trajectory profile of a three stage solid propellant small launch vehicle ... small satellite of 80 kg to a low earth orbit (LEO) of 660 km altitude. This hybrid ...
Small Launch Vehicle Trajectory Profile Optimization Using Hybrid Algorithm Fredy M. Villanueva*, He Linshu*, Amer Farhan Rafique**, and Tawfiqur Rahman*
School of Astronautics, Beihang University, Beijing 100191, China, Mechanical Engineering Department, Mohammad Ali Jinnah University, Islamabad, Pakistan *
**
Abstract- A hybrid optimization approach combining a genetic algorithm (GA) with sequential quadratic
programming (SQP) has been used for optimization of the trajectory profile of a three stage solid propellant small launch vehicle configured from existing solid rocket motors. The selected launch vehicle (LV) is capable of delivering a small satellite of 80 kg to a low earth orbit (LEO) of 660 km altitude. This hybrid optimization approach combines the advantage of GA as a global optimizer and complemented with SQP to find the local optimum. The vertical flight time, launch maneuver variable, maximum angle of attack, coasting time between the first and second stage and the second coasting time between the second and third stage were optimized. It is shown that the proposed hybrid optimization approach was able to find the convergence of the optimal solution with very acceptable values. 1.
INTRODUCTION
The trajectory profile optimization of a LV is one of the key cost-effective ways to put a payload into its prescribed orbit. A well-defined trajectory allows reduction of mass of auxiliary equipment, as well as lessens the fuel requirement for the attitude control system, and thus increases payload capacity. In recent years, hybrid optimization strategies had been widely accepted to solve complex aerospace problems [I], [2], its advantages resides in finding the optimum by using the global and local optimizations sequentially. The hybridization of heuristic search algorithm that is considered as non derivative with a gradient search algorithm gives several advantages resulting in a better convergence of the objective function and reduced processing time. Our research effort has been oriented to apply a hybrid genetic algorithm (GA) sequential quadratic programming (SQP) approach to optimize the trajectory profile of a small launch vehicle (LV) composed from existing solid rocket motors (SRM). Reference [3] have shown the effectiveness of GA in finding the global optimum in a ramjet application, references [4], [5] used GA for LV multidisciplinary design optimization. Heuristic optimization methods, are considered as the preferred choice in finding the global optimum, they search through all the design space selecting random values of design variables, and do not require an initial value to perform the optimization [6], [7]. Their independence of gradients gives an additional advantage avoiding converging in a local optimum, but requires more calculation as the algorithm searches at every point
inside the search space in order to guarantee convergence. Reference [8] proposed a hybridization of GA with SQP to solve a reusable reentry vehicle trajectory optimization. Additionally, [9], [10], and [II] provides a comprehensive implementation of GA in LV multidisciplinary design and optimization. References [12] and [13] have applied hybrid methods based on GA and gradient-based methods in LV design optimization. 11.
HYBRID OPTIMIZATION STRATEGY
A. Optimization Approach
There are several methods to solve the trajectory optimization problem, the most used is the collocation method, however for the present research effort a hybrid approach is considered. This hybridization approach intended to use the advantages of a global optimizer as well as the advantages of a local optimizer in a sequentially stacked optimization problem. The GA optimization method is one of the most used global optimizer in the aerospace industry; its main advantage is that there is not necessary an initial point to perform the optimization, additionally it operates in the whole search space fmding the best solution. The optimization process considers several operations: population initialization, selection, crossover, mutation, insertion, working in a routine until the optimum solution was reached. The drawback of this method could be the lack of accuracy in finding the solution. The SQP optimization method is a gradient based
I
I
Variables (X)
I
+ GENETIC ALGORITHM
Find Salis!)
.,
Optimul1 Variables Constraint
GA Optimal XC;A
I
:Minimizc �X)
Tr�iectory Analysis
f--+ �
SEQUENTIAL QUADRATIC PROGRAMMING
Find Satis!}'
I+-lhlitial SolutionXaAI
Optimun Variables Constraint
Minimi7.t� t(X)
I
"" Hybrid Optimal Xc;;\. SQP
I
Fig 1. Hybrid Optimization approach
optimization method, its advantages are its numerical
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perfonnance and accuracy, and represent one of the best nonlinear programming method. The Hybrid Optimization Approach (HOA) presented herein in Fig. 1, considers a stacked optimization procedure with GA as a global optimizer and SQP as local optimizer. This approach takes the best solution from GA and passed on as initial variables to SQP, resulting in the best solution. Ill.
L
LAUNCH VEHICLE MODEL
= CLqSrel
(2)
D = CDqSrel
A. Launch Vehicle Definition
A three stage solid propellant LV configured from existing solid rocket motors (SRM) is considered for this research. Each SRM has a predefined specific characteristics and constraints, and are shown in table 1. Tt is important to mention that because of using existing SRM the design of the selected LV is not optimized, and several parameters could not be inside the required margin compared to a newly LV design, as a consequence, it can be used only for specific payload that meet the LV performance limitations. The mission is to deliver a small payload of 80 kg. into a low earth orbit (LEO) altitude of 660 km. B.
a high fidelity aerodynamic calculation. Tn that way, the Missile DATCOM 1997 provides a easier, economical, practical and quickly result [16]. DATCOM had been widely adopted in conceptual and preliminary design phases. References [12] and [17] applied DATCOM in LV aerodynamic calculations. The aerodynamic forces are calculated as:
Where, L is the lift, D drag, q dynamic pressure,
Srel vehicle reference area, CL lift coefficient, and CD
the drag coefficient. The aerodynamic coefficients were calculated for Mach numbers ranged from 0.1 to 8 and angles of attack from -6 to +1 degrees. The density and gravity variations with altitude can be represented as: P
(-hl fJ)
= Po e
(3) (4)
Propulsion Analysis
The propulsion analysis considers the calculation of thrust with the variation of altitude for each SRM. Sutton and Biblarz [14] and He LinShu [15] provided a comprehensive method for calculation of thrust T from a given SRM parameters. (1) Where, I,p is the specific impulse,
mgn
mass flow,
Pa atmospheric pressure, and Ae nozzle exit area.
Where, Po is the sea level density, fJ density scale height, and f..l the earth gravitational parameter. D. Trajectory Analysis
For trajectory analysis, a MatIab STMULINK was used. A three degree of freedom (3DOF) model has been implemented [18], [19]. The input data for trajectory were the aerodynamics coefficients, mass and thrust of each SRM. The LV flies in a 2D coordinate system as a point mass in a spherical non-rotating earth model. Fig 2 illustrates the forces acting on a LV and below the governing equations of motion [15].
C. Aerodynamic Analysis
For aerodynamic analysis, an analytical software based calculation was selected; our intention was not to conduct
y
TABLE!. LAUNCH VEHICLE CHARACTERISTICS
Launch vehicle Gross launch mass Payload Deployment module Stage III Gross mass Propellant mass Thrust Burn time Specific impulse'" Stage II Gross mass Propellant mass Thrust Burn time Specific impulse'" Stage I Gross mass Propellant mass Thrust Burn time Specific impulse
kg kg kg
35520 80 50
kg kg kN s N.s/kg
3650 3300 169. 4 55 2824
kg kg kN s N.s/kg
10950 9800 422. 9 65 2805
kg kg kN s N.s/kg
20790 18400 669. 2 65 2364
x
Fig 2. Forces actin g on a launch vehicle.
Proceedings of 2013 10th International Bhurban Conference on Applied Sciences & Technology (IBCAST) Islamabad, Pakistan, 15th - 19th January, 2013
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dV Tcosa-D - g SlllO' m dt d.9 Tsina+L g cos.9 V cosO' = + dt mV V Re +h dh -=VSlll O' dt dl � = Vcos.9 dt Re +h a=1]+rp-.9 -----
.
phase (third stage SRM burning time) are the required circular orbital velocity, and the flight path angle equal zero degrees.
n
F.
n
.
n
(5)
Flight Profile Formulation
The flight profile dictates the performance characteristics of the LV, and has substantial influence on the mission accomplishment (orbit injection), Fig. 3 explains the variation of the angle of attack with time during the pitch over phase of the LV. [15], [20]: aprox (t)
=- 4amaxz (1- z) z = e -am
I
(6)
(1-11)
(7)
Where, amax is the maximum angle of attack,
1]=Re
a= aprox(t) Where, V is the velocity, m vehicle mass, !) flight path angle, a angle of attack, rp pitch angle, B trajectory angle, 17 range angle, Re radius of earth,
am
launch maneuver variable, t time of flight, and tJ the time of start of pitch over phase, coincident in value with time tv'
h
O��---r---�-----�
height above the ground, I range, and aprog (t) the programmed angle of attack. E.
alll l'lxl-__----"'-'--' -"
Mission Profile
The trajectory profile was optimized considering the particularity of the LV configuration, and was oriented to ensure the integrity of the LV. The trajectory is composed of seven flight phases, and are described next [15], [17]: Vertical launch phase: This phase starts from the time of ignition of the first stage SRM, until the end of vertical flight time tv' the condition for this phase is that the flight
Fig 3. Pitch over ascent phase of the launch vehicle.
The body axial and normal overloads mathematically expressed as:
n, =
T + L sina - Dcosa
mg
path angle should be maintained in 90 degrees. Pitch over phase: This phase starts from time
ny =
tv' until
the start of transonic phase (Mach 0.8), during this phase, the angle of attack is detennined by the flight program and constrained to approach zero at the end of phase. Powered first stage phase: Starts from the end of pitch over phase (Mach 0.8), and continue until the shutdown of the first stage SRM. During transonic phase (0.8
of attack, am launch maneuver variable, and are listed in Table 2.
2
TABLE II. TRAJECTORY PROFILE VARJABLES
X
Variables
0
Symbol
XI
Vertical flight time
tv
X2
Coasting time I (between I sl and 2nd stage) Coasting time 2 ( between 2nd and 3'd stage)
tel
X3 X4
Maximum angle of attack (absolute)
X5
Launch maneuver variable
Units
a
100
200
400
500
600
300 Time (s)
400
500
600
300 (s)
400
500
600
300 Time
(s)
700 600
te2 amax
E deg
500
� 400 Q) "0
.= 300
""
am
«
200 100
C.
Trajectory Constraints
0
For the present analysis, several constraints had been considered, and are listed in Table 3. TABLE III. Constraints
Cl
Orbit insertion velocity
C2
Axial overload (l sl and 2nd stages)
C3
Normal overload
C4
Maximum dynamic pressure
C5
Maximum angle of attack (absolute)
C6
Angle of attack (0. 8 :s Mach :s 1.3)
C7
Orbit insertion angle
100
200
a
100
200
100
TRAJECTORY CONSTRAINTS
C
0
Valne
Vr?:7525mls nx�12 ny�2
Oi Q) � Q) c;, c .,
60
iii
40
L: Ql u::
20
.0
0
0
20
40
Time (s)
60
80
100
Fig 4. Trajectory profile of launch vehicle
Proceedings of 2013 10th International Bhurban Conference on Applied Sciences & Technology (IBCAST) Islamabad, Pakistan, 15th - 19th January, 2013
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CONCLUSION
2
1.5 " '" � .t:.
0.5
0
c;; Ql
10
0
20
40
30
Time(s)
a
E. -1 '" "
,l!l !
0 Ql C> c:
