A method for consistent collaborative control between fixed time axes applied to multiple robotic arms

Abstract

The present invention relates to a high-precision, fast following consistency control method applied to multiple robotic arms, and belongs to the technical field of collaborative control of multiple robotic arm systems. The present invention designs a fixed-time observer applied to the follower robotic arm, which is used to estimate the state information of the leader robotic arm within a fixed time, and adopts an adjacent cross-coupling method to perform state transformation on the multi-robotic arm system to obtain a robotic arm control model corresponding to the error state; designs a controller for the multi-system robotic arm, and designs corresponding control inputs, and based on the corresponding control inputs and the robotic arm control model, enables the follower robotic arm to track the movement of the leader robotic arm. Under the influence of robotic arm uncertainty and external interference, the present invention breaks through the shortcomings of traditional multi-robotic arm collaborative methods in control accuracy, response speed and anti-interference, and solves the problem of high-precision, fast and stable control of the position and speed of the joints of multiple robotic arms under the influence of comprehensive interference.

Description

Technical Field

[0001] The present invention relates to the technical field of collaborative control of multiple robotic arm systems, and in particular to a high-precision fast following consistency control method applied to multiple robotic arms. Background Art

[0002] Fixed-time consensus control of multi-agent systems has emerged as an important research topic in recent years. Polyakov first proposed the concept of fixed-time stability. Compared to conventional asymptotic consistency control, fixed-time consensus control offers advantages such as faster convergence rate, stronger robustness, and higher control accuracy. Furthermore, the system's convergence time is independent of initial conditions and depends solely on the design parameters. In practical applications, even if the system's operating environment changes, there is no need to readjust the design parameters to ensure system convergence time. In recent years, an increasing number of researchers have investigated the problem of fixed-time consensus control of multi-agent systems. As core equipment in intelligent manufacturing processes, robotic arms are increasingly used. However, the synchronization and coordination between their axes is a key factor affecting their high-precision tracking. Furthermore, to perform more complex and high-precision control tasks, the coordinated control of multiple robotic arms is often required. Compared to single-arm systems, multi-arm systems have stronger nonlinear characteristics, making multi-arm coordinated control more complex. Therefore, studying distributed multi-arm fixed-time coordinated control has practical significance.

[0003] However, the traditional multi-manipulator fixed-time collaborative control method has certain shortcomings, which are mainly reflected in the following aspects:

[0004] (1) Traditional multi-manipulator collaborative control methods usually only consider the coordination between manipulators. However, since the actual manipulator system is often composed of multiple joints, the synchronization and coordination between the manipulator joints will also affect the control accuracy of the manipulator system. In addition, considering the synchronization and coordination between joints can effectively avoid problems such as overload of a certain joint of the manipulator and loss of vision of the manipulator. Therefore, multi-manipulator collaborative control should consider the coordination between manipulators and the coordination between manipulator joints at the same time.

[0005] (2) Traditional multi-manipulator fixed-time collaborative control methods suffer from algebraic loop problems. The topology of a multi-agent system is usually complex, and the connections between agents are diverse. In this case, traditional multi-manipulator control algorithms can easily form loops in the control loop. These loops are mathematically manifested as algebraic loop problems, affecting the stability and performance of the system.

[0006] (3) Traditional multi-manipulator collaborative control methods do not consider the uncertainty of the multi-manipulator model and external disturbances. However, in actual control, model uncertainty and external disturbances will inevitably appear and affect the stability of the system. If these uncertainties are not considered and control is still performed according to the designed control algorithm, it will cause system errors and affect the normal operation of the system. Summary of the Invention

[0007] In view of the above problems, the present invention provides a high-precision, fast following consistency control method for multiple robotic arms, designs a fixed-time observer for the follower robotic arm, and estimates the state information of the leader robotic arm within a fixed time, and adopts the adjacent cross-coupling method to perform state transformation on the multi-robotic arm system to obtain a robotic arm control model corresponding to the error state; designs a controller for the multi-system robotic arm, and designs corresponding control inputs, and based on the corresponding control inputs and the robotic arm control model, enables the follower robotic arm to track the movement of the leader robotic arm. The present invention takes into account the requirements for safe, stable and reliable operation of multiple robotic arms in a collaborative control environment, and under the influence of robotic arm uncertainty and external interference, breaks through the shortcomings of traditional multi-robotic arm collaborative methods in control accuracy, response speed and anti-interference, solves the problem of high-precision, fast and stable control of the position and speed of the joints of multiple robotic arms under the influence of comprehensive interference, and achieves the purpose of improving the control performance of multiple robotic arms.

[0008] The present invention provides a high-precision fast following consistency control method applied to multiple robotic arms, comprising:

[0009] Step S1: determining multiple robotic arms and their related parameters; the multiple robotic arms include a leader robotic arm and multiple follower robotic arms; and establishing a multi-robotic arm system based on the related parameters of the multiple robotic arms;

[0010] Step S2: Obtain the error state 1 of the state variable of each follower manipulator in the multi-manipulator system; design a manipulator fixed time observer based on the error value of the state variable of each follower manipulator and the leader manipulator;

[0011] Determine the upper bound of the convergence time of the fixed-time observer to obtain a convergence time of one;

[0012] Step S3: Considering the synchronization relationship between the joints of each robotic arm, the state of each robotic arm in the multi-robotic arm system is transformed to obtain the second error state of the multi-robotic arm;

[0013] Step S4: designing sliding mode variables for each follower manipulator arm based on the error state 2 of the multiple manipulator arms; establishing a controller for the multi-manipulator system based on the sliding mode variables of each manipulator arm; and obtaining the time for each follower manipulator arm sliding mode variable to converge to zero after the fixed time observer converges, thereby obtaining the convergence time 2.

[0014] After the fixed time observer converges and the sliding mode variables of each follower manipulator converge, the time for each follower manipulator state variable to converge to zero is obtained, and the convergence time three is obtained;

[0015] Step S5: determining an upper limit of the multi-manipulator convergence time based on the convergence time 1, the convergence time 2, and the convergence time 3; and obtaining a control input of each manipulator based on a controller of the multi-manipulator system;

[0016] Based on the control input of each robotic arm and the upper limit of the convergence time of the multi-robotic arms, the follower robotic arm tracks the movement of the leader robotic arm within a predetermined time range, thereby achieving the goal of consistent collaborative control of the multi-robotic arms.

[0017] Preferably, the specific steps of establishing the multi-robotic arm system in step S1 include:

[0018] Determining multiple robotic arms; the multiple robotic arms include a leader robotic arm and multiple follower robotic arms;

[0019] Determining relevant parameters of the plurality of robotic arms; the relevant parameters of the plurality of robotic arms include an inertia matrix, a Coriolis force / centrifugal force matrix, a gravity matrix, and a comprehensive interference of each robotic arm;

[0020] A multi-robotic arm system is established based on relevant parameters of the multiple robotic arms.

[0021] Preferably, the specific steps of obtaining the fixed time observer of the multi-manipulator in step 2 include:

[0022] Design a corresponding original fixed-time observer for each follower robotic arm to observe the state of the leader robotic arm;

[0023] Obtaining the error state 1 of the state variable of each follower manipulator in the multi-manipulator system;

[0024] Perform vector rewriting on the state variables of the navigator manipulator corresponding to each original fixed-time observer to obtain rewriting vector 1;

[0025] Perform vector rewriting on the state variables of each follower manipulator corresponding to each original fixed-time observer to obtain rewriting vector 2;

[0026] Based on the Lipschitz continuity condition of the nonlinear term of the robotic arm, the error state one, vector one and vector two of the state variables of each follower robotic arm in the multi-robotic arm system, a fixed-time observer of each follower robotic arm is obtained to observe the state of the leader robotic arm; the fixed-time observer of each follower robotic arm is characterized as a fixed-time observer of the multi-robotic arm.

[0027] Preferably, the specific steps of obtaining the convergence time 1 are: designing a Lyapunov function 1 and obtaining a fixed time estimation parameter using Lyapunov's second method;

[0028] Using the fixed time theory and the fixed time estimation parameters, the convergence time of the fixed time observer is determined to obtain the convergence time 1; the convergence time 1 is the upper bound of the time for each follower manipulator to estimate the leader manipulator arm;

[0029] Preferably, the multi-manipulator system is expressed as:

[0030]

[0031] in, is the speed of the i-th robot arm in state x, is the position of the i-th robotic arm in state x, The velocity derivative of the i-th robot arm in state x, is the state of the i-th robotic arm, is a real number, n is the degree of freedom of the robot arm, g i is the nonlinear term of the i-th robotic arm, u i is the control input of the i-th robotic arm, d i is the comprehensive interference of the i-th robotic arm, i = 0, 1, 2, 3…N. When i = 0, it is the leader robotic arm, and when i = 1, 2, 3…N, it is the follower robotic arm.

[0032] Preferably, the Lipschitz continuity condition of the nonlinear term of the robotic arm is expressed as:

[0033]

[0034] in, is the nonlinear term of the i-th robot arm in state x, is the nonlinear term of the i-th robot in the x′ state, is the position of the i-th robotic arm in state x, is the speed of the i-th robot arm in state x, is the position of the i-th robot arm in the x′ state, is the velocity of the i-th manipulator in the x′ state, ψ1 is the Lipschitz constant related to the position state of the manipulator, ψ2 is the Lipschitz constant related to the velocity state of the manipulator; Ψ is the minimum value between the two, Ψ = min(Ψ1,Ψ2).

[0035] Preferably, the original fixed time observer of the multi-manipulator is expressed as:

[0036]

[0037] in, is the derivative of the position state variable of the i-th follower manipulator with respect to the leader manipulator, is the derivative of the velocity state variable estimate of the i-th follower manipulator with respect to the leader manipulator, The follower manipulator estimates the velocity state variable of the leader manipulator, sgn(·) represents the sign function, l1 is the low-power constant one, l2 is the low-power constant two, is the symbolic function constant one, The symbolic function term is constant 2, m1 is the high power term constant 1, m2 is the high power term constant 2, represents the communication interaction error between the manipulators associated with the i-th follower manipulator, represents the communication interaction error between the manipulators related to the velocity state of the i-th follower manipulator, Representation and variables The related nonlinear terms of the manipulator, p represents the low-power term parameter, and q represents the high-power term parameter.

[0038] Furthermore, the inertia g of the i-th robotic arm i The expression is:

[0039]

[0040] in, is the nonlinear term of the i-th robot arm in state x, is the nominal value of the inertia matrix of the i-th manipulator in the x state, is the nominal value of the Coriolis force and / or centrifugal force of the i-th robot arm in state x, is the nominal value of gravity of the i-th robot arm in state x, is the position of the i-th robotic arm in state x, is the state of the i-th robotic arm.

[0041] Furthermore, the nominal value of the inertia matrix of the i-th manipulator in the x state is Nominal value of the Coriolis force and / or centrifugal force of the i-th robot arm in state x Nominal value of gravity of the i-th robot arm in state x The expressions are:

[0042]

[0043] in, is the actual inertia value of the i-th robot arm in state x, is the inertia error value of the i-th robotic arm when it is in the x state, is the actual value of the Coriolis force and / or centrifugal force of the i-th robot arm in state x, is the error value of the Coriolis force and / or centrifugal force of the i-th robot arm in the x state, is the actual value of gravity of the i-th robot arm in state x, is the gravity error of the i-th robot arm in the x state.

[0044] Furthermore, the comprehensive interference d of the i-th robot arm i , the expression is:

[0045]

[0046] Among them, d i is the comprehensive disturbance of the i-th robot arm, δ is the sum of the robot arm model uncertainty and external disturbance, is the external disturbance, including mechanical friction, air resistance, sensor actuator error, i = 0, 1, 2, 3...N, the comprehensive disturbance satisfies ||d i || ∞ ≤ρ, ρ is the upper limit of the comprehensive interference and is a known positive constant.

[0047] Preferably, when i=0, the i-th robot is the navigator, and the control input of the i-th robot as the navigator is bounded, that is,

[0048] Preferably, the specific steps of obtaining the error state 1 of the state variable of each follower manipulator in the multi-manipulator system include:

[0049] Define the actual value of parameter k of the multi-manipulator system in state x as x k ,in, When i=1,2,3…N, the i-th robot is the follower robot; when i=0, the i-th robot is the leader robot; k=1,2; when k=1, x k Indicates the actual value of the position of the multi-manipulator system in the x state. When k = 2, x k Indicates the actual value of the speed of the multi-manipulator system in the x state;

[0050] Define the estimated value of parameter k of the multi-manipulator system in state x as

[0051]

[0052] Define the actual value of parameter k when the robot arm as the navigator is in state x

[0053] The error value of the parameter k of the multi-manipulator system in the x state is obtained based on the estimated value of the parameter k of the multi-manipulator system in the x state and the actual value of the parameter k of the manipulator as the navigator in the x state.

[0054] The error value of parameter k of the multi-manipulator system in state x The expression is:

[0055]

[0056] It can be understood that the error value of the i-th robotic arm in the x state is the error value between the estimated value of the i-th robotic arm and the actual value of the robotic arm serving as the navigator in the x state;

[0057] Preferably, in step 2, the state variables of the navigator manipulator corresponding to each original fixed-time observer are rewritten by vectors to obtain a rewritten vector set 1;

[0058] The specific steps of rewriting the state variables of each follower manipulator corresponding to each original fixed-time observer into vectors to obtain the second rewriting vector set include:

[0059] Rewrite the control input of the navigator manipulator into a set of control input vectors u 0 Provides control input for the pilot's robotic arm;

[0060] Rewrite the comprehensive interference of the navigator manipulator into a set of comprehensive interference vectors d 0 Comprehensive interference for the navigator's robotic arm;

[0061] Rewrite the inertia vector of the navigator manipulator into the inertia vector set 1 g 0 is the inertia of the navigator's robotic arm;

[0062] Rewrite the inertia vector of each follower robot arm into inertia vector set 2 is the inertia of the Nth follower robot arm;

[0063] Furthermore, the expression of the fixed time observer of the multi-manipulator is:

[0064]

[0065] in, is the derivative of the position state variable error vector of the follower manipulator and the leader manipulator, is the derivative of the velocity state variable error vector of the follower manipulator and the leader manipulator, is the error vector of the velocity state variables of the follower manipulator and the leader manipulator, e1 represents the communication interaction error vector between the manipulators related to the position state of the manipulator, and e2 represents the communication interaction error vector between the manipulators related to the velocity state of the manipulator, Representation and variables and The associated set of nonlinear vectors of the manipulator, represents the nonlinear vector set of the navigator manipulator, p = 1-2 / μ, q = 1+2 / μ, μ is a positive constant three, μ>2; Input vector set for the navigator controller; The integrated interference vector for the navigator;

[0066] Furthermore, the expression of the convergence time 1 is:

[0067]

[0068] Where T0 is the convergence time, χ is the design parameter, χ>0, μ is a positive constant, is the upper bound of the fixed time.

[0069] Preferably, the specific steps of obtaining the second error state of multiple robotic arms in step 3 include:

[0070] Obtain the error state between the state of each follower manipulator in the multi-manipulator system and the state of the leader manipulator, and obtain the error state 1 of the multiple manipulators;

[0071] Based on the synchronization relationship between the joints of each robotic arm, the adjacent cross-coupling method is introduced into the error state one of multiple robotic arms to obtain the error state two of multiple robotic arms.

[0072] Preferably, the error state expression of the multiple robotic arms is:

[0073]

[0074] in, is the difference between the actual value of the position state variable of the i-th follower manipulator and the estimated value of the position state of the leader manipulator, is the difference between the actual value of the velocity state variable of the i-th follower manipulator and the estimated value of the velocity state of the leader manipulator.

[0075] The error state 2 of the multiple robotic arms is expressed as:

[0076]

[0077] Among them, Teq is a symmetric positive definite matrix, To use the matrix T eq Error status The state change of To use the matrix T eq Error status state change.

[0078] The expression of the robotic arm control model is:

[0079]

[0080] in, is the position-related error state vector of the follower robot after state transformation; is the speed-related error state vector of the follower robot after state transformation.

[0081] Furthermore, the symmetric positive definite matrix T eq The expression is:

[0082] T eq =I+λT,λ>0,

[0083] Where T is the parameter matrix, I is the identity matrix of the corresponding dimension, λ is a positive constant, λ>0.

[0084]

[0085] Preferably, the specific steps of obtaining the second convergence time in step S4 include:

[0086] A second Lyapunov function is designed according to the robot control model; and according to the second Lyapunov function and the fixed time theory, after the fixed time observer converges, the time for the sliding mode variables of each robot to converge to zero is obtained, and the convergence time 2 is obtained.

[0087] Preferably, the controller expression of the multi-manipulator system is:

[0088]

[0089] in, is the control input of the i-th robotic arm, is the auxiliary control item of the i-th robotic arm, is the non-singular fixed-time control term of the i-th robot arm, is the position error compensation item of the i-th robotic arm, is the speed error compensation term of the i-th robot arm, is the interference compensation term of the i-th robotic arm.

[0090] Preferably, the sliding mode variables of each robotic arm are expressed as follows:

[0091]

[0092] Among them, s i is the sliding mode variable of the i-th robotic arm, is the system variable related to the position state of the i-th robot arm, is the system variable related to speed, diag(·) is the diagonal function, κ(·) is the designed positive definite vector function, q1 is the high-power constant four, and p1 is the low-power constant four.

[0093] Preferably, the second convergence time includes: the convergence time T1 of the sliding mode variables of each manipulator in the controller of the multi-manipulator system after the fixed time observer converges and the time ε(τ) when the sliding mode variables of each manipulator enter the sliding mode region δ1 from the saturation critical region δ2;

[0094] The convergence time 3 includes the convergence time of the state variables of each manipulator after the fixed time observer and the sliding mode variables of each manipulator converge;

[0095] Furthermore, the convergence time T1 of the sliding mode variables of each robot arm is expressed as:

[0096]

[0097] Among them, α″2 is the design parameter one, β2 is the design parameter two and is a positive constant, n2 is the high-power term performance parameter two, m2 is the high-power term constant two, q2 is the high-power term parameter two, and p2 is the low-power term parameter two.

[0098] Furthermore, the expression for the time from the saturation critical region δ2 to the sliding mode region δ1 is:

[0099]

[0100] Where ε(τ) is the time when the sliding mode variable of the manipulator enters the sliding mode region δ1 from the saturation critical region δ2;

[0101] k represents the number of manipulators in the saturation critical region δ2, 1≤k≤N, 1≤j≤k, ε j (τ) represents the time when the j-th robot arm enters the sliding mode region δ1 from the saturation critical region δ2.

[0102] The expression of the convergence time T2 is:

[0103]

[0104] Among them, n1 is the high-power performance parameter one, m1 is the high-power constant one, β1 is the design parameter three, α1 is the design parameter four, and it is a positive constant.

[0105] Furthermore, the expression of the auxiliary control item of the i-th robotic arm is:

[0106]

[0107] in, represents the auxiliary control item of the i-th robot arm, γ=m1 / n1-p1 / q1>1, α1 is a positive constant, η=q1 / p1-1∈(0,1), and κ is a positive definite vector.

[0108] The expression of the non-singular fixed-time control term of the i-th manipulator is:

[0109]

[0110] Among them, μ τ (·) Compensation function designed to avoid non-singular terms, α2 is the design parameter five.

[0111] Preferably, the error compensation term of the i-th robotic arm is expressed as:

[0112]

[0113] Among them, u e1 is the error compensation term, and e1 represents the communication interaction error between the manipulators related to the position state of the manipulators.

[0114] Preferably, the expression of the position error compensation term of the i-th robotic arm is:

[0115]

[0116] Among them, u e2 It is also the error compensation term, is the nonlinear term of the manipulator related to the observer observation value, is a low-power constant of two, l2 is a low-power constant of one, and e2 represents the communication interaction error between the manipulators related to the velocity state of the manipulator.

[0117] Preferably, the interference compensation term of the robotic arm is expressed as:

[0118]

[0119] Among them, u d represents the interference compensation term, g i is the nonlinear term of the manipulator, and ρ is the upper limit of the comprehensive interference of the manipulator.

[0120] According to the fixed time theory, after the fixed time observer of the multi-manipulator and the sliding mode variables of each manipulator converge, the state variables of each manipulator will be The inner convergence to zero can ensure that the entire multi-manipulator system will achieve consistent tracking control of the leader within a fixed time.

[0121] Preferably, the upper limit of the multi-manipulator convergence time in step S5 is expressed as:

[0122] T≤T max =T o +T1+T2+ε(τ)

[0123] Where T is the actual convergence time of the state variables of the multi-manipulator system, T max is the upper bound of the convergence time, T0 is the convergence time one of the fixed time observer, T1+ε(τ) is the convergence time two, ε(τ) is the time when the sliding mode variable of each robot arm enters the sliding mode region from the saturation critical region, and T2 is the convergence time three.

[0124] Compared with the prior art, the present invention has at least the following beneficial effects:

[0125] (1) Under the multi-manipulator fixed time axis consistent collaborative control method proposed in the present invention, the multi-manipulator system can not only achieve the coordination between the manipulators and the fixed time convergence characteristics of the state error, but also achieve the synchronization and coordination between the joints of each manipulator, thereby improving the control accuracy and robustness of the manipulator system and avoiding the situation where a joint is suddenly subjected to a large load;

[0126] (2) In the proposed observer-based framework for consistent time axes between multiple manipulators, the output value of the observer is used as the expected value of the controller. As the output value of the observer changes, the controller is adjusted accordingly, which can effectively avoid the algebraic loop problem existing in numerical simulations.

[0127] (3) The present invention takes into account situations with model uncertainty and external interference. The considered situations are more complex and more consistent with the robot arm system model that can be obtained in practice, so that the uncertain robot arm system can still operate normally and reach the expected state under the controller proposed by the present invention, and the system stability and reliability are stronger. BRIEF DESCRIPTION OF THE DRAWINGS

[0128] The drawings are only for purposes of illustrating particular embodiments and are not to be considered limiting of the invention.

[0129] Figure 1 A schematic diagram of a flow chart of Example 1 of the present invention;

[0130] Figure 2Schematic diagram of the multi-manipulator system control framework of Example 1 of the present invention.

[0131] Figure 3 Schematic diagram of a communication network topology diagram of a multi-manipulator system according to embodiment 1 of the present invention;

[0132] Figure 4 Schematic diagram of position observation values ​​of two joints of the multi-manipulator system relative to the navigator in Example 1 of the present invention;

[0133] Figure 5 Schematic diagram of the velocity observation values ​​of two joints of the multi-manipulator system relative to the navigator in Example 1 of the present invention;

[0134] Figure 6 Schematic diagram of the position trajectory diagram of two joints of the multi-manipulator system of Example 1 of the present invention;

[0135] Figure 7 Schematic diagram of the velocity trajectory diagram of two joints of the multi-manipulator system of Example 1 of the present invention. DETAILED DESCRIPTION

[0136] In order to more clearly understand the above-mentioned objects, features and advantages of the present invention, the present invention is further described in detail below with reference to the accompanying drawings and specific embodiments. It should be noted that, in the absence of conflict, the embodiments of the present invention and the features in the embodiments can be combined with each other. In addition, the present invention can also be implemented in other ways different from those described herein. Therefore, the scope of protection of the present invention is not limited by the specific embodiments disclosed below.

[0137] A specific embodiment of the present invention, as Figure 1-7 , discloses a high-precision fast following consistency control method for multiple robotic arms. In order to illustrate the effectiveness of the method proposed by the present invention, the above technical solution of the present invention is described in detail through a specific embodiment below. The specific implementation steps are as follows:

[0138] The present invention provides a high-precision fast following consistency control method applied to multiple robotic arms, comprising:

[0139] Step S1: determining multiple robotic arms and their related parameters; the multiple robotic arms include a leader robotic arm and multiple follower robotic arms; and establishing a multi-robotic arm system based on the related parameters of the multiple robotic arms;

[0140] Step S2: Obtain the error state 1 of the state variable of each follower manipulator in the multi-manipulator system; design a manipulator fixed time observer based on the error value of the state variable of each follower manipulator and the leader manipulator;

[0141] Determine the upper bound of the convergence time of the fixed-time observer to obtain a convergence time of one;

[0142] Step S3: Considering the synchronization relationship between the joints of each robotic arm, the state of each robotic arm in the multi-robotic arm system is transformed to obtain the second error state of the multi-robotic arm;

[0143] Step S4: designing sliding mode variables for each follower manipulator arm based on the error state 2 of the multiple manipulator arms; establishing a controller for the multi-manipulator system based on the sliding mode variables of each manipulator arm; and obtaining the time for each follower manipulator arm sliding mode variable to converge to zero after the fixed time observer converges, thereby obtaining the convergence time 2.

[0144] After the fixed time observer converges and the sliding mode variables of each follower manipulator converge, the time for each follower manipulator state variable to converge to zero is obtained, and the convergence time three is obtained;

[0145] Step S5: determining an upper limit of the multi-manipulator convergence time based on the convergence time 1, the convergence time 2, and the convergence time 3; and obtaining a control input of each manipulator based on a controller of the multi-manipulator system;

[0146] Based on the control input of each robotic arm and the upper limit of the convergence time of the multi-robotic arms, the follower robotic arm tracks the movement of the leader robotic arm within a predetermined time range, thereby achieving the goal of consistent collaborative control of the multi-robotic arms.

[0147] Preferably, the specific steps of establishing the multi-robotic arm system in step S1 include:

[0148] Determining multiple robotic arms; the multiple robotic arms include a leader robotic arm and multiple follower robotic arms;

[0149] Determining relevant parameters of the plurality of robotic arms; the relevant parameters of the plurality of robotic arms include an inertia matrix, a Coriolis force / centrifugal force matrix, a gravity matrix, and a comprehensive interference of each robotic arm;

[0150] A multi-robotic arm system is established based on relevant parameters of the multiple robotic arms.

[0151] Preferably, the specific steps of obtaining the fixed time observer of the multi-manipulator in step 2 include:

[0152] Design a corresponding original fixed-time observer for each follower robotic arm to observe the state of the leader robotic arm;

[0153] Obtaining the error state 1 of the state variable of each follower manipulator in the multi-manipulator system;

[0154] Perform vector rewriting on the state variables of the navigator manipulator corresponding to each original fixed-time observer to obtain rewriting vector 1;

[0155] Perform vector rewriting on the state variables of each follower manipulator corresponding to each original fixed-time observer to obtain rewriting vector 2;

[0156] Based on the Lipschitz continuity condition of the nonlinear term of the robotic arm, the error state one, vector one and vector two of the state variables of each follower robotic arm in the multi-robotic arm system, a fixed-time observer of each follower robotic arm is obtained to observe the state of the leader robotic arm; the fixed-time observer of each follower robotic arm is characterized as a fixed-time observer of the multi-robotic arm.

[0157] Preferably, the specific steps of obtaining the convergence time 1 are: designing a Lyapunov function 1 and obtaining a fixed time estimation parameter using Lyapunov's second method;

[0158] Using the fixed time theory and the fixed time estimation parameters, the convergence time of the fixed time observer is determined to obtain the convergence time 1; the convergence time 1 is the upper bound of the time for each follower manipulator to estimate the leader manipulator arm;

[0159] Preferably, the multi-manipulator system is expressed as:

[0160]

[0161] in, is the speed of the i-th robot arm in state x, is the position of the i-th robotic arm in state x, The velocity derivative of the i-th robot arm in state x, is the state of the i-th robotic arm, is a real number, n is the degree of freedom of the robot arm, g i is the nonlinear term of the i-th robotic arm, u i is the control input of the i-th robotic arm, d i is the comprehensive interference of the i-th robotic arm, i = 0, 1, 2, 3…N. When i = 0, it is the leader robotic arm, and when i = 1, 2, 3…N, it is the follower robotic arm.

[0162] Preferably, the Lipschitz continuity condition of the nonlinear term of the robotic arm is expressed as:

[0163]

[0164] in, is the nonlinear term of the i-th robot arm in state x, is the nonlinear term of the i-th robot in the x′ state, is the position of the i-th robotic arm in state x, is the speed of the i-th robot arm in state x, is the position of the i-th robot arm in the x′ state, is the velocity of the i-th manipulator in the x′ state, ψ1 is the Lipschitz constant related to the position state of the manipulator, ψ2 is the Lipschitz constant related to the velocity state of the manipulator; Ψ is the minimum value between the two, Ψ = min(Ψ1,Ψ2).

[0165] Preferably, the original fixed time observer of the multi-manipulator is expressed as:

[0166]

[0167] in, is the derivative of the position state variable of the i-th follower manipulator with respect to the leader manipulator, is the derivative of the velocity state variable estimate of the i-th follower manipulator with respect to the leader manipulator, The follower manipulator estimates the velocity state variable of the leader manipulator, sgn(·) represents the sign function, l1 is the low-power constant one, l2 is the low-power constant two, is the symbolic function constant one, The symbolic function term is constant 2, m1 is the high power term constant 1, m2 is the high power term constant 2, represents the communication interaction error between the manipulators associated with the i-th follower manipulator, represents the communication interaction error between the manipulators related to the velocity state of the i-th follower manipulator, Representation and variables The related nonlinear terms of the manipulator, p represents the low-power term parameter, and q represents the high-power term parameter.

[0168] Furthermore, the inertia g of the i-th robotic arm i The expression is:

[0169]

[0170] in, is the nonlinear term of the i-th robot arm in state x, is the nominal value of the inertia matrix of the i-th manipulator in the x state, is the nominal value of the Coriolis force and / or centrifugal force of the i-th robot arm in state x, is the nominal value of gravity of the i-th robot arm in state x, is the position of the i-th robotic arm in state x, is the state of the i-th robotic arm.

[0171] Furthermore, the nominal value of the inertia matrix of the i-th manipulator in the x state is Nominal value of the Coriolis force and / or centrifugal force of the i-th robot arm in state x Nominal value of gravity of the i-th robot arm in state x The expressions are:

[0172]

[0173] in, is the actual inertia value of the i-th robot arm in state x, is the inertia error value of the i-th robotic arm when it is in the x state, is the actual value of the Coriolis force and / or centrifugal force of the i-th robot arm in state x, is the error value of the Coriolis force and / or centrifugal force of the i-th robot arm in the x state, is the actual value of gravity of the i-th robot arm in state x, is the gravity error of the i-th robot arm in the x state.

[0174] Furthermore, the comprehensive interference d of the i-th robot arm i , the expression is:

[0175]

[0176] Among them, d i is the comprehensive disturbance of the i-th robot arm, δ is the sum of the robot arm model uncertainty and external disturbance, is the external disturbance, including mechanical friction, air resistance, sensor actuator error, i = 0, 1, 2, 3...N, the comprehensive disturbance satisfies ||d i || ∞ ≤ρ, ρ is the upper limit of the comprehensive interference and is a known positive constant.

[0177] Preferably, when i=0, the i-th robot is the navigator, and the control input of the i-th robot as the navigator is bounded, that is,

[0178] Preferably, the specific steps of obtaining the error state 1 of the state variable of each follower manipulator in the multi-manipulator system include:

[0179] Define the actual value of parameter k of the multi-manipulator system in state x as x k ,in, When i=1,2,3…N, the i-th robot is the follower robot; when i=0, the i-th robot is the leader robot; k=1,2; when k=1, x k Indicates the actual value of the position of the multi-manipulator system in the x state. When k = 2, xk Indicates the actual value of the speed of the multi-manipulator system in the x state;

[0180] Define the estimated value of parameter k of the multi-manipulator system in state x as

[0181] Define the actual value of parameter k when the robot arm as the navigator is in state x

[0182] The error value of the parameter k of the multi-manipulator system in the x state is obtained based on the estimated value of the parameter k of the multi-manipulator system in the x state and the actual value of the parameter k of the manipulator as the navigator in the x state.

[0183] The error value of parameter k of the multi-manipulator system in state x The expression is:

[0184]

[0185] It can be understood that the error value of the i-th robotic arm in the x state is the error value between the estimated value of the i-th robotic arm and the actual value of the robotic arm serving as the navigator in the x state;

[0186] Preferably, in step 2, the state variables of the navigator manipulator corresponding to each original fixed-time observer are rewritten by vectors to obtain a rewritten vector set 1;

[0187] The specific steps of rewriting the state variables of each follower manipulator corresponding to each original fixed-time observer into vectors to obtain the second rewriting vector set include:

[0188] Rewrite the control input of the navigator manipulator into a set of control input vectors u 0 Provides control input for the pilot's robotic arm;

[0189] Rewrite the comprehensive interference of the navigator manipulator into a set of comprehensive interference vectors d 0 Comprehensive interference for the navigator's robotic arm;

[0190] Rewrite the inertia vector of the navigator manipulator into the inertia vector set 1 g 0 is the inertia of the navigator's robotic arm;

[0191] Rewrite the inertia vector of each follower robot arm into inertia vector set 2 is the inertia of the Nth follower robot arm;

[0192] Furthermore, the expression of the fixed time observer of the multi-manipulator is:

[0193]

[0194] in, is the derivative of the position state variable error vector of the follower manipulator and the leader manipulator, is the derivative of the velocity state variable error vector of the follower manipulator and the leader manipulator, is the error vector of the velocity state variables of the follower manipulator and the leader manipulator, e1 represents the communication interaction error vector between the manipulators related to the position state of the manipulator, and e2 represents the communication interaction error vector between the manipulators related to the velocity state of the manipulator, Representation and variables and The associated set of nonlinear vectors of the manipulator, represents the nonlinear vector set of the navigator manipulator, p = 1-2 / μ, q = 1+2 / μ, μ is a positive constant three, μ>2; Input vector set for the navigator controller; The integrated interference vector for the navigator;

[0195] Furthermore, the expression of the convergence time 1 is:

[0196]

[0197] Where T0 is the convergence time, χ is the design parameter, χ>0, μ is a positive constant, is the upper bound of the fixed time.

[0198] Preferably, the specific steps of obtaining the second error state of multiple robotic arms in step 3 include:

[0199] Obtain the error state between the state of each follower manipulator in the multi-manipulator system and the state of the leader manipulator, and obtain the error state 1 of the multiple manipulators;

[0200] Based on the synchronization relationship between the joints of each robotic arm, the adjacent cross-coupling method is introduced into the error state one of multiple robotic arms to obtain the error state two of multiple robotic arms.

[0201] Preferably, the error state expression of the multiple robotic arms is:

[0202]

[0203] in, is the difference between the actual value of the position state variable of the i-th follower manipulator and the estimated value of the position state of the leader manipulator, is the difference between the actual value of the velocity state variable of the i-th follower manipulator and the estimated value of the velocity state of the leader manipulator.

[0204] The error state 2 of the multiple robotic arms is expressed as:

[0205]

[0206] Among them, T eq is a symmetric positive definite matrix, To use the matrix T eq Error status The state change of To use the matrix T eq Error status state change.

[0207] The expression of the robotic arm control model is:

[0208]

[0209] in, is the position-related error state vector of the follower robot after state transformation; is the speed-related error state vector of the follower robot after state transformation.

[0210] Furthermore, the symmetric positive definite matrix T eq The expression is:

[0211] T eq =I+λT,λ>0,

[0212] Where T is the parameter matrix, I is the identity matrix of the corresponding dimension, λ is a positive constant, λ>0.

[0213]

[0214] Preferably, the specific steps of obtaining the second convergence time in step S4 include:

[0215] A second Lyapunov function is designed according to the robot control model; and according to the second Lyapunov function and the fixed time theory, after the fixed time observer converges, the time for the sliding mode variables of each robot to converge to zero is obtained, and the convergence time 2 is obtained.

[0216] Preferably, the controller expression of the multi-manipulator system is:

[0217]

[0218] in, is the control input of the i-th robotic arm, is the auxiliary control item of the i-th robotic arm, is the non-singular fixed-time control term of the i-th robot arm, is the position error compensation item of the i-th robotic arm, is the speed error compensation term of the i-th robot arm, is the interference compensation term of the i-th robotic arm.

[0219] Preferably, the sliding mode variables of each robotic arm are expressed as follows:

[0220]

[0221] Among them, s i is the sliding mode variable of the i-th robotic arm, is the system variable related to the position state of the i-th robot arm, is the system variable related to speed, diag(·) is the diagonal function, κ(·) is the designed positive definite vector function, q1 is the high-power constant four, and p1 is the low-power constant four.

[0222] Preferably, the second convergence time includes: the convergence time T1 of the sliding mode variables of each manipulator in the controller of the multi-manipulator system after the fixed time observer converges and the time ε(τ) when the sliding mode variables of each manipulator enter the sliding mode region δ1 from the saturation critical region δ2;

[0223] The convergence time 3 includes the convergence time of the state variables of each manipulator after the fixed time observer and the sliding mode variables of each manipulator converge;

[0224] Furthermore, the convergence time T1 of the sliding mode variables of each robot arm is expressed as:

[0225]

[0226] Among them, α″2 is the design parameter one, β2 is the design parameter two and is a positive constant, n2 is the high-power term performance parameter two, m2 is the high-power term constant two, q2 is the high-power term parameter two, and p2 is the low-power term parameter two.

[0227] Furthermore, the expression for the time from the saturation critical region δ2 to the sliding mode region δ1 is:

[0228]

[0229] Where ε(τ) is the time when the sliding mode variable of the manipulator enters the sliding mode region δ1 from the saturation critical region δ2;

[0230] k represents the number of manipulators in the saturation critical region δ2, 1≤k≤N, 1≤j≤k, ε j (τ) represents the time when the j-th robot arm enters the sliding mode region δ1 from the saturation critical region δ2.

[0231] The expression of the convergence time T2 is:

[0232]

[0233] Among them, n1 is the high-power performance parameter one, m1 is the high-power constant one, β1 is the design parameter three, α1 is the design parameter four, and it is a positive constant.

[0234] Furthermore, the expression of the auxiliary control item of the i-th robotic arm is:

[0235]

[0236] in, represents the auxiliary control item of the i-th robot arm, γ=m1 / n1-p1 / q1>1, α1 is a positive constant, η=q1 / p1-1∈(0,1), and κ is a positive definite vector.

[0237] The expression of the non-singular fixed-time control term of the i-th manipulator is:

[0238]

[0239] Among them, μ τ (·) Compensation function designed to avoid non-singular terms, α2 is the design parameter five.

[0240] Preferably, the error compensation term of the i-th robotic arm is expressed as:

[0241]

[0242] Among them, u e1 is the error compensation term, and e1 represents the communication interaction error between the manipulators related to the position state of the manipulators.

[0243] Preferably, the expression of the position error compensation term of the i-th robotic arm is:

[0244]

[0245] Among them, u e2 It is also the error compensation term, is the nonlinear term of the manipulator related to the observer observation value, is a low-power constant of two, l2 is a low-power constant of one, and e2 represents the communication interaction error between the manipulators related to the velocity state of the manipulator.

[0246] Preferably, the interference compensation term of the robotic arm is expressed as:

[0247]

[0248] Among them, u d represents the interference compensation term, g i is the nonlinear term of the manipulator, and ρ is the upper limit of the comprehensive interference of the manipulator.

[0249] According to the fixed time theory, after the fixed time observer of the multi-manipulator and the sliding mode variables of each manipulator converge, the state variables of each manipulator will be The inner convergence to zero can ensure that the entire multi-manipulator system will achieve consistent tracking control of the leader within a fixed time.

[0250] Preferably, the upper limit of the multi-manipulator convergence time in step S5 is expressed as:

[0251] T≤T max =T o +T1+T2+ε(τ)

[0252] Where T is the actual convergence time of the state variables of the multi-manipulator system, T max is the upper bound of the convergence time, T0 is the convergence time one of the fixed time observer, T1+ε(τ) is the convergence time two, ε(τ) is the time when the sliding mode variable of each robot arm enters the sliding mode region from the saturation critical region, and T2 is the convergence time three.

[0253] Example 1

[0254] Build Figure 3 The multi-manipulator system shown here consists of a leader and five follower manipulators.

[0255] Each robot arm is simulated using a 2-DOF robot arm. The dynamic model of each robot arm is the same, as shown below:

[0256]

[0257] in are the masses of the two connecting rods of the robotic arm respectively, and it is assumed that the two connecting rods have the same length, and l represents the length of the connecting rod.

[0258] The state variable of the i-th robot arm is

[0259] The controller parameters and robot arm parameters are shown in the following table:

[0260] Table 1. Controller parameters and robot arm parameters

[0261]

[0262]

[0263] where p i ,i=1,...,5 represents the actual parameter value of the robot arm, Indicates the nominal parameter value of the robot arm. The initial value of the robot arm system is set to: x 10 =[020.51.5111.50.520] T The initial values ​​of the two joint positions of the five follower manipulators are represented by x. The initial values ​​of the two joint velocities of the five follower manipulators are represented by x. 20 =[1.50.11.50.120.12.50.12.50.1] T , the initial position and velocity of the navigator's manipulator are set to

[0264] The fixed time observer designed by the present invention observes the position of the two joints of the navigator's manipulator as follows: Figure 4 As shown in the figure, the position observation results converge at about 0.02s, and the speed observation results of the two joints of the navigator's manipulator are as follows: Figure 5 As shown in the figure, it can be seen that the velocity observation results converge at about 0.04s.

[0265] By adopting the multi-manipulator fixed time axis consistent cooperative controller of the present invention, the position tracking trajectory of the two joints of the multi-manipulator system is as follows: Figure 6 As shown in the figure, the position tracking of the five followers completes convergence in about 0.1s, and the five agents catch up with the leader almost at the same time. The speed tracking trajectory of the two joints of the multi-manipulator system is as follows: Figure 7 As shown in the figure, the speed tracking of the five followers completes convergence in about 0.2s, and the five agents catch up with the leader almost at the same time, which verifies the correctness of the fixed time axis collaborative control algorithm of the present invention.

[0266] The above description is only a preferred specific embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any changes or substitutions that can be easily thought of by any technician familiar with this technical field within the technical scope disclosed by the present invention should be covered by the scope of protection of the present invention.

Claims

1. A method for consistent coordinated control of fixed time axes applied to multiple robotic arms, characterized in that: include: Step S1: determining multiple robotic arms and related parameters thereof; the multiple robotic arms include a leader robotic arm and multiple follower robotic arms; Establishing a multi-manipulator system based on relevant parameters of the multiple manipulators; Step S2: Obtain the error state 1 of the state variable of each follower manipulator in the multi-manipulator system; design a manipulator fixed time observer based on the error value of the state variable of each follower manipulator and the leader manipulator; Determine the upper bound of the convergence time of the fixed-time observer to obtain a convergence time of one; Step S3: Considering the synchronization relationship between the joints of each robotic arm, the state of each robotic arm in the multi-robotic arm system is transformed to obtain the second error state of the multi-robotic arm; Step S4: designing the sliding mode variables of each follower robotic arm based on the error state 2 of the multiple robotic arms; Establishing a controller for the multi-manipulator system based on the sliding mode variables of each manipulator, and obtaining the time for each follower manipulator sliding mode variable to converge to zero after the fixed time observer converges, thereby obtaining convergence time 2; After the fixed time observer converges and the sliding mode variables of each follower manipulator converge, the time for each follower manipulator state variable to converge to zero is obtained, and the convergence time three is obtained; Step S5, determining an upper limit of the multi-manipulator convergence time based on the convergence time 1, the convergence time 2, and the convergence time 3; The controller based on the multi-manipulator system obtains the control input of each manipulator; Based on the control input of each robotic arm and the upper limit of the convergence time of the multi-robotic arms, the follower robotic arm tracks the movement of the leader robotic arm within a predetermined time range, thereby achieving the goal of consistent collaborative control of the multi-robotic arms.

2. The method for consistent collaborative control between fixed time axes according to claim 1, characterized in that: The specific steps of obtaining the fixed-time observer of the multi-manipulator described in step 2 include: Design a corresponding original fixed-time observer for each follower robotic arm to observe the state of the leader robotic arm; Obtaining the error state 1 of the state variable of each follower manipulator in the multi-manipulator system; Perform vector rewriting on the state variables of the navigator manipulator corresponding to each original fixed-time observer to obtain a rewriting vector set 1; Perform vector rewriting on the state variables of each follower manipulator corresponding to each original fixed-time observer to obtain a second rewriting vector set; Based on the Lipschitz continuity condition of the nonlinear term of the robotic arm, the error state one, vector one and vector two of the state variables of each follower robotic arm in the multi-robotic arm system, a fixed-time observer of each follower robotic arm is obtained to observe the state of the leader robotic arm; the fixed-time observer of each follower robotic arm is characterized as a fixed-time observer of the multi-robotic arm.

3. The method for consistent collaborative control between fixed time axes according to claim 1, characterized in that: The expression of the fixed time observer of the multi-manipulator is: in, is the derivative of the position state variable error vector of the follower manipulator and the leader manipulator, is the derivative of the velocity state variable error vector of the follower manipulator and the leader manipulator, is the error vector of the velocity state variables of the follower manipulator and the leader manipulator, e1 represents the communication interaction error vector between the manipulators related to the position state of the manipulator, and e2 represents the communication interaction error vector between the manipulators related to the velocity state of the manipulator, Represents a set of nonlinear vectors related to the robotic arm, represents the nonlinear vector set of the navigator manipulator, Input vector set for the navigator controller; is the integrated interference vector of the navigator; sgn(·) represents the sign function, l1 is the low power constant one, l2 is the low power constant two, is the symbolic function constant one, The symbolic function term constant is two, m1 is the high-power term constant one, m2 is the high-power term constant two, p represents the low-power term parameter, and q represents the high-power term parameter.

4. The method for consistent collaborative control between fixed time axes according to claim 1, characterized in that: The specific steps of obtaining the error state 2 of multiple robotic arms in step 3 include: Obtain the error state between the state of each follower manipulator in the multi-manipulator system and the state of the leader manipulator, and obtain the error state 1 of the multiple manipulators; Based on the synchronization relationship between the joints of each robotic arm, the adjacent cross-coupling method is introduced into the error state one of multiple robotic arms to obtain the error state two of multiple robotic arms.

5. The method for consistent collaborative control between fixed time axes according to claim 4, characterized in that: Step S3 further includes: obtaining a robotic arm synchronization control model based on a fixed time observer and error states 2 of the plurality of robotic arms.

6. The method for consistent coordinated control between fixed time axes according to claim 5, characterized in that: The specific steps of obtaining the second convergence time in step S4 include: A second Lyapunov function is designed according to the robot control model; and according to the second Lyapunov function and the fixed time theory, after the fixed time observer converges, the time for the sliding mode variables of each robot to converge to zero is obtained, and the convergence time 2 is obtained.

7. The method for consistent coordinated control between fixed time axes according to claim 1, wherein: The controller expression of the multi-manipulator system in step S4 is: Among them, ~u i is the control input of the i-th robotic arm, is the auxiliary control item of the i-th robotic arm, is the non-singular fixed-time control term of the i-th robot arm, is the position error compensation item of the i-th robotic arm, is the speed error compensation term of the i-th robot arm, is the interference compensation term of the i-th robotic arm.

8. The method for consistent coordinated control between fixed time axes according to claim 7, characterized in that: The sliding mode variables of each manipulator are expressed as follows: Among them, s i is the sliding mode variable of the i-th robotic arm, is the system variable related to the position state of the i-th robot arm, is the system variable related to speed, diag(·) is the diagonal function, κ(·) is the designed positive definite vector function, q1 is the high-power constant four, and p1 is the low-power constant four.

9. The method for consistent coordinated control between fixed time axes according to claim 8, characterized in that: The second convergence time is the time it takes for the sliding mode variables of each manipulator in the controller of the multi-manipulator system to enter the sliding mode region from the saturation critical region after the fixed time observer converges, and the convergence time of the sliding mode variables of each manipulator; The convergence time three is the convergence time of the state variables of each robotic arm obtained after the fixed time observer and the sliding mode variables of each robotic arm converge.

10. The method for consistent coordinated control between fixed time axes according to claim 9, characterized in that: The upper limit of the multi-manipulator convergence time is expressed as: T≤T max =T o +T1+T2+e(t) Where T is the actual convergence time of the state variables of the multi-manipulator system, T max is the upper bound of the convergence time, T0 is the convergence time one of the fixed-time observer, T1+ε(τ) is the convergence time two, ε(τ) is the time when the sliding mode variable of each robot arm enters the sliding mode region from the saturation critical region, T2 is the convergence time three, and T1 is the convergence time of the sliding mode variable of each robot arm in the controller of the multi-robot system.

Images