1. Introduction
The bridge crane is an essential piece of hoisting equipment in a hydropower plant, and plays an important role in unit installation and maintenance. Currently, the operation of a hydropower plant bridge mainly relies on manual operation, and the positioning accuracy is mainly based on empirical judgment. Moreover, due to the large inertia of the load on large components, it is difficult to precisely control the braking position and speed of the crane. When the crane is approaching the centerline position of the unit, the driver operates with minor adjustments. During this process, the crane will experience small oscillations due to the velocity changes caused by step changes.
Extensive research has been conducted by scholars on the positioning and anti-sway control of bridge cranes, mainly focusing on the closed-loop control composed of input, feedback and control algorithms and the open-loop control strategy that generates control signals from the input signals. Among them, research on closed-loop control is mainly focused on theoretical studies, such as fuzzy PID control [
1,
2,
3], sliding mode control [
4,
5], linear control [
6,
7], etc. In [
8,
9], the parameters of the PID controller were adjusted based on generalized predictive control (GPC), hybrid particle swarm optimization (PSO) and simulated annealing (SA) algorithm, respectively, and the optimal parameters of PID controller were obtained, effectively shortening the parameter adjustment time and eliminating the hook swing. In [
10], based on the variable universal fuzzy multi-parameter self-tuning PID (VUFMS-PID) control strategy, the number of fuzzy control rules was dynamically adjusted according to the system error and changing error rate. In [
11], an adaptive sliding mode control (SMC) method based on a neural network was designed, which compensates for stable drive and underdrive state variables through the neural network. In [
12], a prescribed-time sliding mode controller was designed for the positioning and anti-swing time of the bridge crane under different initial conditions. The overshoot time and load swing angle amplitude of the controller were reduced. According to most studies, closed-loop control mainly adjusts the controller parameters and searches for the optimal controller parameters in combination with the algorithm. This solution has good anti-interference ability, but its performance mainly depends on the sensor performance and the structural design and parameter optimization of the feedback system. At the same time, the effect of sliding mode control at high frequencies is limited, and it is difficult to effectively suppress the natural flutter. Some nonlinear factors, such as friction and elasticity, will cause the system response to deviate from the linear model, which will affect the control effect.
The open-loop control strategy based on input shaping [
13,
14] and trajectory planning [
15,
16,
17] is simple, practical and has a fast response speed. More and more scholars have applied this strategy to crane control. Singhose et al. [
18,
19] proposed an input shaping control method to reduce load swing and residual vibration for gantry crane and bridge crane, respectively. Masoud et al. [
20] proposed a frequency modulation (FM) input shaping technique, which utilized a single-mode input shaper to achieve the swing suppression of a multimode crane system. Huang et al. [
21,
22] analyzed the swing angle, acceleration amplitude and acceleration of the crane based on the phase plane trajectory and designed the input signal of the relationship between the acceleration and swing angle by using the input shaper. Then, T-shaped velocity trajectory planning [
23] and S-shaped velocity trajectory planning [
24] were applied to the crane control, and the uniform acceleration and variable acceleration control of bridge crane were studied to reduce the inertia impact of the crane. Tho et al. [
25] adopted the dynamic coupling relationship between the crane motion and load swing angle and introduced an S-shaped curve to establish a non-swing trajectory. Zhang et al. [
26] used particle swarm optimization (PSA) to solve the optimal parameter optimization of the S-shaped trajectory to maximize the performance of the motion trajectory. In [
27,
28], the travel path tree of the crane was constructed based on the rapidly exploring random tree (RRT) algorithm. In [
29,
30], under the constraint conditions, the trajectory planning scheme with optimal energy consumption was proposed; this can achieve the transportation task and reduce the energy consumption to the maximum extent. In general, most scholars have focused primarily on optimizing single trajectory planning algorithms through trial and error or optimization algorithms. However, the complex working conditions of the bridge crane of a hydropower station require the algorithm to iterate repeatedly to find the optimal path when the system parameters or constraints change, and the effect, in practical engineering applications, has certain limitations.
In this paper, four trajectory planning algorithms, T-shaped trajectory planning, S-shaped trajectory planning, cubic polynomial trajectory and quintic polynomial trajectory, are selected as the evaluation and application object of a multi-trajectory control strategy. First of all, the hydropower plant bridge has strict operating procedures, such as travel speed, load, hoisting range, etc. Under these constraints, Matlab (version R2023a) is used to realize each trajectory planning algorithm and obtain the trajectory curve. Secondly, in order to verify the practicability of the simulation data, a simple mechanical model of the hydropower plant crane was established and imported into the Adams motion simulation platform; a motion pair was added to the crane model, the track curve was imported into the motion pair and the load swing angle data of the crane during the movement was obtained, which was used as an evaluation index and recorded. Based on the entropy weight TOPSIS evaluation method and data, we evaluated each trajectory curve on the premise of the walking displacement range of the crane and obtained the optimal trajectory curve of each displacement range. Finally, we set up a simple crane walking experiment platform to verify the accuracy of the simulation data and prove the effectiveness of the method. The main contributions of this paper are outlined as follows:
- (1)
The multi-trajectory integrated control strategy of the crane is proposed to ensure that the load swing angle, residual swing angle, braking distance and travel time, and other indicators are optimized when the crane reaches different desired positions.
- (2)
Compared with the optimization process of the traditional trajectory algorithm, the trajectory evaluation method based on the entropy weight TOPSIS evaluation algorithm takes the actual operation results of the crane as the evaluation data, without searching for the optimal parameters, and avoids the accumulation of errors caused by changes in system parameters during the walking process.
- (3)
Compared with the closed-loop control strategy, the control method is simple in structure and easy to implement. At the same time, the response time delay and frequent parameter adjustment of the feedback mechanism are avoided.
The rest of this article consists of the following.
Section 2 introduces the classical trajectory planning algorithm and the evaluation process of multi-trajectory application.
Section 3 introduces the entropy weight TOPSIS evaluation algorithm, carries on the kinematics analysis of the crane and constructs the trajectory evaluation index system. In
Section 4, the results of simulation, experiment and comparison are given.
Section 5 is the conclusion.
2. Comprehensive Evaluation of Trajectory Based on Entropy Weight—TOPSIS Method
T-shaped trajectories, S-shaped trajectories, cubic polynomials and quintic polynomials trajectories are commonly used trajectory generation methods in travelling trajectory planning, and have a wide range of applications in the fields of industrial robotics, unmanned vehicles and automated systems. The acceleration formula of the four trajectories is shown in (1)–(4), and the generated acceleration curve is shown in
Figure 1. T-shaped trajectory accelerates and decelerates with constant acceleration and has smooth acceleration and deceleration transition segments but does not provide higher order of smoothness. S-shaped trajectory introduces a continuous microscopic S-curve during acceleration and deceleration, which has smoother acceleration and deceleration transitions and reduces shocks and vibrations compared to trapezoidal trajectory. Both cubic polynomial and quintic polynomial trajectory planning use polynomial functions to generate smooth curved trajectories with continuous velocity and acceleration at the starting and target points. The quintic polynomial trajectory planning algorithm is a higher-order trajectory planning algorithm that can provide higher smoothness and curvature continuity than cubic polynomials. Therefore, the motion characteristics and application scenarios of different trajectory planning algorithms are different, and the bridge crane in hydropower plants is usually used for moving, installing and transporting heavy equipment or materials, which has the characteristics of large moving range and complex working conditions, and is in line with the application of multi-trajectory planning control strategy.
Equations (1)–(4) are the acceleration expressions of T-type trajectory, S-type trajectory, cubic polynomial trajectory and quintic polynomial trajectory.
The centerline positions of the hydroelectric power plant units are fixed. Assuming there are N units, the distances from the starting position of the crane to the different central line positions of the units can be classified as short distance, medium-short distance, medium-long distance and long distance. Therefore, the adaptability of each trajectory curve is discussed within different distance ranges. As shown in
Figure 2, the evaluation process of the trajectory is mainly divided into three parts: establishment of trajectory evaluation index, acquisition of trajectory evaluation data and trajectory evaluation. In the process of establishing the trajectory evaluation index, the motion process of the crane is first analyzed to obtain the relationship between the change of the trajectory curve acceleration and the change of the swing angle. Based on this, the swing angle change, the total time and the anti-interference ability of the crane are taken as the evaluation index. In the process of acquiring trajectory evaluation data, the maximum velocity/acceleration/jerk and walking displacement of the crane are taken as constraint conditions, and trajectory curves under different displacements are obtained through Matlab simulation. Then a crane model is established based on the parameters of the hydropower plant crane. On the basis of the model, the index data of the crane walking process under different trajectory curves are obtained by Adams simulation. After the above two processes, the index data are processed and calculated based on the entropy weight TOPSIS evaluation method, and the trajectory curve with the highest score under different displacements is obtained, completing the selection of the optimal trajectory of the crane in different distance ranges.
5. Conclusions
Based on the entropy weight TOPSIS evaluation method, a multi-index evaluation system including the total time, maximum load swing angle and load residual swing angle is constructed. The simulation of four trajectory planning algorithms is carried out, and the optimal matching trajectory in different walking displacement ranges of the bridge crane is obtained. The method provides accurate and reliable evaluation results for the walking path of the bridge by objectively assigning weights and evaluating indexes. Compared with a single trajectory, the comprehensive multi-trajectory control strategy based on this evaluation method is more suitable for complex working conditions of hydropower plant bridges, giving full play to the performance advantages of each trajectory planning algorithm, and effectively controlling the walking process and accurate positioning of bridges.
However, in practical applications, bridge cranes usually use only one trajectory planning algorithm for control, and the application of multi-trajectory integrated control strategy requires a complex electrical system design, which requires sufficient equipment and a suffient number of researchers to conduct research. At the same time, the simulation and experiment results prove the effectiveness of the control method, but the real working environment of the crane is lacking for verification. Therefore, our next major work will be to build a more perfect experimental platform, committed to promoting the evaluation-based multi-trajectory control strategy for the hydropower plant bridge crane, and to carry out a feasibility analysis of the optimization results.