Asymptotic analysis of the angular motion of a projectile based on Lyapunov artificial-small-parameter perturbation

In the traditional analysis of the angular motion of a projectile, the coefficient freezing method is often used to simplify the differential equation of angular motion. However, this method gives a poor approximation for long time scales. In this paper, based on Lyapunov artificial-small-parameter perturbation theory, an asymptotic analytical solution for the angular motion of a projectile is derived by constructing artificial small parameters for the variable coefficient terms in the original equation. In an example, the angular motion of a single-channel rotating projectile with two-dimensional trajectory correction was analyzed. The method proposed in this paper was compared with the coefficient freezing method. The results show that the proposed method can give a good approximation over a long time scale. This study may be significant for the solution of differential equations with variable coefficients and may be useful for certain engineering applications.


Introduction
Analyzing the angular motion of a projectile is important for the stability and maneuverability of the projectile and is also essential for navigation and guidance.The accuracy of the results directly affects the flight performance of the projectile.In particular, single-channel rotating projectiles use a periodic control force to correct the trajectory.If there is no correction, the projectile will swing through a cone under the action of a rotating aerodynamic force.The angular motion of this kind of projectile is directly related to the design of the correction control force, so its angular motion must be known accurately [1][2][3][4][5].
Many scholars have analyzed the angular motion of such projectiles.There are two main methods for solving their angular motion.The first is numerically.Wang et al. established an eccentric motion model of a projectile with a fixed canard wing, two-dimensional trajectory correction, and a fuse.Using a numerical approach, these authors studied the method for deviation correction and verified the guidance and control strategy [6].Guan et al. established a trajectory model with seven degrees of freedom for a guided double-spin projectile [7].To verify the trackless Kalman filter algorithm for the trajectory, the trajectory and attitude of the projectile were analyzed numerically.Such numerical methods can accurately determine the whole trajectory and the calculation speed is fast, but the results are only for discrete time points and the influence of various parameters on the results is unclear, so they are suitable only for verification.
The second method for solving the equation of angular motion is the approximate analytical method.To establish the stability criterion for a double-spin missile, Zhu et al. derived an equation of motion with seven degrees of freedom in a fixed-plane coordinate system [8].A differential equation for the complex angle of attack was derived using linearization theory, and an analytical solution for a stable pitch boundary was obtained.Wu et al. established an equation of angular motion for the angle of attack and velocity of a projectile with two-dimensional trajectory correction under fixed rudder action [9].The coefficient freezing method can give analytical solutions for the forced motion of the angle of attack when a fixed rudder rotates uniformly and for the transient and steady-state response of the angle of attack when the fixed rudder generates a step excitation.However, when finding an analytical solution of the equation of angular motion, the above methods all ignore the variable coefficients.This leads to the following problems.First, the analytical solution has high accuracy only on the arc near a characteristic point on the trajectory.Second, the accuracy decreases for longer time scales.
The equation of angular motion for a projectile is a second-order non-homogeneous ordinary differential equation with variable coefficients.For this kind of equation, research has shown that it is possible to obtain an accurate analytical solution only when the equation is in a specific form [10][11][12].However, the angular motion equation is not in a specific form.Thus, it has not been possible to obtain an accurate analytical solution.In this paper, we use the Lyapunov artificial-small-parameter perturbation method [13][14][15], which is effective for finding asymptotic analytical solutions of differential equations with nonlinear and variable coefficients.We consider the equation of angular motion for a projectile and construct small parameter for the variable coefficients in the original equation.An asymptotic analytical solution of the original equation was derived based on parameter perturbation theory, and the accuracy of this method was verified in an example.

Mathematical description of the angular motion of a projectile
To analyze the attitude and stability of a projectile during flight, it is usually necessary to solve the equation for the angular motion of the projectile.In general, to assess the stability of the flight of a projectile, the angle of attack δ as a function of time t needs to be obtained.The actual trajectory of a projectile deviates slightly from the ideal trajectory.Normally, the velocity line of the ideal trajectory is taken as a reference.The pitching and yawing of the projectile around this reference is called the angular motion of the projectile.
The unit spherical surface description of the angular motion of a projectile is shown in Figure 1.The figure shows a unit spherical surface with the center of mass c of the projectile at the center.The unit vector 0  r is the ideal ballistic velocity direction, and its intersection with the unit sphere is o.The plane cox is the horizontal plane of the ballistic coordinate system.The plane coy is the vertical plane of the ballistic coordinate system.The unit vector 1  r is the velocity direction of the projectile, and its intersection with the unit sphere is T. The unit vector 2  r is the direction of the projectile axis, and its intersection with the unit sphere is B. The small sphere at xoy in Figure 1 can be expanded into the plane shown in Figure 2. If the x-axis is the imaginary axis and the y-axis is the real axis, then the vectors  oT ,  oB , and  TB can be represented by the complex numbers   The equation for the angle of attack of a projectile can be simplified by ignoring some small quantities: .Where 0 v represents the initial velocity of the projectile and n represents the attenuation factor of velocity [16].
Equation ( 1) is a second-order non-homogeneous differential equation with variable coefficients.Generally, there are no accurate analytical solutions for this kind of equation.When they do exist, they have complex terms.Therefore, they are usually solved by a numerical method or after further simplification.Next, the problem was constructed as a parameter perturbation problem, and an asymptotic analytical solution was obtained using the direct expansion method.

Direct expansion method for parameter perturbations
This type of practical physical problem can generally be described mathematically in the form of the following differential equations and boundary conditions: where x is a scalar or a vector, ε is a small parameter, and L is a differential operator, which can be for ordinary differentiation or partial differentiation.Analytical solutions cannot be obtained for more general differential equations, such as Equation (1).However, if there was a value ε 0 , when ε = ε 0 , the above problem can be solved precisely or at least the analytical solution is more convenient.Then, u(x,ε) can be approximated by finding the power series expansion of ε: This is a power series in ε.To determine the specific forms of u 0 (x), u 1 (x), …, u n (x), a series of recursive equations can be obtained by substituting Equation (3) into Equation ( 2) and combining terms with the same power of ε: The series solution of u(x,ε) was obtained by substituting the solution of Equation ( 4) into Equation (3).The sum of a finite number of terms is an approximate solution of u(x,ε) within a certain accuracy range.Next, this idea was used to solve Equation (1).

Asymptotic analytical solution of the angular motion of a projectile
Since there is no perturbation in Equation ( 1), it is necessary to construct a perturbation parameter ε artificially, so that it is framed as a parameter perturbation problem.According to the Lyapunov artificial-small-parameter method, Equation (1) can be transformed as follows: The initial conditions are   . Here v 0 is the initial velocity of the projectile.Thus, the effect of variable coefficient v on the solution of δ(t) is transferred to the right end of Equation (5).By multiplying the term containing  and   on the right-hand side of Equation ( 5) by the parameter ε, we can construct the following equation: Thus, Equation ( 6) becomes the original equation when ε = 1.When ε = 0, Equation ( 6) degenerates into a second-order non-homogeneous differential equation with constant coefficients.Next, we expand the angle of attack  into the first-order approximate expansion of ε, i.e.: ) Substituting Equation ( 7) into Equation (6) gives the following: We can rearrange Equation ( 8) according to the form of the order of ε: 9) and solving it by the constant variation method gives: t C e C e Ke (11) where Here, C 1 and C 2 are undetermined coefficient, which can be found from the initial conditions Next, to solve Equation ( 10), we substitute Equation (11) into Equation (10).Note that the order of magnitude of   0  A v v is 10 0 and that of   The undetermined coefficients C 3 and C 4 can be determined from the initial conditions   . We substitute Equations ( 11) and ( 12) into Equation (7).Then, with ε = 1, we obtain an asymptotic analytical solution of Equation (1).

Example analysis
Taking a single-channel rotating projectile with two-dimensional trajectory correction as an example, the angular motion of the projectile was solved for a stable rotational speed.The relevant parameters are shown in Table 1.These were substituted into Equation (7) to give the asymptotic analytical solution of Equation ( 1): Equation (1) was also solved by the coefficient freezing method, as follows.
First, we fixed the coefficient v at v 0 , and then the solution is   To verify the accuracy of the two methods, the results were compared with the numerical solution It can be seen from Figures 3 and 4 that the swing frequency of the angle of attack δ of the projectile is the same as the frequency of rotation ω.The amplitude fluctuated in the first 0.5 s and then decayed gradually.Comparing the two analytical methods, the analytical solution for the coefficient freezing method was a better approximation to the numerical solution in the initial 0.5 s, but it does not account properly for the attenuation of the amplitude, which results in a larger error for the amplitude over time.The analytical solution based on a Lyapunov artificial-small-parameter perturbation has a slightly lower accuracy in the first 0.5 s compared to the numerical solution.However, over time, due to the coefficient D 3 in Equation (12), this analytical solution accounts for the amplitude attenuation, resulting in a better approximation, with values basically the same as those for the numerical solution.As can be seen from Figure 5, the error of the artificial-small parameter perturbation method fluctuates significantly in the first 1 s, but did not exceed 100%, and the relative error decreased continuously to less than 10% after 2 s.For the coefficient freezing method, only the relative error within the first 0.5 s was less than 20%, and then increased continuously with time.It was nearly 300% in the 5th second.In conclusion, the asymptotic analytical solution based on the Lyapunov artificialsmall-parameter perturbation method, as presented in this paper, gave a better approximation for long time scales than the coefficient freezing method.

Conclusions
This study addressed the difficulty in obtaining accurate analytical solutions of the equation of angular motion for a projectile due to the variable coefficients.An asymptotic analytical solution for the equation of angular motion was derived based on Lyapunov artificial-small-parameter perturbation theory.This method has the following advantages: 1. Compared with the numerical method, the effects of various parameters on the angular motion can be obtained and the stability conditions can be analyzed.
2. Compared with the coefficient freezing method, it has better accuracy for long time scales.
3. The first n terms in the series for the asymptotic solution can be taken according to the accuracy requirements.Generally, an acceptable accuracy can be obtained by taking the first two terms.The analytical expression is not complex in form.
 i , respectively, where δ 1 indicates whether the angle of attack is high or low and δ 2 is the direction of the angle of attack.Moreover, they satisfy the relation b a    .

Figure 1 .
Figure 1.Unit sphere description of the angular motion of a projectile.

Figure 2 .
Figure 2. Complex plane description of the angular motion of a projectile.
constants.The parameters ρ, S, l, m, ξ,  zz m ,  z m ,  y c , α, and ω are the air density, characteristic area, characteristic length, mass, equatorial moment of inertia, derivative of the equatorial damping moment coefficient, derivative of the static moment coefficient, derivative of the lift coefficient, aerodynamic eccentricity angle, and projectile rotation speed, respectively.The velocity v is a variable whose magnitude decreases exponentially with time, i.e., 0 nt v v e   is 10 3 .We can simplify by omitting the small value  0  A v v :MEIE-2023 Journal of Physics: Conference Series 2591 (2023) 012001 IOP Publishing doi:10.1088/1742-6596/2591/1/0120015

Figure 3 compares the heights of the angle of attack   1  j t and   1  x t , and Figure 4 2  j t and   2 
Figure 4 compares the directions of the angle of attack

Figure 3 . 1  1 
Figure 3.Comparison of the heights of the angle of attack   1  j t and

Figure 4 . 2  2 .
Figure 4. Comparison of the directions of the angle of attack   2  j t and

Figure 5 .
Figure 5. Amplitudes of the relative errors for the two methods.