CN113377016B - Slip-considered multi-mobile-robot cooperative formation buffeting-free sliding mode control method - Google Patents
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Abstract
The invention discloses a slip-form control method for multi-mobile-robot cooperative formation without buffeting in consideration of slipping, and solves the formation control problem of multi-mobile-robots under the condition of slipping. The method comprises the following specific steps: establishing a mathematical model of a multi-mobile-robot formation control system considering a slipping condition; designing a formation buffeting-free sliding mode controller based on the established formation system mathematical model, and realizing that each mobile robot in the formation can form a formation with a pilot robot according to a preset formation; and (5) stability analysis of the formation controller. The multi-mobile-robot formation control method can eliminate the adverse effect of wheel slip on the formation control effect, and can ensure the rapid and stable control of formation even under the condition of slip.
Description
Technical Field
The invention relates to the field of multi-mobile robot control, in particular to a slip-form control method for multi-mobile robot cooperative formation without buffeting, which considers slip.
Background
In order to complete related tasks, the efficiency and the success rate of task completion can be greatly improved through the cooperative cooperation of the multiple mobile robots. For example, when complex tasks are executed, the tasks can be decomposed and then completed by different mobile robots, so that the tasks can work in parallel, and the time for completing the tasks can be greatly shortened. In addition, when a certain mobile robot or some mobile robots are in fault in the task execution process, other mobile robots can still complete the given task through mutual cooperation. Based on this, the problem of cooperative control of multiple mobile robots has become a hot research problem in the field of robot application. As one of the important research directions of cooperative control, formation control of multiple mobile robots also has an important research value.
With the wide application of mobile robots, the working environment is more complicated, and the multi-mobile robot formation system is interfered by more uncertainties. The problem of slipping of the wheels of the mobile robot is a problem to be considered, especially when the convoy system is applied to icy or slippery road surfaces.
The sliding mode control has strong robustness on the uncertainty of system matching. However, the conventional sliding mode control has a serious buffeting problem, which not only is easy to accelerate the abrasion of the system actuator, but also has a serious negative effect on the control precision of the system.
Disclosure of Invention
The invention aims to solve the problem that the wheel slip of a mobile robot affects the formation control effect of multiple mobile robots, provides a slip-mode control method for the cooperative formation without buffeting of the multiple mobile robots considering the slip, and realizes the rapid and stable control of the formation of the multiple mobile robots even under the condition that the wheel slip exists.
The invention adopts a continuous approach law to design a multi-mobile robot formation controller, and solves the buffeting problem of sliding mode control signals. In addition, the interference observer can be designed to effectively estimate the uncertainty of the system, so that the control effect of the formation controller can be further improved.
The invention is realized by the following technical scheme:
and (1) establishing a mathematical model of the multi-mobile-robot formation control system considering the slipping condition.
(1.1) for under-actuated wheeled mobile robots, the under-actuated wheeled mobile robots are generally composed of two driving wheels and one following wheel, and the kinematic model of the under-actuated wheeled mobile robots can be expressed as follows without considering wheel slip
In the formula, q i ＝[x i ,y i ,θ i ] T ∈R 3 A pose vector representing the i (i ═ 1,2,3, …, N) th mobile robot; [ v ] of i ,ω i ] T ∈R 2 The control input velocity vector indicating the i-th mobile robot is composed of a control input linear velocity and a control input angular velocity of the mobile robot.
(1.2) when the mobile robot works on wet and slippery ground and other complex application occasions, the wheels of the mobile robot skid with the ground, and the actual control input speed of the mobile robot is not equal to the control input speed of the mobile robot, so that a certain deviation exists between the actual control input speed and the control input speed. When considering the case where the wheels of the mobile robot slip, the kinematic model of the ith mobile robot can be described as the following equation:
in the formula, ρ 1i ，ρ 2i The wheel slip parameter directly acting on the control input linear speed and the control input angular speed can be understood as the actual control input speed and the control input speed of the mobile robotThe ratio of degrees.
(1.3) further, the control input linear velocity and the control input angular velocity of the mobile robot and the angular velocity omega output by the two driving wheels li 、ω ri The relationship between can be expressed as:
in the formula, r i Radius of a driving wheel of the mobile robot; b is a mixture of i The distance between two driving wheels of the mobile robot; omega li The angular velocity of the left driving wheel of the mobile robot; omega ri Is the angular velocity of the right drive wheel of the mobile robot.
(1.4) the piloting mobile robot plays a role in piloting in the formation control, and the motion track of the piloting mobile robot is described by the following equation:
in the formula, q 0 ＝[x 0 ,y 0 ,θ 0 ] T ∈R 3 The pose coordinates of the piloted mobile robot are obtained; v. of 0 Is the linear velocity of the piloted mobile robot; omega 0 Is the angular velocity of the piloted mobile robot.
(1.5) defining the expected relative distance and direction angle of formation as l id ，θ id And further, an expected track which is formed into a certain formation and is required to be tracked by following a reference point of the mobile robot i can be obtained through coordinate transformation:
wherein [ x ] id ,y id ] T ∈R 2 Representing the coordinate representation in the global coordinate system XOY following the desired trajectory of the mobile robot i.
(1.6) selecting a local coordinate system x of the following mobile robot bi A point p in the forward direction of the axis i As a reference point to follow the mobile robot, its coordinates may be expressed in the global coordinate system XOY as
L i Representing the distance between the reference point and the origin of the local coordinate system of the following mobile robot.
(1.7) to p xi ，p yi Derivation can be obtained:
in the formula (I), the compound is shown in the specification,
for systematic uncertainty, assume f i And its first derivative satisfy the relationship: | f i ||≤d 1i ，
Wherein d is 1i And d 2i As unknown normal numbers.
(1.8) the error equation for the formation control can be defined as
In the formula, E i ＝[e xi ,e yi ] T 。
(1.9) the error dynamic equation for the formation control of the multiple mobile robots can be obtained from the equations (5), (6) and (7):
in the formula (I), the compound is shown in the specification,
(1.10) designing a buffeting-free sliding-mode formation controller under the condition that the wheels of the formation mobile robot slip with the ground, so that the formation error of the multi-mobile robot system can overcome the influence of wheel slip, a more expected formation control effect is obtained, and meanwhile, the control signal of the formation mobile robot is ensured to be smooth.
And (2) designing a formation buffeting-free sliding mode controller in order to realize a formation control target.
(2.1) defining an integral slip-form surface as follows:
in the formula, S i ＝[s 1i ,s 2i ] T ；k 1i And more than 0 is an integral sliding mode surface design parameter.
(2.2) the derivation of equation (10) can be:
(2.3) the approximation rule is defined as
In the formula, the design parameter k of the approach law 2i ，k 3i ，k 4i ，α 1i And alpha 2i The relationship needs to be satisfied: k is a radical of 2i ＞0，k 3i ＞0，k 4i ＞0，0＜α 1i ＜1，α 2i ＞1；
sign (·) is a sign function.
(2.4) is obtained from the formulae (11) and (12):
(2.5) the first part of the controller may be designed to
In the formula (I), the compound is shown in the specification,
is f i An estimate of (d). Uncertainty term f i The estimated value of (c) can be obtained by designing the disturbance observers equation (15) and equation (16):
in the formula, K oi ＝diag{k 01i ,k 02i The value of 0 is a gain matrix of the interference observer to be designed; beta is a i Is the intermediate vector of the disturbance observer.
Defining a system uncertainty estimation error of
From equations (7), (15) and (16), the dynamic equation of the estimation error can be obtained:
thus, the uncertainty estimation error can be determined by simply selecting the appropriate observer gain matrixConverge to a very small area near the origin, i.e. satisfy the relation
ε i An upper bound for the error is estimated for the disturbance observer.
(2.6) the second part of the controller may be designed to
(2.7) the complete formation controller expression is obtained from equations (14) and (18):
and (3) analyzing the stability of the formation controller.
(3.1) designing a Lyapunov function:
(3.2) is obtained from the formula (11), the formula (20):
(3.3) substitution of formula (19) for formula (21) gives:
in the formula, beta i ＝min{k 2i ,k 3i ,k 4i }。
(3.4) rewriting the formula (22) as
(3.5) As is clear from formula (23), provided that s ji One of the following conditions is satisfied:
the following equation can be made:
(3.6) therefore, the system is stable as seen by Lyapunov's stability theory and can be controlled by selecting the appropriate controller parameter k 2i ，k 3i ，k 4i Can make the sliding mode variable s ji And the method converges to a tiny area near the zero point, so that a good formation error control effect is obtained.
The invention has the advantages that: the method can eliminate the adverse effect of wheel slip on the formation control effect, can ensure the rapid and stable control of formation even under the condition of wheel slip, and simultaneously, the control signal output by the formation controller is smooth.
Drawings
Fig. 1 is a diagram illustrating a formation structure of a multi-mobile robot considering a slip situation according to the present invention.
Fig. 2 shows the formation error of the following mobile Robot Follower Robot1 according to the present invention.
Fig. 3 shows the formation error of the following mobile Robot Follower Robot2 according to the present invention.
FIG. 4a shows the uncertainty f of the following mobile Robot Follower Robot1 disturbance observer to the system 11 Is estimated.
FIG. 4b shows uncertainty f of following mobile Robot Follower Robot1 disturbance observer to system 21 Is estimated.
FIG. 5a shows the uncertainty f of the following mobile Robot Follower Robot2 disturbance observer to the system 12 Is estimated.
FIG. 5b shows uncertainty f of following mobile Robot Follower Robot2 disturbance observer to system 22 Is estimated.
Fig. 6 shows control signals of a following mobile Robot, namely a Follower Robot 1.
Fig. 7 shows control signals of a following mobile Robot, namely a Follower Robot 2.
Fig. 8 is a schematic structural diagram of the controller according to the present invention.
Detailed Description
In order to more intuitively illustrate the technical solution and technical advantages of the present invention, the following description of the technical solution of the present invention with reference to specific embodiments is made clearly and completely with reference to fig. 2 to 7.
A slip-considered multi-mobile-robot collaborative formation buffeting-free sliding mode control method comprises the following specific technical steps:
and (1) establishing a mathematical model of the multi-mobile-robot formation control system considering the slipping condition.
(1.1) for the wheel-type mobile robot under-actuated, the mobile robot is generally composed of two driving wheels and one following wheel, and the kinematic model of the mobile robot can be expressed as the following wheel without considering the wheel slip
In the formula, q i ＝[x i ,y i ,θ i ] T ∈R 3 A pose vector representing the i (i ═ 1,2,3, …, N) th mobile robot; [ v ] of i ,ω i ] T ∈R 2 The control input velocity vector representing the ith mobile robot is composed of a control input linear velocity and a control input angular velocity of the mobile robot.
(1.2) when the mobile robot works on wet and slippery ground and other complex application occasions, the wheels of the mobile robot skid with the ground, and the actual control input speed of the mobile robot is not equal to the control input speed of the mobile robot, so that a certain deviation exists between the actual control input speed and the control input speed. When considering the case where the wheels of the mobile robot slip, the kinematic model of the ith mobile robot can be described as the following equation:
in the formula, ρ 1i ，ρ 2i The wheel slip parameter, which is indicative of the direct action on the control input linear and control input angular velocities, is understood to be the ratio of the actual control input velocity of the mobile robot to its control input velocity.
(1.3) further, the control input linear velocity and the control input angular velocity of the mobile robot and the angular velocity omega output by the two driving wheels li 、ω ri The relationship between can be expressed as:
in the formula, r i Radius of a driving wheel of the mobile robot; b i The distance between two driving wheels of the mobile robot; omega li The angular velocity of the left driving wheel of the mobile robot; omega ri Is the angular velocity of the right drive wheel of the mobile robot.
(1.4) the piloting mobile robot plays a role in piloting in the formation control, and the motion track of the piloting mobile robot is described by the following equation:
in the formula, q 0 ＝[x 0 ,y 0 ,θ 0 ] T ∈R 3 The pose coordinates of the piloted mobile robot are obtained; v. of 0 Is the linear velocity of the piloted mobile robot; omega 0 To pilot the angular velocity of a mobile robot。
(1.5) defining the expected relative distance and direction angle of formation as l id ，θ id And further, expected tracks which form a certain formation and need to be tracked along a reference point of the mobile robot i can be obtained through coordinate transformation:
wherein [ x ] id ,y id ] T ∈R 2 Representing the coordinate representation in the global coordinate system XOY following the desired trajectory of the mobile robot i.
(1.6) selecting a local coordinate system x of the following mobile robot bi A point p in the forward direction of the axis i As a reference point to follow the mobile robot, its coordinates may be expressed in the global coordinate system XOY as
L i Representing the distance between the reference point and the origin of the local coordinate system of the following mobile robot.
(1.7) to p xi ，p yi The derivation can be:
in the formula (I), the compound is shown in the specification,
for systematic uncertainty, assume f i And its first derivative satisfy the relationship: i f i ||≤d 1i ，
Wherein d is 1i And d 2i As unknown normal numbers.
(1.8) the error equation for the formation control can be defined as
In the formula, E i ＝[e xi ,e yi ] T 。
(1.9) the error dynamic equation for the formation control of the multiple mobile robots can be obtained from the equations (5), (6) and (7):
in the formula (I), the compound is shown in the specification,
(1.10) designing a buffeting-free sliding-mode formation controller under the condition that the wheels of the formation mobile robot slip with the ground, so that the formation error of the multi-mobile robot system can overcome the influence of wheel slip, a more expected formation control effect is obtained, and meanwhile, the control signal of the formation mobile robot is ensured to be smooth.
And (2) designing a formation buffeting-free sliding mode controller for realizing a formation control target, as shown in fig. 8.
(2.1) defining an integral slip-form surface as follows:
in the formula, S i ＝[s 1i ,s 2i ] T ；k 1i And more than 0 is an integral sliding mode surface design parameter.
(2.2) derivation of equation (10) can be found:
(2.3) the approximation rule is defined as
In the formula, the design parameter k of the approach law 2i ，k 3i ，k 4i ，α 1i And alpha 2i The relationship needs to be satisfied: k is a radical of 2i ＞0，k 3i ＞0， k 4i ＞0，0＜α 1i ＜1，α 2i ＞1；
sign (·) is a sign function.
(2.4) is obtained from the formulae (11) and (12):
(2.5) the first part of the controller may be designed to be
In the formula (I), the compound is shown in the specification,
is f i An estimate of (d). Uncertainty term f i The estimated value of (c) can be obtained by designing the disturbance observers equation (15) and equation (16):
in the formula, K oi ＝diag{k 01i ,k 02i The value of 0 is a gain matrix of the interference observer to be designed; beta is a beta i Is the intermediate vector of the disturbance observer;
defining a system uncertainty estimation error as
From equations (7), (15) and (16), the dynamic equation of the estimation error can be obtained:
therefore, the uncertainty estimation error can be converged to a very small area near the origin, i.e. the relation is satisfied, by selecting a suitable observer gain matrix
ε i An upper bound for the error is estimated for the disturbance observer.
(2.6) the second part of the controller may be designed to
(2.7) the complete formation controller expression is obtained from equations (14) and (18):
and (3) analyzing the stability of the formation controller.
(3.1) designing a Lyapunov function:
(3.2) is obtained from the formula (11), the formula (20):
(3.3) the formula (19) may be substituted for the formula (21):
in the formula, beta i ＝min{k 2i ,k 3i ,k 4i }。
(3.4) rewriting formula (22) as
(3.5) As is clear from formula (23), provided that s ji One of the following conditions is satisfied:
the following equation can be made:
(3.6) therefore, the system is stable as known from Lyapunov's stability theory and can be controlled by selecting the appropriate controller parameter k 2i ，k 3i ，k 4i Namely, the sliding mode variable s ji And the method converges to a tiny area near the zero point, so that a good formation error control effect is obtained.
In the embodiment, three mobile robots are provided, namely a Leader Robot, a Follower Robot1 and a Follower Robot 2.
Linear velocity v of piloted mobile robot 0 Is set as v 0 0.3m/s, angular velocity ω 0 Set to ω 0 0.1rad/s, with initial pose coordinate set to q 0 ＝[0,0,0] T . The physical parameter of the following mobile Robot Robot1 is set as r 1 ＝0.08m，b 1 ＝0.3m，L 1 1m, the reference point initial coordinate is set to
The physical parameter of the following mobile Robot2 is set as r 2 ＝0.08m， b 2 ＝0.3m，L 2 1m, the reference point initial coordinate is set to
The controller parameters for the Follower Robot1 are set to: k is a radical of 11 ＝2，k 21 ＝0.4，k 31 ＝0.1， k 41 ＝0.1，α 11 ＝0.8，α 21 1.8; observer gain matrix parameter set to K o1 Diag { -6, -6 }. The controller parameters for the Follower Robot1 are set to: k is a radical of 12 ＝2，k 22 ＝0.4，k 32 ＝0.1，k 42 ＝0.1，α 12 ＝0.8，α 22 1.8; observer gain matrix parameter set to K o2 ＝diag{-6,-6}。
Simulation experiment results of the control method are shown in fig. 2-7, and fig. 2 and 3 are schematic diagrams of formation errors of following mobile robots, namely, a Follower Robot1 and a Follower Robot2 respectively; FIGS. 4 and 5 are schematic diagrams of the following mobile robots Follower Robot1 and Follower Robot2 for estimating the system uncertainty by the disturbance observer, respectively; fig. 6 and 7 are control signals of following mobile robots, Follower Robot1 and Follower Robot2, respectively. As can be seen from fig. 2 and 3, the controller can well realize the control of the formation system of the multiple mobile robots with the slipping condition between the wheels and the ground, when the slipping condition occurs between the wheels and the ground within 15 seconds, the formation error fluctuates to a certain extent, but the formation error can still obtain a good control effect due to the timely intervention of the designed interference observer. As can be seen from fig. 6 and 7, the control signal of the controller is smooth, which is advantageous for practical application of the controller.
Claims (1)
1. A slip-considered multi-mobile-robot collaborative formation buffeting-free sliding mode control method is characterized by comprising the following steps: the method specifically comprises the following steps:
(1) establishing a mathematical model of a multi-mobile-robot formation control system considering a slipping condition;
(2) designing a formation buffeting-free sliding mode controller based on the established mathematical model, and realizing that each mobile robot in the formation can form a formation with a pilot robot according to a preset formation;
(3) analyzing the stability of the designed formation buffeting-free sliding mode controller;
establishing a mathematical model of the multi-mobile-robot formation control system considering the slipping condition in the step (1), which comprises the following specific steps:
(1.1) the under-actuated wheeled mobile robot is composed of two driving wheels and one following wheel, and its kinematic model is expressed as the following wheel without considering the wheel slip
In the formula, q i ＝[x i ,y i ,θ i ] T ∈R 3 A pose vector representing the ith mobile robot, i ═ 1,2,3, …, N; [ v ] of i ,ω i ] T ∈R 2 The control input velocity vector of the ith mobile robot is represented and is composed of a control input linear velocity and a control input angular velocity of the mobile robot;
(1.2) when the mobile robot works on a slippery ground, slipping occurs between wheels of the mobile robot and the ground, the actual control input speed of the mobile robot is not equal to the control input speed of the mobile robot, and a certain deviation exists between the actual control input speed and the control input speed, and when the situation that the wheels of the mobile robot slip is considered, the kinematic model of the ith mobile robot is described as the following equation:
in the formula, ρ 1i ，ρ 2i Indicating wheel slip parameters acting directly on control input linear and control input angular velocities, i.e. mobile machinesThe ratio of the actual human control input speed to its control input speed;
(1.3) control input linear velocity and control input angular velocity of mobile robot and angular velocity omega output by two driving wheels li 、ω ri The relationship between them is expressed as:
in the formula, r i Radius of a driving wheel of the mobile robot; b i The distance between two driving wheels of the mobile robot; omega li Is the angular velocity of the left drive wheel of the mobile robot; omega ri The angular velocity of the right driving wheel of the mobile robot;
(1.4) the motion trajectory of the piloted mobile robot is described by the following equation:
in the formula, q 0 ＝[x 0 ,y 0 ,θ 0 ] T ∈R 3 The pose coordinates of the piloted mobile robot are obtained; v. of 0 The linear velocity of the piloted mobile robot; omega 0 Is the angular velocity of the piloted mobile robot;
(1.5) defining the expected relative distance and direction angle of formation as l id ，θ id And obtaining an expected track which forms a certain formation and needs to be tracked along with a reference point of the mobile robot by coordinate transformation:
wherein [ x ] id ,y id ] T ∈R 2 A coordinate representation in the global coordinate system XOY representing a desired trajectory for following the mobile robot i;
(1.6) selecting a local coordinate system x of the following mobile robot bi A point p in the forward direction of the axis i As a reference point to follow the mobile robot, its coordinates are represented in the global coordinate system XOY as
L i Representing the distance between the reference point and the origin of the local coordinate system of the following mobile robot;
(1.7) to p xi ，p yi And (5) obtaining a derivative:
in the formula (I), the compound is shown in the specification,
for systematic uncertainty, assume f i And its first derivative satisfy the relationship: | f i ||≤d 1i ，
Wherein d is 1i And d 2i Is an unknown normal number;
(1.8) formation control error equation is defined as
In the formula, E i ＝[e xi ,e yi ] T ；
(1.9) error dynamic equation for formation control of multiple mobile robots by equation (5), equation (6) and equation (7):
in the formula (I), the compound is shown in the specification,
designing a formation buffeting-free sliding mode controller based on the established mathematical model in the step (2), and realizing that each mobile robot in the formation can form a formation with a pilot robot according to a preset formation, wherein the formation comprises the following specific steps:
(2.1) defining an integral slip-form surface as follows:
in the formula, S i ＝[s 1i ,s 2i ] T ；k 1i More than 0 is an integral sliding mode surface design parameter;
(2.2) deriving equation (10):
(2.3) the approximation rule is defined as
In the formula, the design parameter k of the approach law 2i ，k 3i ，k 4i ，α 1i And alpha 2i The relationship needs to be satisfied: k is a radical of formula 2i ＞0，k 3i ＞0，k 4i ＞0，0＜α 1i ＜1，α 2i ＞1；
sign (·) is a sign function;
(2.4) is obtained from the following formulae (11) and (12):
(2.5) the first part of the formation buffeting-free sliding mode controller is designed to be
In the formula (I), the compound is shown in the specification,
is f i Estimated value of (a), uncertainty term f i The estimated value of (c) is obtained by the designed disturbance observers equation (15) and equation (16):
in the formula, K oi ＝diag{k 01i ,k 02i The value of 0 is a gain matrix of the interference observer to be designed; beta is a beta i Is the intermediate vector of the disturbance observer;
defining a system uncertainty estimation error as
The dynamic equation of the estimation error is obtained from equations (7), (15) and (16):
satisfy the relationship
ε i Estimating an upper bound of error for the disturbance observer;
(2.6) second part of the formation buffeting-free sliding mode controller is designed to be
(2.7) the complete formation controller expression is obtained from equations (14) and (18):
analyzing the stability of the designed formation buffeting-free sliding mode controller in the step (3), and concretely comprising the following steps:
(3.1) designing a Lyapunov function:
(3.2) is obtained from the following formulae (11) and (20):
(3.3) substitution of formula (19) for formula (21):
in the formula, beta i ＝min{k 2i ,k 3i ,k 4i }；
(3.4) rewriting formula (22) as
(3.5) As long as | s is given by the formula (23) ji L satisfies one of the following conditions:
that is, the following equation holds:
(3.6) the system is stable from Lyapunov's stability theory and by choosing the appropriate controller parameter k 2i ，k 3i ，k 4i So that the sliding mode variable s ji And the method converges to a tiny area near the zero point, so that a good formation error control effect is obtained.
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