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An automated method to compute orbital re-entry 
trajectories with heating constraints 



Curtis Zimmerman 

Greg Dukeman 

Dr. John Hanson 

NASA Marshall Space Flight Center 

Huntsville, AL 35812 



Determining how to properly manipulate the controls of a re-entering re-usable launch vehicle (RLV) so that 
it is able to safely return to Earth and land involves the solution of a two-point boundary value problem 
(TPBVP). This problem, which can be quite difficult, is traditionally solved on the ground prior to flight. If 
necessary, a nearly unlimited amount of time is available to find the "best" solution using a variety of 
trajectory design and optimization tools. The role of entry guidance during flight is to follow the pre- 
determined reference solution while correcting for any errors encountered along the way. This guidance 
method is both highly reliable and very efficient in terms of onboard computer resources. There is a growing 
interest in a style of entry guidance that places the responsibility of solving the TPBVP in the actual entry 
guidance flight software. Here there is very limited computer time. The powerful, but finicky, mathematical 
tools used by trajectory designers on the ground cannot in general be converted to do the job. Non- 
convergence or slow convergence can result in disaster. The challenges of designing such an algorithm are 
numerous and difficult Yet the payoff (in the form of decreased operational costs and increased safety) can 
be substantial. This paper presents an algorithm that incorporates features of both types of guidance 
strategies. It takes an initial RLV orbital re-entry state and finds a trajectory that will safely transport the 
vehicle to Earth*. During actual flight, the computed trajectory is used as the reference to be flown by a 
more traditional guidance method. 



Overview 

In the design of entry trajectories, two control 
variables are normally available: alpha (angle of 
attack) and phi (bank angle) 1 . The entry guidance 
algorithm developed in this paper (referred to as 
EGuide) primarily attempts to find and adjust bank 
angle profiles to meet final state constraints while 
maintaining a constant or pre-determined angle of 
attack profile. EGuide contains a classic shooting 
method as its solver. When solving a specific entry 
problem, EGuide adjusts parameters to meet 
specified goals using a Newton method that has been 
configured to generate its Jacobian matrix by flying 
predictive simulations. For orbital entry, most of the 
heavy computational load of the Newton process can 
occur prior to the de-orbit burn. This is what is 
meant by an orbital entry planner. The solution 
derived on-board can be used to supply trajectory 
information to a more traditional profile-following 
guidance law such as Dukeman' s LQR 2 . It is also a 
natural setup to run as a predictor/corrector guidance. 
EGuide contains a planning stage and functions both 



as a predictor/corrector and as a profile follower using 
LQR. 

Development and testing of EGuide has been carried 
out using the MAVERIC vehicle simulation. MAVERIC 
is a full 6-DOF simulation developed to test GNC flight 
software for the X33. 

Shooting Method 

EGuide solves the TPBVP using mnewt 3 . Mnewt** 
generates its Jacobian matrix by measuring the effect of 
control changes on the final state of simulated flights. 
EGuide uses a self-contained 3-DOF-trajectory simulation 
(independent from MAVERIC) that models motion over a 
rotating oblate Earth with a US62 standard atmosphere. 
The equations of motion are integrated using a 4 th order 
Runge-Kutta algorithm with a fixed step size of 1 second. 

Sub-orbital Re-entry Guidance 

Part of EGuide is dedicated to solving the TPBVP of a 
sub-orbital entry trajectory. Specifically, it tries to figure 
out how to deliver a vehicle from a variety of widely 



dispersed sub-orbital entry conditions to a Terminal 
Area Energy Management (TAEM) interface box 
within an acceptable tolerance of altitude, range, and 
heading. 

To solve the sub-orbital problem, a linear 
equation is used to define the bank angle. At each 
time point during an EGuide trajectory simulation the 
commanded bank angle is computed using the 
following formulation: 



<Pcmd=Bl+B2(tc 



-t init ) 



(1) 



The sign of (p cmd is assigned to maintain the heading 

angle within a specified corridor. Alpha angle is 
predefined as a function of mach number. Through 
shooting, EGuide identifies the individual values that 
parameters Biand B 2 must take so that the vehicle 
will fly to TAEM. 

Solving a sub-orbital entry trajectory problem is 
essential in the process that EGuide uses to solve for 
an orbital re-entry trajectory. It is also a required 
function to participate in the Advanced Guidance and 
Control project 4 . This project uses the MAVERIC 
vehicle simulation loaded with an X33 vehicle model 
as its testing platform and provides a scored 
evaluation to the participating algorithms. Nominal 
and off-nominal X33 missions are included in the 
testing criteria. For a nominal X33 flight, the EGuide 
planning phase takes place shortly after main engine 
cutoff (MECO). Once parameters E^ and B 2 from 
equation (1) have been found, the trajectory is 
recorded and LQR is activated to guide the vehicle to 
TAEM. 

Sub-Orbital Results 



From the 3-DOF equations of motion for a vehicle flying 
over a spherical Earth we have 5 : 



f = V sin y 
V=-D--^-siny 
v 2_£|cosY + 2wVcos(pcos()) . (2) 



Y = — <Lcosct + 

V ' 



For a vehicle entering a planetary atmosphere, the time 
rate of average heat input per unit area can be estimated 



with the expression 



Q=cVpv n 



(3) 



where n = 3.15, and C is a constant. Heat-rate tracking 
guidance begins with the definition of an error term 



e = Q-Q 



ref 



(4) 



and the intent for this term to exhibit the behavior of a 
stable second order feedback system. To accomplish this, 
it is substituted into the following classical second order 
system 



e + 2£co n e + co;;e = 



(5) 



The three preceding equations along with the equations of 
motion yield the following bank angle formulation: 



<Pcmd = COS 



Ycmd + ~ 

r v 



Alternative bank angle formulations 



-a-2Cco n Q-oJ(Q-Q ref ) 



Tcmd 



(6) 



Constant heat-rate tracking 

Although equation (1) can be used to generate valid 
trajectories from orbital re-entry states, it does not 
contain enough flexibility to consistently return 
practical solutions. Heat and dynamic pressure 
constraints, which have previously been ignored, 
must somehow be addressed to assure safe flight 
conditions. Heating in particular can be effectively 
controlled using a bank angle formulation designed to 
maintain a constant heat rate during flight. 



where a and b are expressions that are determined in the 
second time derivative of equation (3). Thus, given a 

reference heat-rate Q re f and using equation (6) to 
generate bank angle control commands, the vehicle is 
expected to track a constant heat rate. 

Orbital re-entry planning 

In an orbital re-entry, high heating starts near the 
beginning of the flight back into the atmosphere when 
speeds are still close to orbital velocity. Re-entry 
guidance can begin as soon as there is enough dynamic 



pressure to maintain adequate control of the vehicle. 
The EGuide planning simulation is configured to use 
heat rate tracking guidance at the onset of orbital re- 
entry. Once the vehicle has been safely transported 
through high heating, heat rate tracking guidance is 
deactivated and the sub-orbital guidance formulation 
is used for the remainder of the flight. The trajectory 
is fully characterized by four parameters. The first 
three Q ref , Bl, and B2 are from the heat rate 
tracking and sub-orbital guidance formulations. The 
fourth parameter is the time chosen to terminate heat 
rate tracking and switch to sub-orbital guidance (Fig. 
2). 



Bank Angle vs. Time 



De -orbit 



TAEM 



Landing 




Fig. 2 The four parameters of an EGuide orbital re-entry 
trajectory 

Selection of the guidance switch time is based on 
observed bank angle behavior during heat rate 
tracking. As illustrated in Figure 3, in the very first 
part of an orbital re-entry, heat rate tracking guidance 
may modulate the bank angle quickly as the control 
"latches on" to the specified reference heat rate. This 
initial transient behavior gives way to a slower 
varying bank angle which is decreasing in magnitude 
as the control tracks the reference. If heat rate 
tracking is allowed to remain active indefinitely, the 
magnitude of the bank angle eventually begins to 
increase and finally becomes excessive as the control 
struggles to track a reference heat rate it can no 
longer sustain. The main feature of interest in the 
bank angle vs. time plot of Figure 3 is the local 
minimum that occurs at approximately 800 seconds 
(Point A). Point A is designated as the guidance 
switch time. In actuality, good solutions to the 
orbital re-entry problem may exist by switching 
guidance formulations anywhere along the bank 
angle profile between points A and B. However, 
Point A offers the advantage of being an event that is 
easily detectable. EGuide is programmed to find 
Point A by simulating an orbital re-entry (using heat 
rate tracking guidance exclusively) and looking for 
the last occurrence of a bank angle minimum. 
Additionally, the fact that the bank angle magnitude 
must increase after Point A to maintain the reference 



200 




200 400 600 800 1000 1200 
Time (sec) 

Heat Rate vs. Time 



60 
50 
40 
30 
20 
10 



" i T 

- t : • r « 

: : ; I o 

■ i : ■ ° 

i - i : ° 
: : I ; o 
_ ; i I ; o 

: : i : O 

■ ; O 
: : : o 

- ' r- f : " - 

> ; 8 



200 400 600 800 1000 1200 
Time (sec) 

Fig. 3 Heat rate tracking guidance in action 

heat rate indicates that the high heating portion of the 
flight has past. 



Orbital re-entry results 



References 

[1] Harpold, 3.C., and Graves, C.A. Jr., "Shuttle Entry 
Guidance," Journal of the Astronautical Sciences, Vol. 
XXVII, No. 3, pp 239-267, July-September, 1979. 
[2] Dukeman, G.A., Profile following Entry Guidance 
using Linear Quadratic Regulator Theory, AIAA-xxxx. 

[3] Press, W.H., Teukolsky, S.A., Vetterling, W.H., and 
Flannery, B.P., "Numerical Recipes in C." Second 
Edition., Cambridge Univ. Press, New York, 1993, pp 
379-389. 

[4] Hanson, J.M., "Advanced Guidance and Control 
Project for Reusable Launch Vehicles," AIAA-2000- 
3957. August 14, 2000. 



[5] Vinh, N.X., "Optimal Trajectories in Atmospheric [6] Vinh, N.X., et al.. Hypersonic and Planetary 

Flight", Elsevier, New York, 1981, pp 47-62. Entry Flight Mechanics, The University of Michigan 

Press, Ann Arbor, MI, 1980.