In order to make a systematic and accurate analysis of the problem to be solved in this paper, it is necessary to establish corresponding coordinate systems in combination with the actual situation, and put the problem in the appropriate coordinate systems for analysis and derivation. Therefore, firstly, the necessary coordinate systems are systematically defined, and then the conversion formulas of the corresponding coordinate systems are deduced according to the geometric relationship.

1) Ground coordinate system: take the exterior trajectory measurement equipment as the origin, the y-axis is the direction of the reverse plumb line, and the x-axis is perpendicular to the y-axis and points to the true north. The established right-hand system is called the ground coordinate system (also known as the station center coordinate system), which is represented by O r − X r Y r Z r .

2) Body coordinate system: take the projectile body centroid as the origin, the direction from the origin to the warhead is the positive direction of the x-axis, and the normal direction of the plane where the trajectory is located in the y-axis. The right-hand system is called the body coordinate system, which is represented by O m − X m Y m Z m .

3) Reference coordinate system: take the origin of the body coordinate system as the origin, the x, y, z-axis of reference coordinate system and the x, y, z-axis of ground coordinate system are established respectively, which is called the reference coordinate system and represented by O − U V W .

4) Transponder coordinate system: the origin and x-axis are defined the same as body coordinate system, the direction of the main response axis of the transponder antenna as the positive direction of the y-axis. The established right-hand system is called the transponder coordinate system and is represented by O b − X b Y b Z b .

According to the above definition, O r − X r Y r Z r and O − U V W are translation relations, O − U V W and O m − X m Y m Z m are rotation relations, O r − X r Y r Z r and O m − X m Y m Z m are rotation relations after translation, O m − X m Y m Z m and O b − X b Y b Z b are rotation relations. Then, the formulas of translation transformation and rotation transformation of coordinate systems are derived respectively (See Figure 1 & Figure 2).

1) Translation transformation: without losing generality, take the two-dimensional case as shown in Figure 3(a), (x, y) is the original coordinate, (u, v) is the coordinate after translation, and (a, b) is the amount of translation. According to the geometric relationship:

{ u = x − a v = y − b (1)

Similarly, for the three-dimensional case, if the original coordinate is (x, y, z), the translated coordinate is (u, v, w), and the amount of translation is (a, b, c), the following relationship can be obtained:

{ u = x − a v = y − b w = z − c (2)

Written in matrix form:

[ u v w ] = [ x y z ] − [ a b c ] (3)

2) Rotation transformation: the rotation transformation of any coordinate system can be decomposed into the following three simple rotation transformations: the rotation of the x-axis in the y-o-z plane, the rotation of the y-axis in the x-o-z plane, and the rotation of the z-axis in the x-o-y plane. Without losing generality, the expression of coordinate rotation transformation is derived below for the rotation of the x-axis in they-o-z plane. As shown in Figure 3(b), Let (u, v, w) be the coordinate of any point P before rotation, and (x, y, z) be the coordinate after rotation, α is the counterclockwise rotation angle. According to the geometric relationship:

{ x = u y = v cos ( α ) + w sin ( α ) z = − v sin ( α ) + w cos ( α ) (4)

Written in matrix form:

[ x y z ] = [ 1 0 0 0 cos α sin α 0 − sin α cos α ] [ u v w ] (5)

In this section, the mathematical model is established to obtain the transponder power at the exterior trajectory measurement equipment by using the flight trajectory information obtained from the exterior trajectory measurement equipment. The derivation of the model is generally divided into the following five steps: 1) Convert the spherical coordinate flight trajectory data obtained by the external measuring equipment into rectangular coordinate data. 2) Convert the flight trajectory data of rectangular coordinates into the body coordinate system. 3) Convert the body coordinate system data into the transponder coordinate system, and obtain the elevation and azimuth. 4) Figureup the antenna gain by introducing the antenna pattern. 5) The flight trajectory data and antenna pattern gain are brought into the radar formula to obtain the received power of the exterior trajectory measurement equipment.

1) Flight trajectory spherical coordinate to rectangular coordinate

Let the spherical coordinate of the projectile in the O r − X r Y r Z r at a certain time be ( ρ r o m , ϕ r o m , θ r o m ) , then its rectangular coordinate ( x r o m , y r o m , z r o m ) is:

{ x r o m = ρ r o m sin θ r o m cos ϕ r o m y r o m = ρ r o m sin θ r o m sin ϕ r o m z r o m = ρ r o m cos θ r o m (6)

2) Ground coordinate system to body coordinate system

Set the exterior trajectory measurement equipment in the O m − X m Y m Z m is ( x m o r , y m o r , z m o r ) , and the angles of rotation along the x, y, and z axes from the O − U V W coordinate system to the O m − X m Y m Z m coordinate system are α , β , γ . According to the previous definition, the translation of O − U V W relative to the O r − X r Y r Z r coordinate system is ( x r o m , y r o m , z r o m ) . Because the coordinate of the exterior trajectory measurement equipment in the O r − X r Y r Z r coordinate system is (0, 0, 0), the expression obtained according to the translation and rotation formula of the coordinate system is:

[ x m o r y m o r z m o r ] = [ 1 0 0 0 cos α sin α 0 − sin α cos α ] [ cos β 0 sin β 0 1 0 − sin β 0 cos β ]     × [ cos γ sin γ 0 − sin γ cos γ 0 0 0 1 ] [ 0 − x r o m 0 − y r o m 0 − z r o m ] (7)

Substitute the Formula (6):

[ x m o r y m o r z m o r ] = [ 1 0 0 0 cos α sin α 0 − sin α cos α ] [ cos β 0 sin β 0 1 0 − sin β 0 cos β ]     × [ cos γ sin γ 0 − sin γ cos γ 0 0 0 1 ] [ − ρ r o m sin θ r o m cos ϕ r o m − ρ r o m sin θ r o m sin ϕ r o m − ρ r o m cos θ r o m ] (8)

3) Body coordinate system is converted to transponder coordinate system to gain elevation and azimuth

According to the definition of the coordinate system, the relationship between O b − X b Y b Z b and O m − X m Y m Z m is a rotation transformation along the x-axis. Let the rotation angle be α r e s and the coordinate of the exterior trajectory measurement equipment in O b − X b Y b Z b be ( x b o r , y b o r , z b o r ) . According to the rotation formula of coordinate system:

[ x b o r y b o r z b o r ] = [ 1 0 0 0 cos α r e s sin α r e s 0 − sin α r e s cos α r e s ] [ x m o r y m o r z m o r ] (9)

Substitute the Formula (8):

[ x b o r y b o r z b o r ] = [ 1 0 0 0 cos α r e s sin α r e s 0 − sin α r e s cos α r e s ] [ 1 0 0 0 cos α sin α 0 − sin α cos α ]     × [ cos β 0 sin β 0 1 0 − sin β 0 cos β ] [ cos γ sin γ 0 − sin γ cos γ 0 0 0 1 ] [ − ρ r o m sin θ r o m cos ϕ r o m − ρ r o m sin θ r o m sin ϕ r o m − ρ r o m cos θ r o m ] (10)

According to the geometric relationship, the elevation angle θ b o r and azimuth angle ϕ b o r of the exterior trajectory measurement equipment in the O b − X b Y b Z b can be expressed as:

{ θ b o r = arctan ( x b o r 2 + z b o r 2 y b o r ) ϕ b o r = arctan z b o r x b o r (11)

4) Calculate the antenna gain of transponder

By substituting (11) into the expression of transponder antenna pattern, the transponder antenna gain G is:

G = B ( θ b o r , ϕ b o r ) = B ( arctan ( x b o r 2 + z b o r 2 y b o r ) , arctan z b o r x b o r ) (12)

5) Obtain the received power of exterior trajectory measurement equipment by the formula of radar

Received power of exterior trajectory measurement equipment can be obtained by substituting flight trajectory data and transponder antenna gain into the secondary radar formula:

S r i = P r e s G A e 4 π R 2 = P r e s B ( arctan ( x b o r 2 + z b o r 2 y b o r ) , arctan z b o r x b o r ) A e 4 π ( x r o m 2 + y r o m 2 + z r o m 2 ) (13)

Therein, A e is effective aperture of antenna of exterior trajectory measurement equipment, P r e s is transmitting power of the transponder, R is radial distance between transponder and exterior trajectory measurement equipment.

In the simulation process of this paper, due to the direct generation of data in rectangular coordinate system, there is no conversion process in Formula (6). Bring the flight trajectory data into Formula (7) to obtain the trajectory of the exterior trajectory measurement equipment in the O m − X m Y m Z m coordinate system, as shown in Figure 5(a).

According to common conditions, let the transponder antenna angle be α r e s = − 1.5184 ( rad ) substituted into Formula (10) to obtain the trajectory of the exterior trajectory measurement equipment in the O b − X b Y b Z b coordinate system, as shown in Figure 5(b).

Call Formula (11) to obtain the elevation and azimuth of the equipment in the O b − X b Y b Z b , as shown in the following Figure 6.

It can be seen from Figure 6 that within all flight trajectories of the target, the variation range of elevation is large. In some measuring segments, the exterior trajectory measurement equipment will leave the main lobe of the transponder antenna, resulting in very low reception or failure to receive the response power.

In addition, the exterior trajectory measurement equipment practically only observes part of the flight trajectory, so only a given measurement segment in the flight trajectory is analyzed subsequently. As shown in Figure 7, the measuring segment marked in blue is the subsequent simulation measuring segment.

Bring the measuring segment data into the above model to obtain the elevation and azimuth of the exterior trajectory measurement equipment in the O b − X b Y b Z b , as shown in the following Figure 8.

Assuming the antenna of transponder is a uniformly weighted 7*7 array antenna, its pattern is shown in Figure 9. According to Formula (12), the change of transponder antenna gain is shown in Figure 9.

Bring the antenna gain and flight trajectory information into Formula (13) to obtain the response power change curve received by the equipment in the measuring segment, as shown in Figure 10.

Convert to received power: (Figure 11).

According to the simulation results can be seen that, in the process of flying projectile target position and attitude change constantly, the direction of the transponder antenna mounted on the body changes accordingly, resulting in changes in the receiving power of the exterior trajectory measurement equipment. When the transponder antenna is in an unfavorable position relative to the exterior trajectory measurement equipment, so that the receiving power is lower than the receiving sensitivity, the measurement target will be lost. Therefore, it is necessary to comprehensively consider the flight track and the position and capability of the station on the route. Reasonable arrangement of station relay measurement, selection of adequate power transponder, and installation in an appropriate position are important guarantees for effective control of target flight information.