Strict safety track optimization control method and system for under-actuated bridge crane

Abstract

The invention discloses a strict safety trajectory optimization control method and system for an under-actuated bridge crane. The method comprises the following steps: constructing a variable rope length kinetic equation model of the bridge crane; setting an end point constraint, a speed constraint and an acceleration constraint for a given initial trolley and rope length expected running track; respectively optimizing a trolley reference speed and a rope length reference speed based on the end point constraint, the speed constraint and the acceleration constraint; performing angle constraint by using a first control obstacle function, and setting an upper bound and a lower bound of a swing angle state variable; performing obstacle avoidance constraint by using a second control obstacle function; obtaining a cost function according to the first control barrier function and the second control barrier function; and solving the QP problem to obtain the corrected control input mu meeting the angle constraint and the obstacle avoidance constraint. According to the strict safe trajectory optimization control method and system for the under-actuated bridge crane, the moving trajectory is optimized, meanwhile, obstacles can be effectively avoided, and swing of the load is reduced.

Description

Technical Field

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

[0002] In actual bridge crane operations, the trolley must accurately reach the designated target location. Therefore, the desired trajectory must be optimized in real time to ensure it reaches the target quickly and accurately. Furthermore, obstacles such as trucks and containers often exist in the crane's operating yard during movement, potentially hindering the crane's normal operation. Therefore, optimizing the desired trajectory also requires obstacle avoidance capabilities.

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

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

[0005] In a first aspect, the present application provides a strict safety trajectory optimization control method for an underactuated bridge crane, comprising the following steps:

[0006] Construct the dynamic equation model of the bridge crane system with variable rope length;

[0007] For the given initial trolley and rope length, set the endpoint constraint, velocity constraint, and acceleration constraint for the desired trajectory.

[0008] Based on the endpoint constraint, speed constraint and acceleration constraint, the trolley reference speed and the rope length reference speed are optimized respectively;

[0009] Use the first control barrier function to perform angle constraint and set the upper and lower bounds of the swing angle state variable;

[0010] Use the second control obstacle function to perform obstacle avoidance constraints;

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

[0012] By solving the QP problem, the control input μ that satisfies the angle constraint and obstacle avoidance constraint is obtained after correction.

[0013] Preferably, the kinetic equation model is:

[0014]

[0015] Where M is the mass of the trolley, m is the mass of the load, l is the length of the rope, θ is the swing angle, and F x is the driving force, F l is the rope tension.

[0016] Preferably, the endpoint constraint is expressed as:

[0017]

[0018] Among them, x d is the target position of the car, l d is the target position of the rope length, is the speed of the car, is the speed of the rope, t f1 ,t f2 It is a preset bounded time;

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

[0020]

[0021] Among them, v max ,a max is the maximum value of the car's speed and acceleration, L vmax is the maximum value of the rope length velocity.

[0022] Preferably, the optimizing of the trolley reference speed and the rope length reference speed comprises:

[0023]

[0024] Derivative the above formula, we can get the reference acceleration of the car The formula is:

[0025]

[0026] Among them, ρ x ,ρ l ,ρ θ is a normal number, p and q are positive odd numbers, and 2q>p>q.

[0027] Preferably, the using the first control obstacle function to perform angle constraint includes:

[0028] Using sinθ≈θ,cosθ≈1, Approximate processing, we get:

[0029]

[0030] The Bring in, define Simplifying it into the following form:

[0031]

[0032] in, μ r is the reference acceleration of the car based on the reference trajectory and rope length reference speed

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

[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] Among them, γ1,γ2,δ1,δ2 are all positive numbers, k,k η are all positive numbers, η ∞ It is the preset maximum permissible swivel angle constant.

[0039] Preferably, the using the second control obstacle function to perform obstacle avoidance constraint includes:

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

[0041] h(x,y)=(x p (t)-x bos ) 2 +(y p (t)-y obs ) 2 -r s 2

[0042] Among them, x p (t) and y p (t) is the spatial position coordinate of the load, (x obs ,y obs ) is the geometric center of the obstacle, r s Equal to the expansion coefficient multiplied by the actual physical radius of the obstacle, x p (t)=x(t)+l(t)sinθ(t),y p (t) = Ll(t);

[0043] The following inequality is obtained:

[0044]

[0045] Here, α is a positive constant.

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

[0047]

[0048] Where μ is the control input, μ r is the reference control input, μ r is the reference acceleration of the car based on the reference trajectory and rope length reference speed

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

[0050]

[0051] Where I2 is the second-order identity matrix.

[0052] In a second aspect, the present application also provides a strict safety trajectory optimization control system for an under-actuated 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, the processor implementing the above method when executing the program.

[0053] Compared with the prior art, the present application has the following beneficial effects: the present application provides a strict safety trajectory optimization control method and system for an under-actuated bridge crane, the method comprising the following steps: constructing a variable rope length dynamic equation model of a bridge crane system; setting end point constraints, speed constraints and acceleration constraints for a given initial trolley and rope length expected running 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 obstacle function for angle constraints, and setting upper and lower bounds of the swing angle state variable; using a second control obstacle function for obstacle avoidance constraints; obtaining a cost function based on the first and second control obstacle functions; and obtaining a control input μ that satisfies the angle constraints and obstacle avoidance constraints after correction by solving the QP problem, which can effectively avoid obstacles and reduce load swing while optimizing the running trajectory. BRIEF DESCRIPTION OF THE DRAWINGS

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

[0055] Figure 1This is a schematic diagram of the dynamic analysis of the bridge crane system in the embodiment of the present application;

[0056] Figure 2 Flowchart of a strict safety trajectory optimization control method for an under-actuated bridge crane in an embodiment of the present application; Figure 3 This is a simulation diagram of an algorithm comparison based on the trolley position of the strict safety trajectory optimization control method of the under-actuated bridge crane in an embodiment of the present application;

[0057] Figure 4 This is a simulation diagram of an algorithm comparison based on trolley speed of a strict safety trajectory optimization control method for an under-actuated bridge crane in an embodiment of the present application;

[0058] Figure 5 This is a simulation diagram comparing algorithms based on load swing angle of a strict safety trajectory optimization control method for an under-actuated bridge crane in another embodiment of the present application;

[0059] Figure 6 This is a simulation diagram of algorithm comparison based on load spatial position of the strict safety trajectory optimization control method of the under-actuated bridge crane in an embodiment of the present application;

[0060] Figure 7 This is a simulation diagram comparing rope-length-based algorithms for a strict safety trajectory optimization control method for an under-actuated bridge crane in another embodiment of the present application.

[0061] The above drawings illustrate specific embodiments of the present application, which will be described in more detail below. These drawings and the textual description are not intended to limit the scope of the present application in any way, but rather to illustrate the concepts of the present application to those skilled in the art by reference to specific embodiments. DETAILED DESCRIPTION

[0062] Exemplary embodiments will be described in detail herein, with examples illustrated in the accompanying drawings. In the following description, when referring to the drawings, identical numerals in different figures represent identical or similar elements, unless otherwise indicated. The embodiments described in the following exemplary embodiments are not intended to represent all embodiments consistent with the present application. Rather, they are merely examples of apparatus and methods consistent with certain aspects of the present application, as detailed in the appended claims.

[0063] In order to solve the above problems, the embodiments provided in this application provide a strict safety trajectory optimization control method and system for an under-actuated bridge crane. According to the actual working conditions on site, the initial expected operating trajectory is first optimized online using state feedback. For the optimized trajectory, combined with the design of the control obstacle function, the final load trajectory can be limited to an acceptable range, while also ensuring that the load can successfully avoid obstacles.

[0064] The research subject mentioned in this application is the trolley, the load generally refers to a sling or a container, etc., and the trajectory optimization mentioned is aimed at the running trajectory of the trolley and the changing trajectory of the rope length.

[0065] (1) System model

[0066] Figure 1 This is a schematic diagram of the dynamic analysis of the bridge crane system in the embodiment of this application. Figure 2 Flowchart of the strict safety trajectory optimization control method for under-actuated bridge crane in the embodiment of the present application. Figure 1 and Figure 2 The embodiment of the present application provides a strict safety trajectory optimization control method for an under-actuated bridge crane, comprising the following steps:

[0067] S101: Construct a dynamic equation model for a bridge crane with variable rope length;

[0068] S102: Setting endpoint constraints, velocity constraints, and acceleration constraints for the given initial trolley and rope length desired trajectory;

[0069] S103: Optimizing the trolley reference speed and the rope length reference speed based on the endpoint constraint, the speed constraint, and the acceleration constraint respectively;

[0070] S104: Using the first control obstacle function to perform angle constraint, setting the upper and lower bounds of the swing angle state variable;

[0071] S105: Using the second control obstacle function to perform obstacle avoidance constraint;

[0072] S106: Obtaining a cost function according to the first control obstacle function and the second control obstacle function;

[0073] S107: By solving the QP problem, a control input μ that satisfies the angle constraint and the obstacle avoidance constraint after correction is obtained.

[0074] Specifically, the reference trolley acceleration and the reference rope speed are used as the reference inputs of the system. Based on the above reference input, it is necessary to solve a minimum correction amount based on the reference control input, that is, the change amount, to simultaneously satisfy the above angle constraint and obstacle avoidance constraint. This minimum correction amount is regarded as the optimization target to obtain the corrected control input.

[0075] By combining the first control obstacle function designed based on the angle constraint and the second control obstacle function designed based on the obstacle avoidance constraint, a set of inequality constraints are constructed. Based on this set of inequalities, with the minimization of the correction amount of the control input as the optimization goal, a quadratic programming optimization problem, namely the QP (Quadratic Programming, referred to as QP) problem, is established, and the Hildreth algorithm is used for efficient solution. This algorithm can achieve rapid convergence to the optimal solution without directly solving complex linear systems through primal-dual transformation and iterative algorithms. In this way, the control input that satisfies the angle constraint and obstacle avoidance constraint after correction can be solved.

[0076] The dynamic equation modeling of the bridge crane system with variable rope length is as follows:

[0077]

[0078] Where M is the mass of the trolley, m is the mass of the load, l is the length of the rope, θ is the swing angle, and F x is the driving force, F l is the rope tension.

[0079] Rewriting (3) yields the following expression:

[0080]

[0081] in, M θ =ml 2 ,M x = mlcosθ,

[0082] G=mglsinθ, and we have

[0083] By definition Write (3) in the following form:

[0084]

[0085] in,

[0086]

[0087] (2) Trajectory optimization

[0088] For a given initial trolley and rope length, the expected trajectory x i ,l i , the following constraints need to be met:

[0089] 1) End point constraint:

[0090]

[0091] Among them, x d ,l d is the target position of the trolley and the rope length, is the speed of the cart and the rope, t f1 ,t f2 It is a bounded time set by humans.

[0092] 2) Velocity / acceleration constraints:

[0093]

[0094] Among them, v max ,a max is the maximum value of the car's speed and acceleration, L vmax is the maximum value of the rope length velocity.

[0095] Based on the above constraints, the reference speed of the car is given as and rope length reference speed Optimization method:

[0096]

[0097] Among them, ρ x ,ρ l ,ρ θ is a normal number with a value range of (0,1), p and q are positive odd numbers, and 2q>p>q. Usually q is 3, 5, or 7, and p is 5, 7, or 9.

[0098] Derivative (6) and (7) yields the reference acceleration of the car: The expression:

[0099]

[0100] Define the error signal:

[0101] e x =x r -x i ,e l =l r -l i

[0102] Substituting the error signal into (7), we can obtain:

[0103]

[0104] According to the non-smooth control theorem, e l , It will converge to 0 in a finite time, that is, the rope length can converge to the desired position in a finite time.

[0105] Theorem 1 gives the optimization method to achieve the reachability of the car position converging to the target position.

[0106] Theorem 1: Design the following Lyapunov function

[0107]

[0108] in, It can also be noted that V(t)≥0 always holds.

[0109] Taking the derivative of V(t),

[0110]

[0111] Substituting (4) and the error signal inference into the above formula, we can obtain:

[0112]

[0113] Through (9), we can further obtain:

[0114]

[0115] Finally, we can simplify to the following inequality:

[0116]

[0117] Through the LaSalle invariant set principle, we can finally derive the following equation:

[0118]

[0119] like but Substituting it into (4) we get G = 0, that is, l r sinθ=0 always holds true; therefore, substituting it into (6), it can be proved that the car position can reach the target position within a finite time.

[0120] (3) Control Barrier Function Design

[0121] Angle Constraint

[0122] By using sinθ≈θ, cosθ≈1, we can approximate (3) and obtain:

[0123]

[0124] The trajectory optimization part is obtained Bring in, define Simplifying it into the following form:

[0125]

[0126] in, μ r is the reference acceleration of the car based on the reference trajectory and rope length reference speed

[0127] The time-varying function of the positive upper bound of the first control barrier function set in this application is k,k η are all positive numbers, η ∞ The advantage of using an exponential function as a bound is that the payload will eventually converge to a smaller region due to the imposed safety limit.

[0128] For the upper bound of the swing angle state variable, the first control barrier function is designed as follows:

[0129]

[0130] For the lower bound of the swing angle state variable, the first control barrier function is designed as follows:

[0131]

[0132] Among them, γ1,γ2,δ1,δ2 are all positive numbers with a value range of (0,15), k,k η are all positive numbers with a value range of (0,15), η ∞ It is the preset maximum permissible swivel angle constant.

[0133] Lemma 1: If h 1,2 (θ)≥0, and h 2,2 If (θ)≥0 is satisfied at the same time, h can be guaranteed 1,0 (θ)≥0 and h 2,0 (θ)≥0, and both are satisfied.

[0134] II Obstacle Avoidance Constraints

[0135] 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 known.

[0136] Definition (x obs ,y obs ) is the geometric center of the obstacle, r obs is the radius of the obstacle. In addition, for safety reasons, the expansion coefficient r is introduced s A safety boundary that serves as the shape of the obstacle.

[0137] Therefore, the second control obstacle function for obstacle avoidance is designed as follows:

[0138] h(x,y)=(x p (t)-x obs ) 2 +(y p (t)-y obs ) 2 -r s 2

[0139] Among them, x p (t) and y p (t) is the spatial position coordinate of the load, (x obs ,y obs ) is the geometric center of the obstacle, r s Equal to the expansion coefficient multiplied by the actual physical radius of the obstacle, x p (t)=x(t)+l(t)sinθ(t),y p (t) = Ll(t), thereby converting the spatial coordinates into the trolley position-rope length (xl) coordinates.

[0140] According to Lemma 2, we can get the following inequality:

[0141]

[0142] Among them, α is a positive constant with a value range of (0,10), and the expansion coefficient r s The value of is based on engineering experience and ranges from (1.1, 1.8).

[0143] According to the first control barrier function and the second control barrier function, the following cost function can be obtained:

[0144]

[0145] Where μ is the control input, μ r is the reference control input, μ r is the reference acceleration of the car based on the reference trajectory and rope length reference speed

[0146] The cost function is converted into a vector form, which is as follows:

[0147]

[0148] Where I2 is the second-order identity matrix.

[0149] By solving the QP problem, the corrected control input μ that satisfies the angle constraint and obstacle avoidance constraint is obtained, so as to ensure that the angle constraint and obstacle avoidance constraint can be simultaneously guaranteed during the optimization of the trajectory.

[0150] The algorithm verification of the above control method and system is carried out below.

[0151] The simulation time of the algorithm is set to 30 seconds, and the initial state of the system is set to The target position of the car is set to x d =10[m], the target position of the rope length is set to l d =8[m].

[0152] The initial trajectory is defined as follows:

[0153]

[0154] The optimization parameter is designed as ρ x =0.72,ρ θ =0.8,ρ l =0.8,p=11,q=9.

[0155] The parameters in the design of the first and second control barrier functions are selected as k = 0.3, k η =0.15,η ∞ =2[deg].γ1=δ1=10, γ2=δ2=3, α=0.8.

[0156] The acceleration due to gravity is g = 9.8 [m / s 2 ].

[0157] The trajectory optimization method without considering the first control obstacle function or the combination of the second control obstacle function is abbreviated as OATP in the simulation legend. The other methods only consider the angle constraint or the obstacle avoidance constraint. The method that combines the two constraints with the trajectory optimization method is abbreviated as OATP with both CBF.

[0158] See now Figure 3 Compared with other methods, the proposed method has excellent reachability to the designated location. The robot exhibits efficient motion performance, quickly reaching the desired location in approximately 15 seconds, accompanied by smooth trajectory control, characterized by minimal oscillation and extremely short settling time, demonstrating the effectiveness of the proposed method in achieving fast and stable robot positioning.

[0159] See now Figure 4 ,The car ’s speed will initially decrease due to the active obstacle avoidance measures.,Despite the initial speed reduction, the dynamic characteristics of the system,result in less overshoot during the steady-state period.

[0160] See now Figure 5 ,The payload swing angle always remains within the specified range, while under other algorithms, the angle state is obviously outside the specified range for about 3 seconds, with a deviation of about 1.5 degrees. In addition, during steady-state operation, the proposed algorithm shows that the load converges to a much smaller area than other methods.

[0161] See now Figure 6 As evident from the payload trajectory depicted, the proposed method effectively avoids obstacles while minimizing payload sway. This phenomenon highlights the trade-off between avoiding obstacles and maintaining accurate trajectory control, demonstrating the adaptive capability of the proposed method.

[0162] See now Figure 7 , the rope length converges quickly to the desired length with minimal oscillation. In contrast, the method combined with the first control barrier function exhibits significant overshoot and takes about 6 seconds to converge to a specific point.

[0163] In summary, the embodiments of the present application provide a strict safety trajectory optimization control method and system for an under-actuated bridge crane, and the method includes the following steps: constructing a variable rope length dynamic equation model of a bridge crane system; setting end point constraints, speed constraints and acceleration constraints for a given initial trolley and rope length expected running 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 obstacle function to perform angle constraints and set upper and lower bounds of the swing angle state variable; using a second control obstacle function to perform obstacle avoidance constraints; obtaining a cost function based on the first control obstacle function and the second control obstacle function; and obtaining a control input μ that satisfies the angle constraints and obstacle avoidance constraints after correction by solving the QP problem, which can effectively avoid obstacles and reduce the swing of the load while optimizing the running trajectory.

[0164] Those skilled in the art will readily appreciate other embodiments of the present application after considering the specification and practicing the invention disclosed herein. This application is intended to cover any variations, uses, or adaptations of the present application that follow the general principles of the present application and include common knowledge or customary techniques in the art not disclosed herein. The description and examples are to be considered merely as exemplary, and the true scope and spirit of the present application are indicated by the claims.

[0165] It should be understood that the present application is not limited to the exact structure described above and shown in the drawings, and that various modifications and changes may be made without departing from the scope thereof. The scope of the present application is limited only by the appended claims.

Claims

1. A strict safety trajectory optimization control method for an underactuated bridge crane, characterized in that: The steps include: Construct the dynamic equation model of the bridge crane system with variable rope length; For the given initial trolley and rope length, set the endpoint constraint, velocity constraint, and acceleration constraint for the desired trajectory. Based on the endpoint constraint, speed constraint and acceleration constraint, the trolley reference speed and the rope length reference speed are optimized respectively; Use the first control barrier function to perform angle constraint and set the upper and lower bounds of the swing angle state variable; Use the second control obstacle function to perform obstacle avoidance constraints; Obtaining a cost function according to the first control obstacle function and the second control obstacle function; By solving the QP problem, the control input μ that satisfies the angle constraint and obstacle avoidance constraint is obtained after correction.

2. The strict safety trajectory optimization control method for an underactuated overhead crane according to claim 1, characterized in that: The kinetic equation model is: Where M is the mass of the trolley, m is the mass of the load, l is the length of the rope, θ is the swing angle, and F x is the driving force, F l is the rope tension.

3. The strict safety trajectory optimization control method for an underactuated overhead crane according to claim 2, characterized in that: The endpoint constraint is expressed as: Among them, x d is the target position of the car, l d is the target position of the rope length, is the speed of the car, is the speed of the rope, t f1 ,t f2 It is a preset bounded time; The velocity and acceleration constraints are expressed as: Among them, v max ,a max is the maximum value of the car's speed and acceleration, l vmax is the maximum value of the rope length velocity.

4. The strict safety trajectory optimization control method for an underactuated overhead crane according to claim 3, characterized in that: The optimization of the trolley reference speed and the rope length reference speed includes: Derivative the above formula, we can get the reference acceleration of the car The formula is: Among them, ρ x ,ρ l ,ρ θ is a normal number, p and q are positive odd numbers, and 2q>p>q.

5. The strict safety trajectory optimization control method for an underactuated overhead crane according to claim 4, characterized in that: The using the first control obstacle function to perform angle constraint includes: Using sinθ≈θ, cosθ≈1, Approximate processing, we get: The Bring in, define Simplifying it into the following form: in, μ r is the reference acceleration of the car based on the reference trajectory and rope length reference speed 6. The strict safety trajectory optimization control method for an underactuated overhead crane according to claim 5, characterized in that: The upper and lower bounds of the swing angle state variable are set as follows: 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: Among them, γ1,γ2,δ1,δ2 are all positive numbers, k,k η are all positive numbers, η ∞ It is the preset maximum permissible swivel angle constant.

7. The strict safety trajectory optimization control method for an underactuated overhead crane according to claim 2, characterized in that: The use of the second control obstacle function to perform obstacle avoidance constraint includes: The second control barrier function is designed as follows: h(x,y)=(x p (t)-x obs ) 2 +(y p (t)-y obs ) 2 -r s 2 Among them, x p (t) and y p (t) is the spatial position coordinate of the load, (x obs ,y obs ) is the geometric center of the obstacle, r s Equal to the expansion coefficient multiplied by the actual physical radius of the obstacle, x p (t)=x(t)+k(t)sinθ(t),y p (t) = Ll(t); The following inequality is obtained: Here, α is a positive constant.

8. The strict safety trajectory optimization control method for an underactuated overhead crane according to claim 7, characterized in that: The cost function is obtained according to the first control obstacle function and the second control obstacle function as follows: Where μ is the control input, μ r is the reference control input, μ r is the reference acceleration of the car based on the reference trajectory and rope length reference speed 9. The strict safety trajectory optimization control method for an underactuated overhead crane according to claim 8, characterized in that: Convert the cost function into a vector form, which is as follows: Where I2 is the second-order identity matrix.

10. A strict safety trajectory optimization control system for an underactuated bridge crane, characterized in that: include: A server comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor implements the method according to any one of claims 1 to 9 when executing the program.

Images