THE INVERSE PROBLEM OF EXTERNAL BALLISTICS FOR IDENTIFICATION OF AERODYNAMIC COEFFICIENTS OF A SPIN-STABILIZED PROJECTILE WITHIN THE MODIFIED POINT-MASS TRAJECTORY MODEL

The aim of this paper is to develop a technique for the identifi cation of the projectile aerodynamic coefficients using the free-fl ight-test measurements. An algebraic method for solving the inverse problem of external ballistics is proposed. As the initial mathematical model of the projectile fl ight, a simplifi ed version of the modifi ed point-mass trajectory model in explicit form is used. For all aerodynamic coefficients of the model, exact explicit analytic expressions for their dependence on the experimentally measurable trajectory parameters are derived. Importantly, within the proposed approach, the solution to the inverse problem is unique.


INTRODUCTION
Артилерійське та стрілецьке озброєння ISSN 2414-0651 (друк) were based on rather rough approximate solutions of the equations of projectile motion [2,5]. Nowadays, in a signifi cant part of works on this topic, the problem is considered in the form of a "black box or gray box" with the use of considerable simplifi cations, for example, replacing the sought functions with approximating or interpolating polynomials, etc. Several studies on this topic have been conducted and most of them employ the minimization of the difference between the measured and the calculated data [6−14].
PRELIMINARY REMARKS This paper develops the idea expressed in [15], where an attempt was made to identify the drag coeffi cient from the trajectory data of the free fl ight of the projectile. The mathematical basis of the proposed method is the approach described in [16].
To clarify the essence of the method and the range of issues associated with it, let us consider the simplest model where   t y is the scalar function describing the time dependence of a certain "trajectory",   to identify the unknown parameter   t  from the particular "solution"   To be explicit, let us list several features in the physical setting of the considered problem. First, as a rule, the dependence   t ŷ is represented in tabular form by its values at discrete time moments. Second, the registration process is always accompanied by experimental (hardware) errors, which, in turn, will be transferred to the identifi ed function   t  . Since expression (2) contains the derivative   t ŷ of the experimental discrete function, the use of numerical differentiation can lead to destructive consequences for the method. In such a case, the numerical differentiation operation is an ill-conditioned problem. Our experience reveals that for such problems, the preliminary approximation of the initial discrete experimental data by one expression of certain smooth functions over the entire essentially solves the problem of differentiation since in this case continuous functions that are already smoothed during approximation are analytically differentiated.
Often solution (2) is not unique. To choose the correct branch of solutions, one has to use a priori conditions or refer to the physical meaning of the identifi ed parameter. Now let's complicate the formulation of the problem. Consider a more realistic model which differs from (1) for two variables and the required parameters will be the solution of this algebraic system with respect to     The rest of the recovery procedure remains unchanged.
Finally, let us consider an additional version, in which the derivative of the identifi ed parameter stands within the mathematical model At fi rst glance, the problem of recovering the parameter   t  from a known particular solution becomes much more complicated, since (4) is a differential equation for   We emphasize that in all the cases considered above, it is assumed that the model is adequate in relation to the corresponding physical phenomenon, and it is this phenomenon that is investigated experimentally. For this reason, in particular, the replacement of t by t during the identification of t does not affect the physical phenomenon, since it still occurs in accordance with (4). Then, after reconstruction, it should automatically turn out that t t with a certain degree of accuracy. The simplest example where this trick does not work is a model of the form t y , is a single parameter.
Consequently, the differential equation cannot be reduced to an algebraic problem. We will meet such a situation further below. One can notice that above we chose the number of necessary particular (independent) solutions (experimental functions) according to the number of parameters to be identified. Then, a question arises, what if we use more experimental functions? The answer is simple: if and only if the mathematical model is adequate to the physical process, then using different experimental functions we will get the same results, of course, taking into account the natural errors of both the model and the experiment. This almost obvious result opens up possibilities, simultaneously with the reconstruction of the model, to establish its adequacy due to the excess of the experimental data array.
An increase in the number of equations in the mathematical model, the number of parameters to be identified, and the differential order of equations hardly affects the described approach. Only the cumbersomeness of the calculations increases. Fortunately, such a formulation of external ballistics problems, as a rule, does not require real-time data processing.

SYSTEM OF EQUATIONS FOR MPMTM IN EXPLICIT FORM
In this section we present the description of the identification process of the aerodynamic coefficients of a spin stabilized projectile. The simplest currently accepted model is the Modified Point-Mass Trajectory Model (MPMTM) [3] recommended by the STANAG 4355 NATO standard. A significant drawback of this model, from the point of view of the approach we propose, is that this model is implicit. Recently, in [4], an equivalent MPMTM modification of the system of equations was proposed in an explicit form, which is most suitable to our purposes. The main difference of this latter form of the model in comparison with the MPMTM is that functions for the transverse angles of the projectile location are accurately deduced from the system of equations.
For the sake of clarity of the presentation, we will use a slightly simplified version of this system, which does not take into account the presence of wind, i.e. u , Coriolis acceleration, etc. We follow the system of notation adopted in [3] and partially in [4], though the index was excluded from the notation in the places where it does not lead to misunderstandings.
is the three-dimensional position vector, u is the velocity of the projectile with respect to the ground-fixed axes, is the speed of the projectile or rocket with respect to air, p is the axial spin rate of Артилерійське та стрілецьке озброєння ISSN 2414-0651 (друк) The dimensionless, i.e. "hatted", coeffi cients were given as in [4]: Again, it is worth to start with a simplifi ed problem. Suffi ciently accurate experimental free fl ight data of the trajectory parameters are most often obtained only at the initial part of the trajectory, where the change in the projectile rotation speed is small and can be neglected. For where c t is the twist of rifl ing at muzzle. To proceed we have to transform the system of equations (7)-(9) into a form suitable for identifi cation of the aerodynamic coeffi cients according to the scheme described above in Section 2.
Importantly, we do not introduce special notations for the experimental parameters, since this difference follows from the context. The Cartesian coordinates of the center of mass of the projectile are converted into spherical coordinates by the conventional transformations: and we will assume that such a transformation has already been done. It should be noted, that there is no principal problem to perform the reconstruction procedure directly from Cartesian coordinates, but this does not bring anything new, while results in more cumbersome expressions.
Note that equation (8) , after their approximation, one can integrate the second of equations (5a) and obtain   t y and, then, to restore  .
Equations (6) and (7) allow one to identify L Ĉ and f mag Ĉ  by an algebraic method from one set of experimental is the data time interval.
Since expressions (10) and (11) contain the derivatives of the measured parameters, then, as we mentioned above, for such problems, preliminary approximation of the initial discrete experimental data by certain smooth functions is required over the entire interval   1 0 t , t . By the substitution of the above mentioned smooth functions into expressions (6) and (7), one obtains the dependence of the corresponding aerodynamic coeffi cient on time. However, also one needs their dependence on the speed  . The easiest way to do this for the discrete time is to relate i t in the tabular form with the corresponding value of the speed   i t  . In addition to this, it is convenient to use the approximation of the obtained aerodynamic coeffi cient by a single smooth function over the entire range of ʋ values, as we proposed in [17]. The remarks given here are valid for all cases considered below in this paper.
Equation (8), an analogue of equation (3), contains two parameters to be recovered. Note that all aerodynamic coeffi cients depend only on the velocity modulus      t (or, more precisely, on the Mach number, but we will not complicate the expressions, assuming that such a transformation has been already done) while being independent of the quadrant elevation, etc. In other words, for any initial conditions of projectile, aerodynamic coeffi cients depend on the speed (Mach number) in the same way. Similarly to case (  are independent of i , which in turn means, that they are the same in all equations. In the latter expressions, we use the "hat" symbol over the speed variable only to emphasize that this parameter is related to the synchronization of the j t variables when using several particular solutions.
It should be emphasized that the derivatives of the velocities i   depend on i .
The solution for system (9) is Therefore, we have demonstrated in a simplifi ed manner how, based on the experimental trajectory data of the projectile fl ight and the system of equations (6), (7), (10), (11), one can identify the aerodynamic coeffi cients by algebraic methods. Importantly, the solutions are unique.
Note that if the function   t p can be recorded experimentally, then the above expressions and (5b) are suitable for identifi cation all aerodynamic coeffi cients without the assumption that spin C is small.

SOLUTION OF THE INVERSE PROBLEM OF PROJECTILE AERODYNAMIC COEFFICIENTS IDENTIFICATION BASED ON MPMTM
Now let the speed of rotation of the projectile be variable in time, implying   t p p  in equations (5c-e). Experimental measurement of the speed of rotation of the projectile with a required accuracy is of considerable diffi culty and, thus, to avoid them, we use the method of excluding "unmeasurable" parameters.
To proceed we solve algebraically the system of two equations (5d) and (5e) for p and f mag Ĉ  . The solution is unique and is of the form By substitution of (12)-(13) into (5c), we obtain the equation   (17) We emphasize that the system of equations (12)- (14) and (17) is equivalent to the system (5b)-(5e). We only excluded the function p from the equations for the identifi cation of the aerodynamic coeffi cients. Note that in fact, which is a combination of two parameters in (13) is a single parameter, the same concerns the ratios (14) and (17). Note also that equation (13) depends solely on one actual parameter, while (14) and (17) depend on two parameters.
To solve equations (14) and (17) (19), thereby obtaining all the aerodynamic coeffi cients considered here. Note that solutions to the inverse problem based on experimental trajectory data of the projectile fl ight are unique in this section as well.

CONCLUDING REMARKS
In this work, exact explicit algebraic expressions for the dependences of all aerodynamic coeffi cients MPMTM on the experimentally measured parameters are obtained.
Despite the fact that the equations obtained in this paper for the identifi cation of the aerodynamic coeffi cients are equivalent forms of the original equations, the results of the identifi cation from the experimental data of the projectile fl ight, even in an ideal case, cannot be exact. First, any mathematical model is an approximation to the real process. Second, the MPMTM used in this paper is already simplifi ed. It is known, for example, that the projectile fl ight range depends on the initial projectile nutation angles, but this condition is completely ignored in MPMTM. In the course of a real fl ight and, accordingly, in the experimental data, this dependence is retained, which will affect the accuracy of the identifi cation aerodynamic coeffi cients.