**1. Introduction**

An intercontinental ballistic missile (ICBM) is a ballistic missile with a long range (some greater than 10000 km) and great firepower typically designed for nuclear weapons delivery, such as PeaceKeeper (PK) missile (Shattuck, 1992), Minutesman missile (Tony C. L., 2003). Due to the long-distance flight, the requirement for navigation system is rigorous and only gimbaled inertial navigation system (INS) is presently competent, such as the advanced inertial reference sphere (AIRS) used in the PK missile (John L., 1979), yet the strapdown inertial navigation system is generally not used on the intercontinental ballistic missile because of the poor accuracy (Titterton & Weston, 1997). The gimbaled inertial navigation system typically contains three single-degree-of-freedom rate integrating gyros, three mutually perpendicular single-axis accelerometers, a loop system and other auxiliary system, providing an orientation of the inertial navigation platform relative to inertial space. Due to system design and production technology there exist a lot of errors referred as guidance instrumentation systematic errors (IEEE Standards Committee, 1971; IEEE Standards Board, 1973), which have an important effect on impact accuracy of ballistic missile. Before the flight of ballistic missile, the guidance instrumentation systematic errors are need to calibrate, and then the calibration results are used to compensate the instrumental errors, which has been discussed in depth by Thompson (Thompson, 2000), Eduardo and Hugh (Eduardo & Hugh, 1999), Jackson (Jackson, 1973), Coulter and Meehan (Coulter & Meehan , 1981). Some content discussed has been issued as IEEE standard (IEEE Standards Committee, 1971; IEEE Standards Board, 1973).

However, the guidance instrumentation systematic errors cannot be completely compensated by using the calibration results. Therefore, flight test of ballistic missile is usually performed to qualify the performance. Because of different objectives of test or some other reasons specific testing trajectory is sometimes adopted, and herein the flight test cannot reflect the actual situation of ballistic missile in the whole trajectory. Consequently, it is necessary to analyze the landing errors resulted from guidance instrumentation systematic errors in the specific trajectory and convert them into those landing errors in the case of the whole trajectory.

In fact, there are many factors affecting the impact accuracy of ballistic missile, such as gravity anomaly, upper atmosphere, electromagnetic force, etc. Forsberg and Sideris has

Error Separation Techniques Based on Telemetry and Tracking Data for Ballistic Missile 445

coordinate system, the other is to subtract the gravitational acceleration from tracking data or to add gravitational acceleration into telemetry data. The difference between telemetry and tracking data can be reckoned in either launch inertial coordinate system or launch coordinate system. A typical method is to convert the tracking data into launch inertial coordinate system and then to subtract the gravitational acceleration. In fact, guidance instrumentation systematic errors are contained in the telemetry data while initial launch parameter errors are generated in the case of the conversion for tracking data and the

The apparent velocity and position in the launch inertial coordinate system can be

The transformation matrix from geocentric coordinate system to launch coordinate system

sin sin cos sin cos sin cos cos sin sin cos cos cos cos cos sin sin cos sin sin sin cos cos cos sin sin sin sin cos

− − <sup>−</sup> <sup>=</sup> − + + −

λλ

λλ

where subscript *e* denotes geocentric coordinate system and superscript *g* denotes launch

respectively. Also, the transformation matrix relating launch coordinate system to launch

cos cos sin sin cos 1 0 0 cos sin cos sin sin , 0 cos sin sin 0 cos 0 sin cos

 <sup>−</sup> = − <sup>=</sup> <sup>−</sup> **A B** (3)

<sup>0</sup> 2 2

0 0 (1 sin ) *Na B* = + *e e* α

Thus, the components of launch site in the geocentric coordinate system are written as

α α

<sup>−</sup> <sup>=</sup> − −

(1 ) 1 (2 )sin *e e e e*

α

2

*TT T TT e e T T e e*

*AB B AB t t A A t t*

*TT TT T T T TTT T T T T T T T TT TT T T T TTT T T*

 λ

(1)

 λ

 λ

*<sup>T</sup>* are astronomical azimuth, latitude and longitude,

*a TT* **C ABA** *<sup>g</sup>* = (2)

ω

ω

ω

*<sup>e</sup>* is the earth flattening, *B*<sup>0</sup> is the geodetic latitude.

0

*B*

 ω

 ω

(4)

(5)

*<sup>e</sup>* is the earth rate, *t* is the

*A AB A AB AB B BB A AB A AB AB*

computation of gravity acceleration, so the sources of them are absolutely different.

λ

*TT T TT*

*<sup>a</sup> <sup>N</sup>*

α

*AB B AB*

where superscript *a* denotes launch inertial coordinate system,

The radius of prime vertical circle of launch site is given by

2. Radius vector from earth center to launch site

where *<sup>e</sup> a* is the earth semimajor axis,

Ignoring higher-order terms yields

computed as follows. 1. Transformation matrix

can be represented by

*g*

with

in-flight time.

0 0 <sup>2</sup> 1 3 [ (90 )] [ ] [ (90 )]

λ

λ

*e TT T*

*A B*

=−+ − −

λ

**CM M M**

coordinate system; *AT* , *BT* ,

λ

inertial coordinate system is given by

taken into account the effect of gravity anomaly and presented the analysis method (Forsberg & Sideris, 1993). The effect of upper atmosphere and electromagnetic force is considered by Zheng (Zheng, 2006), but these error factors are so small compared to guidance instrumentation systematic errors that they are capable of not being considered when analyzing the impact accuracy. The analysis of guidance instrumentation systematic errors is generally performed using telemetry data and tracking data. Telemetry data are the angular velocity and acceleration information measured by inertial navigation system on the ballistic missile and transmitted by telemetric equipment, while tracking data are those information measured by radar and optoelectronic device in the test range. 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, which is independent of instrumentation errors (Liu et al, 2000). Comparison of telemetry data and tracking data is used to obtain the velocity and position errors resulted from guidance instrumentation systematic errors. It is noticeable that the telemetry data are measured in the inertial coordinate system and exclude gravitational acceleration information while tracking data usually measured in the horizontal coordinate system. The conversion of two types of data into identical coordinate system is necessary.

Maneuvering launch manners are commonly adopted such as road-launched and submarine-launched manners to improve the viability and strike capacity for ballistic missile. Maneuvering launch ballistic missile especially for submarine-launched ballistic missile is often affected by ocean current, wave, and vibration environment, etc. Obviously, there are measurement errors in the initial launch parameters including location and orientation parameters as well as carrier's velocity. Theoretical analysis and numerical simulation indicate that initial launch parameter errors are equivalent in magnitude to the guidance instrumentation systematic errors (Zheng, 2006; Gore, ). Since the landing errors due to initial launch parameter errors and guidance instrumentation systematic errors are coupled, the error separation procedure for those two types of errors must be performed using telemetry and tracking data.

The error separation model can be simplified as a linear model using telemetry and tracking data (Yang et al, 2007). It is noted that the linear model is directly obtained by telemetry and tracking data and is independent of the flight of ballisitc missile. The remarkable features of this linear model is high dimension and collinearity, which is a severe problem when one wishes to perform certain types of mathematical treatment such as matrix inversion. These categories of problem can be treated many advanced methods, such as improved regression estimation (Barros & Rutledge, 1998; Cherkassky & Ma, 2005), partial least square (PLS) method (Wold et al, 2001), and support vector machines (SVM) (Cortes & Vapnik, 1995), however, these analysis methods are of no interest in this chapter. This chapter mainly focuses on the modeling of separation of instrumentation errors based on telemetry and tracking data and presents a novel error separation technique.

#### **2. Calculation of difference between telemetry and tracking data**

Telemetry and tracking data are known as important information sources in the error separation procedure. Two key problems are needed to be solved when computing the difference between telemetry and tracking data, since they are described in different coordinate systems. One is to convert the telemetry and tracking data into the same coordinate system, the other is to subtract the gravitational acceleration from tracking data or to add gravitational acceleration into telemetry data. The difference between telemetry and tracking data can be reckoned in either launch inertial coordinate system or launch coordinate system. A typical method is to convert the tracking data into launch inertial coordinate system and then to subtract the gravitational acceleration. In fact, guidance instrumentation systematic errors are contained in the telemetry data while initial launch parameter errors are generated in the case of the conversion for tracking data and the computation of gravity acceleration, so the sources of them are absolutely different.

The apparent velocity and position in the launch inertial coordinate system can be computed as follows.

#### 1. Transformation matrix

The transformation matrix from geocentric coordinate system to launch coordinate system can be represented by

$$\begin{aligned} \mathbf{C}\_{\varepsilon}^{\mathcal{S}} &= \mathbf{M}\_{2} [- (90^{0} + A\_{T})] \mathbf{M}\_{1} [B\_{\mathrm{T}}] \mathbf{M}\_{3} [- (90^{0} - A\_{\mathrm{T}})] \\ &= \begin{bmatrix} -\sin A\_{T} \sin \mathcal{A}\_{\mathrm{T}} - \cos A\_{T} \sin B\_{\mathrm{T}} \cos \mathcal{A}\_{\mathrm{T}} & \sin A\_{T} \cos \mathcal{A}\_{\mathrm{T}} - \cos A\_{\mathrm{T}} \sin B\_{\mathrm{T}} \sin \mathcal{A}\_{\mathrm{T}} & \cos A\_{T} \cos B\_{\mathrm{T}} \\ \cos B\_{\mathrm{T}} \cos \mathcal{A}\_{\mathrm{T}} & \cos B\_{\mathrm{T}} \sin \mathcal{A}\_{\mathrm{T}} & \sin B\_{\mathrm{T}} \end{bmatrix} \\ &- \cos A\_{T} \sin \mathcal{A}\_{\mathrm{T}} + \sin A\_{T} \sin B\_{\mathrm{T}} \cos \mathcal{A}\_{\mathrm{T}} & \cos A\_{T} \cos \mathcal{A}\_{\mathrm{T}} + \sin A\_{T} \sin B\_{\mathrm{T}} \sin \mathcal{A}\_{\mathrm{T}} & - \sin A\_{T} \cos B\_{\mathrm{T}} \end{bmatrix} \tag{1} \\ \begin{bmatrix} \text{1} & \text{1} \end{bmatrix} \end{aligned}$$

where subscript *e* denotes geocentric coordinate system and superscript *g* denotes launch coordinate system; *AT* , *BT* , λ*<sup>T</sup>* are astronomical azimuth, latitude and longitude, respectively. Also, the transformation matrix relating launch coordinate system to launch inertial coordinate system is given by

$$\mathbf{C}\_{\mathcal{S}}^{a} = \mathbf{A}^{T}\mathbf{B}^{T}\mathbf{A} \tag{2}$$

with

444 Modern Telemetry

taken into account the effect of gravity anomaly and presented the analysis method (Forsberg & Sideris, 1993). The effect of upper atmosphere and electromagnetic force is considered by Zheng (Zheng, 2006), but these error factors are so small compared to guidance instrumentation systematic errors that they are capable of not being considered when analyzing the impact accuracy. The analysis of guidance instrumentation systematic errors is generally performed using telemetry data and tracking data. Telemetry data are the angular velocity and acceleration information measured by inertial navigation system on the ballistic missile and transmitted by telemetric equipment, while tracking data are those information measured by radar and optoelectronic device in the test range. 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, which is independent of instrumentation errors (Liu et al, 2000). Comparison of telemetry data and tracking data is used to obtain the velocity and position errors resulted from guidance instrumentation systematic errors. It is noticeable that the telemetry data are measured in the inertial coordinate system and exclude gravitational acceleration information while tracking data usually measured in the horizontal coordinate system. The conversion of two types of data into identical coordinate system is

Maneuvering launch manners are commonly adopted such as road-launched and submarine-launched manners to improve the viability and strike capacity for ballistic missile. Maneuvering launch ballistic missile especially for submarine-launched ballistic missile is often affected by ocean current, wave, and vibration environment, etc. Obviously, there are measurement errors in the initial launch parameters including location and orientation parameters as well as carrier's velocity. Theoretical analysis and numerical simulation indicate that initial launch parameter errors are equivalent in magnitude to the guidance instrumentation systematic errors (Zheng, 2006; Gore, ). Since the landing errors due to initial launch parameter errors and guidance instrumentation systematic errors are coupled, the error separation procedure for those two types of errors must be performed

The error separation model can be simplified as a linear model using telemetry and tracking data (Yang et al, 2007). It is noted that the linear model is directly obtained by telemetry and tracking data and is independent of the flight of ballisitc missile. The remarkable features of this linear model is high dimension and collinearity, which is a severe problem when one wishes to perform certain types of mathematical treatment such as matrix inversion. These categories of problem can be treated many advanced methods, such as improved regression estimation (Barros & Rutledge, 1998; Cherkassky & Ma, 2005), partial least square (PLS) method (Wold et al, 2001), and support vector machines (SVM) (Cortes & Vapnik, 1995), however, these analysis methods are of no interest in this chapter. This chapter mainly focuses on the modeling of separation of instrumentation errors based on telemetry and

Telemetry and tracking data are known as important information sources in the error separation procedure. Two key problems are needed to be solved when computing the difference between telemetry and tracking data, since they are described in different coordinate systems. One is to convert the telemetry and tracking data into the same

necessary.

using telemetry and tracking data.

tracking data and presents a novel error separation technique.

**2. Calculation of difference between telemetry and tracking data** 

$$\mathbf{A} = \begin{bmatrix} \cos A\_T \cos B\_T & \sin B\_T & -\sin A\_T \cos B\_T \\ -\cos A\_T \sin B\_T & \cos B\_T & \sin A\_T \sin B\_T \\ \sin A\_T & 0 & \cos A\_T \end{bmatrix}, \mathbf{B} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos \alpha\_t t & \sin \alpha\_t t \\ 0 & -\sin \alpha\_t t & \cos \alpha\_t t \end{bmatrix} \tag{3}$$

where superscript *a* denotes launch inertial coordinate system, ω*<sup>e</sup>* is the earth rate, *t* is the in-flight time.

2. Radius vector from earth center to launch site

The radius of prime vertical circle of launch site is given by

$$N\_0 = \frac{a\_\varepsilon (1 - \alpha\_\varepsilon)}{\sqrt{1 - (2\alpha\_\varepsilon - \alpha\_\varepsilon^2) \sin^2 B\_0}}\tag{4}$$

where *<sup>e</sup> a* is the earth semimajor axis,α *<sup>e</sup>* is the earth flattening, *B*<sup>0</sup> is the geodetic latitude. Ignoring higher-order terms yields

$$N\_0 = a\_e (1 + \mathcal{Q}\_e \sin^2 B\_0) \tag{5}$$

Thus, the components of launch site in the geocentric coordinate system are written as

Error Separation Techniques Based on Telemetry and Tracking Data for Ballistic Missile 447

arcsin *<sup>g</sup> <sup>e</sup>*

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*

0

= − <sup>−</sup>

ω

<sup>−</sup>

*eay eax*

and position of missile in the launch coordinate system provided by tracking data, respectively. **V**0*<sup>a</sup>* is the initial velocity of launch site with respect to launch coordinate

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

The telemetric apparent velocity can be obtained by the integration of telemetric apparent

tele tele <sup>0</sup> () ( ) *<sup>t</sup> a a t d* <sup>=</sup>

tele tele <sup>0</sup> () ( ) *<sup>t</sup> a a t d* <sup>=</sup>

The difference between telemetry data and tracking data is obtained by subtracting synchronous tracking data and compensation from telemetry data, namely, we can have the

τ τ

τ τ

δ**X** *t* .

 ω

*eaz eax*

*eax eay eaz* are three components of **ω***ea* , respectively; **V***<sup>g</sup>* and **ρ***<sup>g</sup>* are the velocity

ω= ⋅ +⋅ ⋅ + − −

0

ω

ω

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

Hence, gravitational acceleration in the launch inertial coordinate system is written as

φ

5. Calculation of apparent velocity and position of tracking data

*a*

ω

6. Calculation of apparent velocity and position of telemetry data

7. Calculation of difference between telemetry and tracking data

difference between telemetry velocity and tracking velocity, ( ) *<sup>v</sup>*

Integrating Eq.(20) gives the telemetric apparent position

between telemetry velocity and tracking velocity, ( ) *<sup>r</sup>*

The tracking apparent velocity is given by

system due to earth rotation, written as

Likewise, the tracking apparent position is given by

with

where , , ωωω

acceleration, given by

*g e*

**W CV <sup>Ω</sup> C R <sup>ρ</sup> V g** (16)

*eaz eay*

 ω

> ω

> > 0

*d du* **W CR <sup>ρ</sup> RV g** (19)

**Ω** (17)

0 0 (0) **V** *a ea a* = × **ω R** (18)

τ τ

**W W** (20)

**W W** (21)

δ

**X** *t* , and the difference

(14)

τ τ

*<sup>a</sup>* **g Cg** *<sup>a</sup>* = ⋅ *g g* (15)

**r ω r ω**

$$\mathbf{R}\_{0\epsilon} = \begin{bmatrix} (N\_0 + H\_0) \cos B\_0 \cos \mathcal{A}\_0 \\ (N\_0 + H\_0) \cos B\_0 \sin \mathcal{A}\_0 \\ \left( N\_0 \left( 1 - \alpha\_\epsilon^2 \right) + H\_0 \right) \sin B\_0 \end{bmatrix} \tag{6}$$

where λ<sup>0</sup> , *B*<sup>0</sup> , *H*<sup>0</sup> are the geodetic longitude, geodetic latitude and geodetic height of launch site, respectively. Using coordinate transformation we can write the radius vector from earth center to launch site in the launch coordinate system as

$$\mathbf{R}\_{0g} = \mathbf{C}\_{\varepsilon}^{\mathcal{S}} \mathbf{R}\_{0e} \tag{7}$$

#### 3. Earth rate

The components of earth rate expressed in the launch coordinate system are given by

$$\mathbf{u}\mathbf{o}\_{\text{eg}} = \mathbf{C}\_{\epsilon}^{\mathcal{S}} \begin{bmatrix} 0\\0\\0\\o\_{\epsilon} \end{bmatrix} = a\mathbf{o}\_{\epsilon} \begin{bmatrix} \cos B\_T \cos A\_T\\\sin B\_T\\-\cos B\_T \sin A\_T \end{bmatrix} \tag{8}$$

The angular velocity of launch coordinate system with respect to launch inertial coordinate system is the earth rate, so earth rate expressed in the launch inertial coordinate system is given by

$$
\mathfrak{so}\_{ca} = \mathbf{C}\_{\mathcal{X}}^{\mathfrak{a}} \cdot \mathfrak{so}\_{\mathfrak{e}\mathfrak{x}} \tag{9}
$$

#### 4. Gravitational acceleration

The radius vector from earth center to center of mass of missile in the launch coordinate system is given by

$$\mathbf{r}\_{\mathcal{g}} = \mathbf{R}\_{0\mathcal{g}} + \mathbf{p}\_{\mathcal{g}} \tag{10}$$

where **ρ***<sup>g</sup>* is the missile location provided by tracking data.

The gravitational acceleration taking into account the 2*J* term in the launch coordinate system is given by

$$\mathbf{g}\_{\mathcal{S}} = \mathbf{g}\_r \cdot \frac{\mathbf{r}\_{\mathcal{S}}}{\left| \mathbf{r}\_{\mathcal{S}} \right|} + \mathbf{g}\_{\mathcal{o}o} \cdot \frac{\mathbf{o}\_{\mathcal{eq}}}{o\_e} \tag{11}$$

where

$$\mathcal{g}\_r = -\frac{\mu}{r\_\mathcal{g}^2} \cdot \left[1 + \mathcal{I}\_2 \cdot \left(\frac{a\_e}{r\_\mathcal{g}}\right)^2 \cdot \left(1 - 5\sin^2\phi\right)\right] \tag{12}$$

$$\mathcal{g}\_{\alpha\nu} = -2\frac{\mu}{r\_{\mathcal{S}}^2} \cdot \mathcal{J}\_2 \cdot (\frac{a\_e}{r\_{\mathcal{S}}})^2 \cdot \sin\phi \tag{13}$$

and the geocentric latitudeφcan be computed as follows

$$\phi = \arcsin \frac{\mathbf{r}\_{\text{g}} \cdot \mathbf{o}\_{\text{e}}}{\left| \mathbf{r}\_{\text{g}} \cdot \mathbf{o}\_{\text{e}} \right|} \tag{14}$$

Hence, gravitational acceleration in the launch inertial coordinate system is written as

$$\mathbf{g}\_a = \mathbf{C}\_\mathcal{g}^a \cdot \mathbf{g}\_\mathcal{g} \tag{15}$$

5. Calculation of apparent velocity and position of tracking data The tracking apparent velocity is given by

$$\mathbf{W}\_{\rm tra}^{a}(t) = \mathbf{C}\_{\rm g}^{a}(t) \cdot \mathbf{V}\_{\rm g}(t) + \mathbf{Q}\_{\rm o}^{a} \cdot \mathbf{C}\_{\rm g}^{a}(t) \cdot \left(\mathbf{R}\_{0} + \mathbf{p}\_{\rm g}\right) - \mathbf{V}\_{0a} - \int\_{0}^{t} \mathbf{g}\_{a}(\tau) d\tau \tag{16}$$

with

446 Modern Telemetry

00 0 0

1 sin

<sup>0</sup> , *B*<sup>0</sup> , *H*<sup>0</sup> are the geodetic longitude, geodetic latitude and geodetic height of launch

*N HB*

<sup>+</sup> = + − +

cos cos cos sin λ

λ

**R** (6)

*<sup>g</sup>* **R CR** *<sup>g</sup>* <sup>=</sup> *e e* (7)

*<sup>a</sup>* **ω***ea* = ⋅ **C***<sup>g</sup>* **ω***eg* (9)

*g gg* = + <sup>0</sup> **r R ρ** (10)

(11)

(12)

(13)

( ) ( )

α

*e*

site, respectively. Using coordinate transformation we can write the radius vector from earth

0 0

0 cos cos 0 sin

= = <sup>−</sup>

cos sin

*e T T*

*T T*

**ω C** (8)

*B A B B A*

The components of earth rate expressed in the launch coordinate system are given by

*eg e e T*

ω

The angular velocity of launch coordinate system with respect to launch inertial coordinate system is the earth rate, so earth rate expressed in the launch inertial coordinate system is

The radius vector from earth center to center of mass of missile in the launch coordinate

The gravitational acceleration taking into account the 2*J* term in the launch coordinate

*g g*

**r**

=⋅ + ⋅ **r ω**

<sup>2</sup> <sup>2</sup> [1 ( ) (1 5sin )] *<sup>e</sup> <sup>r</sup>*

*g r*

*<sup>a</sup> g J r r*

μ

ω

φ

*g g*

*<sup>a</sup> g J r r*

can be computed as follows

μ=− ⋅ ⋅ ⋅

=− ⋅ + ⋅ ⋅ −

**g**

*g eg*

ω ω

2 2

φ

2 <sup>2</sup> <sup>2</sup> 2 ( ) sin *<sup>e</sup> g g*

φ

*g e*

*g*

ω

0 00 0 0 2 0 00

*NH B NH B*

( ) ( )

*e*

center to launch site in the launch coordinate system as

where **ρ***<sup>g</sup>* is the missile location provided by tracking data.

where λ

3. Earth rate

given by

4. Gravitational acceleration

system is given by

system is given by

and the geocentric latitude

where

$$
\boldsymbol{\Delta}\_{\boldsymbol{\alpha}}^{a} = \begin{bmatrix}
0 & -\boldsymbol{\alpha}\_{\boldsymbol{\alpha}\boldsymbol{x}} & \boldsymbol{\alpha}\_{\boldsymbol{\alpha}\boldsymbol{y}} \\
\boldsymbol{\alpha}\_{\boldsymbol{\alpha}\boldsymbol{x}} & 0 & -\boldsymbol{\alpha}\_{\boldsymbol{\alpha}\boldsymbol{x}} \\
\end{bmatrix}
\tag{17}
$$

where , , ωωω *eax eay eaz* are three components of **ω***ea* , respectively; **V***<sup>g</sup>* and **ρ***<sup>g</sup>* are the velocity and position of missile in the launch coordinate system provided by tracking data, respectively. **V**0*<sup>a</sup>* is the initial velocity of launch site with respect to launch coordinate system due to earth rotation, written as

$$\mathbf{V}\_{0a} = \boldsymbol{\mathfrak{os}}\_{en}(\mathbf{0}) \times \mathbf{R}\_{0a} \tag{18}$$

Likewise, the tracking apparent position is given by

$$\mathbf{W}\_{\rm tra}^{a}\left(t\right) = \mathbf{C}\_{\mathcal{g}}^{a}\left(t\right) \cdot \left(\mathbf{R}\_{0} + \mathbf{p}\_{\mathcal{g}}\right) - \mathbf{R}\_{0a} - \mathbf{V}\_{0a} \cdot t - \int\_{0}^{t} \int\_{0}^{u} \mathbf{g}\_{a}\left(\tau\right) d\tau du\tag{19}$$

6. Calculation of apparent velocity and position of telemetry data The telemetric apparent velocity can be obtained by the integration of telemetric apparent acceleration, given by

$$\mathbf{W}\_{\text{tele}}^{a}(t) = \int\_{0}^{t} \dot{\mathbf{W}}\_{\text{tele}}^{a}(\boldsymbol{\tau}) \, d\boldsymbol{\tau} \tag{20}$$

Integrating Eq.(20) gives the telemetric apparent position

$$\mathbf{W}\_{\text{tlele}}^{a}(t) = \int\_{0}^{t} \mathbf{W}\_{\text{tlele}}^{a}(\boldsymbol{\tau}) \, d\boldsymbol{\tau} \tag{21}$$

7. Calculation of difference between telemetry and tracking data

The difference between telemetry data and tracking data is obtained by subtracting synchronous tracking data and compensation from telemetry data, namely, we can have the difference between telemetry velocity and tracking velocity, ( ) *<sup>v</sup>* δ**X** *t* , and the difference between telemetry velocity and tracking velocity, ( ) *<sup>r</sup>* δ**X** *t* .

Error Separation Techniques Based on Telemetry and Tracking Data for Ballistic Missile 449

gravitational force, the velocity and position errors of trajectory are the errors of apparent velocity and position respectively. The apparent acceleration error arisen from guidance

**WW W W M M M W** = − = − − − −⋅ −

where **<sup>W</sup>***<sup>p</sup>* is the apparent acceleration measured by inertial navigation platform, **W***<sup>a</sup>* is the real apparent acceleration; 3 **M** ( )⋅ , <sup>2</sup> **M** ( )⋅ , <sup>1</sup> **M** ( )⋅ are the rotation matrices about *z* , *y* , *x* axis,

small values; **Δ** is the error vector measured by accelerometer. Since the true value of **W***<sup>a</sup>* is not available, the substitution of **W***<sup>a</sup>* is generally obtained by converting the tracking data.

1

α α

−

Rearranging Eq.(25) and ignoring the second-order small values yield

0 0

 α

 α

 α 0

<sup>321</sup> ( ) ( ) ( )( ) *pap z y x p*

**W** is the difference of apparent acceleration between telemetry and tracking data.

*z y*

0

0 11 12 0 11 12

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

0 11 12

 

*pz py <sup>x</sup>*

*pz xp y x*

α α

αα

0

*W W*

*W W*

 

*pz px y py px z*

> α *<sup>x</sup>* ,α *<sup>y</sup>* ,α

*W W*

<sup>−</sup>

= −+ <sup>−</sup> **δW Δ**

*g x g x ax g x ay x x t t y y g y g y ay g y ax z z g z g z az g z ay*

 

*k kW k kW k kW*

*x a x a x ax y a y a y ay z a z a z az*

Note that , , *WWW ax ay az* are the apparent accelerations in the launch inertial coordinate system, unfortunately we cannot obtain the measurements in practice. Since the values of , , *WWW px py pz* are given by the telemetry data, so we can approximately substitute , , *WWW px py pz* for , , *WWW ax ay az* during the error separation process. Hence, Eqs.(27) and

 Δ + Δ= + <sup>Δ</sup> <sup>+</sup>

+ +

= =+ + + +

<sup>−</sup> = − −⋅ −

ααα

*<sup>z</sup>* are the drift angles along the three directions, which are assumed as

1 () 1

α

α

α

**W W W Δ** (25)

**Δ** (24)

(26)

(27)

*<sup>z</sup>* are the drift angles of gyroscope

(28)

instrumentation systematic error is represented by

Neglecting the second-order term, Eq.(24) is changed to

δ

where*Wpx* , *Wpy* , *Wpz* are the components of **W***<sup>p</sup>* ;

α

α

α

By the accelerometer error model, we can have

and obtained by integrating Eq.(22)

(28) can be rewritten respectively as

δ

α *<sup>x</sup>* ,α *<sup>y</sup>* ,α

respectively;

δ

Thereby

### **3. Separation model of guidance instrumentation systematic errors**

There are many reasons influencing the landing errors of ICBM, which can be fallen into two categories: 1) guidance instrumentation systematic errors, and 2) initial launch parameter errors. Guidance instrumentation systematic errors primarily consist of accelerometer, gyroscope and platform systematic errors. Before the flight test ground calibration test is usually performed for inertial navigation system and then the estimates of instrumentation error coefficients are compensated in flight, which can reduce the landing errors and the difference between telemetry and tracking data effectively. However, because of the residual between the calibrated values and the actual values of instrumentation errors, the separation of the behaved values of the instrumentation error coefficients from telemetry and tracking data is need to perform.

#### **3.1 Model of guidance instrumentation systematic errors**

Since the determination of error model is correlated with the performance of inertial platform, there are many error coefficients required to separate for inertial platform with high accuracy while a minority of primary error terms for general inertial platform with poor accuracy. The gyroscope error model of inertial platform is given by

$$\begin{cases} \dot{\alpha}\_x(t) = k\_{g0x} + k\_{g11x} \dot{\mathcal{W}}\_x(t) + k\_{g12x} \dot{\mathcal{W}}\_x(t) \\ \dot{\alpha}\_y(t) = k\_{g0y} + k\_{g11y} \dot{\mathcal{W}}\_y(t) + k\_{g12y} \dot{\mathcal{W}}\_x(t) \\ \dot{\alpha}\_z(t) = k\_{g0z} + k\_{g11z} \dot{\mathcal{W}}\_z(t) + k\_{g12z} \dot{\mathcal{W}}\_y(t) \end{cases} \tag{22}$$

and accelerometer error model is given by

$$\begin{cases} \Delta\_x(t) = k\_{a0x} + k\_{a1x} \dot{W}\_x(t) \\ \Delta\_y(t) = k\_{a0y} + k\_{a1y} \dot{W}\_y(t) \\ \Delta\_z(t) = k\_{a0z} + k\_{a1z} \dot{W}\_z(t) \end{cases} \tag{23}$$

Where α *<sup>x</sup>* ,α *<sup>y</sup>* ,α *<sup>z</sup>* are angular velocity drifts of three gyroscopes, respectively; *Wx* , *Wy* , *<sup>z</sup> <sup>W</sup>* are apparent accelerations of vehicle; *g*0*<sup>x</sup> k* , *g y*<sup>0</sup> *k* , *<sup>g</sup>*0*<sup>z</sup> k* are zero biases of three gyroscopes, *<sup>g</sup>*11*<sup>x</sup> k* , *g y* <sup>11</sup> *k* , *<sup>g</sup>*11*<sup>z</sup> k* are proportional error coefficients, *g*12*<sup>x</sup> k* , *g y* <sup>12</sup> *k* , *<sup>g</sup>*12*<sup>z</sup> k* are first-order error coefficients; *a x*<sup>0</sup> *k* , *<sup>a</sup>*0*<sup>y</sup> k* , *a z* <sup>0</sup> *k* are zero biases and *a x*<sup>1</sup> *k* , *<sup>a</sup>*1*<sup>y</sup> k* , *a z*<sup>1</sup> *k* are proportional error coefficients of three accelerometers. Model of guidance instrumentation systematic errors contains 15 error coefficients in total.

The accurate velocity, position and orientation information of ballistic missile are not available due to the errors resulted from maneuvering of ballistic missile and measurements, which generates the initial launch parameter errors. The initial launch parameter errors primarily consist of geodetic longitude, geodetic latitude, geodetic height, astronomical longitude, astronomical latitude and astronomical azimuth errors of launch site, and initial velocity errors of ballistic missile about three directions, amounting to 9 terms.

#### **3.2 Separation model of instrumentation errors**

Guidance instrumentation systematic errors can affect telemetric apparent acceleration so as to affect apparent velocity and position. Without regard to the calculation error of

There are many reasons influencing the landing errors of ICBM, which can be fallen into two categories: 1) guidance instrumentation systematic errors, and 2) initial launch parameter errors. Guidance instrumentation systematic errors primarily consist of accelerometer, gyroscope and platform systematic errors. Before the flight test ground calibration test is usually performed for inertial navigation system and then the estimates of instrumentation error coefficients are compensated in flight, which can reduce the landing errors and the difference between telemetry and tracking data effectively. However, because of the residual between the calibrated values and the actual values of instrumentation errors, the separation of the behaved values of the instrumentation error coefficients from

Since the determination of error model is correlated with the performance of inertial platform, there are many error coefficients required to separate for inertial platform with high accuracy while a minority of primary error terms for general inertial platform with

> 0 11 12 0 11 12 0 11 12

*<sup>z</sup>* are angular velocity drifts of three gyroscopes, respectively; *Wx*

*x ax ax x y ay ay y z az az z*

are apparent accelerations of vehicle; *g*0*<sup>x</sup> k* , *g y*<sup>0</sup> *k* , *<sup>g</sup>*0*<sup>z</sup> k* are zero biases of three gyroscopes, *<sup>g</sup>*11*<sup>x</sup> k* , *g y* <sup>11</sup> *k* , *<sup>g</sup>*11*<sup>z</sup> k* are proportional error coefficients, *g*12*<sup>x</sup> k* , *g y* <sup>12</sup> *k* , *<sup>g</sup>*12*<sup>z</sup> k* are first-order error coefficients; *a x*<sup>0</sup> *k* , *<sup>a</sup>*0*<sup>y</sup> k* , *a z* <sup>0</sup> *k* are zero biases and *a x*<sup>1</sup> *k* , *<sup>a</sup>*1*<sup>y</sup> k* , *a z*<sup>1</sup> *k* are proportional error coefficients of three accelerometers. Model of guidance instrumentation systematic errors

The accurate velocity, position and orientation information of ballistic missile are not available due to the errors resulted from maneuvering of ballistic missile and measurements, which generates the initial launch parameter errors. The initial launch parameter errors primarily consist of geodetic longitude, geodetic latitude, geodetic height, astronomical longitude, astronomical latitude and astronomical azimuth errors of launch site, and initial velocity errors

Guidance instrumentation systematic errors can affect telemetric apparent acceleration so as to affect apparent velocity and position. Without regard to the calculation error of

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

 

*t k k Wt t k k Wt t k k Wt*

(22)

(23)

 , *Wy* , *<sup>z</sup> <sup>W</sup>*

*x gx g x x g x z y gy g y y g y x z gz g z z g z y*

 

=+ +

=+ +

=+ +

Δ= +

Δ= +

Δ= +

of ballistic missile about three directions, amounting to 9 terms.

**3.2 Separation model of instrumentation errors** 

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

*t k k Wt k Wt t k k Wt k Wt t k k Wt k Wt*

**3. Separation model of guidance instrumentation systematic errors** 

telemetry and tracking data is need to perform.

**3.1 Model of guidance instrumentation systematic errors** 

α

α

and accelerometer error model is given by

contains 15 error coefficients in total.

Where

α *<sup>x</sup>* ,α *<sup>y</sup>* ,α α

poor accuracy. The gyroscope error model of inertial platform is given by

gravitational force, the velocity and position errors of trajectory are the errors of apparent velocity and position respectively. The apparent acceleration error arisen from guidance instrumentation systematic error is represented by

$$\delta \dot{\mathbf{W}} = \dot{\mathbf{W}}\_p - \dot{\mathbf{W}}\_a = \dot{\mathbf{W}}\_p - \mathbf{M}\_3 (-\alpha\_z) \mathbf{M}\_2 (-\alpha\_y) \mathbf{M}\_1 (-\alpha\_x) \cdot (\dot{\mathbf{W}}\_p - \Delta) \tag{24}$$

where **<sup>W</sup>***<sup>p</sup>* is the apparent acceleration measured by inertial navigation platform, **W***<sup>a</sup>* is the real apparent acceleration; 3 **M** ( )⋅ , <sup>2</sup> **M** ( )⋅ , <sup>1</sup> **M** ( )⋅ are the rotation matrices about *z* , *y* , *x* axis, respectively; α *<sup>x</sup>* ,α *<sup>y</sup>* ,α *<sup>z</sup>* are the drift angles along the three directions, which are assumed as small values; **Δ** is the error vector measured by accelerometer. Since the true value of **W***<sup>a</sup>* is not available, the substitution of **W***<sup>a</sup>* is generally obtained by converting the tracking data. Thereby δ **W** is the difference of apparent acceleration between telemetry and tracking data. Neglecting the second-order term, Eq.(24) is changed to

$$
\delta \dot{\mathbf{W}} = \dot{\mathbf{W}}\_p - \begin{bmatrix} 1 & -\alpha\_z & \alpha\_y \\ \alpha\_z & 1 & -\alpha\_x \\ -\alpha\_y & \alpha\_x & 1 \end{bmatrix} \cdot (\dot{\mathbf{W}}\_p - \Delta) \tag{25}
$$

Rearranging Eq.(25) and ignoring the second-order small values yield

$$
\mathbf{\dot{\sigma}} \mathbf{\dot{W}} = \begin{bmatrix} 0 & -\dot{\mathcal{W}}\_{pz} & \dot{\mathcal{W}}\_{py} \\ \dot{\mathcal{W}}\_{pz} & 0 & -\dot{\mathcal{W}}\_{px} \\ -\dot{\mathcal{W}}\_{py} & \dot{\mathcal{W}}\_{px} & 0 \end{bmatrix} \begin{bmatrix} \alpha\_x \\ \alpha\_y \\ \alpha\_z \end{bmatrix} + \Delta \tag{26}
$$

where*Wpx* , *Wpy* , *Wpz* are the components of **W***<sup>p</sup>* ;α *<sup>x</sup>* ,α *<sup>y</sup>* ,α *<sup>z</sup>* are the drift angles of gyroscope and obtained by integrating Eq.(22)

$$
\begin{bmatrix}
\alpha\_x\\\alpha\_y\\\alpha\_z
\end{bmatrix} = \int\_0^t \dot{\alpha}\_y \begin{bmatrix}
\dot{\alpha}\_x\\\dot{\alpha}\_y\\\dot{\alpha}\_z
\end{bmatrix} dt = \int\_0^t \begin{vmatrix}
k\_{g0x} + k\_{g11x}\dot{\mathcal{W}}\_{ax} + k\_{g12x}\dot{\mathcal{W}}\_{ay}\\k\_{g0y} + k\_{g11y}\dot{\mathcal{W}}\_{ay} + k\_{g12y}\dot{\mathcal{W}}\_{ax}\\k\_{g0z} + k\_{g11z}\dot{\mathcal{W}}\_{az} + k\_{g12z}\dot{\mathcal{W}}\_{ay}
\end{bmatrix} dt
\tag{27}
$$

By the accelerometer error model, we can have

$$\begin{bmatrix} \Delta\_x\\ \Delta\_y\\ \Delta\_z \end{bmatrix} = \begin{bmatrix} k\_{a0x} + k\_{a1x} \dot{V} \dot{V}\_{ax} \\ k\_{a0y} + k\_{a1y} \dot{V} \dot{V}\_{ay} \\ k\_{a0z} + k\_{a1z} \dot{V} \dot{V}\_{az} \end{bmatrix} \tag{28}$$

Note that , , *WWW ax ay az* are the apparent accelerations in the launch inertial coordinate system, unfortunately we cannot obtain the measurements in practice. Since the values of , , *WWW px py pz* are given by the telemetry data, so we can approximately substitute , , *WWW px py pz* for , , *WWW ax ay az* during the error separation process. Hence, Eqs.(27) and (28) can be rewritten respectively as

Error Separation Techniques Based on Telemetry and Tracking Data for Ballistic Missile 451

where *<sup>v</sup>* **ε** and *<sup>r</sup>* **ε** are the random errors. It is seen from Eq.(35) that the separation model of

Actually, the apparent velocity and position errors are computed by the telemetry and tracking data. When taking no account of the random errors, the tracking data can be

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

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*

base of controlling the attitude motion will also change. At this point, the reference plane

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

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

In the case of maneuvering launch, the initial missile velocity in the launch inertial

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

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

0 0

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

= + = +

*v v r r*

ε

(35)

′ ′′ − plane, simultaneously the reference

*<sup>a</sup> V a ea a s* =×+ *ω R V* (36)

′ ′′ − plane. Due to the

′ ′′′ − , the

ε

() () () ()

*t t t t*

**δW SD δW SD**

instrumentation errors can be simplified as a linear model.

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

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

plane *O xy a aa* − controlled by yaw angle changes to *O xy a aa*

considered as the true values of ballistic data.

mechanism of initial errors is analyzed thereinafter.

1. Effect to trajectory in the geocentric coordinate system

*O xz a aa* − controlled by pitch angle changes to *O xz a aa*

factors.

error.

ballistic missile.

coordinate system is given by

3. Effect to the stress of missile

$$
\begin{bmatrix}
\alpha\_x\\\alpha\_y\\\alpha\_z\\\alpha\_z
\end{bmatrix} = \stackrel{\circ}{\mathbf{J}} \begin{bmatrix}
\dot{\alpha}\_x\\\dot{\alpha}\_y\\\dot{\alpha}\_z
\end{bmatrix} dt = \stackrel{\circ}{\mathbf{J}} \begin{bmatrix}
k\_{g\otimes x} + k\_{g\cdot 11x} \dot{\mathcal{W}}\_{px} + k\_{g\cdot 12x} \dot{\mathcal{W}}\_{py}\\k\_{g\otimes y} + k\_{g\cdot 11y} \dot{\mathcal{W}}\_{py} + k\_{g\cdot 12y} \dot{\mathcal{W}}\_{px}\\k\_{g\otimes z} + k\_{g\cdot 11z} \dot{\mathcal{W}}\_{pz} + k\_{g\cdot 12z} \dot{\mathcal{W}}\_{py}
\end{bmatrix} dt
\tag{29}
$$

$$\begin{bmatrix} \Delta\_x\\ \Delta\_y\\ \Delta\_z \end{bmatrix} = \begin{bmatrix} k\_{a0x} + k\_{a1x} \dot{W}\_{px} \\ k\_{a0y} + k\_{a1y} \dot{W}\_{py} \\ k\_{a0z} + k\_{a1z} \dot{W}\_{pz} \end{bmatrix} \tag{30}$$

Herein we select 0 0 0 11 11 11 12 12 12 0 0 0 1 1 0 *<sup>T</sup> 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>** , then apparent acceleration error **δW** and instrumentation error coefficients **D** are written in linear relation as

$$
\delta \dot{\mathbf{W}} = \mathbf{S}\_a \cdot \mathbf{D} \tag{31}
$$

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

$$\mathbf{S}\_a = \begin{bmatrix} \mathbf{S}\_e \cdot \mathbf{S}\_{Ag} & \mathbf{S}\_{Aa} \end{bmatrix} \tag{32}$$

where

$$\begin{aligned} \mathbf{S}\_{\varepsilon} &= \begin{bmatrix} 0 & \dot{W}\_{zp} & -\dot{W}\_{yp} \\ -\dot{W}\_{zp} & 0 & \dot{W}\_{zp} \\ \dot{W}\_{yp} & -\dot{W}\_{zp} & 0 \end{bmatrix}, \quad \mathbf{S}\_{Aa} = \begin{bmatrix} 1 & 0 & 0 & \dot{W}\_{px} & 0 & 0 \\ 0 & 1 & 0 & 0 & \dot{W}\_{py} & 0 \\ 0 & 0 & 1 & 0 & 0 & \dot{W}\_{pz} \end{bmatrix}, \\ \mathbf{S}\_{A\xi} &= \begin{bmatrix} t & 0 & 0 & \dot{W}\_{zp} & 0 & 0 & \dot{W}\_{yp} & 0 & 0 \\ 0 & t & 0 & 0 & \dot{W}\_{zp} & 0 & 0 & \dot{W}\_{zp} & 0 \\ 0 & 0 & t & 0 & 0 & \dot{W}\_{zp} & 0 & 0 & \dot{W}\_{yr} \end{bmatrix} \end{aligned}$$

Integrating Eq.(31) gives the apparent velocity error

$$\mathbf{\dot{S}W(t) = \int\_{0}^{t} \mathbf{S}\_{a}(\tau)dt \cdot \mathbf{D} = \mathbf{S}\_{v}(t)\mathbf{D} \tag{33}$$

where ( ) *<sup>v</sup>* **S** *t* is the environmental function matrix of instrumental error of apparent velocity. Taking the integration of Eq.(33) again gives the apparent position error

$$\mathbf{\dot{S}} \mathbf{V}(t) = \int\_{0}^{t} \mathbf{S}\_{v}(\tau) dt \cdot \mathbf{D} = \mathbf{S}\_{r}(t) \mathbf{D} \tag{34}$$

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 consideration of random errors are represented by

$$\begin{aligned} \boldsymbol{\mathfrak{dW}}(t) &= \mathbf{S}\_v(t)\mathbf{D} + \mathbf{e}\_v \\ \boldsymbol{\mathfrak{dW}}(t) &= \mathbf{S}\_r(t)\mathbf{D} + \mathbf{e}\_r \end{aligned} \tag{35}$$

where *<sup>v</sup>* **ε** and *<sup>r</sup>* **ε** are the random errors. It is seen from Eq.(35) that the separation model of instrumentation errors can be simplified as a linear model.

Actually, the apparent velocity and position errors are computed by the telemetry and tracking data. When taking no account of the random errors, the tracking data can be considered as the true values of ballistic data.
