3D nonlinear guidance law for bank-to-turn missile

Shuangchun Peng, Liang Pan, Tianjiang Hu, Lincheng Shen College of Mechatronic Engineering and Automation, National University of Defense Technology,

Changsha 410073, P.R.China psc1212001@yahoo.com.cn

Keywords: bank-to-turn missile, three-dimensional, guidance Law, Lyapunov theory, twist

Abstract. A new three-dimensional (3D) nonlinear guidance law is proposed and developed for

bank-to-turn (BTT) with motion coupling. First of all, the 3D guidance model is established. In detail,

the line-of-sight (LOS) rate model is established with the vector description method, and the

kinematics model is divided into three terms of pitching, swerving and coupling, then by using the

twist-based method, the LOS direction changing model is built for designing the guidance law with

terminal angular constraints. Secondly, the 3D guidance laws are designed with Lyapunov theory,

corresponding to no terminal constraints and terminal constraints, respectively. And finally, the

simulation results show that the proposed guidance law can effectively satisfy the guidance precision

requirements of BTT missile.

Introduction

BTT missile has overwhelming advantages and better performance than skid-to-turn (STT) missile in

aerodynamic efficiency, maneuverability, controllability, et al[1][2]

. As a result, increasingly severe

mission requirements and the potential for improved performance offered by BTT steering are

generating renewed interest in BTT missiles. However, there are several challenges in BTT missile

flight control as well as guidance law design. In detail, large roll angle and rolling rate are necessary in

BTT mode [3]

. On the contrary, pre-existing traditional decoupling methods [4][5]

almost are

established on the assumption that the roll angle and its rolling rate are nearly zeros. In this way,

theoretical error would be produced if traditional guidance law is used into BTT missiles due to the

information loss caused by decoupling. To solve this problem, new methods considering motion

coupling should be studied and new guidance law adapting to BTT missiles should be designed.

Aiming at such motion coupling in BTT missiles, Han [6]

, Shi [7]

, She [8]

, Yuan [9]

and Tyan [10]

designed a three-dimensional non-decoupling guidance law respectively based on spherical

coordinates. Zhang [11]

, Chiou [12]

and Li [13]

equally designed a non-decoupling guidance law

respectively based on differential geometry method. These two methods could effectively avoid the

guidance information loss, however, there still faced some challenges caused by the terminal angular

constraints, so the above-mentioned guidance laws could not be adapted to the situation with terminal

angular constraints. Furthermore, Han [3]

and Peng[14]

both designed Lie-group based non-decoupling

guidance law for the three-dimensional motion coupling, which can satisfy the situation with terminal

angular constraints, but the result is too complex to be imported into engineering applications.

Inherent coupling phenomenon exists in BTT missiles generally, to solve this problem, this paper

proposed a 3D nonlinear guidance law considering motion coupling, which not only lost any guidance

information, but also can satisfy the situation with both azimuth and impact angular constraints. The

proposed guidance law has simple expression , so it’s convenient for applications in practical

engineering.

Problem formulation

Generally, missile guidance could be divided into the control problems of line-of-sight (LOS) rate

and direction. Therefore, we can establish the guidance model from the above-mentioned two aspects.

Advanced Materials Research Vols. 317-319 (2011) pp 727-733Online available since 2011/Aug/16 at www.scientific.net© (2011) Trans Tech Publications, Switzerlanddoi:10.4028/www.scientific.net/AMR.317-319.727

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Modeling for the LOS rate

Consider the representative 3D attack geometry in Fig.1 with a fixed target T. The coordinate system

Oxyz has its origin O at the target. The missile is denoted by M and its velocity is denoted by v whose

magnitude is v. The LOS vector is denoted by r whose magnitude is r. The LOS elevation and azimuth

are denoted by qd and qt. the LOS azimuth in the swerving plane is denoted by qtt.

In conventional decoupling method[4][5]

, the missile’s 3D motion can be decoupled into two 2D

motions, namely pitching and swerving. And then, the guidance laws are designed independently, but

it will cause information loss during decoupling and have a negative influence on guidance precision.

In order to avoid it, five unit vectors: er, et, ec, ed and ett are introduced. Among them, er is aligned with

the vector r, et is aligned with the y-axis, ec is aligned with the projection of vector r. Define de as

d c t= ×e e e (1)

Thereby the three unit vectors ec, et and ed constitute one orthogonal system. Define ett as

tt d r= ×e e e (2)

Accordingly the three unit vectors ed , er and ett constitute another orthogonal system in the same way.

Based on the theorem of angular velocity addition [15]

, the LOS rotational velocity ω can be

denoted by [14]

d d t tq + q=ω e e� � (3)

From Eq. (3) we can easily see that the direction of LOS rotational velocity ω is not always

perpendicular to the LOS vector r. considering that the component ω� parallel to the direction of LOS

will not affect the LOS direction, indeed the factor which has influence on LOS direction is the

component ⊥ω perpendicular to r. Therefore here we only consider the control aiming to the

component ⊥ω . Form Eq. (3) and Fig.1, we have

cosd d t d ttq q q⊥ = +ω e e� � (4)

Note that

costt t dq q q=� � (5)

so

d d tt ttq q⊥ = +ω e e� � (6)

Considering

cos sin

( ) sin

d

tt

d d t t d t c t d r t d tt

tt tt t t d d tt t d d

q q q q q q

q q q q

= × = × = = −

= × = − × =

e

e

e ω e e e e e e

e ω e e e e e

� � � � �

� � � � (7)

The theorem of angular velocity addition is used in the second equation of (7), taking the derivative

on both sides of (6) , we have

cos sin sind d tt tt d t d r d t d tt tt t d dq + q q q q q q q q q q⊥ = + − +ω e e e e e� �� �� � � � � � � (8)

where, d dq e�� and tt ttq e�� are the components of LOS angular acceleration corresponding to pitching

and swerving plane respectively, cos sin sind t d r d t d tt tt t d dq q q q q q q q q− +e e e� � � � � � is the coupling component

caused by missile’s rolling in the course of decoupling. In BTT mode, the coupling component is

biggish in the right of Eq. (8), so it can not be ignored simply, otherwise theoretic error will be brought

into the guidance course.

728 Equipment Manufacturing Technology and Automation

Because of [4]

2 1

2 1

d d d

g g

tt tt t

g g

q q θT T

q q θT T

= +

= +

��� �

��� �

(9)

where θd and θt are the azimuth angles of v in pitching and swerving plane respectively.

Substitute Eqs. (9) into Eq. (8) and define d d t ttθ θ= +u e e� � , we have

2 1cos sin sind t d r d t d tt tt t d d

g g

q q q q q q q q qT T

⊥ ⊥= + + − +ω ω u e e e� � � � � � � (10)

The Eq.(10) depicts the LOS rate model well and makes a good base for the coming

three-dimensional guidance law design.

Fig 1. Sketch map of LOS rate

Fig.2 Euler angles and vectors determined by LOS

Modeling for LOS direction changing

Although the Eq. (10) reflect the LOS rate nicely, it can’t reflect the LOS direction changing straight,

so the design of guidance law with terminal angular constraints will face challenge if only based on

the LOS rate model. The Ref. [3] and [14] made the defect up through Lie-group method, however it

brought much complex computation into guidance course. Therefore there is necessary to study the

changing rule of LOS direction. In this paper a twist is designed refer to Ref. [16] and [17], and it

depict the LOS direction well.

If a LOS vector ( ) [ , , ]Ts x y z=r between the missile M and the target T is ascertained, the

corresponding Euler angles: 1ξ , 2ξ , 3ξ , 1σ , 2σ , 3σ can be determined by r. As shown in Fig.2, 1ξ , 2ξ , 3ξ

denote the angles between r and its projections in the three planes oy z , oz x , ox y respectively. 1σ , 2σ , 3σ

denote the angles between y, z, x and the projections of r in oy z , oz x , ox y respectively.

The twist is defined as

(11)

Here atan2 is four-quadrant inverse tangent function.

Taking the derivative of Eq.(11), we have

2 2

2 2 1

2 2

1 ( ) 0 0 0

0 1 ( ) 0 0

00 0 1 ( )

y z z y x

z x z x y

y x zx y

−

+ − = + − × −+

rσ P r r

�

�� � �

�

(12)

where

2 2 2 2 2 2diag( + , + , + )y z z x x y=rP (13)

Advanced Materials Research Vols. 317-319 729

Note that

r r= ×e ω e� (14)

therefore

(15)

and then

1ξ−

⊥=σ Q ω� (16)

where

2 2 21 2 3diag(cos ,cos ,cos )ξ ξ ξ ξ=Q (17)

The Eq. (16) founded the relationship between LOS direction changing and LOS rate, Based on

Eq. (16), the guidance law with terminal angular constraints can be designed expediently.

3D Nonlinear Guidance Law

Solution for no terminal angular constraint

For the case of no terminal angular constraint, A common criterion is that LOS rate tends to zero, so

we can design the Lyapunov function as follow

1

2V ⊥ ⊥= ⋅ω ω (18)

Using the Eq. (10) and taking the derivative of Eq. (18), we have

2 1( cos sin sin )d t d r d t d tt tt t d d

g g

V q q q q q q q q qT T

⊥ ⊥= ⋅ + + − +ω ω u e e e� � � � � � � (19)

Note that

0r⊥ ⋅ =ω e (20)

so the coupling component along with er direction needn’t to be considered. Design

1 ( sin sin )g tt t d d d t d ttk T q q q q q q⊥= − − −u ω e e� � � � (21)

Substitute Eqs. (21) into Eq. (19), we can see: if 1 2k > , then 0V <� , the stability of Lyapunov function

can be guaranteed.

So the Eq. (21) ( 1 2k > ) is the guidance law for no terminal angular constraint, where 1k ⊥− ω is

control item for inhibiting the rotation of LOS, ( sin sin )g tt t d d d t d ttT q q q q q q− −e e� � � � is the item for making

the influence of coupling up. If the coupling item is ignored and order 1 3k = , the guidance law will be

accordant with the traditional guidance law [4]

.

Solution for terminal angular constraints

Within the many cases the terminal constraints are necessary, in many documents, the impact angle

constraint is considered weightily, but the azimuth angle constraint is usually left out of account.

However in modern precise guidance, the pure impact angle control is insufficient, and the azimuth

angle should be also controlled to a right value at the same time, because only in both fit impact and

azimuth angle the capability of missiles can be exerted fully. Therefore the study on guidance law

aiming at both impact and azimuth angular constraints is needed.

It’s easy to know that the terminal impact and azimuth angular constraints can be transformed into

a twist constraint Dσ . So in this situation the Lyapunov function can be designed as follow

1 1∆ ∆

2 2

TV ⊥ ⊥= ⋅ +ω ω σ Λ σ (22)

where: ∆ D= −σ σ σ , Dσ denotes the terminal angular constraints, 11 22 33diag( , , )λ λ λ=Λ is a positive

constant diagonal matrix.

730 Equipment Manufacturing Technology and Automation

Taking the derivative of Eq. (22) by using the Eq. (10) and (16), we have

12 1[ cos sin sin ∆ ]d t d r d t d tt tt t d d ξ

g g

V q q q q q q q q qT T

−⊥ ⊥= ⋅ + + − + +ω ω u e e e Q Λ σ� � � � � � � (23)

In the same way, order

12 ∆ ( sin sin )g ξ g tt t d d d t d ttk T T q q q q q q

−⊥= − − − −u ω Q Λ σ e e� � � � (24)

Therefore, if 2 2k > , then 0V <� will be guaranteed, so the stability of Lyapunov function can be

guaranteed in the same way.

The Eq. (24) ( 2 2k > ) is the guidance law for the situation with both azimuth and impact

constraints, where: 2k ⊥− ω is control item for inhibiting the rotation of LOS, 1∆g ξT

−− Q Λ σ is the item for

controlling LOS vector to a demanded direction, ( sin sin )g tt t d d d t d ttT q q q q q q− −e e� � � � is the item for making

the influence of coupling up. If the coupling item is ignored and order 22 4, 2 /ξ gk T= =Λ Q , the guidance

law will be similar to the traditional guidance law [4]

.

Simulation and discussion

To validate the effectiveness of the proposed guidance law, in this section we compared the new

guidance law with the traditional method in [4], within a similar guidance problem. The simulation

object is a BTT missile and the simulation conditions are listed in Tab.1. Tab.1 Simulation conditions

Conditions Numerical value Unit

Missile Initial Position (Long., Lat., ht) (0.4, 0.4, 22) (deg., deg., km)

Target Position (Long., Lat., ht) (0, 0, 0) (deg., deg., km)

Missile Initial Velocity (in Geocentric Coordinate System) (0, -1050, -328) m/s

Attack Angle Range [-6, 12] deg.

Roll Angle Range [-45, 45] deg.

The guidance laws (21) and (24) are used respectively in solving the situation with no terminal

angular constraint and with both azimuth and impact angle constraints .The parameters corresponding

to (21) and (24) respectively valued as follows

1 3k = (25)

2

2

4

2 /ξ g

k

T

=

=Λ Q (26)

Simulation 1: no terminal angular constraint

Here, to the situation of no terminal angular constraint, the comparison experiment between the

proposed method and traditional method is done. The miss distance is 0.87m in proposed method

while it’s 3.52m in traditional method. The trajectories are shown in Fig.3 and the guidance

commands are shown in Fig.4~5.

As indicated in Fig.3, the trajectory in this method is flatter than in traditional method; it means

that fewer maneuvers are needed and less energy is consumed in the guidance course when in the

proposed method. As indicated in Fig.4~5, both the attack and roll angle commands converge to small

values in the proposed method when in terminal phase, however the corresponding values in

traditional method have slight tendency of transpiration which may has influence on the guidance

stability.

Simulation 2: both azimuth and impact angle constraints

Similarly, aiming to the situation of both azimuth and impact angle constraints, the comparison

experiment between the proposed method and traditional method is done too.

Advanced Materials Research Vols. 317-319 731

Set that the desired impact and azimuth angle are 70º and 30º respectively when attacking the

target. The simulation results are shown as in Fig.6~8. The miss distance is 1.31m in proposed

method while 4.62m in traditional method. The impact and azimuth angle are 70.03º and 29.62º

respectively in proposed method while they are 68.37º and 32.03º in traditional method.

As indicated in Fig.6, the trajectory can adjust to the desired direction quicker in the proposed

method than in traditional method. Therefore the proposed method is more adaptive to both impact

and azimuth angular constraints. From Fig.7~8, we can see that the guidance commands converge to

small values in the proposed method in terminal phase, while the corresponding values have tendency

of transpiration which may affect the guidance stability. Simulation shows that the proposed guidance

law has higher guidance precision and more robust stability, which can be adapted to maneuvering

BTT missiles with terminal angular constraints.

0

0.2

0.4 00.1

0.20.3

0.4

0

0.5

1

1.5

2

2.5

x 104

Lat.(°)Long.(°)

Height(m)

This method

Traditional method

Fig.3 Comparison between 3D trajectories without

terminal angular constraint

0 20 40 60 80-6

-4

-2

0

2

4

6

t(s)

Attack angle( °)

This method

Traditional method

Fig.4 Comparison between attack angle commands

without terminal angular constraint

0 20 40 60 80-50

0

50

t(s)

Roll angle( °)

This method

Traditional method

Fig.5 Comparison between roll angle commands without

terminal angular constraint

0

0.1

0.2

0.3

0.4 0

0.1

0.2

0.3

0.4

0

0.5

1

1.5

2

2.5

x 104

Lat.(°)Long.(°)

Height(m)

This method

Tradional method

Fig.6 Comparison between 3D trajectories with terminal

angular constraints

0 20 40 60 80-5

0

5

10

15

t(s)

Attack angle ( °)

This method

Traditional method

Fig.7 Comparison between attack angle commands with

terminal angular constraints

0 20 40 60 80-50

0

50

t(s)

Roll angle ( °)

This method

Traditional method

Fig.8 Comparison between roll angle commands with

terminal angular constraints

Discussion

Through guidance simulations, the proposed method is validated to satisfy the precision guidance of

BTT missiles in the conditions of motion coupling. Moreover, the proposed method has relatively

strong adaptability to motion coupling. The guidance law considered the coupling comprehensively

and depicted it well, so the information loss which is usually generated in traditional decoupling

method can be avoided. Therefore, the guidance precision is higher, to some extent.

Besides, in the aspect of information integrality, the proposed method has coherence with the

method in [3] and [14], and significantly, its geometrical meaning is relatively clearer.

732 Equipment Manufacturing Technology and Automation

Conclusions

Aiming at motion coupling in guidance of BTT missiles, this paper proposed a new non-decoupling

3D nonlinear guidance law. In detail, the 3D guidance model is established through vector depiction

and twist depiction. Then based on Lyapunov theory, the 3D guidance laws are designed respectively

corresponding to no terminal constraint and terminal constraints. Finally the simulation results show

that the proposed guidance law can satisfy the guidance precision requirements of BTT missile.

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Advanced Materials Research Vols. 317-319 733

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