A joint module robust control method based on non-cooperative game

Abstract

The application relates to the technical field of robot joint module control, in particular to a joint module robust control method based on non-cooperative game. The application realizes accurate control of the collaborative robot joint module, achieves optimal control performance, effectively avoids the beneficial effects of the uncertainty caused by external interference, manufacturing errors, working environment and the like, and the specific steps are as follows: constructing a dynamics equation of the collaborative robot joint module; determining the uncertainty boundary of the dynamics equation of the collaborative robot joint module; designing a trajectory tracking controller of the collaborative robot joint module; and calculating optimal control parameters of the trajectory tracking controller through a game algorithm.

Description

Technical Field

[0001] This invention relates to the field of robot joint module control technology, and in particular to a robust control method for joint modules based on non-cooperative game theory. Background Technology

[0002] Uncertainty inevitably exists in mechanical systems due to external interference, manufacturing errors, and the working environment. Uncertainty makes it difficult for engineers to accurately control systems. In the presence of uncertainty, nominal control may no longer meet some practical control requirements. Robust control maintains certain performance even when the system is disturbed. As a control method for handling uncertainty, it has always been a research hotspot in the field of control. Currently, within the scope of robust control for joint modules, with the goal of optimizing control accuracy, there is no relevant research or complete solution on how to calculate the control parameters. Summary of the Invention

[0003] This invention discloses a robust control method for joint modules based on non-cooperative game theory, the specific method of which is as follows:

[0004] Constructing the dynamic equations of the joint module of the collaborative robot;

[0005] Determine the uncertainty boundary of the dynamic equations of the joint module of the collaborative robot;

[0006] Design a trajectory tracking controller for the joint module of a collaborative robot;

[0007] To reduce the tracking error of the trajectory tracking controller, the optimal control parameters of the trajectory tracking controller are calculated using a game theory algorithm.

[0008] Furthermore, the power structure of the collaborative robot joint includes a surface-mount permanent magnet synchronous motor and a harmonic reducer. The dynamic equations of the collaborative robot joint module are constructed by the dynamic equations of the surface-mount permanent magnet synchronous motor and the harmonic reducer.

[0009] Furthermore, the dynamic equations of the surface-mount permanent magnet synchronous motor are constructed, and the specific method is as follows:

[0010] Constructing a mathematical model for a permanent magnet synchronous motor:

[0011]

[0012] In the formula, u d u q L d L q Representing stator current, stator voltage, stator inductance, d-axis, and q-axis respectively, n pLet ω be the number of pole pairs, ω be the rotor angular velocity, and ψ be the number of pole pairs. f R is the rotor flux linkage, and R is the stator resistance; the mechanical system parameters J, B, and T of the permanent magnet synchronous motor. e T L These are rotational inertia, coefficient of viscous friction, electromagnetic torque, and load torque, respectively.

[0013] There is a nonlinear coupling between the speed and armature current of a permanent magnet synchronous motor, and the electromagnetic torque T e It can be represented as:

[0014]

[0015] Based on the FOC principle, control the three-phase stator current synthesis vector i s Make i on the q-axis d i is zero q equals i s i d Equal to zero, which makes i q Direct control of electromagnetic torque, and individual control of electromagnetic torque, can achieve better dynamic performance. Surface-mount permanent magnet synchronous motors, i.e., L-type motors, are commonly used in laboratories. d =L q =L, then the mathematical model of the surface-mount permanent magnet synchronous motor is:

[0016]

[0017] Electromagnetic torque T e for

[0018]

[0019] Where K is the torque coefficient;

[0020] The dynamic equation of a surface-mount permanent magnet synchronous motor is:

[0021] Furthermore, the dynamic equations of the harmonic reducer are constructed using the following method:

[0022] The simplified dynamic equation of the harmonic reducer can be expressed as follows:

[0023] T D =ληT L

[0024] Among them, T D Let T be the output torque of the harmonic reducer, λ be the reduction ratio, and η be the transmission efficiency, considering friction. f The impact, and can The dynamic equations of the collaborative robot joint module can then be expressed as follows:

[0025]

[0026] or

[0027]

[0028] Furthermore, the dynamic equations of the collaborative robot joint module are expressed as follows:

[0029]

[0030] Where M is the inertia matrix, C is the Coriolis and centrifugal force matrix, F is the friction force matrix, and τ is the generalized force vector, and we have

[0031] M(q,σ,t)=J

[0032]

[0033]

[0034] The frictional forces in the joint modules of collaborative robots are divided into Coulomb friction and viscous friction.

[0035]

[0036] Among them, f c For Coulomb friction, f v η is viscous friction, and r is dry friction;

[0037] If trajectory tracking control is applied to this mechanical system, then the output error is:

[0038] e(t)=q(t)-q d (t)

[0039]

[0040] The dynamic equations of the collaborative robot joint module can be expressed as follows:

[0041]

[0042] Furthermore, the uncertainty boundary of the dynamic equations of the collaborative robot joint module is determined, specifically by the following method:

[0043] Because of the uncertainty in the parameters of the dynamic equations of the joint modules of collaborative robots, the dynamic model can be divided into a nominal part and an uncertain part.

[0044]

[0045]

[0046]

[0047] in, The nominal portion; ΔM, ΔC, and ΔF are the uncertain portions;

[0048] Assume the uncertainty boundary is determined by ρ.

[0049]

[0050] in

[0051]

[0052] Where S>0 and is a constant, considering that the collaborative robot's joint module operates at low speed, the uncertainties of inertia, Coriolis force, and centrifugal force can be ignored, and only the uncertainty of friction force can be considered. Therefore, the uncertainty boundary can be expressed as follows:

[0053]

[0054] If there is no uncertainty, then Φ = 0, meaning there is no interference from the uncertainty term of friction.

[0055] Furthermore, a trajectory tracking controller for the joint module of the collaborative robot is designed, specifically using the following method:

[0056] Design a trajectory tracking controller such that the trajectory tracking error e(t) is less than a threshold. This trajectory tracking controller is...

[0057]

[0058] Where P and D are traditional PID control parameters, γ>0 is a constant, and the selection of parameters F=(0,∞) and ρ=(0,0.5] affects the control performance of the trajectory tracking controller;

[0059] Considering that the collaborative robot's joint modules operate at low speeds, the uncertainties of inertia, Coriolis force, and centrifugal force can be ignored, and only the uncertainty of friction force needs to be considered. Since ρ determines the uncertainty boundary, the uncertainty of the trajectory tracking controller can be expressed as:

[0060] ΔF = 0.1|sin(2t)|

[0061] ρ=2|cos(4t)|

[0062] Furthermore, the optimal control parameters of the trajectory tracking controller are calculated using a game theory algorithm, as follows:

[0063] The parameters ΔF and ρ of the trajectory tracking controller are analogous to the parameters ψ and κ, and the weight coefficients are normalized, i.e., χ1, χ2, χ3 are 1. With the goal of ensuring that the tracking error of the trajectory tracking controller is less than the threshold, the optimal control parameters ΔF and ρ are selected through a game theory method.

[0064] Using sinusoidal signals as the target motion for the collaborative robot's joint modules:

[0065] Solve for ω1, ω2, ω3, ω4, ω5, ω6, k, V s

[0066] The cost functions for player 1ΔF and player 2ρ are then obtained as follows:

[0067]

[0068]

[0069] Based on the specific cost function, calculate the Nash equilibrium strategy combination for the collaborative robot joint module under given conditions. Substitute the optimal parameter ΔF * ,ρ *

[0070]

[0071]

[0072]

[0073] Due to the adoption of the above solutions, the present invention has the following beneficial effects:

[0074] 1. Due to external interference, manufacturing errors, working environment, and other factors, uncertainty is always unavoidable in mechanical systems. Uncertainty makes it difficult for engineers to accurately control the system. This invention uses a robust controller to control the joint module, which can effectively avoid the impact of uncertainty caused by external interference, manufacturing errors, working environment, and other factors.

[0075] 2. This invention uses non-cooperative game theory to optimally select the parameters of the trajectory tracking controller of the collaborative robot joint module platform, and obtains the optimal control parameters that are minimized through the Nash equilibrium solution, thereby further improving the control accuracy.

[0076] Other advantages, objectives, and features of the invention will be set forth in part in the description which follows, and in part will be apparent to those skilled in the art from the following examination, or may be learned from practice of the invention. The objectives and other advantages of the invention can be realized and obtained from the following description and claims. Attached Figure Description

[0077] The accompanying drawings of this invention are described below.

[0078] Figure 1 This is a schematic diagram of the composition of the collaborative robot joint module in the embodiment.

[0079] Figure 2 This is a cross-sectional view of the collaborative robot joint module in the embodiment.

[0080] Figure 3 This is a schematic diagram of an experimental platform for collaborative robot joint modules.

[0081] Figure 4 This is a comparison chart of sine trajectory tracking errors in the embodiment.

[0082] Figure 5 This is a schematic diagram of the control flow for an example. Detailed Implementation

[0083] The present invention will be further described below with reference to the accompanying drawings and embodiments.

[0084] Collaborative robot joint module structure such as Figure 1 and Figure 2 As shown, it consists of a servo driver, a frameless torque motor, a harmonic reducer, a photoelectric encoder before the reducer, an absolute encoder after the reducer for measuring the absolute angle of the robotic arm's output axis, and a relay brake, all integrated into a single joint. This makes it well-suited for assembling into a multi-axis robotic arm and can also be used as other automated equipment. It features higher transmission efficiency, a compact structure, and the ability to achieve greater continuous and instantaneous torque output, characterized by its small size, powerful performance, and high precision.

[0085] Through such Figure 3 The experimental platform for the collaborative robot joint module shown below was used to obtain experimental data for the joint module, as shown in the table below:

[0086]

[0087]

[0088] A robust control method for joint modules based on non-cooperative game theory, such as Figure 5 As shown: The mathematical model of the permanent magnet synchronous motor is as follows:

[0089]

[0090] Here u d u q L d L q Representing stator current, stator voltage, stator inductance, d-axis, and q-axis respectively, n p Let ω be the number of pole pairs, ω be the rotor angular velocity, and ψ be the number of pole pairs. fR is the rotor flux linkage, and R is the stator resistance; the mechanical system parameters J, B, and T of the permanent magnet synchronous motor. e T L These are rotational inertia, viscous friction coefficient, electromagnetic torque, and load torque, respectively.

[0091] There is a nonlinear coupling between the speed and armature current of a permanent magnet synchronous motor, and the electromagnetic torque T e It can be represented as

[0092]

[0093] Based on the FOC principle, control the three-phase stator current synthesis vector i s Make i on the q-axis d i is zero q equals i s i d Equal to zero, which makes i q Direct control of electromagnetic torque, and individual control of electromagnetic torque, can achieve better dynamic performance. Surface-mount permanent magnet synchronous motors, i.e., L-type motors, are commonly used in laboratories. q =L q =l, then the mathematical model of the surface-mount permanent magnet synchronous motor is:

[0094]

[0095] Electromagnetic torque T e for

[0096]

[0097] Where K is the torque coefficient.

[0098] The dynamic equation of a surface-mount permanent magnet synchronous motor is:

[0099]

[0100] The output shaft of the motor is connected to a harmonic reducer to control the rotation of the robotic arm joints. During operation, the transmission ratio and efficiency of the harmonic reducer must be considered. The simplified dynamic equation of the harmonic reducer can be expressed as follows:

[0101] T D =ληT L

[0102] Among them, T D Let T be the output torque of the harmonic reducer, λ be the reduction ratio, and η be the transmission efficiency, considering friction. f The impact, and can The dynamic equations of the collaborative robot joint module can then be expressed as follows:

[0103]

[0104] or

[0105]

[0106] Here, the dynamic system of the collaborative robot joint module can be represented as:

[0107]

[0108] Where M is the inertia matrix, C is the Coriolis and centrifugal force matrix, F is the friction force matrix, and τ is the generalized force vector, and we have

[0109] M(q,σ,t)=J

[0110]

[0111]

[0112] The frictional forces in the joint modules of collaborative robots are divided into Coulomb friction and viscous friction.

[0113]

[0114] Among them, f c For Coulomb friction, f v η is the viscous friction force, and r is the dry friction force, both of which are usually very large.

[0115] If trajectory tracking control is applied to this mechanical system, then the output error is:

[0116] e(t)=q(t)-q d (t)

[0117]

[0118] The dynamic equations of the collaborative robot joint module can be expressed as follows:

[0119]

[0120] Due to the uncertainty of the parameters, dynamic models can be divided into nominal and uncertain parts.

[0121]

[0122]

[0123]

[0124] in, ΔM, ΔC, and ΔF represent the nominal portion; ΔM, ΔC, and ΔF represent the uncertain portion.

[0125] Assume the uncertainty boundary is determined by ρ.

[0126]

[0127] in

[0128]

[0129] Where S>0 and is a constant, considering that the collaborative robot's joint module operates at low speed, the uncertainties of inertia, Coriolis force, and centrifugal force can be ignored, and only the uncertainty of friction force can be considered. Therefore, the uncertainty boundary can be expressed as follows:

[0130]

[0131] If there is no uncertainty, then Φ = 0, meaning there is no interference from the uncertainty term of friction.

[0132] Design a controller such that the trajectory tracking error e(t) is sufficiently small (control objective). This controller is...

[0133]

[0134] Where P and D are traditional PID control parameters, γ>0 is a constant, and the selection of parameters F=(0,∞) and ρ=(0,0.5] affects the control performance of the controller.

[0135] Considering that the collaborative robot's joint modules operate at low speeds, the uncertainties of inertia, Coriolis force, and centrifugal force can be ignored, and only the uncertainty of friction force needs to be considered. Since ρ determines the uncertainty boundary, the uncertainty of the trajectory tracking controller can be expressed as being jointly determined by ΔF = 0.1|sin(2t)| and ρ = 2|cos(4t)|. Referring to the optimal selection of parameters ψ and κ in non-cooperative game theory, the parameters ΔF and ρ of the trajectory tracking controller are analogous to parameters ψ and κ, and the weight coefficients are normalized, i.e., χ1, χ2, χ3 are 1. With the goal of minimizing the tracking error of the trajectory tracking controller, the optimal control parameters ΔF and ρ are selected through game theory.

[0136] The specific parameters of the dynamic equations for the collaborative robot's joint module are M = J = 0.00165 kg.m 2 , C=B=0.008Nm / rad / sec, T D =0.21Nm, eta = 0.95, λ = 101, n p =6, ψ f =0.025, P=150, D=10, S=1, γ=3, f c =0.15, f v =0.25, r=100.

[0137] For time-varying, uncertain mechanical systems and robust control, balancing performance and control costs to achieve the optimal state of the system by adjusting ψ and κ is a multi-objective optimization problem. Since the choices of ψ and κ are independent, non-cooperative game theory is used to analyze and solve this problem.

[0138] In game theory, each player's decision affects their own costs and those of other players. Each participant aims to minimize their costs and maximize their gains; that is, there exists a set of decisions that allows all participants to simultaneously incur the minimum costs, and this "minimum" is global in meaning. Relationships between players in a game can be categorized into cooperative and non-cooperative types. Unlike cooperative relationships, in a non-cooperative environment, players are in competition, and all players' decisions are independent and unrelated to other players. In the non-cooperative scenario, there exists a set of decisions that maximizes the actual benefit of each participant; this is called a Nash equilibrium.

[0139] If for all i = 1, 2, 3, ..., N and s i ∈S i ,have Then the decision combination S * ∈S1×S2×…×S i ×…×S n To achieve a Nash equilibrium, the problem that a Nash equilibrium needs to solve is...

[0140] If decision combination If it is not a Nash equilibrium, then the i-th participant can reduce costs by changing their strategy.

[0141] The parameters ψ and κ affect control performance and control cost, so ψ and κ can be viewed as two players in a game, with their decision sets S1 = (0, ∞) and S2 = (0, 0.5). A cost function is designed for each player.

[0142]

[0143]

[0144] These two cost functions have similar structures, consisting of three parts. J 11 (ψ,κ), J 21 (ψ,κ) represents the value from t s The average total transient energy at time t; J 12 (ψ,κ), J 22 (ψ,κ) represents the average value of the overall steady-state performance; J 13 (ψ), J 23(κ) represents the control cost, and χ1, χ2, and χ3 are weighting coefficients.

[0145] in,

[0146]

[0147] Its differential,

[0148]

[0149] in,

[0150] Therefore, the two cost functions can also be expressed as

[0151]

[0152] From the cost function above, we can see that the decisions of participants ψ and κ (i.e., the values ​​of ψ and κ) will affect the system's performance and control costs.

[0153] According to Nash equilibrium, the strategy combination of the two cost functions is equivalent to...

[0154]

[0155] Take J1(ψ,κ) * ) for the first derivative of ψ

[0156]

[0157] Take J2(ψ) * ,κ) for the first derivative of κ

[0158]

[0159] When (ψ) * ,κ * For a Nash equilibrium to be achieved, the following extreme value conditions must be satisfied.

[0160] and

[0161] The above are the necessary conditions for the existence of Nash equilibrium. They can be solved numerically using modern mathematical calculation software. When D[σ]≠0, the necessary conditions for the existence of Nash equilibrium have a solution, that is, the Nash equilibrium exists.

[0162] Transform the Nash equilibrium problem into an extremum problem of a multivariate function, and design the following function.

[0163]

[0164] Consider the extremum problem of multivariable functions

[0165]

[0166] Assume that the extreme value problem must satisfy the following conditions

[0167]

[0168] It can be known And the solution to the above equation is the same. The solution is...

[0169] When the assumed extreme value problem has no solution to the condition, it is assumed that the solution exists on the boundary between S1 and S2.

[0170] When κ = 0.5, Find the first derivative of ψ in this expression.

[0171]

[0172] make Then there is

[0173] 2χ3ψ 4 +χ1ω2ω3 2 ψ-ω5 2 (χ1ω4+χ2ω6)=0

[0174] in

[0175]

[0176] Since η>0, D[σ]≠0, therefore make It is continuous with respect to ψ and takes values ​​in the range [0, +∞), therefore 2χ3ψ 4 +χ1ω2ω3 2 ψ-ω5 2 The solution (χ1ω4+χ2ω6)=0 always exists.

[0177] in addition

[0178]

[0179] That is, minimizing ψ' make

[0180] The solution to the Nash equilibrium satisfies the following conditions

[0181]

[0182] Using a sinusoidal signal as a reference trajectory, the joint module is expected to perform the following actions: Given the above conditions, ω1, ω2, ω3, ω4, ω5, ω6, k, and V can be solved using modern mathematical software. s Then we obtain the cost functions for player 1ΔF and player 2ρ.

[0183]

[0184] Based on the specific cost function, the Nash equilibrium strategy combination of the collaborative robot joint modules under given conditions can be calculated. Substitute the optimal parameter ΔF * ,ρ *

[0185]

[0186]

[0187]

[0188] To achieve optimal control for sinusoidal trajectory tracking, i.e., to minimize the sinusoidal trajectory tracking error, a comparison of sinusoidal trajectory tracking errors is shown in the figure below. Figure 4 As shown.

[0189] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and not to limit it. Although the present invention has been described in detail with reference to the above embodiments, those skilled in the art should understand that modifications or equivalent substitutions can still be made to the specific implementation of the present invention. Any modifications or equivalent substitutions that do not depart from the spirit and scope of the present invention should be covered within the scope of protection of the claims of the present invention.

Claims

1. A joint module robust control method based on non-cooperative game, characterized in that, The specific method is as follows: The dynamics equation of the collaborative robot joint module is constructed; The uncertainty boundary of the dynamics equation of the collaborative robot joint module is determined; The trajectory tracking controller of the collaborative robot joint module is designed; The optimal control parameters of the trajectory tracking controller are calculated through a game algorithm for the purpose of reducing the tracking error of the trajectory tracking controller; The trajectory tracking controller of the collaborative robot joint module is designed, and the specific method is as follows: A trajectory tracking controller is designed such that a trajectory tracking error is less than a threshold value, the trajectory tracking controller is wherein, , are traditional PID control parameters, is a constant, parameters and the selection of which affects the control performance of the trajectory tracking controller; Considering that the collaborative robot joint module is in a low-speed running state when working, the uncertain terms of inertia, Coriolis and centrifugal force can be ignored, and only the uncertain term of friction is considered, and because The uncertainty boundary is determined, and the uncertainty of the trajectory tracking controller can be expressed as ， ， The optimal control parameters of the trajectory tracking controller are calculated through a game algorithm, and the specific method is as follows: The parameters of trajectory tracking controller are determined by the game method and The parameters of trajectory tracking controller are determined by the game method and the weight coefficient is unitized, i.e. , , 1, the optimal control parameters are selected by the game method for the purpose of tracking error of trajectory tracking controller being less than a threshold value and ; A sine signal is used as the desired motion of the joint module of the collaborative robot: Solving the equation The cost function of player 1 and player 2 is: According to the specific cost function, the Nash equilibrium strategy combination of the collaborative robot joint module under the given condition is calculated , the optimal parameters are brought in , , .

2. The non-cooperative game based robust control method of joint module according to claim 1, characterized in that, The power structure of the collaborative robot joint includes a surface-mounted permanent magnet synchronous motor and a harmonic reducer, and the dynamics equation of the collaborative robot joint module is constructed through the dynamics equation of the surface-mounted permanent magnet synchronous motor and the dynamics equation of the harmonic reducer.

3. The non-cooperative game based robust control method of joint module according to claim 2, characterized in that, The dynamics equation of the surface-mounted permanent magnet synchronous motor is constructed, and the specific method is as follows: The mathematical model of the permanent magnet synchronous motor is constructed: wherein, , , , , , respectively represent stator current, stator voltage, stator inductance, shaft and shaft, is the number of pole pairs, is the rotor angular velocity, is the rotor flux, is the stator resistance; mechanical system parameters of permanent magnet synchronous motor , , , respectively represent moment of inertia, viscous friction coefficient, electromagnetic torque and load torque; The permanent magnet synchronous motor has nonlinear coupling between the rotation speed and the armature current, and the electromagnetic torque may be expressed as: According to the FOC principle, control the three-phase stator current synthesis vector Make Zero, Equal to , Equal to zero, can make Direct control of electromagnetic torque, individual control of electromagnetic torque can obtain better dynamic performance, commonly used in the laboratory surface mount permanent magnet synchronous motor, namely Then the mathematical model of the surface mount permanent magnet synchronous motor is Electromagnetic torque To wherein, is the torque coefficient; The dynamic equation of the surface-mounted permanent magnet synchronous motor is .

4. The non-cooperative game based robust control method of joint module according to claim 3, characterized in that, The dynamics equation of the harmonic reducer is constructed, and the specific method is as follows: The simplified dynamics equation of the harmonic reducer can be expressed as wherein, is the harmonic reducer output torque, is the reduction ratio, is the transmission efficiency, taking into account the effect of friction and can be expressed as , the dynamics equation of the collaborative robot joint module can be expressed as or 。 5. The non-cooperative game based robust control method of joint module according to claim 4, characterized in that, The dynamics equation of the collaborative robot joint module is expressed as: wherein, is the inertia matrix, is the Coriolis and centrifugal force matrix, is the friction matrix, is the generalized force vector, and has The friction of the collaborative robot joint module is divided into coulomb friction and viscous friction wherein, is the Coulomb friction force, is the viscous friction force, is the dry friction; Considering trajectory tracking control of the mechanical system, the output error is , The dynamics equation of the cooperative robot joint module can be expressed as 。 6. The non-cooperative game based robust control method of joint module according to claim 5, characterized in that, The uncertainty boundary of the dynamics equation of the collaborative robot joint module is determined, and the specific method is as follows: Due to the uncertainty of the parameters of the dynamics equation of the collaborative robot joint module, the dynamic model can be divided into a nominal part and an uncertain part wherein , , is a nominal part; , , is an uncertainty part; Assume that the uncertainty boundary is determined by the decision wherein where and is a constant, considering that the collaborative robot joint module works in a low-speed running state, the uncertainty terms of inertia, Coriolis and centrifugal force can be ignored, only the uncertainty term of friction is considered, and the uncertainty boundary can be expressed as If there is no uncertainty, then i.e. no disturbance of the friction uncertainty term.

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