AbstractA predictor-corrector guidance method that tracks the optimized trajectory of hypersonic reentry glide process is presented. First, aiming at the minimum heating rate problem with multiple constraints, the hp-adaptive pseudospectral method generates optimized trajectory rapidly. Then a BPNN (Back-Propagation neural network) is trained by parameter profiles of optimized trajectory considering different dispersions to simulate the nonlinear mapping relationship between the current flight states and terminal states. Hence, the predictor algorithm substituted by the BPNN can be more efficient and the guidance is achieved by nullifying the terminal errors. Simulation examples show that the guidance method based on trajectory optimization and neural network can well satisfy both path and terminal constraints and has good validity and robustness.

Keywords Predictor-corrector Guidance; Neural Network; Optimized Trajectory; hp-adaptive Pseudospectral Method

I. INTRODUCTION

With the rapid rise of hypersonic vehicle technology, the demand for researching reentry guidance also increases. The NASA Marshall Space Flight Center started the AG&C (Advanced Guidance and control) program at the end of 1999[1]. Since then, the research of reentry guidance with adaptive and robustness began to form a new round of upsurge[2-6].

Guidance methods can be divided into standard trajectory method and predictor-corrector guidance. The standard trajectory method generates a reference trajectory profile by optimization theory, and then tracks it; predictor-corrector guidance focusing on the prediction of terminal state, evaluates the errors between predictor value and theoretical value at first, and then generates the guidance command to achieve a desired flight termination. Both two methods have advantages and disadvantages. This paper blending trajectory optimization idea in predictor-corrector guidance by BPNN, proposes a novel guidance method. The effectiveness and robustness of the method is verified by simulations.

II. TRAJECTORY OPTIMIZATION METHOD

The trajectory optimization method is presented in three sections. The first section incorporates the equation of motion, path constraints and terminal constraints of the optimization problem. The second section proposes the optimization objective. The third section discusses the pseudospectral

*Resrach supported by Aeronautical Science Foundation of China under

grant #20130788001. 1 Science and Technology on Scramjet Laboratory, National University

of Defense Technology, Changsha 410073, China. 2 Science and Technology on Aircraft Control Laboratory, Xian 710065,

China.

method which is involved to solve the optimization problem faster.

A. Reentry Dynamics Modeling Ignoring the earth rotation, and making the vehicle sideslip

angle is 0, the 3 DOF (degrees of freedom) equations of motion of reentry process over a spherical Earth in an Earth-centered and Earth-fixed coordinate system are given by

sincos sin

coscos cos

sin

cos coscos

tan cos sinsincos

D m

L m

L m

r vv

rv

rC qSv g

mC qS g V

mv v rC qS v

mv r

=

=

=

=

= +

= +

(1)

Where [r, , , v, , ] is the flight state, r is the radial distance from the center of the earth to the vehicle, and are the longitude and latitude. v is the Earth-relative velocity, and are the flight path angle and heading angle. [,] is the control state, is the angle of attack, is the bank angle.

For the reentry flight, the path constraints such as heating rate, aerodynamic load and dynamic pressure factor are given by

0.5 3.15 maxQ C v Q= (2)

maxcos sin

zL Dn n

mg += (3)

2 max12

q v q= (4)

Here, C is a constant, and is the atmospheric density, L and D are the aerodynamic lift and drag.

Besides, the terminal constraints need to be satisfied, including longitude and latitude, altitude and speed constraints. The requirements of the terminal condition are expressed as follows:

, , ,f ref f ref f ref f refh h v v = = = = (5)

Neural Predictor-Corrector Guidance Based on Optimized Trajectory*

Zhang Kai1, 2, Guo Zhenyun1, 2

523

Proceedings of 2014 IEEE Chinese Guidance, Navigation and Control Conference August 8-10, 2014 Yantai, China

978-1-4799-4699-0/14/$31.002014 IEEE

B. Optimization Objective According to the characteristics of reentry vehicle and

difference of flight tasks, the optimization targets of reentry trajectory are various. To the optimization problem of reaching the specified target point or region, the total heating rate is a desired optimization objective, and it can be represented by heating rate integral item. The optimization objective related to flight path angle is also introduced by considering the stability and trajectory smoothness of flight. Hence, taking the weighted sum of the above two objectives in account, the function of the optimal control problem is defined as:

0 0

2f ft t

t tJ a Qdt b dt= + (6)

Here, a and b are constant coefficients.

C. Pseudospectral Optimization Strategy Duo to the great advantage in computational efficiency,

the pseudospectral method has gradually become a powerful tool for solving optimal control problems [7-9]. There are many kinds of pseudospectral method which use various numerical fitting methods based on diverse nodes, distribution points and polynomial functions. Hp-adaptive pseudospectral method is an excellent solution of optimization problem which takes the advantages of Radau pseudospectral method and the hp-finite element method [10-11]. With simple implementation, excellent accuracy and efficiency, this method has a wide application prospect in the field of aerospace trajectory optimization.

Figure 1. hp-adaptive pseudospectral calculation process.

As shown in Figure 1, hp-adaptive pseudospectral calculation process is described as follows:

Present the initial grid time segments, polynomial orders and Lagrange discrete errors roughly, and initialize the state variables and constraints.

Transform the time variable, and discretizes the optimization equation by Radau pseudospectral method, including dynamic equations, path constraints and boundary conditions. Then the optimization problem is transformed into solving the NLP problem.

Utilize the hp-finite element method to choose whether h-method or p-method: h-method fractionizes the interpolation unit length by segmenting finite element mesh progressively; p-method increases the degrees of Lagrange polynomial in each segment which has large error.

Solve the NLP problem, if errors meet the requirements, stop the calculation process; and if not, go back to fore step.

According to the calculation process above and dynamic model given in Section A, the state variables and control variable profiles can be obtained by solving the reentry trajectory optimization problem, which is of great importance to train the BPNN in Chapter III.

III. PREDICTOR-CORRECTOR GUIDANCE ALGORITHM

The guidance algorithm is presented in five sections. The first section discusses a BPNN predictor algorithm which can obtain the terminal errors faster instead of the traditional integral process of predictor-corrector guidance. The second section discusses how to train the BPNN. The third section proposes the corrector algorithm which adjusts two control variables to nullify the terminal errors by feedback. The forth section shows the lateral guidance briefly. The fifth section summarizes the guidance algorithm strategy.

A. BPNN Predictor Algorithm Predictor-corrector guidance predicts terminal states based

on current states, and adjusts control variables to eliminate terminal error. The control variable profiles have been obtained by optimization method in Chapter II, thereby terminal states can be obtained by integrating dynamics equation to track the profiles based on the real-time position and velocity vector. It means that a nonlinear mapping relationship is indicated between current states and terminal states of the optimized trajectory as follow:

{ , } { , }Ff fh v H V

(7)

Here, H

is the position vector andV

is the velocity vector of the current flight states. hf is terminal altitude, vf is terminal velocity.

To facilitate the burden of the predictor process onboard, its urgent to find a method to predict the terminal states more quickly. The BPNN has excellent characteristics of fitting a nonlinear function with any precision, which is used in guidance gradually in recent years [12-14], so the nonlinear mapping relationship referred above can be represented by the BPNN.

Variables discretization

Interpolate unit length

Solve NLP problem

Increase order of polynomial

Satisfy terminal constraint? End

Satisfy the precision?

Begin

Yes

Yes

No

No

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Figure 2. BPNN predictor sturcture.

To sum up, it indicates that the terminal states can be obtained by the BPNN, and the terminal state errors are generated online by comparison between BPNN predictor terminal states and the terminal constraints. Hence, the integral process of traditional predictor-corrector guidance can be taken over by the BPNN which can greatly alleviate the burden of computer onboard.

B. BPNN Design According to the control parameter profiles of optimized

trajectory obtained in Chapter II, the terminal states can be obtained by dynamics integral of the control parameter profiles. Thereby the terminal states with errors can also be obtained by integration of the optimized profiles referred above if the current flight states with errors. Hence, the training samples are chosen as follow: (1) the flight states of sampling nodes are obtained by taking every nth diverse nodes of the optimized trajectory at first, and then input training samples of BPNN are generated uniformly and randomly by Monte Carlo method based on the flight states of sampling nodes. So the resulting terminal altitude and speed can be obtained by integration of the optimized profiles referred above respectively, which are taken as output training samples.

Six flight state parameters are presented as parameters of the input training samples: [r, , , v, , ]. In the practical case, because of the change of launch point and target point, and are not suitable to be the input training samples, therefore, the longitude and latitude are transformed into downrange R and crossrange L. In addition, because the change of geocentric distance r is so smooth that r is not suitable to training the BPNN, altitude h is substituted for geocentric distance. Hence, parameters of the input training samples are presented as follows: [h, L, R, v, , ].

Thereby the BPNN is trained by the training samples, and the BPNN predictor algorithm to predict the terminal states is achieved now.

C. Corrector Algorithm The corrector algorithm reduces or nullifies the terminal

errors by adjusting the control variables. In general, traditional guidance realizes the trajectory by only adjustment of bank angle . In this paper, is considered as the primary control parameter for modulating the trajectory. In addition, is

considered as the accessory control parameter, which consists of the predefined profile 0(v) with a limited variation termed as . 0(v) is obtained by optimized trajectory profile in Chapter II, is adjusted slightly online by feedback. The reason why control variables are considered as the aforementioned design is that: (1) the terminal states need to meet the landing position, speed and altitude constraints, only adjusting the bank angle cannot satisfy all terminal constraints accurately;(2) affects the aerodynamic characteristics of the aircraft directly, because the predefined profile of angle of attack has been optimized, the limited variation can avoid large variations on the profile in case of increasing the burden of control system.

The corrector algorithm used for feedback linearization is given by

0 1 max

max max

0 0( )f refk v v

(8)

0 2 max

max max

0 0( ) ( )f refv k h h

(9)

Here, k1 and k2 are feedback coefficients, and 0 is a constant. max is the maximum bank angle, max is the maximum angle of attack.

With the corrector algorithm the terminal states will meet the terminal constraints, only leaving the path constraints which are not considered here. Actually, the predictor-corrector guidance will modulate the trajectory to track the optimized trajectory, its clear that the trajectory optimization has taken the path constraints into account. In view of this, the corrector algorithm can meet the path constraints without special consideration.

D. Lateral Guidance Logic The lateral guidance is based on the bank-reversal criterion

presented as follows:

1 ( )

( ) 1 ( )( ) ( )

th

th

th th

vsign v

hold v v

= < <

(10)

Where the (v) is the heading error, th(v) is a variable versus velocity. The sign of the bank angle is maintained until Eq. (10) is again violated.

E. Predictor-Corrector Guidance Strategy As shown in Figure 3, the BPNN predictor-corrector

guidance method is proposed as follows:

According to BPNN predictor algorithm which has been involved the idea of optimization, terminal velocity and altitude is predicted corresponding to current flight state by the BPNN.

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Terminal errors are obtained by comparison between BPNN predictor terminal states and the terminal constraints as the feedback to corrector algorithm, bank angle is generated by terminal velocity error, and the sign of is determined by lateral guidance; the limited variation is generated by terminal altitude error.

When the derivative of Leave-to-go 0S = , terminate the reentry process.

In view of this, it is seen that the terminal longitude and latitude are achieved implicitly, which can meet the terminal constraints at every cycle of BPNN predictor algorithm, simultaneously.

Figure 3. Predictor-corrector guidance strategy process.

IV. SIMULATION ANALYSIS

A. Simulation of Nominal Case A 3-DOF MATLAB simulation for the reentry vehicle

CAV-H of Lockheed Martin is used for the conceptual simulation model of the predictor-corrector guidance [15]. The path constraints are: Qmax= 2.6MW/m2, nmax= 4, qmax= 200kPa. Table 1 shows the initial and terminal conditions.

The simulation results with the conditions and constraints above are shown in Figs. 4-11, where Case 1 is the guidance trajectory parameters, and Case 2 is the optimized trajectory parameters.

TABLE I. INITIAL AND TERMINAL CONDITIONS

Parameters Initial value Terminal value h/(km) 48 20 /() 0 70 /() 0 0

Parameters Initial value Terminal value v/( ms-1) 6000 2000 /() 0 / /() -60 /

Figure 4 shows the flight altitude versus velocity. It is apparent that the altitude deviation is mainly in the initial stage of reentry flight, where aerodynamic force is too small to guarantee the optimized flight profile. With the descent of vehicle altitude, the aerodynamic increases, thereby the actual flight states can approximate standard flight profile and the deviation turns smaller relatively. It can be seen that the phugoid motion between the guidance trajectory and optimized trajectory is caused by the deviation. Besides, the terminal states meet the terminal constraints relatively under the guidance.

100020003000400050006000

20

30

40

50

60

Velocity/(m/s)

Alti

tude

/(km

)

Case 1

Case 2

Figure 4. Altitude profile.

100020003000400050006000

10

12

14

16

Velocity/(m/s)

Ang

le o

f atta

ck/()

Case 1

Case 2

Figure 5. Angle of attack profile.

BPNN predictor

Begin

Calculate bank angle

Sign of bank angle

Dynamic equation

Satisfy terminal conditions?

End

Yes

No

Calculate angle of attack

526

100020003000400050006000-50

0

50

Velocity/(m/s)

Ban

k an

gle/

( )

Case 1

Case 2

Figure 6. Bank angle profile.

Figure 5-6 show the angle of attack and bank angle versus velocity, the change happened between the guidance trajectory and optimized trajectory is because there is an interaction between the angle of attack and bank angle, which affect the aerodynamic characteristics simultaneously.

Figure 7-9 show heating rate, aerodynamic load and curve dynamic pressure versus velocity. It is apparent the path constraints are well obeyed.

1000200030004000500060000

0.2

0.4

0.6

0.8

1

Velocity/(m/s)

Hea

ting

rate

/(M

W/m

2 )

Case 1

Case 2

Figure 7. Heating rate profile.

100020003000400050006000

0.5

1

1.5

Velocity/(m/s)

Aer

odyn

amic

load

Case 1

Case 2

Figure 8. Aerodynamic load profile.

A simulation between traditional predictor process and BPNN predictor process has also been carried out. Result reveals that traditional predictor process spends about 12~15 seconds every predictor cycle. As a matter of fact, vehicle

computer may spend more time because its performance is weaker than desktop computer. However, the BPNN predictor process spends about 0.0002 seconds every predictor cycle as it almost needs no computing capability.

100020003000400050006000

20

40

60

80

100

120

Velocity/(m/s)

Dyn

amic

pre

ssur

e/(k

Pa)

Case 1

Case 2

Figure 9. Dynamic pressure profile.

Hence, the simulations above indicate that the guidance method is good to meet the flight mission requirements.

B. Simulations of Dispersed Cases To further verify the robustness and reentry range

adaptability of the guidance method proposed in this paper, two additional reentry simulations are carried out as follow: (1) the first simulation shown in table II is under initial dispersed environment considering random deviation factors including velocity, altitude, heading angle and flight path angle deviation; (2) the second simulation shown in table III and Figure 10-12 is under varying reentry ranges.

TABLE II. . PERFORMANCE SUMMARY UNDER VARYING INITIAL REENTRY CONDITIONS

Dispersion factor Dispersion value Velocity Latitude Distance errorNominal / 1505 m/s 20.1 km 620 m Velocity +50m/s 1490 m/s 20.2 km 507 m Velocity -50m/s 1520 m/s 20.1 km 1452 m Latitude +2km 1504 m/s 20.1 km 415 m Latitude -2km 1514 m/s 20.1 km 1113 m Flight path angle +0.5 1479 m/s 20.3 km 504 m Flight path angle -0.5 1513 m/s 20.0 km 1124 m Heading angle +5 1505 m/s 20.1 km 243 m Heading angle -5 1514 m/s 20.3 km 1453 m

TABLE III. . PERFORMANCE SUMMARY UNDER VARYING REENTRY RANGES

Example Range Velocity Latitude Distance error Case 4 6500 km 1467 m/s 20.2 km 966 m Case 5 7000 km 1515 m/s 20.1 km 1087 m Case 6 7500 km 1547 m/s 19.9 km 1873 m

It can be seen in table II that the speed and altitude errors are small and in a limited range. Besides, position error is also within 2km. The terminal states show that the predictor-corrector guidance method has good robustness. Besides, as is shown in Figure 10-12, the flight trajectory curve is smooth, and the change of angle of attack and bank

527

angle performs slowly. The results in table III show that the guidance method also has good reentry range adaptability.

-20

0

20

020

4060

80

20

30

40

50

60

Longitude/()Latitude/()

Alti

tude

/(km

)

Case 4

Case 5

Case 6

Figure 10. Flight trajectory profile.

1000200030004000500060008

10

12

14

16

18

20

Velocity/(m/s)

Ang

le o

f atta

ck/()

Case 4

Case 5Case 6

Figure 11. Angle of attack profile.

100020003000400050006000-50

0

50

100

Velocity/(m/s)

Ban

k an

gle/

( )

Case 4

Case 5Case 6

Figure 12. Bank angle profile.

V. CONCLUSION

A predictor-corrector guidance based on BPNN for reentry glide process is presented in this paper. With the BPNN

predictor-corrector algorithm, the terminal errors can be obtained online quickly, and then the guidance trajectory can track the optimized trajectory by adjusting the control variables with high precision. The guidance method combines advantages of the standard trajectory guidance and predictor-corrector guidance, which has a strong engineering application. Simulation results in nominal case and dispersed cases show that this method has good robustness and reentry range adaptability.

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[8] F. Fahroo, D.B. Doman, A. D. Ngo, Footprint generation for reusable launch vehicles using a direct pseudospectral method, in Proceedings of American Control Conference, 2003.

[9] Divya Garg, Advances in Global Pseudospectral Methods for Optimal Control, Ph.D. dissertation, University of Florida, 2011.

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