A Control Method for Time-Delay Dual-Arm Cooperative Robots Based on Udwadia-Kalaba and Cooperative Game Theory

Abstract

The present invention relates to the field of robot control technology, and particularly to a control method for a time-delay dual-arm collaborative robot based on the Udwadia-Kalaba and cooperative game theories, which mainly includes the following steps: According to the uncertainty of the system, obtain the control input for which the dual-arm collaborative robot system obeys the servo constraint, and adopt the method of time-delay estimation to simplify the dynamic model to achieve time-delay estimation and dual-arm collaborative impedance control. According to the time-varying uncertainty existing in the system and combined with the adaptive update law, establish an adaptive robust controller, and obtain the optimal control parameters of the robust controller through the method of cooperative game. The method recorded in the present invention can optimize the control parameters and achieve performance maximization, not only ensuring that the robot can effectively balance between different performance objectives, but also promoting the optimization of the overall control performance of the system.

Description

Technical Field

[0001] The present invention relates to the technical field of robot control, and particularly to a control method for a time-delay dual-arm collaborative robot based on the Udwadia-Kalaba and cooperative game theories. Background Art

[0002] In recent years, robotic manipulators have played an important role in industrial applications. Compared with a single manipulator, multiple manipulators have a greater load-bearing capacity and can perform tasks with more functions. When multiple manipulators jointly operate an object, a set of kinematic and dynamic constraints are imposed on the motion of the manipulators due to the formation of a closed kinematic chain. Therefore, compared with the control of a single robotic manipulator, the control of multiple robotic manipulators is much more complex.

[0003] To solve the problems of complete and nonholonomic trajectory tracking, the U-K theory is adopted to solve this problem. The remarkable feature of the U-K theory is that it can solve both complete constraint problems and nonholonomic constraint problems. In addition, no linearization of the nonlinear system is required, nor are quasi-variables and auxiliary variables. For these reasons, we apply the U-K theory to the tracking control of multiple manipulators. However, only controlling the overall system through the U-K theory will result in system instability problems because the inevitable uncertainties in the system are not taken into account, and the existence of uncertainties will limit the control performance of the system.

[0004] In the design process of robust control, the description of uncertainties is a crucial process. Generally speaking, the commonly used uncertainty description methods can be classified into three types: deterministic methods, probabilistic methods, and fuzzy methods. Among them, the fuzzy method is another uncertainty description method. So far, the application of the fuzzy method has mainly focused on the control design based on fuzzy logic theory. In order to further optimize the control of the dual-arm collaborative robot, a control method for a time-delay dual-arm collaborative robot based on the Udwadia-Kalaba and cooperative game theories is also required. Summary of the Invention

[0005] In view of the deficiencies of the existing technology, the present invention provides the following technical solutions:

[0006] A control method for a time-delay dual-arm collaborative robot based on the Udwadia-Kalaba and cooperative game theories includes the following steps:

[0007] S10. Establish a dynamic model of the dual-arm collaborative robot system.

[0008] S20. Based on Udwadia-Kalaba, transform the trajectory tracking control into a servo constraint on the dual-arm collaborative robot system.

[0009] S30. Obtain the control input for which the dual-arm collaborative robot system conforms to the servo constraint according to the uncertainty of the system.

[0010] S40. Simplify the dynamic model by using the time delay estimation method to achieve time delay estimation-based and dual-manipulator collaborative impedance control.

[0011] S50. Establish an adaptive robust controller according to the time-varying uncertainty existing in the system in combination with the adaptive update law.

[0012] S60. Obtain the optimal control parameters of the robust controller through the method of cooperative game.

[0013] As an improvement of the above technical solution, the dynamic model includes the following formula:

[0014]

[0015] where t is the time in the dual-arm collaborative robot system, β represents the position of the joints in the dual-arm collaborative robot system, represents the velocity of the joints in the dual-arm collaborative robot system, represents the acceleration of the joints in the dual-arm collaborative robot system, γ represents the uncertain parameters in the mechanical system, H is the inertia matrix of the system, C is the Coriolis / centrifugal force of the joint module system, g is the gravity term of the system, and u is the control torque.

[0016] As an improvement of the above technical solution, the step S20 includes the following steps:

[0017] S21: Obtain the motion trajectory of two planar two-degree-of-freedom manipulators grasping an object, convert the trajectory into motion constraints and represent them in matrix form.

[0018] S22: Obtain the first-order constraints and the second-order constraints according to the motion constraints in matrix form of the motion trajectory.

[0019] As an improvement of the above technical solution, the motion trajectory of the two planar two-degree-of-freedom manipulators grasping an object includes the following formula:

[0020]

[0021] The first-order constraint is expressed as the following formula:

[0022]

[0023] The second-order constraint is expressed as the following formula:

[0024]

[0025] Among them, Ψ is the constraint matrix, t is the time in the dual-arm collaborative robot system, and β represents the positions of the joints in the dual-arm collaborative robot system. represents the velocity of the joints in the dual-arm collaborative robot system. represents the acceleration of the joints in the dual-arm collaborative robot system, a = [a1, a2, …, a h T is the desired second-order constraint, b = [b1, b2, …, b h T is the desired first-order constraint.

[0026] As an improvement to the above technical solution, the acquisition of the control input depends on the following formula:

[0027]

[0028] Among them, the symbol “+” represents the Moore-Penrose inverse, H is the inertia matrix of the system, C is the Coriolis / centrifugal force of the joint module system, and g is the gravity term of the system.

[0029] As an improvement to the above technical solution, the step S40 includes the following steps:

[0030] S41: Introduce an estimated value of the inertia matrix and obtain the nonlinear part in the dynamic model according to this estimated value.

[0031] S42: When the sampling time interval is small enough, estimate the state of the system according to the actual angular acceleration of the manipulator joints under the action of the control torque at the previous moment, and obtain the dynamic equation based on time-delay estimation.

[0032] Among them, the dynamic equation based on time-delay estimation includes the following formula:

[0033]

[0034] Among them, is a positive definite constant diagonal matrix, ΔN is defined as the time-delay estimation error, represents 's estimated value, β represents the positions of the joints in the dual-arm collaborative robot system, represents the velocity of the joints in the dual-arm collaborative robot system, represents the acceleration of the joints in the dual-arm collaborative robot system, is the nonlinear part in the dynamic model, L is the sampling time interval, t is the time in the dual-arm collaborative robot system, and u is the control torque.

[0035] As an improvement to the above technical solution, the step S50 includes the following steps:

[0036] ​​S51: Obtain the servo constraint following error value, and obtain the adaptive update law by means of estimation.

[0037] S52: Based on the adaptive update law, establish an adaptive robust controller.

[0038] As an improvement of the above technical solution, the obtaining of the servo constraint following error value depends on the following formula:

[0039]

[0040] where, Ψ is the constraint matrix, t is the time in the dual-arm cooperative robot system, β represents the position of the joints in the dual-arm cooperative robot system, represents the velocity of the joints in the dual-arm cooperative robot system, and b is the second-order constraint.

[0041] The obtaining of the adaptive update law depends on the following formula:

[0042]

[0043] where, k1, ξ1, ξ2 are all control parameters in the adaptive law, are positive constants, is a preset function, χ is the servo constraint following error, and β represents the position of the joints in the dual-arm cooperative robot system, represents the velocity of the joints in the dual-arm cooperative robot system, and t is the time in the dual-arm cooperative robot system.

[0044] The adaptive robust controller includes the following formula:

[0045]

[0046] where, τ0 and τ1 are the control terms of the controller established based on the Udwadia-Kalaba principle, τ2 is the control term of the controller obtained from time delay estimation and dual-arm cooperative impedance control, τ3 is the control term for adaptively compensating system uncertainties, is the adaptive update law.

[0047] As an improvement of the above technical solution, the step S60 includes the following steps:

[0048] S61: Establish a cost function model of the control parameters at any time;

[0049] S62: Perform D-operation on the cost function model;

[0050] S63: Optimize the control parameters of the cost function model by using the method of cooperative game, and obtain the minimum performance index;

[0051] S64: Obtain the optimal control parameter corresponding to the minimum performance metric.

[0052] As an improvement to the above technical solution, the cost function model in step S61 includes the following formula:

[0053]

[0054] Among them, J1 reflects the cumulative error of the system from the initial moment to infinity, J2 reflects the error level of the system at infinity, reflects the transient performance of the system, reflects the steady-state performance of the system, t S is an arbitrary initial moment, k1 and k2 are constant control parameters, is the adaptive update law.

[0055] The acquisition method of the minimum performance metric depends on the following formula:

[0056]

[0057] Among them, γ1 > 0 and γ2 > 0, and γ1 + γ2 = 1, is the optimal control parameter

[0058] Advantages of the present invention:

[0059] The method recorded in the present invention can optimize control parameters to achieve maximum performance. The performance metric constructed based on cooperative game theory uses the "Pareto optimal" method to find the optimal control parameter that can balance two key performance metrics. This process not only ensures effective trade-offs between different performance goals of the robot but also promotes the optimization of the overall control performance of the system. Compared with traditional methods, the present invention can provide more refined control parameter settings, enabling the robot to perform optimally under various operating conditions. Description of the Drawings

[0060] Figure 1 is a schematic structural diagram of a two-degree-of-freedom planar manipulator for controlling the same object in the present invention. Detailed Embodiments

[0061] In order to make the objectives, technical solutions, and advantages of the present invention clearer, the present invention will be further described in detail below. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not used to limit the present invention.

[0062] In the design process of robust control, the description of uncertainty is a crucial process. Generally speaking, the commonly used uncertainty description methods can be classified into three types: deterministic method, probability method, and fuzzy method. Among them, the fuzzy method is another uncertainty description method. So far, the application of the fuzzy method has mainly focused on the control design based on fuzzy logic theory. In order to further optimize the control of the dual-arm collaborative robot, a control method for the time-delay dual-arm collaborative robot based on Udwadia-Kalaba and cooperative game theory is also required.

[0063] Before explaining the specific scheme of this embodiment, it is also necessary to calculate the control parameters for two planar two-degree-of-freedom manipulators to control a common object along the expected trajectory to illustrate the performance of the proposed controller.

[0064] Suppose the base positions of the two robotic manipulators are (0, 0) and (1, 0) respectively. Figure 1 where l ij , and I ij (i, j = 1, 2) represent the length, mass, and inertia of the j-th link of the i-th robotic manipulator respectively.

[0065] Define β ij as the joint angle of the j-th link of the i-th robotic manipulator, and the position and orientation of the centroid of the grasped object are p o = [x o , y o , θ] T , and the position of the end effector of the i-th robotic manipulator is p i = [x i , y i T . The kinematic relationship between the joint angle of each robotic manipulator in the joint space and the position of the end effector in the Cartesian space is:

[0066]

[0067] where [d i , 0] T is the base position of the i-th robotic manipulator, x i is the x-coordinate position of the i-th robotic manipulator, and y i is the y-coordinate position of the i-th robotic manipulator.

[0068] The kinematic relationship between the position and orientation p i of the end effector and the position and orientation p o of the centroid of the grasped object is:

[0069] ​

[0070] Based on the above theory, the specific solution of this embodiment includes the following steps:

[0071] S10. Establish the dynamic model of the dual-arm collaborative robot system.

[0072] The dynamic model includes the following formula:

[0073]

[0074] where t is the time in the dual-arm collaborative robot system, β represents the position of the joints in the dual-arm collaborative robot system, represents the velocity of the joints in the dual-arm collaborative robot system, represents the acceleration of the joints in the dual-arm collaborative robot system, γ represents the uncertain parameters in the mechanical system, H is the inertia matrix of the system, C is the Coriolis / centrifugal force of the joint module system, g is the gravity term of the system, and u is the control torque.

[0075] S20. Based on Udwadia-Kalaba, transform the trajectory tracking control into the servo constraint of the dual-arm collaborative robot system.

[0076] Specifically, the step S20 includes the following steps:

[0077] S21: Obtain the motion trajectory of two planar two-degree-of-freedom manipulators grasping an object, transform the trajectory into a motion constraint and represent it in matrix form.

[0078] S22: Obtain the first-order constraint and the second-order constraint according to the motion constraint in matrix form of the motion trajectory.

[0079] As Figure 1 shown, the motion trajectory of the two planar two-degree-of-freedom manipulators grasping an object includes the following formula:

[0080]

[0081] The first-order constraint is expressed as the following formula:

[0082]

[0083] Derive formula 2 to obtain its second-order constraint, which is specifically expressed as the following formula:

[0084]

[0085] where Ψ is the constraint matrix, t is the time in the dual-arm collaborative robot system, β represents the position of the joints in the dual-arm collaborative robot system, represents the velocity of the joints in the dual-arm collaborative robot system, Denote the acceleration of the joints in the dual-arm collaborative robot system, \(a = [a_1, a_2, \ldots, a h T is the desired second-order constraint, \(b = [b_1, b_2, \ldots, b h T is the desired first-order constraint.

[0086] S30. Obtain the control input for the dual-arm collaborative robot system to comply with the servo constraint according to the uncertainty of the system.

[0087] If it is assumed that when \(\varPsi(\beta, t)\) and are known, then Equation 3 is also known, or if \(\varPsi(\beta, t)\) and are known, then there is at least one solution in Equation 3

[0088] Considering the case where the unconstrained system is subject to constraints, the acquisition of the control input depends on the following formula:

[0089]

[0090] where the symbol \("+\)" represents the Moore-Penrose inverse, \(H\) is the inertia matrix of the system, \(C\) is the Coriolis / centrifugal force of the joint module system, and \(g\) is the gravity term of the system.

[0091] S40: Simplify the dynamic model by using the time-delay estimation method to achieve time-delay estimation and dual-arm collaborative impedance control.

[0092] Specifically, it includes the following steps:

[0093] S41: Introduce an estimated value of the inertia matrix and obtain the non-linear part in the dynamic model according to this estimated value.

[0094] To obtain the relationship between the control torque \(u\) and the joint angular acceleration, an estimated value of an inertia matrix is introduced Substitute it into Equation 1 and deform it to get:

[0095]

[0096] In the formula, is a positive definite constant diagonal matrix, \(\beta\) represents the position of the joints in the dual-arm collaborative robot system, represents the velocity of the joints in the dual-arm collaborative robot system, represents the acceleration of the joints in the dual-arm collaborative robot system, is the non-linear part in the dynamic model, which can be set by the user in practical applications and satisfies the following formula:

[0097] ​​

[0098] For the non - linear part in the dynamic model Utilize its partially linear characteristics and adopt time - delay estimation to make a relatively accurate estimation of this part.

[0099] When the sampling time interval \(L\) is small enough, the non - linear part of the manipulator dynamic model at a certain moment can be approximately regarded as:

[0100]

[0101] where, denotes the estimated value of.

[0102] According to the dynamic model in step S10, we can obtain

[0103]

[0104] where, is a positive definite constant diagonal matrix, \(\Delta N\) is defined as the time - delay estimation error, \(\beta\) represents the position of the joints in the dual - arm collaborative robot system, represents the velocity of the joints in the dual - arm collaborative robot system, represents the acceleration of the joints in the dual - arm collaborative robot system, is the non - linear part in the dynamic model, \(L\) is the sampling time interval, and \(t\) is the time in the dual - arm collaborative robot system.

[0105] S42: When the sampling time interval is small enough, estimate the system state based on the actual angular acceleration of the manipulator joints under the action of the control torque at the previous moment, and obtain the dynamic equation based on time - delay estimation.

[0106] Among them, the dynamic equation based on time - delay estimation includes the following formula:

[0107]

[0108] where, is defined as the time - delay estimation error.

[0109] Based on the above reasons, the time - delay estimation error compensation controller based on desired velocity feedback can be designed as:

[0110]

[0111] where, \(\beta\) d 、 are respectively the desired position and desired acceleration of the joints in the multi - arm robot system, \(\lambda\in R\) 2n×2nis a constant diagonal matrix, v is an intermediate quantity, e is the tracking error of the joint angle, and e = β d -β.

[0112] Based on the above method, it is also necessary to consider the time-varying uncertainties existing in the system. Such uncertainties are bounded and can be characterized by fuzzy set theory to achieve the uniform boundedness and uniform ultimate boundedness of the tracking error. The specific implementation steps are S50.

[0113] S50: Establish an adaptive robust controller according to the time-varying uncertainties existing in the system in combination with the adaptive update law.

[0114] Generally, due to load variations, measurement errors, and external disturbances, it is difficult to accurately obtain the physical parameters of the cooperative robot system. Therefore, the matrices H(β, γ, t), and g(β, γ, t) can be decomposed as:

[0115] H(β, γ, t) = H0(β, t) + H Δ (β, γ, t)

[0116]

[0117] g(β, γ, t) = g0(β, t) + g Δ (β, γ, t)

[0118] where H0, C0, and g0 represent the nominal parts of the dual-arm cooperative robot, and H Δ , C Δ and g Δ represent the uncertain parts of the dual-arm cooperative robot, and H0, C0, g0, H Δ , C Δ and g Δ are all continuous.

[0119] For the above reasons, specifically, the step S50 includes the following steps:

[0120] S51: Obtain the servo constraint following error value and obtain the adaptive update law by estimation;

[0121] S52: Establish an adaptive robust controller based on this adaptive update law.

[0122] The acquisition of the servo constraint following error value depends on the following formula:

[0123]

[0124] where Ψ is the constraint matrix, t is the time in the dual-arm cooperative robot system, and β represents the position of the joints in the dual-arm cooperative robot system, Denote the speed of the joints in the dual-arm collaborative robot system, and \(b\) is the desired first-order constraint.

[0125] Adaptive update law is obtained depending on the following formula:

[0126]

[0127] where \(k_1, \xi_1, \xi_2\) are all control parameters in the adaptive law, and are positive constants. is a preset function, \(\chi\) is the servo constraint following error, and

[0128] The adaptive robust controller includes the following formula:

[0129]

[0130] Specifically,

[0131]

[0132]

[0133] where,

[0134]

[0135] where \(t\) is the time in the dual-arm collaborative robot system, \(H_0\) represents the inertia matrix of the nominal part of the dual-arm collaborative robot, \(C_0\) represents the Coriolis / centrifugal force of the nominal part of the joint module system of the dual-arm collaborative robot, and \(g_0\) represents the gravity term of the nominal part of the dual-arm collaborative robot system. \(s, \omega\) are intermediate variables derived during the calculation process, \(\tau_0\) and \(\tau_1\) are control terms of the controller established based on the Udwadia-Kalaba principle, \(\tau_2\) is the control term of the controller obtained from time delay estimation and dual-arm collaborative impedance control, \(\tau_3\) is the control term for adaptively compensating system uncertainties, \(Q \in R\) 2×n is a positive definite matrix, \(\Theta\) is a preset function, \(k_1\) and \(k_2\) are constant control parameters, and \(k_1 > 0\).

[0136] S60. Obtain the optimal control parameters of the robust controller through the method of cooperative game.

[0137] According to Equation 4 and its derivation process, it can be obtained that the magnitude of the control input is affected by the control parameters \(v\) and \(\in\). In addition, the size of the uniformly bounded region is also affected by the control parameters \(k_1\) and \(k_2\). Therefore, in order to achieve an acceptable balance between the control effort and the system performance, it is necessary to seek the optimal \(k_1\) and \(k_2\).

[0138] Specifically, step S60 includes the following steps:

[0139] S61: Establish a cost function model of control parameters at any time.

[0140] The cost function model in step S61 includes the following formula:

[0141]

[0142] Among them, J1 reflects the cumulative error of the system from the initial moment to infinity, and J2 reflects the error level of the system at infinity. reflects the transient performance of the system, reflects the steady-state performance of the system, t S is any initial moment, k1 and k2 are constant control parameters, is the adaptive update law.

[0143] S62: Perform a D-operation on the cost function model.

[0144] S63: Optimize the control parameters of the cost function model using the method of cooperative game and obtain the minimum performance index.

[0145] S64: Obtain the optimal control parameters corresponding to this minimum performance index.

[0146] Since there is a fuzzy part in the cost function, a D-operation is adopted (the D-operation can be regarded as a defuzzification method). By performing a D-operation on the above formula, we can get:

[0147]

[0148] Among them, the performance index J1 reflects the cumulative error of the system from the initial moment t s to infinity, considering the deviation of the system state, such as position, speed, etc., and the influence of control parameters k1 and k2 on these deviations, where σ1, σ2, σ3, σ4, σ5, σ6, σ7, σ8, σ9, σ 10 are adaptive parameters that satisfy uniform boundedness;

[0149] Similarly, we can get

[0150]

[0151] From this, it can be obtained that the performance index J2 reflects the error level of the system at infinity, that is, the final error state of the system, which is usually related to the steady-state performance of the system and measures whether the system can reach or approach the desired state after long-term operation. Among them, λ1, λ2, λ3, λ4, λ5, λ6 are adaptive parameters that satisfy uniform boundedness;

[0152] The problem of optimizing control parameters can be solved by the following cooperative game.

[0153] First, the optimal For any There is:

[0154]

[0155] To find Use Lemma 1: The decision pair is Pareto optimal if there exist weighting factors γ1>0 and γ2>0, and γ1 + γ2 = 1, such that:

[0156]

[0157] where the cost function is as follows:

[0158]

[0159] Therefore, the problem of optimizing control parameters is transformed into a constrained optimization problem, that is

[0160] For any time t s and the given decision set Satisfy

[0161]

[0162] To obtain this solution, apply the partial differential operator to J(k1,k2) to get:

[0163]

[0164] Next, make

[0165]

[0166]

[0167] where k1 ∈ D1, k2 ∈ D2.

[0168] Subsequently, Equation 5 is minimized by the solutions of Equations 6 and 7, and this solution is the Pareto optimal solution;

[0169] Based on the above analysis, the minimum performance index obtained after calculation is expressed as follows:

[0170]

[0171] where γ1>0 and γ2>0, and γ1 + γ2 = 1, are the optimal control parameters.

[0172] After calculating the minimum performance indicators, the optimal control parameters among them are applied to the controller to complete the uncertainty control of the cooperative robot system.

[0173] The above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit it.

Claims

1. A time-delay dual-arm collaborative robot control method based on Udwadia-Kalaba and cooperative game theory, characterized in that: The steps include: S10. Establish a dynamic model of the dual-arm collaborative robot system; S20, based on Udwadia-Kalaba, the trajectory tracking control is converted into a servo constraint for the dual-arm collaborative robot system; S30, obtaining a control input of the dual-arm collaborative robot system that obeys the servo constraint according to the uncertainty of the system; S40, simplifying the dynamic model by using a time delay estimation method to achieve dual-manipulator coordinated impedance control based on time delay estimation; S50, establishing an adaptive robust controller according to the time-varying uncertainty existing in the system and combining the adaptive update law; S60, obtaining optimal control parameters of the robust controller through a cooperative game method; The step S60 includes the following steps: S61: Establishing a cost function model of control parameters at any time; S62: performing D-operation on the cost function model; S63: Using cooperative game method to optimize the control parameters of the cost function model and obtain the minimum performance index; S64: Obtaining the optimal control parameter corresponding to the minimum performance index; The cost function model in step S61 includes the following formula: in, It reflects the cumulative error of the system from the initial moment to infinite time. Reflects the error level of the system at infinity, It reflects the transient performance of the system. It reflects the steady-state performance of the system. For any initial time, and Constant control parameters, Adaptive update law; The method for obtaining the minimum performance index depends on the following formula: ； in, and ,and , , is the optimal control parameter.

2. The time-delay dual-arm collaborative robot control method based on Udwadia-Kalaba and cooperative game theory according to claim 1 is characterized in that: The kinetic model includes the following formula: in, is the time in the dual-arm collaborative robot system, represents the position of the joints in the dual-arm collaborative robot system, represents the speed of the joints in the dual-arm collaborative robot system, represents the acceleration of the joints in the dual-arm collaborative robot system, represents the uncertain parameters in the mechanical system, is the inertia matrix of the system, is the Coriolis / centrifugal force of the joint module system, is the gravity term of the system, It is the control torque.

3. The time-delay dual-arm collaborative robot control method based on Udwadia-Kalaba and cooperative game theory according to claim 1 is characterized in that: The step S20 comprises the following steps: S21: obtaining the motion trajectory of two planar two-degree-of-freedom manipulators grasping an object, converting the trajectory into motion constraints and expressing it in matrix form; S22: Obtain first-order constraints and second-order constraints according to the motion constraints in matrix form of the motion trajectory.

4. The time-delay dual-arm collaborative robot control method based on Udwadia-Kalaba and cooperative game theory according to claim 3 is characterized in that: The motion trajectories of the two planar two-degree-of-freedom manipulators grasping objects include the following formula: The first-order constraint is expressed as follows: The second-order constraint is expressed as follows: in, is the constraint matrix, is the time in the dual-arm collaborative robot system, represents the position of the joints in the dual-arm collaborative robot system, represents the speed of the joints in the dual-arm collaborative robot system, represents the acceleration of the joints in the dual-arm collaborative robot system, is the expected second-order constraint, is the expected first-order constraint.

5. The time-delay dual-arm collaborative robot control method based on Udwadia-Kalaba and cooperative game theory according to claim 4 is characterized in that: The acquisition of the control input depends on the following formula: The symbol "+" indicates the Moore-Penrose inverse. is the inertia matrix of the system, is the Coriolis / centrifugal force of the joint module system, is the gravity term of the system.

6. The time-delay dual-arm collaborative robot control method based on Udwadia-Kalaba and cooperative game theory according to claim 1 is characterized in that: The step S40 includes the following steps: S41: introducing an estimated value of the inertia matrix and obtaining the nonlinear part of the dynamic model based on the estimated value; S42: When the sampling time interval is small enough, the state of the system is estimated according to the actual angular acceleration of the manipulator joint under the control torque at the previous moment, and a dynamic equation based on time delay estimation is obtained; wherein the dynamic equation based on time delay estimation includes the following formula: in, is a positive constant diagonal matrix, is defined as the time delay estimation error, express The estimated value of represents the position of the joints in the dual-arm collaborative robot system, represents the speed of the joints in the dual-arm collaborative robot system, represents the acceleration of the joints in the dual-arm collaborative robot system, is the nonlinear part of the dynamic model, is the sampling time interval, is the time in the dual-arm collaborative robot system.

7. The time-delay dual-arm collaborative robot control method based on Udwadia-Kalaba and cooperative game theory according to claim 1, characterized in that: The step S50 comprises the following steps: S51: obtaining a servo constraint following error value, and obtaining an adaptive update law by estimating the value; S52: Establishing an adaptive robust controller based on the adaptive update law.

8. The time-delay dual-arm collaborative robot control method based on Udwadia-Kalaba and cooperative game theory according to claim 7, characterized in that: The acquisition of the servo constraint following error value depends on the following formula: in, is the constraint matrix, is the time in the dual-arm collaborative robot system, represents the position of the joints in the dual-arm collaborative robot system, represents the speed of the joints in the dual-arm collaborative robot system, is a second-order constraint; The acquisition of the adaptive update law depends on the following formula: in, are all control parameters in the adaptive law and are positive constants. Preset Functions ， is the servo-constrained following error, and , represents the position of the joints in the dual-arm collaborative robot system, represents the speed of the joints in the dual-arm collaborative robot system, is the time in the dual-arm collaborative robot system.

9. The time-delay dual-arm collaborative robot control method based on Udwadia-Kalaba and cooperative game theory according to claim 1, characterized in that: The adaptive robust controller includes the following formula: in, and It is the control item of the controller based on the Udwadia-Kalaba principle. is the control item of the controller obtained by time delay estimation and dual manipulator coordinated impedance control, is the control term for adaptively compensating system uncertainty, is the adaptive update law.

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