Modeling and Analysis of Dual Motor Precision Transmission Mechanism

Aiming at the difficulty of fine modeling of dual-motor precision transmission mechanism, the modeling of friction and clearance of the system is studied. Firstly, the structure of the double-motor precision transmission mechanism is introduced, the system is simplified to a three-mass system, and a linear model of the system is established. On this basis, the improved stribeck model is used to describe the friction nonlinearity inside the two-way transmission chain and the large ring gear, and the dead zone model is used to describe the gap nonlinearity between the pinion and the large ring gear, thus forming a complete dual-motor precision transmission mechanism. kinetic model. An experimental device was built, and different excitation signals were used to verify the accuracy of the model. The experimental results show that the Pearson correlation coefficient between the established dynamic model and the actual system is >99% under various excitation signals, which shows the accuracy of the modeling method.


Introduction
In high-precision servo devices such as aerospace equipment, precision machine tools, industrial robots, and pointing mechanisms, transmission clearance has always been one of the most important factors affecting system performance [1,2]. When the drive part loses direct contact with the load, backlash occurs in the servo system, causing the movement of the load to be autonomous, i.e. "uncontrollable". Therefore, the nonlinearity of the gap often leads to steady-state errors and even oscillation instability, which seriously reduces the control performance and stability of the equipment. In recent years, more and more researchers have used dual-motor precision transmission mechanisms to eliminate backlash in high-precision servo devices. This method can completely eliminate the gap on the basis of ensuring the system servo accuracy [3,4].
Before 2010, most scholars focused on the anti-backlash method and synchronous control of the dual-motor transmission mechanism, and most of them stayed in the linear model in terms of modeling analysis, which could not meet the requirements of system characteristic analysis. In recent years, some scholars have begun to explore the modeling method of dual-motor precision transmission mechanism. Wen [5] established the dynamic model of the double pinion system considering the complex influence of nonlinear mesh stiffness and reverse impact between the pinion and the bull gear. The obtained model is helpful to design the intelligent controller of the double pinion system. Masahiko [6] established a four-mass model of a dual-motor mechanism. The model considered the rigidity of the connecting shaft between the large ring gear and the load, and used the model to estimate the load speed converted to the motor shaft, and dynamically calculated the relationship between the load speed and the motor speed. The difference value is added to the speed command to suppress the transient vibration generated at the load, and a better control effect is achieved. Zhao [7] proposed a model of the linear part of the system for the control problem of a dual-motor drive servo system with dead-band nonlinearity and used the description function method to analyze the dead-band nonlinearity. The simulation results show that due to the existence of dead-band nonlinearity, The error curve peaks at the commutation instant of the sinusoidal response. FENG [8] established a complex dynamic model of the dual-motor system by considering the effects of backlash nonlinearity, periodic time-varying stiffness of the gear train, random wind disturbance torque and motor cogging torque, and designed for the dynamic optimization of the bias torque. Provides a model reference. Wei [9] established the overall dynamic model of The dualmotor coupling drive system (DCDS) considering planetary gears, differential bevel gears and transmission shafts by using the transfer matrix method, and studied the effects of mesh stiffness and excitation source on the dynamic characteristics. A detailed theoretical analysis is carried out, and the validity of the dynamic model is verified by using the DCDS experimental platform. Jiang [10] established a dynamic model of a dual-servo motor drive system to analyze the nonlinearity of backlash, wear and backlash in the drive system.
The above research provides a reference for the optimization of the model of the dual-motor transmission mechanism, but ignores the importance of the nonlinear friction of the system, and simplifies the friction as viscous damping for analysis, resulting in a low degree of fitting between the model and the actual system, which cannot be truly described The influence of nonlinearities such as clearance and friction on system characteristics. In this paper, the friction characteristics of the two-way transmission chain and the large ring gear of the dual-motor precision transmission mechanism are described by the improved stribeck model, and the dead zone model is used to describe the gap nonlinearity of the system. The dynamic model of the dual-motor precision transmission mechanism was established, and finally an experimental device was built to verify the high accuracy of the model.

Dynamic Model of Double Motor Precision Transmission Mechanism
The dual-motor precision transmission mechanism is usually driven by two groups of permanent magnet synchronous motors and planetary reducers with the same nominal parameters, respectively, to drive pinions with the same module and number of teeth. "Large gear ring") on both sides of the outer ring, the outer ring of the large gear ring is fixedly installed on the base, and the inner ring is fixedly connected with the rotating tower base. After the two groups of pinions mesh with the outer ring gear of the large ring gear, they complete the rotation and drive the rotating tower with the load to revolve. The schematic diagram of the transmission structure is shown in Figure 1. The planetary reducer selected for the firststage transmission chain of the dual-motor precision transmission mechanism is a precision backlash, and the transmission accuracy is high. The transmission with the large ring gear is analyzed emphatically.
According to the dual-motor precision transmission mechanism shown in Figure 1, it is simplified to the three-inertia dynamic model shown in Figure 2. The meaning of the variables in the figure will be explained later. The two-way transmission chain in the dual-motor precision transmission mechanism belongs to parallel connection. relationship, the meaning of the parameters is exactly the same, therefore, one of the transmission chains can be analyzed as an example.   (1) where u is the armature voltage of the motor; i is the armature current of the motor; R is the armature resistance of the motor; L is the armature inductance of the motor; e K is the back EMF coefficient of the motor; m  is the rotation angle of the motor. The drivers of the two motors are set to the current loop mode, and the bandwidth of the current loop of the driver is more than 1KHz, which is much higher than the response bandwidth of the system speed loop, so the driver can be approximately regarded as a proportional link.
(2) where, d K is the driver conversion factor. The electromagnetic torque of the permanent magnet synchronous motor is proportional to the armature current, so there is T is the electromagnetic torque of the motor; M K is the torque coefficient of the motor. Assuming that the planetary reducer is an ideal transmission link, the force analysis of the pinion can be obtained: (4) where, 1 N is the reduction ratio of the planetary reducer, m J is the moment of inertia of the motor, 1 J is the moment of inertia of the planetary reducer, x  is the rotation angle of the pinion at the output end of the planetary reducer, and  is the output torque of the gear at the output end of the planetary reducer.
w are the positive and negative static friction, Coulomb friction, stribeck velocity, viscous damping coefficient and pre-slip zone, respectively.
The dead-band model is widely used to describe the backlash nonlinearity of the control system [12][13][14][15][16]. Let the backlash width be 2, as shown in Figure 3, then the gap between the output gear of the planetary reducer and the outer ring gear of the large ring gear The gap can be described as: Driving gear where K is the meshing stiffness of the pinion gear and the large ring gear; is the transmission error between the output gear of the planetary reducer and the large ring gear; 2 N is the gear ratio of the pinion gear and the large ring gear; L  is the rotation angle of the rotating tower base. According to the force of the rotating tower, the dynamic equation of the rotating tower is: (8) where 1 2 ,   are the torques output by the two transmission chains to the large ring gear, L J is the moment of inertia of the rotating tower base, and fL T is the friction of the large ring gear, which can also  (8) can obtain the dynamic model of the dual-motor precision transmission mechanism including the gap between the output gear of the planetary reducer and the large ring gear, and draw the model block diagram as shown in Figure 4.

Model Experiment Verification
In order to verify the accuracy of the established model, an experimental test device for a dual-motor precision transmission mechanism was built, and the dSPACE hardware-in-the-loop simulation platform was used to carry out model verification experiments.

Experimental device
A dual-motor precision transmission mechanism experimental device as shown in Figure 6   By referring to the specific model and key parameters of the experimental equipment, the device was tested and identified, and the system parameter values shown in Table 1 and the friction model parameter  values shown in Table 2 were obtained. Table 1

Experimental results and analysis
In order to fully verify the accuracy of the model, the difference between the model response and the actual speed signal under different excitation signals is compared. The excitation signals include sinusoidal signals and square wave signals with different amplitudes and frequencies. The experimental results are shown in Figure 6.
It can be seen from the figure that the overall fit between the model response and the actual speed curve is high, but there is a large jitter in the model response during the commutation process, while the actual system is relatively smooth. The reason is that the model adopts the dead zone model. Describing the gap nonlinearity, the disadvantage of this model is that the driving part and the driven part do not transmit torque at all in the range of the gap. During the simulation analysis, it is found that the output torque of the dead zone model in the low-speed section of the driving part and the driven part after crossing the gap is not equal. This phenomenon is different from the actual system, so it is necessary to correct the dead zone model to correctly describe the nonlinear characteristics of the system gap.  Figure 7 Experimental results In order to quantify the accuracy of the model, the Pearson correlation coefficient was used to evaluate the fitting degree of the model response curve and the actual speed signal curve. The Pearson correlation coefficient is the ratio of the product of the covariance and the standard deviation of two variables, which is used to describe the degree of correlation between the two sets of data X and Y. The more correlated the data, the closer the value of the Pearson correlation coefficient is to 1. The expression for the Pearson correlation coefficient is: