CN113703451B - Self-adaptive fault-tolerant control method for formation of multiple mobile robots with preset performance - Google Patents
Self-adaptive fault-tolerant control method for formation of multiple mobile robots with preset performance Download PDFInfo
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Abstract
The invention discloses a self-adaptive fault-tolerant control method for formation of multiple mobile robots with preset performance, which comprises the following specific steps of: establishing a mathematical model of a multi-mobile robot formation system considering the faults of an actuating mechanism; designing a preset performance multi-mobile robot formation speed control law; designing a self-adaptive fault-tolerant torque control law of the formation mobile robots; and (3) stability analysis of the self-adaptive fault-tolerant control method for the formation of the multiple mobile robots with preset performance. The multi-mobile-robot formation control method disclosed by the invention can realize the preset performance control of the formation error, effectively eliminate the adverse effects on the formation control caused by the system execution mechanism fault and the system external disturbance uncertainty, and realize stable and reliable formation control.
Description
Technical Field
The invention relates to the field of robot control methods, in particular to a self-adaptive fault-tolerant control method for multi-mobile robot formation with preset performance.
Background
With the complication of the work environment of the mobile robot and the more diversified work tasks, the single mobile robot is no longer suitable for bearing some special task requirements, and the multi-mobile robot system has many advantages compared with the single robot.
Faults, uncertain dynamics and external interference of an actuating mechanism generally bring great influence on transient performance and steady-state performance of a mobile robot formation system, and in severe cases, the formation system may be unstable, and even collision among robots may occur. However, current research on multi-mobile robot formation control mainly aims to improve the steady-state control accuracy of the robot formation system, and the transient response performance and output constraint problems of the formation system are rarely considered.
The method can ensure that the steady-state error of the formation closed-loop system is converged to a preset allowable range, and simultaneously ensure that the convergence speed and the overshoot are less than a certain preset value.
Disclosure of Invention
The invention aims to provide a self-adaptive fault-tolerant control method for multi-mobile-robot formation with preset performance, so as to solve the problem of unstable control of the mobile-robot formation in the prior art.
In order to achieve the purpose, the technical scheme adopted by the invention is as follows:
the self-adaptive fault-tolerant control method for the formation of the multiple mobile robots with preset performance, wherein the formation of the multiple mobile robots is formed by a plurality of mobile robots, comprises the following steps:
step 1, establishing a multi-mobile-robot formation system, wherein a kinematics and dynamics model of any mobile robot i is shown as a formula (1):
in the formula (1), q i =[x i ,y i ,θ i ] T ∈R 3 Represents a pose vector of the mobile robot i, where x i ,y i Respectively representing the coordinates theta of the mobile robot i in the X-axis direction and the Y-axis direction of the global coordinate system i Represents a direction angle of the mobile robot;u i =[v i ,ω i ] T ∈R 2 a velocity vector representing the mobile robot i, consisting of the linear velocity and the angular velocity of the mobile robot i, v i Is the linear velocity, omega, of the mobile robot i i Is the angular velocity of the mobile robot i;positive definite inertia matrix, m, representing mobile robot i i ,I i Respectively representing the mass and inertia of the mobile robot i; f i ∈R 2 Representing a ground friction force dynamic vector; tau is di ∈R 2 Representing an external time-varying bounded perturbation;inputting a moment transformation array for the mobile robot i, wherein r i Radius of the driving wheel, b i For moving robotsThe width of (d); tau is i =[τ 1i ,τ 2i ] T ∈R 2 Representing input torque vectors of two driving wheels of the mobile robot i;
in the formula (2), τ 1i =Δ 1i τ 11i +σ 1i ,τ 2i =Δ 2i τ 22i +σ 2i Wherein, τ 11i 、τ 22i Control torque signal output by controller, 0 < delta 1i ≤1,0<Δ 2i A coefficient of unknown ≦ 1 for indicating the degree of failure, σ, of the ith mobile robot actuator 1i ,σ 2i Indicating an unknown additive fault of the i-th mobile robot actuator. Delta of 1i ,Δ 2i ,σ 1i And sigma 2i The magnitude of the value is determined by the degree of malfunction of the mobile robot actuator, e.g. when Δ 1i =Δ 2i 1 and σ 1i =σ 2i If =0, it indicates that the actuator of the i-th mobile robot has no fault;
in the formula (3), the first and second groups,suppose G i Bounded and satisfies the relation G i ||≤c i Wherein c is i Is an unknown normal number;wherein, tau 11i 、τ 22i A control torque signal output for the controller;
in the formula (4), q 0 =[x 0 ,y 0 ,θ 0 ] T ∈R 3 To pilot the movement track, x, of the mobile robot 0 ,y 0 Respectively representing the coordinates theta of the piloted robot in the X-axis direction and the Y-axis direction of the global coordinate system 0 Representing a direction angle of the piloted robot; v. of 0 The linear velocity of the piloted mobile robot; omega 0 Is the angular velocity of the piloted mobile robot.
To obtain the desired formation, first, the formation parameters are defined in the local coordinate system of the piloted mobile robot: expected relative distance to formation l id And relative direction angle theta id (ii) a Then, obtaining the desired trajectory of the reference point of any one of the following mobile robots i through coordinate transformation is shown in formula (5):
in the formula (5), [ x ] id ,y id ] T ∈R 2 I =1,2,3, …, N, representing a desired trajectory to follow mobile robot i, where x id As X-axis coordinate, y, of the desired trajectory id Is the Y-axis coordinate of the desired trajectory. Sending the expected track to a following mobile robot i which forms a formation with the expected track by a pilot robot; l id A desired formation relative distance; theta id A desired formation relative azimuth;
thus, the multi-mobile robot formation system control error equation can be defined as shown in equation (6):
in the formula (6), e xi The control error of the formation system in the X axial direction is obtained; e.g. of the type yi The control error of the Y-axis of the formation system is shown.
The time derivative is calculated for the formula (6), and the formula (1) and the formula (5) are combined, so that the dynamic equation of the formation control error of the multiple mobile robots can be obtained as shown in the formulas (7) and (8):
the invention also comprises a step 5: designing a preset performance multi-mobile robot formation speed control law, wherein the process comprises the following steps:
step 5.1, the design of the preset performance function is shown as a formula (9):
μ ji (t)=(μ ji,0 -μ ji,∞ )exp(-k ji t)+μ ji,∞ ,j=x,y (9),
in the formula (9), μ ji,0 0 is the initial value of the predetermined performance function, mu ji,∞ > 0 is the steady state value of the preset performance function when t → ∞ k ji Setting parameters for the convergence speed of the preset performance function if the convergence speed is more than 0;
to achieve the preset performance control requirement, the formation controller is designed to make the formation control error meet the requirement of the formula (10):
-μ ji (t)<e ji <μ ji (t), j = x, y (10), and in equation (10), parameter μ ji,0 Is set to satisfy-mu ji,0 <e ji (0)<μ ji,0 ;
And 5.2, defining an error transformation function as shown in a formula (11):
derivation of equation (11) yields equation (12):
by defining the error transformation function, only delta is ensured in the design of the queue controller ji Bounded, i.e. guaranteeing a formation control error e xi 、e yi Meet a predetermined performance requirement, i.e., -mu ji (t)<e ji <μ ji (t);
Step 5.3, designing the Lyapunov function as shown in formulas (13) and (14):
wherein, delta xi An error transformation function selected for X-axis error control; delta yi An error transformation function selected for Y-axis error control;
equations (15) and (16) can be obtained by deriving equations (13) and (14):
design u from equations (15), (16) vi ,u ωi As shown in equations (17) and (18):
wherein u is vi ,u ωi Design parameter k 1i >0,k 2i >0;
The formula (17) and the formula (18) may be substituted for the formula (15) and the formula (16), respectively:
combining the formula (8), the control law of the formation speed of the multi-mobile robot with preset performance can be obtained as shown in the formula (21):
in the formula (21), v ci A pre-set performance multi-mobile robot formation linear speed control law; omega ci And forming an angular speed control law for the multi-mobile robot with preset performance.
The invention also comprises a step 6: designing a self-adaptive fault-tolerant torque control law of the formation mobile robot, wherein the process is as follows:
step 6.1, after the design of the speed control law of the formation mobile robot is finished, the design of the self-adaptive fault-tolerant moment control law is required to be carried out by combining a dynamic model of the mobile robot so as to obtain the actual control moment input of the following mobile robot; first, a velocity tracking error vector is defined as shown in equation (22):
in the formula (22), e 1i Tracking error for the linear velocity following the mobile robot i; e.g. of the type 2i Tracking error for following the angular velocity of the mobile robot i;
step 6.2, in order to improve the control precision of the speed tracking error, the invention designs an integral sliding mode surface shown as a formula (23):
in the formula (23), the first and second groups,S i =[s 1i ,s 2i ] T wherein s is 1i ,s 2i Respectively represent the vectors S i The first and second components of (a); k vi =diag{k v1i ,k v2i The value is more than 0, which is the design parameter of the integral sliding mode surface;
the time derivative of equation (23) can be derived as equation (24):
D i Satisfies the relationship:wherein, b i =max{1,c i The (x) is an unknown constant which,can be obtained by on-line calculation;
step 6.3 for unknown item b i The invention adopts a parameter self-adaptive method for processing, and designs a self-adaptive fault-tolerant moment control law following the mobile robot i by combining a formula (24) as shown in a formula (25):
in the formula (25), the first and second groups,k 3i >0,k 4i >0,k 5i a control gain parameter that is positive > 0; design parameter beta i The conditions need to be satisfied: beta is more than 0 i <1;sign (·) is a sign function;is an unknown constant b i Is determined by the estimated value of (c),the adaptive update law of (2) is shown in equation (26):
in the formula, k 6i A constant greater than zero.
Compared with the prior art, the invention has the advantages that:
the invention has the advantages that: the multi-mobile-robot formation control method can realize the preset performance control of the formation error, effectively eliminate the adverse effects of system execution mechanism faults and system external disturbance uncertainty on formation control, and realize reliable and stable formation control.
Drawings
Fig. 1 is a schematic diagram of a formation structure of multiple mobile robots according to the present invention.
Fig. 2a is a schematic diagram of a queuing error of the adaptive fault-tolerant control method Follower 1 with preset performance according to the present invention.
Fig. 2b is a schematic diagram of a queuing error of the adaptive fault-tolerant control method Follower 1 with preset performance according to the present invention.
Fig. 3a is a schematic diagram of a Follower 2 formation error of the adaptive fault-tolerant control method with preset performance according to the present invention.
Fig. 3b is a schematic diagram of a Follower 2 queuing error of the adaptive fault-tolerant control method with preset performance according to the present invention.
Fig. 4a is a schematic diagram of a queuing error of the adaptive fault-tolerant control method Follower 1 without the preset performance according to the present invention.
Fig. 4b is a schematic diagram of a queuing error of the adaptive fault-tolerant control method Follower 1 without the preset performance according to the present invention.
Fig. 5a is a schematic diagram of a queuing error of the adaptive fault-tolerant control method Follower 2 without the preset performance according to the present invention.
Fig. 5b is a schematic diagram of a queuing error of the adaptive fault-tolerant control method Follower 2 without the preset performance according to the present invention.
Fig. 6a shows control signals of a method C1 according to Follower 1 of the present invention.
Fig. 6b shows the control signals of the method C2 used by the Follower 1 according to the present invention.
Fig. 7a shows control signals of a method C1 used by Follower 2 according to the present invention.
Fig. 7b shows control signals of Follower 2 according to the present invention using method C2.
Fig. 8 is a block diagram of the control system according to the present invention.
Detailed Description
The invention is further illustrated with reference to the following figures and examples.
As shown in fig. 1, the invention relates to a self-adaptive fault-tolerant control method for formation of multiple mobile robots with preset performance, wherein the formation of multiple mobile robots is composed of multiple mobile robots, and the method comprises the following steps:
step 1, establishing a kinematics and dynamics model of any one mobile robot i in a multi-mobile-robot formation system as shown in a formula (1):
in the formula (1), q i =[x i ,y i ,θ i ] T ∈R 3 Represents the pose vector of the mobile robot i, where x i ,y i Represents the coordinates of the mobile robot i in the X-axis direction and the Y-axis direction of the global coordinate system, respectively, and θ i Representing a directional angle of the mobile robot;u i =[v i ,ω i ] T ∈R 2 a velocity vector representing the mobile robot i, consisting of the linear velocity and the angular velocity of the mobile robot i, v i Is the linear velocity, omega, of the mobile robot i i Is the angular velocity of the mobile robot i;represents the positive definite inertia matrix, m, of the mobile robot i i ,I i Respectively representing the mass and inertia of the mobile robot i; f i ∈R 2 Representing a ground friction force dynamic vector; tau is di ∈R 2 Representing an external time-varying bounded perturbation;inputting a moment transformation array for the mobile robot i, wherein r i Radius of the driving wheel, b i For moving robotsThe width of (d); tau is i =[τ 1i ,τ 2i ] T ∈R 2 Representing input torque vectors of two driving wheels of the mobile robot i;
and 2, in the working process of the queuing mobile robot, the fault problem of an actuating mechanism is inevitable. Once an executing mechanism breaks down, a light person can have an adverse effect on the formation control effect, and a failure of the formation control task of the multiple mobile robots is caused. Therefore, the problem of the failure of the mobile robot actuator needs to be considered in the formation control.
Considering the problem of actuator failure, the dynamic model of the mobile robot i can be modeled as shown in formula (2):
in the formula (2), τ 1i =Δ 1i τ 11i +σ 1i ,τ 2i =Δ 2i τ 22i +σ 2i Wherein, τ 11i 、τ 22i Control force output for controllerMoment signal, 0 < delta 1i ≤1,0<Δ 2i A coefficient of unknown ≦ 1 for indicating the degree of failure, σ, of the ith mobile robot actuator 1i ,σ 2i Indicating an unknown additive fault of the i-th mobile robot actuator. Delta 1i ,Δ 2i ,σ 1i And sigma 2i The magnitude of the value is determined by the degree of malfunction of the mobile robot actuator, e.g. when Δ 1i =Δ 2i 1 and σ 1i =σ 2i If =0, it indicates that the actuator of the i-th mobile robot has no fault;
in the formula (3), the first and second groups,suppose G i Bounded and satisfies the relation G i ||≤c i Wherein c is i Is an unknown normal number;wherein, tau 11i 、τ 22i A control torque signal output for the controller;
In the invention, firstly, a pilot robot in a multi-mobile-robot formation system is determined, and an expected track generated by the pilot robot is determined by the following formula (4):
in the formula (4), q 0 =[x 0 ,y 0 ,θ 0 ] T ∈R 3 To pilot the movement track, x, of the mobile robot 0 ,y 0 Respectively representing the coordinates theta of the piloted robot in the X-axis direction and the Y-axis direction of the global coordinate system 0 Representing a direction angle of the piloted robot; v. of 0 The linear velocity of the piloted mobile robot; omega 0 Is the angular velocity of the piloted mobile robot.
To obtain the desired formation form, first, the formation parameters are defined in the local coordinate system of the piloting mobile robot: expected relative distance to formation l id And relative direction angle theta id (ii) a Then, obtaining a desired trajectory of a reference point of any one following mobile robot i through coordinate transformation is as shown in formula (5):
in the formula (5), [ x ] id ,y id ] T ∈R 2 I =1,2,3, …, N, representing a desired trajectory to follow mobile robot i, where x id As X-axis coordinate, y, of the desired trajectory id Is the Y-axis coordinate of the desired trajectory. Sending the expected track to a following mobile robot i which forms a formation with the expected track by a pilot robot; l id A desired relative distance to formation; theta id A desired formation relative azimuth;
thus, the multi-mobile robot formation system control error equation can be defined as shown in equation (6):
in the formula (6), e xi The control error of the formation system in the X axial direction is obtained; e.g. of the type yi The control error of the Y-axis of the formation system is shown.
The time derivative is calculated for the formula (6), and the formula (1) and the formula (5) are combined, so that the dynamic equation of the formation control error of the multiple mobile robots can be obtained as shown in the formulas (7) and (8):
step 5.1, the design of the preset performance function is shown as a formula (9):
μ ji (t)=(μ ji,0 -μ ji,∞ )exp(-k ji t)+μ ji,∞ ,j=x,y (9),
in the formula (9), μ ji,0 0 is the initial value of the predetermined performance function, mu ji,∞ > 0 is the steady state value of the preset performance function when t → ∞ k ji And > 0 is a parameter for setting the convergence speed of the preset performance function.
To achieve the preset performance control requirement, the formation controller is designed to make the formation control error meet the requirement of the formula (10):
-μ ji (t)<e ji <μ ji (t),j=x,y (10),
in the formula (10), the parameter μ ji,0 Is set to satisfy-mu ji,0 <e ji (0)<μ ji,0 。
And 5.2, defining an error transformation function as shown in a formula (11):
derivation of equation (11) yields equation (12):
by defining the error transformation function, only delta is ensured in the design of the queue controller ji Bounded, i.e. guaranteeing a formation control error e xi 、e yi Meet a predetermined performance requirement, i.e., -mu ji (t)<e ji <μ ji (t)。
Step 5.3, designing the Lyapunov function as shown in formulas (13) and (14):
wherein, delta xi An error transformation function selected for X-axis error control; delta yi An error transfer function is selected for Y-axis error control.
Equations (15) and (16) can be obtained by deriving equations (13) and (14):
design u from equations (15), (16) vi ,u ωi As shown in equations (17) and (18):
wherein u is vi ,u ωi Design parameter k 1i >0,k 2i >0。
The formula (17) and the formula (18) may be substituted for the formula (15) and the formula (16), respectively:
combining the formula (8), the control law of the formation speed of the multi-mobile robot with the preset performance is shown as a formula (21):
in the formula (21), v ci A pre-set performance multi-mobile robot formation linear speed control law; omega ci And forming an angular speed control law for the multi-mobile robot with preset performance.
and 6.1, after the design of the speed control law of the formation mobile robot is finished, designing a self-adaptive fault-tolerant moment control law by combining a dynamic model of the mobile robot so as to obtain the actual control moment input of the following mobile robot. First, a velocity tracking error vector is defined as shown in equation (22):
in the formula (22), e 1i Tracking error for the linear velocity following the mobile robot i; e.g. of the type 2i To follow the angular velocity of the mobile robot i.
Step 6.2, in order to improve the control precision of the speed tracking error, the invention designs an integral sliding mode surface as shown in a formula (23):
in the formula (23), S i =[s 1i ,s 2i ] T Wherein s is 1i ,s 2i Respectively represent the vectors S i The first and second components of (a); k vi =diag{k v1i ,k v2i And 0 is a design parameter of an integral sliding mode surface.
The time derivative of equation (23) can be derived as equation (24):
D i Satisfies the relationship:wherein, b i =max{1,c i The (x) is an unknown constant which,can be obtained by on-line calculation.
Step 6.3 for unknown item b i The invention adopts a parameter self-adaptive method for processing, and designs a self-adaptive fault-tolerant moment control law following the mobile robot i by combining a formula (24) as shown in a formula (25):
in the formula (25), the first and second groups,k 3i >0,k 4i >0,k 5i a control gain parameter that is positive > 0; design parameter beta i The conditions need to be satisfied: beta is more than 0 i Less than 1; sign (·) is a sign function;is an unknown constant b i Is determined by the estimated value of (c),the adaptive update law of (2) is shown in equation (26):
in the formula, k 6i A constant greater than zero.
The stability analysis of the self-adaptive fault-tolerant control method for the formation of the multiple mobile robots with the preset performance is as follows:
(1) The Lyapunov function is designed as shown in formula (27):
in the formula (27), β min > 0 is a matrixThe minimum eigenvalue of (c);as an unknown parameter b i The estimation error of (2).
(2) Formula (28) is obtained by obtaining a time derivative of formula (27) and substituting formula (19), formula (20), formula (24), and formula (26):
(3) Formula (29) can be obtained by substituting formula (25) for formula (28):
in equation (29):
(4) From V i As shown in the definition of (1) and the formula (29), δ xi ,δ yi ,S i ,Is consistently bounded, so the formation control error e xi ,e yi And the preset performance requirement is met.
Examples
This example presents the comparison results for two different control methods: self-adaptive fault-tolerant control method (C1), u, for formation of multiple mobile robots with preset performance vi ,u ωi Designing formulas (17) and (18), and designing control moment as formula (25); self-adaptive fault-tolerant control method (C2), u, for formation of multiple mobile robots without preset performance vi ,u ωi Is designed as u vi =-k 7i e xi -Γ 1i ,u ωi =-k 8i e yi -Γ 2i ,k 7i >0,k 8i And (4) if the fault-tolerant torque control law is greater than 0, the self-adaptive fault-tolerant torque control law is designed as shown in the formula (25). In this embodiment there are three mobile robots, leader Robot, follower 1, follower 2.
Linear velocity v of Leader Robot 0 Is set as v 0 =0.5m/s, angular velocity ω 0 Set to ω 0 =0.1rad/s, initial pose coordinate set to q 0 =[0,0,0] T . Follower 1 has physical parameter set to m 1 =10.5kg,I 1 =3.11kg·m 2 ,r 1 =0.25m,b 1 =0.4m,L 1 =1.6m; the initial pose coordinate is set to [ -2-1 π/8 ]] T (ii) a Follower 2 has a physical parameter set to m 2 =10.5kg,I 2 =3.11kg·m 2 ,r 2 =0.25m,b 2 =0.4m,L 2 =1.6m; the initial pose coordinate is set to [ -3-2 π/9 [)] T 。
For better comparisonIn the study, the same controller parameter settings were the same in both control methods. The controller parameters for Follower 1 are set to: k is a radical of 11 =1.5,k 21 =1.6,k 31 =8,k 41 =1,k 51 =0.01,k 61 =0.03,K v1 =diag{0.4,0.5};k 71 =1.5,k 81 =1.6;μ x1,0 =4,μ y1,0 =3,μ x1,∞ =μ y1,∞ =0.1,k x1 =k y1 And (5) =2. The controller parameters for Follower 2 are set to: k is a radical of 12 =1.5,k 22 =1.6,k 32 =8,k 42 =1,k 52 =0.01,k 62 =0.03,K v2 =diag{0.4,0.5};k 72 =1.5,k 82 =1.6;μ x2,0 =4,μ y2,0 =3,μ x2,∞ =μ y2,∞ =0.1,k x2 =k y2 =2。
Suppose that the dynamic vector of the ground friction force borne by the 2 following mobile robots is F 1 =F 2 =[sin(t)+2.4,cos(t)+2.4] T (ii) a Exposed external time-bounded perturbation τ d1 =τ d2 =[0.1sin(t)+2,0.5cos(t)+0.4] T (ii) a The failure of the actuator is assumed to be
The simulation experiment results of the two control methods are shown in fig. 2a, fig. 2b, fig. 3a, fig. 3b, fig. 4a, fig. 4b, fig. 5a, fig. 5b, fig. 6a, fig. 6b, fig. 7a, fig. 7b, and fig. 8-and fig. 2a, fig. 2b, fig. 3a, and fig. 3b are schematic diagram of the formation errors of the adaptive fault-tolerant control methods Follower 1 and Follower 2 with preset performance according to the present invention, respectively; fig. 4a, 4b, 5a, and 5b are schematic diagrams of the formation errors of adaptive fault-tolerant control methods Follower 1 and Follower 2, respectively, without preset performance according to the present invention; fig. 6a, 6b and fig. 7a, 7b are control signals of Follower 1 and Follower 2, respectively, according to two control methods of the present invention. As can be seen from fig. 2a, 2b, 3a, and 3b, the adaptive fault-tolerant control method with the preset performance can effectively handle the adverse effects of system execution mechanism faults and system external disturbance uncertainty on formation control, the variation of formation errors is limited within a preset performance boundary, the convergence rate of the formation errors is not less than a preset convergence rate, and the steady-state errors of the formation are not greater than a preset steady-state value. As can be seen from fig. 4a, 4b, 5a, and 5b, under the condition of using the same controller parameters, the formation error of the adaptive fault-tolerant control method without the preset performance has a large fluctuation, and the change process breaks through the preset performance boundary, so that the formation control effect is obviously inferior to the adaptive fault-tolerant control method with the preset performance. Fig. 6a, 6b, 7a, and 7b show the control signals of Follower 1 and Follower 2, respectively, for the two control methods of the present invention, and it can be seen from the figure that the smoothness of the control signal with the preset performance control method is not as good as that without the preset performance control method, but remains substantially smooth and does not affect the practical application of the control method.
The described embodiments of the present invention are only for describing the preferred embodiments of the present invention, and do not limit the concept and scope of the present invention, and the technical solutions of the present invention should be modified and improved by those skilled in the art without departing from the design concept of the present invention, and the technical contents of the present invention which are claimed are all described in the claims.
Claims (1)
1. The self-adaptive fault-tolerant control method for the formation of the multiple mobile robots with preset performance is characterized by comprising the following steps of:
step 1, establishing a multi-mobile-robot formation system, wherein a kinematics and dynamics model of any mobile robot i is shown as a formula (1):
in the formula (1), q i =[x i ,y i ,θ i ] T ∈R 3 Represents a pose vector of the mobile robot i, where x i ,y i Respectively representing the coordinates theta of the mobile robot i in the X-axis direction and the Y-axis direction of the global coordinate system i Represents a direction angle of the mobile robot;u i =[v i ,ω i ] T ∈R 2 a velocity vector representing the mobile robot i, consisting of the linear velocity and the angular velocity of the mobile robot i, v i Is the linear velocity, omega, of the mobile robot i i Is the angular velocity of the mobile robot i;positive definite inertia matrix, m, representing mobile robot i i ,I i Respectively representing the mass and inertia of the mobile robot i; f i ∈R 2 Representing a ground friction force dynamic vector; tau is di ∈R 2 Representing an external time-varying bounded perturbation;inputting a moment transformation array for the mobile robot i, wherein r i Radius of the driving wheel, b i For moving robotsThe width of (d); tau is i =[τ 1i ,τ 2i ] T ∈R 2 Representing input torque vectors of two driving wheels of the mobile robot i;
step 2, considering the problem of the fault of the executing mechanism, the dynamic model of the mobile robot i can be modeled as shown in a formula (2):
in the formula (2), τ 1i =Δ 1i τ 11i +σ 1i ,τ 2i =Δ 2i τ 22i +σ 2i Wherein, τ 11i 、τ 22i Control torque signal output by controller, 0 < delta 1i ≤1,0<Δ 2i A coefficient not more than 1 is unknown and is used for expressing the failure degree of the actuator of the ith mobile robot, sigma 1i ,σ 2i Indicating an unknown additive fault of an ith mobile robot actuator; delta 1i ,Δ 2i ,σ 1i And sigma 2i The magnitude of the value is determined by the degree of malfunction of the mobile robot actuator, e.g. when Δ 1i =Δ 2i 1 and σ 1i =σ 2i If =0, it means that the actuator of the i-th mobile robot is not in failure;
in the formula (3), the first and second groups,suppose G i Bounded and satisfies the relation G i ||≤c i Wherein c is i Is an unknown normal number;wherein, tau 11i 、τ 22i A control torque signal output by the controller;
step 3, determining a pilot robot in the multi-mobile-robot formation system, wherein an expected track generated by the pilot robot is determined by the following formula (4):
in the formula (4), q 0 =[x 0 ,y 0 ,θ 0 ] T ∈R 3 To pilot the movement track, x, of the mobile robot 0 ,y 0 Respectively representing the coordinates theta of the piloted robot in the X-axis direction and the Y-axis direction of the global coordinate system 0 Representing a direction angle of the piloted robot; v. of 0 The linear velocity of the piloted mobile robot; omega 0 Is the angular velocity of the piloted mobile robot;
to obtain the desired formation form, first, the formation parameters are defined in the local coordinate system of the piloting mobile robot: expected relative distance to formation l id And relative direction angle theta id (ii) a Then, obtaining the desired trajectory of the reference point of any one of the following mobile robots i through coordinate transformation is shown in formula (5):
in the formula (5), [ x ] id ,y id ] T ∈R 2 I =1,2,3, …, N, representing a desired trajectory to follow mobile robot i, where x id As X-axis coordinate, y, of the desired trajectory id Sending the expected track to a following mobile robot i which forms a formation with the expected track by a pilot robot for the Y-axis coordinate of the expected track; l id A desired relative distance to formation; theta.theta. id A desired formation relative azimuth;
step 4, selecting a local coordinate system x of the following mobile robot i bi A point p in the forward direction of the axis i (p xi ,p yi ) As a reference point for the following mobile robot i, the coordinates of the following mobile robot i may be represented in the global coordinate system XOY asWherein p is xi To follow the coordinates of the i-reference point of the mobile robot in the X-axis direction, p yi For following the i reference point of the mobile robot in the Y axial directionCoordinate of (a), L i The distance between the reference point and the origin of coordinates of the local coordinate system of the following mobile robot;
thus, the multi-mobile robot formation system control error equation can be defined as shown in equation (6):
in the formula (6), e xi The control error of the formation system in the X axial direction is obtained; e.g. of the type yi The control error of the formation system in the Y-axis direction is obtained;
the time derivative is calculated for the formula (6), and the formula (1) and the formula (5) are combined, so that a dynamic equation of the formation control error of the multiple mobile robots can be obtained as shown in the formulas (7) and (8):
and 5: designing a preset performance multi-mobile robot formation speed control law, wherein the process comprises the following steps:
step 5.1, the design of the preset performance function is shown as a formula (9):
μ ji (t)=(μ ji,0 -μ ji,∞ )exp(-k ji t)+μ ji,∞ ,j=x,y (9),
in the formula (9), μ ji,0 0 is the initial value of the predetermined performance function, mu ji,∞ > 0 is the steady state value of the preset performance function when t → ∞ k ji Setting parameters for the convergence speed of the preset performance function if the convergence speed is more than 0;
to achieve the preset performance control requirement, the formation controller is designed to make the formation control error meet the requirement of the formula (10):
-μ ji (t)<e ji <μ ji (t),j=x,y (10),
in the formula (10), the parameter μ ji,0 Is set to satisfy-mu ji,0 <e ji (0)<μ ji,0 ;
And 5.2, defining an error transformation function as shown in a formula (11):
derivation of equation (11) yields equation (12):
by defining the error transformation function, only delta is ensured in the design of the queue controller ji Bounded, i.e. guaranteeing a formation control error e xi 、e yi Meet a predetermined performance requirement, i.e., -mu ji (t)<e ji <μ ji (t);
Step 5.3, designing the Lyapunov function as shown in formulas (13) and (14):
wherein, delta xi An error transformation function selected for X-axis error control; delta yi An error transformation function selected for Y-axis error control;
derivation of equations (13) and (14) yields equations (15) and (16):
design u from equations (15), (16) vi ,u ωi As shown in equations (17) and (18):
wherein u is vi ,u ωi Design parameter k 1i >0,k 2i >0;
The formula (17) and the formula (18) may be substituted for the formula (15) and the formula (16), respectively:
combining the formula (8), the control law of the formation speed of the multi-mobile robot with the preset performance is shown as a formula (21):
in the formula (21), v ci A pre-set performance multi-mobile robot formation linear speed control law; omega ci Forming an angular speed control law for the multi-mobile robot with preset performance;
step 6: designing a self-adaptive fault-tolerant torque control law of the formation mobile robot, wherein the process is as follows:
step 6.1, after the design of the speed control law of the formation mobile robot is finished, the design of the self-adaptive fault-tolerant moment control law is required to be carried out by combining a dynamic model of the mobile robot so as to obtain the actual control moment input of the following mobile robot; first, a velocity tracking error vector is defined as shown in equation (22):
in the formula (22), e 1i Tracking error for the linear velocity following the mobile robot i; e.g. of the type 2i Tracking error for following the angular velocity of the mobile robot i;
step 6.2, in order to improve the control precision of the speed tracking error, designing an integral sliding mode surface as shown in a formula (23):
in the formula (23), S i =[s 1i ,s 2i ] T Wherein s is 1i ,s 2i Respectively represent the vectors S i The first and second components of (a); k vi =diag{k v1i ,k v2i The value is more than 0, which is the design parameter of the integral sliding mode surface;
the time derivative of equation (23) can be derived from equation (24):
D i Satisfies the relationship:wherein, b i =max{1,c i The (x) is an unknown constant which,can be obtained by on-line calculation;
step 6.3 for unknown item b i The self-adaptive method of parameters is adopted for processing, and the self-adaptive fault-tolerant moment control law of the following mobile robot i is designed by combining a formula (24) and is shown as a formula (25):
in the formula (25), the first and second groups,k 3i >0,k 4i >0,k 5i a control gain parameter which is more than 0 and positive; design parameter beta i The conditions need to be satisfied: beta is more than 0 i Is less than 1; sign (·) is a sign function;is an unknown constant b i Is determined by the estimated value of (c),the adaptive update law of (2) is shown in equation (26):
in the formula, k 6i Is a constant greater than zero.
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