Recognition method for the flight parameters of multiple projectiles with light-screen array sensor

Abstract

In the dispersion measurements of multi-barrel rapid-firing guns (MBRFGs), the conventional measurement methods fail because multiple projectiles pass through the same detection plane. In this article, a method for measuring the flight parameters of multiple projectiles and recognizing their trajectory lines using a seven-light-screen array sensor (seven-LSAS) is proposed. The proposed seven-LSAS integrates two three-light-screen modules and one non-orthogonal single-light-screen to enhance multi-projectile detection capability. Based on ballistic principles, a mathematical model for the flight parameters (such as impact coordinate, flying velocity, azimuth and pitch angle) of multiple projectiles is established. The trajectory lines are calculated by the moment data of the projectile passing through the seven-LSAS. A full-moment vector space is constructed to calculate the potential trajectory lines corresponding to each moment vector. Using the spatial collinearity constraint inherent to trajectory lines of the same projectile, an association recognition algorithm for the trajectory line based on the collinearity judgment is proposed to achieve accurate recognition of the trajectory lines of multiple projectiles. The experimental results demonstrate that compared to traditional testing methods, the proposed recognition method can accurately recognize and calculate the flight parameters of multiple projectiles. Furthermore, the measurement error of the impact coordinates for recognized projectiles does not exceed 5 mm, with 50% improvement over existing recognition methods, validating the feasibility and effectiveness of the proposed recognition method.

Introduction

The parameters of flying projectiles^1,2,3,4,5,6, such as the impact coordinates, flying velocity, azimuth and pitch, core foundation for evaluating weapon system performance, combat effectiveness, and design optimization, directly determining the hit accuracy, damage effectiveness, and mission success rate. Currently, the measurement methods mainly include image measurements (such as the dual linear-array⁷, area array camera intersection measurements⁸, high-speed photography⁹, acoustic measurements^10,11 and light-screen array measurements (such as the 6-light-screen sensors¹², 4-light-screen sensors¹³. However, these methods are generally limited to measure single projectile. When applied to the multi-barrel rapid-firing guns (MBRFGs)^14,15,16,17characterized by high rates of fire and small caliber, they exhibit significant limitations. In such systems, multiple projectiles often pass through the detection plane simultaneously, leading to the failure of traditional testing methods. For example, in acoustic measurements, shock wave interference between projectiles obscures individual shockwave arrival times, preventing accurate impact coordinate calculation. In image and light-screen array measurements, overlapping signals severely prevent the effective distinguishing of individual projectile information. Consequently, research on the method for recognizing and measuring the multi-projectile parameters is critical for the continuous development, design optimization, and performance evaluation of the MBRFGs.

Compared with other non-contact optical measurement methods, the light-screen array sensor (LSAS) method demonstrates better adaptability to projectiles of varying velocities, featuring a rapid response, a large effective detection area, high sensitivity, and strong anti-interference capabilities¹⁸. The LSAS calculates the flight parameters—including impact coordinates, velocity, azimuth, and pitch angles—using the moment of the projectile passing through each light screen (LS) and the spatial structural relationship of the LSs¹⁹. The LSAS remains effective for both single-shot mode and burst mode with large firing intervals²⁰. However, the method fails under high-frequency burst mode or simultaneous firing conditions. When multiple projectiles pass through the LS, the behind projectile may pass through the LS earlier than the front projectile because of the spatial positional relationship of the LSs, leading to confusion in the moment sequence of multiple projectiles. Consequently, the LSA method cannot reliably extract correct projectile information, resulting in erroneous calculations of the flight parameters.

To effectively measure the flight parameters of multiple projectiles, Ni et al. investigated the moment vector recognition algorithm for calculating the impact location dispersion of a double-barrel cannon using the principle that the firing position is on the trajectory line²¹. However, the measurement error increases with the increase of firing distance in this method. To eliminate measurement errors caused by the firing distance, some researchers^22,23 have introduced the correlation between parallel or symmetrical LSs to recognize associated signals or trajectory lines from the same projectile. Nevertheless, these methods are only applicable to parallel or symmetrical LSs. To enhance universality, some researchers²⁴²⁵ 错误!未找到引用源 have integrated linear array cameras into the LSAS or employed multiple linear array cameras to measure multiple projectiles. While this improvement overcomes limitations of specific structures, it certainly has increased operational costs. Additionally, there have also been investigations of recognition methods using the sky screens. Song et al. used two sky screens to form an 8-light-screen device^25,26. Any six LSs as a group form a 6-light-screen array, for a total of sixteen 6-light-screen arrays. Each moment vector of multiple projectiles needs to be calculated in sixteen arrays. If a set of identical calculation results exists in each 6-light-screen array, the corresponding trajectory line is the trajectory line of the projectile. However, this method is relatively complex and very computationally intensive.

To overcome the limitations of the traditional LSAS in measuring the flight parameters of multiple projectiles, a new method of recognizing and measuring the flight parameters of multiple projectiles is proposed using the seven-light-screen array sensor (seven-LSAS). The proposed seven-LSAS is constructed by two three-light-screens and one non-orthogonal single-light-screen, and the mathematical model for calculating the flight parameters of the projectile is established based on the moment values of the projectile passing through each LS and the plane equation of each LS. Then, a full-moment vector space is obtained using the moment values of the projectile passing through the seven-LSAS, and the trajectory lines corresponding to each moment vector are calculated. Finally, the trajectory lines of the same projectile are recognized using the principle that the trajectory lines of each projectile passing through the seven-LSAS are collinear in space, thus achieving the purpose of recognizing multiple projectiles.

The structure of this article is organized to systematically address the core research regarding the measurement and recognition of multiple projectiles flight parameters using the seven-LSAS. To this end, Sect. 2 “Methods” details the measurement principles of the seven-LSAS and the associated recognition algorithm for the trajectory lines of the multiple projectiles, thereby establishing the theoretical foundation of this article. Section 3 “Experiments results and analysis” evaluates the effectiveness of the proposed algorithm through the experimental data and analyses the measurement errors in the impact coordinates after recognition. Finally, Sect. 4 “Conclusions” concludes the principal findings and proposes future work directions to advance the technology.

Methods

To recognize the correct moment sequences of multiple projectiles passing through the light-screen array, this article simplifies the two six-light-screen array sensors (six-LSAS) by using one six-LSAS and one non-orthogonal LS to form a seven-LSAS. A new six-LSAS is constructed by replacing a specified LS in the original six-LSAS with the non-orthogonal LS. In this article, two six-LSASs of them are selected as examples, the consistency of the flight parameters of the same projectile in the two six-LSASs is utilized to study the method for recognizing the correct moment sequences of multiple projectiles.

Measurement principles

Structural components

The structure of the seven-LSAS is shown in Fig. 1 (a). It mainly consists of two identical three-light-screen sensor and a single-light-screen sensor, forming a total of seven LSs, denoted as ${G_1}$, ${G_2}$, ${G_3}$, ${G_4}$, ${G_5}$, ${G_6}$, ${G_7}$, respectively. The O and $O^{\prime}$ are the intersection points of the main optical axes of the lens in the light screen sensor, $\overline {{OO^{\prime}}} =S$. Figure 1 (b) and Fig. 1 (c) represent the projections of the seven-LSAS onto the $xoy$ plane and $xoz$ plane, respectively. $\alpha$ is the angle between ${G_1}$ and the y-axis, and between ${G_3}$ and the y-axis. $\beta$ is the angle between ${G_2}$ and the z-axis. ${G_7}$ is formed by the single light-screen, which is neither parallel nor orthogonal to other LSs. Supposing that ${\mathbf{n}}=({n_x},{n_y},{n_z})$ is the normal vector to the plane of the ${G_7}$, then ${n_x} \ne 0$, ${n_y} \ne 0$ and ${n_z} \ne 0$. ${\alpha _0}$ is the angle between ${G_7}$ and the y-axis, ${\beta _0}$ is the angle between ${G_7}$ and the normal line of the z-axis. ${O_S}$ is the principal point of the lens, $\overline {{O{O_S}}} ={S_0}$. Additionally, in Fig. 1 (b) and Fig. 1 (c), the dashed lines represent the projections of the corresponding LS onto the coordinate plane. Taking ${G_2}$ as an example, its projection onto the $xoy$ plane is actually a plane. To clearly demonstrate the spatial geometric relationships among the LSs, the spatial position of ${G_2}$ is indicated by a dashed line. The other dashed lines serve the same purpose.

Fig. 1

Schematic diagram of the seven-light-screen array sensor: (a) the structure of the seven-LSAS, (b) projection onto the $xoy$ plane, (c) projection onto the $xoz$ plane.

When one projectile passes through the seven-LSAS, a projection of the projectile is created on the photosensitive plane of the photodetector, and a signal of the projectile is output by the signal processing circuit, as shown in Fig. 2. The light-screen sensor uses a highly sensitive photodetector to capture the signal of the projectile. Through a slit diaphragm and optical lens, it forms a fan-shaped detection LS in space. Its detection field of view is determined by the optical lens, slit diaphragm and photodetector. In the obtained the signal waveform, the horizontal axis represents time t in ms, and the vertical axis represents the signal amplitude U in V. It is captured by the multi-channel data acquisition instruments and the moment values ${t_i}$, $i=1,2, \cdots ,7$ when the projectile passes through the LS are calculated using the extraction algorithm. The ${t_i}$ represents the moment value that the base of the projectile passes through the center plane of the LS, which is regarded as an ideal plane in geometry²⁷.

Mathematical model

The distance S of the seven-LSAS is relatively short, typically approximately 10 m. With respect to high-velocity projectiles, the influence of air resistance and gravity on the flying velocity of the projectile can be ignored in this range. Thus, the flying velocity of the projectile is constant in the detection area²⁸.

According to Ref²⁹., only six LSs are needed to calculate the flight parameters of a single projectile. Therefore, any six of the seven LSs in this article can be selected for calculation. Here, ${G_1}$, ${G_2}$, ${G_3}$, ${G_4}$, ${G_5}$, and ${G_6}$ are used as examples. In the seven-LSAS, a moment sequence of the LS represents a natural ordering of the moment values when a projectile passes through one LS in sequence. A moment vector of a projectile represents a set of moment values when the projectile passes through different LSs of the seven-LSAS. The parameters of the flying projectile are calculated based on the moment vector of the projectile and the plane equations of the LSs.

Fig. 2

Schematic diagram of the light screen.

The velocity vector of the flying projectile is $\vec {v}=({v_x},{v_y},{v_z})$, and the coordinate position on the ${G_1}$ is $({x_1},{y_1},{z_1})$. The moment vector ${\mathbf{T}}$ of the projectile is composed of the moment when the projectile passes through the six LSs, expressed as ${\mathbf{T}}=\left[ {{t_1},{t_2},{t_3},{t_4},{t_5},{t_6}} \right]$.

On the basis of the definition of the linear equation of space³⁰, the trajectory line L of the projectile can be expressed as

$$L:\frac{{x - {x_1}}}{{{v_x}}}=\frac{{y - {y_1}}}{{{v_y}}}=\frac{{z - {z_1}}}{{{v_z}}}$$

(1)

We define vector ${\mathbf{L}}={\left[ {\begin{array}{*{20}{c}} {{x_1}}{{y_1}}{{z_1}}{{v_x}}{{v_y}}{{v_z}} \end{array} } \right]^{\text{T}}}$ as the parameter of the trajectory line, which can uniquely determine a line L.

Figure 3 demonstrates the relationship between the flying direction and angle of the projectile. $\theta$ and $\gamma$ are the azimuth and pitch of the flying velocity of the projectile, which can be expressed as

$$\left\{ {\begin{array}{*{20}{c}} {v=\sqrt {v_{x}^{2}+v_{y}^{2}+v_{z}^{2}} } \\ {\theta =\arctan (\frac{{{v_z}}}{{{v_x}}})} \\ {\gamma =\arctan (\frac{{{v_y}}}{{\sqrt {v_{x}^{2}+v_{z}^{2}} }})} \end{array} } \right.$$

(2)

In engineering, the dispersion of projectiles is calculated as the impact coordinates of projectiles passing through a certain plane. If the plane is $x={x_0}$, the coordinates of a projectile in that plane can be expressed as

$$\left\{ {\begin{array}{*{20}{c}} {x={x_0} - {x_1}} \\ {y={y_1}+{v_y} \cdot t} \\ {z={z_1}+{v_z} \cdot t} \end{array} } \right.,\quad t=\frac{x}{{{v_x}}}$$

(3)

The impact coordinates of the projectile in ${G_i}$, $i=2,3, \cdots ,6$ can be expressed as

$$\left\{ {\begin{array}{*{20}{c}} {{x_i}={x_1}+{v_x} \cdot ({t_i} - {t_1})} \\ {{y_i}={y_1}+{v_y} \cdot ({t_i} - {t_1})} \\ {{z_i}={z_1}+{v_z} \cdot ({t_i} - {t_1})} \end{array}} \right.$$

(4)

The general plane equation³¹ of six LSs can be expressed as

$$\left\{ {\begin{array}{*{20}{l}} {{G_1}:{a_1}x+{b_1}y+{c_1}z={d_1}} \\ {{G_2}:{a_2}x+{b_2}y+{c_2}z={d_2}} \\ {{G_3}:{a_3}x+{b_3}y+{c_3}z={d_3}} \\ {{G_4}:{a_4}x+{b_4}y+{c_4}z={d_{\text{4}}}+S \cdot {a_4}} \\ {{G_5}:{a_5}x+{b_5}y+{c_5}z={d_{\text{5}}}+S \cdot {a_5}} \\ {{G_6}:{a_6}x+{b_6}y+{c_6}z={d_{\text{6}}}+S \cdot {a_6}} \end{array}} \right. \Rightarrow \left[ {\begin{array}{*{20}{l}} {{a_1}}&{{b_1}}&{{c_1}} \\ {{a_2}}&{{b_2}}&{{c_2}} \\ {{a_3}}&{{b_3}}&{{c_3}} \\ {{a_4}}&{{b_4}}&{{c_4}} \\ {{a_5}}&{{b_5}}&{{c_5}} \\ {{a_6}}&{{b_6}}&{{c_6}} \end{array}} \right] \cdot \left[ {\begin{array}{*{20}{c}} x \\ y \\ z \end{array}} \right]=\left[ {\begin{array}{*{20}{c}} {{d_1}} \\ {{d_{\text{2}}}} \\ {{d_{\text{3}}}} \\ {{d_{\text{4}}}+S \cdot {a_4}} \\ {{d_{\text{5}}}+S \cdot {a_{\text{5}}}} \\ {{d_{\text{6}}}+S \cdot {a_{\text{6}}}} \end{array}} \right]$$

(5)

where ${a_i}$, ${b_i}$, ${c_i}$, ${d_i}$ ($i=1,2, \cdots ,6$) are the coefficients of the general plane equations, all of which are determined by the structural parameters of the seven-LSAS.

According to (4) and (5), combined with the structural parameters of the seven-LSAS, the vector ${\mathbf{L}}$ can be expressed as

$${\mathbf{L}}={{\mathbf{M}}^{ - 1}}{\mathbf{P}}$$

(6)

where,

$${\mathbf{M}}=\left[ {\begin{array}{*{20}{c}} {\cos \alpha }&{\sin \alpha }&0&{({t_1} - {t_1}) \cdot \cos \alpha }&{({t_1} - {t_1}) \cdot \sin \alpha }&0 \\ {\cos \beta }&0&{\sin \beta }&{({t_2} - {t_1}) \cdot \cos \beta }&0&{({t_2} - {t_1}) \cdot \sin \beta } \\ { - \cos \alpha }&{\sin \alpha }&0&{ - ({t_3} - {t_1}) \cdot \cos \alpha }&{({t_3} - {t_1}) \cdot \sin \alpha }&0 \\ {\cos \alpha }&{\sin \alpha }&0&{({t_4} - {t_1}) \cdot \cos \alpha }&{({t_4} - {t_1}) \cdot \sin \alpha }&0 \\ {\cos \beta }&0&{\sin \beta }&{({t_5} - {t_1}) \cdot \cos \beta }&0&{({t_5} - {t_1}) \cdot \sin \beta } \\ { - \cos \alpha }&{\sin \alpha }&0&{ - ({t_6} - {t_1}) \cdot \cos \alpha }&{({t_6} - {t_1}) \cdot \sin \alpha }&0 \end{array}} \right]$$

(7)

$${\mathbf{P}}={[\begin{array}{*{20}{l}} 0&0&0&{\cos \alpha \cdot S}&{\cos \beta \cdot S}&{ - \cos \alpha \cdot S} \end{array}]^{\text{T}}}$$

(8)

Due to $r({\mathbf{M}})$=$r({\mathbf{M}},{\mathbf{P}})$=6, the (6) must have a unique solution. As a result, given a moment vector ${\mathbf{T}}$ of the projectile, the trajectory line L of the projectile must be uniquely determined, so that the parameters ${x_1}$, ${y_1}$, ${z_1}$, v, $\theta$ and $\gamma$ are obtained.

Fig. 3

The vector direction of the flying velocity.

Multi-projectile test analysis

In the MBRFGs, it is extremely possible for 2 or 3 projectiles to pass simultaneously through the LS³². We denote ${T_{Gi}}=({t_{i1}},{t_{i2}}, \cdots ,{t_{i{N_i}}})$, $i=1,2, \cdots ,6$ as the moment sequence of the i-th LS, where ${N_i}$ is the total number of the moment values in the i-th LS, ${N_i} \leqslant N$, and N is the total number of the projectile, N= 2 or 3. It is particularly emphasized that the ${t_{i1}}$ represents the first moment value in the i-th LS, but cannot be the moment value of the first projectile passing through the i-th LS.

Defined ${T_0}$ as the threshold of the firing interval between projectiles ${{\text{d}}_1}$ and ${{\text{d}}_2}$, the length of the projectile as l. In the single-shot mode, there is only one moment value in moment sequence of every LS, and only one moment vector can be obtained. In the burst mode with large firing intervals, the flying time between two projectiles is greater than the threshold. Thus, the elements in the moment vector of the projectile correspond to the elements in the moment sequence of the LS. The burst mode with the small firing intervals is analyzed in this article.

Defined ${T_1}$ as the maximum threshold of the firing interval in the ${G_2}$, ${T_2}$ as the maximum threshold of the firing interval in the ${G_3}$. The threshold conditions for two projectiles passing through the ${G_2}$ and ${G_3}$ are shown in Fig. 4. Two projectiles passing through the ${G_1}$ are similar to ${G_2}$. ${G_2}$ and ${G_3}$ are projected in the $xoy$ plane and $xoz$ plane, respectively. The threshold ${T_0}$ can be expressed as

$${T_0}=\hbox{max} ({T_1},{T_2})$$

(9)

where

$$\left\{ \begin{gathered} {T_1}=\frac{{l\cos {\theta _{\hbox{min} }}+\Delta z \cdot \tan \beta }}{{{v_{\hbox{min} }}}},\;xoz\;plane \hfill \\ {T_2}=\frac{{l\cos {\gamma _{\hbox{min} }}+\Delta y \cdot \tan \alpha }}{{{v_{\hbox{min} }}}},\;xoy\;plane \hfill \\ \end{gathered} \right.$$

(10)

where $\Delta y$ is the high difference between two projectiles on the y-axis, $\Delta z$ is the high difference between two projectiles on the z-axis, ${\theta _{\hbox{min} }}$ is the minimum azimuth, ${\gamma _{\hbox{min} }}$ is the minimum pitch, ${v_{\hbox{min} }}$ is the minimum velocity of the flying projectiles.

Fig. 4

The threshold conditions for two projectiles passing simultaneously through the ${G_2}$ and ${G_3}$.

When the firing intervals is greater than the threshold ${T_0}$, the two projectiles pass through the LS successively, then ${N_1}$=${N_2}$=${N_3}$=${N_4}$=${N_5}$=${N_6}$=2. Moment vectors of the projectile are obtained by the natural ordering of the moment sequence of the LS. In contrast, ${N_1}$, ${N_2}$, ${N_3}$, ${N_4}$, ${N_5}$ and ${N_6}$ are not completely equal. Therefore, the numbers of the same projectile in the moment sequences of every LS could differ.

For example, we assumed that projectiles ${{\text{d}}_1}$ and ${{\text{d}}_2}$ with velocities of ${\vec {v}_1}$ and ${\vec {v}_2}$ pass through the ${G_1}$, ${G_2}$ and ${G_3}$ parallel to the ballistic line, as shown in Fig. 5. Projectiles ${{\text{d}}_1}$ and ${{\text{d}}_2}$ pass through the ${G_1}$ at the same time, which means that ${t_{11}}={t_{12}}={t_1}$. According to the position of the two projectiles and the geometric relationship of the structure of the ${G_1}$, ${G_2}$ and ${G_3}$, projectile ${{\text{d}}_1}$ passed through the ${G_2}$ before projectile ${{\text{d}}_2}$ and after projectile ${{\text{d}}_2}$ passed through the ${G_3}$. Projectiles ${{\text{d}}_1}$ and ${{\text{d}}_2}$ passed through another three-light-screen detector in the same way, thus two projectiles passing through a three-light-screen detector were analyzed.

The moment sequences of every LS are denoted as ${T_{G1}}=({t_1})$, ${T_{G2}}=({t_{21}},{t_{22}})$, ${T_{G3}}=({t_{31}},{t_{32}})$, respectively. The moment vectors arranged by a natural moment sequences of each LS are denoted as ${{\mathbf{T}}_{\mathbf{1}}}=({t_1},{t_{21}},{t_{31}})$ and ${{\mathbf{T}}_{\mathbf{2}}}=({t_1},{t_{22}},{t_{32}})$. As shown in Fig. 5, the moment vectors of the two projectiles are ${{\mathbf{T}}_{{\mathbf{d1}}}}=({t_1},{t_{21}},{t_{32}})$ and ${{\mathbf{T}}_{{\mathbf{d2}}}}=({t_1},{t_{22}},{t_{31}})$, and obviously, ${{\mathbf{T}}_{\mathbf{1}}} \ne {{\mathbf{T}}_{{\mathbf{d1}}}}$ and ${{\mathbf{T}}_{\mathbf{2}}} \ne {{\mathbf{T}}_{{\mathbf{d2}}}}$.

In fact, the two projectiles passed through the ${G_1}$ simultaneously, but did not necessarily pass through the ${G_4}$ at the same time. Therefore,${T_{G4}}=({t_{41}},{t_{42}})$. Taking any one element from every moment sequence of six LSs, there are eight moment vectors using the permutations and combinations. Every moment vector corresponds to a line, which can be expressed as.

Fig. 5

Schematic of the signals of the two projectiles passing through the three-light-screen sensor.

$$\left\{ {\begin{array}{*{20}{c}} {{{\mathbf{T}}_{\mathbf{1}}}=({t_1},{t_{21}},{t_{31}},{t_{41}},{t_{51}},{t_{61}}) \to {L_1}} \\ {{{\mathbf{T}}_{\mathbf{2}}}=({t_1},{t_{21}},{t_{32}},{t_{41}},{t_{51}},{t_{62}}) \to {L_2}} \\ {{{\mathbf{T}}_{\mathbf{3}}}=({t_1},{t_{22}},{t_{31}},{t_{41}},{t_{52}},{t_{61}}) \to {L_3}} \\ {{{\mathbf{T}}_{\mathbf{4}}}=({t_1},{t_{22}},{t_{32}},{t_{41}},{t_{52}},{t_{62}}) \to {L_4}} \\ {{{\mathbf{T}}_{\mathbf{5}}}=({t_1},{t_{21}},{t_{31}},{t_{42}},{t_{51}},{t_{61}}) \to {L_5}} \\ {{{\mathbf{T}}_{\mathbf{6}}}=({t_1},{t_{21}},{t_{32}},{t_{42}},{t_{51}},{t_{62}}) \to {L_6}} \\ {{{\mathbf{T}}_{\mathbf{7}}}=({t_1},{t_{22}},{t_{31}},{t_{42}},{t_{52}},{t_{61}}) \to {L_7}} \\ {{{\mathbf{T}}_{\mathbf{8}}}=({t_1},{t_{22}},{t_{32}},{t_{42}},{t_{52}},{t_{62}}) \to {L_8}} \end{array} } \right.$$

(11)

Among these eight lines, we do not know which one is the trajectory line of the projectile. We need to investigate a method to determine ${L_{{\text{d}}1}}$ and ${L_{{\text{d2}}}}$, and then determine ${{\mathbf{T}}_{{\mathbf{d1}}}}$ and ${{\mathbf{T}}_{{\mathbf{d2}}}}$. For the recognized moment vectors, the flight parameters of every projectile can be calculated using the (6).

Principles of recognition algorithms

In accordance with the measurement principles of the flight parameters, the flight parameters of the flying projectile can be calculated using any six LSs of the seven-LSAS. In this article, we use array A={${G_1}$, ${G_2}$, ${G_3}$, ${G_4}$, ${G_5}$, ${G_6}$} and array B={${G_2}$, ${G_3}$, ${G_7}$, ${G_4}$, ${G_5}$, ${G_6}$} as examples. Note that other combinations are also feasible. The fact that the trajectory line of the flying projectile passing through the seven-LSAS is unique. In other words, the trajectory lines calculated by the two arrays are collinear. Based on this principle, the association recognition algorithm for the trajectory lines of multiple projectiles is proposed in this article. A detailed description follows.

Establishing a system of equations

In the proposed seven-LSAS, the plane equation of the ${G_7}$ can be expressed as

$$\begin{gathered} \cos ({\beta _0})\cos ({\alpha _0}) \cdot x+\cos ({\beta _0})\sin ({\alpha _0}) \cdot y \hfill \\ \quad \quad \quad \quad \quad \quad \quad +\sin ({\beta _0}) \cdot z=\cos ({\beta _0})\cos ({\alpha _0}) \cdot {S_0} \hfill \\ \end{gathered}$$

(12)

Based on the measurement principles in Sect. 2.1.2, the formula for calculating the trajectory line of the A and B groups array is expressed as

$$\left\{ {\begin{array}{*{20}{c}} {{{\mathbf{L}}_{\mathbf{A}}}{\mathbf{=M}}_{{\mathbf{A}}}^{{-1}}{{\mathbf{P}}_{\mathbf{A}}}} \\ {{{\mathbf{L}}_{\mathbf{B}}}{\mathbf{=M}}_{{\mathbf{B}}}^{{-1}}{{\mathbf{P}}_{\mathbf{B}}}} \end{array}} \right.$$

(13)

where, ${{\mathbf{L}}_{\mathbf{A}}}$ denotes the trajectory line calculated by the array A, ${{\mathbf{M}}_{\mathbf{A}}}$ is the coefficient matrix composed of the moment vector of the projectile passing through the array A and the structural parameters of the array A. ${{\mathbf{P}}_{\mathbf{A}}}$ is a constant matrix composed of the structural parameters of the array A. ${{\mathbf{L}}_{\mathbf{B}}}$ denotes the trajectory line calculated by array B, ${{\mathbf{M}}_{\mathbf{B}}}$ is the coefficient matrix composed of the moment vector of the projectile passing through the array B and the structural parameters of the array B. ${{\mathbf{P}}_{\mathbf{B}}}$ is a constant matrix composed of the structural parameters of the array B.

Constructing the moment vectors and trajectory lines

The moment sequences of two projectiles passing through the seven LSs are expressed as

$${T_{Gi}}=\left\{ {{t_{i1}},{t_{i2}}} \right\},i=1,2, \cdots ,7$$

(14)

If two projectiles pass through a LS simultaneously, ${t_{i1}}={t_{i2}}$.

Because the structures of the two three-light screens are completely identical, there are m moment vectors based on the permutations and combinations, where $m=C_{2}^{1}C_{2}^{1}C_{2}^{1}=8$. The moment vector space ${{\mathbf{T}}_{\mathbf{A}}}$ on array A can be expressed as

$${{\mathbf{T}}_{\mathbf{A}}}=\{ {{\mathbf{T}}_{{\mathbf{Aj}}}}\} ,j=1,2, \cdots ,8$$

(15)

where${{\mathbf{T}}_{{\mathbf{Aj}}}}=\left\{ {({t_{1i^{\prime}}},{t_{2i^{\prime}}},{t_{3i^{\prime}}},{t_{4i^{\prime}}},{t_{5i^{\prime}}},{t_{6i^{\prime}}})|{t_{ii^{\prime}}} \in {T_{Gi}}} \right\}$, $i^{\prime}=1,2$. Each ${{\mathbf{T}}_{{\mathbf{Aj}}}}$ can calculate a trajectory line ${L_{{\text{A}}j}}$.

Similarly, the moment vector space ${{\mathbf{T}}_{\mathbf{B}}}$ and the corresponding trajectory line ${L_{Bk}}$ in array B can be expressed as

$$\begin{gathered} {{\mathbf{T}}_{\mathbf{B}}}=\left\{ {{{\mathbf{T}}_{{\mathbf{Bk}}}}} \right\},k=1,2, \cdots ,8 \hfill \\ {{\mathbf{T}}_{{\mathbf{Bk}}}}=\left\{ {({t_{1i^{\prime}}},{t_{2i^{\prime}}},{t_{3i^{\prime}}},{t_{4i^{\prime}}},{t_{5i^{\prime}}},{t_{6i^{\prime}}})|{t_{ii^{\prime}}} \in {T_{Gi}}} \right\} \to {L_{Bk}} \hfill \\ \end{gathered}$$

(16)

Collinearity judgments

Every ${{\mathbf{T}}_{{\mathbf{Aj}}}}$ or ${{\mathbf{T}}_{{\mathbf{Bk}}}}$ corresponds to the lines ${L_{{\text{A}}j}}$ or ${L_{Bk}}$, respectively. The parameter vectors are denoted as ${{\mathbf{L}}_{{\mathbf{Aj}}}}$=[${x_{{\text{A}}j}}$, ${y_{{\text{A}}j}}$, ${z_{{\text{A}}j}}$, ${v_{x{\text{A}}j}}$, ${v_{y{\text{A}}j}}$,${v_{z{\text{A}}j}}$]^T, ${{\mathbf{L}}_{{\mathbf{Bk}}}}$=[${x_{{\text{B}}k}}$, ${y_{{\text{B}}k}}$, ${z_{{\text{B}}k}}$, ${v_{x{\text{B}}k}}$, ${v_{y{\text{B}}k}}$, ${v_{z{\text{B}}k}}$]^T, $j,k=1,2, \cdots ,8$. The condition³³ under which the lines ${L_{{\text{A}}j}}$ or ${L_{Bk}}$ are collinear can be expressed as

$$\begin{gathered} {v_{x{\text{A}}j}}:{v_{y{\text{A}}j}}:{v_{z{\text{A}}j}}={v_{x{\text{B}}k}}:{v_{y{\text{B}}k}}:{v_{z{\text{B}}k}} \hfill \\ \quad \quad \quad \quad \quad \;\;=({x_{{\text{B}}k}} - {x_{{\text{A}}j}}):({y_{{\text{B}}k}} - {y_{{\text{A}}j}}):({z_{{\text{B}}k}} - {z_{{\text{A}}j}}) \hfill \\ \end{gathered}$$

(17)

The collinear coefficients are described as

$$\delta {v_{xjk}}=\frac{{{v_{x{\text{A}}j}}}}{{{v_{x{\text{B}}k}}}},\delta {v_{yjk}}=\frac{{{v_{y{\text{A}}j}}}}{{{v_{y{\text{B}}k}}}},\delta {v_{zjk}}=\frac{{{v_{z{\text{A}}j}}}}{{{v_{z{\text{B}}k}}}}$$

(18)

$$\delta {x_{{\text{A}}jk}}=\frac{{{x_{{\text{B}}k}} - {x_{{\text{A}}j}}}}{{{v_{x{\text{A}}j}}}},\delta {y_{{\text{A}}jk}}=\frac{{{y_{{\text{B}}k}} - {y_{{\text{A}}j}}}}{{{v_{y{\text{A}}j}}}},\delta {z_{{\text{A}}jk}}=\frac{{{z_{{\text{B}}k}} - {z_{{\text{A}}j}}}}{{{v_{z{\text{A}}j}}}}$$

(19)

$$\delta {x_{{\text{B}}jk}}=\frac{{{x_{{\text{B}}k}} - {x_{{\text{A}}j}}}}{{{v_{x{\text{B}}k}}}},\delta {y_{{\text{B}}jk}}=\frac{{{y_{{\text{B}}k}} - {y_{{\text{A}}j}}}}{{{v_{y{\text{B}}k}}}},\delta {z_{{\text{B}}jk}}=\frac{{{z_{{\text{B}}k}} - {z_{{\text{A}}j}}}}{{{v_{z{\text{B}}k}}}}$$

(20)

If

$$\delta {v_{xjk}}=\delta {v_{yjk}}=\delta {v_{zjk}}$$

(21)

Then, the lines ${L_{{\text{A}}j}}$ or ${L_{Bk}}$ are parallel to each other.

If

$$\left. {\begin{array}{*{20}{c}} {\delta {x_{{\text{A}}j}}=\delta {y_{{\text{A}}j}}=\delta {z_{{\text{A}}j}}} \\ {\delta {x_{{\text{B}}k}}=\delta {y_{{\text{B}}k}}=\delta {z_{{\text{B}}k}}} \end{array}} \right\}={\text{constant}}$$

(22)

Then, the lines ${L_{{\text{A}}j}}$ or ${L_{Bk}}$ are collinear.

As a result, if the ${{\mathbf{T}}_{{\mathbf{Aj}}}}$ and ${{\mathbf{T}}_{{\mathbf{Bk}}}}$ satisfy the (21) or (22), then the corresponding ${L_{{\text{A}}j}}$ or ${L_{Bk}}$ are parallel to each other, and are the trajectory lines of the projectiles.

Under ideal conditions, the spatial lines ${L_{{\text{A}}j}}$ or ${L_{Bk}}$ are perfectly collinear for a single projectile trajectory, satisfying the condition expressed in (21) or (22). However, owing to inherent timing and ranging measurement errors in the seven-LSAS, the flight parameters calculated by (6) exhibit errors. Consequently, lines ${L_{{\text{A}}j}}$ or ${L_{Bk}}$ deviate from strict collinear. To quantify this deviation, we define the parallelism index between two spatial lines to describe the level of collinearity, expressed as

$$\left\{ {\begin{array}{*{20}{c}} {\theta =\arccos (\frac{{\left| {{{\vec {v}}_A} \cdot {{\vec {v}}_B}} \right|}}{{\left\| {{{\vec {v}}_A}} \right\|\left\| {{{\vec {v}}_B}} \right\|}})} \\ {D=\frac{{\left| {({{\text{P}}_B} - {{\text{P}}_A}) \cdot ({{\vec {v}}_A} \times {{\vec {v}}_B})} \right|}}{{\left\| {{{\vec {v}}_A} \times {{\vec {v}}_B}} \right\|}}} \end{array} } \right.$$

(23)

where $\theta$ and D are the angle and the vertical distance between lines ${L_{{\text{A}}j}}$ or ${L_{Bk}}$, respectively, ${{\text{P}}_B}$ and ${{\text{P}}_A}$ are the point on the lines ${L_{{\text{A}}j}}$ or ${L_{Bk}}$, respectively, ${\vec {v}_A}$ and ${\vec {v}_B}$ are the direction vectors.

Let ${\theta _{th}}$ represent the angular deviation threshold and ${D_{th}}$ represent the spatial separation metric between the lines ${L_{{\text{A}}j}}$ or ${L_{Bk}}$. Then, the relationship between the lines ${L_{{\text{A}}j}}$ or ${L_{Bk}}$ is expressed as

$$\left\{ {\begin{array}{*{20}{c}} {Parallelism({L_A},{L_B})=\left\{ {\begin{array}{*{20}{c}} {True}&{if\;\theta <{\theta _{th}}} \\ {False}&{otherwise} \end{array}} \right.} \\ {Collinear({L_A},{L_B})=\left\{ {\begin{array}{*{20}{l}} {True}&{if\;\theta <{\theta _{th}}\; \wedge D<{D_{th}}} \\ {False}&{otherwise} \end{array}} \right.} \end{array} } \right.$$

(24)

In summary, the steps for recognizing the trajectory lines of the projectiles using the seven-LSAS are as follows.

Step 1

Obtain the moment sequences ${T_{Gi}}$=$({t_{i1}},{t_{i2}}, \cdots ,{t_{i{N_i}}})$, $i=1,2, \cdots ,7$ of every LS.

Step 2

Generate all the moment vectors ${{\mathbf{T}}_{{\mathbf{Aj}}}}$ and ${{\mathbf{T}}_{{\mathbf{Bk}}}}$, $j,k=1,2, \cdots ,m$ by permutation, and calculate the lines ${L_{{\text{A}}j}}$ or ${L_{Bk}}$ corresponding to the moment vectors by (6).

Step 3

Calculate the collinear coefficients by (18) ~ (20), and initially determine the spatial relationship between the lines ${L_{{\text{A}}j}}$ or ${L_{Bk}}$ using the (21) or (22).

Step 4

Calculate the parallelism index $\theta$ and D using the (23), the spatial relationship between the lines ${L_{{\text{A}}j}}$ or ${L_{Bk}}$ is further determined by (24), so that the trajectory lines of the projectile can be recognized.

Similarly, the trajectory lines of three or more projectiles are obtained using this method.

Experimental results and analysis

Experimental setup

According to Fig. 1, to verify the feasibility of the proposed recognition method, a seven-LSAS was constructed using two three-light-screens and one single-light-screen. The experimental setup is shown in Fig. 6. To expand the detection field of view, each LS is formed by the coplanar splicing of two lenses. The structural parameters of the constructed seven-LSAS are listed in Table 1.

Fig. 6

The experimental setup.

Table 1 Structural parameters of the seven-LSAS.

Traditional methods for testing multiple projectiles

To further verify the proposed recognition algorithms for the projectiles, we conducted a comparative analysis using two projectiles passing through the seven-LSAS as an example. The analysis process is as follows.

The launcher fires two projectiles simultaneously, and the original signals of the two projectiles passing through the seven-LSAS are collected. Using the moment extraction algorithm of the projectile, we obtained the moment sequences of the two projectiles passing through each LS, as shown in Table 2.

In the conventional measurement methods, only six LSs are needed, as exemplified by the array A in the seven-LSAS. This method directly utilizes moment values in natural order as the moment sequence for each projectile passing through the array A. Therefore, each row in Table 2 is treated as the moment vector for calculating the flight parameters. To verify the correctness of this conventional method, the calculation results are compared with the coordinates of the wooden witness plate. The comparison results are presented in Table 3.

Table 2 The measured moment sequences of each LS (Values are expressed in milliseconds (ms)).

Table 3 Test results of the conventional measurement method (Values are expressed in millimeters (mm)).

When the two projectiles pass through the array A, the conventional measurement method yields significantly larger deviations between the calculated impact coordinates and the reference coordinates from the wooden witness plate in Table 3. These results cannot be regarded as valid impact coordinates, indicating that the naturally ordered moment vectors are inaccurate.

The proposed methods for testing multiple projectiles

Two projectiles passing through the seven-LSAS are processed using the recognition method proposed in this article, as follows.

Step 1

Obtain the moment sequences of the two projectiles passing through the seven-LSAS, as shown in Table 2.

Step 2

Construct the moment vector space. Because the assignment of moment values in each LS is uncertain, each moment value in the arrays A and B needs to be arranged using the method in Sect. 3.2, and the resulting moment vectors are shown in Tables 4 and 5, respectively.

Table 4 Eight moment vectors in the array A (Values are expressed in milliseconds (ms)).

Table 5 Eight moment vectors in the array B (Values are expressed in milliseconds (ms)).

Step 3

Calculate the collinear coefficients by (18) ~ (20), and initially determine the spatial relationships between lines ${L_{{\text{A}}j}}$ and ${L_{Bk}}$ using (21) or (22).

Substitute the moment vectors from Tables 4 and 5 into the measurement Eq. (6) to calculate the trajectory lines in the arrays A and B. The calculation results are shown in Tables 6 and 7.

According to the trajectory lines of the two projectiles in Tables 6 and 7, the results of the collinear coefficients calculated using the equations (18) ~ (20) were shown in Table 8. Table 8 represents the calculated results for two moment vectors for which the values of the ${G_2}$, ${G_3}$, ${G_4}$, ${G_5}$, and ${G_6}$ are the same, because the calculated lines could be the trajectory lines of the projectiles only if these five moments are the same in the arrays A and B, i.e., $j=k$.

Every row in Table 8 represents the result calculated in equations (18) ~ (20) for the same rows in Tables 6 and 7. For example, the first row in Table 8 indicates the recognition results between ${L_{{\text{A1}}}}$ and ${L_{B1}}$, and the others are similar. The values of each row are not equal in Table 8, which indicates that it is impossible to directly determine whether the trajectory lines are parallel or coincidence.

Table 6 Eight trajectory lines in the array A.

Table 7 Eight trajectory lines in the array B.

Table 8 Calculated results of the collinear coefficients ($j=k$).

Step 4

Calculate the parallelism index $\theta$ and D using (23), and recognize the trajectory lines of the projectiles.

The reference values for the parallelism indexes are calculated from single-projectile test data using Eq. (23). The test data can be found in the next section. Then, according to (23), ${\theta _{th}}=0.15^\circ$ and ${D_{th}}=0.05{\text{mm}}$ are calculated. Here, we take the average value of ten calculation results as the threshold.

To further recognize the trajectory lines of the projectiles, it is necessary to examine the parallelism indexes of each line in Tables 6 and 7. The calculation results are shown in Table 9.

Table 9 Parallelism indexes of eight trajectory lines ($j=k$).

The calculation results in Table 9 demonstrate that only groups 2 and 7 satisfy the parallel and coincident judgment in (24), with values below the predefined threshold. This finding indicates that the corresponding trajectory lines (${L_{{\text{A2}}}}$/${L_{B2}}$ and ${L_{{\text{A7}}}}$/${L_{B7}}$) are spatially coincident and belong to the same projectile. The recognized moment vectors and the corresponding flight parameters are shown in Table 10.

Table 10 The recognition results for the two projectiles.

Compared with the moment vectors arranged in natural order in Table 2, there are significant differences in the recognized vectors. For example, compared with the natural order, the moment order of ${G_3}$ and ${G_6}$ is reversed in the recognized vectors. Meanwhile, these results (Tables 9 and 10) also confirm that the proposed recognition method can reliably recognize the correct moment sequences of multiple projectiles passing simultaneously through the LS.

Measurement error analysis

The measurement errors of the recognized flight parameters of multiple projectiles are equivalent to those of a single projectile. For single-shot testing, only the array A of the seven-LSAS is needed. The measurement errors include the impact coordinate measurement errors and velocity measurement errors. The velocity measurement errors were validated through comparative testing with a calibrated velocity-measuring sky screen. The impact coordinate measurement errors were verified against a wooden witness plate. Since the wooden witness plate was fixed at a predetermined position along the ballistic line, the measurement error of the x-coordinate was not considered.

The relative velocity measurement error is expressed as

$$\delta v=\frac{{{v_G} - {v_{ce}}}}{{{v_{ce}}}} \times 100\%$$

(25)

where ${v_G}$ is the velocity measured by the seven-LSAS, ${v_{ce}}$ is the velocity measured by the calibrated velocity measuring sky screen.

The impact coordinate measurement errors are defined as the difference between the impact coordinates measured by the seven-LSAS and the wooden witness plate, expressed as

$$\left\{ {\begin{array}{*{20}{c}} {\Delta z=\left| {{z_G} - {z_p}} \right|} \\ {\Delta y=\left| {{y_G} - {y_p}} \right|} \end{array}} \right.$$

(26)

where $({z_G},{y_G})$ is the impact coordinate calculated by the seven-LSAS, $({z_p},{y_p})$ denotes the coordinate manually acquired from the wooden witness plate.

Ten firing tests for a single projectile were conducted, with the flight velocities and impact coordinates recorded. Table 11 presents the reference velocities measured by the calibrated sky screen and the corresponding velocities obtained by the seven-LSAS. Table 12 compares the impact coordinates acquired from the wooden witness plate against those measured by the seven-LSAS.

Table 11 Velocity comparison results (Values are expressed in meters per second (m/s)).

Table 12 Impact coordinate comparison results (Values are expressed in millimeters (mm)).

In Table 11, compared with the calibrated sky screen, the relative velocity measurement error of the seven-LSAS is no greater than 1‰, meeting the test accuracy requirements.

In Table 12, the maximum error of the impact coordinate measured by the seven-LSAS is 5.4 mm, the average errors for z and y coordinates are 3.8 mm and 4.0 mm, respectively. Traditional six-light-screen arrays were employed in Refs²⁹. and³⁴for measuring the single projectile. The measurement error of the proposed method is very close to that reported in Ref²⁹., while it is reduced by 50% compared with Ref³⁴.. Ref²⁴. used a combination of a light-screen array and a linear array camera for measuring multiple projectiles, and the measurement error of the proposed method is 50% lower than that of this method.

In the testing of multiple projectiles, compared with the reference coordinates from the wooden witness plate in Table 3, the errors of the impact coordinates calculated from the recognized two moment vectors (${{\mathbf{T}}_{\mathbf{1}}}$ and ${{\mathbf{T}}_{\mathbf{2}}}$) are $\Delta {z_1}=3.3{\text{mm}}$, $\Delta {y_1}=4.7{\text{mm}}$, and $\Delta {z_2}=2.4{\text{mm}}$, $\Delta {y_2}=3.9{\text{mm}}$, which fall within the acceptable error range for a single projectile measurements using the seven-LSAS, meeting the test accuracy requirement.

When multiple projectiles pass through the seven-LSAS, the flight parameters calculated from the measured moment sequences exhibit significant inaccuracies (Table 3). Due to the moment of the projectile passing through the seven-LSAS directly determining the calculation results, the primary source of error can be attributed to inaccuracies in the moment vectors of the projectile. This is attributable to the combined effects of the velocity variations between projectiles and the tilted LSs. Owing to these two factors, the inconsistency in the moment sequence of two projectiles passing through differently LSs prevents accurate assignment of moment values to each projectile, ultimately leading to measurement inaccuracies.

By applying the proposed recognition method to process the original moment sequences, the recognition results (Table 10) demonstrate that the deviation between the calculated impact coordinates and the reference coordinates (measured by the wooden witness plate in Table 3) remains within 5 mm. This error is consistent with the errors of impact coordinates observed in single-projectile tests (Table 12). A direct comparison between the original moment sequences and the processed moment vectors clearly reveals distinct differences in the moment vectors for different projectiles passing through the seven-LSAS, verifying the validity and feasibility of the proposed recognition method.

Conclusions

This article focuses on the recognition problem of multiple projectiles fired by the multi-barrel rapid-firing guns. A test method for multiple projectiles using a seven-LSAS is proposed, and a mathematical model of the flight parameters of the projectiles is established based on the plane equation of the detection light screen. According to the principle that the trajectory lines of the same projectile is collinear, an association recognition algorithm of the trajectory lines of multiple projectiles is proposed. Through the experimental verification, we obtain the following conclusions.

• 1) The measurement error of the impact coordinates of the proposed seven-LSAS is no greater than 5 mm, 50% improvement over existing measurement methods of multiple projectiles.

• 2) The proposed recognition method can recognize the correct trajectory lines of multiple projectiles passing through the seven-LSAS, meeting the requirements of multiple projectiles testing.

• 3) The proposed recognition method can also be used for three or more projectiles.

The proposed method can recognize and measure the flight parameters of multiple projectiles, making it play a significant role in the field of the weapon performance evaluation and design optimization, particularly for the multi-barrel rapid-firing guns. Notably, owing to experimental constraints, experiments were performed only for two projectiles. Although the proposed method is theoretically extendable to cases involving three or more projectiles, the computational data increase exponentially with the number of projectile. Thus, developing efficient data compression algorithms will be a key focus of future research. Additionally, the measurement accuracy of moments is limited by signal overlap when multiple projectiles pass through the same LS simultaneously. Resolving this signal overlap—i.e., accurately separating and recognizing overlapping signals—will be critical for enhancing system robustness in high-density projectile environments.