2.1. System Model
The system model of the EMA system is presented in
Figure 2.
The brushless direct current motor (BLDCM) (shown in
Figure 2, represented by the symbol M) is used to drive the mechanism of the EMA system. The transmitted torque and the motion relationship between the actual position and backlash is expressed by Equation (1).
where
represents the equivalent moments of inertia.
,
,
,
denote the pulse width modulation (PWM) coefficient, the armature resistance, the armature inductance, and the armature current, respectively.
,
,
and
denote the rotor speed, the motor torque, the torque constant, and the electrical constant, respectively.
is the friction torque,
is the load torque,
is the equivalent backlash of the output shaft,
is the actual position with backlash, and
is the position without backlash. As shown in Equation (1), if the load of the EMA system is determined, the motor selection is determined, and the model of nonlinear characteristics is clear, meaning the dynamic characteristics of the EMA system can be analyzed. In
Section 2.2, the nonlinear model will be analyzed.
2.2. Nonlinear Factors
In the EMA system, nonlinear factors are inevitable, and they seriously degrade system performance. It is urgent to carry out research on the nonlinear model. Introduction and identification of the friction model are first presented. The LuGre model is a popular friction model. The schematic of the LuGre model is illustrated in
Figure 3.
Friction torque is as follows [
35]:
where
is the friction torque,
is the coulomb friction torque,
is the static friction torque,
is the bristle stiffness,
is the bristle damping,
is the viscous friction coefficient,
is the Stribeck velocity,
is the velocity of the system,
is the dynamics of the deformation of bristles.
also appears in Equation (1). If the parameters of the friction model in Equation (2) of the system can be determined, the value of
can be obtained. When the value of
is obtained, it can be combined with Equation (1) to calculate the performance of the EMA system. Static parameters, such as
,
,
and
, are obtained by identification with the constant speed motion method. Thus,
(as a result
), and
can be described by the following equation:
Equation (3) is the intermediate equation for friction parameter identification. Substituting Equations (2) and (3) into Equation (1),
(shown in Equation (1)) is set as 0, and the friction torque is as follows:
where the subscript
denotes the steady state, that is, the speed is the data measured at a constant speed. It can be seen from Equation (4) that we can measure the armature current
to obtain the friction torque. The test bench is shown in
Figure 4. The test bench consists of a PC and the EMA system. The PC is used to send commands to the EMA system and receive the sensors’ output from the EMA system. Then, the PC processes the data to identify the friction parameters.
According to the type selection and constituent units of the EMA system, shown in
Figure 1, ignoring the high-order factors, this paper uses the second-order model in Equation (1) to study the characteristics of the system. Therefore, in the next identification process, our physical model is consistent with the mathematical model shown in Equation (1).
The identification process is as follows: run the EMA system at a uniform speed and measure the corresponding speed and current to obtain a set of values, and then the speed increases continuously from a small value to a large value (the curve type shown in
Figure 5). Then, the static parameters in the corresponding Equation (4) can be obtained by the least square method. Static parameters are identified by using the least square method and are shown in
Table 1.
The identification of dynamic parameters is also very important. The dynamic parameters of the LuGre model can be obtained as follows: firstly, model linearization is adopted at
and
, and then
, and
should be replaced in Equation (3),
shown in Equation (1) can be written as:
In order to deduce how to obtain dynamic parameters, set
and combine Equation (1), Equation (2), and Equation (5) by Laplace transform, and we can get:
According to the form of Equation (6), we can use the second-order system theory to analyze the dynamic parameters. Set
,
, Equation (6) can be described as:
As shown in Equation (7), the theory of second-order damping systems can be used to deal with the system’s friction problem. Using the experimental platform shown in
Figure 4, the input command is the position command. By obtaining the step response curve of the system, then, according to the principle of a second-order system,
,
is obtained. Based on classical second-order control theory,
and
are obtained. Dynamic parameters are presented in
Table 2.
Introduction and identification of the backlash model are presented. The Hysteresis model is applied to describe backlash and is shown in Equation (8). The inverse model is presented in Equation (9) [
36]:
where
is the desired output of the EMA system,
is Dirac function, which compensates the backlash instantaneously when the backlash is
,
is the reduction ratio.
The schematic diagrams of the Hysteresis and Hysteresis inverse model are shown in
Figure 6. The backlash characteristic is shown in
Figure 6a. The backlash inverse shown in
Figure 6b is used to cancel the effect of the backlash [
16].
The total length of the screw is 40 mm, and the total length of the nut is 16 mm. Thus, the nut needs to move 24 mm, since the backlash is different at each point, with eight points of 10 mm, 12 mm, 14 mm, 16 mm, 18 mm, 20 mm, 22 mm, 24 mm. They are set to measure the backlash. Finally, the average value of the backlash, obtained above, is the system backlash. The test bench shown in
Figure 4 is applied to implement the experiment to measure the backlash. Finally, the system backlash is obtained by collecting and processing the position information.
The backlash is measured in the following steps. Firstly, rotate the motor in a clockwise direction until the nut cannot be rotated due to reaching the bracket wall. Here, the backlash is eliminated. Meanwhile, stop rotating the motor, and the value of the potentiometer is recorded at this position. Secondly, fix the output shaft of the EMA system using the chucking appliance, and then rotate the motor in an anticlockwise direction with a slow speed. The potentiometer records the position at that moment. The difference between the position at that moment and the position of the last time is the backlash at this point. The backlash of the different point is measured by the above method. Finally, the average of the backlash is 0.142°, namely the system backlash.