Fixed-time ship berthing control method based on translation transformation BLF under asymmetric constraints

Abstract

The invention discloses a ship fixed-time berthing control method using a translation transformation BLF under asymmetric constraints, comprising: establishing a ship motion model; calculating an allowable error range for trajectory tracking according to a waterway range and a desired trajectory; performing a translation transformation on the tracking error of the ship trajectory to convert the asymmetric constraint into a symmetric constraint; designing a control rate of a tracking controller with fixed-time convergence; designing a barrier Lyapunov function to prove that the system converges in fixed time and satisfies the asymmetric constraint, and under the action of the tracking controller with fixed-time convergence, the ship trajectory tracking closed-loop system is fixed-time stable at the origin; transmitting the control quantity to the ship actuator to adjust the ship motion; the invention utilizes translation transformation to construct a BLF, converts the asymmetric constraint into a symmetric constraint, and achieves fixed-time convergence, thereby ensuring safety performance under the conditions of model uncertainty and bounded tracking errors of the ship trajectory and speed in response to uncertain external interference.

Description

Technical Field

[0001] The present invention belongs to the technical field of ship control, and in particular relates to a fixed-time ship berthing control method using a translation transformation BLF under asymmetric constraints. Background Art

[0002] Currently, fixed-time convergence control of ships remains a key topic in the field of ship control. Fixed-time convergence means that the system state converges to zero within a bounded fixed time, T. In many practical processes, control systems must complete the control process within a finite time, ensuring rapid control system response and ensuring that the system state variables reach a small neighborhood of the equilibrium point within a finite time.

[0003] In practice, most ships and vehicles are often subject to various state constraints to ensure their safety, such as limited workspace and speed. Once these constraints are violated, dynamic systems may suffer from inaccurate control, system instability, and accidents. Barrier Lyapunov functions (BLFs) can be used to address a class of constrained problems with both safety and stability in mind. The BLF tends to zero or infinity when approaching a safety boundary. By ensuring the boundedness of the BLF, the system state is constrained, allowing the system trajectory to approach the boundary of the safety set without leaving it. The BLF theoretically allows for the development of safe controllers for nonlinear systems. However, it also presents some challenges.

[0004] (1) To ensure the safety of nonlinear systems, it is necessary to study robustness under model uncertainty, system noise, measurement noise, and input disturbances. Due to simplified models of complex interactions within components, there are differences between deterministic system models with precise observations of system states and the real system. If these differences are not accounted for in the control design, unsafe behavior may result under critical conditions. This may also lead to sensitivity to noise and external disturbances when approaching the safety margin.

[0005] (2) Traditional BLFs have disadvantages such as complex structure, difficult differential operations, and symmetric convergence region rules. There is a lack of general methods to establish effective BLFs for use in specific applications. To achieve simultaneous stability and safety, a set of control invariants is required, in which any trajectory of the dynamic system can be maintained indefinitely. For general nonlinear dynamic systems, the Hamilton-Jacobi partial differential equation is a standard tool for computing the invariant set. However, it has exponential complexity and is difficult to calculate.

[0006] (3) In practical situations, various constraints may not satisfy symmetric relationships. For example, for ship berthing, the constraint on trajectory tracking error is based on the special terrain of the dock. This problem has not been addressed in some existing studies. In addition, some existing asymmetric BLFs use discontinuous switching functions.

[0007] Therefore, constructing an asymmetric carrier function with an internal safety set while ensuring its invariance in the presence of interference is a challenging task to achieve both stability and security. Summary of the Invention

[0008] The purpose of this invention is to provide a fixed-time ship berthing control method based on translation transformation BLF under asymmetric constraints. A new asymmetric barrier Lyapunov function (BLF) is constructed by using translation transformation to convert the asymmetric constraint into a symmetric constraint. The continuously differentiable BLF is smooth and has high-order derivatives that converge in fixed time. The designed BLF is C 0 Continuous and first-order derivatives are available. For uncertain external disturbances, an adaptive sliding mode controller based on asymmetric BLF is designed, which can ensure safety performance under model uncertainty and bounded tracking errors of ship trajectory and speed.

[0009] The ship fixed-time berthing control method of the present invention using a translation transformation BLF under asymmetric constraints includes the following steps:

[0010] Step 1: Establish a ship motion model;

[0011] Step 2: Calculate the allowable error range of trajectory tracking based on the water area of ​​the channel and the expected trajectory;

[0012] Step 3: Perform translation transformation on the tracking error of the ship trajectory to convert the asymmetric constraint into a symmetric constraint;

[0013] Step 4: Design the control rate of the tracking controller with fixed time convergence;

[0014] Step 5: Design the barrier Lyapunov function and prove that the system is fixed-time convergent and satisfies the asymmetric constraints. That is, under the action of the fixed-time convergent tracking controller, the ship trajectory tracking closed-loop system is fixed-time stable at the origin.

[0015] Step 6: Transmit the control quantity to the ship's actuator to adjust the ship's motion until the accuracy meets the requirements.

[0016] The step 1 is specifically as follows:

[0017] Assuming that η is the true value of the ship's posture, υ is the velocity vector, u is the forward velocity, v is the yaw velocity, r is the yaw velocity, x is the forward position, y is the yaw position, and ψ is the bow angle, the ship motion model is given by the following equation:

[0018] η&=R(ψ)υ (1)

[0019] Mυ&+C(υ)+Dυ+Δ=τ+τ d (2)

[0020] η=[xy ψ] T (3)

[0021] υ=[uvr] T (4)

[0022] Where M is the weight and hydrodynamic inertia, D is the linear hydrodynamic damping parameter matrix, τ d is the disturbance, τ is the input force and torque, Δ is the uncertainty model, C is the Coriolis and centripetal moment matrix, and R is the rotation matrix, which is given by:

[0023]

[0024]

[0025] The step 2 is specifically as follows:

[0026] In order to ensure the safety of the ship, the tracking error of the ship when berthing is limited to the bounded area near the dock. Assume that the quadrilateral ABCD is the bounded area near the dock. If (x d +x e ,y d +y e )∈quadrilateral ABCD, where x d is the expected value of the forward position, y d is the expected value of the swing position, and the error tolerance range x is obtained based on the bounded area near the dock. e and y e ;

[0027] Ship trajectory tracking refers to tracking and controlling the ship's position within a specified time, reducing the ship's trajectory tracking error to Expressed as:

[0028]

[0029] Among them, η d is the expected value of the ship's posture, and η is the true value of the ship's posture;

[0030] According to the water range of the channel and the expected trajectory, the allowable error range of trajectory tracking is calculated, that is, the tracking error of the ship's trajectory is controlled by the full-state tracking controller at the same time to Limited to:

[0031] a=[ x e , y e , ψ e ] T (8)

[0032]

[0033]

[0034] Among them, a represents the lower limit of the allowable error of the ship's posture, and b represents the upper limit of the allowable error of the posture. x e represents the lower bound of the longitudinal position error, represents the upper bound of the longitudinal position error, y e represents the lower bound of the lateral position error, represents the upper bound of the lateral position error, ψ e represents the lower bound of the heading angle error, Indicates the upper bound of the heading angle error.

[0035] The step 3 is specifically as follows:

[0036] For the established ship motion model, according to the linear transformation of formula (11), the asymmetric constraint of formula (10) is replaced by the symmetric constraint of formula (12), and the tracking error of the ship trajectory in formula (10) is Perform the following translation transformation:

[0037]

[0038] Substituting equation (11) into equation (10), the inequality is written as:

[0039]

[0040] The step 4 is specifically as follows:

[0041] The calculation of constructed dummy variables is as follows:

[0042]

[0043] in, represents the change law of the desired posture, k1 represents the coefficient, and z1 represents the vector after the translation transformation of formula (11);

[0044] remember:

[0045]

[0046] Where α1 represents the dummy variable in formula (14);

[0047] The calculation results of the time derivative of formula (14) are as follows:

[0048]

[0049] Substituting formula (2) into formula (15) yields the following result:

[0050]

[0051] The uncertainty model Δ is expressed as:

[0052] δ=RM -1 τ d -RM -1 Δ (17)

[0053] The estimated error is given by:

[0054]

[0055] The control rate of the full-state tracking controller, i.e., the input force and torque τ, is designed as follows:

[0056]

[0057] Among them, k1>0, k2>0, k3>0, k4>0, 0<α3<1, β3>1, 0<α2<1, β2>1; these are all coefficient parameters in the control law, and γ is also a coefficient parameter in the control law.

[0058] The step 5 is specifically as follows:

[0059] The designed barrier Lyapunov function BLF is:

[0060]

[0061] Based on inequality (12), the following holds:

[0062]

[0063]

[0064] According to the definition, when the function variable approaches the constraint boundary, the BLF value will tend to infinity, which also shows that when the BLF is bounded, the function variable will always remain within the constraint boundary, that is, the constraint is satisfied;

[0065] Take the time derivative of the barrier Lyapunov function BLF:

[0066]

[0067] Substituting formula (16) into formula (23) yields:

[0068]

[0069] Substituting formula (19) into formula (24) yields:

[0070]

[0071] Substituting formula (17) into formula (25) yields:

[0072]

[0073] Substituting formula (18) into formula (26) yields:

[0074]

[0075] From formula (14), we can infer that:

[0076]

[0077] Substituting formula (28) into formula (27) yields:

[0078]

[0079] It can be seen that:

[0080]

[0081] remember:

[0082]

[0083]

[0084] Then formula (30) is transformed into:

[0085]

[0086] in:

[0087]

[0088]

[0089] thus:

[0090]

[0091] in:

[0092]

[0093] A device comprising a processor and a memory; the memory is used to store a computer program; the processor is used to execute any one of the above-mentioned ship fixed-time berthing control methods using a translational transformation BLF under asymmetric constraints according to the computer program.

[0094] A computer-readable storage medium is used to store a computer program, wherein the computer program is used to execute any one of the above-mentioned ship fixed-time berthing control methods using a translational transformation BLF under an asymmetric constraint.

[0095] A chip for running instructions, which is used to execute any of the above-mentioned ship fixed-time berthing control methods using a translation transformation BLF under asymmetric constraints.

[0096] This paper uses translation transformation to construct a new asymmetric barrier Lyapunov function, namely a continuously differentiable barrier Lyapunov function (BLF). It is smooth and has high-order derivatives that converge in a fixed time. The resulting control law has Lipschitz continuity. The designed barrier Lyapunov function is C0 continuous and can be first-order differentiated.

[0097] This paper proposes a controller for continuous systems that reduces tracking error within the bounds of an internal safety set in the presence of uncertain models and disturbances. In this scenario, the controller ensures the safety of the system state and high fixed-time tracking performance. To ensure vessel safety, the tracking error during berthing is confined to a bounded region near the pier. This robust control approach ensures high safety performance despite model uncertainty and bounded tracking error for the vessel trajectory.

[0098] The present invention demonstrates fixed-time convergence using Lyapunov theory. The results demonstrate strong anti-interference capabilities, robustness, and simplicity in adverse sea conditions. It can also handle unknown interference, ensuring safe navigation, and maintains good tracking of the route under wind and current interference in random sea conditions. BRIEF DESCRIPTION OF THE DRAWINGS

[0099] Figure 1 is a flow chart of the present invention;

[0100] Figure 2 This is a working principle diagram of the present invention;

[0101] Figure 3 Schematic diagram of the ship motion model in the present invention;

[0102] Figure 4 This is a schematic diagram of the bounded area near the dock in the present invention;

[0103] Figure 5 Schematic diagram of translation transformation in the present invention;

[0104] Figure 6 This is a posture tracking error diagram in the simulation experiment of the present invention. DETAILED DESCRIPTION

[0105] To make the objectives, technical solutions, and advantages of the present invention more apparent, the present invention will be further described in detail below with reference to the accompanying drawings. It is apparent that the embodiments described are only some, not all, of the present invention. All other embodiments derived by persons of ordinary skill in the art based on the embodiments of the present invention without inventive effort are intended to fall within the scope of protection of the present invention.

[0106] Example 1

[0107] like Figure 1-2 As shown, the first embodiment of the present invention provides a fixed-time berthing control method for a ship using a translation transformation BLF under an asymmetric constraint, comprising the following steps:

[0108] Step 1: Establish a ship motion model

[0109] like Figure 3 As shown in Figure 2, assuming that η is the true value of the ship's posture, υ is the velocity vector, u is the forward velocity, v is the yaw velocity, r is the yaw velocity, x is the forward position, y is the yaw position, and ψ is the bow angle, the ship motion model is given by the following formula:

[0110] η&=R(ψ)υ (1)

[0111] Mυ&+C(υ)+Dυ+Δ=τ+τ d (2)

[0112] η=[xy ψ] T (3)

[0113] υ=[uvr] T (4)

[0114] Where M is the weight and hydrodynamic inertia, D is the linear hydrodynamic damping parameter matrix, τ d is the disturbance, τ is the input force and torque, Δ is the uncertainty model, C is the Coriolis and centripetal moment matrix, and R is the rotation matrix, which is given by:

[0115]

[0116]

[0117] Step 2: Calculate the allowable error range of trajectory tracking based on the water area of ​​the channel and the expected trajectory

[0118] In order to ensure the safety of the ship, the tracking error of the ship when berthing is limited to a bounded area near the dock, such as Figure 4 As shown, quadrilateral ABCD is a bounded area near the pier. In order to make (x d +x e ,yd +y e )∈quadrilateral ABCD, where x d is the expected value of the forward position, y d is the expected value of the swing position, and the error tolerance range x is obtained based on the bounded area near the dock. e and y e ;

[0119] Ship trajectory tracking refers to tracking and controlling the ship's position within a specified time, reducing the ship's trajectory tracking error to Expressed as:

[0120] Among them, η d is the expected value of the ship's posture, and η is the true value of the ship's posture;

[0121] The present invention uses a full-state tracking controller to simultaneously control the position and bow angle of the ship, and reduces the tracking error of the ship's trajectory to Limited to:

[0122] a=[ x e , y e , ψ e ] T (8)

[0123]

[0124]

[0125] Among them, a represents the lower limit of the allowable error of the ship's posture, and b represents the upper limit of the allowable error of the posture. x e represents the lower bound of the longitudinal position error, represents the upper bound of the longitudinal position error, y e represents the lower bound of the lateral position error, represents the upper bound of the lateral position error, ψ e represents the lower bound of the heading angle error, Indicates the upper bound of the heading angle error;

[0126] Step 3: Perform translation transformation on the tracking error of the ship trajectory to convert the asymmetric constraint into a symmetric constraint

[0127] For the nonlinear kinematic and dynamic equations of ship trajectory tracking, that is, the ship motion model is given in step 1. According to the linear transformation of formula (11), the asymmetric constraint of formula (10) is replaced by the symmetric constraint of formula (12), as follows: Figure 5As shown, the tracking error of the ship trajectory in formula (10) is Perform the following translation transformation:

[0128]

[0129] Substituting equation (11) into equation (10), the inequality is written as:

[0130]

[0131] Step 4: Design the control rate of the tracking controller with fixed time convergence

[0132] The calculation of constructed dummy variables is as follows:

[0133]

[0134] in, represents the change law of the desired posture, k1 represents the coefficient, and z1 represents the vector after the translation transformation of formula (11);

[0135] remember:

[0136]

[0137] Where α1 represents the dummy variable in formula (14);

[0138] The calculation results of the time derivative of formula (14) are as follows:

[0139]

[0140] Substituting formula (2) into formula (15) yields the following result:

[0141]

[0142] The uncertainty model Δ is expressed as:

[0143] δ=RM -1 τ d -RM -1 Δ (17)

[0144] The estimated error is given by:

[0145]

[0146] The control rate of the full-state tracking controller, i.e., the input force and torque τ, is designed as follows:

[0147]

[0148] Among them, k1>0, k2>0, k3>0, k4>0, 0<α3<1, β3>1, 0<α2<1, β2>1; these are all coefficient parameters in the control law, and γ is also a coefficient parameter in the control law;

[0149] Step 5. According to the barrier Lyapunov function BLF, it is inferred from formula (36) that the system is fixed-time convergent and satisfies the asymmetric constraint. That is, under the action of the fixed-time convergent tracking controller, the ship trajectory tracking closed-loop system is fixed-time stable at the origin. The barrier Lyapunov function BLF is designed as:

[0150]

[0151] Based on inequality (12), the following holds:

[0152]

[0153]

[0154] By definition, the BLF value tends to infinity when the function variable approaches the constraint bounds. This also shows that when the BLF is bounded, the function variable will always remain within the constraint bounds, that is, the constraints are satisfied. This is also the key feature of the BLF in constrained control design.

[0155] Take the time derivative of the barrier Lyapunov function BLF:

[0156]

[0157] Substituting formula (16) into formula (23) yields:

[0158]

[0159] Substituting formula (19) into formula (24) yields:

[0160]

[0161] Substituting formula (17) into formula (25) yields:

[0162]

[0163] Substituting formula (18) into formula (26) yields:

[0164] From formula (14), we can infer that:

[0165]

[0166] Substituting formula (28) into formula (27) yields:

[0167]

[0168] It can be seen that:

[0169]

[0170] remember:

[0171]

[0172]

[0173] Then formula (30) is transformed into:

[0174]

[0175] in:

[0176]

[0177]

[0178] thus:

[0179]

[0180] in:

[0181]

[0182] Step 6: Transmit the control quantity to the ship's actuator to adjust the ship's motion until the accuracy meets the requirements.

[0183] In order to verify the feasibility of the proposed method, the present invention provides the simulation results of the control method on the MATLAB platform. The parameters are given as follows: the asymmetric constraint is: the error lower bound a=[-300,-50,-1.5] T , error upper bound b = [50, 250, 1.5] T The results are as follows Figure 6 As shown, the shadow on the horizontal axis represents the upper limit of the error b, and the shadow on the lower axis represents the lower limit of the error a.

[0184] Example 2

[0185] A second embodiment of the present invention provides an electronic device, which may be the aforementioned terminal device or server, or a terminal device or server connected to the aforementioned terminal device or server for implementing the method of the first embodiment of the present invention.

[0186] The electronic device may include: a processor (e.g., a CPU), a memory, and a data acquisition device; the processor is connected to and controls the data acquisition device. The memory may store various instructions for performing various processing functions and implementing the processing steps described in the method of the first embodiment.

[0187] Example 3

[0188] The third embodiment of the present invention further provides a computer-readable storage medium, in which instructions are stored. When the computer-readable storage medium is run on a computer, the computer is caused to execute the processing steps described in the method of the first embodiment.

[0189] Example 4

[0190] The fourth embodiment of the present invention further provides a chip for executing instructions, which is used to execute the processing steps described in the method of the first embodiment.

[0191] Professionals should also be further aware that the units and algorithm steps of each example described in conjunction with the embodiments disclosed in the present invention can be implemented in electronic hardware, computer software, or a combination of the two. In order to clearly illustrate the interchangeability of hardware and software, the above description has generally described the components and steps of each example according to their functions. Whether these functions are performed in hardware or software depends on the specific application and design constraints of the technical solution. Professionals and technicians can use different methods to implement the described functions for each specific application, but such implementation should not be considered to be beyond the scope of the present invention.

[0192] The specific implementation methods described above further illustrate the objectives, technical solutions and beneficial effects of the present invention in detail. It should be understood that the above description is only a specific implementation method of the present invention and is not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of the present invention should be included in the scope of protection of the present invention.

Claims

1. A ship fixed-time berthing control method based on translation transformation BLF under asymmetric constraints, characterized by The steps include: Step 1: Establish a ship motion model; Step 2: Calculate the allowable error range of trajectory tracking based on the water area of ​​the channel and the expected trajectory; Step 3: Perform translation transformation on the tracking error of the ship trajectory to convert the asymmetric constraint into a symmetric constraint; Step 4: Design the control rate of the tracking controller with fixed time convergence; Step 5: Design the barrier Lyapunov function and prove that the system is fixed-time convergent and satisfies the asymmetric constraints. That is, under the action of the fixed-time convergent tracking controller, the ship trajectory tracking closed-loop system is fixed-time stable at the origin. Step 6: Transmit the control quantity to the ship's actuator to adjust the ship's motion until the accuracy meets the requirements.

2. The ship fixed time berthing control method of the asymmetric constrained translation transformation BLF according to claim 1 is characterized in that: The step 1 is specifically as follows: Assuming that η is the true value of the ship's posture, υ is the velocity vector, u is the forward velocity, v is the yaw velocity, r is the yaw velocity, x is the forward position, y is the yaw position, and ψ is the bow angle, the ship motion model is given by the following equation: η=[xy ψ] T (3) υ=[u v r] T (4) Where M is the weight and hydrodynamic inertia, D is the linear hydrodynamic damping parameter matrix, τ d is the disturbance, τ is the input force and torque, Δ is the uncertainty model, C is the Coriolis and centripetal moment matrix, and R is the rotation matrix, which is given by:

3. The ship fixed time berthing control method of the asymmetric constrained translation transformation BLF according to claim 2 is characterized in that: The step 2 is specifically as follows: In order to ensure the safety of the ship, the tracking error of the ship when berthing is limited to the bounded area near the dock. Assume that the quadrilateral ABCD is the bounded area near the dock. If (x d +x e ,y d +y e )∈quadrilateral ABCD, where x d is the expected value of the forward position, y d is the expected value of the swing position, and the error tolerance range x is obtained based on the bounded area near the dock. e and y e ; Ship trajectory tracking refers to tracking and controlling the ship's position within a specified time, reducing the ship's trajectory tracking error to Expressed as: Among them, η d is the expected value of the ship's posture, and η is the true value of the ship's posture; According to the water range of the channel and the expected trajectory, the allowable error range of trajectory tracking is calculated, that is, the tracking error of the ship's trajectory is controlled by the full-state tracking controller at the same time to Limited to: a＝[ x e ， y e ， ψ e ] T (8) Among them, a represents the lower limit of the allowable error of the ship's posture, and b represents the upper limit of the allowable error of the posture. x e represents the lower bound of the longitudinal position error, represents the upper bound of the longitudinal position error, y e represents the lower bound of the lateral position error, represents the upper bound of the lateral position error, ψ e represents the lower bound of the heading angle error, Indicates the upper bound of the heading angle error.

4. The ship fixed time berthing control method of the asymmetric constrained translation transformation BLF according to claim 3 is characterized in that: The step 3 is specifically as follows: For the established ship motion model, according to the linear transformation of formula (11), the asymmetric constraint of formula (10) is replaced by the symmetric constraint of formula (12), and the tracking error of the ship trajectory in formula (10) is Perform the following translation transformation: Substituting equation (11) into equation (10), the inequality is written as:

5. The ship fixed time berthing control method of the asymmetric constrained translation transformation BLF according to claim 4 is characterized in that: The step 4 is specifically as follows: The calculation of constructed dummy variables is as follows: in, represents the change law of the desired posture, k1 represents the coefficient, and z1 represents the vector after the translation transformation of formula (11); remember: Where α1 represents the dummy variable in formula (14); The calculation results of the time derivative of formula (14) are as follows: Substituting formula (2) into formula (15) yields the following result: The uncertainty model Δ is expressed as: δ=RM -1 τ d -RM -1 Δ (17) The estimated error is given by: The control rate of the full-state tracking controller, i.e., the input force and torque τ, is designed as follows: Among them, k1>0, k2>0, k3>0, k4>0, 0<α3<1, β3>1, 0<α2<1, β2>1; these are all coefficient parameters in the control law, and γ is also a coefficient parameter in the control law.

6. The ship fixed time berthing control method of the asymmetric constrained translation transformation BLF according to claim 5 is characterized in that: The step 5 is specifically as follows: The designed barrier Lyapunov function BLF is Based on inequality (12), the following holds: According to the definition, when the function variable approaches the constraint boundary, the BLF value will tend to infinity, which also shows that when the BLF is bounded, the function variable will always remain within the constraint boundary, that is, the constraint is satisfied; Take the time derivative of the barrier Lyapunov function BLF: Substituting formula (16) into formula (23) yields: Substituting formula (19) into formula (24) yields: Substituting formula (17) into formula (25) yields: Substituting formula (18) into formula (26) yields: From formula (14), we can infer that: Substituting formula (28) into formula (27) yields: It can be seen that: remember: Then formula (30) is transformed into: in: thus: in:

7. A device, characterized in that: The device includes a processor and a memory; the memory is used to store a computer program; the processor is used to execute any one of claims 1-6 of the ship fixed-time berthing control method of translation transformation BLF under asymmetric constraints according to the computer program.

8. A computer-readable storage medium, characterized in that: The computer-readable storage medium is used to store a computer program, and the computer program is used to execute the ship fixed-time berthing control method of translation transformation BLF under asymmetric constraints according to any one of claims 1 to 6.

9. A chip for executing instructions, characterized in that: The chip is used to execute any one of claims 1-6 of the ship fixed-time berthing control method of translation transformation BLF under asymmetric constraints.

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