CN109857100A - A kind of complex track tracking control algorithm based on the method for inversion and fast terminal sliding formwork - Google Patents
A kind of complex track tracking control algorithm based on the method for inversion and fast terminal sliding formwork Download PDFInfo
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Abstract
本发明公开了一种基于反演法和快速终端滑模的复合轨迹跟踪控制算法，属于机器人技术领域，包括以下步骤：(一)移动机器人运动学模型，(二)控制算法的设计，(三)对误差ye的收敛性证明。本发明解决了传统全局快速终端滑模技术设计的轨迹跟踪控制器中存在难以兼顾收敛速度和精确度的问题，不仅提高了系统误差和输出的收敛速度，也减小了系统输出的误差；并且消除了传统滑模结构中具有的不连续项，避免了抖振现象，保证了输出的稳定性。
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
Technical field
The present invention relates to robotic technology field more particularly to a kind of composite rails based on the method for inversion and fast terminal sliding formwork
Mark tracking control algorithm.
Background technique
With the rapid development of robot industry, the requirement to robot working index is also higher and higher, therefore moving machine
The tracking accuracy and stability of device people is also more important.Mobile robot is a kind of control system by nonholonomic constraint,
The tracking error system of its track following is often a Coupling nonlinear system, is unsatisfactory for the necessary condition of Brockett, right
It controls and seems complex the problems such as planning.Therefore, many scholars propose a variety of methods and are used to solve mobile robot
Track following problem.The robustness of traditional pid algorithm controller is poor, and parameter tuning insensitive to extraneous interference is tired
It is difficult.Sliding moding structure method, which has, responds fast, good robustness, but the discontinuous term in control law is transferred directly to by it
Output item causes system inevitable chattering phenomenon occur.Iterative Learning Control Algorithm can be actual tracking error convergence
In scheduled error locus, but it generally requires initial position on desired trajectory, and the number of iterations influences final
It practises as a result, being limited for practical application.Self adaptive control can constantly obtain system input, state, output and performance
Parameter simultaneously makes corresponding adjustment to control law, is optimal control performance, but its parameter selection is complicated.Fuzzy control
Method has certain robustness, but the subjective factor that fuzzy control rule will receive people influences.For mobile robot system
On the one hand system allows system that can receive in finite time herein using the thought of the method for inversion and global fast terminal sliding mode technology
Equilibrium state is held back, the discontinuous term having in traditional sliding-mode structure is on the other hand eliminated, avoids chattering phenomenon, ensure that
The stability of output.
Summary of the invention
Technical problems based on background technology, the invention proposes a kind of based on the method for inversion and fast terminal sliding formwork
Complex track tracking control algorithm.
The technical solution adopted by the present invention is that:
A kind of complex track tracking control algorithm based on the method for inversion and fast terminal sliding formwork, which is characterized in that including with
Lower step:
(1) moveable robot movement model
Using two degrees of freedom wheeled mobile robot as research object, position and attitude error coordinate diagram is established；
In position and attitude error coordinate diagram, M, M' are the axis midpoint of two driving wheels；(x,y),(xr,yr) it is mobile robot
Position；θ,θrFor the angle of mobile robot direction of advance and x-axis；xe,ye,θeFor the plane coordinates error of mobile robot
With deflection error；Enable P=(x, y, θ)T, q=(v, w)T, v and w are respectively the linear velocity and angular speed of mobile robot；
The kinematical equation of mobile robot are as follows:
Use Pr=(xr,yr,θr)TAnd qr=(vr,wr)TTo indicate position command and speed command with reference to mobile robot；
In position and attitude error coordinate diagram, mobile robot is from pose P=(x, y, θ)TIt is moved to pose Pr=(xr,yr,θr)T, moving machine
Device people is in new coordinate system Xe-YeIn coordinate are as follows: Pe=(xe,ye,θe)T, wherein θe=θr-θ；
If new coordinate Xe-YeAngle between coordinate system X-Y is θ, according to coordinate transform formula, can must describe mobile position
The error equation of appearance are as follows:
The position and attitude error differential equation can be obtained in joint type (1) (2) are as follows:
From the above analysis, the track following of moveable robot movement model finds control input q=(v, w)T, make pair
Arbitrary initial error, system is in the case where controlling input action, Pe=(xe,ye,θe)TBounded, and
(2) design of control algolithm
If
Wherein α1>0,β1>0,α2>0,β2> 0, and p1,q1,p2,q2For positive odd number and meet p1>q1,p2>q2
It can be obtained by formula (3)
Linear first-order differential equation formula (4) are solved it is found that in finite time t1It is interior, θe=0, and
Similarly, solution linear first-order differential equation formula (5) is it is found that in finite time t2It is interior, xe=0, and
For mobile-robot system, as long as t > max { t1,t2, then xe=0, θe=0；
According to the method for inversion (Back-stepping), take
(a) work as θeWhen=0, v=v is taken1, then formula (3) can be expressed as
It is as follows to introduce new virtual feedback variable:
Wherein k11>0；
Take Lyapunov function
It can be obtained in conjunction with error differential equation (8):
So v1Design are as follows:
Wherein k12>0；By control rate v1(11) formula of substitution, convolution (9) can obtain:
By Barbalat theorem it is found thatIt goes to zero respectively；BecauseI.e.By
Control rate knows that w is nonidentical in zero, andIt goes to zero, obtains ye→0；Further byKnow xe→0；
(b) work as xeWhen=0, w=w is taken2, then formula (3) can be expressed as:
It is as follows to introduce new virtual feedback variable:
Take Lyapunov function
WhereinIt can be obtained in conjunction with error differential equation (14)
So w2It is designed as
Wherein k21>0；Control rate is substituted into formula (17), convolution (15) can obtain
By Barbalat theorem it is found that ye,Level off to zero；And becauseBy
ThenThe formula withIt is of equal value；
(c) the comprehensively control rate is taken to be
Substitution formula (6) (7) (12) (18)
Wherein k11,k12,k21,α1,β1,α2,β2It is the positive number greater than zero, p1,q1,p2,q2For positive odd number and meet p1>
q1,p2>q2。
(3) to error yeConvergence prove
(a) for mobile-robot system, as long as t > max { t1,t2, then xe=0, θe=0, it substitutes into formula (4) (5) and knowsConvolution (3), formula (21) are available
yew-v+vr=0 (22)
wr- w=0 (23)
Convolution (22) (23) (24) can obtain
Because system expectation inputs vr,wrIt cannot simultaneously be zero, then wrIt is nonidentical in zero, then obtain system balancing point be ye
=0；
(b) Lyapunov function is taken
By formula (13), formula (19) is knownLevel off to zero, convolution (9) (15) can obtain
It can obtain
It follows that pose yeGlobally asymptotical convergence is to zero.
The invention has the advantages that
The present invention solves simultaneous in the presence of being difficult in the contrail tracker of traditional global fast terminal sliding mode technology design
The problem of caring for convergence rate and accuracy, not only increases the convergence rate of systematic error and output, also reduces system output
Error；And the discontinuous term having in traditional sliding-mode structure is eliminated, chattering phenomenon is avoided, ensure that the stabilization of output
Property.
Detailed description of the invention
Fig. 1 is mobile robot position and attitude error coordinate diagram.
Fig. 2 is robotic tracking's control system block diagram.
Specific embodiment
Following will be combined with the drawings in the embodiments of the present invention, and technical solution in the embodiment of the present invention carries out clear, complete
Site preparation description, it is clear that described embodiments are only a part of the embodiments of the present invention, instead of all the embodiments.
Embodiment.
Such as Fig. 2, a kind of complex track tracking control algorithm based on the method for inversion and fast terminal sliding formwork, including following step
It is rapid:
(1) moveable robot movement model
Using two degrees of freedom wheeled mobile robot as research object, position and attitude error coordinate diagram is as shown in Figure 1.
In fig. 1, M, M' are the axis midpoint of two driving wheels；(x,y),(xr,yr) be mobile robot position；θ,
θrFor the angle of mobile robot direction of advance and x-axis；xe,ye,θeIt is missed for the plane coordinates error of mobile robot and direction
Difference；Enable P=(x, y, θ)T, q=(v, w)T, v and w are respectively the linear velocity and angular speed of mobile robot；
The kinematical equation of mobile robot are as follows:
Use Pr=(xr,yr,θr)TAnd qr=(vr,wr)TTo indicate position command and speed command with reference to mobile robot；
In Fig. 1, mobile robot is from pose P=(x, y, θ)TIt is moved to pose Pr=(xr,yr,θr)T, mobile robot is in new coordinate
It is Xe-YeIn coordinate are as follows: Pe=(xe,ye,θe)T, wherein θe=θr-θ；
If new coordinate Xe-YeAngle between coordinate system X-Y is θ, according to coordinate transform formula, can must describe mobile position
The error equation of appearance are as follows:
The position and attitude error differential equation can be obtained in joint type (1) (2) are as follows:
From the above analysis, the track following of moveable robot movement model finds control input q=(v, w)T, make pair
Arbitrary initial error, system is in the case where controlling input action, Pe=(xe,ye,θe)TBounded, and
(2) design of control algolithm
If
Wherein α1>0,β1>0,α2>0,β2> 0, and p1,q1,p2,q2For positive odd number and meet p1>q1,p2>q2
It can be obtained by formula (3)
Linear first-order differential equation formula (4) are solved it is found that in finite time t1It is interior, θe=0, and
Similarly, solution linear first-order differential equation formula (5) is it is found that in finite time t2It is interior, xe=0, and
For mobile-robot system, as long as t > max { t1,t2, then xe=0, θe=0；
According to the method for inversion (Back-stepping), take
(a) work as θeWhen=0, v=v is taken1, then formula (3) can be expressed as
It is as follows to introduce new virtual feedback variable:
Wherein k11>0；
Take Lyapunov function
It can be obtained in conjunction with error differential equation (8):
So v1Design are as follows:
Wherein k12>0；By control rate v1(11) formula of substitution, convolution (9) can obtain:
By Barbalat theorem it is found thatIt goes to zero respectively；BecauseI.e.By controlling
Rate processed knows that w is nonidentical in zero, andIt goes to zero, obtains ye→0；Further byKnow xe→0；
(b) work as xeWhen=0, w=w is taken2, then formula (3) can be expressed as:
It is as follows to introduce new virtual feedback variable:
Take Lyapunov function
WhereinIt can be obtained in conjunction with error differential equation (14)
So w2It is designed as
Wherein k21>0；Control rate is substituted into formula (17), convolution (15) can obtain
By Barbalat theorem it is found that ye,Level off to zero；And becauseBy
ThenThe formula withIt is of equal value；
(c) the comprehensively control rate is taken to be
Substitution formula (6) (7) (12) (18)
Wherein k11,k12,k21,α1,β1,α2,β2It is the positive number greater than zero, p1,q1,p2,q2For positive odd number and meet p1>
q1,p2>q2；
(3) to error yeConvergence prove
(a) for mobile-robot system, as long as t > max { t1,t2, then xe=0, θe=0, it substitutes into formula (4) (5) and knowsConvolution (3), formula (21) are available
yew-v+vr=0 (22)
wr- w=0 (23)
Convolution (22) (23) (24) can obtain
Because system expectation inputs vr,wrIt cannot simultaneously be zero, then wrIt is nonidentical in zero, then obtain system balancing point be ye
=0；
(b) Lyapunov function is taken
By formula (13), formula (19) is knownLevel off to zero, convolution (9) (15) can obtain
It can obtain
It follows that pose yeGlobally asymptotical convergence is to zero.
The foregoing is only a preferred embodiment of the present invention, but scope of protection of the present invention is not limited thereto,
Anyone skilled in the art in the technical scope disclosed by the present invention, according to the technique and scheme of the present invention and its
Inventive concept is subject to equivalent substitution or change, should be covered by the protection scope of the present invention.
Claims (1)
1. a kind of complex track tracking control algorithm based on the method for inversion and fast terminal sliding formwork, which is characterized in that including following
Step:
(1) moveable robot movement model
Using two degrees of freedom wheeled mobile robot as research object, position and attitude error coordinate diagram is established；
In position and attitude error coordinate diagram, M, M' are the axis midpoint of two driving wheels；(x,y),(xr,yr) be mobile robot position
It sets；θ,θrFor the angle of mobile robot direction of advance and x-axis；xe,ye,θeFor the plane coordinates error of mobile robot and side
To error；Enable P=(x, y, θ)T, q=(v, w)T, v and w are respectively the linear velocity and angular speed of mobile robot；
The kinematical equation of mobile robot are as follows:
Use Pr=(xr,yr,θr)TAnd qr=(vr,wr)TTo indicate position command and speed command with reference to mobile robot；In place
In appearance error coordinate diagram, mobile robot is from pose P=(x, y, θ)TIt is moved to pose Pr=(xr,yr,θr)T, mobile robot
In new coordinate system Xe-YeIn coordinate are as follows: Pe=(xe,ye,θe)T, wherein θe=θr-θ；
If new coordinate Xe-YeAngle between coordinate system X-Y is θ, according to coordinate transform formula, can must describe moving position gesture
Error equation are as follows:
The position and attitude error differential equation can be obtained in joint type (1) (2) are as follows:
From the above analysis, the track following of moveable robot movement model finds control input q=(v, w)T, make to any
Initial error, system is in the case where controlling input action, Pe=(xe,ye,θe)TBounded, and
(2) design of control algolithm
If
Wherein α1>0,β1>0,α2>0,β2> 0, and p1,q1,p2,q2For positive odd number and meet p1>q1,p2>q2
It can be obtained by formula (3)
Linear first-order differential equation formula (4) are solved it is found that in finite time t1It is interior, θe=0, and
Similarly, solution linear first-order differential equation formula (5) is it is found that in finite time t2It is interior, xe=0, and
For mobile-robot system, as long as t > max { t1,t2, then xe=0, θe=0；
According to the method for inversion (Back-stepping), take
(a) work as θeWhen=0, v=v is taken1, then formula (3) can be expressed as
It is as follows to introduce new virtual feedback variable:
Wherein k11>0；
Take Lyapunov function
It can be obtained in conjunction with error differential equation (8):
So v1Design are as follows:
Wherein k12>0；By control rate v1(11) formula of substitution, convolution (9) can obtain:
By Barbalat theorem it is found thatIt goes to zero respectively；BecauseI.e.By control rate
Knowing, w is nonidentical in zero, andIt goes to zero, obtains ye→0；Further byKnow xe→0；
(b) work as xeWhen=0, w=w is taken2, then formula (3) can be expressed as:
It is as follows to introduce new virtual feedback variable:
Take Lyapunov function
Wherein
It can be obtained in conjunction with error differential equation (14)
So w2It is designed as
Wherein k21>0；Control rate is substituted into formula (17), convolution (15) can obtain
By Barbalat theorem it is found that ye,Level off to zero；And becauseByThenThe formula withIt is of equal value；
(c) the comprehensively control rate is taken to be
Substitution formula (6) (7) (12) (18)
Wherein k11,k12,k21,α1,β1,α2,β2It is the positive number greater than zero, p1,q1,p2,q2For positive odd number and meet p1>q1,p2>
q2；
(3) to error yeConvergence prove
(a) for mobile-robot system, as long as t > max { t1,t2, then xe=0, θe=0, it substitutes into formula (4) (5) and knowsConvolution (3), formula (21) are available
yew-v+vr=0 (22)
wr- w=0 (23)
Convolution (22) (23) (24) can obtain
Because system expectation inputs vr,wrIt cannot simultaneously be zero, then wrIt is nonidentical in zero, then obtain system balancing point be ye=0；
(b) Lyapunov function is taken
By formula (13), formula (19) is knownLevel off to zero, convolution (9) (15) can obtain
It can obtain
It follows that pose yeGlobally asymptotical convergence is to zero.
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| 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 |
| CN113377115A (en) * | 2021-07-05 | 2021-09-10 | 沈阳工业大学 | Stability control method for autonomous learning transient motion time of service robot |
| CN113671950A (en) * | 2021-07-28 | 2021-11-19 | 长安大学 | Vehicle trajectory tracking control method based on pose convergence algorithm |
| CN114115261A (en) * | 2021-11-22 | 2022-03-01 | 集美大学 | Time-adjustable robot tracking sliding mode control method |
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| 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 |
| CN113377115A (en) * | 2021-07-05 | 2021-09-10 | 沈阳工业大学 | Stability control method for autonomous learning transient motion time of service robot |
| 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 |
| CN113671950A (en) * | 2021-07-28 | 2021-11-19 | 长安大学 | Vehicle trajectory tracking control method based on pose convergence algorithm |
| CN113671950B (en) * | 2021-07-28 | 2024-02-02 | 长安大学 | Vehicle track tracking control method based on pose convergence algorithm |
| CN114115261A (en) * | 2021-11-22 | 2022-03-01 | 集美大学 | Time-adjustable robot tracking sliding mode control method |
| CN114115261B (en) * | 2021-11-22 | 2024-03-29 | 集美大学 | Robot tracking sliding mode control method with adjustable time |
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