Measurement principle and precision analysis of chamber parameters on images

Abstract

A new kind of measuring principle of reading data across the diameter of a target image is proposed for high-accuracy measurements of artillery-chamber parameters. A new kind of ruler target mechanism is designed to read the cross-diameter data. The ruler target mechanism is pushed into the chamber by a driver mechanism. It moves along the axis of the chamber, and the sequence images are collected. The radial displacement of the cross-measurement claw can be obtained by processing the change of inscribed lines on the ruler target, and the current diameter, volume and conical degree of the chamber can all be acquired by referring to the demarcating annulus size. Measurement precision is analysed based on error propagation theory, with the diameter and volume parameters as examples. According to this principle, the measurement prototype is designed, and experiments are conducted on it. We show that the measurement error of the diameter is no more than 10 µm, and the volume error of the artillery chamber is less than 2‰, which satisfies the practical engineering requirement.

Introduction

An artillery chamber is a space that accommodates the gunpowder case, obturator and propellant powder and ensures that the powder burns. The chamber volume is a static parameter for artillery, which affects loading density, breech pressure and the value of the initial velocity of the artillery projectile and also affects the range of artillery, the trail of trajectory, the accuracy of the ammunition’s fire point and the safety of the artillery’s launch.^1–3 With an increase in the size of the projectile, the diameter and length of the chamber will increase with the repeated rapid expansion pressure of gunpowder burning gases, and the volume of the chamber will also increase. Therefore, the parameters of the chamber must be measured directly. At present, the methods of chamber parameter measurement include saline injection, sonic testing^4–6 and laser light projection,^7–10 inter alia. However, these methods have the disadvantages of complex operation, low efficiency, low accuracy or one single parameter of measurement, so they cannot meet the requirement of artillery firing range for high-performance measurement.

Structural features of the chamber and measurement system

Structural features of the chamber

As shown in Figure 1, the structure of chamber is a multistage vertebral body that includes several cones and one transition cone. With bullet launch, the transition cone will extend in the barrel direction, which makes the chamber volume increase. Due to its irregular shape, the diameter, length and volume are all static parameters that should be measured.

Figure 1.

Schematic diagram of chamber section.

Measurement system

As can be seen from Figure 2, the system mainly constitutes a host computer, an axial displacement precision driving and controller device, a measuring section (that includes a charge-coupled device (CCD) camera, a three-jaw centring mechanism, a ruler target for image and a scene illumination system) and an auxiliary device of the barrel-rear. The measurement section is pushed by an axial displacement precision driving mechanism and is moved forward along the chamber axis in certain steps, according to the order of host computer, and collects intermittently field optical image information. The host computer processes the acquired optical image information and then obtains the diameter value of the corresponding location and displays the result for user interface.

Figure 2.

System function block diagram of chamber parameter measurement.

The principle structure diagram of the chamber measurement system is shown in Figure 3. After the measurement begins, the measurement section is driven by the axial displacement actuating device and the cross-measurement contacts enter into the calibration ring. Calibration is performed in the calibration ring in order to acquire the measurement reference dimension. Then, the measurement section enters into the chamber from the bottom of the chamber, followed by step-by-step measurement. The axial displacement distance of the grating ruler is $L_i$ . The host computer collects the scale target image sequence with a camera, and diameter data are obtained from image processing. According to the definition of various parameters, the corresponding chamber static parameters are acquired by fusion processing (as formula (6) in section “Calculating the chamber volume”) with the calculation of diameter and axial displacement data.

Figure 3.

Principal structural diagram of chamber parameter measurement system.

Structure of the measurement section for vision reading

The measurement section is the core component for diameter vision reading. The front end and back end of the measurement section correspond to the three-jaw elastic centring mechanism, which can ensure high-precision centring of measurement at different diameters of the chamber. Dual-cross-variable diameter ruler targets are installed on the middle part of the measurement section. As can be seen from Figure 4, two pairs of measurement claws are on one side. There is a mechanical spring inside the measurement claw to ensure a glued contact to the inner wall of the chamber. The measurement claws can glide along guide rail. There is a fixed high-precision ruler at the other end of the measurement claw. The rulers of the two measurement claws are in different planes. During the measurement, contact claws move under the acting force of the pipe inner wall, then the rulers also move, so that the target ruler can transmit slight variations of chamber diameter to relative motions of the ruler image through mechanical conveyance, in order to obtain accurate diameter information.

Figure 4.

Dual-cross-variable diameter ruler target of measurement section.

The images of the measurement ruler target are acquired by a CCD camera. The light-emitting diode (LED) distribution illuminating system with diffuse reflection is used around the measurement target surface, which ensures the quality of the images. One of the sequence images of the measurement scale target is shown in Figure 5, which was acquired from the measurement target on the scene. The actual interval between two inscribed lines is 1 mm.

Figure 5.

Ruler target plane image.

The measurement principle and obtaining static parameters

As described earlier, the host computer can process the sequence images of the ruler target scale acquired by the CCD imaging unit. The exact section diameter information of the current position can be obtained through image processing. ¹¹ The axial displacement information can be measured by the circular grading mount at the axial displacement drive device. The static parameters such as volume, inflection point, taper and length of chamber can all be calculated through processing the axial displacement and diameter. We will describe the calculation of the parameters in this section.

Obtaining chamber diameter and depth

The cross-section diameter data are divided into two groups, and the final value of diameter is the mean of the two directions, so that the error is eliminated. The diameter is composed of the reference diameter and relative diameter. The cross-section diameter $D_i$ is expressed by

$$D_i=D_b+\Delta D_i$$

(1)

where D_i is the average diameter of the cross section i, D_b is the reference diameter and ΔD_i is the relative change between the diameter of the cross section and that of calibration ring, which can be expressed as

$$\Delta D_i=\frac{\Delta {D_h}_i+\Delta {D_v}_i}{2}$$

(2)

where $\Delta {D_h}_i$ and $\Delta {D_v}_i$ are the diameter variations of the cross directions separately. All of the measurement data from image processing are calculated in accordance with equal precision, each for N times for both efficiency and accuracy, we take N = 5, that is

$$\Delta D_{\mathrm{hi}}=\frac{\sum _{j=1}^n\Delta h_j}{n}=\frac{\sum _{j=1}^5\Delta h_j}{5},\Delta D_{\mathrm{vi}}=\frac{\sum _{j=1}^n\Delta v_j}{n}=\frac{\sum _{j=1}^5\Delta v_j}{5}$$

(3)

where $\Delta h_j$ and $\Delta v_j$ are the ruler image displacements in millimetres after excising the maximum and minimum values from the result. The depth of the chamber can be obtained directly by the axial-direction displacement measurement grating that is

$$L=L_m-L_0$$

(4)

where $L_0$ is the start length, that is, a fixed point of the axial-direction displacement measurement grating, and $L_m$ is the end length of the chamber, which can be restricted by the diameter measurement result

$$L_i=L_m\begin{array}{c}\hfill \hfill \end{array}\mathrm{when}\begin{array}{c}\hfill \hfill \end{array}{D}_i=D_{\mathrm{barrel}}+0.05$$

(5)

Calculating the chamber volume

According to the diameter and axial displacement of every measured section, the volume of every frustum can be calculated. The volume of frustum i can be expressed by

$$\begin{array}{c}\multicolumn{1}{c}{V_i=\frac{\pi }{12}(L_{i+1}-L_i)(D_i^2+D_{i+1}^2+D_i{D}_{i+1})}\\ \multicolumn{1}{c}{=\frac{\pi }{12}{H}_i(D_i^2+D_{i+1}^2+D_i{D}_{i+1})}\end{array}$$

(6)

where L_i is the axial-direction displacement of circular section i, D_i and D_{i+ 1} are the upper and lower diameters of the frustum i. The volume of the chamber can be obtained by accumulation of the volume of each frustum

$$V=\sum _{i=1}^m{V}_i$$

(7)

Calculation of inflection point and taper angle

Because of the repeated bomb shooting, the inner wall of the chamber wears down seriously, the inflection and taper can be obtained by practical measurement. As can be seen from Figure 6, a kind of intersection of fitting lines is presented, in which the neighbourhood taper function can be obtained by line fitting and the inflection point is the intersection of neighbour line of ¹² S_i and S _(i+1).

Figure 6.

Calculation of inflection and taper angle.

Precision analysis of chamber parameters

There is error for any measurement system inevitably, so analysing the measurement precision is important to reduce the error.

Precision analysis of diameter

From equation (1) we can get

$${\sigma }_D=\sqrt{{\sigma }_{\mathrm{Db}}^2+{\sigma }_{\Delta D_i}^2}$$

(8)

From the Bessel formula ¹³

$$\begin{array}{c}\multicolumn{1}{c}{{\sigma }_{(\mathrm{Db})}=\sqrt{\frac{\sum _{k=1}^n{\delta }_{{(\mathrm{Db})}_k}^2}{n-1}}=\sqrt{\frac{\sum _{k=1}^5{\delta }_{{(\mathrm{Db})}_k}^2}{4}},}\\ \multicolumn{1}{c}{{\sigma }_{(\Delta \mathrm{Di})}=\sqrt{\frac{\sum _{k=1}^n{\delta }_{{(\Delta \mathrm{Di})}_k}^2}{n-1}}=\sqrt{\frac{\sum _{k=1}^5{\delta }_{{(\Delta \mathrm{Di})}_k}^2}{4}}}\end{array}$$

(9)

For each measurement of the diameter, several parameters will affect the measurement accuracy. So, accuracy analysis is presented as follows:

1. Uncertainty of calibration ring manufacturing, which can reach ${\delta }_{(\mathrm{Cr})}=0.001(\mathrm{mm})$ .

2. Uncertainty of sequence image processing, decided by optical and CCD camera, which is ${\delta }_{(\Delta \mathrm{Di})}=0.005(\mathrm{mm})$ .

In general, as shown in Figure 7, the image plane of CCD is not parallel to the target plane. Without loss of generality, the angle between the target plane and the image plane is θ, the current angle of view is ${\alpha }_i$ , the distance of neighbouring pixels is $\mathrm{Lp}$ , the angle of field view is $2\alpha$ , $f$ is the focal length of the lens and the number of pixels included in the field of view is 2n. The distance from the optic centre to the target plane is $H_0$ (CP). Then, the distance $\mathrm{di}{s}_i$ which the position of the target direction corresponds to the ith row pixels to the point of $o\prime$ is

$$\mathrm{di}{s}_i=\frac{H_0·\sin {\alpha }_i}{\cos \theta ·\cos (\theta -{\alpha }_i)}=\frac{H_0·\mathrm{Lp}·i}{f\cos \theta +\mathrm{Lp}·i·\sin \theta }$$

(10)

Figure 7.

Camera imaging error by target plane non-orthogonal and pixels uniform.

It can be seen that the distance $\mathrm{di}{s}_i$ depends nonlinearly on the pixel number $i$ . The actual distance in the target corresponding to the pixel $i$ is

$$l_i=\mathrm{di}{s}_i-\mathrm{di}{s}_{i-1}$$

(11)

According to the theory of error propagation, set $\mathrm{Lp}·i=x$ , then

$${\sigma }_{\mathrm{di}{s}_x}=\sqrt{{\left(\frac{\partial \mathrm{di}{s}_x}{\partial x}\right)}^2{\delta }_x^2}=\frac{H_0·f·\cos \theta }{{(f\cos \theta +x\sin \theta )}^2}{\delta }_x$$

(12)

In general, θ is a small value, then

$${\sigma }_{\mathrm{di}{s}_x}=\frac{H_0·f·\cos \theta }{{(f\cos \theta +x\sin \theta )}^2}{\delta }_x<H_0{\delta }_x$$

(13)

While ${\delta }_x$ is the manufacture error of CCD and image process, it is no more than 0.1 µm, $H_0<50\mathrm{mm}$ , then we can obtain ${\sigma }_{\mathrm{di}{s}_x}<0.005\mathrm{mm}$ .

• 3. Incline of the target mechanism.

As can be seen from Figure 8, there is a small angle θ between the axis of the chamber and the measurement target because of the mechanical assembly. Assuming the taper angle is α, then we can obtain

$$\mathrm{OA}\prime =\frac{\mathrm{OA}\cos \alpha }{\cos (\alpha -\theta )},\mathrm{OB}\prime =\frac{\mathrm{OB}\cos \alpha }{\cos (\alpha +\theta )}$$

(14)

Figure 8.

Inclination of the measurement touch.

Then

$$\begin{array}{c}\multicolumn{1}{c}{{\delta }_c=A\prime B\prime -\mathrm{AB}=\mathrm{AB}(\cos \theta -1)}\\ \multicolumn{1}{c}{+\sin \theta [\mathrm{OB}\tan (\alpha +\theta )-\mathrm{OA}\tan (\alpha -\theta )]>0}\end{array}$$

(15)

Without a loss of generality, θ < 0.6°, α < 10° and AB < 200 mm; then ${\delta }_c<0.012$ .

• 4. Un-centring mechanism

$$\begin{array}{c}\multicolumn{1}{c}{{\delta }_e=\frac{L_1+L_2}{2}-\varphi =\left(\frac{\sqrt{{\varphi }^2-4{d_1}^2}+\sqrt{{\varphi }^2-4d_2^2}}{2}\right)}\\ \multicolumn{1}{c}{-\varphi ⩽\sqrt{{\varphi }^2-2d^2}-\varphi ⩽0}\end{array}$$

(16)

In general, 70 mm ≤ ϕ ≤ 200 mm, d ≤ 0.05, so |δ_e | ≤ 0.005 mm. Equation (16) also demonstrates that the cross-diameter measurement method can compensate the error of eccentricity of centre (Figure 9). We can assume the correlation coefficient is ρ = −0.5, so we can combine the error of δ_c and δ_e as

$${\delta }_{\mathrm{ce}}=\sqrt{{\delta }_c^2+{\delta }_e^2+2\rho {\delta }_c{\delta }_e}=0.0092\mathrm{mm}$$

Figure 9.

Eccentricity of centre for assistant centring.

From earlier analysis, we can obtain the precision analysis of the diameter as

$$\begin{array}{c}\multicolumn{1}{c}{{\delta }_d=\sqrt{{\delta }_{\mathrm{ce}}^2+{\delta }_{\mathrm{cr}}^2+{\delta }_{\Delta \mathrm{Di}}^2}}\\ \multicolumn{1}{c}{=\sqrt{{0.001}^2+{0.005}^2+{0.0092}^2}=0.105\mathrm{mm}}\end{array}$$

Precision analysis of volume

From equation (6) we know that

$${\sigma }_{V_i}=\sqrt{{\left(\frac{\partial V_i}{\partial H_i}\right)}^2{S}_{H_i}^2+{\left(\frac{\partial V_i}{\partial D_i}\right)}^2{S}_{{}_{D_i}}^2+{\left(\frac{\partial V_i}{\partial D_{i+1}}\right)}^2{S}_{{}_{D_{i+1}}}^2}$$

(17)

Taking a sub-set of data from the actual measurement, that is, $D_1=140\mathrm{mm}$ , $D_2=136\mathrm{mm}$ , $H_1=2\mathrm{mm}$ , because the depth of the chamber can be measured by the circular grating ruler that is $S_{\mathrm{Hi}}⩽0.003\mathrm{mm}$ , then

$$\frac{\partial V_i}{\partial H_i}{S}_H{}_i=44.8743,\frac{\partial V_i}{\partial D_i}{S}_{Di}=22.8592,\frac{\partial V_i}{\partial D_{i-1}}{S}_{D_{i-1}}=22.6509$$

Then, ${\sigma }_{V_i}=55.2205$ , and we can compute the relative error with ${\sigma }_{V_i}/V_i=1.847‰$ ‰, which is no more than the strict requirement of 2‰. ¹³

Experimental results

The measurement model machine is shown in Figure 10. The measurement claws are tested by the precise displacement device. In measurement, every measurement claw is calibrated by taking the high-precision grating scale as a calibration reference. As the test scheme shows in Figure 11, moving the measurement claw and grating by 30 mm, the displacement for the first and third rulers is drawn in Table 1, which shows that the measurement error is less than 10 µm for every measurement claw. This fact satisfies the requirements of the system.

Figure 10.

Measurement system for chamber parameters.

Figure 11.

Claw calibration experimental scheme.

Table 1.

Measurements of error of diameter direction cross claws for rulers 1 and 3 used in the model machine.

Measurement pot and number	Diameter direction error of ruler 1			Diameter direction error of ruler 3
	Standardization grating data (mm)	Our system data (mm)	Error (µm)	Standardization grating data (mm)	Our system data (mm)	Error (µm)
1	0.301	0.300	1	0.301	0.299	2
2	0.606	0.605	1	0.604	0.603	1
3	1.201	1.201	0	1.201	1.205	−4
4	1.800	1.798	2	1.802	1.804	−2
5	2.402	2.400	2	2.402	2.403	−1
6	3.001	2.999	2	3.005	3.006	−1
7	3.600	3.597	3	3.604	3.608	−4
8	4.201	4.200	1	4.203	4.205	−2
9	5.700	5.698	2	5.714	5.716	−2
10	7.200	7.198	2	7.205	7.207	−2
11	8.101	8.100	1	8.103	8.105	−2
12	9.000	9.002	−1	9.004	8.998	−6

The upper computer software is realized by VC++6.0. The radial displacement of the diameter is drawn on the home screen as shown in Figure 12.

Figure 12.

Home interface of the chamber parameter measurement system.

Discussion

From the precision analysis and experiments, we can find that the shorter the step, the higher the volume measurement precision. But short step means lower measuring efficiency. Therefore, the research of the self-adaption step measurement algorithm which is characterized by short step at inflection point and long step at smooth place can meet the necessary requirements for high precision and high efficiency. We can focus on the self-step measurement and high-resolution image measurement algorithm in the future research.

Conclusion

A kind of chamber measurement system based on machine vision is proposed through a research chamber measurement technique, and the system measurement precision is analysed. Experimental results on the prototype are presented. The system skilfully combined the image and scale targets, and the measurement values are obtained accurately, which provides a reference value for other similar systems with promising potential applications.