3.1. Experimental Setup
The test platform to measure output torque of the PMSM is shown in
Figure 6. It mainly consists of a drive circuit, a control circuit, a computer, and a MEMS gyroscope sensor. The MPU6050 is chosen as the core component of the MEMS gyroscope sensor and is used to obtain rotor dynamics data which is then processed by the computer.
Figure 7 shows the installation location of the MEMS gyroscope sensor and the initial position of the rotor in the experiment. The MEMS gyroscope sensor is mounted on the top of the output shaft as shown in
Figure 7a. As shown in
Figure 7b, the demo board containing MPU6050 is placed inside the sensor housing. The sensor housing is mounted on the top of the output shaft and two screws are used to hold it in place. The reset key of the demo board is on the sensor housing. When the rotor is adjusted to the initial position, the reset key will be pressed to avoid drift errors and to ensure that the current position is the initial position for the sensor. The initial position of the rotor shown in
Figure 7c is a relative position between the rotor and the stator, which is consistent with that in the coordinate system of the simulation model in
Figure 2. A detachable limit device shown in
Figure 7b is utilized to ensure that the rotor can be adjusted to the initial position before the experiment is conducted. The stator static coordinate OXYZ is coincided with the simulation coordinate. When the PMSM is driven, the relevant dynamic data are measured by the MEMS gyro sensor and are kept in the computer.
In the experiment, the oscilloscope was connected to the AC/DC current probe of A622 100 Amp type and then used to monitor the current of each coil winding. The PWM current waveform of No. 2 stator coil with a duty cycle of 90% is shown in
Figure 8, and the measured current data are shown in
Table 1.
3.2. Principles of Experimentation
The coordinate system O
pX
pY
pZ
p of the rotor is defined as a reference to describe the position of the rotor. It is a right-hand dynamic coordinate system. The origin O is always coincident with the origin of the stator coordinate system OXYZ, which is defined in
Figure 2. As shown in
Figure 9, the coordinate system OXYZ is coincident with O
pX
pY
pZ
p when the rotor does not rotate.
When the rotor of the PMSM is moving to one direction, without considering friction force its dynamics can be expressed according to Lagrange’s Equations of second kind [
17,
18], and its output torque can be derived as follows:
where
is the output torque vector of the PMSM.
is the rotor angular displacement vector;
is the rotor inertia matrix.
is the Gothic and centrifugal force matrix and can be ignored under the condition of rotation around the constant axis.
is denoted as:
where
,
and
are the moments of inertia that are relative to each coordinate axis of OXYZ. Substituting Equation (8) into Equation (7) the relationship between the output torque and the rotor angular acceleration is expressed as:
,
and
are rotor angular acceleration components decomposed into X-, Y-, Z-axis, respectively. Moreover, the rotor angular acceleration vector
can be expressed as
. Obviously,
can be obtained if the moment of inertia and angular acceleration vector
can be measured on the basis of the known rotor material, size and structure. The moments of inertia can be calculated by using the ADAMS software. The simulation results of the moment of inertia of each coordinate axis in the coordinate system are respectively obtained as
,
,
.
The remaining challenge is to obtain the components of rotor angular acceleration
,
, and
using the dynamic information obtained by the MEMS gyroscope sensor. Define
,
and
as components of the unit position vector of the rotor
on each coordinate axis of OXYZ.
represents the time sequence number of the rotor position vector that is collected by the sensor, and
represents the sampling time. By using the MPU6050 gyroscope sensor, the components
,
, and
can be measured during the sampling time [
19]. The formula for calculating
,
, and
via Euler angles can be obtained as follows:
where
,
,
, and
are Euler angles shown in
Figure 9.
Based on the definition of linear velocity, the components of unit linear velocity vector
,
, and
during the
i-th sampling time are obtained by taking the first derivative of
,
, and
as follows:
where
.
Similarly, based on the definition of linear acceleration, the components of unit linear acceleration vector
,
, and
during the
i-th sampling time are obtained by taking the first derivative of
,
, and
as follows:
where
.
In the three-dimensional space, angular acceleration vector in the fixed coordinate system OXYZ is expressed according to the rigid body kinematics as follows:
where
is the rotor position vector,
is the rotor linear acceleration,
represents cross product. According to the definition of the unit position vector and unit linear acceleration vector in Equation (9) and Equation (11), it can be obtained that:
where
,
. By taking Equation (13) into Equation (12), the angular acceleration is rewritten as follows:
Furthermore, the components of
during the
i-th sampling time are derived as:
By taking Equation (9), (10), and (11) into Equation (15), the functional relation of the angular acceleration vector and Euler angles can be established as Equation (16).
The flow chart for calculating the output torque is shown in
Figure 10.
3.3. Experimental Results
As shown in
Figure 9, under the condition that the stator winding current is set as 5.42 A, the computer draws the trajectory diagram of the tilt motion mapped on the sphere with a radius of 65 mm. The corresponding components of the unit linear velocity vector, unit linear acceleration vector, angular acceleration and output torque vector in the stator coordinate system OXYZ, are shown in
Figure 11,
Figure 12,
Figure 13 and
Figure 14, respectively.
It can be seen from
Figure 11 that the electromagnetic torque will quickly drive the rotor to reach the forward maximum velocity which is in the direction of the negative OX axis, then the rotor decelerates to zero in the influence of friction. As shown in
Figure 12, the linear acceleration component of the rotor increases to the maximum positive value first driven by the electromagnetic force, and then decreases to zero due to the friction. This process corresponds to the stage that the velocity increases from zero to the maximum. In the influence of the friction, the linear acceleration component reaches the inverse maximum value and finally becomes zero. This process corresponds to the stage that the velocity increases from the maximum value to zero.
Figure 13 shows that the change of the component of the angular acceleration vector on the OY axis is consistent with the component of the linear acceleration vector on the OX axis in
Figure 12. As shown in
Figure 14, the output torque increases rapidly from zero to the maximum value and then it gradually decreases to zero under the influence of the friction. In this stage, the rotor speed increases continuously under the influence of the output torque. When the electromagnetic force is less than the friction force, the rotor will slow down to the final stillness.
3.4. Comparison with Simulated Results
In order to compare with the simulated torque by the FEM, we take the output torque vector at the moment that the torque component on the OX axis reaches the maximum value at the first time as an example. This is the vector corresponding to the output torque at the starting time of the rotor. The torque components on the OX, OY and OZ axis of the coordinate system OXYZ at the moment are represented by
,
and
, respectively. Thus, the output torque vector
and its amplitude
can be respectively written as:
Two motion cases are considered in the comparison with simulated results. In the first case, the initial rotor potion is stationary, and the total current of coils increased from 1.5 to 5.42 A. In the second case, the initial rotor position rotates around the OZ axis from 0 to 180 degrees and the total current of coils is fixed 4 A. In this case, each coil is energized by unit current. The current directions of No. 2 coil and No. 20 coil are positive as the same as shown in
Figure 2, while the current directions of No. 8 coil and No. 20 coil are negative, which are opposite to that in
Figure 2. The experimental results of the two cases are shown in
Figure 15.
Taking the simulation torque as a reference, we define the measurement error
as:
where
denotes the simulation torque. The measurement errors of the two motion cases are shown in
Table 2 and
Table 3.
The output torque measured by the MEMS gyroscope sensor has a wide range of error from 9.48% to 53.54% in the coordinate system OXYZ.
3.5. Error Analysis and Compensation
The rotor of the PMSM is heavy, which causes great pressure on the contact surface of the support structure. Therefore, it inevitably brings error for measuring output torque during the start-up process of the rotor. When the rotor moves, the error caused by the friction will reduce the accuracy of torque measurement. This type of error can be estimated by repetitive experiments and the targeted error compensation can be conducted to improve the accuracy of the measurement.
In order to study the influence of friction torque on the measurement error of the output torque, the maximum static friction in the initial position of rotor is measured by a tension meter.
Figure 16 shows the measurement of the maximum static friction. The distance from the top of the rotor output shaft to the center of the rotor is 0.1 m, that is, the force arm of the maximum static friction force is 0.1 m. The tension meter and the top of the output shaft are connected by a thin nylon wire. A spirit level is used to ensure that the tension is perpendicular to the output shaft. Then, the tension meter is slowly pulled along the negative direction of the OX axis and the data displayed in the tension meter are recorded at the moment of rotor startup. The measurement for static friction is repeatedly conducted by 20 times and the measured results are listed in
Table 4.
The average value of the static friction measured by the above 20 groups of experiments is used as the estimated maximum static friction for the compensation. The average is 69.7 mN·m. The output torques are compensated and are shown in
Figure 17.
Table 5 and
Table 6 show the measurement errors after the compensation, and it can be seen that the measurement errors decrease to 0.39–23.47%.
After the compensation of the maximum static friction, the remaining error can be analyzed in two ways. Firstly, since the initial position of the rotor is manually adjusted, the motor stator and rotor in the coordinate system OXYZ cannot be perfectly aligned, which will change the experimental motion conditions and affect the measurement of the output torque. Secondly, only the maximum static friction is considered in this paper for compensation because of the limitation of the experimental method. In fact, other complex friction which is not estimated will affect the effectiveness of the compensation.