**4. Error separation model of initial launch parameters**

450 Modern Telemetry

 

*k kW k kW k kW*

*ax ax px <sup>x</sup> y a y a y py z az az pz*

 <sup>+</sup> <sup>Δ</sup> Δ= + <sup>Δ</sup> <sup>+</sup>

Herein we select 0 0 0 11 11 11 12 12 12 0 0 0 1 1 0 *<sup>T</sup>*

then apparent acceleration error **δW** and instrumentation error coefficients **D** are written in

0 100 0 0

<sup>−</sup>

*yp xp pz*

*zp yp px e zp xp Aa py*

= − <sup>=</sup> <sup>−</sup>

*W W W W W W W W W*

 

00 0 0 0 0 0 00 0 0 0 00 0 0 0 0

*tW W*

() ( ) ()

*a v t dt t* = ⋅= τ

where ( ) *<sup>v</sup>* **S** *t* is the environmental function matrix of instrumental error of apparent velocity.

() ( ) ()

*v r t dt t* = ⋅= τ**<sup>δ</sup>W S DS D**

where ( ) *<sup>r</sup>* **S** *t* is the environmental function matrix of instrumental error of apparent position. In the actual situation, the apparent velocity and position error models with the

*xp yp*

 

*tW W*

0

0

*t*

Taking the integration of Eq.(33) again gives the apparent position error

*t*

*Ag yp xp*

*tW W*

**S S**

=

Integrating Eq.(31) gives the apparent velocity error

consideration of random errors are represented by

**S**

0 , 010 0 0 , 0 001 0 0

*zp yp*

where **S***<sup>a</sup>* is the environmental function matrix of apparent acceleration, given by

*g x g x px g x py x x t t y y g y g y py g y px z z g z g z pz g z py*

 + + = =+ + + +

0 0

 α

 α

 α

α

α

α

linear relation as

where

0 11 12 0 11 12

*k kW kW dt k k W k W dt k kW kW*

(29)

(30)

**<sup>δ</sup>WSD** = ⋅ *<sup>a</sup>* (31)

*a e Ag Aa* = ⋅ **S SS S** (32)

**<sup>δ</sup>W S DS D** (33)

(34)

0 11 12

 

*gx gy gz g x g y g z g x g y g z ax ay az ax ay az* <sup>=</sup> *kk kk k k k k k kkkkkk* **<sup>D</sup>** ,

The initial launch parameter errors not only affect the apparent position and velocity and stress of ballistic missile, but also the airborne computer guidance calculation. The mechanism of initial errors is analyzed thereinafter.

#### **4.1 Effect to landing error of ballistic missile caused by initial errors**

1. Effect to trajectory in the geocentric coordinate system

The localization and orientation parameters directly determine the foundation of coordinate system. When the launch inertial coordinate system *O xyz a a aa* − changes to *O xyz a a aa* ′ ′′′ − , the base of controlling the attitude motion will also change. At this point, the reference plane *O xz a aa* − controlled by pitch angle changes to *O xz a aa* ′ ′′ − plane, simultaneously the reference plane *O xy a aa* − controlled by yaw angle changes to *O xy a aa* ′ ′′ − plane. Due to the noncoincidence of the two pairs of planes, the shape and azimuth of the in-flight trajectory are not the same with respect to the "real earth". Also, the location of trajectory is determined by the initial localization and orientation parameters. Therefore, the position of landing point of ballistic missile in the geocentric coordinate system will offset the objective point when the parameters are not error-free, in despite of taking no account of other error factors.

2. Effect to the initial velocity of missile in the launch inertial coordinate system

The launch site coordinate *R*0*<sup>a</sup>* and the earth rate *ωea* are determined by the initial localization and orientation parameters, which affect the initial velocity and stress of ballistic missile.

In the case of maneuvering launch, the initial missile velocity in the launch inertial coordinate system is given by

$$\mathcal{V}\_{0a} = \boldsymbol{\alpha}\_{aa} \times \mathbf{R}\_{0a} + \boldsymbol{\mathcal{V}}\_{s}^{a} \tag{36}$$

where *<sup>a</sup> V<sup>s</sup>* is the carrier's instantaneous velocity with respect to the ground. Obviously, the initial velocity is largely related to the initial localization and orientation parameters and the velocity of carrier. When these parameters are with errors, the initial velocity of missile is in error.

3. Effect to the stress of missile

The acceleration of gravity of missile is determined by the angular velocity of the Earth and the coordinates of launch point in the launch inertial coordinate system and launch coordinate system. Due to the difference of stress of missile, the flight height and velocity

Error Separation Techniques Based on Telemetry and Tracking Data for Ballistic Missile 453

where *<sup>e</sup> C<sup>n</sup>* is the rotation matrix mapping horizontal coordinate system to geocentric coordinate system. The precise Euler angles are available since the geodetic coordinates of the observation station are accurate. However, there are errors in the Euler angles of rotation

**4.3 Relationship between initial orientation errors and alignment errors of platform**  Before work the levelling and aligning are need to perform for inertial platform. For the maneuvering-launch-based missile, there may exist errors in the process of levelling and

*a Ce*

*X* ′

*X* ′′

ϕ *y* α*x*

*AT*

Fig. 2. The relationship between orientation errors and alignment errors of platform.

As shown in Fig.2, *N* is true north direction, *N*′ is north direction measured by the vehicle, and *ATn* Δ ′ is the northing error. *X* is the ideal direction of fire, *X*′ is the direction

information measured in the frame involved in *X*′′ axis while tracking data provides the information measured in the frame involved in *X* axis. Therefore, the azimuth

*AA A T T Tn*

*E*

*<sup>y</sup>* , *X*′′ is the actual direction provided by INS due to the

*<sup>y</sup>* = +Δ + ′ ′ (38)

*<sup>x</sup>* . In fact, telemetry data provides the apparent acceleration

ϕ

*X*

*e Cn*

matrix *<sup>a</sup> Ce* and then the orientation errors are introduced.

*ATn* Δ ′

*O*

*AT* ′

ϕ

*<sup>N</sup> <sup>N</sup>*′

Fig. 1. The conversion of tracking data.

aligning for onboard platform system.

contaminated by alignment error

α

from *X* direction to true north direction is given by

platform drift angle

are different, which indirectly causes the variation of thrust and aerodynamic forces. When computing the thrust forces, the effect of atmospheric pressure is considered, which is known as a function of height. At the same time, the calculation of thrust vector is related to the deflection angle of rudder, of which calculation is also affected by the height. In addition, the aerodynamic coefficients, velocity head and velocity are related to the height.

4. Effect to airborne guidance calculation

At present, the real velocity and position are commonly adopted for the calculation of guidance. Firstly, the integration of the apparent acceleration measured is performed to obtain apparent velocity; secondly, the real velocity and position are computed by the recursion formulas according to the computed apparent velocity and acceleration of gravity. When the true velocity and position satisfy the cut-off equations, the engines of missile shut down.

When there exist localization and orientation errors, on the one hand, the guidance coordinate system is different from the actual flight coordinate system, thereby the fact that the cut-off equations are satisfied cannot ensure that the missile hit the target; on the other hand, the initial values of recursion formulas involved real velocity and position and the calculation of gravitational acceleration are different from those of actual conditions, which induces that the computed real velocity and position don't agree with those under the actual situations.

For the closed-loop guidance case, the required commanded missile velocity is determined by the onboard computer in real time. Specifically, the required velocity is a function of current velocity and position of missile, location of launch point and target point, angular velocity of the Earth and orientation parameters, that is

$$\boldsymbol{V}\_{a\mathbb{R}} = \boldsymbol{V}\_{a\mathbb{R}}(\boldsymbol{V}\_{a}, \mathbf{R}\_{a}, \mathbf{R}\_{\alpha bj}, \mathbf{R}\_{0a}, \boldsymbol{\text{op}}\_{\alpha a}, \boldsymbol{\hat{\mathcal{A}}}\_{\boldsymbol{T}}, \boldsymbol{B}\_{\boldsymbol{T}}, \boldsymbol{A}\_{\boldsymbol{T}}) \tag{37}$$

It is obvious that the errors of localization and orientation parameters directly influence the calculation of required velocity and the cut-off of missile.

#### **4.2 Sources of errors of initial localization and orientation parameters**

In fact, the telemetry data should reflect the acceleration information of ballistic missile provided that the guidance instrumentation systematic errors are not taken into account. Tracking data are obtained in the horizontal coordinate system by measurement devices and then converted into geocentric coordinate system. Since the precise data in the local horizontal coordinate system are available, the tracking data measured in the geocentric coordinate system don't contain the initial errors and are precise.

The difference between telemetry and tracking data is generally reckoned in the launch inertial coordinate system. The launch inertial coordinate system is determined by the initial location and orientation parameters, and the launch inertial coordinate system is inaccurate if those parameters are with errors. It is necessary to convert the tracking data in the geocentric coordinate system into the launch inertial coordinate system. The location parameters are required for the calculation of initial velocity and position while orientation parameters are demanded for the calculation of the Euler angle mapping the geocentric coordinate system into launch inertial coordinate system, which generates the initial location and orientation parameter errors. The conversion of the tracking data is described as follows:

Fig. 1. The conversion of tracking data.

452 Modern Telemetry

are different, which indirectly causes the variation of thrust and aerodynamic forces. When computing the thrust forces, the effect of atmospheric pressure is considered, which is known as a function of height. At the same time, the calculation of thrust vector is related to the deflection angle of rudder, of which calculation is also affected by the height. In addition, the aerodynamic coefficients, velocity head and velocity are related to the

At present, the real velocity and position are commonly adopted for the calculation of guidance. Firstly, the integration of the apparent acceleration measured is performed to obtain apparent velocity; secondly, the real velocity and position are computed by the recursion formulas according to the computed apparent velocity and acceleration of gravity. When the true velocity and position satisfy the cut-off equations, the engines of missile shut

When there exist localization and orientation errors, on the one hand, the guidance coordinate system is different from the actual flight coordinate system, thereby the fact that the cut-off equations are satisfied cannot ensure that the missile hit the target; on the other hand, the initial values of recursion formulas involved real velocity and position and the calculation of gravitational acceleration are different from those of actual conditions, which induces that the computed real velocity and position don't agree with those under the actual

For the closed-loop guidance case, the required commanded missile velocity is determined by the onboard computer in real time. Specifically, the required velocity is a function of current velocity and position of missile, location of launch point and target point, angular

<sup>0</sup> (, , , , ,, , ) *V V VRR R aR aR a a obj a ea T T T* = *ω*

It is obvious that the errors of localization and orientation parameters directly influence the

In fact, the telemetry data should reflect the acceleration information of ballistic missile provided that the guidance instrumentation systematic errors are not taken into account. Tracking data are obtained in the horizontal coordinate system by measurement devices and then converted into geocentric coordinate system. Since the precise data in the local horizontal coordinate system are available, the tracking data measured in the geocentric

The difference between telemetry and tracking data is generally reckoned in the launch inertial coordinate system. The launch inertial coordinate system is determined by the initial location and orientation parameters, and the launch inertial coordinate system is inaccurate if those parameters are with errors. It is necessary to convert the tracking data in the geocentric coordinate system into the launch inertial coordinate system. The location parameters are required for the calculation of initial velocity and position while orientation parameters are demanded for the calculation of the Euler angle mapping the geocentric coordinate system into launch inertial coordinate system, which generates the initial location and orientation parameter errors. The conversion of the tracking data is described

λ

*B A* (37)

height.

down.

situations.

as follows:

4. Effect to airborne guidance calculation

velocity of the Earth and orientation parameters, that is

calculation of required velocity and the cut-off of missile.

**4.2 Sources of errors of initial localization and orientation parameters** 

coordinate system don't contain the initial errors and are precise.

where *<sup>e</sup> C<sup>n</sup>* is the rotation matrix mapping horizontal coordinate system to geocentric coordinate system. The precise Euler angles are available since the geodetic coordinates of the observation station are accurate. However, there are errors in the Euler angles of rotation matrix *<sup>a</sup> Ce* and then the orientation errors are introduced.

#### **4.3 Relationship between initial orientation errors and alignment errors of platform**

Before work the levelling and aligning are need to perform for inertial platform. For the maneuvering-launch-based missile, there may exist errors in the process of levelling and aligning for onboard platform system.

Fig. 2. The relationship between orientation errors and alignment errors of platform.

As shown in Fig.2, *N* is true north direction, *N*′ is north direction measured by the vehicle, and *ATn* Δ ′ is the northing error. *X* is the ideal direction of fire, *X*′ is the direction contaminated by alignment error ϕ*<sup>y</sup>* , *X*′′ is the actual direction provided by INS due to the platform drift angleα *<sup>x</sup>* . In fact, telemetry data provides the apparent acceleration information measured in the frame involved in *X*′′ axis while tracking data provides the information measured in the frame involved in *X* axis. Therefore, the azimuth from *X* direction to true north direction is given by

$$A\_T = A\_T' + \Delta A\_{Tn}' + \varphi\_y \tag{38}$$

Error Separation Techniques Based on Telemetry and Tracking Data for Ballistic Missile 455

not taken into account, then both **V***e* and *er* are precise. The tracking velocity **V***<sup>e</sup>* , consisting

where **V***eg* is the incremental velocity due to gravitational acceleration, **V***es* is the velocity of maneuverable carrier, **V***ew* is the tracking apparent velocity which has removed the effect of

where *eg* **r** is the incremental position due to gravitational acceleration, *es* **r** is the incremental position due to the velocity of maneuverable carrier, *ew***r** is the apparent tracking position getting rid of the effect of gravity force and initial velocity of carrier, **R**0*<sup>e</sup>* denotes the radius vector of origin of north-east-down coordinate system in the

The tracking missile position in the launch coordinate system can be written in vector

0

 π

00 0 0

**R C** (45)

1 sin

*N eH B*

cos cos

λ

2 13 ( ) () ( ) 2 2

( ) ( )

*TT T* ′′ ′ *B A* are the orientation parameters contaminated by random errors, 000 *H B* ′′′ , ,

(,, ) cos sin

<sup>+</sup> ′ ′′ = + ′′ ′ ′ ′ ′ − + ′ ′

*BA N H B*

( )

*NH B*

2 0 0

*e TT T A B*

0 00 0 0

The initial errors are introduced due to the localization and orientation parameters contaminated by random errors when computing transformation matrix *<sup>g</sup>* **<sup>C</sup>***e* and position

The tracking velocity expressed in the launch coordinate system is represented by

**CM M M** = −− ′′ ′ − +

π

**4.4.1 Analysis of tracking data in the launch coordinate system** 

*g*

*g g eTT T*

are localization parameters contaminated by random errors.

vector **R**0*<sup>g</sup>* , although precise **V***<sup>e</sup>* and *<sup>e</sup>***r** are available.

λ

**VV V V** *e eg es ew* =++ (41)

*e eg es ew e* = ++ + <sup>0</sup> **rr r r R** (42)

*<sup>g</sup>* **<sup>ρ</sup>***<sup>g</sup>* = ⋅− **Cr R** *e e <sup>g</sup>* (43)

λ

*<sup>g</sup>* **V CV** *<sup>g</sup>* = ⋅ *e e* (46)

 λ

(44)

λ

of three terms is written as

the gravity forces and initial velocity of carrier.

The position vector *e***r** is given by

geocentric coordinate system.

form

with

and

where , , λ

and the initial azimuth error is defined as

$$
\Delta A\_T' = \Delta A\_{Tn}' + \varphi\_y \tag{39}
$$

The above analysis gives an indication of linear correlation between the northing error and alignment errors of INS. Similarly, the relationship between astronomical latitude and levelling error is linear correlation.

Fig. 3. The relationship between levelling errors and orientation parameters.

As can be seen in Fig.3, , , *a aa xyz* are the coordinate axes of launch inertial frame, *p*0*<sup>x</sup> k* and *<sup>p</sup>*0*<sup>z</sup> k* are the levelling errors along *<sup>a</sup> x*′ and *<sup>a</sup> z*′ axes, respectively. Thus, the levelling errors can be converted into the astronomical latitude errors in the following form

$$\begin{aligned} \Delta B\_p &= -k\_{p0x} \sin A\_T' - k\_{p0z} \cos A\_T' \\ \Delta \mathcal{A}\_p &= k\_{p0x} \cos A\_T' - k\_{p0z} \sin A\_T' \end{aligned} \tag{40}$$

It is shown from the above analysis that the relationship between initial errors and levelling and alignment errors of guidance instrumentation systematic errors is linear correlation. Therefore, those errors cannot be separated merely using the telemetry and tracking data. Thereinafter the levelling and alignment errors are not included in the simulated cases.

#### **4.4 Preliminary analysis of tracking data**

In order to obtain the tracking data with sufficient precision, the incorporated measurement of multiple observation stations is generally used. It is pointed out in the previous section that the horizontal coordinate system of observation station is known exactly and the mapping relation with the geocentric coordinate system can be precisely described. To simplify the definition, the tracking velocity in the geocentric coordinate system is denoted by **V***<sup>e</sup>* , and the position vector from the earth center expressed in the geocentric coordinate system is denoted as *e***r** . Obviously, provided that the random errors of exterior devices are not taken into account, then both **V***e* and *er* are precise. The tracking velocity **V***<sup>e</sup>* , consisting of three terms is written as

$$\mathbf{V}\_e = \mathbf{V}\_{cg} + \mathbf{V}\_{es} + \mathbf{V}\_{ew} \tag{41}$$

where **V***eg* is the incremental velocity due to gravitational acceleration, **V***es* is the velocity of maneuverable carrier, **V***ew* is the tracking apparent velocity which has removed the effect of the gravity forces and initial velocity of carrier.

The position vector *e***r** is given by

$$\mathbf{r}\_e = \mathbf{r}\_{cg} + \mathbf{r}\_{es} + \mathbf{r}\_{env} + \mathbf{R}\_{0e} \tag{42}$$

where *eg* **r** is the incremental position due to gravitational acceleration, *es* **r** is the incremental position due to the velocity of maneuverable carrier, *ew***r** is the apparent tracking position getting rid of the effect of gravity force and initial velocity of carrier, **R**0*<sup>e</sup>* denotes the radius vector of origin of north-east-down coordinate system in the geocentric coordinate system.

#### **4.4.1 Analysis of tracking data in the launch coordinate system**

The tracking missile position in the launch coordinate system can be written in vector form

$$\mathbf{p}\_{\mathcal{K}} = \mathbf{C}\_{e}^{\mathcal{S}} \cdot \mathbf{r}\_{e} - \mathbf{R}\_{0\mathcal{g}} \tag{43}$$

with

454 Modern Telemetry

ϕ

*az*

*<sup>y</sup>* Δ =Δ + ′ ′ (39)

*<sup>a</sup> <sup>a</sup> <sup>y</sup> <sup>y</sup>*′

*<sup>p</sup>*0*<sup>z</sup> k*

*AT* ′

*a x*

*a x*′

*A A T Tn*

*a x*

Fig. 3. The relationship between levelling errors and orientation parameters.

can be converted into the astronomical latitude errors in the following form

λ

As can be seen in Fig.3, , , *a aa xyz* are the coordinate axes of launch inertial frame, *p*0*<sup>x</sup> k* and *<sup>p</sup>*0*<sup>z</sup> k* are the levelling errors along *<sup>a</sup> x*′ and *<sup>a</sup> z*′ axes, respectively. Thus, the levelling errors

> 0 0 0 0

 *k Ak A* Δ =− − ′ ′

It is shown from the above analysis that the relationship between initial errors and levelling and alignment errors of guidance instrumentation systematic errors is linear correlation. Therefore, those errors cannot be separated merely using the telemetry and tracking data. Thereinafter the levelling and alignment errors are not included in the simulated cases.

In order to obtain the tracking data with sufficient precision, the incorporated measurement of multiple observation stations is generally used. It is pointed out in the previous section that the horizontal coordinate system of observation station is known exactly and the mapping relation with the geocentric coordinate system can be precisely described. To simplify the definition, the tracking velocity in the geocentric coordinate system is denoted by **V***<sup>e</sup>* , and the position vector from the earth center expressed in the geocentric coordinate system is denoted as *e***r** . Obviously, provided that the random errors of exterior devices are

sin cos cos sin *p p x T p z T p p x T p z T B k Ak A*

Δ= − ′ ′ (40)

*<sup>a</sup> <sup>y</sup> <sup>a</sup> y*′

*AT* ′

**4.4 Preliminary analysis of tracking data** 

The above analysis gives an indication of linear correlation between the northing error and alignment errors of INS. Similarly, the relationship between astronomical latitude and

and the initial azimuth error is defined as

levelling error is linear correlation.

*<sup>p</sup>*0*<sup>x</sup> k*

*a z*

*a z*′

$$\mathbf{C}\_{\varepsilon}^{\mathcal{S}} = \mathbf{M}\_2(-\frac{\pi}{2} - A\_T')\mathbf{M}\_1(\mathcal{B}\_T')\mathbf{M}\_3(-\frac{\pi}{2} + \mathcal{X}\_T') \tag{44}$$

and

$$\mathbf{R}\_{0g} = \mathbf{C}\_{e}^{g} \begin{pmatrix} \lambda\_{\Gamma}^{\prime} \,\_{\nu} B\_{\Gamma}^{\prime} \,\_{\nu} A\_{\Gamma}^{\prime} \end{pmatrix} \begin{pmatrix} N\_{0} + H\_{0}^{\prime} \, \right\vert \cos B\_{0}^{\prime} \cos \lambda\_{0}^{\prime} \\\\ \left( N\_{0} + H\_{0}^{\prime} \right) \cos B\_{0}^{\prime} \sin \lambda\_{0}^{\prime} \\\\ \left[ N\_{0} \left( 1 - e^{2} \right) + H\_{0}^{\prime} \right] \sin B^{\prime} \end{pmatrix} \tag{45}$$

where , , λ*TT T* ′′ ′ *B A* are the orientation parameters contaminated by random errors, 000 *H B* ′′′ , ,λ are localization parameters contaminated by random errors.

The tracking velocity expressed in the launch coordinate system is represented by

$$\mathbf{V}\_{\mathcal{g}} = \mathbf{C}\_{e}^{\mathcal{g}} \cdot \mathbf{V}\_{e} \tag{46}$$

The initial errors are introduced due to the localization and orientation parameters contaminated by random errors when computing transformation matrix *<sup>g</sup>* **<sup>C</sup>***e* and position vector **R**0*<sup>g</sup>* , although precise **V***<sup>e</sup>* and *<sup>e</sup>***r** are available.

Error Separation Techniques Based on Telemetry and Tracking Data for Ballistic Missile 457

*I P a a aa*

*<sup>a</sup>* **P PP** = −′ (52)

τ τ τ τ

> π

′ ′′ − − + (57)

 λ

′ (58)

(51)

() () () () () () () ()

**δX δX δX**

where **δX***<sup>I</sup>* is the difference of telemetry data and tracking data due to guidance instrumentation systematic errors, and **δX***P* is the difference of telemetry data and tracking

where **P**′ are the known binding values of initial launch parameters consisting of 9 terms

tra ( ) <sup>0</sup> <sup>0</sup> <sup>0</sup> () () () () ( ) *<sup>t</sup> a a a a gg g g a a t tt t d*

tra ( ) ( ) <sup>0</sup> 0 0 0 0 ( ) ( ) *t u a a g g aa a t t* = ⋅ + − − ⋅−*t*

and the tracking velocity expressed in the launch inertial coordinate system is given by

where *e***r** and **V***<sup>e</sup>* are the error-free tracking position and velocity expressed in the geocentric

*<sup>g</sup> e TT eT T T B A tB B*

*e ge T T A B t T e*

tra ( ) <sup>0</sup> <sup>0</sup> () () ( ) ( ) *<sup>t</sup> a a aa e ee e e a a tt t* = ⋅+ × ⋅− −

tra ( ) 0 0 0 0 ( ) ( ) *t u a a ee a a a t t* = ⋅ − − ⋅−*t*

**C CC M M M M** = ⋅ = − − − −+ − ′ ′

Substituting Eq.(58) into Eqs. (55) and (56) yields the tracking apparent velocity

 **CC M M M M M M M** ⋅ = ′ ′ −ω

( ) 3 2 10 3 2 1 3 () ( ) ( ) () ( ) () ( ) 2 2

2 3 23 0 ( )( )()( ) 2 2

π

*<sup>d</sup>* **W CV <sup>ω</sup> C rV g** (59)

*d du* **W C rR V g** (60)

*tra* **<sup>W</sup>** *<sup>t</sup>*

00 0 ()( ) () *a a*

**W CV <sup>Ω</sup> C R <sup>ρ</sup> V g** (53)

*d du* **W CR <sup>ρ</sup> RV g** (54)

*<sup>a</sup> g g <sup>a</sup> g g <sup>a</sup>* **ρ** = ⋅ + − = ⋅− **C R** *t t* **ρ R C rR** (55)

*<sup>a</sup> g gg e e* **V C V C CV** = ⋅= ⋅⋅ *t t* (56)

π

 π

τ τ

λ ω

> τ τ

*tra* **W** *t* and ( ) *<sup>a</sup>*

ω= ⋅ +⋅ ⋅ + − −

The tracking position in the launch inertial coordinate system can be written as

*tt tt tt tt* − − <sup>=</sup> <sup>−</sup> ≡ − − −

tele tra0 tra tra0 tele tra0 tra tra0

*a a aa*

**W W WW**

**W W WW**

data due to initial errors. Define the initial errors as

coordinate system.

mentioned above, **P** is the unknown true value.

() () *a a <sup>g</sup>*

By the definition of transformation matrix, we can have

*a g T*

*a a g*

Simplifying the Eq.(57) results in

and the tracking apparent position

Recalling Eqs.(16) and (19) gives ( ) *<sup>a</sup>*

#### **4.4.2 Effect of maneuverable carrier's velocity**

The carrier's velocity is generally expressed in the body frame and the measurement is denoted as *<sup>s</sup>* **V**′ , which is represented in the north-east-down (NED) coordinate system by

$$\mathbf{V}\_s^n = \mathbf{M}\_2(A\_s)\mathbf{M}\_3(-\varphi\_s)\mathbf{M}\_1(-\gamma\_s)\mathbf{V}\_s^\prime \tag{47}$$

where *As* is the flight-path angle, which is measured from the north and is clockwise about the body axes , is positive; ϕ*<sup>s</sup>* is the pitch angle, upward direction is positive; *<sup>s</sup>* γ is the roll angle, and is clockwise about the body axes, is positive. Herein assume *As* ,ϕ*<sup>s</sup>* and *<sup>s</sup>* γ are known exactly. Letting 2( ) *<sup>g</sup>* **C M** *n T* = −*A*′ , thus,

$$\mathbf{V}\_s^d = \mathbf{C}\_{\mathcal{g}}^d \cdot \mathbf{V}\_s^{\mathcal{g}} = \mathbf{C}\_{\mathcal{g}}^d \cdot \mathbf{C}\_n^{\mathcal{g}} \cdot \mathbf{V}\_s^n \tag{48}$$

where *<sup>g</sup>* **<sup>C</sup>***<sup>n</sup>* is coordinate transformation matrix relating horizontal coordinate system to launch coordinate system. It is seen that the carrier's velocity is related to the launch azimuth. The carrier's velocity is known as a portion of initial velocity of missile, yet the tracking velocity and position reflect the real velocity and position of missile if the random errors are not taken into account, therefore, the tracking velocity contains the information of the carrier's velocity.

The position variation of missile due to the initial velocity is represented by

$$\mathbf{r}\_s^a = \mathbf{C}\_\mathcal{g}^a \cdot \mathbf{r}\_s^\mathcal{g} = \mathbf{C}\_\mathcal{g}^a \cdot \mathbf{V}\_s^\mathcal{g} \cdot t = \mathbf{C}\_\mathcal{g}^a \cdot \mathbf{C}\_n^\mathcal{g} \cdot \mathbf{V}\_s^n \cdot t \tag{49}$$

It follows from Eq.(36) that the carrier's velocity is contained in the initial velocity of missile and the incurred position variation is also contained in the tracking data.

#### **4.5 Separation model of initial errors**

It follows from the foregoing analysis that the guidance instrumentation systematic errors are contained in the telemetry data while the initial errors are primarily introduced during the data processing for tracking data. Therefore, the separation of these two types of errors can be performed respectively. The difference between telemetry and tracking data is written in the following form

$$\mathbf{6X} = \begin{bmatrix} \mathbf{W}\_{\text{tele}}^{\boldsymbol{a}}(t) - \mathbf{W}\_{\text{tra}}^{\boldsymbol{a}}(t) \\ \mathbf{W}\_{\text{tele}}^{\boldsymbol{a}}(t) - \mathbf{W}\_{\text{tra}}^{\boldsymbol{a}}(t) \end{bmatrix} = \begin{bmatrix} \left( \mathbf{W}\_{\text{tele}}^{\boldsymbol{a}}(t) - \mathbf{W}\_{\text{tra0}}^{\boldsymbol{a}}(t) \right) - \left( \mathbf{W}\_{\text{tra}}^{\boldsymbol{a}}(t) - \mathbf{W}\_{\text{tra0}}^{\boldsymbol{a}}(t) \right) \\\\ \left( \mathbf{W}\_{\text{tele}}^{\boldsymbol{a}}(t) - \mathbf{W}\_{\text{tra0}}^{\boldsymbol{a}}(t) \right) - \left( \mathbf{W}\_{\text{tra}}^{\boldsymbol{a}}(t) - \mathbf{W}\_{\text{tra0}}^{\boldsymbol{a}}(t) \right) \end{bmatrix} \tag{50}$$

where tele **W***<sup>a</sup>* and tele **W***<sup>a</sup>* are apparent velocity and position provided by telemetry data, respectively; tra **W***<sup>a</sup>* and tra **W***<sup>a</sup>* are apparent velocity and position provided by tracking data, respectively; tra0 **W***<sup>a</sup>* and tra0 **W***<sup>a</sup>* are the tracking information which don't contain the initial errors. The term on the right-hand side of Eq.(50) comprises two parts of information, one is the effect of guidance instrumentation systematic errors, and the other is the effect of initial errors. Thus, the difference between the telemetry data and tracking data can be rewritten as

$$\mathbf{\tilde{\bf \bf \bf \bf X}} = \begin{bmatrix} \mathbf{W}\_{\text{tele}}^{a}(t) - \mathbf{W}\_{\text{tra0}}^{a}(t) \\ \mathbf{W}\_{\text{tele}}^{a}(t) - \mathbf{W}\_{\text{tra0}}^{a}(t) \end{bmatrix} - \begin{bmatrix} \mathbf{W}\_{\text{tra}}^{a}(t) - \mathbf{W}\_{\text{tra0}}^{a}(t) \\ \mathbf{W}\_{\text{tra}}^{a}(t) - \mathbf{W}\_{\text{tra0}}^{a}(t) \end{bmatrix} \equiv \mathbf{\tilde{\bf }} \mathbf{X}\_{I} - \mathbf{\tilde{\bf }} \mathbf{X}\_{P} \tag{51}$$

where **δX***<sup>I</sup>* is the difference of telemetry data and tracking data due to guidance instrumentation systematic errors, and **δX***P* is the difference of telemetry data and tracking data due to initial errors.

Define the initial errors as

456 Modern Telemetry

The carrier's velocity is generally expressed in the body frame and the measurement is denoted as *<sup>s</sup>* **V**′ , which is represented in the north-east-down (NED) coordinate system by

> 23 1 () ( ) ( ) *<sup>n</sup>* **VM M M V** *s s s ss* = −− *A* ϕ

where *As* is the flight-path angle, which is measured from the north and is clockwise about

where *<sup>g</sup>* **<sup>C</sup>***<sup>n</sup>* is coordinate transformation matrix relating horizontal coordinate system to launch coordinate system. It is seen that the carrier's velocity is related to the launch azimuth. The carrier's velocity is known as a portion of initial velocity of missile, yet the tracking velocity and position reflect the real velocity and position of missile if the random errors are not taken into account, therefore, the tracking velocity contains the information of

*aa a a n gg g*

It follows from Eq.(36) that the carrier's velocity is contained in the initial velocity of missile

It follows from the foregoing analysis that the guidance instrumentation systematic errors are contained in the telemetry data while the initial errors are primarily introduced during the data processing for tracking data. Therefore, the separation of these two types of errors can be performed respectively. The difference between telemetry and tracking data is

> tele tra0 tra tra0 tele tra <sup>a</sup> tele tra tele tra0 tra tra0

*a a aa a a a a a aa*

**W W W W WW**

errors. The term on the right-hand side of Eq.(50) comprises two parts of information, one is the effect of guidance instrumentation systematic errors, and the other is the effect of initial errors. Thus, the difference between the telemetry data and tracking data can be rewritten

() () () () () () () () () () () ()

*tt tt t t t t tt tt* − −− <sup>−</sup> = = <sup>−</sup> − −− **W W W W WW**

angle, and is clockwise about the body axes, is positive. Herein assume *As* ,

The position variation of missile due to the initial velocity is represented by

and the incurred position variation is also contained in the tracking data.

 γ

*<sup>s</sup>* is the pitch angle, upward direction is positive; *<sup>s</sup>*

*aa a n g g* **V CV CCV** *<sup>s</sup>* =⋅ =⋅⋅ *<sup>g</sup> <sup>s</sup> <sup>g</sup> n s* (48)

*s gs g s g n s* **r Cr CV CCV** = ⋅ = ⋅ ⋅= ⋅ ⋅ ⋅ *t t* (49)

( ) ( )

( ) ( )

are apparent velocity and position provided by tracking data,

are the tracking information which don't contain the initial

are apparent velocity and position provided by telemetry data,

′ (47)

γ

ϕ*<sup>s</sup>* and *<sup>s</sup>* γare

is the roll

(50)

**4.4.2 Effect of maneuverable carrier's velocity** 

ϕ

known exactly. Letting 2( ) *<sup>g</sup>* **C M** *n T* = −*A*′ , thus,

**4.5 Separation model of initial errors** 

written in the following form

**δX**

where tele **W***<sup>a</sup>* and tele **W***<sup>a</sup>*

as

respectively; tra **W***<sup>a</sup>* and tra **W***<sup>a</sup>*

respectively; tra0 **W***<sup>a</sup>* and tra0 **W***<sup>a</sup>*

the body axes , is positive;

the carrier's velocity.

$$\mathbf{P}\_a = \mathbf{P}' - \mathbf{P} \tag{52}$$

where **P**′ are the known binding values of initial launch parameters consisting of 9 terms mentioned above, **P** is the unknown true value.

Recalling Eqs.(16) and (19) gives ( ) *<sup>a</sup> tra* **W** *t* and ( ) *<sup>a</sup> tra* **<sup>W</sup>** *<sup>t</sup>*

$$\mathbf{W}\_{\rm tra}^{a}(t) = \mathbf{C}\_{\rm g}^{a}(t) \cdot \mathbf{V}\_{\rm g}(t) + \mathbf{Q}\_{\rm o}^{a} \cdot \mathbf{C}\_{\rm g}^{a}(t) \cdot \left(\mathbf{R}\_{0} + \mathbf{p}\_{\rm g}\right) - \mathbf{V}\_{0a} - \int\_{0}^{t} \mathbf{g}\_{a}(\tau) d\tau \tag{53}$$

$$\mathbf{W}\_{\rm tra}^{a}\left(t\right) = \mathbf{C}\_{\rm g}^{a}\left(t\right) \cdot \left(\mathbf{R}\_{0} + \mathbf{p}\_{\rm g}\right) - \mathbf{R}\_{0a} - \mathbf{V}\_{0a} \cdot t - \int\_{0}^{t} \int\_{0}^{u} \mathbf{g}\_{a}\left(\tau\right) d\tau du\tag{54}$$

The tracking position in the launch inertial coordinate system can be written as

$$\mathbf{p}\_a = \mathbf{C}\_\mathcal{g}^a(t) \cdot (\mathbf{R}\_0 + \mathbf{p}\_\mathcal{g}) - \mathbf{R}\_{0a} = \mathbf{C}\_\mathcal{g}^a(t) \cdot \mathbf{r}\_\mathcal{g} - \mathbf{R}\_{0a} \tag{55}$$

and the tracking velocity expressed in the launch inertial coordinate system is given by

$$\mathbf{V}\_a = \mathbf{C}\_g^a(t) \cdot \mathbf{V}\_g = \mathbf{C}\_g^a(t) \cdot \mathbf{C}\_e^g \cdot \mathbf{V}\_e \tag{56}$$

where *e***r** and **V***<sup>e</sup>* are the error-free tracking position and velocity expressed in the geocentric coordinate system.

By the definition of transformation matrix, we can have

$$\mathbf{C}\_{\mathcal{g}}^{a} \cdot \mathbf{C}\_{\varepsilon}^{\mathcal{g}} = \left(\mathbf{M}\_{3}(\mathcal{B}\_{\Gamma}')\mathbf{M}\_{2}(A\_{\Gamma}')\right)^{\mathrm{T}}\mathbf{M}\_{1}(-a\partial\_{0}t)\mathbf{M}\_{3}(\mathcal{B}\_{\Gamma}')\mathbf{M}\_{2}(-\frac{\pi}{2})\mathbf{M}\_{1}(\mathcal{B}\_{\Gamma}')\mathbf{M}\_{3}(-\frac{\pi}{2} + \lambda\_{\Gamma}') \tag{57}$$

Simplifying the Eq.(57) results in

$$\mathbf{C}\_{\varepsilon}^{a} = \mathbf{C}\_{g}^{a} \cdot \mathbf{C}\_{\varepsilon}^{g} = \mathbf{M}\_{2}(-A\_{T}^{\prime})\mathbf{M}\_{3}(-B\_{T}^{\prime})\mathbf{M}\_{2}(-\frac{\pi}{2})\mathbf{M}\_{3}(-\frac{\pi}{2} + \lambda\_{T}^{\prime} - a\_{0\varepsilon}t) \tag{58}$$

Substituting Eq.(58) into Eqs. (55) and (56) yields the tracking apparent velocity

$$\mathbf{W}\_{\rm tra}^{a}(t) = \mathbf{C}\_{\varepsilon}^{a}(t) \cdot \mathbf{V}\_{\varepsilon} + \mathfrak{w}\_{\varepsilon}^{a} \times \left(\mathbf{C}\_{\varepsilon}^{a}(t) \cdot \mathbf{r}\_{\varepsilon}\right) - \mathbf{V}\_{0a} - \int\_{0}^{t} \mathbf{g}\_{a}(\tau) d\tau \tag{59}$$

and the tracking apparent position

$$\mathbf{W}\_{\rm tra}^{a}\left(t\right) = \mathbf{C}\_{e}^{a}\left(t\right) \cdot \mathbf{r}\_{e} - \mathbf{R}\_{0a} - \mathbf{V}\_{0a} \cdot t - \int\_{0}^{t} \int\_{0}^{u} \mathbf{g}\_{a}\left(\tau\right) d\tau du\tag{60}$$

Error Separation Techniques Based on Telemetry and Tracking Data for Ballistic Missile 459

( () )

*A*

<sup>2</sup> ( () ) ( () )

The third term on the right-hand side of Eq.(61) can be written in expended form

(0) (0)

*a a n*

3 0 ( )( ) <sup>0</sup>

*B A*

**C C** (0) (0) *a*

*B A*

000

00 0

*aaa*

000

*B H*

 ∂ ∂∂ ∂∂ ∂ − × +× × +× × +× ∂ ∂∂ ∂∂ ∂ ′ ′′ ′′ ′

*e a aaa e a e a a e a e ae t T T T T T T*

′′ ′ *B H* , *<sup>T</sup>*

Because the telemetry data don't contain the effect of gravitational acceleration, the effect of gravitational acceleration of tracking data is necessary to drop when computing the

**<sup>ω</sup> <sup>R</sup> <sup>ω</sup> <sup>R</sup> <sup>ω</sup> <sup>R</sup> <sup>R</sup> <sup>ω</sup> <sup>R</sup> <sup>ω</sup> <sup>R</sup> <sup>ω</sup> <sup>P</sup>**

*T T*

(0) (0) (0) (0)

**CCC V V VPC P***<sup>v</sup>*

*nnn n n na sss t n*

*B A*

*<sup>T</sup>* **<sup>P</sup>***<sup>s</sup>* ≡Δ Δ Δ λ

∂∂∂ <sup>−</sup> − ⋅ ∂∂∂ ′′′

*B A* <sup>∂</sup> Δ + Δ − ⋅Δ ′ ′ ∂ ∂ ′ ′

*aaa aaa Pv e <sup>e</sup> <sup>e</sup> <sup>s</sup>*

**RRR <sup>X</sup> ωωω <sup>P</sup>**

 ∂∂∂ =− × × × ⋅ ∂∂∂ ′′ ′

*Pv a e a ns*

**X V ω R CV**

= −Δ = −Δ × − Δ ⋅

*eee T T Ta*

*B A*

**ωωω <sup>R</sup>**

*TTT*

*t tt*

*aaa aa a eee a a ee e T T T ee e T T TTT TT T*

 ∂∂∂ ∂∂ ∂ = Δ + Δ + Δ × + × Δ+ Δ + Δ ′′ ′ ′′ ′ ∂∂∂ ′′ ′ ∂∂ ∂ ′′ ′

*B A B A*

*<sup>e</sup> <sup>e</sup> T*

*a a a a <sup>a</sup> <sup>a</sup> e e a a e e a a <sup>e</sup> pv e e e e e e e e e et*

At launch, launch coordinate system coincides with launch inertial coordinate system, so the initial velocity expressed in the launch inertial coordinate system can be substituted for the

0

0 0 0 000

*B A B H*

*n na n T Ts n s*

000

*B BA A*

*v sx sy sz* ≡Δ Δ Δ *VVV* **<sup>P</sup>** .

 λ

**<sup>C</sup> VC V**

*<sup>a</sup> a a a aaa e TT T*

∂∂ ∂ ∂∂∂ − × Δ+ Δ + Δ + Δ+ Δ+ Δ ′′ ′ ′′ ′ ∂ ∂ ∂ ∂∂∂ ′′ ′ ′′ ′

**RR R RRR**

 ∂ ∂ ∂ ∂ ∂ = × × +× × +× ∂ ∂ ′ ′ ∂ ∂ ′ ′ <sup>∂</sup> ′ **<sup>C</sup> <sup>ω</sup> <sup>C</sup> <sup>ω</sup> <sup>C</sup> <sup>X</sup> <sup>ω</sup> r Cr <sup>ω</sup> r Cr <sup>ω</sup> r P** (66)

*T T T T T t t B B A A*

ω

**ωωω CC C C r <sup>ω</sup> <sup>r</sup>** (65)

*B At B A*

 λ

cos sin 0 cos cos

*a T T*

 <sup>−</sup> ′ ′ <sup>∂</sup> <sup>=</sup> <sup>∂</sup> ′ <sup>−</sup> ′ ′ **<sup>ω</sup>** .

 λ

*T T*

00 0

00 0

*BA BH*

 λ

*B A*

*B A*

**e**

(67)

(68)

<sup>2</sup> ( ) ( () ) ( () ) ( () )

=Δ × =Δ × + ×Δ

sin cos cos sin sin

*T T*

initial velocity expressed in the launch coordinate system.

*aaa*

∂∂∂ =− Δ + Δ + Δ × ′′ ′ ∂∂∂ ′′ ′

*TTT*

*B A*

*B A B*

*a T T <sup>e</sup> e T*

 <sup>−</sup> ′ ′ <sup>∂</sup> <sup>=</sup> ′ <sup>∂</sup> ′ ′ ′ **<sup>ω</sup>** , and

Thus, Eq.(65) can be rewritten as follows

λ

(0) (0)

Similarly, Eq.(67) can be rewritten in the form

*aaa*

where (0) (0) *<sup>a</sup> <sup>g</sup>* **C C** *n n* <sup>=</sup> , [ ] 00 0

λ

*TTT*

 λ

λ

*a a n n <sup>T</sup>*

λ

λ λ

λ

*T*

∂ ∂ − Δ+′ <sup>∂</sup> ′

λ

3

λ

λ

δ

4. Fourth term

**X ω C r ω C r ω C r**

δ

where

λ

*T*

ω

*B*

λ

0 *a e* λ*T* <sup>∂</sup> <sup>=</sup> <sup>∂</sup> ′ **<sup>ω</sup>** ,

> δ

3. Third term

δ

**ω**

*aa aa a a pv e e e e e e e e e*

Taking the total differentiation of Eq.(59), thus apparent velocity error is given by

$$\begin{split} \delta \mathbf{X}\_{P\boldsymbol{v}} &= \Delta \Big( \mathbf{C}\_{\varepsilon}^{a}(t) \cdot \mathbf{V}\_{\varepsilon}(t) + \mathfrak{w}\_{\varepsilon}^{a} \times \Big( \mathbf{C}\_{\varepsilon}^{a}(t) \cdot \mathbf{r}\_{\varepsilon} \Big) - \mathbf{V}\_{0a} - \int\_{0}^{t} \mathbf{g}\_{a}(\tau) d\tau \Big) \\ &= \Delta \Big( \mathbf{C}\_{\varepsilon}^{a}(t) \cdot \mathbf{V}\_{\varepsilon}(t) \Big) + \Delta \Big( \mathfrak{w}\_{\varepsilon}^{a} \times \Big( \mathbf{C}\_{\varepsilon}^{a}(t) \cdot \mathbf{r}\_{\varepsilon} \Big) \Big) - \Delta \mathbf{V}\_{0a} - \Delta \Big[ \int\_{0}^{t} \mathbf{g}\_{a}(\tau) d\tau \end{split} \tag{61}$$

Similarly, taking the total differentiation of Eq.(60) gives apparent position error

$$\begin{split} \delta \mathbf{X}\_{\mathbf{Pr}} &= \Delta \left( \mathbf{C}\_{\epsilon}^{a}(t) \cdot \mathbf{r}\_{\varepsilon} - \mathbf{R}\_{0a} - \mathbf{V}\_{0a} \cdot t - \int\_{0}^{t} \int\_{0}^{u} \mathbf{g}\_{a}(\tau) d\tau du \right) \\ &= \Delta \left( \mathbf{C}\_{\epsilon}^{a}(t) \cdot \mathbf{r}\_{\varepsilon} \right) - \Delta \mathbf{R}\_{0a} - \Delta \mathbf{V}\_{0a} \cdot t - \Delta \int\_{0}^{t} \int\_{0}^{u} \mathbf{g}\_{a}(\tau) d\tau du \end{split} \tag{62}$$

#### **4.5.1 Error analysis of apparent velocity**

It follows From Eq.(61) that the tracking apparent velocity is related to initial localization and orientation parameters, initial velocity and the calculation of attraction. To separate the initial errors, the relationship between them is needed to be analyzed. Four terms contained in Eq.(61) are taken into account as follows.

#### 1. First term

The first term on the right-hand side of Eq.(61) can be written in expended form

$$
\delta \mathbf{X}\_{pv1} = \Delta \mathbf{C}\_{\varepsilon}^{a}(t) \cdot \mathbf{V}\_{\varepsilon}(t) = \left(\frac{\partial \mathbf{C}\_{\varepsilon}^{a}}{\partial \mathcal{A}\_{T}^{\prime}} \Delta \mathcal{A}\_{T}^{\prime} + \frac{\partial \mathbf{C}\_{\varepsilon}^{a}}{\partial \mathcal{B}\_{T}^{\prime}} \Delta \mathcal{B}\_{T}^{\prime} + \frac{\partial \mathbf{C}\_{\varepsilon}^{a}}{\partial \mathcal{A}\_{T}^{\prime}} \Delta \mathcal{A}\_{T}^{\prime}\right) \cdot \mathbf{V}\_{\varepsilon}(t) \tag{63}
$$

where

cos( )sin sin( )sin cos cos( )sin cos sin( )sin 0 sin( )cos cos( )cos 0 cos( )cos sin( )sin sin cos( )sin sin *a Te T Te T T Te T Te T <sup>e</sup> Te T Te T T Te T Te T T Te T tA tB A tB A tA <sup>C</sup> t B t B t A tB A tB* λ ω λ ω λ ω λ ω λ ω λ ω λ λ ω λ ω λ ω −− +− ′ ′ ′ ′′ ′ ′′ ′ ′ −− −− <sup>∂</sup> = −−′ ′ ′ ′ <sup>−</sup> <sup>∂</sup> ′ −− −− ′ ′ ′ ′′ ′ ′ <sup>−</sup> sin( )cos 0 *A tA T Te T* λ ω ′′ ′ − − cos( )cos cos sin( )cos cos sin cos *tB A tB A B A* λωλω

$$\frac{\partial \mathbf{C}\_{\varepsilon}^{a}}{\partial B\_{\Gamma}^{\prime}} = \begin{pmatrix} -\cos(\lambda\_{\Gamma}^{\prime} - a\_{\Gamma}t)\cos B\_{\Gamma}^{\prime}\cos A\_{\Gamma}^{\prime} & -\sin(\lambda\_{\Gamma}^{\prime} - a\_{\Gamma}t)\cos B\_{\Gamma}^{\prime}\cos A\_{\Gamma}^{\prime} & -\sin B\_{\Gamma}^{\prime}\cos A\_{\Gamma}^{\prime} \\ -\cos(\lambda\_{\Gamma}^{\prime} - a\_{\Gamma}t)\sin B\_{\Gamma}^{\prime} & -\sin(\lambda\_{\Gamma}^{\prime} - a\_{\Gamma}t)\sin B\_{\Gamma}^{\prime} & \cos B\_{\Gamma}^{\prime} \\ \cos(\lambda\_{\Gamma}^{\prime} - a\_{\Gamma}t)\cos B\_{\Gamma}^{\prime}\sin A\_{\Gamma}^{\prime} & \sin(\lambda\_{\Gamma}^{\prime} - a\_{\Gamma}t)\cos B\_{\Gamma}^{\prime}\sin A\_{\Gamma}^{\prime} & \sin B\_{\Gamma}^{\prime}\sin A\_{\Gamma}^{\prime} \end{pmatrix}$$
 
$$\begin{bmatrix} -\sin(\lambda\_{\Gamma}^{\prime} - a\_{\Gamma}t)\cos A\_{\Gamma}^{\prime} + \cos(\lambda\_{\Gamma}^{\prime} - a\_{\Gamma}t)\sin B\_{\Gamma}^{\prime}\sin A\_{\Gamma}^{\prime} & \sin(\lambda\_{\Gamma}^{\prime} - a\_{\Gamma}t)\sin B\_{\Gamma}^{\prime}\sin A\_{\Gamma}^{\prime} + \cos(\lambda\_{\Gamma}^{\prime} - a\_{\Gamma}t)\cos A\_{\Gamma}^{\prime} & -\cos B\_{\Gamma}^{\prime}\sin A\_{\Gamma}^{\prime} \end{bmatrix}$$

$$\frac{\partial \underline{C}\_{\tau}}{\partial A\_{\tau}'} = \begin{bmatrix} -\sin(\lambda\_{\mathsf{T}}' - a\underline{\mu}\_{\mathsf{T}})\cos A\_{\mathsf{T}}' + \cos(\lambda\_{\mathsf{T}}' - a\underline{\mu}\_{\mathsf{T}})\sin B\_{\mathsf{T}}'\sin A\_{\mathsf{T}}' & \sin(\lambda\_{\mathsf{T}}' - a\underline{\mu}\_{\mathsf{T}})\sin B\_{\mathsf{T}}'\sin A\_{\mathsf{T}}' + \cos(\lambda\_{\mathsf{T}}' - a\underline{\mu}\_{\mathsf{T}})\cos A\_{\mathsf{T}}' & -\cos B\_{\mathsf{T}}'\sin A\_{\mathsf{T}}'\\ 0 & 0 & 0\\ \sin(\lambda\_{\mathsf{T}}' - a\underline{\mu}\_{\mathsf{T}})\sin A\_{\mathsf{T}}' + \cos(\lambda\_{\mathsf{T}}' - a\underline{\mu}\_{\mathsf{T}})\sin B\_{\mathsf{T}}'\cos A\_{\mathsf{T}}' & \sin(\lambda\_{\mathsf{T}}' - a\underline{\mu}\_{\mathsf{T}})\sin B\_{\mathsf{T}}'\cos A\_{\mathsf{T}}' - \cos(\lambda\_{\mathsf{T}}' - a\underline{\mu}\_{\mathsf{T}}')\sin B\_{\mathsf{T}}\sin A\_{\mathsf{T}}' & \sin(\lambda\_{\mathsf{T}}' - a\underline{\mu}\_{\mathsf{T}})\sin B\_{\mathsf{T}}\\ \end{bmatrix}$$

Therefore, *pv*<sup>1</sup> δ**X** can be rewritten as follows

$$
\delta \mathbf{X}\_{pv1} = \begin{bmatrix}
\frac{\partial \mathbf{C}\_{\varepsilon}^{a}}{\partial \mathcal{A}\_{\Gamma}^{\prime}} \cdot \mathbf{V}\_{\varepsilon} & \frac{\partial \mathbf{C}\_{\varepsilon}^{a}}{\partial \mathcal{B}\_{\Gamma}^{\prime}} \cdot \mathbf{V}\_{\varepsilon} & \frac{\partial \mathbf{C}\_{\varepsilon}^{a}}{\partial \mathcal{A}\_{\Gamma}^{\prime}} \cdot \mathbf{V}\_{\varepsilon} \\
\end{bmatrix} \cdot \mathbf{P}\_{t} \tag{64}$$

where [ ]*<sup>T</sup>* **<sup>P</sup>***t TT T* ≡Δ Δ Δ λ′′ ′ *B A* .

#### 2. Second term

The second term on the right-hand side of Eq.(61) can be written in expended form

$$
\begin{split}
\delta\mathbf{X}\_{\mathcal{V}\mathcal{V}} &= \Delta\Big(\mathbf{o}\mathbf{o}^{a}\_{\varepsilon}\times\big(\mathbf{C}^{a}\_{\varepsilon}(t)\mathbf{r}\_{\varepsilon}\big)\Big) = \Delta\mathbf{o}^{a}\_{\varepsilon}\times\big(\mathbf{C}^{a}\_{\varepsilon}(t)\mathbf{r}\_{\varepsilon}\big) + \mathbf{o}\mathbf{o}^{a}\_{\varepsilon}\times\Delta\big(\mathbf{C}^{a}\_{\varepsilon}(t)\mathbf{r}\_{\varepsilon}\big) \\
&= \left(\frac{\partial\mathbf{o}^{a}\_{\varepsilon}}{\partial\mathcal{A}^{\prime}\_{T}}\Delta\mathcal{I}^{\prime}\_{T} + \frac{\partial\mathbf{o}^{a}\_{\varepsilon}}{\partial\mathcal{B}^{\prime}\_{T}}\Delta\mathcal{B}^{\prime}\_{T} + \frac{\partial\mathbf{o}^{a}\_{\varepsilon}}{\partial\mathcal{A}^{\prime}\_{T}}\Delta\mathcal{A}^{\prime}\_{T}\right)\times\big(\mathbf{C}^{a}\_{\varepsilon}(t)\mathbf{r}\_{\varepsilon}\big) + \mathbf{o}\mathbf{o}^{a}\_{\varepsilon}\times\Big(\frac{\partial\mathbf{C}^{a}\_{\varepsilon}}{\partial\mathcal{A}^{\prime}\_{T}}\Delta\mathcal{A}^{\prime}\_{T} + \frac{\partial\mathbf{C}^{a}\_{\varepsilon}}{\partial\mathcal{B}^{\prime}\_{T}}\Delta\mathcal{B}^{\prime}\_{T} + \frac{\partial\mathbf{C}^{a}\_{\varepsilon}}{\partial\mathcal{A}^{\prime}\_{T}}\Delta\mathcal{A}^{\prime}\_{T}\Big)\mathbf{r}\_{\varepsilon}\tag{65}
$$

where

458 Modern Telemetry

( ) ( )

() () ( ) ( )

**C V ω Cr V g**

*tt t d*

( )

( ) ( )

0 0 0 0

*t t d du*

( ) ( )

It follows From Eq.(61) that the tracking apparent velocity is related to initial localization and orientation parameters, initial velocity and the calculation of attraction. To separate the initial errors, the relationship between them is needed to be analyzed. Four terms contained

> <sup>1</sup> () () ( ) *aaa <sup>a</sup> eee pv e e <sup>T</sup> <sup>T</sup> T e*

cos( )sin sin( )sin cos cos( )sin cos sin( )sin 0

′′ ′ − −

cos( )cos cos sin( )cos cos sin cos cos( )sin sin( )sin cos cos( )cos sin sin( )cos sin sin sin

λ ω

sin( )cos cos( )sin sin sin( )sin sin cos( )cos cos sin

λ ω

λ ω

*aaa eee pv e e et TTT B A*

 ∂∂∂ = ⋅ ⋅ ⋅⋅ ∂∂∂ ′′ ′

*a Te T Te T T Te T T Te T T T*

*tA tB A tB A tA*

−− + − ′ ′ ′ ′′ ′ ′′ ′ ′ ′′ − +− − <sup>∂</sup> <sup>=</sup> <sup>∂</sup> ′ ′ ′ ′ ′′ ′ ′′ ′ ′ − +− − −− − os cos *B A T T* ′ ′

λ ω

*a Te T T Te T T T T <sup>e</sup> Te T Te T <sup>T</sup>*

<sup>−</sup> ′ ′′ ′ ′′ ′′ − − − − <sup>∂</sup> =− − ′′ ′′ ′ − − <sup>∂</sup> ′ ′ ′′ ′ ′′ ′′ − −

*<sup>C</sup> t B tB B <sup>B</sup> tBA tBA BA*

sin( )sin cos( )sin cos sin( )sin cos cos( )sin c

*Te T Te T T Te T T Te T*

*tA tB A tB A tA*

 λ λ

*a Te T Te T T Te T Te T*

−− +− ′ ′ ′ ′′ ′ ′′ ′ ′ −− −− <sup>∂</sup> = −−′ ′ ′ ′ <sup>−</sup> <sup>∂</sup> ′ −− −− ′ ′ ′ ′′ ′ ′ <sup>−</sup> sin( )cos 0 *A tA T Te T*

∂∂∂ =Δ ⋅ = Δ + Δ + Δ ⋅ ′′ ′ ∂∂∂ ′′ ′

*TT T t t B At*

*B A*

λ ω

> λ ω

λ ω

**CCC X CV <sup>V</sup>** (63)

sin( )cos cos( )cos 0

*tB A tB A B A*

0 0 0

**CCC X V V VP** (64)

λ ω

λ ω

*Te T T Te T T T T*

*t A tB A tB A t A B A*

λ ω

*t u a ee a a a*

**Cr R V g**

*t u a ee a a a*

0 0 0 0

*t t d du*

*t a aa Pv e e e e e a a t a a a e e eee a a*

() () () ( )

*tt t d*

<sup>0</sup> <sup>0</sup>

τ τ

> τ τ

(61)

(62)

λ ω

λ ω

<sup>0</sup> <sup>0</sup>

τ τ

τ τ

Taking the total differentiation of Eq.(59), thus apparent velocity error is given by

=Δ ⋅ + × ⋅ − −

**X CV ω C rV g**

=Δ ⋅ +Δ × ⋅ −Δ −Δ

Similarly, taking the total differentiation of Eq.(60) gives apparent position error

=Δ ⋅ − − ⋅ −

**X C rR V g Pr**

=Δ ⋅ −Δ −Δ ⋅ −Δ

The first term on the right-hand side of Eq.(61) can be written in expended form

cos( )cos sin( )sin sin cos( )sin sin

λ ω

λ ω

λ ω

λ ω

> λ ω

λ ω

δ

′′ ′ *B A* .

**X** can be rewritten as follows

1

λ

The second term on the right-hand side of Eq.(61) can be written in expended form

λ ω

λ ω

*Te T Te T T Te T*

*<sup>C</sup> t B t B t A tB A tB*

*<sup>e</sup> Te T Te T*

( ) () ( )

( )

δ

> δ

**4.5.1 Error analysis of apparent velocity** 

in Eq.(61) are taken into account as follows.

δ

λ ω

λ ω

*T*

λ ω

λ ω

where [ ]*<sup>T</sup>* **<sup>P</sup>***t TT T* ≡Δ Δ Δ λ

Therefore, *pv*<sup>1</sup> δ

2. Second term

1. First term

where

*T*

λ

*e T*

*C A*

$$\frac{\partial \mathbf{o}\_{\varepsilon}^{a}}{\partial \boldsymbol{A}\_{\boldsymbol{T}}^{\prime}} = \boldsymbol{0} \; \prime \; \frac{\partial \mathbf{o}\_{\varepsilon}^{a}}{\partial \boldsymbol{B}\_{\boldsymbol{T}}^{\prime}} = a \boldsymbol{\varepsilon} \begin{pmatrix} -\sin \boldsymbol{B}\_{\boldsymbol{T}}^{\prime} \cos \boldsymbol{A}\_{\boldsymbol{T}}^{\prime} \\ \cos \boldsymbol{B}\_{\boldsymbol{T}}^{\prime} \\ \sin \boldsymbol{B}\_{\boldsymbol{T}}^{\prime} \sin \boldsymbol{A}\_{\boldsymbol{T}}^{\prime} \end{pmatrix} \; \text{and} \; \frac{\partial \mathbf{o}\_{\varepsilon}^{a}}{\partial \boldsymbol{A}\_{\boldsymbol{T}}^{\prime}} = a \boldsymbol{\varepsilon} \begin{pmatrix} -\cos \boldsymbol{B}\_{\boldsymbol{T}}^{\prime} \sin \boldsymbol{A}\_{\boldsymbol{T}}^{\prime} \\ \boldsymbol{0} \\ -\cos \boldsymbol{B}\_{\boldsymbol{T}}^{\prime} \cos \boldsymbol{A}\_{\boldsymbol{T}}^{\prime} \end{pmatrix}.$$

Thus, Eq.(65) can be rewritten as follows

$$\delta \mathbf{X}\_{pv2} = \left[ \mathbf{o}\_{\varepsilon}^{a} \times \frac{\partial \mathbf{C}\_{\varepsilon}^{a}}{\partial \mathcal{A}\_{\Gamma}^{\prime}} \mathbf{r}\_{\varepsilon} \quad \frac{\partial \mathbf{o}\_{\varepsilon}^{a}}{\partial B\_{\Gamma}^{\prime}} \times (\mathbf{C}\_{\varepsilon}^{a}(t) \mathbf{r}\_{\varepsilon}) + \mathbf{o}\_{\varepsilon}^{a} \times \frac{\partial \mathbf{C}\_{\varepsilon}^{a}}{\partial B\_{\Gamma}^{\prime}} \mathbf{r}\_{\varepsilon} \quad \frac{\partial \mathbf{o}\_{\varepsilon}^{a}}{\partial A\_{\Gamma}^{\prime}} \times (\mathbf{C}\_{\varepsilon}^{a}(t) \mathbf{r}\_{\varepsilon}) + \mathbf{o}\_{\varepsilon}^{a} \times \frac{\partial \mathbf{C}\_{\varepsilon}^{a}}{\partial A\_{\Gamma}^{\prime}} \mathbf{r}\_{\varepsilon} \right] \mathbf{P}\_{t} \quad \text{(66)}$$

#### 3. Third term

At launch, launch coordinate system coincides with launch inertial coordinate system, so the initial velocity expressed in the launch inertial coordinate system can be substituted for the initial velocity expressed in the launch coordinate system.

The third term on the right-hand side of Eq.(61) can be written in expended form

3 0 ( )( ) <sup>0</sup> 0 0 0 0 000 00 0 00 0 (0) (0) (0) (0) *a a n Pv a e a ns aaa eee T T Ta TTT <sup>a</sup> a a a aaa e TT T TTT a a n n <sup>T</sup> T B A B A BA BH B A B H* δ λ λ λ λ λ λ λ λ = −Δ = −Δ × − Δ ⋅ ∂∂∂ =− Δ + Δ + Δ × ′′ ′ ∂∂∂ ′′ ′ ∂∂ ∂ ∂∂∂ − × Δ+ Δ + Δ + Δ+ Δ+ Δ ′′ ′ ′′ ′ ∂ ∂ ∂ ∂∂∂ ′′ ′ ′′ ′ ∂ ∂ − Δ+′ <sup>∂</sup> ′ **X V ω R CV ωωω <sup>R</sup> RR R RRR ω C C** (0) (0) *a n na n T Ts n s T T B A B A* <sup>∂</sup> Δ + Δ − ⋅Δ ′ ′ ∂ ∂ ′ ′ **<sup>C</sup> VC V** (67)

Similarly, Eq.(67) can be rewritten in the form

000 3 00 0 000 000 (0) (0) (0) (0) *aaa aaa Pv e <sup>e</sup> <sup>e</sup> <sup>s</sup> aaa e a aaa e a e a a e a e ae t T T T T T T aaa nnn n n na sss t n TTT B H B BA A B A* δ λ λ λ λ ∂∂∂ =− × × × ⋅ ∂∂∂ ′′ ′ ∂ ∂∂ ∂∂ ∂ − × +× × +× × +× ∂ ∂∂ ∂∂ ∂ ′ ′′ ′′ ′ ∂∂∂ <sup>−</sup> − ⋅ ∂∂∂ ′′′ **RRR <sup>X</sup> ωωω <sup>P</sup> <sup>ω</sup> <sup>R</sup> <sup>ω</sup> <sup>R</sup> <sup>ω</sup> <sup>R</sup> <sup>R</sup> <sup>ω</sup> <sup>R</sup> <sup>ω</sup> <sup>R</sup> <sup>ω</sup> <sup>P</sup> CCC V V VPC P***<sup>v</sup>* (68)

where (0) (0) *<sup>a</sup> <sup>g</sup>* **C C** *n n* <sup>=</sup> , [ ] 00 0 *<sup>T</sup>* **<sup>P</sup>***<sup>s</sup>* ≡Δ Δ Δ λ′′ ′ *B H* , *<sup>T</sup> v sx sy sz* ≡Δ Δ Δ *VVV* **<sup>P</sup>** .

4. Fourth term

Because the telemetry data don't contain the effect of gravitational acceleration, the effect of gravitational acceleration of tracking data is necessary to drop when computing the

Error Separation Techniques Based on Telemetry and Tracking Data for Ballistic Missile 461

*aa a*

*TT T*

**RRR <sup>C</sup>**

∂∂ ∂ =− Δ + Δ + Δ ⋅ ′′ ′ ∂∂ ∂ ′′ ′

λ λ

( ) ( )

( )

0 2

− ++ ′ ′′ ′ <sup>∂</sup> <sup>=</sup> ′ ′′ + + ′ <sup>∂</sup> ′

λ

*<sup>e</sup> e e*

= − +− + ′′ ′ ′ <sup>∂</sup> ′

*<sup>e</sup> <sup>e</sup> e e*

0

*H*

*e*

0

λ

λ

λ

0 00 0 0 2 0 00

*NH B NH B*

000

*<sup>a</sup> eee <sup>e</sup>*

∂∂∂ − ⋅ Δ+ Δ+ Δ ′′ ′ ∂∂∂ ′′ ′

00 0 0

1 sin

0 00 0

*e e*

 α

**<sup>R</sup>** (78)

 α

 λ

**X** = −Δ × − Δ ⋅ ⋅ **ω R CV** *t t* (82)

**<sup>R</sup>** (80)

000

00 0

 α

cos sin [ (1 sin )] cos cos [ (1 sin )] 0

*B Ha B B Ha B*

0 00 0

00 0 0

*Ba H a B*

 α

*Ba H a B*

 α

**<sup>R</sup>** (79)

<sup>1</sup> cos sin [ (1 3cos2 )] <sup>2</sup>

*e e e*

− +− + ′′ ′ ′

<sup>1</sup> sin sin [ (1 3cos2 )] <sup>2</sup>

00 0 0

0 0 0

0 0

λ

λ

0 0

0

*B*

<sup>3</sup> cos [ 2 ( 1 2 )(1 cos2 )] <sup>2</sup>

′ ′ + − − −+ − ′

*Ba H a a B*

*e ee ee e*

αα

cos cos cos sin sin

*B B*

*eee aaa eee Pr e e et e e e s TTT B A B H*

Similarly, launch coordinate system coincides with launch inertial coordinate system at launch moment, so the radius of earth center in the launch inertial coordinate system can be

> 3 0 ( )( ) (0) *a an Pr e a n s*

 ∂∂∂ ∂∂∂ = − ⋅ − <sup>⋅</sup> ∂∂∂ ∂∂∂ ′′ ′ ′′ ′ **CCC RRR X R R R PC C C P** (81)

′ ′ <sup>∂</sup> <sup>=</sup> ′ ′ <sup>∂</sup> ′ ′

*N eH B*

<sup>+</sup> ′ ′′ = + ′ ′′ − + ′ ′

cos cos cos sin

*ee e T Te*

*B A*

**CC C <sup>R</sup>**

*B A*

00 0

00 0

λ

λ

**R** (77)

2

*B H*

*B H*

0

(76)

The second term on the right-hand side of Eq.(73) can be written in expended form

2 00

= −Δ ⋅ − ⋅ Δ

**X CR C R**

*e*

0

2 000

represented by that in the launch coordinate system, thus,

δ

*aaa*

λ

0

0

*B*

∂

Thus, Eq.(76) can be rewritten as

λ

δ

3. Third term

λ λ

*a a Pr eee e*

δ

It follows from the previous section that

Therefore, we can have that

2. Second term

difference between telemetry data and tracking data. Integrating gravitational acceleration one can obtain the velocity and perform the integration again to obtain the position. It is noted that the tracking data is used to calculate the gravitational acceleration. It follows from the previous section that the gravitational acceleration in the launch inertial coordinate system is given by

$$\mathbf{g}\_{a} = g\_{r}\frac{\mathbf{C}\_{\varepsilon}^{a} \cdot \mathbf{r}\_{\varepsilon}}{r} + g\_{a\boldsymbol{\theta}} \frac{\mathbf{C}\_{\varepsilon}^{a} \cdot \mathbf{o}\_{c0}}{a\boldsymbol{\theta}\_{\varepsilon}} = \mathbf{C}\_{\varepsilon}^{a} \cdot \left(g\_{r}\frac{\mathbf{r}\_{\varepsilon}}{r} + g\_{a\boldsymbol{\theta}}\frac{\mathbf{o}\_{c0}}{a\boldsymbol{\theta}\_{\varepsilon}}\right) \tag{69}$$

By examining Eq.(69) we can find that the main reason introducing the computational error of gravitational acceleration is that there exist errors in the Euler angles of transformation matrix *<sup>a</sup>* **C***<sup>e</sup>* , whereas the bracketed term on the right-hand side of Eq.(69) is error-free. It is noted that

$$\sin\varphi\_{\varepsilon} = \frac{\mathbf{r}\_{\varepsilon} \cdot \mathbf{o}\_{\varepsilon a}}{r o\_{\varepsilon}} = \frac{\left(\mathbf{C}\_{\varepsilon}^{a}\mathbf{r}\_{\varepsilon}\right) \cdot \left(\mathbf{C}\_{\varepsilon}^{a}\mathbf{o}\_{\varepsilon 0}\right)}{r o\_{\varepsilon}} = \frac{\left(\mathbf{C}\_{\varepsilon}^{a}\mathbf{r}\_{\varepsilon}\right)^{\intercal} \left(\mathbf{C}\_{\varepsilon}^{a}\mathbf{o}\_{\varepsilon 0}\right)}{r o\_{\varepsilon}} = \frac{\mathbf{r}\_{\varepsilon} \cdot \mathbf{o}\_{\varepsilon 0}}{r o\_{\varepsilon}}\tag{70}$$

which can be computed exactly, thus, the error of gravitational acceleration is given by

$$\Delta \mathbf{g}\_a = \Delta \mathbf{C}\_e^a \cdot \left( \mathbf{g}\_r \frac{\mathbf{r}\_e}{r} + \mathbf{g}\_{ao} \frac{\mathbf{o}\_{e0}}{a\boldsymbol{\alpha}\_e} \right) = \begin{bmatrix} \frac{\partial}{\partial \lambda\_\Gamma'} \mathbf{g}\_e & \frac{\partial \mathbf{C}\_e^a}{\partial B\_\Gamma'} \mathbf{g}\_e & \frac{\partial \mathbf{C}\_e^a}{\partial A\_\Gamma'} \mathbf{g}\_e \end{bmatrix} \mathbf{P}\_t = \mathbf{G}\_g \cdot \mathbf{P}\_t \tag{71}$$

where *e e*<sup>0</sup> *e r e g g r* ω ω = + **r ω g** .

The error of tracking apparent velocity is given by

$$
\delta \mathbf{X}\_{Pv4} = -\int\_0^t \Delta \mathbf{g}\_a(\tau) d\tau \cdot \mathbf{P}\_t = -\int\_0^t \mathbf{G}\_{\mathcal{S}}(\tau) d\tau \cdot \mathbf{P}\_t \tag{72}
$$

#### **4.5.2 Error analysis of apparent position**

Recalling Eq.(62) gives apparent position error

$$\begin{split} \delta \mathbf{X}\_{\mathbf{Pr}} &= \Delta \Big( \mathbf{C}\_{\varepsilon}^{a}(t) \cdot \mathbf{r}\_{\varepsilon} - \mathbf{R}\_{0a} - \mathbf{V}\_{0a} \cdot t - \int\_{0}^{t} \int\_{0}^{u} \mathbf{g}\_{a}(\tau) d\tau du \Big) \\ &= \Delta \Big( \mathbf{C}\_{\varepsilon}^{a}(t) \cdot \mathbf{r}\_{\varepsilon} \Big) - \Delta \mathbf{R}\_{0a} - \Delta \mathbf{V}\_{0a} \cdot t - \Delta \int\_{0}^{t} \int\_{0}^{u} \mathbf{g}\_{a}(\tau) d\tau du \end{split} \tag{73}$$

In the similar manner four terms contained in Eq.(73) are analyzed as follows.

1. First term

The first term on the right-hand side of Eq.(73) can be written in expended form

$$
\delta \mathbf{X}\_{Pr1} = \Delta \mathbf{C}\_{\varepsilon}^{a} \cdot \mathbf{r}\_{\varepsilon} = \left( \frac{\partial \mathbf{C}\_{\varepsilon}^{a}}{\partial \mathcal{A}\_{T}^{\prime}} \Delta \mathcal{X}^{\prime} + \frac{\partial \mathbf{C}\_{\varepsilon}^{a}}{\partial \mathcal{B}\_{T}^{\prime}} \Delta \mathcal{B}\_{T}^{\prime} + \frac{\partial \mathbf{C}\_{\varepsilon}^{a}}{\partial \mathcal{A}\_{T}^{\prime}} \Delta \mathcal{A}\_{T}^{\prime} \right) \cdot \mathbf{r}\_{\varepsilon} \tag{74}$$

Rearranging Eq.(74) gives

$$
\delta \mathcal{S} \mathbf{X}\_{p\_{\mathrm{I}}} = \left[ \frac{\partial \mathbf{C}\_{\mathrm{e}}^{a}}{\partial \mathcal{A}\_{\mathrm{T}}^{\prime}} \cdot \mathbf{r}\_{\mathrm{e}} \quad \frac{\partial \mathbf{C}\_{\mathrm{e}}^{a}}{\partial \mathcal{B}\_{\mathrm{T}}^{\prime}} \cdot \mathbf{r}\_{\mathrm{e}} \quad \frac{\partial \mathbf{C}\_{\mathrm{e}}^{a}}{\partial \mathcal{A}\_{\mathrm{T}}^{\prime}} \cdot \mathbf{r}\_{\mathrm{e}} \right] \cdot \mathbf{P}\_{\mathrm{i}} \tag{75}
$$

#### 2. Second term

460 Modern Telemetry

difference between telemetry data and tracking data. Integrating gravitational acceleration one can obtain the velocity and perform the integration again to obtain the position. It is noted that the tracking data is used to calculate the gravitational acceleration. It follows from the previous section that the gravitational acceleration in the launch inertial coordinate

*ee e e <sup>a</sup> e e a r e r*

By examining Eq.(69) we can find that the main reason introducing the computational error of gravitational acceleration is that there exist errors in the Euler angles of transformation matrix *<sup>a</sup>* **C***<sup>e</sup>* , whereas the bracketed term on the right-hand side of Eq.(69) is error-free. It is

> 0 00 ( )( ) ( )( ) sin *a a aT a a ea e e e e e e e e e e <sup>e</sup> ee ee rr rr*

*aaa <sup>a</sup> ee e e e a er e e et <sup>g</sup> <sup>t</sup>*

*r B A*

<sup>4</sup> 0 0 ( ) ( ) *t t Pv a t g t*

 =− Δ ⋅ =− ⋅ ττ

 λ

 ∂∂∂ Δ =Δ ⋅ + = = ⋅ ∂∂∂ ′′ ′ **r ω CCC**

*eTTT*

( )

( ) ( )

0 0 0 0

*t t d du*

( ) ( )

*t u a ee a a a*

**Cr R V g**

 λ λ

=Δ ⋅ − − ⋅ −

**X C rR V g Pr**

=Δ ⋅ −Δ −Δ ⋅ −Δ

In the similar manner four terms contained in Eq.(73) are analyzed as follows.

The first term on the right-hand side of Eq.(73) can be written in expended form

λ

*t u a ee a a a*

0 0 0 0

*aa a <sup>a</sup> ee e e e T Te TT T*

∂∂ ∂ =Δ ⋅ = Δ + Δ + Δ ⋅ ′′ ′ ∂∂ ∂ ′′ ′

*aaa eee Pr e e et TTT B A*

∂∂∂ = ⋅ ⋅ ⋅⋅ ∂∂∂ ′′′

*t t d du*

*B A*

**CC C X Cr <sup>r</sup>** *Pr* (74)

**g C ggg P GP** (71)

which can be computed exactly, thus, the error of gravitational acceleration is given by

*g g gg r r*

ω

*a a*

ωω

ω ω

*g g*

*e*

The error of tracking apparent velocity is given by

δ

( )

1

δ

ω ω

**4.5.2 Error analysis of apparent position**  Recalling Eq.(62) gives apparent position error

> δ

> > 1

δ

 = + **r ω g** . 0

ω

0 0

 ττ*d d* **X g PG P** (72)

> τ τ

> > (73)

τ τ

*B A*

*CCC <sup>X</sup> r r rP* (75)

⋅ ⋅ <sup>⋅</sup> == = = **<sup>r</sup> <sup>ω</sup> Cr C <sup>ω</sup> Cr C <sup>ω</sup> <sup>r</sup> <sup>ω</sup>** (70)

**Cr C <sup>ω</sup> <sup>r</sup> <sup>ω</sup> g C** (69)

 ω

 ω

ωω

*e e*

⋅ ⋅ = + =⋅ +

system is given by

noted that

where *e e*<sup>0</sup> *e r*

1. First term

Rearranging Eq.(74) gives

*g g r*

ϕ

The second term on the right-hand side of Eq.(73) can be written in expended form

$$\begin{split} \delta \mathbf{X}\_{p\_{T}2} &= -\Delta \mathbf{C}\_{\varepsilon}^{a} \cdot \mathbf{R}\_{0\varepsilon} - \mathbf{C}\_{\varepsilon}^{a} \cdot \Delta \mathbf{R}\_{0\varepsilon} \\ &= -\left(\frac{\partial \mathbf{C}\_{\varepsilon}^{a}}{\partial \mathcal{A}\_{T}'} \Delta \mathcal{A}' + \frac{\partial \mathbf{C}\_{\varepsilon}^{a}}{\partial B\_{T}'} \Delta B\_{T}' + \frac{\partial \mathbf{C}\_{\varepsilon}^{a}}{\partial A\_{T}'} \Delta A\_{T}'\right) \cdot \mathbf{R}\_{0\varepsilon} \\ &- \mathbf{C}\_{\varepsilon}^{a} \cdot \left(\frac{\partial \mathbf{R}\_{0\varepsilon}}{\partial \mathcal{A}\_{0}'} \Delta \mathcal{A}\_{0}' + \frac{\partial \mathbf{R}\_{0\varepsilon}}{\partial B\_{0}'} \Delta B\_{0}' + \frac{\partial \mathbf{R}\_{0\varepsilon}}{\partial H\_{0}'} \Delta H\_{0}'\right) \end{split} \tag{76}$$

It follows from the previous section that

$$\mathbf{R}\_{0e} = \begin{bmatrix} (N\_0 + H\_0') \cos B\_0' \cos \lambda\_0' \\ (N\_0 + H\_0') \cos B\_0' \sin \lambda\_0' \\ \left[ N\_0 \left( 1 - e^2 \right) + H\_0' \right] \sin B\_0' \end{bmatrix} \tag{77}$$

Therefore, we can have that

$$\frac{\partial \mathbf{R}\_{0\varepsilon}}{\partial \lambda\_0'} = \begin{bmatrix} -\cos B\_0' \sin \lambda\_0' [H\_0' + a\_\varepsilon (1 + \alpha\_\varepsilon \sin^2 B\_0')] \\ \cos B\_0' \cos \lambda\_0' [H\_0' + a\_\varepsilon (1 + \alpha\_\varepsilon \sin^2 B\_0')] \\ 0 \end{bmatrix} \tag{78}$$

$$\frac{\partial \mathbf{R}\_{0e}}{\partial \mathcal{S}\_{0}'} = \begin{pmatrix} -\cos \mathcal{X}\_{0}' \sin \mathcal{B}\_{0}' [a\_{e} + H\_{0}' - \frac{1}{2} a\_{e} \alpha\_{e} (1 + 3 \cos 2B\_{0}')] \\\\ -\sin \mathcal{X}\_{0}' \sin \mathcal{B}\_{0}' [a\_{e} + H\_{0}' - \frac{1}{2} a\_{e} \alpha\_{e} (1 + 3 \cos 2B\_{0}')] \\\\ \cos \mathcal{B}\_{0}' [a\_{e} + H\_{0}' - 2a\_{e} \alpha\_{e} - \frac{3}{2} a\_{e} \alpha\_{e} (-1 + 2\alpha\_{e}) (1 - \cos 2B\_{0}')] \end{pmatrix} \tag{79}$$

$$\frac{\partial \mathbf{R}\_{0e}}{\partial H\_{0}'} = \begin{pmatrix} \cos \mathcal{B}\_{0}' \cos \mathcal{A}\_{0}' \\\\ \cos \mathcal{B}\_{0}' \sin \mathcal{A}\_{0}' \\\\ \sin \mathcal{B}\_{0}' \end{pmatrix} \tag{80}$$

Thus, Eq.(76) can be rewritten as

$$
\delta \mathbf{X}\_{P\tau2} = -\left[\frac{\partial \mathbf{C}\_{\varepsilon}^{d}}{\partial \mathcal{A}\_{\Gamma}^{\prime}} \mathbf{R}\_{0\varepsilon} \quad \frac{\partial \mathbf{C}\_{\varepsilon}^{d}}{\partial \mathcal{B}\_{\Gamma}^{\prime}} \mathbf{R}\_{0\varepsilon} \quad \frac{\partial \mathbf{C}\_{\varepsilon}^{d}}{\partial \mathcal{A}\_{\Gamma}^{\prime}} \mathbf{R}\_{0\varepsilon}\right] \cdot \mathbf{P}\_{t} - \left[\mathbf{C}\_{\varepsilon}^{d} \frac{\partial \mathbf{R}\_{0\varepsilon}}{\partial \mathcal{A}\_{0}^{\prime}} \quad \mathbf{C}\_{\varepsilon}^{d} \frac{\partial \mathbf{R}\_{0\varepsilon}}{\partial \mathcal{B}\_{0}^{\prime}} \quad \mathbf{C}\_{\varepsilon}^{d} \frac{\partial \mathbf{R}\_{0\varepsilon}}{\partial \mathcal{H}\_{0}^{\prime}}\right] \cdot \mathbf{P}\_{s} \tag{81}
$$

#### 3. Third term

Similarly, launch coordinate system coincides with launch inertial coordinate system at launch moment, so the radius of earth center in the launch inertial coordinate system can be represented by that in the launch coordinate system, thus,

$$\Delta \delta \mathbf{X}\_{Pr3} = -\Delta \left( \mathbf{o} \mathbf{o}\_e^a \times \mathbf{R}\_{0a} \right) \mathbf{t} - \Delta \left( \mathbf{C}\_n^a(\mathbf{0}) \cdot \mathbf{V}\_s^n \right) \cdot \mathbf{t} \tag{82}$$

Error Separation Techniques Based on Telemetry and Tracking Data for Ballistic Missile 463

 λ

∂∂ ∂ ∂ ∂

(0) *aa a <sup>a</sup> ee e a n a n st <sup>e</sup> <sup>e</sup> a e <sup>s</sup> TT T T T*

**C C <sup>ω</sup> R C G rR R <sup>ω</sup> <sup>V</sup>** (93)

**C C <sup>ω</sup> R C G rR R <sup>ω</sup> <sup>V</sup>** (94)

**C C <sup>ω</sup> R C G rR R <sup>ω</sup> <sup>V</sup>** (95)

*BB HH*

00 00 00 00 00 00 *aa aa aa ea ea ea sse e e e e e ttt*

= ⋅ − − × + × ⋅− ⋅ ∂∂ ∂ ∂ ∂ ′′ ′ ′ ′

(0) *aa a <sup>a</sup> ee e a n a n st <sup>e</sup> <sup>e</sup> a e <sup>s</sup> TT T T T*

= ⋅ − − × + × ⋅− ⋅ ∂∂ ∂ ∂ ∂ ′′ ′ ′ ′

(0) *aa a <sup>a</sup> ee e a n a n st <sup>e</sup> <sup>e</sup> a e <sup>s</sup> TT T T T*

= ⋅ − − × + × ⋅− ⋅ ∂∂ ∂ ∂ ∂ ′′ ′ ′ ′

 ∂∂ ∂∂ ∂∂ =− − × − − × − − × ∂∂ ∂∂ ∂∂ ′′ ′′ ′′ *RR RR RR G C <sup>ω</sup> <sup>C</sup> <sup>ω</sup> <sup>C</sup> <sup>ω</sup>* (96)

**X XX** <sup>=</sup> , [ ]*<sup>T</sup>* **P PPP** *a tsv* <sup>=</sup> , then the difference between the telemetry data

*vt g vs vv <sup>v</sup> <sup>P</sup> a a t u <sup>s</sup> st <sup>g</sup> ss sv*

 <sup>−</sup> <sup>=</sup> ⋅= ⋅ <sup>−</sup>

By examining the above model, we can find that the correlation of the environmental function column corresponding to the geodetic latitude and height in the velocity domain,

them is not easy. But in the position domain, the property of initial error environmental function matrix is good therefore, the separation of initial errors is needed to perform in the

It is pointed out in the previous section that the guidance instrumentation systematic errors are contained in the telemetry data and the initial errors are primarily introduced during the data processing of tracking data. Consequently, in addition to the alignment errors and levelling errors of inertial platform and initial error parameters, the other error coefficients are separated. It follows from Eqs.(51) and (98) that the relationship involved in instrumentation error coefficients and initial errors as well as the difference between

where **S** is the environmental function matrix of instrumentation errors and **G** is the environmental function matrix of initial errors. This model is known as the separation

**G G GG G X P P <sup>G</sup> G G GG**

*BB B B B* ∂∂ ∂ ∂ ∂

*AA A A A* ∂∂ ∂ ∂ ∂

1 00

2 00

3 00

 λ

0

*t*

*dt*

*d du*

τ

0 0

0 *<sup>a</sup> <sup>a</sup> <sup>e</sup> <sup>H</sup>* ∂ − ×

∂ ′ **R**

**4.6 Separation model of instrumentation errors and initial errors** 

telemetry data and tracking data, which can be described as follows

model of instrumentation errors and initial errors and it is a linear model.

λλ

λ

Let [ ]*<sup>T</sup> P Pv Pr*

 δδ

namely, <sup>0</sup>

δ

(0) , *<sup>a</sup>*

and tracking data can be written in matrix form

δ

**ω** and <sup>0</sup>

position domain or velocity-position domain.

0 *<sup>a</sup> <sup>a</sup> <sup>e</sup> <sup>B</sup>* ∂ − ×

∂ ′ **R**

0

0

0

*sv n g g* **G C sv** =− ⋅ Δ =Δ *t* (97)

**ω** in the **G***vs* matrix, is large and the separation between

*<sup>a</sup>* **δX SD** = ⋅ −⋅ + **G P ε** (99)

 λ *t t*

*t t*

*t t*

(98)

 λ

Combing the analysis of apparent velocity gives

$$
\delta \mathbf{X}\_{Pr3} = \delta \mathbf{X}\_{Pr3} \cdot t \tag{83}
$$

4. Fourth term

The fourth term is the gravitational acceleration term, which can be obtained by integrating the error of apparent tracking velocity, written as

$$
\delta X\_{Pr4} = -\int\_0^t \delta X\_{Pv4} d\tau = -\int\_0^t \int\_0^u G\_{\mathcal{S}}(\tau) d\tau d\mu \tag{84}
$$

#### **4.5.3 Relationship of the difference between telemetry data, tracking data and initial errors**

According to the above analysis, the relationship of the difference between telemetry velocity and tracking velocity and initial errors can be concluded as follows

$$\begin{split} \delta \mathbf{X}\_{Pv} &= \delta \mathbf{X}\_{Pv1} + \delta \mathbf{X}\_{Pv2} + \delta \mathbf{X}\_{Pv3} + \delta \mathbf{X}\_{Pv4} \\ &= \mathbf{G}\_{vt} \cdot \mathbf{P}\_t + \mathbf{G}\_{vs} \cdot \mathbf{P}\_s + \mathbf{G}\_{vv} \cdot \mathbf{P}\_v + \Delta \mathbf{v}\_\chi \end{split} \tag{85}$$

where

$$\mathbf{G}\_{vt1} = \frac{\partial \mathbf{C}\_{\varepsilon}^{a}}{\partial \mathcal{A}\_{\Gamma}^{\prime}} \cdot \mathbf{V}\_{\varepsilon} + \mathbf{o} \boldsymbol{\omega}\_{\varepsilon}^{a} \times \frac{\partial \mathbf{C}\_{\varepsilon}^{a}}{\partial \mathcal{A}\_{\Gamma}^{\prime}} \mathbf{r}\_{\varepsilon} - \left(\frac{\partial \mathbf{o}\_{\varepsilon}^{a}}{\partial \mathcal{A}\_{\Gamma}^{\prime}} \times \mathbf{R}\_{0a} + \mathbf{o} \boldsymbol{\varepsilon}\_{\varepsilon}^{a} \times \frac{\partial \mathbf{R}\_{0a}}{\partial \mathcal{A}\_{\Gamma}^{\prime}}\right) - \frac{\partial \mathbf{C}\_{n}^{a}(0)}{\partial \mathcal{A}\_{\Gamma}^{\prime}} \mathbf{V}\_{s}^{n} \tag{86}$$

$$\mathbf{G}\_{vt2} = \frac{\partial \mathbf{C}\_{\varepsilon}^{a}}{\partial \mathbf{B}\_{\Gamma}^{\prime}} \cdot \mathbf{V}\_{\varepsilon} + \frac{\partial \mathbf{o}\_{\varepsilon}^{a}}{\partial \mathbf{B}\_{\Gamma}^{\prime}} \times (\mathbf{C}\_{\varepsilon}^{a} \mathbf{r}\_{\varepsilon}) + \mathbf{o}\_{\varepsilon}^{a} \times \frac{\partial \mathbf{C}\_{\varepsilon}^{a}}{\partial \mathbf{B}\_{\Gamma}^{\prime}} \mathbf{r}\_{\varepsilon} - \left(\frac{\partial \mathbf{o}\_{\varepsilon}^{a}}{\partial \mathbf{B}\_{\Gamma}^{\prime}} \times \mathbf{R}\_{0a} + \mathbf{o}\_{\varepsilon}^{a} \times \frac{\partial \mathbf{R}\_{0a}}{\partial \mathbf{B}\_{\Gamma}^{\prime}}\right) - \frac{\partial \mathbf{C}\_{\imath}^{a}(0)}{\partial \mathbf{B}\_{\Gamma}^{\prime}} \mathbf{V}\_{s}^{\prime} \tag{87}$$

$$\mathbf{G}\_{\mathrm{ret}3} = \frac{\partial \mathbf{C}\_{\varepsilon}^{a}}{\partial A\_{\Gamma}^{\prime}} \cdot \mathbf{V}\_{\varepsilon} + \frac{\partial \mathbf{o}\_{\varepsilon}^{a}}{\partial A\_{\Gamma}^{\prime}} \times (\mathbf{C}\_{\varepsilon}^{a} \mathbf{r}\_{\varepsilon}) + \mathbf{o}\_{\varepsilon}^{a} \times \frac{\partial \mathbf{C}\_{\varepsilon}^{a}}{\partial A\_{\Gamma}^{\prime}} \mathbf{r}\_{\varepsilon} - \left(\frac{\partial \mathbf{o}\_{\varepsilon}^{a}}{\partial A\_{\Gamma}^{\prime}} \times \mathbf{R}\_{0a} + \mathbf{o}\_{\varepsilon}^{a} \times \frac{\partial \mathbf{R}\_{0a}}{\partial A\_{\Gamma}^{\prime}}\right) - \frac{\partial \mathbf{C}\_{\pi}^{a}(0)}{\partial A\_{\Gamma}^{\prime}} \mathbf{V}\_{s}^{\pi} \tag{88}$$

$$\mathbf{G}\_{\rm cs} = \begin{bmatrix} -\mathbf{o}\mathbf{o}\_{\varepsilon}^{a} \times \frac{\partial \mathbf{R}\_{0a}}{\partial \mathcal{J}\_{0}^{\prime}} & -\mathbf{o}\mathbf{o}\_{\varepsilon}^{a} \times \frac{\partial \mathbf{R}\_{0a}}{\partial B\_{0}^{\prime}} & -\mathbf{o}\mathbf{o}\_{\varepsilon}^{a} \times \frac{\partial \mathbf{R}\_{0a}}{\partial H\_{0}^{\prime}} \end{bmatrix} \tag{89}$$

(0) *<sup>a</sup>* **G C** *vv n* = − (90)

$$
\Delta \dot{\mathbf{v}}\_{\mathcal{g}} = -\Delta \mathbf{g}\_{a} = -\begin{bmatrix}
\frac{\partial \mathbf{C}\_{\varepsilon}^{a}}{\partial \mathcal{A}\_{\Gamma}^{\prime}} \mathbf{g}\_{\varepsilon} & \frac{\partial \mathbf{C}\_{\varepsilon}^{a}}{\partial \mathcal{B}\_{\Gamma}^{\prime}} \mathbf{g}\_{\varepsilon} & \frac{\partial \mathbf{C}\_{\varepsilon}^{a}}{\partial \mathcal{A}\_{\Gamma}^{\prime}} \mathbf{g}\_{\varepsilon} \end{bmatrix} \mathbf{P}\_{t} \tag{91}
$$

In the same manner the relationship of the difference between telemetry position and tracking position and initial errors can be concluded as follows

$$\begin{split} \delta \mathbf{X}\_{Pr} &= \delta \mathbf{X}\_{Pr1} + \delta \mathbf{X}\_{Pr2} + \delta \mathbf{X}\_{Pr3} + \delta \mathbf{X}\_{Pr4} \\ &= \mathbf{G}\_{st} \cdot \mathbf{P}\_t + \mathbf{G}\_{ss} \cdot \mathbf{P}\_s + \mathbf{G}\_{sv} \cdot \mathbf{P}\_v + \Delta \mathbf{s}\_{\mathcal{g}} \end{split} \tag{92}$$

where

*Pr Pv* 3 3

The fourth term is the gravitational acceleration term, which can be obtained by integrating

4 4 <sup>0</sup> 0 0 ( ) *<sup>t</sup> t u X X d G d du Pr Pv <sup>g</sup>*

 τ

**4.5.3 Relationship of the difference between telemetry data, tracking data and initial** 

According to the above analysis, the relationship of the difference between telemetry

*Pv Pv Pv Pv Pv*

 =+++ = ⋅ + ⋅ + ⋅ +Δ **XX X X X**

∂ ∂ ∂ ∂∂ = ⋅+ × − × + × − ∂ ∂∂ ∂∂ ′ ′′ ′′

 δ

λλ

1234

*vt t vs s vv v g*

(0) *a aa <sup>a</sup> e ee a an a n vt ee e <sup>e</sup> <sup>s</sup>*

**C C <sup>ω</sup> R C G V <sup>ω</sup> r R <sup>ω</sup> <sup>V</sup>** (86)

(0) ( )

(0) ( )

**RRR <sup>G</sup> ωωω** (89)

**CCC v g g g gP** (91)

 δ

**GPGPGP s** (92)

000 00 0

 *B H* ∂∂∂ =− × − × − × ∂∂∂ ′′ ′

*aaa*

*TTT B A* ∂∂∂

1234

*st t ss s sv v g*

 δ

*T TT TT*

*a a a a a e e a a e e a a n n*

*a a a a a e e a a e e a a n n*

*vt <sup>e</sup> ee e e a e <sup>s</sup> A A AA AA T T TT TT* ∂ ∂ ∂ ∂ ∂∂ = ⋅+ × + × − × + × − ∂ ∂ ∂∂ ∂∂ ′ ′ ′′ ′′ **<sup>C</sup> <sup>ω</sup> <sup>C</sup> <sup>ω</sup> R C G V Cr <sup>ω</sup> r R <sup>ω</sup> <sup>V</sup>** (88)

*aaa aaa vse e e*

(0) *<sup>a</sup>* **G C** *vv n* = − (90)

*eee <sup>g</sup> <sup>a</sup> e e et*

λ

 δ

Δ = −Δ = − ∂∂∂ ′′ ′

In the same manner the relationship of the difference between telemetry position and

*Pr Pr Pr Pr Pr*

 =+++ = ⋅ + ⋅ + ⋅ +Δ **XX X X X**

*vt <sup>e</sup> ee e e a e <sup>s</sup> B B BB BB T T TT TT* ∂ ∂ ∂ ∂ ∂∂ = ⋅+ × + × − × + × − ∂ ∂ ∂∂ ∂∂ ′ ′ ′′ ′′ **<sup>C</sup> <sup>ω</sup> <sup>C</sup> <sup>ω</sup> R C G V Cr <sup>ω</sup> r R <sup>ω</sup> <sup>V</sup>** (87)

 δ

**X X** = ⋅*t* (83)

 ττ=− =− (84)

> δ

**GPGPG P v** (85)

0

λλ

0

0

 δ

δ

 δ

velocity and tracking velocity and initial errors can be concluded as follows

Combing the analysis of apparent velocity gives

the error of apparent tracking velocity, written as

δ

λ

δδ

1 0a

2 0

3 0

tracking position and initial errors can be concluded as follows

δδ

λ

4. Fourth term

**errors** 

where

where

$$\mathbf{G}\_{\text{s}t1} = \frac{\partial \mathbf{C}\_{\varepsilon}^{a}}{\partial \mathcal{A}\_{T}^{\prime}} \cdot \mathbf{r}\_{\varepsilon} - \frac{\partial \mathbf{C}\_{\varepsilon}^{a}}{\partial \mathcal{A}\_{T}^{\prime}} \mathbf{R}\_{\varepsilon 0} - \left(\frac{\partial \mathbf{o}\_{\varepsilon}^{a}}{\partial \mathcal{A}\_{T}^{\prime}} \times \mathbf{R}\_{0a} + \mathbf{o}\_{\varepsilon}^{a} \times \frac{\partial \mathbf{R}\_{0a}}{\partial \mathcal{A}\_{T}^{\prime}}\right) \cdot \mathbf{t} - \frac{\partial \mathbf{C}\_{n}^{a}(0)}{\partial \mathcal{A}\_{T}^{\prime}} \mathbf{V}\_{s}^{n} \cdot \mathbf{t} \tag{93}$$

$$\mathbf{G}\_{st2} = \frac{\partial \mathbf{C}\_{\varepsilon}^{a}}{\partial \mathbf{B}\_{T}^{\prime}} \cdot \mathbf{r}\_{\varepsilon} - \frac{\partial \mathbf{C}\_{\varepsilon}^{a}}{\partial \mathbf{B}\_{T}^{\prime}} \mathbf{R}\_{\varepsilon 0} - \left(\frac{\partial \mathbf{o}\_{\varepsilon}^{a}}{\partial \mathbf{B}\_{T}^{\prime}} \times \mathbf{R}\_{0a} + \mathbf{o}\_{\varepsilon}^{a} \times \frac{\partial \mathbf{R}\_{0a}}{\partial \mathbf{B}\_{T}^{\prime}}\right) \cdot t - \frac{\partial \mathbf{C}\_{n}^{a}(0)}{\partial \mathbf{B}\_{T}^{\prime}} \mathbf{V}\_{s}^{n} \cdot t \tag{94}$$

$$\mathbf{G}\_{st3} = \frac{\partial \mathbf{C}\_{\varepsilon}^{a}}{\partial A\_{T}^{\prime}} \cdot \mathbf{r}\_{\varepsilon} - \frac{\partial \mathbf{C}\_{\varepsilon}^{a}}{\partial A\_{T}^{\prime}} \mathbf{R}\_{\varepsilon 0} - \left(\frac{\partial \mathbf{o}\_{\varepsilon}^{a}}{\partial A\_{T}^{\prime}} \times \mathbf{R}\_{0a} + \mathbf{o}\_{\varepsilon}^{a} \times \frac{\partial \mathbf{R}\_{0a}}{\partial A\_{T}^{\prime}}\right) \cdot t - \frac{\partial \mathbf{C}\_{n}^{a}(0)}{\partial A\_{T}^{\prime}} \mathbf{V}\_{s}^{n} \cdot t \tag{95}$$

$$\mathbf{G}\_{ss} = \begin{bmatrix} -\mathbf{C}\_{\varepsilon} \frac{\partial \mathbf{R}\_{0\varepsilon}}{\partial \lambda\_{0}^{\prime}} - \alpha\_{\varepsilon}^{a} \times \frac{\partial \mathbf{R}\_{0a}}{\partial \lambda\_{0}^{\prime}} t & -\mathbf{C}\_{\varepsilon}^{a} \frac{\partial \mathbf{R}\_{0\varepsilon}}{\partial B\_{0}^{\prime}} - \alpha\_{\varepsilon}^{a} \times \frac{\partial \mathbf{R}\_{0a}}{\partial B\_{0}^{\prime}} t & -\mathbf{C}\_{\varepsilon}^{a} \frac{\partial \mathbf{R}\_{0\varepsilon}}{\partial H\_{0}^{\prime}} - \alpha\_{\varepsilon}^{a} \times \frac{\partial \mathbf{R}\_{0a}}{\partial H\_{0}^{\prime}} t \end{bmatrix} \tag{96}$$

$$\mathbf{G}\_{sv} = -\mathbf{C}\_n^t(0) \cdot t \; \; \; \Delta \dot{\mathbf{s}}\_g = \Delta \mathbf{v}\_g \tag{97}$$

Let [ ]*<sup>T</sup> P Pv Pr* δ δδ **X XX** <sup>=</sup> , [ ]*<sup>T</sup>* **P PPP** *a tsv* <sup>=</sup> , then the difference between the telemetry data and tracking data can be written in matrix form

$$
\delta \mathbf{X}\_P = \begin{vmatrix} \mathbf{G}\_{vt} - \int\_0^t \mathbf{G}\_{g} dt & \mathbf{G}\_{vs} & \mathbf{G}\_{vv} \\\\ \mathbf{G}\_{st} - \int\_0^t \int\_0^u \mathbf{G}\_{g} d\tau du & \mathbf{G}\_{ss} & \mathbf{G}\_{sv} \end{vmatrix} \cdot \mathbf{P}\_a = \begin{bmatrix} \mathbf{G}\_v \\\\ \mathbf{G}\_s \end{bmatrix} \cdot \mathbf{P}\_a \tag{98}$$

By examining the above model, we can find that the correlation of the environmental function column corresponding to the geodetic latitude and height in the velocity domain, namely, <sup>0</sup> 0 *<sup>a</sup> <sup>a</sup> <sup>e</sup> <sup>B</sup>* ∂ − × ∂ ′ **R ω** and <sup>0</sup> 0 *<sup>a</sup> <sup>a</sup> <sup>e</sup> <sup>H</sup>* ∂ − × ∂ ′ **R ω** in the **G***vs* matrix, is large and the separation between

them is not easy. But in the position domain, the property of initial error environmental function matrix is good therefore, the separation of initial errors is needed to perform in the position domain or velocity-position domain.

#### **4.6 Separation model of instrumentation errors and initial errors**

It is pointed out in the previous section that the guidance instrumentation systematic errors are contained in the telemetry data and the initial errors are primarily introduced during the data processing of tracking data. Consequently, in addition to the alignment errors and levelling errors of inertial platform and initial error parameters, the other error coefficients are separated. It follows from Eqs.(51) and (98) that the relationship involved in instrumentation error coefficients and initial errors as well as the difference between telemetry data and tracking data, which can be described as follows

$$
\delta \mathbf{X} = \mathbf{S} \cdot \mathbf{D} - \mathbf{G} \cdot \mathbf{P}\_4 + \varepsilon \tag{99}
$$

where **S** is the environmental function matrix of instrumentation errors and **G** is the environmental function matrix of initial errors. This model is known as the separation model of instrumentation errors and initial errors and it is a linear model.

Error Separation Techniques Based on Telemetry and Tracking Data for Ballistic Missile 465

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
