A strict safety trajectory optimization control method and system for an underactuated bridge crane

Abstract

The application discloses a strict safety trajectory optimization control method and system of an underactuated bridge crane, and the method comprises the following steps: constructing a bridge crane system variable rope length dynamics equation model; setting end point constraints, speed constraints and acceleration constraints for a given initial trolley and rope length desired operation trajectory; optimizing the trolley reference speed and the rope length reference speed based on the end point constraints, the speed constraints and the acceleration constraints; using a first control barrier function to perform angle constraints, setting the upper limit and the lower limit of the swing angle state variable; using a second control barrier function to perform obstacle avoidance constraints; obtaining a cost function according to the first control barrier function and the second control barrier function; and obtaining the control input mu that satisfies the angle constraints and the obstacle avoidance constraints after correction through the solution of a QP problem. The strict safety trajectory optimization control method and system of the underactuated bridge crane provided by the application can effectively avoid obstacles and reduce the swing of the load while optimizing the operation trajectory.

Description

TECHNICAL FIELD

[0001] The present application relates to the field of hoisting system transportation, in particular to a strict safety trajectory optimization control method and system for under-actuated bridge cranes. BACKGROUND

[0002] In the actual operation of a bridge crane, the trolley needs to accurately reach the specified target position, therefore, the given desired running trajectory needs to be optimized in real time to ensure that the trolley can quickly and accurately reach the target position. Meanwhile, during the movement, there are usually obstacles such as truck vehicles and containers in the yard where the crane is operating, which may affect the normal operation of the crane. Therefore, the desired running trajectory optimization also needs to have the ability to avoid obstacles.

[0003] Therefore, it is necessary to provide a strict safety trajectory optimization control method and system for under-actuated bridge cranes to solve the above problems. SUMMARY

[0004] The present application provides a strict safety trajectory optimization control method and system for under-actuated bridge cranes, which can effectively avoid obstacles and reduce the swing of the load while optimizing the running trajectory.

[0005] In a first aspect, the present application provides a strict safety trajectory optimization control method for under-actuated bridge cranes, comprising the following steps:

[0006] Constructing a bridge crane system variable rope length dynamics equation model;

[0007] Setting end point constraints, speed constraints and acceleration constraints for the given initial trolley and rope length desired running trajectory;

[0008] Optimizing the trolley reference speed and rope length reference speed based on the end point constraints, speed constraints and acceleration constraints, respectively;

[0009] Using a first control barrier function for angle constraint, setting the upper and lower bounds of the swing angle state variable;

[0010] Using a second control barrier function for obstacle avoidance constraint;

[0011] Obtaining a cost function according to the first control barrier function and the second control barrier function;

[0012] Obtaining the control input after correction which satisfies the angle constraint and the obstacle avoidance constraint through the solution of the QP problem .

[0013] Preferably, the dynamics equation model is:

[0014]

[0015] where M is the mass of the trolley, m is the mass of the load, l is the rope length, is the swing angle, is the driving force, is the rope tension.

[0016] Preferably, the end-point constraint is expressed as:

[0017]

[0018] where, is the target position of the trolley, is the target position of the rope length, is the velocity of the trolley, is the velocity of the rope, is the preset bounded time;

[0019] The velocity and acceleration constraints are expressed as:

[0020]

[0021] where, is the maximum value of the trolley velocity and acceleration, is the maximum value of the rope length velocity.

[0022] Preferably, the optimization of the trolley reference velocity and the rope length reference velocity comprises:

[0023]

[0024] The above formula is differentiated to obtain a formula about the trolley reference acceleration :

[0025]

[0026] where, is a normal number, is a positive odd number, and satisfies .

[0027] Preferably, the angle constraint using the first control barrier function comprises:

[0028] Using the approximation processing, we obtain:

[0029]

[0030] The is brought in, and is defined as , which is simplified to the following form:

[0031]

[0032] wherein, , is the trolley reference acceleration obtained based on the reference trajectory and the rope length reference velocity .

[0033] Preferably, the setting the upper bound and lower bound of the swing angle state variable comprises:

[0034] For the upper bound of the swing angle state variable, the first control barrier function is designed as follows:

[0035]

[0036] For the lower bound of the swing angle state variable, the first control barrier function is designed as follows:

[0037]

[0038] wherein, are both normal numbers, are both normal numbers, is a preset allowed maximum swing angle constant.

[0039] Preferably, the obstacle avoidance constraint using the second control barrier function comprises:

[0040] The second control barrier function is designed as follows:

[0041]

[0042] wherein, is the spatial position coordinate of the load, is the geometric center of the obstacle, is equal to the inflation coefficient multiplied by the physical radius of the actual obstacle, , ;

[0043] The following inequality is obtained:

[0044]

[0045] wherein, is a normal number, is the position of the trolley, is the rope length, is the overall height of the gantry in the yard.

[0046] Preferably, the cost function is obtained according to the first control barrier function and the second control barrier function as follows:

[0047]

[0048] wherein, is a control input, , is a reference control input, , is a trolley reference acceleration based on the reference trajectory and a rope length reference velocity .

[0049] Preferably, the cost function is converted into a vector form, which is as follows:

[0050]

[0051] wherein, is a second-order identity matrix.

[0052] In a second aspect, the present application also provides a strict safety trajectory optimization control system of an underactuated bridge crane, comprising: a server, the server comprising a memory, a processor and a computer program stored in the memory and executable on the processor, and the processor implements the above method when executing the program.

[0053] The present application has the following beneficial effects compared with the prior art: the strict safety trajectory optimization control method and system of the underactuated bridge crane provided by the present application, the method comprising the following steps: constructing a bridge crane system variable rope length dynamics equation model; setting end point constraints, velocity constraints and acceleration constraints for a given initial trolley and rope length desired operation trajectory; optimizing the trolley reference velocity and the rope length reference velocity based on the end point constraints, the velocity constraints and the acceleration constraints respectively; using a first control barrier function to perform angle constraint, setting the upper and lower bounds of the swing angle state variable; using a second control barrier function to perform obstacle avoidance constraint; obtaining a cost function according to the first control barrier function and the second control barrier function; obtaining the control input after correction which satisfies the angle constraint and the obstacle avoidance constraint through solving of a QP problem , while optimizing the operation trajectory, the obstacles can be effectively avoided and the swing of the load can be reduced. BRIEF DESCRIPTION OF DRAWINGS

[0054] The accompanying drawings, which are incorporated in and constitute a part of the specification, illustrate embodiments consistent with the present application and, together with the description, serve to explain the principles of the present application.

[0055] Figure 1 It is a schematic diagram of dynamics analysis of the bridge crane system in the embodiments of the present application;

[0056] Figure 2 It is a flowchart of the strict safety trajectory optimization control method of the underactuated bridge crane in the embodiments of the present application;

[0057] Figure 3The simulation diagram of the algorithm based on the position of the trolley for the strict safety trajectory optimization control method of the underactuated bridge crane in the embodiment of the present application is compared;

[0058] Figure 4 The simulation diagram of the algorithm based on the speed of the trolley for the strict safety trajectory optimization control method of the underactuated bridge crane in the embodiment of the present application is compared;

[0059] Figure 5 The simulation diagram of the algorithm based on the load swing angle for the strict safety trajectory optimization control method of the underactuated bridge crane in another embodiment of the present application is compared;

[0060] Figure 6 The simulation diagram of the algorithm based on the position of the load space for the strict safety trajectory optimization control method of the underactuated bridge crane in the embodiment of the present application is compared;

[0061] Figure 7 The simulation diagram of the algorithm based on the length of the rope for the strict safety trajectory optimization control method of the underactuated bridge crane in another embodiment of the present application is compared.

[0062] Through the above-mentioned drawings, the specific embodiments of the present application have been shown, and more detailed descriptions will be given hereinafter. These drawings and textual descriptions are not intended to limit the scope of the concept of the present application in any way, but to illustrate the concept of the present application to those skilled in the art by referring to specific embodiments. DETAILED DESCRIPTION

[0063] The exemplary embodiments will be described in detail herein with reference to the accompanying drawings. In the following description, unless otherwise indicated, the same numbers in different drawings represent the same or similar elements. The implementations described in the following exemplary embodiments are not meant to represent all implementations consistent with the present application. Rather, they are merely examples of apparatuses and methods consistent with some aspects of the present application as detailed in the appended claims.

[0064] To solve the above-mentioned problems, the embodiments provided by the present application provide a strict safety trajectory optimization control method and system for an underactuated bridge crane. In view of the actual field working condition requirements, firstly, the state feedback is used to perform online optimization for the initial desired running trajectory. In combination with the design of the control barrier function, the final obtained load trajectory can be limited in an acceptable range, and meanwhile, it can also ensure that the load can successfully avoid the obstacles.

[0065] The research subject mentioned in the present application is the trolley, the load generally refers to the spreader or container, and the trajectory optimization mentioned refers to the running trajectory of the trolley and the change trajectory of the length of the rope.

[0066] (1) System model

[0067] Figure 1 Fig. 1 is a schematic diagram of a bridge crane system for the dynamic analysis in the embodiments of the present application, Figure 2 Fig. 2 is a flowchart of a strict safety trajectory optimization control method for an underactuated bridge crane in the embodiments of the present application. Now referring to Figure 1 and Figure 2 The embodiments of the present application provide a strict safety trajectory optimization control method for an underactuated bridge crane, comprising the following steps:

[0068] S101: constructing a rope length dynamics equation model of the bridge crane system;

[0069] S102: setting end point constraints, speed constraints and acceleration constraints for a given initial trolley and rope length desired operation trajectory;

[0070] S103: optimizing the trolley reference speed and rope length reference speed based on the end point constraints, speed constraints and acceleration constraints, respectively;

[0071] S104: using a first control barrier function for angle constraints, setting the upper and lower bounds of the swing angle state variable;

[0072] S105: using a second control barrier function for obstacle avoidance constraints;

[0073] S106: obtaining a cost function according to the first control barrier function and the second control barrier function;

[0074] S107: obtaining the control input after correction that satisfies the angle constraints and obstacle avoidance constraints through the solution of a QP problem .

[0075] Specifically, the reference trolley acceleration and the reference rope speed are taken as the reference input of the system . On the basis of the above reference input, a minimum correction amount based on the reference control input, i.e. the change amount, needs to be solved to simultaneously satisfy the above angle constraints and obstacle avoidance constraints, and the minimum correction amount is taken as the optimization objective to obtain the corrected control input.

[0076] By combining the first control barrier function designed based on angle constraint and the second control barrier function designed based on obstacle avoidance constraint, a set of inequality constraints is constructed. Based on the inequality set, a quadratic programming (QP) optimization problem is established by taking the minimum correction of control input as the optimization objective, and the Hildreth algorithm is used for efficient solution. The algorithm can realize the fast convergence of the optimal solution without directly solving the complex linear system through the original-dual transformation and iterative algorithm. Thus, the control input after correction can be solved to meet the angle constraint and obstacle avoidance constraint .

[0077] The bridge crane system is modeled as follows:

[0078]

[0079] where M is the mass of the trolley, m is the mass of the load, l is the rope length, is the swing angle, is the driving force, is the rope tension.

[0080] Rewriting (3) can obtain the following expression:

[0081]

[0082] where , .

[0083] By defining , (3) can be written as follows:

[0084]

[0085] where

[0086] (2) Trajectory optimization

[0087] For a given initial trolley and rope length desired trajectory , the following constraints need to be met:

[0088] 1) End point constraint:

[0089]

[0090] where is the target position of the trolley and rope length, is the trolley and rope speed, is the artificially set bounded time.

[0091] 2) Velocity / acceleration constraints:

[0092]

[0093] in, It represents the maximum value of the car's speed and acceleration. It is the maximum value of the speed along the rope length.

[0094] Based on the above constraints, the reference speed for the car is given. and rope length reference speed Optimization methods:

[0095]

[0096] in, It is a positive integer, and its value range is (0,1). It is a positive odd number and satisfies .generally The value can be 3, 5, or 7. The value can be 5, 7, or 9.

[0097] Differentiating (6) and (7) yields the reference acceleration of the car. The expression:

[0098]

[0099] Define the error signal:

[0100]

[0101] Substituting the error signal into (7), we get:

[0102]

[0103] According to the nonsmooth control theorem It will converge to 0 in a finite amount of time, meaning the rope length can converge to the desired position in a finite amount of time.

[0104] Theorem 1 suggests that the optimization method can be the reachability of the vehicle's position to the target position.

[0105] Theorem 1: Design the following Lyapunov function

[0106]

[0107] in, At the same time, it can be noted Heng was established.

[0108] right Perform differentiation.

[0109]

[0110] Substituting (4) and the error signal into the above equation, we have

[0111]

[0112] Further, we have

[0113]

[0114] Finally, we have the following inequality

[0115]

[0116] By LaSalle invariance principle, we have

[0117]

[0118] If , then ; substituting it into (4), we have which is always true; thus, substituting it into (6), we can prove that the trolley position can reach the target position in finite time.

[0119] (3) Control barrier function design

[0120] I Angle constraint

[0121] By using , we have

[0122]

[0123] Substituting the above equation into (3), we define , which can be simplified as

[0124] where

[0125] , , is the trolley reference acceleration based on the reference trajectory and the rope length reference velocity .

[0126] The time-varying function of the upper bound of the first control barrier function in this application is , which are all positive constants, is a positive constant, the negative lower bound and the positive upper bound have opposite signs. The advantage of using the exponential function as the bound is that, due to the safety limit imposed, the payload will eventually converge to a smaller region.

[0127] For the upper bound of the swing angle state variable, the first control barrier function is designed as follows:

[0128]

[0129] For the lower bound of the swing angle state variable, the first control barrier function is designed as follows:

[0130]

[0131] wherein, are all positive constants, are all positive constants, is a preset allowed maximum swing angle constant.

[0132] Lemma 1: If are satisfied, it can be guaranteed that are satisfied.

[0133] II Obstacle Avoidance Constraint

[0134] Assumption 1: The obstacles considered in this application are circular obstacles with fixed shape, fixed size, and fixed position, and the parameters of these obstacles are all known.

[0135] Definition is the geometric center of the obstacle, is the radius of the obstacle. In addition, for safety considerations, an inflation coefficient is introduced as a safety boundary for the shape of the obstacle.

[0136] Therefore, the second control barrier function for obstacle avoidance is designed as follows:

[0137]

[0138] wherein, is the spatial position coordinate of the load, is the geometric center of the obstacle, is equal to the inflation coefficient multiplied by the physical radius of the actual obstacle, , , so as to convert the spatial coordinate into the trolley position-rope length coordinate.

[0139] According to Lemma 2, the following inequality can be obtained:

[0140]

[0141] wherein, is a normal number, and the value range is (0, 10), the expansion coefficient The value is based on engineering experience, and the value range is (1.1, 1.8), is the position of the trolley, is the rope length, is the overall height of the gantry in the yard.

[0142] According to the first control barrier function and the first control barrier function, the following cost function can be obtained:

[0143]

[0144] wherein, is the control input, , is the reference control input, , is the reference acceleration of the trolley based on the reference trajectory and the rope length reference speed .

[0145] The cost function is converted into a vector form, and the vector form is as follows:

[0146]

[0147] wherein, is a second-order unit matrix.

[0148] By solving the QP problem, the control input after correction that satisfies the angle constraint and the obstacle avoidance constraint is obtained to ensure that the angle constraint and the obstacle avoidance constraint can be guaranteed at the same time during the optimization of the trajectory.

[0149] The following algorithm verification is performed on the above control method and system.

[0150] The simulation time of the algorithm is set to 30 seconds, the initial state of the system is set to , the target position of the trolley is set to , and the target position of the rope length is set to .

[0151] The initial trajectory is defined as follows:

[0152]

[0153] The optimization parameter is designed as .

[0154] The parameters in the design of the first control barrier function and the second control barrier function are selected as .

[0155] The acceleration due to gravity is .

[0156] The trajectory optimization method considering the combination of the first control barrier function or the second control barrier function is denoted as OATP in the simulation legend, and the rest of the methods only consider the angle constraint or the obstacle avoidance constraint. The method combining the two constraints with the trajectory optimization method is denoted as OATP with both CBF.

[0157] Now referring to Figure 3 Compared with other methods, the method proposed in the present application has excellent reachability of the specified position. The trolley shows high efficient motion performance, and can quickly reach the required position in about 15 seconds, accompanied by stable trajectory control, with minimal oscillation and extremely short settling time, which shows the effectiveness of the proposed method in achieving fast and stable positioning of the trolley.

[0158] Now referring to Figure 4 Due to the active obstacle avoidance measures taken, the speed of the trolley will initially decrease. Despite the initial speed reduction, the dynamic characteristics of the system result in a small overshoot during the steady state period.

[0159] Now referring to Figure 5 The payload swing angle is always maintained within the specified range, while under other algorithms, the angle state significantly exceeds the specified range for about 3 seconds, with a deviation of about 1.5 degrees. In addition, during the steady state operation, the proposed algorithm shows that the region to which the load converges is much smaller than that of other methods.

[0160] Now referring to Figure 6 As can be clearly seen from the load trajectory described therein, the proposed method effectively avoids obstacles while minimizing the swing of the payload. This phenomenon highlights the trade-off between avoiding obstacles and maintaining precise trajectory control, demonstrating the adaptive ability of the method.

[0161] Now referring to Figure 7 The rope length quickly converges to the required length with minimal oscillation. In contrast, the method combining the first control barrier function exhibits a significant overshoot, taking about 6 seconds to converge to the specified point.

[0162] In summary, the strict safety trajectory optimization control method and system of the under-actuated bridge crane provided by the embodiments of the present application, the method comprises the following steps: constructing a bridge crane system variable rope length dynamics equation model; setting end point constraints, speed constraints and acceleration constraints for a given initial trolley and rope length desired operation trajectory; optimizing the trolley reference speed and the rope length reference speed based on the end point constraints, the speed constraints and the acceleration constraints; using a first control barrier function to perform angle constraints, setting the upper and lower bounds of the swing angle state variable; using a second control barrier function to perform obstacle avoidance constraints; obtaining a cost function according to the first control barrier function and the second control barrier function; obtaining the control input after correction which satisfies the angle constraints and the obstacle avoidance constraints through the solution of the QP problem, while optimizing the operation trajectory, the obstacles can be effectively avoided and the swing of the load can be reduced.

[0163] Other embodiments of the application will be apparent to those skilled in the art from consideration of the specification and practice of the application disclosed herein. It is intended that the specification and examples be considered as exemplary only, with the true scope and spirit of the application being indicated by the following claims.

[0164] It should be understood that the application is not limited to the precise construction that has been described above and shown in the accompanying drawings, and that various modifications and changes can be made by those skilled in the art without departing from the scope of the application. The scope of the application is limited only by the appended claims.

Claims

1. A method for strict safety trajectory optimization control of an underactuated gantry crane, characterized in that, The method comprises the following steps: constructing a bridge crane system variable rope length dynamics equation model; setting end point constraints, speed constraints and acceleration constraints for a given initial trolley and rope length desired operation trajectory; optimizing the trolley reference speed and rope length reference speed based on the end point constraints, speed constraints and acceleration constraints respectively; using a first control barrier function to perform angle constraints, setting upper and lower bounds of the swing angle state variable; using a second control barrier function to perform obstacle avoidance constraints; obtaining a cost function according to the first control barrier function and the second control barrier function; obtaining a control input that satisfies the angle constraints and the obstacle avoidance constraints after correction through solution of a quadratic programming optimization problem; the angle constraints using the first control barrier function comprise: By using For the approximation process, we obtain: ， The Bringing in, defining , which is reduced to the following form: ， wherein, is a trolley reference acceleration obtained based on the reference trajectory and a rope length reference velocity ; the upper and lower bounds of the swing angle state variable comprise: for the upper bound of the swing angle state variable, the first control barrier function is designed as follows: ， for the lower bound of the swing angle state variable, the first control barrier function is designed as follows: ， wherein, are all normal numbers, are all normal numbers, is a preset allowed maximum swing constant; the obstacle avoidance constraints using the second control barrier function comprise: the second control barrier function is designed as follows: ， wherein, and are spatial position coordinates of the load, is the geometric center of the obstacle, is equal to the expansion coefficient multiplied by the physical radius of the actual obstacle, , ; the following inequality is obtained: ， wherein, is a normal number, is the position of the trolley, is the rope length, L is the overall height of the gantry in the yard.

2. The strict safety trajectory optimization control method for the under-actuated bridge crane according to claim 1, characterized in that, the dynamics equation model is: ， where M is the mass of the trolley, m is the mass of the load, l is the rope length, is the swing angle, is the driving force, is the rope tension, is the gravitational acceleration.

3. The strict safety trajectory optimization control method for the under-actuated bridge crane according to claim 2, characterized in that, the end point constraints are expressed as: ， wherein, is a target position of the trolley, is a target position of the rope length, is a velocity of the trolley, is a velocity of the rope, is a preset bounded time; the speed and acceleration constraints are expressed as: ， wherein is the maximum value of the trolley speed and acceleration, is the maximum value of the rope length speed.

4. The method of strict security trajectory optimization control of an underactuated gantry crane according to claim 3, characterized in that, the optimization of the trolley reference speed and the rope length reference speed comprises: ， Taking the derivative of the above equation, the equation for the trolley reference acceleration is obtained: ， wherein, is the velocity of the trolley, is the reference trajectory position of the trolley, is the position of the trolley, is the velocity of the rope, is the reference trajectory position of the rope, is the position of the rope, M is the mass of the trolley, is a positive constant, is a positive odd integer, and satisfies .

5. The method of strict security trajectory optimization control of an underactuated gantry crane according to claim 1, characterized in that, the cost function obtained according to the first control barrier function and the second control barrier function is as follows: ， wherein is a control input, , is a reference control input, , is a trolley reference acceleration based on the reference trajectory and a rope length reference velocity .

6. The method of strict security trajectory optimization control of an underactuated gantry crane according to claim 5, characterized in that, the cost function is converted into a vector form, and the vector form is as follows: ， wherein is a second order identity matrix.

7. A strict safety trajectory optimization control system for an underactuated gantry crane, characterized in that, comprise: a server comprising a memory, a processor and a computer program stored on the memory and executable on the processor, the processor implementing the method of any one of claims 1-6 when executing the program.

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