaping is used for moving suspended objects and/or flexible systems [3]. Along this line of research, Starr [7] developed a strategy for swing-free transport of suspended objects. His work dealt with systems in which the duration of acceleration is small compared to the period of oscillation of the suspended object. Strip [8] showed that for simply suspended objects, there is a family of acceleration strategies that results in a swing-free motion at a chosen velocity. His control strategy was based on accelerating a suspended object for one full period of oscillation to reach a zero vibration condition at the final velocity. To finish the maneuver, the system was decelerated at the same rate as it was accelerated. He also showed that accelerating a suspended object for any period of time then waiting for half the period of free oscillation, then accelerating at the same rate produces swing free motion. This is known as the double-step command shaping. Input shaping is one of the most commonly used command shaping techniques. Input shaping is defined as a method of reducing residual vibrations by convolving a sequence of carefully timed impulses with a general reference command signal. For a constant reference command, the resulting shaped command becomes sequence of input step functions. Smooth profiles, such as S-curves, trigonometric transition functions, Gaussians, and cam polynomials are also used for command shaping. These profiles introduce a low-pass filtering effect. It is well known that low-pass filtering can reduce residual vibration, however it incurs a large rise-time penalty [9,10]. Time-delay filters where successfully implemented [11] to reduce jerks in input shapers for undamped and damped systems. Singhose et al. [12] compared smooth and non-smooth commands by interpreting smooth commands as input-shaped functions. They concluded that S-curves smooth function must be four times slower than step commands shaped with zero-vibration shapers to eliminate vibrations in single-mode systems. By imposing limits on the first and second time derivatives of the feed rate, Erkorkmaz and Altintas [13] presented continuous position, velocity, and acceleration profiles for high speed CNC systems using quintic spline trajectory generation algorithm. Alhazza and Masoud [1] introduced a continuous wave-form command profile to eliminate residual vibrations in single mode systems. Their work went into details of the effect of discretization of continuous commands on the system performance. The work showed that timing sensitivity of command shapers became evident when timing precision was compromised by the sampling rates of the control hardware used. They showed that discrete wave-form commands were also sensitive to the precision of switching times that depended on the speed and processing power of the control hardware. To overcome that problem, they incorporated a simulated feedback system in their command shaping algorithms. The continuous wave-form command shaping strategy was further extended by including the effect of damping [14,15]. To avoid the complexity of including hoisting in crane controllers’ design, researchers tend to assume constant cable length operating conditions. Such control strategies depend on raising the payload, moving it horizontally, and lowering it to the final position, sequentially. Although this approach is effective, it results in a large time penalty. Implementing constant length input shapers in maneuvers that involve hoisting can lower the residual vibrations but cannot eliminate them [16]. However, recent research incorporated simultaneous travel and hoist in command shaping. Techniques such as graphical profile generation [17], wave-form profiles [18], and iterative learning control [19] were used to produce shaped commands for crane maneuvers involving simultaneous travel and hoist. In mathematics, discretization is the process of transferring continuous models into discrete counterparts. This process is usually carried out as a first step toward making these models suitable for implementation on digital computers. In digital control systems, digitally generated continuous profiles are converted into analog voltage or current signals. These digital to analog (D/A) converters are known to be relatively slow, which results in a step discrete command signal. Faster D/A convertors are available in the high-end control hardware, however, these systems tend to be relatively expensive. Even with these systems, the power drainage in the D/A convertors, combined with intensive signal processing might significantly slow down the conversion process. Masoud et al. [5] used a high-end data acquisition and control system for the control of an experimental model of a container crane. The design conversion time of the used hardware was in the order of microseconds. However, when physically connecting the system to the hardware circuits of the crane, the conversion rate dropped unexpectedly to 50 Hz. Similar observation was made when their control system was implemented on a superpanamax container crane. One of the most advanced stand-alone National Instruments control hardware (Compact-RIO) was used. The Compact-RIO was capable of a conversion rate in an order of microseconds. However, the conversion rate dropped to approximately 10 Hz when the system was physically connected to the drive circuits of the crane. This indicates that the real world implementation of continuous commands results in discrete signals. This increases the discrepancies between the design continuous command and the actual command, leading to significant unpredictable change in the system response. In this work, an optimization strategy based on a finite step segments technique is used to generate shaped acceleration commands for an overhead crane. The strategy aims at introducing the freedom of choice of the sampling frequency in the command profiles. The segments are integrated in a matrix, which is coupled with a response matrix, through the system's
K.A. Alghanim et al. / Journal of Sound and Vibration 345 (2015) 47–57
49
Fig. 1. Simple-pendulum model of an overhead crane.
equation of motion, to produce swing free motion. The proposed technique takes into consideration several parameters such as maneuver time, time step, maximum allowable oscillation amplitude, cruising velocity, and hoisting speed. It also includes the effect of damping on the system response. The effects of discretization and actuation time step of different shaped profiles are discussed. Several examples are presented and validated experimentally to demonstrate the effectiveness of the proposed technique.
2. Mathematical model An overhead crane can be modeled as a two-dimensional pendulum with a massless variable length hoisting cable. The jib of the crane is modeled as a horizontal slider to which the suspension point of the pendulum is attached, Fig. 1. Assuming linear viscous damping at the pendulum–jib joint, the nonlinear equation of motion of the model is lθ€ þ ð2_l þ cÞθ_ þg sin ðθÞ ¼ u€ cos ðθÞ
(1)
where θ is the oscillation angle of the hoisting cable, g is the gravitational acceleration, l is the cable length, c is the damping coefficient, and u€ is the jib acceleration. Assuming small oscillation angles, the equation of motion, Eq. (1), can be linearized as ! _l u€ € (2) θ þ2 þ ζωn θ_ þ ω2n θ ¼ l l where ωn ¼
pffiffiffiffiffiffiffi g=l is the linear natural frequency of the pendulum and
ζ is the damping ratio.
3. Optimization technique In this section, the acceleration profile of the jib of the crane is derived for a variable length of the hoisting cable. As can be observed in the equation of motion of the crane model, Eq. (2), the rate of change of the length can act as a positive or a negative damper, depending on whether the payload is being hoisted up or lowered down. When raising a payload, the rate of change of the hoisting cable length is negative, which is equivalent to introducing negative damping to the system. Whereas in the case of payload lowering, the rate of change in the cable length is positive. This leads to an additional positive damping in the system. The acceleration of the jib of the crane is subjected to several constraints in order to meet a desired system response. Boundary conditions on both input and output sides of the system, and limitations on the input acceleration are also considered. Considering an input vector representing the jib acceleration at different time steps, where the total number of time steps is p and the acceleration is constant for the duration of each time step, this acceleration vector can be presented as € ¼ ½u€ 1 u€ 2 … u€ k … u€ p T U
(3)
where u€ k is the input acceleration at the time interval k. The time rate of change hoisting cable length is assumed linear as lðtÞ ¼ li þst
(4)
where li is the initial length, and s is a constant representing the hoisting rate. Assuming small variations in the cable length at each time step, the cable length can be treated as constant at each time step Δt, but varying over the whole period of acceleration stage τ ¼ pΔt. Therefore, Eq. (2) for each time step is given by s u€ θ€ k þ 2 þ ζωnk θ_ k þ ω2nk θk ¼ k ; 0 r t r Δt and k ¼ 1; 2; …; p (5) lk lk
50
K.A. Alghanim et al. / Journal of Sound and Vibration 345 (2015) 47–57
where ωnk and lk are the natural frequency of the system and cable length at the time step k, respectively, and are defined by rffiffiffiffi 1 g (6) lk ¼ li þ k sΔt and ωnk ¼ 2 lk The initial conditions are
θ_ 1 ð0Þ ¼ 0;
θ1 ð0Þ ¼ 0;
θk ð0Þ ¼ θik ;
θ_ k ð0Þ ¼ θ_ ik
(7)
For simplicity, Eq. (5) can be written as
θ€ k þak θ_ k þ bk θk ¼ ck where
ak ¼ 2
s þ ζωnk ; lk
bk ¼ ω2nk ;
(8)
ck ¼
u€ k lk
Assuming a small enough damping ratio, ζ, the term s=lk becomes dominant and therefore, the sign of s would represent the sign of the constant ak. bk is always positive, and ck is a constant that needs to be determined to satisfy the system's design requirements. Assuming underdamped response, the term 14a2k bk is always negative, and the solution of Eq. (8) is c θk ¼ e ð1=2Þak t Ak sin ðωnk tÞ þ Bk cos ðωnk ktÞ þ k (9) bk where Ak ¼
1
ωnk
a c θ_ ik þ k θik k ; 2 bk
c Bk ¼ θik k bk
and
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 ωnk ¼ bk k 4
Notice that the solution given by Eq. (9) is valid for both raising and lowering cases. The sign of the constant ak differentiates the two cases. Evaluating the solution at the end of each time step yields
1 _ a c c c θk Δt ¼ e ð1=2Þak Δt θ ik þ k θik k sin ωnk Δt þ θik k cos ωnk Δt þ k (10) ωnk 2 bk bk bk
θ_ k Δt ¼ e ð1=2Þak Δt
θik
ck 1 _ 1 c θ ik þ ak θik k ak K 2 K1 þ 2 bk ωnk 2bk
(11)
where
1 K 1 ¼ ak cos ωnk Δt ωnk sin ωnk Δt 2
and
1 K 2 ¼ ak sin ωnk Δt þ ωnk cos ωnk Δt 2
are constants for each time step. Rearranging the solution, Eqs. (10) and (11), yields
θk ðΔtÞ ¼ K 3 θik þ K 4 θ_ ik þ K 5 ck and θ_ k ðΔtÞ ¼ K 6 θik þK 7 θ_ ik þK 8 ck where
(12)
a K 3 ¼ e ð1=2Þak Δt cos ωnk Δt þ k sin ωnk Δt 2ωnk
1 ð1=2Þak Δt K4 ¼ e sin ωnk Δt
ωnk
1 K 5 ¼ ð1 K 3 Þ bk a K 6 ¼ e ð1=2Þak Δt K 1 þ k K 2 2ωnk K2 ð1=2Þak Δt K7 ¼ e
ωnk
K6 K8 ¼ bk It is clear that the initial conditions of interval k, θik and θ_ ik , are the system response of the previous interval θk 1 ðΔtÞ and θ_ k 1 ðΔtÞ, respectively. Therefore, Eq. (12) can be written as
θk ðΔtÞ ¼ K 3 θk 1 ðΔtÞ þ K 4 θ_ k 1 ðΔtÞ þ K 5 ck and θ_ k ðΔtÞ ¼ K 6 θk 1 ðΔtÞ þ K 7 θ_ k 1 ðΔtÞ þK 8 ck
(13)
Expanding Eq. (13) for all time intervals yields _ ¼ NC Θ ¼ MC and Θ
(14)
K.A. Alghanim et al. / Journal of Sound and Vibration 345 (2015) 47–57
51
where
Θ ¼ ½θ1 θ2 θ3 ⋯ θp 1 θp T and C ¼ ½c1 c2 c3 ⋯ cp 1 cp T and where the matrices M and N depict the recurrence in Eq. (13), and where the elements of M and N matrices are M ij ¼ K 3 Mi 1;j þ K 4 Ni 1;j
N ij ¼ K 6 M i 1;j þ K 7 Ni 1;j ;
and
M ij ¼ K 5
and
N ij ¼ K 8 ;
M ij ¼ 0
and
N ij ¼ 0;
i4j
i¼j ioj
At this stage, the input vector C has p number of independent variables c1 ; c2 ; …; cp . Nevertheless, by introducing the system constraints defined by final conditions of the system
θ_ p ¼ 0;
θp ¼ 0;
u_ p ¼ vf
(15)
the number of independent variables is reduced to p 3 , with 3 dependent variables that satisfy system constraints given by Eq. (15), where vf is the required final speed of the jib of the crane. However, the last condition of Eq. (15) can be satisfied by € Given that input acceleration for each interval is considered as a step function, the numerically integrating the vector U. € summation of U multiplied by Δt gives an exact result. Therefore u_ p ¼ Δt
p X
u€ k ¼ Δt
k¼1
p X
ck lk
(16)
k¼1
By simple mathematical manipulations, the system constraints can be simplified as c ¼ J1 J2 C
(17)
where c ¼ ½c1 c2 c3 T C ¼ ½c4 c5 ⋯ cp T 3 2 3 mp;1 mp;2 mp;3 1 0 6n 7 6 7 J1 ¼ 4 p;1 np;2 np;3 5 4 0 5 vf l1 l2 l3 Δt 3 2 mp;2 mp;3 1 mp;4 mp;5 ⋯ mp;p 1 6 np;2 np;3 7 5 4 np;4 np;5 ⋯ np;p 1 l4 l5 ⋯ lp 1 l2 l3 2
2
mp;1 6 J2 ¼ 4 np;1 l1
mp;p
3
np;p 7 5 lp
Now, the model is designed to satisfy system's constraints, therefore, minimizing the norm of the input acceleration vector becomes simpler. The minimization process implies minimizing the norms of vectors c and C. Consider an optimality function q that represents the summation of the input acceleration squares as q¼
p X
T
c2k ¼ c T c þ C C
(18)
k¼1
Optimal C that minimizes function q can be obtained by setting ∂q ∂C
¼0
(19)
Substituting Eq. (18) into Eq. (19) and solving for C yields C ¼ ðJT2 J2 þIÞ 1 JT2 J1
(20)
where I is an identity matrix of size p 3. Once C is determined, c can be determined from Eq. (17). The optimal input acceleration that minimizes the input vector amplitudes, and satisfies the system's constraints can be determined by € ¼ ½c1 l1 c2 l2 ⋯ ck lk T U
(21)
4. Parametric analysis Since the proposed technique offers a freedom of choice of profile generation parameters, as well as the ability to handle crane operations involving simultaneous travel and hoist maneuvers, it is important to study the effect of varying such parameters on the system response. In the following simulations, the effect of damping is considered negligible for cranes.
52
K.A. Alghanim et al. / Journal of Sound and Vibration 345 (2015) 47–57
It is important to emphasize that the effect of damping in this work is included for the sake of completeness, since this technique can be used for any second-order system.
4.1. Step size and acceleration time Two sets of numerical simulation are used to demonstrate the performance of the proposed technique for various time step sizes and acceleration times. In the first set of simulations, the duration of the acceleration phase, τ, is set to 1 s, the length of the hoisting cable is set to 0.6 m and the final velocity is set to 0.3 m/s. Three different discretization time step sizes, Δt, are tested: 0.3333 s, 0.1 s, and 0.01 s, Fig. 2. These simulations reveal a significant advantage of the proposed technique, which is that the system maintains ideal response regardless of the discretization step size used for generating the acceleration profiles. This advantage is extremely significant since the time steps of these profiles can now be selected according to the sampling speed of hardware used and not the other way around. In the second set of simulations, the time step of the acceleration profiles is set to Δt ¼ 0:05 s. Three simulations for acceleration times of τ ¼ 0.8 s, τ ¼0.95 s, and τ ¼1.05 s are presented, Fig. 3. The length of the hoisting cable is set to 0.4 m. This set of simulations reveals another important feature of the proposed technique, which is that the time length of these profiles is independent of the natural period of the model. However, it is important to choose acceleration times that are multiples of time step size. The above simulations are further verified through experiments on a scaled model of a 3D overhead crane in the Advanced Vibration Lab at Kuwait University, Fig. 4. The jib of the crane is controlled by a DC motor. Three quadrature incremental-encoders are used to measure the jib motion, the length of the hoisting cable, and the oscillation angle of the hoisting cable. The resolution of each encoder is 1024 pulses per revolution. An interface between a PC and the crane setup
0.07
1
Δ t = 1/3 Δ t =0.1 Δ t = 0.01
0.05
Payload Angle [rad]
2
Jib Acceleration [m/s ]
0.8 0.6 0.4 0.2 0
0.04 0.03 0.02 0.01
−0.2 −0.4
Δ t = 1/3 Δ t =0.1 Δ t = 0.01
0.06
0
0
0.2
0.4
0.6
0.8
1
−0.01
0
0.2
0.4
0.6
0.8
1
1.2
Time [s]
Time [s]
Fig. 2. Acceleration profiles and corresponding system responses for three different time step sizes of Δt ¼ 0:3333 s, Δt ¼ 0:1 s, and Δt ¼ 0:01 s (l ¼0.6 m and τ¼ 1 s): (a) command profiles and (b) angle of the hoisting cable.
1.2
0.8 0.6 0.4 0.2
τ = 0.8 τ = 0.95 τ =1.05
0.07 0.06
Payload Angle [rad]
1 2 Jib Acceleration [m/s ]
0.08
τ = 0.8 τ = 0.95 τ =1.05
0.05 0.04 0.03 0.02 0.01
0 −0.2
0 0
0.2
0.4
0.6
Time [s]
0.8
1
−0.01
0
0.2
0.4
0.6
0.8
1
Time [s]
Fig. 3. Acceleration profiles and corresponding system responses for three different acceleration times of τ ¼0.8 s, τ ¼ 0.95 s, and τ¼ 1.05 s (l ¼0.4 m and Δt ¼ 0:05 s): (a) command profiles and (b) angle of the hoisting cable.
K.A. Alghanim et al. / Journal of Sound and Vibration 345 (2015) 47–57
53
Fig. 4. The experimental crane setup.
is achieved using RT-DAC/PCI multipurpose digital I/O board. The interface software uses MATLAB's real-time Simulink environment. The sampling rate of the control hardware is 100 Hz. The maximum jib speed and acceleration are 0.3 m/s and 0.9 m/s2, respectively. The crane has a 0.6 m usable track. A light weight cable is used with a mass of 0.092 kg attached to its end representing the crane payload. The maximum hoisting speed is approximately 0.134 m/s. The first set of simulations, Fig. 2, are verified on the setup using the same operating conditions, Fig. 5. The acceleration profile is repeated to decelerate the jib to a stop, Fig. 5(a). Results show agreement between the simulations and the experiments except for small deviations due to modeling uncertainties and unmodeled nonlinearities such as backlash and jib friction. Several other tests are conducted using arbitrary crane and profile parameters, Table 1. Parameters are chosen arbitrarily to verify the two major advantages revealed in simulations. First, the independence of the acceleration time from the natural period of oscillation demonstrated by using arbitrary hoisting cable lengths. Second, performance robustness to different time step sizes used for profile generation. Results of three of those tests are shown in Fig. 6. The figure shows the acceleration profiles generated by the proposed technique. Hoisting angle measurements, Fig. 6(b) indicates that the proposed technique is capable of minimizing residual vibrations at the end of the maneuver with very small discrepancies attributed to many unmodeled dynamics including friction and backlash in the cranes actuators. For completeness, a parametric sensitivity analysis is conducted. An acceleration profile is generated assuming a hoisting cable length of 0.60 m. This profile is then applied to models with different cable lengths. Residual oscillation angles are plotted against the actual cable length, Fig. 7. Examining the results shows that the performance of the proposed technique is robust to small change in the estimated cable length, and therefore, relatively small residual oscillations occur. In fact, a 10 percent measurement error in the cable length yields a maximum of 0.012 rad residual oscillation angle. As expected, small discrepancies between numerical and the experimental results exist.
4.2. Hardware sampling frequency Here, we will try to answer an obvious question, What is the effect of hardware sampling frequency on the performance of the shaped command? As indicated in the Introduction, operations of cranes using digital control lead to discrete commands. The time step size of these commands is highly dependent on the system hardware and its sampling frequency capabilities [5]. The time step size can greatly compromise the system performance by distorting the shape of the input profile [1]. A test model is chosen with a hoisting cable length of l¼0.60 m. An acceleration profile is generated using a discretization time step of Δt ¼ 0:1 s. To analyze the effect of different sampling frequencies of control hardware, several simulations are performed by sampling the already discrete command profile at different sampling rates, δt, representing the sampling rates of the control hardware used. The profile sampling rate is varied between 0.01 s and 0.3 s. Results of these simulations are presented in Fig. 8. Fig. 8 shows a detrimental impact of the actual sampling rate on the performance of shaped commands. It is clear that the command profile suffers serious degradation in performance in terms of system response. Simulations reveal the fact that it is imperative that a shaped command must have the same time step size as the control hardware. Faster hardware can be used as long as the profile discretization step remains an integer multiple of the sampling time of the control hardware used. Simulations in Fig. 8 also show significant impact on the jib speed reached at the end of the acceleration time. In most of the simulations, the final speed is well above the design speed of the jib, which limits the ability of the crane actuators to track profiles of the shaped commands.
54
K.A. Alghanim et al. / Journal of Sound and Vibration 345 (2015) 47–57
1
0.08 Case 1 Case 2 Case 3
0.8
Payload Angle [rad]
Jib Acceleration [m/s2]
0.6
Case 1 Case 2 Case 3
0.06
0.4 0.2 0 −0.2 −0.4
0.04 0.02 0 −0.02 −0.04
−0.6 −0.06
−0.8 −1
0
1
2
3
4
−0.08
5
0
1
2
3
Time [s]
4
5
Time [s]
Fig. 5. Acceleration profiles and corresponding experimental system responses for three different time step sizes of Δt ¼ 0:3333 s, Δt ¼ 0:1 s, and Δt ¼ 0:01 s (l ¼0.6 m and τ ¼1 s): (a) command profiles and (b) angle of the hoisting cable. Table 1 Crane and profile parameters. Case no.
Cable length (m)
Acceleration time (s)
Step size (s)
1 2 3
0.4 0.35 0.2
0.67 0.9 1.2
0.01 0.1 0.3
1
0.1 Case 1 Case 2 Case 3
0.8
0.06
Payload Angle [rad]
Jib Acceleration [m/s2]
0.6 0.4 0.2 0 −0.2 −0.4
0.04 0.02 0 −0.02 −0.04
−0.6
−0.06
−0.8
−0.08
−1
0
1
2
3
Time [s]
4
Case 1 Case 2 Case 3
0.08
5
−0.1
0
1
2
3
4
5
Time [s]
Fig. 6. Experimental results using profile and crane parameters in Table 1: (a) command profiles and (b) angle of the hoisting cable.
5. Simultaneous travel and hoist maneuvers Simultaneous travel and hoist maneuvers add a great complexity to the design of shaped commands. Hoisting converts the equation of motion into a complex type of ordinary differential equation with time varying coefficients, Eq. (1). This type of equations, usually, has no exact solution. For this reason, researchers tend to avoid command shaping for maneuvers involving hoisting. Optimal control can be used to generate commands that result in zero residual oscillations [20,21]. However, when hoisting is involved, the difficulty of generating optimal command becomes more challenging because the system becomes nonlinear [16]. To demonstrate the ability of the proposed technique of handling simultaneous travel and hoist maneuvers, two test cases are numerically simulated and tested experimentally. The hoisting speed is kept within the maximum hoisting speed of the experimental setup which is 0.134 m/s. Maneuver and profile parameters of these two tests are in Table 2. Simulations indicate the ability of the proposed technique to produce acceleration commands capable of handling simultaneous travel and hoisting maneuvers, Figs. 9 and 10. A close match between simulations and experimental tests is obtained.
K.A. Alghanim et al. / Journal of Sound and Vibration 345 (2015) 47–57
55
0.06 Simulated Experimental
Maximum Angle [rad]
0.05
0.04
0.03
0.02
0.01
0 0.45
0.5
0.55
0.6
0.65
0.7
0.75
L0 [m] Fig. 7. Simulated and experimental sensitivity analysis when l ¼0.60 and τ ¼ 1.00 and Δt ¼ 0:01.
0.55
0.05
0.5
Maximum Velocity [m/s]
Residual Angle [rad]
0.04
0.03
0.02
0.45
0.4
0.35
0.01
0
0.3
0
0.05
0.1
0.15
0.2
0.25
0.3
0.25 0
0.05
0.1
Step Size δ t [s]
0.15
0.2
0.25
0.3
Step Size δ t [s]
Fig. 8. The effect of discretization on the payload angle, velocity and displacement when Δt ¼ 0:1 s as a function of the discretization step time δt: (a) residual oscillation and (b) terminal jib velocity. Table 2 Simultaneous travel and hoist test parameters. Test no.
1 2
Cable length (m) Initial
Intermediate
Final
0.4 0.35
0.3 0.4
0.4 0.3
Step time (s)
Acceleration time (s)
0.05 0.3
0.8 0.9
6. Conclusions An optimization strategy for generating shaped commands for overhead cranes is presented. The optimization technique used produces discrete-time shaped commands. The effectiveness and the performance robustness of the shaped commands were demonstrated using both numerical simulations and experiments on a scaled model of an overhead crane. Simulations and experiments demonstrate the ability of the command profiles to eliminate residual payload oscillations in the presence of simultaneous travel and hoist maneuvers that are fundamental for time optimal operations. Using this strategy, a profile designer has the freedom to change several important parameters of the generated profile. In addition to the usual parameters in all command shaping techniques, such as maximum velocity and acceleration, the presented technique adds two more selectable parameters: the size of the time steps and the time length of the command profile, regardless of the system natural period of oscillation. The size of the profile's time steps is freely selectable without any performance compromise. This feature is very crucial as it was shown that a good match between profile steps and control hardware sampling rate is essential for achieving desired outcome. The selectable acceleration time gives the strategy its independence from the natural period of oscillation of the system.
56
K.A. Alghanim et al. / Journal of Sound and Vibration 345 (2015) 47–57
0.8
0.08
0.6
Payload Angle [rad]
Amplitude
0.4 0.2 0 −0.2 Acceleration [m/s2] Displacement [m] Velocity [m/s] Length[m]
−0.4 −0.6 −0.8
Simulated Experimental
0.06
0
0.5
1
1.5
0.04 0.02 0 −0.02 −0.04 −0.06
2
2.5
3
−0.08
3.5
0
1
2
Time [s]
3
4
5
Time [s]
Fig. 9. Simultaneous travel and hoisting maneuver test case 1: (a) jib command and hoist profiles and (b) angle of the hoisting cable.
0.8
0.08
0.6
Payload Angle [rad]
Amplitude
0.4 0.2 0 −0.2 Acceleration [m/s2] Displacement [m] Velocity [m/s] Length[m]
−0.4 −0.6 −0.8
Simulated Experimental
0.06
0
0.5
1
1.5
0.04 0.02 0 −0.02 −0.04 −0.06
2
Time [s]
2.5
3
3.5
−0.08
0
1
2
3
4
5
Time [s]
Fig. 10. Simultaneous travel and hoisting maneuver test case 2: (a) jib command and hoist profiles and (b) angle of the hoisting cable.
References [1] K. Alhazza, Z. Masoud, A novel wave-form command shaper for overhead cranes, Journal of Engineering Research 1 (2013) 181–209. [2] E.M. Abdel-Rahman, A.H. Nayfeh, Z.N. Masoud, Dynamics and control of cranes: a review, Journal of Vibration and Control 9 (2003) 863–908. [3] W. Singhose, Command shaping for flexible systems: a review of the first 50 years, International Journal of Precision Engineering and Manufacturing 10 (2009) 153–168. [4] Z.N. Masoud, A.H. Nayfeh, D.T. Mook, Cargo pendulation reduction of ship-mounted cranes, Nonlinear Dynamics 35 (2004) 299–311. [5] Z.N. Masoud, A.H. Nayfeh, N.A. Nayfeh, Sway reduction on quay-side container cranes using delayed feedback controller: simulations and experiments, Journal of Vibration and Control 11 (2005) 1103–1122. [6] S.-K. Cho, H.-H. Leeb, A fuzzy-logic antiswing controller for three-dimensional overhead cranes, ISA Transactions 41 (2002) 235–243. [7] G.P. Starr, Swing-free transport of suspended objects with a path-controlled robot manipulator, Journal of Dynamic Systems, Measurement, and Control 107 (1995) 97–100. [8] D.R. Strip, Swing-free transport of suspended objects: a general treatment, IEEE Transactions on Robotics and Automation 5 (1989) 234–236. [9] M.A. Ahmad, R.M.T.R. Ismail, M.S. Ramli, N.M. Abd Ghani, N. Hambali, Investigations of feed-forward techniques for anti-sway control of 3-D gantry crane system, Proceedings of the IEEE Symposium on Industrial Electronics Applications, ISIEA 2009, Vol. 1, Piscataway, New Jersey, USA, 2009, pp. 265– 270. [10] N. Singer, W. Singhose, W. Seering, Comparison of filtering methods for reducing residual vibration, European Journal of Control 5 (1999) 208–218. [11] T. Singh, Jerk limited input shapers, Journal of Dynamic Systems, Measurement, and Control 126 (2004) 215–219. [12] W. Singhose, R. Eloundou, J. Lawrence, Command generation for flexible systems by input shaping and command smoothing, Journal of Guidance, Control, and Dynamics 33 (2010) 1697–1707. [13] K. Erkorkmaz, Y. Altintas, High speed CNC system design. Part I: jerk limited trajectory generation and quintic spline interpolation, International Journal of Machine Tools and Manufacture 41 (2001) 1323–1345. [14] K. Alhazza, A. Al-Shehaima, Z. Masoud, A continuous modulated wave-form command shaping for damped overhead cranes, Proceedings of the ASME 2011 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, Washington, DC, USA, 2011, pp. 361–366. [15] K. Alhazza, Experimental validation on a continuous modulated wave-form command shaping applied on damped systems, Topics in Dynamics of Civil Structures, Vol. 4, Springer, New York, 2013, pp. 445–451. [16] W. Singhose, L. Porter, M. Kenison, E. Kriikku, Effects of hoisting on the input shaping control of gantry cranes, Control Engineering Practice 8 (2000) 1159–1165.
K.A. Alghanim et al. / Journal of Sound and Vibration 345 (2015) 47–57
57
[17] Z.N. Masoud, M.F. Daqaq, A graphical approach to input-shaping control design for container cranes with hoist, IEEE Transactions on Control Systems Technology 14 (2006) 1070–1077. [18] K.A. Alhazza, Experimental and numerical validation on a continuous modulated wave-form command shaping control considering the effect of hoisting, Proceedings of the ASME 2013 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, Portland, Oregon, USA, 2013, pp. V07AT10A057. [19] K.A. Alhazza, A.M. Hassan, K.A. Alghanim, Z.N. Masoud, An iterative learning control technique for point-to-point maneuvers applied on an overhead crane, Shock and Vibration 2014 (2014) 11. (Article ID 261509), /http://dx.doi.org/10.1155/2014/261509S. [20] Y. Sakawa, Y. Shindo, Optimal control of container cranes, Automatica 18 (1982) 257–266. [21] J.W. Auernig, H. Troger, Time optimal control of overhead cranes with hoisting of the load, Automatica 23 (1987) 437–447.
Contents lists available at ScienceDirect
Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi
Discrete-time command profile for simultaneous travel and hoist maneuvers of overhead cranes Khaled A. Alghanim a, Khaled A. Alhazza a,n, Ziyad N. Masoud b a b
Department of Mechanical Engineering, Kuwait University, P.O.Box 5969, Safat 13060, Kuwait Department of Mechatronics Engineering, German Jordanian University, Amman 11180, Jordan
a r t i c l e i n f o
abstract
Article history: Received 31 March 2014 Received in revised form 12 January 2015 Accepted 20 January 2015 Handling Editor: W. Lacarbonara Available online 2 March 2015
A strategy for generating optimal shaped command-profile for the reduction of residual vibrations in rest-to-rest crane maneuvers is proposed. The technique used is based on generating discrete-time shaped acceleration profile. This profile accommodates simultaneous travel and hoist maneuvers in the presence of damping. The discrete-time acceleration profile is derived analytically using finite step segments. These segments are integrated into an input matrix coupled with a response matrix through a time varying equation of motion of the system. The input acceleration matrix is then optimized to satisfy specific rest-to-rest maneuver conditions. Several parameters are incorporated in the proposed strategy including maneuver time, time step, hoisting speed, damping, and maximum travel velocity and acceleration. Unlike traditional command shapers, the proposed strategy matches the discrete command signal steps to the machine minimum time step of the control hardware used. To achieve an optimum maneuvering time, the shaped command utilizes the full system acceleration and velocity capabilities. The resulting shaped command profile is capable of eliminating travel and residual oscillations. Successful performance of the proposed command shaping strategy is validated through numerical simulations and experiments on a scaled model of an overhead crane. & 2015 Elsevier Ltd. All rights reserved.
1. Introduction Digital control systems and associated hardware are becoming more common in large scale industrial applications. Crane industry is one application that is advancing rapidly towards automation. This trend commands a shift from continuoustime to discrete-time control design. The need for this shift becomes evident and crucial in open-loop control strategies such as command shaping. Timing is the key to successfully shaped commands. The mismatch between continuous commands and discrete commands compromises performance effectiveness and may even render a shaped command useless. Several examples that illustrate the effect of this mismatch on the performance of the shaped commands are presented in [1]. Examples demonstrate that such mismatch has a direct impact on the carefully timed acceleration profiles. The mismatch led to variations in terminal velocity and significant residual payload oscillations. Shaped commands are widely used for rest-to-rest maneuvers. Industrial cranes are one of the most important applications of such maneuvers. Dynamics and control of cranes have been the focus of vast theoretical and experimental
n
Corresponding author. E-mail addresses: [email protected] (K.A. Alghanim), [email protected] (K.A. Alhazza), [email protected] (Z.N. Masoud).
http://dx.doi.org/10.1016/j.jsv.2015.01.042 0022-460X/& 2015 Elsevier Ltd. All rights reserved.
48
K.A. Alghanim et al. / Journal of Sound and Vibration 345 (2015) 47–57
research [2,3]. The main objectives of research were reducing or eliminating residual vibrations, reducing maneuver time, and enhancing safety in operating conditions. Many control techniques were developed to achieve those goals. Feedback control [4–6] was used because of its robustness to modeling uncertainties. Feedback control systems may outperform open-loop control systems. However, this comes at a price of adding extra hardware components, including jib motion sensors and oscillations angle sensors. Payload sway sensing systems are particularly challenging and expensive. In many overhead crane systems, this price is too expensive for the required operation tasks. Open-loop control, on the other hand, does not require changes or additions to the existing crane hardware, which makes it more attractive. Command shaping is one open-loop approach that was developed to eliminate the extra requirements of feedback control. Command shaping is used for moving suspended objects and/or flexible systems [3]. Along this line of research, Starr [7] developed a strategy for swing-free transport of suspended objects. His work dealt with systems in which the duration of acceleration is small compared to the period of oscillation of the suspended object. Strip [8] showed that for simply suspended objects, there is a family of acceleration strategies that results in a swing-free motion at a chosen velocity. His control strategy was based on accelerating a suspended object for one full period of oscillation to reach a zero vibration condition at the final velocity. To finish the maneuver, the system was decelerated at the same rate as it was accelerated. He also showed that accelerating a suspended object for any period of time then waiting for half the period of free oscillation, then accelerating at the same rate produces swing free motion. This is known as the double-step command shaping. Input shaping is one of the most commonly used command shaping techniques. Input shaping is defined as a method of reducing residual vibrations by convolving a sequence of carefully timed impulses with a general reference command signal. For a constant reference command, the resulting shaped command becomes sequence of input step functions. Smooth profiles, such as S-curves, trigonometric transition functions, Gaussians, and cam polynomials are also used for command shaping. These profiles introduce a low-pass filtering effect. It is well known that low-pass filtering can reduce residual vibration, however it incurs a large rise-time penalty [9,10]. Time-delay filters where successfully implemented [11] to reduce jerks in input shapers for undamped and damped systems. Singhose et al. [12] compared smooth and non-smooth commands by interpreting smooth commands as input-shaped functions. They concluded that S-curves smooth function must be four times slower than step commands shaped with zero-vibration shapers to eliminate vibrations in single-mode systems. By imposing limits on the first and second time derivatives of the feed rate, Erkorkmaz and Altintas [13] presented continuous position, velocity, and acceleration profiles for high speed CNC systems using quintic spline trajectory generation algorithm. Alhazza and Masoud [1] introduced a continuous wave-form command profile to eliminate residual vibrations in single mode systems. Their work went into details of the effect of discretization of continuous commands on the system performance. The work showed that timing sensitivity of command shapers became evident when timing precision was compromised by the sampling rates of the control hardware used. They showed that discrete wave-form commands were also sensitive to the precision of switching times that depended on the speed and processing power of the control hardware. To overcome that problem, they incorporated a simulated feedback system in their command shaping algorithms. The continuous wave-form command shaping strategy was further extended by including the effect of damping [14,15]. To avoid the complexity of including hoisting in crane controllers’ design, researchers tend to assume constant cable length operating conditions. Such control strategies depend on raising the payload, moving it horizontally, and lowering it to the final position, sequentially. Although this approach is effective, it results in a large time penalty. Implementing constant length input shapers in maneuvers that involve hoisting can lower the residual vibrations but cannot eliminate them [16]. However, recent research incorporated simultaneous travel and hoist in command shaping. Techniques such as graphical profile generation [17], wave-form profiles [18], and iterative learning control [19] were used to produce shaped commands for crane maneuvers involving simultaneous travel and hoist. In mathematics, discretization is the process of transferring continuous models into discrete counterparts. This process is usually carried out as a first step toward making these models suitable for implementation on digital computers. In digital control systems, digitally generated continuous profiles are converted into analog voltage or current signals. These digital to analog (D/A) converters are known to be relatively slow, which results in a step discrete command signal. Faster D/A convertors are available in the high-end control hardware, however, these systems tend to be relatively expensive. Even with these systems, the power drainage in the D/A convertors, combined with intensive signal processing might significantly slow down the conversion process. Masoud et al. [5] used a high-end data acquisition and control system for the control of an experimental model of a container crane. The design conversion time of the used hardware was in the order of microseconds. However, when physically connecting the system to the hardware circuits of the crane, the conversion rate dropped unexpectedly to 50 Hz. Similar observation was made when their control system was implemented on a superpanamax container crane. One of the most advanced stand-alone National Instruments control hardware (Compact-RIO) was used. The Compact-RIO was capable of a conversion rate in an order of microseconds. However, the conversion rate dropped to approximately 10 Hz when the system was physically connected to the drive circuits of the crane. This indicates that the real world implementation of continuous commands results in discrete signals. This increases the discrepancies between the design continuous command and the actual command, leading to significant unpredictable change in the system response. In this work, an optimization strategy based on a finite step segments technique is used to generate shaped acceleration commands for an overhead crane. The strategy aims at introducing the freedom of choice of the sampling frequency in the command profiles. The segments are integrated in a matrix, which is coupled with a response matrix, through the system's
K.A. Alghanim et al. / Journal of Sound and Vibration 345 (2015) 47–57
49
Fig. 1. Simple-pendulum model of an overhead crane.
equation of motion, to produce swing free motion. The proposed technique takes into consideration several parameters such as maneuver time, time step, maximum allowable oscillation amplitude, cruising velocity, and hoisting speed. It also includes the effect of damping on the system response. The effects of discretization and actuation time step of different shaped profiles are discussed. Several examples are presented and validated experimentally to demonstrate the effectiveness of the proposed technique.
2. Mathematical model An overhead crane can be modeled as a two-dimensional pendulum with a massless variable length hoisting cable. The jib of the crane is modeled as a horizontal slider to which the suspension point of the pendulum is attached, Fig. 1. Assuming linear viscous damping at the pendulum–jib joint, the nonlinear equation of motion of the model is lθ€ þ ð2_l þ cÞθ_ þg sin ðθÞ ¼ u€ cos ðθÞ
(1)
where θ is the oscillation angle of the hoisting cable, g is the gravitational acceleration, l is the cable length, c is the damping coefficient, and u€ is the jib acceleration. Assuming small oscillation angles, the equation of motion, Eq. (1), can be linearized as ! _l u€ € (2) θ þ2 þ ζωn θ_ þ ω2n θ ¼ l l where ωn ¼
pffiffiffiffiffiffiffi g=l is the linear natural frequency of the pendulum and
ζ is the damping ratio.
3. Optimization technique In this section, the acceleration profile of the jib of the crane is derived for a variable length of the hoisting cable. As can be observed in the equation of motion of the crane model, Eq. (2), the rate of change of the length can act as a positive or a negative damper, depending on whether the payload is being hoisted up or lowered down. When raising a payload, the rate of change of the hoisting cable length is negative, which is equivalent to introducing negative damping to the system. Whereas in the case of payload lowering, the rate of change in the cable length is positive. This leads to an additional positive damping in the system. The acceleration of the jib of the crane is subjected to several constraints in order to meet a desired system response. Boundary conditions on both input and output sides of the system, and limitations on the input acceleration are also considered. Considering an input vector representing the jib acceleration at different time steps, where the total number of time steps is p and the acceleration is constant for the duration of each time step, this acceleration vector can be presented as € ¼ ½u€ 1 u€ 2 … u€ k … u€ p T U
(3)
where u€ k is the input acceleration at the time interval k. The time rate of change hoisting cable length is assumed linear as lðtÞ ¼ li þst
(4)
where li is the initial length, and s is a constant representing the hoisting rate. Assuming small variations in the cable length at each time step, the cable length can be treated as constant at each time step Δt, but varying over the whole period of acceleration stage τ ¼ pΔt. Therefore, Eq. (2) for each time step is given by s u€ θ€ k þ 2 þ ζωnk θ_ k þ ω2nk θk ¼ k ; 0 r t r Δt and k ¼ 1; 2; …; p (5) lk lk
50
K.A. Alghanim et al. / Journal of Sound and Vibration 345 (2015) 47–57
where ωnk and lk are the natural frequency of the system and cable length at the time step k, respectively, and are defined by rffiffiffiffi 1 g (6) lk ¼ li þ k sΔt and ωnk ¼ 2 lk The initial conditions are
θ_ 1 ð0Þ ¼ 0;
θ1 ð0Þ ¼ 0;
θk ð0Þ ¼ θik ;
θ_ k ð0Þ ¼ θ_ ik
(7)
For simplicity, Eq. (5) can be written as
θ€ k þak θ_ k þ bk θk ¼ ck where
ak ¼ 2
s þ ζωnk ; lk
bk ¼ ω2nk ;
(8)
ck ¼
u€ k lk
Assuming a small enough damping ratio, ζ, the term s=lk becomes dominant and therefore, the sign of s would represent the sign of the constant ak. bk is always positive, and ck is a constant that needs to be determined to satisfy the system's design requirements. Assuming underdamped response, the term 14a2k bk is always negative, and the solution of Eq. (8) is c θk ¼ e ð1=2Þak t Ak sin ðωnk tÞ þ Bk cos ðωnk ktÞ þ k (9) bk where Ak ¼
1
ωnk
a c θ_ ik þ k θik k ; 2 bk
c Bk ¼ θik k bk
and
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 ωnk ¼ bk k 4
Notice that the solution given by Eq. (9) is valid for both raising and lowering cases. The sign of the constant ak differentiates the two cases. Evaluating the solution at the end of each time step yields
1 _ a c c c θk Δt ¼ e ð1=2Þak Δt θ ik þ k θik k sin ωnk Δt þ θik k cos ωnk Δt þ k (10) ωnk 2 bk bk bk
θ_ k Δt ¼ e ð1=2Þak Δt
θik
ck 1 _ 1 c θ ik þ ak θik k ak K 2 K1 þ 2 bk ωnk 2bk
(11)
where
1 K 1 ¼ ak cos ωnk Δt ωnk sin ωnk Δt 2
and
1 K 2 ¼ ak sin ωnk Δt þ ωnk cos ωnk Δt 2
are constants for each time step. Rearranging the solution, Eqs. (10) and (11), yields
θk ðΔtÞ ¼ K 3 θik þ K 4 θ_ ik þ K 5 ck and θ_ k ðΔtÞ ¼ K 6 θik þK 7 θ_ ik þK 8 ck where
(12)
a K 3 ¼ e ð1=2Þak Δt cos ωnk Δt þ k sin ωnk Δt 2ωnk
1 ð1=2Þak Δt K4 ¼ e sin ωnk Δt
ωnk
1 K 5 ¼ ð1 K 3 Þ bk a K 6 ¼ e ð1=2Þak Δt K 1 þ k K 2 2ωnk K2 ð1=2Þak Δt K7 ¼ e
ωnk
K6 K8 ¼ bk It is clear that the initial conditions of interval k, θik and θ_ ik , are the system response of the previous interval θk 1 ðΔtÞ and θ_ k 1 ðΔtÞ, respectively. Therefore, Eq. (12) can be written as
θk ðΔtÞ ¼ K 3 θk 1 ðΔtÞ þ K 4 θ_ k 1 ðΔtÞ þ K 5 ck and θ_ k ðΔtÞ ¼ K 6 θk 1 ðΔtÞ þ K 7 θ_ k 1 ðΔtÞ þK 8 ck
(13)
Expanding Eq. (13) for all time intervals yields _ ¼ NC Θ ¼ MC and Θ
(14)
K.A. Alghanim et al. / Journal of Sound and Vibration 345 (2015) 47–57
51
where
Θ ¼ ½θ1 θ2 θ3 ⋯ θp 1 θp T and C ¼ ½c1 c2 c3 ⋯ cp 1 cp T and where the matrices M and N depict the recurrence in Eq. (13), and where the elements of M and N matrices are M ij ¼ K 3 Mi 1;j þ K 4 Ni 1;j
N ij ¼ K 6 M i 1;j þ K 7 Ni 1;j ;
and
M ij ¼ K 5
and
N ij ¼ K 8 ;
M ij ¼ 0
and
N ij ¼ 0;
i4j
i¼j ioj
At this stage, the input vector C has p number of independent variables c1 ; c2 ; …; cp . Nevertheless, by introducing the system constraints defined by final conditions of the system
θ_ p ¼ 0;
θp ¼ 0;
u_ p ¼ vf
(15)
the number of independent variables is reduced to p 3 , with 3 dependent variables that satisfy system constraints given by Eq. (15), where vf is the required final speed of the jib of the crane. However, the last condition of Eq. (15) can be satisfied by € Given that input acceleration for each interval is considered as a step function, the numerically integrating the vector U. € summation of U multiplied by Δt gives an exact result. Therefore u_ p ¼ Δt
p X
u€ k ¼ Δt
k¼1
p X
ck lk
(16)
k¼1
By simple mathematical manipulations, the system constraints can be simplified as c ¼ J1 J2 C
(17)
where c ¼ ½c1 c2 c3 T C ¼ ½c4 c5 ⋯ cp T 3 2 3 mp;1 mp;2 mp;3 1 0 6n 7 6 7 J1 ¼ 4 p;1 np;2 np;3 5 4 0 5 vf l1 l2 l3 Δt 3 2 mp;2 mp;3 1 mp;4 mp;5 ⋯ mp;p 1 6 np;2 np;3 7 5 4 np;4 np;5 ⋯ np;p 1 l4 l5 ⋯ lp 1 l2 l3 2
2
mp;1 6 J2 ¼ 4 np;1 l1
mp;p
3
np;p 7 5 lp
Now, the model is designed to satisfy system's constraints, therefore, minimizing the norm of the input acceleration vector becomes simpler. The minimization process implies minimizing the norms of vectors c and C. Consider an optimality function q that represents the summation of the input acceleration squares as q¼
p X
T
c2k ¼ c T c þ C C
(18)
k¼1
Optimal C that minimizes function q can be obtained by setting ∂q ∂C
¼0
(19)
Substituting Eq. (18) into Eq. (19) and solving for C yields C ¼ ðJT2 J2 þIÞ 1 JT2 J1
(20)
where I is an identity matrix of size p 3. Once C is determined, c can be determined from Eq. (17). The optimal input acceleration that minimizes the input vector amplitudes, and satisfies the system's constraints can be determined by € ¼ ½c1 l1 c2 l2 ⋯ ck lk T U
(21)
4. Parametric analysis Since the proposed technique offers a freedom of choice of profile generation parameters, as well as the ability to handle crane operations involving simultaneous travel and hoist maneuvers, it is important to study the effect of varying such parameters on the system response. In the following simulations, the effect of damping is considered negligible for cranes.
52
K.A. Alghanim et al. / Journal of Sound and Vibration 345 (2015) 47–57
It is important to emphasize that the effect of damping in this work is included for the sake of completeness, since this technique can be used for any second-order system.
4.1. Step size and acceleration time Two sets of numerical simulation are used to demonstrate the performance of the proposed technique for various time step sizes and acceleration times. In the first set of simulations, the duration of the acceleration phase, τ, is set to 1 s, the length of the hoisting cable is set to 0.6 m and the final velocity is set to 0.3 m/s. Three different discretization time step sizes, Δt, are tested: 0.3333 s, 0.1 s, and 0.01 s, Fig. 2. These simulations reveal a significant advantage of the proposed technique, which is that the system maintains ideal response regardless of the discretization step size used for generating the acceleration profiles. This advantage is extremely significant since the time steps of these profiles can now be selected according to the sampling speed of hardware used and not the other way around. In the second set of simulations, the time step of the acceleration profiles is set to Δt ¼ 0:05 s. Three simulations for acceleration times of τ ¼ 0.8 s, τ ¼0.95 s, and τ ¼1.05 s are presented, Fig. 3. The length of the hoisting cable is set to 0.4 m. This set of simulations reveals another important feature of the proposed technique, which is that the time length of these profiles is independent of the natural period of the model. However, it is important to choose acceleration times that are multiples of time step size. The above simulations are further verified through experiments on a scaled model of a 3D overhead crane in the Advanced Vibration Lab at Kuwait University, Fig. 4. The jib of the crane is controlled by a DC motor. Three quadrature incremental-encoders are used to measure the jib motion, the length of the hoisting cable, and the oscillation angle of the hoisting cable. The resolution of each encoder is 1024 pulses per revolution. An interface between a PC and the crane setup
0.07
1
Δ t = 1/3 Δ t =0.1 Δ t = 0.01
0.05
Payload Angle [rad]
2
Jib Acceleration [m/s ]
0.8 0.6 0.4 0.2 0
0.04 0.03 0.02 0.01
−0.2 −0.4
Δ t = 1/3 Δ t =0.1 Δ t = 0.01
0.06
0
0
0.2
0.4
0.6
0.8
1
−0.01
0
0.2
0.4
0.6
0.8
1
1.2
Time [s]
Time [s]
Fig. 2. Acceleration profiles and corresponding system responses for three different time step sizes of Δt ¼ 0:3333 s, Δt ¼ 0:1 s, and Δt ¼ 0:01 s (l ¼0.6 m and τ¼ 1 s): (a) command profiles and (b) angle of the hoisting cable.
1.2
0.8 0.6 0.4 0.2
τ = 0.8 τ = 0.95 τ =1.05
0.07 0.06
Payload Angle [rad]
1 2 Jib Acceleration [m/s ]
0.08
τ = 0.8 τ = 0.95 τ =1.05
0.05 0.04 0.03 0.02 0.01
0 −0.2
0 0
0.2
0.4
0.6
Time [s]
0.8
1
−0.01
0
0.2
0.4
0.6
0.8
1
Time [s]
Fig. 3. Acceleration profiles and corresponding system responses for three different acceleration times of τ ¼0.8 s, τ ¼ 0.95 s, and τ¼ 1.05 s (l ¼0.4 m and Δt ¼ 0:05 s): (a) command profiles and (b) angle of the hoisting cable.
K.A. Alghanim et al. / Journal of Sound and Vibration 345 (2015) 47–57
53
Fig. 4. The experimental crane setup.
is achieved using RT-DAC/PCI multipurpose digital I/O board. The interface software uses MATLAB's real-time Simulink environment. The sampling rate of the control hardware is 100 Hz. The maximum jib speed and acceleration are 0.3 m/s and 0.9 m/s2, respectively. The crane has a 0.6 m usable track. A light weight cable is used with a mass of 0.092 kg attached to its end representing the crane payload. The maximum hoisting speed is approximately 0.134 m/s. The first set of simulations, Fig. 2, are verified on the setup using the same operating conditions, Fig. 5. The acceleration profile is repeated to decelerate the jib to a stop, Fig. 5(a). Results show agreement between the simulations and the experiments except for small deviations due to modeling uncertainties and unmodeled nonlinearities such as backlash and jib friction. Several other tests are conducted using arbitrary crane and profile parameters, Table 1. Parameters are chosen arbitrarily to verify the two major advantages revealed in simulations. First, the independence of the acceleration time from the natural period of oscillation demonstrated by using arbitrary hoisting cable lengths. Second, performance robustness to different time step sizes used for profile generation. Results of three of those tests are shown in Fig. 6. The figure shows the acceleration profiles generated by the proposed technique. Hoisting angle measurements, Fig. 6(b) indicates that the proposed technique is capable of minimizing residual vibrations at the end of the maneuver with very small discrepancies attributed to many unmodeled dynamics including friction and backlash in the cranes actuators. For completeness, a parametric sensitivity analysis is conducted. An acceleration profile is generated assuming a hoisting cable length of 0.60 m. This profile is then applied to models with different cable lengths. Residual oscillation angles are plotted against the actual cable length, Fig. 7. Examining the results shows that the performance of the proposed technique is robust to small change in the estimated cable length, and therefore, relatively small residual oscillations occur. In fact, a 10 percent measurement error in the cable length yields a maximum of 0.012 rad residual oscillation angle. As expected, small discrepancies between numerical and the experimental results exist.
4.2. Hardware sampling frequency Here, we will try to answer an obvious question, What is the effect of hardware sampling frequency on the performance of the shaped command? As indicated in the Introduction, operations of cranes using digital control lead to discrete commands. The time step size of these commands is highly dependent on the system hardware and its sampling frequency capabilities [5]. The time step size can greatly compromise the system performance by distorting the shape of the input profile [1]. A test model is chosen with a hoisting cable length of l¼0.60 m. An acceleration profile is generated using a discretization time step of Δt ¼ 0:1 s. To analyze the effect of different sampling frequencies of control hardware, several simulations are performed by sampling the already discrete command profile at different sampling rates, δt, representing the sampling rates of the control hardware used. The profile sampling rate is varied between 0.01 s and 0.3 s. Results of these simulations are presented in Fig. 8. Fig. 8 shows a detrimental impact of the actual sampling rate on the performance of shaped commands. It is clear that the command profile suffers serious degradation in performance in terms of system response. Simulations reveal the fact that it is imperative that a shaped command must have the same time step size as the control hardware. Faster hardware can be used as long as the profile discretization step remains an integer multiple of the sampling time of the control hardware used. Simulations in Fig. 8 also show significant impact on the jib speed reached at the end of the acceleration time. In most of the simulations, the final speed is well above the design speed of the jib, which limits the ability of the crane actuators to track profiles of the shaped commands.
54
K.A. Alghanim et al. / Journal of Sound and Vibration 345 (2015) 47–57
1
0.08 Case 1 Case 2 Case 3
0.8
Payload Angle [rad]
Jib Acceleration [m/s2]
0.6
Case 1 Case 2 Case 3
0.06
0.4 0.2 0 −0.2 −0.4
0.04 0.02 0 −0.02 −0.04
−0.6 −0.06
−0.8 −1
0
1
2
3
4
−0.08
5
0
1
2
3
Time [s]
4
5
Time [s]
Fig. 5. Acceleration profiles and corresponding experimental system responses for three different time step sizes of Δt ¼ 0:3333 s, Δt ¼ 0:1 s, and Δt ¼ 0:01 s (l ¼0.6 m and τ ¼1 s): (a) command profiles and (b) angle of the hoisting cable. Table 1 Crane and profile parameters. Case no.
Cable length (m)
Acceleration time (s)
Step size (s)
1 2 3
0.4 0.35 0.2
0.67 0.9 1.2
0.01 0.1 0.3
1
0.1 Case 1 Case 2 Case 3
0.8
0.06
Payload Angle [rad]
Jib Acceleration [m/s2]
0.6 0.4 0.2 0 −0.2 −0.4
0.04 0.02 0 −0.02 −0.04
−0.6
−0.06
−0.8
−0.08
−1
0
1
2
3
Time [s]
4
Case 1 Case 2 Case 3
0.08
5
−0.1
0
1
2
3
4
5
Time [s]
Fig. 6. Experimental results using profile and crane parameters in Table 1: (a) command profiles and (b) angle of the hoisting cable.
5. Simultaneous travel and hoist maneuvers Simultaneous travel and hoist maneuvers add a great complexity to the design of shaped commands. Hoisting converts the equation of motion into a complex type of ordinary differential equation with time varying coefficients, Eq. (1). This type of equations, usually, has no exact solution. For this reason, researchers tend to avoid command shaping for maneuvers involving hoisting. Optimal control can be used to generate commands that result in zero residual oscillations [20,21]. However, when hoisting is involved, the difficulty of generating optimal command becomes more challenging because the system becomes nonlinear [16]. To demonstrate the ability of the proposed technique of handling simultaneous travel and hoist maneuvers, two test cases are numerically simulated and tested experimentally. The hoisting speed is kept within the maximum hoisting speed of the experimental setup which is 0.134 m/s. Maneuver and profile parameters of these two tests are in Table 2. Simulations indicate the ability of the proposed technique to produce acceleration commands capable of handling simultaneous travel and hoisting maneuvers, Figs. 9 and 10. A close match between simulations and experimental tests is obtained.
K.A. Alghanim et al. / Journal of Sound and Vibration 345 (2015) 47–57
55
0.06 Simulated Experimental
Maximum Angle [rad]
0.05
0.04
0.03
0.02
0.01
0 0.45
0.5
0.55
0.6
0.65
0.7
0.75
L0 [m] Fig. 7. Simulated and experimental sensitivity analysis when l ¼0.60 and τ ¼ 1.00 and Δt ¼ 0:01.
0.55
0.05
0.5
Maximum Velocity [m/s]
Residual Angle [rad]
0.04
0.03
0.02
0.45
0.4
0.35
0.01
0
0.3
0
0.05
0.1
0.15
0.2
0.25
0.3
0.25 0
0.05
0.1
Step Size δ t [s]
0.15
0.2
0.25
0.3
Step Size δ t [s]
Fig. 8. The effect of discretization on the payload angle, velocity and displacement when Δt ¼ 0:1 s as a function of the discretization step time δt: (a) residual oscillation and (b) terminal jib velocity. Table 2 Simultaneous travel and hoist test parameters. Test no.
1 2
Cable length (m) Initial
Intermediate
Final
0.4 0.35
0.3 0.4
0.4 0.3
Step time (s)
Acceleration time (s)
0.05 0.3
0.8 0.9
6. Conclusions An optimization strategy for generating shaped commands for overhead cranes is presented. The optimization technique used produces discrete-time shaped commands. The effectiveness and the performance robustness of the shaped commands were demonstrated using both numerical simulations and experiments on a scaled model of an overhead crane. Simulations and experiments demonstrate the ability of the command profiles to eliminate residual payload oscillations in the presence of simultaneous travel and hoist maneuvers that are fundamental for time optimal operations. Using this strategy, a profile designer has the freedom to change several important parameters of the generated profile. In addition to the usual parameters in all command shaping techniques, such as maximum velocity and acceleration, the presented technique adds two more selectable parameters: the size of the time steps and the time length of the command profile, regardless of the system natural period of oscillation. The size of the profile's time steps is freely selectable without any performance compromise. This feature is very crucial as it was shown that a good match between profile steps and control hardware sampling rate is essential for achieving desired outcome. The selectable acceleration time gives the strategy its independence from the natural period of oscillation of the system.
56
K.A. Alghanim et al. / Journal of Sound and Vibration 345 (2015) 47–57
0.8
0.08
0.6
Payload Angle [rad]
Amplitude
0.4 0.2 0 −0.2 Acceleration [m/s2] Displacement [m] Velocity [m/s] Length[m]
−0.4 −0.6 −0.8
Simulated Experimental
0.06
0
0.5
1
1.5
0.04 0.02 0 −0.02 −0.04 −0.06
2
2.5
3
−0.08
3.5
0
1
2
Time [s]
3
4
5
Time [s]
Fig. 9. Simultaneous travel and hoisting maneuver test case 1: (a) jib command and hoist profiles and (b) angle of the hoisting cable.
0.8
0.08
0.6
Payload Angle [rad]
Amplitude
0.4 0.2 0 −0.2 Acceleration [m/s2] Displacement [m] Velocity [m/s] Length[m]
−0.4 −0.6 −0.8
Simulated Experimental
0.06
0
0.5
1
1.5
0.04 0.02 0 −0.02 −0.04 −0.06
2
Time [s]
2.5
3
3.5
−0.08
0
1
2
3
4
5
Time [s]
Fig. 10. Simultaneous travel and hoisting maneuver test case 2: (a) jib command and hoist profiles and (b) angle of the hoisting cable.
References [1] K. Alhazza, Z. Masoud, A novel wave-form command shaper for overhead cranes, Journal of Engineering Research 1 (2013) 181–209. [2] E.M. Abdel-Rahman, A.H. Nayfeh, Z.N. Masoud, Dynamics and control of cranes: a review, Journal of Vibration and Control 9 (2003) 863–908. [3] W. Singhose, Command shaping for flexible systems: a review of the first 50 years, International Journal of Precision Engineering and Manufacturing 10 (2009) 153–168. [4] Z.N. Masoud, A.H. Nayfeh, D.T. Mook, Cargo pendulation reduction of ship-mounted cranes, Nonlinear Dynamics 35 (2004) 299–311. [5] Z.N. Masoud, A.H. Nayfeh, N.A. Nayfeh, Sway reduction on quay-side container cranes using delayed feedback controller: simulations and experiments, Journal of Vibration and Control 11 (2005) 1103–1122. [6] S.-K. Cho, H.-H. Leeb, A fuzzy-logic antiswing controller for three-dimensional overhead cranes, ISA Transactions 41 (2002) 235–243. [7] G.P. Starr, Swing-free transport of suspended objects with a path-controlled robot manipulator, Journal of Dynamic Systems, Measurement, and Control 107 (1995) 97–100. [8] D.R. Strip, Swing-free transport of suspended objects: a general treatment, IEEE Transactions on Robotics and Automation 5 (1989) 234–236. [9] M.A. Ahmad, R.M.T.R. Ismail, M.S. Ramli, N.M. Abd Ghani, N. Hambali, Investigations of feed-forward techniques for anti-sway control of 3-D gantry crane system, Proceedings of the IEEE Symposium on Industrial Electronics Applications, ISIEA 2009, Vol. 1, Piscataway, New Jersey, USA, 2009, pp. 265– 270. [10] N. Singer, W. Singhose, W. Seering, Comparison of filtering methods for reducing residual vibration, European Journal of Control 5 (1999) 208–218. [11] T. Singh, Jerk limited input shapers, Journal of Dynamic Systems, Measurement, and Control 126 (2004) 215–219. [12] W. Singhose, R. Eloundou, J. Lawrence, Command generation for flexible systems by input shaping and command smoothing, Journal of Guidance, Control, and Dynamics 33 (2010) 1697–1707. [13] K. Erkorkmaz, Y. Altintas, High speed CNC system design. Part I: jerk limited trajectory generation and quintic spline interpolation, International Journal of Machine Tools and Manufacture 41 (2001) 1323–1345. [14] K. Alhazza, A. Al-Shehaima, Z. Masoud, A continuous modulated wave-form command shaping for damped overhead cranes, Proceedings of the ASME 2011 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, Washington, DC, USA, 2011, pp. 361–366. [15] K. Alhazza, Experimental validation on a continuous modulated wave-form command shaping applied on damped systems, Topics in Dynamics of Civil Structures, Vol. 4, Springer, New York, 2013, pp. 445–451. [16] W. Singhose, L. Porter, M. Kenison, E. Kriikku, Effects of hoisting on the input shaping control of gantry cranes, Control Engineering Practice 8 (2000) 1159–1165.
K.A. Alghanim et al. / Journal of Sound and Vibration 345 (2015) 47–57
57
[17] Z.N. Masoud, M.F. Daqaq, A graphical approach to input-shaping control design for container cranes with hoist, IEEE Transactions on Control Systems Technology 14 (2006) 1070–1077. [18] K.A. Alhazza, Experimental and numerical validation on a continuous modulated wave-form command shaping control considering the effect of hoisting, Proceedings of the ASME 2013 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, Portland, Oregon, USA, 2013, pp. V07AT10A057. [19] K.A. Alhazza, A.M. Hassan, K.A. Alghanim, Z.N. Masoud, An iterative learning control technique for point-to-point maneuvers applied on an overhead crane, Shock and Vibration 2014 (2014) 11. (Article ID 261509), /http://dx.doi.org/10.1155/2014/261509S. [20] Y. Sakawa, Y. Shindo, Optimal control of container cranes, Automatica 18 (1982) 257–266. [21] J.W. Auernig, H. Troger, Time optimal control of overhead cranes with hoisting of the load, Automatica 23 (1987) 437–447.