Qirong Zhang， Xin Jin,2,， Kai Zhou， Zhongqing Zhang and Dongmei Liu

（1. School of Mechatronic Engineering, Beijing Institute of Technology, Beijing 100081, China；2. Lu’nan Research Institute of Beijing Institute of Technology, Tengzhou, Shandong 277500,China；3. The 31st Research, Third Academy of CASIC, Beijing 100074, China；4. State Key Laboratory of Intelligent Manufacturing System Technology for Complex Products, Beijing Institute of Electronic Engineering, Beijing 100854, China）

Abstract: Center of gravity (COG) is an important parameter of projectiles and rockets, for which supporting reaction method(or support reaction method) is an important COG measurement method. Based on this supporting reaction method a novel design method is proposed to determine the key design parameters of the COG measurement system. The method can quantitatively analyze the influence of the design parameters on the COG accuracy, in the measurement system designed with supporting reaction method. Using the principle of static balance, the error propagation theory, and the system accuracy analysis method, the equal-range required sensor precision (RSP) surface and non-equal-range required sensor pair precision (RSPP) adapted surface are adopted. The influence of random errors(like sensor accuracy and distance calibration accuracy) is analyzed. The selection strategy of equal-range and non-equal-range sensors is chosen, and then the recommended calibration accuracy values are obtained. For the quality detection accuracy of ±0.6 kg and the axial COG detection accuracy of ±1.5 mm, the RSP surface is drawn by the proposed method, and the force sensor with ±0.23 kg detection accuracy is selected. The experimental verification meets the accuracy requirements and verifies the effectiveness of the proposed design method for the system parameters of the COG measurement equipment.

Key words: center of gravity(COG)；supporting reaction method；required sensor precision(RSP)surface；accuracy analysis；parameter design

Mass and center of gravity(COG) are important parameters of rocket and missile engines,which are of great significance to the flight stability, operation safety and strike accuracy of missiles. It is difficult to determine the geometric position of the actual COG accurately by theoretical calculation, therefore it is necessary to measure the mass and COG by measuring equipment.

There are many ways to measure COG, such as swing method, hanging method, zero-position method, supporting reaction method, mass reaction method, unbalanced moment method, rotation balance method, and inertial method[1], etc.

The mass measurement accuracy of static multi-point supporting reaction method is between ±0.3% and 0.01%[2]. Due to the simple data processing and the high measurement accuracy, the supporting reaction method is widely used in the field of aerospace and other COG measurement tasks. Tab.1 lists the mass and COG measurement accuracy of the supporting reaction method in the measurement of the COG of rockets, missiles, etc.

Experts and scholars have conducted a lot of research work on the accuracy evaluation of COG measurement. Guo Zhicheng, et al.[18]used an error theory to analyze the effect of positioning errors on the accuracy of the COG measurement of tactical missiles. Chang Xiaodong, et al.[19]did a measurement uncertainty analysis of COG measurement equipment. Zhong Jiang, et al.[20]analyzed the influence of the mass of the detected object on the accuracy of the threepoint method for COG measurement. Zhang Yinghua, et al.[16]analyzed the effect of different sensor ranges on the COG measurement error.Ding Junhui, et al.[21]used the calculation method of the probability density function of multi-dimensional independent random variables to analyze the probability distribution of the COG measurement error.

Sensor accuracy and measurement system calibration errors are important error sources for COG measurement errors.

Tab. 1 Accuracy of COG via supporting reaction method

1) When designing a COG measurement system, the choice of sensor accuracy and system calibration accuracy during the design process,often depends on experience and lacks a unified method for accurate selection based on design parameters.

2) In the evaluation of COG measurement errors, it is often and only possible to perform a quantitative analysis based on design parameters,or to give only a quantitative relationship between some design parameters and COG errors. There is a lack of a COG error evaluation method which comprehensively considers all design parameters.

3) Existing research mostly uses equal-range sensors. For the selection of non-equal-range sensors, there are no corresponding selection methods available.

Regarding the above problems, we propose an equal-range required sensor precision (RSP)selection surface, which is used to quantitatively describe and analyze the relationship between various design parameters and the accuracy of COG detection. The method can accurately guide the selection of sensor accuracy and system calibration accuracy when designing a COG measurement equipment. Additionally the non-equalrange required sensor pair precision (RSPP) adaptive curved surface is proposed. The use of unequal-range sensor combinations can improve the measurement accuracy of the COG via the supporting reaction method. The method can be used to quantitatively select the sensor combination that meets the design indicators, and can guide the design of error devices such as COG in all directions.

1 Theoretical Analysis

1.1 Calculation model of measurement domain and error domain

The calculation model is established, based on the principle of the supporting reaction method[22?25]. The arrangement of the sensors in the measurement system is shown in Fig.1.

Fig. 1 Measurement scheme using equal-range sensors

The measurement domain of the system is defined as the mass and COG measurement range that meets the design accuracy requirements. The calculation of the system’s mass measurement domain is presented as follows

whereis the mass obtained by the sensorSiduring thekth measurement.is the mass information of the tare read out by the sensorSi.

According to the system torque balance,each force takes the coordinate axis to obtain the COG coordinates. System COG measurement domain can be calculated as follows

where (xi,yi) is the coordinate of the detecting point of the sensorSi.

The error propagation formula for calculating errors in the measurement domain is

wherefis a function ofxi, andσxiis the error ofxi.

The error domain of the system is calculated according to the error propagation formula

whereσxi,σyiare the position calibration errors of the detecting point of the sensorSi.

1.2 Equal range RSP selection surface

There are many parameters in Eq. (4). For the convenience of analysis, the moment of the coordinate line whereS2is located at is taken when the moment is balanced. In order to unify the calculation formulas of the coordinate components of the COG, the sensors are categorized into two groups according to the labels, and divided into [(1, 2), (3, 4)] when solving the COG in thexdirection, and [(2, 3), (1,4)]. The COG measurement domain is calculated by

wherel0is the coordinate component ofS2,ηis the sum of the mass of the sensor group withoutS2,lis the distance between two sets of sensors.

Noticed thatσM2=2ση2=8σS2, and letk=η/M, combining Eqs.(5) and (3), we got the COG error domain as where the parameterkis defined as the COG shift rate. For example, the conditionk∈[0,1],k=0.5 indicates that the COG is at the center of the two groups of sensors. Andkincreases as the distance between COG andS2increases.

It can be seen from Eq.(6) that the COG error increases withkincreasing. To ensure that the COG error domain covers the COG measurement domain,k≥ 0.5 needs to be satisfied. For example, ifk=0.6,η/M≤ 0.6 should be satisfied in actual measurement. If the calculatedη/M>0.6,S3should be selected as the reference of the moment projection coordinate line. Because whenk> 0.5, we can take the moment balance solution for the other side of the sensor, so whenk>0.5, we can use 1–kinstead ofK. Therefore, whenk= 0.5, the maximum error can be obtained.

When designing a measurement system, the COG error domain is often considered as a design parameter. It is necessary to select an appropriate sensor quality measurement error and sensor measurement point position calibration error according to the design requirements.

The accuracy of the sensor that satisfies the Eq. (7) is to ensure the accuracy range of the sensor in the measurement domain and the error domain. The upper boundary surface is the collection of the maximum sensor errors, and it represents the minimum accuracy of the required sensor. We name the surface as the equal-range RSP selection surface.

where the needed COG detection accuracyσ, the sensor position calibration accuracyσl, and the COG offset ratekare known. RSP is a function of the distance between the measuring points and the total mass of the testing.

1.3 Non-equal range RSPP fitting surface

RSP decreases askbecomes smaller from Eq.(8).

Decreasing the value ofkmeans that the force acting on the force sensor group that mainly acts when the calculating COG error is reduced. The smaller the range of the force sensor of the same level, the higher the absolute accuracy of the sensor. Therefore, the corresponding sensor range can be reduced to improve the accuracy. The accuracy requirements of a set of force sensors can be appropriately reduced. In order to improve the radial COG measurement accuracy, the non-equal-range sensor arrangement shown in Fig.2 can be used.S2andS3use a large range force sensors,S1andS4use a smallrange force sensor, starting from the COG offset positionS1orS4. The action constitutes a 3-point support.

Fig. 2 Measurement scheme using unequal-range sensors

Noticed thatσM2=ση2+2σγ2, the COG measurement domain and error domain are

Considering design indicatorsσy=σ, we have

The above equation can obtain the precision selection relationship of high and low precision sensors. The distance between radial measuring points is usually a fixed value, so the equation contains only three variables, namely the accuracy of the sensor pair and the total mass. The required sensor pair precision (RSPP) adaptation surface is defined as

2 Simulation Analysis

2.1 RSP surface of high-precision measurement system

Equal-range RSP surface can be used to guide the sensor’s accuracy selection based on the design parameters, when designing a high-precision COG measurement system.

For the design parameters of a certain type of high-precision COG measurement system as shown in Tab.2, the RSP surface can be used to guide the sensor precision selection design, design parameter optimization, and so on. From the design parameters, it can be seen that the detection object has a relatively long diameter and needs to detect the COG in three directions, so the radial (y-z) COG error and the axial (x)COG error need to be analyzed separately.

Tab. 2 Design parameters of high-precision measurement system

2.1.1 Radial RSP surface

Letk=0.6,σl=0.02 mm, the radial RSP surface is shown in Fig.3.

Fig. 3 Radial RSP curves (k=0.6, σl =0.02 mm)

According to the radial RSP surface contour projection shown in Fig.3, the following comments can be illustrated:

① The contour line is a straight line atz=0.

② A contour projection line divides thez=0 plane into two parts, and the larger part ofMis the accuracy guarantee area. Sensor accuracy selection can be performed according to different combinations ofMand l of actual detection conditions.

③ Contour lines become sparse as the RSP value decreases, which means that when the accuracy of the sensor is very high, continuing to increase a small amount of accuracy will obtain a large area of accuracy guarantee area. As can be seen from the two contour lines shown by the thick black solid line in Fig.3, when the RSP increases from 0.02 to 0.01, the accuracy guarantee area has been greatly improved.

④ To ensure that the measurement states of all combinations ofMand l meet the COG accuracy index, the absolute accuracy of the required sensor needs to be 0.001 kg.

The RSP surface clusters withk=0.6 andσltaking different values, shown in Fig.4. According to the contour projection line, no matter what the value of RSP is, when the sensor position calibration accuracyσl= 0.02 mm, the improvement of the position calibration accuracy has little effect on the movement of the projection line, which means that it has little effect on the improvement of the accuracy guaranteed area.

2.2 RSPP surface of high-precision measurement system

Since the accuracy of the radial COG is more important for the missile, it is necessary to improve the accuracy of the radial COG detection. According to the radial RSP surface, to ensure the accuracy of the entire measurement domain, the absolute detection accuracy of the sensor needs to be very high, as shown in Fig.3,which requires 0.001 kg.

The combination of non-equal-range sensors can reduce sensor accuracy. The RSPP surface is drawn according to the design parameters in Tab.2 as shown in Fig.7.

Fig. 7 RSPP curves

The lower right side of the RSPP surface is the solution area, and the upper left purple area is the non-solution area. For example, when the mass of the detection part is 100 kg and the accuracy of the long-range sensor isσγ= 0.04 kg,from Fig.7 we can getση= 0.02 kg, which means that the accuracy of the small-range sensor at this time can meet the COG detection accuracy requirement.

2.3 RSP surface of medium precision measurement system

According to the requirements of detection accuracy, the contour projection lines of equalrange RSP curved surfaces can be drawn to realize the division of measurement space. According to the requirements of specific inspection tasks, the selection of system parameters of COG measurement equipment can be guided according to the RSP surface. The design of a COG measurement system of a certain model shown in Tab.3 is taken as an example.

Tab. 3 Design parameters of medium precision measurement system

Letσl=0.1 mm,k=0.5, the RSP surface is shown as Fig.8.

Fig. 8 RSP curves (k=0.5)

According to the division of the measurement domain by the contour projection of the RSP surface shown in Fig.8, it is found that the sensor can basically meet the entire measurement space with an accuracy of 0.1 kg. For specific inspection tasks, a lower-precision sensor can be selected to reduce equipment costs while meeting the requirements of COG measurement. For example, when the measuring point spacing is 600 mm and the detection mass is 200 kg, selecting the sensor accuracy of 0.2 kg can meet the requirements of COG detection accuracy.

3 Experiment

In order to verify that the proposed RSP surface has a correct guiding effect on the accuracy selection of the sensor, according to the design parameters in Tab.3, a four-point support COG measurement device is designed as shown in Fig.9, as well as the actual COG measurement device is shown in Fig.10.

Fig. 9 Centroid measuring equipment model

Fig. 10 Centroid measuring equipment

The test sample is a steel cylinder. According to our previous calibration, its mass is 231.26 kg, and in the sample coordinate system,the center of mass coordinate is [499.86, –0.67,–0.16] mm. The calibration results of the mass and the COG parameters are shown in Tab.4.

Tab. 4 Parameters of the calibrated sample

Seven COG measurement tests were conducted. The distance between the measuring points during measurement is 614.3 mm. The experiment value of the mass is 230.21±0.66 kg, and the COG is 499.48±1.49 mm. Tab.5 shows the statistical mass and the mean and unbiased standard deviations of the COG.

Tab. 5 Experimental data

According to the experimental data, the measurement accuracy of a single sensor is

Because the measured object is close to the homogeneous cylinder,k= 0.5 is taken in the experiment. According to the previous analysis, the measured error of the center of mass is the maximum value of the measured error of the system center of mass.

According to the RSP surface in Fig.8,m=231 kg,l=614 mm, the required accuracy of a single sensor is 0.23 kg, which is consistent with the experiment.

The experiment shows that for the given design parameters, our method can be used to design the measurement system (mainly including the selection of force sensor accuracy, the distance of force sensor measuring point, the selection of force sensor measuring point distance calibration accuracy, etc.), which can meet the measurement accuracy required in the design.

4 Conclusions

In order to quantitatively analyze the influence of various system parameters on the COG measurement accuracy in the supporting reaction method, the equal range RSP selection surface and non-equal-range RSPP adaptive surface are proposed. Through theoretical analysis, simulation analysis and experimental verification, the following main results are obtained.

1) The equal range RSP surface is used to analyze the required sensor accuracy in the highprecision measurement system in which each COG component meets the design detection accuracy requirement.

2) Using equal-range RSP curved surface analysis, it is found that when the calibration accuracy of the sensor measurement point position is 0.02 mm, further improving the position calibration accuracy has little effect on improving the accuracy of the COG detection.

3) Using the equal-range RSP surface, the effects of the COG shift rate, the total mass of the detected object, and the distance between the measuring points of the sensors on the COG measurement accuracy are analyzed quantitatively. The COG accuracy depends to a large extent on the absolute accuracy of the sensor.

4) Using the proposed RSPP adaptive surface, a non-equal-range sensor accuracy selection strategy is obtained, which can reduce the requirements for sensor accuracy and reduce costs.

5) According to the given measurement index, the system parameters are designed by the proposed method, through which the design results meet the design index requirements. When it is applied to the parts whose center of mass deviates greatly from the center of the measuring point of the sensor, or the parts whose center of mass deviates greatly from the center of geometry (complex shape objects), it still has such a high precision.