The ability of the projectile to strike accurately in the air depends on the accurate measurement of its flight distance. Due to the influence of the external environment, it is very difficult to measure the distance of high-speed projectiles in real time with ranging equipment such as radar and laser. Based on this, a real-time measurement model of the horizontal flight distance of the projectile was designed, which combines trajectory equation and discrete acceleration. First, the angle of inclination of the projectile at each position was calculated according to the six-degree-of-freedom rigid outer trajectory model. Then, two kinds of distance measurement models based on discrete acceleration integration were introduced, and the accuracy simulation was carried out. Simulation results show that the Simpson integral method is more accurate. The flight arc length of the projectile was calculated by the Simpson integral method. Finally, the horizontal flight distance of the projectile was obtained by combining the flight arc length and inclination angle of the projectile. Finally, the flight distance measurement model was simulated, and the simulation results show that the model has high accuracy. The simulation results show that the horizontal flight distance of the projectile can be measured by the trajectory equation and discrete acceleration according to the set conditions. This method can be used to measure the distance of the projectile by carrying the sensor and is less affected by the external environment. It can provide a theoretical basis and reference for the measurement of the horizontal distance of the projectile.

## I. INTRODUCTON

The accurate control of projectile flight distance has always been a hot topic in the field of equipment accurate control. The realization of flight control can make space equipment more intelligent and safe.^{1} Researchers are keen on using simulation methods to simulate the flight trajectories of various types of flight devices.^{2} Flight devices such as rockets often use methods such as radar or satellite positioning to measure their flight paths.^{3–5} Furthermore, it can realize the control of its own attitude, trajectory, and position through autonomous correction. This method would be expensive for lower cost flight equipment. In addition, for the small internal space of the projectile, complex equipment is obviously impossible to implant.

In many cases, we need to control flight equipment based on flight distance. When a projectile is flying freely in a certain initial state in the air, we usually do not care about its flight path. Since the explosion control signal or some action needs to be performed at a predetermined horizontal distance, we usually need to know the horizontal flight distance of the projectile from the starting point to a certain location. For example, in the field of aerial target detection and real-time control of flying targets,^{6} it is necessary to know the horizontal flight distance of flying objects in many cases. It is difficult to measure the horizontal flight distance of projectile in real time.

Laser rangefinder and radar are a commonly used distance measuring instrument, which have a high measuring accuracy when measuring static objects.^{7} For the distance measurement of the flying object, it is difficult for the laser rangefinder to obtain the dynamic feedback signal in real time. Because it is sensitive to light, laser rangefinder cannot adapt to the harsh environment and cannot measure the horizontal flight distance of flying objects in real time. There are other types of distance-measuring instruments, but these are all external measuring devices for the projectile itself. On the one hand, these external measuring instruments are limited by the natural environment and affect the measurement accuracy. On the other hand, these distance-measuring equipment cannot interact with the information of the flying object itself. Obviously, these external measuring instruments cannot carry out real-time monitoring and signal control of the flight information of the projectile.

To measure the horizontal distance of a projectile in real time during the flight, a method combining projectile equation and acceleration integral was proposed. In view of the narrow space inside the projectile, we propose a method to measure the projectile acceleration in real time by the micro acceleration sensor and further realize the distance measurement by integrating.^{8} This part of the work is described in detail in our previous work. In the aspect of algorithm, the method of distance calculation by acceleration integral is also proved to be effective. As early as the 1960s, Berg and Housner proposed that there is a simple transformation relationship between the acceleration, velocity, and displacement signals of various points during the motion of an object.^{9} For example, after the acceleration signal is obtained, the speed or displacement signal can be further obtained by integrating.^{10,11} Therefore, acceleration-based distance measurement methods are used to measure ground deformation, vibration velocity, and displacement during earthquakes. The effectiveness of the distance measurement system and algorithm designed by us has also been verified on the linear slide.

According to the ballistic equation, the trajectory of the projectile in the air is an arc, while what we need to determine is the horizontal flight distance of the projectile. Therefore, a computational model was proposed to convert the arc length into the horizontal flight distance of the projectile.

The model combines the inclination of the projectile during flight with the acceleration of the projectile, where the inclination can be solved by the ballistic equation and the acceleration can be obtained by the measurement system. This paper mainly does the following three aspects of work: First, the trajectory equation is constructed and the influence of the angle of attack on the range measurement accuracy is analyzed. Second, the accuracy of the two methods based on acceleration integration was compared. Third, the method of constructing a horizontal distance measurement model was introduced in detail and simulation was carried out.

## II. ESTABLISHMENT OF SIX-DEGREE-OF-FREEDOM RIGID BODY TRAJECTORY MODEL

To investigate the variation law of the angle of attack and inclination of the projectile on the full trajectory, it is necessary to establish a trajectory model. The trajectory model constructed can simulate the relationship between the influencing factors and provide a theoretical basis for the construction of the horizontal distance model. To obtain the trajectory parameters of the projectile during flight, a six-degree-of-freedom rigid trajectory model of the projectile was constructed using equations.^{12} The following basic assumptions are made in the trajectory model-building process:

The projectile is regarded as a rigid body and an axisymmetric body.

The surface is flat, the gravitational acceleration value is 9.8

*m/s*^{2}, and the influence of earth curvature is ignored.The ground coordinate system is used as the inertial coordinate system.

^{13}

*F*

_{x2},

*F*

_{y2}, and

*F*

_{z2}are the forces on each axis of the projectile’s center of mass in the ballistic coordinate system.

*ψ*

_{2}is the velocity direction angle, and

*θ*

_{a}is the velocity elevation angle.

*m*is the projectile mass, and

*v*is the projectile centroid velocity.

*A*is the lateral moment of inertia, and

*C*is the axial moment of inertia.

*M*

_{ξ},

*M*

_{ζ}, and

*M*

_{η}represent the components of the moment of momentum of the external force on the center of mass in the coordinate system of the projectile axis.

*φ*

_{2}is the direction angle of the projectile axis, and

*ω*stands for angular velocity.

Finally, the six-degree-of-freedom rigid ballistic model of the projectile can be obtained immediately by combining the above equations. The parameters such as angle of attack, arc length, displacement, and acceleration can be obtained by solving the trajectory equation.

## III. SIMULATION OF THE ANGLE OF ATTACK

Due to the effect of air resistance and gravity on the projectile when it flies at high speed, the longitudinal acceleration direction of the projectile and the tangential trajectory of the projectile usually do not coincide. The angle of attack of the projectile is shown in Fig. 1.

According to the flight path of the projectile, it is necessary to measure the tangential acceleration of the projectile to accurately calculate the arc length of the projectile’s flight, but the acceleration sensor can only be arranged in the axial direction to measure the axial acceleration of the projectile. Because of the existence of the angle of attack, there is a certain error in calculating the trajectory arc length using the axial acceleration integral. Therefore, it is necessary to explore the variation of the angle of attack of the projectile during the whole flight.

Before formula derivation and simulation, the definition of each variable is as follows:

*ϕ*: Space azimuth.*ψ*: Spatial azimuth of velocity.*δ*: Angle of attack.*S*: Maximum cross-sectional area of a projectile.*m*_{z}: Derivative of static torque coefficient.*m*_{zz}: Equatorial damping torque coefficient.*ρ*: Air density.*w*_{⊥}: Lateral wind.*l*: Full length of projectile.*c*_{x},*c*_{y},*c*_{z}: Aerodynamic coefficient or its derivative.

^{14}

Taking a certain type of projectile as the simulation research object, combining the angle of attack equation and the equations of six-degree-of-freedom rigid ballistic model of the projectile for simulation, the variation law of the angle of attack with the flight arc length of the projectile can be obtained, as shown in Fig. 2.

As can be seen from the figure above, when the shot angle is 23.36°, the angle of attack of the shot will reach the maximum. The maximum angle of attack is only 0.6°, which means that the axial acceleration of the projectile and the tangential acceleration of the center of mass almost coincide during the projectile's flight. To improve the range of the projectile, the angle of fire is generally not more than 20°. Therefore, the simulation results are most adequate to meet our requirements.

A certain type of projectile has a range of 1500* m* and an angle of fire of 11.7°. Now, the maximum angle of attack for the entire flight of the projectile is set at 0.6°, and the arc length of the projectile flight is calculated by integrating the axial and tangential acceleration of the projectile. The specific integral errors of the two calculation methods are shown in Table I.

Projectile range (m) . | S_{tangential} (m)
. | S_{Axial direction} (m)
. | Error (m) . |
---|---|---|---|

500 | 500.385 | 500.385 | ≈0 |

1000 | 1002.284 | 1002.284 | ≈0 |

1500 | 1518.176 | 1518.378 | 0.202 |

2000 | 2080.327 | 2080.636 | 0.309 |

Projectile range (m) . | S_{tangential} (m)
. | S_{Axial direction} (m)
. | Error (m) . |
---|---|---|---|

500 | 500.385 | 500.385 | ≈0 |

1000 | 1002.284 | 1002.284 | ≈0 |

1500 | 1518.176 | 1518.378 | 0.202 |

2000 | 2080.327 | 2080.636 | 0.309 |

From the error comparison of the integral results, it can be seen that when the shooting distance of the projectile is less than 2000 *m*, the integral results obtained by using the two accelerations have little difference and can be ignored. Therefore, the arc length can be obtained by integrating the projectile’s axial acceleration.

## IV. CONSTRUCTION OF HORIZONTAL DISTANCE MEASUREMENT MODEL

After the acceleration measurement system is implanted in the projectile, the system begins to collect the acceleration of the flying projectile after the projectile is launched. However, the direct integral of acceleration is not the horizontal distance but the length of the arc of the flying object. Due to the sampling frequency of the accelerometer reaching the millisecond level, the arc length of the projectile flying over within a sampling cycle can be approximated as the length of a straight line. To obtain the horizontal flight distance of the projectile, the flight arc length needs to be further processed. The horizontal flight distance of the projectile can be obtained by multiplying the flight arc length of the projectile in one sampling period by the cosine of the angle of the projectile at that time. The calculation relationship between the flight arc length and the horizontal distance of projectile is shown in Fig. 3.

The meaning of the variables in the model:

*t*_{n}: Total flight time of the object.Δ

*t*: The time interval between two adjacent acceleration points.*θ*_{n}: Flight inclination at time*t*_{n}.*ds*_{n}: The arc length of flight at time*t*_{n}.*dl*_{n}: Horizontal distance of movement between*t*_{n}*t*_{n+1}.

*L*is

The inclination angles of each position in the horizontal distance measurement model can be obtained by solving the ballistic equation. The obtained angle values at each moment are installed into the software program in sequence. Inclination angle can be called in a software program when the horizontal distance is calculated from the arc length. First, the arc length of flight in a certain period of time is obtained by acceleration integral. Second, the horizontal flight distance of this period can be obtained by the flight arc length and the flight inclination during this period. Finally, the total flight distance of the flying object can be obtained by summing up the horizontal flight distance of each time period.

## V. CALCULATION METHOD OF FLIGHT ARC LENGTH

In this section, two distance-solving algorithms based on discrete acceleration integration are derived. Then, the accuracy of these two algorithms is simulated.

### A. Trapezoidal formula integral method

According to the calculus definition, the velocity increment can be taken as the trapezoid area formed by the two adjacent discrete acceleration values and the time axis. Similarly, the displacement value of any discrete time can be obtained by further integrating the discrete velocity. The principle of the trapezoidal formula method is shown in Fig. 4, where the horizontal axis represents the time and the vertical axis is the acceleration axis.

*v*(

*t*

_{i}) is the velocity of each discrete moment. Since the sampling period of the acceleration sensor is a fixed value, the following transformation can be obtained:

^{15}

### B. Simpson formula integration method

The principle of Simpson’s integral method is to first solve a parabolic function from three discrete acceleration points. Second, the integral result of the quadratic function over this interval is taken as the velocity increment in this time period. Thus, the velocity sequence of discrete time can be obtained, and then the displacement sequence of discrete time can be obtained. The principle of Simpson's integral method is shown in Fig. 5.

*t*

_{2n−2},

*a*

_{2n-2}), (

*t*

_{2n−1},

*a*

_{2n−1}), (

*t*

_{2n},

*a*

_{2n}), then a ternary system of equations can be obtained as follows:

*k*,

*m,*and

*n*of the quadratic function can be determined. After the quadratic function is determined, the definite integral on the interval $(t2n\u22122,t2n)$ can be regarded as the velocity increment,

### C. Precision simulation of two range algorithms

First, the above formula is integrated in the time domain to obtain the displacement signal with the time variable.

Then, it is obtained that the true displacement of the moving object in half a period is 5.7299, where the displacement value is no longer set as a unit. Next, the time domain sampling of the analog acceleration signal is performed at the sampling frequency of 10, 25, and 100 *Hz* so that the discrete acceleration signal of the moving object in the time domain can be obtained. Also, the motion displacement of the object is recalculated by trapezoid formula integral method and Simpson integral method. Finally, compared with the actual displacement, the accuracy of the two algorithms can be obtained, as shown in Table II.

Integral method . | Displacement/10 Hz . | Error . | Displacement/25 Hz . | Error . | Displacement/100 Hz . | Error . |
---|---|---|---|---|---|---|

Trapezoidal formula | 5.5980 | 0.1319 | 5.7086 | 0.0213 | 5.7282 | 0.0017 |

Simpson method | 5.6524 | 0.0775 | 5.7179 | 0.0120 | 5.7295 | 0.0004 |

Integral method . | Displacement/10 Hz . | Error . | Displacement/25 Hz . | Error . | Displacement/100 Hz . | Error . |
---|---|---|---|---|---|---|

Trapezoidal formula | 5.5980 | 0.1319 | 5.7086 | 0.0213 | 5.7282 | 0.0017 |

Simpson method | 5.6524 | 0.0775 | 5.7179 | 0.0120 | 5.7295 | 0.0004 |

The following conclusions can be drawn from the above table:

With the increase in sampling frequency of the same algorithm, the displacement accuracy obtained is also higher.

Because the trapezoidal formula integral method simply connects two discrete acceleration values to form a trapezoid in the integration process, the characteristics of smooth gradient between discrete acceleration points are ignored. Therefore, the integral accuracy of the trapezoidal formula method is lower than that of Simpson integral method when the sampling frequency is constant.

Therefore, the Simpson integral method should be used to integrate the flight arc length of the projectile. In addition, the two integration algorithms derived in this paper will eventually become addition and subtraction operations between discrete acceleration points. Therefore, when the algorithm is implanted into the system, it will not increase the processor's overhead.

## VI. SIMULATION OF HORIZONTAL DISTANCE MEASUREMENT MODLE

The acceleration, inclination, and other parameters of the projectile can be calculated by constructing the trajectory equation. The simulation of angle of attack shows that the tangential acceleration can be replaced by axial acceleration. By comparing the accuracy of the algorithm based on discrete acceleration, we choose the Simpson integral method to integrate the arc length of the projectile flight. On this basis, we further simulate the constructed horizontal distance measurement model and verify whether the model is feasible.

According to the above method, the flight arc length of the projectile in each time period can be obtained from the acceleration. The horizontal flight distance of the projectile can be further obtained from the flight arc length and the flight angle. After comparing the set horizontal flight distance with the simulated horizontal flight distance, the accuracy of the theoretical model can be verified.

Under the conditions of standard atmospheric pressure and emission speed of 250* m/s*, the launch angle is set to 11.7°, 30°, and 60°, and the simulation distance is set to 500, 1000, and 1500 *m*, respectively. The simulation results of horizontal distance are shown in Table III.

Launch angle . | Simulation results/m . | Set horizontal distance/m . | Simulation error/m . |
---|---|---|---|

11.7° | 500.2 | 500 | 0.2 |

1000.4 | 1000 | 0.4 | |

1500.8 | 1500 | 0.8 | |

30° | 500.2 | 500 | 0.2 |

1000.3 | 1000 | 0.3 | |

1500.5 | 1500 | 0.5 | |

60° | 500.3 | 500 | 0.3 |

1000.5 | 1000 | 0.5 | |

1501.1 | 1500 | 1.1 |

Launch angle . | Simulation results/m . | Set horizontal distance/m . | Simulation error/m . |
---|---|---|---|

11.7° | 500.2 | 500 | 0.2 |

1000.4 | 1000 | 0.4 | |

1500.8 | 1500 | 0.8 | |

30° | 500.2 | 500 | 0.2 |

1000.3 | 1000 | 0.3 | |

1500.5 | 1500 | 0.5 | |

60° | 500.3 | 500 | 0.3 |

1000.5 | 1000 | 0.5 | |

1501.1 | 1500 | 1.1 |

The simulation results show that the horizontal distance obtained by the theoretical model has high precision under the set conditions. In addition, it can be seen that although the simulation error increases with the increase in the simulation distance, the maximum simulation error is only 1.1 *m*. Therefore, the simulation results show that the above theoretical analysis method is feasible and the constructed horizontal distance simulation model is effective.

## VII. CONCLUSION

To measure the horizontal distance of the projectile, this paper innovatively constructs a horizontal distance measurement model based on discrete acceleration and ballistic equations. A six-degree-of-freedom rigid trajectory model is constructed, and the angle of attack of the projectile is simulated based on this model. It is concluded that the arc length integration can be performed by axial acceleration instead of tangential acceleration. The algorithm of arc length integration based on acceleration is deduced and simulated, and a conclusion is drawn that the Simpson integral method is more accurate. Based on the above work, the precision simulation of the horizontal distance model of the flying projectile is carried out. The simulation results show that the model has high precision, which can provide a certain basis for the experimental and theoretical research in this field. In addition, to achieve the accurate measurement of the horizontal distance of the projectile, the initial velocity measurement is also an additional problem that needs to be considered.

Based on the research in this paper, we can continue to study different types of models and try to optimize the models. On the other hand, we can further discuss the type of acceleration sensor and how to improve the measurement accuracy by restraining the temperature error and random drift error of the sensor.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

All authors contributed equally to this work.

**Yonglei Shi**: Conceptualization (equal); Data curation (equal); Investigation (equal); Writing – original draft (equal). **Tiebi Zhang**: Conceptualization (equal); Resources (equal). **Guangfei Jia**: Formal analysis (equal); Investigation (equal). **Yuejing Zhao**: Methodology (equal); Supervision (equal). **Zhanpu Xue**: Validation (equal). **Zhiying Qin**: Methodology (equal); Project administration (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.