Finite-time tracking control method for underactuated bridge cranes

Abstract

The present application relates to the technical field of control of underactuated bridge crane, and discloses a finite time tracking control method of underactuated bridge crane, which is characterized by the following steps: installing an incremental encoder at the tail of the motor of the bridge crane, measuring the displacement of the trolley by the incremental encoder, calculating the required displacement of the trolley and the signal size of the load swing by the finite time tracking controller of the bridge crane, and then moving the trolley to the target position by the motor of the bridge crane, so as to indirectly control the load swing. The establishment method of the finite time tracking controller comprises the following steps: proposing the acceleration motion trajectory of the bridge crane system, and then using the sliding mode control to perform the finite time tracking control on the bridge crane system, so that the bridge crane can maintain good control performance even under adverse conditions, and the control target is achieved.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of control of underactuated bridge cranes, in particular to a finite time tracking control method for underactuated bridge cranes. BACKGROUND

[0002] Bridge cranes are large engineering handling equipment, which have become a research hotspot in the field of modern industry in recent years and are widely used in many fields such as construction sites, ports, factories, workshops and the like. The main work of bridge cranes is to transport materials, which can accurately and quickly transport the load from the initial position to the target position while suppressing the swing of the load. Although the bridge crane has a series of advantages such as high transportation efficiency, low energy consumption, simple mechanical structure and low hardware cost, as a typical underactuated device, its underactuated characteristics seriously increase the research difficulty of the control problem. The so-called underactuation means that the number of control inputs is less than the number of controlled degrees of freedom, so there will be a control target that is indirectly controlled, thereby increasing the design difficulty of the control method.

[0003] At present, for the control problem of underactuated bridge crane systems, according to whether feedback signals are needed, the research results of control methods can be mainly divided into two aspects: open-loop control method and closed-loop control method. The open-loop control method mainly utilizes the coupling relationship between the trolley displacement and the load swing. The existing open-loop control methods mainly include input shaping method, optimal control method, trajectory planning method and the like. The open-loop control method does not need feedback information and has a simple structure, which is convenient for practice, so it has achieved great success in the automatic control field of bridge cranes. The closed-loop control method mainly suppresses and eliminates the swing of the load through the measurement and estimation of the system state, so as to obtain accurate positioning. The existing closed-loop control methods mainly include energy-based control method, adaptive control method, sliding mode control method and the like. In addition, some scholars have also proposed some intelligent control methods, such as fuzzy control method, neural network control method and the like. However, in actual application, the robustness of the open-loop control method is poor, and once it is disturbed by the outside world, the control performance will be greatly reduced. Compared with the open-loop control method, the closed-loop control method has better and stronger robustness and is more suitable for bridge cranes working outdoors. However, most of the existing closed-loop control methods can only guarantee the asymptotic stability of the closed-loop system, which is far from enough in the transportation task with high precision requirements.

[0004] In view of the above control method and its problems, many scholars have carried out relevant research on the tracking control of the bridge crane. The control problem of the trolley displacement is converted into a tracking control problem, and a new type of trajectory tracking control method is designed, so that the operation and control quantity of the trolley are more smooth. The purpose of tracking control of the bridge crane is to enable it to quickly move to the target position and ensure that the load swings within a small angle during movement. At present, many algorithms have achieved good results in the tracking control problem of most bridge cranes, which can ensure that the system is asymptotically stable or uniformly ultimately bounded stable. However, the existing control methods can only achieve the control target when time tends to infinity, and cannot achieve tracking effect in finite time, which is certainly not desirable in practice.

[0005] In view of the above problems, the prior art also proposes some solutions, for example, the application name is: a bridge crane control method based on finite time compound, application number: CN 202111611271.7. The dynamics model is linearized, so it can only be effective near the equilibrium point. In addition, due to the complex expression of the designed controller, noise is easily generated in practice. The application name is: crane finite time trajectory tracking controller and method with uncertain dynamics, application number: CN 201611160077.0, the first terminal sliding mode designed for the swing angle contains the third derivative of the state variable. Generally speaking, when designing a controller, if the feedback information needs to use the acceleration signal, in addition to the difficulty in obtaining the acceleration signal, the control input of the system will change sharply due to the high-order noise contained in the acceleration signal, thereby reducing the control effect. Therefore, the third derivative information (jerk signal) of the state variable used in this algorithm is more difficult to obtain in practice, and the noise generated is more violent, and obviously the control effect is also worse, which is unrealistic and undesirable.

[0006] Therefore, in view of the problems existing in the prior art, the present application designs a finite time tracking control design method. SUMMARY

[0007] The technical problem to be solved by the present application is to provide a finite time tracking control method for an underactuated bridge crane, which uses sliding mode control to design a finite time tracking controller for the underactuated bridge crane system, to achieve accurate tracking of the acceleration trajectory in finite time.

[0008] To solve the above technical problems, the present application provides a finite time tracking control method for an underactuated bridge crane, comprising:

[0009] The incremental encoder is installed at the tail of the motor of the bridge crane, the displacement of the trolley is measured by the incremental encoder, then the trolley displacement required to be controlled and the signal size of the load swing are calculated by the finite time tracking controller of the bridge crane, and then the trolley is moved to the target position by controlling the motor of the bridge crane, so as to indirectly control the load swing;

[0010] The method for establishing the finite time tracking controller comprises the following steps:

[0011] S1, establishing the trolley dynamics equation of the system model of the bridge crane;

[0012] S2, establishing the smooth acceleration motion trajectory of the trolley of the bridge crane;

[0013] S3, the control target of the finite time trajectory tracking control method is to ensure that the position, speed and acceleration of the trolley of the bridge crane converge to the target position, speed and acceleration in a finite time, and quickly eliminate the load swing angle, angular velocity and angular acceleration;

[0014] S4, determining the actual parameters of the acceleration motion trajectory of the trolley;

[0015] S5, establishing the finite time tracking controller based on the finite time convergence sliding mode surface, and realizing accurate tracking of the smooth acceleration motion trajectory established in step S2 in a finite time.

[0016] Improvements of the finite time tracking control method of the under-actuated bridge crane of the application:

[0017] The trolley dynamics equation in step S1 is:

[0018]

[0019] Wherein, M is the mass of the trolley, m is the mass of the load, l is the length of the hoisting rope, x is the displacement of the trolley, is the speed of the trolley, is the acceleration of the trolley, θ is the angle of the load swing, is the angular velocity of the load swing, is the angular acceleration of the load swing, F is the driving force acting on the trolley, and g is the acceleration of gravity;

[0020] Divide equation (2) by ml to obtain equation:

[0021]

[0022] Further, we have:

[0023]

[0024] And the above formula (4) is substituted into formula (1), the equation is as follows:

[0025]

[0026] Wherein, m(θ) and Both are auxiliary functions, the specific expression is as follows:

[0027] m(θ)=M+msin 2 θ (6)

[0028]

[0029] As a further improvement of the finite time tracking control method of the underactuated bridge crane of the application:

[0030] The smooth acceleration motion trajectory of the bridge crane trolley in step S2 is:

[0031]

[0032] Wherein, τ∈(0, T1 / 4), t1=τ, t2=τ+t a , t3=2τ+t a , t4=2τ+t a +t c , t5=3τ+t a +t c , t6=3τ+2t a +t c , t7=4τ+2t a +t c , a max For the maximum acceleration used in the actual transport process, τ, t a , t c Respectively represent the variable acceleration (variable deceleration), uniform acceleration (uniform deceleration), uniform speed time constant, T1 is the vibration period of the load under constant acceleration;

[0033] The ideal acceleration trajectory should meet the following performance indicators:

[0034]

[0035] |θ(t)|≤θ ub

[0036] Wherein Is the target position, T is the trolley arrival time, Respectively the maximum speed and maximum acceleration that the trolley can reach, Is the maximum load swing angle that the system can allow.

[0037] As a further improvement of the finite time tracking control method of the underactuated bridge crane of the present application:

[0038] The control target in step S3 is specifically:

[0039]

[0040] Wherein,

[0041] x r , are respectively the preset position motion trajectory, the speed motion trajectory, and the acceleration motion trajectory;

[0042] To achieve the control target formula (9), the tracking error of the trolley is defined as follows:

[0043]

[0044] Wherein are respectively the first-order derivative and the second-order derivative of the trolley tracking error with respect to time.

[0045] Then, the control target of the finite time trajectory tracking control method can be written as:

[0046]

[0047] As a further improvement of the finite time tracking control method of the underactuated bridge crane of the present application:

[0048] The actual parameters in step S4 are specifically:

[0049] When t = τ, the initial value of θ r (t), is:

[0050]

[0051] θ r (t), are respectively the load swing angle and the angular velocity of the trolley when uniformly accelerated, ω n is the natural frequency of the system; the time used for uniform acceleration of the trolley:

[0052]

[0053] Wherein:

[0054]

[0055] Wherein The time used for uniform speed of the trolley:

[0056]

[0057] Maximum speed of the trolley:

[0058]

[0059] wherein Maximum acceleration of the trolley:

[0060] As a further improvement of the finite time tracking control method of the underactuated bridge crane of the application, the finite time convergent sliding mode surface in step S5 is:

[0061]

[0062] wherein, 0<σ2<1, sgn(·) is a sign function, and · can represent any function:

[0063] It can be obtained from equation (5) that:

[0064] Derivate the sliding mode surface s and substitute equation (24) into it:

[0065] The driving force F of the trolley should satisfy the approaching condition: In order to satisfy condition (26), As follows:

[0066]

[0067] wherein, is a positive control gain.

[0068] By combining equations (25) and (27), we get:

[0069] According to equation (28), the finite time tracking controller is obtained:

[0070]

[0071] The beneficial effects of the application mainly include:

[0072] Firstly, a smooth acceleration trajectory for the underactuated bridge crane system is proposed; then, the finite-time tracking control method is designed for the underactuated bridge crane system by using the sliding mode control, so that the bridge crane can maintain good control performance and achieve the control target even under adverse conditions. Because the sliding mode control method is not sensitive to model errors, parameter uncertainties and other disturbances, it has attracted widespread attention and is widely used in bridge cranes. The finite-time tracking control design method proposed in this paper can accurately track the preset trajectory in finite time without approximating and linearizing the bridge crane model, and can achieve the trolley positioning and load swing elimination targets. BRIEF DESCRIPTION OF DRAWINGS

[0073] The specific embodiments of the present application will be further described below in combination with the drawings.

[0074] Figure 1 The figure is a schematic diagram of the system model of the bridge crane.

[0075] Figure 2 The figure is a simulation result diagram of the finite-time tracking control method of the present application under zero initial conditions.

[0076] Figure 3 The figure is a simulation result diagram of the finite-time tracking control method of the present application under system model parameter changes. DETAILED DESCRIPTION

[0077] The present application will be further described below in combination with specific examples, but the protection scope of the present application is not limited to this:

[0078] Example 1, the finite-time tracking control method of the underactuated bridge crane, specifically:

[0079] Step one, establish the dynamic model:

[0080] The system model of the bridge crane is shown in Figure 1 A trolley with a mass of M is arranged on the bridge, and a load with a mass of m is hung below the trolley by a length of l of the hoisting rope to move, and the dynamic equation of the bridge crane system can be obtained by Euler-Lagrange equation:

[0081]

[0082] Wherein, M is the mass of the trolley, unit kg; m is the mass of the load, unit kg; l is the length of the hoisting rope, unit m; x, θ、 is the state variable of the system, x is the displacement of the trolley, is the speed of the trolley, is the acceleration of the trolley, and θ is the angle of the load swing, is the angular velocity of the load swing, is the angular acceleration of the load swing; F is the driving force acting on the trolley, in N; g is the gravitational acceleration, in m / s 2 .

[0083] In order to facilitate the subsequent description and control method design, divide both sides of formula (2) by ml to obtain the equation:

[0084]

[0085] Further, we obtain:

[0086]

[0087] And substitute the above formula (4) into formula (1) to obtain the equation as follows:

[0088]

[0089] Wherein, m(θ) and are auxiliary functions, and the specific expressions are as follows:

[0090] m(θ)=M+msin 2 θ (6)

[0091]

[0092] Step two, trajectory planning:

[0093] In the trajectory planning of the bridge crane, the traditional design method is to use the three-section acceleration trajectory, that is, the uniform acceleration-uniform speed-uniform deceleration trajectory for transmission. However, the traditional three-section acceleration trajectory has discontinuity, so it will cause certain damage to the bridge crane in actual work and cannot be reversed. In addition, the traditional three-section acceleration trajectory has too strict requirements, and it is very challenging to follow the traditional three-section acceleration trajectory in the running process. Therefore, the present application adopts a smooth acceleration trajectory, and its expression is as follows:

[0094]

[0095] Wherein, τ∈(0,T1 / 4)、t1=τ、t2=τ+t a 、t3=2τ+t a 、t4=2τ+t a +t c 、t5=3τ+t a +t c 、t6=3τ+2t a +t c 、t7=4τ+2t a +tc , a max is the maximum acceleration adopted in the actual transport process, τ, t a , t c respectively represent variable acceleration (variable deceleration), uniform acceleration (uniform deceleration), uniform speed time constant, and T1 is the vibration period of the load under constant acceleration.

[0096] The ideal acceleration trajectory should meet the following performance indicators:

[0097]

[0098] |θ(t)|≤θ ub

[0099] wherein is the target position, T is the trolley arrival time, are respectively the maximum speed and the maximum acceleration that the trolley can reach, is the maximum load swing angle that the system can allow.

[0100] Step three, control target:

[0101] The present application is directed to a bridge crane system, directly controlling the movement of the trolley by the motor, transporting the load to the target position through the trolley, and indirectly controlling the swing of the load through the movement of the trolley, so that there is no residual swing of the load after the trolley reaches the target position. The control target of the bridge crane system is to ensure that the position, speed and acceleration of the trolley converge to the target position, speed and acceleration within a limited time, while quickly eliminating the load swing angle, angular velocity and angular acceleration, and the mathematical expression is:

[0102]

[0103] x r , are respectively the preset position motion trajectory, the speed motion trajectory and the acceleration motion trajectory.

[0104] In addition, considering the actual working condition, without proof, the present application makes the following conditional assumptions, that is, the load is always located below the bridge:

[0105]

[0106] To achieve the control target formula (9), the tracking error of the trolley is defined as follows:

[0107]

[0108] wherein are respectively the first order derivative and the second order derivative of the trolley tracking error with respect to time.

[0109] Then, the control objective of the finite-time trajectory tracking control design method can be written as:

[0110]

[0111] Step 4, determine the actual parameters:

[0112] In order to obtain the specific expression of the acceleration trajectory, some actual parameters need to be determined. By solving the dynamic equation of the bridge crane system, analyzing the coupling relationship between the trolley acceleration and the load swing, and calculating the actual uniform acceleration time t a , uniform speed time t c , maximum speed v max , maximum acceleration a max and other parameters, an ideal acceleration trajectory can be obtained.

[0113] When t = τ, the initial value of θ r (t), is:

[0114]

[0115] θ r (t), are the load swing angle and angular velocity when the trolley is uniformly accelerated, respectively.

[0116] The time used for trolley uniform acceleration:

[0117]

[0118] Where ω n is the natural frequency of the system.

[0119]

[0120] Where

[0121] The time used for trolley uniform speed:

[0122]

[0123] The maximum speed of the trolley:

[0124]

[0125] Where

[0126] The maximum acceleration of the trolley:

[0127]

[0128] Step five, design the finite-time convergent sliding mode surface:

[0129] Based on the dynamic equation and the control objective, the following sliding mode surface is established:

[0130]

[0131] where 0 < σ2< 1, sgn(·) is the sign function, and · can represent any function:

[0132]

[0133] From equation (5) in step one, we have:

[0134]

[0135] Taking the derivative of the sliding mode surface s and substituting equation (24) into it, we get:

[0136]

[0137] After selecting the sliding mode surface, the next step is to select the driving force F of the trolley that allows the error vector to reach the sliding mode surface. To do this, the design of the driving force of the trolley should satisfy the following condition, also known as the reaching condition:

[0138]

[0139] In order to satisfy condition (26), The traditional design method is usually chosen, as follows:

[0140]

[0141] where is a positive control gain.

[0142] Step six, design the finite-time tracking controller:

[0143] In order to select the driving force F of the trolley that allows the error vector to reach the sliding mode surface, i.e. to design the corresponding finite-time tracking controller F that allows the error vector to reach the sliding mode surface, equations (25) and (27) in step five are combined to get:

[0144]

[0145] The finite-time tracking controller is designed according to (28) as follows:

[0146]

[0147] This controller can achieve precise tracking of the trajectory in step two in finite time.

[0148] Step seven, analysis of the finite time convergence of the present application:

[0149] For the present application, the system is analyzed for finite time convergence to prove that the designed control method makes the trolley move to the specified position in finite time and effectively eliminates the load swing, that is, achieves the control goal.

[0150] The following Lyapunov function is introduced:

[0151]

[0152] Take the first-order derivative of the above formula (30) and substitute formula (25) of step five and formula (29) of step six, to obtain:

[0153]

[0154] From this, it can be concluded that s tends to 0 in finite time. Further, the following conclusions can be drawn:

[0155] In finite time

[0156] When , the following can be obtained:

[0157]

[0158] In summary, in finite time:

[0159]

[0160] Further, it can be concluded that in finite time:

[0161]

[0162] Finite time tracking can be achieved.

[0163] The above conclusions can be transformed into:

[0164]

[0165] In finite time, the trolley moves to the specified position and effectively eliminates the load swing, that is, achieves the control goal.

[0166] Step eight, implementation of the control method:

[0167] An incremental 4000 PPR encoder is added to the motor tail of the bridge crane. The incremental encoder is a sensor that can measure the position of rotary or linear displacement movement, and can convert the displacement of the trolley into a periodic electrical signal, and then convert the electrical signal into a counting pulse. The encoder can generate 4000 pulse signals, and the number of pulses represents the size of the displacement, which can convey very accurate position, angle or speed information. The size of the displacement of the trolley on the bridge is measured by the incremental encoder, and then the required control of the trolley displacement and the signal size of the load swing are calculated by the finite time tracking controller (formula (29)) of the bridge crane, and then the trolley is moved by the motor control of the bridge crane, thereby indirectly controlling the load swing, thereby completing the control target. It should be noted that the model and installation method of the above-mentioned incremental encoder are prior art and can be easily obtained from the market, and the structure and implementation principle are not described here. In addition, when using the incremental encoder, the absolute position cannot be known at the start, so there may be a starting position error. In order to solve this problem, a zero point calibrator or other absolute position encoder auxiliary calibration is required.

[0168] Experiment:

[0169] In order to test the control performance of the control method proposed in the present application, two groups of simulation experiments were carried out: the first group, simulation 1, is the trolley positioning control with zero initial condition; the second group, simulation 2, is the robustness test of uncertain parameters.

[0170] Simulation 1, trolley positioning control simulation with zero initial condition

[0171] The simulation parameter values are set as follows: M = 20 kg, m = 2 kg, l = 0.5 m, g = 9.8 m / s 2 , p dx = 2 m, k s = 310, σ1 = 3, σ2 = 0.03, θ ub = 4°, v ub = 0.4 m / s, a ub = 0.5 m / s 2 .

[0172] The simulation 1 result of the bridge crane system model is shown in Figure 2 , the trolley reaches the target position within 7s, the load swing amplitude is about 3°, and the load stops swinging within 7s. It can be seen that the control algorithm of the present application well meets the control requirements, and can quickly make the trolley reach the specified position along the preset trajectory and effectively suppress the swing of the load.

[0173] Simulation 2, robustness test of uncertain parameters

[0174] The system parameters are different from simulation 1. Compared with simulation 1, the parameters of the bridge crane system model are changed as M = 22 kg, m = 2.3 kg, and l = 0.6 m, and other parameters are not changed, and the simulation results are shown in Figure 3 The trajectory of the trolley within 7 s meets the constraints of the system to reach the designated position, and the load swings within the preset range and has no residual swing. Therefore, in the case of changing the parameters of the bridge crane system model, the finite time tracking control method of the present application can still achieve the control target well.

[0175] In summary, the method designed by the present application has good control effect on the positioning control of the trolley and the suppression of load swing, and has good robustness in the case of parameter uncertainty.

[0176] Finally, it should be noted that the above enumeration is only a few specific embodiments of the present application. Obviously, the present application is not limited to the above embodiments, but can also have many variations. All variations that can be directly derived or inferred from the disclosed content by those of ordinary skill in the art should be considered as falling within the scope of the present application.

Claims

1. A finite-time tracking control method for an underactuated bridge crane, characterized in that... The process is as follows: An incremental encoder is installed at the tail of the bridge crane motor. The incremental encoder measures the displacement of the trolley. Then, the finite-time tracking controller of the bridge crane calculates the required trolley displacement and the magnitude of the load swing signal. The bridge crane motor then controls the trolley to move to the target position, thereby indirectly controlling the load swing. The method for establishing the finite-time tracking controller includes: S1. Establish the trolley dynamics equations for the system model of the bridge crane; S2. Establish a smooth acceleration motion trajectory for the bridge crane trolley; S3. The control objective of the finite-time trajectory tracking control method is to ensure that the position, speed and acceleration of the bridge crane trolley converge to the target position, speed and acceleration within a finite time, while quickly eliminating the load swing angle, angular velocity and angular acceleration. S4. Determine the actual parameters of the trolley's acceleration trajectory; S5. Establish a finite-time tracking controller based on the finite-time convergent sliding surface to accurately track the smooth acceleration motion trajectory established in step S2 within a finite time. The finite-time convergence sliding surface mentioned in step S5 is: Where 0 < σ² < 1, sgn(·) is a symbolic function, and · can represent any function: From equation (5), we can derive: Differentiate the sliding surface s and substitute equation (24) into: The driving force F of the trolley that allows the error vector to reach the sliding surface should satisfy the convergence condition: In order to satisfy condition (26), as follows: in, It is a positive control gain; Combining equations (25) and (27), we get: The finite-time tracking controller is obtained according to equation (28):

2. The finite-time tracking control method for an underactuated bridge crane according to claim 1, characterized in that: The trolley dynamics equations mentioned in step S1 are as follows: Where M is the mass of the trolley, m is the load mass, l is the length of the lifting rope, and x is the displacement of the trolley. For the speed of the trolley, Let θ be the acceleration of the trolley, and θ be the angle of the load's swing. The angular velocity of the load swing. Let F be the angular acceleration of the load swinging, F be the driving force acting on the trolley, and g be the gravitational acceleration. Dividing both sides of equation (2) by ml, we get the equation: Therefore, we get: Substituting equation (4) into equation (1), we obtain the following equation: Where, m(θ) and These are all auxiliary functions, specifically expressed as follows: m(θ)=M+msin 2 i (6) 3. The finite-time tracking control method for an underactuated bridge crane according to claim 2, characterized in that: The smooth acceleration trajectory of the bridge crane trolley in step S2 is as follows: Where τ∈(0,T1 / 4), t1=τ, t2=τ+t a t3=2τ+t a t4=2τ+t a +t c t5=3τ+t a +t c t6=3τ+2t a +t c t7=4τ+2t a +t c a max τ and t represent the maximum acceleration used in the actual transportation process. a t c These represent the time constants for variable acceleration (variable deceleration), uniform acceleration (uniform deceleration), and uniform velocity, respectively, while T1 is the vibration period of the load under constant acceleration. An ideal acceleration trajectory should meet the following performance indicators: in T represents the target location, and T represents the arrival time of the vehicle. These are the maximum speed and maximum acceleration that the trolley can achieve, respectively. It is the maximum load swing angle that the system can allow.

4. The finite-time tracking control method for an underactuated bridge crane according to claim 3, characterized in that: The control objective mentioned in step S3 is specifically: in, x r , These are respectively the preset position motion trajectory, velocity motion trajectory, and acceleration motion trajectory; To achieve the control objective (9), the tracking error of the trolley is defined as follows: in These are the first and second derivatives of the trolley tracking error with respect to time, respectively. The control objective of the finite-time trajectory tracking control method is written as:

5. The finite-time tracking control method for an underactuated bridge crane according to claim 4, characterized in that: The actual parameters mentioned in step S4 are as follows: When t = τ, we can obtain θ when the trolley accelerates uniformly. r (t), The initial value is: θ r (t), These are: the load swing angle and angular velocity of the trolley during uniform acceleration, ω n The natural frequency of the system; Time taken for the trolley to accelerate uniformly: in: in Time taken for the trolley to maintain a constant speed: Maximum speed of the trolley: in Maximum acceleration of the trolley:

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