A Generic Reentry Trajectory Planning Algorithm ... - Semantic Scholar

NSC 2005 Conference Paper No.39

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 Drag and lift coefficients respectively CD , CL Acceleration due to gravity g Second gravitational harmonics J2 LQR Linear Quadratic Regulator m Vehicle mass Maximum allowable load factor n max NLP Nonlinear Programming q max 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

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Q& , Q& max r RN RLV TAEM V Vcir

α β γ λ µ ρ ρ0 σ φ ψ Ωe

Heat rate and its maximum allowable value Radial distance Nose radius Reusable Launch Vehicle Terminal Area Energy Management Relative velocity Circular velocity Angle of attack Scale height Flight path angle Vehicle longitude Gravitational constant, Atmospheric density Sea level density Bank angle Vehicle latitude Relative yaw angle of the vehicle 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,

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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 predictorcorrector 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.

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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.  D V& = −   + g R sin γ − gφ cos γ cosψ m

+ rΩe 2 cos φ (− cos γ cosψ sin φ + sin γ cos φ )

(1)

1L 1 V2 cos γ ]  cos σ + [ g R cos γ + gφ sin γ cosψ + V m V r

γ& = 

rΩe 2 cos φ (sin γ cosψ sin φ + cos γ cos φ ) V 1 L 1 V ψ& = ( gφ sinψ ) + cos γ sinψ tan φ   sin σ + V cos γ  m  V cos γ r + 2Ωe sinψ cos φ +

rΩ e 2 cos φ sinψ sin φ V cos γ V cos γ sinψ λ& = r cos φ

+ 2Ωe (sin φ − tan γ cosψ cos φ ) +

r& = V sin γ ;

gR =−

φ& =

V cos γ cosψ ; r

µ 

R 3  1 − J 2 ( e ) 2 ( 3 sin 2 φ − 1 )  2  2 r  r

(3) (4) (5)

2

R  gφ = J  e  sin φ cos φ 2 2 r  r 1 L= ρ ( r )V 2 SC L ( α , M , r ) ; 2m 3µ

(2)

(6) D=

1 ρ ( r )V 2 SC D ( α , M , r ) 2m

(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

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defined in terms of structural load factor, while dynamic pressure limit arises from actuator capabilities. These path constraints are as given below.  D L V2 cos( σ ) ≤  g − nz = [1 + ( L / D ) 2 ]1 / 2 ≤ n max (8)  cos( γ ) ; m r  m  11030  ρ  Q& =   R N  ρ0 

1/ 2

 V    Vcir 

3.15

≤ Q& max ;

q = ( 1 / 2 ) ρ V 2 ≤ q max

(9)

2.3 Reentry Guidance Problem Definition In its most general form the guidance problem can be states as: Given initial state X 0 , the guidance objective is to estimate α ( t ) and σ ( t ) histories without violating the path constraints given by Eq. 8 and 9 so that at final time t f , the terminal state vector X f 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, E f , the target conditions are given by,

φ( E f ) − φd = 0

;

λ( E f ) − λd = 0

(10)

ψ ( E f ) −ψ d = 0

;

γ ( E f )− γ d =0

(11)

where, φ d , λ d are desired latitude and longitude of TAEM interface location and ψ d , γ d 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,

Y = X f − Xd

(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 tf

δ ci = ∫ ∆C 2 ci dt ; where, ∆Cci = 

∆ Ci

 0,

t0

if ∆Ci = Ci − Cl > 0   otherwise 

(13)

It is to be noted that if the ith constraint is satisfied then δ ci = 0 . Similarly, the constraint violation can be computed for all the path constraints. The constraint error vector C is defined with components of ∆C ci . Therefore, in general, the reentry guidance problem can be restated as an optimization problem where the following performance index is to be minimized. tf T

J = Y ( X f )HY ( X f ) +

∫C

T

( X ( t ), uc ( t ))ZC ( X ( t ), uc ( t ))dt

t0

5

(14)


subject to the differential constraints given by Eq. 1 to Eq. 4 with the prescribed initial conditions t0 and X0, where uc ( t ) = {α ( t ) σ ( t )} T is the control vector history. Assuming the target error as,

et ( uc ( t )) = Y T ( X f ) H Y ( X f )

(15)

and the constraint error as tf

∫C

ec ( uc ( t )) =

T

( X ( t ), uc ( t ))ZC ( X ( t ), uc ( t ))dt

(16)

t0

the performance index can be rewritten as

J = et ( uc ( t )) + ec ( uc ( t ))

(17)

This optimal control problem can now be converted into a NLP problem by parameterizing the control time history uc ( t ) , as given below.

uc ( t ) = u k +

( uk +1 − uk )( t − tk ) , tk < t < tk +1 ( tk +1 − tk )

(18)

where u k are the control parameters. If u denotes the vector of control parameters, then the performance index and the corresponding dynamics are written as,

J ( u ) = et ( u ) + ec ( u ) ;

X& = f ( X ( t ), u )

(19)

It is to be noted that, the guidance solution is achieved when J ( u ) = 0 . Since the terms in Eq. 19 are quadratic, this is possible only when

et ( u ) = 0.0 ;

ec ( u ) = 0.0

(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  e ( u* )  control vector u = u ∗ which ensures that error vector e( u* ) =  t *  = 0 , subject to the ec ( u ) system dynamics, X& = f ( X ( t ), u ) and the initial conditions t0 and X 0 . 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, u0 and the error vector for this reference control vector, e( u0 ) is computed. Next, the increment on the control vector, ∆u is found so that the control vector, u* = u0 + ∆ u ensures e( u* ) = 0. The gradient vectors of the target and constraint errors are defined as,

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 ∂e ∇ et T ( u0 ) =  t  ∂u1

 ∂e ∂et  ∂ec ∂ec  ; ∇ ec T ( u0 ) =  c K (21)   ∂u k  u ∂u1 ∂u 2 ∂u k  u  0 0 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 u0 is taken to be linear, then the unique optimum search direction ∆u is the one, which satisfies the following linearized error equation S ( u0 )∆u + e( u0 ) = 0 and minimizes the length of ∆u . The solution to the above equation defines a linear manifold C ( u0 ) (in this case, a straight line) as given in Fig-1. ∂et ∂u 2

K

The

Fig-1 Minimum-Norm Correction Direction

The desired minimum norm correction, ∆u is then the vector of minimum length from u0 to the linear manifold C ( u0 ) . Analytically, this is given as

∆u = − S T ( u0 ) [ S ( u0 )S T ( u0 )] −1 e( u0 )

(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 onedimensional 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, 2 J 0 ( δ ) = e( u0 + δ ∆u ) 2 ; with J 0 ( 0 ) = e( u0 ) 2 ; J 0 ( 1 ) = e( u0 + ∆u )

(23)

Differentiating via chain rule yields

J 0′ ( 0 ) = 2e T ( u0 )S ( u0 )∆u ;

(24)

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

J 0' ( 0 ) . The quadratic polynomial along with the optimum step length and the optimum control vector is given below:

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J 0 ( δ ) = a0 + a1δ + a 2δ 2 ;

δ* =−

a1 ; u* = u0 + δ * ∆u 2a2

(25)

The optimum solution is achieved through iteration with updated value of u = u* 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

φ ( E f ) − φd = 0 ;

λ ( E f ) − λd = 0

(26)

where, φ ( E f ) , λ ( E f ) are the final latitude and longitude at the defined TAEM energy level, E f and φ d , λd 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 Q& l and is as given below.

Q& − Q& 1 ≤ 0 ;

where

11030  ρ  Q& =   RN  ρ0 

1/ 2

 V    Vcir 

3.15

(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 8


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 t < t a ,

α (t ) = α n ;

for t ≥ t a ,

&

α (t ) = α n − α& (t − t a )

(28)

where predefined values are used for α n , 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. σd σ1 σ2 σ(deg)

σ3

σ4 σ5 σ6

t1

t2

t3

t4

t5

t6

t

Fig-2 Angle of Attack History

Fig-3 Bank Angle History

At any instant of time, the bank angle is computed as for t < t1 , σ (t ) = 0 for t ≥ ti ,

 i −1



 j =1 

 

σ ( t ) = σ d +  ∑ ( t j +1 − t j )σ& j  + ( t − t i )σ& i

(29)

(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

u = [σ d

σ& 1 σ& 2 σ& 3 σ& 4 σ& 5 σ& 6 ] T

(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

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φ ( E f ) − φ d  −T Y= HY ;  ; et = Y λ ( E ) − λ f d   tf

ec =

∫ ∆Q& c

2

H H =  11  H 21

H 12  H 22 

∆Q& if ∆Q& = Q& − Q& 1  ;  0 , otherwise 

e  e( u ) =  t  e c 

∆Q& c = 

dt ;

t0

(32)

(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 d  ∂X  ∂f ( X , u )  = (34) X& = f ( X, u ) ; dt  ∂u i  ∂u i where, u is the control vector and ∂ X is the sensitivity of system state vector with respect to ∂u i

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

 ∂V

T

T

∂σ d

∂γ ∂σ d

∂ψ ∂σ d

∂r ∂σ d

∂φ ∂σ d

 ∂V ∂λ   ;ξ2 =  ∂σ d  ∂σ& 1

 ∂V ∂σ& 2

∂γ ∂σ& 2

∂ψ ∂σ& 2

∂r ∂σ& 2

∂φ ∂σ& 2

∂λ   ; ξ4 ∂σ& 2 

T

 ∂V = ∂σ& 3

∂γ ∂σ& 3

∂ψ ∂r ∂φ ∂λ   ∂σ& 3 ∂σ& 3 ∂σ& 3 ∂σ& 3 

 ∂V ∂σ& 4

∂γ ∂σ& 4

∂ψ ∂σ& 4

∂r ∂σ& 4

∂φ ∂σ& 4

T

ξ5 = 

∂λ   ; ξ6 ∂σ& 4 

 ∂V = ∂σ& 5

∂γ ∂σ& 5

∂ψ ∂σ& 5

 ∂V ξ7 =  ∂σ&6

∂γ ∂σ&6

∂ψ ∂σ&6

∂r ∂σ&6

∂φ ∂σ&6

∂λ   ∂σ&6 

ξ1 = 

ξ3 = 

∂γ ∂σ& 1

∂ψ ∂σ& 1

∂r ∂σ& 1

∂φ ∂σ& 1

∂λ   ∂σ& 1  T

∂r ∂σ& 5

∂φ ∂σ& 5

T

∂λ   ∂σ& 5 

T

(35)

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

ξ&i = Lξ i + Μ i

(36)

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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.

 ∂e η = c  ∂σ d

∂ec ∂σ& 1

∂ec ∂σ& 2

∂ec ∂σ& 3

∂ec ∂σ& 4

∂ec ∂σ& 5

∂ec   ∂σ& 6 

T

(37)

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

β β  3.15   3.15  ξ 11 − ξ 14  ; η& 2 = 2Q& * ∆Q& c  ξ 21 − ξ 24  2 2  V   V 

η&1 = 2Q& * ∆Q& c 

β β  3.15   3.15  ξ 31 − ξ 34  ; η&4 = 2Q& * ∆Q& c  ξ 41 − ξ 44  2 2  V   V 

η&3 = 2Q& * ∆Q& c 

β β  3.15   3.15  ξ 51 − ξ 54  ; η&6 = 2Q& * ∆Q& c  ξ 61 − ξ 64  2 2  V   V 

η&5 = 2Q& * ∆Q& c 

β  3.15  ξ71 − ξ74  V 2  

η&7 = 2Q& * ∆Q& c 

(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 =0 t0 = 121.809 km , = 7635.7 m/s , = -2.1 o h V γ = 100 o , = 15 o , = 0o Az φ λ

11


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 E d = 1234457 m2/s2 is defined as

φd

= -16o ;

λd

= 64.7o

The trajectory is propagated up to E d 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

= -0.0767o/s and t a = 1030 s is used in The fixed α profile defined by α N = 45o, α& the trajectory planning algorithm. The initial guess for σ profile is given by = -0.01o/s t1 = 240 s, σ d = 82o, σ&1 t2 = 500 s, = -0. 03o/s σ& 2 t3 = 650 s, σ& 3 = -0. 12o/s t4 = 800 s, σ& 4 = -0. 1o/s t5 = 950 s, = -0. 1o/s σ& 5 t6 = 1200 s, = -0. 05o/s σ& 6 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

0 36000000  H = 0 36000000   5.2.3 Termination conditions

The algorithm execution is stopped under one of the following conditions: (i) Number of iterations > 50 (ii) et < 10 (corresponding to 0.03 o tolerance) and ec < 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 12


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 -15o 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 Reentry Interface h (km) V (m/s) γ (deg) Az (deg) φ (deg) λ (deg) Target Aimed φ d (deg) λd (deg) Constraint Imposed Q& (W/cm2) Target Achieved φ f (deg)

λ f (deg)

Case-1

Case-2

Case-3

Case-4

Case-5

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

-16 64.7

-16 64.7

-16 64.7

-16 64.7

-16 64.7

< 70

< 70

< 70

< 70

< 70

-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.

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NSC 2005 Conference Paper No.39 6

100 C ASE-2

1.0x10

6

0.5x10

6

50

CASE-2

C ASE-4 C ASE-3 0

0

2

4

6

8

0

10

30000

CASE-1 CASE-4

CASE-5

-50

CASE -5 20000 CASE -1, CASE -2 10000

0

C A S E -4

-100

CASE-3

C AS E -3

0

2

4

6

8

10

-150

0

400

800

1200

1600

ITER ATIO N

TIME (s)

Fig-4 Convergence Errors for Different Trajectory Planning Cases

Fig-5 Bank Angle Profile Generated by Trajectory Planning Algorithm for Different Cases

80

140

60

2

HEAT FLUX (W/cm )

120

100

Altitude (km)

WEIGHTED CONSTRAINT ERROR

C ASE-5

C ASE-1

σ (deg)

WEIGHTED TARGET ERROR

1.5x10

80

60

20

20

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40

10 0 20 0

Latitude(deg) 10

-10 20

30

40

Longitude(deg)

50

60

70

-20

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Fig-6 Trajectory Profiles for Different Cases Generated by Trajectory Planning Algorithm

Fig-7 Heat Flux Profiles for Different Planning Cases

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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. REFERENCES 1

Harpold, J.C., and Graves, C.A., “Shuttle Entry Guidance”, Jl. of Astronautical Sciences, Vol 27, No.3, pp.239-268, July-Sept., 1979

2

Ishimoto, S., Takizawa, M., Suzuki, H., and Morito, T., “Flight Control System of Hypersonic Flight Experiment Vehicle”, AIAA Paper No. AIAA-96-3403, Atmospheric Flight Mechanics Conference, July 1996.

3

Hanson, J. M., Coughlin, D. J., Dukeman, G.A., and Malqueen, J.A, and McCarter, J.W., “Ascent, Transition, Entry and Abort Guidance Algorithm design for the X-33 Vehicle”, AIAA Paper No. AIAA-98-4409, Guidance Navigation and Control Conference, 1998.

15


4

Dukemann, G. A., Gallaher, M. W., “Guidance and Control Concepts for the X-33 Technology Demonstrator”, AAS Paper No. AAS-98-026.

5

Lu, P., Hanson, J. M., Dukeman, G.A., and Bhargava, S., “An Alternative Entry Guidance Scheme for the X-33”, AIAA Paper No. AIAA-98-4255, Atmospheric Flight Mechanics Conference, 1998.

6

Lu, P., “Entry Guidance and Trajectory Control for Reusable Launch Vehicle”, Jl. of Guidance, Control and Dynamics, Vol 20, No.1, pp.143-149, Nov-Dec 1997.

7

Lu, P., “Regulation about Time-Varying Trajectories: Precision Entry Guidance Illustrated”, AIAA Paper No. AIAA-99-4070.

8

Lu, P., Shen, Z., Dukeman, G.A., Hanson, J. M., “Entry Guidance by Trajectory Regulation”, AIAA Paper No. AIAA-2000-3958, Guidance, Navigation and Control Conference and Exhibit 2000.

9

Mease, K. D., Chen, D.T., Tandon, S., Young, D.H., and Kim, S., “A Three Dimensional Predictive Entry Guidance Approach”, AIAA Paper No. AIAA-2000-3959, Guidance, Navigation and Control Conference and Exhibit 2000.

10

Bharadwaj, S., Rao, A. V., and Mease, K. D., “Entry Trajectory Tracking Law via Feedback Linearization”, Jl. of Guidance, Control and Dynamics, Vol 21, No.5, Sept. – Oct. 1998.

11

Youssef, H., Choudhry, R.S., Lee, H., Rody, P., and Zimmermann, C., “PredictorCorrector Entry Guidance for Reusable Launch Vehicle”, AIAA Paper No. AIAA-20014043, Guidance, Navigation and Control Conference, 2001.

12

Ishizuka, K., Shimura, K., and Izhimoto, S., “A Re-entry Guidance Law Employing Simple Real-time Integration”, AIAA Paper No. AIAA-98-4329.

13

Fuhry, D. P., “Adaptive Atmospheric Reentry Guidance for the Kistler K-1 Orbital Vehicle”, AIAA Paper No. AIAA-99-4211.

14

Dukemann, G. A., “Profile-Following Entry Guidance Using Quadratic Regular Theory”, AIAA Paper No. AIAA-2002-4457.

15

Zimmerman, C., Dukeman, G., Hanson, J., “An Automated Method to Compute Orbital Reentry Trajectories with Heating Constraints”, AIAA Paper No. AIAA-2002-4454.

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Ashok Joshi, Sivan, K., “Reentry Guidance for Generic RLV Using Optimal Perturbations and Error Weights”, Paper No. AIAA-2005-6438, AIAA Guidance, Navigation and Control Conference, San Fransisco, USA, August 15-18, 2005.

17

Ashok Joshi, Sivan, K., “Modeling and Open Loop Simulation of Reentry Trajectory for RLV Applications”, Proceedings of the 4th International Symposium on Atmospheric Reentry Vehicle and Systems, March 21-23, 2005, Arcachon, France.

18

Eussel, W.R.,(Ed.), “Aerodynamic Design Data Book, Volume-I, Orbiter Vehicle SIS1”, Report No. SD72-SH-0060, Space Systems Group, Rockwell International, California, 1980.

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