CN109857100B - Composite track tracking control algorithm based on inversion method and fast terminal sliding mode - Google Patents

Abstract

The invention discloses a compound trajectory tracking control algorithm based on an inversion method and a fast terminal sliding mode, which belongs to the field of robotics and includes the following steps: (1) a kinematics model of a mobile robot, (2) design of a control algorithm, (3) ) to the proof of the convergence of the error y _e . The invention solves the problem that it is difficult to take into account the convergence speed and accuracy in the trajectory tracking controller designed by the traditional global fast terminal sliding mode technology, not only improves the system error and the output convergence speed, but also reduces the system output error; and The discontinuous term in the traditional sliding mode structure is eliminated, the chattering phenomenon is avoided, and the stability of the output is ensured.

Description

Composite track tracking control algorithm based on inversion method and fast terminal sliding mode

Technical Field

The invention relates to the technical field of robots, in particular to a composite track tracking control algorithm based on an inversion method and a fast terminal sliding mode.

Background

With the rapid development of the robot industry, the requirements on the working indexes of the robot are higher and higher, and therefore the tracking precision and stability of the mobile robot are more and more important. The mobile robot is a control system which is not completely constrained, a tracking error system of the track tracking is often a coupled nonlinear system, does not meet the necessary condition of Brockett, and is more complex to the problems of control, planning and the like. Therefore, many scholars have proposed various methods for solving the problem of trajectory tracking of mobile robots. The traditional PID algorithm controller has poor robustness, is insensitive to external interference and has difficult parameter setting. The sliding mode variable structure method has fast response and good robustness, but the discontinuous items in the control law are directly transferred to the output items, so that the inevitable buffeting phenomenon of the system is caused. The iterative learning control algorithm can be that the actual tracking error converges on a predetermined error track, but it generally requires that the initial position is on the predetermined track and the number of iterations affects the final learning result, which has a limit for practical application. The adaptive control can continuously acquire system input, state, output and performance parameters and correspondingly adjust the control law, so that the control performance is optimal, but the parameter selection is complex. The fuzzy control method has certain robustness, but the fuzzy control rule can be influenced by subjective factors of people. Aiming at a mobile robot system, the idea of an inversion method and a global fast terminal sliding mode technology is adopted, so that the system can be converged to a balanced state within limited time, discontinuous items in a traditional sliding mode structure are eliminated, the phenomenon of buffeting is avoided, and the stability of output is ensured.

Disclosure of Invention

Based on the technical problems in the background art, the invention provides a composite track tracking control algorithm based on an inversion method and a fast terminal sliding mode.

The technical scheme adopted by the invention is as follows:

a composite track tracking control algorithm based on an inversion method and a fast terminal sliding mode is characterized by comprising the following steps:

kinematic model of mobile robot

Establishing a pose error coordinate graph by taking a two-degree-of-freedom wheeled mobile robot as a research object;

in the attitude error coordinate diagram, M and M' are the shaft middle points of the two driving wheels; (x, y), (x)_r,y_r) Is the position of the mobile robot; theta, theta_rIs the included angle between the advancing direction of the mobile robot and the x axis; x is the number of_e,y_e,θ_eThe plane coordinate error and the direction error of the mobile robot are obtained; let P be (x, y, theta)^T，q＝(v,w)^TV and w are the linear velocity and angular velocity of the mobile robot, respectively;

the kinematic equation of the mobile robot is as follows:

by P_r＝(x_r,y_r,θ_r)^TAnd q is_r＝(v_r,w_r)^TTo represent a position command and a velocity command of the reference mobile robot; in the pose error coordinate diagram, the mobile robot sets the pose P as (x, y, theta)^TMove to pose P_r＝(x_r,y_r,θ_r)^TMoving robot in new coordinate system X_e-Y_eThe coordinates in (1) are: p_e＝(x_e,y_e,θ_e)^TWherein theta_e＝θ_r-θ；

Setting new coordinate X_e-Y_eThe included angle between the coordinate system X and the coordinate system Y is theta, and according to a coordinate transformation formula, an error equation for describing the moving pose can be obtained as follows:

the differential equation of the attitude error obtained by the joint type (1) and (2) is as follows:

from the above analysis, the trajectory tracking of the kinematic model of the mobile robot, i.e. the seek control input q ═ (v, w)^TSo that for any initial error, the system is under the control input, P_e＝(x_e,y_e,θ_e)^TIs bounded, an

Design of control algorithm

Is provided with

Wherein alpha is₁＞0,β₁＞0,α₂＞0,β₂> 0, and p₁,q₁,p₂,q₂Is positive odd and satisfies p₁＞q₁,p₂＞q₂

From the formula (3)

By solving the first order linear differential equation (4), it can be known that the time t is finite₁Inner, theta_eIs equal to 0, and

similarly, solving the first order linear differential equation (5) can know that the time t is finite₂Inner, x_eIs equal to 0, and

for a mobile robotic system, as long as t > max { t₁,t₂Is then x_e＝0,θ_e＝0；

Taking according to the inversion method (Back-stepping)

(a) When theta is_eWhen v is 0, v is equal to v₁Then the formula (3) can be expressed as

The introduction of the new virtual feedback variables is as follows:

wherein k is₁₁＞0；

Taking Lyapunov function

Combining the error differential equation (8) to obtain:

so v₁The design is as follows:

wherein k is₁₂Is greater than 0; will control the rate v₁Substituting equation (11) into equation (9) can obtain:

as can be seen from the barkalat theorem,

respectively tend to zero; because of the fact that

Namely, it is

From the control rate, w is not constantly equal to zero, and

go to zero to obtain y_e→ 0; further comprises

Knowing x_e→0；

(b) When x is_eWhen the value is equal to 0, the value w is equal to w₂Then, equation (3) can be expressed as:

the introduction of the new virtual feedback variables is as follows:

taking Lyapunov function

Wherein

Combining the error differential equation (14) to obtain

Therefore w₂Is designed as

Wherein k is₂₁Is greater than 0; the control rate is substituted for formula (17) or formula (15) to obtain

According to the Barbalt's theorem, y_e,

Approaching to zero; and because of

By

Then

The formula is

Equivalence;

(c) taking the comprehensive control rate as

Substituted into the formulae (6), (7), (12) and (18)

Wherein k is₁₁,k₁₂,k₂₁,α₁,β₁,α₂,β₂Are all positive numbers greater than zero, p₁,q₁,p₂,q₂Is positive odd and satisfies p₁＞q₁,p₂＞q₂。

(III) pairs of errors y_eDemonstration of convergence

(a) For a mobile robotic system, as long as t > max { t₁,t₂Is then x_e＝0,θ_e0, substituted by formula (4) or (5)

The combination of formula (3) and formula (21) can give

y_ew-v+v_r＝0 (22)

w_r-w＝0 (23)

The combined type (22), (23) and (24) can be obtained

Because the system expects an input v_r,w_rCannot be simultaneously zero, w_rIf the value is not equal to zero, the system balance point is y_e＝0；

(b) Taking Lyapunov function

From the formulas (13) and (19)

Approaching zero, the combined formulas (9) and (15) can be obtained

Can obtain the product

From this, the error y_eThe global asymptote converges to zero.

The invention has the advantages that:

the invention solves the problem that the convergence speed and the accuracy are difficult to be considered in the track tracking controller designed by the traditional global fast terminal sliding mode technology, not only improves the system error and the output convergence speed, but also reduces the system output error; and the discontinuous items in the traditional sliding mode structure are eliminated, the buffeting phenomenon is avoided, and the output stability is ensured.

Drawings

FIG. 1 is a diagram of a pose error coordinate of a mobile robot.

Fig. 2 is a block diagram of a robot tracking control system.

Detailed Description

The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments.

Examples are given.

Referring to fig. 2, a composite trajectory tracking control algorithm based on an inversion method and a fast terminal sliding mode includes the following steps:

kinematic model of mobile robot

A two-degree-of-freedom wheeled mobile robot is taken as a research object, and a pose error coordinate graph of the two-degree-of-freedom wheeled mobile robot is shown in figure 1.

In FIG. 1, M, M' is the axle midpoint of the two drive wheels; (x, y), (x)_r,y_r) Is the position of the mobile robot; theta, theta_rIs the included angle between the advancing direction of the mobile robot and the x axis; x is the number of_e,y_e,θ_eThe plane coordinate error and the direction error of the mobile robot are obtained; let P be (x, y, theta)^T，q＝(v,w)^TV and w are the linear velocity and angular velocity of the mobile robot, respectively;

the kinematic equation of the mobile robot is as follows:

by P_r＝(x_r,y_r,θ_r)^TAnd q is_r＝(v_r,w_r)^TTo represent a position command and a velocity command of the reference mobile robot; in FIG. 1, a mobile machineMan-in-the-pose P ═ (x, y, θ)^TMove to pose P_r＝(x_r,y_r,θ_r)^TMoving robot in new coordinate system X_e-Y_eThe coordinates in (1) are: p_e＝(x_e,y_e,θ_e)^TWherein theta_e＝θ_r-θ；

Setting new coordinate X_e-Y_eThe included angle between the coordinate system X and the coordinate system Y is theta, and according to a coordinate transformation formula, an error equation for describing the moving pose can be obtained as follows:

the differential equation of the attitude error obtained by the joint type (1) and (2) is as follows:

from the above analysis, the trajectory tracking of the kinematic model of the mobile robot, i.e. the seek control input q ═ (v, w)^TSo that for any initial error, the system is under the control input, P_e＝(x_e,y_e,θ_e)^TIs bounded, an

Design of control algorithm

Is provided with

Wherein alpha is₁＞0,β₁＞0,α₂＞0,β₂> 0, and p₁,q₁,p₂,q₂Is positive odd and satisfies p₁＞q₁,p₂＞q₂

From the formula (3)

By solving the first order linear differential equation (4), it can be known that the time t is finite₁Inner, theta_eIs equal to 0, and

similarly, solving the first order linear differential equation (5) can know that the time t is finite₂Inner, x_eIs equal to 0, and

for a mobile robotic system, as long as t > max { t₁,t₂Is then x_e＝0,θ_e＝0；

Taking according to the inversion method (Back-stepping)

(a) When theta is_eWhen v is 0, v is equal to v₁Then the formula (3) can be expressed as

The introduction of the new virtual feedback variables is as follows:

wherein k is₁₁＞0；

Taking Lyapunov function

Combining the error differential equation (8) to obtain:

so v₁The design is as follows:

wherein k is₁₂Is greater than 0; will control the rate v₁Substituting equation (11) into equation (9) can obtain:

as can be seen from the barkalat theorem,

respectively tend to zero; because of the fact that

Namely, it is

From the control rate, w is not constantly equal to zero, and

go to zero to obtain y_e→ 0; further comprises

Knowing x_e→0；

(b) When x is_eWhen the value is equal to 0, the value w is equal to w₂Then, equation (3) can be expressed as:

the introduction of the new virtual feedback variables is as follows:

taking Lyapunov function

Wherein

Combining the error differential equation (14) to obtain

Therefore w₂Is designed as

Wherein k is₂₁Is greater than 0; the control rate is substituted for formula (17) or formula (15) to obtain

Determined by BarbaltIn principle, y is_e,

Approaching to zero; and because of

By

Then

The formula is

Equivalence;

(c) taking the comprehensive control rate as

Substituted into the formulae (6), (7), (12) and (18)

Wherein k is₁₁,k₁₂,k₂₁,α₁,β₁,α₂,β₂Are all positive numbers greater than zero, p₁,q₁,p₂,q₂Is positive odd and satisfies p₁＞q₁,p₂＞q₂；

(III) pairs of errors y_eDemonstration of convergence

(a) For a mobile robotic system, as long as t > max { t₁,t₂Is then x_e＝0,θ_e0, substituted by formula (4) or (5)

The combination of formula (3) and formula (21) can give

y_ew-v+v_r＝0 (22)

w_r-w＝0 (23)

The combined type (22), (23) and (24) can be obtained

Because the system expects an input v_r,w_rCannot be simultaneously zero, w_rIf the value is not equal to zero, the system balance point is y_e＝0；

(b) Taking Lyapunov function

From the formulas (13) and (19)

Approaching zero, the combined formulas (9) and (15) can be obtained

Can obtain the product

From this, the error y_eThe global asymptote converges to zero.

The above description is only for the preferred embodiment of the present invention, but the scope of the present invention is not limited thereto, and any person skilled in the art should be considered to be within the technical scope of the present invention, and the technical solutions and the inventive concepts thereof according to the present invention should be equivalent or changed within the scope of the present invention.

Claims (1)

1. a composite trajectory tracking control algorithm based on inversion method and fast terminal sliding mode, is characterized in that, comprises the following steps: (1) Kinematics model of mobile robot Taking the two-degree-of-freedom wheeled mobile robot as the research object, the coordinate diagram of the pose error is established; In the pose error coordinate diagram, M, M' are the axis midpoints of the two driving wheels; (x, y), (x _r , y _r ) are the positions of the mobile robot; θ, θ _r are the forward directions of the mobile robot The included angle with the x-axis; x _e , y _e , θ _e are the plane coordinate error and direction error of the mobile robot; let P=(x, y, θ) ^T , q=(v, w) ^T , v and w are the linear velocity and angular velocity of the mobile robot, respectively; The kinematic equation of the mobile robot is:

Use P _r =(x _r , y _r , θ _r ) ^T and q _r =(v _r ,w _r ) ^T to represent the position command and speed command of the reference mobile robot; in the pose error coordinate diagram, the mobile robot starts from Pose P=(x, y, θ) ^T moves to pose P _r =(x _r , y _r , θ _r ) ^T , the coordinates of the mobile robot in the new coordinate system X _e -Y _e are: P _e =( x _e , y _e , θ _e ) ^T , where θ _e =θ _r -θ; Let the angle between the new coordinate X _e -Y _e and the coordinate system XY be θ, according to the coordinate transformation formula, the error equation describing the moving pose can be obtained as:

Combining equations (1) and (2), the differential equation of the pose error can be obtained as:

From the above analysis, the trajectory tracking of the kinematic model of the mobile robot is to find the control input q=(v,w) ^T , so that for any initial error, under the action of the control input, P _e =(x _e , y _e , θ ) _e ) ^T is bounded, and

(2) Design of control algorithm Assume

where α ₁ >0, β ₁ >0, α ₂ >0, β ₂ >0, and p ₁ , q ₁ , p ₂ , q ₂ are positive odd numbers and satisfy p ₁ >q ₁ , p ₂ >q ₂ From formula (3), we can get

Solving the first-order linear differential equation (4), we know that in the finite time t ₁ , θ _e = 0, and

Similarly, by solving the first-order linear differential equation (5), it can be known that in the finite time t ₂ , x _e = 0, and

For the mobile robot system, as long as t>max{t ₁ , t ₂ }, then x _e =0, θ _e =0; According to the inversion method (Back-stepping), take

(a) When θ _e = 0, take v = v ₁ , then formula (3) can be expressed as

A new dummy feedback variable is introduced as follows:

where k ₁₁ >0; Take the Lyapunov function

Combining the error differential equation (8), we can get:

So v1 is designed as _:

where k ₁₂ >0; Substitute the control rate v ₁ into the formula (11), and combine the formula (9), we can get:

According to Barbalat's theorem,

tend to zero, respectively; because

which is

From the control rate, w is not always equal to zero, and

tends to zero, and y _e → 0 is obtained; further by

Knowing x _e → 0; (b) When x _e =0, take w = w ₂ , then formula (3) can be expressed as:

A new dummy feedback variable is introduced as follows:

Take the Lyapunov function

in

Combining the error differential equation (14), we can get

So w ₂ is designed as

where k ₂₁ >0; substituting the control rate into equation (17), combined with equation (15), we can get

According to Barbalat's theorem, y _e ,

approaching zero; and because

Depend on

but

This formula and

equivalence; (c) Take the comprehensive control rate as

Substitute into formula (6)(7)(12)(18) to get

where k ₁₁ , k ₁₂ , k ₂₁ , α ₁ , β ₁ , α ₂ , β ₂ are all positive numbers greater than zero, p ₁ , q ₁ , p ₂ , q ₂ are positive odd numbers and satisfy p ₁ >q ₁ , p ₂ >q ₂ ; (3) Convergence proof for error y _e (a) For the mobile robot system, as long as t>max{t ₁ , t ₂ }, then x _e =0, θ _e =0, substituting into equations (4) and (5) to know

Combined with formula (3), formula (21) can be obtained y _e w-v+v _r =0 (22) w _r -w=0 (23)

Combining formulas (22) (23) (24), we can get

Because the system expects that the input v _r and w _r cannot be zero at the same time, then w _r is not always equal to zero, and the equilibrium point of the system is y _e =0; (b) Take the Lyapunov function

From formula (13), formula (19) know

approaching zero, combined with equations (9) and (15), we can get

Available

It can be seen that the error y _e converges to zero globally asymptotically.

Images