A method for pre-timed synchronous cooperative control of heterogeneous multi-robot systems

Abstract

The present invention relates to a method for time-scheduled synchronous collaborative control of a heterogeneous multi-robot system. First, a heterogeneous multi-robot system model and network communication topology are constructed, a time-scheduled synchronous collaborative control target is established, and a time-scheduled distributed observer is constructed for each follower robot to estimate the expected position vector estimate and expected velocity vector estimate for achieving the time-scheduled synchronous collaborative control target. Next, an interval observer is constructed for the velocity vector of the follower robot to obtain the algebraic relationship between the velocity vector and the unknown disturbance. An algebraic reconstruction mechanism for the unknown disturbance is constructed that is decoupled from the control input of the follower robot to obtain an estimate of the unknown disturbance. Finally, a time-scheduled synchronous control protocol for the multi-robot system is constructed by combining the expected position vector estimate, the expected velocity vector estimate, the unknown disturbance estimate, and the switching sliding mode variable. This technical solution improves the system's resistance to external disturbances.

Description

Technical Field

[0001] The present invention belongs to the technical field of heterogeneous multi-robot control, and in particular relates to a predetermined time synchronous collaborative control method for a heterogeneous multi-robot system. Background Art

[0002] With the widespread application of robotic systems in industries such as industry, services, and the military, scheduled collaborative control is gaining increasing attention as a key technology for achieving efficient multi-robot collaborative operations. This technology demonstrates superior collaborative control performance and metrics, and is of great significance for improving the efficiency and collaborative capabilities of robotic systems. However, existing technologies for collaborative control of multi-robot systems primarily focus on improving the convergence performance of the system. For example:

[0003] Literature [1]: Hierarchical Predefined-Time Control for Time-VaryingFormation Tracking of Multiple Heterogeneous Euler-Lagrange Agents.

[0004] Document [2]: Predefined-Time Bipartite Time-Varying Formation TrackingControl of Networked Autonomous Surface Vehicles via Hierarchical ControlApproach.

[0005] Reference [1] proposed a time-varying function-based scheduled time control scheme to solve the time-varying formation control problem of a multi-robot system. Reference [2] established a time-varying binary formation control protocol under the hierarchical control design framework, ensuring that the multi-robot system completes the desired binary formation within the scheduled time.

[0006] In reality, the performance of a control system depends not only on when the system state converges but also, to a significant extent, on when and how each element of the system state converges. In some operations, scheduled coordinated control is insufficient. For example, to achieve optimal defensive performance, a group of military robots must simultaneously and synchronously form a tactical formation. Otherwise, the first robot to arrive would be vulnerable and easily attacked. Therefore, research on scheduled synchronized coordinated control methods for heterogeneous multi-robot systems has important theoretical and practical value.

[0007] Traditional control methods for heterogeneous multi-robot systems with Euler-Lagrangian dynamics are difficult to effectively apply due to the differences in the dynamic models of different robot types. Therefore, the development of new control strategies is urgently needed. Furthermore, existing technologies often fail to fully consider the complex environments and dynamic changes faced by multi-robot systems in practical applications, and they often fail to adequately handle external disturbances in the system, resulting in unsatisfactory results in real-world scenarios. Summary of the Invention

[0008] The technical problem to be solved by the present invention is to make up for the deficiencies of the prior art and provide a method for predetermined time synchronization collaborative control of a heterogeneous multi-robot system.

[0009] To solve the above technical problems, the technical solution of the present invention is:

[0010] A method for pre-determined time synchronization collaborative control of a heterogeneous multi-robot system, comprising the following steps:

[0011] S1: Construct a heterogeneous multi-robot system model based on the Euler-Lagrange equation. The heterogeneous multi-robot system model includes the dynamic system model of N follower robots and the dynamic system model of 1 leader robot, where N>0;

[0012] S2: Based on the heterogeneous multi-robot system model constructed in step S1, a cooperative-adversarial network communication topology between the follower robots and a network communication topology between the follower robots and the leader robot are constructed, and a predetermined time synchronization collaborative control goal of the multi-robot system is established;

[0013] S3: Based on the predetermined time synchronization cooperative control target established in step S2, a predetermined time distributed observer is constructed for each follower robot, so that each follower robot estimates an expected position vector estimate and an expected velocity vector estimate that achieve the predetermined time synchronization cooperative control target;

[0014] S4: To achieve compensation control for unknown disturbances, first, an interval observer is constructed for the velocity vector of the follower robot. An algebraic relationship between the velocity vector of the follower robot and the unknown disturbance is established based on the properties of the interval observer. Then, by combining a predetermined time differentiator and the algebraic relationship, an unknown disturbance algebraic reconstruction mechanism is constructed that is decoupled from the control input of the follower robot. An estimated value of the unknown disturbance is obtained through the unknown disturbance algebraic reconstruction mechanism.

[0015] S5: Combining the expected position vector estimate and the expected velocity vector estimate in step S3, the unknown disturbance estimate in step S4, and the switching sliding mode variable based on the normed symbolic function, a multi-robot system predetermined time synchronization control protocol is constructed to achieve the predetermined time synchronization collaborative control goal of the heterogeneous multi-robot system in the presence of unknown disturbances.

[0016] Furthermore, in step S1, the dynamic system model of the follower robot is:

[0017]

[0018] Where i represents the i-th follower robot; q i 、 Represent the position vector, velocity vector and acceleration vector of the i-th follower robot joint space, respectively. p i is the dimension of the joint space vector, represents the inertia matrix, is the centripetal Coriolis matrix, is the gravity vector, represents the control input, represents the unknown external disturbance vector, x i and represent the position vector and velocity vector of the i-th follower robot’s task space, respectively. n is the dimension of the task space vector, represents the forward kinematics function, is a Jacobian matrix that satisfies and Represents p i ×p i Dimensional sum n×p i -dimensional Euclidean matrix space, and Represents p i dimensional and n-dimensional vector spaces;

[0019] The dynamic system model of the navigator robot is:

[0020]

[0021] in, represents the position vector of the leader robot, represents the velocity vector of the leader robot, Represents the acceleration vector of the leader robot.

[0022] Furthermore, in step S2, the cooperation-adversarial network communication topology between the follower robots is represented by a connectivity graph To indicate that, represents the set of follower robot nodes, represents the set of follower robot edges, represents the weight matrix of the follower robot, a ijis the connection weight between the ith follower robot and the jth follower robot, j = 1, ..., N, and j ≠ i; if the ith follower robot can obtain the information of the jth follower robot, and if there is a cooperative relationship between the two, then a ij =1, if the relationship between the two is antagonistic, then a ij = -1; if the i-th follower robot cannot obtain the information of the j-th follower robot, a ij =0; Represents the N×N dimensional Euclidean matrix space; defines the i-th follower's favorite node υ i The in-degree is Further definition and diagram The associated Laplace matrix is

[0023] The network communication topology between the follower robot and the leader robot is represented by a connectivity graph To indicate that, υ0 represents the node of the leader robot, represents the edge set consisting of follower robots and leader robots; define is the weight matrix of the navigator robot, b i is the connection weight between the leader robot and the ith follower robot; if the information of the leader robot is available to the ith follower robot, then b i =1; otherwise, b i =0;

[0024] For connected graphs Will Divide into a binary set Make and And satisfy when When a ij =1, when When a ij =0; if Define σ i =1, if Define σ i =-1; define the matrix σ = diag(σ1,…,σ N ), σ satisfies The function |·| represents a matrix whose elements are composed of the absolute values ​​of its elements; the leader-follower matrix is ​​defined as matrix Among them, the elements Represents a vector The i-th element, y iRepresents a vector The i-th element of N Represents an N-dimensional column vector whose elements are all 1.

[0025] Furthermore, in step S2, the predetermined time synchronization collaborative control target of the multi-robot system is:

[0026]

[0027] in, is the desired formation vector, t is the time variable, T f is the time constant for the actual convergence of the synchronized formation, satisfying T f ≤T p , T p is the predetermined synchronization convergence time set manually; in addition, for any <t2≤T f The two time constants t1 and t2 have:

[0028]

[0029] Among them, x ik and Represents the task space position vector x of the i-th follower robot i and vector The kth element in Represents the final value of the bipartite formation of the multi-robot system.

[0030] Furthermore, in step S3, the predetermined time distributed observer is:

[0031]

[0032] in, and They represent the expected position vector estimate and expected velocity vector estimate of the i-th follower robot respectively; is the distributed position observation error, is the distributed velocity observation error, and

[0033]

[0034] Represents the formation vector f i The first time derivative of ;

[0035] is the upper bound of the acceleration of the navigator robot, satisfying Represents the formation vector f i The second-order time derivative of ; the classical symbolic function sgn cThe definition of (·) is sgn c (ξ)=col(sgn(ξ1),…,sgn(ξ n ), improved symbolic function The definition of The definition of the sign function sgn(·) is μ1, μ2, η1, and η2 are positive constants satisfying 0 < μ1 < 1, 0 < μ2 < 1, η1 > 1, and η2 > 1, respectively. T1 represents the scheduled time for the estimated value of the desired position vector of the i-th follower robot to converge, and T2 represents the scheduled time for the estimated value of the desired velocity vector of the i-th follower robot to converge. γ1 and γ2 are two design parameters of the scheduled time distributed observer, as follows:

[0036]

[0037] in, r max =max{r1,…,r N}, Θ is a symmetric positive definite matrix, λ min (Θ) represents the minimum eigenvalue of the matrix Θ, and the function Γ(·) represents the gamma function, which is in the form of

[0038]

[0039] in,

[0040] Furthermore, step S4 includes the following sub-steps:

[0041] S4.1: First, according to the dynamic system model of the i-th follower robot, the velocity vector The time derivative of The first form of expression:

[0042]

[0043] For velocity vector Construct an interval observer:

[0044]

[0045] in, and v i Represents the upper and lower bound states of the interval observer, satisfying d i and represents the unknown disturbance di The upper and lower bounds of Gain matrix F i Satisfies both the Metzner matrix and the Hurwitz matrix;

[0046] The upper bound state of the interval observer Nether state v i and velocity vector Respectively expressed as v i =col( v i1 ,…, v in )and Then there must be a time-varying variable ω ik ∈[0,1], satisfying Thus we get:

[0047]

[0048] Among them, ω i =col(ω i1 ,…,ω in ), time-varying variable ω ik It can be expressed as:

[0049]

[0050] Then, The second form of expression:

[0051]

[0052] in, Represents ω i The time derivative of is an unknown quantity;

[0053]

[0054] Combine Two expressions of velocity vector The algebraic relationship between and the unknown disturbance:

[0055]

[0056] S4.2: To reconstruct the unknown perturbation d i , using a predetermined time differentiator to To estimate, the predetermined time differentiator is:

[0057]

[0058] Among them,ik,0 and ζ ik,1 are the two states of the predetermined time differentiator, T c is the differentiator's predetermined time convergence parameter, and It's about T c The two correction functions are defined as:

[0059]

[0060] Among them, α is a positive scalar, and the function L1(t) is selected to satisfy The function L2(t) is defined as L2(t)=L1(t)(α(T c -t)) 2 , The selection satisfies The selection satisfies function and is defined as follows:

[0061]

[0062] In T c The first state of the predetermined time differentiator ζ ik,0 yes An accurate estimate of

[0063] S4.3: Based on the velocity vector The algebraic relationship between the unknown disturbance and An estimated value of the unknown disturbance is obtained by establishing a predetermined time disturbance reconstruction mechanism to asymptotically reconstruct the unknown disturbance.

[0064] The scheduled time disturbance reconstruction mechanism is:

[0065]

[0066] in represents the unknown disturbance d i The estimated value of i,1 express The estimated time of booking, The scheduled time disturbance reconstruction mechanism satisfies the c Internal reconstruction.

[0067] Furthermore, step S5 includes the following sub-steps:

[0068] S5.1: For the i-th follower robot, define its auxiliary velocity vector and the auxiliary acceleration vector for:

[0069]

[0070] Among them, T3 is and The time constant of the predetermined time stabilization, It is p i ×p i dimensional identity matrix, γ3 is and The design parameters in

[0071]

[0072] in, Represents the gradient of the performance indicator; the matrix The expression is:

[0073]

[0074] in, and Represents the matrix J i The standard inverse matrix and generalized inverse matrix of ;

[0075] Switching law θ i and its time derivative for:

[0076]

[0077] Among them, e i is x i The estimation error of ι>0;

[0078] Normized symbolic function The definition of

[0079] Represents the trigger sliding mode variable, and its expression is:

[0080]

[0081] Variables h1, h2, ρ i1 and ρ i2 They are defined as:

[0082]

[0083] Where η3 is a scalar greater than 1, in and All are odd numbers;

[0084] S5.2: In order to achieve time synchronization convergence, for the i-th follower robot, construct a switching sliding mode variable:

[0085]

[0086] S5.3: Based on the expected position vector estimate, the expected velocity vector estimate, and the unknown disturbance estimate, combined with the switching sliding mode variables, a predetermined time synchronization control protocol is constructed:

[0087]

[0088] Among them, τ i1 represents the dynamic compensation term, τ i2 represents the synchronization control term, T4 is the predetermined time constant, 0<μ4<1, η4>1; γ4 is the predetermined time synchronization control protocol design parameter, expressed as:

[0089]

[0090] in,

[0091] The beneficial effects that can be achieved by the present invention are:

[0092] (1) The present invention realizes the accurate estimation of the desired position vector and velocity vector in the cooperative control of heterogeneous multi-robot systems by designing a predetermined time distributed observer; by establishing an interval observer, the algebraic relationship between the unknown disturbance and the system state is obtained, and based on this relationship, a predetermined time disturbance reconstruction mechanism is established. The disturbance reconstruction mechanism can provide an accurate estimation of the unknown disturbance. At the same time, the reconstruction mechanism realizes decoupling from the control input of the robot system, so that the disturbance estimation value can be introduced when designing the control protocol, thereby realizing compensatory control of the unknown disturbance, overcoming the limitations of the traditional predetermined time cooperative control method in the face of uncertainty, and thus improving the system's resistance to external disturbances.

[0093] (2) The proposed time synchronization control scheme utilizes the estimated values ​​provided by the time-scheduled distributed observer and the time-scheduled disturbance reconstruction mechanism, and combines the time synchronization control protocol established by the switching sliding mode technology based on the normalized symbolic function. It not only meets the performance requirements of the traditional time-scheduled cooperative control in references [1] and [2], but also achieves the control goal of all state components of all follower robot systems reaching the equilibrium point simultaneously and synchronously. This preset time synchronization control method not only improves the time efficiency of task execution, but also enhances the adaptability of the multi-robot system in a dynamically changing environment, which helps to improve the reliability and stability of the overall task completion. BRIEF DESCRIPTION OF THE DRAWINGS

[0094] Figure 1 This is a flow chart of the predetermined time synchronization collaborative control method for a heterogeneous multi-robot system of the present invention.

[0095] Figure 2 Schematic diagram of the predetermined time synchronization collaborative control architecture of the heterogeneous multi-robot system of the present invention.

[0096] Figure 3 4 is a network communication topology diagram of a heterogeneous multi-robot system in an embodiment of the present invention.

[0097] Figure 4 Schematic diagram of the collaborative formation trajectory and snapshots of a heterogeneous multi-robot system in an embodiment of the present invention.

[0098] Figure 5 Schematic diagram of the error in the collaborative formation of a heterogeneous multi-robot system in an embodiment of the present invention.

[0099] Figure 6 Schematic diagram of the collaborative formation error of a heterogeneous multi-robot system under a comparison method in an embodiment of the present invention.

[0100] Figure 7 Schematic diagram of position observation error, velocity observation error and reconstruction error in an embodiment of the present invention. DETAILED DESCRIPTION

[0101] The present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0102] A method for pre-determined time synchronization collaborative control of a heterogeneous multi-robot system comprises the following steps:

[0103] Step 1: In order to realize the scheduled time synchronous cooperative control of heterogeneous multi-robots, a dynamic system model of N (N>0) follower robots and 1 leader robot in the heterogeneous multi-robot system is given according to the Euler-Lagrange equation.

[0104] The following sub-steps are included:

[0105] Step 1.1: Collect robot mechanical structure data. For a heterogeneous multi-robot system with N followers, the Euler-Lagrangian dynamic system model of the i-th (i=1,...,N) follower robot is as follows:

[0106]

[0107] in, represents the position, velocity and acceleration vector of the i-th follower robot joint space, p i is the dimension of the joint space vector, represents the inertia matrix, is the centripetal Coriolis matrix, is the gravity vector, represents the control input, represents the unknown external disturbance vector, represents the position vector and velocity vector of the task space of the i-th follower robot, n is the dimension of the task space vector, represents the forward kinematics function, is a Jacobian matrix that satisfies and Represents p i ×p i Dimensional sum n×p i -dimensional Euclidean matrix space, and Represents p i dimensional and n-dimensional vector spaces.

[0108] Step 1.2: The dynamic system model of the navigator robot is as follows:

[0109]

[0110] in, represents the position vector of the leader robot, represents the velocity vector of the leader robot, Represents the acceleration vector of the leader robot.

[0111] Step 2: Based on the heterogeneous multi-robot system model established in step 1, construct the cooperative-adversarial network communication topology between follower robots and the network communication topology between follower robots and leader robots, and establish the multi-robot system's predetermined time synchronization collaborative control goal.

[0112] The following sub-steps are included:

[0113] Step 2.1: Consider the heterogeneous multi-robot system with N followers described in step 1.1. The cooperative-adversarial network communication topology between the follower robots is represented by a connectivity graph To indicate that represents the set of follower robot nodes, represents the set of follower robot edges, Represents the weight matrix of the follower robot. If the i-th follower robot can obtain the information of the j-th (j=1,...,N and j≠i) follower robot, and if there is a cooperative relationship between the two, then the connection weight a ij =1, if there is an adversarial relationship between the two, then the connection weight a ij = -1, if the i-th follower robot cannot obtain the information of the j-th follower robot, the connection weight a ij=0; Represents the N×N dimensional Euclidean matrix space; defines the i-th follower's favorite node υ i The in-degree is Further definition and diagram The associated Laplace matrix is The function diag(·) is defined as It is an element is a diagonal matrix consisting of diagonal elements.

[0114] Construct a network communication topology consisting of the N follower robots described in step 1.1 and the 1 navigator robot described in step 1.2. Assuming that at least one follower robot can obtain the information of the navigator robot, the network communication topology between the N follower robots and the 1 navigator robot is shown in Figure 1. To indicate that, and are the node set and edge set consisting of N follower robots and 1 leader robot, respectively, υ0 represents the node of the leader robot; define is the weight matrix of the leader robot, where if the information of the leader robot is available to the i-th follower robot, the connection weight b of the leader robot is i =1, otherwise the connection weight b of the leader robot i =0;

[0115] For the graph Follower robot node collection Can be divided into a binary set Make and And satisfy when When a ij =1, when When a ij =0; if Define σ i =1, if Define σ i =-1; at this time, define σ=diag(σ1,…,σ N ), the diagonal matrix σ satisfies The function |·| represents a matrix whose elements are composed of the absolute values ​​of its elements; the leader-follower matrix is ​​defined as matrix Among them, the elements Represents a vector The i-th element, y i Represents a vector The i-th element of NRepresents an N-dimensional column vector with all elements equal to 1; define a symmetric positive definite matrix Θ with the form

[0116] Step 2.2: Based on the heterogeneous multi-robot dynamic system model obtained in Steps 1.1 and 1.2, and considering the heterogeneous multi-robot system cooperative-adversarial network communication topology constructed in Step 2.1, establish the multi-robot system scheduled time synchronization collaborative control objectives as follows:

[0117]

[0118] in, is the desired formation vector, t is the time variable, T f is the time constant for the actual convergence of the synchronized formation, satisfying T f ≤T p , T p is the predetermined synchronization convergence time set manually; in addition, for any <t2≤T f The two time constants t1 and t2 have:

[0119]

[0120] Among them, x ik and Represents the task space position vector x of the i-th follower robot i and vector The kth element in represents the final value of the bipartite formation of the multi-robot system. This indicates that the follower robots will be divided into two groups based on cooperative and antagonistic relationships. The first group of follower robots will form a geometric shape around the leader robot, while the second group of follower robots will move in the opposite direction of the first group to form a geometric shape. At the same time, all follower robots will converge simultaneously and synchronously to their equilibrium point for achieving the bipartite formation. That is, no follower robot will reach and remain at its equilibrium point before achieving the final formation.

[0121] Step 3: In order to enable the follower robot to obtain its desired position vector and velocity vector, a distributed position observation error and a distributed velocity observation error are established according to the predetermined time-synchronized collaborative control target, and then a predetermined time distributed observer is designed for each follower robot, so that each follower robot obtains the desired position vector and velocity vector to achieve the time-synchronized collaborative control target.

[0122] The steps include:

[0123] Step 3.1: Based on the predetermined time synchronization cooperative control goal of the heterogeneous multi-robot system established in step 2, define the distributed position observation error for the i-th follower robot: and distributed velocity observation errors

[0124]

[0125] in, Represents the formation vector f i The first time derivative of and Denote the expected position vector estimate and the expected velocity vector estimate of the ith follower robot, respectively, which are provided by the following scheduled time distributed observer:

[0126]

[0127] in, is the upper bound of the acceleration of the navigator robot, satisfying Represents the formation vector f i The second-order time derivative of; the classical symbolic function sgn c The definition of (·) is sgn c (ξ)=col(sgn(ξ1),…,sgn(ξ n ), improved symbolic function The definition of The definition of the sign function sgn(·) is The function col(·) represents the column vector formed by the vertical concatenation of its elements; μ1, μ2, η1, and η2 are positive constants satisfying 0<μ1<1, 0<μ2<1, η1>1, and η2>1, respectively. T1 represents the scheduled time for the estimated value of the expected position vector of the i-th follower robot to converge, T2 represents the scheduled time for the estimated value of the expected velocity vector of the i-th follower robot to converge, and γ1 and γ2 are two design parameters of the scheduled time distributed observer.

[0128] Step 3.2: According to the pre-determined time stability theory, the first design parameter γ1 of the pre-determined time distributed observer has the following form:

[0129]

[0130] in, r max =max{r1,…,r N},λ min(Θ) represents the minimum eigenvalue of the matrix Θ, which is a symmetric positive definite matrix calculated according to step 2.1; the function Γ(·) represents the gamma function, which is

[0131] According to the time-determining stability theory, the second design parameter γ2 of the time-determining distributed observer has the following form:

[0132]

[0133] in,

[0134] The verification process of the scheduled time distributed observer achieving scheduled time stability is as follows:

[0135] Defining position observation error and velocity observation error for:

[0136]

[0137] According to the scheduled time distributed observer in step 3.1, we can further obtain:

[0138]

[0139] Furthermore, the following compact form can be derived:

[0140]

[0141] in, I n Represents the n×n-dimensional identity matrix.

[0142] The first step is to prove that the velocity observation error achieves the predetermined time stability.

[0143] First, define an auxiliary velocity observation error where ε=col(ε1,…,ε N ). Further, we get:

[0144]

[0145] Next, define a Lyapunov function as:

[0146]

[0147] Its time derivative satisfies the following inequality:

[0148]

[0149] Furthermore, since 0<μ2<1<η2, the following inequality holds:

[0150]

[0151] According to the above inequality, we can get the function V ε The time derivative of satisfies the inequality At this time, the auxiliary speed observation error ε will converge to zero within the predetermined time T1, indicating that when the time variable t satisfies t≥T1, the speed observation error Converges to zero.

[0152] The second step is to prove that the position observation error achieves the predetermined time stability.

[0153] First, when the time variable t satisfies t≥T1, δ x The dynamic equation has the following form:

[0154]

[0155] Define an auxiliary position observation error where z=col(z1,…,z N ), whose time derivative is as follows:

[0156]

[0157] Next, define a Lyapunov function as:

[0158]

[0159] Furthermore, we can get the function V z The time derivative of satisfies the inequality At this time, the auxiliary position observation error z will be e =T1+T2 converges to zero, indicating that when the time variable t satisfies t≥T e When the position observation error of the distributed observer at the predetermined time is Converges to zero.

[0160] Therefore, the scheduled time distributed observer can accurately provide the estimated values ​​of the position vector and velocity vector expected by the follower robot to achieve cooperative formation within the scheduled time.

[0161] Step 4: To achieve compensation control for unknown disturbances, first, an interval observer is designed for the velocity vector of the follower robot. Based on the properties of the interval observer, an algebraic relationship between the velocity vector of the follower robot and the unknown disturbance is established. Then, using an existing time-determined differentiator and the established algebraic relationship between the velocity vector and the unknown disturbance, an algebraic reconstruction mechanism for the unknown disturbance is constructed that is decoupled from the follower robot's control input.

[0162] The following sub-steps are included:

[0163] Step 4.1: For the velocity vector of the i-th follower robot, establish an interval observer to obtain the algebraic relationship between the velocity vector and the unknown disturbance. The design process is as follows:

[0164] First, according to the system model of the follower robot in step 1.1, the velocity vector The time derivative of :

[0165]

[0166] For velocity vector Construct an interval observer:

[0167]

[0168] in, and v i Represents the upper and lower bound states of the interval observer, satisfying d i and represents the unknown disturbance d i The upper and lower bounds of Gain matrix F i It is both a Metzner matrix and a Hurwitz matrix, that is, all off-diagonal elements of the matrix are non-negative, and all eigenvalues ​​of the matrix have negative real parts.

[0169] The upper bound state of the interval observer Nether state v i and velocity vector Respectively expressed as v i =col( v i1 ,…, v in )and Then there must be a time-varying variable ω ik ∈[0,1], satisfying Thus we get

[0170]

[0171] Among them, ω i =col(ω i1 ,…,ω in ), time-varying variable ω ik The specific form can be expressed as:

[0172]

[0173] Furthermore, the velocity vector The time derivative of can also be expressed as:

[0174]

[0175] in Represents ω i The time derivative of , which is unknown,

[0176] Combined velocity vector The time derivative of The two expressions of velocity vector can be obtained The algebraic relationship between and the unknown disturbance is as follows:

[0177]

[0178] Step 4.2: To reconstruct the unknown perturbation d i , using a predetermined time differentiator to provide The estimated time differentiator is as follows:

[0179]

[0180] Among them, ik,0 and ζ ik,1 are the two states of the predetermined time differentiator, T c is the differentiator's predetermined time convergence parameter, and It is about the differentiator's predetermined time convergence parameter T c The two correction functions are defined as follows:

[0181]

[0182] Among them, α is a positive scalar, and the function L1(t) is selected to satisfy The function L2(t) is defined as L2(t)=L1(t)(α(T c -t)) 2 , The selection satisfies The selection satisfies function and is defined as follows:

[0183]

[0184] Among them, the function and function According to the improved classical symbol function in step 3.1 and the classic symbolic function sgn c The definition of (·) is obtained. At the predetermined time T c The first state of the predetermined time differentiator ζ ik,0 will be accurate estimate of .

[0185] Step 4.3: Based on the algebraic expression for the unknown perturbation obtained in step 4.1 and the one provided in step 4.2 An estimated value of is used to establish a predetermined time disturbance reconstruction mechanism to asymptotically reconstruct the unknown disturbance, which is in the following form:

[0186]

[0187] in represents the unknown disturbance d i The estimated value of i,1 express The estimated time of the reservation is The predetermined time disturbance reconstruction mechanism satisfies the predetermined time T c Internal reconstruction is achieved, that is, when the time variable t satisfies t≥T c When the perturbation reconstruction error Converges to zero.

[0188] Step 5: Based on the expected position vector estimate and expected velocity vector estimate provided in step 3, the unknown disturbance estimate provided in step 4, and the switching sliding mode variable based on the normed symbolic function, a multi-robot system scheduled time synchronization control protocol is designed to achieve the scheduled time synchronization collaborative control goal of the heterogeneous multi-robot system in the presence of unknown disturbances. The architecture is as follows: Figure 2 shown.

[0189] The steps include:

[0190] Step 5.1: For the i-th follower robot, define its auxiliary velocity vector and the auxiliary acceleration vector for:

[0191]

[0192] Where T3 is the auxiliary velocity vector and the auxiliary acceleration vector The time constant of the predetermined time stabilization, It is p i ×p i dimensional identity matrix, γ3 is the auxiliary velocity vector and the auxiliary acceleration vector The design parameters in are expressed as

[0193]

[0194] in, Represents the gradient of the performance indicator; the matrix The expression is:

[0195]

[0196] in, and Represents the matrix J i The standard inverse matrix and generalized inverse matrix of ;

[0197] Design switching law θ i and its time derivative Has the following form:

[0198]

[0199] Among them, e i Represents the position vector x i The estimation error is defined as ι>0 is a very small constant; the symbolic function of the norm The definition of Represents the trigger sliding mode variable, and its expression is:

[0200]

[0201] In addition, the variables h1, h2, ρ i1 and ρ i2 They are defined as:

[0202]

[0203] Where η3 is a scalar greater than 1, in and are two odd numbers;

[0204] Step 5.2: To achieve time synchronization convergence, construct the following switching sliding mode variables for the i-th follower robot:

[0205]

[0206] Step 5.3: Based on the desired position vector and desired velocity vector provided in Steps 3 and 4 and the estimated value of the unknown disturbance, combined with the switching sliding mode variables based on the normalized sign function established in Steps 5.1 and 5.2, the following predetermined time synchronization control protocol is constructed:

[0207]

[0208] Among them, τ i1 represents the dynamic compensation term, τ i2 represents the synchronization control term, T4 is the predetermined time constant, μ4 and η4 are positive constants satisfying 0 < μ4 < 1 and η4 > 1 respectively; γ4 is the design parameter of the predetermined time synchronization control protocol, expressed as:

[0209]

[0210] in,

[0211] Under the action of the scheduled time synchronization cooperative control protocol, the verification process of the heterogeneous multi-robot system to achieve scheduled time synchronization cooperative formation control is as follows:

[0212] The first step is to prove that the switching sliding mode variable s i At the scheduled time T c +T4 converges asymptotically to zero.

[0213] First, switch the sliding mode variable s i The time derivative of has the following form:

[0214]

[0215] Secondly, define a Lyapunov function as Its time derivative has the following form:

[0216]

[0217] When the time variable t satisfies t≥T c When the perturbation reconstruction error Function V s The time derivative of can be further described as:

[0218]

[0219] Due to 0 <k 41 <1 <k42 , the following inequality holds:

[0220]

[0221] According to the above inequality, we can get the function V s The time derivative of satisfies the inequality At this time, switch the sliding mode variable s i The predetermined time stability is achieved, which means that when the time variable t satisfies t≥T c At +T4, switch the sliding mode variable s i converges asymptotically to zero.

[0222] The second step is to prove the bisection formation error of each follower robot All components are at the scheduled time T e +T c +T3+T4 converges to zero synchronously.

[0223] First, switch the sliding surface s i =0, we can get:

[0224]

[0225] When the time variable t satisfies t≥T e +T c +T4, binary formation error The dynamic equations have the following form:

[0226]

[0227] Secondly, define a Lyapunov function as Its time derivative has the form

[0228]

[0229] We can further obtain the function The time derivative of satisfies the inequality At this time, the error of the two-division formation has achieved the predetermined time stability, which means that when the time variable t satisfies t≥T e +T c +T3+T4, the binary formation error variable of each follower robot converges to zero; in addition, the bisection formation error Also satisfies the equation Explain the error of the two-division formation The ratio between any two elements in is constant, that is, the bisection formation error of each follower robot All components of converge to zero simultaneously and synchronously.

[0230] In the third step, we use the proof by contradiction to prove that all follower robots reach their equilibrium points synchronously.

[0231] Assume that the mth follower robot first moves at t = t m >0 reaches the equilibrium point and remains at the equilibrium point, while the hth follower robot reaches the equilibrium point at t m The equilibrium point is not reached at this moment; since the cooperation-adversarial communication network topology between the follower robots is connected, this indicates that there is a path between the hth follower robot and the mth follower robot; without loss of generality, assume that there are p follower robots (labeled as p1, ..., p p ); then, the position vector of the p1th follower robot directly connected to the hth follower robot is It must not be at its equilibrium point, because: 1) If t m At the moment p1th follower robot is at its equilibrium point, then That is t m The hth follower robot is also at its equilibrium point at time t m At time h, the follower robot is not at its equilibrium point. 2) If t m At the moment p1th follower robot is at its false equilibrium point, then:

[0232]

[0233] Before the final formation is achieved, each follower robot will not stay on its false equilibrium point, that is, its false equilibrium points are sparse, which further indicates that at t m At time p1, the follower robot has not reached its true equilibrium point.

[0234] Similarly, the position vectors of other follower robots on this path can be inferred are not at their equilibrium points; for the mth follower robot, we can also get and exists at most at an isolated point in time; therefore, the mth follower robot exists at t m Position vector at time is not its true equilibrium point, because the follower robot p connected to it p In t m is not at its true equilibrium point at the moment; this is obviously different from assuming that the mth follower robot is at t m It is contradictory to always reach the equilibrium point first.

[0235] Therefore, under the action of the scheduled time synchronous cooperative control protocol, the heterogeneous multi-robot system can achieve the cooperative formation control goal simultaneously and synchronously within the scheduled time.

[0236] Simulation Verification

[0237] A heterogeneous multi-robot system consisting of a leader robot (marked as 0) and eight follower robots (marked as 1,...,8) with scheduled time synchronization formation is simulated to verify the feasibility and effectiveness of the proposed control scheme. The network communication topology of the heterogeneous multi-robot system is shown in Figure 2. Figure 3 As shown in the figure, the third and seventh follower robots are 3-DOF manipulators, and the remaining follower robots are 2-DOF manipulators. The dynamic model of the 2-DOF manipulator is described as follows:

[0238]

[0239] and Represents the position vector of the joint and task space, the inertia matrix Centripetal Coriolis Matrix Gravity vector and the Jacobian matrix The expression and parameters are consistent with those in the literature [1], and the unknown disturbance is selected as d i =col(sin(t+0.2i),cos(t+0.2i)), the upper and lower bounds of the disturbance are selected as d i =col(-2,-2), the gain matrix of the interval observer is selected as F i =diag(-0.5,-1).

[0240] The dynamic model of the 3-DOF manipulator is described as follows:

[0241]

[0242] and Represents the position vector of the joint and task space, the inertia matrix Centripetal Coriolis Matrix Gravity vector and the Jacobian matrix The expression and parameters are also consistent with those in the literature [1], and the unknown disturbance is selected as d i =col(sin(t+0.2i),cos(t+0.2i),0.5(sin(t+0.2i)+cos(t+0.2i))), the upper and lower bounds of the disturbance are selected as d i =col(-2,-2,-2), the gain matrix of the interval observer is selected as F i=diag(-0.5,-1,-1.5).

[0243] The formation signal expected by the follower robot is selected as f i =col(3cos(2t+(2i-1)π / 4),3sin(2t+(2i-1)π / 4)), the acceleration vector of the navigator robot is set to

[0244]

[0245] Other design parameters are selected as μ1=μ2=μ3=μ4=0.6,η1=η2=η3=η4=1.2,T1=T2=T3=T4=1,T c =0.5,ι=0.001, After calculation, we can get γ1=γ2=24.6633, γ3=γ4=21.4702. Figure 3 It can be obtained that σ1=σ2=σ3=σ4=1, σ5=σ6=σ7=σ8=-1.

[0246] The simulation results are as follows Figure 4-7 shown. Figure 4 The motion trajectories of all robots are shown, including snapshots of the collaborative formation positions over 5s and 10s. This is to further demonstrate the superiority of the proposed scheduled time synchronization control algorithm. Figure 5 and Figure 6 The error trajectories of the cooperative formation under the control method of this embodiment and the traditional predetermined time control method are depicted respectively, where the two control methods share the same control parameters to provide a fair comparison. Figure 5 and Figure 6 It can be observed that, unlike the traditional scheduled time control method, the scheduled time synchronization control method designed in this embodiment can not only achieve scheduled time stability, but also ensure that the followers complete the two-part formation synchronously. Figure 7 The figure depicts the position observation error, velocity observation error and reconstruction error of the distributed observer. It can be seen that the position observation error, velocity observation error and reconstruction error can all converge to zero asymptotically within the predetermined time.

[0247] The above is only one embodiment of the present invention. The protection scope of the present invention is not limited to the above embodiment. It should be pointed out that for ordinary technicians in this technical field, several improvements and modifications made without departing from the concept of the present invention are all within the protection scope of the present invention.

Claims

1. A method for pre-determined time synchronization and collaborative control of a heterogeneous multi-robot system, characterized by: The following steps are involved: S1: Construct a heterogeneous multi-robot system model based on the Euler-Lagrange equation. The heterogeneous multi-robot system model includes the dynamic system model of N follower robots and the dynamic system model of 1 leader robot, where N>0; S2: Based on the heterogeneous multi-robot system model constructed in step S1, a cooperative-adversarial network communication topology between the follower robots and a network communication topology between the follower robots and the leader robot are constructed, and a predetermined time synchronization collaborative control goal of the multi-robot system is established; S3: Based on the predetermined time synchronization cooperative control target established in step S2, a predetermined time distributed observer is constructed for each follower robot, so that each follower robot estimates an expected position vector estimate and an expected velocity vector estimate that achieve the predetermined time synchronization cooperative control target; S4: To achieve compensation control for unknown disturbances, first, an interval observer is constructed for the velocity vector of the follower robot. An algebraic relationship between the velocity vector of the follower robot and the unknown disturbance is established based on the properties of the interval observer. Then, by combining a predetermined time differentiator and the algebraic relationship, an unknown disturbance algebraic reconstruction mechanism is constructed that is decoupled from the control input of the follower robot. An estimated value of the unknown disturbance is obtained through the unknown disturbance algebraic reconstruction mechanism. S5: Combining the expected position vector estimate and the expected velocity vector estimate in step S3, the unknown disturbance estimate in step S4, and the switching sliding mode variable based on the normed symbolic function, a multi-robot system predetermined time synchronization control protocol is constructed to achieve the predetermined time synchronization collaborative control goal of the heterogeneous multi-robot system in the presence of unknown disturbances.

2. The method for predetermined time synchronization and cooperative control of a heterogeneous multi-robot system according to claim 1 is characterized in that: In S1, the dynamic system model of the follower robot is: Where i represents the i-th follower robot; q i 、 They represent the position vector, velocity vector and acceleration vector of the i-th follower robot joint space, q i , p i is the dimension of the joint space vector, represents the inertia matrix, is the centripetal Coriolis matrix, is the gravity vector, represents the control input, represents the unknown disturbance, x i and represent the position vector and velocity vector of the i-th follower robot’s task space, respectively. n is the dimension of the task space vector, represents the forward kinematics function, is a Jacobian matrix that satisfies and Represents p i ×p i Dimensional sum n×p i -dimensional Euclidean matrix space, and Represents p i dimensional and n-dimensional vector spaces; The dynamic system model of the navigator robot is: in, represents the position vector of the leader robot, represents the velocity vector of the leader robot, Represents the acceleration vector of the leader robot.

3. The method for predetermined time synchronization and coordinated control of a heterogeneous multi-robot system according to claim 2, wherein: In step S2, the cooperation-adversarial network communication topology between the follower robots is connected using a graph To indicate that, represents the set of follower robot nodes, represents the set of follower robot edges, represents the weight matrix of the follower robot, a ij is the connection weight between the ith follower robot and the jth follower robot, j = 1, ..., N, and j ≠ i; if the ith follower robot can obtain the information of the jth follower robot, and if there is a cooperative relationship between the two, then a ij =1, if the relationship between the two is antagonistic, then a ij = -1; if the i-th follower robot cannot obtain the information of the j-th follower robot, a ij =0; Represents the N×N dimensional Euclidean matrix space; defines the i-th follower robot node υ i The in-degree is Further definition and connected graph The associated Laplace matrix is The network communication topology between the follower robot and the leader robot is represented by a connectivity graph To indicate that, υ0 represents the node of the leader robot, represents the edge set consisting of follower robots and leader robots; define is the weight matrix of the navigator robot, b i is the connection weight between the leader robot and the ith follower robot; if the information of the leader robot is available to the ith follower robot, then b i =1; otherwise, b i =0; For connected graphs Will Divide into a binary set Make and And satisfy when When a ij =1, when When a ij =0; if Define σ i =1, if Define σ i =-1; define the matrix σ = diag(σ1,...,σ N ), σ satisfies The function |·| represents a matrix whose elements are composed of the absolute values ​​of its elements; the leader-follower matrix is ​​defined as matrix Among them, the elements Represents a vector The i-th element, y i Represents a vector The i-th element of N Represents an N-dimensional column vector whose elements are all 1.

4. The method for predetermined time synchronization and coordinated control of a heterogeneous multi-robot system according to claim 3, wherein: In step S2, the predetermined time synchronization collaborative control target of the multi-robot system is: in, is the desired formation vector, t is the time variable, T f is the time constant for the actual convergence of the synchronized formation, satisfying T f ≤T p , T p is the predetermined synchronization convergence time set manually; in addition, for any condition satisfying 0≤t1<t2≤T f The two time constants t1 and t2 have: Among them, x ik and Represents the task space position vector x of the i-th follower robot i and vector The kth element in , k=1,...,n, Represents the final value of the bipartite formation of the multi-robot system.

5. The method for predetermined time synchronization and coordinated control of a heterogeneous multi-robot system according to claim 4, characterized in that: In step S3, the predetermined time distributed observer is: in, and They represent the expected position vector estimate and expected velocity vector estimate of the i-th follower robot respectively; is the distributed position observation error, is the distributed velocity observation error, and Represents the formation vector f i The first time derivative of ; is the upper bound of the acceleration of the navigator robot, satisfying Represents the formation vector f i The second-order time derivative of ; the classical symbolic function sgn c The definition of (·) is sgn c (ξ)=col(sgn(ξ1),...,sgn(ξ n ), improved symbolic function The definition of The definition of the sign function sgn(·) is col(·) represents the column vector formed by the vertical concatenation of its elements; μ1, μ2, η1, and η2 are positive constants satisfying 0<μ1<1, 0<μ2<1, η1>1, and η2>1, respectively; T1 represents the scheduled time for the estimated value of the expected position vector of the i-th follower robot to converge, and T2 represents the scheduled time for the estimated value of the expected velocity vector of the i-th follower robot to converge; γ1 and γ2 are two design parameters of the scheduled time distributed observer, as follows: in, r max =max{r1,...,r N }, Θ is a symmetric positive definite matrix, λ min (Θ) represents the minimum eigenvalue of the matrix Θ, and the function Γ(·) represents the gamma function, which is in the form of in, 6. The method for predetermined time synchronization and coordinated control of a heterogeneous multi-robot system according to claim 5, characterized in that: Step S4 includes the following sub-steps: S4.1: First, according to the dynamic system model of the i-th follower robot, the velocity vector The time derivative of The first form of expression: For velocity vector Construct an interval observer: in, and Represents the upper and lower bound states of the interval observer, satisfying and represents the unknown disturbance d i The upper and lower bounds of Gain matrix F i Satisfies both the Metzner matrix and the Hurwitz matrix; The upper bound state of the interval observer Nether state and velocity vector Respectively expressed as and Then there must be a time-varying variable ω ik ∈[0,1], satisfying Thus we get: Among them, ω i =col(ω i1 ,...,ω in ), time-varying variable ω ik It can be expressed as: Then, The second form of expression: in, Represents ω i The time derivative of is an unknown quantity; Combine Two expressions of velocity vector The algebraic relationship between and the unknown disturbance: S4.2: To reconstruct the unknown perturbation d i , using a predetermined time differentiator to To make an estimate, the predetermined time differentiator is: Among them, ik,0 and ζ ik,1 are the two states of the predetermined time differentiator, T c is the differentiator's predetermined time convergence parameter, and It's about T c The two correction functions are defined as: Among them, α is a positive scalar, and the function L1(t) is selected to satisfy The function L2(t) is defined as L2(t)=L1(t)(α(T c -t)) 2 , The selection satisfies The selection satisfies for function and The definition is as follows: In T c The first state of the predetermined time differentiator ζ ik,0 yes An accurate estimate of S4.3: Based on the velocity vector The algebraic relationship between the unknown disturbance and An estimated value of the unknown disturbance is obtained by establishing a predetermined time disturbance reconstruction mechanism to asymptotically reconstruct the unknown disturbance. The scheduled time disturbance reconstruction mechanism is: in represents the unknown disturbance d i The estimated value of i,1 express The estimated time of booking, The scheduled time disturbance reconstruction mechanism satisfies the c Internal reconstruction.

7. The method for predetermined time synchronization and coordinated control of a heterogeneous multi-robot system according to claim 6, characterized in that: Step S5 includes the following sub-steps: S5.1: For the i-th follower robot, define its auxiliary velocity vector and the auxiliary acceleration vector for: Among them, T3 is and The time constant of the predetermined time stabilization, It is p i ×p i dimensional unit vector, γ3 is and The design parameters in in, Represents the gradient of the performance indicator; the matrix The expression is: in, and Represents the matrix J i (q i )’s standard inverse and generalized inverse; Switching law θ i and its time derivative for: Among them, e i is x i The estimation error of ι>0; Normized symbolic function The definition of Represents the trigger sliding mode variable, and its expression is: Variables h1, h2, ρ i1 and ρ i2 They are defined as: Where η3 is a scalar greater than 1, μ3 = l1 / l2∈(0.5,1), where l1 and l2 are both odd numbers; S5.2: In order to achieve time synchronization convergence, for the i-th follower robot, construct a switching sliding mode variable: S5.3: Based on the expected position vector estimate, the expected velocity vector estimate, and the unknown disturbance estimate, combined with the switching sliding mode variables, a predetermined time synchronization control protocol is constructed: Among them, τ i1 represents the dynamic compensation term, τ i2 represents the synchronous control term, T4 is a predetermined time constant, 0<μ4<1, η4>1; γ4 is a parameter of the scheduled time synchronization control protocol, and its expression is: in,

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