A NMPC method for mobile robots based on fixed-time stability theory

Abstract

The invention discloses a mobile robot NMPC method based on fixed time stability theory, comprising the following steps: establishing a kinematic model of a two-wheel differentially driven mobile robot; describing a nonlinear model predictive control problem as an optimal control problem, introducing a Hamiltonian and discretizing a continuous system to obtain a discrete necessary condition for optimality and an optimization problem; iterating a solution to the optimization problem using a fixed time convergence rate, and using the obtained solution as an initial solution of a continuous generalized minimum residual method to control a wheeled mobile robot, thereby achieving point stabilization of the wheeled mobile robot, providing better closed-loop performance and interference suppression performance, improving the real-time performance of the system, enabling the system state to converge within a fixed time, and overcoming the problem of poor real-time performance of nonlinear model predictive control.

Description

Technical Field

[0001] The invention relates to the technical field of mobile robots, in particular to a mobile robot NMPC method based on fixed time stability theory. Background Art

[0002] In recent years, due to the shortage of labor and rising labor costs, more and more factories and enterprises have chosen to use robots to replace some of the labor to help speed up production. With the continuous improvement and development of electronic communications, control theory, power electronics, artificial intelligence and other technologies, the research and application of robot systems has become a hot spot in the industry and has received widespread attention. Among them, wheeled mobile robots have been used in logistics and transportation, convenience services, etc., such as cargo transportation robots in express delivery companies, guided robots in banks, and even food delivery robots in Haidilao. They are all wheeled mobile robots. Because their mechanical structure and control model are relatively simple and intuitive, wheeled mobile robots can be widely used in daily life.

[0003] Among the problems of wheeled mobile robots, there are usually point stabilization problems, path tracking problems and trajectory tracking problems, which are classic problems of mobile robots. Among them, the point stabilization problem is a classic problem in wheeled mobile robots due to the existence of Brockett's theorem, that is, there is no continuous and smooth pure state feedback control rate. This has attracted widespread attention from many scholars. Many scholars have proposed various methods to overcome this difficulty. However, when designing the control domain, they did not take into account the constraints on the robot itself, such as state constraints, speed constraints, etc. Therefore, a mobile robot NMPC (nonlinear model predictive control) method based on fixed time stability theory is urgently needed to solve the above problems. Summary of the invention

[0004] The present invention provides a mobile robot NMPC method based on fixed time stability theory, which improves the real-time performance of the system, enables the system state to converge within a fixed time, and overcomes the problem of poor real-time performance of nonlinear model predictive control.

[0005] To achieve the above object, the present invention provides the following technical solution: a mobile robot NMPC method based on fixed time stability theory, comprising the following steps:

[0006] S1. Establish the kinematic model of a two-wheel differentially driven mobile robot;

[0007] S2, introduce Hamiltonian and discretize the continuous system to obtain the necessary discrete conditions for optimality F(U,x,t);

[0008] S3. Based on the fixed-time stability theory, a controller is designed to control the wheeled mobile robot to achieve point stabilization of the wheeled mobile robot, wherein a fixed-time convergence rate is used to solve F(U,x,t).

[0009] Preferably, in step S1, the kinematic model is: Among them, x∈R and y∈R are the positions of the mobile robot, θ is the angle between the robot's moving direction and the positive direction of X, v and ω are the linear velocity and angular velocity of the mobile robot, and the matrix form of the state quantity is [xy θ] T , the matrix form of the control quantity is [v ω] T .

[0010] Preferably, the kinematic model of the mobile robot sets constraint information:

[0011] Preferably, in step S2, a nonlinear model is established, and its state equation and equality constraints are:

[0012]

[0013] C(x(t),u(t))=0;

[0014] Where x(t)∈R n is the state vector, u(t)∈R m is the control input vector;

[0015] The nonlinear model predictive control problem can be described as an optimal control problem. The objective function of the optimal control problem is:

[0016]

[0017]

[0018] in, is the terminal constraint, L(·) is the loss function, x * (κ, t) is the trajectory of x(t) starting from the time κ = 0, and the optimal control input is the solution of the optimal control problem in each time domain t;

[0019]

[0020]

[0021] Among them, P and Q are weight matrices, x f The expected state.

[0022] Preferably, the integrand is expressed by quoting the Hamiltonian:

[0023] H(x,u,σ,λ)=L(x,u)+σ T f(x,u)+λ T C(x,u);

[0024] Among them, σ∈R n and λ∈R m are covariates and Lagrange multipliers.

[0025] Preferably, in step S2, a discretization process is performed: the prediction time domain is divided into N steps, and the necessary conditions after the optimal discretization are:

[0026]

[0027]

[0028] definition is the vector consisting of the input vector sequence and the Lagrange multiplier sequence:

[0029]

[0030] The necessary condition for optimality is:

[0031]

[0032] Preferably, in step S3, the fixed time convergence rate is:

[0033]

[0034] Among them, And a>0, define "F" a =|F| a sign(F); α1>0, α2>0, If β∈(0,1), then 0<β1<1<β2;

[0035] If F U is non-singular, then:

[0036] Initialize U(0) and solve the equation as: Solve U through U(0) iteration.

[0037] Preferably, the forward difference approximation method and the continuous / generalized minimum residual method are used to calculate U, specifically:

[0038] a. Use the Jacobian matrix and vector product to perform forward difference approximation, giving the Jacobian matrix and O∈R mN , O∈R nN , the product of O∈R:

[0039] b.

[0040] Where h is the forward difference step size, then:

[0041]

[0042] c. Calculate U using the continuous / generalized minimum residual method.

[0043] Compared with the prior art, the beneficial effects of the present invention are as follows: the present invention uses a nonlinear model predictive control method to perform point stabilization control on a wheeled mobile robot, and adopts a fixed-time stability theory to provide better closed-loop performance and interference suppression performance, further improving the real-time performance of the system, so that the system state can converge within a fixed time, overcoming the problem of poor real-time performance of nonlinear model predictive control. The method of the present invention can better achieve point stabilization of a wheeled mobile robot, and can also be better used in actual industry. BRIEF DESCRIPTION OF THE DRAWINGS

[0044] The accompanying drawings are used to provide further understanding of the present invention and constitute a part of the specification. They are used to explain the present invention together with the embodiments of the present invention and do not constitute a limitation of the present invention.

[0045] In the attached picture:

[0046] Figure 1 It is a schematic diagram describing the reference system of the wheeled mobile robot of the present invention;

[0047] Figure 2 is a schematic diagram of a motion curve of the wheeled mobile robot of the present invention;

[0048] Figure 3 is a linear velocity variation curve diagram of the wheeled mobile robot of the present invention;

[0049] Figure 4 It is a curve diagram of angular velocity variation of the wheeled mobile robot of the present invention. DETAILED DESCRIPTION

[0050] The preferred embodiments of the present invention are described below in conjunction with the accompanying drawings. It should be understood that the preferred embodiments described herein are only used to illustrate and explain the present invention, and are not used to limit the present invention.

[0051] Example: Figure 1 As shown, a mobile robot NMPC (nonlinear model predictive control) method based on fixed time stability theory includes the following steps:

[0052] S1. Establish the kinematic model of a two-wheel differentially driven mobile robot;

[0053] The kinematic model is: Among them, x∈R and y∈R are the positions of the mobile robot, θ is the angle between the robot's moving direction and the positive direction of X, v and ω are the linear velocity and angular velocity of the mobile robot, and the matrix form of the state quantity is [xy θ] T , the matrix form of the control quantity is [v ω] T .

[0054] Discretize the kinematic model of the wheeled mobile robot:

[0055]

[0056] Where ΔT is the sampling time.

[0057] The kinematic model of the mobile robot sets the constraint information:

[0058] S2, introduce Hamiltonian and discretize the continuous system to obtain the necessary discrete conditions for optimality F(U,x,t);

[0059] A nonlinear model is established, and its state equation and equality constraints are:

[0060]

[0061] C(x(t),u(t))=0;

[0062] Where x(t)∈R n is the state vector, u(t)∈R m is the control input vector;

[0063] The nonlinear model predictive control problem can be described as an optimal control problem. The objective function of the optimal control problem is:

[0064]

[0065]

[0066] in, is the terminal constraint, L(·) is the loss function, x * (κ, t) is the trajectory of x(t) starting from the time κ = 0, and the optimal control input is the solution of the optimal control problem in each time domain t;

[0067]

[0068]

[0069] Among them, P and Q are weight matrices, x f is the desired state;

[0070] To solve the optimal control problem, the Hamiltonian is used to represent the integrand:

[0071] H(x,u,σ,λ)=L(x,u)+σ T f(x,u)+λ T C(x,u);

[0072] Among them, σ∈R n and λ∈R m are the covariates and Lagrange multipliers;

[0073] In order to realize computer control, the control input is expressed by finite parameters. The system needs to be discretized and the prediction time domain is divided into N steps. The necessary conditions for the optimal discretization are:

[0074]

[0075]

[0076] definition is the vector consisting of the input vector sequence and the Lagrange multiplier sequence:

[0077]

[0078] The necessary condition for optimality is:

[0079]

[0080] S3. Based on the fixed-time stability theory, a controller is designed to control the wheeled mobile robot to achieve point stabilization of the wheeled mobile robot, wherein a fixed-time convergence rate is used to solve F(U,x,t).

[0081] Among them, fixed time convergence can provide better closed-loop performance and interference suppression performance, and can accelerate the convergence speed. The fixed time convergence rate is:

[0082]

[0083] Among them, And a>0, define "F" a =|F| a sign(F); α1>0, α2>0, If β∈(0,1), then 0<β1<1<β2;

[0084] If F U is non-singular, then:

[0085] Initialize U(0) and solve the equation as: Solve U through U(0) iteration.

[0086] In order to reduce the computational cost, the forward difference approximation method and the continuous / generalized minimum residual method (C / GMRES) algorithm are used to calculate U, specifically:

[0087] a. Use the Jacobian matrix and vector product to perform forward difference approximation, giving the Jacobian matrix and O∈R mN , O∈R nN , the product of O∈R:

[0088] b.

[0089] Where h is the forward difference step size, then:

[0090]

[0091] c. Calculate U using the continuous / generalized minimum residual method.

[0092] In a simulation experiment, the basic parameters are:

[0093] (1)

[0094] (2) Weight matrix: P = diag{0.1 0.1 0.01}, Q = diag{1 5 0.1}, R = diag{0.5 0.05};

[0095] (3) The initial pose is: x0 = [0 0 0] T , the final value posture is: x f =[1.5 1.5 0] T ;

[0096] (4) The values ​​of the parameters are α1=30, α2=20, β=0.6, then β1=3 / 7, β2=11 / 7. For the time parameters in the simulation, set the total simulation time t_end=20, the control period dt=0.001, the forward difference step h=0.002, the prediction step N=10, and the prediction time domain length T f =1, iterative upper limit k max =10;

[0097] The above parameters are numerically simulated in the matlab2016a environment to obtain the following motion curve diagram of the wheeled mobile robot: Figure 2 As shown in the figure, the linear velocity variation curve of the wheeled mobile robot is as follows: Figure 3 As shown in the figure, the angular velocity variation curve of the wheeled mobile robot is as follows: Figure 4 As shown in the figure, the simulation results show that the method proposed in this paper has good real-time performance, which verifies the effectiveness of the method proposed in this paper.

[0098] Finally, it should be noted that the above description is only a preferred example of the present invention and is not intended to limit the present invention. Although the present invention has been described in detail with reference to the aforementioned embodiments, those skilled in the art can still modify the technical solutions described in the aforementioned embodiments or replace some of the technical features therein by equivalents. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present invention shall be included in the protection scope of the present invention.

Claims

1. A mobile robot NMPC method based on fixed time stability theory, characterized by: The steps include: S1. Establish the kinematic model of a two-wheel differentially driven mobile robot; S2. Introduce Hamiltonian and discretize the continuous system to obtain the necessary discrete conditions for optimality ; A nonlinear model is established, and its state equation and equality constraints are: ； ； in, is the state vector, is the control input vector; The nonlinear model predictive control problem can be described as an optimal control problem. The objective function of the optimal control problem is: ； ； in, is the terminal constraint, is the loss function, For The moment starts The optimal control input is the solution of the optimal control problem in each time domain t; ； ； Among them, P and Q are weight matrices, is the desired state; S3. Design a controller based on the fixed time stability theory to control the wheeled mobile robot and achieve point stabilization of the wheeled mobile robot. To solve; The fixed time convergence rate is: ； Among them, and ,definition ; , , , ;like ,but ; like is non-singular, then: ; initialization The solution equation is: ;pass Iterate and solve U.

2. The mobile robot NMPC method based on fixed time stability theory according to claim 1, characterized in that: In step S1, the kinematic model is: ;in, and is the position of the mobile robot, is the angle between the robot's motion direction and the positive X direction, v and is the linear velocity and angular velocity of the mobile robot, and the matrix form of the state quantity is , the matrix form of the control quantity is .

3. The mobile robot NMPC method based on fixed time stability theory according to claim 2, characterized in that: The kinematic model of the mobile robot sets the constraint information: .

4. The mobile robot NMPC method based on fixed time stability theory according to claim 1, characterized in that: Express the integrand by quoting the Hamiltonian: ； in, and are covariates and Lagrange multipliers.

5. The mobile robot NMPC method based on fixed time stability theory according to claim 4 is characterized in that: In step S2, discretization processing is performed: the prediction time domain is divided into N steps, and the necessary conditions for the optimal discretization are: ； ； definition is the vector consisting of the input vector sequence and the Lagrange multiplier sequence: ； The necessary condition for optimality is: 。 6. The mobile robot NMPC method based on fixed time stability theory according to claim 5 is characterized by: The forward difference approximation method and the continuous / generalized minimum residual method are used to calculate U, specifically: a. Use the Jacobian matrix and vector product to perform forward difference approximation, giving the Jacobian matrix and , , The product of: b、 ； Where h is the forward difference step size, then: ； c. Calculate U using the continuous / generalized minimum residual method.

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