In many mechatronic systems, gear transmission chains are often used to transmit motion and power between motors and loads, especially for light, small but large torque output systems. Gear transmission chains will inevitably bring backlash as well as elasticity of shafts and meshing teeth. All of these nonlinear factors will affect the performance of mechatronic systems. Anti-backlash gear systems can reduce the transmission error, but elasticity has to be considered too. The aim of this paper is to find the key parameters affecting the resonance and anti-resonance frequencies of anti-backlash gear systems and then to give the design optimization methods of improving performance, both from element parameters and mechanical designing. The anti-backlash geared servo system is modeled using a two-inertia approximate model; a method of computing the equivalent stiffness of anti-backlash gear train is proposed, which comprehensively considers the total backlash of transmission chain, gear mesh stiffness, gear shaft stiffness and torsional spring stiffness. With the
As a primary method of mechanical transmission, gear transmission is widely used in the precision servo mechanism field, such as robot, seeker and inertially stabilized platforms, which have the requirements of fast response, high positioning accuracy and good stability (Chung et al., 2010; Hilkert, 2008; Slamani and Bonev, 2013; Baek et al., 2003a). However, the traditional gear transmission mechanisms have the disadvantages of backlash, friction and elasticity, which limit the performance of response, positioning accuracy and stability, and they are seriously restricted in the application of high-precision servo systems.
To solve these problems, researchers and engineers proposed several solutions, one of which is to choose the principle of the transmission mechanism. Using the harmonic gear can eliminate backlash effectively, but friction and flexibility of the transmission chain are increased (Vassileva et al., 2011; Masoumi and Alimohammadi, 2013), and using direct drive principle can omit the transmission chain. The motor and load are connected directly. Hence, the friction is reduced, backlash is eliminated, and the mechanical stiffness is very high (Asada et al., 1989; Su et al., 2002). However, the direct drive is not suitable for small, light but large output torque systems.
The elasticity of transmission chain is also an important problem which influences the bandwidth of servo systems. There are several mechanical components that have elasticity which can influence the transmission error and performance of the gear transmission chain: transmission shaft elasticity (Dwivedula and Pagilla, 2012), support bearing elasticity (Walha et al., 2009; Wang et al., 2011, 2016), gear meshing elasticity caused by gear tooth stiffness (Jia et al., 2003; Li et al., 2020) and even the gear meshing stiffness vary with time (Shi et al., 2020). All these factors make the transmission chain not ideally rigid, but flexible, under specific dynamic conditions, the systems will generate undesirable vibration. Moreover, the elasticity and backlash will have combined influence on the servo system.
Some researchers tried to solve these problems directly from the control aspect without changing any mechanical design. Kolnik and Agranovich (2012) estimated the equivalent backlash and torsional torque disturbances using disturbance observer (DOB). The torque is then compensated in the inner feedback loop, which decreased the vibrations caused by the backlash and the transmission shaft elasticity, but this compensation algorithm is just simulated using a transmission system model with a very small backlash. Yang et al. (2016) proposed a robust shaft torque compensator in a two-inertia elastic system with backlash; this compensator can make the servo system behave like a rigid control system with single moment of inertia by adjusting the feedback coefficient, and this control method can suppress mechanical resonance significantly.
Another method to solve the backlash problem is using anti-backlash mechanisms. Different anti-backlash gear transmissions were designed to decrease the influence of backlash in some research. Most of them mainly focus on mechanism structure design, transmission precision analysis and application. Several anti-backlash mechanisms were investigated; the advantages and disadvantages of these methods were analyzed (Hale and Slocum, 1994). Shi and Yang (2000) enumerated several types of the anti-backlash gear train mechanisms, and the preload of torsional spring of double gear mechanism was calculated. Allan and Levy (1980) deduced the formula of minimum preload torque in the design of spring-loaded anti-backlash gear. Yang et al. (2013) deduced the dynamic model of anti-backlash gear considering friction and time-varying mesh stiffness. These mentioned studies provide theories for us about the calculation of the comprehensive equivalent stiffness of the anti-backlash gear train.
Some researchers modeled the multistage general spur-gear trains and analyzed the frequency response characteristics of them. Baek et al. (2003b) considered the dynamics of a two-stage general spur-gear train as a two-inertia approximate model; the influences of motor input voltage and total backlash on the resonance and anti-resonance frequencies of system were analyzed. Especially the contribution rate of each stage backlash to the resonance and anti-resonance frequencies were obtained. Zhou et al. (2009) built the model of multistage general spur-gear train based on two-inertia approximate model, which contained the backlash, gear shaft stiffness and gear mesh stiffness. The influences of backlash and reduction ratio on frequency response characteristics of system were analyzed. Based on the modeling method of multistage general spur-gear train, some researchers studied the model of multistage anti-backlash gear train. Kwon et al. (2004) built a two-inertia model of three-stage anti-backlash geared servo system. A normalized describing function was used to describe the backlash; the formulas of resonance and anti-resonance frequencies were deduced. However, the model did not consider the torsional spring. The relation between the main gears and freewheel gears was not clear, and only the motor output torque influence on the frequency response characteristics of system was analyzed.
In the anti-backlash geared servo systems, the resonance and anti-resonance frequencies of system may be close to the servo bandwidth, which can limit the performance of response and stability of the servo system (Hale and Slocum, 1994; Shim et al., 2008; Jesper, 2004). Acceleration feedback loop and notch filter are used in some research to suppress the mechanism resonance (Lee et al., 2012; Szabat and Orlowska-Kowalska, 2007; George and Gao, 2001; George, 2004; Hoogendijk et al., 2014). However, If the notch filter's center frequency is different from the actual resonant frequency, the closed-loop system could exhibit increased oscillations or even become unstable (Bahn et al., 2016). The center frequency of the notch filter must be the same as the resonance frequency accurately, which is difficult when the resonance frequency is approximately equal to the crossover frequency of the speed loop; the acceleration feedback also increases the cost of system. Therefore, the problem needs to be solved by other methods. The best one is to make the resonance and anti-resonance frequencies of the system far from the bandwidth of the speed loop. The literature shows that current research mainly focuses on the multistage general spur-gear servo system, and frequency response characteristic analyses are concentrating on the total backlash of gear train, gear shaft stiffness, gear mesh stiffness and motor input voltage. However, the literature about the anti-backlash gear train model is rare; the key factors affecting frequency response characteristics are not completely clear.
In this paper, a multistage torsional spring-loaded anti-backlash gear transmission chain is designed to eliminate the backlash and to achieve a large output torque from a small input torque. Based on this transmission chain, this paper aims to find the key parameters affecting the resonance and anti-resonance frequencies of anti-backlash gear systems and to propose design optimization methods of improving performance, both from element parameters and mechanical designing. The two-inertia model of torsional spring-loaded anti-backlash gear train will be built based on the model of the general spur-gear train; a new method of computing equivalent stiffness of the anti-backlash gear train will be proposed, which considers the total backlash of transmission chain, gear mesh stiffness, gear shaft stiffness and torsional spring stiffness comprehensively. Then the
The structure of this paper is as follows. The model of anti-backlash geared servo system is established, the proposed equivalent stiffness computation method of anti-backlash gear train is analyzed, the formulas of key parameters are deduced, and the
The anti-backlash geared servo system is composed of a DC motor, four-stage anti-backlash gear train, load, sensors, motor driver and motion controller, which is shown in Fig. 1. This mechanism is applied in a two-axis inertially stabilized platform; the gyroscope closed loop is designed to isolate the carrier disturbances.
Schematic of the anti-backlash geared servo system. V is a tachometer, P is a potentiometer as load position sensor, and G is a gyroscope.
In Fig. 1, the anti-backlash gear train is mainly composed of driving gear (1), main gears (2, 3, 4, 5, 6, 7, 8), freewheel gears (
In the multistage anti-backlash gear train, the transmission components can be treated as a single, composite, equivalent torsional spring that interconnects with the motor and load, which is shown in Fig. 2.
Equivalent model of the anti-backlash geared servo system.
Where
The output torque of the motor is controlled. Namely, the motor driver is operated in the current closed-loop mode, and the controller of current loop is a proportional plus integral (PI). The composite model of motor driver and motor can be written as
In the spur-gear train, if the backlash is not considered, the transmission torque can be calculated by
However, the backlash can not be neglected in the system. If the transmission angle error
Compared with the general spur-gear train, the motor and load always contact in the anti-backlash gear train because of the preload torsional spring. The transmission torque is not zero in the backlash area, and the equivalent stiffness of the backlash area should be considered, which is different with the spur-gear train. Assuming that the equivalent stiffness of main gear train is
Stiffness model of anti-backlash gear train.
There are two states in the anti-backlash gear motion process. One is the motion in the backlash area. The main gear train does not transfer the torque directly. The torque is transferred through the freewheel gear train and the torsional spring. The equivalent stiffness of the anti-backlash gear train is the series of
Stiffness to transmission angle error.
Compared with the general spur-gear train, the equivalent stiffness is enhanced by using the anti-backlash gear, especially when the transmission angle error is larger than the backlash. Based on Fig. 4b, compared with Eq. (3a), the equivalent stiffness is not zero, but the combination of the freewheel gear train and the torsional spring stiffness when the motion is in the backlash area (
Block diagram of the anti-backlash geared servo system in
The model has been established, but the key parameters of the model are unknown. At the mechanism design stage, the parameters can only be obtained through theoretical analysis. The formulas of
In the anti-backlash geared servo system model, the parameter values of motor, motor driver and bearing friction can be obtained from their handbooks. However, for the anti-backlash gear train, researchers only know the size and material of gears and springs. The equivalent moment of inertia and stiffness can not be obtained directly. Without considering the backlash, the main gear train and the freewheel gear train may be considered as a spur-gear train. The calculation methods of equivalent stiffness of the spur-gear train can be used (Baek et al., 2003b). The anti-backlash gear train is composed of the main gear train and the freewheel gear train, which is connected together by a torsional spring, as shown in Fig. 6.
Schematic of the anti-backlash gear train.
In the two-inertia system, the motor is one mass, and the load is another one. The total mass of the system is focused on the motor and load. The equivalent moment of inertia of the motor can be calculated by
Then, the equivalent moment of inertia of load can be calculated by
From Fig. 6, the
Parameters values of the anti-backlash geared servo system.
Structure of the arc torsional spring.
Assuming that a force
Equation (17) can be rewritten as
Based on the above analysis, the values of key parameters of the system –
The simulation model of the anti-backlash geared servo system is built in Fig. 5. The values of basic parameters in the model are shown in Table 1, and parameter values of computed variables are shown in Table 2. As the input of system, the excitation signal is a chirp sinusoidal signal; the frequency ranges from 1 to 300
Parameters values of computed variables.
In the anti-backlash geared servo system, if changing the magnitude of backlash only and other model parameters shown in Tables 1 and 2, the backlash ranges from 0.01 to 0.5
Influence of backlash variation on the frequency response characteristics of system.
Influence of torsional spring stiffness variation on the frequency response characteristics of system.
The anti-resonance and resonance frequencies decrease with the increase of backlash. When the backlash ranges from 0.01 to 0.5
In the anti-backlash geared servo system, only changing the stiffness of the torsional spring, the influence of torsional spring stiffness variation on the frequency response characteristics of system is analyzed. Due to the size restriction of the last stage main and freewheel gears, the radius of torsional spring can not be changed. Only the width or height of rectangular section can be adjusted. In this paper, four kinds of torsional spring are chosen. Their stiffnesses are [24.80, 49.91, 102.50, 239.57]
In Fig. 9, the anti-resonance and resonance frequencies increase with the increase of stiffness of the torsional spring. When the stiffness is [24.80, 49.91, 102.50, 239.57]
The anti-backlash gear train is applied in the inertially stabilized platform, which should remain the same control precision in different load conditions. While the resonance frequency of system is one of the main factors which influence the control precision, the influence of load variation on the frequency response characteristics of system is analyzed. In the simulation model, only changing the load moment of inertia, the load moment of inertia ranging from 483 to 2000
Influence of load variation on the frequency response characteristics of system.
From Fig. 10, the resonance and anti-resonance frequencies decrease with the increase of load moment of inertia. When the moment of inertia ranges from 483 to 2000
In the anti-backlash gear train, the size and material of each stage freewheel gear are the same as the main gear of the same stage, so
Bode diagrams of anti-backlash geared servo system with the gears and shafts stiffness variation.
From Fig. 11, when the
The experimental setup of anti-backlash geared servo system is shown in Fig. 12, which is composed of an anti-backlash gear mechanism, motor driver (Maxon-ADS50V/5A), dSPACE1103 hardware-in-loop simulation system and a computer. The tachometer is used to measure the angular velocity of the motor, and the sensitivity is 0.1
Experimental setup of the anti-backlash geared servo system: (1) dSPACE1103, (2) motor driver, (3) motor, (4) tachometer and (5) anti-backlash gear train, six-load.
The initial state of the experimental setup containing the magnitude of backlash is approximately 0.15
Comparative analysis between the simulation and experiment results.
From Figs. 13 and 14, the resonance and anti-resonance frequencies of system increase with the increase of torsional spring stiffness and decrease with the increase of load moment of inertia. Compared with the experiment results, the maximum error of simulation results is less than 10
Bode diagram of the experimental setup with different torsional spring stiffnesses.
Bode diagram of the experimental setup with different loads.
The total backlash, torsional spring stiffness and load moment of inertia are the main factors that influence the frequency response characteristics of the anti-backlash geared servo system. The resonance and anti-resonance frequencies increase with the decrease of the total backlash and the load moment of inertia and increase with the increase of the torsional spring stiffness.
The simulation results show that when the total backlash ranges from 0.01 to 0.5
From the mechanical parameter point of view, increasing the stiffness of torsional spring can improve the frequency response characteristics of system. This target is easy to realize by changing the width or height parameters of rectangular section of the torsional spring. Reducing the total backlash of anti-backlash gear train is another method of system improvement. The magnitude of backlash is influenced by the manufacture and assembly technology. From the mechanical design aspect, designing the center distance adjusting mechanism can decrease the initial total backlash, and the load lightweight design can also improve the frequency response characteristics of system, but the stiffness of the load should be considered simultaneously. These measures will be applied in our future work.
All data generated or analyzed during this study are included in this published article.
LZ and HL designed the methodology, created the model, simulated the model and designed the experimental setup of servo system. DF provided the ideas, financial support and supervised the research activity planning and execution. SF designed the experimental software and collected the experimental data. JZ designed the mechanical structure of anti-backlash geared setup and the arc springs. LZ designed the experiments and carried them out. LZ and HL prepared the manuscript with contributions from all co-authors.
The authors declare that they have no conflict of interest.
The authors would like to express their thanks to the supporting agencies.
This research has been supported by the National Key R&D Program of China (grant no. 2019YFB2004700) and the Preliminary Research Project of National University of Defense Technology (grant no. ZN2019-7).
This paper was edited by Dario Richiedei and reviewed by two anonymous referees.