Flexible joint parameter identification optimization method based on anti-disturbance sliding mode observer

Abstract

The invention relates to the technical field of flexible joint mechanical arm control, in particular to an optimization method for flexible joint parameter identification based on an anti-disturbance sliding-mode observer, which comprises the following steps of: deducing a torque mathematical expression of a connecting rod side according to a typical flexible joint kinematic model, and further performing position decoupling on the torque expression to obtain a flexible joint parameter identification model; and obtaining reference position input of the motor side. In the design of a torque observer, a sliding mode disturbance observer is used for estimating feedback torque and unknown disturbance of a dual-mass system of a motor and a connecting rod, and the disturbance comprises external contact torque, friction torque and the like, and also comprises system internal parameter fluctuation and the like. And finally, in consideration of irreversibility and ill-conditioned matrix problems existing when parameter identification is carried out by using a traditional least square method, a least square method identification process based on a ridge regression method is designed, and kinetic parameter identification is realized.

Description

Technical Field

[0001] The present invention relates to the technical field of flexible joint manipulator control, and particularly to an optimization method for flexible joint parameter identification based on an anti-disturbance sliding mode observer. Background Art

[0002] With the increasing in-depth research on the modeling and robust control of joint robots, more and more robots are equipped with flexible joints, enabling them to move and operate flexibly in complex environments while reducing the collision damage between the robot and the surrounding environment. Compared with rigid joints, flexible joints are more adaptable to irregular shapes and surfaces, thus increasing the adaptability and safety of robots in human-robot interaction.

[0003] However, irregular parameter variations and model-free disturbances pose challenges to flexible joint control. At the same time, traditional torque control strategies based on torque sensor feedback increase the difficulty of joint integration, which is not conducive to the requirements of lightweight and cost control. Therefore, a comprehensive and efficient optimization method is needed to solve the problems of irregular disturbances and high-precision trajectory tracking faced in current flexible joint control.

[0004] Traditional solutions based on PD position controllers have achieved certain results in solving joint position tracking problems. However, the input of the control end is only the position signal feedback from the link side, and there is a certain flexibility in the system, resulting in the inability of the drive motor to achieve precise position control. In addition, the existence of irregular disturbances also increases the difficulty of joint trajectory tracking. For this reason, an optimization method for flexible joint parameter identification based on an anti-disturbance sliding mode observer is proposed. Summary of the Invention

[0005] In view of this, the purpose of the present invention is to propose an optimization method for flexible joint parameter identification based on an anti-disturbance sliding mode observer to solve the problems that the input of the control end is only the position signal feedback from the link side, and there is a certain flexibility in the system, resulting in the inability of the drive motor to achieve precise position control. In addition, the existence of irregular disturbances also increases the difficulty of joint trajectory tracking.

[0006] Based on the above purpose, the present invention provides an optimization method for flexible joint parameter identification based on an anti-disturbance sliding mode observer, including the following steps:

[0007] S1. Establish a motion model of the flexible joint dual-mass system and decouple the motor input torque;

[0008] S2. Design a sliding mode disturbance observer, that is, SDMO;

[0009] Specifically, step S2 includes the following steps:

[0010] S2.1. Establish a perturbation estimation model: By considering the perturbation caused by irregular parameter changes and external disturbances as the total perturbation acting on the flexible joint, a mathematical model derivation is carried out to obtain the mathematical model of the total perturbation.

[0011] S2.2. Design a sliding mode observer: By designing a motor-link double-side angular velocity observation error control law based on the integral sliding mode surface and reducing the observation error of the angular velocity, an accurate feedback torque observation value is obtained.

[0012] S2.3. Stability analysis: To ensure the effectiveness of the above-designed sliding mode perturbation observer, the observer should meet the following two objectives:

[0013] Objective 1: The designed SDMO can make the state converge to zero within a finite time by selecting appropriate parameters.

[0014] Objective 2: The designed SDMO can meet the stability constraints.

[0015] S3. Adopt the Fourier series trajectory as the excitation trajectory and perform the least square parameter identification process based on the ridge regression method to achieve the dynamic parameter identification of the flexible joint motor-link double side.

[0016] Preferably, in S1, establishing the motion model of the flexible joint double-mass system includes the following steps:

[0017] S1.1. Describe the flexible joint as a double-mass system composed of a motor, a link load, and a flexible transmission component that plays a connecting role. Ignoring the inertial coupling between the link and the motor, the mathematical model of the flexible joint is expressed as follows:

[0018]

[0019] Among them, η represents the angular deformation that occurs when the elastic element acts. Let η = q / λ – θ, where q, represent the rotational position, angular velocity, and angular acceleration on the link side respectively, and θ, represent the rotational position, angular velocity, and angular acceleration of the motor rotor respectively. λ is the reduction ratio of the harmonic reducer, m represents the link side, M represents the motor side, J, D, g are the inertia, viscous friction, and gravity torque coefficients respectively, and τ m and τ M are the elastic force causing the deformation of the flexible part of the joint and the output torque of the motor respectively;

[0020] S1.2. Perform closed-loop control by the method of giving the joint motion trajectory. The position error is defined as e m =(q r -q), e M =(θ r -θ), with e mWith e M Design a motion controller for the input, and the trajectory tracking error is:

[0021]

[0022] where, Λ m and Λ M are weight coefficients;

[0023] S1.3. The control law is expressed as:

[0024]

[0025] where, k m and k M are the gain coefficients of the PD controller. Combining (1) and (3), the flexible joint motion model is rewritten as:

[0026]

[0027] where, θ, are the angle and speed feedback values measured by the motor respectively, q, are the angle and speed feedback values measured by the connecting rod respectively, the desired connecting rod flexible torque τ em and the desired motor torque τ eM are the control inputs calculated based on the output of the PD controller and the feedback torque.

[0028] Preferably, in S1, the decoupling of the motor input torque specifically includes the following steps:

[0029] S1.4. Since the motion trajectory on the connecting rod side of the flexible joint generates a torque command through the inverse kinematics model and acts on the motor, essentially, the movement of the connecting rod is controlled by the output of the motor. Therefore, it is necessary to design a decoupling model for the connecting rod flexible torque to convert the flexible torque τ em into the angle and speed signals on the motor side;

[0030] S1.5. The harmonic reducer, as a flexible component in the flexible joint, provides a reduction ratio to increase the joint output. The joint flexibility function is expressed as:

[0031] τ em = K s (θ r / λ - q) (34)

[0032] By solving the inverse function of the above formula, we get:

[0033]

[0034] where, K s is the joint stiffness, θ ris the reference value of the motor position, for θ r Taking the derivative to obtain the reference value of the angular velocity

[0035] S1.6. Let K d be the damping coefficient. After introducing K d the torque command τ on the connecting rod side emd is expressed as:

[0036]

[0037] Then, the reference angular velocity on the motor side can be obtained by solving the inverse function of the above formula:

[0038]

[0039] Preferably, in S2.1, it specifically includes the following sub-steps:

[0040] S2.11. The torque commands on the connecting rod side and the motor side consist of two parts, namely the output of the PD controller and the feedback torque used to compensate for disturbances. The torque command control rate is expressed as:

[0041]

[0042] where, v m and v M are the disturbance compensation torques calculated by the disturbance observer;

[0043] S2.12. In actual motion, flexible joints often face internal parameter changes and external irregular torque disturbances caused by operating conditions. Therefore, formula (4) is expressed as:

[0044]

[0045] and

[0046] ΔJ and ΔD are the mismatch values of the corresponding parameters during operation.

[0047] Preferably, in S2.2, it specifically further includes the following sub-steps:

[0048] S2.21. By modeling the disturbance as an extended state variable of the system, the sliding mode observer is designed in the following form:

[0049]

[0050] where, is the observation error of the connecting rod motion angular velocity and the motor angular velocity, is the observation error of the disturbance, is The observed estimated value, l is the observer control gain, and g(z) is the control law designed according to the angular velocity observation error;

[0051] S2.22. To eliminate the observation error, an integral sliding mode surface is selected for the design of the observer. The integral sliding mode surface is designed as:

[0052]

[0053] where the parameter c > 0. To make the observation error be able to reach the sliding mode surface within a finite time and converge to zero along the sliding mode surface, the sliding mode control law SMC is designed as follows:

[0054] g(z) = cz + γsign(s) (43)

[0055] where sign(s) is the sign function with respect to s;

[0056] To avoid the discontinuity of the sign function at the zero point, the saturation function sat(s / κ) is used for substitution;

[0057] The g(z) of the improved SMO is expressed as:

[0058] g(z) = cz + γsat(s / κ) (44)

[0059]

[0060] Preferably, in S2.3, the verification process of the convergence of the sliding mode motion system is as follows:

[0061] Goal 1: The designed SDMO can make the state converge to zero within a finite time by selecting appropriate parameters;

[0062] Proof 1: The system state is stable and the disturbance sliding mode observer has converged. At this time

[0063] From (12), we get:

[0064]

[0065] Further solving gives:

[0066]

[0067] where a is a constant. To ensure that z f can converge to 0 within a finite time, l needs to satisfy l < 0. The disturbance estimation error decreases with the increase of time t and finally tends to zero. The convergence rate is determined by the selection of the switching gain parameter l.

[0068] Preferably, in S2.3, the process of verifying the stability of the sliding mode motion system is as follows:

[0069] Objective 2: The designed SDMO can meet the stability constraints:

[0070] Proof 2: To prove the stability, the Lyapunov function is designed as follows:

[0071]

[0072] Combining Equation (12), Equation (13) and Equation (15), the derivative form of Equation (19) is expressed as:

[0073]

[0074] Let To meet the Lyapunov stability theory, it can be obtained that:

[0075]

[0076] Generally, in practical applications, the parameter γ needs to satisfy:

[0077]

[0078] where L is the stability coefficient of ISMO and L>1.

[0079] Preferably, in S3, the Fourier series trajectory is adopted as the excitation trajectory, and the specific process is as follows:

[0080] S3.1. The Fourier series trajectory is as follows:

[0081]

[0082] where, q r (t) is the angle of the joint at time t, is the angular velocity of the joint at time t, is the angular acceleration of the joint at time t, q0 is the angle compensation of the joint at time t, ω0 is the basic angular frequency, ω f t is the interval angular frequency, n is the number of harmonic terms of the Fourier series, a k and b k are the amplitudes of the sine term and cosine term of the joint. Each Fourier series contains 2n+1 parameters;

[0083] S3.2. Ignoring the influence of the feedback torque estimation error, rewrite Equation (10) into the following form:

[0084]

[0085] where,

[0086] p m is a row vector composed of a reference trajectory, a feedback position, a weight coefficient, etc. The reference trajectory can be obtained from formula (23), and the feedback position information is measured by a position sensor; β M is a column vector composed of parameters to be identified. m and β M is a column vector composed of parameters to be identified.

[0087] Preferably, in S3, the parameter identification process specifically includes the following steps:

[0088] S3.3. Move the machine joint along a preset trajectory, sample the joint position information at N moments during the movement and the feedback torque v observed by ISMO, where N≥3, and then substitute the data into formula (24) to obtain:

[0089]

[0090] where, Г is a column vector of N×1, P is a matrix of N×r, and r is the number of rows of the vector p;

[0091] S3.4. When X T X is non-invertible or X T X is an ill-conditioned matrix, the solution formula cannot be calculated, resulting in the lack of stability and reliability of the least squares method in this case. Therefore, the ridge regression method is obtained by improving the traditional least squares method. Let the analytical solution of the ridge regression method be:

[0092] θ = (X T X + ζI) -1 X T y (57)

[0093] where, ζ is a hyperparameter called the ridge coefficient, and I is the identity matrix;

[0094] Compared with the traditional least squares method, by adding a perturbation matrix ζI, (X T X + ζI) must be invertible. Therefore, the physical parameters of the flexible joint can be solved by the ridge regression method, and the obtained parameter identification analytical formula is as follows:

[0095]

[0096] The beneficial effects of the present invention are as follows:

[0097] First, based on the link kinematics principle, the present invention decouples the link required torque into the motor position input, realizing the design of the motor-link bilateral control system, which is beneficial to further improving the accuracy of the flexible joint position tracking.

[0098] 2. The sliding mode disturbance observer proposed in the present invention regards the feedback torque, external irregular disturbance and internal parameter changes as a whole for dynamic estimation, freeing the joint from dependence on torque sensors, which is more conducive to reducing costs and integration difficulty. Test results show that even under dynamic working conditions, the disturbance observer can still achieve overall accurate estimation of the torque.

[0099] 3. Based on the traditional parameter identification based on the least squares method, the present invention takes into account the irreversible and ill-conditioned matrix problems and designs a least squares parameter identification process based on the ridge regression method to realize the bilateral dynamic parameter identification of the flexible joint motor-connecting rod. BRIEF DESCRIPTION OF THE DRAWINGS

[0100] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings required for use in the embodiments or the description of the prior art will be briefly introduced below. Obviously, the drawings described below are only embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative work.

[0101] Figure 1 is a flow chart of an embodiment of the present invention;

[0102] Figure 2 This is a control principle diagram of the motor-connecting rod double-side series system according to an embodiment of the present invention;

[0103] Figure 3 It is a schematic diagram of the flexible joint position tracking accuracy result according to an embodiment of the present invention;

[0104] Figure 4 It is a schematic diagram of the accuracy result of the estimated torque calculation value according to an embodiment of the present invention;

[0105] Figure 5 It is a schematic diagram of the calculation results of parameter estimation errors on both sides of the motor-connecting rod according to an embodiment of the present invention. DETAILED DESCRIPTION

[0106] In order to make the objectives, technical solutions and advantages of the present invention more clearly understood, the present invention is further described in detail below in conjunction with specific embodiments and with reference to the accompanying drawings.

[0107] It should be noted that, unless otherwise defined, the technical terms or scientific terms used in the embodiments of the present invention should have the ordinary meanings understood by those with ordinary skills in the field to which the present invention belongs. The "first", "second" and similar terms used in the present invention do not denote any order, quantity or importance, but are only used to distinguish different components. The terms such as "including" or "comprising" mean that the elements or objects appearing before this word cover the elements or objects listed after this word and their equivalents, without excluding other elements or objects. The terms such as "connected" or "coupled" are not limited to physical or mechanical connections, but may include electrical connections, whether direct or indirect. The terms such as "upper", "lower", "left", "right" are only used to represent relative positional relationships, and when the absolute position of the object being described changes, the relative positional relationship may also change accordingly.

[0108] As Figure 1 , Figure 2 , Figure 3 , Figure 4 , Figure 5 shown, an optimization method for flexible joint parameter identification based on an anti-disturbance sliding mode observer includes the following steps:

[0109] S1. Establishing the mathematical model of the flexible joint is the basic step to implement the method of the present invention, which includes the following key links:

[0110] A. Design of the double-mass system motion model; when constructing the flexible joint control system, it is necessary to comprehensively consider the mechanical motion equations on both the motor-link sides and analyze the torque transmission process, and select appropriate control laws and gain coefficients for the design of the trajectory tracking controller to ensure the closed-loop tracking of the joint trajectory.

[0111] In S1, establishing the motion model of the flexible joint double-mass system includes the following steps:

[0112] S1.1. Describe the flexible joint as a double-mass system composed of a motor, a link load, and a flexible transmission component for connection. Ignoring the inertial coupling between the link and the motor, the flexible joint mathematical model is expressed as follows:

[0113]

[0114] where η represents the angular deformation that occurs when the elastic element acts, let η = q / λ – θ, q, represent the rotational position, angular velocity, and angular acceleration of the link side respectively, θ, represent the rotational position, angular velocity, and angular acceleration of the motor rotor respectively, λ is the reduction ratio of the harmonic reducer, m represents the link side, M represents the motor side, J, D, g are the inertia, viscous friction, and gravity torque coefficients respectively, τ mand τ M are the elastic force causing the deformation of the flexible part of the joint and the output torque of the motor, respectively;

[0115] S1.2. Perform closed-loop control by the method of specifying the joint motion trajectory. The position error is defined as e m =(q r -q), e M =(θ r -θ). Design a motion controller with e m and e M as inputs. The trajectory tracking error is:

[0116]

[0117] where Λ m , Λ M are weight coefficients;

[0118] S1.3. The control law is expressed as:

[0119]

[0120] where k m and k M are the gain coefficients of the PD controller. Combining (1) and (3), the flexible joint motion model is rewritten as:

[0121]

[0122] The double-ended cascade control structure of the flexible joint is as shown in Figure 2 . Among them, θ, are the angle and speed feedback values measured by the motor respectively, q, are the angle and speed feedback values measured by the link respectively. The desired link flexible torque τ em and the desired motor torque τ eM are control inputs calculated based on the output of the PD controller and the feedback torque.

[0123] B. Decoupling of the motor input torque; During the torque transmission process, the link required torque needs to be input into the motor trajectory controller in the form of specific position information. Therefore, it is necessary to consider parameters such as joint stiffness and damping to design a torque decoupling model to realize the transformation from the transmitted torque to specific position information.

[0124] In S1, the decoupling of the motor input torque specifically includes the following steps:

[0125] S1.4, as an internal drive system, since the motion trajectory of the flexible joint link side generates a torque command to act on the motor through the inverse kinematics model, in essence, the movement of the link is controlled by the output of the motor. Therefore, it is necessary to design a decoupling model of the link's flexible torque to convert the flexible torque τ em Convert into angle and speed signals on the motor side;

[0126] S1.5. The harmonic reducer is a flexible component in the flexible joint, providing a reduction ratio to increase the joint output. The joint flexibility function is expressed as:

[0127] τ em =K s (θ r / λ-q) (63)

[0128] By solving the inverse function of the above formula, we can get:

[0129]

[0130] Among them, K s is the joint stiffness, θ r is the reference value of the motor position, r Derivative to get the reference value of angular velocity

[0131] S1.6. Due to the existence of flexible components such as harmonic drive reducers, the influence of flexibility must be considered in the calculation. Selecting appropriate damping can enhance the stability of the joint and prevent instability, especially when subjected to external interference. In addition, in practical applications, the calculation process of angular velocity and angular acceleration is very sensitive to noise. Assume K d is the damping coefficient, and K is introduced d After that, the connecting rod side torque command τ emd It is expressed as:

[0132]

[0133] Then, the reference angular velocity on the motor side is By solving the inverse function of the above formula, we can get:

[0134]

[0135] S2. Design a sliding mode disturbance observer (SDMO);

[0136] like Figure 2As shown in the figure, in the designed flexible joint control system, the torque commands on the link side and the motor side are composed of two parts, namely the output of the PD controller and the feedback torque used to compensate for disturbances. And in actual motion, flexible joints often face internal parameter changes caused by operating conditions and external irregular torque disturbances. Therefore, v (disturbance compensation torque) is regarded as a whole without considering its specific internal structure for estimation. A sliding mode observer is used to observe the change of v in real time, and the estimated disturbance observation value is used as feedback compensation. It includes the following key links:

[0137] S2.1. Establish a disturbance estimation model: By regarding the disturbances caused by irregular parameter changes and external disturbances as the total disturbances suffered by the flexible joint, a mathematical model derivation is carried out on it to obtain the mathematical model of the total disturbances;

[0138] Specifically, the following sub-steps are included in S2.1:

[0139] S2.11. The torque commands on the link side and the motor side are composed of two parts, namely the output of the PD controller and the feedback torque used to compensate for disturbances. The torque command control rate is expressed as:

[0140]

[0141] where, v m and v M are the disturbance compensation torques calculated by the disturbance observer;

[0142] S2.12. In actual motion, flexible joints often face internal parameter changes caused by operating conditions and external irregular torque disturbances. Therefore, formula (4) is expressed as:

[0143]

[0144] And

[0145] ΔJ and ΔD are the mismatch values of the corresponding parameters during operation.

[0146] The existence of internal parameter mismatch values and unknown system feedback torques has a great impact on the stability of the motion system. Therefore, v is regarded as a whole without considering its specific internal structure for estimation. A sliding mode observer SMO is used to observe the change of v in real time, and the estimated disturbance observation value is used as feedback compensation.

[0147] S2.2. Design a sliding mode observer: By designing a motor-link double-side angular velocity observation error control law based on the integral sliding mode surface, and by reducing the observation error of the angular velocity, an accurate feedback torque observation value is obtained; in addition, in the design of the control law, the significant chattering problem caused by the traditional sign function also needs to be considered.

[0148] Specifically, S2.2 further includes the following sub-steps:

[0149] S2.21. By modeling the disturbance as an extended state variable of the system, the sliding mode observer is designed in the following form:

[0150]

[0151] where is the observation error between the angular velocity of the connecting rod movement and the angular velocity of the motor, is the observation error of the disturbance, is the observation estimated value of , l is the observer control gain, and g(z) is the control law designed based on the angular velocity observation error;

[0152] S2.22. To eliminate the observation error, an integral sliding mode surface is selected for the design of the observer. The integral term in the integral sliding mode surface can enhance the robustness of the system to parameter changes, external disturbances, and modeling errors, thereby improving the stability and performance of the control system. The integral sliding mode surface is designed as:

[0153]

[0154] where the parameter c > 0. To make the observation error reach the sliding mode surface within a finite time and converge to zero along the sliding mode surface, the sliding mode control law SMC is designed as follows:

[0155] g(z) = cz + γsign(s) (72)

[0156] where sign(s) is the sign function with respect to s;

[0157] To avoid the discontinuity of the sign function at the zero point, the saturation function sat(s / κ) is used for substitution; the g(z) of the improved SMO is expressed as:

[0158] g(z) = cz + γsat(s / κ) (73)

[0159]

[0160] S2.3. Stability analysis: To ensure the effectiveness of the above-designed sliding mode disturbance observer, the observer should meet the following two objectives:

[0161] Objective 1. By selecting appropriate parameters, the designed SDMO can make the state converge to zero within a finite time;

[0162] In S2.3, the verification process of the convergence of the sliding mode motion system is as follows;

[0163] Proof 1: The system state is stable and the disturbance sliding mode observer has converged. At this time

[0164] From (12), we have:

[0165]

[0166] Further solution gives:

[0167]

[0168] where a is a constant. To ensure that z f can converge to 0 within a finite time, l needs to satisfy l < 0. The disturbance estimation error decreases as time t increases and finally approaches zero. The convergence rate is determined by the choice of the switching gain parameter l.

[0169] Objective 2: The designed SDMO can meet the stability constraints.

[0170] In S2.3, the stability verification process of the sliding mode motion system is as follows:

[0171] Proof 2: To prove the stability, the Lyapunov function is designed as follows:

[0172]

[0173] Combining equations (12), (13) and (15), the derivative form of equation (19) is expressed as:

[0174]

[0175] Let To satisfy the Lyapunov stability theory, we have:

[0176]

[0177] Generally, in practical applications, the parameter γ needs to satisfy:

[0178]

[0179] where L is the stability coefficient of the ISMO and L > 1.

[0180] There is a quantitative relationship between the physical parameters of the flexible joint and the dynamic response of the kinematic model. Therefore, in the process of calculating the transmitted torque, the influence of these parameters is very significant. The present invention proposes a flexible joint parameter identification method based on the improved least square method theory.

[0181] S3. Use the Fourier series trajectory as the excitation trajectory and implement the parameter identification of the bilateral dynamics of the flexible joint motor-link based on the least squares parameter identification process of the ridge regression method.

[0182] A. Fourier series trajectory planning. The purpose of designing the excitation trajectory is to fully stimulate the movement of the link, so as to obtain the joint physical parameters more accurately. The joint trajectory excitation includes two aspects. The type of excitation trajectory uses the finite-term Fourier series trajectory, and the parameters in the excitation trajectory need to be optimized according to the actual situation. The Fourier series trajectory has the following advantages:

[0183] a. It can avoid the natural frequency of the robotic arm and set the motion frequency of the joint;

[0184] b. It is convenient to select the appropriate filtering frequency in data processing;

[0185] C. The sine and cosine functions are highly differentiable and will not cause additional errors due to sudden changes in speed or acceleration.

[0186] In S3, due to the many advantages of the Fourier series, this application uses the Fourier series trajectory as the excitation trajectory, and the specific process is as follows:

[0187] S3.1. The Fourier series trajectory is as follows:

[0188]

[0189] where, q r (t) is the angle of the joint at time t, is the angular velocity of the joint at time t, is the angular acceleration of the joint at time t, q0 is the angle compensation amount of the joint at time t, ω0 is the base angular frequency, ω f t is the interval angular frequency, n is the number of harmonic terms of the Fourier series, a k and b k are the amplitudes of the sine and cosine terms of the joint. Each Fourier series contains 2n + 1 parameters;

[0190] S3.2. Ignoring the influence of the feedback torque estimation error, rewrite formula (10) into the following form:

[0191]

[0192] where,

[0193] p m and p M are row vectors composed of the reference trajectory, feedback position, weight coefficient, etc. The reference trajectory can be obtained from formula (23), and the feedback position information is measured by the position sensor; β mand β M is a column vector composed of parameters to be identified.

[0194] B. Parameter identification method based on ridge regression. The general expression form of the analytical solution of the traditional least squares method is:

[0195] θ = (X T X) -1 X T Y (84)

[0196] where X is the feature matrix with observed samples arranged by rows, with size m×n, m representing the number of samples, and n representing the number of features of the samples. However, this expression will face a problem in practical applications: when X T X is non - invertible or X T X is an ill - conditioned matrix, the solution formula cannot be calculated, resulting in the lack of stability and reliability of the least squares method in this case. Therefore, the traditional least squares method is improved to obtain the ridge regression method, making the system more stable and reliable.

[0197] In S3, the parameter identification process specifically includes the following steps:

[0198] S3.3. Move the machine joint along the preset trajectory, sample the joint position information at N moments during the movement and the feedback torque v observed by ISMO, where N≥3, and then substitute the data into Equation (24), we can get:

[0199]

[0200]

[0201] where, Г is a column vector of N×1, P is a matrix of N×r, and r is the number of rows of vector p;

[0202] S3.4. When X T X is non - invertible or X T X is an ill - conditioned matrix, the solution formula cannot be calculated, resulting in the lack of stability and reliability of the least squares method in this case. Therefore, the traditional least squares method is improved to obtain the ridge regression method. Let the analytical solution of the ridge regression method be:

[0203] θ = (X T X + ζI) -1 X T y (87)

[0204] where, ζ is a hyperparameter called the ridge coefficient, and I is the identity matrix;

[0205] Compared with the traditional least squares method, by adding a perturbation matrix ζI, it makes (X T(X + ζI) must be invertible. Therefore, the physical parameters of the flexible joint can be solved using the ridge regression method, and the obtained parameter identification analytical formula is as follows:

[0206]

[0207] In the parameter identification strategy, a perturbation matrix ζI is introduced to complete the construction of an invertible matrix. When the sample matrix is invertible, that is, there is no ill-conditioned matrix or non-invertible matrix, the calculation matrix constructed by the present invention is still valid because ζI does not change the characteristics of the original matrix.

[0208] C. Evaluation and estimation of the identification accuracy. The relative error percentage (REP) of the parameter estimation is introduced to evaluate the performance of the estimation.

[0209]

[0210] where and α j represent the estimated value and the nominal value respectively. The following convergence criterion is designed to evaluate the convergence iteration of the REP-estimated parameters:

[0211]

[0212] where a determines the length of the observation window [k - a + 1, k], and κ is a predefined convergence tolerance threshold.

[0213] The following convergence rules need to be satisfied:

[0214] a. Stop the iteration when REP < κ;

[0215] b. Stop the iteration when the maximum number of iterations is reached.

[0216] The estimation results of the method proposed by the present invention are shown below. The motion trajectory tracking results are as shown in Figure 3 and the torque estimation results are as shown in Figure 4 and the parameter estimation results are as shown in Figure 5 It can be seen that the method proposed by the present invention realizes the precise tracking of the joint position, and both the torque estimation and the joint parameter identification results are relatively accurate. Table 1 gives the detailed results of the dynamic parameter estimation of the proposed method.

[0217] Joint parameters <![CDATA[J m > <![CDATA[D m > g <![CDATA[J M > <![CDATA[D M > <![CDATA[K s > <![CDATA[K d > Traditional method (REP) 5.3 6.7 1.1 13.6 17.1 1.5 1.9 Proposed method (REP) 0.7 0.5 0.8 5.1 3.5 1.7 1.4

[0218] Table 1 Estimation errors of each parameter of the flexible joint

[0219] Those of ordinary skill in the art should understand that any discussion of the above embodiments is merely exemplary and is not intended to imply that the scope of the present invention is limited to these examples; under the concept of the present invention, the technical features in the above embodiments or different embodiments can also be combined, the steps can be implemented in any order, and there are many other variations in different aspects of the present invention as described above, which are not provided in detail for the sake of brevity.

[0220] Embodiments of the present invention are intended to cover all such substitutions, modifications, and variations that fall within the broad scope of the appended claims. Therefore, any omissions, modifications, equivalent substitutions, improvements, etc., made within the spirit and principle of the present invention shall be included within the protection scope of the present invention.

Claims

1. An optimized method for flexible joint parameter identification based on an anti-disturbance sliding mode observer, characterized in that It includes the following steps: S1. Establish a motion model of a flexible joint dual-mass system and decouple the motor input torque; S2. Design a sliding mode disturbance observer, namely the SDMO; Specifically, S2 includes the following steps: S2.

1. Establish a disturbance estimation model: By regarding the disturbance caused by irregular parameter changes and external disturbances as the total disturbance suffered by the flexible joint, and conducting mathematical model derivation on it, the mathematical model of the total disturbance is obtained; S2.

2. Design a sliding mode observer: By designing a control law for the observation error of the bilateral angular velocities of the motor-link based on an integral sliding mode surface, and reducing the observation error of the angular velocity, an accurate feedback torque observation value is obtained; S2.

3. Stability analysis: To ensure the effectiveness of the above-designed sliding mode disturbance observer, the observer should meet the following two objectives: Objective 1. The designed SDMO can make the state converge to zero within a finite time by selecting appropriate parameters; Objective 2. The designed SDMO can meet the stability constraints; S3. Adopt a Fourier series trajectory as the excitation trajectory and a least-square parameter identification process based on the ridge regression method to realize the dynamic parameter identification of the bilateral motor-link of the flexible joint.

2. The optimization method for flexible joint parameter identification based on an anti-disturbance sliding mode observer according to claim 1, wherein In S1, establishing a motion model of a flexible joint dual-mass system includes the following steps: S1.

1. Describe the flexible joint as a dual-mass system composed of a motor, a link load, and a flexible transmission component for connection. Ignoring the inertial coupling between the link and the motor, the mathematical model of the flexible joint is expressed as follows: where η represents the angular deformation that occurs when the elastic element acts, and let η = q / λ – θ, q, represent the rotational position, angular velocity, and angular acceleration on the link side respectively, θ, represent the rotational position, angular velocity, and angular acceleration of the motor rotor respectively, λ is the reduction ratio of the harmonic reducer, m represents the link side, M represents the motor side, J, D, g are the inertia, viscous friction, and gravitational torque coefficients respectively, τ m and τ M are the elastic force causing the deformation of the flexible part of the joint and the output torque of the motor respectively; S1.

2. Perform closed-loop control by specifying the joint motion trajectory. The position error is defined as e m =(q r -q), e M =(θ r -θ). Design a motion controller with e m and e M as inputs. The trajectory tracking error is as follows: where, Λ m and Λ M are weighting coefficients; S1.

3. The control law is expressed as: where k m and k M are the gain coefficients of the PD controller. Combining (1) and (3), the flexible joint motion model is rewritten as: where θ, are the angle and speed feedback values measured by the motor respectively, q, are the angle and speed feedback values measured by the connecting rod respectively, and the desired flexible torque τ em of the connecting rod and the desired motor torque τ eM are control inputs calculated based on the output of the PD controller and the feedback torque.

3. An optimization method for flexible joint parameter identification based on an anti-disturbance sliding mode observer according to claim 2, characterized in that, In S1, specifically decoupling the motor input torque includes the following steps: S1.

4. Since the motion trajectory on the flexible joint link side generates torque commands through the inverse kinematic model and acts on the motor, essentially, the movement of the link is controlled by the output of the motor. Therefore, it is necessary to design a decoupling model for the flexible torque of the link to convert the flexible torque τ em into the angle and speed signals on the motor side; S1.

5. The harmonic reducer, as a flexible component in the flexible joint, provides a reduction ratio to increase the joint output. The joint flexibility function is expressed as: τ em = K s (θ r / λ - q)(5) Obtained by solving the inverse function of the above formula: Among them, K s is the joint stiffness, θ r is the reference value of the motor position. Taking the derivative of θ r yields the reference value of the angular velocity S1.

6. Set K d as the damping coefficient. After introducing K d , the link side torque command τ emd is expressed as: Then, the reference angular velocity on the motor side can be obtained by solving the inverse function of the above formula:

4. An optimization method for flexible joint parameter identification based on an anti-disturbance sliding mode observer according to claim 3, characterized in that Specifically, S2.1 includes the following sub-steps: S2.

11. The torque commands on the link side and the motor side consist of two parts, namely the output of the PD controller and the feedback torque used to compensate for the disturbance. The torque command control rate is expressed as: where, v m and v M are the disturbance compensation torques calculated by the disturbance observer; S2.

12. In actual motion, the flexible joint often faces internal parameter changes and external irregular torque disturbances caused by operating conditions. Therefore, formula (4) is expressed as: With ΔJ and ΔD are the mismatch values of the corresponding parameters during the operation process.

5. An optimization method for flexible joint parameter identification based on an anti-disturbance sliding mode observer according to claim 4, characterized in that, Specifically, S2.2 also includes the following sub-steps: S2.

21. By modeling the disturbance as an extended state variable of the system, the sliding mode observer is designed as: Among them, is the observation error of the angular velocity of the connecting rod movement and the angular velocity of the motor, is the observation error of the disturbance, is the observation estimated value of, l is the observer control gain, and g(z) is the control law designed according to the observation error of the angular velocity; S2.

22. To eliminate the observation error, an integral sliding mode surface is selected for the design of the observer. The integral sliding mode surface is designed as: where the parameter c > 0. In order to make the observation error reach the sliding surface within a finite time and converge to zero along the sliding surface, the sliding mode control law SMC is designed as follows: g(z) = cz + γsign(s) (14) where sign(s) is the sign function with respect to s; To avoid the discontinuity of the sign function at the zero point, the saturation function sat(s / κ) is used for substitution; The g(z) of the improved SMO is expressed as: g(z) = cz + γsat(s / κ) (15) 6. According to the optimized method for parameter identification of a flexible joint based on an anti-disturbance sliding mode observer as claimed in claim 5, characterized in that In S2.3, the verification process of the convergence of the sliding mode motion system is as follows: Objective 1: The designed SDMO can make the state converge to zero within a finite time by selecting appropriate parameters; Proof 1: The system state is stable and the disturbance sliding mode observer has converged. At this time From (12), we have: Further solving gives: where a is a constant. To ensure that z f can converge to 0 within a finite time, l needs to satisfy l < 0. The disturbance estimation error decreases as time t increases and finally approaches zero. The convergence rate is determined by the choice of the switching gain parameter l.

7. An optimization method for flexible joint parameter identification based on an anti-disturbance sliding mode observer according to claim 5, characterized in that, In S2.3, the verification process of the stability of the sliding mode motion system is as follows: Objective 2: The designed SDMO can meet the stability constraints: Proof 2: To prove the stability, the Lyapunov function is designed as follows: Combining equations (12), (13) and (15), the derivative form of equation (19) is expressed as: Suppose To satisfy the Lyapunov stability theory, it can be obtained that: Generally, in practical applications, the parameter γ needs to satisfy: where L is the stability coefficient of the ISMO and L>1.

8. An optimization method for flexible joint parameter identification based on an anti-disturbance sliding mode observer according to claim 7, characterized in that In S3, the Fourier series trajectory is used as the excitation trajectory, The specific process is as follows: S3.1 The Fourier series trajectory is as follows: where q r (t) is the angle of the joint at time t, is the angular velocity of the joint at time t, is the angular acceleration of the joint at time t, q0 is the angle compensation of the joint at time t, ω0 is the basic angular frequency, ω f t is the interval angular frequency, n is the number of harmonic terms of the Fourier series, a k and b k are the amplitudes of the sine and cosine terms of the joint. Each Fourier series contains 2n + 1 parameters; S3.2 Ignoring the influence of the feedback torque estimation error, rewrite formula (10) into the following form: Among them, p m is associated with p M is a row vector composed of a reference trajectory, a feedback position, a weight coefficient, etc. The reference trajectory can be obtained from formula (23), and the feedback position information is measured by a position sensor; β m and β M is a column vector composed of parameters to be identified.

9. An optimization method for flexible joint parameter identification based on an anti-disturbance sliding mode observer according to claim 8, characterized in that, In S3, the parameter identification process specifically includes the following steps: S3.3 Move the machine joint along the preset trajectory, sample the joint position information at N moments during the movement and the feedback torque v observed by the ISMO, and N≥3. Then substitute the data into equation (24) to get: where Г is an N×1 column vector, P is an N×r matrix, and r is the number of rows of the vector p; S3.

4. When X T is non-invertible or X T is an ill-conditioned matrix, the solution formula cannot be calculated, resulting in the lack of stability and reliability of the least squares method in this case. Therefore, the ridge regression method is obtained by improving the traditional least squares method. Let the analytical solution of the ridge regression method be: θ=(X T X+ζI) -1 X T y (28) where ζ is a hyperparameter called the ridge coefficient, and I is the identity matrix; Compared with the traditional least squares method, by adding a perturbation matrix ζI, it is ensured that (X T X + ζI) is surely invertible. Therefore, the physical parameters of the flexible joint can be solved by the ridge regression method, and the obtained parameter identification analytical formula is as follows:

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