Air Force Research Laboratory, Kirtland AFB, NM 87117
This paper investigates gliding guidance of hypersonic vehicles with high lift-to-drag(L/D) ratios. The application scenarios envisioned are largely those that may be calledsuborbital entry, although orbital entry cases are not excluded. An adaptive predictor-corrector algorithm is used for guidance purposes. For high L/D vehicles a unique chal-lenge is found to be large phugoid oscillations in the gliding trajectory. A simple andeffective feedback mechanism is developed for trajectory damping control in conjunctionwith the predictor-corrector guidance algorithm to eliminate these phugoid oscillations.Another unique capability illustrated in this paper is end-to-end mission planning, wherethe hypersonic gliding vehicle is launched by a multi-stage rocket, and hypersonic glidingfollowed. This paper shows that shaping the ascent trajectory differently from conventionalballistic launch can bring significant benefits to the gliding phase in terms of dramaticallyreduced the heating and dynamic load stresses.
I. Introduction
Gliding guidance for a hypersonic vehicle with aerodynamic lifting capability is concerned with steeringthe vehicle, so that it will reach the designated termination point with prescribed condition while satisfying allthe necessary path constraints for safety and operational considerations. In essence, hypersonic gliding flightis a controlled process of dissipating high energy at well-timed pace to cover the required range. Physically,this objective is achieved by controlling the direction of the aerodynamic lift force vector to maintain anappropriate three-dimensional gliding trajectory. In this sense, it is the same as entry guidance for liftingre-entry spacecraft. A notable difference though lies in the initial condition: for most cases of recent interestconsidered in this paper the initial velocity and/or altitude are lower than the typical values for entry flightfrom an orbit. Another consequential difference is that the hypersonic lift-to-drag (L/D) ratios of the vehiclesconsidered in this paper are significantly higher than those for lifting orbiters such as the Space Shuttle. By“high L/D”, it is meant in this paper to refer to hypersonic L/D ratios that are considerably greater than 1.
Traditional lifting entry guidance technology is exemplified by the Space Shuttle entry guidance design,developed in the early 1970s.1 In this approach a reference trajectory is defined by a drag-vs-velocity (andversus energy at lower speeds) profile. This reference is designed off-line, although minor on-line adjustmentsin response to small range dispersions are allowed. A linearized time-varying trajectory tracking control lawis used to track the reference and provide guidance commands. While highly successful for the Shuttle, thisapproach requires significant pre-mission planning, and lacks the adaptive capability to respond to large off-nominal conditions. For hypersonic vehicles intended for responsive missions with little pre-planning time,or test vehicles for which large uncertainty may exist on the system modeling and performance capabilitythat could result in significantly dispersed flight conditions, a more adaptive approach has clear potentialadvantages. When advanced capabilities such as in-flight update of mission objective are sought, the guidancesystem will have to be both adaptive and autonomous.
Great strides have been made since the 1980s in the research of other entry guidance methods andnumerical techniques beyond the Shuttle entry guidance.2–12 In particular the class of numerical predictor-corrector guidance algorithms2–11 has emerged to become a popular choice in various applications. In thiswork we propose a gliding guidance algorithm in this class for high L/D hypersonic vehicles. This approachrequires no pre-planned reference trajectory of any kind. A feasible gliding trajectory is generated byguidance algorithm based on the current condition and specified terminal condition which may be updatedin flight. The tasks of on-line trajectory planning and closed-loop guidance are performed with the samecore algorithm. There is no need for separate trajectory tracking law. The algorithm applies to a wide rangeof initial flight conditions, including any suborbital entry and orbital entry conditions. For the same vehicle,there is no need for any mission-dependent tuning of guidance parameters or gain scheduling.
A phenomenon unique to high L/D hypersonic vehicles as compared to medium L/D vehicles such as theShuttle is that the gliding trajectory of a high L/D vehicle is prone to having large phugoid oscillations. Theselarge oscillations tend to cause high heating and load stresses, and pose difficulty to the flight control systemrelying on aerodynamic control effectors. In this paper a simple but very effective technique is developed toeliminate the phugoid oscillations. This is accomplished by introducing in the trajectory control appropriatefeedback compensation proportional to the difference between the actual and a desired altitude rate, wherethe baseline trajectory control comes conveniently from the predictor-corrector guidance algorithm. Theresulting gliding trajectory is essentially in equilibrium glide for the most part.
Preliminary investigation of planning complete end-to-end missions from launch of the hypersonic vehicleby a multi-stage rocket to the termination of its gliding flight is conducted. This capability is enabledby combining an existing rapid ascent trajectory optimization tool and the ability to reliably and quicklygenerate feasible gliding trajectories afforded by the predictor-corrector algorithm. The findings stronglysuggest that the launch ascent for delivering a hypersonic gliding vehicle should take a much differenttrajectory than a typical ballistic launch ascent, in order to avoid inducing high peak heating and load onthe gliding vehicle during gliding phase.
II. Preliminary
The 3-dimensional equations of motion of a gliding vehicle over a spherical, rotating Earth in terms ofnon-dimensional variables are
r = V sin γ (1)
θ = V cos γ sin ψ
r cos φ(2)
φ = V cos γ cos ψ
r(3)
V = −D −(
sin γ
r2
)+ Ω2r cos φ(sin γ cos φ − cos γ sin φ cos ψ) (4)
γ = 1V
[L cos σ +
(V 2 − 1
r
) (cos γ
r
)+ 2ΩV cos φ sin ψ (5)
+ Ω2r cos φ(cos γ cos φ + sin γ cos ψ sin φ)]
ψ = 1V
[L sin σ
cos γ+ V 2
rcos γ sin ψ tan φ − 2ΩV (tan γ cos ψ cos φ − sin φ) (6)
+ Ω2r
cos γsin ψ sin φ cos φ
]
where r is the radial distance from the Earth center to the vehicle, θ and φ the longitude and latitude, V theEarth-relative velocity, γ the flight-path angle of the Earth-relative velocity vector, and ψ the heading angleof the same velocity vector, measured clockwise in the local horizontal plane from the north. In dimensionlessform, length is normalized by the equatorial radius of the Earth R0=6,378,135m, time normalized by tscale =√
R0/g0 where g0=9.81 m/s2. The choices of distance and time normalization leads to velocity to benormalized by Vscale=
√g0R0. The differentiations in above equations are with respect to the dimensionless
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time τ = t/tscale. The terms L and D are the aerodynamic lift and drag acceleration in g0, respectively. Thebank angle σ is the roll angle of the vehicle about the relative velocity vector, positive to the right. FinallyΩ is the (dimensionless) Earth self-rotation rate. Since the time is not a critical parameter in our problem,an energy-like variable e will be used as the independent variable in guidance algorithm development
e = 1r
− V 2
2(7)
It is clear that e defined above is the negative of the specific mechanical energy used in orbital mechanics.If the Earth-rotation term in Eq. (4) is ignored, it can be readily shown that
de
dτ= DV > 0 (8)
Therefore e is a monotonically increasing variable. Let x = (r θ φ γ ψ)T be the independent state vector,whereas the velocity V is determined by the values e and r from Eq. (7). The angle of attack α enters inL and D through the dependence on α by the lift and drag coefficients CL and CD. As in most hypersonicgliding guidance developments, the angle of attack profile is considered to be fixed as a given function ofMach number. This profile is determined by considerations in ranging capability (in downrange and/orcrossrange direction), thermal protection, and flight control. The primary means for trajectory control isthe bank angle σ which is to be determined by the guidance algorithm. The full 3DOF equations of motion(1)-(6), excluding Eq. (4), may be re-written with e as the independent variable
x′ =dx
de= f (x, σ), x(e0) = x0 (9)
The initial condition of the system (9) is taken to be the current state x0 at the current energy e0. Thetypical final constraints are that the trajectory reaches a position at a specified distance s∗
f (s∗f can be zero)
from the target location at specified final altitude r∗f and velocity V ∗
f . In other words
r(τf ) = r∗f (10)
V (τf ) = V ∗f (11)
s(θ(τf ), φ(τf )) = s∗f (12)
where s denotes the range to the location of final destination which is a function of longitude and latitude. Thefirst two conditions in Eqs. (10) and (11) may be combined to define a specified final energy ef = 1/r∗
f −V ∗f /2.
Thus the single terminal constraint iss(ef ) = s∗
f (13)
The problem of planning a feasible trajectory is to find the bank-angle profile so that the correspondingtrajectory of the system Eqs. (9) will satisfy the boundary conditions (on x(e0) and (13)). For simplicity ofpresentation trajectory inequality constraints are not included in this paper. An innovative and systematicapproach is developed most recently in Ref. 13 to enforce common inequality constraints in hypersonic glidesuch as those on heating rate and load factor. Whenever necessary, the method in Ref. 13 can be appliedconveniently together with the guidance method presented in this paper for inequality constraints.
III. Predictor-Corrector Guidance
The predictor-corrector guidance algorithm used to determine the magnitude of the bank angle profileis similar to the “longitudinal mode” approach presented in Ref. 5, with some meaningful differences. Let sdenote the range-to-go along the great circle connecting the current location of the vehicle and the site offinal destination. The differential equation that governs s is
s = ds
dτ= −V cos γ
r(14)
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When the Earth rotational effects are ignored, the longitudinal trajectory dynamics comprising of Eqs. (1),(5) and (14) with e in Eq. (7) as the independent variable are
ds
de= −cos γ
rD(15)
dr
de= sin γ
D(16)
dγ
de= 1
DV 2
[L cos σ +
(V 2 − 1
r
) (cos γ
r
)](17)
where V =√
2(1/r − e) is used wherever V is needed. The bank angle magnitude profile sought is parame-terize by a linear function of e:
|σ(e)| = σ0 + e − e0ef − e0
(σf − σ0) (18)
where σ0 ≥ 0 is a parameter to be found, and σf > 0 is a prescribed constant. Reference 5 contains adiscussion on how an appropriately large σf may be selected to reserve sufficient energy toward the end ofthe flight for accommodating unusually large uncertainties. If σf = σ0 is taken, then the parameterizedσ-profile is a constant.
In each guidance cycle, a constant σ0 is to be found so that the longitudinal trajectory of Eqs. (15)–(17)leading from the current condition to the final energy ef , under bank angle defined in Eq. (18), satisfies theterminal constraint
z(σ0) = s(ef ) − s∗f = 0 (19)
In Ref. 5 condition (19) is treated as a zero-finding problem. Here an alternate perspective is taken: we seekσ0 to minimize the error function
f(σ0) =12
z2(σ0) =12
[s(ef ) − s∗
f
]2 (20)
At the current e0 and for the current iterate σ(k)0 , the predictor step of the algorithm will numerically
integrate Eqs. (15)–(17) from the current state to ef , using σ in Eq. (18). The value of the error functionf(σ(k)
0 ) is evaluated. If the stopping criterion in Eq. (22) later is not yet met, the corrector step of thealgorithm seeks to update σ0 by a step-controlled Gauss-Newton method
σ(k+1)0 = σ
(k)0 − λk
∂f(σ(k)0 )/∂σ0
[∂z(σ(k)0 )/∂σ0]2
(21)
where step-size parameter λk = 1/2i, i is chosen to the smallest integer (including 0) such that f(σ(k+1)0 ) <
f(σ(k)0 ). The partial derivative in Eq. (23) is computed by finite difference. The predictor-corrector process
repeats until ∣∣∣∂f(σ(k+1)0 )
∂σ0
∣∣∣ =
∣∣∣∣∣z(σ(k+1)0 )∂z(σ(k+1)
0 )∂σ0
∣∣∣∣∣ ≤ ε (22)
for a pre-selected small ε > 0. Note that condition (22) is met when either |z| = |s(ef) − s∗f | is small, or
f = z2/2 reaches its minimum (so |∂f/∂σ0| is close to zero). The option of the latter is not in Ref. 5.This is a rather useful refinement in the case when the required range requirement cannot be met by the(insufficient) energy of the vehicle. In such a case this option will allow the algorithm to find a solution withthe closest approach to the destination. Note that the minimization of the cost in Eq. (20) is different fromfull-fledge trajectory optimization as in Ref. 14 which requires much more intensive computation.
A closer examination reveals that the update equation (21) can be simplified to
σ(k+1)0 = σ
(k)0 − λk
z(σ(k)0 )
∂z(σ(k)0 )/∂σ0
(23)
This equation is the same as the Newton-Raphson method applied to find the zero of Eq. (19)! This isbecause for a single univariate function, the Gauss-Newton method is exactly the same as the Newton-Raphson method. An immediate benefit of this realization is that we are assured that the Gauss-Newton
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iteration in (23) enjoys quadratic convergence as Newton-Raphson method does, which is not true for Gauss-Newton method in other cases. If the derivative in Eq. (23) is approximated by a secant scheme based onthe two consecutive previous iterates, the update becomes
σ(k+1)0 = σ
(k)0 − λk
z(σ(k)0 )[
z(σ(k)0 ) − z(σ(k−1)
0 )] (
σ(k)0 − σ
(k−1)0
)(24)
Similar to a secant scheme, iteration (24) will enjoy super-linear convergence. But unlike in a root-findingproblem, the interpretation of minimizing f(σ0) in Eq. (20) together with the stopping criterion in Eq. (22)offers a reassurance: that even in the case when the terminal constraint (19) cannot be met (because of ashortage in energy, for instance), the algorithm will always find a trajectory that ends the closest to theterminal condition in (19).
Once σ0 has been found, the magnitude of the current bank angle command σcmd will have the magnitudeof σ0. The sign of σcmd will be determined by a bank reversal logic similar to that used by the Shuttle.1Define the current heading off-set
δψ = ψ − Ψ (25)
where Ψ is the azimuth angle at the current location along the great circle connecting the current locationand the target location, and ψ current actual heading angle. A velocity-dependent deadzone Δazmth(V ) isdesigned. As soon as |δψ| increases from one direction and exceeds Δazmth(V ), the bank angle is commandedto change its sign to reduce |δψ|. An appropriate design of Δazmth(V ) should not require too many bankreversals for low-crossrange missions but still ensure success in large-crossrange missions. The specifics ofsuch a Δazmth(V ) will be dependent on the lifting capability of the vehicle and the specified terminal velocity.For the hypersonic gliding vehicle to be described in the next section and V ∗
f = 2000 m/s, the following isused: for V2 = V ∗
f + 1000 and V3 = V ∗f + 200 (m/s)
Δazmth =
⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩
10 deg, 6000 m/s ≤ V
18 deg, V2 < V < 6000 m/s
18 − 10(
V −V2V3−V2
)deg, V3 ≤ V ≤ V2
8 deg, V < V3
(26)
IV. Application to a High L/D Vehicle
The model for a generic hypersonic gliding vehicle called CAV-H in Ref. 15 is used in this paper. Figure 1shows the shape of the vehicle. In Fig. 2 the maximum L/D of the vehicle and the corresponding angle ofattack profile for the vehicle at different Mach numbers are plotted. It is seen that the vehicle has themaximum L/D ratios as high as 3.5 at hypersonic speeds. The angle of attack profile in the second subplotof Fig. 2 will serve as the nominal α-profile for all the numerical results in this paper.
The predictor-corrector guidance algorithm presented in the preceding section is applied to the CAV-Hvehicle. All the simulation results in this paper are closed-loop simulations, in that the guidance algorithmis called at 1 HZ rate to provide the guidance commands for the bank angle and angle of attack (which isthe nominal profile in Fig. 2) for trajectory simulations. The 3DOF equations of motion in Eqs. (1)–(6)are used. In addition, the rate and acceleration of the bank angle are limited by 10 deg/s and 5 deg/s2,respectively, so that no unrealistically rapid changes in the bank angle are allowed (such as instantaneousbank reversals). The nominal conditions for terminating the gliding phase are set to be at an altitude of 30km and velocity of 2000 m/s, when the vehicle is at a distance of 50 nm from the target.
Four different mission scenarios are set up with the same target. But the initial conditions differ in initialrange s0, altitude h0 and velocity V0. They are, respectively,
• Mission 4: V0 = 6400 m/s (21,000 ft/s), h0 = 50 km, s0 = 4000 nm;
Mission 1 may qualify to be an orbital entry case, whereas Missions 2-4 are definitively suborbital entry dueto the suborbital initial velocities and altitudes. The predictor-corrector guidance is called every second ifthe current condition is such that the total aerodynamic acceleration
√L2 + D2 ≥ 1.83 m/s2 (6 ft/s2); if
not, the dynamic pressure is too low for the aerodynamic control to be effective, so the bank angle commandis held to be the same as the current bank angle.
Figure 3 shows the altitude histories of the 4 missions. The ground tracks are plotted in Fig. 4, andthe corresponding bank angle histories in Fig. 5. The algorithm works equally well for all 4 missions. Thevehicle accurately reaches the required terminal distance to the target at the specified final energy level inall cases, despite the very different initial conditions. A phenomenon that clearly stands out in Fig. 3 is thephugoid oscillations in the altitude. Figure 6 illustrates the corresponding flight path angle profiles. Theseoscillations are present in all trajectories, even when the initial altitude is already relatively low. Such acharacteristic behavior will become progressively more pronounced when the L/D ratio of the vehicle getshigher. The immediate adverse effects such phugoid oscillations tend to induce include repeated high peakheating rate, as demonstrated in Fig. 7. Figure 7 also shows that the dynamic pressure in some cases oscillatesthrough cycles where at the bottom the dynamic pressure is too low for the aerodynamic control effectorto be effective. It should be stressed that these oscillations are not a necessary trait of the trajectories inorder to cover the required ranges. Rather, they are a result of the trajectory control (bank angle) not beingsynchronized with the altitude rate (so that the trajectory pulls too hard at the bottom of a dive). This isno fault of the guidance algorithm because such is not a part of the design objective in the parameterizationof the bank angle magnitude in the predictor-corrector algorithm. For lifting vehicles with medium to lowL/D ratios like the Shuttle, the persistent large phugoid oscillations as in Fig. 3 are not an issue.
0 1000 2000 3000 4000 5000 600020
30
40
50
60
70
80
90
100
110
120
130
range (nm)
altit
ude
(km
)
V0 = 7400 m/s, s
0 = 6000 nm (Mission 1)
V0 = 7010 m/s, s
0 = 6000 nm (Mission 2)
V0 = 7010 m/s, s
0 = 6000 nm (Mission 3)
V0 = 6400 m/s, s
0 = 4000 nm (Mission 4)
Figure 3. Altitude histories of Missions 1–4
V. Elimination of Phugoid Oscillations
The objective in this section is to develop a simple and effective method, within the framework of thepredictor-corrector guidance algorithm, to eliminate the large phugoid oscillations in hypersonic glidingtrajectories of high L/D vehicles. The idea is to introduce an appropriate feedback term proportional to thealtitude rate in the commanded vertical component of the aerodynamic lift force. To this end, let us considerthe equilibrium glide condition which is obtained by setting γ = 0 and ignoring the Earth-rotation terms inEq. (5)
L cos σ +(
V 2 − 1r
)cos γ
r= 0 (27)
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In the above equation, set σ = σQEGC = a constant (less than 90 deg), approximate r ≈ 1 and γ ≈ 0. Theresult may be called the quasi-equilibrium glide condition (QEGC)
L cos σQEGC +(V 2 − 1
)= 0 (28)
Differentiate Eq. (28) once using the expression L = R0Sref CDρV 2/2m (where the dimensional atmosphericdensity ρ is a function of the dimensionless r) and Eqs. (1) and (4) where the Earth-ration term is omitted.The lift coefficient CL is treated as a constant. The result gives rise to the required flight path angle to flythe QEGC in Eq. (28):
γQEGC = V 2 + L cos σQEGC
(V 2/2)(βr cos σQEGC)
(1
CL/CD
)(29)
where sin γQEGC ≈ γQEGC is used and
βr(r) = ∂ρ/∂r
ρ(30)
For the Earth the dimensionless parameter βr is in the range of −800 ∼ −1000 between the altitude of 60and 25 km where equilibrium glide can take place. If Eq. (28) is employed again to simplify the numeratorof Eq. (29), it reduces to
γQEGC = 1(V 2/2)(βr cos σQEGC)
(1
CL/CD
)(31)
For σQEGC = 0, the γQEGC in Eq. (31) is exactly the first-order solution for the motion in the vertical planefor shallow entry derived in Ref. 16. At high speeds, the values of γQEGC produced by Eqs. (29) and (31)are very close. They differ only at lower speeds. For our subsequent purpose, any of the two equations canbe used with equally good effect. Figure 8 compares the two histories of γQEGC computed from Eq. (29)and (31) along the same hypersonic gliding trajectory of CAV-H.
0 200 400 600 800 1000 1200 1400 1600 1800 2000−4
−3.5
−3
−2.5
−2
−1.5
−1
−0.5
0
γQ
EG
C (
deg)
time (sec)
γQEGC
from Eq. (28)
γQEGC
from Eq. (30)
Figure 8. γQEGC computed from Eq. (29) and Eq. (31) along the same hypersonic gliding trajectory
Define the altitude rate required to fly the QEGC by
hQEGC = V sin γQEGC (32)
where V is the current actual velocity. In each guidance cycle, denote the current bank angle computedby the predictor-corrector algorithm by σnom (as the nominal). The nominal vertical aerodynamic lift forcecomponent in the vertical direction will be L cos σnom where L is the current total lift force. The commandedvertical lift then is computed from
L cos σcmd = L cos σnom − k(h − hQEGC) (33)
where h is the current altitude rate from the navigation system. The gain k > 0 may be scheduled as alinear function of velocity
k =
⎧⎨⎩
k0 +(
V0−VV0−V1
)(k1 − k0), V0 ≤ V ≤ V1
0, V < V1
(34)
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where k0 > k1 ≥ 0, and V1 > V ∗f . In this work we take V1 = V ∗
f + 1000 m/s. The appropriate value fork0 can very easily determined by simulations for a given vehicle, and any k1 = 0 ∼ 0.5k0 is just fine. Theperformance is not sensitive to the selection of k1 at all (more on this later). The value of k1 = 20 (innondimensional form) is used in this paper. All the simulation results hereafter in this paper use Eq. (33)to compute the bank angle command.
The magnitude of the commanded bank angle σcmd is now computed from Eq. (33). The commandedbang angle now will steer the trajectory to fly with an altitude rate close to that of a quasi-equilibriumglide trajectory which will be free of large phugoid oscillations. The gradually diminishing gain designed inEq. (34) is based on the observation that the most important period in which the altitude rate compensationin Eq. (33) is needed is the initial phase of the gliding flight. Once the trajectory has entered a nearlyequilibrium glide for some time, it will not start the phugoid oscillations again. This is when the feedbackterm in Eq. (33) is no longer required and the trajectory control should focus on the ranging requirementwhich is the basis how σnom is generated. Note that the objective of the feedback term in Eq. (33) is not toclosely track hQEGC , but to provide sufficient trajectory damping. This is the reason why the selection ofthe value for k in Eq. (33) is not critical. Also, different choices of σQEGC in Eq. (29) or (31) do not havepractically significant influence on the actual trajectory, because the γQEGC computed is always small forσQEGC < 80 deg.
Missions 1-4 are re-executed under the bank control in Eq. (33). Figure 9 shows the altitude-vs-rangehistories. The sharp contrast between Figs. 3 and 9 stands out: the phugoid oscillations in all 4 missionsare now effectively damped out. The high terminal accuracy (not shown) remains the same. It is instructiveto look closely at the comparison of other aspects of the trajectories. Figure 10 compares the bank anglehistories of Mission 2, with and without the use of the compensation in Eq. (33). It is seen that themain difference occurs during the period when the vehicle first descends into the dense atmosphere. Thecompensation in Eq. (33) generates a larger bank angle magnitude in this period, so the trajectory pulls upless hard, resulting a smooth and monotonically decreasing altitude profile after the initial pull-up maneuveras seen in Fig. 9. Without the compensation in Eq. (33), the trajectory pulls up too high after the initialdive, and then continues the oscillatory cycle for the rest of the trajectory as the dash-dot line in Fig. 3depicts. The comparison of the flight path angle histories for Mission 2 is illustrated in Fig. 11 where γQEGC
from Eq. (31) is also plotted. As a result of Eq. (33), the actual flight path angle closely follows γQEGC afterthe initial dive and before the last part of the trajectory where the feedback term in Eq. (33) is eased out.The trajectory essentially flies on an equilibrium glide for most of the trajectory with a small negative flightpath angle. The small spikes in the flight path angle correspond to the bank reversals in Fig. 10 when thevertical lift component temporarily increases as the bank angle rotates through the attitude of zero bankangle.
Finally, flying on an equilibrium glide path significantly improves the heating rate and dynamic pressureprofiles as seen in Fig. 12. While the first peak heating rate is only reduced moderately because it islargely a function of the initial condition, the subsequent peaks are eliminated. The nearly flat profile of theheating rate after the peak is quite typical with the compensation in Eq. (33), and it suggests that imposingadditional heating rate inequality constraint would be ineffectual and unnecessary because the heating rateprofile already has close to the lowest possible peak value. Similarly, the dynamic pressure profile is nowmuch more suitable to render the aerodynamic control effectors effective throughout the trajectory after theinitial descent.
In Refs. 9 and 12, considerable efforts are spent in search for an automated and effective way to dampout the altitude pull-up sometimes present for an entry vehicle of Shuttle-class right after the descent fromthe entry interface for orbital entry. The approach presented in Eq. (33) is much simpler and should beequally effective in such a case (see the improvement made on the orbital entry case of Mission 1 in Fig. 9).
In Ref. 17 a much more tedious expression for an equilibrium glide flight path angle γEG is producedand the angle of attack is augmented with a feedback term proportional to (γ − γEQ) for reducing phugoidoscillations in two-dimensional (vertical plane) flight. If the vehicle flies an angle of attack near the valuefor maximum lift, this approach will fail, as there will be no room for further increase of the lift evenif the compensation calls for it. Since any practical hypersonic gliding/entry flight is three-dimensional,α-modulation (in a limited range) would provide just a fraction of intended compensation in the vertical
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without compensation in Eq. (32)with compensation in Eq. (32)
Figure 12. Heating rate, load and dynamic pressure histories of Missions 2, with the trajectory dampingcompensation in Eq. (33)
direction with a nonzero bank angle, hence will be very ineffective in general.
VI. Maximum Crossrange Capability
A distinct characteristic of the high L/D hypersonic gliding vehicles is their capability to cover largecrossranges. Flying large-crossrange missions can pose challenges to the guidance algorithm in two fronts: 1)some algorithms rely on the assumption of small crossranges as an underlying foundation; 2) large-crossrangemissions by definition push toward the boundary of the vehicle’s capability, thus demand high robustnessand reliability of the guidance algorithm. The guidance algorithm presented in this paper is tested againstsome large-crossrange missions.
For suborbital Mission 3, the initial heading angle is first varied to generate a large initial left crossrangeof 2300 nm, and then right crossrange of 2500 nm which is close to the maximum right crossrange possible forthe initial conditions and the vehicle capability. The guidance algorithm is applied without any adjustmentsspecific to any case, just to show how well the guidance algorithm still works in these stressful situations.For comparison a third case is selected where the initial crossrange is near zero.
Figure 13 shows the ground tracks of the 3 cases (keep in mind the deformation of the projection ofspherical curves in a Cartesian coordinate system in long distances). Notice that the major portion of thelarge left-crossrange trajectory flies along the 87-deg parallel. Plotted in Fig. 14 are the altitude and velocityhistories. The bank angle histories are given in Fig. 15. It is noted in Fig. 14 that the altitude actuallyincreases initially for the trajectory with large right crossrange. This is because of the initial zero bank anglethe trajectory flies for an extended period as can be seen in Fig. 15. From Fig. 15 the fact that there arebarely two bank reversals toward the very end for the large right-crossrange case suggests that it is indeedvery close to the maximum right-crossrange boundary for the vehicle. Conversely, the bank angle historyfor the trajectory with left-crossrange of 2300 nm indicates that this is not quite near the left-crossrangeboundary yet. Physically the vehicle can still fly even larger left crossrange. Judging from Fig. 13, thismeans that the trajectory would fly over the North Pole. Simulating such a case would require a change ofcoordinate system because of the singularity in Eq. (3) at φ = 90 deg.
If Mission 1 is tested for crossrange capability, not surprisingly the maximum crossrange will be evenlarger because of the higher energy (velocity mostly) in the orbital entry case. For instance, the maximumright-crossrange will reach 3000 nm. Other than that, the rest of the behaviors of the trajectories are similar
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Figure 13. Ground tracks for suborbital entry Mission 3, with minimum, large left and large right initialcrossranges
0 500 1000 1500 2000 2500 300020
30
40
50
60
70
time (sec)
altit
ude
(km
)
0 500 1000 1500 2000 2500 30001000
2000
3000
4000
5000
6000
7000
8000
time (sec)
V(m
/s)
min crossrangeleft crossrange=2300 nmright crossrange=2500 nm
Figure 14. Altitude and velocity for Mission 3, with minimum, large left and large right initial crossranges
VII. End-to-End Mission Planning
When the initial condition for a hypersonic gliding vehicle is from the ascent trajectory by a rocket launchvehicle, the end-to-end trajectory from launch lift-off to the termination of the gliding phase becomes anintegral part of the overall mission. As we shall see, the launch ascent will impact the gliding trajectory, mostgreatly not in the ranging capability (a somewhat surprising finding) but on the heating and load stresses.This section examines some of the interconnection between launch ascent and gliding flight.
For our purpose of generating complete trajectories from launch to termination of gliding phase, the
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Figure 15. Bank angle for Mission 3, with minimum, large left and large right initial crossranges
Minotaur IV launch vehicle model is used.18, 19 The Minotaur IV can have four stages, but for suborbitalmissions only the first 3 solid-rocket stages are employed in this paper. The payload is the CAV-H. Againall simulations are done in closed-loop guided mode in this section including the launch ascent. The launchascent guidance of the first stage utilizes the optimal endo-atmospheric ascent guidance algorithm in Refs. 20and 21. During the ascent of the first stage, a bending moment constraint is imposed on the trajectory
|qα| ≤ mαmax (35)
where q is the dynamic pressure, α the angle of attack, and mαmax > 0 a given constant. In the followingmαmax = 1500 psf-deg is used, unless stated otherwise. This constraint ensures that the launch vehicledoes not fly large α when dynamic pressure is high, so as not to induce excessive bending moment on thevehicle. Whether or not the constraint (35) is enforced changes the ascent trajectory, and the performance,significantly. More will be discussed on this point later.
The second and third stages are guided by the optimal exo-atmospheric ascent guidance algorithm inRefs. 22 and 23. After the burn-out of the third stage of the launch vehicle, the CAV-H in is free flight ifthe total aerodynamic acceleration is less than 1.83 m/s2. The closed-loop gliding guidance described in thepreceding sections takes over when the total aerodynamic acceleration exceeds 1.83 m/s2. The destinationlocation is the same as before. The range for hypersonic unpowered gliding is about 4000 nm.
The first case is a typical ballistic launch in that the launch vehicle burn-out occurs at an altitude of 90km with a flight path angle of 7 deg. The final Earth-relative velocity reached at that point is 6500 m/s. Asthe payload, CAV-H continues the climb to apogee at about 205 km in altitude after the burnout before itbegins the descent. The altitude and velocity profiles along the complete end-to-end trajectory are plottedin Fig 16. Because of the condition induced by loft ascent trajectory, the gliding vehicle dips deeply intothe atmosphere, reaching an altitude about 30 km at the bottom. There is little the unpowered glidingvehicle can do to reduce this deep dive, for it is dictated by the condition before the vehicle enters the denseatmosphere. At such a low altitude with a velocity of about 6500 m/s, the heating and dynamic load stresswill be tremendous as will be seen shortly.
The second case is an unconventional launch, called equilibrium-glide-insertion launch. The intention isto directly insert the gliding vehicle into an initial state of lower altitude and nearly zero flight path anglesuch that it can start hypersonic gliding in equilibrium glide right away. By doing so the vehicle will be ableto avoid the deep penetration of the atmosphere due to condition created by typical ballistic launch as seenin Fig. 16. At an altitude where the atmosphere density just begins to reach sufficient levels (say, 50∼60km), the equilibrium glide flight path angle can be readily estimated by Eq. (29). At such an altitude theaerodynamic lift acceleration L is still much smaller than V 2 (all non-dimensional). Therefore the term Lin the numerator of Eq. (29) may be ignored to get
γQEGC = 1(1/2)(βr cos σQEGC)
(1
CL/CD
)(36)
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This equation gives an adequate targeting condition for the flight path angle at the rocket burn-out for anygiven gliding vehicle and its nominal α-profile.
For case 2 of our test, the engine cut-off condition for the 3-stage Minotaur IV is set at 60 km in altitudeand a flight path angle of -0.1 deg. The final Earth-relative velocity attained at the 3rd stage burnout isabout 6400 m/s (compared to 6500 m/s for the ballistic launch in Fig. 16). Figure 17 shows the historiesof altitude and velocity along the complete trajectory. Since the rocket cut-off condition is conducive forsmooth gliding, the unpowered trajectory goes on equilibrium glide quickly. Notice the different scales inaltitude in the first subplot of Fig. 17 from that in the first subplot of Fig. 16 to realize that the trajectoryin Fig. 17 is a much shallower one.
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Figure 16. End-to-end mission case 1: typical ballistic launch followed by gliding flight
The comparison of altitude and velocity along the gliding trajectories from the two different launch modesis plotted in Fig. 18. The first halves of the trajectories cannot be more different, given that this is fromthe same launch vehicle, launch site, gliding vehicle, and terminal condition of glide. Yet it is interesting tonotice that the second halves of the trajectories are really close to each other, probably because of the sameterminal condition and the same vehicle. Figure 19 demonstrates the bank angle profiles during the glide.They are also rather similar in the second half of the flight. The first halves are very different. Following theballistic launch ascent, the unpowered CAV-H flies nearly half of its trajectory outside the dense atmosphere,where trajectory control by aerodynamic force is ineffective thus not performed (with bank angle set at zero).With equilibrium-glide-insertion launch, the gliding vehicle quickly captures the near equilibrium-glide pathand begins to utilize aerodynamic (bank-angle) control shortly after the rocket burnout. Although not apart of objective here, the latter trajectory is expected to be much more robust with respect to system andenvironment uncertainty. This is simply because the trajectory is under closed-loop guidance for almost allof the flight time, hence will be able to respond more effectively to larger dispersions.
The most dramatic difference, however, lies in the heating and aerodynamic loads in gliding. Figure 20
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Figure 17. End-to-end mission case 2: equilibrium-glide-insertion launch followed by gliding flight
compares the heating rate, load factor and dynamic pressure of the two gliding trajectories. The peakheating rate, load factor, and dynamic pressure encountered in the glide following the ballistic launch are,respectively, 3.9, 7.4 and 9.5 times of those following equilibrium-glide-insertion launch.
Figure 18. Altitude histories of the unpowered trajectories from ballistic launch and equilibrium-glide-insertionlaunch
Now the advantages of the equilibrium-glide-insertion launch for the subsequent hypersonic gliding havebeen demonstrated, more on such unconventional ascent of rockets should be discussed. One critical aspectfor achieving this type of low-altitude, high-velocity and nearly horizontal insertion condition is the safetyof the launch vehicle. For example, the bending moment constraint in Eq. (35) will have great impact onhow the ascent trajectory and the performance will be. For the same insertion altitude at 60 km and flightpath angle of -0.1 deg, Fig. 21 shows the 3 optimal ascent trajectories of the 3-stage Minotaur IV, two ofwhich are with the constraint (35) enforced with mαmax = 1500 and 4500 psf-deg, respectively, and onewithout. Without the constraint (35), the ascent trajectory has a monotonically increasing altitude profile,
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Figure 20. Heating rate, load factor and dynamic pressure histories of the unpowered trajectories from ballisticlaunch and equilibrium-glide-insertion launch
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and the final insertion Earth-relative velocity at the burnout reaches 7600 m/s. With the constraint (35)and mαmax = 1500 psf-deg, the first-stage burnout altitude is 10 km higher, and ascent trajectory reaches itsapex at 95 km in altitude at the burnout of the 2nd stage before it descends to the required final altitude of60 km. The final insertion velocity is about 6400 m/s. With the constraint (35) and mαmax = 4500 psf-deg,the apex altitude is just 69.3 km and the final insertion velocity is 7400 m/s. Figure 22 reveals the reason forthe differences: the magnitude of the angle of attack flown during the first stage without constraint (35) isabout 5-6 times that with constraint (35) and mαmax = 1500 psf-deg, and 2 times with mαmax = 4500 (Notethat the angle of attack is negative because of the low insertion altitude). Consequently the ascent trajectoryis more depressed without constraint (35), and the resulting peak magnitude of |qα| is about 12100 psf-deg,8 times of the constrained value of 1500 psf-deg and 2.7 times of 4500 psf-deg, as seen in the second subplotof Fig. 22.
Figures 21 and 22 clearly suggest clear trade-off between performance and constraints in the launchascent. Although not shown in the paper, a similar difference in terms of the final velocity is observed in theballistic launch case in Fig. 16. So such trade-off is not limited to equilibrium-glide-insertion ascent.
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Figure 21. Optimal launch ascent trajectories for equilibrium-glide insertion with and without the bendingmoment constraint in Eq. (35)
VIII. Conclusions
The predictor-corrector hypersonic gliding guidance algorithm proposed in this paper is able to reliablyand rapidly generate a feasible trajectory from the current condition to the specified final condition. The sim-plicity and robustness of the algorithm could allow its real-time applications for a wide range of vehicles. Forhigh L/D vehicles, the common phenomenon of large phugoid oscillations in the hypersonic gliding trajectoryis found to be undesirable from viewpoint of both thermal management and effective flight control. Thispaper develops a simple and very effective feedback compensation in conjunction with the predictor-correctorguidance algorithm to eliminate the phugoid oscillations. A unique capability demonstrated in this paperis the combination of a state-of-the-art rapid launch ascent optimization tool with the predictor-correctorgliding guidance algorithm to perform rapid end-to-end missions planning and trade studies, involving thelaunch by a multi-stage rocket, followed by hypersonic gliding to the designated destination.
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Figure 22. Variations of the 1st-stage angle of attack and the product |qα| for equilibrium-glide insertion launchwith and without the bending moment constraint in Eq. (35)
Acknowledgment
The author at Iowa State University gratefully acknowledges the support to this research by the SpaceVehicles Directorate, Air Force Research Laboratory under Contract FA9453-12-1-0239.
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References[1] Harpold, J. C., and Graves, C. A., “Shuttle Entry Guidance,” The Journal of the Astronautical Sciences,
Vol. XXXVII, No. 3, 1979, pp. 239-268.
[2] Gamble, J. D., Cerimele, C. J., Moore, T. E., and Higgins, J., “Atmospheric Guidance Concepts for anAeroassisted Flight Experiment,” The Journal of the Astronautical Sciences, Vol. 36, No. 1, 1988, pp.45-71.
[3] Braun, R. D., and Powell, R. W., “Predictor-Corrector Guidance Algorithm for Use in High-EnergyAerobraking System Studies,” Journal of Guidance, Control and Dynamics, Vol. 15, No. 3, 1992, pp.672-678.
[4] Fuhry, D. P., “Adaptive Atmospherics Reentry Guidance for the Kistler K-1 Orbital Vehicle,” AIAAPaper 99-4211, Aug. 1999.
[5] Lu, P., “Predictor-Corrector Entry Guidance for Low Lifting Vehicles,” Journal of Guidance Controland Dynamics, Vol. 31, No. 4, 2008, pp. 1067-1075.
[6] Putnam, Z. R., Braun, R. D., Bairstow, S. H., and Barton, G. H., “Improving Lunar Return EntryFootprint Using Enhanced Skip Trajectory Guidance,” AIAA Paper 2006-7438, Sept. 2006.
[7] Tigges, M. A., Crull, T., and Rea, J. R., “Numerical Skip-Entry Guidance,” AAS Paper 07-076, Feb.2007
[8] Brunner, C., and Lu, P.,“Skip Entry Trajectory Planning and Guidance”, Journal of Guidance, Control,and Dynamics, Vol., 31, No. 5, 2008, pp. 1210–1219.
[9] Xue, S., and Lu, P., “Constrained Predictor-Corrector Entry Guidance”, Journal of Guidance, Control,and Dynamics, Vol. 33, No. 4, 2010, pp. 1273–1280.
[10] Brunner, C., and Lu, P.,“Comparison of Fully Numerical Predictor-Corrector and Apollo Skip EntryGuidance Algorithms”, to appear in The Journal of the Astronautical Sciences, 2013
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[12] Shen, Z., and Lu, P., “On-Board Generation of Three-Dimensional Constrained Entry Trajectories”,Journal of Guidance, Control, and Dynamics, Vol. 26, No. 1, 2003, pp. 111–121.
[13] Lu, P., “Entry Guidance: A Unified Method”, accepted for publication in Journal of Guidance, Control,and Dynamics, 2013
[14] Jorris, T. R., Common Aero Vehicle Autonomous Reentry Trajectory Optimization Satisfying Waypointand No-Fly Zone Constraints, Ph. D. dissertation, Air Force Institute of Technology, 2007
[15] Phillips, T. H., A Common Aero Vehicle (CAV) Model, Description, and Employment Guide. TechnicalReport, Schafer Corporation for AFRL and AF- SPC, 27 January 2003.
[16] Vinh, N. X., Busemann, A., and Culp, R. D., Hypersonic and Planetary Entry Flight Mechanics, TheUniversity of Michigan Press, Ann Arbor, MI, 1980, Chapter 7.
[17] Yu, W., and Chen, W., “Guidance Scheme for Glide Range Maximization of a Hypersonic Vehicle”,AIAA Paper 2011-6714, Aug., 2011.
[18] “Minotaur 4 Data Sheet”, Space Launch Report, http://www.spacelaunchreport.com/mintaur4.html
[19] “Minotaur IV Space Launch Vehicle”,http://www.orbital.com/NewsInfo/Publications/Minotaur IV Fact.pdf
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