**2.2 Calibration**

At present, the principle of the mainstream measuring platform is almost based on the effect of force. The thrust is obtained by determining the basic parameters of the vibration system and measuring the displacement response of the vibration components and then by inversion according to the dynamic equation. The determination of the basic parameters of the vibration system is the calibration process, which applies a constant calibration force or calibration impulse to the measuring platform and then determines the system parameters according to the obtained system response and dynamic relationship [5]. Although the dynamic function can be calculated theoretically by obtaining the correlation coefficient of the measuring platform, it is more convenient, intuitive, and reliable to determine the system parameters through the calibration process.

In the process of micro-thrust and impulse measurement, the calibration process is indispensable. The first is to establish the functional relationship between thrust or impulse and displacement through the calibration process. The second is to determine the accuracy, repeatability, stability, and sensitivity of the measuring device by repeatedly loading the standard known force. The third is to avoid system error by comparing and correcting the measured value and theoretical value. Therefore, the repeatability, operability, adjustability, stability, and accuracy of the calibration device is one of the key technologies to ensure the performance of the micro-thrust measurement system and also an important basis to verify its measurement level.

According to whether the calibration force generation device is in direct contact with the measuring device, the calibration method can be divided into contact type and non-contact type. The contact calibration method includes the weight method and the impact hammer. The principle of the weight method is simple and easy to operate, but it is easy to be affected by sliding friction, air resistance, elastic expansion, and drag of the rope and limited by the minimum weight; it is difficult to provide high-precision micro-newton calibration force. The pendulum method uses a pendulum with a known mass to impact the measuring bench from a certain height to form an impulse with a known size. This method is also simple and easy to operate, but the error is large and difficult to control [12, 13, 21]. Non-contact calibration methods include the electrostatic comb method and the electromagnetic force method. The electrostatic comb method is usually composed of a group of interlocking non-contact comb teeth separated at very small intervals. The electrostatic repulsion force generated when approaching is used as the calibration force. It can not only provide a stable thrust but can also generate a high-accuracy impulse by controlling the number of comb charges and voltage application time. The electromagnetic force calibration method uses the magnetic effect of a current or the theorem of ampere force to generate a stable electromagnetic force, including the combination of a coil with a permanent magnet and the combination of an electromagnet with an energized wire. Similar to the electrostatic comb method, the electromagnetic force method can also generate stable calibration force and can also accurately generate calibration impulses of known size. In addition, the electromagnetic force is not easily affected by the external power supply line, and its accuracy is higher than that of the electrostatic comb method [22–24].

## **2.3 Performance metrics**

The main technical indicators of the micro-thrust and impulse measurement platform include sensitivity, stability, accuracy, resolution, and response time.

### *2.3.1 Sensitivity*

As the core technical index of the measuring platform, sensitivity, accuracy, and resolution are closely related, the definition of sensitivity in micro-thrust and impulse measurement is slightly different. Generally speaking, the sensitivity of steady-state micro-thrust measurement is the offset that the platform can achieve under a given thrust. As shown in Formula (2.7), the sensitivity is related to the length of the moment arm and the elastic coefficient and is regarded as the key index of the mechanical design of the measuring bench [5, 9].

### *2.3.2 Stability*

In the process of micro-thrust and impulse measurement, it is necessary to ensure that the response of the measuring bench is consistent and repeatable; otherwise, the calibration will be meaningless, and the accuracy and accuracy of the measurement cannot be guaranteed. In addition, long-term measurement needs to ensure the longterm stability of the bench. There are two main factors that affect the repeatability of measurement. One is the zero drift, that is, the change of zero point or the actual position of the pendulum, and the other is the gain drift, that is, the change of response coefficient or elasticity coefficient. The temperature change or friction of mechanical or electronic components and elastic components will cause drift. In order to pursue the accuracy of measurement, the generation of drift should be avoided as much as possible [9, 21].

### *2.3.3 Resolution*

Resolution is defined as the minimum difference in the response of the measuring platform after being loaded by two different thrusts or impulses. Resolution is the ability of the measuring platform to distinguish the mechanical effects of loading and is also a measure of the minimum value of thrust and impulse change of the measuring platform. The noise level of the measuring platform is directly related to the resolution, and high resolution can be achieved by minimizing the noise. In the process of micro-thrust and impulse measurement, the noise sources usually include the electrical noise of the sensor, the mechanical noise caused by environmental vibration, and the response change caused by the periodic change of temperature.

The resolution of the measuring bench can be quantified by continuously changing the magnitude of the loading force until the response cannot be resolved. However, in the actual measurement process, the error of this method is relatively large. Usually, the resolution is determined according to the measurement noise level, and the resolution can be specified to be twice the noise signal. The noise signal can be characterized by the ratio between amplitude and frequency or power spectral density [5, 8, 9, 25].

In the measurement process of an electric thruster, unless some measurement methods specifically use the resonance principle, the natural frequency between the electric thruster and the measuring platform should generally be avoided, so as to prevent the generation of resonance. In addition, to ensure the accuracy of measurement, the noise of the bench must be far less than the range of the thruster that can be produced by the thruster.

### *2.3.4 Response time*

The response time of the measuring bench is an important indicator of dynamic measurement, which can be characterized by many parameters, such as rise time (the time required for the response to reach 100% of the steady-state value), peak time (the time required for the response to reach the peak value), and stability time (the time required for the response to change less than 2% near the steady-state value) [9].

### *2.3.5 Accuracy*

Accuracy is a measure of the error between the measured value and the true value of the measuring bench. For a high-performance measuring bench, it is not only required to be sensitive enough, the results have obvious repeatability, not affected by random errors, but also to ensure the measurement results [16]. In the measurement process, the system error is usually minimized to improve the measurement accuracy, and the accuracy needs to be calibrated through standard thrust and impulse. In addition, the calibration method needs to be strictly designed to avoid introducing new system errors.
