Modified attitude pursuit guidance law for low-cost missiles using strap-down seekers

SONG Tao(宋韬), LIN De-fu(林德福)， LI Chuan(李川)

(1.School of Aerospace Engineering, Beijing Institute of Technology, Beijing 100081, China; 2.Equipment Institute of Second Artillery, Beijing 100085, China)

Modified attitude pursuit guidance law for low-cost missiles using strap-down seekers

SONG Tao(宋韬), LIN De-fu(林德福)1， LI Chuan(李川)2

(1.School of Aerospace Engineering, Beijing Institute of Technology, Beijing 100081, China; 2.Equipment Institute of Second Artillery, Beijing 100085, China)

Attitude pursuit guidance law is suitable for low cost missiles. A strap-down seeker is used to achieve this guidance law. The additional angles of attack or sideslip caused by wind and by control system are considered as two disturbing factors which make attitude pursuit law impossible. Therefore, general attitude pursuit guidance law did not account for this two disturbing factors, because with those disturbing factors, it is difficult to apply. To solve the problem, the principle of strap-down seeker detecting target is investigated, the mathematical control model is established, then a modified attitude pursuit guidance law which employs the angular correction for those two disturbing factors is presented. It is proved that the modified attitude pursuit guidance law is appropriated to both in the presence of the additional angle of attack or sideslip via the simulations with the mathematical control model and Monte-Carlo method.

modified attitude pursuit guidance law; strap-down seeker; angle of attack; additional angel of attack or sideslip; Monte-Carlo method

Attitude pursuit guidance (APG), velocity pursuit guidance (VPG) and proportional navigation (PN) are the three most commonly-used guidance laws for fire-and-forget missiles. For VPG guidance law, it is usually implemented with an air-vane mounted seeker. A gimbaled seeker is necessary for the missile using PN guidance law[1]. For APG guidance law, it could be implemented by a strap-down seeker[2]. However both an air-vane mounted seeker or a gimbaled seeker are expensive for low cost missiles. APG guidance law is appropriate for cost consideration by using a strap-down seeker. With this guidance law, a strap-down seeker detects the angleεqdifference between the body axis and the LOS, then the guidance command proportional to the angleεqis produced by the control system.

A strap-down seeker is fixedon the missile’s body and has the same degrees of freedom as the body, therefore the output of the seeker is interrelated with the motion of the body. When a missile rotates to change its angle of attack or sideslip in response to guidance command, the output of the strap-down seeker contains errors caused by the angle of attack or sideslip. In this paper, the angle of attack expressed byζ1is considered as a disturbing factor. It has been proved that attitude pursuit guidance is impossible withoutζ1sensors by Gano B[1]. However, there is nosuch a sensor to measureζ1directly. Usuallyζ1is obtained by inertial navigation system (INS) which is too expensive for low cost missiles. It means that for a missile using a strap-down seeker a new method should be developed to estimateζ1.

With the interference of wind the additional angle of attack appears, which is expressed byζ2, and it has become the dominant factor to reduce the guidance precision[3].

In the presence ofζ1andζ2, the axis of body rotates an angle ofεin the pitch plane which is shown in Fig.1.αis the angle of attack, Δαis the additional angle of attack,θis flight path angle, ϑ is the pitch attitude angle,Vis the missile velocity andVWis the wind velocity.

Fig.1 Low cost missile with strap-down seeker

In this paper, the mathematical model of a strap-down seeker is established, and a modified attitude pursuit law is designed for guidance. To compensate wind velocity, the additional angle of attack or sideslip Δαis calculated by the difference between the wind velocity vector and the missile velocity vector. The wind velocity is measured by an anemometer mounted on the launch tube. The dual-accelerometers are fixed on the missile to estimate the angleζ1, then the velocity vector angle can be approximately obtained by integrating this estimated angle with the attitude angle. By this way, the modified guidance law can obtain the purpose of velocity pursuit. It is proved that with these improvements the modified attitude pursuit guidance law performs appropriate in the presence ofζ1andζ2by Monte-Carlo simulation[4].

1 Mathematical model of strap-down seeker

A strap-down seeker is fixed on the missile body, and the motion can be defined with these parameters: pitch attitude angle ϑ, yaw attitude angleψand the missile’s coordinates (Xm,Ym,Zm) in earth-fixed coordinate systemoxeyeze[5]. It is assumed that target coordinates in the earth-fixed coordinate system are (XT,YT,ZT). The mathematical model of the strap-down seeker can be obtained as follows:

Firstly, a earth-fixed coordinate systemoxeyezeis transformed to a coordinate systemo1x1y1z1. Coordinates transformations between the earth-fixed coordinate systemoxeyezeand the coordinate systemo1x1y1z1are shown as

(1)

Thenbyfixingtheaxiso1y1in the coordinate systemo1x1y1z1, the coordinate systemo2x2y2z2can be obtained by rotating the planex1o1z1on the axiso1y1for the angle ofψ. Coordinates transformations are shown as

(2)

Aftertransmutationofcoordinates,theseeker’scoordinatesareM(0,Ym,0), the coordinates of target in coordinate systemo2x2y2z2areT(X2T,Y2T,Z2T). Fig.2 shows the principle of strap-down seeker detecting target. In Fig.2,Tis the target,Mis the seeker,Dis the center of the seeker plane D-EGH, the axis of body intersects ground at the pointB,Fis the image of target projecting on the seeker plane, the lineMTis the LOS,δis the angle difference between the LOS and the axis of missile, the angleλandτreflect the azimuth of target relative to the strap-down seeker[6].

λ,τ,δcan be calculated by the relative position between the target and missile which is shown in Fig.2.

Fig.2 Strap-down seeker model of target detection

(3)

λ=

(4)

τ=

(5)

In Eqs.(4)-(6),λ,τ∈(0°-180°). Azimuth angleσcan be obtained byλ,τ. Whenτ>90°,σ=360°-λ. Whenτ≤90°,σ=λ.

2 Angle of attack or sideslip identification based on dual-accelerometer

In the presence ofζ1, an attitude pursuit guidance system may become unstable. Furthermore, a strap-down seeker outputs boresight errors with respect to the body. Therefore theζ1sensor is absolutely necessary for attitude pursuit guidance, however for low cost missile, the INS is too expensive. Therefore a new way to estimateζ1by using two accelerometers is studied[5].

The fixed position of two accelerometers on the body is shown in Fig.3. In Fig.3ois the center of gravity, the body-fixed coordinate system isoxbybzb,A,Bare two accelerometers and has the same sensitivity axisoyaandozawhich superpose the axisoybandozb,lAis the distance between accelerometer A and center of gravity (CG),lBis the distance between accelerometer B and center of gravity andris the rolling attitude angle.

Fig.3 Fixing position of two accelerometers

It is assumed that the whole angle of attackαTis defined as

(6)

whereαis the angle of attack andβis the angle of sideslip. The missile’s oscillation angle is equivalent toαTapproximately, which can be described as

αT=Ksin(ωt)，

(7)

whereKis the swing of the missile,ωis the natural oscillation frequency.

The outputs of the dual-accelerometers can be written as

a=a1+a2,

(8)

wherea1is the acceleration acting on the center of mass caused by aerodynamics anda2is the acceleration produced by body angular acceleration.a1can be expressed as

(9)

(10)

wherelis the distance between the accelerator and CG, if the accelerator is fixed before CG,l>0.

The accelerations measured by the axisoyaandozaof accelerator A can be expressed as

(11)

(12)

The accelerations measured by the axisoybandozbof accelerator B can be expressed as

(13)

(14)

whereγ′ is the roll angle between the oscillation direction and the axisoya.

Then, by substituting the appropriate values oflA,lBandω, the whole angle of attackKsin(ωt) can be described as

(15)

Theprojectionvaluesofthebodyoscillationontheaxisofoybandozbin body-fixed coordinate system are

(16)

(17)

The angle of attack and sideslip can be expressed by the following equations

(18)

(19)

3 Estimation of additional angle of attackor sideslip caused by wind

Generally, the wind vectorVWcan be decomposed into two parts: the crosswind vectorWzwhich is perpendicular to trajectory plane and the vertical wind vectorWxwhich is paralleled with the trajectory plane (Fig.4). With the effect ofWx, the missile velocity vector becomesV1fromVand engenders the additional angle of attack ΔαW. In the presence of crosswind, the velocity vector becomesV2and brings the additional angle of sideslip ΔβW[7-8].

The additional angle of attack and sideslip can be estimated as

(20)

(21)

where by using an anemometer mounted on the launch tube,WxandWzare measured, ϑ can be obtained by gyroscope andVis the missile’s average velocity.

Fig.4 Additional angle of attack and sideslip caused by wind

4 Modified attitude pursuit guidance law

It is known that with the effect ofζ1andζ2, the axis of body rotates an angleεfrom the velocity vector and the angelεconstitutes of two parts: one part is the angle of attack or sidelip, the other part is the additional angle of attack or sideslip. With this angleε, the output of strap-down seeker has become inaccurate. The attitude pursuit guidance may become instable with this output error. With the modified attitude pursuit guidance law,ζ1is identified by Dual-accelerometer andζ2is estimated by using an anemometer. Then by compensating the angleε, the flight path angleθand flight path deflection angleψVare approximately shown as

θ≈ϑ-α-ΔαW,

(22)

ψV≈ψ-β-ΔβW.

(23)

The relative position between missile and target can be confirmed by the angleδandσwhich are related with the parametersX2T,Ym,Z2T, ϑ,εin the model of strap-down seeker.Ymis the missile flight altitude measured by altimeter; ϑ is the pitching angle of missile which can be obtained by a gyroscope.

Combining Eq.(3) and Eq.(4),X2TandZ2Tin coordinate systemo2x2y2z2can be calculated, shown as

X2T=Ymtan(90°+ϑ)(tan2λ·cos2δ+1)±Ym·

[2tan2(90°+ϑ)tan2λ·cos2δ-tan4(90°+

ϑ)cos4δ-2tan2(90°+ϑ)cos4δ-

tan4(90°+ϑ)cos4δ·tan2λ+2tan2(90°+

ϑ)cos2δ-cos4δ-2tan2(90°+ϑ)cos4δ·tan2λ+

cos2δ+tan4(90°+ϑ)cos2δ+tan4(90°+ϑ)cos2δ·

[tan2(90°+ϑ)cos2δ+cos2δ-

tan2(90°+ϑ)+cos2δ·tan2λ]

(24)

Z2T=Ymtan(90°+ϑ)tanλ·[1-tan2(90°+

ϑ)cos2δ-cos2δ+tan2(90°+ϑ)]±

Ymtanλ·[2tan2(90°+ϑ)tan2λ·cos2δ-

tan4(90°+ϑ)cos4δ+cos2δ-2tan2(90°+

ϑ)cos4δ-cos4δ-tan4(90°+ϑ)cos4δ·

tan2λ+2tan2(90°+ϑ)cos2δ+cos2δ·tan2λ-

2tan2(90°+ϑ)cos4δ·tan2λ+tan4(90°+

ϑ)cos2δ-cos4δ·tan2λ+tan4(90°+ϑ)cos2δ·

(25)

The signs in Eqs.(30) (31) are defined as follows:

① Whenσ∈(0,90°), the sign in the numerator ofX2Tis positive and the sign in the numerator ofZ2Tis positive.

② Whenσ∈(90°,180°), the sign in the numerator ofX2Tis minus and the sign in the numerator ofZ2Tis minus.

③ Whenσ∈(180°,270°), the sign in the numerator ofX2Tis minus and the sign in the numerator ofZ2Tis positive.

④ Whenσ∈(270°,360°), the sign in the numerator ofX2Tis positive and the sign in the numerator ofZ2Tis minus.

The coordinates transformations from coordinate systemo2x2y2z2to coordinate systemo3x3y3z3are shown as

(26)

Thecorrectedoutputsδ1,λ1,τ1of the seeker by using modified attitude pursuit guidance law are shown as

δ1=

(27)

(28)

τ1=

(29)

Whenτ1>90°,σ1=360°-λ1. Whenτ1≤90°,σ1=λ1. The angleδ1andσ1are transported to the control system as the outputs of the strap-down seeker.

5 Results of Monte-Carlo simulation

To evaluate the performance of the proposed guidance law, the simulation is performed, which uses Monte-Carlo method against stationary targets both in the presence and absence ofζ1andζ2.

Fig.5 shows the simulation results in a lateral plane with attitude pursuit guidance law. It is assumed that at the time zero, yaw attitude angle is -3°, the initial relative cross displacement between missile and target is 5 m, and the wind velocity is 8 m/s. Fig.5b is the contrastive curve of the target projecting on the seeker in the presence and absence of wind. It shows that the position of target image which projects on the strap-down seeker becomes inaccuracy and this error information which is produced by the seeker will reduce the precision.

Fig.6a is the scatter of missile using attitude pursuit guidance law in the absence of additional angle of attack and circular error probable (CEP) is 3.8 m. In Fig.6b, the modified attitude pursuit guidance law is adopted to compensatingζ1, the result of Monte-Carlo simulation shows that CEP is 2.2 m which means the modified attitude pursuit guidance law can enhance the precision by eliminating the impact of angle of attack.

Fig.5 Simulation result with 8 m/s crosswind

Fig.6 Simulation result in the presence of ζ1

Fig.7 Simulation result in the presence of ζ1and ζ2

The influence of wind is introduced into the Monte-Carlo simulation model, in the simulation, the 8m/s crosswind is in the same direction as the angle of attack. In the presence of crosswind, the additional angle of sideslip occurs and this angle combined with the angle of attack will result in unacceptable guidance precision by using attitude pursuit guidance law, CEP is 11.2 m, shown in Fig.7a. For the same case, Fig.7b presents the result of simulation using the modified attitude pursuit guidance law, CEP is 2.6 m. It evaluates that the modified attitude pursuit guidance law can compensateζ1andζ2.

6 Conclusion

Attitude pursuit guidance law will fail in the presence of the additional angle of attack and the angle of attack. The results of Monte-Carlo simulation indicate that the modified attitude pursuit guidance law performs very well for low cost missiles with strap-down seekers. The present formulation eliminates the disturbing factors by employing an angular correction. This angle is the sum of angle of attack and the additional angle of attack. The angle of attack may be obtained by the identification based on a dual-accelerometer. The additional angle of attack can be estimated by using an anemometer mounted on the launch tube.

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(Edited by Wang Yuxia)

2012- 12- 19

TJ 765.3 Document code: A Article ID: 1004- 0579(2014)02- 0158- 07

E-mail: 10901034@bit.edu.cn