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Procedía Engineering 99 (2015) 600 - 606

Procedía Engineering

www.elsevier.com/locate/procedia

"APISAT2014", 2014 Asia-Pacific International Symposium on Aerospace Technology,

AP1SAT2014

Six-DOF Modeling and Simulation for Generic Hypersonic Vehicle

in Reentry Phase

Wang Chao*, Liu Xinyu, Li Feng

China Academy of Aerospace Aerodynamics(CAAA),Beijing 7201box 16box, BeiJing 100074, China

Abstract

The purpose of this research is to model a hypersonic vehicle in the pull-up phase and provide an insight into the inherent dynamics. Based on Newtonian mechanics, the theoretical flight dynamic model of the pull-up phase of reentry motion is established; Based on MATLAB/S1MUL1NK, simulation model for reentry hypersonic vehicle was modeled. Lifting body hypersonic vehicle aerodynamic data is obtained by high-accuracy CFD code which copyright by CAAA. We adopted feedback linearization control theory, pitching angle holding control laws in order to keep maximum lift-drag ratio glider, the six degree of freedom (DOF) simulation validate the control law. The simulation results show some hypersonic flight characteristics, including static unstable, earth radius and rotation effects and high-coupled characteristics. The simulation model and results provide a useful reference for a hypersonic vehicle's dynamics analysis, control law design and overall design. © 2015Published byElsevierLtd.Thisisanopen access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Peer-review under responsibility of Chinese Society of Aeronautics and Astronautics (CSAA)

Keywords: Six-DOF flight dynamic models; on-linear equations; Centrifugal force; feedback linearization simulation

Nomenclature

CD total drag coefficient, non-dimensional (n. d.) CL total lift coefficient for basic vehicle, n. d. CC total side coefficient for basic vehicle, n. d. Cmx total roll moment coefficient, n. d.

* Corresponding author. Tel.: +86-10-68742511; E-mail address: caaawangchao@163.com

1877-7058 © 2015 Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Peer-review under responsibility of Chinese Society of Aeronautics and Astronautics (CSAA) doi: 10.1016/j.proeng.2014.12.577

C total yaw moment coefficient, n. d.

C total pitch moment coefficient, n. d.

sr rudder angle, degrees

3 a aileron angle, degrees

elevon angle, degrees

a angle of attack, degrees

ß angle of sideslip, degrees

0 trajectory angle, degrees

q dynamic pressure

S reference area, m2

Lr reference length, m

DOF degrees of freedom

Ix,Iy,Iz roll, yaw, and pitch moments of inertia respectively, kg-m2

Ixy products of inertia, kg-m2

X,Y,Z total aerodynamic forces (in body coordinate x, y, and z)

1. Introduction

The purpose of this research is to model a hypersonic vehicle in the reentry phase. Due to variations of wide speed and altitude range, there are some unique features of hypersonic vehicles and complex flight dynamic characteristics, which are not existent with conventional airplanes, such as strong nonlinearity, strong coupling and large envelope. While flight dynamics modeling of hypersonic vehicles is of prime importance for the design and simulation of control, the wide-range maneuver, the complex aerodynamic characteristics and the mutual coupling of aerodynamics and motion of a hypersonic vehicle contribute greatly to the strong uncertainty of its aerodynamics, which is a great challenge to the dynamic analysis and control law design. So we need to establish a set of 6-DOF nonlinear dynamic model for the purpose of better reflecting the flight characteristics and dynamic quality, making a comprehensive of the over-all characteristics of hypersonic vehicles.

Many researchers study the hypersonic vehicle's dynamic model for stability analyses and control. Reference 1 and 3 developed a GHV dynamic model based on 'flat earth'. Reference 4 developed a longitudinal hypersonic dynamic model integrated with Centrifugal force.

In this paper we design a matlab simulation model for hypersonic vehicle reentry dynamic analysis. The sphere rotational earth approximation is used for this study . Based on Generic hypersonic vehicle model and an Lifting body hypersonic vehicle, we showed some results of the dynamic simulation.

2. The Propulsion, Gravity and Aerodynamic Forces & Moments

During the motion of the hypersonic vehicle, it will be acted by engine thrust, aerodynamic forces, earth gravity, solar energy, magnetic force, buoyancy etc. We only consider the engine thrust, aerodynamic forces and earth gravity to simplify the research.

2.1. Gravity

The gravity g model 2

g = g0

(Re + H )2

Which g0=9.8204m/s2 is the gravity on the earth surface, Re=637100m is Earth's even radius, H is flight altitude.

2.2. Aerodynamic Forces & Moments

There are two methods to obtain the aerodynamic coefficient, the first one is to set up the aerodynamic model, then identify the coefficient base on CFD database. We use this method on GHV example. The total lift, drag and side force coefficient is described by the expression:

The second method is directly interpolation. We use this method on lifting body hypersonic vehicle. The aerodynamic data is obtained by high-accuracy CFD code which copyright by CAAA, and the aerodynamic coefficient is modeled by 4D lookup table.

2.3. Propulsion

The propulsion is different from vehicles. The scramjet hypersonic vehicle can be written as a function of Mach number, reference [2] presented a thrust model for GHV. The hypersonic reentry glide vehicle has no propulsion.

3. Dynamic models

Based on Newtonian mechanics, the theoretical flight dynamic model of the reentry phase of reentry motion is established. Firstly the simplified assumptions are as follows:

(1) The hypersonic vehicle is a ideal rigid body, the wing^ body and tail are considered without elastic freedom.

(2) The moment of inertia of the control surface is 0.

(3) The earth coordination is inertial coordinate frame, and the aerodynamic and thrust forces along with gravitational acceleration g are under the approximations of a "sphere-Earth".

3.1. Translational dynamic equation

Usually the centroid dynamical equation of the flight vehicle can be written as:

CL = CL 0 + CaLa + C?0+ CsLeSe

CD = CD 0 + C"Da + C№ + Cs¿Se + Cs¿ Sa + Cs¿8,.

Cc = CC o + + C%8r + Csc-8a

The coefficient of the model can be seen in reference [1].

C = Cm o+O+Cmxt3+Ciôe+Ctsa+c^+C2ëy < Cmy = C^+C:va+Csn;ySr +Csn;Sa + CÍy¡5+C%8r (2)

= Co + c> + C'lfi + C%8, + C*má + ci¿, + C2ëz

Which m is the instantaneous mass, V is the centroid velocity vector under inertial coordinate frame, dv/dt and 5v/5v are time derivatives of centroid velocity vector under inertial coordinate frame and velocity coordinate frame,

ro is the angle velocity vector of velocity coordinate relative to inertial coordinate. Fi is the external force including gravity, aerodynamic forces, thrust etc.

The centroid dynamical equation be deduced under trajectory coordinate frame

cos [<pp + a)cos P

mV— =P cos/Vsin(ipp + a)+sin7Vcos(^, + aJsinP + dyV sin yV sin(+ a)- cos yV cos (<pp + a)sin P

-mV cosfr

-D -mgsin^

L cos yV-C sinyV + -mg cos0 L sin yV - C cos yV 0

— cos 0 - cos2 9cosi^V tanp

+ 2mV®r

cos ^ cos <J/V sin^cos^-cos^sin^sin^

+ mcoDr cos <p

sin Ocosy-sin pcos 0sin <J/V cos0cosp+sinpsin0sini^V sin^cos^V

Which P is the propulsion, L, D, C are the lift force ^ drag force and side force. 3.2. Rotational dynamic equation

The rotational dynamic equations are as follows:

dt V, - ¡1

dt ¡I

drnz - M +

Um, + i, K+(4 + - iz )-(/i + ¡i - I,A

' ,xy i 2 2 \

+ — \ax ~m, )

Where cox,coy,az is the angular velocity of roll, yaw, and pitch (in body coordinate x, y, and z). 3.3. Kinematical equation equation

The translational and rotational kinematical equation is

dr t/ a — = V sin 9 dt

dp V cos d sin dt r

dX V cosdcos^v dt r cos ^

— = rn siny + ® cosy dt

— = aTb - {wb_d_, cosY- a>zt sin y)tan &

dy _ 1 / dt cos 3 ^

rny cosy-rnz sin y

Where r, A, (p are the distance of the radius to the center of the earth, latitude and longitude separately. The 3, y, y/ are the pitch, roll and yaw angle separately. if/V is flight -path azimuth angle.

3.4. Geometric relationship equation

sin ß = cos#[ cos/sin ) + sin$sin/cos )]-sin#cos$sin/

sin a = {cos#[ sinocos/cos (y/-yv)- sin /sin (y/~WV)]_ sin 0 cos 3 cos /J / cos ß

sin yv = (cos öt sin ß sin 3 - sin a sin ß cos y cos 3 + cos ß sin y cos 3) / cos 6

3.5. Numerical method

A MATLAB program was written in order to prepare the initial data, and a Simulink model was developed for trajectory and attitude angular simulation. The simulation adopted fourth-order variable step-size Runge-Kutta numerical integration method. Figure. 1 shows the modelling and simulation flow and SIMULINK model. For unsteady longitude lifting body hypersonic vehicle, an enforced stability PID controller is designed for the flight simulation.

Initialization H Mt,o,ß,rAv

Forces and Moments

qSCL " ~ qSbCmx '

F = \T"¡ ] qSCD , Mt = qScCm

qSCC qsbCmy _

Newton's Equation

^-f 1+ m [»' J J = + V + m [* ]'

ÍH ]B 1=M}B

Kinematics Equation

f = v,[3,y,y,] = f{a>r,<».,<»=)

Fig. 1. (a) Modeling and Simulation Procedure; (b) Matlab Simulink Model ;

4. Results and discussion

Figure. 2 shows Centrifugal force by earth radius effects, Coriolis inertia force, and Centrifugal force by earth rotation effects. From the chart, we get the centrifugal force should not be ignored at hypersonic flight. While the Coriolis inertia force, and Centrifugal force by earth rotation effects can be ignored at preliminary analysis.

- —— H=30km H=40km -■-H=50km ■■■■H=60km i

/A ///

4 ✓V >yy

3.55 — 3.54 — 3.53.. % 3.52 — 5 3.51 — ä 3.5 — 3.49 — 3.48 —

-H=30km

H=40km --■-H=50km ■■■ H=60km T

Fig. 2. (a) Centrifugal force by earth radius to gravity; (b) Coriolis force to gravity ; (c) Centrifugal force by earth rotation to gravity;

4.1. GHV

The simulation status is altitude 60km, Ma=10. Figure. 2 shows the earth radius and rotation effects and hypersonic high-coupled characteristics.

Fig. 3. Dynamics of uncontrolled open cycle response of GHV

4.2. Lifting body hypersonic vehicle

The simulation status is altitude 50km, Ma=l5. The simulation assess the dynamic characteristics of new lifting

hypersonic concept, which consists open cycle natural response and closed cycle controlled response.

trajectory mj«t«y

Fig. 4. (a) Uncontrolled trajectory ; (b) PID control glider trajectory ;

-4+r ,W fw

I AAAW 05 V

1 № V v

0 10 20 JO 40 50

Times.s

0 10 20 JO 40 50

Times.s

0 10 20 30 40 50

Times.s

10 20 30 40

Times.s

10 20 J0 40

Times.s

0 10 20 J0 40 50

Times.s

Fig. 5. Dynamics of the attitude parameter with PID controller

Figure. 3 shows two trajectory, which consists a uncontrolled unsteady hypersonic trajectory and a PID enforced stability trajectory. Figure 4 shows the response of six attitude parameter dynamics.

5. Conclusion

Based on MATLAB/SIMULINK and the theoretical model, simulation model for reentry hypersonic vehicle was modeled. The earth radius and the angular rate of earth rotation effects to the hypersonic dynamic model was analysed, the earth radius effects should not ignored at hypersonic flight and the earth rotation can ignore at some preliminary design. The simulation results shows that the six DOF flight dynamic model can reflect the nonlinear and strong coupling effects, and test the control law in reentry phase.

References

[1]Keshmiri, S., R. Colgren et al. Six DoF Nonlinear Equations of Motion for a Generic Hypersonic Vehicle. 2007, AIAA 2007-6626.

[2]Mirmirani, M., et al., Ramjet and Scramjet Engine Cycle Analysis for a Generic Hypersonic Vehicle. 2006, AIAA 2006-8158.

[3]Keshmiri, S., R. Colgren and M. Mirmirani, Six-DOF Modeling and Simulation of a Generic Hypersonic Vehicle for Control and Navigation Purposes. 2006, AIAA 2006-6694.

[4]Wen, B., Study on Longitudinal Modeling for Integrated Centrifugal/ AeroForce LiftingDbody Hypersonic Vehicles. Journao of Astronautics, 2009. 30(1): p. 128-133