Research on radial velocity and airspace position calculations of target in airborne active radar simulation

The detection of the target in the airborne active radar simulation is simplified to the calculations of the radial velocity, the maximum detecting range of the airborne active radar corresponding to the RCS, and the airspace position of the target. However, the coordinate transformations related to the calculations of the radial velocity and the airspace position are usually bewildering due to the different referenced coordinate systems. This paper introduces the radial velocity and airspace position calculations of the target in the airborne active radar simulation to illuminate the transformation process containing the specific rotation matrices, suggesting the unifying transformation to the north-up-east (NUE) rectangular coordinate system of the aircraft for both the radial velocity calculation and airspace position calculation of the target.


Introduction
Active radars, like the early warning radar and the fire-control radar, are used to detect the approximate or specific position of the moving target.There are two main active radar systems, the continuous-wave (CW) radar system, and the pulsed Doppler (PD) radar system.The CW radar can be divided into the simple CW radar, the frequency-modulated CW radar [1,2], and the phase-modulated CW radar [3,4].The PD radars work on detecting objects based on measuring the Doppler shift of radar echoes through a coherent sequence of pulses and have been widely applied in applications requiring the detection of moving targets in a severely cluttered environment [5].A tactical flight simulation usually consists of the state simulation, the flying environment simulation, the task system simulation, and the pilot controlling simulation to validate the availability of the tactical simulation system or the feasibility of the operational strategy.The airborne active radar simulation is a part of the task system simulation.It is an important simulation to test the usability of the airborne active radar system before the actual design and manufacturing.The detection of the target is usually confirmed by the reflection wave of the active radar which is too difficult to implement the simulation due to the complex physical environment and signal processing [6,7].The reflection wave from the target will be detected by the airborne active radar when the radial velocity between the target and the aircraft is in the detecting range of the radial velocity of the airborne active radar [8].The relative distance between the target and the aircraft should be below the maximum detecting range of the airborne active radar corresponding to the radar cross-section (RCS) of the target [9].Furthermore, the prerequisite of the reflection wave is that the target is in the detectable airspace of the airborne active radar.Therefore, the detection of the target in the airborne active radar simulation should be simplified to the calculations of the radial velocity, the maximum detecting range of the airborne active radar corresponding to the target RCS, and the airspace position, just for supporting the tactical flight simulation system.However, the coordinate transformations related to the radial velocity and the airspace position calculations are usually bewildering due to the different referenced coordinate systems.Thus, this paper introduces the radial velocity and airspace position calculations of the target in the airborne active radar simulation to illuminate the transformation process containing the specific rotation matrices, suggesting the unifying transformation to the north-up-east (NUE) rectangular coordinate system of the aircraft for both the radial velocity calculation and the airspace position calculation of the target.

Calculation data flow
The calculation data flow of our airborne active radar simulation is shown in Figure 1.The inputting parameters include the aircraft information, the target information list, and the antenna information.The aircraft information contains the aircraft's position, attitude, and north-up-east velocity.The target information list inputs each target's serial number, position, attitude, and speed scalar in every processing period of the calculation.The serial number is only used for marking the different targets with no contribution to any calculation.The aircraft's position contains its longitude, latitude, and altitude under the geographic coordinate system.The NUE velocity is the vector composed of the components of its speed scalar in the three directions of the north-up-east system.The implication of the target's position is the same as the aircraft.The speed scalar refers to the moving direction of the target with no consideration of the angle of attack.If the head direction of the target is defined as the direction X then the component of the velocity vector in the direction X is equal to the speed scalar value, meaning the components in other directions are zeroes.The radial velocity between the aircraft and the target can be calculated by the above inputting information.The airspace position is closely related to the orientation of the airborne radar since the airborne active radar is not all omnidirectional radar.The target will reflect the active radar wave only under the current scanning airspace of the antenna of the airborne active radar.The aircraft's position and attitude, the target's position, the antenna's attitude including the azimuth center and the pitching center, and its corresponding scanning range including the azimuth and the pitching range, are necessary for the airspace position calculation.The antenna radiation status notes the working mode of the left or the right radar of the aircraft, it does not participate in the airspace position calculation.The calculation of the maximum detecting range of the airborne active radar corresponding to the target RCS is omitted in the data flow diagram because there is no coordinate transformation involved.The final calculations of each target would be confirmed for target detection.The detectable target would be outputted to the target traces list for the subsequent process of the data flow.(1) In equation set (1), , , and  represent the longitude, latitude, and altitude of the transformed object respectively.The calculated coordinate (, , ) is the coordinate transformed from the geographic coordinate system to the WGS-84 system.

The transformation from the WGS-84 system to the NUE system
The coordinate axis of the WGS-84 system can't be transformed to the NUE system of the aircraft by rotations of the longitude and latitude merely, as shown in Figure 2.There should be an axis aligning operation before the rotations of the longitude and latitude.
Matrix  is the coordinate axis aligning matrix from the WGS-84 system to the NUE system.
Matrix  is the longitude rotation matrix from the WGS-84 system to the NUE system where the  is the longitude.
Matrix  is the latitude rotation matrix from the WGS-84 system to the NUE system where the  is the latitude.
(  ,   ,   )  = (, , ) The calculated coordinate (  ,   ,   ) is the final coordinate transformed from the WGS-84 system to the NUE system.The translation from the earth's core to the aircraft centroid is not considered yet, because this translation would be offset during the subsequent calculation of the relative position and velocity between the target and the aircraft.

The transformation from the NUE system to the body axis system
Matrix   is the rotation matrix around the axis X.
Matrix   is the rotation matrix around the axis Y.
Matrix   is the rotation matrix around the axis Z.
The relationships among the value of , , , and the angle of azimuth, pitch, roll depend on the definition of the body axis system.The general definition shown in Figure 3 is that the axis OX is located in the aircraft reference plane, parallel to the fuselage axis and pointing to the front of the aircraft, and the axis OY is perpendicular to the aircraft reference plane and pointing to the right side of the aircraft, then the axis OZ is perpendicular to the XOY plane in the reference plane, pointing below the aircraft.Considering the transforming order of yawing, pitching and rolling, the value of , φ, ξ would correspond to the angle of roll, pitch, and azimuth, respectively.
The axis of symmetry of the aircraft It should be noted that whether the NUE system can be transformed into the body axis system merely through the three rotation matrices depends on the definition of the body axis system.If the body axis system is defined as in Figure 3, there should be the additional aligned transformation matrix   from the NUE system to the body axis system of the aircraft as equation (9).
Nevertheless, if the definition of the body axis system is correctly changed (meet the right-hand rule) by exchanging the axis OZ with the axis OY as Figure 4.The   should be the identity matrix as equation (10).
The axis of symmetry of the aircraft The coordinate under the NUE system should be transformed by the additional transformation matrix   firstly before the subsequent axis rotations.

Radial velocity and airspace position calculation
The body axis systems of the target, the aircraft, and the antenna in our airborne active radar simulation are both defined as Figure 3.

Radial velocity calculation
Firstly, the speed scalar of the target should be transformed to the velocity vector under the body axis system of the target.If the target's speed scalar is  then the velocity vector  under the body axis system of the target could be equation ( 12). = (, 0, 0)  (12) The NUE velocity vector  ′ transformed from the body axis system to the NUE system of the target can be calculated by equation ( 13).The angle of , ,  in corresponding   ,   ,   are the attitude angle of roll, pitch, azimuth of the target, respectively.
The transformation matrix   from the WGS-84 system to the NUE system of the target is equation ( 14).λ  and   represent the latitude and the longitude of the target respectively.
The The transformation matrix   from the WGS-84 system to the NUE system of the aircraft is equation (15).λ  and   represent the latitude and the longitude of the aircraft respectively.
Then the NUE velocity vector  ′′ of the target under the NUE system of the aircraft should be calculated by equation ( 16).
The target's position coordinate (  ,   ,   ) and the aircraft's position coordinate (  ,   ,   ) under the WGS-84 system can be calculated by the equation set (1) respectively.Then the relative position vector of the target under the WGS-84 system of the aircraft  can be obtained by equation ( 17).

𝒑 = (𝑥
The relative position vector of the target under the NUE system of the aircraft   can be calculated by equation (18).
If the NUE velocity vector of the aircraft  = (  ,   ,   )  , then the velocity vector difference   can be calculated by equation (19).
Finally, the radial velocity scalar   can be calculated by equation (20) where the  is the intersection angle between   and   which is also called the aspect angle.Howbeit, it's not necessary to calculate .
The target is approaching the aircraft if the value of   is negative whereas it's leaving the aircraft if the value of   is positive.The target will be defined as radial velocity detectable if the calculated   is in the detectable range of the radial velocity in our simulation system.

Airspace position calculation
The angle of the aircraft's attitude should be defined before calculating the airspace position of the target.In the vertical view, the clockwise angle of the azimuth is defined as positive whereas the counterclockwise angle is defined as negative with the range of -π to π.In the right-side view, the counterclockwise angle of the pitch is defined as positive whereas the clockwise angle is defined as negative with the range of -π/2 to π/2.In the head front view, the counterclockwise angle of the roll is defined as positive whereas the clockwise angle is defined as negative with the range of -π to π.The definition of the aircraft attitude angle is presented in Figure 5.
The target coordinate under the body axis system of the antenna   can be calculated by equation set (22) where the   ,   correspond to the angle of pitch and azimuth of the antenna respectively with no necessity to consider the rolling angle and the tilt angle during the installation of the antenna.The target will be defined as airspace detectable if the calculated  and  are in the detectable range of the azimuth and pitch of the antenna respectively in our simulation system.

Conclusion
The detection of the target in the airborne active radar simulation is simplified to the calculations of the radial velocity, the maximum detecting range of the airborne active radar corresponding to the RCS, and the airspace position of the target in this paper.The coordinate transformation related to the calculations of the radial velocity and the airspace position of the target is proposed.It suggests that the transformation should be unified to coordinate under the NUE system of the aircraft for both the radial velocity calculation and the airspace position calculation of the target.There is an additional aligned transformation matrix from the NUE system to the body axis system of the aircraft during the airspace position calculation, and the specific additional matrix depends on the definition of the body axis system of the aircraft.
The proposed coordinate transformation method can be used not only in the active radar simulation but also in the actual system containing the real-time calculation of the relative position of the target, for instance, the flight control in missile guidance, the target tracking in the Track-While-Scan (TWS) radar [10], etc.

Figure 1 .
Figure 1.Calculation data flow diagram.3.Coordinate transformation3.1.The transformation from the geographic coordinate system to the WGS-84 system

Figure 2 .
Figure 2. The transformation from the WGS-84 system to the NUE system.

Figure 3 .
Figure 3.The general definition of the body axis system.It should be noted that whether the NUE system can be transformed into the body axis system merely through the three rotation matrices depends on the definition of the body axis system.If the body axis system is defined as in Figure3, there should be the additional aligned transformation matrix   from the NUE system to the body axis system of the aircraft as equation (9).

Figure 4 .
Figure 4. Another definition of the body axis system.

Figure 5 .
Figure 5.The definition of the attitude angle from the three views of the aircraft.The target coordinate under the body axis system of the aircraft   can be calculated by equation set (21) where the   ,   ,   correspond to the angle of roll, pitch, and azimuth of the aircraft, respectively.
Then the angle of azimuth and pitch of the target under the body axis system of the antenna should be calculated after obtaining the target coordinate   (  ,   ,   ).The azimuth angle  and the pitch angle  would be calculated by equation set (23) and equation set (24) respectively according to the definition of the aircraft attitude angle.