For the randomness of crane working load leading to the decrease of load spectrum prediction accuracy with time, an adaptive TSSA-HKRVM model for crane load spectrum regression prediction is proposed. The heterogeneous kernel relevance vector machine model (HKRVM) with comprehensive expression ability is established using the complementary advantages of various kernel functions. The combination strategy consisting of refraction reverse learning, golden sine, and Cauchy mutation + logistic chaotic perturbation is introduced to form a multi-strategy improved sparrow algorithm (TSSA), thus optimizing the relevant parameters of HKRVM. The adaptive updating mechanism of the heterogeneous kernel RVM model under the multi-strategy improved sparrow algorithm (TSSA-HKMRVM) is defined by the sliding window design theory. Based on the sample data of the measured load spectrum, the trained adaptive TSSA-HKRVM model is employed to complete the prediction of the crane equivalent load spectrum. Applying this method to QD20/10 t × 43 m × 12 m general bridge crane, the results show that: compared with other prediction models, although the complexity of the adaptive TSSA-HKRVM model is relatively high, the prediction accuracy of the load spectrum under long periods has been effectively improved, and the completeness of the load information during the whole life cycle is relatively higher, with better applicability.

The hoisting machinery is widely available for equipment manufacturing, transportation, aerospace, nuclear power construction, and other pillar industries of the national economy. It plays a vital role in economic development. The safety of its product service process is crucial to ensure the smooth implementation of the project [

In the field of mechanical equipment, the load spectrum description methods generally include three categories: the method based on measured load information, based on simulation, and based on intelligent optimization algorithm and machine learning prediction technology [

The description method based on the measured load information is a simple, efficient, and intuitionistic load spectrum acquisition method. This method directly uses sensors to measure generalized load information, such as load changes and stress changes, which is widely available for the life assessment of various structures. In [

The description method based on simulation is the theory or finite element simulation to simulate the actual working state of equipment, which is easy to operate. The load spectrums of diversity speed levels, line conditions, fault conditions, and tread conditions are analyzed in [

The description method based on intelligent optimization algorithm and machine learning prediction technology is a technology integrating mathematical statistics, machine learning, intelligent optimization, etc. And it is applicable to the implementation scheme with few measured load samples, obviously data nonlinearity, and good popularization. The prediction method of the crane load spectrum is constructed in [

Thus, it can be seen that, with the rapid development of computer technology and the continuous expansion of the field of artificial intelligence, machine learning technology combined with intelligent optimization algorithms has been widely used in various fields of target prediction. The former excavates the potential laws between data through data statistics and learning and gives a mapping mechanism between input and output. The latter solves the nonconvex optimization and combinatorial optimization problems in machine learning. The combination of the two has higher accuracy and robustness for nonlinear data processing and has gradually become a new hotspot in load spectrum and other related prediction research fields [

1) For the kernel function, subjective selection based on experience tends to lead to errors in kernel function selection, which is too random and limited, and the single-core and multi-core selection mechanisms are fuzzy. 2) For kernel parameter optimization, the intelligent algorithm has few parameters and is simple to operate, which can achieve rapid optimization of RVM kernel function parameters. However, diversity optimization algorithms have different applications. Inappropriate optimization algorithms will lead to premature convergence in the later iteration, resulting in reduced reliability of RVM parameters and affecting the optimization accuracy. 3) For load spectrum prediction, the load spectrum describing the whole life cycle of the equipment is predicted by relying on the measured load information of the equipment in a fixed period, which will lead to the difference between the prediction results and the actual load conditions being larger with time. And the accuracy of long-term prediction decreases.

Given the above problems, if a prediction method of crane equivalent load spectrum with high prediction accuracy and a good fit with the actual working process can be created, it is of great significance to judge whether the equipment can be used safely from the perspective of fatigue life. Therefore, this paper proposes the adaptive TSSA-HKRVM model for load spectrum regression prediction of the crane. Firstly, four frequently-used kernels are adopted to analyze the performance of single or mixed kernels. And the complementary advantages of various kernels are used to build HKRVM with comprehensive expression ability. Secondly, the combination strategy of ‘refraction reverse learning, golden sine, and Cauchy variation + logistic chaotic perturbation’ is introduced to form the multi-strategy improved sparrow algorithm (TSSA), thus obtaining relevant parameters of the HKRVM prediction model through optimization. Then, the TSSA-HKMRVM model is defined. Thirdly, an adaptive updating mechanism of the prediction model is designed by the sliding window design theory from the standpoint of ‘prediction + monitoring’. Finally, using the QD20/10 t × 43 m × 12 m general bridge crane as an example, the applicability and validity of the proposed method are confirmed by comparison.

It is difficult to carry out large-scale load spectrum tests of equal-scale cases in the whole life cycle of general bridge cranes, which will reduce the accuracy of prediction results to predict large samples only through fixed measured small samples over time. Considering this, the adaptive TSSA-HKRVM model for the load spectrum regression prediction of the crane is constructed from the aspects of the construction of HKRVM (see

As given in

In the description of the crane load spectrum, the machine learning algorithm is an emerging technology to solve regression prediction problems at present, mainly including BP neural network, support vector machine (SVM), and correlation vector machine (RVM) [

RVM model defines that under the condition of the input vector

where

RVM model has

To solve the weight vector, the prior probability distribution with the mean value of 0 and variance of

where

In RVM, as a bridge from low-dimensional to high-dimensional space, the kernel function is the pivot of transforming the nonlinear relationship into the linear relationship, which is the critical link affecting its regression prediction performance. As described in [

where,

Based on the regression prediction theory of RVM (see

The type of kernel function and the value of the kernel parameter have an influence at different levels on the prediction performance of HKRVM. The kernel parameters determine the ability of kernel function mapping, which indirectly affects the prediction performance to a greater extent. The traditional generalized root means square method, subjective assignment method, cross-validation method, and gradient descent method all have problems, such as higher calculation cost, greater training error, and stronger human interference in the optimization of kernel parameters and weight parameters. Therefore, the combination strategy is introduced to improve the sparrow optimization algorithm (SSA [

SSA is a novel intelligent optimization algorithm that mimics sparrow hunting and anti-hunting operations. Discoverers, followers, and scouters are the population members. The basic principle is to abstract the foraging and anti-predation processes of sparrows, such as discoverers searching for food, followers joining foraging, and scouts deciding whether to change the safe area [

In SSA, the position of the sparrow population corresponds to the effective solution of the search space. The sparrow population position

where,

When the energy reserve of the discoverer is higher, the corresponding fitness value of the individual is larger, and the updated equation is:

where,

The remaining individuals of the sparrow population are followers, and the updated equation is:

where,

For

where

For

Although SSA has a fast convergence speed and strong search ability, as a swarm intelligence optimization algorithm, its global search ability is poor, and it is easy to fall into local optimization. The following improvement strategies are introduced to overcome these disadvantages.

1) Refraction reverse learning strategy

For the refraction reverse learning strategy [

Assuming that the current feasible solution is

where,

2) Golden sine strategy

In

where,

3) Cauchy variation + logistic chaotic perturbation strategy

In the algorithm iteration process of SSA, the early-stage deviates from the target value, and various local optimizations in the later stage are nearly invalid. The simple and high-sensitive logistic chaotic map [

where,

The combination strategy of ‘refraction reverse learning, golden sine, and Cauchy variation + logistic chaotic perturbation’ is introduced to improve SSA, thus forming TSSA. With the kernel parameters and weight parameters as the design variables and the root mean square error of HKRVM prediction results as the objective function, the optimization of parameters is completed through TSSA within the range of corresponding parameter settings. On this basis, the TSSA-HKMRVM model is constructed to forecast the crane load spectrum.

The existing crane load prediction methods based on machine learning all predict large samples with fixed small samples. The optimization of relevant parameters of the machine learning model is realized by an intelligent optimization algorithm, which reflects the adaptability of the model parameters. But the increase in prediction steps for the long-term prediction process makes it difficult to ensure prediction accuracy. The reason is the diversity, randomness, and uncertainty of the load of the general bridge crane in the working cycle. Under various working cycle processes (as shown in

As illustrated in

In ① stage, the measured load spectrum of small samples is as the sample data set _{1}+_{2}, in which _{1} groups are randomly selected for TSSA-HKRVM model training, and the remaining _{2} groups are applied for model testing.

In ② stage, the TSSA-HKRVM model constructed in ① stage is employed to predict the

In ③ stage, since

The definition of crane load spectrum currently refers to the relationship between the load on the metal structure and the cumulative frequency of the load under the corresponding working cycle in the actual service process [

Given this, through structural fatigue life analysis from the aspect of safe service of the crane bridge structure, in accordance with ‘GB/T30024-2020 cranes—proof of competence of steel structures’ [_{1}’, ‘whether the loaded trolley passes the mid-span position _{2}’ and the corresponding ‘number of working cycles _{1}, _{2}, and

Before forecasting the crane equivalent load spectrum under the periodic inspection cycle, the load information collection system has been built as shown in

As illustrated in

Based on the measured load spectrum of the small sample

1) Sample data without rigid quantification processing.

Normalize the small sample measured load spectrum _{1} as one part, and the remaining group _{2} as the other part, so as to complete the data preparation.

2) TSSA-HKRVM construction, including HKRVM construction and parameter optimization.

Step 1. Based on the linear kernel function

Step 2. Initialization of population position and parameters of TSSA, such as population number _{f} and _{g}, and safety value

Step 3. According to the refraction reverse learning strategy (see

Step 4. Determine the number of discoverers, followers, and scouts in the population. According to the golden sine strategy (see

Step 5. Calculate the fitness of the individual population, determine the location of the best individual, and compare it with the fitness of the first five times. If the fitness does not change at this time, it is judged that it has fallen into the local optimum, and Step 6 is executed. If the fitness is optimal at this time, the iteration termination condition shall be judged. For the error accuracy or the maximum number of iterations meeting the requirements, output the specific values of the core parameters and weight parameters. For the iteration termination conditions being not met, execute Step 3.

Step 6. In case of falling into local optimum, Step 3 shall be executed after generating a new individual position with

3) Equivalent load spectrum prediction based on adaptive TSSA-HKRVM.

Step 1. The sample distribution information with ‘lifting load _{1}’ and ‘whether the trolley passes the mid-span position _{2}’ as the input variables are obtained by analyzing the sample data of the characteristic parameters of the load spectrum. According to the distribution characteristics, the input variables for the load spectrum to be predicted are generated through LHS sampling. And the trained TSSA-HKRVM model is used to predict the equivalent load spectrum.

Step 2. The accuracy of prediction results is checked by the root mean square error. For

Step 3. For

Taking the QD20/10 t × 43 m × 12 m general bridge crane as the research object, the equivalent load spectrum prediction of the crane under the regular inspection cycle (one year) is completed by the above method.

Using the crane load information acquisition system (see _{1}’ and ‘whether the trolley passes through the mid-span position or not _{2}’ is taken as the input variables, that is, the characteristic parameters of the load spectrum. And taking the number of working cycles as the feedback object, the small sample measured load spectrum is compiled, as shown in

No. | _{c} |
No. | _{c} |
No. | _{c} |
No. | _{c} |
No. | _{c} |
||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 0.3 | 0 | 1 | 7 | 2.7 | 1 | 3 | 13 | 4.1 | 1 | 10 | 19 | 6.0 | 1 | 16 | 25 | 7.7 | 1 | 25 |

2 | 1.3 | 1 | 1 | 8 | 3.1 | 1 | 6 | 14 | 4.5 | 1 | 10 | 20 | 6.4 | 1 | 15 | 26 | 7.8 | 0 | 26 |

3 | 1.4 | 1 | 1 | 9 | 3.2 | 0 | 4 | 15 | 5.1 | 0 | 11 | 21 | 6.5 | 1 | 15 | 27 | 8.0 | 1 | 27 |

4 | 1.9 | 1 | 3 | 10 | 3.4 | 1 | 7 | 16 | 5.4 | 1 | 13 | 22 | 6.9 | 0 | 17 | 28 | 8.1 | 0 | 28 |

5 | 2.2 | 1 | 4 | 11 | 3.7 | 1 | 5 | 17 | 5.7 | 1 | 12 | 23 | 7.1 | 1 | 20 | ||||

6 | 2.7 | 0 | 5 | 12 | 3.9 | 0 | 8 | 18 | 6.0 | 0 | 15 | 24 | 7.5 | 1 | 18 | 73 | 19.8 | 1 | 37 |

Notes: 1) _{c} is the corresponding number of working cycles. 2) 0 indicates that the trolley has not crossed the mid-span, and 1 indicates that the trolley has crossed the mid-span.

After normalizing the data in

It can be seen from

The distribution of each input variable is obtained by analyzing the data in _{2}’ meets the 0–1 distribution

It can be seen from

No. | _{c} |
No. | _{c} |
No. | _{c} |
No. | _{c} |
No. | _{c} |
||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 0.4 | 0 | 1 | 7 | 2.7 | 1 | 3 | 13 | 4.0 | 1 | 10 | 19 | 6.1 | 1 | 16 | 25 | 7.6 | 1 | 18 |

2 | 1.2 | 1 | 1 | 8 | 3.0 | 1 | 5 | 14 | 4.4 | 1 | 10 | 20 | 6.4 | 0 | 14 | 26 | 7.8 | 1 | 20 |

3 | 1.3 | 1 | 1 | 9 | 3.2 | 0 | 5 | 15 | 5.1 | 0 | 12 | 21 | 6.5 | 1 | 15 | 27 | 8.0 | 1 | 21 |

4 | 1.8 | 0 | 3 | 10 | 3.5 | 1 | 8 | 16 | 5.5 | 0 | 13 | 22 | 6.8 | 1 | 18 | 28 | 8.1 | 0 | 22 |

5 | 2.2 | 1 | 3 | 11 | 3.7 | 1 | 5 | 17 | 5.6 | 1 | 12 | 23 | 7.1 | 1 | 20 | ||||

6 | 2.5 | 0 | 4 | 12 | 3.9 | 0 | 9 | 18 | 6.0 | 0 | 15 | 24 | 7.4 | 1 | 19 | 73 | 19.8 | 1 | 37 |

Notes: 1) _{c} is the corresponding number of working cycles. 2) 0 indicates that the trolley has not crossed the mid-span, and 1 indicates that the trolley has crossed the mid-span.

The applicability of the method proposed in this paper is verified by discussing the influence of various influencing factors on the prediction results from four aspects, including the construction of kernel function, optimization of kernel parameters and weight parameters, adaptive control methods, and load spectrum prediction methods.

To demonstrate the reasonableness of the selection of kernel function of the TSSA-HKRVM model, a single kernel function, two heterogeneous kernel functions and three heterogeneous kernel functions are contrasted and studied. The uniformity of the comparison test is considered and the TSSA algorithm is employed for relevant parameter optimization of any kernel function. The data in

Kernel function | Training time/s | Testing Times/s | Maximum relative error _{max} |
Root mean square error _{mes} |
Goodness of fit ^{2} |
---|---|---|---|---|---|

66.089 | 0.00326 | 6.4375 | 1.6636 | 0.9327 | |

5.0679 | 0.000867 | 7.7555 | 1.2581 | 0.7600 | |

6.730 | 0.00062 | 3.9111 | 1.0545 | 0.8830 | |

14.108 | 0.000614 | 3.2052 | 0.8570 | 0.9550 | |

20.344 | 0.000658 | 0.4581 | 0.1638 | 0.9638 | |

23.777 | 0.001216 | 1.0367 | 0.3244 | 0.9270 | |

17.798 | 0.000665 | 0.0897 | 0.0784 | 0.9921 | |

13.433 | 0.000688 | 4.6671 | 1.2129 | 0.9158 | |

16.155 | 0.001568 | 0.1093 | 0.1217 | 0.9488 | |

13.1414 | 0.00319 | 4.7664 | 1.2676 | 0.8663 | |

18.006 | 0.000843 | 0.2809 | 0.1046 | 0.9880 | |

23.965 | 0.000696 | 0.5214 | 0.1782 | 0.9667 | |

18.411 | 0.000797 | 0.45048 | 0.1382 | 0.9896 | |

13.923 | 0.000795 | 0.4937 | 0.3624 | 0.9271 |

It can be seen from

1) For a single kernel function, the results for the goodness of fit of

2) For the binomial heterogeneous kernel functions, the model accuracies have been significantly improved. For the heterogeneous combination of

3) For the trinomial heterogeneous kernel functions, the increase in the number of kernel functions leads to an increase in the model complexity. The fitting performance after mixing is considerably enhanced when compared to the original single kernel function, but not when compared to the two heterogeneous kernel functions.

To sum up, the heterogeneous kernel function benefits from diverse kernel functions, higher learning capacity, and better promotion ability. Considering the complexity, fitting degree, and calculation efficiency of the model comprehensively, the heterogeneous kernel function composed of

The effectiveness of the TSSA algorithm in parameter optimization of the HKRVM prediction model is verified by comparing TSSA with SSA, ISSA1 (the refraction inversion improved SSA), ISSA2 (the refraction inversion + golden sine hybrid improved SSA), ISSA3 (the refraction direction + golden sine + Cauchy strategy hybrid improved SSA).

The initial parameter settings are consistent, and the prediction model is HKRVM. On this basis and in combination with sample data 2 (

Prediction model | SSA-HKRVM | ISSA1-HKRVM | ISSA2-HKRVM | ISSA3-HKRVM | TSSA-HKRVM |
---|---|---|---|---|---|

Maximum relative error | 0.29059 | 0.2712 | 0.6099 | 0.5304 | 0.0897 |

Root mean square error | 0.1055 | 0.1092 | 0.1750 | 0.1701 | 0.0784 |

Goodness of fit | 0.9730 | 0.9820 | 0.9854 | 0.9901 | 0.9920 |

Running time/s | 15.8048 | 16.1192 | 16.8429 | 17.4715 | 17.7981 |

As can be seen from

The feasibility of the adaptive mechanism proposed in this paper is validated in the prediction of the equivalent load spectrum of general bridge cranes by comparing it with the traditional adaptive predictive control methods [^{5}I. The load prediction control process is described in

In the load prediction control curve in

In

The reason for the above phenomenon is that:

1) The predictive control method based on neural network is composed of neural unit adaptive PID controller and Smith predictor based on neural network. The predictor performs multi-step prediction on the output. And the controller acts ahead to eliminate the impact of time delay on the system. The adaptive PID controller realizes its weight adjustment through the supervised Hebb learning algorithm, and the adaptive control is achieved through the online adjustment of the weight coefficient. The degree of nonlinearity is good, and the robustness is high. However, there is no uniform standard for the selection of the neural network hierarchy. The designer makes a tentative choice according to his own experience. The subjective interference factors are stronger, and the prediction efficiency and accuracy are greatly affected by the network hierarchy.

2) The implicit generalized predictive control (IGPC) algorithm does not need to identify the model parameters of the object. And it uses input/output data to directly identify the controller parameters to solve the optimal control increment, which has the characteristics of small computation and high real-time performance. But its degree of nonlinear description is poor.

3) TSSA-HKRVM is based on small measured samples, directly controlling the prediction of load spectrum rather than the traditional control of the load change process. It is combined with the TSSA algorithm to carry out rolling optimization of relevant parameters of HKRVM. It uses the sliding window design and the sample update to carry out adaptive feedback correction from the perspective of ‘prediction + monitoring’. The load information is complete and the method has strong robustness, high prediction accuracy, and strong adaptive ability.

The adaptive TSSA-HKRVM in the equivalent load spectrum prediction of general bridge cranes is demonstrated by comparing the method in the paper with the current swarm intelligence optimization algorithm improved relevance vector machine prediction technology (ILON-RVM [

Population size | Number of hunters | Nomadic lion ratio | Number of settled lions | Ratio of settled female lions | |
---|---|---|---|---|---|

30 | 10 | 0.2 | 10 | 0.8 | |

ILOA-RVM | Migration ratio | Territory wandering ratio | Mating probability | Mutation probability | Iterations |

0.4 | 0.2 | 0.8 | 0.3 | 20 | |

Initial value of step size | Step end value | Iterations | Adjustment coefficient | Step size of jumping out of local solution | |

3 | 7.9 × 10^{−5} |
100 | 6 | 0.03 | |

IBAS-LSSVM | Iteration accuracy | Parameter |
Parameter |
||

10^{−4} |
10^{−4} |
0 | 0.036 | 6.985 |

Prediction model | Model complexity | Time consuming degree | Accuracy of results | Completeness of load information | Update mechanism | Adaptive situation | Subsequent prediction steps |
---|---|---|---|---|---|---|---|

TSSA-HKRVM | High | Long | Higher | Strong | Possess | Have | Long |

IBAS-LSSVM | Low | Low | High | Weak | Defect | Defect | Short |

ILOA-RVM | Medium | Medium | High | Weak | Defect | Defect | Short |

It can be seen from

According to

An adaptive TSSA-HKRVM model for the regression prediction of crane load spectrum is proposed in the paper. It overcomes the problem that the randomness of the crane working load leads to the reduction of load spectrum prediction accuracy over time, which provides a basis for the compilation of load spectrum for crane matching design and retirement mechanism. The specific conclusions are as follows:

1) For the adaptive TSSA-HKRVM prediction model, single and mixed kernel functions are tested with load spectrum samples based on RVM to obtain the optimal combination of the heterogeneous kernel function

2) The proposed method is applied to QD20/10 t × 43 m × 12 m general bridge crane. The results show that: compared with the traditional adaptive predictive control method, when the load samples are based on 3000 working cycles (one-year regular inspection cycle), the TSSA-HKRVM prediction has a higher fitting degree, a smaller root mean square error and higher predictive control performance of the model. Compared with the existing load spectrum prediction methods (ILOA-RVM and IBAS-LSSVM), TSSA-HKRVM has a higher model complexity, but it provides an adaptive update mechanism of the prediction model by updating samples, which improves the prediction accuracy of long periods, ensures the completeness of load information throughout the life cycle of the crane and has strong applicability and popularization.

This paper is based on the extensive investigation and testing results finished by experts and testers from Zhuzhou Tianqiao Crane Co., Ltd., and Weite Technologies Co., Ltd. The author cordially thanks experts and testers for on-site survey and real-time detection. This paper is sponsored by the

The authors declare that they have no conflicts of interest to report regarding the present study.