Performance comparison of guided mortar projectiles with fixed and moving fins Sabit ve hareketli kanatlar ile güdümlü havan mermilerinin performans karşılaştırması

Guidance munition has become one of the popular subjects in both the theoretical and applicable studies since they could find a wide field of use in recent years because of their high performance and lower collateral damage capabilities as per the improving defence concept. The use of smaller and lighter guided munition makes the stated advantages increase without relinquishing the effectiveness. In this study, the design of a guidance kit which makes the mortar projectiles become guided when released from aerial platforms and the relevant computer simulations performed upon a selected projectile model are investigated. Here, two different configurations are considered based on the rotational degree of freedom of a pair of fins mounted on a rotary ring. In the simulations in which it is assumed that the guided projectile is released from an unmanned aerial vehicle, the different values of the fin deflection, autopilot switching duration, and side wind are considered for both of the mentioned geometries. Finally, the final miss distance and time of flight values obtained for all the designated cases are compared


INTRODUCTION
In accordance with the improvements in the technology, the mass demolition approach has been replaced by the point destruction. This way, it has become possible to attain the cost effectiveness and minimum colateral damage requirements apart from the high performance demand. In this context, guided munitions have been developed by regarding guidance and control algorithms designed as per certain technical specifications [1]. Actually, the guidance and control problem constitutes a wide area for the control of aerial platforms as well as munitions [2,3,4]. Depending on whether the design of the explosive part is included in the entire system, the guided munitions can be divided into two main groups. In this sense, missiles constitute the first class since their explosive parts are designed within the whole munition development process. The second group involves the smart bombs built by mounting specifically designed guidance kits upon existing general purpose bombs. Actually, guided projectiles are involved in the latter class because they are the composition of the guidance kits with unguided projectiles [1].
The guided projectiles are usually lighter in mass and smaller in size than the missiles and smart bombs. This provides the users with releasing more munitions towards designated target points. Looking at the available guided projectiles around the world, it is seen that they are intended to be fired from launchers deployed on ground against prescribed stationary surface targets. In those munitions, one of the control strategies below is considered in order to reach the desired guidance and control effectiveness [1,5]:  Reaction jet control,  Control with high frequency piezoelectric actuators,  Use of internal components,  Control with reverse rotation,  Use of nose actuation kits. In the selection of the most convenient control approach, the first attempt is upon the establishing a convenient mathematical model for the projectile under consideration [6][7][8]. In this extent, one of the most significant considerations is the endurance of the relevant munition against the high acceleration loads occurring in firing through the launcher [1,[7][8][9][10]. Actually, this issue is more critical for the surface-to-surface guided projectiles than the air-to-surface configurations due to the effect of the booster resulting high amount of linear acceleration [11,12]. In the sense of control of the guided projectiles, several schemes are developed regarding the effectiveness of the application [13][14][15][16][17]. In this extent, certain control approaches including robust algorithms are designed for the elimination of the diverting effects of the disturbances and noise [18,19]. The roll control of guided mortars comes into the the picture a specific implementation of this problem [20]. Also, several guidance laws are proposed in accordance with the considered engagement problem in the sense of trajectory planning of the guided projectiles [21][22][23][24]. Regarding the guided projectiles, the munitions with fixed fins in canard type are also examined in addition to the rotating ones [25]. Among these works, the guidance and control schemes regarding sensor and actuator constraints are also suggested [26]. The fuzzy logicbased integrated guidance and control schemes constitutes another class of the projectile guidance [27]. One of the consequences of the inclusion of the unmanned air vehicles into the defence scheme appears as the need of lighter but effective munitions. In this extent, the guided projectiles seem to be viable candidates for air-to-surface applications. It is obvious that the success of such munitions is related to the performance characteristics of the chosen control approach [1,10]. The present examples to the air-tosurface guided projectiles can be given as in Figure 1, In this study, the engagement capabilities of two guided projectiles composed of a 120 mm mortar projectile and guidance kit mounted on the nose part of the projectile are investigated. Here, the control of the projectile is performed by means of a rotary ring upon which a pair of fixed fins is mounted at a certain configuration whereas the fins have a degree of freedom around their hinge lines in the second group. Selecting the maximum angular deflection of the fins and autopilot switching duration as the comparison criteria, the quantities consisting of the final miss distance from the designated target point and time of flight are calculated for the designated simulation cases according to different engagement scenarios. In the related computer simulations, the linear homing guidance (LHG) law is chosen as the guidance strategy and the effect of the side wind is also taken into consideration.

GUIDANCE KIT GEOMETRY AND CONTROL APPROACH
The guidance kit which is mounted upon the nose of the unguided mortar projectile is composed of a fuze, rotor outer ring having a pair of control fins, i.e. fins, actuator rotating the rotor outer ring, sensors, electronic cards, and battery as depicted in Figure 4. In this configuration, a brushless alternating current (BLAC) type electrical motor is considered.

Figure 4. Considered Guidance Kit Geometry
Using the guidance kit introduced above, the motion control of the mortar projectile is planned to be conducted. In the designed scheme, the angular positions of the fins are supplied by the rotation of the rotor outer ring. Namely, once the fin pair is deployed in a horizontal manner, the correponding control of the guided projectile is carried out in its pitch plane. In a similar way, the vertical orientation of the fins indicates the yaw plane control. As implied, the pitch and yaw controls are not performed simultaneously but they follow a sequence each section of which is called switching duration. In the present study, two different situations are dealt with in the sense of the movement of the fins. In the first case, the fins are taken to be fixed on the rotor outer ring. On the contrary, the second configuration regards the degree of freedom of the fins around their hinge lines. The performance characteristics of both of these configurations are evaluated for a number of projectiletarget engagement scenarios.

DYNAMIC MODELING OF THE GUIDED PROJECTILE
The equations of motion of the guided mortar projectile under consideration can be ensured with the following way [30]:  5) and (6) that describe the dynamic behavior of the guided projectile can be simplified to the following equations in the pitch and yaw planes [30]: The aerodynamic moment and force components Y, Z, M, and N given in equations (7) Respectively q, SM, dM represent the dynamic pressure acting on the projectile, cross-sectional area of the projectile, and projectile diameter. In the above equations, the aerodynamic parameters represented by Cz, Cy, Cn, and Cm can be formulated depending on the side-slip angle (), angle of attack (), elevator angle (e), rudder angle (r), q, and r in the next manner [30]: Here, as vM demonstrates the absolute value of vector corresponding to the linear velocity of the projectile, the stability derivatives expressed by , , , , and depending on the mach number (M) are instantaneously updated during the computer simulations [30].

MORTAR PROJECTILE GUIDANCE LAW
The guided mortar projectile is oriented to the intended target point using the LHG law which aims at keeping the projectile on a triangle, namely the collision triangle, designated by the predicted intercept point of the missile and intended target. Denoting the duration from the initial time (t0) to the end of the intercept (tF) as t, the guidance command for the flight path angle component of the missile in the yaw plane ( ) is determined in the following manner so that ( ) ≠ 0 [30]: Likewise, the pitch plane form of the guidance command (γ m c ) can be derived as given below [30]: In the expressions above x, y, and z demonstrate components of the relative position vector between the missile and target. Moreover, and vTx, vTy, and vTz stand for the components of the target velocity vector [30].

PROJECTILE CONTROL SYSTEM
The projectile control system, i.e. autopilot, designed to convert the guidance commands of the LHG law into physical motion is designed such that it operates in a separate manner in the pitch and yaw planes in a sequential manner. Hence, the closed loop transfer function between the desired and actual flight path angles in both the pitch plane (md and m) can be obtained in the following manner [30]: where the parameters n1, n2, n3, d1, d2, d3, and d4 involve the autopilot gains, projectile diameter, components of the moment of inertia terms of the projectile, components of the velocity vector, q, and aerodynamic gains. From equation (21), the characteristic polynomial happens to be in the forthcoming fashion: ( ) = 4 4 + 3 3 + 2 2 + 1 + 1 (22) Thus, the autopilot gains can be obtained by means of the well-known pole placement approach in which the next fourth-order Butterworth polynomial is utilized by regarding the damping ratio () to be 0.707: where c stands for the desired bandwidth of the autopilot. In a similar case, the transfer function can be adapted from the transfer function attained for the pitch plane by introducing n1, n2, n3, d1, d2, d3, and d4 in the yaw plane [30] :

ENGAGEMENT MODEL
The following relationships can be written for the distance corresponding to the line-of-sight between the projectile and the intended target (rT/M) and for the orientation angles to the pitch and yaw planes (p and y) [30]: Assuming that the component of rT/M in vertical sense occurs almost zero (z=0) since it is a surface target, the final miss distance from the target at the termination of the engagement (dmiss) can be determined by the use of the equation below [30]: where tF stands for the time value at the end on the engagement.

COMPUTER SIMULATIONS
In the present study, a mortar projectile released towards a stationary surface target at a specified altitude (zP0) from an unmanned aerial platform crusing at a constant low speed (vP0) is taken into considerations. As LM and xTF denote the total length of the guided projectile and the longitudinal distance to the target when the projectile is dropped from the platform, respectively, the related simulations are carried out in the computer environment by regarding the data presented in Table 1 [31].
The miss distance from the target and time of flight values of the guided projectile are given in Table 2 and Table 3 according to the angular deflection of the fin and autopilot switching quantities as well as the side wind value for all the 18 specified cases. In Table 2 and Table  3, the fixed and rotating conditions of the fins are also considered. The sample plots belonging to the engagement geometries, projectile speed, nose kit commands, and motor angle are shown in Figure 5 through Figure 13.

DISCUSSION AND CONCLUSION
The simulation data submitted in Table 2 and Table 3 indicate that the rotating find leads smaller miss distance for the shortest autopilot switching duration considered for both windless and windy cases. Once the switching duration increases, the smaller miss distance is attained with the fixed fins. An interesting point shown from the acquired data belonging to the cases with the rotating fins is that the final miss distance quantities do not change for the same switching duration when the fin angle becomes different. However, the guided projectile with the fixed fins has a growing pattern as the fin angle gets larger. Also, although the results are obtained for the initial speed of 0.8 Mach in the present study, the proposed method is applicable up to the projectile speed of 0.2 Mach for the unmanned aerial platforms.
 Comparing these two configurations in the sense of the time of flight, it can be stated that the configuration with the rotating fins yields smaller durations except cases 1 and 10.
 As expected, the results reveal that both the miss distance and time of flight become larger when the wind acts on the projectile.
 Consequently, the guided projectile geometries with rotating fins on a rotary rind result in small final miss distance and time of flight quantities in general.

DECLARATION OF ETHICAL STANDARDS
The author(s) of this article declare that the materials and methods used in this study do not require ethical committee permission and/or legal-special permission.