NASA Technical Reports Server (NTRS) 20000060812: Guidance and Control for Mars Atmospheric Entry: Adaptivity and Robustness

6 Pages·0.35 MB·English

by  NASA Technical Reports Server (NTRS)

GUIDANCE AND CONTROL FOR MARS
ATMOSPHERIC ENTRY: ADAPTIVITY AND
ROBUSTNESS 1
Wei-Min Lu* and David S. Bayard **
"AMRL, AB5/_82, IBM Corporation, San Jose, CA 95193;
Bmail: t_nluOu_.ib'm.com
"*JPL 198-326, CaITech, Padadena, CA 91109
Abstract: In this paper, we address the atmospheric entry guidance and control
problem for Mars precision landing. The guidance and control design is based on
the principle of tracking a reference drag versus velocity profile in the entry flight
corridor, which is determined by physical constraints of the flight. An integrated
adaptive/robust control approach to atmospheric entry guidance and control is
introduced to deal with different uncertainties.
Keywords: Aerospace Systems, Adaptive Control,Guidance and Control,Nonlinear
Systems, PlanetaryEntry,Robust Control
I. INTRODUCTION winds/gusts, air density dispersions, and measure-
ment and knowledge errors; in addition, some sim-
To facilitate guidance, navigation, and control de- plification assumption on mathematical model made
sign for Mars precision landing, the Mars landing to facilitate the control design also generates a gap
process is divided into three phases, i.e., atmo- the real system and the employed model, which is
spheric entry phase, parachute phase, and terminal convenientlytreatedas uncertainty.Basically,the
landing phase. In this study, we only focus on guid- above uncertaintiesareclassifieadsexternaldistur-
ance and control design for the entry phase, i.e., bances and parametricuncertainty.
the phase from atmospheric entry of Mars orbiter
In thispaper, we willaddress the guidance and
to parachute deployment. Feedback aerodynamic
controlproblem from the view point of adaptive
maneuvering during the entry phase is used to re-
and robust controldesign.The followingquestions
duce the error to a required level at the parachute
willbe answered to provide an integrateddesign
deployment. Hindering the achievement of this goal
method: givenan a priorierrorbound, does there
are uncertainties in the guidance and control sys-
existany guidance and controllaw to achieve an
tems. The main sources of the uncertainties include
errorlevelnot greaterthan the givenbound under
alladmissibledisturbances and vehicleevolution
I Pa_-ti=lly supported by IBM and NASA; part of the work parameter changes,and how does one design such
wLs done when the first author was with JPL
a control law? In particular, we consider the lon- main idea for the guidance strategy is to control
gitudinal landing guidance following McEneaney- the vehicle to follow a given drag-velocity profile.
Mease's approach [6]. The objective is to achieve
a certain specified range during entry by control of
the lift/drag ratio component in position-velocity 2.1 Equations of Motion
plane, which is achieved by modulation of the Mars
The equationsoftranslationalmotion used inthis
landing vehicle's angle of attack and/or bank angle.
paper aredevelopedina coordinatesystem withone
The guidance approach is based on the principle
axisorientedalongtheMars-relativevelocityvector,
of tracking a reference drag-velocity profile in the
one axisperpendicularto the plane formed by the
entry flight corridor, which is determined in the
positionand Mars relativevelocityvectors,and
drag-velocity plane by physical constraints of the
thethirdaxiscompletingtheright-handcoordinate
flight; those constraints include the limitations of
system.The Mars-relativetranslationalstateofthe
the thermal protection system surface temperatures,
landing vehicleisrepresented by the variablesR
normal load factor, dynamic pressure, and equilib-
(range),h (altitude),V (velocityrelativeto the
rium glide flight (see [2] for details). A guidance
Mars), 7 (flight path angle), and _ (heading angle).
approach based on drag reference tracking has been
used in the NASA Space Shuttle re-entry guidance These equations of motion are as follows [10]:
law [2,3].Basing guidance and controllaw on drag i/= V cos_/ (1)
is advantageous because drag is essentially directly
h= vsixn (2)
measured by the accelerometer, and the range to be
flown during entry can be decided by the drag ac-
II= -D - gsinv (3)
celeration profile maintained throughout the entry
flight and can be predicted using simple analytical 1 .(V 2
techniques. "_= _ ( _ - g) cos 7 -t-L cos a) (4)
1 V 2
There exists several guidance and control techniques
¢- vJo .i( ; cos2 7 tan _bsin ¢ + L sin a) (5)
under ideal conditions, i.e., assuming perfect mea-
surement and no uncertainty [2,6,7], but without
guarantee for error tolerance level due to distur- where a is the bank (or roll) angle and _bis vehicle
bances. In this paper, basing on previous guidance Mars-relative latitude, r the distance between the
and control results, we will develop an integrated lander and the center of Mars, g = /_/r 2 with
guidance and control law to minimize the impact /_ the Martian GM is the Martian gravitational
of uncertainties. This 2-part paper is organized as acceleration. These equations of motion neglect the
follows: in section 2, an uncertain drag dynamical Coriolis and centripetal accelerations due to the
model is derived from the basic equations of motion Mars' rotation because they are relatively small. If
with the consideration of atmospheric disturbances the relative motion of the atmosphere to Mars is
and some operating parameter variations; the guid- ignored, the specific drag and lift are given by the
ance and control problem for Mars atmospheric following:
entry is formulated as a problem of tracking a drag-
1 S,., V2 1 S,., V2
velocity profile. In section 3, the guidance and con- O = _pmt_D ;L = _pmt_L , (6)
trol problem is solved as an adaptive disturbance
rejection problem where the parametric uncertainty where CD and CL are drag and liftcoefficients,
and atmospheric disturbances are treated separately respectively;they change with the change of the
so as to improve system performance. angleofattackand the magnitude of the velocity.
An exponentialatmospheric densityp model isused
inthefollowing.
2. UNCERTAIN DYNAMICS OF MARS
ATMOSPHERIC ENTRY
2.2 Drag Reference and Range Prediction
The objective of the longitudinal guidance is to For the Mars atmospheric entry, we employ a drag
control the range to be flown during entry, which profile in the drag acceleration / Mars-relative ve-
is a function of the drag-velocity profile [2]. The locity plane. The drag profile is chosen to stay
within the entry flight corridor due to some physical of the atmospheric uncertainties, including surface
constraints and to minimize the accumulate aerody- winds/gusts and atmospheric density dispersions.
namic heat load (see [2] for details). A typical drag
profile, Dr = f(V), is shown in Fig. 1, composed
Impact o/ Atmospheric Uncertainties
Co_r_m Din#
, The atmospheric uncertaintiesconsidered in the
followingincludethe airdensitydispersionand the
1, winds/gusts.When uncertaintiesare presented,D
!
and L intheequationsofmotion are not drag and
liftr,athertheyaretheaerodynamic forcesparallel
and perpendicularinthe plane ofsymmetry to the
Equlllb_um Glide _ OmJdma¢ Din8
vehiclevelocityvectorrelativetoMars, respectively.
The airflowvelocitydue to winds and gustscan
be representedas Vw = V + AV, where V isthe
velocityof the landing vehicle relativeto Mars,
I , I , I . I
I 3 3 4 AV is thedisturbancedue to the winds. The drag
and liftcoefllcientcshange with the change of the
angleofattackaswellasthemagnitude ofvelocity.
Fig. 1. Typical Drag-Velocity Profile
The influenceofthe surfacewinds/gusts,which are
viewed as disturbances,can alsocontributeto the
of a quadratic drag segment, an equilibriumglide
changesinspecificdragand liftcoefficientsi,nwhich
(linear)segment, and a constantdragsegment [2,6].
case:
Each ofthesegments istakenasageneralquadratic
profile:f(V) = Co+ ciV + c2V2.The objectivefor co = +aco,
CL = COL "t- ACL
the entryguidance law istoachieveadesiredrange
ata certainspecifiedvelocity.Infact,therangecan
where _ and _L arethe actualdrag and liftcoef-
be approximated by thefollowingrelation:
ficientsa,nd ACt) and ACL are the corresponding
v vl deviationsdue tothe above treatment.
R(v)= - V dV_- -vqsdV The atmospheric density can be modeled as the
Vo Vo idealexponentialdensitymodel (8)plusuncertain
dispersionAp (see Fig. 2), i.e.,it can be well
at the earlier stage of the flight as the flight path representedby thefollowing:
angle is small. During the latter part of the entry
p= poe-_ + Ap. (7)
when the landing vehicle is at lower speed, one can
use an energy variable E = gh + ½V2 to replace
the velocity variable V in the drag expression [2]. Io"
Therefore, the range to be flown is determined by
the drag-velocity (or drag-energy) profile. In the
10e
following, the control and guidance law is designed
so that the actual drag will track the given drag-
velocity profile.
2.3 Drsg Dlmamics with Uncertainty
I01
To facilitate the control design for entry guid-
ance based on the drag-velocity profile, one should
_0_
directly consider the drag dynamics, instead of ioI IOt loI io_
(rural
the dynamical equations considered earlier in (1)-
(5). We are particularly interested in the impact Fig.2.Mars Atmospheric Density Dispersion
The influenceoftheatmospheric uncertaintiescon-
tributesto thechanges inspecificdragand lifti,n = -H( '_ DD2z + 2VD + 2_D--2_- C - 6,).(11)
which case:D = Do + AD and L = Lo + AL, where
Do and L0 aredrag and liftwithout disturbances, From (I0),one can solve h; replace the h and
and AD and AL arethe resultingperturbations. defined by (9) in (II),one has the following
equation:
From thedensitymodel (7),one has that
b +go(D,£),V) +gl(D,£),V,u)O
p- Ap H' = w + g2(D, D, V)u, (12)
for some smooth functions go, g,, and g_ of/),D,
or _ = -_ + 6d for some 6d.
and V, where 0 is a parameter vector defined by
From the drag formulation, one has
-_ = -_ "_'D= - -_ + 2p + C + 6'_"(8)
0= 1-;cos _7 ; (13)
1-cos7
sina 7
where C := L_° and $. =
w is the combined disturbances:
,Sd+2 + (CO+DACD)OOD
49H D 2
w = V(V2 +2gH)6a +Dis;
Uncertain Drag Dynamical Model On the other hand, from (3) and (2), one has
Now we are ready to derive the drag dynamics lk =-D- gsin7 =-D- 9"--h
v
for control design. The control of the translational
motion is performed by adjusting u := _cosa,
which can be achieved by the modulation of either = _o(D, D, V) + _, (D, D, V)C + _2(D,/), V)6a(14)
angle of attack or the bank angle. =: _(D,/), V, C, 6a),
In the following, we assume the gravity acceleration
g = #/f with f being the .reference radius is for smooth functions _o, _,, and _ of D,/), V, C, and
constant. The actual variation of g can be taken 6a.
as a part of the disturbance to be discussed later. The above equations give a model for drag dynamics
From the equation (2), (3), and (4), one has with uncertainty. Note that the model (12)-(14) is
fairly general to cover many other sources of un-
d . certainties. In particular, the measurement noises,
= _(Vsm7)= VcosT_+ 1?sin7 estimation errors, as well as the gravitational ac-
celeration variations can be conveniently considered
V2 h __
= ('7 - g) - D_ - (1 - rcos27) + Dcosvu.(9_s part of the combined disturbance w. If we take
r x, = D, x2 = D, and z3 = V, then complete
dynamics governing the drag evolution is as follows:
With the consideration of atmospheric uncertain-
ties, from (8), one has
•_2= -go(Z1,z2,z3) - gl(=l, z2,z3,u)O
+w + g2(xl, z2, x3)u (15)
(1+ -V-_-)•h2=gH-H(_---+ 2-_ - 6'- 6,). (10)
z•,_3== Z_2(zL,z2,z3,C,_,,)
Z = 21
Take derivativeson both sides,one has
In the following, we will use the above dynamical
(1 + 2gT+_VH)2h 6_AgHD" + 4g2H7s_i-n_2 equation to design a guidance law.
3. GUIDANCE AND CONTROL DESIGN
s
I D ÷ •
3.1 Guidance as a Tracking Problem with Disturbance ,,
Rejection
As the drag profile is given by the range-velocity
requirement, to reach certain range requirement, it
is sufficient for the drag governed by the dynamics
i
(12)-(14) to follow the corresponding drag-velocity i
i
profile in the presence of disturbances. Therefore, l '
theguidance problem can be formulatedasa track- P I
L V"
ingproblem with disturbancerejection.
Guidance Problem: Consider the uncertain sys- tJ_ L_ _
tem (15), suppose 0, E O, where O is a convex dosed
bounded admissible parameter set. Given a drag-
velocity profile, Dr =/(V); design a feedback con- Fig. 3. Adaptive Disturbance Rejection
troller with measurement of D, D, and V, as well as
zt = _:l - D,. = e, z2 = :2 - br;
the profile information, such that for all admi_ble
energy-bounded unknown parameters, D governed
then equation (15) can be represented as:
by (12) satisfies the following requirements:
• Tracking:. if w = 0,
13= ((z_ + D_, z2 + b_, x3,6', 6o) (17)
lira i z = Az+ Bid+ B2v
t_oo{D(t) -/(V(t))} = 0. e=zl
• Disturbance Rejection:
T T where A = 0
f liD(t) - D_(t)ll 2dt < A2f IIw(t)ll2dt + e [1 0]. The above (error) system is input-output
linear with linear party state z; we will use a linear
o o
technique to derive the control v.
forgivene> 0, and some A > 0.
In the followingsection,we willprovide an inte-
grated adaptive/robust controlapproach to deal 3.3 74oo-Control Design
with the above problem based on the dynamical
equation(15).The controlarchitectureisillustrated Consider the linear part of the equation (17) with
in Fig. 3. The techniques of linearization,74oo- state z. Note that the error system is independent
control,and dissipationtheory-basedadaptivecon- of the state variable zs = V and observable from
trolwill be used ([5]). output e = 11. Suppose P > 0 is a solution to the
Riccati equation for the state-feedback 74oo-control
problem:
3.2 Feedback Linearization
PA + ATp + P(LB_B_ - B2Br2 )P + CTC O.
A"
The startingpointforthisproblem isto designa
controllerforthedisturbancerejectionproblem with
If A > 1, then P > 0. In this case, the 74oo-control
assumption thattheparameter 8isknown. Consider
solution gives feedback controller:
system (15),take
v = Fz := -B_Pz.
go(z_, 12,13) + g°(x_, 12,:rs)_ + v (16) 2
(1 + 8s)g_(z_, 12,13)
with foT(lle(t)ll2)dt < A2foT(lld(t)ll_)dt. One can
recover the following controller:
where v is the new input variable. Let z = z2 '
where a = K(O;D_,Dr,D,,zt,x2,z3)
Fz - b_ + go(zl, z2,xs) + _°(zl, z2,x.) Therefore,if one chooses
°_
(1+ 0_)_(xl,=2,z3) =_Ip,Q-_[0-g_(_1_,,_.,_)]Pz),(20)
toachievethe disturbancerejection.
where 7rQ is the vector projection (see [5]), then
(p- o)_'Q(p+ Q-' [0 oT(_,_,_, _)] P_) <o.
3.4 Adaptive Control Law
Thus, we have
However, as 0 isnot known, theabove controllaw T T
can not be directlyused.One can usean estimatep
/liell2 dt ___A2/(llwll2)dt+e.
of0 instead,and the controllaw would be
0 o
u = K(p;i)r,br, Dr,Zl,Z2,Z3) (18)
Let's look at the tracking issue if w = 0. We still use
W(z,p) as the Lyapunov function. Then with the
where p is given by an update law:
above adaptive law, one has P_(z,p) < -[lel[ 2 _<O.
Therefore, z and p- 0 are bounded. It can be shown
that fo Ilell2_ < oo, and e(t)-*O as t-too.
Therefore, the guidance and control system has the Therefore, the adaptive control law given by (18)-
structure as illustrated in Fig. 3. The error equation (20) achieves adaptive disturbance rejection.
can be equivalently represented as:
ACKNOWLEDGEMENT: The authors would like
to thank D. Farless, and Drs. D. Boussalis, K.
0 Mease, and R. Smith for their helpful inputs.
+ gl(zl, x2,x3,u) (p- O)
_3 = _(zx + D., z2 + b_, z., C,6o)
4. REFERENCES
e=zl
[1] D. Boussalis, "Investigation of Longitudinal
where F is a state feedback matrix defined as F = Motion of Low-Lift Entry Vehicles," JPL Engi.
_1B 2TP and P satisfies the following equation: Memo., No.3456-96-002, JPL, 1996.
[2] J.C. Harpold and C.A. Graves, Jr., J. of Astro.
P(A + B2F) ÷ (A + B_F)Tp
Sciences, Vol.27(3), pp.239-268, 1979.
[3] J.C. Harpold and D. E. Gavert, AIAA J. Guid-
+_PB1BTp + cTc = O.
ance, Vol.6(6), pp.442--447, 1983.
If V(z) = zTPz, then completion of square implies [4] R.N. Ingoldby, AIAA J. Guidance and Control,
Vol.l(3), pp.189-196, 1978.
_((A + B2F)z + Blw) <_As llwll2- Ilell2(19) [5] W.-M. Lu and A. Packard, "Adaptive Hog-
Control of Nonlinear Uncertainty Systems,"
Proc. 1996 ACC, Albuquerque, NM, 1997.
To design the update law, we take the following
[6] W.M. McEneaney and K.D. Mease, J. of Astro.
functionas a storagefunction[5]:
Sciences, Vol.39(4), pp.423-445, 1991.
W(z,p) = V (z) + (p - O)r Q(p - e) [7] K.D. Mease and J-P. Kremer, AIAA J. Guid-
ance, Contr.. Dynamics, Vol.17(6), pp.1350-
1356, 1994.
where Q > 0 such that
[8] R. Smith, "Closed-Loop Aeromaneuvering for a
max {(p- O)rQ(p - 0)} < e. Mars Precision Landing," CCEC Tech. Report,
p,OEO No.CCEC-96-0205, UCSB, 1996.
[9] D.G. Tuckness, AIAA J. Spacecraft and Rock-
Therefore,by completion ofsquare,one has
ets, Vol.32(1), pp.142-148, 1995.
[10] N.X. Vinh, A. Busemann, and R.D. Culp, Hy-
W(z,p) <__2ilwll2_ ilell=+
personic and Planetary Entry Flight Mechanics,
2(p- O)TQ(p+Q-' [ogr(_, _, _.,.) ] Pz) Ann Arbor: University of Michigan Press, 1980.