**5.1 Verification of separation model of initial errors**

The telemetry and tracking data are obtained using the six-degree-of-freedom ballistic program. For the certain trajectory with 10000 kilometers of range, the initial errors are listed in Table 1.


Table 1. The true values of initial errors.

During the simulation process, all the guidance instrumentation systematic errors are set to zero therefore, the difference between telemetry data and tracking data merely contain initial errors. Herein, define **Y GP** = ⋅ *<sup>a</sup>*<sup>0</sup> , namely, **Y** is the difference between telemetry data and tracking data, which is calculated using the product of environmental function matrix of initial errors **G** and true values of initial errors **P***<sup>a</sup>*<sup>0</sup> . Define **δX***P* is the difference between telemetry data and tracking data obtained by the simulation data. Now, define **δY** = − **δX Y** *<sup>P</sup>* is the residual of the difference between telemetry data and tracking data. Simulation results are shown in the following figures, Fig.4 shows the difference between telemetry velocity and tracking velocity, *Pv δX* ; Fig.5 shows the difference between telemetry position and tracking position, **δX***Ps* ; Fig.6 shows the residual of the difference between telemetry velocity and tracking velocity, **δY***<sup>v</sup>* ; and Fig.7 shows the residual of the difference between telemetry position and tracking position, **δY***<sup>s</sup>* .

Fig. 4. The difference between telemetry and tracking velocity.

In the previous section, the separation model of initial errors based on telemetry and tracking data and the separation model of instrumentation errors and initial errors are deduced in detail. In this section, numerical examples are given to verify the separation

The telemetry and tracking data are obtained using the six-degree-of-freedom ballistic program. For the certain trajectory with 10000 kilometers of range, the initial errors are listed in Table 1.

Parameter Error Value Parameter Error Value Parameter Error Value

During the simulation process, all the guidance instrumentation systematic errors are set to zero therefore, the difference between telemetry data and tracking data merely contain initial errors. Herein, define **Y GP** = ⋅ *<sup>a</sup>*<sup>0</sup> , namely, **Y** is the difference between telemetry data and tracking data, which is calculated using the product of environmental function matrix of initial errors **G** and true values of initial errors **P***<sup>a</sup>*<sup>0</sup> . Define **δX***P* is the difference between telemetry data and tracking data obtained by the simulation data. Now, define **δY** = − **δX Y** *<sup>P</sup>* is the residual of the difference between telemetry data and tracking data. Simulation results are shown in the following figures, Fig.4 shows the difference between telemetry velocity and tracking velocity, *Pv δX* ; Fig.5 shows the difference between telemetry position and tracking position, **δX***Ps* ; Fig.6 shows the residual of the difference between telemetry velocity and tracking velocity, **δY***<sup>v</sup>* ; and Fig.7 shows the residual of the difference between telemetry




Initial Velocity *Vx*

Initial Velocity *Vy*

Initial Velocity *Vz* -0.1m/s


0.1m/s

λ0

model of initial errors and instrumentation errors and initial errors.

Geodetic Longitude

Geodetic Latitude *B*<sup>0</sup>

Geodetic Height *H*<sup>0</sup>

**5.1 Verification of separation model of initial errors** 

30 arcsec

30 arcsec

120 arcsec

Table 1. The true values of initial errors.

position and tracking position, **δY***<sup>s</sup>* .

Fig. 4. The difference between telemetry and tracking velocity.

**5. Simulated cases** 

Astronomical Longitude

Astronomical Latitude *BT*

Astronomical Azimuth *AT*

λ*T*

Fig. 5. The difference between telemetry and tracking position.

Fig. 6. The residual of the difference between telemetry and tracking velocity.

Fig. 7. The residual of the difference between telemetry and tracking position.

It is clearly seen from Figs. 4 and 6 that the differences between telemetry velocity and tracking velocity obtained by the two methods agree well. When the third stage engine shut down, the difference between telemetry velocity and tracking velocity is ( ) 0.52, 0.53, 4.1 m/s − − , while

Error Separation Techniques Based on Telemetry and Tracking Data for Ballistic Missile 467

are seen in Table 1. The model of guidance instrumentation systematic errors are given by

Similarly, environmental function matrix of instrumentation error, **S** , and environmental function matrix of initial errors, **G** , are obtained to compute the difference between telemetry data and tracking data, **Y** , which is defined as **Y SD GP** =⋅ − ⋅ 0 0*<sup>a</sup>* . Simultaneously, **δX** is the difference between telemetry and tracking data obtained by the simulation data. Likewise, define **δY** = − **δX Y** is the residual. Simulation results are shown in the following figures, Fig.8 shows the difference between telemetry velocity and tracking velocity, **δX***<sup>v</sup>* ; Fig.9 shows the difference between telemetry position and tracking position, **δX***<sup>s</sup>* ; Fig.10 shows the residual of the difference between telemetry velocity and tracking velocity, **δY***<sup>v</sup>* , and Fig.11 shows the residual of the difference between telemetry position

Eqs.(22) and (23) and the levelling and alignment errors are not included.

Fig. 8. The difference between telemetry and tracking velocity.

Fig. 9. The difference between telemetry and tracking position.

and tracking position, **δY***<sup>s</sup>* .

the largest residual computed by the two methods is ( ) −− − 0.0015, 0.0013, 0.0002 m/s , which is quite smaller than the difference between telemetry velocity and tracking velocity. Similarly, as seen in Figs. 5 and 7, when the third stage engine shut down, the difference between telemetry and tracking position is ( ) − − 400, 25, 672 m , while the largest residual computed by the two methods is ( ) 0.29, 0.19, 0.2 m , which is quite smaller than the difference between telemetry position and tracking position. It follows that the separation model of initial errors are exact and the accuracy is fine.

Therefore, the initial errors can be estimated by using the computed difference between telemetry data and tracking data and the environmental function matrix of initial errors. In position domain, using the least-square estimation method we can have

$$\hat{\mathbf{P}}\_a = (\mathbf{G}\_s^{\ \ T} \cdot \mathbf{G}\_s)^{-1} \mathbf{G}\_s^{\ \ T} \boldsymbol{\delta} \mathbf{X}\_p \tag{100}$$

The estimates of initial errors are given in Table 2.


Table 2. The estimates of initial errors. Notes: True denotes the true value of parameter and Est denotes the estimates of parameter.

As it is seen from Table 2, the estimated accuracy of initial errors is high, such as astronomical longitude and latitude, azimuth, geodetic longitude and latitude, among others, of which the estimated relative error is smaller than 0.1%. Simultaneously, the estimated relative error of initial velocity is smaller than 1% and the estimated relative error of geodetic height is 4.2%.

It is necessary to point out that the variation of apparent acceleration due to the uncertainty of initial parameters in the error separation model mentioned above is not taken into consideration. In effect, the state of missile will change as the initial launch parameters change, subsequently the thrust and aerodynamic forces acting on the missile will vary. Simulation results indicate that the assumption that the factor is neglected is rational in the most cases. However, if there are errors in the position of vertical direction, then the large errors may be caused, for example, the variation of geodetic height will affect the shape of the trajectory severely. Although the error of geodetic height affect the apparent acceleration, this deviation of apparent acceleration can be measured onboard and reflected in both telemetry data and tracking data, which can be offset when computing the difference between telemetry data and tracking data. In the practical project, the ballistic missile is generally equipped with guidance system. Under the ideal situation, the error of apparent acceleration due to the initial errors can be completely offset therefore, the error of apparent acceleration will do no effect on the impact point.

#### **5.2 Verification of separation model of instrumentation errors and initial errors**

In the same manner the telemetry data and tracking data are generated by using the sixdegree-of-freedom ballistic program with 10000 kilometers of range, and the initial errors

the largest residual computed by the two methods is ( ) −− − 0.0015, 0.0013, 0.0002 m/s , which is quite smaller than the difference between telemetry velocity and tracking velocity. Similarly, as seen in Figs. 5 and 7, when the third stage engine shut down, the difference between telemetry and tracking position is ( ) − − 400, 25, 672 m , while the largest residual computed by the two methods is ( ) 0.29, 0.19, 0.2 m , which is quite smaller than the difference between telemetry position and tracking position. It follows that the separation model of initial errors

Therefore, the initial errors can be estimated by using the computed difference between telemetry data and tracking data and the environmental function matrix of initial errors. In

> ˆ <sup>1</sup> ( ) *T T a ss s p*

True 30 30 120 -20 -20 -5 -0.1 -0.05 0.1 Est 30.013 29.987 120.01 -19.988 -20.007 -5.210 -0.0993 -0.049 0.1001 Table 2. The estimates of initial errors. Notes: True denotes the true value of parameter and

As it is seen from Table 2, the estimated accuracy of initial errors is high, such as astronomical longitude and latitude, azimuth, geodetic longitude and latitude, among others, of which the estimated relative error is smaller than 0.1%. Simultaneously, the estimated relative error of initial velocity is smaller than 1% and the estimated relative error

It is necessary to point out that the variation of apparent acceleration due to the uncertainty of initial parameters in the error separation model mentioned above is not taken into consideration. In effect, the state of missile will change as the initial launch parameters change, subsequently the thrust and aerodynamic forces acting on the missile will vary. Simulation results indicate that the assumption that the factor is neglected is rational in the most cases. However, if there are errors in the position of vertical direction, then the large errors may be caused, for example, the variation of geodetic height will affect the shape of the trajectory severely. Although the error of geodetic height affect the apparent acceleration, this deviation of apparent acceleration can be measured onboard and reflected in both telemetry data and tracking data, which can be offset when computing the difference between telemetry data and tracking data. In the practical project, the ballistic missile is generally equipped with guidance system. Under the ideal situation, the error of apparent acceleration due to the initial errors can be completely offset therefore, the error of apparent

**5.2 Verification of separation model of instrumentation errors and initial errors**  In the same manner the telemetry data and tracking data are generated by using the sixdegree-of-freedom ballistic program with 10000 kilometers of range, and the initial errors

*B*0 (arcsec)

<sup>−</sup> **P GG G** = ⋅ **δX** (100)

*Vx* ( m/s )

*Vy* ( m/s )

*Vz* ( m/s )

*H*0 ( m )

position domain, using the least-square estimation method we can have

λ0 (arcsec)

are exact and the accuracy is fine.

λ*T* (arcsec)

The estimates of initial errors are given in Table 2.

acceleration will do no effect on the impact point.

*AT* (arcsec)

*BT* (arcsec)

Est denotes the estimates of parameter.

of geodetic height is 4.2%.

are seen in Table 1. The model of guidance instrumentation systematic errors are given by Eqs.(22) and (23) and the levelling and alignment errors are not included.

Similarly, environmental function matrix of instrumentation error, **S** , and environmental function matrix of initial errors, **G** , are obtained to compute the difference between telemetry data and tracking data, **Y** , which is defined as **Y SD GP** =⋅ − ⋅ 0 0*<sup>a</sup>* . Simultaneously, **δX** is the difference between telemetry and tracking data obtained by the simulation data. Likewise, define **δY** = − **δX Y** is the residual. Simulation results are shown in the following figures, Fig.8 shows the difference between telemetry velocity and tracking velocity, **δX***<sup>v</sup>* ; Fig.9 shows the difference between telemetry position and tracking position, **δX***<sup>s</sup>* ; Fig.10 shows the residual of the difference between telemetry velocity and tracking velocity, **δY***<sup>v</sup>* , and Fig.11 shows the residual of the difference between telemetry position and tracking position, **δY***<sup>s</sup>* .

Fig. 8. The difference between telemetry and tracking velocity.

Fig. 9. The difference between telemetry and tracking position.

Error Separation Techniques Based on Telemetry and Tracking Data for Ballistic Missile 469

The estimates of error coefficients of gyroscope and accelerometer are given in Tables 3 and

*<sup>g</sup>*0*<sup>x</sup> k g y*<sup>0</sup> *k <sup>g</sup>*0*<sup>z</sup> k <sup>g</sup>*11*<sup>x</sup> k g y* <sup>11</sup> *k <sup>g</sup>*11*<sup>z</sup> k <sup>g</sup>*12*<sup>x</sup> k g y* <sup>12</sup> *k <sup>g</sup>*12*<sup>z</sup> k*

<sup>−</sup> **K HH H** = **δX** (101)


0.0113 0.0200 0.0201

0.0172

*H*0 ( m )

*Vx* ( m/s )

*Vy* ( m/s )

*Vz* ( m/s )

*a x*<sup>0</sup> *<sup>k</sup>* ( <sup>2</sup> m/s ) *<sup>a</sup>*0*<sup>y</sup> <sup>k</sup>* ( <sup>2</sup> m/s ) *a z* <sup>0</sup> *<sup>k</sup>* ( <sup>2</sup> m/s )

*a x*<sup>1</sup> *k <sup>a</sup>*1*<sup>y</sup> k a z*<sup>1</sup> *k*

*B*0 (arcsec)

True 30 30 120 -20 -20 -5 -0.1 -0.05 0.1 Est 30.586 30.198 116.25 -19.989 -20.007 -5.195 -0.0998 -0.049 0.1002

It is seen from Tables 3 through 5 that the instrumentation error coefficients and initial errors are well estimated in the position domain by using the separation model of

In this chapter, the separation model of initial launch parameter errors and guidance instrumentation systematic errors are formulated based on telemetry and tracking data. The calculation of difference between telemetry and tracking data is discussed in detail. It is generally considered that the telemetry data contain instrumentation errors while tracking data contain systematic errors and random measurement errors of exterior measurement equipment. Numerical examples are given for the verification of the separation by using sixdegree-of-freedom trajectory program. Simulation results indicate that the separation model of initial errors and guidance instrumentation systematic errors can estimate the error

ˆ <sup>1</sup> ( ) *T T ss s*

True 0.5 0.3 -0.5 -0.01 0.01 0.01 -0.01 0.02 0.02

0.0139 0.0051 -

4, respectively. The estimates of initial errors are given in Table 5.

Table 3. The estimates of gyroscope error coefficients. (Units: deg/hour)

True <sup>3</sup> 2.0 10<sup>−</sup> − × <sup>3</sup> 2.0 10<sup>−</sup> − × <sup>3</sup> 1.0 10<sup>−</sup> × Est <sup>3</sup> 2.0374 10<sup>−</sup> − × <sup>3</sup> 2.0086 10<sup>−</sup> − × <sup>3</sup> 1.0826 10<sup>−</sup> ×

True <sup>4</sup> 5.0 10<sup>−</sup> × <sup>4</sup> 5.0 10<sup>−</sup> × <sup>4</sup> 5.0 10<sup>−</sup> × Est <sup>4</sup> 5.0042 10<sup>−</sup> × <sup>4</sup> 5.0055 10<sup>−</sup> × <sup>4</sup> 3.8726 10<sup>−</sup> ×

> λ0 (arcsec)

Table 4. The estimates of accelerometer error coefficients.

*AT* (arcsec)

instrumentation errors and initial errors mentioned above.

Est 0.632 0.368 -0.503 -

λ*T* (arcsec)

**6. Conclusions** 

coefficient well and is exact.

*BT* (arcsec)

Table 5. The estimates of initial errors.

Fig. 10. The residual of the difference between telemetry and tracking velocity.

Fig. 11. The residual of the difference between telemetry and tracking position.

It is clearly seen from Figs. 8 and 10 that the differences between telemetry velocity and tracking velocity obtained by the two methods agree well. When the third stage engine shut down, the difference between telemetry velocity and tracking velocity is ( ) 3.46, 1.34, 0.90 m/s − , while the largest residual computed by the two methods is ( ) −− − 0.0015, 0.0019, 0.006 m/s , which is quite smaller than the difference between telemetry velocity and tracking velocity. Similarly, as seen in Figs. 5 and 7, when the third stage engine shut down, the difference between telemetry position and tracking position is ( ) − − 400, 25, 672 m , while the largest residual computed by the two methods is ( ) 0.29, 0.19, 0.19 m , which is quite smaller than the difference between telemetry position and tracking position. It follows that the separation model of instrumentation errors and initial errors are exact and precise.

The instrumentation errors and initial errors are estimated by using the above data. Selecting vector = *<sup>T</sup> T T <sup>a</sup>* **KDP** and letting *<sup>T</sup> T T ss s* = − **HS G** , the in the position domain, the

Error Separation Techniques Based on Telemetry and Tracking Data for Ballistic Missile 469

$$\hat{\mathbf{K}} = (\mathbf{H}\_s^T \mathbf{H}\_s)^{-1} \mathbf{H}\_s^T \mathbf{\hat{\mathbf{G}}} \mathbf{X} \tag{101}$$

The estimates of error coefficients of gyroscope and accelerometer are given in Tables 3 and 4, respectively. The estimates of initial errors are given in Table 5.


Table 3. The estimates of gyroscope error coefficients. (Units: deg/hour)


Table 4. The estimates of accelerometer error coefficients.


Table 5. The estimates of initial errors.

It is seen from Tables 3 through 5 that the instrumentation error coefficients and initial errors are well estimated in the position domain by using the separation model of instrumentation errors and initial errors mentioned above.
