Four-legged collaborative robot preset performance tracking control method based on nash game

Abstract

The present application relates to the field of robot control, especially to a preset performance tracking control method for quadruped collaborative robots based on Nash game, comprising the following steps: establishing a dynamic model of a six-axis robot arm system, and introducing fuzzy set theory to deal with bounded uncertainty in the system; using cycloid curve to generate a smooth desired trajectory, and proposing a preset performance optimal tracking control method to express the desired trajectory in the form of an equality servo constraint; obtaining the control parameters of the system according to the constraint, and establishing a fuzzy adaptive controller according to the control parameters to realize accurate tracking of the error; converting the design of the control parameters into a two-person non-cooperative game, and balancing the system indicators through the Nash equilibrium method to obtain the optimal control parameters. The present application improves the system by adding the Nash game method on the basis of accurate tracking control, improves the fuzzy-based performance through optimal design, and seeks the optimal value of the control parameters by constructing a two-player Nash game.

Description

Technical Field

[0001] The present invention relates to the field of robot control, and in particular to a preset performance tracking control method of a quadruped collaborative robot based on Nash game. Background Art

[0002] Quadruped collaborative robots are capable of adapting to various complex terrains, including uneven surfaces, soft soil, and rubble. Their multi-degree-of-freedom design enables them to perform more sophisticated operations and adapt to complex and ever-changing situations. This adaptability and flexibility make them important in military, emergency rescue, and public service operations. Manipulator control is a crucial component in the design of quadruped collaborative robots. By designing appropriate control algorithms and systems, the manipulator's end effector can precisely move along a pre-defined trajectory or path. Considering that quadruped manipulators may operate on uneven terrain and in the presence of obstacles, traditional control methods can meet the manipulator's trajectory tracking requirements to a certain extent. However, due to the complex environment and numerous interference factors, it is not only difficult to control the manipulator to operate along the intended trajectory, but also requires extensive parameter adjustments and manual adjustments to find the optimal parameters to achieve the best results. However, due to the difficulty in adapting to complex working environments and changing task requirements, the manipulator cannot effectively cope with these requirements, significantly reducing the control accuracy of the manipulator.

[0003] The precise tracking control technology of the robotic arm ensures that the robotic arm moves along a predetermined trajectory and largely meets the need for precise control of the robotic arm by accurately measuring the state information of the robotic arm (such as position, speed, acceleration, etc.) and adjusting the output of the actuator in real time. However, in different environments, parameters still need to be adjusted to adapt to the environment according to the influence of interference factors. How to find the optimal parameters to adapt to different environments to achieve the best effect remains a problem. Summary of the Invention

[0004] The present invention aims to solve the problems existing in the prior art and provides the following technical solutions:

[0005] A preset performance tracking control method for a quadruped collaborative robot based on Nash game theory includes the following steps:

[0006] S10: Establish a dynamic model of the six-axis robotic arm system and introduce fuzzy set theory to deal with bounded uncertainty in the system.

[0007] S20: A cycloid curve is used to generate a smooth desired trajectory, and a preset performance optimal tracking control method constraint is proposed to express the desired trajectory as an equation servo constraint.

[0008] S30: Obtaining control parameters of the system according to the constraint, and establishing a fuzzy adaptive controller according to the control parameters to achieve accurate tracking of the error.

[0009] S40: Convert the design of control parameters into a two-player non-cooperative game, and balance the system indicators through the Nash equilibrium method to obtain the optimal control parameters.

[0010] As an improvement to the above technical solution, the dynamic model of the six-axis robotic arm system is as follows:

[0011]

[0012] Where θ∈R 6 represents the state vector, represents the velocity vector, represents the acceleration vector, t∈R represents time; δ∈R n represents the uncertainty vector, M represents the mass matrix, which is determined by the geometric parameters and mass distribution of each manipulator joint, C represents the Coriolis force and centripetal force matrix, which characterizes the nonlinear force generated by the rotation effect during the system movement, G is the gravity vector, which reflects the influence of gravity on the torque of each joint of the system, u∈R 6 Indicates control input.

[0013] The method of introducing fuzzy set theory to process bounded uncertainty in the system includes the following formula:

[0014] S δi ={(δ i ,μ δi (δ i ))|δ i ∈Ω δi},i=1,2,...,m

[0015] S θi ={(θ 0i ,μ θi (θ 0i ))|θ 0i ∈Ω θi},i=1,2,3,4,5,6

[0016] Among them, S δi Represents δ i Fuzzy set, Ω δi ∈R represents δ i The domain of discourse, with fuzzy boundaries, μ δi →[0,1] is S δi The membership function, S θi represents θ i Fuzzy set, Ω θi ∈R represents θ ithe domain of discourse, with fuzzy boundaries, μ θi →[0,1] is a membership function of S θi .

[0017] As an improvement of the above technical solution, the step S20 comprises the following steps:

[0018] S21: generating a smooth desired trajectory by using a cycloid curve.

[0019] S22: obtaining a tracking error in the system according to the desired trajectory, and expressing the tracking error in the form of an AC power system performance.

[0020] S23: obtaining a final convergence area decision parameter of the tracking error expressed in the form of the AC power system performance, and expressing the desired trajectory as an equality servo constraint according to the decision parameter.

[0021] As an improvement of the above technical solution, the desired trajectory is as follows:

[0022] P(t)=[P1(t),P2(t),P3(t),P4(t),P5(t),P6(t)]

[0023]

[0024] P6(t)=ρ-P1(t)-P2(t)-P3(t)-P4(t)-P5(t)-P6(t)

[0025] where θ is and θ id are the initial value and target value of the state vector of the i th mechanical arm shaft, respectively, t d is the set time for each mechanical arm shaft to reach the target angle, and P i (t) is the desired trajectory of the i th mechanical arm shaft in the time interval t<t d , wherein i=1, 2, 3, 4, 5, 6, is a linear time proportion item, is a sine correction item, and ρ is a fixed angle constant between the end of the mechanical arm and the platform.

[0026] After the tracking error is expressed in the form of the AC power system performance, the following formula is obtained:

[0027] Ξ i (θ,t):=θ i (t)-P i (t)

[0028] where θ i represents the angle of the i th mechanical arm joint, and Ξ i(θ, t) is the tracking error, i = 1, 2, 3, 4, 5, 6, and for Ξ i The expected value of is 0.

[0029] The final convergence region determination parameter is obtained as follows:

[0030]

[0031] in, Represents an exponential decay function, which is used to describe the convergence process of tracking error and to obtain the preset performance function The middle value, h i is a positive constant that determines the convergence speed. In order to make i The initial value of (θ,t) satisfies The parameters of Determine the final convergence region;

[0032] The equation servo constraint is expressed as follows:

[0033]

[0034] Among them, x i (θ, t) represents the tracking error variable after state conversion, where i = 1, 2, 3, 4, 5, and 6, corresponding to the pitch angle error of each axis of the manipulator, respectively. By integrating the equality constraint of the desired trajectory with the inequality constraint of the preset performance index, a new unified equality constraint is formed to describe the dynamic deviation between the actual state of the system and the target trajectory.

[0035] As an improvement of the above technical solution, step S30 includes the following steps:

[0036] S31: Performing state conversion based on the equational servo constraint and the dynamic model of the six-axis robotic arm system to obtain a conversion model of the converted six-axis robotic arm system under the equational servo constraint.

[0037] S32: Servo constraints are applied to the conversion model and the nominal control parameters of the system are established through the UK method.

[0038] S33: Obtain compensation parameters of the initial state deviation according to the converted system performance indicators, and establish control parameters of the uncertain part in the control system through the adaptive law.

[0039] S34: Establish a fuzzy adaptive controller according to the nominal control parameters, the compensation parameters, and the control parameters of the uncertain part to achieve accurate tracking of the error.

[0040] As an improvement to the above technical solution, the conversion model of step S31 includes the following formula:

[0041]

[0042]

[0043] Where t represents time, x(t) represents the state vector after conversion, Indicates the rate of change of the state vector after conversion, reflecting the dynamic change of the tracking error. represents the acceleration of the converted state vector, reflecting the dynamic response of the system under control, δ(t) represents the uncertainty vector, M is the inertia matrix, Θ is the state-related function matrix, characterizing the inertia and coupling effects in this dynamic model; Υ is the nonlinear coupling matrix, used to describe dynamic effects such as centrifugal force and Coriolis force; Γ is the gravity and external interference term, u(t) represents the control input, and And y1,y2,y3,y4,y5,y6 and x i , P i (t) and the length and mass of each robot arm's rotating shaft.

[0044] The acquisition of the nominal control parameters depends on the following formula:

[0045]

[0046] Where τ1 is the nominal control parameter of the system, D(x,t)=(Θ N (x,t)) -1 ,Θ N (x, t) is the mass matrix under the system name, and D(x, t) is the inverse matrix of the mass matrix under the system name. The superscript N represents the determination part. is the control gain moment.

[0047] The control parameters of the uncertain part are obtained based on the following formula:

[0048]

[0049] Among them, τ2 is the compensation parameter of the initial state deviation, Represents the tracking error performance indicator of the system.

[0050]

[0051] Among them, τ3 is the control parameter of the uncertain part of the system, α is the fuzzy vector, is an estimate of α, and γ is the adjustment function, k1 is the adjustable control parameter, and Z is a known function.

[0052] As an improvement of the above technical solution, step S40 includes the following steps:

[0053] S41: Transform the optimal choice of control parameters into minimizing the impact of the decision set on the cost function in the player game.

[0054] S42: Obtain cost functions of two players in the game, and obtain pairing conditions under a Nash equilibrium state based on the cost functions.

[0055] S43: Obtain a solution for the minimum performance index according to the pairing condition, and use it as the optimal control parameter.

[0056] As an improvement to the above technical solution, the cost functions of the two players in step S42 include the following formula:

[0057]

[0058] Among them, J1 and J2 are two independent parameter cost functions used to quantify the comprehensive index of system performance and control energy consumption, while k1 and k2 are adjustable control parameters, corresponding to the decision variables of the two players respectively, and their square terms represent the penalty terms of control efforts. Indicates that from time t s The overall transient performance value of the start A value indicating steady-state performance.

[0059] As an improvement to the above technical solution, the method for obtaining the solution of the minimum performance index in step S43 includes the following steps:

[0060] S431: If a solution to the matching condition exists, a search is performed to find a solution that meets the preset condition.

[0061] S432: Obtain a local minimum point according to the solution, where the minimum value is the solution of the minimum performance indicator.

[0062] As an improvement to the above technical solution, the pairing condition in step S43 is as follows:

[0063]

[0064]

[0065] in, represents the partial derivative of the cost function J1 or J2 with respect to the control parameter k1 or k2, When k2 is a fixed equilibrium value When , the partial derivative of J1 with respect to k1 is 0, that is, is the extreme point of J1, When k1 is a fixed equilibrium value When , the partial derivative of J2 with respect to k2 is 0, that is, is the extreme point of J2, and Ensure that the extreme point is the minimum value.

[0066] The preset condition is as follows:

[0067]

[0068] Wherein, And Reflect the local curvature of the cost of the parameters, And Indicate the cross effect between parameters, used to evaluate the coupling effect.

[0069] The beneficial effects of the present application are:

[0070] The present application adds Nash game method to improve the system on the basis of precise tracking control, improves the performance based on fuzzy by optimal design, and seeks the optimal value of the control parameter by constructing a double-player Nash game. This is different from the traditional optimization of combining multiple goals into one through weight, and the optimal value of the adjustable parameter can be determined by Nash equilibrium. This not only improves the control accuracy but also improves the control performance. BRIEF DESCRIPTION OF DRAWINGS

[0071] Figure 1 The overall structure diagram of the four-legged collaborative robot for the application of the present application;

[0072] Figure 2 The design flowchart of the present application. DETAILED DESCRIPTION

[0073] The embodiments of the present application will be described below through specific specific examples, and those skilled in the art can easily understand other advantages and effects of the present application from the disclosure of the present specification. The present application can also be implemented or applied in other different specific embodiments, and the details in the specification can be modified or changed based on different views and applications without departing from the spirit of the present application.

[0074] The precise tracking control technology of the robot arm can ensure the robot arm to move according to the predetermined trajectory and meet the needs of the control precision of the robot arm to a great extent by accurately measuring the state information of the robot arm (such as position, speed, acceleration, etc.) and adjusting the output of the actuator in real time. However, it is still a problem to find the best parameters to adapt to different environments to achieve the best effect under the influence of different environmental disturbances.

[0075] In order to solve the above problems, please refer to Figure 1 And Figure 2 A preset performance tracking control method for four-legged collaborative robots based on Nash game is provided, comprising the following steps:

[0076] S10: Establish a dynamic model of the six-axis robotic arm system and introduce fuzzy set theory to deal with bounded uncertainty in the system.

[0077] The dynamic model of the six-axis robotic arm system is as follows:

[0078]

[0079] Where θ∈R 6 represents the state vector, represents the velocity vector, represents the acceleration vector, t∈R represents time; δ∈R n represents the uncertainty vector, M represents the mass matrix, which is determined by the geometric parameters and mass distribution of each manipulator joint; C represents the Coriolis force and centripetal force matrix, which characterizes the nonlinear force generated by the rotation effect during the system movement; G is the gravity vector, which reflects the influence of gravity on the torque of each joint of the system; where M∈R 6×6 , C∈R 6×6 and G∈R 6 and θ(t) or the uncertain δ(t), u∈R 6 Indicates control input.

[0080] in,

[0081] θ=[θ1, θ2, θ3, θ4, θ5, θ6] T , G=[G1, G2, G3, G4, G5, G6] T ,

[0082] u=[u1,u2,u3,u4,u5,u6] T

[0083] Assuming that the uncertainty δ(t) and the initial state θ0 are bounded and lie in a known fuzzy set, when introducing fuzzy set theory to deal with bounded uncertainty in the system, it can be expressed as follows:

[0084] S δi ={(δ i ,μ δi (δ i ))|δ i ∈Ω δi},i=1,2,...,m

[0085] S θi ={(θ 0i ,μ θi (θ 0i ))|θ0i ∈Ω θi},i=1,2,3,4,5,6

[0086] Among them, S δi Represents δ i Fuzzy set, Ω δi ∈R represents δ i The domain of discourse, with fuzzy boundaries, μ δi →[0,1] is S δi The membership function of S is such that the membership values ​​“1” and “0” represent the most likely event and the least likely event respectively. θi The characters in its expression are the same as S δi have the same meaning.

[0087] S20: A cycloid curve is used to generate a smooth desired trajectory, and a preset performance optimal tracking control method constraint is proposed to express the desired trajectory as an equation servo constraint.

[0088] Specifically, step S20 includes the following steps:

[0089] S21: Generate a smooth desired trajectory using a cycloid curve.

[0090] The expected trajectory is as follows:

[0091] P(t)=[P1(t),P2(t),P3(t),P4(t),P5(t),P6(t)]

[0092]

[0093]

[0094] P6(t)=ρ-P1(t)-P2(t)-P3(t)-P4(t)-P5(t)-P6(t)

[0095] Among them, θ is and θ id are the starting value and target value of the i-th robot arm axis state vector, t d is the set time for each robot axis to reach the target angle, and P i (t) is the time interval t <t d The expected trajectory of the i-th robot axis, where i = 1, 2, 3, 4, 5, 6, is the linear time scaling term, is a sinusoidal correction term, and ρ is a fixed angle constant between the end of the manipulator and the platform.

[0096] S22: Obtaining a tracking error in the system according to the expected trajectory, and expressing the tracking error in terms of AC system performance.

[0097] For the desired trajectory in step S21, the following formula is obtained:

[0098]

[0099] in, Indicates the speed of each robot axis trajectory curve, and Indicates the acceleration of each robot arm axis trajectory curve.

[0100] And because of the following conditions:

[0101]

[0102] Then we can assume that the expected trajectory is second-order continuously differentiable in the entire time range, so let It can be considered as the system performance of AC power and i The expected value of is 0, where Ξ i (θ, t) is the tracking error, and i = 1, 2, 3, 4, 5, 6.

[0103] The determination of the tracking error can be achieved by using a preset performance optimal tracking control method, where the preset performance optimal tracking control method represents preset transient and steady-state performance, and its implementation method is to ensure that the convergence speed is greater than a certain specific value, and the tracking error eventually converges to a predefined area close to 0. However, it is difficult to meet the preset performance optimal tracking control method in actual control, so step S23 is executed.

[0104] S23: Obtain a final convergence region determination parameter of the tracking error expressed in terms of AC system performance, and express the desired trajectory as an equation servo constraint based on the determination parameter.

[0105] By designing a control quantity u∈R 6 To drive the state θ∈R of the fuzzy adaptive controller 6 Track the desired trajectory P∈R 6 , and satisfied by The preset performance optimal tracking control method represented by is implemented as follows:

[0106]

[0107] in, is a continuously differentiable, bounded, strictly positive and decreasing function, as shown below:

[0108]

[0109] in, Represents an exponential decay function, which is used to describe the convergence process of tracking error and to obtain the preset performance function The middle value, h i is a positive constant that determines the convergence speed. In order to make i The initial value of (θ,t) satisfies Parameters, and by to determine the final convergence region.

[0110] In order to deal with the number of preset performance optimal tracking control methods, a state conversion method is proposed to combine Equation 1 and Equation 2 constraints into an equality constraint. The state after this conversion is not subject to inequality constraints. The expression of the converted equality servo constraint is as follows:

[0111]

[0112] Among them, x i (θ, t) represents the tracking error variable after state conversion, where i = 1, 2, 3, 4, 5, and 6, corresponding to the pitch angle error of each axis of the manipulator, respectively. By integrating the equality constraint of the desired trajectory with the inequality constraint of the preset performance index, a new unified equality constraint is formed to describe the dynamic deviation between the actual state of the system and the target trajectory.

[0113] In the above formula, there are:

[0114]

[0115] Among them, θ i Represents the angle of the i-th robotic arm joint, when When, i → x i →-∞, if Ξ i =0 i ,x i = 0, which means that the transformed state x i Not subject to inequality constraints. In addition, this state transition is smooth and bijective. Based on this state transition, if a control can ensure that x i (t) is bounded, then the system motion will track the desired trajectory, that is, converge to a point close to Ξ i = 0, and at the same time meet the requirements of the preset optimal performance tracking control method, that is, meet For this reason, step S30 is performed.

[0116] S30: Obtaining control parameters of the system according to the constraint, and establishing a fuzzy adaptive controller according to the control parameters to achieve accurate tracking of the error.

[0117] Specifically, step S30 includes the following steps:

[0118] S31: Performing state conversion based on the equational servo constraint and the dynamic model of the six-axis robotic arm system to obtain a conversion model of the converted six-axis robotic arm system under the equational servo constraint.

[0119] The conversion model of step S31 includes the following formula:

[0120]

[0121]

[0122] Among them, x(t) represents the state vector after conversion, Indicates the rate of change of the state vector after conversion, reflecting the dynamic change of the tracking error. Represents the acceleration of the state vector after conversion, reflecting the dynamic response of the system under control. M is the inertia matrix, Θ is the state-related function matrix, which characterizes the inertia and coupling effects in this dynamic model; Y is the nonlinear coupling matrix, which is used to describe dynamic effects such as centrifugal force and Coriolis force; Γ is the gravity and external interference term, and And y1,y2,y3,y4,y5,y6 and x i , P i (t) and the length and mass of each robot arm's rotating shaft.

[0123] S32: Servo constraints are applied to the conversion model and the nominal control parameters of the system are established through the UK method.

[0124] Specifically, it includes making the transformed AC system comply with the servo constraint:

[0125]

[0126] Or in matrix form as Where c =

[0127] [-x1,-x2.-x3,-x4,-x5,-x6] T The second-order form can be expressed as in

[0128] Where c(x) is the nonlinear dynamic function of the system, describing the state changes caused by natural forces such as the Coriolis force; is the control gain matrix, which represents the mapping relationship to the system acceleration.

[0129] The decomposition of Θ, i, and Γ in the conversion model is as follows:

[0130]

[0131] Where, the superscript N and Δ represent the determined part and the uncertain part respectively. N >0 is always possible, because the deterministic part is at the designer's discretion, and the nominal transformation system can be expressed as follows:

[0132]

[0133] Where τ1 is the nominal control parameter of the system, Θ N (x,t) is the nominal mass matrix of the system.

[0134] For all It is controllable with respect to the servo constraints of Equation 3, and the corresponding control can be designed by the UK method with nominal control parameters as follows:

[0135]

[0136] Where D(x,t)=(Θ N (x,t)) -1 , and D(x,t) is the inverse matrix of the nominal mass matrix of the system.

[0137] The nonlinear tracking control problem is viewed through the lens of servo constraints. Unlike passive constraints (which specify what the environment should do to the system), servo constraints specify what the control should do to the system. Therefore, the constraint force is the control force, and the servo constraint is the control target. Based on this, step S33 is executed.

[0138] S33: Obtain compensation parameters of the initial state deviation according to the converted system performance indicators, and establish control parameters of the uncertain part in the control system through the adaptive law.

[0139] Let the performance index of the converted system be:

[0140]

[0141] in, Represents the tracking error performance indicator of the system.

[0142] The control parameters of the uncertain part are obtained based on the following formula:

[0143]

[0144] Among them, τ2 is the compensation parameter of the initial state deviation.

[0145] definition:

[0146] E(x,δ,t)=Θ N(x,t)×(Θ(x,δ,t)) -1 -I

[0147] ΔD(x,δ,t)=(Θ(x,δ,t)) -1 -(Θ(x,t)) -1 =D(x,t)E(x,δ,t)

[0148] Where E(x,δ,t) is the error matrix between the nominal dynamic matrix and the actual dynamic matrix, and ΔD(x,δ,t) is the difference matrix between the nominal inverse matrix and the actual inverse matrix.

[0149] According to the above formula, we can make:

[0150]

[0151] Among them, W is the stability index.

[0152] There exists a fuzzy number σ>-1 such that for all

[0153] W≥σ

[0154] Among them, λ min represents the smallest eigenvalue of the matrix. N and The terms in are either constants, states and velocities, or quadratic terms thereof, so there exists a fuzzy vector i∈(0,∞) 3 and a known function: So that for all There is the following formula:

[0155]

[0156] Fuzzy numbers σ and α i Related to δ. Their membership functions can be calculated based on μ i (δ), fuzzy arithmetic and decomposition theorem.

[0157] Therefore, the control parameters for dealing with the uncertain part can be proposed as follows:

[0158]

[0159] Among them, τ3 is the control parameter of the uncertain part of the system, is an estimate of α, and

[0160] in:

[0161]

[0162] Among them, γ is the adjustment function, ζ is the adaptive adjustment term, k1, k2∈(0,∞) are adjustable parameters, is an estimate of α, and i=1,2,3. Control The adaptive law gives the following formula:

[0163]

[0164] S34: Establish a fuzzy adaptive controller according to the nominal control parameters, the compensation parameters, and the control parameters of the uncertain part to achieve accurate tracking of the error.

[0165] Among them, the fuzzy adaptive controller is as follows:

[0166] u=τ1+τ2+τ3

[0167] S40: Convert the design of control parameters into a two-player non-cooperative game, and balance the system indicators through the Nash equilibrium method to obtain the optimal control parameters.

[0168] Specifically, step S40 includes the following steps:

[0169] S41: Transform the optimal choice of control parameters into minimizing the impact of the decision set on the cost function in the player game.

[0170] Let J i (k1,k2) and are the cost function and the decision set for players k1 and k2 respectively. Each cost function consists of the average fuzzy system performance indicator and the control effort of the relevant player. This problem is a two-player game in which the two players can only change their own decisions. The first player wants to In the example above, player 2 wants to minimize J1(k1,k2) by choosing k1 in the example above. The second player wants to minimize J1(k1,k2) by choosing k1 in the example above. Choose k2 to minimize J2(k1,k2).

[0171] The optimal design problem of control parameters can be expressed as follows: In a player game, there exists the following equation:

[0172]

[0173]

[0174] The solution to this problem is to find a pair And make it satisfy the following solution, which is the Nash equilibrium:

[0175]

[0176] Based on the above reasons, step S42 is executed.

[0177] S42: Obtain cost functions of two players in the game, and obtain pairing conditions under a Nash equilibrium state based on the cost functions.

[0178] The cost functions of the two players in step S42 include the following formula:

[0179]

[0180] Among them, J1 and J2 are cost functions of two independent parameters, which are used to quantify the comprehensive indicators of system performance and control energy consumption, while k1 and k2 are adjustable control parameters, corresponding to the decision variables of the two players respectively, and their square terms represent the penalty terms of control efforts.

[0181] In the above formula, Indicates that from time t s The overall transient performance value of the start represents the value of steady-state performance, players k1 and k2 aim to minimize the average fuzzy system performance and control effort simultaneously, which means to strike a balance between system performance and control effort. In addition, and It’s about control strength.

[0182] For the cost function, we can let

[0183]

[0184] Therefore, combined with the above formula, the cost function can be rewritten as

[0185]

[0186] Among them, J * The benchmark cost item representing the system performance includes transient and steady-state performance indicators of the tracking error and the uncertainty influence described by fuzzy set theory, based on which step S43 is executed.

[0187] S43: Obtain a solution for the minimum performance index according to the pairing condition, and use it as the optimal control parameter.

[0188] To get the solution of Equation 4, we need to find The first-order derivative with respect to k1, and With respect to the first derivative of k2, any pairing that satisfies the following conditions will be in a Nash equilibrium:

[0189]

[0190] in, represents the partial derivative of the cost function J1 or J2 with respect to the control parameter k1 or k2, J1k1=0, i.e. is the extreme point of J1, J2k2=0, i.e. is the extreme point of J2, and ensures that the extreme point is the minimum value.

[0191] Assume:

[0192]

[0193] Then:

[0194]

[0195] Equation 5

[0196] The solution of equation 5 always exists, and the solution of equation 5 is the solution of equation 4. The solution of equation 5 can be obtained by the following method:

[0197]

[0198] The method for obtaining the solution of the minimum performance index in step S43 includes the following steps:

[0199] S431: If the solution of the pairing condition exists, find the solution satisfying the preset condition by searching.

[0200] It can be found that the solution of equation 6 is actually the same equation group as the pairing condition. If the solution of equation 6 exists and is represented as: (k 1i ,k 2i )(i∈Ω i ), the solution satisfying the preset condition can be found by searching:

[0201] The preset condition is as follows:

[0202]

[0203] wherein, and reflect the local curvature of the parameters to the cost, and represent the cross effects between parameters, used to evaluate the coupling effect.

[0204] S432: Obtain a local minimum point according to the solution, and the minimum value is the solution of the minimum performance index.

[0205] ​​When the solution found meets the preset conditions, it can be used as the local minimum point of J(k1,k2). Therefore, the minimum value of J, that is, the minimum performance index, is:

[0206]

[0207] Let the above solution be This solution is both the solution of Equation 4 and the solution of Equation 5. In the conclusion, since the solution of Equation 5 always exists, the Nash equilibrium of Equation 4 always exists. If the solution satisfies Problem 5, then it is the Nash equilibrium of the optimal design Problem 4.

[0208] Finally, the parameters Applied to the six-axis robotic arm control of a quadruped manipulator, achieving optimized precise tracking control.

[0209] The above embodiments are intended only to illustrate the technical solutions of the present invention and are not intended to limit the same. Anyone skilled in the art may modify or alter the above embodiments without departing from the spirit and scope of the present invention. Therefore, all equivalent modifications or alterations made by one of ordinary skill in the art without departing from the spirit and technical concepts disclosed herein are intended to be covered by the claims of the present invention.

Claims

1. A preset performance tracking control method for a quadruped collaborative robot based on Nash game, characterized in that: The steps include: S10: Establish a dynamic model of the six-axis robotic arm system and introduce fuzzy set theory to deal with bounded uncertainty in the system; S20: Generate a smooth desired trajectory using a cycloid curve, and propose a preset performance optimal tracking control method constraint to express the desired trajectory as an equation servo constraint; S30: obtaining control parameters of the system according to the constraint, and establishing a fuzzy adaptive controller according to the control parameters to achieve accurate tracking of the error; S40: Convert the design of control parameters into a two-player non-cooperative game, and balance the system indicators through the Nash equilibrium method to obtain the optimal control parameters; The step S30 includes the following steps: S31: performing state conversion according to the equational servo constraint and the dynamic model of the six-axis robotic arm system, and obtaining a conversion model of the converted six-axis robotic arm system under the equational servo constraint; S32: Servo constrain the conversion model and establish the nominal control parameters of the system through the UK method; S33: Obtain compensation parameters for the initial state deviation according to the converted system performance indicators, and establish control parameters for the uncertain parts of the control system through an adaptive law; S34: Establish a fuzzy adaptive controller according to the nominal control parameters, the compensation parameters, and the control parameters of the uncertain part to achieve accurate tracking of the error.

2. The preset performance tracking control method of a quadruped collaborative robot based on Nash game according to claim 1, characterized in that: The dynamic model of the six-axis robotic arm system is as follows: in, ∈R 6 represents the state vector, ∈R 6 represents the velocity vector, ∈R 6 represents the acceleration vector, t∈R represents time; ∈R n represents the uncertainty vector, M represents the mass matrix, which is determined by the geometric parameters and mass distribution of each manipulator joint, C represents the Coriolis force and centripetal force matrix, which characterizes the nonlinear force generated by the rotation effect during the system movement, G is the gravity vector, which reflects the influence of gravity on the torque of each joint of the system, u∈R 6 represents the control input; The method of introducing fuzzy set theory to process bounded uncertainty in the system includes the following formula: in, express The fuzzy set of ∈R represents The domain of discourse, with fuzzy boundaries, →[0,1] is The membership function of express The fuzzy set of ∈R represents The domain of discourse, with fuzzy boundaries, →[0,1] is The membership function of .

3. The preset performance tracking control method of a quadruped collaborative robot based on Nash game according to claim 1, characterized in that: The step S20 includes the following steps: S21: Generate smooth desired trajectory using cycloid curve; S22: Obtaining a tracking error in the system according to the expected trajectory, and expressing the tracking error in terms of AC system performance; S23: Obtain a final convergence region determination parameter of the tracking error expressed in terms of AC system performance, and express the desired trajectory as an equation servo constraint based on the determination parameter.

4. The preset performance tracking control method of a quadruped collaborative robot based on Nash game according to claim 3, characterized in that: The expected trajectory is as follows: in, and are the starting value and target value of the i-th robot arm axis state vector, is the set time for each robot axis to reach the target angle, and (t) is the time interval The expected trajectory of the i-th robot arm axis, where , is the linear time scaling term, is the sine correction term, is the fixed angle constant between the end of the robot arm and the platform; The tracking error is expressed in terms of AC system performance as follows: in, represents the angle of the i-th robotic arm joint, is the tracking error, , and for The expected value of ; The final convergence region determination parameter is obtained as follows: in, Represents an exponential decay function, which is used to describe the convergence process of tracking error and to obtain the preset performance function The median value of is a positive constant that determines the convergence speed. In order to make The initial value satisfies The parameters of Determine the final convergence region; The equation servo constraint is expressed as follows: in, represents the tracking error variable after state conversion, , which correspond to the pitch angle errors of each axis of the manipulator respectively. By integrating the equality constraint of the desired trajectory with the inequality constraint of the preset performance index, a new unified equality constraint is formed to describe the dynamic deviation between the actual state of the system and the target trajectory.

5. The preset performance tracking control method of a quadruped collaborative robot based on Nash game according to claim 1, characterized in that: The conversion model of step S31 includes the following formula: Where t represents time, represents the state vector after conversion, Indicates the rate of change of the state vector after conversion, reflecting the dynamic change of the tracking error. Represents the acceleration of the state vector after conversion, reflecting the dynamic response of the system under control. represents the uncertainty vector, M is the inertia matrix, is the state correlation function matrix, which characterizes the inertia and coupling effects in this dynamic model; It is a nonlinear coupling matrix used to describe dynamic effects such as centrifugal force and Coriolis force; is gravity and external disturbance, represents the control input, and ,and and , , (t) and the length and mass of each manipulator axis; The acquisition of the nominal control parameters depends on the following formula: in, is the nominal control parameter of the system, , is the nominal mass matrix of the system, and is the inverse matrix of the mass matrix under the system name, and the superscript N represents the determined part. is the control gain moment; The control parameters of the uncertain part are obtained based on the following formula: in, is the compensation parameter of the initial state deviation, Represents the tracking error performance index of the system; in, is the control parameter of the uncertain part of the system, is the fuzzy vector, for The estimated value of , is the adjustment function, is an adjustable control parameter, is a known function.

6. The preset performance tracking control method of a quadruped collaborative robot based on Nash game according to claim 1, characterized in that: The step S40 includes the following steps: S41: Transform the optimal choice of control parameters into the minimization of the impact of the decision set on the cost function in the player game; S42: Obtaining a cost function of two players in the game, and obtaining a pairing condition under a Nash equilibrium state based on the cost function; S43: Obtain a solution for the minimum performance index according to the pairing condition, and use it as the optimal control parameter.

7. The preset performance tracking control method of a quadruped collaborative robot based on Nash game according to claim 6, characterized in that: The cost functions of the two players in step S42 include the following formula: in, and As a cost function of two independent parameters, it is used to quantify the comprehensive index of system performance and control energy consumption. and As adjustable control parameters, they correspond to the decision variables of the two players, and their square terms represent the penalty terms of control efforts. Indicates time The overall transient performance value of the start A value indicating steady-state performance.

8. The preset performance tracking control method of a quadruped collaborative robot based on Nash game according to claim 6, characterized in that: The method for obtaining the solution of the minimum performance index in step S43 includes the following steps: S431: If a solution to the pairing condition exists, a search is performed to find a solution that meets the preset condition; S432: Obtain a local minimum point according to the solution, where the minimum value is the solution of the minimum performance indicator.

9. The preset performance tracking control method of a quadruped collaborative robot based on Nash game according to claim 7, characterized in that: The pairing condition in step S43 is as follows: in, Represents the cost function or Control parameters or The partial derivative, =0 is when A fixed equilibrium value hour, right The partial derivative of is 0, that is for The extreme point of For the time A fixed equilibrium value hour, right The partial derivative of is 0, that is for The extreme point of Ensure that the above extreme points are minimum values; The preset conditions are as follows: in, and reflects the local curvature of the parameter to the cost, and It represents the cross-influence between parameters and is used to evaluate coupling effects.

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