A nonholonomic mobile robot formation control method

Abstract

This paper discloses a method for controlling a nonholonomic mobile robot formation. This method first models the nonholonomic mobile robot system and then, by constructing reasonable assumptions, designs a simple and computationally inexpensive control strategy. This control strategy is suitable for robotic systems subject to modeling uncertainty, unknown external interference, and unpredictable actuator failures. It also supports one-to-one communication between adjacent robots while maintaining a safe distance.

Description

Technical Field

[0001] The present invention relates to a mobile robot formation control method, and in particular to a non-holonomic mobile robot formation control method. Background Art

[0002] Over the past few decades, considerable research has been devoted to the distributed control of multi-agent systems. This is due to the difficulty of information exchange between agents in multi-agent systems, yet multi-agent systems are widely used in practical engineering applications, such as multi-robot systems, energy systems, and biological systems. Formation control is a popular research topic in multi-agent systems and can be categorized into three key areas: position-based formation control, distance-based formation control, and direction-based formation control. However, early research in this area primarily focused on simplified dynamic models, such as single-integral and double-integral models. However, due to factors such as system uncertainty, external disturbances, and unpredictable actuator failures, single-integral and double-integral models are not applicable to multi-agent systems with multiple nonholonomic mobile robots.

[0003] While research on the control of nonholonomic mobile robot formations has yielded substantial results, early control methods were limited to mobile robot formation control systems with known models. While there have also been corresponding research findings on distributed control methods for mobile robot formation control systems with unknown dynamic models, these methods fail to consider the limited communication distance between robots and the potential for collisions between them. In practical applications, the effective communication distance between robots is often limited by the communication equipment, and collisions can occur if the distance between robots is too small. Therefore, ensuring effective communication while avoiding collisions within the framework of multi-robot formation control is a promising research direction. While some control methods currently exist that can ensure effective communication or avoid collisions, none can meet both requirements simultaneously.

[0004] To achieve the aforementioned control objectives, one distributed control approach involves introducing certain potential functions into the Lyapunov function. However, this approach can lead to conflicts when selecting design parameters. Related literature also mentions a distributed formation control method based on a unified error transformation mechanism and a distributed adaptive formation control scheme. While these two approaches can address the aforementioned issues, both are based on backstepping, and therefore, the controller design becomes complex due to the introduction of the time derivative of the virtual controller. Furthermore, the distributed formation control method based on the unified error transformation mechanism requires online updating of additional adaptive parameters and a large number of neurons capable of real-time computation, which also increases the complexity of the control method. Summary of the Invention

[0005] In view of the above problems existing in the prior art, the technical problem to be solved by the present invention is: how to design a non-holonomic mobile robot formation control method with low complexity and low computational cost.

[0006] To solve the above technical problems, the present invention adopts the following technical solution: a non-holonomic mobile robot formation control method, comprising the following steps:

[0007] S1: Establish mathematical model:

[0008] S11: Establish a dynamic model of the robot in the nonholonomic mobile robot system, wherein the nonholonomic mobile robot system includes N two-wheeled mobile robots. The dynamic model of the kth robot in the nonholonomic mobile robot system is shown in formula (2):

[0009]

[0010] in, (x k ,y k ),θ k represent the position and orientation angle of the kth robot respectively; represents the angular velocity vector of the left and right wheels of the kth robot; represents the output vector of the actuator, i.e., the control torque vector applied to the wheel; represents the unknown external disturbance of the left and right wheels of the k-th robot;

[0011] J k , M k , C k and D k It has no actual physical meaning and is an intermediate variable. represents a third-order vector, represents a second-order vector.

[0012] S12: Establish the robot system dynamics model as shown in formula (3):

[0013]

[0014] Among them, r k and b k are the wheel radius and half-width of the kth robot respectively; d k,1 and d k,2 are two positive constants representing its damping coefficient; m k,1 and m k,2 It has no actual physical meaning and is an intermediate variable. k,c and mk,w are the masses of the kth robot and wheel respectively; c k is the mass center P of the kth robot c,k To the midpoint P of the line connecting the two wheels o,k Distance I k,c represents the moment of inertia about the vertical direction passing through the mass center of the kth robot, I k,w Represents the moment of inertia of the wheel with the motor around the wheel axis, I k,m The moment of inertia of the wheel with the motor rotor around the wheel diameter, I k represents the total moment of inertia of the kth robot.

[0015] S2: Assumptions of distributed control:

[0016] Assumption 1: The leader robot's trajectory η L is measurable, is piecewise continuous, and both are bounded; Assumption 1 cancels the requirement for η L The constraint is that the trajectory η L It only requires real-time measurability, and does not need to be known a priori, and its time derivative is not required to be known. Therefore, the solution designed by the present invention is suitable for a wider range of practical applications.

[0017] Assumption 2: There is an unknown constant, s k,l and s k,m , and satisfy 0< s k,l ≤σ k,l (t)≤1,0< s k,m ≤σ k,m (t)≤1, k=1,...,N. Assumption 2 allows the unknown time-varying function representing the “health index” to be piecewise continuous, which shows that the control scheme proposed in this invention considers both initial and sudden faults of the multiplicative PLOE actuator.

[0018] Assumption 3: Desired relative distance and desired azimuth Satisfy respectively And the initial conditions of both satisfy d l,k <d k (0) <d m,k ,|β k (0)|<β m,k , Assumption 3 is necessary for the control scheme to solve the potential CVCPFTFC problem, and this assumption is also consistent with the actual situation.

[0019] S3: The design process of the distributed controller is as follows:

[0020] S31: Distributed virtual controllers must meet the conditions of formulas (13a, 13b) and (15a, 15b):

[0021] η L =[x L ,y L ,θ L ] T Indicates the leader robot R L The posture is as follows:

[0022]

[0023] Among them, θ L ∈[-π,π),v L and w L Represents robot R L Linear velocity and angular velocity;

[0024] d k and β k Represent the relative distance and azimuth to the adjacent robots, respectively, where k = 1, ..., N, d k and β k It is expressed as follows:

[0025]

[0026] When k=1, x k-1 =x0,y k-1 =y0, at this time x0, y0 represents the leader robot R L posture;

[0027]

[0028] To avoid collisions and maintain communication between robots, d k and β k The following constraints need to be met:

[0029]

[0030] Among them, d l,k d m,k and are all positive constants.

[0031] The tracking error vector is expressed as The specific definitions are as follows:

[0032]

[0033] Among them, d k (t) and β k(t) represents the relative distance and azimuth of adjacent robots at time t, d k * (t) and β k * (t) represents the expected relative distance and expected azimuth at time t respectively; e d,k,1 (t) and e β,k,1 (t) represent the relative distance error and azimuth error between adjacent robots, respectively. The performance range of these two values ​​is:

[0034]

[0035] in, and i∈{d,β} is the performance function, which can be obtained from formulas (16a, 16b):

[0036]

[0037] Among them, k i,k >0,

[0038]

[0039] They represent the upper and lower bounds of the performance function of the relative distance error, They represent the upper and lower initial values ​​of the azimuth error performance function, d m,k , d l,k , β m,k , β l,k represents a positive constant value and satisfies formula (13).

[0040] S32:

[0041]

[0042] in v r,k-1 and θ k-1 Represents the k-1th robot R k-1 Linear velocity and direction angle, v r,0 =v l , θ0=θ l Next, the control scheme is designed using a backstepping-like design process.

[0043] They represent the derivatives of the relative distance error and azimuth error between the kth robot and its adjacent robots, respectively, and w r,k , v r,k Represents the kth robot Rk angular velocity and linear velocity.

[0044] S33: Design of distributed virtual controller, virtual control signal for the kth robot Designed to: The specific design method is as follows:

[0045] ξ k,1 (t) represents the normalized error vector, which can be expressed as follows:

[0046]

[0047] According to equations (15) and (19), we can get ξ d,k,1 (t)∈(-1,1) and ξ β,k,1 (t)∈(-1,1), then, through equations (14) and (19), d k and β k Defined as:

[0048]

[0049] Next, use Represents an intermediate variable vector, which can be expressed as:

[0050]

[0051] Virtual control signal of the kth robot Designed to:

[0052]

[0053] In the above formula, μ k,1 =diag{μ d,k,1 ,μ β,k,1}, where μ d,k,1 and μ β,k,1 is a positive design parameter,

[0054] S34: Design a distributed real controller for the kth robot R k The actual input signal of the controller It is expressed as follows: The specific design method is as follows:

[0055] The virtual tracking error vector is expressed as e k,2 (t) is expressed as follows:

[0056]

[0057] Among them, e d,k,2(t), e β,k,2 (t) represents the virtual error of the relative distance and azimuth angle between the kth robot and its adjacent robots, ε d,k (t), ε β,k (t) represents the virtual control signals of the relative distance and azimuth angle between the kth robot and its adjacent robots, respectively.

[0058] Formula (23) must meet the following preset performance boundaries:

[0059] |e d,k,2 (t)|<ρ d,k,2 (t),|e β,k,2 (t)|<ρ β,k,2 (t)(24)

[0060] ρ d,k,2 (t) and ρ β,k,2 (t) are expressed as e d,k,2 (t) and e β,k,2 (t) and the corresponding performance function, both are determined according to the following formula:

[0061] ρ d,k,2 (t)=(ρ d,k,0 -ρ d,k,∞ )exp(-k d,k t)+ρ d,k,∞ (25a)

[0062] ρ β,k,2 (t)=(ρ β,k,0 -ρ β,k,∞ )exp(-k β,k t)+ρ β,k,∞ (25b)

[0063] where ρ d,k,0 >e d,k,2 (0),ρ β,k,0 >e β,k,2 (0),ρ d,k,∞ ∈(0,ρ d,k,0 ],ρ β,k,∞ ∈(0,ρ β,k,0 ], k d,k >0 and k β,κ >0 are design parameters, respectively e d,k,2 (t), e β,k,2 (t) Pre-allocate transient performance indicators and steady-state performance indicators.

[0064] Same as designing a virtual controller, k,2 (t) represents the normalized error vector, which is expressed by the formula:

[0065]

[0066] The corresponding intermediate variable vector can be expressed as It can be expressed as:

[0067]

[0068] The kth robot R k The actual input signal of the controller Defined as:

[0069]

[0070] where ρ k,2 =diag{ρ d,k,2 ,ρ β,k,2},μ k,2 =diag{μ d,k,2 ,μ β,k,2}, and μ d,k,2 and μ β,k,2 is a positive design parameter,

[0071] Preferably, the S1 further includes S13 coordinate transformation, and the specific steps are as follows:

[0072] ω k =ζ k B k ,τ k =H k -1 u k (5)

[0073] in,

[0074] ζ k =[v r,k ,w r,k ] T represents the linear / angular velocity vector of the kth robot, u k =[u k,1 ,u k,2 ] T is an auxiliary variable. B k and H k is a reversible matrix. Substituting Equation (5) into Equation (2) and then using Equation (4), the dynamics of the k-th robot can be expressed as follows:

[0075]

[0076] in:

[0077]

[0078] According to formula (7a), we can deduce

[0079]

[0080] Among them, k represents the linear / angular velocity vector of the kth robot, u k It is an auxiliary variable and can be regarded as the control input that needs to be designed. k It is a diagonal matrix composed of time-varying scalars reflecting the effectiveness of the k-th robot actuator. K (·), Γ k , Δ k It has no actual physical meaning and is an intermediate variable.

[0081] The above formula shows that the kth robot moves only on the axis perpendicular to the driving wheel, which means that the speed of the robot in the direction of the wheel axis is 0. Therefore, formula (9) is called a non-holonomic constraint.

[0082] Preferably, the ρ d,k,2 (t),ρ β,k,2 The selection method of (t) is as follows:

[0083] Based on assumption 3 and according to formula (17), ρ d,k,0 ,ρ β,k,0 , and satisfy |e d,k,2 |<ρ d,k,0 ,|e β,k,2 (0)|<ρ β,k,0 ; In this way, the initial conditions can be met And the relative distance and azimuth constraints (i.e., Equation (13)) will not be violated.

[0084] Compared with the prior art, the present invention has at least the following advantages:

[0085] 1. Compared with formation control schemes based on backstepping or neural networks, the control method designed by the present invention has the advantages of simple controller structure, low computational cost, and low communication requirements. This is because the control method designed by the present invention does not use prior knowledge of system nonlinearity, nor does it use nonlinear approximators to process prior knowledge. At the same time, the controller design does not involve the time derivative of the virtual controller or the trajectory of the leader robot. In addition, the control method designed by the present invention does not use the velocity information of neighboring robots, nor does it require actuator fault detection or diagnostic units. These features make the control scheme designed by the present invention more direct and convenient to execute. In addition, since all closed-loop signals are bounded based on the Lyapunov theorem and no potential function is used, this scheme also avoids the singularity problem.

[0086] 2. Compared with the prior art, the method designed by the present invention is a control scheme with a simple structure and low computational cost, which makes the design and use of this scheme more direct and convenient. The control scheme designed by the present invention only uses the posture information of the neighboring robots, while the distributed formation control methods mentioned in other documents also require the speed information of the neighboring robots. Therefore, in comparison, the control scheme designed by the present invention has lower requirements for communication between robots. In addition, when designing the controller, the present invention takes into account the situation where the system has modeling uncertainty, unknown external interference and unpredictable actuator failure. Therefore, this scheme can still achieve the control target when the system has the above-mentioned situations.

[0087] 3. When designing the controller, an appropriate performance function is introduced to constrain the relative distance and azimuth angle between adjacent mobile robots. This ensures the reliability of communication between robots while also solving the problem of collisions between robots caused by too small a distance. BRIEF DESCRIPTION OF THE DRAWINGS

[0088] Figure 1 is the adjacent mobile robot model.

[0089] Figure 2 Communication topology for mobile robots. DETAILED DESCRIPTION

[0090] The present invention is described in further detail below.

[0091] A nonholonomic mobile robot formation control method comprises the following steps:

[0092] S1: Establish mathematical model:

[0093] S11: Establish a dynamic model of the robot in the nonholonomic mobile robot system, wherein the nonholonomic mobile robot system includes N two-wheeled mobile robots. The dynamic model of the kth robot in the nonholonomic mobile robot system is shown in formula (2):

[0094]

[0095] in, (x k ,y k ),θ k represent the position and orientation angle of the kth robot respectively; represents the angular velocity vector of the left and right wheels of the kth robot; represents the output vector of the actuator, i.e., the control torque vector applied to the wheel; represents the unknown external disturbance of the left and right wheels of the k-th robot;

[0096] Jk , M k , C k and D k It has no actual physical meaning and is an intermediate variable. represents a third-order vector, represents a second-order vector.

[0097] S12: Establish the robot system dynamics model as shown in formula (3):

[0098]

[0099] Among them, r k and b k are the wheel radius and half-width of the kth robot respectively; d k,1 and d k,2 are two positive constants representing its damping coefficient; m k,1 and m k,2 It has no actual physical meaning and is an intermediate variable. k,c and m k,w are the masses of the kth robot and wheel respectively; c k is the mass center P of the kth robot c,k To the midpoint P of the line connecting the two wheels o,k Distance I k,c represents the moment of inertia about the vertical direction passing through the mass center of the kth robot, I k,w Represents the moment of inertia of the wheel with the motor around the wheel axis, I k,m The moment of inertia of the wheel with the motor rotor around the wheel diameter, I k represents the total moment of inertia of the kth robot.

[0100] S2: Assumptions of distributed control:

[0101] Assumption 1: The leader robot's trajectory η L is measurable, is piecewise continuous, and both are bounded; Assumption 1 cancels the requirement for η L The constraint is that the trajectory η L It only requires real-time measurability, and does not need to be known a priori, and its time derivative is not required to be known. Therefore, the solution designed by the present invention is suitable for a wider range of practical applications.

[0102] Assumption 2: There is an unknown constant, σ k,l and σ k,m , and satisfy 0<σ k,l ≤σ k,l (t)≤1,0<σ k,m ≤σ k,m(t)≤1, k=1,…,N. Assumption 2 allows the unknown time-varying function representing the “health index” to be piecewise continuous, which means that the control scheme proposed in this invention takes into account both the initial and sudden faults of the multiplicative PLOE actuator.

[0103] Assumption 3: Desired relative distance and desired azimuth Satisfy respectively And the initial conditions of both satisfy d l,k <d k (0) <d m,k ,|β k (0)|<β m,k , Assumption 3 is necessary for the control scheme to solve the potential CVCPFTFC problem, and this assumption is also consistent with the actual situation.

[0104] S3: The design process of the distributed controller is as follows:

[0105] S31: Distributed virtual controllers must meet the conditions of formulas (13a, 13b) and (15a, 15b):

[0106] η L =[x L ,y L ,θ L ] T Indicates the leader robot R L The posture is as follows:

[0107]

[0108] Among them, θ L ∈[-π,π),v L and w L Represents robot R L Linear velocity and angular velocity;

[0109] d k and β k Represent the relative distance and azimuth to the adjacent robots, respectively, where k = 1, ..., N, d k and β k It is expressed as follows:

[0110]

[0111] When k=1, x k-1 =x0,y k-1 =y0, at this time x0, y0 represents the leader robot R L posture.

[0112]

[0113] To avoid collisions and maintain communication between robots, d k and β k The following constraints need to be met:

[0114]

[0115] Among them, d l,k d m,k and are all positive constants;

[0116] The tracking error vector is expressed as The specific definitions are as follows:

[0117]

[0118] Among them, d k (t) and β k (t) represents the relative distance and azimuth of adjacent robots at time t, d k * (t) and β k * (t) represents the expected relative distance and expected azimuth at time t respectively; e d,k,1 (t) and e β,k,1 (t) represents the relative distance error and azimuth error between adjacent robots, e d,k,1 and e d,k,1 (t) means the same thing as e β,k,1 and e β,k,1 (t) means the same thing as e d,k,1 (t) and e β,k,1 (t) The performance range for these two values ​​is:

[0119]

[0120] in, and i∈{d,β} is the performance function, which can be obtained from formulas (16a, 16b):

[0121]

[0122] Among them, k i,k >0,

[0123]

[0124] They represent the upper and lower bounds of the performance function of the relative distance error, They represent the upper and lower initial values ​​of the azimuth error performance function. m,k , d l,k , β m,k , β l,k represents a positive constant value and satisfies formula (13).

[0125] S32:

[0126]

[0127] in v r,k-1 and θ k-1 Represents the k-1th robot R k-1 Linear velocity and direction angle, v r,0 =v l , θ0=θ l Next, the control scheme is designed using a backstepping-like design process.

[0128] They represent the derivatives of the relative distance error and azimuth error between the kth robot and its adjacent robots, w r,k , v r,k Represents the kth robot R k Angular velocity and linear velocity;

[0129] S33: Design of distributed virtual controller, virtual control signal for the kth robot Designed to: The specific design method is as follows:

[0130] ξ k,1 (t) represents the normalized error vector, which can be expressed as follows:

[0131]

[0132] According to equations (15) and (19), we can get ξ d,k,1 (t)∈(-1,1) and ξ β,k,1 (t)∈(-1,1), then, through equations (14) and (19), d k and β k It is expressed as follows:

[0133]

[0134] Next, use Represents an intermediate variable vector, which is expressed as follows:

[0135]

[0136] Virtual control signal of the kth robot Designed to:

[0137]

[0138] In the above formula, μ k,1 =diag{μ d,k,1 ,μ β,k,1}, where μ d,k,1 and μ β,k,1 is a positive design parameter,

[0139] S34: Design a distributed real controller for the kth robot R k The actual input signal of the controller It is expressed as follows: The specific design method is as follows:

[0140] The virtual tracking error vector is expressed as e k,2 (t) is expressed as follows:

[0141]

[0142] Among them, e d,k,2 (t), e β,k,2 (t) represents the virtual error of the relative distance and azimuth angle between the kth robot and its adjacent robots, ε d,k (t), ε β,k (t) represents the virtual control signal of the relative distance and azimuth angle between the kth robot and its adjacent robots. r,k (t) and w r,k The same meaning as v r,k (t) and v r,k have the same meaning.

[0143] Formula (23) must meet the following preset performance boundaries:

[0144] |e d,k,2 (t)|<ρ d,k,2 (t),|e β,k,2 (t)|<ρ β,k,2 (t)(24)

[0145] ρ d,k,2 (t) and ρ β,k,2 (t) are respectively expressed as e d,k,2 (t) and e β,k,2 (t), both are determined according to the following formula:

[0146] ρ d,k,2 (t)=(ρ d,k,0-ρ d,k,∞ )exp(-k d,k t)+ρ d,k,∞ (25a)

[0147] ρ β,k,2 (t)=(ρ β,k,0 -ρ β,k,∞ )exp(-k β,k t)+ρ β,k,∞ (25b)

[0148] where ρ d,k,0 >e d,k,2 (0),ρ β,k,0 >e β,k,2 (0),ρ d,k,∞ ∈(0,ρ d,k,0 ],ρ β,k,∞ ∈(0,ρ β,k,0 ], k d,k >0 and k β,κ >0 are design parameters, respectively e d,k,2 (t), e β,k,2 (t) pre-allocating transient performance indicators and steady-state performance indicators;

[0149] Same as designing a virtual controller, k,2 (t) represents the normalized error vector, which is expressed by the following formula:

[0150]

[0151] The corresponding intermediate variable vector can be expressed as It can be expressed as:

[0152]

[0153] The kth robot R k The actual input signal Defined as:

[0154]

[0155] where ρ k,2 =diag{ρ d,k,2 ,ρ β,k,2},μ k,2 =diag{μ d,k,2 ,μ β,k,2}, and μ d,k,2 and μ β,k,2 is a positive design parameter,

[0156] Specifically, it includes S13 coordinate transformation, and the specific steps are as follows:

[0157] ω k =ζ k B k ,τ k =H k -1 u k (5)

[0158] in,

[0159] ζ k =[v r,k ,w r,k ] T represents the linear / angular velocity vector of the kth robot, u k =[u k,1 ,u k,2 ] T is an auxiliary variable. B k and H k is a reversible matrix. Substituting Equation (5) into Equation (2) and then using Equation (4), the dynamics of the k-th robot can be expressed as:

[0160]

[0161] in:

[0162]

[0163] According to formula (7a), we can deduce

[0164]

[0165] Among them, k represents the linear / angular velocity vector of the kth robot, u k It is an auxiliary variable and can be regarded as the control input to be designed. k It is a diagonal matrix composed of time-varying scalars reflecting the effectiveness of the k-th robot actuator. K (·), Γ k , Δ k It has no actual physical meaning and is an intermediate variable.

[0166] The above formula shows that the kth robot moves only on the axis perpendicular to the driving wheel, which means that the speed of the robot in the direction of the wheel axis is 0. Therefore, formula (9) is called a non-holonomic constraint.

[0167] Specifically, the ρ d,k,2 (t),ρ β,k,2 The selection method of (t) is as follows:

[0168] Based on assumption 3 and according to formula (17), ρ d,k,0 ,ρ β,k,0 , and satisfy |e d,k,2 |<ρ d,k,0 ,|e β,k,2 (0)|<ρ β,k,0 ; In this way, the initial conditions can be met And the relative distance and azimuth constraints (Equation (13)) will not be violated.

[0169] By adjusting k i,k and ρ i,k,∞ The relative distance tracking error e can be preset respectively. d,k,1 , azimuth tracking error e β,k,1 Asymmetric transient and steady-state performance limits can also be set in advance for the virtual tracking error e k,2 The convergence speed and steady-state error of k i,k >0,

[0170] Because the control gain μ d,k,1 >0,μ β,k,1 >0,μ d,k,2 >0,μ β,k,2 >0 no longer dominates the performance of the closed-loop system, so the control gains can be chosen freely.

[0171] Specifically, S1 also includes actuator fault analysis: the present invention takes into account the situation where an unknown fault occurs in the actuator. In order to make the control method more accurate, it is necessary to perform actuator fault analysis and incorporate it into the design of the control method.

[0172] Since mobile robots often work in dangerous and complex environments, the actuators may experience unpredictable failures. In this case, the actual control torque τ a,k ,k=1,…,N and the desired control input The two are no longer the same. But they can be linked by formula (4):

[0173] τ a,k =σ k (t)τ k +ε k (t)(4)

[0174] in, represents the unknown but bounded part of the actuator, σ k (t) = diag{σ k,l (t),σ k,m (t)} is a diagonal matrix, and σ k,l (t)∈(0,1] and σ k,m(t)∈(0,1] is a time-varying scalar that reflects the effectiveness of the k-th robot actuator, and is therefore also called a “health indicator”. In particular, when σ k,i =1,ε k,i = 0, i∈{l,m}, the actuator is effective. k,i ∈(0,1), the actuator has partial failure (POLE) 2 .

[0175] In order to analyze the feasibility of the method of the present invention, the stability analysis of the system is then performed.

[0176] Lemma 1: Let Ω be An open set in , consider the function And the function satisfies the following conditions:

[0177] a) For any Defined in Ω t : = The function t→g(z,t) on {t:(z,t)∈Ω} is measurable;

[0178] b) For any Defined in Ω z :=The function z→g(z,t) on {z:(z,t)∈Ω} is continuous;

[0179] c) For any compact set There exist constants c0 and l0 such that

[0180]

[0181] So for the initial value problem z0=z(t0),(z0,t0)∈Ωin the range [t0,t max )(where t max >t0) there is a unique maximum solution, that is

[0182] Lemma 2: Let all (z, t)∈Ω satisfy the conditions in Lemma 1, and the initial value problem z0=z(t0) in the range t∈[t0,t max ) has a unique maximum solution. From this we can get t max =∞ or

[0183] Lemma 3: H k M k B k is a diagonally positive definite matrix;

[0184] Lemma 4: H k σ k Hk -1 is a symmetric positive definite matrix;

[0185] Performance function: The present invention uses a performance function when designing the controller. For ease of understanding, a brief description of the performance function is given here. When the tracking error of the system strictly converges to a pre-set range, it can be expressed as follows:

[0186]

[0187] Among them, ρ j (t)>0,j∈{l,u} is the performance function, and the performance function satisfies the following conditions:

[0188] a)ρ j (t) is smooth and bounded for any t ≥ 0;

[0189] b)ρ j The first-order derivative of (t) is bounded for any t ≥ 0.

[0190] The commonly used performance function form is in

[0191] Lemmas are formulas or conditions that are first proven for the convenience of controller design and stability analysis.

[0192] Step 1: Since the time derivative expression of the virtual controller is too complex to facilitate stability analysis, it is necessary to simplify the mathematical expression of the closed-loop dynamic system of the mobile robot.

[0193] Taking the derivative of both sides of equation (19) and substituting equation (18) into equation (20), (23) and (26), we can directly derive

[0194]

[0195] Where, ε 0,1 (·)=[0,0] T ,Ξ k-1,1 ,Θ k,1 and Ψ k,1 The specific expression of is shown in the above formula (*). Obviously, Θ k,1 Under the constraint of azimuth, it is negative and positive. Similarly, by taking the derivative of both sides of equation (26) and substituting equation (7), we can directly obtain

[0196]

[0197] in is the time derivative of the virtual controller, which can be expressed as follows

[0198]

[0199] It can be seen from the above formula that is complex, and due to h k,1 The existence of is uncertain, so it cannot be used directly in controller design. It is worth noting that, unlike the backstepping method, this design does not require No prior knowledge of , and no nonlinear approximator is needed to compensate

[0200] make and The closed-loop dynamic system of the mobile robot can be expressed concisely as follows:

[0201]

[0202] The above formula is defined on the set Among them

[0203]

[0204] The theoretical results can be summarized as the following theorem.

[0205] Theorem 1: Under assumptions 1-3, consider the nonholonomic mobile robot dynamics system (see Equation (2)) after coordinate transformation (see Equation (7)). By appropriately choosing the performance function, we can ensure that the initial condition ξ is satisfied. i (0)∈Ω ξ ,i=1,2, the distributed control scheme composed of Equation (22) and Equation (28) can solve the CVCPFTFC problem.

[0206] Step 2: Prove that Equation (32) is valid for the time interval [0,t max ) exists a unique maximum solution (ξ1,ξ2). It can be seen from formula (33) that ξ i (t), the attraction domain of i=1,2 is a non-empty open set. And, by choosing a suitable performance function ρ d,k,2 (t),ρ l,k,2 (t) can satisfy the initial condition ξ i (0)∈Ω ξ Because the system is nonlinear, external disturbance, performance function and its first-order derivative are piecewise continuous, and the distributed control signal ε k (·) and u k (·) in Ωξ are smooth, so it is easy to verify that the right side of Equation (32) satisfies all the conditions in Lemma 1. Therefore, the closed-loop dynamic expression (32) is max ) exists a unique maximum solution (ξ1,ξ2), namely

[0207]

[0208] In a later section, it is proved that under Eq. (34), all closed-loop signals in the time interval [0,t max ) are all bounded.

[0209] Step 3: Prove that under Equation (34), all closed-loop signals in the time interval [0, t max ) are bounded. In this section, we will analyze the stability using a simple form. First, we introduce the following augmented vector / matrix: i =col{ε 1,i …,ε N,i},Ψ1=col{Ψ 1,1 ,…,Ψ N,1}, ε=col{ε1,…,ε N}, Λ1=diag{Λ 1,1 ,…,Λ N,1}, P1=diag{ρ 1,1 ,…,ρ N,1},U1=diag{μ 1,1 ,…μ N,1},κ=1,…,Ν. Then it can be easily verified that for all t≥0, U1, Λ1, and P1 are all diagonally positive matrices. Furthermore, U1 is designed to be a constant matrix. The rest of this section consists of the following two steps.

[0210] Step 3.1: Prove that every quantity in (29) is bounded and that is also bounded. Consider the following Lyapunov function:

[0211]

[0212] Through formula (21) and formula (29), we can deduce

[0213]

[0214] in,

[0215] From formula (*), it can be verified that for any The matrix Q1 is positive definite because Θ k,1Under the azimuth constraint (see equation (13b)), it is negative. Substituting equation (22) into equation (36) yields Since U1, Λ1 and P1 are all diagonal matrices, U1Λ1P1 -1 =P1 -1 Λ1U1. In addition, it is obvious that Where, It is obliquely symmetrical. is positive definite. And formula (22) can be further deduced:

[0216]

[0217] It is worth noting that in formula (*) ρ k,i and By construction or assumption, they are all bounded. Under the constraint of formula (34), ξ k,1 and ξ k,2 For all t∈[0,t max ) are all bounded. Then, applying extreme value theory, we can easily get from (*) that there is a positive constant Make

[0218]

[0219] At the same time, it is easy to

[0220]

[0221] in, and is a positive constant. Through formula (38) and formula (39), You can further narrow the scope.

[0222] This shows that when hour, Therefore, we can get for all t∈[0,t max ), z1 are both bounded, and through formula (21) we can further obtain the existence of a positive constant So that for any t∈[0,t max )

[0223]

[0224] This shows that Λ k,1 (·) and ε in formula (22) k are all bounded. The boundedness of the performance function can be obtained from Equations (14), (20), (26) and (34) for any t∈[0,tmax ) all satisfy e k,1 and e k,2 is bounded. Then, from equations (5) and (23), we can get ζ k and ω k is bounded. In addition, from equations (14)-(17), it can be inferred that the designed control scheme does not exceed the constraints of relative distance and azimuth (see equation (13)). Since under assumption 1, η L According to the boundedness of , we can get η according to formula (11a) k is also bounded. From the above conclusions, it can be obtained that each quantity in formula (29) is bounded and the quantity in formula (31) is It is also bounded.

[0225] Step 3.2: Prove that for any t∈[0,t max )All closed-loop signals are bounded. Let u=col{u1,…,u N}, B=diag{B1,…,B N}, H=diag{H1,…H N}, σ=diag{σ1,…σ N}, σ=diag{σ1,…σ N}, C=diag{C1,…,C N}, D=diag{D1,…,D N}, ε=diag{ε1,…,ε N},δ=diag{δ1,…,δ N}, Λ2=diag{Λ 1，2 ,…,Λ N,2}, P2=diag{ρ 1,2 ,…,ρ N,2},U2=diag{μ 1,2 ,…,μ N,2}, k=1,…,N. For any and t≥0, U2, Λ2 and P2 are all symmetric positive definite matrices. In addition, U2, H, M, B are all constant matrices. By Lemma 3, we can further obtain that HMB and U2HMB are also symmetric positive definite matrices. Therefore, we can easily get

[0226]

[0227] Furthermore, consider the following Lyapunov function:

[0228]

[0229] From equations (27), (31) and (42), we can get

[0230]

[0231] in Matrix B, matrix M and matrix D are all constant matrices, C and ζ k , k=1,…,N are all bounded. In addition, P2, ε and δ are bounded by construction or assumption. Consider ξ2, ε and The boundedness of , we can find a positive constant So that for any t∈[0,t max )

[0232]

[0233] Substituting equations (28) and (45) into equation (44), The bounded range of can be expressed as

[0234] in, is symmetric positive definite. Similar to the analysis of equations (38) to (40), The boundedness of can be directly expressed as

[0235]

[0236] in, λ 2,1 =min{eig{γ}} are all positive constants. From formula (46), we can get that as long as There is Therefore, we can further get that for any t∈[0,t max ), z2 are bounded, which shows that there is a positive constant and So that for any t∈[0,t max ),satisfy:

[0237]

[0238] From formula (28), we can deduce Λ k,2 and u k is bounded. Further from formula (5), we can get τ k is bounded. Therefore, we can get max ), all closed-loop signals are bounded.

[0239] Step 3.3: Prove that Equation (13) is satisfied. According to Equations (41) and (47), we can obtain that for any t∈[0,t max ),have From Lemma 2, we can get tmax =∞, which means that all signals in the closed-loop system are uniformly bounded. And under equations (21) and (27), z k,i The boundedness of κ = 1,…,N, and i = 1,2 ensures that for any t ≥ 0, the performance limit will not be exceeded (see Equations (15) and (24)). Furthermore, under Equations (14)-(17), the relative distance and azimuth angle can always be maintained within the corresponding constraint range (see Equation (13)). This completes the stability proof.

[0240] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and are not limiting. Although the present invention has been described in detail with reference to the preferred embodiments, those skilled in the art should understand that the technical solutions of the present invention may be modified or replaced by equivalents without departing from the purpose and scope of the technical solutions of the present invention, which should all be included in the scope of the claims of the present invention.

Claims

1. A nonholonomic mobile robot formation control method, characterized by: The steps include: S1: Establish mathematical model: S11: Establish a dynamic model of the robot in the nonholonomic mobile robot system, wherein the nonholonomic mobile robot system includes N two-wheeled mobile robots. The dynamic model of the kth robot in the nonholonomic mobile robot system is shown in formula (2): in, (x k ,y k ),θ k represent the position and orientation angle of the kth robot respectively; represents the angular velocity vector of the left and right wheels of the kth robot; represents the output vector of the actuator, i.e., the control torque vector applied to the wheel; represents the unknown external disturbance of the left and right wheels of the k-th robot; J k , M k , C k and D k It has no actual physical meaning and is an intermediate variable. represents a third-order vector, represents a second-order vector; S12: The specific description of the established robot system dynamics model formula is shown in formula (3): Among them, r k and b k are the wheel radius and half-width of the kth robot respectively; d k,1 and d k,2 are two positive constants representing its damping coefficient; m k,1 and m k,2 It has no actual physical meaning and is an intermediate variable. k,c and m k,w are the masses of the kth robot and wheel respectively; c k is the mass center P of the kth robot c,k To the midpoint P of the line connecting the two wheels o,k distance; I k,c represents the moment of inertia about the vertical direction passing through the mass center of the kth robot, I k,w Represents the moment of inertia of the wheel with the motor around the wheel axis, I k,m The moment of inertia of the wheel with the motor rotor around the wheel diameter, I k represents the total moment of inertia of the kth robot; S2: Assumptions of distributed control: Assumption 1: The leader robot's trajectory η L is measurable, are piecewise continuous, and both are bounded; Assumption 2: There exists an unknown constant, σ k,l and σ k,m , and satisfy 0<σ k,l ≤σ k,l (t)≤1,0<σ k,m ≤σ k,m (t)≤1, k=1,...,N; Assumption 3: Desired relative distance and desired azimuth Satisfy respectively And the initial conditions of both satisfy d l,k <d k (0) <d m,k ,|β k (0)|<β m,k ; S3: The design process of the distributed controller is as follows: S31: Distributed virtual controllers must meet the conditions of formulas (13a, 13b) and (15a, 15b): η L =[x L ,y L ,θ L ] T Indicates the leader robot R L The posture is as follows: Among them, θ L ∈[-π,π),v L and w L Represents robot R L Linear velocity and angular velocity; d k and β k Represent the relative distance and azimuth to the adjacent robots, respectively, where k = 1, ..., N, d k and β k It is expressed as follows: When k=1, x k-1 =x0,y k-1 =y0, at this time x0, y0 represents the leader robot R L posture; To avoid collisions and maintain communication between robots, d k and β k The following constraints need to be met: Among them, d l,k d m,k and are all positive constants; and Denote the desired relative distance and the desired azimuth respectively, and the tracking error vector is expressed as The specific definitions are as follows: Among them, d k (t) and β k (t) represents the relative distance and azimuth of adjacent robots at time t, and They represent the expected relative distance and expected azimuth at time t respectively; e d,k,1 (t) and e β,k,1 (t) represent the relative distance error and azimuth error between adjacent robots, respectively. The performance range of these two values ​​is: in, and i∈{d,β} is the performance function, which is obtained from formulas (16a, 16b): Among them, k i,k >0, They represent the upper and lower bounds of the performance function of the relative distance error, They represent the upper and lower initial values ​​of the azimuth error performance function, d m,k , d l,k , β m,k , β l,k represents a positive constant value and satisfies formula (13); S32： in v r,k-1 and θ k-1 Represents the k-1th robot R k-1 Linear velocity and direction angle, v r,0 =v l , θ0=θ l ,Next, the control scheme is designed using a design process similar to ,the backstepping method; They represent the relative distance error and the first-order derivative of the azimuth error between the kth robot and its adjacent robots, respectively. r,k , v r,k Represents the kth robot R k Angular velocity and linear velocity; S33: Design of distributed virtual controller, virtual control signal for the kth robot Designed to: The specific design method is as follows: ξ k,1 (t) represents the normalized error vector, which is expressed as follows: According to equations (15) and (19), we can get ξ d,k,1 (t)∈(-1,1) and ξ β,k,1 (t)∈(-1,1), then, through equations (14) and (19), d k and β k It can be expressed as: Next, use represents an intermediate variable vector, defined as: Virtual control signal of the kth robot Designed to: In the above formula, μ k,1 =diag{μ d,k,1 ,μ β,k,1 }, where μ d,k,1 and μ β,k,1 is a positive design parameter, S34: Design a distributed real controller for the kth robot R k The actual input signal It is expressed as follows: The specific design method is as follows: The virtual tracking error vector is expressed as e k,2 (t) represents, defined as: Among them, e d,k,2 (t), e β,k,2 (t) represents the virtual error of the relative distance and azimuth angle between the kth robot and its adjacent robots, ε d,k (t), ε β,k (t) represents the virtual control signal of the relative distance and azimuth angle between the kth robot and its adjacent robots; The virtual tracking error must meet the following preset performance boundaries: |e d,k,2 (t)|<ρ d,k,2 (t),|e β,k,2 (t)|<ρ β,k,2 (t)(24) ρ d,k,2 (t) and ρ β,k,2 (t) represents e d,k,2 (t) and e β,k,2 (t) corresponds to the preset performance function, both of which are determined according to the following formula: r d,k,2 (t)=(ρ d,k,0 -r d,k,∞ )exp(-k d,k t)+r d,k,∞ (25a) r β,k,2 (t)=(ρ β,k,0 -r β,k,∞ )exp(-k β,k t)+r β,k,∞ (25b) where ρ d,k,0 >e d,k,2 (0),ρ β,k,0 >e β,k,2 (0),ρ d,k,∞ ∈(0,ρ d,k,0 ],ρ β,k,∞ ∈(0,ρ β,k,0 ], k d,k >0 and k β,κ >0 are design parameters, respectively e d,k,2 (t), e β,k,2 (t) Presetting transient performance indicators and steady-state performance indicators; Same as designing a virtual controller, k,2 (t) represents the normalized error vector, which is expressed by the following formula: The corresponding intermediate variable vector is expressed as Defined as: The kth robot R k The actual input signal of the controller Expressed as: where ρ k,2 =diag{ρ d,k,2 ,ρ β,k,2 },μ k,2 =diag{μ d,k,2 ,μ β,k,2 }, and μ d,k,2 and μ β,k,2 is a positive value, 2. A nonholonomic mobile robot formation control method according to claim 1, characterized in that: The S1 also includes S13 coordinate transformation, and the specific steps are as follows: oh k =ζ k B k ,t k =H k -1 you k (5) in, ζ k =[v r,k ,w r,k ] T represents the linear / angular velocity vector of the kth robot, u k =[u k,1 ,u k,2 ] T is an auxiliary variable, B k and H k is a reversible matrix. Substituting Equation (5) into Equation (2) and then using Equation (4), the dynamics of the k-th robot is expressed as follows: in: According to formula (7a), we can deduce Among them, k represents the linear / angular velocity vector of the kth robot, u k It is an auxiliary variable, which is regarded as the control input that needs to be designed. k is a diagonal matrix composed of time-varying scalars reflecting the effectiveness of the k-th robot actuator, S K (·), Γ k , Δ k It has no actual physical meaning and is an intermediate variable.

3. A nonholonomic mobile robot formation control method according to claim 1 or 2, characterized in that: described The selection method is as follows: Based on assumption 3 and according to formula (17), ρ d,k,0 ,ρ β,k,0 , and satisfy |e d,k,2 |<ρ d,k,0 ,|e β,k,2 (0)|<ρ β,k,0 ; Thus, the initial conditions are met And the relative distance and azimuth constraints will not be violated.

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