NSC 2005 Conference Paper No.39

1

A Generic Reentry Trajectory Planning Algorithm for Reusable

Launch Vehicle Missions

Ashok Joshi *, K. Sivan

@

Indian Institute of Technology Bombay, Mumbai – 400 076, India

and

B. N. Suresh$, S. Savithri Amma

#

Vikram Sarabhai Space Centre, Thiruvananthapuram-695 022, India

ABSTRACT

A generic reentry guidance trajectory planning algorithm of a Reusable Launch Vehicle is

presented in this paper. The study is aimed at the development of a trajectory planning

algorithm, which is accurate, efficient, robust and reconfigurable. In order to achieve the

above objectives, the proposed reentry trajectory planning algorithm generates the optimum

reference trajectory from any reentry interface to a specific target location meeting all the

path constraints real time. The applicable optimal control problem of reentry guidance is

converted into an equivalent targeting problem in Nonlinear Programming and a simplified

solution methodology is devised to solve the three dimensional trajectory planning without

using bank reversals. The necessary mathematical models are represented in polar coordinate

system and a judicious selection of control parameters, coupled with an efficient method of

computation of sensitivity matrix elements, ensures better mission planning and faster

convergence. The performance results establish the mission flexibility of steering the vehicle

from widely separated reentry locations, while meeting the path constraints. The guidance

law and the models are general and adaptable to any RLV mission and have the potential for

onboard implementation.

Reentry, Guidance, Trajectory Planning, Heat Rate, Dynamic Pressure, Latitude,

Longitude, Target

NOMENCLATURE

DC , LC Drag and lift coefficients respectively

g Acceleration due to gravity

2J Second gravitational harmonics

LQR Linear Quadratic Regulator

m Vehicle mass

maxn Maximum allowable load factor

NLP Nonlinear Programming

maxq Maximum allowable dynamic pressure

___________________________________________________________________________

* Professor, Department of Aerospace Engineering @ Research Student, Department of Aerospace Engineering, Group Director, Mission

Synthesis & Simulation Group, Vikram Sarabhai Space Centre, Thiruvananthapuram $

Director # Engineer, Mission Synthesis & Simulation Group

NSC 2005 Conference Paper No.39

2

maxQ,Q && Heat rate and its maximum allowable value

r Radial distance

NR Nose radius

RLV Reusable Launch Vehicle

TAEM Terminal Area Energy Management

V Relative velocity

cirV Circular velocity

α Angle of attack

β Scale height

γ Flight path angle

λ Vehicle longitude

µ Gravitational constant,

ρ Atmospheric density

0ρ Sea level density

σ Bank angle

φ Vehicle latitude

ψ Relative yaw angle of the vehicle

eΩ Rotational velocity of Earth

1. INTRODUCTION

It has been established in literature that the atmospheric reentry of a Reusable Launch

Vehicle (RLV) is the most critical part of the overall return mission and therefore, reentry

guidance algorithm is an important component of the overall strategy for steering the vehicle

safely through the dispersed environment, while dissipating the large amount of energy and

meeting the mission requirements. Further, low cost operation of future RLVs and crew/cargo

return vehicles is strongly dependent on the availability of a re-configurable and robust

reentry guidance scheme. Reentry guidance strategy consists of two components: (1)

generation of a feasible reference trajectory to reach the specified target, satisfying all the

path constraints, (2) generation of steering commands, angle of attack and bank angle to track

the reference trajectory. The flight proven reentry guidance algorithms to date have been

implemented by storing onboard, pre-computed reference trajectory profile1,2

. This reference

trajectory profile is modified onboard, using sensor data, and is tracked using a control

system to meet the desired range requirements. In view of the above, the guidance algorithm

is required to be computationally efficient and the algorithm used for space shuttle missions

is considered to be not suitable for the new generation RLVs. This is mainly because of wide

dynamic range of operation and a need to have minimal aerodynamic characterization at

ground for such RLVs. Hence, recent studies in this context have focused on the

development of efficient and robust guidance algorithms.

These algorithms mainly address improvements such as nonlinear control laws to track the

reference trajectory profile, advanced linear control laws with least sensitivity to reference

trajectory, optimum method of updating reference profile onboard to increase reentry

capability and generation of reference trajectory onboard to add mission flexibility. In ref .3

and ref.4, feed back linearization and in ref.5 and ref.6, analytical parameter optimization

procedures are used to generate a reference drag acceleration profile. Another approach treats

the nonlinear trajectory tracking as a regulator problem about the reference trajectory7,8

,

NSC 2005 Conference Paper No.39

3

which requires no online integrations or iterations and needs only very few off-line parameter

adjustments. At the core of this method is a new closed form approximate receding-horizon

control law that guarantees closed loop stability when designed properly. In three

dimensional predictive reentry guidance approach9, the space shuttle’s two-dimensional

reentry trajectory planning method is extended to three-dimensions to achieve the desired

down range and cross range through drag and lateral acceleration profiles and a nonlinear

tracking control law10

.

In a predictor-corrector based reentry guidance11

, a six element state vector is propagated to

the target through the integration of equations of motion, whereas numerical optimization

methods are used to nullify the altitude, heading angle and range errors with use of bank

angle, angle of attack and time of roll reversal as control variables. In the “free-form”

guidance with flexibility in trajectory guidance law12

, simple real time integration with an

iterative predictor-corrector technique is used to generate reference trajectory, which reduces

the computational requirements and the problem of convergence. Kistler K-1 orbital vehicle13

uses an efficient predictor-corrector method to compute the bank angle and the start time of a

single bank angle reversal required to null the predicted target position miss. A simple and

efficient profile tracking reentry guidance algorithm is presented in ref.14, which is based on

optimal Linear Quadratic Regulator (LQR) theory. Another automated reentry guidance

algorithm15

uses the predictor-corrector method to generate the reference trajectory with

heating constraint and uses LQR technique for the trajectory control law.

It is to be mentioned here that the reentry guidance algorithms based on numerical predictor-

corrector methods yield reduced maneuvering with good targeting accuracy and robustness to

dispersions. Even though these algorithms have built-in mission flexibility and are adaptable

to any flight conditions without any pre-computations, full potential of these schemes

remains untapped due to computational complexity and risk of no convergence inherent in all

such numerical procedures. In our earlier study16

, a real time planning reentry guidance

algorithm based on predictor-corrector method was presented to achieve mission flexibility,

faster solution, improved accuracy and robustness. In the above study, bank angle and angle

of attack to meet the down range and cross range requirements at target are computed at

every guidance cycle during reentry by solving numerical predictor-corrector algorithm using

Newton-Raphson method. The Jacobian required for the control update is generated by

numerical finite differencing method. The path constraints are implemented at the guidance

output level, the dispersion due to this will be taken care by guidance computations in the

next cycle. In the present study, an alternate guidance algorithm is developed to generate the

reference trajectory onboard. This paper attempts to evolve a generic reentry guidance

planning algorithm which generates a feasible reference trajectory onboard before initiation

of the actual reentry for proving solution that best combines the solution methodologies and

onboard computing capabilities. Here also, numerical predictor-corrector method is used; but

the reference trajectory is achieved by solving guidance problem through a targeting problem

in Nonlinear Programming (NLP) Approach.

This paper gives the reentry guidance problem definition along with the vehicle constraints,

proposed generic reentry trajectory planning guidance and applications of the generic

trajectory planning algorithm for a typical RLV mission and performance results of the

proposed algorithm.

NSC 2005 Conference Paper No.39

4

2. REENTRY GUIDANCE PROBLEM

2.1 Reentry dynamics

In the present study, a strategy for trajectory planning is presented that aims to reduce

computational load on the onboard computers. In view of the enhanced accuracy

requirements for future reentry missions, a generalized model for the three degrees of

freedom dynamics about an oblate rotating planet is required. The system dynamic equations

are expressed in polar coordinate system with respect to an oblate rotating planet, which also

provide ease of computation of sensitivity coefficients, as described below.

ψγγ φ coscosgsingm

DV R −+

−=&

)cossinsincoscos(cos2 φγφψγφ +−Ω+ er (1)

]coscossincos[1

cos1 2

γψγγσγ φr

Vgg

Vm

L

VR +++

=&

)coscossincos(sincos

cossin22

φγφψγφ

φψ +Ω

+Ω+V

r ee (2)

φψγψγ

σγ

ψ φ tansincos)sin(cos

1sin

cos

1

r

Vg

Vm

L

V++

=&

φψφγ

φψγφ sinsincoscos

)coscostan(sin22

V

r ee

Ω+−Ω+ (3)

γsinVr =& ; r

coscosV ψγφ =& ;

φ

ψγλ

cosr

sincosV=& (4)

−−−= )1sin3()

r

R(J

2

31

rg 22e

22R φµ

(5)

φφµ

φ cossin3

2

22

=

r

RJ

rg e

(6)

)r,M,(SCV)r(m2

1L L

2 αρ= ; )r,M,(SCV)r(m2

1D D

2 αρ= (7)

2.2 Vehicle and control constraints

The objective of any reentry guidance system is to steer the vehicle from reentry interface to

a specified Terminal Area Energy Management (TAEM) interface, within the prescribed

error bands and with adequate robustness to dispersed flight environment. The dispersions of

significance include are: (1) vehicle characteristics, (2) environmental characteristics, (3)

initial reentry state vector and (4) propagation error in navigation state. In order to achieve

the mission objectives under such conditions, the vehicle needs to fly within the reentry

corridor defined by the thermal, structural and hinge moment constraints along with the

vehicle maneuvering capability constraints. While, the maneuverability constraint appears in

terms of the requirement to fly in equilibrium glide condition, the thermal constraint is

generally expressed in terms of heat rate at stagnation point. Further, the structural limit is

NSC 2005 Conference Paper No.39

5

defined in terms of structural load factor, while dynamic pressure limit arises from actuator

capabilities. These path constraints are as given below.

)cos(r

Vg)cos(

m

L 2

γσ

−≤ ; max

2/12 ])/(1[ nDLm

Dnz ≤+= (8)

max

15.3

cir

2/1

0N

QV

V

R

11030Q && ≤

=

ρ

ρ ; max

2 qV)2/1(q ≤= ρ (9)

2.3 Reentry Guidance Problem Definition

In its most general form the guidance problem can be states as: Given initial state 0X , the

guidance objective is to estimate )t(α and )t(σ histories without violating the path constraints

given by Eq. 8 and 9 so that at final time ft , the terminal state vector fX satisfies the TAEM

interface conditions. The terminal target conditions are mainly to achieve TAEM interface

location with specified energy level and velocity heading. By propagating the trajectory up to

TAEM energy level, fE , the target conditions are given by,

0)E( df =−φφ ; 0)E( df =− λλ (10)

0=− df )E( ψψ ; 0)E( df =− γγ (11)

where, dφ , dλ are desired latitude and longitude of TAEM interface location and dψ , dγ

are the desired azimuth and flight path angle at TAEM interface.

3. GENERIC REENTRY GUIDANCE THEORY

3.1 Trajectory planning guidance law

The target error vector is defined in terms of the difference in required and achieved states as,

df XXY −= (12)

The path constraint error can be defined as an integral of square of the amount by which the

constraint is violated. Therefore, for the ith constraint, the constraint error is defined as

∫=

f

i

t

t

ic dtcC

0

2∆δ ; where,

>−=

=otherwise0

0CCCifCC

liiici ,

∆∆∆ (13)

It is to be noted that if the ith constraint is satisfied then 0

ic=δ . Similarly, the constraint

violation can be computed for all the path constraints. The constraint error vector C is defined

with components of ic

C∆ . Therefore, in general, the reentry guidance problem can be

restated as an optimization problem where the following performance index is to be

minimized.

∫+=

ft

t

ccT

ffT dt))t(),t(())t(),t(()()(J

0

uXZCuXCXHYXY (14)

NSC 2005 Conference Paper No.39

6

subject to the differential constraints given by Eq. 1 to Eq. 4 with the prescribed initial

conditions t0 and X0, where Tc ttt )()()(u σα= is the control vector history. Assuming the

target error as,

)()())t((e ffT

ct XYHXYu = (15)

and the constraint error as

∫=

ft

t

ccT

cc dt))t(),t(())t(),t(())t((e

0

uXZCuXCu (16)

the performance index can be rewritten as

))(u())(u( teteJ ccct += (17)

This optimal control problem can now be converted into a NLP problem by parameterizing

the control time history )(u tc , as given below.

1kkk1k

kk1kkc ttt

tt

ttuuutu +

+

+ <<−

−−+= ,

)(

)()()( (18)

where ku are the control parameters. If udenotes the vector of control parameters, then the

performance index and the corresponding dynamics are written as,

)u()u()u( ct eeJ += ; )u),(X(fX t=& (19)

It is to be noted that, the guidance solution is achieved when 0J =)u( . Since the terms in Eq.

19 are quadratic, this is possible only when

00et .)u( = ; 00ec .)u( = (20)

which states that the specified target conditions are achieved without violating the path

constraints. This problem can be viewed as targeting problem in NLP for computing the

control vector ∗= uu which ensures that error vector 0u

uue =

=

)(e

)(e)(

*c

*t* , subject to the

system dynamics, )u),(X(fX t=& and the initial conditions 0t and 0X . The solution

methodology developed for this problem is given below.

3.1.1 Solution Methodology

The solution is started by the a priori knowledge of the reference control vector, 0u and the

error vector for this reference control vector, )u(e 0 is computed. Next, the increment on the

control vector, u∆ is found so that the control vector, uuu* ∆+= 0 ensures )u(e

* = 0.

The gradient vectors of the target and constraint errors are defined as,

NSC 2005 Conference Paper No.39

7

0

tk

t

2

t

1

t0

Te

u

e

u

e

u

e

u

)u(

∂

∂

∂

∂

∂

∂= K∇∇∇∇ ;

0u

)u(

∂

∂

∂

∂

∂

∂=

k

c

2

c

1

c0

Te

u

e

u

e

u

ec

K∇∇∇∇ (21)

and the solution to the above problem is obtained through a simple NLP algorithm, in which

the optimum incremental control vector u∆ is obtained iteratively by finding optimum search

direction and optimum step length along the search direction, in order to drive the error

vector to zero. If the control vector size be k (k > 2) and the variation of target error and

constraint error about the reference control vector 0u is taken to be linear, then the unique

optimum search direction u∆ is the one, which satisfies the following linearized error

equation 000 =+ )u(eu)u(S ∆ and minimizes the length of u∆ . The solution to the above

equation defines a linear manifold )u( 0C (in this case, a straight line) as given in Fig-1.

The

The desired minimum norm correction, u∆ is then the vector of minimum length from 0u to

the linear manifold )u( 0C . Analytically, this is given as

)u(e)]u(S)u(S[)u(Su 01

0T

00T −−=∆ (22)

If the errors are actually linear, then the optimum minimum norm correction computed in the

Eq. (22) ensures zero errors. For the realistic case, an optimum step length along the

minimum-norm direction is required to minimize the error. This is computed by one-

dimensional minimization method. The study indicates that quadratic interpolation method is

sufficient to find the optimum step length. The function to be minimized along the search

direction, u∆ is the sum of the squares of the errors namely,

200J )uu(e)( ∆δδ += ; with 2

00 0J )u(e)( = ;2

00 1J )uu(e)( ∆+= (23)

Differentiating via chain rule yields

u)u()u(e)( ∆00T

0 S20J =′ ; (24)

A quadratic polynomial is fitted for the function )(J0 δ with the values of )0(J0 , )1(J0 and

)0(J'

0 . The quadratic polynomial along with the optimum step length and the optimum

control vector is given below:

Fig-1 Minimum-Norm Correction Direction

NSC 2005 Conference Paper No.39

8

22100 aaa)(J δδδ ++= ;

2

1*

a2

a−=δ ; uuu

** ∆δ+= 0 (25)

The optimum solution is achieved through iteration with updated value of *uu = till the error

vector is within the allowable tolerances.

3.2 Generic planning algorithm

The trajectory planning algorithm is a numerical iterative predictor-corrector method.

Assuming nominal vehicle data and environmental parameters and with initial guess for the

control vector, the predictor numerically propagates the trajectory from reentry interface to

the terminal energy. The error and gradient vectors and sensitivity matrix are computed for

the predicted trajectory and using this information, the corrector computes the optimum

search direction, step length and updates the control vector.

This procedure is iterated till the optimum solution is achieved. With the optimum control

vector history, the reference trajectory profiles are generated which ensures that the achieved

target conditions satisfy all the path constraints. These computations are carried out prior to

reentry and the reference profiles are stored for further use in the profile tracking algorithm

during actual reentry.

4. APPLICATION OF THE REALTIME PLANNING ALGORITHM TO A

TYPICAL RLV MISSION

The generic reentry guidance planning algorithm described above is applied to a typical RLV

mission. The objective of the guidance function is to steer the vehicle from any reentry

interface point to the defined TAEM interface location with specified energy level, Ef,

without violating the heat flux constraint. The necessary models and formulations to meet the

above objectives as applicable to the generic guidance law are given in this section. The

development involves the proper selection of control vector for three dimensional trajectory

planning, evolution of different methods and optimum & efficient strategy for the

computation of sensitivity matrix. The terminal target conditions are defined as

0E df =− φφ )( ; 0E df =− λλ )( (26)

where, )( fEφ , )( fEλ are the final latitude and longitude at the defined TAEM energy

level, fE and dφ , dλ are the targeted latitude and longitude values respectively. The path

constraint chosen is the stagnation point heat flux which is required to be less than the

allowable limit lQ& and is as given below.

0QQ 1 ≤− && ; where

15.3

cir

2/1

0N V

V

R

11030Q

=

ρ

ρ& (27)

4.1 Control vector

Three dimensional trajectory planning is achieved by steering through α and σ profiles.

With regard to the control history, an approach using only the bank angle (σ ) modulation

without reversal, is employed while keeping the predefined profile for angle of attack (α ).

Usage of predefined profile for α provides a major advantage in that it ensures the

satisfaction of vehicle and mission constraints such as trim limits and better management of

NSC 2005 Conference Paper No.39

9

thermal constraint. In the present planning algorithm development, the fixed angle of attack

profile is used and at any instant of time, the angle of attack (figure 2) is computed as,

for ,tt a< nt αα =)( ; & for ,tt a≥ )()( an ttt −−= ααα & (28)

where predefined values are used for nα , and α& . The bank angle history is assumed to be

zero during the initial phase of reentry while vehicle flies with high angle of attack, which

ensures better thermal management in the high velocity regime. Non-zero bank angle profile

is initiated after a defined time and from this time onwards till the TAEM interface, the bank

angle modulations are considered sufficient for achieving the desired trajectory, while

meeting the target and path constraints. The σ history is parameterized as given in Fig-3.

At any instant of time, the bank angle is computed as

for ,1tt < 0)( =tσ (29)

for ,tt i≥

ii

i

j

jjjd )tt()tt()t( σσσσ && −+

−+= ∑

−

=+

1

1

1 (30)

The parameters considered in the present study are; the initial bank angle at the time of bank

initiation and bank angle rates during the remaining phase of the flight where the time of

initiation of these rates are predefined values. Therefore, the control vector assumed for this

study is given as

[ ]T654321d σσσσσσσ &&&&&&=u (31)

It is to be noted here that larger numbers of parameters, along with a selection of proper time

instants ensure faster solution. In the present study, the time instants assumed are arbitrary

and only the seven parameters are considered for the model development. As per the mission

requirements, the parameters can be increased and the general model developed in this

section can be used for the extended case also.

4.2 Target and constraint error

Using Eq. 26 and Eq. 27, the target error vector, constraint error, weighting matrix and error

vector are defined as

Fig-2 Angle of Attack History

t1 t2 t3 t4 t5 t6

σd

σ1

σ2

σ3

σ4

σ5

σ6

σ(d

eg)

t

Fig-3 Bank Angle History

NSC 2005 Conference Paper No.39

10

−

−=

df

df

E

E

λλ

φφ

)(

)(Y ; YHY

Tte

−= ;

=

2221

1211

HH

HHH (32)

dtQe

ft

t

cc ∫∆=

0

2& ;

−=

=otherwise,0

QQQifQQ 1

c

&&&&&

∆∆∆ ;

=

c

t

e

e)u(e (33)

4.3 Sensitivity matrix computation

In gradient based methods, the efficiency and accuracy of the solution depends on the

correctness of the sensitivity matrix. In this regard, the literature shows that the sensitivity

matrix computations are carried out using finite differencing and in these methods the

correctness depends on the perturbation levels used for finite differencing16

. It is found that

due to the wide range of trajectory parameters during the entire reentry phase, the

perturbation levels required for a particular control variable also vary and the correct level of

these perturbations is a strong function of the flight environment.

In addition, perturbation selection is severely affected by the resolutions of the computations

in a particular computer system and the perturbation level used in one computer system may

not produce correct results in another computer system for the same flight environment. A

new approach proposed in the present study aims to simplify the sensitivity matrix

computations and is described below.

The system dynamics and corresponding sensitivity of system state are given as

)uX,(fX =& ; ii uudt

d

∂

∂=

∂

∂ )u,X(fX (34)

where, u is the control vector and iu∂

∂X is the sensitivity of system state vector with respect to

the control parameter, iu . The basic states for the sensitivity coefficients are given below:

T

dddddd

rV

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂=

σ

λ

σ

φ

σσ

ψ

σ

γ

σ1ξξξξ ;

T

1111112

rV

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂=

σ

λ

σ

φ

σσ

ψ

σ

γ

σ &&&&&&ξξξξ

T

2222223

rV

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂=

σ

λ

σ

φ

σσ

ψ

σ

γ

σ &&&&&&ξξξξ ;

T

3333334

rV

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂=

σ

λ

σ

φ

σσ

ψ

σ

γ

σ &&&&&&ξξξξ

T

4444445

rV

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂=

σ

λ

σ

φ

σσ

ψ

σ

γ

σ &&&&&&ξξξξ ;

T

5555556

rV

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂=

σ

λ

σ

φ

σσ

ψ

σ

γ

σ &&&&&&ξξξξ

T

6666667

rV

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂=

σ

λ

σ

φ

σσ

ψ

σ

γ

σ &&&&&&ξξξξ (35)

The corresponding dynamic equations for the sensitivity coefficients are given below.

iii ΜL += ξξξξξξξξ& (36)

NSC 2005 Conference Paper No.39

11

where, the elements of the matrix L and Mi are derived using the Eq. (34) from the dynamics

given in Eq. (1) to Eq. (4).

The states for the constraint error sensitivity are given below.

T

6

c

5

c

4

c

3

c

2

c

1

c

d

c eeeeeee

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂=

σσσσσσσ &&&&&&ηηηη (37)

The corresponding dynamic equations for the sensitivity coefficients are given below:

−= 1411c1

2V

153QQ2 ξξξξξξξξ

β∆η

.* &&& ;

−= 2421c2

2V

153QQ2 ξξξξξξξξ

β∆η

.* &&&

−= 3431c3

2V

153QQ2 ξξξξξξξξ

β∆η

.* &&& ;

−= 4441c4

2V

153QQ2 ξξξξξξξξ

β∆η

.* &&&

−= 5451c5

2V

153QQ2 ξξξξξξξξ

β∆η

.* &&& ;

−= 6461c6

2V

153QQ2 ξξξξξξξξ

β∆η

.* &&&

−= 7471c7

2V

153QQ2 ξξξξξξξξ

β∆η

.* &&& (38)

5. PERFORMANCE OF THE GENERIC REENTRY GUIDANCE LAW

In order to evaluate the performance of the generic guidance algorithm, detailed simulation

studies are carried out for the reentry mission objectives of achieving the TAEM interface

location with specified energy level without violating heat flux constraint using a general

RLV simulation tool17

. The performance measures for the trajectory planning algorithm are

the faster convergence, accuracy of achieving the target and constraint violation error. The

robustness and mission flexible capability of the planning algorithm are established by

ensuring nominal performance of the algorithm under widely varying reentry interface

conditions. Typical data available in literature is used for the vehicle characteristics18

whereas

standard models are used for simulating environment. The detailed vehicle data along with

dispersion are used for the simulator, whereas, to improve the guidance execution time,

simplified aerodynamic model in terms of curve fits and exponential model for density are

used in predictor part of the trajectory planning algorithm.

The initial conditions, target parameters, constraint limit and design parameters used to

evaluate the performance of the algorithm are purely arbitrary and do not belong to any

specific RLV mission. The purpose of such arbitrary values is to test the algorithm and its

implementation under limiting conditions.

5.1 Reentry interface and target

The simulations are carried out from the reentry interface to the target TAEM interface point

and the corresponding nominal reentry interface point assumed for the study is given by

0t = 0

h = 121.809 km , V = 7635.7 m/s , γ = -2.1 o

Az = 100 o , φ = 15

o , λ = 0

o

NSC 2005 Conference Paper No.39

12

The velocity, flight path angle and velocity azimuth are the relative values. The guidance

target location at 7800 km down range and 1800 km cross range with respect to reentry

interface location is assumed. Correspondingly, the target TAEM interface location at energy

level of dE = 1234457 m2/s

2 is defined as

dφ = -16o ; dλ = 64.7

o

The trajectory is propagated up to dE and hence target conditions are only the latitude and

longitude defined above. The limit on the heat flux rate is assumed as 70 W/cm2.

5.2 Design parameters

5.2.1 Initial guess for the control history in trajectory planning algorithm

The fixed α profile defined by Nα = 45o, α& = -0.0767

o/s and at = 1030 s is used in

the trajectory planning algorithm. The initial guess for σ profile is given by

t1 = 240 s, dσ = 82o, 1σ& = -0.01

o/s

t2 = 500 s, 2σ& = -0. 03o/s

t3 = 650 s, 3σ& = -0. 12o/s

t4 = 800 s, 4σ& = -0. 1o/s

t5 = 950 s, 5σ& = -0. 1o/s

t6 = 1200 s, 6σ& = -0. 05o/s

The initial guess as given above is used for all the trajectory planning cases.

5.2.2 Weighting matrix

The weighing matrix elements are selected such that the maximum allowable dispersion on

target latitude and longitude is 0.01o. Accordingly the weighing matrix elements are given as

=

360000000

036000000H

5.2.3 Termination conditions

The algorithm execution is stopped under one of the following conditions:

(i) Number of iterations > 50

(ii) te < 10 (corresponding to 0.03 o tolerance) and ce < 100.

(iii) Optimum step length computed by quadratic interpolation algorithm is δ <

1.0x10-04

. This criteria indicates that the solution achieved is very close to the

previous instant value.

5.3 Performance Results

Returning from the orbit for different missions, the vehicle may initiate reentry flight at same

altitude and velocity, but different reentry conditions, if the orbits are very similar8.

Therefore, in order to measure the performance of the trajectory planning algorithm,

simulations are carried out with varying reentry interface conditions to achieve the specified

NSC 2005 Conference Paper No.39

13

target without violating the mean heat flux value. The reentry flights simulated in the present

study is with reentry location latitude varying from +20o to -15

o whereas nominal reentry

latitude assumed is at 15o. Table-1 provides these results.

Table-1

Reentry Interface Conditions and Performance of Trajectory Planning Algorithm

Parameters Case-1 Case-2 Case-3 Case-4 Case-5

Reentry Interface

h (km)

V (m/s)

γ (deg)

Az (deg)

φ (deg)

λ (deg)

121.8

7635.7

-2.1

100

15

0

121.8

7635.7

-2.1

100

20

0

121.8

7635.7

-2.1

100

0

0

121.8

7635.7

-2.1

100

-10

0

121.8

7635.7

-2.1

100

-15

0

Target Aimed

dφ (deg)

dλ (deg)

-16

64.7

-16

64.7

-16

64.7

-16

64.7

-16

64.7

Constraint Imposed

Q& (W/cm2)

< 70

< 70

< 70

< 70

< 70

Target Achieved

fφ (deg)

fλ (deg)

-16.005

64.707

-15.78

64.73

-16.05

64.696

-15.999

64.68

-16.004

64.75

The performance of trajectory planning algorithm in terms of solution convergence is given

in Fig-4. It is seen from the figure that for all the cases, the trajectory planning algorithm

drives both target and errors to zero within 10 iterations. For all the cases, the predefined

angle of attack profile is used whereas bank angle profiles generated by the trajectory

planning algorithm for different cases are given in Fig-5. It is again seen that the control

profiles are altered significantly in order to meet the target conditions without violating heat

flux constraint from different initial conditions. From the trajectory profile generated by the

planning algorithm for different cases as given in Fig-6, it is seen that the algorithm steers the

vehicle from different reentry interface points to the specified TAEM interface point. The

target conditions aimed and achieved show that the accurate solution is achieved for all the

reentry interface conditions as given in Table-1 and Fig-6. The heat flux profiles given in

Fig-7 show that for all the cases, the maximum value is limited to 70 W/cm2.

5.4 Algorithm execution time

The software modules are executed in Intel Pentium-4 PC and WINDOWS FORTRAN

compiler. The execution time for trajectory planning algorithm for 10 iterations is about 9s.

NSC 2005 Conference Paper No.39

14

Fig-4 Convergence Errors for Different

Trajectory Planning Cases

-150

-100

-50

0

50

100

0 400 800 1200 1600

CASE-4

CASE-5

CASE-3

CASE-1

CASE-2

TIME (s)

σσ σσ (

de

g)

Fig-5 Bank Angle Profile Generated by

Trajectory Planning Algorithm

for Different Cases

010

2030

4050

6070

-20

-10

0

10

20

20

40

60

80

100

120

140

Latitude (deg)

Longitude (deg)

Alt

itu

de

(k

m)

0

20

40

60

80

0 400 800 1200 1600

TIME (s)

HE

AT

FL

UX

(W

/cm

2)

Fig-6 Trajectory Profiles for Different

Cases Generated by

Trajectory Planning Algorithm

Fig-7 Heat Flux Profiles for

Different Planning Cases

0

0.5x106

1.0x106

1.5x106

0 2 4 6 8 10

CASE-1

CASE-4

CASE-5

CASE-2

CASE-3WE

IGH

TE

D T

AR

GE

T E

RR

OR

0

10000

20000

30000

0 2 4 6 8 10

CASE-1,CASE-2

CASE-3

C A SE-4

CASE-5

ITERATION

WE

IGH

TE

D C

ON

ST

RA

INT

ER

RO

R

NSC 2005 Conference Paper No.39

15

6. CONCLUSIONS

In this paper, a generic reentry trajectory planning guidance law is presented to steer the

vehicle from any reentry interface point to the specified target meeting all the path constraints

without the need for initial feasible reference trajectory. The planning algorithm solves the

three dimensional trajectory planning problem without the bank reversals and is capable of

handling any reentry interface condition, thus having in-built mission flexibility. In this

algorithm, the optimal control problem is converted into an equivalent targeting problem in

NLP and a simple solution methodology is developed to solve the resulting reentry guidance

problem and the trajectory planning algorithm can be executed onboard during the time

available between the end of de-boost manoeuvre and start of the actual reentry, without any

difficulties. This generic guidance law is then applied to a generic RLV mission with the

objective of steering the vehicle with heat flux constraint from any reentry interface point to

the target of a TAEM interface location with specified energy level. In order to increase

guidance accuracy and to reduce the computational load and for the ease of computation of

sensitivity matrix, the general vehicle model moving about an oblate rotating planet is

developed in polar coordinate system for three dimensional trajectory planning and profile

tracking. The bank angle profile modulation without roll reversal, along with a predefined

angle of attack profile meeting all the vehicle constraints, is used as control histories for the

trajectory planning, which is considered to be an advantage from the mission planning point

of view. The control vector components assumed for the planning algorithm are initial bank

angle and parameterized rates of bank angle history. The sensitivity matrix coefficients are

derived as additional states and are computed through numerical integration, which avoid the

problems related to numerical computation of the gradients.

The robustness of the reentry trajectory planning algorithm is established by steering the

vehicle only through bank angle modulation without any roll reversal from different reentry

interface points physically separated by 3900 km to a specified target, meeting the vehicle

constraints. The computational load for this algorithm can be handled easily by the present

day computers. The algorithm developed is independent of any mission specific data and

therefore fully generic and adaptable to any RLV mission. The output of the trajectory

planning algorithm can be used for profile tracking algorithms during actual reentry. The

integrated guidance algorithm with the proposed trajectory planning algorithm along with a

profile tracking algorithm available in literature forms a good candidate for any RLV

mission.

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