Automatic parameter selection ZVD shaping algorithm for crane vibration suppression based on particle swarm optimisation

According to the difficulty of parameters selection in the traditional ZVD shaping algorithm, we proposed an APS-ZVD shaping algorithm for crane vibration suppression based on particle swarm optimisation. The APS-ZVD shaping algorithm comprises three parts: signal acquisition, evaluation and particle swarm optimisation, as shown in Figure 1. This algorithm comprises the following steps:

A square-wave speed command V(t) is entered to the transfer function of system G(t) (shown in Equation 5), the output pendulum angles A(t) are recorded;

We input a square-wave speed signal to the system to let the trolley move for a short time, and collect the pendulum angle during the movement. The speed and acceleration of the trolley's movement should be as large as possible so as to do not damage the system. Because a short-time motion with a large acceleration contains enormous low-frequency vibration energy, under this circumstance the low-order resonance frequency of the system can be fully stimulated. In this way, the pendulum angle can be used to translate the characteristics more accurately.

As shown in Equations (1–6), there are five parameters to be optimised, A₁, A₂, A₃, t₂ and t₃. We assume that K=e−ξπ1−ξ2 K = {e^{ - {{\xi \pi } \over {\sqrt {1 - {\xi ^2}} }}}} and T=πω1−ξ2 T = {\pi \over {\omega \sqrt {1 - {\xi ^2}} }} and then the optimal solution of ZVD input shaper can be obtained by optimising parameters T and K. Determination of parameter ranges significantly influences the speed and accuracy of parameter selection. Since the crane model is a second-order system (no zero poles cancellation), we know that the vibration of the second-order system is determined by pole distribution. The key parameter which determines the distribution of poles is the damping ratio ξ. In the case of ξ = 0, the second-order system is in an underdamped status and the pendulum keeps oscillating. While, when, ξ = 1 the second-order system is in a critical damping status, the pendulum does not swing. The swing of the pendulum is an attenuated oscillation, so 0 < <ξ 1. Therefore, the range of parameter K is [0, 1]. Since T=2πω {\rm{T}} = {{2\pi } \over \omega } , ω=g/l \omega = \sqrt {{\rm{g}}/{\rm{l}}} , the swing frequency can be estimated roughly according to the pendulum length. The range of T can be assigned to [0,2πlg] \left[ {0,{{2\pi \sqrt {\rm{l}} } \over {\sqrt {\rm{g}} }}} \right] .

The evaluation function should be able to evaluate the impact of adjustment time, amplitude and overshooting on the crane system. Considering the above factors, the design evaluation function can be written as Equation 8: (8) {e(t)=si(t)−se(t)Mp=max(e(t))J(t)=∫0⧜η1|e(t)|dt+η2|Mp| \left\{ {\matrix{ {e\left( {\rm{t}} \right) = {{\rm{s}}_{\rm{i}}}\left( t \right) - {{\rm{s}}_{\rm{e}}}\left( t \right)} \cr {{{\rm{M}}_{\rm{p}}} = max\left( {e\left( {\rm{t}} \right)} \right)} \cr {J\left( t \right) = {\int _0^\infty } {\eta _1}\left| {{\rm{e}}\left( {\rm{t}} \right)} \right|dt + {\eta _2}\left| {{{\rm{M}}_{\rm{p}}}} \right|} \cr } } \right. where, s_i (t) is the value of the input signal at time t, s_e (t) is the expected value of the signal at t, e(t) is the error between the input signal and the expected signal at time t, M_p is the maximum error between the input signal and the expected signal, η₁ and η₂ are the weighted values, and J (t) is the final value of the evaluation function. The integral of e(t) is used to describe the time and average amplitude of adjustment. is M_p used to reflect the maximum magnitude of the amplitude. The penalty factors η₁ and η₂ are used to reflect the relation between e(t) and M_p. The overshoot can be reduced by increasing η₂, but the response time will be extended.

The particle swarm optimisation algorithm is described as: (9) vidk+1=ωvidk+c1r1(pidk−xidk)+c2r2(pgdk−xidk) {\rm{v}}_{{\rm{id}}}^{{\rm{k}} + 1} = \omega {\rm{v}}_{{\rm{id}}}^{\rm{k}} + {{\rm{c}}_1}{{\rm{r}}_1}\left( {{\rm{p}}_{{\rm{id}}}^{\rm{k}} - {\rm{x}}_{{\rm{id}}}^{\rm{k}}} \right) + {{\rm{c}}_2}{{\rm{r}}_2}\left( {{\rm{p}}_{{\rm{gd}}}^{\rm{k}} - {\rm{x}}_{{\rm{id}}}^{\rm{k}}} \right) (10) xidk+1=xidk+vidk+1 {\rm{x}}_{{\rm{id}}}^{{\rm{k}} + 1} = {\rm{x}}_{{\rm{id}}}^{\rm{k}} + {\rm{v}}_{{\rm{id}}}^{{\rm{k}} + 1} where vidk+1 v_{id}^{k + 1} is the velocity of each particle in the next iteration, vidk v_{id}^k is its current velocity, pidk {\rm{p}}_{{\rm{id}}}^{\rm{k}} is the personal best particle, and pgdk {\rm{p}}_{{\rm{gd}}}^{\rm{k}} is the global best particle. Furthermore, xidk+1 x_{id}^{k + 1} represents the new location of the particle, ω represents the inertial weight, which is the effect of the current velocity on the next iteration, and c₁ and c₂ are study factors that represent the information exchange between each particle in the whole population; r₁ and r₂ are acceleration coefficients, which are random numbers that are uniformly distributed in the range of [0,1] and are used to increase the randomness of particle movement. Equation 17 is the fitness function of the PSO. By finding the minimum value of Equation 17, the optimal values of K and T can be reached, and then by putting them back into Equations 7–12, the optimal Ai and ti can be obtained. Finally, the ZVD shaper C(t) can be structured by the optimised Ai and ti, and then be put behind the speed command to suppress vibration by shaping the speed command. (11) fit(t)=A(t)*C(t)−se(t) fit\left( t \right) = A\left( t \right){\rm{*}}C\left( t \right) - {s_e}\left( t \right)

In order to simulate the real crane structure, an experimental platform was built, as shown in Figure 8. The platform comprises a servo motor, a pendulum (14.5 mm) and a payload (0.05 kg). The encoder records the swing angle of the pendulum.

Figure 3 shows the swing angles of pendulum recorded at the trolley's speed of 480 mm/s. It can be seen that the swing angle of the pendulum was very large (35 o) when the trolley stops at the beginning (at 4.5 s), then it decreased gradually until 0 o at 12.4 s.

The recorded swing angles of pendulum were input to APS-ZVD (as described in Figure 2). The key parameters of APS-ZVD algorithm K and T are automatically optimised by the PSO algorithm. The initial parameters of PSO are as shown in Table 1.

Fig. 2

Fig. 3

Fig. 4

Figure 5 shows the fitness curve of PSO. After 10 iterations, the final fitness value stabilised at 26.68. Compared with 44.6 in the first iteration, the fitness value decreased by 40.17%.

Fig. 5

The optimisation curves of parameter K and T are shown in Figure 7. After 18 iterations, K and T are gradually tending to stable, and finally reached to K = 0.8952, T = 0.3626.

Fig. 6

Fig. 7

Putting K and T back to Equations 7–12, the ZVD parameters are achieved as A1 = 0.3708, A2 = 0.4985, A3 = 0.2231, t₁ = 0, t₂ = 0.3626, t₃ = 0.7251, and then the ZVD shaper equation can be written as: (12) C(t)=0.3708δ(t)+0.4985×δ(t−0.3626)+0.2231×δ(t−0.7251) {\rm{C}}\left( {\rm{t}} \right) = 0.3708\delta \left( {\rm{t}} \right) + 0.4985 \times \delta \left( {{\rm{t}} - 0.3626} \right) + 0.2231 \times \delta \left( {{\rm{t}} - 0.7251} \right)

We put the achieved C(t) in front of the model system to verify the effect of APS-ZVD, the swing angles of pendulum with APS-ZVD and without APS-ZVD as shown in Figure 8. With APS-ZVD, the time to stop swinging is shortened from 7.9 s to 1.26 s, and the maximum swing angle is decreased from 35° to 6.6°.

Fig. 8

We next quantified the swing energy of the pendulum with and without the APS-ZVD using Equation 13: (13) P(t)=∫0⧜|e(t)|dt {\rm{P}}\left( {\rm{t}} \right) = {\int _0^\infty } \left| {{\rm{e}}\left( {\rm{t}} \right)} \right|{\rm{dt}} where e(t) is the difference between the actual swing angle of pendulum and expected swing angle (zero). The results show that the swing energy of pendulum has been decreased from 262.06 to 26.60 with the APS-ZVD. The amplitude of attenuation is 89.85%.

We did the Fourier transform for the swing of the pendulum and found the resonance frequency to be 1.4 Hz (shown in Figure 9, blue). We also detected the sensitivity curve of APS-ZVD (shown in Figure 9, green), and found the optimal frequency of the suppression is 1.38 Hz, which is matched very well with the resonance frequency of the pendulum. The Fourier transform for the swing of the pendulum with APS-ZVD is also plotted in Figure 7, shown in the red, where the swing amplitude has been sharply reduced by the order of three.

In the actual hoisting, usually the position of trolley and the length of cable are changed frequently, which affects the vibration suppression effect. Thus, we changed the pendulum length to detect the suppression effect of optimal parameter found by APS-ZVD, and the results are as shown in Table 2. In Table 2, Pendulum 1 is the original one which is used to generate the optimal parameters. The length of Pendulum 2 is extended by 6.9% compared to 1, and the length of Pendulum 3 is shortened by 13.8% compared to 1. The parameters found by the APS-ZVD algorithm based on the length of Pendulum 1 were used in all three experiments (A₁ = 0.3708, A₂ = 0.4985, A₃ = 0.2231, t₁ = 0, t₂ = 0.3626, t₃ = 0.7251), and the vibration suppression results are obtained.

In the case of Pendulum 3, the pendulum length changed by 13.8% compared to 1; the damping percentage of APS-ZVD shaper can still achieve 62%, and the peak swing angle and the adjusting time of residual vibration did not change significantly. The comparison shows that the robustness of the selected parameters can be ensured by our designed APS-ZVD algorithm.