Available online at www.sciencedirect.com
ScienceDirect
Procedia Engineering 00 (2014) 000–000
www.elsevier.com/locate/procedia
“APISAT2014”, 2014 Asia-Pacific International Symposium on Aerospace Technology,
APISAT2014
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/SIMULINK, 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.
© 2014 The Authors. Published by Elsevier Ltd.
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
CL
CC
Cmx
total drag coefficient, non-dimensional (n. d.)
total lift coefficient for basic vehicle, n. d.
total side coefficient for basic vehicle, n. d.
total roll moment coefficient, n. d.
* Corresponding author. Tel.: +86-10-68742511;
E-mail address: caaawangchao@163.com
1877-7058 © 2014 The Authors. Published by Elsevier Ltd.
Peer-review under responsibility of Chinese Society of Aeronautics and Astronautics (CSAA).
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Wang Chao / Procedia Engineering 00 (2014) 000–000
Cmy
Cmz
r
a
e

total yaw moment coefficient, n. d.
total pitch moment coefficient, n. d.
rudder angle, degrees
aileron angle, degrees
elevon angle, degrees
angle of attack, degrees


angle of sideslip, degrees
q
S
Lr
DOF
Ix,Iy,Iz
Ixy
X,Y,Z
trajectory angle, degrees
dynamic pressure
reference area, m2
reference length, m
degrees of freedom
roll, yaw, and pitch moments of inertia respectively, kg-m2
products of inertia, kg-m2
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
g  g0
Re2
 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.
(1)
Wang Chao / Procedia Engineering 00 (2014) 000–000
3
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:
CL  CL 0  CL   CL   CLe  e






CD  CD 0  CD  CD   CDe  e  CDa  a  CDr  r

a
r

CC  CC 0  CC   CC  r  CC  a
(1)
The coefficient of the model can be seen in reference [1].

e
a
x


Cmx  Cmx 0  Cmx
  Cmx
  Cmx
 e  Cmx
 a  Cmx
 x  Cmxy  y

y







Cmy  Cmy   Cmy  Cmyr  r  Cmya  a  Cmy   Cmyr  r  Cmy  y  Cmyx  x

e
e
z



Cmz  Cmz 0  Cmz  Cmz   Cmz e  Cmz  Cmz e  Cmz  z
(2)
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:
m
dV
 V

 m
   V    Fi
dt
 t

(3)
Which m is the instantaneous mass, V is the centroid velocity vector under inertial coordinate frame, dv/dt and
δv/δv are time derivatives of centroid velocity vector under inertial coordinate frame and velocity coordinate frame,
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Wang Chao / Procedia Engineering 00 (2014) 000–000
ω 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
dV


m




cos  p    cos 
dt
D
   mg sin  



 
d

 mV
  P cos  sin      sin  cos     sin     L cos   C sin     mg cos  
V
p
V
p
V
V 



dt

 
 L sin  V  C cos  V  

0




sin

si
n




cos

cos



sin

p  V p  
V

 mV cos  d V 
dt 

(4)


0
0


sin  cos   sin  cos  sin V 
mV 2 


  m 2 r cos  cos  cos   sin  sin  sin 


c
o
s


2
mV

cos

co
s

D
V
D
V



r 
  cos 2  cos tan  
sin  cos   cos  sin  sin V 


sin  cos V
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:
 d
1
 x 
I y M x  I xy M y   I x  I y  I z  xz   I xy2  I y2  I y I z   y z
2
dt
I
I

x y  I xy

1
 d y

I x M y  I xy M x   I x  I y  I z   y z   I xy2  I x2  I x I z  xz

2
dt
I
I
x y  I xy


 d z  M z   I x  I y     I xy  2   2
 x y
x y
 dt
Iz
Iz
Iz








(5)
Where  x ,  y ,  z 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
 dt  V sin 

 d V cos  sin v


r
 dt
 d  V cos  cos v
 dt 
r cos 

(6)
 d
 dt   yb sin   zb cos 

 d
 xb  (b d  yb cos   zb sin  ) tan 

 dt
1
 d
 dt  cos   yb cos    zb sin 



(7)
Where r ,  ,  are the distance of the radius to the center of the earth, latitude and longitude separately. The
,  , are the pitch, roll and yaw angle separately.  V is flight –path azimuth angle.
Wang Chao / Procedia Engineering 00 (2014) 000–000
5
3.4. Geometric relationship equation
sin   cos  cos  sin   V   sin  sin  cos   V    sin  cos  sin 

sin   cos  sin  cos  cos   V   sin  sin   V    sin  cos  cos  / cos 

sin  V   cos  sin  sin   sin  sin  cos  cos   cos  sin  cos   / cos 

(8)

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, Ma,, ,,
,
Forces and Moments
Newton’s Equation
qSbCmx 
qSCL 


F  T VB  qSCD  , Mi  qScCmz 
qsbCmy 
qSCC 


P  f (Ma)
B
B
 dV E 
B
m  B   m  E   V E   P B  FaB  m  g 
 dt 
 dH BE 
B
E B
E B

      H    M i 
 dt 
Kinematics Equation
r  v,[ ,  , ]  f x ,  y , z 
Results
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.
120
1.2
1
ac / g (%)
are / g (%)
80
60
40
3.55
H=30km
H=40km
H=50km
H=60km
x 10
3.54
3.53
awe / g (%)
100
1.4
H=30km
H=40km
H=50km
H=60km
0.8
0.6
-3
H=30km
H=40km
H=50km
H=60km
3.52
3.51
3.5
0.4
3.49
20
0
0
0.2
5
10
15
Ma
20
25
0
0
3.48
5
10
15
Ma
20
25
3.47
0
5
10
15
20
25
Ma
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.
6
Wang Chao / Procedia Engineering 00 (2014) 000–000
Fig. 3. Dynamics of uncontrolled open cycle response of GHV
4.2. Lifting body hypersonic vehicle
The simulation status is altitude 50km, Ma=15. The simulation assess the dynamic characteristics of new lifting
hypersonic concept, which consists open cycle natural response and closed cycle controlled response.
Fig. 4. (a) Uncontrolled 偏航角速度
trajectory
; (b) PID control glider trajectory
;
 随时间变化的历程
滚转角速度 随时间变化的历程
俯仰角速度z 随时间变化的历程
俯仰角速度z 随时间变化的历程
y
y(°/s)
y(°/s)
0.1
0.05
0.05
0
0
-0.05
-0.05
0
10.5
10
20
30
40
1
0.5
0.4
0.2
0.5
0
0
-0.5
Times,s30
40
Times,s
俯仰姿态角 随时间变化的历程
10
20
-1
-1.5
50
-1.5
50
10.4
10.3
0
10
20
30
40
Times,s30
40
Times,s
偏航姿态角 随时间变化的历程
10
20
 (°) (°)
10.2
10.1
10
-0.8
50
1
0.8
0.5
0.8
0.6
0
0
-0.5
0
10
20
0
10
20
30
40
50
Times,s30
40
50
Times,s
-1
-1
0
0
1
偏航姿态角 随时间变化的历程
1
10
20
30
40
Times,s30
40
Times,s
滚转姿态角 随时间变化的历程
10
20
50
50
滚转姿态角 随时间变化的历程
0.6
0.4
0.4
0.2
0.2
0
0
-0.2
-0.5
10.1
10
-0.2
-0.4
-0.6
-0.8
50
0.5
10.3
10.2
0.2
0
0
-0.2
-0.4
-0.6
0
1
俯仰姿态角 随时间变化的历程
10.5
10.4
(°)(°)
0.6
0.4
-0.5
-1
0
滚转角速度x 随时间变化的历程
0.6
1.5
1
(°) (°)
z(°/s)
z(°/s)
0.15
0.1
x
偏航角速度y随时间变化的历程
1.5
0.2
0.15
x(°/s)
x(°/s)
0.2
0
10
20
0
10
20
30
40
50
Times,s30
40
50
-0.2
-0.4
-0.4
0
10
20
0
10
20
Times,s
Fig. 5. Dynamics of the attitude parameter with PID controller
30
40
50
Times,s30
40
50
Times,s
Wang Chao / Procedia Engineering 00 (2014) 000–000
7
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 Lifting�body Hypersonic Vehicles. Journao of Astronautics,
2009. 30(1): p. 128-133