Multi-Trajectory Planning Control Strategy for Hydropower Plant Bridge Crane Based on Evaluation Algorithm

Abstract

Currently, the research on crane trajectory planning mostly aims to, first, plan the trajectories of the crane and the trolley, and then to use a trial-and-error method or optimization algorithm to iteratively calculate the optimal trajectory parameters under the control of the optimal trajectory parameters to achieve the suppression of the swing angle. However, research on the fusion application of multi-trajectory planning algorithms is very rare. In addition, the existing methods are not suitable for the special operation control of hydropower plant bridge cranes. Based on the application scenario of hydropower plant bridge cranes, this paper proposes a comprehensive multi-trajectory control strategy based on the entropy weight technique for order preference, similarly to the ideal solution (TOPSIS) evaluation method. Specifically, the kinematic analysis of the crane is carried out and the trajectory evaluation index system is established. Secondly, under the walking constraint condition, four different trajectory planning algorithms are used to obtain the crane trajectory curve. In order to ensure the accuracy and comprehensiveness of the evaluation, the evaluation data are obtained through the Adams motion simulation platform. Finally, based on the entropy weight TOPSIS evaluation method, the optimal walking trajectory for each displacement is selected. The simulation and experimental results show that the evaluation method can select the optimal trajectory based on the motion characteristics of the trajectory algorithm in different displacement conditions, effectively reducing the load swing during the walking process of the crane and improving the positioning accuracy.

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.

$$\{\begin{array}{c}\begin{array}{c}{a}_a,t\le t_1\\ 0,t_1<t\le t_2\end{array}\\ -a_d,t_2<t\end{array}$$

(1)

$$\{\begin{array}{c}{j}_{max}{\tau }_1,0\le t<t_1\\ a_{max},t_1\le t\le t_2\\ a_{max}-j_{max}{\tau }_3,t_2\le t\le t_3\\ 0,t_3\le t\le t_4\\ -j_{max}{\tau }_5,t_4\le t\le t_5\\ -a_{max},t_5\le t\le t_6\\ -a_{max}+j_{max},t_6\le t\le t7\end{array}$$

(2)

$$\{\begin{array}{c}q\left(t\right)=k_0+k_{1}t+k_2{t}^2+k_3{t}^3\\ \dot{q}\left(t\right)=k_1+2k_{2}t+3k_3{t}^2\\ \ddot{q}\left(t\right)=2k_2+6k_{3}t\end{array}$$

(3)

$$\{\begin{array}{c}q\left(t\right)=k_0+k_{1}t+k_2{t}^2+k_3{t}^3+k_4{t}^4+k_5{t}^5\\ \dot{q}\left(t\right)=k_1+2k_{2}t+3k_3{t}^2+4k_4{t}^3+5k_5{t}^4\\ \ddot{q}\left(t\right)=2k_2+6k_{3}t+124k_4{t}^2+20k_5{t}^3\end{array}$$

(4)

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.

3. Entropy Weight TOPSIS Evaluation and Construction of Evaluation Index System

3.1. Entropy Weight TOPSIS

TOPSIS method is a sorting method approaching ideal solutions. The basic idea of this method is to define ideal and negative ideal solutions in decision-making problems, and the main goal is to select an optimal solution among all feasible solutions. Entropy weight method is also an algorithm used to assign weights; unlike subjective assignment methods such as hierarchical analysis and fuzzy evaluation, entropy weighting method is based on the attributes of the data themselves. In this paper, the entropy weight method is used to determine the weights of different indicators, and TOPSIS method is combined to add weights and evaluate and sort different trajectory planning algorithms of the bridge crane and select the optimal walking trajectory of bridge crane under different working conditions. Specific calculation steps are as follows:

(1)

Normalization and standardization

The evaluation indexes of the track curve of bridge crane are all negative indexes and the dimensions of each index are very different; they need to be normalized and standardized. It is assumed that i travel curve of bridge crane has been planned, and each trajectory planning scheme has a total of j indicators, X_ij (i = 1, 2, … n, j = 1, 2, … m). Normalization and standardization are calculated as

$$x_{i{j}^{\prime }=max\left(X_j\right)}-x_{ij}$$

(5)

$$x_{i{j}^{\prime }}={\displaystyle \frac{x_{ij}}{\sqrt{{{\displaystyle \sum }}_{i=1}^n{x}_{ij}^2}}}$$

(6)

(2)

Calculation of entropy weights

Indicators after normalization and standardization can constitute a decision matrix R = ($r_{ij}$) m × n; for a certain indicator $r_j$, information entropy is Ej, then the entropy weight calculation formula is

$$E_j=-{\displaystyle \frac{1}{Inm}}{{\displaystyle \sum }}_{i=1}^m{p}_{ij}In{p}_{ij},p_{ij}={\displaystyle \frac{r_{ij}}{{{\displaystyle \sum }}_{i=1}^m{r}_{ij}}}$$

(7)

$$w_j={\displaystyle \frac{\left(1-E_j\right)}{{{\displaystyle \sum }}_{j=1}^n\left(1-E_j\right)}}$$

(8)

(3)

Determine positive/negative ideal solutions

In the constructed matrix R = ($r_{ij}$) m × n, define each index, i.e., the best value $r_j^{+}$ and the worst value $r_j^{-}$ for each column:

$$r_j^{+}=max\left(r_{1j},r_{2j}\cdots ,r_{nj}\right)$$

(9)

$$r_j^{-}=min\left(r_{1j},r_{2j}\cdots ,r_{nj}\right)$$

(10)

(4)

Calculate the distance from each solution to the positive/negative ideal solution

Calculate the distance between the i evaluation object of matrix R and the optimal value $d_i^{+}$ and the worst value $d_i^{-}$, respectively:

$$d_i^{+}=\sqrt{{{\displaystyle \sum }}_{j=1}^n{w}_j{\left(r_{j^{+}}-r_{i^j}\right)}^2 }$$

(11)

$$d_i^{-}=\sqrt{{{\displaystyle \sum }}_{j=1}^n{w}_j{\left(r_{j^{-}}-r_{ij}\right)}^2}$$

(12)

(5)

Score_i is calculated to calculate the closeness between each trajectory scheme and the optimal scheme:

$$Scor{e}_i={\displaystyle \frac{d_i^{-}}{d_i^{+}-d_i^{-}}}$$

(13)

${\mathrm{Score}}_i$ determines the order of trajectory planning algorithms, and the higher the value, the higher the applicability of the scheme to a certain working condition.

3.2. Crane Kinematics Analysis

The main reason for the sliding and small swing of the bridge crane during the walking and braking of the hydropower plant is that the movement of the crane and the hoisting weight are not synchronized due to the acceleration and deceleration during the walking process and the huge load of the hoisting object. The actual bridge crane model is very complex. For the convenience of theoretical analysis, the system model is further simplified to a two-dimensional model here, assuming the following conditions:

(1)

The elastic deformation of the system is ignored, and the quality and length changes of the wire rope (hanging rope) are ignored;

(2)

Factors such as wind, air resistance and friction at the connection between the wire rope and the trolley are ignored;

(3)

The hoisting weight is treated as a particle without volume.

The two-dimensional simplified model of the crane is obtained as shown in Figure 3, where the mass of the trolley is M, the mass of the hoisting weight is m, the friction coefficient between the trolley and the track is µ, the length of the wire rope is l, the horizontal driving force of the trolley is F and the swing angle of the hoisting weight (the angle between the wire rope and the vertical direction) is θ.

Based on the two-dimensional model of the trolley–crane system and Lagrange equation, the dynamic model of crane operating system is established. The Lagrange equation is an important equation in analytical mechanics. Based on the principle of least action, the equation is as follows:

$$L\left(q,\dot{q}\right)=T\left(q,\dot{q}\right)-V\left(q,\dot{q}\right)$$

(14)

In the formula, L is the Lagrange operator, T is the kinetic energy function of the system, V is the potential energy function of the system and $q$, $\dot{q}$ are the Lagrange variables and, respectively, are the generalized coordinates and generalized velocities. If the position coordinate of the trolley is (x, 0), the position relation of the weight relative to the origin is

$$\{\begin{array}{c}{x}_m=x_M+l\sin \theta \\ y_m=-l\cos \theta \end{array}$$

(15)

The velocity components of the weight along the x and y axes are

$$\{\begin{array}{c}{\dot{\mathrm{x}}}_{\mathrm{m}}={\dot{\mathrm{x}}}_{\mathrm{m}}+\dot{\mathrm{l}}\sin \mathsf{\theta }+\mathrm{l}\dot{\mathsf{\theta }}\cos \mathsf{\theta }\\ {\dot{\mathrm{y}}}_{\mathrm{m}}=-\dot{\mathrm{l}}\cos \mathsf{\theta }+\mathrm{l}\dot{\mathsf{\theta }}\sin \mathsf{\theta }\end{array}$$

(16)

The equations of displacement, rope length and swing angle established by Lagrange equation are

$$\{\begin{array}{c}\left(M+m\right){\ddot{x}}_M+m\ddot{l}\sin \theta +ml\ddot{\theta }\cos \theta +2m\dot{l}\dot{\theta }\cos \theta -ml{\dot{\theta }}^2\sin \theta +\mu {\dot{x}}_M=F \\ {\ddot{x}}_M\cos \theta +l\ddot{\ddot{\theta }+2\dot{l}}\dot{\theta }co{s}^2\theta +g\sin \theta =0 \end{array}$$

(17)

Formula (1) is the driving part of the system, and Formula (2) is the underdriving part of the system. M and m are the masses of the car and the load, respectively; l is the cable length, x is the car position, θ is the load swing angle, F is the driving force and ${\mathsf{\mu }\dot{\mathrm{x}}}_{\mathrm{M}}$ is the friction between the car and the track.

3.3. Establishment of Trajectory Evaluation Index

The index evaluation system, as shown in Figure 4, is established according to the actual working conditions of hydropower plant bridge crane. When the bridge crane is hoisting the rotor, due to the large mass of the rotor itself, the large load swing angle during the walking process will produce excessive stress on the structure of the hoisting equipment, resulting in fatigue, damage and even load tilt. Since the moment of inertia of the load is larger than the braking moment of the brake, the bridge crane will continue to travel forward for a small distance after reaching the position, which is uncertain. Therefore, the maximum swing angle and the load residual swing angle of the bridge crane under different trajectory curves are important evaluation indexes. In addition, the selection and evaluation of trajectory smoothness and anti-interference ability indexes are based on the following:

(1)

Trajectory Smoothing

It can be seen from Equation (13) that there is a dynamic coupling relationship between trolley acceleration and load swing angle θ. The movement of the trolley has nothing to do with the mass of the trolley and the load. Hydropower plants are not allowed to have a large swing during rotor hoisting, so the swing angle can be approximately equal to 0 within a certain range, and further simplification can be obtained:

$$\{\begin{array}{c}\left(M+m\right){\ddot{x}}_M+ml\ddot{\theta }+\mu {\dot{x}}_M=F\\ {\ddot{x}}_M+l\ddot{\theta }+g\theta =0\end{array}$$

(18)

where M is the bridge mass, m is the load mass, θ is the load swing angle and ${\ddot{x}}_M$ is the acceleration. Therefore, there is a dynamic coupling relationship between the acceleration change of the bridge crane and the load swing angle change. Comparing the load swing angle change under different trajectory algorithms means comparing the acceleration change of different curves, that is, the trajectory smoothness.

(2)

Anti-interference

Due to the increase of track defects, uneven friction resistance, structural deformation of the bridge, vibration of the bridge itself and other reasons, there will be small disturbances during the walking process of the bridge, which will affect the walking performance and positioning accuracy. Therefore, the interference is simulated by adding Gaussian noise to the trajectory curve. The formula is as follows:

Y = y + k × randn(size)

(19)

where y represents the original data, size represents the dimension of data y, randn is a random scalar obtained from the standard normal distribution (mean 0, variance 1) and k is the horizontal coefficient of relative error. The anti-interference ability of different trajectory curves can be compared and evaluated by simulating the trajectory curves and measuring the load swing angle.

4. Simulation and Experiment Results

4.1. Simulation Modeling Based on Adams and Matlab

Taking four generator hydropower plants as an example, the simplified model of double-beam bridge crane for hydropower plant is established in SolidWorks, as shown in Figure 5. The main parameters are as follows: the full weight of the bridge crane is 1200 t; the hoisting weight is 350 t; the hoisting height is 8 m; the hoisting positions of the No. 1 generator, the No. 2 generator, the No. 3 generator and the No. 4 generator are 10 m, 25 m, 40 m and 55 m, respectively; the crane travelling displacement is 0~80 m; the crane travelling maximum and minimum speed is 0.33 m/s and 0.03 m/s, respectively; and the maximum acceleration of the crane travelling is 0.028 m/s². The model established by SolidWorks parameterization is imported into the ADAMS software (Adams view 2020) for kinematic analysis, and its Adams model is shown in Figure 6. Kinematic pairs and constraints were added to the walking direction (X-axis direction) of the crane main beam and damping and stiffness during horizontal movement were added as shown in Figure 7. In order to validate the motion characteristics of the crane under different trajectory curves, as shown in Figure 8, the time–displacement data of the trajectory curve are imported into the motion vice, so as to simulate the real working environment and motion characteristics of the rotor hoisted by the hydropower plant bridge crane.

According to the minimum initial speed, maximum speed and constraints stipulated by the bridge crane in the hydropower plant, at the same time, the positions of the four generators in the hydropower plant are, respectively, located at 10 m, 25 m, 40 m and 55 m relative to the starting point, and the positions of the four generators are regarded as the walking displacement of the crane. The algorithm models of the T-shaped trajectory, S-shaped trajectory, cubic polynomial trajectory and quintic polynomial trajectory were established by Matlab, and the displacement, velocity and acceleration curves of the four algorithms under different displacements were obtained. The walking displacement curves of different target locations are shown in Figure 9. Under the same displacement, the time of the four algorithms are different, and the quintic polynomial displacement curve is more continuous and smoother than the T-shaped curve and S-shaped curve displacement. Among them, in Figure 9a, the four curves are closest when the displacement is 10 m; in Figure 9d, when the displacement is 55 m, the time taken by the four tracks is 181.5 s, 191.3 s, 241 s and 297 s, respectively.

The velocity and acceleration curves of the four algorithms under different displacements are shown in Figure 10 and Figure 11. The T and S velocity curves are mainly divided into three stages: acceleration, uniform speed and deceleration. The T-shaped curve adopts a uniformly variable speed for transition during acceleration and deceleration, while the cubic and quintic polynomial velocity curves are mainly divided into variable acceleration and variable deceleration without a uniform speed stage. The time for the four algorithms in Figure 10d to reach the maximum speed of 0.33 m/s is 14.9 s, 25.1 s, 110.9 s and 155.2 s, respectively. The acceleration profiles of the four algorithms are shown in Figure 11. It can be seen from Equations (1)–(4) that the quintic polynomial algorithm has a higher order acceleration expression, so the acceleration curve is smooth and continuous, while the acceleration curves of the T-trajectory, S-trajectory and cubic polynomial algorithm show linear changes, and the T-shaped trajectory has obvious abrupt change during the three stages of uniform acceleration, uniform speed and uniform deceleration. The maximum acceleration under the different displacements of the cubic and quintic polynomials is different. In Figure 11a,d, the maximum acceleration of the cubic and quintic polynomial trajectories is 0.027 m/s² and 0.017 m/s², and 0.005 m/s² and 0.003 m/s², respectively.

4.2. Simulation and Analysis of Dynamic Characteristics of Bridge Cranes

It can be seen from Formula (18) that when the hydropower plant bridge crane is hoisting the generator rotor and other large components, the acceleration and deceleration process of the trolley will lead to a certain amplitude of the swing angle. According to practical experience, even if there is a counterbalance beam, there will still be a small swing angle due to the huge load, resulting in the positioning misalignment of the bridge crane. The displacement–time curves simulated by Matlab under different displacement and algorithm conditions are imported into Adams to generate the drive sub-spline curves of the bridge crane model, and the load swing angle of the bridge crane under different algorithm conditions can be obtained after simulation, as shown in Figure 12. Among the four algorithms, the T-shaped trajectory has a larger swing amplitude than the other three algorithms at different displacements, and the S-shaped trajectory has a better swing angle variation than the cubic polynomial trajectory and is close to the quintic polynomial trajectory at displacement = 10 m in Figure 12a. As shown in Figure 12c,d, the S-shaped trajectory is better than the cubic polynomial trajectory and the quintic polynomial trajectory in the process of uniform velocity, but there are large changes in the acceleration and deceleration phases, and the T-shaped trajectory in Figure 12d reaches the maximum value of the pendulum angle of 0.91 and 0.71 at t = 15.3 s and t = 176.8 s, respectively, which are much higher than the swing angle of the polynomial trajectory in the same moment.

The actual operation of the bridge crane in the hydropower plant is accompanied by irregular disturbances. From Formula (19) of the original trajectory curve, the result of adding the error level of 0.05 perturbation and displacement = 25 m, as an example, after adding the perturbation of the displacement–time curve, is shown in Figure 13. The displacement–time curve after adding the disturbance is imported into Adams, and the variation of the swing angle of the bridge load under different displacements and algorithms is shown in Figure 14. The magnitude of the swing increases for all the algorithms after adding the disturbance, the magnitude of the T-shaped and S-shaped trajectory changes significantly after adding the disturbance in the uniform velocity phase and the third and fifth degree polynomials still have large swings in the deceleration phase after adding the disturbance.

4.3. Evaluation Process

Based on the Adams and Matlab model simulation results, the total time, trajectory smoothness, emergency braking, maximum load swing angle and residual swing angle and anti-interference ability of each trajectory under different displacements and algorithms were recorded. The average values of the swing angle and speed of each trajectory were, respectively, taken for trajectory smoothness and emergency braking distance over time. The anti-interference ability takes the difference between the mean swing angle after adding interference and the mean swing angle without adding interference, and the evaluation indicators constituted are shown in

Table 1. Evaluation index of each trajectory algorithm at different displacements are obtained by simulation.

S	Algorithms	A	B	C	D	E	F
10 m	T-shaped trajectory	45.2 s	0.63754°	0.2207 m/s	1.83091°	0.55649°	0.14435°
	S-shaped trajectory	55 s	0.40612°	0.1818 m/s	0.91391°	0.15685°	0.13813°
	Cubic polynomial	44 s	0.64204°	0.2265 m/s	1.59733°	0.28710°	0.13429°
	Quintic polynomial	55 s	0.38279°	0.1815 m/s	0.70261°	0.18120°	0.14560°
25 m	T-shaped trajectory	90.6 s	0.46856°	0.2756 m/s	1.32222°	0.68613°	0.01970°
	S-shaped trajectory	100.4 s	0.25440°	0.2490 m/s	0.91319°	0.05730°	0.14250°
	Cubic polynomial	110 s	0.26144°	0.2270 m/s	0.87451°	0.28433°	0.17904°
	Quintic polynomial	135 s	0.18973°	0.1850 m/s	0.54503°	0.02005°	0.20475°
40 m	T-shaped trajectory	136.1 s	0.38615°	0.2936 m/s	1.39457°	0.67396°	0.21340°
	S-shaped trajectory	145.9 s	0.19217°	0.2741 m/s	0.91319°	0.06804°	0.20333°
	Cubic polynomial	176 s	0.17479°	0.2271 m/s	0.70404°	0.24208°	0.14475°
	Quintic polynomial	216 s	0.13636°	0.1851 m/s	0.47270°	0.02936°	0.19624°
55 m	T-shaped trajectory	181.5 s	0.32149°	0.3028 m/s	1.32723°	0.63671°	0.17438°
	S-shaped trajectory	191.3 s	0.15243°	0.2875 m/s	0.91319°	0.00673°	0.19844°
	Cubic polynomial	241 s	0.13860°	0.1386 m/s	0.63600°	0.17977°	0.19865°
	Quintic polynomial	297 s	0.10683°	0.1068 m/s	0.46911°	0.00152°	0.22573°

In the table, S represents the walking displacement of the crane, and A–F is the evaluation index, comprising the total time (s), trajectory smoothness (θ), emergency braking (m/s), maximum swing angle (θ), residual swing angle (θ) and anti-interference (θ), respectively.

. Among them, all indicators are negative indicators, that is, the smaller the value of the index, the better the performance. It can be seen that the index performance of each algorithm is different under different displacements.

The evaluation index in

Table 2. Normalization results of evaluation indexes for each trajectory algorithm at different displacements.

S	Algorithms	A	B	C	D	E	F
10 m	T-shaped trajectory	0.66521	0.01284	0.09106	0.00000	0.00000	0.09484
	S-shaped trajectory	0.00000	0.67289	0.70181	0.62271	0.65431	0.54714
	Cubic polynomial	0.74666	0.00000	0.00000	0.15863	0.44096	0.83165
	Quintic polynomial	0.00000	0.73963	0.70652	0.76620	0.61436	0.00000
25 m	T-shaped trajectory	0.72088	0.00000	0.00000	0.00000	0.00000	0.93970
	T-shaped trajectory	0.56177	0.52484	0.25048	0.41493	0.62867	0.31611
	S-shaped trajectory	0.40590	0.50757	0.45764	0.45417	0.40169	0.13056
	Quintic polynomial	0.00000	0.68331	0.85313	0.78840	0.66590	0.00000
40 m	T-shaped trajectory	0.70353	0.00000	0.00000	0.00000	0.00000	0.00000
	S-shaped trajectory	0.61724	0.50995	0.15146	0.38561	0.61548	0.14089
	Cubic polynomial	0.35221	0.55564	0.51653	0.55315	0.43870	0.96047
	Quintic polynomial	0.00000	0.65667	0.84276	0.73847	0.65477	0.24008
55 m	T-shaped trajectory	0.69462	0.00000	0.00000	0.00000	0.00000	0.80050
	S-shaped trajectory	0.63568	0.51416	0.05974	0.35174	0.62712	0.42543
	Cubic polynomial	0.33678	0.55624	0.64109	0.58723	0.45487	0.42215
	Quintic polynomial	0.00000	0.65287	0.76514	0.72900	0.63231	0.00000

In the table, S represents the walking displacement of the crane, A–F is the evaluation index, and the corresponding column data are the normalized and standardized data.

is positively normalized and standardized by Formulas (5) and (6), and the obtained data are shown in

Table 3. Evaluation index weighting data.

Displacement	Total Time	Trajectory Smoothness	Emergency Braking	Maximum Swing Angle	Residual Swing Angle	Anti-Interference
5 cm	0.2225	0.2088	0.1630	0.1409	0.0969	0.1679
15 cm	0.1416	0.1340	0.1814	0.1493	0.1401	0.2536
25 cm	0.1414	0.1272	0.2007	0.1392	0.1309	0.2607
35 cm	0.1544	0.1366	0.2593	0.1533	0.1394	0.1570

. After standardization, the size of all data is between 0 and 1 within the same dimension, and after normalization, all indicators change from negative to positive, that is, the greater the corresponding value of the indicator, the better the performance. By converting the data in Table 2 into a radar chart, as shown in Figure 15, the differences and advantages of each algorithm at each index point under different displacement curves can be visually compared. As can be seen from Figure 15, when displacement = 10 m, the S-shaped and quintic polynomials are optimized in terms of anti-interference, smoothness, residual swing angle and maximum swing angle, while the T-shaped trajectory is optimized only in terms of total time. When displacement = 25 m, the performance of each index of quintic polynomial is best according to all factors except for the total time, and the performance of each index of cubic multinomial is the most balanced when displacement = 40 and displacement = 55 m.

According to the positive and standardized data, the information entropy and weight of each indicator are calculated by Formulas (7) and (8), and weights are assigned to indicators with different differences. The weights obtained are shown in Table 3. The index change degree of each algorithm is different under different displacements. For example, when displacement = 10 m, the difference of each algorithm focuses on trajectory smoothing and total time. Under the condition of displacement = 20 m, the difference is mainly manifested in anti-interference ability and emergency braking distance.

After the weight of each index is obtained, the distance between each index’s data and the positive and negative ideal solutions is determined by Formulas (11) and (12), and the evaluation scores of each algorithm under different displacements are obtained by Formula (13). The evaluation results are shown in Figure 16. Under displacement = 40 m and displacement = 55 m, the trajectory score of the cubic polynomial algorithm is significantly higher than that of other three algorithms. When the displacement is 25 m, the score of the quintic polynomial algorithm is the highest and the score difference with other algorithms is small. When displacement = 10 m, the S-shaped trajectory has the highest score. The comprehensive evaluation results show that the S-shaped trajectory is optimal when the displacement is within 10 m. The displacement range is the best cubic polynomial trajectory in the range of 40 m~55 m. The displacement range is 25 m, and the algorithm can be selected according to the merits of a certain index or the quintic polynomial trajectory can be selected directly.

4.4. Experiment

In this section, in order to further prove the effectiveness of the proposed control method, we built a simple experimental crane platform on which two groups of experimental tests were carried out, as shown in the Figure 17. The embedded development tool MDK5 (MDK-ARM Version 5) is used to implement four trajectory planning algorithms through C language, compile and burn them into MCU to generate control commands, and send the pulse converted by MCU to the driver to control the motor. When the crane platform is running, the acceleration change generated by the trolly walking will cause the load to swing slightly. The load rope is fixed on the coupling of the encoder, and the coupling drives the absolute encoder to change the angle. Through the UART (Universal Asynchronous Receiver/Transmitter) communication protocol, the encoder can communicate with the MCU and the MCU to the PC, respectively, so as to record the data of the swing angle at the PC.

In this experiment study, the physical parameters are configured as

M = 1540 g, m = 200 g, v_max = 3.3 cm/s, a_max = 0.28 cm/s² and s = 0~35 cm.

Under the condition that the above physical parameters are unchanged, the walking displacement s is set to be 5 cm, 15 cm, 25 cm and 35 cm, respectively. Four trajectory planning algorithms in the simulation are used to control the motor, and the swing angle changes during the walking process are recorded, respectively, under the condition that the hoisting rope length is 12 cm and 24 cm. Take the variation of swing angle under T-shaped trajectory and S-shaped trajectory as an example, as shown in Figure 18 and Figure 19. Figure 18a and Figure 19a, respectively, show the swing angle data of the experimental platform when the rope length L = 12 cm and L = 24 cm, and Figure 18b and Figure 19b show the swing angle data in the simulation environment. It can be seen that the swing angle–time curve obtained is basically consistent with the variation of the swing angle in the simulation. Due to the large difference between the simulated process and the actual load mass and lifting weight, the overall swing angle is less than 1 degree compared with the simulation data. The swing angle of T-shape and S-shape trajectory presents a step process during acceleration and deceleration, and a simple swing motion during uniform velocity. Meanwhile, compared with Figure 18a and Figure 19a, the change of rope length causes the change in the swing angle amplitude range, but the overall change amplitude is consistent.

Based on the experimental results of the experimental platform, the specific data of each index under different displacements and algorithms are recorded. Take the experimental group data when the rope length is 12 cm as an example. As shown in

Table 4. Evaluation index of each trajectory algorithm at different displacements are obtained by experimental platform.

S	Algorithms	A	B	C	D	E	F
5 cm	T-shaped trajectory	3.4 s	0.110°	2.207 cm/s	0.29°	0.03°	0.14435°
	S-shaped trajectory	4 s	0.080°	1.818 cm/s	0.24°	0.01°	0.13813°
	Cubic polynomial	3.3 s	0.120°	2.265 cm/s	0.27°	0.06°	0.13429°
	Quintic polynomial	4 s	0.078°	1.815 cm/s	0.21°	0.04°	0.14560°
15 cm	T-shaped trajectory	6.3 s	0.101°	2.756 cm/s	0.26°	0.1°	0.01970°
	S-shaped trajectory	7.6 s	0.064°	2.490 cm/s	0.31°	0.02°	0.14250°
	Cubic polynomial	9.4 s	0.065°	2.270 cm/s	0.28°	0.07°	0.17904°
	Quintic polynomial	12.5 s	0.054°	1.850 cm/s	0.18°	0.015°	0.20475°
25 cm	T-shaped trajectory	9.4 s	0.105°	2.936 cm/s	0.26°	0.16°	0.21340°
	S-shaped trajectory	13.9 s	0.030°	2.741 cm/s	0.19°	0.04°	0.20333°
	Cubic polynomial	15.4 s	0.020°	2.271 cm/s	0.15°	0.01°	0.14475°
	Quintic polynomial	19.3 s	0.021°	1.851 cm/s	0.12°	0.015°	0.19624°
35 cm	T-shaped trajectory	11.8 s	0.137°	3.028 cm/s	0.41°	0.4°	0.17438°
	S-shaped trajectory	15.6 s	0.043°	2.2875 cm/s	0.23°	0.03°	0.19844°
	Cubic polynomial	18.3 s	0.038°	1.386 cm/s	0.18°	0.15°	0.19865°
	Quintic polynomial	22 s	0.030°	1.068 cm/s	0.14°	0.001°	0.22573°

In the table, S represents the walking displacement of the crane, and A–F is the evaluation index, including the total time (s), trajectory smoothness (θ), emergency braking (cm/s), maximum swing angle (θ), residual swing angle (θ) and anti-interference (θ), respectively.

, the load mass, displacement and speed during the experiment are different from the simulation data, and the amplitude of data change such as time and maximum swing angle are also different. For further processing of specific data, first of all, the experimental data of the two groups of rope length 12 cm and 24 cm were positive and standardized, and then the information entropy and weight of each index were calculated, and the obtained weights were shown in the

Table 5. Evaluation index weighting data (L = 12 cm).

Displacement	Total Time	Trajectory Smoothness	Emergency Braking	Maximum Swing Angle	Residual Swing Angle	Anti-Interference
5 cm	0.2378	0.1466	0.1730	0.1422	0.1219	0.1786
15 cm	0.1340	0.1216	0.1660	0.1936	0.1528	0.2320
25 cm	0.1559	0.1238	0.1982	0.1391	0.1251	0.2579
35 cm	0.1700	0.1342	0.2575	0.1397	0.1423	0.1564

and

Table 6. Evaluation index weighting data (L = 24 cm).

Displacement	Total Time	Trajectory Smoothness	Emergency Braking	Maximum Swing Angle	Residual Swing Angle	Anti-Interference
5 cm	0.2220	0.2003	0.1615	0.1483	0.1012	0.1667
15 cm	0.1441	0.1360	0.1785	0.1477	0.1440	0.2496
25 cm	0.1504	0.1193	0.1913	0.1265	0.1636	0.2488
35 cm	0.1761	0.1550	0.1720	0.1548	0.1802	0.1620

. It can be seen that under different rope lengths, the degree of change of each index is different. For example, as shown in Table 5, when the rope length is 12 cm and the displacement is 5 cm, the difference between the algorithms is the time and anti-interference ability. As shown in Table 6, when the rope length is 24 cm and the displacement is 5 cm, the time and trajectory smoothness are different.

After the weight data are obtained, the trajectory planning algorithm is evaluated, and the evaluation results are shown in Figure 20. Among them, Figure 20a,b show the evaluation results when the rope length is 12 cm and 24 cm, respectively. Taking the rope length of 12 cm as an example, when the displacement is 5 cm, 15 cm, 25 cm and 35 cm, compared with other trajectory control strategies, the comprehensive performance of the multi-trajectory control strategy can be improved by up to 32.4%, 33.8%, 57.7% and 30.7%, respectively. From the comparison results of the two groups of experiments, it can be seen that the final evaluation results of the experiment and the simulation are consistent, and the curves corresponding to the highest scores of each displacement are the S-shaped trajectories, quintic polynomial trajectories, cubic polynomial trajectories and cubic polynomial trajectories. At the same time, when the rope length is 24 cm and the displacement is 15 cm, the scores of T-shaped trajectory, S-shaped trajectory and quintic polynomial trajectory are similar, indicating that in this case, one of the three algorithms can be selected according to the actual situation.

4.5. Analysis and Discussion

From the final trajectory evaluation results, it can be seen that under the same displacement range and identical constraint conditions, the data of the evaluation indicators, such as the maximum load swing angle, residual swing angle, and time, are different for the crane under different trajectory planning algorithms. The simulation and experimental results of this paper all illustrate this point. Based on this, we propose a multi-trajectory integrated control strategy based on the evaluation algorithm, which is different from the trajectory planning control of most current studies. Instead of optimizing the trajectory parameters through finding the optimal trajectory parameters [26,27,28], input shaping control [29,30] or designing closed-loop controllers [8,9,10] to achieve optimal control effects, we directly evaluate and select the optimal trajectory based on the evaluation indicators and evaluation algorithm. At the same time, the fusion application of the proposed method with artificial intelligence algorithms [31,32] can be a new research direction. We also need to acknowledge the limitations of this study. First, the establishment of the crane evaluation indicators in this paper only considers the six most important indicators in the actual operation of the crane, and other factors such as the sliding distance of the crane brake after braking have not been considered. Second, there is a lack of experimental results for similar methods. In addition, further research is needed for the design and implementation of the actual electrical control system.

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.