Pointing Algorithm for Reentry Orbit Injection with Uncontrolled Last Stage

An essencial requirement for a launcher with uncontrolled last stage is to compute the ignition time of that stage and its attitude, in order to obtain a proper orbit transfer from a nonpropelled trajectory to a desired orbit, achieving the proper conditions to inject a vehicle with uncontrolled last stage into a descent Keplerian trajectory, in injection. Impulsive ignition time and necessary attitude for the last stage are obtained based on orbital parameters, having a match between the nonpropelled and reentry orbits, and boundary conditions at the initial state and at the entry interface. Impulsive ignition time was corrected for compensating the impulsive assumption and the calculated with uncontrolled last stage, as the Sharp Edge Flight Experiment (SHEFEX) program.


INTRODUCTION
Some reentry experiments for investigating aerothermodybeen conducted to provide design guidelines for future reusable space transportation systems or hypersonic vehicles.Such studies evaluated the design aspects of aerothermodynamics The Sharp Edge Flight Experiment (SHEFEX) program is the main instance of such reentry missions.A launch vehicle for a reentry object is typically composed of two or more stages.
propellant rocket  Missions are programmed with sequenced events and they are guided based on nominal trajectories.In an uncontrolled Only between the penultimate stage burnout and the last maneuver) and spin-up maneuvers shall happen.The ignition vehicle into a descent one for the reentry.
The pointing algorithm described here is based on the developed an algorithm to obtain a proper transfer from a

Uncontrolled Last Stage
Flávio Eler de Melo 1, *, Waldemar de Castro Leite Filho 2 , Hilton Cleber Pietrobom 2 1 2 Abstract: An essencial requirement for a launcher with uncontrolled last stage is to compute the ignition time of that stage and its attitude, in order to obtain a proper orbit transfer from a nonpropelled trajectory to a desired orbit, achieving the proper conditions to inject a vehicle with uncontrolled last stage into a descent Keplerian trajectory, in injection.Impulsive ignition time and necessary attitude for the last stage are obtained based on orbital parameters, having a match between the nonpropelled and reentry orbits, and boundary conditions at the initial state and at the entry interface.Impulsive ignition time was corrected for compensating the impulsive assumption and the calculated with uncontrolled last stage, as the Sharp Edge Flight Experiment (SHEFEX) program.
injection altitude of the satellite corresponded to the perigee pointing strategy to permit the reentry of the satellite through A pointing algorithm can be run in the beginning of the coast phase to obtain a proper transfer from the actual orbit to the desired one.The orbit transfer is assumed as an impulsive increment of position and velocity due to its burning) and reentry that the last stage must be pinpointed in this time are calculated.Impulsive ignition time is then corrected for compensating the mate stage separation to avoid that the increasing inertial sensor Some tradeoffs must be taken into consideration when selecting one of the possible orbit transfers.The impulsive orbit transfer may occur in the ascent section of the coast For preventing disturbances that may degrade the accuracy injection occurs in the ascent section of the coast phase orbit.which means that less propellant would be necessary for an injection from the descent section.

GENERAL FORMULATION
The general problem formulation to establish the necessary conditions for an orbit transfer at t=t I will be recalled from Leite Filho (1994).The instant t I is the ignition time of the last stage for an hypothetical impulsive orbit transfer.
The motion of the vehicle's center of gravity during the as in Eq. 1: where g v is the gravitational acceleration and is the unit is the propulsive acceleration given by Eq. 2: where (t-t ig m s is the structural mass and m p (t -t ig ) is the propellant mass of the last stage with ignition time t ig .(.) and m p (.) are known functions.The solution to are increments of velocity and position due and are increments The increment of angular momentum due to the last stage burning is provided by Eq. 9: et al.

Re-entry trajectory 2
Figure 1.Impulsive orbit transfer for a reentry mission.
the same direction as R ig and the distance travelled during the t f is the burnout time of the last stage.Equation 10 can be rearranged as Eq.11: Equation 11 is interpreted as an orbital change due to an impulsive thrust applied at R I .By the principle of conservation Equation 12 shows that R I will occur seconds after t ig and it shows that the impulsional shot in R I is equivalent to the actual thrust started in t ig with the same direction.This equivalence is corrected by the factor compensates the impulsional assumption for the ignition time t I time (Eqs.14 and 15):

CONDITIONS FOR IMPULSIVE ORBIT TRANSFER
last stage from the coast phase to the reentry orbit might be V in a direction described by condition of radius (R 0 V 0 0 ) at the begining of the coast phase and to the reentry conditions ).When applying the increment of velocity in the direction described by vis viva equation and the angular momentum conservation principle produces The parameter is the product of the Earth's mass with the gravitational constant.The resulting reentry orbit is also manipulated by proceeding the substitutions R I .This range is obtained based on the fact that R I cannot be smaller than the radius of than the solution where the reentry orbit locus has only one coincident point with the coast phase orbit locus.The practical range for the ^h depend on h 2 at the limit radius for a successful orbit transfer.h 2 and maximum h 2 at the limitradius can be calculated by a R I be carried out at it to inject the last stage into the reentry 2 .The direction of the impulsive shot is trajectories that satisfy the same conditions.One solution is for the other one is for the transfer from the descent section.It is worth to note that this study can support analyses for selecting V the entry interface depends on the energy of the launch vehicle V V possible radius for the orbit transfer can be obtained by using in the ascent section of the coast phase trajectory or in the descent impulsive shot in the ascent section would be the best option.

IGNITION TIME
when the impulsive shot is carried out is related to the time interval that the vehicle last stage takes from R 0 to R I .Expressing the ignition time in terms of the eccentric anomaly E (Cornelisse et al.
f -t ig is the burning time of the last stage.The the eccentricity of the coast phase ascent trajectory is showed by Eq. 40: and its semi-major axis is described by Eq. 41:

RELATION TO THE INERTIAL FRAME
The vehicle pitch angle as Eq.42: The angle between the local vertical direction at the launch site and the one at the the coast phase beginning can be calcu- t ^h is the radius (position) at the launch site considering the mean Earth's radius R e the launch site h launch x at the i x t .
By calculating the angular argument variation of the can be obtained by Eq. 44: (44) relation between both frames must be considered as follows.
to which the

POINTING ALGORITHM
t ig and the direction ( and ) of the shot for injecting the vehicle listed as following: t 0 R 0 V 0 in navigation frame and assignment of reentry conditions R R V R and R ; 0 a 1 e 1 E I a 2 (Eq.41 for R R V R ); R I t ig frame and (Eq.45 to 49).
This algorithm shall transfer the vehicle to the required reentry orbit meeting conditions R R V R and R at the entry interface.The results clearly show that the pointing algorithm produces ensuring that the required conditions are met at the entry ascent or descent sections of the coast phase trajectory.

Simulations
The calculated limits for R I R V provide means to during the preliminary mission design and programming to Figure 4 shows the simulation results for injections into conditions established in Table 1. Figure 5 shows the orbit transfer for M R = 11 and R established initial conditions at the beginning of the coast phase.

CONCLUSIONS
Differently of the pointing algorithms described in Leite this paper presented an algorithm that permits the reentry of a vehicle path angle at the entry interface.
Simulations showed that the pointing algorithm provides of the algorithm by providing important engineering results

SHEFEX research program.
Other important results are the condition analyses and limits for the orbit transfer.These analyses allowed a preliminary evaluation during the mission design and provided a launch vehicle.permit its real time implementation into the onboard computer and to provide a successful mission.
based on the hyperbolic-cosine tripleleading to the solution described by Eq. 29.
to verify the compliance of orbit injection performed for three different sets of established reentry condi-The initial radius R 0 et al.V 0 0 were determined by recalling three simulation cases executed by similar to the ones used for the SHEFEX-2 launch vehicle.characteristics of the reentrying object determine the velocity V R and the reentry angle R at the entry interface.The reentry et al.
of the SHEFEX-2 mission: Mach (M R to test the effectivity of the algorithm: R V was estimated from temporal mass and thrust curves for similar to the second stage of the SHEFEX-2 experiment.The results of the algorithm application for calculating the radius to the orbit transfer R I t I into reentry trajectories by the application of the algorithm