CN108614426B - Multi-mobile-robot formation robust control method based on disturbance observer - Google Patents

Multi-mobile-robot formation robust control method based on disturbance observer Download PDF

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CN108614426B
CN108614426B CN201810576368.0A CN201810576368A CN108614426B CN 108614426 B CN108614426 B CN 108614426B CN 201810576368 A CN201810576368 A CN 201810576368A CN 108614426 B CN108614426 B CN 108614426B
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formation
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CN108614426A (en
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郭一军
孙剑
赵年顺
黄辉
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Huangshan University
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Abstract

The invention discloses a multi-mobile-robot formation robust control method based on an interference observer, which comprises the steps of firstly establishing a multi-mobile-robot formation control system model under the condition of considering the influence of uncertainty factors such as model parameter perturbation and external disturbance; then designing an auxiliary speed controller for stabilizing the formation error for the following robot according to the formation error dynamic model; and finally, taking the output of the auxiliary speed controller as a speed given signal following the robot dynamic model, and designing an adaptive sliding mode controller based on a disturbance observer on the dynamic level. The control method can effectively inhibit the adverse effect of the total disturbance in the multi-mobile-robot formation system, accelerate the response speed of the formation system and improve the stability and robustness of the system.

Description

Multi-mobile-robot formation robust control method based on disturbance observer
Technical Field
The invention relates to the field of robot control methods, in particular to a multi-mobile-robot formation robust control method based on an interference observer.
Background
Compared with a single robot, the multi-robot system has many potential advantages including strong flexibility, adaptability and robustness, so that the multi-mobile robot formation control research gradually becomes an important research field of the mobile robots. The formation control is to control a group of robots to form a formation according to a desired geometric shape and maintain the formation in the process of movement according to tasks to be completed by formation.
Currently, the formation control methods of mobile robots can be mainly classified into a master-slave structure method, a behavior control method and a virtual structure method, and although there are related researches on the three methods, the researches mainly focus on the kinematics level of the formation system, and the researches on the system dynamics level are relatively few. In addition, in the process of controlling the formation of the mobile robots, the influence of uncertain factors such as system parameter perturbation, load change, unknown ground friction force and external disturbance inevitably exists, and how to effectively eliminate the adverse effect of the uncertain factors is a problem to be solved urgently in the control of the formation of the multiple mobile robots.
The sliding mode control algorithm has the advantages that the dynamic response speed is high, the response performance can be flexibly designed according to a specific system, and the system is insensitive to parameter perturbation and external disturbance once entering a sliding mode, so that the sliding mode control algorithm is widely applied to the field of nonlinear uncertain system control. As an important branch of sliding mode control, the self-adaptive sliding mode control can obtain better control performance while maintaining the dynamic performance of the traditional sliding mode control.
Disclosure of Invention
The invention aims to provide a multi-mobile-robot formation robust control method based on a disturbance observer, so as to make up for the defects of the existing multi-mobile-robot formation control method.
In order to achieve the purpose, the technical scheme adopted by the invention is as follows:
a multi-mobile-robot formation robust control method based on a disturbance observer is characterized by comprising the following steps: the method comprises the following steps:
(1) establishing a multi-mobile-robot formation control system model under the condition of considering the influence of uncertainty factors including model parameter perturbation and external disturbance, and specifically comprising the following steps of:
(1.1), the moment required by the forward and rotary motion of the mobile robot is provided by two driving wheels arranged on the same shaft, the front wheels only play a supporting role, and the kinematic equation of the mobile robot can be described as follows:
Figure BDA0001687202280000021
in the formula (1), x, y and theta are respectively the coordinate and the direction angle of the mobile robot in the global coordinate system, and the counterclockwise direction of the direction angle is set as positive;
Figure BDA0001687202280000022
u＝[v ω]Tis a vector composed of a linear velocity and an angular velocity of the mobile robot, wherein v represents the linear velocity and ω represents the angular velocity;
from Lagrange modeling, the kinetic equations of a mobile robot can be described as
Figure BDA0001687202280000023
In formula (2), q ═ x y θ]T∈R3×1；
Figure BDA0001687202280000024
Positively determining an inertia matrix for the system, wherein m and I represent the mass and inertia of the mobile robot, respectively; c is belonged to R3×3Is the centrifugal and coriolis force matrices of the system; g is belonged to R3×1Is the gravity term of the system, which is zero for a mobile robot moving in a plane;
Figure BDA0001687202280000025
is an unknown ground friction term; tau isd∈R3×1An externally bounded perturbing term;
Figure BDA0001687202280000026
converting an array for a control signal, wherein r and 2L respectively represent the radius of a driving wheel of the mobile robot and the distance between the two driving wheels; τ ═ τ [ τ ]r τl]T∈R2×1Controlling signal vectors for the two driving wheels;
Figure BDA0001687202280000027
is a matrix related to the system incomplete constraint;
Figure BDA0001687202280000028
is Lagrange multiplier;
substituting the formula (1) and its first derivative into the formula (2) and multiplying by ST(q) available:
Figure BDA0001687202280000029
from STAT(q) ═ 0, available:
Figure BDA0001687202280000031
wherein the content of the first and second substances,
Figure BDA0001687202280000032
due to the influence of uncertainty factors such as load variation, measurement error and system external disturbance, it is difficult to obtain an accurate model of the mobile robot, and therefore, the model of the actual system of the mobile robot can be expressed as:
Figure BDA0001687202280000033
in the formula (5), the first and second groups,
Figure BDA0001687202280000034
M0is composed of
Figure BDA0001687202280000035
Can be determined empirically, Δ M is
Figure BDA0001687202280000036
The indeterminate portion of (a);
Figure BDA0001687202280000037
can be seen as a sum disturbance of the system;
(1.2) the Leader robot in the formation system plays a role in navigation, and the Follower robot follows the Leader robot according to a preset expected relative distance and an expected relative direction angle to form an expected formation; the expected parameter for a given formation is LrlfAnd
Figure BDA0001687202280000038
the reference pose and the actual pose of the following robot can be obtained through the geometric position relationship between the master-slave robots as shown in formulas (6) and (7), respectively:
Figure BDA0001687202280000039
Figure BDA00016872022800000310
in the formulas (6) and (7), [ x ]fr yfr θfr]TRepresenting the pose of the Follower robot reference; [ x ] off yf θf]TRepresenting the actual pose of the Follower robot; [ x ] ofl yl θl]TRepresenting the pose of the Leader robot; thetarA reference direction angle is set for the Follower robot;
subtracting the equation (6) from the equation (7), and forming an error vector [ x ] by coordinate transformationef yef θef]TCan be expressed as follows in the local coordinate system of the robot
Figure BDA0001687202280000041
Wherein, thetalf＝θl-θf；
The derivation of the formula (8) along the formulas (6) - (7) can obtain a dynamic model of the formation error of the mobile robot kinematics layer:
Figure BDA0001687202280000042
wherein the content of the first and second substances,
Figure BDA0001687202280000043
(2) the design of the multi-mobile robot formation auxiliary speed controller comprises the following specific processes:
(2.1) because the direction angles of the following robots in the formation process are not completely the same due to the incomplete constraint of the mobile robot and the limitation of the master-slave formation control target, setting the reference direction angles of the formation following robots as follows:
Figure BDA0001687202280000044
d is the distance between the middle point of the driving wheel shaft and a reference point for calculating the relative distance of formation; k is a radical ofθDesigning parameters for an auxiliary speed control law to be designed;
thus, the kinematic formation error dynamic model (9) of the mobile robot can be rewritten as
Figure BDA0001687202280000045
In order to stabilize the system shown in the formula (11), according to the Lyapunov stability theory, the following robot-assisted speed control law is designed as follows:
Figure BDA0001687202280000046
wherein k isx>0、ky>0、kθ>0 is an auxiliary speed control law parameter;
to facilitate kinematic controller parameter kx，kyAnd kθSubstituting a control law formula (12) into a formation error dynamic equation (11) and linearizing the control law formula near a balance point to obtain:
Figure BDA0001687202280000051
wherein the content of the first and second substances,
Figure BDA0001687202280000052
from equation (15), the characteristic equation is:
a3s3+a2s2+a1s+a0(iii) 0(16), wherein,
Figure BDA0001687202280000053
therefore, the parameter k is known from the Laus-Holwitz stability criterionx，kyAnd kθThe design of (c) also needs to satisfy the condition: a isi>0(i ═ 0,1,2,3) and a1a2-a0a3>0；
(3) Designing a formation dynamics controller, and specifically comprising the following processes:
(3.1) designing a formation disturbance observer: in order to estimate f, the model equation (5) needs to be expanded to
Figure BDA0001687202280000054
Wherein the content of the first and second substances,
Figure BDA0001687202280000055
representing the rate of change of the system's sum disturbance;
is provided with
Figure BDA0001687202280000056
And
Figure BDA0001687202280000057
the estimated values of u and f respectively, and the system state estimation error is defined as
Figure BDA0001687202280000058
The disturbance observer of equation (17) can be designed as:
Figure BDA0001687202280000059
wherein, K1，K2An observer gain matrix to be designed;
the observation error dynamic equation obtained by subtracting equation (17) from equation (18) is:
Figure BDA0001687202280000061
wherein the content of the first and second substances,
Figure BDA0001687202280000062
representing the sum total disturbance estimation error;
by designing the observer gain matrix K1、K2Make the observation error system matrix
Figure BDA0001687202280000063
The characteristic values of (a) are all located in the left half-open complex plane;
(3.2) on the basis of the design of the auxiliary speed controller, in order to make the design closer to the practical system application, the design of a formation dynamics controller is also needed:
defining the auxiliary velocity tracking error state vector for the formation as:
Figure BDA0001687202280000064
wherein u isc＝[vf ωf]TAnd u are the given velocity vector and the actual velocity vector of the formation dynamics system, respectively;
the derivation of equation (20) and substitution of equation (5) can be obtained:
Figure BDA0001687202280000065
the slip form surface is designed as follows:
Figure BDA0001687202280000066
wherein λ ═ diag (λ)1,λ2)>0 is a constant matrix to be designed;
the sliding mode surface is subjected to time derivation to obtain:
Figure BDA0001687202280000067
when the system is in the sliding mode
Figure BDA0001687202280000068
Under the condition of not considering the disturbance of the system sum, the equivalent control law of the system is obtained as follows:
Figure BDA0001687202280000069
once the system error state variable moves to the sliding mode surface, only the lambda is seti(i＝1,2)>0 can make
Figure BDA0001687202280000071
However, the initial state of the system is usually not exactly located on the sliding mode surface, and the state running on the sliding mode surface of the system is also deviated from the sliding mode surface due to the influence of uncertainty factors such as system model parameter change and external disturbance, so that the control performance of the system is deteriorated, and a switching control item needs to be designed in order to eliminate the influence of uncertainty factors:
Figure BDA0001687202280000072
wherein, K3＝diag(k31,k32)>0 is the gain matrix of the switching control term to be designed, whose elements are derived fromThe adaptive law is adjusted, and the adaptive law is designed as follows:
Figure BDA0001687202280000073
wherein, γ1,γ2>0 is the adaptive gain, so the adaptive sliding mode control law based on the disturbance observer is designed as:
Figure BDA0001687202280000074
the invention has the advantages that: the control method can effectively inhibit the adverse effect of the total disturbance in the multi-mobile-robot formation system, accelerate the response speed of the formation system and improve the stability and robustness of the system.
Drawings
FIG. 1 is a schematic diagram of a Leader-follower robot formation system according to the present invention;
FIG. 2 is a schematic diagram of the formation control system estimating the total disturbance. Wherein (a) is a schematic diagram of the estimation of the total perturbation by Follow1, and (b) is a schematic diagram of the estimation of the total perturbation by Follow 2;
FIG. 3 is a schematic diagram of a comparison of tracking errors of the formation control system. Wherein, the graph (a) is a tracing error comparison schematic diagram of Follower1, and the graph (b) is a tracing error comparison schematic diagram of Follower 2;
FIG. 4 is a schematic diagram comparing control signals of the formation control system. Wherein, the graph (a) is a control signal comparison schematic diagram of Follower1, and the graph (b) is a control signal comparison schematic diagram of Follower 2.
Detailed Description
The invention is further illustrated with reference to the following figures and examples.
As shown in fig. 1, a method for controlling robustness of formation of multiple mobile robots based on a disturbance observer includes the following steps:
(1) establishing a multi-mobile-robot formation control system model under the condition of considering the influence of uncertainty factors including model parameter perturbation and external disturbance, and specifically comprising the following steps of:
(1.1), the moment required by the forward and rotary motion of the mobile robot is provided by two driving wheels arranged on the same shaft, the front wheels only play a supporting role, and the kinematic equation of the mobile robot can be described as follows:
Figure BDA0001687202280000081
in the formula (1), x, y and theta are respectively the coordinate and the direction angle of the mobile robot in the global coordinate system, and the counterclockwise direction of the direction angle is set as positive;
Figure BDA0001687202280000082
u＝[v ω]Tis a vector composed of a linear velocity and an angular velocity of the mobile robot, wherein v represents the linear velocity and ω represents the angular velocity;
from Lagrange modeling, the kinetic equations of a mobile robot can be described as
Figure BDA0001687202280000083
In formula (2), q ═ x y θ]T∈R3×1；
Figure BDA0001687202280000084
Positively determining an inertia matrix for the system, wherein m and I represent the mass and inertia of the mobile robot, respectively; c is belonged to R3×3Is the centrifugal and coriolis force matrices of the system; g is belonged to R3×1Is the gravity term of the system, which is zero for a mobile robot moving in a plane;
Figure BDA0001687202280000085
is an unknown ground friction term; tau isd∈R3×1An externally bounded perturbing term;
Figure BDA0001687202280000086
for control signal transformation matrix, where r and 2L respectively denote mobile stationsThe radius of the driving wheels of the robot and the distance between the two driving wheels; τ ═ τ [ τ ]r τl]T∈R2×1Controlling signal vectors for the two driving wheels;
Figure BDA0001687202280000087
is a matrix related to the system incomplete constraint;
Figure BDA0001687202280000088
is Lagrange multiplier;
substituting the formula (1) and its first derivative into the formula (2) and multiplying by ST(q) available:
Figure BDA0001687202280000091
from STAT(q) ═ 0, available:
Figure BDA0001687202280000092
wherein the content of the first and second substances,
Figure BDA0001687202280000093
due to the influence of uncertainty factors such as load variation, measurement error and system external disturbance, it is difficult to obtain an accurate model of the mobile robot, and therefore, the model of the actual system of the mobile robot can be expressed as:
Figure BDA0001687202280000094
in the formula (5), the first and second groups,
Figure BDA0001687202280000095
M0is composed of
Figure BDA0001687202280000096
Can be determined empiricallyΔ M is
Figure BDA0001687202280000097
The indeterminate portion of (a);
Figure BDA0001687202280000098
can be seen as a sum disturbance of the system;
(1.2) the Leader robot in the formation system plays a role in navigation, and the Follower robot follows the Leader robot according to a preset expected relative distance and an expected relative direction angle to form an expected formation; the expected parameter for a given formation is LrlfAnd
Figure BDA0001687202280000099
the reference pose and the actual pose of the following robot can be obtained through the geometric position relationship between the master-slave robots as shown in formulas (6) and (7), respectively:
Figure BDA00016872022800000910
Figure BDA00016872022800000911
in the formulas (6) and (7), [ x ]fr yfr θfr]TRepresenting the pose of the Follower robot reference; [ x ] off yf θf]TRepresenting the actual pose of the Follower robot; [ x ] ofl yl θl]TRepresenting the pose of the Leader robot; thetarA reference direction angle is set for the Follower robot;
subtracting the equation (6) from the equation (7), and forming an error vector [ x ] by coordinate transformationef yef θef]TCan be expressed as follows in the local coordinate system of the robot
Figure BDA0001687202280000101
Wherein, thetalf＝θl-θf；
The derivation of the formula (8) along the formulas (6) - (7) can obtain a dynamic model of the formation error of the mobile robot kinematics layer:
Figure BDA0001687202280000102
wherein the content of the first and second substances,
Figure BDA0001687202280000103
(2) the design of the multi-mobile robot formation auxiliary speed controller comprises the following specific processes:
(2.1) because the direction angles of the following robots in the formation process are not completely the same due to the incomplete constraint of the mobile robot and the limitation of the master-slave formation control target, setting the reference direction angles of the formation following robots as follows:
Figure BDA0001687202280000104
d is the distance between the middle point of the driving wheel shaft and a reference point for calculating the relative distance of formation; k is a radical ofθDesigning parameters for an auxiliary speed control law to be designed;
thus, the kinematic formation error dynamic model (9) of the mobile robot can be rewritten as
Figure BDA0001687202280000105
In order to stabilize the system shown in equation (11), according to the Lyapunov stability theory, the following robot-assisted speed control is designed as follows:
Figure BDA0001687202280000111
wherein k isx>0、ky>0、kθ>0 is an auxiliary speed control law parameter;
to facilitate kinematic controller parameter kx，kyAnd kθSubstituting a control law formula (12) into a formation error dynamic equation (11) and linearizing the control law formula near a balance point to obtain:
Figure BDA0001687202280000112
wherein the content of the first and second substances,
Figure BDA0001687202280000113
from equation (15), the characteristic equation is:
a3s3+a2s2+a1s+a0＝0 (16)，
wherein the content of the first and second substances,
Figure BDA00016872022800001111
therefore, the parameter k is known from the Laus-Holwitz stability criterionx，kyAnd kθThe design of (c) also needs to satisfy the condition: a isi>0(i ═ 0,1,2,3) and a1a2-a0a3>0；
(3) Designing a formation dynamics controller, and specifically comprising the following processes:
(3.1) designing a formation disturbance observer: in order to estimate f, the model equation (5) needs to be expanded to
Figure BDA0001687202280000115
Wherein the content of the first and second substances,
Figure BDA0001687202280000116
representing the rate of change of the system's sum disturbance;
is provided with
Figure BDA0001687202280000117
And
Figure BDA0001687202280000118
the estimated values of u and f respectively, and the system state estimation error is defined as
Figure BDA0001687202280000119
The disturbance observer of equation (17) can be designed as:
Figure BDA00016872022800001110
wherein, K1，K2An observer gain matrix to be designed;
the observation error dynamic equation obtained by subtracting equation (17) from equation (18) is:
Figure BDA0001687202280000121
wherein the content of the first and second substances,
Figure BDA0001687202280000122
representing the sum total disturbance estimation error;
by designing the observer gain matrix K1、K2Make the observation error system matrix
Figure BDA0001687202280000123
The characteristic values of (a) are all located in the left half-open complex plane;
(3.2) on the basis of the design of the auxiliary speed controller, in order to make the design closer to the practical system application, the design of a formation dynamics controller is also needed:
defining the auxiliary velocity tracking error state vector for the formation as:
Figure BDA0001687202280000124
wherein u isc＝[vf ωf]TAnd u are the given velocity vector and the actual velocity vector of the formation dynamics system, respectively;
the derivation of equation (20) and substitution of equation (5) can be obtained:
Figure BDA0001687202280000125
the slip form surface is designed as follows:
Figure BDA0001687202280000126
wherein λ ═ diag (λ)1,λ2)>0 is a constant matrix to be designed;
the sliding mode surface is subjected to time derivation to obtain:
Figure BDA0001687202280000127
when the system is in the sliding mode
Figure BDA0001687202280000128
Under the condition of not considering the disturbance of the system sum, the equivalent control law of the system is obtained as follows:
Figure BDA0001687202280000129
once the system error state variable moves to the sliding mode surface, only the lambda is seti(i＝1,2)>0 can make
Figure BDA0001687202280000131
However, the initial state of the system is usually not exactly located on the sliding mode surface, and the state running on the sliding mode surface of the system is also deviated from the sliding mode surface due to the influence of uncertainty factors such as the parameter change of the system model and the external disturbance, so that the control performance of the system is deteriorated, and a switching control item needs to be designed to eliminate the influence of the uncertainty factors：
Figure BDA0001687202280000132
Wherein, K3＝diag(k31,k32)>0 is a switching control item gain matrix to be designed, the elements of which are adjusted by an adaptive law, and the adaptive law is designed as follows:
Figure BDA0001687202280000133
wherein, γ1,γ2>0 is the adaptive gain, so the adaptive sliding mode control law based on the disturbance observer is designed as:
Figure BDA0001687202280000134
in order to verify the effectiveness of the designed multi-mobile-robot formation control method, simulation comparison research is respectively carried out on an adaptive sliding mode control method (ASMC + DOB) and a common sliding mode control method (SMC) based on a disturbance observer. The sliding mode surface design of the common sliding mode control method is as the formula (22), and the control law is designed as follows:
Figure BDA0001687202280000135
wherein the content of the first and second substances,
Figure BDA0001687202280000136
for the switching control gain matrix to be designed, its values are usually determined empirically and need to be satisfied
Figure BDA0001687202280000137
Suppose that the multi-mobile robot formation system consists of 3 mobile robots, one mobile robot is taken as a Leader, and the other 2 mobile robots are respectively taken as followers 1Follower 2. The formation main track is generated by a Leader, and the initial pose of the Leader is [ 310 ]]Linear and angular velocities are v, respectivelyl4m/s and ω l1 rad/s. Following the robot Follower1, the initial pose of Follower2 is [ 22 pi/10 ] respectively]，[2 -0.5 -π/9]The parameter of formation with Leader is set to [ 1.52 pi/3 respectively]，[1.5 -2π/3]. The physical parameters of 3 mobile robots in the formation system are assumed to be the same and are taken as follows: the mass m is 3 kg; inertia I1.5 kg · m2(ii) a The radius r of the driving wheel is 0.031 m; the distance L between the left driving wheel and the right driving wheel and the central point of the rear shaft of the mobile robot is 0.3 m; the distance d between the middle point of the driving wheel shaft and the reference point of the relative distance of the calculation formation is 0.06 m.
For comparative studies, the auxiliary speed controller, disturbance observer and sliding mode surface parameters of Follow1 and Follow2 in 2 control methods were set to be the same: k is a radical ofx＝27，ky＝14，kθ＝18；
Figure BDA0001687202280000141
Figure BDA0001687202280000142
λ1＝λ 220. Adaptive gain parameters of Folow 1 and Folow 2 in the adaptive sliding mode control method based on the disturbance observer are set as gamma1＝γ20.005; switching control gain settings for Folow 1 and Folow 2 in a conventional slip molding process
Figure BDA0001687202280000143
Assuming that the change of the model parameters of the two-following robot system is delta M0.01M0The external disturbance torque is [0.2sin (t) 0.1cos (t) ]]TAnd [0.2sin (t) + 0.010.1 cos (t) +0.02]T。
The control effects of the 2 control methods are shown in fig. 2-4. As can be seen from fig. 2, there is an obvious adjustment process in the initial stage of the sum-disturbance estimation, in which the sum-disturbance estimation error has a certain overshoot, but after a short adjustment process, the sum-disturbance estimation error can converge to zero quickly with high accuracy. This means that the disturbance observers of the two following robots can observe the sum disturbance signal well as long as the parameters of the disturbance observers are chosen reasonably, which provides conditions for enhancing the robustness of the proposed control method. It can be seen from fig. 3 that, under the condition that the total disturbance exists in the formation system, the formation tracking errors of the 2 control methods are smaller, and the formation accuracy is higher. Observing the control signals of the two following robots in the figure 4, it is easy to find that the control signals in the ASMC + DOB method have obvious buffeting in the initial stage of formation due to the existence of the overshoot of the disturbance observer in the initial adjustment stage, but the buffeting is almost eliminated by the control signals after the formation system enters a steady state, and the control signals in the SMC method have obvious buffeting in the whole formation control process.

Claims (1)

1. A multi-mobile-robot formation robust control method based on a disturbance observer is characterized by comprising the following steps: the method comprises the following steps:
(1) establishing a multi-mobile-robot formation control system model under the condition of considering the influence of uncertainty factors including model parameter perturbation and external disturbance, and specifically comprising the following steps of:
(1.1), the moment required by the forward and rotary motion of the mobile robot is provided by two driving wheels arranged on the same shaft, the front wheels only play a supporting role, and the kinematic equation of the mobile robot can be described as follows:
Figure FDA0002738345440000011
in the formula (1), x, y and theta are respectively the coordinate and the direction angle of the mobile robot in the global coordinate system, and the counterclockwise direction of the direction angle is set as positive;
Figure FDA0002738345440000012
u＝[v ω]Tis a vector composed of a linear velocity and an angular velocity of the mobile robot, wherein v represents the linear velocity and ω represents the angular velocity;
from Lagrange modeling, the kinetic equations of a mobile robot can be described as
Figure FDA0002738345440000013
In formula (2), q ═ x y θ]T∈R3×1；
Figure FDA0002738345440000014
Positively determining an inertia matrix for the system, wherein m and I represent the mass and inertia of the mobile robot, respectively; c is belonged to R3×3Is the centrifugal and coriolis force matrices of the system; g is belonged to R3×1Is the gravity term of the system, which is zero for a mobile robot moving in a plane;
Figure FDA0002738345440000015
is an unknown ground friction term; tau isd∈R3×1An externally bounded perturbing term;
Figure FDA0002738345440000016
converting an array for a control signal, wherein r and 2L respectively represent the radius of a driving wheel of the mobile robot and the distance between the two driving wheels; τ ═ τ [ τ ]r τl]T∈R2×1Controlling signal vectors for the two driving wheels;
Figure FDA0002738345440000017
is a matrix related to the system incomplete constraint;
Figure FDA0002738345440000018
is Lagrange multiplier;
substituting the formula (1) and its first derivative into the formula (2) and multiplying by ST(q) available:
Figure FDA0002738345440000021
from STAT(q) ═ 0, available:
Figure FDA0002738345440000022
wherein the content of the first and second substances,
Figure FDA0002738345440000023
Figure FDA0002738345440000024
due to the influence of uncertainty factors such as load variation, measurement error and system external disturbance, it is difficult to obtain an accurate model of the mobile robot, and therefore, the model of the actual system of the mobile robot can be expressed as:
Figure FDA0002738345440000025
in the formula (5), the first and second groups,
Figure FDA0002738345440000026
M0is composed of
Figure FDA0002738345440000027
Can be determined empirically, Δ M is
Figure FDA0002738345440000028
The indeterminate portion of (a);
Figure FDA0002738345440000029
can be seen as a sum disturbance of the system;
(1.2) the Leader robot in the formation system plays a role in navigation, and the Follower robot follows the Leader robot according to a preset expected relative distance and an expected relative direction angle to form an expected formation; given period of formationThe observation parameter is LrlfAnd
Figure FDA00027383454400000210
the reference pose and the actual pose of the following robot can be obtained through the geometric position relationship between the master-slave robots as shown in formulas (6) and (7), respectively:
Figure FDA00027383454400000211
Figure FDA00027383454400000212
in the formulas (6) and (7), [ x ]fr yfr θfr]TRepresenting the pose of the Follower robot reference; [ x ] off yf θf]TRepresenting the actual pose of the Follower robot; [ x ] ofl yl θl]TRepresenting the pose of the Leader robot; thetarA reference direction angle is set for the Follower robot;
subtracting the equation (6) from the equation (7), and forming an error vector [ x ] by coordinate transformationef yef θef]TCan be expressed as follows in the local coordinate system of the robot
Figure FDA0002738345440000031
Wherein, thetalf＝θl-θf；
The derivation of the formula (8) along the formulas (6) - (7) can obtain a dynamic model of the formation error of the mobile robot kinematics layer:
Figure FDA0002738345440000032
wherein the content of the first and second substances,
Figure FDA0002738345440000033
(2) the design of the multi-mobile robot formation auxiliary speed controller comprises the following specific processes:
(2.1) because the direction angles of the following robots in the formation process are not completely the same due to the incomplete constraint of the mobile robot and the limitation of the master-slave formation control target, setting the reference direction angles of the formation following robots as follows:
Figure FDA0002738345440000034
d is the distance between the middle point of the driving wheel shaft and a reference point for calculating the relative distance of formation;
thus, the kinematic formation error dynamic model (9) of the mobile robot can be rewritten as
Figure FDA0002738345440000035
In order to stabilize the system shown in the formula (11), according to the Lyapunov stability theory, the following robot-assisted speed control law is designed as follows:
Figure FDA0002738345440000041
wherein k isx＞0、ky＞0、kθThe auxiliary speed control law parameter is more than 0;
to facilitate kinematic controller parameter kx，kyAnd kθSubstituting a control law formula (12) into a formation error dynamic equation (11) and linearizing the control law formula near a balance point to obtain:
Figure FDA0002738345440000042
(15) wherein, in the step (A),
Figure FDA0002738345440000043
from equation (15), the characteristic equation is:
a3s3+a2s2+a1s+a0＝0 (16)，
wherein, a3＝1；
Figure FDA0002738345440000044
Figure FDA0002738345440000045
Therefore, the parameter k is known from the Laus-Holwitz stability criterionx，kyAnd kθThe design of (c) also needs to satisfy the condition: a isi> 0(i ═ 0,1,2,3) and a1a2-a0a3＞0；
(3) Designing a formation dynamics controller, and specifically comprising the following processes:
(3.1) designing a formation disturbance observer: in order to estimate f, the model equation (5) needs to be expanded to
Figure FDA0002738345440000046
Wherein the content of the first and second substances,
Figure FDA0002738345440000047
representing the rate of change of the system's sum disturbance;
is provided with
Figure FDA0002738345440000051
And
Figure FDA0002738345440000052
the estimated values of u and f respectively, and the system state estimation error is defined as
Figure FDA0002738345440000053
The disturbance observer of equation (17) can be designed as:
Figure FDA0002738345440000054
wherein, K1，K2An observer gain matrix to be designed;
the observation error dynamic equation obtained by subtracting equation (17) from equation (18) is:
Figure FDA0002738345440000055
(19) wherein, in the step (A),
Figure FDA0002738345440000056
representing the sum total disturbance estimation error;
by designing the observer gain matrix K1、K2Make the observation error system matrix
Figure FDA0002738345440000057
The characteristic values of (a) are all located in the left half-open complex plane;
(3.2) on the basis of the design of the auxiliary speed controller, in order to make the design closer to the practical system application, the design of a formation dynamics controller is also needed:
defining the auxiliary velocity tracking error state vector for the formation as:
Figure FDA0002738345440000058
wherein u isc＝[vf ωf]TAnd u are the given velocity vector and the actual velocity vector of the formation dynamics system, respectively;
the derivation of equation (20) and substitution of equation (5) can be obtained:
Figure FDA0002738345440000059
the slip form surface is designed as follows:
Figure FDA00027383454400000510
wherein λ ═ diag (λ)1,λ2) More than 0 is a constant matrix to be designed;
the sliding mode surface is subjected to time derivation to obtain:
Figure FDA0002738345440000061
when the system is in the sliding mode
Figure FDA0002738345440000062
Under the condition of not considering the disturbance of the system sum, the equivalent control law of the system is obtained as follows:
Figure FDA0002738345440000063
once the system error state variable moves to the sliding mode surface, only the lambda is seti(i is 1,2) > 0 can make
Figure FDA0002738345440000064
However, the initial state of the system is usually not exactly located on the sliding mode surface, and the state running on the sliding mode surface of the system is also deviated from the sliding mode surface due to the influence of uncertainty factors such as system model parameter change and external disturbance, so that the control performance of the system is deteriorated, and a switching control item needs to be designed in order to eliminate the influence of uncertainty factors:
Figure FDA0002738345440000065
wherein, K3＝diag(k31,k32) The gain matrix of the switching control item to be designed is more than 0, the elements of the gain matrix are adjusted through an adaptive law, and the adaptive law is designed as follows:
Figure FDA0002738345440000066
wherein, γ1,γ2Adaptive gain is > 0, therefore, the adaptive sliding mode control law based on the disturbance observer is designed as:
Figure FDA0002738345440000067
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