CN109857100B - Composite track tracking control algorithm based on inversion method and fast terminal sliding mode - Google Patents
Composite track tracking control algorithm based on inversion method and fast terminal sliding mode Download PDFInfo
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Abstract
The invention discloses a composite track tracking control algorithm based on an inversion method and a quick terminal sliding mode, which belongs to the technical field of robots and comprises the following steps: the method comprises the following steps of (I) mobile robot kinematic model, (II) control algorithm design, and (III) error yeIs verified by convergence. 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.
Description
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,yr) Is the position of the mobile robot; theta, thetarIs the included angle between the advancing direction of the mobile robot and the x axis; x is the number ofe,ye,θeThe 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 Pr=(xr,yr,θr)TAnd q isr=(vr,wr)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 Pr=(xr,yr,θr)TMoving robot in new coordinate system Xe-YeThe coordinates in (1) are: pe=(xe,ye,θe)TWherein thetae=θr-θ;
Setting new coordinate Xe-YeThe 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, Pe=(xe,ye,θe)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, thetaeIs equal to 0, and
similarly, solving the first order linear differential equation (5) can know that the time t is finite2Inner, xeIs equal to 0, and
for a mobile robotic system, as long as t > max { t1,t2Is then xe=0,θe=0;
(a) When theta iseWhen 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 thatNamely, it isFrom the control rate, w is not constantly equal to zero, andgo to zero to obtain ye→ 0; further comprisesKnowing xe→0;
(b) When x iseWhen 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
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, ye,Approaching to zero; and because ofByThenThe formula isEquivalence;
(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 yeDemonstration of convergence
(a) For a mobile robotic system, as long as t > max { t1,t2Is then xe=0,θe0, substituted by formula (4) or (5)The combination of formula (3) and formula (21) can give
yew-v+vr=0 (22)
wr-w=0 (23)
The combined type (22), (23) and (24) can be obtained
Because the system expects an input vr,wrCannot be simultaneously zero, wrIf the value is not equal to zero, the system balance point is ye=0;
(b) Taking Lyapunov function
Can obtain the product
From this, the error yeThe 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,yr) Is the position of the mobile robot; theta, thetarIs the included angle between the advancing direction of the mobile robot and the x axis; x is the number ofe,ye,θeThe 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 Pr=(xr,yr,θr)TAnd q isr=(vr,wr)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 Pr=(xr,yr,θr)TMoving robot in new coordinate system Xe-YeThe coordinates in (1) are: pe=(xe,ye,θe)TWherein thetae=θr-θ;
Setting new coordinate Xe-YeThe 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, Pe=(xe,ye,θe)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, thetaeIs equal to 0, and
similarly, solving the first order linear differential equation (5) can know that the time t is finite2Inner, xeIs equal to 0, and
for a mobile robotic system, as long as t > max { t1,t2Is then xe=0,θe=0;
(a) When theta iseWhen 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 thatNamely, it isFrom the control rate, w is not constantly equal to zero, andgo to zero to obtain ye→ 0; further comprisesKnowing xe→0;
(b) When x iseWhen 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
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 ise,Approaching to zero; and because ofByThenThe formula isEquivalence;
(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 yeDemonstration of convergence
(a) For a mobile robotic system, as long as t > max { t1,t2Is then xe=0,θe0, substituted by formula (4) or (5)The combination of formula (3) and formula (21) can give
yew-v+vr=0 (22)
wr-w=0 (23)
The combined type (22), (23) and (24) can be obtained
Because the system expects an input vr,wrCannot be simultaneously zero, wrIf the value is not equal to zero, the system balance point is ye=0;
(b) Taking Lyapunov function
Can obtain the product
From this, the error yeThe 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 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,yr) Is the position of the mobile robot; theta, thetarIs the included angle between the advancing direction of the mobile robot and the x axis; x is the number ofe,ye,θeThe 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 Pr=(xr,yr,θr)TAnd q isr=(vr,wr)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 Pr=(xr,yr,θr)TMoving robot in new coordinate system Xe-YeThe coordinates in (1) are: pe=(xe,ye,θe)TWherein thetae=θr-θ;
Setting new coordinate Xe-YeThe included angle between the coordinate system X and the coordinate system Y is theta, and description can be obtained according to a coordinate transformation formulaThe error equation of the moving pose is 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, Pe=(xe,ye,θe)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, thetaeIs equal to 0, and
similarly, solving the first order linear differential equation (5) can know that the time t is finite2Inner, xeIs equal to 0, and
for a mobile robotic system, as long as t > max { t1,t2Is then xe=0,θe=0;
(a) When theta iseWhen 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 thatNamely, it isFrom the control rate, w is not constantly equal to zero, andgo to zero to obtain ye→ 0; further comprisesKnowing xe→0;
(b) When x iseWhen 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
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, ye,Approaching to zero; and because ofByThenThe formula isEquivalence;
(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 yeDemonstration of convergence
(a) For a mobile robotic system, as long as t > max { t1,t2Is then xe=0,θe0, substituted by formula (4) or (5)Combination formula (3) and formula (2)1) Can obtain
yew-v+vr=0 (22)
wr-w=0 (23)
The combined type (22), (23) and (24) can be obtained
Because the system expects an input vr,wrCannot be simultaneously zero, wrIf the value is not equal to zero, the system balance point is ye=0;
(b) Taking Lyapunov function
Can obtain the product
From this, the error yeGlobalAsymptotically converging to zero.
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CN110865641B (en) * | 2019-10-30 | 2022-06-24 | 吉首大学 | Track tracking method of wheeled mobile robot controlled by inversion sliding mode |
CN111103798B (en) * | 2019-12-20 | 2021-03-30 | 华南理工大学 | AGV path tracking method based on inversion sliding mode control |
CN112286229A (en) * | 2020-10-22 | 2021-01-29 | 上海海事大学 | Moving robot finite time trajectory tracking control method based on recursive sliding mode |
CN113377115B (en) * | 2021-07-05 | 2023-10-20 | 沈阳工业大学 | Stable control method for service robot to autonomously learn transient movement time |
CN113671950B (en) * | 2021-07-28 | 2024-02-02 | 长安大学 | Vehicle track tracking control method based on pose convergence algorithm |
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