Probability-dependent underwater robot Backstepping control method

Abstract

The invention discloses a probability-dependent underwater robot Backstepping control method which is used for solving the problems that a control system of an existing underwater robot control method is inaccurate and the control conservative property is too high. The method comprises the following steps: firstly, carrying out stress analysis on the underwater robot to obtain a kinematic equation, carrying out coordinate transformation, unifying coordinate parameters, and then generalizing the kinematic equation into a complex nonlinear control system. A Backstepping method is used for decomposing the system, and a Lyapunov function is selected from a first-stage subsystem to ensure the stability of each subsystem. Finally, a controller of the whole control system is obtained through recursion, a Bernoulli probability gain controller is introduced, and the conservative property of the control system is reduced. According to the method, the underwater robot nonlinearly and quickly reaches a stable state, and meanwhile, the method does not have too high control conservative property, so that the method has practical significance in reducing energy consumption.

Description

Technical Field

[0001] The invention relates to an underwater robot control method, in particular to a probability-dependent underwater robot Backstepping control method. Background Art

[0002] Backstepping, first proposed by Kanellakopoulos and Kokotovic in 1991, represents a systematic recursive approach to design controllers for systems with uncertain parameters. The method cleverly decomposes complex nonlinear systems into simple subsystems that are easily controlled and do not exceed the order of the original system, thereby achieving a stepwise design process. In each step, Lyapunov functions are applied to virtual controller design to ensure the stability of each subsystem and, by extension, the entire system. Backstepping control methods are particularly effective for nonlinear systems with strict feedback structures, parameter uncertainties, and unpredictable conditions. In the field of control, traditional strategies often adopt highly conservative situations to ensure robustness, which paradoxically can lead to suboptimal performance due to excessive energy consumption. Since real systems usually do not require such highly conservative control measures to achieve control effects, low-conservative methods can significantly reduce energy consumption without compromising stability and performance. Summary of the invention

[0003] In order to overcome the problems of inaccuracy and high conservatism of current underwater robot control systems, the present invention provides a probability-dependent underwater robot Backstepping control method. The method is based on the six degrees of freedom of the underwater robot's own coordinate system and the earth's fixed coordinate system, builds a dynamic model of the underwater robot, and unifies the state variables of the two coordinate systems through coordinate transformation. Further coordinate fusion and generalization are performed to obtain the state equation of the underwater robot with three degrees of freedom:

[0004] dl1=l2dt+ξf1(t;l(t);u)dt+g1(t;l(t);u)dt

[0005] dl2=u+ξf2(t;l(t);u)dt+g2(t;l(t);u)dt

[0006] y=l1

[0007] This state equation takes into account the nonlinear state of the underwater robot itself, and also takes into account the impact of nonlinear interference that occurs in probability on the control system. First, the complex nonlinear state equation is decomposed into a simple nonlinear system that is easy to design: dl1=l2dt+ξf1(t;l(t);u)dt+g1(t;l(t);u)dt. And assume that the unmeasurable state l2 in the original state equation is the controller to be designed. Select the first Lyapunov function V(l1) to design the controller of this subsystem. The controller obtained from the subsystem is used as the input of the next-level subsystem, and the second Lyapunov function is selected to design the next-level system controller, and so on to obtain the final controller of the system. In order to reduce the conservatism of system control, a probabilistic gain controller is introduced: u(l)=K(P)l(t) to find the boundary conditions of control, thereby reducing energy consumption. The specific steps are as follows:

[0008] Step 1: Get the kinematic equation by analyzing the force of the underwater robot:

[0009]

[0010] M is the inertia matrix which contains the AUV additional mass, C(υ) represents the Coriolis centripetal force of the underwater robot, D(υ) represents the fluid damping of the underwater robot, g(∈) represents the restoring force (torque) generated by the gravity and buoyancy of the underwater robot during operation, and F is the resultant force and torque. The whole equation involves two coordinate system parameters, namely the position and direction of the fixed coordinate system ∈=[xyz α βγ] T and the linear velocity and angular velocity of its own coordinate system ν=[u ν ω pqr] T .

[0011] Step 2: For better calculation, introduce coordinate transformation μ to unify the variables of the two coordinate systems. The transformation form is as follows: The kinematic equation is further transformed into:

[0012]

[0013] According to the characteristics of the unified dynamic equation, the dynamic model is generalized into a three-degree-of-freedom state equation:

[0014] dl1=l2dt+ξf1(t;l(t);u)dt+g1(t;l(t);u)dt

[0015] dl2=u+ξf2(t;l(t);u)dt+g2(t;l(t);u)dt

[0016] y=l1

[0017] l=(l1;...;l n )T ∈R n ;u∈R;y∈R represents the unmeasurable nonlinear state of the system and the input and output of the system. i (0) = 0; g i (0) = 0 represents nonlinear interference and the nonlinear state of the system itself. ξf1(t; l(t); u)dt represents nonlinear random disturbance that occurs according to probability.

[0018] Step 3: Use the Backstepping design method to decompose the complex nonlinear model into simple nonlinear models. First, select the 0th Lyapunov equation: Find its differential:

[0019]

[0020] Step 4, after z1; z2 state transformation:

[0021]

[0022] Choose the first Lyapunov equation:

[0023]

[0024] Find its differential:

[0025]

[0026] Select the virtual controller for the first nonlinear system: Make the system stable.

[0027] Step 5: Use the virtual controller of the first nonlinear system as the controller input of the next nonlinear system. Similarly, select the second Lyapunov equation to obtain the controller of the entire nonlinear system:

[0028]

[0029] Step six, introduce the Bernoulli probability gain controller u(l)=K(P)l(t) to reduce the conservatism of the control system without affecting the control effect.

[0030] Compared with the prior art, the present invention has the following beneficial effects: starting from the actual kinematic equation of the underwater robot, the present invention considers a type of nonlinear disturbance that appears according to probability in the actual situation. In addition, a probability gain controller is introduced, and on the basis of not affecting the final control effect, the Bernoulli probability gain controller is adopted to reduce energy consumption and find the control boundary conditions, thereby providing a new perspective and method for solving the control problem in the underwater vehicle, which has practical significance.

[0031] The present invention is described in detail below based on the accompanying drawings and specific embodiments. BRIEF DESCRIPTION OF THE DRAWINGS

[0032] Figure 1 The present invention is a probability-dependent underwater robot Backstepping control method controller design process.

[0033] Figure 2 These are the two coordinate reference systems of the underwater robot. DETAILED DESCRIPTION

[0034] According to the accompanying drawings, the specific steps of the probability-dependent underwater robot Backstepping control method of the present invention are as follows:

[0035] Step 1: Based on Newton’s second law F=Ma, the kinematic equation of the underwater robot is obtained by force analysis:

[0036]

[0037] M is the inertia matrix which contains the AUV additional mass, C(υ) represents the Coriolis centripetal force of the underwater robot, D(υ) represents the fluid damping of the underwater robot, g(∈) represents the restoring force (torque) generated by the gravity and buoyancy of the underwater robot during operation, and F is the resultant force and torque. The whole equation involves two coordinate system parameters, namely the position and direction of the fixed coordinate system ∈=[xyz α βγ] T and the linear velocity and angular velocity of its own coordinate system ν=[u ν ω pqr] T .

[0038] Step 2: For better calculation, the six-degree-of-freedom model is decomposed into two kinematic models according to the moving coordinate system, namely the linear velocity model and the angular velocity model; and decomposed into two kinematic models according to the fixed coordinate system, namely the position model and the angle model. At the same time, the coordinate transformation μ is introduced to unify the variables of the two coordinate systems. The relationship between the position and velocity in the fixed coordinate system and the moving coordinate system is Where J1(∈) is the dynamic transfer matrix used for transformation between coordinate systems.

[0039] The coordinate system transformation is as follows:

[0040]

[0041] According to the above coordinate changes, the kinematic equation can be further transformed into:

[0042]

[0043] The unified dynamic equation has the characteristics of nonlinear strong coupling, generalizing the dynamic model into a three-degree-of-freedom state equation with unknown control direction:

[0044] dl1=l2dt+ξf1(t;l(t);u)dt+g1(t;l(t);u)dt

[0045] dl2=u+ξf2(t;l(t);u)dt+g2(t;l(t);u)dt

[0046] y=l1

[0047] where l=(l1;...;l n ) T ∈R n ;u∈R;y∈R represent the unmeasurable nonlinear state of the system and the input and output of the system. i (0) = 0; g i (0) = 0 represents the nonlinear interference and the nonlinear state of the system itself, which satisfies the Lipschitz continuity condition and is assumed to satisfy the linear growth condition. ξf1(t; l(t); u)dt represents the nonlinear random disturbance that occurs in actual situations according to probability.

[0048] Step 3: Use the Backstepping design method to decompose the complex nonlinear model into multiple simple nonlinear models. First, select the 0th Lyapunov equation: The differential is:

[0049]

[0050] Step 4: Design the Backstepping controller for the nonlinear system. First, introduce the z1;z2 state transformation:

[0051]

[0052] in is the controller to be designed, z2 is the error between the actual control quantity and the ideal controller control quantity, and the controller needs to be designed to make the error approach 0.

[0053] Choose the first Lyapunov equation:

[0054]

[0055] Find its differential:

[0056]

[0057] Select a real number H ≥ 4nc1 and the inequality (a + b) 2 2a 2 +2b2 , the virtual controller of the first nonlinear system can be obtained as LV1 can be made negative and the system can be stable.

[0058] Step 5: Use the virtual controller of the first nonlinear system as the controller input of the next nonlinear system. Similarly, select the second Lyapunov equation to obtain the controller of the entire nonlinear system:

[0059]

[0060] in,

[0061]

[0062] Step 6: Introduce the Bernoulli probability gain controller u(l)=K(P)l(t) to reduce the conservatism of the control system without affecting the control effect, where K(P)=K0+P(t)K u is the probability gain. The probability in the probability gain satisfies the Bernoulli distribution:

[0063]

[0064] Therefore, the final controller designed is in the following form to make the nonlinear system reach a stable state:

[0065]

Claims

1. A probability-dependent underwater robot backstepping control method, which is characterized in that it includes the following steps: Step 1: Perform force analysis on the underwater robot and establish the dynamic equations of the underwater robot's own coordinate system and the fixed coordinate system: Step 2: Introduce coordinate transformation to unify the parameters in the kinematic equation relative to two different coordinate systems into one coordinate system: The unified dynamic equation is obtained: According to the characteristics of the unified dynamic equation, the system of the underwater robot is generalized into a nonlinear system state equation: dl1=l2dt+ξf1(t;l(t);u)dt+g1(t;l(t);u)dt dl2=u+ξf2(t;l(t);u)dt+g2(t;l(t);u)dt y＝l1 Step 3: Decompose the system using the Backstepping method. Design the controller of each subsystem from the first simple nonlinear system and use it as the controller input of the next subsystem. Step 4: Change the state: Convert the unmeasurable state into the error between the virtual controller and the actual ideal control state, and then select the first Lyapunov equation: Find its differential LV0 and get the virtual controller when this differential reaches negative definite: Step 5: Use the virtual controller obtained from the first-level nonlinear system as the controller input of the next-level system, and recursively obtain the actual controller u(l) of the entire system.

2. The probability-dependent underwater robot Backstepping control method according to claim 1 is characterized in that: Introduce a Bernoulli probability gain to the controller: u(l)=K(P)l(t) The probability gain is K(P) = K0 + P(t)K u , P(t) is a time-varying quantity.

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