Research on ballistic missile laser SIMU error propagation mechanism

Journal of Systems Engineering and Electronics Vol. 19, No. 2, 2008, pp.356–362

Research on ballistic missile laser SIMU error propagation mechanism Wei Shihui & Xiao Longxu Equipment Inst. of Second Artillery, Beijing 100085, P. R. China (Received November 10, 2006)

Abstract: It is necessary that the laser inertial system is used to further improve the fire accuracy and quick reaction capability in the ballistic missile strapdown inertial navigation system. According to the guidance controlling method and the output and error model of ballistic missile laser SIMU, the mathematical model of error propagation mechanism is set up and any transfer environmental function of error coefficient that affects the fire accuracy is deduced. Also, the missile longitudinal/lateral impact point is calculated using MATLAB. These establish the technical foundation for further researching the dispersion characteristics of impact point and reducing the laser guidance error.

Keywords: ballistic missile, laser SIMU, instrument error, impact point deviation, environmental function.

1. Introduction

2. Laser SIMU instrument error

Future operation will be the multi-arms of services united association model. Both ourselves and the enemy will use electronic jamming each other. In the complex battlefield environment, a GPS/GLONASS navigation locating system may be interfered, which will affect the operational use. Being the self-cont ained guidance system, inertial navigation has the characteristic of preventing ambient electronic interference. With domestic relative technical breakthrough and maturation, the laser inertial unit has already turned to practicality. Compared with the flexible inertial unit, it has several advantages that include high precision, heating stability, and rapid test [1−5] . Using the laser inertial unit, the requirements that further improve the impact accuracy and quick reaction capability can be met in the ballistic missile strapdown inertial navigation system. Therefore, by building up the laser inertial unit instrument error model and by predominating its propagation mechanical schematic, the impact point deviation affected by guidance error and the missile impact point dispersion characteristic can be researched as well as a stable supporting technology for laser inertial unit instrument error coefficient compensation can be established.

2.1

Laser gyro error model

The missile strapdown system is constructed by fixing three laser gyroscopes and three quartz accelerometers of the laser SIMU to three missile-body axes. It is measured by the laser gyroscope for rates of angular rotation θ˙x1 , θ˙y1 , θ˙z1 in flight missile rotation of three axes of the missile coordinate system[6] . The results of the measurement are transformed to its proportional pulses by V/F. Owing to the equipment manufacture and installation error, its output value is not the real missile rate of angular rotation. The relational formula between the gyro output value and the missile rate of angular rotation is ⎧ G G ˙ G ˙ ⎪ N Bx = D0x + KxG θ˙x1 + Eyx θy1 + Ezx θz 1 ⎪ ⎪ ⎪ ⎨ G G ˙ G ˙ (1) N By = D0y + Exy θx1 + KyG θ˙y1 + Ezy θz 1 ⎪ ⎪ ⎪ ⎪ ⎩ N B = DG + E G θ˙ + E G θ˙ + K G θ˙ z 0z xz x1 yz y1 z z1 where, N Bx1 , N By1 , N Bz1 are gyro output pulse G G G , D0y , D0z are gyro zero drift values; values; D0x G G G Kx , Ky , Kz are gyro proportionality coefficients; G G G G G G , Ezx , Exy , Ezy , Exz , Eyz are installation error coEyx ˙ x1 , W ˙ y1 , W ˙ z1 should be separately divided efficients; W

Research on ballistic missile laser SIMU error propagation mechanism

357

by g0 when their unit is m/s2 . The results of function (1) θ˙x1 , θ˙y1 , θ˙z1 can be shown as ⎧ G G G 1 E1x ⎪ ˙y − E2x θ˙z − D0x ⎪ θ˙x1 = N B − θ x ⎪ 1 ⎪ KxG Kxa 1 Kxa 1 Kxa ⎪ ⎪ ⎪ ⎪ ⎨ G G G E1y E2y D0y 1 (2) θ˙y1 = G N By1 − G θ˙x1 − G θ˙z1 − G ⎪ Ky Ky Ky Ky ⎪ ⎪ ⎪ ⎪ G G G ⎪ ⎪˙ 1 E1z ⎪ ˙x − E2z θ˙y − D0z ⎩θz1 = N B − θ z 1 1 1 z 1 KzG KzG KzG KzG

where, θ˙x1 , θ˙y1 , θ˙z1 can approximately replace Kx N Bx1 , Ky N By1 , Kz N Bz1 , and therefore, Eq. (5) can also be shown as ⎧ ⎪ δ θ˙x1 = δKx θ˙x1 +δEyx θ˙y1 +δEzx θ˙z1 +δD0x ⎪ ⎪ ⎪ ⎨ (6) δ θ˙y1 = δKy θ˙y1 +δExy θ˙x1 +δEzy θ˙z1 +δD0y ⎪ ⎪ ⎪ ⎪ ⎩ δ θ˙ = δK θ˙ +δE θ˙ +δE θ˙ +δD z1 z z1 xz x1 yz y1 0z

Let

Three accelerometers in the rate strapdown guidance scheme are installed along the three axes of missile body coordinate system, which measure their apparent acceleration component. According to the accelerometer manufacturing and installation error, the output value is not exactly the in-flight missile real acceleration value. The results of measurement are transformed by V/F and output its proportional pulses. The relational expression between the output value and the measurement value is ⎧ a ⎪ ˙ x1 + E a W ˙ y1 + Nx1 = K0x ⎪ + Kxa W ⎪ yx ⎪ ⎪ ⎪ ⎪ ⎪ a ˙ a ˙ 2 ⎪ Ezx Wz1 + K2x Wx1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ˙ y1 + E a W ˙ x1 + ⎨ Ny1 = K a + K a W 0y y xy (7) ⎪ a ˙ a ˙ 2 ⎪ W W + K E ⎪ z 1 y 2y 2y 1 ⎪ ⎪ ⎪ ⎪ ⎪ a a ⎪ ˙ z1 + E a W ˙ x1 + ⎪ N = K0z + Kz W ⎪ xz ⎪ z1 ⎪ ⎪ ⎪ ⎩ ˙ y1 + K a W ˙ 2 Ea W

Kx =

1 , KxG

Ky =

1 , KyG

Kz =

1 KzG

Eyx = −

G Eyx , Kxa

E1y = −

G Exy , Kya

E1z = −

Ezx = −

G Ezx , Kxa

G Exz Kza

E2y = −

G Ezy , Kya

E2z = −

G Eyz Kza

D0 = −

G D0x , Kxa

D0 = −

G D0y , Kya

D0 = −

G D0z Kza

Therefore, we can convert function (2) to ⎧ ⎪ θ˙ = Kx N Bx1 + Eyx θ˙y1 + Ezx θ˙z1 + D0x ⎪ ⎨ x1 θ˙y1 = Ky N By1 + Exy θ˙x1 + Ezy θ˙z1 + D0y ⎪ ⎪ ⎩ ˙ θz1 = Kz N Bz1 + Exz θ˙x1 + Eyz θ˙y1 + D0z

(3)

Then, function (3) is the gyro actual output model. ¯ If we use θ˙ to show the gyro output actual value, then, its model expression is ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

θ¯˙x1 = (1+δKx)Kx N Bx1 +(Eyx +δEyx )θ˙y1+ (Ezx +δEzx )θ˙z +(D0x +δD0x) 1

¯ θ˙y1 = (1+δKy )Ky N By1 +(Exy +δExy )θ˙x1+ ⎪ ⎪ (Ezy +δEzy )θ˙z1 +(D0y +δD0y ) ⎪ ⎪ ⎪ ⎪ ⎪ θ¯˙z = (1+δKz )Kz N Bz +(Exz +δExz )θ˙x + ⎪ 1 1 1 ⎪ ⎪ ⎪ ⎩ ˙ (Eyz +δEyz )θy1 +(D0z +δD0z )

(4)

Subtracting Eq.(3) from Eq.(4), we get the error model of laser gyro, that is, ⎧ ⎪ δ θ˙ = δKx · Kx N Bx1 +δEyx θ˙y1 +δEzx θ˙z1 +δD0x ⎪ ⎨ x1 δ θ˙y1 = δKy · Ky N By1 +δExy θ˙x1 +δEzy θ˙z1 +δD0y ⎪ ⎪ ⎩ ˙ δ θz = δKz · Kz N Bz +δExz θ˙x +δEyz θ˙y +δD0z 1

1

1

1

(5)

2.2

Accelerometer error model

2z

2z

z1

where, Nx1 , Ny1 , Nz1 are accelerometer output pulse a a a rates; K0x , K0y , K0z are accelerometer zero drift vala a a ues; Kx , Ky , Kz are accelerometer proportionality coa a a a a a efficients; E1x , E2x , E1y , E2y , E1z , E2z are acceleroma a a , K2y , K2z eter installation error coefficients; K2x are every accelerometer quadratic drift coefficients; ˙ x1 , W ˙ y1 , W ˙ z1 are every accelerometer output realW time measurement apparent accelerations. Ignoring the quadratic drift, the above relational expression (7) can be shown as ⎧ a a ˙ a ˙ a ˙ ⎪ ⎪ Nx1 = K0x Wx1 + Eyx Wy1 + Ezx Wz1 + K1x ⎪ ⎪ ⎪ ⎨ a a ˙ a ˙ a ˙ Wy1 + E1y Wx1 + E2y Wz1 Ny1 = K0y + K1y (8) ⎪ ⎪ ⎪ ⎪ ˙ z1 + E a W ˙ x1 + E a W ˙ y1 ⎪ ⎩ Nz1 = K a + K a W 0z

1z

1z

2z

Wei Shihui & Xiao Longxu

358 that is ⎧ a a Ea ⎪ ˙ x1 = 1 Nx1 − K0x − yx W ˙ y1 − Ezx W ˙z ⎪ W ⎪ a a a ⎪ K1x K1x K1x K1x 1 ⎪ ⎪ ⎪ ⎪ ⎨ K a Ea Ea ˙ y1 = 1 Ny1 − 0y − 1y W ˙ x1 − 2y W ˙ z1 W a a a a ⎪ K1y K1y K1y K1y ⎪ ⎪ ⎪ ⎪ a a a ⎪ ⎪ ⎪ ˙ z1 = 1 Nz1 − K0z − E1z W ˙ x1 − E2z W ˙ y1 ⎩ W a a a a K1z K1z K1z K1z

(9)

where, let K1x

1 K1y = a , K1y

1 = a , K1x

K0x = − Eyx = − Ezx = −

a K0x , a K1x a Eyx

K0y = −

,

E1y = −

a Ezx , K1x

E2y = −

a K1x

a K0y a K1y a E1y a K1y a E2y

K1y

1 K1z = a K1z ,

K0z = −

a K0z a K1x

,

E1z = −

a E1z a K1z

,

Ea E2z = − 2z K1z

Therefore, function (9) can also be shown as ⎧ ˙ = K1x Nx1 +K0x +Eyx W ˙ y1 +Ezx W ˙ z1 ⎪ W ⎪ ⎨ x1 ˙ y1 = K1y Ny1 +K0y +E1y W ˙ x1 +E2y W ˙ z1 (10) W ⎪ ⎪ ⎩ ˙ ˙ x1 +E2z W ˙ y1 Wz1 = K1z Nz1 +K0z +E1z W Then, function (10) is the accelerometer actual output model. ¯˙ to show the output real value, then, its If we use W model expression is ⎧ ¯˙ = (1+δK )K N +(K +δK )+ ⎪ W ⎪ x1 1x 1x x1 0x 0x ⎪ ⎪ ⎪ ⎪ ˙ y1 +(Ezx +δEzx )W ˙ z1 ⎪ (Eyx +δEyx )W ⎪ ⎪ ⎪ ⎪ ¯ ⎨ W ˙ y1 = (1+δK1y )K1y Ny1 +(K0y +δK0y )+ (11) ⎪ ˙ x1 +(E2y +δE2y )W ˙ z1 ⎪ (E1y +δE1y )W ⎪ ⎪ ⎪ ⎪ ¯˙ = (1+δK )K N +(K + δK )+ ⎪ W ⎪ z1 1z 1z z1 0z 0z ⎪ ⎪ ⎪ ⎩ ˙ ˙ y1 (E1z +δE1z )Wx1 +(E2z +δE2z )W Subtracting Eq.(10) from Eq.(11), we then get the accelerometer error model, that is, ⎧ ˙ x1 = δK1x · K1x Nx1 + δK0x + ⎪ δW ⎪ ⎪ ⎪ ⎪ ⎪ ˙ y1 + δEzx W ˙ z1 ⎪ δEyx W ⎪ ⎪ ⎪ ⎪ ⎨ δW ˙ y1 = δK1y · K1y Ny1 + δK0y + (12) ⎪ ˙ x1 + δE2y W ˙ z1 ⎪ δE1y W ⎪ ⎪ ⎪ ⎪ ⎪ ˙ z1 = δK1z · K1z Nz1 + δK0z + δW ⎪ ⎪ ⎪ ⎪ ⎩ ˙ x1 + δE2z W ˙ y1 δE1z W

˙ y1 , W ˙ z1 can approximately replace ˙ x1 , W where, W K1x Nx1 , K1y Ny1 , K1z Nz1 , then, function (12) can also be shown as ⎧ ˙ x1 +δK0x +δEyx W ˙ y1 +δEzx W ˙ z1 ˙ x1 = δK1x W ⎪ δW ⎪ ⎨ ˙ y1 +δK0y +δE1y W ˙ x1 +δE2y W ˙ z1 ˙ y1 = δK1y W δW ⎪ ⎪ ⎩ ˙ z1 +δK0z +δE1z W ˙ x1 +δE2z W ˙ y1 ˙ z1 = δK1z W δW (13)

3. Missile impact point deviation linear model The missile deviation of impact point includes two parts, which is the longitudinal dispersion ∆L in the firing direction and the lateral dispersion ∆H in the vertical direction of firing sector, respectively[7−9] . 3.1

Range deviation linear expanded formula

According to the ballistics theory, the missile firing range L on the earth is the function of powered phase terminal parameter. When it is represented by inertial parameter, the firing range can be L = L(Via , ria , ti )

(14)

where, rka is missile centroid vector radius related to geocentric at powered phase terminal; Vka is missile powered phase terminal absolute velocity vector related to inertial coordinate system; tk is duration of powered flight; When the motion parameter is shown by launching point inertial coordinate system components x, y, z and Vx , Vy , Vz , then, function (14) is converted to L = L(Vxk , Vyk , Vzk , xk , yk , zk , t)

(15)

Suppose that x1 = Vx , x2 = Vy , x3 = Vz , x4 = xt , x5 = yt , x6 = zt , x7 = t, then, function (15) is reduced to L = L(xik ), i = 1, 2, . . . , 7 (16) If the standard ballistic powered terminal parameter and time is x ˜ik , and the standard firing range is ˜ L, Eq.(16) can be further rewritten as ˜ = L(˜ ˜ xik ), L

i = 1, 2, . . . , 7

(17)

Hence, any interference force and disturbance torque that can affect the missile motion are small,

Research on ballistic missile laser SIMU error propagation mechanism and the firing range deviation is also not large. Therefore, we can expand the actual firing range L to Taylor series when it closes to the moment of standard cutoff time t˜k , we get the firing range deviation. When any high-order term which upwards second-order term be ignored, ˜ = L(xik ) − L(˜ ˜ xik ) = ∆L = L − L

7  i=1

∂L ∆xik ∂xik

that is ∆L =

∂L ∂L ∂L ∆Vxk + ∆Vyk + ∆Vzk + ∂Vxk ∂Vyk ∂Vzk

(18) Lateral deviation linear expanded formula

We can get the impact point lateral deviation linear expanded formula in first-order approximate condition using the similar derivation method as the firing range deviation linear expanded formula. ˜ = H(xik ) − H(˜ ˜ xik ) = ∆H = H − H

7  ∂H ∆xik ∂x ik i=1

that is, ∆H =

∂H ∂H ∂H ∆Vxk + ∆Vyk + ∆Vzk + ∂Vxk ∂Vyk ∂Vzk ∂H ∂H ∂H ∂H ∆xk + ∆yk + ∆zk + ∆tk ∂xk ∂yk ∂zk ∂tk (19)

4. The transfer model of laser inertial instrument error[10] We can only take the effect from error of laser gyro axial X to longitudinal impact point deflection as an example to analyze and predict other derivation of directional error similar to it. It can be seen from Eq.(6) that the error model of axial X direction is δ θ˙x1 = δKx θ˙x1 + δEyx θ˙y1 + δEzx θ˙z1 + δD0x

sector, that is ψ = γ = 0. According to the coordinate transformation matrix, by transferring the missile body coordinate system δ θ˙x1 to the axial of inertial coordinate system, we get δ θ˙x = δ θ˙x1 cos ϕ δ θ˙y = δ θ˙x1 sin ϕ

(21)

The apparent acceleration deviation in the inertial coordinate system caused by δ θ˙x , δ θ˙y is δθx = δ θ˙x τ = δ θ˙x1 cos ϕτ

∂L ∂L ∂L ∂L ∆xk + ∆yk + ∆zk + ∆tk ∂xk ∂yk ∂zk ∂tk

3.2

359

(20)

Since the roll angle and pitch angle are small when the missile is flying, the main effect to the missile longitudinal impact point deflection is the yaw angle; therefore, we only consider the effect in launching

˙ z1 cos ϕτ ˙ y = −W ˙ z δθx = −δ θ˙x1 W δW ˙ x1 sin ϕ cos ϕτ ˙ y δθx = δθx1 W ˙z =W δW δθy = δ θ˙y τ = δ θ˙x1 sin ϕτ ˙ z1 sin ϕτ ˙x =W ˙ z δθy = δ θ˙x1 W δW ˙ x1 cos ϕ sin ϕτ ˙ x δθy = −δ θ˙x1 W ˙ z = −W δW Combining the apparent acceleration deviation caused by δ θ˙x , δ θ˙y , we get the apparent acceleration deviation caused by δ θ˙x1 ⎧ ˙ x = δKx θ˙x1 W ˙ z1 sin ϕτ +δEyx θ˙y1 W ˙ z1 sin ϕτ+ ⎪ ⎪ δW ⎪ ⎪ ⎪ ⎪ ˙ z1 sin ϕτ +δD0x W ˙ z1 sin ϕτ ⎪ δEzx θ˙z1 W ⎪ ⎨ ˙ z1 cos ϕτ −δEyx θ˙y1 W ˙ z1 cos ϕτ− ˙ y = −δKx θ˙x1 W δW ⎪ ⎪ ⎪ ⎪ ˙ z1 cos ϕτ − δD0x W ˙ z1 cos ϕτ ⎪ δEzx θ˙z1 W ⎪ ⎪ ⎪ ⎩ ˙ δ Wz = 0 (22) Integrating (22) within time interval 0∼ t,we get  τ ⎧ ⎪ ˙ z1 sin ϕτ dτ+ ⎪ δWx = δKx θ˙x1 W ⎪ ⎪ ⎪ 0 ⎪ ⎪  τ ⎪ ⎪ ⎪ ˙ z1 sin ϕτ dτ + ⎪ δEyx θ˙y1 W ⎪ ⎪ ⎪ 0 ⎪ ⎪  τ ⎪ ⎪ ⎪ ⎪ ˙ z1 sin ϕτ dτ + ⎪ δEzx θ˙z1 W ⎪ ⎪ ⎪ 0 ⎪  ⎪ τ ⎪ ⎪ ⎪ ˙ z1 sin ϕτ dτ ⎪ δD0 W ⎪ ⎪ ⎪ 0 ⎨  τ ˙ z1 cos ϕτ dτ − δW = − δKx θ˙x1 W ⎪ y ⎪ ⎪ 0 ⎪ ⎪  τ ⎪ ⎪ ⎪ ⎪ ˙ z1 cos ϕτ dτ − ⎪ δEyx θ˙y1 W ⎪ ⎪ ⎪ 0 ⎪ ⎪  τ ⎪ ⎪ ⎪ ⎪ ˙ z1 cos ϕτ dτ − ⎪ δEzx θ˙z1 W ⎪ ⎪ ⎪ 0 ⎪  τ ⎪ ⎪ ⎪ ˙ z1 cos ϕτ dτ ⎪ δD0 W ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎩ δWz = 0 (23)

360 According to dynamics and kinematics, the missile apparent acceleration in the inertial coordinate system is Vx = Wx + gx Vy = Wy + Vy = Wy +

· gy · gy ·

Thereinto, gx , gy , gz are the component integrals of ·

·

·

the gravitational acceleration vector g on the axial of the inertial coordinate. Hence, the error change value of gx , gy , gz is small, ·

·

·

and the error of V is mostly caused by the error of W . Therefore, the velocity error formula is δVx = δWx δVy = δWy δVz = δWz In launching sector



δVx = δKx θ˙x1

τ

0

 δEyx θ˙y1

τ

 0τ

0

 t

˙ z1 cos ϕτ dτ dt− W δEzx θ˙z1 0 0  t τ ˙ z1 cos ϕτ dτ dt W δD0 0

˙ z1 sin ϕτ dτ + W

 (24)

δEzx θ˙z1 

0

τ

0

˙ z1 sin ϕτ dτ + W

˙ z1 sin ϕτ dτ )+ W

 (25)

δV ; therefore, within time interval

0

0

δEzx θ˙z1 

0∼ t, integrating the above formula, we get  t τ ˙ z1 sin ϕτ dτ dt+ W δx = δKx θ˙x1 0 0  t τ ˙ z1 sin ϕτ dτ dt+ W δEyx θ˙y1 0 0  t τ ˙ z1 sin ϕτ dτ dt+ W δEzx θ˙z1 0 0  t τ ˙ z1 sin ϕτ dτ dt W δD0

τ

0

 τ ∂L ˙ z1 cos ϕτ dτ + W (δKx θ˙x1 ∂Vy 0  τ ˙ z1 cos ϕτ dτ + W δEyx θ˙y1

0

 Hence, δx =

δD0

τ

˙ z1 cos ϕτ dτ − W δVy = − δKx θ˙x1 0  τ ˙ ˙ z1 cos ϕτ dτ − W δEyx θy1 0  τ ˙ ˙ z1 cos ϕτ dτ − W δEzx θz1 0  τ ˙ z1 cos ϕτ dτ W δD0

(27)

0

Previously, we mention that the longitudinal impact point deflection is mostly affected by factors within the firing sector; also, according to Eq.(18), the longitudinal impact point deflection formula can simplify to ∂L ∂L ∂L ∂L ∆L = δx + δy (28) δVx + δVy + ∂Vx ∂Vy ∂x ∂y

0



0 τ

The larger the error value, the earlier is the cutoff point, which causes ∆L to deviate to aim closely. Thus, ∆L is negative. In the meantime, put δVx , δVy , δx, δy in the above formula; then,  τ ∂L ˙ ˙ z1 sin ϕτ dτ + W (δKx θx1 ∆L = − ∂Vx 0  τ ˙ ˙ z1 sin ϕτ dτ + W δEyx θy1

˙ z1 sin ϕτ dτ + W

˙ z1 sin ϕτ dτ + W δEzx θ˙z1 0  τ ˙ z1 sin ϕτ dτ W δD0

Wei Shihui & Xiao Longxu  t τ ˙ z1 cos ϕτ dτ dt− W δy = − δKx θ˙x1 0 0  t τ ˙ z1 cos ϕτ dτ dt− W δEyx θ˙y1

δD0

τ

0

0

τ

0

˙ z1 cos ϕτ dτ + W

˙ z1 cos ϕτ dτ )− W

 t τ ∂L ˙ ˙ z1 sin ϕτ dτ dt+ W (δKx θx1 ∂x 0 0  t τ ˙ ˙ z1 sin ϕτ dτ dt+ W δEyx θy1 0

δEzx θ˙z1 (26)

δD0

 t

 t 0

0

0

τ

0

0

τ

˙ z1 sin ϕτ dτ dt+ W

˙ z1 sin ϕτ dτ dt)+ W

Research on ballistic missile laser SIMU error propagation mechanism 361   τ  t τ ∂L ˙ z1 sin ϕτ dτ − ˙ z1 cos ϕτ dτ dt+ W ∆L (δEyx ) = − δEyx kvx W (δKx θ˙x1 ∂y 0 0 0  τ  t τ ˙ z1 cos ϕτ dτ + W k vy ˙ cos ϕτ dτ dt+ W δE θ˙ yx y1

δEzx θ˙z1 δD0

0

 t 0

 t

Thereinto,

0

τ

0

z1

0 τ

0

0

klx

˙ z1 cos ϕτ dτ dt+ W

˙ z1 cos ϕτ dτ dt) W

kly

(29)

∂L ∂L ∂L ∂L , , , are calculated by ∂Vx ∂Vy ∂x ∂y

trajectory, and ϕ is designed from flight program.

0

 Kvy Klx Kly

τ

0

˙ z1 cos ϕτ dτ + W

 t 0

 t 0

τ

0 τ

0

˙ z1 sin ϕτ dτ dt− W  ˙ z1 cos ϕτ dτ dt W

(30)

If we only consider δD0x , then   τ ˙ z1 sin ϕτ dτ − W ∆L (δD0x ) = − δD0x kvx  kvy klx kly

0

0

τ

˙ z1 cos ϕτ dτ + W

 t 0

 t 0

τ 0 τ

0

˙ z1 sin ϕτ dτ dt− W  ˙ z1 cos ϕτ dτ dt W

If we only consider δEyx , then

(31)

0

 t 0

τ

0 τ

0

˙ z1 sin ϕτ dτ dt− W  ˙ z1 cos ϕτ dτ dt W

(32)

If we only consider δEzx , then   τ ˙ z1 sin ϕτ dτ − W ∆L (δEzx ) = − δEzx kvx 

5. The environmental function of impact point deviation effect by laser inertial instrument error For clearly analyzing how the error source affects the impact point as well as its character and size, and for improving the computer operational speed and writing easily, we import the environmental function concept. In Eq.(29), if we only consider the effect from δKx to ∆L and ignore the rest, then   τ ˙ ˙ z1 sin ϕτ dτ − W ∆L(δKx ) = − δKx θx1 Kvx

 t

kvy klx kly

0

0

τ

˙ z1 cos ϕτ dτ + W

 t 0

 t 0

τ

0 τ

0

˙ z1 sin ϕτ dτ dt− W  ˙ z1 cos ϕτ dτ dt W

(33)

Thereinto kvx =

∂L , ∂Vx

kvy =

∂L , ∂Vy

klx =

∂L , ∂x

kly =

∂L ∂y

Based on the above, (30)–(33), we get the error environmental function on laser gyro axial x. The environmental function derivation on other instrument is similar, which is not included here owing to the constraint of article length.

6. The calculated result of laser inertial instrument error environmental function The article uses MATLAB as the calculating tool, according to laser strapdown inertial unit error coefficient and its scattering characteristics, and figures out the impact point deviation theoretical value by the above environmental function calculating method. The environmental function calculating result within certain firing range is listed as Table 1; thereinto, the shadow part of the table is the error item that causes more effect to the impact point. The result above is the same as the calculating result of the flight program simulation.

Wei Shihui & Xiao Longxu

362 Table 1

(L=×××) Environmental function calculating result

Instru−

tong University, 2003, 37(11): 1795–1799. [3] Guo Meifeng, Ten Yunhe, Zhang Yanshen. Research on the static calibration method for a laser inertial navigation sys-

Symbol Dimension 1σ value

∆L/m

∆H/m

ment

tem. Journal of Chinese Inertial Technology, 1997, 5(4):

Gyro

δKx

%

2.00e-04

−0.001

axial

δD0x

◦ /h

0.8

−0.025

[4] Lin Yurong, Deng Zhenglong. Systematic calibration for

X1

δEyx

rad

6.00e-04

−0.007

inertial instruments errors in laser gyro strapdown inertial

δEzx

rad

6.00e-04

−0.007

Gyro

δKy

%

2.00e-04

−0.003

−15.47

axial

δD0y

◦ /h

0.4

−0.015

−85.95

[5] Liu Zhun, Qi Jianyu, Song Zhengyu. Research on the mod-

Y1

δExy

rad

3.00e-04

−0.004

−23.20

δEzy

3.00e-04

−0.004

−23.20

eling and estimating of ring laser gyro. Aerospace Control,

rad

Gyro

δKz

%

2.00e-04

11.03

axial

δD0z

◦ /h

0.4

61.28

Z1

δExz

rad

3.00e-04

16.54

δEyz

rad

3.00e-04

16.54

24–27.

navigation system. Journal of Harbin Institute of Technology, 2001, 33(1): 112–115.

2004, 22(5): 45–49. [6] Sha Yu.

Ballistic missile accuracy analysis conspectus.

Changsha: Defense Science University Publishing Company, 1999: 166–171.

Accele-

δK0x

m/s2

1.00e-03

−29.33

[7] Xiao Longxu. Ballistics and guidance of surface missile.

rometer

δK1x

%

1.00e-04

−93.91

Beijing: Chinese Astronautic Publishing Company, 2003:

Ax

δKyx

rad

1.00e-03

Accele-

85–92.

2.56

δKzx

rad

1.00e-03

δK0y

m/s2

5.00e-04

13.59

[8] Mortenson RE. Strapdown guidance error analysis. IEEE

0.19

rometer

δK1y

%

4.00e-03

−10.76

Ay

δKxy

rad

1.00e-04

84.49 −0.16

Trans. on AES, 1994, 10(4). [9] Crow J. Finite-state analysis of space shuttle contingency guidance requirements. NASA96—CR4741, 1996.

δKzy

rad

1.00e-03

Accele-

δK0z

m/s2

5.00e-04

18.52

racy estimated method research. Dalian Navy Academy,

rometer

δK1z

%

4.00e-03

−0.85

2004.

Az

δKxz

rad

1.00e-04

115.22

δKyz

rad

1.00e-03

−3.45

[10] Wei Shihui. Ballistic missile operational use impact accu-

Note: the shadow is the more effective item.

Wei Shihui was born in 1978. She is a Ph. D. Her main research fields are navigation, guidance, and con-

References

trol. E-mail:[email protected]

[1] Cannon R H. Alignment of inertial guidance systems by gyrocompass-liner theory. Journal of Aerospace Science,

Xiao Longxu was born in 1962.

1961, 28(11).

engineer of the Second Artillery Equipment Institute/researcher, director of Space Navigation College.

[2] Xiong Zhi, Liu Jianye, Lin Xueyuan, et al. Inertial instrument error compensation technology in laser gyro strapdown inertial navigation system. Journal of Shanghai Jiao

He is a chief

His main research fields are launch technique and guidance method research.