Perturbation utilization method-based incomplete trolley formation control strategy
By dividing and utilizing disturbances in incomplete trolley formations, the interference observer and disturbance utilization control strategies are designed, and the problem of insufficient anti-disturbance capabilities of the formation in complex environments is solved, and the stability and control performance of the system are improved.
Patent Information
- Application Number
- CN202411524482.0
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2024-10-30
- Publication Date
- 2025-02-07
AI Technical Summary
In complex environments, the incomplete trolley formation lacks disturbance resistance, resulting in the impact of system stability and control performance.
By dividing the disturbance into perturbations of linear velocity and angular velocity, a composite interference model is constructed, and an interference observer and perturbation utilization control strategy are designed to improve the formation control performance of the system using favorable perturbations.
It improves the stability and adaptability of the system, enhances the formation control performance, and enables multi-robot systems to effectively form in complex environments.
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Figure CN119396153A_ABST
Abstract
Description
Technical Field
[0001] The present invention relates to the field of automatic control technology, and in particular to a disturbance utilization control method. Background Art
[0002] In the practical application of non-complete vehicles, platoon control faces many internal and external disturbances, especially in complex environments, the stability and control performance of the platoon system are easily affected. Therefore, it is particularly important to design an effective anti-disturbance control method, and the disturbance observer-based control method (Disturbance Observer-Based Control, DOBC) has received more and more attention. At present, although some studies have made progress in the control of non-complete vehicle platoons, there is still a lot of room for optimization in the conditional design and practical application of making full use of external disturbances. In addition, disturbances do not always have a negative impact on the system. How to transform external disturbances into favorable factors to enhance the control ability of the system has become an issue worthy of in-depth study.
[0003] Therefore, the effectiveness of the disturbance utilization control method for incomplete vehicle formations in complex environments still needs further research and breakthroughs. Summary of the invention
[0004] In view of the deficiencies in the prior art, the present invention provides a non-complete car formation control strategy based on a disturbance utilization method, which solves the problem of insufficient anti-disturbance ability of the prior art in complex environments. While realizing disturbance estimation, the method utilizes favorable disturbances to improve the formation control performance of the system, thereby improving the stability and adaptability of the system.
[0005] The object of the present invention is achieved in this way: a non-holonomic car formation control method based on a disturbance utilization strategy comprises the following steps:
[0006] Step 1) For the disturbance generated during the motion of the non-holonomic car, it is divided into the linear velocity disturbance caused by the slip phenomenon and the motor disturbance acting on the angular velocity, and a composite disturbance model is constructed;
[0007] Step 2) Based on the angle rigidity graph theory and the composite interference model, a non-holonomic platooning system model with input channel interference is established;
[0008] Step 3) Designing a disturbance observer for the disturbance in the model to effectively estimate the input disturbance in the multi-robot formation system in real time;
[0009] Step 4) Based on the estimation of the disturbance in step 3), a disturbance utilization control strategy is designed, and a disturbance utilization judgment criterion is established to accelerate the formation of the formation;
[0010] Step 5) Using the relative angle information between the robots obtained through communication, combined with the disturbance observer and the disturbance utilization control strategy, a globally stable angle rigid formation control method is designed so that the multi-robot system can achieve the desired formation and there is no collision between the communicating robots during the movement;
[0011] Step 6) Control of the incomplete vehicle formation is achieved through the cascade system stability analysis method.
[0012] Further, step 1) is specifically as follows:
[0013] Establish a mathematical model of the non-holonomic car based on linear velocity and angular velocity interference:
[0014] When the motion plane is relatively smooth or the wheel is squeezed and deformed, the wheel will slip, causing the linear velocity v to deviate, resulting in a linear velocity disturbance d ui , specifically
[0015]
[0016] where r i is the radius of the driving wheel, ζ ri ,ζ li is the disturbance angular velocity caused by the slip of the two driving wheels;
[0017] At the same time, the incomplete car is driven by a motor, introducing the steering angular velocity w i Interference on:
[0018]
[0019] where b i is the viscous friction coefficient, ω τi is the motor rotor speed, J i is the moment of inertia, T ei is the unknown torque disturbance term, T li is the load torque.
[0020] Further, step 2) is specifically as follows:
[0021] The nonholonomic platooning system model with input channel interference is defined as follows:
[0022]
[0023] Where P i =[x i y i ] T represents the position of the robot car, i = 1, 2, ..., n; the car's steering angle is θ i∈(-π,π] represents; the vector of the position and direction of robot i is recorded as and θ=[θ1,θ2,...,θ n ] T , and the input of the translational and rotational velocities of car i is represented as u i and w i , d ui is the linear velocity disturbance, d wi is the angular velocity disturbance.
[0024] Furthermore, the disturbance observer in step 3) is used to estimate and compensate for the external disturbance to the non-holonomic car in real time. The disturbance observer is specifically:
[0025]
[0026] where z ui , z wi is the auxiliary variable of the disturbance observer, L ui , l wi is the gain vector of the designed disturbance observer l wi ≥1 / 4, P i =[x i y i ] T represents the position of the robot car, θ i Represents the steering angle of the car, R i =[cosθ i sinθ i ] T is the rotation matrix, represents the estimated value of the line speed input disturbance, Represents an estimate of the angular velocity input disturbance.
[0027] Further, in step 4):
[0028] Constructing the potential energy function The form is as follows:
[0029]
[0030] definition:
[0031]
[0032] in, I d ∈R d×d is the identity matrix, the edge vector P ij =P i -P j and angle vector and
[0033] The total potential energy function of robot i is:
[0034]
[0035] Potential energy function V ij With smoothness, when the relative angle between robot i and its neighbor j reaches the expected value When V ij is zero; when the relative angle between robot i and any of its neighbors tends to zero or tends to infinity, V ij tends to infinity;
[0036] Design a conditional disturbance utilization judgment including expected angle error and external disturbance
[0037]
[0038] Where R i =[cosθ i sinθ i ] T is the rotation matrix, θ * represents the desired angle, ε i and represents a sufficiently small positive number, The hyperbolic tangent function is defined as follows:
[0039]
[0040] Formulas (9)-(10) are interference utilization control strategies. If and ρ i When >0, the disturbance is defined as an adverse disturbance, and the disturbance observer is used to compensate for this type of disturbance.
[0041] Furthermore, the control law based on interference utilization in step 5) is:
[0042] The communication topology always maintains angular rigidity during the movement; the following formation control law is designed:
[0043]
[0044] where k i ≥1, As shown in formula (9);
[0045]
[0046] Among them l i ≥1,ρ i As shown in formula (10);
[0047] Further, the specific conditions of step 6) are as follows:
[0048] The stability of the nonholonomic system is studied by using the ISS lemma of the cascade system. The precise formation control is achieved by combining the control law and the potential energy function, namely:
[0049]
[0050] The Lyapunov function is selected as follows:
[0051]
[0052] The communication topology graph is an undirected graph with get:
[0053]
[0054] W i The function is differentiated with respect to t to obtain
[0055]
[0056] Substituting formula (9) and (11) into (15), we obtain:
[0057]
[0058] in is non-increasing, and the lower bound is zero, so we get the existence limit W i (∞); let Integrating both sides of the equation,
[0059] get:
[0060]
[0061] exists and is bounded, which proves is consistent and continuous;
[0062] Starting from any starting position, T(b ij (t)) and u i are all bounded, so is also bounded; because b ij =P ij / ||P ij ||, b ij and For i=1,2,…,n,j∈N i are all bounded, which means that b ij exist is uniformly continuous; since T(b ij(t)) is continuous, then it is are all consistent and continuous; therefore, when When T(b ij (t)) The uniform continuity property of can be known, When T(b ij (t)) is also uniformly continuous; therefore, according to Barbalat’s lemma, we can prove that And get:
[0063]
[0064] because so All robots converge to In addition, for each robot i, when According to formula (11), when When So Get θ id =0;
[0065] The Lyapunov function for further analysis of the speed angle system is selected as follows:
[0066]
[0067] Taking the derivative of the above formula with respect to time t, we can get
[0068]
[0069] When we consider formula (12), we get
[0070]
[0071] The steering angle of each robot converges to Right now and is asymptotically stable;
[0072] According to the definition of nonlinear cascade system
[0073]
[0074] In the formula, exist is locally Lipschitz continuous and f2(0)=0, if the subsystem is input-state stable (ISS) from input x2 to state x1, and the subsystem is globally asymptotically stable, then the cascade system (26) globally converges asymptotically to the equilibrium point (x1, x2) = 0;
[0075] Using the ISS theory in (26), we can obtain that the entire formation system is globally asymptotically stable, thus obtaining
[0076]
[0077] So we get
[0078]
[0079] Taking into account the control of the leaderless formation, graph G is an undirected graph, and the edges in graph G are marked in the order of 1 to m, where m is the number of edges; let all the edges in G be E1, E2, …E m , then the corresponding incidence matrix is obtained
[0080] H=H(G)=(h ij )∈R m×n (25)
[0081] The incidence matrix is a matrix composed of 0, 1, and -1;
[0082] Define the side vector and angle vector: e k =P ij =P i -P j , b k =e k / ||e k ||, So we get:
[0083]
[0084] Convert (28) into vector form:
[0085]
[0086] From equation (30) and equation (31), we get the following equation:
[0087]
[0088] Learn At the same time, the linear velocity and angular velocity of the car tend to be stable; according to ISS, the designed formation control system achieves global asymptotic stability, ensuring that all pairs of robots can accurately reach the desired relative angle, combined with the angle rigidity theory, thus achieving the desired formation, that is,
[0089] S0={g ij ,θ i |g ij =g ij *,θ i =θ i*,i=1,2,...,n,j∈N i} (29).
[0090] Because the purpose of the interference utilization technology proposed in the present invention is to speed up the tracking of the system output to the expected output, first in step 1), the disturbance generated during the movement of the non-complete trolley is divided into linear velocity disturbance caused by slipping and motor disturbance acting on angular velocity, and a composite interference model is constructed. On this basis, step 2) based on the angle rigidity graph theory and the composite interference model, a non-complete trolley formation system model with input channel interference is established, followed by step 3) for the disturbance in the model, an interference observer is designed to make real-time and effective estimation of the input interference existing in the multi-robot formation system, step 4) based on the effective estimation of the interference, a disturbance utilization control strategy is designed, an interference utilization judgment criterion is established, and the formation of the formation is accelerated, step 5) using the relative angle information between the robots obtained by communication, combined with the interference observer and the disturbance utilization control strategy, a globally stable angle rigid formation control method is designed, so that the multi-robot system reaches the desired formation, and no collision occurs between the communicating robots during the movement, step 6) through the cascade system stability analysis method, the global stability of the non-complete formation control system is guaranteed.
[0091] Compared with the prior art, the beneficial effects of the present invention are: first, for the incomplete car formation system, a formation control law based on azimuth angle is proposed, and no collision occurs between the communicating robots during the movement; second, compared with the traditional disturbance complete compensation countermeasure, the disturbance utilization scheme of the present invention can utilize some beneficial disturbances, which can effectively improve the control performance of the formation system. BRIEF DESCRIPTION OF THE DRAWINGS
[0092] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings required for use in the embodiments or the description of the prior art will be briefly introduced below. Obviously, the drawings described below are only embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on the provided drawings without paying creative work.
[0093] Figure 1 Flow chart of the control method of the present invention.
[0094] Figure 2 It is a schematic diagram of the disturbance utilization control method based on the non-complete vehicle formation of the present invention. DETAILED DESCRIPTION
[0095] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.
[0096] The present invention proposes a non-holonomic vehicle formation control method based on a disturbance utilization strategy, comprising the following steps:
[0097] Step 1) For the disturbances generated during the motion of the non-holonomic car, the disturbances are divided into linear velocity disturbances caused by slippage and motor disturbances acting on angular velocity, and a composite disturbance model is constructed;
[0098] When the motion plane is relatively smooth or the wheel is squeezed and deformed, the wheel will slip, causing the linear velocity v to deviate, resulting in a linear velocity disturbance d ui , specifically
[0099]
[0100] where r i is the radius of the driving wheel, ζ ri ,ζ li , is the disturbance angular velocity caused by the slip of the two driving wheels.
[0101] at the same time Figure 1 The car shown in the figure is driven by a motor, introducing a steering angular velocity w i Interference on:
[0102]
[0103] where b i is the viscous friction coefficient, ω τi is the motor rotor speed, J i is the moment of inertia, T ei is the unknown torque disturbance term, T li is the load torque.
[0104] Step 2) The non-holonomic platooning system model with input channel interference is defined as follows:
[0105]
[0106] Where P i =[x i y i ] T represents the position of the robot car, i = 1, 2, ..., n. The car's steering angle is θ i∈(-π,π] represents. The vectors of the position and direction of robot i are recorded as and θ=[θ1,θ2,...,θ n ] T , and the input of the translational and rotational velocities of car i is represented as u i and w i , d ui is the linear velocity disturbance, d wi is the angular velocity disturbance.
[0107] Step 3) A state-space based disturbance observer is designed to estimate and compensate for the external disturbance of the non-complete car in real time. The specific design is as follows:
[0108]
[0109] where z ui , z wi is the auxiliary variable of the disturbance observer, L ui , l wi is the gain vector of the designed disturbance observer l wi ≥1 / 4, P i =[x i y i ] T represents the position of the robot car, θ i represents the steering angle of the car,
[0110] R i =[cosθ i sinθ i ] T is the rotation matrix, represents the estimated value of the line speed input disturbance, Represents an estimate of the angular velocity input disturbance.
[0111] Step 4) Design a judgment method for conditional interference utilization including the formation expected angle, expected angle error and external disturbance
[0112] Constructing the potential energy function The form is as follows:
[0113]
[0114] definition:
[0115]
[0116] in, I d ∈R d×d is the identity matrix, the edge vector P ij =Pi -P j and angle vector and So we have:
[0117] The total potential energy function of robot i is:
[0118]
[0119] Obviously, the designed potential energy function V ij With smoothness, when the relative angle between robot i and its neighbor j reaches the expected value When V ij is zero; when the relative angle between robot i and any of its neighbors tends to zero or tends to infinity, V ij Goes to infinity.
[0120] Design a judgment method for conditional interference utilization including the formation expected angle, expected angle error and external disturbance
[0121]
[0122] Where R i =[cosθ i sinθ i ] T is the rotation matrix, θ * represents the desired angle, ε i and represents a sufficiently small positive number, The hyperbolic tangent function is defined as follows:
[0123]
[0124] Formulas (9)-(10) are interference utilization control strategies. Using this strategy, the criterion for judging whether the disturbance has a beneficial or harmful effect on the formation system is given. The interference utilization judgment criterion is as follows: and ρ i ≤0, the disturbance is defined as a beneficial disturbance. Similarly, if the judgment condition and ρ i When > 0, the disturbance is defined as an unfavorable disturbance. In fact, if the disturbance has a positive or beneficial effect on the formation system, there is no need to eliminate the disturbance, but to utilize it; on the contrary, if the disturbance has an unhelpful or even harmful effect on the system, the disturbance observer is used to compensate for the disturbance.
[0125] The specific design of the control law based on interference utilization in step 5) is as follows:
[0126] And from the relevant knowledge of angle rigidity, we can know that the communication topology can always maintain angle rigidity during the movement. The following formation control law is designed:
[0127]
[0128] where k i ≥1, As shown in formula (9).
[0129]
[0130] Among them l i ≥1,ρ i As shown in formula (10).
[0131] Step 6) The stability of the nonholonomic system is studied using the ISS lemma of the cascade system. Combining the control law and the potential energy function, accurate formation control can be achieved, namely:
[0132]
[0133] The Lyapunov function is selected as follows:
[0134]
[0135] The communication topology graph is an undirected graph with get:
[0136]
[0137] W i The function is differentiated with respect to t to obtain
[0138]
[0139] Substituting formula (9) and (11) into (15), we obtain:
[0140]
[0141] in is non-increasing, and the lower bound is zero, so we get the existence limit W i (∞); let Integrating both sides of the equation gives:
[0142]
[0143] exists and is bounded, which proves is consistent and continuous;
[0144] Starting from any starting position, T(b ij (t)) and ui are all bounded, so is also bounded; because b ij =P ij / ||P ij ||, b ij and For i=1,2,…,n,j∈N i are all bounded, which means that b ij exist is uniformly continuous; since T(b ij (t)) is continuous, then it is are all consistent and continuous; therefore, when When T(b ij (t)) The uniform continuity property of can be known, When T(b ij (t)) is also uniformly continuous; therefore, according to Barbalat’s lemma, we can prove that And get:
[0145]
[0146] because so All robots converge to In addition, for each robot i, when According to formula (11), when When So Get θ id =0;
[0147] The Lyapunov function for further analysis of the speed angle system is selected as follows:
[0148]
[0149] Taking the derivative of the above formula with respect to time t, we can get
[0150]
[0151] When we consider formula (12), we get
[0152]
[0153] The steering angles of each robot converge to Right now and is asymptotically stable;
[0154] According to the definition of nonlinear cascade system
[0155]
[0156] In the formula, exist is locally Lipschitz continuous and f2(0)=0, if the subsystem is input-state stable (ISS) from input x2 to state x1, and the subsystem is globally asymptotically stable, then the cascade system (26) globally converges asymptotically to the equilibrium point (x1, x2) = 0;
[0157] Using the ISS theory in (26), we can obtain that the entire formation system is globally asymptotically stable, thus obtaining
[0158]
[0159] So we get
[0160]
[0161] Taking into account the control of the leaderless formation, graph G is an undirected graph, and the edges in graph G are marked in the order of 1 to m, where m is the number of edges; let all the edges in G be E1, E2, …E m , then the corresponding incidence matrix is obtained
[0162] H=H(G)=(h ij )∈R m×n (25)
[0163] The incidence matrix is a matrix composed of 0, 1, and -1;
[0164] Define the side vector and angle vector: e k =P ij =P i -P j , b k =e k / ||e k ||, So we get:
[0165]
[0166] Convert (28) into vector form:
[0167]
[0168] From equation (30) and equation (31), we get the following equation:
[0169]
[0170] Learn At the same time, the linear velocity and angular velocity of the car tend to be stable; according to ISS, the designed formation control system achieves global asymptotic stability, ensuring that all pairs of robots can accurately reach the desired relative angle, combined with the angle rigidity theory, thus achieving the desired formation, that is,
[0171] S0={g ij ,θ i |g ij =g ij *,θ i =θ i *,i=1,2,...,n,j∈N i} (29)
[0172] The above embodiments are only used to help understand the method and core idea of the present invention. It should be noted that, for those skilled in the art, several improvements and modifications can be made to the present invention without departing from the principles of the present invention, and these improvements and modifications also fall within the scope of protection of the claims of the present invention.
Claims
1. A non-holonomic vehicle platoon control method based on disturbance utilization strategy, characterized in that: The following steps are involved: Step 1) For the disturbance generated during the motion of the non-holonomic car, it is divided into the linear velocity disturbance caused by the slip phenomenon and the motor disturbance acting on the angular velocity, and a composite disturbance model is constructed; Step 2) Based on the angle rigidity graph theory and the composite interference model, a non-holonomic platooning system model with input channel interference is established; Step 3) Designing a disturbance observer for the disturbance in the model to effectively estimate the input disturbance in the multi-robot formation system in real time; Step 4) Based on the estimation of the interference in step 3), a disturbance utilization control strategy is designed, and a disturbance utilization judgment criterion is established to accelerate the formation of the formation; Step 5) Using the relative angle information between the robots obtained through communication, combined with the disturbance observer and the disturbance utilization control strategy, a globally stable angle rigid formation control method is designed so that the multi-robot system can achieve the desired formation and there is no collision between the communicating robots during the movement; Step 6) Control of the incomplete vehicle formation is achieved through the cascade system stability analysis method.
2. According to claim 1, a non-holonomic vehicle formation control method based on disturbance utilization strategy is characterized in that: Step 1) is specifically as follows: Establish a mathematical model of the non-holonomic car based on linear velocity and angular velocity interference: When the motion plane is relatively smooth or the wheel is squeezed and deformed, the wheel will slip, causing the linear velocity v to deviate, resulting in a linear velocity disturbance d ui , specifically where r i is the radius of the driving wheel, ζ ri ,ζ li is the disturbance angular velocity caused by the slip of the two driving wheels; At the same time, the incomplete car is driven by a motor, introducing the steering angular velocity w i Interference on: where b i is the viscous friction coefficient, ω τi is the motor rotor speed, J i is the moment of inertia, T ei is the unknown torque disturbance term, T li is the load torque.
3. The non-holonomic vehicle formation control method based on disturbance utilization strategy according to claim 2 is characterized in that: Step 2) is specifically as follows: The nonholonomic platooning system model with input channel disturbance is defined as follows: Where P i =[x i y i ] T represents the position of the robot car, i = 1, 2, ..., n; the car's steering angle is θ i ∈(-π,π] represents; the vector of the position and direction of robot i is recorded as and θ=[θ1,θ2,...,θ n ] T , and the input of the translational and rotational velocities of car i is represented as u i and w i , d ui is the linear velocity disturbance, d wi is the angular velocity disturbance.
4. The non-holonomic vehicle formation control method based on disturbance utilization strategy according to claim 3 is characterized in that: The disturbance observer in step 3) is used to estimate and compensate for the external disturbance of the non-holonomic car in real time. The disturbance observer is specifically: where z ui , z wi is the auxiliary variable of the disturbance observer, L ui , l wi is the gain vector of the designed disturbance observer l wi ≥1 / 4, P i =[x i y i ] T represents the position of the robot car, θ i Represents the steering angle of the car, R i =[cosθ i sinθ i ] T is the rotation matrix, represents the estimated value of the line speed input disturbance, Represents an estimate of the angular velocity input disturbance.
5. The non-holonomic vehicle formation control method based on disturbance utilization strategy according to claim 4 is characterized in that: In step 4): Constructing the potential energy function The form is as follows: definition: in, I d ∈R d×d is the identity matrix, the edge vector P ij =P i -P j and angle vector and The total potential energy function of robot i is: Potential energy function V ij With smoothness, when the relative angle between robot i and its neighbor j reaches the expected value When V ij is zero; when the relative angle between robot i and any of its neighbors tends to zero or tends to infinity, V ij tends to infinity; Design a conditional disturbance utilization judgment including expected angle error and external disturbance Where R i =[cosθ i sinθ i ] T is the rotation matrix, θ * represents the desired angle, ε i and represents a sufficiently small positive number, The hyperbolic tangent function is defined as follows: Formulas (9)-(10) are interference utilization control strategies. If and ρ i When >0, the disturbance is defined as an adverse disturbance, and the disturbance observer is used to compensate for this type of disturbance.
6. The non-holonomic vehicle formation control method based on disturbance utilization strategy according to claim 5 is characterized in that: The control law based on interference utilization in step 5) is: The communication topology always maintains angular rigidity during the movement; the following formation control law is designed: where k i ≥1, As shown in formula (9); Among them l i ≥1,ρ i As shown in formula (10).
7. A method for realizing formation control by utilizing the interference of incomplete vehicles according to claim 6, characterized in that: The specific conditions of step 6) are as follows: The stability of the nonholonomic system is studied by using the ISS lemma of the cascade system. The precise formation control is achieved by combining the control law and the potential energy function, namely: The Lyapunov function is selected as follows: The communication topology graph is an undirected graph with j∈N i ,get: W i The function is differentiated with respect to t to obtain Substituting formula (9) and (11) into (15), we obtain: in is non-increasing, and the lower bound is zero, so we get the existence limit W i (∞); let Integrating both sides of the equation, get: exists and is bounded, which proves is consistent and continuous; Starting from any starting position, T(b ij (t)) and u i are all bounded, so is also bounded; because b ij =P ij / ||P ij ||, b ij and For i=1,2,…,n,j∈N i are all bounded, which means that b ij exist is uniformly continuous; since T(b ij (t)) is continuous, then it is are all consistent and continuous; therefore, when When T(b ij (t)) The uniform continuity property of can be known, When T(b ij (t)) is also uniformly continuous; therefore, according to Barbalat’s lemma, we can prove that And get: because so All robots converge to In addition, for each robot i, when According to formula (11), when When So Get θ id =0; The Lyapunov function for further analysis of the speed angle system is selected as follows: Taking the derivative of the above formula with respect to time t, we can get When we consider formula (12), we get The steering angles of each robot converge to Right now and is asymptotically stable; According to the definition of nonlinear cascade system Where, f1(·): exist is locally Lipschitz continuous and f2(0)=0, if the subsystem is input-state stable (ISS) from input x2 to state x1, and the subsystem is globally asymptotically stable, then the cascade system (26) globally converges asymptotically to the equilibrium point (x1, x2) = 0; Using the ISS theory in (26), we can obtain that the entire formation system is globally asymptotically stable, thus obtaining So we get Taking into account the control of the leaderless formation, graph G is an undirected graph, and the edges in graph G are marked in the order of 1 to m, where m is the number of edges; let all the edges in G be E1, E2, …E m , then the corresponding incidence matrix is obtained H=H(G)=(h ij )∈R m×n (25) The incidence matrix is a matrix composed of 0, 1, and -1; Define the side vector and angle vector: e k =P ij =P i -P j , b k =e k / ||e k ||, So we get: Convert (28) into vector form: From equation (30) and equation (31), we get the following equation: Learn At the same time, the linear velocity and angular velocity of the car tend to be stable; according to ISS, the designed formation control system achieves global asymptotic stability, ensuring that all pairs of robots can accurately reach the desired relative angle, combined with the angle rigidity theory, thus achieving the desired formation, that is, S0={g ij ,θ i |g ij =g ij *,θ i =θ i *,i=1,2,...,n,j∈N i } (29)。