The article establishes a seven-degree-of-freedom projectile trajectory model for a new type of spinning projectile. Based on this model, a numerical analysis is performed on the ballistic characteristics of the projectile, and the trajectory of the dual-spinning projectile is filtered with the unscented Kalman filter algorithm, so that the measurement information of projectile onboard equipment is more accurate and more reliable measurement data are provided for the guidance system. The numerical simulation indicates that the dual-spinning projectile is mainly different from the traditional spinning projectile in that a degree of freedom is added in the direction of the axis of the projectile, the forebody of the projectile spins at a low speed or even holds still to improve the control precision of the projectile control system, while the afterbody spins at a high speed maintaining the gyroscopic stability of the projectile. The trajectory filtering performed according to the unscented Kalman filter algorithm can improve the accuracy of measurement data and eliminate the measurement error effectively, so as to obtain more accurate and reliable measurement data.
With the technical development of weapons, basic requirements of modern weapon systems are low collateral damage and accuracy. Currently, the great stocks of uncontrolled spin-stabilized projectiles in various countries are gradually being converted to precision-guided and controlled projectiles. Spin-stabilized projectiles maintain stability in flight through high-speed spinning. In order to deal with the challenges to measurement and control systems caused by high-speed spinning, drag increasing mechanisms, spinning reducing mechanisms, pulse engines and built-in sliders, and similar components are currently used as actuators to reduce fall point dispersion effectively. However, the drawback is that a continuous control force cannot be applied and precision strikes are difficult to achieve.
At present, a canard-guided dual-spinning projectile [
Structural diagram of dual-spinning projectile.
During flight, since the projectile is affected by factors such as the environment, certain errors will inevitably be caused in the measurement system. With accumulation of these errors, the precision of the control system will be greatly affected. Trajectory filtering based on flight measurement data is an important means to understand the actual flight status of the projectile. It is widely applied in fields such as simulated flight model validation, emplacement reconnaissance amending, and trajectory prediction and filtering [
Since the spinning speeds of the forebody and the afterbody of a dual-spinning projectile are different in the direction of the axis of the projectile, one degree of freedom (DOF) is added to the six-DOF rigid body dynamic model, which is used as a basis. Therefore, a seven-DOF dynamic model of the dual-spinning projectile is established in this article. This model comprises a centroid motion caused by the resultant force and rotation around the centroid caused by the resultant moment.
The coordinate system of the axis of the projectile is as follows: The origin
The schematic diagram of the coordinate system.
This article establishes a dynamic model of the dual-spinning projectile under the coordinate system of the axis of the projectile. This model comprises a dynamic model of the centroid motion and a dynamic model of the rotation around the centroid. The model of the centroid’s motion is shown as (
In (
In the kinematic equation set of the dual-spinning projectile, kinematic equations of the movement of the centroid and the attitude change of the projectile body relative to the ground coordinate system are established. Equation (
In (
Resultant force is calculated as follows:
Aerodynamic force from the projectile is calculated as follows:
Aerodynamic force from the canard wing is calculated as follows:
Magnus force is calculated as follows:
Gravity is calculated as follows:
Resultant moment is calculated as follows:
Moment from the projectile is calculated as follows:
Moment from the canard wing is calculated as follows:
Magnus moment is calculated as follows:
Damping moment is calculated as follows:
Control/friction moment is calculated as follows:
In (
According to the above model, a computer simulation program was developed to perform trajectory simulation. The basic physical parameters and aerodynamic parameters of the projectile are shown in Table
Projectile parameters.
Parameter | Value | Parameter | Value |
---|---|---|---|
45.5 kg | −0.039 | ||
155 mm | 4.102 | ||
909 mm | 0.351 | ||
0.012 kg·m−2 | 0.006 | ||
0.133 kg·m−2 | −0.023 | ||
1.800 kg·m−2 | −0.002 | ||
0.352 | −1.125 | ||
−2.012 | 0.001 | ||
3.121 | 0.00001 |
Figures
Velocity component
Velocity component
Velocity component
Flight trajectory.
Spinning rate properties.
In order to illustrate the distinction between the dual-spinning projectile and the traditional spinning projectile, Figure
The system model of the dual-spinning projectile is strongly nonlinear. If a linear filtering method, for example, KF or EKF, is applied, large linearization error will be caused. Moreover, system equations of seven-DOF are very complex, and it is hard to obtain the Jacobi matrix in analytical form, so in this article, the UKF is used to perform online trajectory reconstruction.
Depending on the actual engineering needs, online data filtering is mainly performed on coordinates
The steps for trajectory reconstruction using the UKF are the following.
The weighting factor
In (
Since approximating the state distribution is easier than approximating the nonlinear function, the nonlinear propagation of random state variables can be completed through a determined sample group. It is assumed that the external stimulation, the estimated value of the state vector
Predicted measurement value
The trajectory filtering method based on the algorithm given in this article was validated. The validation conditions were as follows: the projectile initial speed was 930, the firing angle was 65°, and the projectile weight was 45.5 kg. The measured trajectory value was simulated by adding Gaussian white noise to the theoretical calculating value of the trajectory. Based on this measured value, the effectiveness of the UKF algorithm in the process of trajectory filtering was verified. Figures
Coordinate
Coordinate
Coordinate
The article establishes a seven-DOF projectile trajectory model of a dual-spinning projectile. After analyzing the ballistic characteristics of the dual-spinning projectile with the method of numerical simulation, it can be seen that the major distinction between the dual-spinning projectile and the traditional spinning projectile is that the forebody and the afterbody show different spinning speed characteristics. This article specifically analyzes the reason causing this phenomenon and advantages for using such a structure. Furthermore, in this article, filtering the trajectory of the dual-spinning projectile is performed by using the UKF algorithm. Based on the comparison of the ballistic trajectory data before and after filtering, it can be seen that the precision of the measurement data is further improved, which can provide more accurate and reliable basic data for engineering applications.
The authors declare that they have no competing interests.
This work is supported by the National Natural Science Foundation of China under grant number 11472136.