The Mathematical Framework for Simulating an Air-To-Air Missile Operation on Fighter Aircraft

2.1 Coordinate systems

The inertial coordinate (Fⁱ) is a Cartesian earth-fixed frame whose origin is defined at a given location. The coordinate is sometimes referred to as the north-east-down (NED) reference frame. The vehicle coordinate (F^v) is a frame whose origins is defined at the center of mass of the aircraft. Its axes will be aligned with corresponding axes in the inertial frame as in Figure 1. The body coordinate is a frame that sticks to the aircraft as in Figure 2. Therefore, it will rotate following the jet motion. In general, the relation between the inertial and vehicle coordinate is translation. The body frame, on the other hand, is obtained from the vehicle frame by rotation. The rotation matrix that is derived from rotating the vehicle frame to achieve the body frame is given by Beard and McLain [9].

$R_v^b=\left(\begin{array}{ccc}c_\theta c_\psi & c_\theta s_\psi & -s_\theta \\ s_\phi s_\theta c_\psi-c_\theta s_\psi & s_\phi s_\theta s_\psi+c_\phi c_\psi & s_\phi c_\theta \\ c_\phi s_\theta c_\psi+s_\phi s_\psi & c_\phi s_\theta s_\psi-s_\phi c_\psi & c_\phi c_\theta\end{array}\right)$      (1)

where, c_ϕ = cos(ϕ) and s_ϕ = sin(ϕ). ϕ, θ, and ψ are Euler angles, which are also referred as roll, pitch, and yaw respectively as in Figure 3. The rotation matrix from the body frame to the vehicle frame is given by

$R_b^v=\left(R_v^b\right)^T=\left(\begin{array}{ccc}c_\theta c_\psi & s_\phi s_\theta c_\psi-c_\theta s_\psi & c_\phi s_\theta c_\psi+s_\phi s_\psi \\ c_\theta s_\psi & s_\phi s_\theta s_\psi+c_\phi c_\psi & c_\phi s_\theta s_\psi-s_\phi c_\psi \\ -s_\theta & s_\phi c_\theta & c_\phi c_\theta\end{array}\right)$      (2)

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Figure 1. The definition of initial and vehicle frame [9]

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Figure 2. The definition of the body frame [9]

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Figure 3. The definition of roll ($\phi$), pitch ($\theta$), and yaw ($\psi$) angles and how to obtain the body frame from the vehicle frame [9]

The translation matrix from the vehicle frame to the inertial frame is obtained as follows:

$T_v^i=\left(\begin{array}{cccc}1 & 0 & 0 & x_v^i \\ 0 & 1 & 0 & y_v^i \\ 0 & 0 & 1 & z_v^i \\ 0 & 0 & 0 & 1\end{array}\right)$      (3)

where, $x_v, y_v$, and $z_v$ are the location of vehicle’s origin in the inertial frame. The position of an object converted from the body frame to the inertial frame is obtained by

$\left(\begin{array}{c}x^i \\ y^i \\ z^i \\ 1\end{array}\right)=M_b^i\left(\begin{array}{c}x^b \\ y^b \\ z^b \\ 1\end{array}\right)$      (4)

where, $x^i, y^i$, and $z^i$ are the location of a target in the inertial frame while $x^b, y^b$, and $z^b$ and  are its value in the body frame. The complete transform matrix $M_b^i$ are computed by

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2.2 Target simulation

Target simulation is an important task of the proposed framework. The main goal of this task is to re-create the action of the target following the demand of training. In general, target simulation which is presented in Figure 4, focus on three main aspects: (1) target type, (2) target behaviours [10, 11], and (3) flight dynamics referring to the study [12]. The target type defines the characteristic of target, the dynamical features and the corresponding set of actions. Target behaviours are the set of predefined actions for airborne target, which is classified by the level of task difficulty. In a given scenario, and at the same level of task difficulty, the behaviour of the target is defined by the conditional selection sets of action. In this work, 30 actions and 05 levels are implemented to simulate the common behaviours of the target. For example, steady moving is the simplest action at the lowest level. Kulbit [13] is the most complicated action at the highest level. Target flight dynamics solves a set of derivative equations to calculate the trajectory of target.

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Figure 4. Target simulation diagram

2.3 Avionic systems

2.3.1 Phased array radar

In the fighter aircraft, the phased bar array radar is used to detect targets. Then, targets will be depicted on the HUD to indicate that they are found by the radar as shown in Figure 5. In this study, we simplify the principles of phased bar array radar such that we can reduce the complexity of mathematical modelling, and the computational time.  Figure 6 represents the radar beam. Flying objects are detected if their position is inside, and their relative distance to the radar is smaller than the maximum detection range. In our work, the opening azimuth (α) and elevation angle (β) are 60º and 30º respectively. Several 2D coordinate systems are used to compute the position of targets on the HUD. The first coordinate system is azimuth-radial distance, where the width of HUD is aligned with the azimuth axis varying from -30º to 30º, and the height is aligned with the radial distance axis ranging from 0 to 100km. This coordinate is usually applied for targets moving in the opposite direction to the jet motion. The second coordinate system is azimuth-velocity, in which the height of HUD represents the relative velocity between the jet and the target. It will be employed when targets and jet fly in the same direction. Because it is more difficult for electromagnetic radar to spot objects when the relative velocity between the aircraft and targets are small, in this paper, we will focus on the azimuth-radial distance. Giving the position of the target ($x_t^i, y_t^i, z_t^i$) and the jet ($x_j^i, y_j^i, z_j^i$) in the inertial frame, the location of target ($x_t^b, y_t^b, z_t^b$) in the body frame is computed as follows:

$\left(\begin{array}{c}x_t^b \\ y_t^b \\ z_t^b \\ 1\end{array}\right)=M_i^b\left(\begin{array}{c}x_t^i \\ y_t^i \\ z_t^i \\ 1\end{array}\right)$       (6)

where, the transform matrix ($M_b^i$) is computed by

$M_i^b=\left(M_b^i\right)^{-1}$       (7)

with  is determined from Eq. 5. The azimuthal angle of the target in the body frame is estimated by

$\theta_t^b=\tan ^{-1} \frac{y_t^b}{x_t^b}$        (8)

In order to search in other region in air space, the radar can change the direction as shown in Figure 7. The radar beam can move in both elevation and azimuth plane following the elevation and azimuth coverage bar as in Figure 5. In this study, the azimuth varies from -30º to 30º while the elevation angle range is from -7.5º to 15.0º. Assume that we switch the radar heading direction to -30º on the left as in Figure 5, the new azimuthal angle of target is computed by

$\theta_t^{b, n e w}=\tan ^{-1} \frac{y_t^b}{x_t^b}+30^{\circ}$       （9）

As a result, the target will appear on the right side of HUD if $\theta_t^{b,new}$  is still inside the radar beam.

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Figure 5. The HUD displays the seeking mode with phased array radar, which can operate in several scan modes. If the anonymous flying objects are inside the detected region that is bounded by elevation, azimuth coverage, and expected target range they will be spotted on HUD screen. The target is selected by moving the radar cursor over identified flying object, and then pressing a lock button [14]

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Figure 6. The radar beam in which targets can be tracked down. α is opening azimuth, β is opening elevation angle

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(a) Change the radar beam heading in the elevation plane

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(b) Change the radar beam heading in the azimuth plane

Figure 7. Moving radar direction to change the detection region of radar

2.3.2 Infrared search and track system

Infrared search and track system (IRTS) is a technique for detecting and tracking anonymous objects, which generate infrared radiation as shown in Figures 8 and 9. The IRTS is usually employed when the target transport in the same direction as the jet, because the target’s engine is the main thermal source pointing directly to the jet front. The IRTS will operate in the seeking mode first to depict anonymous flying objects on HUD as represented in Figure 8. Figure 9 shows the identification mode after the seeking mode determine the region where the target is possibly located. The IRTS cannot provide information about the distance between an object and the jet. Therefore, in this work, object positions are computed in the elevation angle-azimuth coordinates to indicate the direction from the jet to an infrared radiation source. The elevation angle can be computed by

$\phi=\tan ^{-1} \frac{-y_t^b}{\sqrt{\left(x_t^b\right)^2+\left(y_t^b\right)^2+\left(z_t^b\right)^2}}$      (10)

where, ($x_t^b, y_t^b, z_t^b$) is location of target in the body frame, which is computed from Eq. 6. The HUD height will coincide with the elevation angle varying from -30º to 30º, while the HUD width will align with azimuth ranging from -30º to 30º. The size of large view cursor is 20º width x 30º height. After moving the cursor to the region containing the suspected target, a button is released to change HUD screen to small view mode such that targets can be distinguished from the other objects. In this mode, the dimension of HUD will coincide with the size of large view cursor depicted in Figure 8. The size of small view cursor in Figure 9 is 7º width x 15º height. In this mode, the target will be locked, then, its location will be used for guiding missile.

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Figure 8. The HUD displays the large view mode with infrared search and track system, where flying objects are detected due to their thermal radiation. They are spotted on HUD screen by target marks. The IRTS large view cursor will move around HUD looking for potential targets. After the suspected target is determined, the large view mode will change to small view mode [14]

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Figure 9. The HUD displays the small view mode with infrared search and track system. In this mode, the target is determined by moving the IRTS mall view cursor over identified flying object, and then pressing a lock button [14]

2.4 Missile dynamic

After the target is locked-on, the pilot will decide to launch the missile when the threat inside the missile range. There are two modes in the missile operation. At first, the missile will slide on the launching rail until it detaches entirely from the jet. Then, the missile will change its direction such that it will follow and destroy the target. The equation describing the motion of missile is given as follows:

$m \boldsymbol{a}_m=\boldsymbol{F}_{\text {thrust }}-\boldsymbol{F}_D$      (11)

where, $\boldsymbol{F}_{\text {thrust }}$ is thrust force produced by missile engine, $\boldsymbol{F}_D$ is the drag force, which can computed by

$F_D=C_D \frac{\rho_{a i r} \,\, v^2}{2} A$      (12)

where, ρ_air is air density, $C_D$ is the drag coefficient, v is the missile velocity vector, and A is the missile reference area, which is evaluated by

$A=\pi \frac{D^2}{4}$        (13)

where, D is the missile diameter. In this paper, we ignore the lift force and gravity.  We only focus on the thrust and drag forces, which are the main forces contributing to the missile acceleration. The missile operation consists of two phases. The first phase is a sliding mode, in which the missile slide on the trailing rail. In this case, the missile steers the same direction as the jet motion. After the missile complete sliding on the trail, the second phase takes place. In this phase, the missile head directly to the target. The position of the missile in the inertial frame ($x_m^i, y_m^i, z_m^i$) can be computed from its location in the body frame ($x_m^b, y_m^b, z_m^b$) by

$\left(\begin{array}{c}x_m^i \\ y_m^i \\ z_m^i \\ 1\end{array}\right)=M_b^i\left(\begin{array}{c}x_m^b \\ y_m^b \\ z_m^b \\ 1\end{array}\right)$      (14)

The velocity of missile in the body frame ($u_m^b, v_m^b, w_m^b$) is determined in the inertial frame ($u_m^i, v_m^i, w_m^i$) as follows:

$\left(\begin{array}{c}u_m^i \\ v_m^i \\ w_m^i \\ 0\end{array}\right)=M_b^i\left(\begin{array}{c}u_m^b \\ v_m^b \\ w_m^b \\ 0\end{array}\right)$      （15）

2.4.1 Sliding mode

In this mode, the missile will slide on the rail. Before the missile is launched, its initial velocity is identical to the jet velocity. The missile velocity in the roll axis $u_m^b$ is updated by

$u_m^{b, n+1}=u_m^{b, n}+a_m^{b, n} \Delta t$      (16)

where, $n+1$ and $n$ represent the time step, $a_m^{b, n}$ is the missile acceleration, and $\Delta t$ is a time step size. In this study, the time step size 0.01 is used. The relative distance $L$ between the missile and the jet in the roll axis is computed by

$L^{n+1}=L^n+\Delta L^{n+1}$     (17)

where, $\Delta L$ is a distance the missile moved in $\Delta t$. The variable is computed by

$\Delta L^{n+1}=\left(u_m^{b, n}-u_j^{b, n}\right) \Delta t+\frac{1}{2} a_m^{b, n} \Delta t^2$      (18)

The position of missile is updated in the body frame by

$\begin{gathered}x_m^{b, n+1}=x_j^{b, n+1}+L^{n+1} \\ y_m^{b, n+1}=y_j^{b, n+1} \\ z_m^{b, n+1}=z_j^{b, n+1}\end{gathered}$      (19)

Inserting Eq. 19 into Eq. 14, the position of the missile in the inertial frame can be computed accordingly. This mode is terminated when the relative distance L exceeds the rail length. Then, the missile will switch to a target heading mode. 

2.4.2 Target heading mode

In this mode, the target position will be used to modify the shift distance, which the missile transport in a time step size $\Delta t$. This mode will mimic the principle of radar guidance. The equation to compute the missile position is given by

$\frac{d^2 x}{d t^2}=a_m$      (20)

In this stage, the missile will head toward the target and change direction following target motion.

$\begin{aligned} & \Delta x_m^{n+1}=u_m^n \Delta t+\frac{1}{2} a_{m, x}^n \Delta t^2 \\ & \Delta y_m^{n+1}=v_m^n \Delta t+\frac{1}{2} a_{m, y}^n \Delta t^2 \\ & \Delta z_m^{n+1}=w_m^n \Delta t+\frac{1}{2} a_{m, z}^n \Delta t^2\end{aligned}$      (21)

The Rodriquez rotation [15] is employed to re-compute the distance that the missile flew in $\Delta t$. It will rotate to the direction from the previous missile position to the most updated target location as in Figure 10. Set $k_{m, t}$ is the vector connecting the target and the missile, $\Delta s^{n+1}\left(\Delta x_m^{n+1}, \Delta y_m^{n+1}, \Delta z_m^{n+1}\right)$ is shifting vector. The new distance is computed based on Rodriquez rotation as follows:

$\begin{gathered}\Delta s_{\text {rot }}^{n+1}=\Delta s^{n+1} \cos \alpha+\left(\boldsymbol{k} \times \Delta s^{n+1}\right) \sin \alpha \\ +\boldsymbol{k}\left(\boldsymbol{k} \cdot \Delta s^{n+1}\right)(1-\cos \alpha)\end{gathered}$      (22)

where, $\boldsymbol{k}$ is determined by

$\boldsymbol{k}=\Delta s \times v_{m j}$      (23)

where, $v_{m j}$  is the vector connecting the missile and jet position at the time step n and n + 1 respectively. Then, the new position of missile can be updated by

$\begin{aligned} & x_m^{n+1}=x_m^n+\Delta x_{\text {rot }}^{n+1} \\ & y_m^{n+1}=y_m^n+\Delta y_{\text {rot }}^{n+1} \\ & z_m^{n+1}=z_m^n+\Delta z_{\text {rot }}^{n+1}\end{aligned}$      (24)

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Figure 10. Redirect the missile such that it keeps heading to the target

In this paper, we have already presented mathematical methods to simulate air-to-air missile operation on fighter aircraft. Methods are flexible and robust, therefore, they can be applied easily on any simulation system. The computed results show that avionic systems can depict the position of anonymous flying objects in both altitude-azimuth and elevation angle-azimuth coordinates. In addition, the missile dynamics equation can predict the missile path, which leads to the target accurately.