A friction compensation method based on LuGre model and terminal sliding mode observer

Abstract

The application discloses a kind of friction compensation methods based on LuGre model and terminal sliding mode observer, for improving the inhibition ability of photoelectric tracking system to friction disturbance.The application includes dynamic friction compensator and friction disturbance compensation based on sliding mode observer, and the friction model approximator constructed by LuGre dynamic friction model is used to estimate the friction torque value offline by adjusting friction parameters, and a part of friction is compensated by feedforward compensation.Secondary compensation uses terminal sliding mode observer, wherein a new type of approach law is designed to ensure that the observer can quickly and accurately compensate friction, and can effectively suppress the influence of external disturbance, effectively improve the tracking accuracy of system.

Description

Technical Field

[0001] This invention belongs to the field of friction suppression, specifically relating to a friction compensation method based on the LuGre model and a terminal sliding mode observer, which is mainly used for friction suppression of photoelectric tracking frames to improve tracking accuracy. Background Technology

[0002] Optical tracking turntables play a crucial role in systems such as ship tracking, unmanned aerial vehicles (UAVs), satellite reconnaissance, astronomical observation, laser communication, and quantum communication. The tracking accuracy of the turntable is a key indicator of its tracking performance. With increasing application demands, frictional torque disturbances have become one of the main factors limiting the improvement of tracking accuracy. Nonlinear friction can cause crawling phenomena in optical tracking equipment at low speeds, waveform distortion when the speed crosses zero, significant errors during turning, and even undesirable limiting cycle oscillations, thus greatly affecting the dynamic and static performance of the system. Therefore, effective friction suppression is essential for improving the tracking accuracy of optical tracking systems.

[0003] The magnitude of frictional torque is mainly related to system structure, lubrication conditions, load size, and speed, and varies with mechanical position and time. Generally, the impact of frictional torque can be reduced by changing the mechanical structure of the equipment, increasing lubrication during operation, and using control strategies for compensation. Among these, the third method, which compensates for friction by modeling the system's friction or through observation, can effectively improve the system's tracking accuracy without increasing economic costs, even with current manufacturing capabilities.

[0004] Currently, friction can be compensated using frictional and non-frictional models. Among these, frictional model-based compensation methods can estimate the frictional torque in advance based on a known model, offering advantages such as strong targeting, clear mechanism, and low computational cost. However, due to the complexity of friction, the suitability of the selected frictional model, the accuracy of the frictional model parameter identification, and whether the model itself changes can all affect the friction compensation effect, leading to significant errors and easily causing undercompensation and overcompensation. Furthermore, frictional force is also affected by changes in position and environment; once the model parameters are identified, they cannot be arbitrarily changed, making it difficult to adapt to changes in external conditions.

[0005] The core idea of ​​model-free compensation methods is to treat friction as an external disturbance or a general nonlinear function, and improve the system's ability to suppress disturbances by changing control parameters or structure. Compared to model-based compensation strategies, this method does not require an accurate friction model and can achieve the purpose of compensating for nonlinear friction without the need for friction parameter estimation. However, model-free methods usually have limited response speed, and their compensation capability is also limited when friction changes dynamically. Therefore, a method that can accurately and effectively compensate for friction is needed. Summary of the Invention

[0006] To more accurately compensate for friction and further reduce its impact on the photoelectric tracking frame, this invention utilizes both a friction model-based compensation method and a non-friction model-based compensation method. First, the LuGre model is used for initial friction compensation. Then, a terminal sliding mode disturbance observation is designed for further compensation. Friction model compensation reduces the friction threshold, while the terminal sliding mode disturbance observer provides secondary compensation for small-scale friction. This combined approach helps reduce compensation errors.

[0007] This invention adopts the following technical solution: a friction compensation method based on the LuGre model and a terminal sliding mode observer, the steps of which are as follows:

[0008] The velocity transfer function of the photoelectric tracking system is described as follows:

[0009]

[0010] Where s represents the Laplace transform, ω n Let ω be the mechanical angular frequency, k be the system gain, and ξ be the system damping.

[0011] The state-space expression of system (1) considering friction is:

[0012]

[0013] in, Let ω be the state vector, and ω be the angular velocity. Angular acceleration, u is the control input, and T is the angular acceleration. f This indicates system friction.

[0014] The system control law is designed as follows:

[0015]

[0016] Among them, u r For the closed-loop controller output, For Lugre friction model estimators; For the terminal sliding mode observer observations.

[0017] The LuGre friction model is designed as follows:

[0018]

[0019] Where σ0 is the bristle stiffness, σ1 is the micro-damping coefficient, σ2 is the viscous friction coefficient, and T c Let T be the Coulomb friction torque. s For the maximum static friction torque, ω sLet ω be the Stribeck angular velocity, z be the average deformation of the stiffener, g(ω) be the frictional Stribeck effect function, and e be the base of the natural logarithm.

[0020] The LuGre friction model (4) contains 6 parameters to be identified, among which T c T s σ2, ω s These are static parameters, while σ0 and σ1 are dynamic parameters. These six parameters are usually identified through experimental methods.

[0021] (4-1) Static parameter identification:

[0022] When the system is in a uniform state, the average value of the electromagnetic thrust from multiple measurements is used as the frictional force value corresponding to the velocity, forming a frictional torque sequence, that is, the microscopic deformation z is a constant value z. ss Then there is Therefore, the second equation (4) can be expressed as:

[0023] z ss =g(ω)sgn(ω) (5)

[0024] Where sgn(·) is the sign function, substituting equation (5) into equation (4) further yields the static friction equation:

[0025]

[0026] in, For static friction torque, the four static parameters T of the LuGre friction model are: c T s σ2, ω s The static identification part of the model was completed by using the particle swarm optimization algorithm based on natural selection.

[0027] (4-2) Dynamic parameter identification:

[0028] When the system's contact surfaces are in the static friction region, the motion state is not obvious, and it is mainly affected by Coulomb friction torque, i.e. z≈x s , Where x s Given a small acceleration signal and the measured displacement before slip of the system in the static friction region, σ0 can be expressed as:

[0029] σ0≈T c sgn(ω) / x s (7)

[0030] From the first equation in (4), we can derive:

[0031]

[0032] Thus, σ1 can be calculated.

[0033] After compensation using the LuGre model, the state space (2) of the system can be redescribed as:

[0034]

[0035] in, This represents the friction compensation error of the LuGre model.

[0036] The sliding mode observer is designed as follows:

[0037]

[0038] in, It is an estimate of x1. This is an estimate of x2, where l1,l2>0 represents the observer gain. For the observation error, subtracting equation (9) from equation (10) yields the observation bias system equation:

[0039]

[0040] in, This represents the error after observer compensation. To ensure that the observer state variables converge to zero in a finite time, the non-singular terminal sliding surface is taken as:

[0041]

[0042] Where α,β>0 is the sliding surface gain, p,q are positive odd numbers, which are parameters related to the convergence time and p>q.

[0043] Differentiating with respect to the sliding surface, we get:

[0044]

[0045] Selecting the exponential approach law:

[0046]

[0047] Where, k c >0 represents the sliding mode convergence rate, and η>0 represents the switching gain.

[0048] The terminal sliding mode observer is designed to measure the following:

[0049]

[0050] To reduce sliding mode chattering caused by the sign function sgn(s) in the exponential reaching law, a novel reaching law is designed as follows:

[0051]

[0052] Where σ and μ are adjustable parameters.

[0053] The present invention has the following advantages:

[0054] (1) First, the LuGre model is used for coarse friction compensation, and then the terminal sliding mode observer is used for secondary friction compensation. Due to the existence of secondary compensation, the sensitivity of the LuGre friction model to dynamic and static parameters can be reduced, that is, the identification parameters do not need to be completely accurate.

[0055] (2) After the LuGre model performs coarse friction compensation, the terminal sliding mode observer is equivalent to compensating for small-amplitude friction. Under the condition that the observation capability of the terminal sliding mode observer is certain, the smaller the friction amplitude, the smaller the final absolute compensation error. Attached Figure Description

[0056] Figure 1 This is a general model block diagram of the friction compensation scheme of the present invention;

[0057] Figure 2 This is a detailed description diagram of the terminal sliding mode observer of the present invention;

[0058] Figure 3 This is a comparison chart of the friction estimation compensation amount and compensation estimation error of the present invention with other methods;

[0059] Figure 4 This is a comparison chart of the velocity tracking response and velocity tracking error of the present invention with other methods. Detailed Implementation

[0060] The specific embodiments of the present invention will be described in detail below with reference to the accompanying drawings.

[0061] A block diagram of a friction compensation method based on the LuGre model and a terminal sliding mode observer is shown below. Figure 1 As shown, it includes a speed-controlled object, a speed regulator, a friction estimation system, and a speed sensor. The friction estimation system consists of a LuGre friction model and a terminal sliding mode observer. To achieve the purpose of this invention, the method steps are as follows:

[0062] Example 1

[0063] Given that the nominal parameters of the photoelectric tracking rack system are ξ = 6.32, ω n =79.48, k=63171.19, the design process and effects of the present invention will be described in detail below:

[0064] The velocity transfer function of the photoelectric tracking system is described as follows:

[0065]

[0066] Its state-space expression is:

[0067]

[0068] Among them, the system friction T f Determined by the rack equipment, given parameters:

[0069] σ0=3000,σ1=25,σ2=0.08,T c =55,T s =555,ω s =0.015

[0070] The system control rate is designed to be:

[0071]

[0072] To implement closed-loop control, the closed-loop controller is designed as follows:

[0073]

[0074] The LuGre friction model is designed as follows:

[0075]

[0076] Static and dynamic parameters were identified separately to achieve better compensation results. The identified parameters are as follows:

[0077] σ0=2500,σ1=10,σ2=0.1,T c =55,T s =140,ω s =0.001

[0078] After friction model compensation, the state space of the system is represented as follows:

[0079]

[0080] The sliding mode observer is designed as follows:

[0081]

[0082] Take the non-singular terminal sliding surface as:

[0083]

[0084] Let's take a new type of reaching law as follows:

[0085]

[0086] The terminal sliding mode observer is designed to measure the following:

[0087]

[0088] according to Figure 2 The simulation structure diagram shown is programmed with a sampling period of h = 0.001s.

[0089] Given velocity reference ω * =5sint, different methods for friction estimation results are as follows Figure 3 As shown, Figure 3 (a) The friction T actually applied to the system f Friction estimates when using the LuGre model alone, when using the terminal sliding mode observer alone, and when using the scheme of this invention. Figure 3 (b) Comparison of friction estimation errors. As can be seen from the figure, the friction estimation error is minimized when using the solution of the present invention, reduced by 89%. This indicates that the friction compensation is more accurate when using the solution of the present invention.

[0090] Figure 4 The speed tracking results are shown when using different friction compensation methods. Figure 4 (a) represents the speed setpoint and speed response value. Figure 4 (b) represents the speed tracking error. As can be seen from the figure, the speed tracking error is minimized when using the present invention near the zero point of the speed. The compensation of the present invention reduces the speed tracking error by 96% compared to the uncompensated version, indicating that the present invention can effectively reduce the turning error and improve the speed tracking capability.

[0091] Those skilled in the art will readily understand that the above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A friction compensation method based on the LuGre model and a terminal sliding mode observer, characterized in that: The implementation steps include the following: Step (1): The velocity transfer function of the photoelectric tracking system is described as follows: （1） in, Represents the Laplace transform. For mechanical angular frequency, For system gain, For system damping; The state-space expression of the system considering friction is: （2） in, For state vectors, Angular velocity, Angular acceleration, To control the input, Indicates system friction; Step (2): Design the system control law as follows: （3） in, For the closed-loop controller output, For LuGre friction model estimators; For the terminal sliding mode observer observations; Step (3): Design the LuGre friction model as follows: （4） in, For the stiffness of the sideburns, The micro damping coefficient, The coefficient of viscous friction, For Coulomb friction torque, For the maximum static friction torque, For Stribeck's angular velocity, For the average deformation of the bristles, Let Stribeck effect function be the friction function. is the base of the natural logarithm; Step (4): Formula (4) representing the LuGre friction model contains 6 parameters to be identified, among which , , , These are static parameters. , These are dynamic parameters, and the following experimental methods were used to identify these six parameters; (4-1) Static parameter identification: When the system is in a uniform velocity state, the average value of the electromagnetic thrust from multiple measurements is used as the frictional force value corresponding to the velocity, forming a frictional torque sequence, i.e., microscopic deformation. constant value Then there is Therefore, the second expression of equation (4) is: （5） in, As a sign function, substituting equation (5) into equation (4) further yields the static friction equation: （6） in, The four static parameters of the LuGre friction model represent the static friction torque. , , , The static identification of the model was completed using the particle swarm optimization algorithm based on natural selection. (4-2) Dynamic parameter identification: When the system's contact surfaces are in the static friction region, the motion state is not obvious, and it is subject to Coulomb friction torque, i.e. , , ,in Given a small acceleration signal and measuring the system's pre-slip displacement as the steady-state displacement in the static friction region, then... Represented as: （7） From the first equation in (4), we can derive: （8） Thus, we can find ; After compensation using the LuGre model in step (5), the state space representation of the system is as follows: （9） in, This indicates the friction compensation error of the LuGre model; Step (6) Design the sliding mode observer as follows: （10） in, yes The estimated value, yes The estimated value, For observer gain, The observation error is represented by equation (9). Subtracting equation (9) from equation (10) yields the observation bias system equation: （11） The error is compensated by the observer. To ensure the observer converges within a finite time, the non-singular terminal sliding surface is taken as: （12） in, For sliding surface gain, Positive odd numbers are parameters related to convergence time and ; Differentiating with respect to the sliding surface, we get: （13） Selecting the exponential approach law: （14） in, Let be the sliding mode convergence rate. To switch the gain; The terminal sliding mode observer is designed to measure the following: （15） To reduce the sign function in the exponential reaching law The resulting sliding mode chattering is addressed by the following approach law: （16） in, This is an adjustable parameter.

2. A LuGre model and terminal sliding mode observer, characterized in that: The friction compensation method of this observer The friction compensation method based on the LuGre model and terminal sliding mode observer is designed according to claim 1.

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