Predetermined time synchronous cooperative control method for heterogeneous multi-robot system

Abstract

The invention relates to a preset time synchronous cooperative control method for a heterogeneous multi-robot system, and the method comprises the steps: firstly constructing a heterogeneous multi-robot system model and a network communication topology, setting a preset time synchronous cooperative control target, constructing a preset time distributed observer for each follower robot, and carrying out the network communication topology; estimating an expected position vector estimated value and an expected speed vector estimated value for realizing a preset time synchronization cooperative control target; secondly, constructing an interval observer according to the velocity vector of the follower robot so as to obtain an algebraic relational expression between the velocity vector and unknown disturbance; constructing an unknown disturbance algebraic reconstruction mechanism which is decoupled from the control input of the follower robot, and obtaining an estimated value of unknown disturbance; and finally, combining the expected position vector estimation value, the expected speed vector estimation value, the unknown disturbance estimation value and the switching sliding mode variable to construct a multi-robot system preset time synchronization control protocol. According to the technical scheme, the resistance of the system to external disturbance is improved.

Description

Technical Field

[0001] The present invention belongs to the technical field of heterogeneous multi-robot control, and particularly relates to a predefined-time synchronization cooperative control method for a heterogeneous multi-robot system. Background Art

[0002] With the wide application of robot systems in fields such as industry, service, and military, predefined-time cooperative control, as a key technology for realizing efficient cooperative operation of multi-robots, has been increasingly emphasized. This technology exhibits superior cooperative control performance and indicators, and is of great significance for improving the working efficiency and cooperative ability of robot systems. However, the existing technologies mainly focus on improving the convergence performance of the multi-robot system in cooperative control, for example:

[0003] Document [1]: Hierarchical Predefined-Time Control for Time-Varying Formation Tracking of Multiple Heterogeneous Euler-Lagrange Agents.

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

[0005] Document [1] proposed a predefined-time control scheme based on time-varying functions to solve the predefined-time formation control problem of multi-robot systems. Document [2] established a predefined-time bipartite formation control protocol under the hierarchical control design framework to ensure that the multi-robot system completes the desired bipartite formation in a predefined time.

[0006] In fact, the performance of the control system not only depends on when the system state converges, but also largely depends on when and how each system state element converges. In some operations, predefined-time cooperative control is not sufficient. For example, in order to obtain the best defense performance, a group of military robots need to form a tactical formation simultaneously and synchronously. Otherwise, the first-arriving robots will become vulnerable and be easily attacked. Therefore, the research on the predefined-time synchronization cooperative control method for heterogeneous multi-robot systems has important theoretical significance and application value.

[0007] For heterogeneous multi-robot systems with Euler-Lagrange dynamics characteristics, traditional control methods are difficult to be effectively applied due to the differences in the dynamic models between different types of robots, and there is an urgent need to develop new control strategies. In addition, existing technologies usually fail to fully consider the complex environment and dynamic changes faced by multi-robot systems in practical applications, and there is a problem of insufficiently handling external disturbances of the system, resulting in unsatisfactory application effects in real 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 predefined-time synchronization and cooperative control method for heterogeneous multi-robot systems.

[0009] To solve the above technical problem, the technical solution of the present invention is as follows:

[0010] A predefined-time synchronization and cooperative control method for heterogeneous multi-robot systems includes the following steps:

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

[0012] S2: According to the heterogeneous multi-robot system model constructed in step S1, construct a cooperation-antagonism network communication topology among the follower robots and a network communication topology between the follower robots and the leader robot, and set a predefined-time synchronization and cooperative control target for the multi-robot system;

[0013] S3: According to the predefined-time synchronization and cooperative control target set in step S2, construct a predefined-time distributed observer for each follower robot, so that each follower robot estimates the expected position vector estimate and the expected velocity vector estimate for achieving the predefined-time synchronization and cooperative control target;

[0014] S4: In order to achieve the compensation control for unknown disturbances, first, construct an interval observer for the velocity vector of the follower robot, and establish an algebraic relationship between the velocity vector of the follower robot and the unknown disturbance according to the properties of the interval observer; then, by combining a predefined-time differentiator and the algebraic relationship, construct an unknown disturbance algebraic reconstruction mechanism decoupled from the control input of the follower robot, and obtain an estimate of the unknown disturbance through the unknown disturbance algebraic reconstruction mechanism;

[0015] S5: Combine the expected position vector estimate and the expected velocity vector estimate in step S3, the estimate of the unknown disturbance in step S4, and a switching sliding mode variable based on the normed sign function to construct a predefined-time synchronization control protocol for the multi-robot system, and achieve the predefined-time synchronization and cooperative control target 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 as follows:

[0017]

[0018] where i represents the i-th follower robot; q i 、 represent the position vector, velocity vector, and acceleration vector of the joint space of the i-th follower robot 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 task space of the i-th follower robot respectively, n is the dimension of the task space vector, represents the forward kinematics function, is the Jacobian matrix, and satisfies and represent p i ×p i dimensional and n×p i dimensional Euclidean matrix spaces respectively, and represent p i dimensional and n-dimensional vector spaces respectively;

[0019] The dynamic system model of the leader robot is as follows:

[0020]

[0021] where, 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-antagonism network communication topology among the follower robots is represented by the connected graph where, represents the set of follower robot nodes, represents the set of follower robot edges, represents the weight matrix of the follower robots, a ijis the connection weight between the $i$-th follower robot and the $j$-th follower robot, $j = 1,\ldots,N$, and $j\neq i$; if the $i$-th follower robot can obtain the information of the $j$-th follower robot, and if the relationship between them is a cooperative one, then $a$ ij $= 1$, if the relationship between them is an adversarial one, 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 $N\times N$-dimensional Euclidean matrix space; define the in-degree of the $i$-th follower robot node $\upsilon$ i as Furthermore, define the Laplacian matrix associated with the graph as

[0023] The network communication topology between the follower robots and the leader robot is represented by a connected graph where, $\upsilon$ 0 represents the node of the leader robot, represents the edge set composed of the follower robots and the leader robot; define as the weight matrix of the leader robot, $b$ i is the connection weight between the leader robot and the $i$-th follower robot; if the information of the leader robot is available to the $i$-th follower robot, then $b$ i $= 1$; otherwise, $b$ i $= 0$;

[0024] For the connected graph Partition into a bipartite set such that and and satisfy that when , $a$ ij $= 1$, when , $a$ ij $= 0$; if Define $\sigma$ i $= 1$, if Define $\sigma$ i $= -1$; define the matrix $\sigma=\text{diag}(\sigma$ 1 ,\ldots,\sigma$ N ), $\sigma$ satisfies where the function $|\cdot|$ represents the matrix composed of the absolute values of each element; define the leader-follower matrix as The matrix where the element represents the vector The i-th element of, y i represents the vector The i-th element of, 1 N represents an N-dimensional column vector with all elements being 1.

[0025] Furthermore, in step S2, the multi-robot system's pre-determined time synchronization and cooperative control objective is:

[0026]

[0027] where, is the desired formation vector, t is the time variable, T f is the time constant for the actual convergence of the synchronous formation, satisfying T f ≤T p , T p is the pre-determined synchronous convergence time set manually; in addition, for any two time constants t 1 <t 2 ≤T f such that: 1 and t 2 , there is:

[0028]

[0029] where, x ik and respectively represent the k-th element in the task space position vector x i of the i-th follower robot and the vector , represents the final value of the binary formation of the multi-robot system.

[0030] Furthermore, in step S3, the pre-determined time distributed observer is:

[0031]

[0032] where, and respectively represent the estimated value of the desired position vector and the estimated value of the desired velocity vector of the i-th follower robot; is the distributed position observation error, is the distributed velocity observation error, and

[0033]

[0034] represents the first-order time derivative of the formation vector f i ;

[0035] is the upper bound of the acceleration of the leader robot, satisfying Denote the formation vector f i as the second-order time derivative; the classical sign function sgn c (·) is defined as sgn c (ξ) = col(sgn(ξ 1 ), …, sgn(ξ n ))), the improved sign function is defined as where the sign function sgn(·) is defined as μ 1 、μ 2 、η 1 and η 2 are positive constants that respectively satisfy 0 < μ 1 < 1, 0 < μ 2 < 1, η 1 > 1 and η 2 > 1, T 1 represents the predetermined time for the estimated value of the desired position vector of the i-th follower robot to converge, T 2 represents the predetermined time for the estimated value of the desired velocity vector of the i-th follower robot to converge; γ 1 and γ 2 are two design parameters of the predetermined time distributed observer, as follows respectively:

[0036]

[0037] where, r max = max{r 1 , …, r N}, Θ is a symmetric positive definite matrix, λ min (Θ) represents the minimum eigenvalue of the matrix Θ, the function Γ(·) represents the gamma function, and the specific form is

[0038]

[0039] where,

[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, obtain the time derivative of the velocity vector and the first expression form:

[0042]

[0043] For the velocity vector Construct an interval observer:

[0044]

[0045] Wherein, and v i represent the upper and lower bound states of the interval observer, satisfying d i and represent the upper and lower bounds of the unknown disturbance d i satisfying The gain matrix F i satisfies being both a Metzner matrix and a Hurwitz matrix;

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

[0047]

[0048] Wherein, ω i = col(ω i1 ,…,ω in ), and the time-varying variable ω ik can be expressed as:

[0049]

[0050] Then, we get the second expression form of

[0051]

[0052] Wherein, represents the time derivative of ω i which is an unknown quantity;

[0053]

[0054] Combined with Two expressions of, to obtain the velocity vector The algebraic relationship between and the unknown disturbance:

[0055]

[0056] S4.2: To reconstruct the unknown disturbance d i , use a predefined time differentiator for to estimate. The predefined time differentiator is:

[0057]

[0058] where ζ ik,0 and ζ ik,1 are two states of the predefined time differentiator, T c is the differentiator predefined time convergence parameter, and are two correction functions with respect to T c and are defined respectively as:

[0059]

[0060] where α is a positive scalar, and the selection of the function L 1 (t) satisfies The definition of the function L 2 (t) is L 2 (t) = L 1 (t)(α(T c -t)) 2 , The selection of satisfies The selection of satisfies The function and are defined as follows:

[0061]

[0062] Within T c , the first state ζ ik,0 of the predefined time differentiator is an accurate estimated value of;

[0063] S4.3: According to the algebraic relationship between the velocity vector and the unknown disturbance and the estimated value of, establish a predefined time disturbance reconstruction mechanism to asymptotically reconstruct the unknown disturbance, and obtain the estimated value of the unknown disturbance;

[0064] The predefined time disturbance reconstruction mechanism is:

[0065]

[0066] wherein represents the estimated value of the unknown disturbance d i , ζ i,1 represents the predetermined time estimate value of The predetermined time disturbance reconstruction mechanism satisfies reconstruction within T c .

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

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

[0069]

[0070] wherein, T 3 is and the time constant of predetermined time stability of is p i ×p i dimensional identity matrix, γ 3 is and the design parameters in

[0071]

[0072] wherein, represents the gradient of the performance index; the matrix has the expression:

[0073]

[0074] wherein, and respectively represent the standard inverse matrix and generalized inverse matrix of the matrix J i ;

[0075] The switching law θ i and its time derivative are:

[0076]

[0077] wherein, e i is the estimation error of x i , ι>0;

[0078] The definition of the normalized sign function is

[0079] represents the triggering sliding mode variable, and its expression is:

[0080]

[0081] The variable h 1 、h 2 、ρ i1 and ρ i2 are respectively defined as:

[0082]

[0083] where η 3 is a scalar greater than 1, where and are both odd numbers;

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

[0085]

[0086] S5.3: According to the estimated values of the desired position vector, the desired velocity vector, and the estimated value of the unknown disturbance, combined with the switching sliding mode variable, construct a predefined time synchronization control protocol:

[0087]

[0088] where τ i1 represents the dynamic compensation term, τ i2 represents the synchronization control term, T 4 is a predefined time constant, 0 < μ 4 < 1, η 4 > 1; γ 4 is the design parameter of the predefined time synchronization control protocol, and its expression is:

[0089]

[0090] where,

[0091] The beneficial effects that the present invention can achieve 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 through a designed pre-determined time distributed observer; an algebraic relationship between the unknown disturbance and the system state is obtained by establishing an interval observer, and a pre-determined time disturbance reconstruction mechanism is established based on this relationship. This disturbance reconstruction mechanism can provide an accurate estimation of the unknown disturbance, and at the same time, this reconstruction mechanism realizes decoupling from the control input of the robot system, enabling the introduction of the disturbance estimation value in the design of the control protocol, thereby realizing the compensation control of the unknown disturbance, overcoming the limitations of traditional pre-determined time cooperative control methods in the face of uncertainties, and thus improving the resistance of the system to external disturbances.

[0093] (2) The pre-determined time synchronization control scheme proposed by the present invention utilizes the estimation values provided by the pre-determined time distributed observer and the pre-determined time disturbance reconstruction mechanism, and combines with the pre-determined time synchronization control protocol established by the switching sliding mode technology based on the normed sign function, which not only meets the performance requirements of traditional pre-determined time cooperative control in references [1] and [2], but also realizes the control goal that all state components of all follower robot systems simultaneously reach the equilibrium point synchronously. This pre-set 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, contributing to improving the reliability and stability of the overall task completion. BRIEF DESCRIPTION OF THE DRAWINGS

[0094] Figure 1 It is a flowchart of the pre-determined time synchronization cooperative control method for the heterogeneous multi-robot system of the present invention.

[0095] Figure 2 It is a schematic diagram of the pre-determined time synchronization cooperative control architecture for the heterogeneous multi-robot system of the present invention.

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

[0097] Figure 4 It is a schematic diagram of the cooperative formation trajectory and snapshot of the heterogeneous multi-robot system in the embodiment of the present invention.

[0098] Figure 5 It is a schematic diagram of the cooperative formation error of the heterogeneous multi-robot system in the embodiment of the present invention.

[0099] Figure 6 It is a schematic diagram of the cooperative formation error of the heterogeneous multi-robot system under the comparative method in the embodiment of the present invention.

[0100] Figure 7 It is a schematic diagram of the position observation error, velocity observation error and reconstruction error in the embodiment of the present invention. DETAILED DESCRIPTION OF THE INVENTION

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

[0102] A predetermined-time synchronization collaborative control method for a heterogeneous multi-robot system includes the following steps:

[0103] Step 1: To achieve the predetermined-time synchronization collaborative control of heterogeneous multi-robots, according to the Euler-Lagrange equation, a dynamic system model of N (N>0) follower robots and 1 leader robot in the heterogeneous multi-robot system is given.

[0104] It includes the following sub-steps:

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

[0106]

[0107] Where, represents the position, velocity, and acceleration vectors of the joint space of the i-th follower robot, 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 the Jacobian matrix, and satisfies and respectively represent the p i ×p i dimensional and n×p i dimensional Euclidean matrix spaces, and respectively represent the p i dimensional and n dimensional vector spaces.

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

[0109]

[0110] Where, 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: According to the heterogeneous multi-robot system model established in Step 1, construct the cooperative-antagonistic network communication topology among the follower robots and the network communication topology between the follower robots and the leader robot, and set the predetermined time synchronization and cooperative control objective of the multi-robot system.

[0112] It includes the following sub-steps:

[0113] Step 2.1: Consider the heterogeneous multi-robot system with N followers described in Step 1.1. The cooperative-antagonistic network communication topology among the follower robots is represented by the connected graph where represents the set of follower robot nodes, represents the set of follower robot edges, represents the weight matrix of the follower robots. If the i-th follower robot can obtain the information of the j-th (j = 1,..., N and j ≠ i) follower robot, and if the relationship between them is cooperative, then the connection weight a ij = 1; if the relationship between them is antagonistic, 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; define the in-degree of the i-th follower robot node υ i as Further define the Laplacian matrix associated with the graph as The definition of the function diag(·) is which is a diagonal matrix with elements as the diagonal elements.

[0114] Construct the network communication topology composed of the N follower robots described in Step 1.1 and the 1 leader robot described in Step 1.2. Assume that at least one follower robot can obtain the information of the leader robot. The network communication topology between the N follower robots and the 1 leader robot is represented by the graph where, and are the set of nodes and the set of edges composed of N follower robots and 1 leader robot respectively. υ 0 represents the node of the leader robot; define is the weight matrix of the leader robot. Among them, if the information of the leader robot is available to the i-th follower robot, the connection weight b of the leader robot i = 1, otherwise the connection weight b of the leader robot i = 0;

[0115] For the graph follower robot node set can be divided into a bipartite set such that and and satisfies that when , a ij = 1, when , a ij = 0; If define σ i = 1, if define σ i = -1; At this time, define σ = diag(σ 1 ,…, σ N ), the diagonal matrix σ satisfies where the function |·| represents a matrix composed of the absolute values of each element; define the leader-follower matrix as matrix where the element represents the i-th element of the vector , y i represents the i-th element of the vector , 1 N represents an N-dimensional column vector with all elements being 1; define the symmetric positive definite matrix Θ to have the form

[0116] Step 2.2: According to the heterogeneous multi-robot dynamic system model obtained in Step 1.1 and Step 1.2, considering the heterogeneous multi-robot system cooperation-antagonism network communication topology constructed in Step 2.1, set the following predefined-time synchronization collaborative control objectives for the multi-robot system:

[0117]

[0118] where is the desired formation vector, t is the time variable, T f is the time constant for the actual convergence of the synchronous formation, satisfying T f ≤ T p , T p is the predefined synchronous convergence time set manually; in addition, for any t satisfying 0 ≤ t 1 < t 2 ≤ T fThe two time constants t 1 and t 2 satisfy:

[0119]

[0120] where x ik and respectively represent the k-th element in the task space position vector x i of the i-th follower robot and the vector . 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 according to the cooperation and confrontation relationships. Among them, the first group of follower robots forms a geometric formation centered on the leader robot, while the second group of follower robots moves in the opposite direction to the first group of follower robots while forming a geometric formation. At the same time, all follower robots converge to their equilibrium points for achieving the bipartite formation simultaneously and synchronously, that is, before achieving the formation of the final formation, no follower robot will arrive and stay at its equilibrium point in advance.

[0121] Step 3: In order to enable the follower robots to obtain their desired position vectors and velocity vectors, according to the predetermined time synchronization cooperative control objective, establish the distributed position observation error and the distributed velocity observation error, and then design a predetermined time distributed observer for each follower robot, so that each follower robot obtains the desired position vector and velocity vector for achieving the time synchronization cooperative control objective.

[0122] It includes the following steps:

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

[0124]

[0125] where represents the first-order time derivative of the formation vector f i , and respectively represent the estimated value of the desired position vector and the estimated value of the desired velocity vector of the i-th follower robot, which are provided by the following predetermined time distributed observer:

[0126]

[0127] where is the upper bound of the acceleration of the leader robot, satisfying Denote the formation vector as $\mathbf{f}$ i the second - order time derivative; the classical sign function $\text{sgn}$ c The definition of $\text{sgn}(\cdot)$ is c $\text{sgn}(\xi)=\text{col}(\text{sgn}(\xi$ 1 $),\cdots,\text{sgn}(\xi$ n $))$, and the improved sign function is defined as where the sign function $\text{sgn}(\cdot)$ is defined as The function $\text{col}(\cdot)$ represents a column vector formed by vertically concatenating its elements; $\mu$ 1 , $\mu$ 2 , $\eta$ 1 and $\eta$ 2 are positive constants satisfying $0\lt\mu$ 1 $\lt1$, $0\lt\mu$ 2 $\lt1$, $\eta$ 1 $\gt1$ and $\eta$ 2 $\gt1$, $T$ 1 represents the predetermined time for the $i$-th follower robot's estimated desired position vector to converge, $T$ 2 represents the predetermined time for the $i$-th follower robot's estimated desired velocity vector to converge, $\gamma$ 1 and $\gamma$ 2 are two design parameters of the predetermined - time distributed observer.

[0128] Step 3.2: According to the predetermined - time stability theory, the first design parameter $\gamma$ 1 of the predetermined - time distributed observer has the following form:

[0129]

[0130] where, $r$ max $=\max\{r$ 1 ,\cdots,r$ N $\}$, $\lambda$ min $(\Theta)$ represents the minimum eigenvalue of the matrix $\Theta$, and $\Theta$ is a symmetric positive - definite matrix calculated according to Step 2.1; the function $\Gamma(\cdot)$ represents the gamma function, and its specific form is

[0131] According to the predetermined - time stability theory, the second design parameter $\gamma$ 2 of the predetermined - time distributed observer has the following form:

[0132]

[0133] where,

[0134] The verification process of achieving pre-specified-time stability by the pre-specified-time distributed observer is as follows:

[0135] Define the position observation error and the velocity observation error as:

[0136]

[0137] According to the pre-specified-time distributed observer in step 3.1, it can be further obtained that:

[0138]

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

[0140]

[0141] where I n represents the n×n dimensional identity matrix.

[0142] In the first step, it is proved that the velocity observation error achieves pre-specified-time stability.

[0143] First, define an auxiliary velocity observation error where ε = col(ε 1 ,…,ε N ). Furthermore, it can be obtained that:

[0144]

[0145] Second, define a Lyapunov function as:

[0146]

[0147] Its time derivative satisfies the following inequality:

[0148]

[0149] Furthermore, since 0 < μ 2 < 1 < η 2 , the following inequality holds:

[0150]

[0151] According to the above inequality, it can be obtained that the time derivative of the function V ε satisfies the inequality At this time, the auxiliary velocity observation error ε will converge to zero within the pre-specified time T 1 , indicating that when the time variable t satisfies t ≥ T 1 , the velocity observation error Converges to zero.

[0152] In the second step, it is proved that the position observation error achieves the pre-specified time stability.

[0153] First, when the time variable t satisfies t ≥ T 1 , δ x has the following dynamic equation:

[0154]

[0155] Define an auxiliary position observation error where z = col(z 1 , …, z N ), and its time derivative is as follows:

[0156]

[0157] Secondly, define a Lyapunov function as:

[0158]

[0159] Furthermore, the time derivative of the function V z satisfies the inequality At this time, the auxiliary position observation error z will converge to zero within the pre-specified time T e = T 1 + T 2 , indicating that when the time variable t satisfies t ≥ T e , the position observation error of the pre-specified time distributed observer converges to zero.

[0160] Therefore, the pre-specified time distributed observer can accurately provide the estimated values of the position vector and velocity vector that the follower robot expects to achieve cooperative formation within the pre-specified time.

[0161] Step 4: To achieve the compensation control for the unknown disturbance, first, design an interval observer for the velocity vector in the follower robot, and establish an algebraic relationship between the velocity vector of the follower robot and the unknown disturbance according to the properties of the interval observer. Then, use the existing pre-specified time differentiator, combined with the established algebraic relationship between the velocity vector and the unknown disturbance, to construct an algebraic reconstruction mechanism for the unknown disturbance that is decoupled from the control input of the follower robot.

[0162] It includes the following sub-steps:

[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 time derivative of the velocity vector is obtained:

[0165]

[0166] For the velocity vector an interval observer is constructed:

[0167]

[0168] where and v i represent the upper and lower state of the interval observer, satisfying d i and represent the upper and lower bounds of the unknown disturbance d i satisfying The gain matrix F i is both a Metzner matrix and a Hurwitz matrix, that is, all non - diagonal elements of the matrix are non - negative, and at the same time all eigenvalues of the matrix have negative real parts.

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

[0170]

[0171] where ω i = col(ω i1 , …, ω in ), and the specific form of the time - varying variable ω ik can be expressed as:

[0172]

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

[0174]

[0175] where denotes the time derivative of ω i , which is unknown,

[0176] Combining the two expressions of the time derivative of the velocity vector gives the algebraic relationship between the velocity vector and the unknown perturbation as follows:

[0177]

[0178] Step 4.2: To reconstruct the unknown perturbation d i , an estimate of is provided by a predefined time differentiator in the form:

[0179]

[0180] where ζ ik,0 and ζ ik,1 are two states of the predefined time differentiator, T c is the predefined time convergence parameter of the differentiator, and are two correction functions with respect to the predefined time convergence parameter T c , and their definitions are as follows:

[0181]

[0182] where α is a positive scalar, and the selection of the function L 1 (t) satisfies The definition of the function L 2 (t) is L 2 (t) = L 1 (t)(α(T c - t)) 2 , The selection of satisfies The selection of satisfies The definitions of the functions

[0183]

[0184] where the functions and the function are respectively based on the improved classical sign function in Step 3.1and the classical sign function sgn cThe definition of (·) is obtained. At the predetermined time T c Within, the first state ζ of the predetermined time differentiator ik,0 will be an accurate estimate of

[0185] Step 4.3: Based on the algebraic expression of the unknown disturbance obtained in Step 4.1 and the estimate of provided in Step 4.2, establish a predetermined time disturbance reconstruction mechanism to asymptotically reconstruct the unknown disturbance, in the form of

[0186]

[0187] where represents the estimate of the unknown disturbance d i and ζ i,1 represents the predetermined time estimate of This predetermined time disturbance reconstruction mechanism satisfies the reconstruction within the predetermined time T c That is, when the time variable t satisfies t ≥ T c the disturbance reconstruction error converges to zero.

[0188] Step 5: Based on the estimated values of the desired position vector and the desired velocity vector provided in Step 3, the estimated value of the unknown disturbance provided in Step 4, and combined with the switched sliding mode variable based on the normalized sign function, design a predetermined time synchronization control protocol for the multi-robot system to achieve the predetermined time synchronization and cooperative control objective of the heterogeneous multi-robot system in the presence of unknown disturbances, and its architecture is as Figure 2 shown.

[0189] It includes the following steps:

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

[0191]

[0192] where T 3 is the time constant for the predetermined time stability of the auxiliary velocity vector and the auxiliary acceleration vector is the p i × p i dimensional identity matrix, and γ 3 is the design parameter in the auxiliary velocity vector and the auxiliary acceleration vector The expression is

[0193]

[0194] Among them, represents the gradient of the performance index; the matrix has the following expression:

[0195]

[0196] Among them, and respectively represent the standard inverse matrix and the generalized inverse matrix of matrix J i ;

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

[0198]

[0199] Among them, e i represents the estimation error of the position vector x i , which is defined as ι > 0 is a very small constant; the normalized sign function is defined as represents the triggering sliding mode variable, and its expression is:

[0200]

[0201] In addition, the variables h 1 , h 2 , ρ i1 and ρ i2 are respectively defined as:

[0202]

[0203] Among them, η 3 is a scalar greater than 1, where and are two odd numbers;

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

[0205]

[0206] Step 5.3: According to the desired position vector and desired velocity vector provided in Steps 3 and 4 and the estimated value of the unknown perturbation, combined with the switched sliding mode variable based on the normalized sign function established in Steps 5.1 and 5.2, construct the following predefined-time synchronization control protocol:

[0207]

[0208] where, τ i1 represents the dynamic compensation term, τ i2 represents the synchronization control term, T 4 is the predefined time constant, μ 4 and η 4 are positive constants satisfying 0 < μ 4 < 1 and η 4 > 1 respectively; γ 4 is the design parameter of the predefined-time synchronization control protocol, and the expression is:

[0209]

[0210] where,

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

[0212] First step, prove that the switched sliding mode variable s i asymptotically converges to zero within the predefined time T c + T 4 .

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

[0214]

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

[0216]

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

[0218]

[0219] Since 0 < k 41 < 1 < k 42 , the following inequality holds:

[0220]

[0221] According to the above inequality, the time derivative of the function V s satisfies the inequality At this time, the switched sliding mode variable s i achieves predefined-time stability, indicating that when the time variable t satisfies t ≥ T c + T 4 the switched sliding mode variable s i asymptotically converges to zero.

[0222] Second, prove that all components of the bipartite formation error of each follower robot synchronously converge to zero within the predefined time T e + T c + T 3 + T 4

[0223] First, on the switched sliding mode surface s i = 0, we can obtain:

[0224]

[0225] When the time variable t satisfies t ≥ T e + T c + T 4 the dynamic equation of the bipartite formation error has the following form:

[0226]

[0227] Second, define a Lyapunov function as whose time derivative has the following form

[0228]

[0229] Furthermore, the time derivative of the function satisfies the inequality At this time, the bipartite formation error achieves predefined-time stability, indicating that when the time variable t satisfies t ≥ T e + T c + T 3 + T 4 the bipartite formation error variable of each follower robot converges to zero; in addition, the bipartite formation error also satisfies the equation indicating that the bipartite formation error ​The ratio between any two elements in is constant, that is, the bipartite formation error of each follower robot All components of converge to zero simultaneously and synchronously.

[0230] In the third step, proof by contradiction is used to show that all follower robots reach their equilibrium points synchronously.

[0231] Suppose the m-th follower robot first reaches the equilibrium point at t = t m > 0 and remains at the equilibrium point, while the h-th follower robot does not reach the equilibrium point at t m ; Since the cooperative-antagonistic communication network topology among the follower robots is connected, this indicates that there is a path between the h-th follower robot and the m-th follower robot; Without loss of generality, assume there are p follower robots (labeled p 1 , …, p p ) on this path; Then, the position vector 1 of the p -th follower robot directly connected to the h-th follower robot must not be at its equilibrium point, because: 1) If the p m -th follower robot is at its equilibrium point at time t 1 , then there is that is, the h-th follower robot is also at its equilibrium point at time t m , which contradicts the fact that the h-th follower robot is not at its equilibrium point at time t m . 2) If the p m -th follower robot is at its virtual equilibrium point at time t 1 , then there is:

[0232]

[0233] And before achieving the final formation, each follower robot does not stay at its virtual equilibrium point, that is, its virtual equilibrium point is sparse, which further indicates that the p m -th follower robot does not reach its true equilibrium point at time t 1 .

[0234] Similarly, it can be inferred that the position vectors of other follower robots on this path are also not at their equilibrium points; For the m-th follower robot, it can also be obtained that and exist at most at isolated time points; Therefore, the position vector m of the m-th follower robot at time t is not its true equilibrium point because the connected follower robot p p is not at its equilibrium point at time t mis not at its true equilibrium point at all times; this is clearly contradictory to the assumption that the m-th follower robot reaches the equilibrium point first at time t m moment.

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

[0236] Simulation verification

[0237] A heterogeneous multi-robot system consisting of 1 leader robot (labeled 0) and 8 follower robots (labeled 1,..., 8) is used for the predetermined-time synchronization formation simulation verification to verify the feasibility and effectiveness of the proposed control scheme. The network communication topology of the heterogeneous multi-robot system is as Figure 3 shown, where the 3rd and 7th follower robots are 3-degree-of-freedom manipulators, and the remaining follower robots are 2-degree-of-freedom manipulators. The dynamic model of the 2-degree-of-freedom manipulator is described as follows:

[0238]

[0239] and respectively represent the position vectors in the joint and task spaces. The expressions and parameters of the inertia matrix centripetal Coriolis matrix gravity vector and Jacobian matrix are consistent with those in the literature [1]. The unknown perturbation is selected as d i = col(sin(t + 0.2i), cos(t + 0.2i)), and the upper and lower bounds of the perturbation are selected as d i = col(-2, -2), and the gain matrix of the interval observer is selected as F i = diag(-0.5, -1).

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

[0241]

[0242] and respectively represent the position vectors in the joint and task spaces. The expressions and parameters of the inertia matrix centripetal Coriolis matrix gravity vector and Jacobian matrix are also consistent with those in the literature [1]. The unknown perturbation is selected as d i= col(sin(t + 0.2i), cos(t + 0.2i), 0.5(sin(t + 0.2i) + cos(t + 0.2i))), and the upper and lower bounds of the perturbation are selected as d i = col(-2, -2, -2), and the gain matrix of the interval observer is selected as F i = diag(-0.5, -1, -1.5).

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

[0244]

[0245] Other design parameters are selected as μ 1 = μ 2 = μ 3 = μ 4 = 0.6, η 1 = η 2 = η 3 = η 4 = 1.2, T 1 = T 2 = T 3 = T 4 = 1, T c = 0.5, ι = 0.001, After calculation, γ can be obtained 1 = γ 2 = 24.6633, γ 3 = γ 4 = 21.4702. According to Figure 3 σ can be obtained 1 = σ 2 = σ 3 = σ 4 = 1, σ 5 = σ 6 = σ 7 = σ 8 = -1.

[0246] The simulation results are as Figures 4-7 shown. Figure 4 It shows the motion trajectories of all robots, including snapshots of the cooperative formation positions at 5s and 10s. To further demonstrate the superiority of the proposed pre-specified time synchronization control algorithm. Figure 5 and Figure 6respectively depict the cooperative formation error trajectories under the control method of this embodiment and the traditional predetermined-time control method, where the two control methods share the same control parameters to provide a fair comparison. From Figure 5 and Figure 6 it can be observed that, different from the traditional predetermined-time control method, the predetermined-time synchronization control method designed in this embodiment can not only achieve predetermined-time stability, but also ensure that the followers complete the binary formation synchronously. Figure 7 The figure depicts the position observation error, velocity observation error of the distributed observer, and the reconstruction error of the disturbance reconstruction mechanism. It can be seen from it that the position observation error, velocity observation error, and reconstruction error can all asymptotically converge to zero within a predetermined time.

[0247] The above is only one implementation manner of the present invention. The protection scope of the present invention is not limited to the above embodiments. It should be noted that for those of ordinary skill in the art, several improvements and refinements made without departing from the idea of the present invention fall within the protection scope of the present invention.

Claims

1. A method for pre-determined time synchronization and cooperative 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, N>0; S2: Based on the heterogeneous multi-robot system model constructed in step S1, a cooperative-adversarial network communication topology between follower robots and a network communication topology between follower robots and a leader robot are constructed, and a predetermined time synchronization collaborative control target of the multi-robot system is established; S3: According to 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 the expected position vector estimate value and the expected velocity vector estimate value for achieving the predetermined time synchronization cooperative control target; S4: In order to realize the compensation control of the unknown disturbance, firstly, an interval observer is constructed for the velocity vector of the follower robot, and an algebraic relationship between the velocity vector of the follower robot and the unknown disturbance is established according to the properties of the interval observer; then, by combining a predetermined time differentiator and the algebraic relationship, an unknown disturbance algebraic reconstruction mechanism decoupled from the control input of the follower robot is constructed, and 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 coordinated control of heterogeneous multi-robot systems 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’s 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 Respectively represent the position vector and velocity vector of the i-th follower robot task space, 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 Dimension and 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, characterized in that: 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 two are in an adversarial relationship, 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 diagram The associated Laplacian 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 ij =1, when When 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 element Representation vector The i-th element of i Representation 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 cooperative control of a heterogeneous multi-robot system according to claim 3, characterized in that: 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 synchronous 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 Represents the final value of the binary 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 pilot 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 ), the 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 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, which are as follows: in, r max =max{r1,…,r N }, Θ is a symmetric positive definite matrix, Θ = RH + H T R,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 cooperative 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 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 satisfies both the Metzner matrix and the Hurwitz matrix; The upper bound state of the interval observer Nether state v i and the 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: 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; Combination 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 estimate, the predetermined time differentiator is: Among them, ik,0 and ik,1 are the two states of the predetermined time differentiator, T c is the differentiator 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 of M1 satisfies The selection of M2 satisfies function and The definition is as follows: In T c Within the predetermined time, the first state of the 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 and obtaining an estimated value of 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 Refactoring within implementation.

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, Yes 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 The standard inverse matrix and generalized inverse matrix of ; 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: Among them, η3 is a scalar greater than 1, in and All are 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 variable, a predetermined time synchronization control protocol is constructed: 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 parameter, and the expression is: in,

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