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# JOURNAL OF GUIDANCE, CONTROL, AND DYNAMICS

## Vol. 28, No. 2, MarchApril 2005

Control of Hovering Spacecraft Near Small Bodies:
Application to Asteroid 25143 Itokawa
Stephen B. Broschart

## University of Michigan, Ann Arbor, Michigan 48109-2140

The small gravitational forces associated with a minor celestial body, such as an asteroid or comet nucleus, allow
visiting spacecraft to implement active control strategies to improve maneuverability about the body. One form of
active control is hovering, where the nominal accelerations on the spacecraft are canceled by a nearly continuous
control thrust. We investigate the stability of realistic hovering control laws in the body-xed and inertial reference
frames. Two implementations of the body-xed hovering thrust solution are numerically tested and compared
with an analytical stability result presented in the previous literature. The perturbation equations for inertial
frame hovering are presented and numerically simulated to identify regions of stability. We nd that body-xed
hovering can be made stable inside a region roughly approximated by the bodys resonance radius and inertial
frame hovering is stable in all regions outside the resonance radius. Acase study of hovering above Asteroid (25143)
Itokawa, target of the Hayabusa mission, is also presented.
Introduction
A
T the Japanese, European, and U.S. space agencies, missions
currently exist to send spacecraft to explore minor celestial ob-
jects such as asteroids, comets, and small planetary satellites. It is
suspected that the elemental composition of these objects, relatively
uncorrupted by time, may offer scientists insight into the early his-
tory of our solar system. Thus, successful missions to such objects
are of critical importance to the ongoing goal of understanding the
formation and evolution of our solar system.
These objects are generally quite small compared to the planets
and moons that have been the targets of most space missions to
date. Because of their small size and correspondingly weak grav-
itational pull, they are also usually quite irregular in shape. These
attributes present great challenges to the traditional orbiter-type
control strategies used to date. Orbital trajectories about small, irreg-
ularly shaped bodies are generally quite complex and nonperiodic.
1
Further, stability for close orbits above many small bodies is guar-
anteed only for a limited range of latitudes,
2
possibly preventing
study of relevant regions of the bodys surface. Also, because the
gravitational forces are small, solar radiation pressure becomes an
important perturbing force, driving orbits to instability for objects
smaller than a particular size.
3
These problems with orbital control strategies for missions to
small bodies necessitate a paradigm shift in the way we control
spacecraft. One solution that has been proposed is hovering.
4,5
This
active control strategy uses thrusters in a nearly continuous manner
to null the nominal accelerations on the spacecraft, creating an equi-
librium point at the desired position. This approach is feasible for
operating near a small body because of the relatively weak gravita-
tional forces involved. Unlike orbital control approaches, hovering
control can be used to explore any size of body at any latitude, lim-
ited only by fuel restrictions. The perturbing forces of solar radiation
pressure and solar gravity are also less of a concern in hovering, be-
cause these forces can be actively nulled by the hovering controller.
publication 12 January 2004. Copyright c 2004 by Stephen B. Broschart.
with permission. Copies of this paper may be made for personal or internal
use, on condition that the copier pay the \$10.00 per-copy fee to the Copyright
Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923; include
the code 0731-5090/05 \$10.00 in correspondence with the CCC.

## Graduate Student, 2016 FXB, Department of Aerospace Engineering;

sbroscha@umich.edu.

## Associate Professor, 3048 FXB, Department of Aerospace Engineering;

scheeres@umich.edu. Associate Fellow AIAA.
In this paper, we investigate two types of hovering: hovering in
the small-body-xed frame and in the inertial frame. Hovering in
the body-xed frame would be ideal for a spacecraft operating very
close to the bodys surface. Because body-xed hovering xes the
spacecrafts position relative to the body, this control could be ad-
vantageous for obtaining high-resolution measurements of a partic-
ular area on the bodys surface. Body-xed hovering also simplies
descent and ascent maneuvers that may be necessary for a sample
return mission. Aspacecraft visiting a small body may also be inter-
ested in hovering in the inertial frame, where the small body rotates
beneath the spacecraft. Inertial-frame hovering may be ideal for a
spacecraft mapping a bodys surface or for use as a holding orbit
from which to stage other maneuvers. For hovering to be a viable
mission option, it must be shown to be stable and robust over a range
of situations and implementable under realistic control constraints.
This paper uses analytic and numerical analysis to identify regions
of stable operation for a spacecraft subject to hovering control.
We begin by formulating the solutions of the two-body problem
corresponding to body-xed and inertial-frame hovering. Next, we
present criteria for stability of body-xed hovering developed in a
previous paper.
5
This is followed by a discussion of the numerical
simulationtool we have developedtoaidour study. We thencompare
results of numerical simulation for two different implementations
of body-xed hovering with the predictions of the analytical stabil-
ity criteria. Next, we present analytical and numerical analysis of
hovering in the inertial frame. Finally, we perform a case study of
both hovering approaches for Asteroid (25143) Itokawa (formerly
1998 SF36), the target of the Japan Aerospace Exploration Agency
(JAXA) Hayabusa (formerly MUSES-C) mission.
6,7
Problem Formulation
In this analysis, we model the spacecraft dynamics near a small
body using a two-body gravitational model in which the spacecraft
has negligible mass. The equations of motion for this system in
the uniformly rotating, small-body-xed Cartesian coordinate frame
with origin at the bodys center of mass are
x 2 y =
V
x
+ T
x
+
2
x (1)
y +2 x =
V
y
+ T
y
+
2
y (2)
z =
V
z
+ T
z
(3)
343
344 BROSCHART AND SCHEERES
where represents the rotation rate of the small body, V is its grav-
itational potential, and T
x
, T
y
, and T
z
represent the component of
the spacecraft thrust in each respective direction. It is assumed that
the small body has constant density and is rotating uniformly about
a xed axis ( z) corresponding to its maximum moment of inertia.
We further orient our coordinate frame by aligning x and y with
the principal axes associated with the bodys minimum and inter-
mediate moments, respectively. A position in this space, [x, y, z]
T
,
is denoted by the vector r. The only forces present in this system
are gravitational forces, inertial forces, and forces from the space-
crafts propulsion system. In this paper, we ignore third-body forces
of smaller magnitude, such as solar radiation pressure and solar
gravity. These additional perturbations will be considered in future
work.
We investigate two solutions to these equations. The rst solu-
tion, which we call body-xed hovering, creates a xed equilibrium
point in the body-xed frame at the desired position, or hovering
point, by enabling thrust. Under this control, the spacecrafts posi-
tion remains xed relative to the rotating small-body when placed
at the nominal hovering point, r
0
=[x
0
, y
0
, z
0
]
T
. Enabling thrust is
dened as
T =
_

V
x

2
x
0
,
V
y

2
y
0
,
V
z

0
_
T
(4)
The second solution to Eqs. (13) that we study is a retrograde,
constant-latitude, circular orbit whose orbital period is equal to the
small bodys rotation period. This solution corresponds to a space-
craft hovering in the inertial reference frame, where the spacecraft
position remains xed in inertial space while the small body rotates
beneath it. In the body-xed frame, this solution is specied as
r
0I
(t ) = r
0I
(0)[cos
0
cos(t +
0
), cos
0
sin(t +
0
), sin
0
]
T
(5)
where
0
and
0
are the initial latitude and longitude (measured from
the positive x axis), respectively. For inertial hovering, the necessary
thrust vector is dened as that necessary to make this circular orbit
a solution to the equations of motion:
T(t ) =
_

V
x

r
0I
(t )
,
V
y

r
0I
(t )
,
V
z

r
0I
(t )
_
T
(6)
Previous Work
Sawai et al.
5
developed a set of criteria analytically that charac-
terize the stability of a hovering trajectory in the body-xed frame.
Through their analysis of the problem, they determined that the
Hessian matrix of the gravitational potential at r
0
determines the
stability of hovering in the body-xed frame subject to their ideal-
ized controller:

2
V
r
2

0
=
_
_
_
_
_
_
_
_
_

2
V
x
2

2
V
xy

2
V
xz

2
V
xy

2
V
y
2

2
V
yz

2
V
xz

2
V
yz

2
V
z
2
_

0
(7)
The conditions for stability they developed are
3
2
v
2
3z
+
2

1
+
2
_

2
+
2
_
0 (8)
_

1
+
2
__

2
+
2
_

2
v
2
1z
_

2
+
2
_

2
v
2
2z
_

1
+
2
_
0
(9)
_

2
_
2
+3
4
v
2
3z
_
v
2
3z
+2
_
8
2
v
2
3z
__

1
+
2
_
+
_

2
+
2
__
> 0 (10)
where
1
,
2
,
3
are the three eigenvalues of the Hessian matrix
at r
0
and v
1
, v
2
, v
3
are the corresponding eigenvectors (subscripts
x, y, and z refer to the components of these eigenvectors). The
eigenvectors and eigenvalues are arranged so that the third set refers
to the eigenvector nearly aligned with the gravitational attraction
direction. Of the remaining two, the eigenvector/eigenvalue pair
with the larger eigenvalue is the rst set.
These stability criteria were developed by linearizing the equa-
0
and assuming one-dimensional, innitely
tight control of motion along v
3
, such that r(t )
T
v
3
0. These three
criteria dene a theoretical region of stability for body-xed hov-
ering above any arbitrary-shaped body. This region roughly cor-
responds to the locus of initial hovering points inside the bodys
r
is dened as the distance
from the bodys rotation axis in the equatorial plane at which, for
a spherical body with the same gravitational parameter , the cen-
trifugal force is equal and opposite to the gravitational attraction:
r
r
= (/
2
)
1
3
(11)
Illustrations of the region dened by these sufciency criteria for
various small-bodies can be seen in Ref. 5.
The assumption of innitely tight control is not realistic, as it is
impossible for an actual spacecrafts control system to implement.
However, similar control can be implemented by use of a dead-band
thrust control on altitude, in which the spacecrafts movement is
conned to a nite region around a target altitude. Allowing motion
across the region dened by such a dead-band may have signi-
cant effects on system dynamics. Thus, an analysis using the full
nonlinear equations of motion is necessary to validate these results.
We further the previous work by numerically simulating the non-
linear equations of motion to determine the stability of hovering
under realistic control constraints. We compare and contrast these
results with the previous ndings to determine in which regions the
analytical stability criteria produce valid results.
Numerical Simulation Tool
Our numerical simulation package was developed to aid us in
the analysis of spacecraft trajectories near arbitrary-shaped rotating
bodies. The basic function of the code is to integrate the body-xed
equations of motion [Eqs. (13)] for a spacecraft in an arbitrary
potential eld subject to a chosen control law. The code for this
simulation was written using Matlab and Simulink software. The
equations of motion are integrated using Matlabs ode45 function
(DormandPrince method) with a relative tolerance of 10
6
(m) and
an absolute tolerance of 10
8
(m).
The simulation was developed to work with two different asteroid
shape and gravity models: the triaxial ellipsoid and the polyhedron.
Triaxial ellipsoids are often good approximations of real small-body
shapes. Simulations using this shape can help us to understand fun-
damental phenomena associated with this system that would be dif-
cult to decouple from other effects in the truly arbitrary-shaped
case. More complex geometries can be specied to the simulation
as n-faced polyhedra. Polyhedra can model a much wider range of
shapes than ellipsoids, allowing depressions, ridges, cliffs, caverns,
and holes on the body. Polyhedron models currently exist for a num-
ber of real asteroids. This shape model can be very accurate, with
resolution increasing with the number of faces used. Assuming the
small body has constant density, previous works exist that exactly
dene the potential around ellipsoidal
8
and polyhedral
9
shapes. The
methods described in these papers are implemented in our simula-
tion to calculate gravitational potential, attraction, the Hessian, and
the Laplacian.
To ensure that the simulation works as intended, we have run a se-
ries of tests with known results and compared the simulation output
to the expected answer. First, we conrmed that the spacecrafts an-
gular momentumand energy above a spherical body were conserved
throughout the simulation if no thrust was applied. Our next test was
to verify that the simulation was computing the correct body forces
on the spacecraft. This was done by analytically solving for the
forces on the spacecraft at a given point and then adding open-loop
BROSCHART AND SCHEERES 345
thrust of equal magnitude and opposite direction to the spacecraft.
If the simulation is computing the forces on the spacecraft correctly,
this open-loop thrust should cancel those forces and the spacecraft
should not move. The simulation was found to correctly produce
this result. Our nal verication test was done to ensure that the
controllers built into the simulator were working as intended. This
can be determined by examining the various simulation outputs.
Thrust should be enabled in the proper direction whenever the cri-
teria for activation of control are met. We performed this test for
many different initial conditions and found the controllers to work
as expected.
Stability of Body-Fixed Hovering
We have investigated the stability of body-xed hovering sub-
ject to two different implementations of the nominal thrust solution
[Eq. (4)]. We compare the numerical regionof stabilityobtainedwith
the region dened by the stability criteria [Eqs. (810)] derived in
the previous work.
5
Because we are working with numerical data, determining
stability characteristics of a trajectory is inherently difcult. With
nite-time simulation data, it is impossible to verify true stabil-
ity or instability. Here, we will be using a modied form of Lya-
punov stability suitable to evaluate our trajectories. Beginning with
a perturbation from equilibrium smaller than a given magnitude,
we quantify stability by the size of the region centered at the initial
hovering point that contains the entire trajectory for a given simu-
lation time. We will dene instability not as unbounded motion but
as motion that extends beyond a specied region during the simula-
tion time. Because of potentially long time constants of instability,
this analysis seeks only to verify stability for a xed duration in the
regions predicted by the analytical stability criteria and not to verify
instability in the complementary space.
Numerical Simulations of Hovering with Gravitational Direction
Thrusting and Sensing with Open Loop Control
The rst body-xed hovering controller we evaluated is a slightly
modied version of the realistic hovering controller suggested in
Ref. 5. This controller, which we call the gravitational direction
thrusting and sensing with open loop (GDTS w/OL) hovering con-
troller, is a combination of open-loop control thrust to eliminate the
spacecrafts nominal acceleration and dead-band control on altitude
measured along v
3
. The dead-band controller is dened by a tar-
get altitude h
0
and a tolerance factor . If the spacecrafts altitude
is outside the band dened by h
0
, thrust is enabled in order
to return the spacecraft into the dead-band. A spacecraft with an
internal model of the small body can compute the Hessian matrix
of the potential and determine v
3
. In Ref. 5, it is suggested that
the open-loop control need only cancel the centrifugal force on the
spacecraft. However, we have chosen to use the open-loop control
to null the full nominal acceleration on the spacecraft, as proposed
in Ref. 4, because we have found this type of control to conne the
spacecrafts range of motion more tightly. The thrust vector T for
the GTDS w/OL controller is
T = T
o
+T
db
(12)
where
T
o
=
_

V
x

2
x
0
,
V
y

2
y
0
,
V
z

0
_
T
(13)
T
db
=
_
_
_
T
m
v
3
, if (h
0
h) >
T
m
v
3
, if (h
0
h) <
0, otherwise (14)
where T
m
is the constant magnitude of the dead-band thruster and
h is the spacecraft altitude. The vector v
3
should be chosen to point
away from the body. This controller mimics the ideal controller
used to derive the conditions for stability of body-xed hovering
[Eqs. (810)] in the previous work. In that ideal case, the dead-band
tolerance factor for this controller is zero. Therefore, a comparison
of the performance of this controller with the analytical stability
conditions is a direct test of the validity of the idealized result for a
spacecraft subject to realistic control.
Using an ellipsoidal small-body shape model, we ran simulations
using the GDTS w/OL controller for random initial positions in the
bodys three symmetric planes (XY, XZ, and Y Z). The ellipsoid
used in these simulations was 15 7 6 km, a rough approximation
of the size of Asteroid (433) Eros, which was visited during NASAs
Near Earth Asteroid Rendezvous (NEAR) mission. A bulk density
of 3 g/cm
3
and a 10-h rotation period were used, which are not
consistent those of with (433) Eros but were chosen to move the
resonance radius further from the body for clarity. (With Eross
true rotation period of 5.27 h and bulk density of 2.4 g/cm
3
, the
resonance radius is roughly 15.7 km, barely beyond the long end
of the body.) The dead-band tolerance factor was 10 m. For each
initial spacecraft position, the simulation was run 10 times with
different velocity error each time. Each Cartesian component of
velocity error was chosen randomly from a uniform distribution
between 1 and 1 cm/s.
Data were collected from independent simulations of two differ-
ent durations, 20,000 and 50,000 s (roughly 5.5 and 13.9 h, respec-
tively). These relatively short simulation times (on the order of one
rotation period) are justied by noting that a spacecraft operating
in the body-xed frame very near the surface would have little rea-
son to remain in one position for long periods of time. This type
of maneuver would likely be used during a descent or a translation
across the surface, whereas longer term hovering station-keeping
maneuvers would more likely be carried out in the inertial frame.
Our simulations showthat in the XZ and Y Z planes the analytical
criteria for stability were valid. There were no regions of numeri-
cally unstable motion that encroached upon the predicted region of
stability. We found that numerical stability extended to some regions
outside the area dened by the stability criteria in these planes but
these trajectories are not necessarily stable in the long term and,
therefore, do not suggest error in the stability conditions.
Figures 1 and 2 present results fromsimulations in the bodys XY
(equatorial) plane for durations of 20,000 and 50,000 s, respectively.
In these gures, each data point represents the average size over 10
trials of the smallest solid angle such that the entire trajectory is
contained within it. For this analysis, we will consider averages of
less than 0.4 deg to be stable. The region inside the bold line is the
locus of initial positions satisfying the analytical stability criteria.
Note that for an ellipsoidal shape model, the equations of motion
exhibit a longitudinal symmetry; that is, results are the same for any
two points 180

longitude apart.
We can see in these gures that the stability criteria do not hold
in the XY plane subject to the GDTS w/OL body-xed hovering
controller. Here, instabilities arise above the bodys leading edges
in regions satisfying the stability conditions. The leading edges are
dened as the two quadrants of longitude on the bodys surface that
extend from the tip of the largest semimajor axis of the ellipsoid
to the intermediate semimajor axis in the direction of z x. As the
simulation duration increases from 20,000 to 50,000 s, we can see
that the area of instability encroaches further upon the region satis-
fying the stability criteria. On the other hand, the region of predicted
stability above the bodys trailing edge is unaffected and remains
stable. We also nd the region of stability off the trailing edge to be
signicantly expanded for shorter duration (20,000 s) hovering.
Analysis of GDTS w/OL Controller Results
We attribute the discrepancy between the ideal region of stability
and our numerical ndings to the Coriolis accelerations introduced
by relaxing the innitely tight control in the gravitational direc-
tion assumed in the analytical work. An interaction between the
dead-band orientation, the control direction, and the Coriolis ac-
celerations causes the degradation of stability above the ellipsoids
leading edges and the improvement in stability above the trailing
edge that are seen in our simulations.
The orientation of the dead-band region for hovering in the XY
plane is shown in Fig. 3. We dene a coordinate systemfor the dead-
band dynamics consisting of the normal vector to the dead-band
346 BROSCHART AND SCHEERES
Fig. 1 Stability results for GDTS w/OL body-xed hovering controller. Angular deviation 20,000 s (XY plane).
Fig. 2 Stability results for GDTS w/OL body-xed hovering controller. Angular deviation 50,000 s (XY plane).
boundary, n
db
, the unit vector alongthe rotationaxis, z, andthe vector
transverse to the dead band, t
db
= z n
db
. The surface normal where
the altitude-sensing-direction vector v
s
intersects the bodys surface
denes n
db
, the local orientation of the dead band. This direction,
in general, is not aligned with the control direction v
c
. Therefore,
when control thrust is applied, it will have some component along
t
db
. This transverse thrust component will be equal and opposite at
the two dead-band boundaries. If the spacecraft hits both boundaries
an equal number of times, this transverse acceleration will have no
net effect on the spacecrafts motion along the dead band. However,
because of Coriolis forces, which effectively rotate the spacecrafts
velocity vector, this may not be the case. This rotation, coupled with
the control thrust component along the dead band, can either cause
the spacecraft to repeatedly hit one boundary of the dead band or
encourage it to bounce back and forth between the boundaries.
Figure 4a illustrates the dynamics above the leading edge. Above
an ellipsoids leading edge subject to the GDTS w/OLhovering con-
troller, the relative orientation of n
db
and v
c
is such that at the mini-
mumaltitude boundary of the dead-band, a component of the control
thrust is applied in the negative t
db
direction and at the maximum
BROSCHART AND SCHEERES 347
altitude boundary, thrust is applied in the positive t
db
direction. The
Coriolis force acting on the spacecraft, F
c
=2( z r), effectively
rotates the spacecrafts velocity vector clockwise in the plane de-
ned by n
db
and t
db
. When the spacecraft reaches a boundary, the
component of control thrust along t
db
will effectively rotate the re-
turn velocity vector clockwise from the direction it would have had
if there had been no thrust component along t
db
. As the spacecraft
moves toward its next boundary crossing, Coriolis forces will ro-
tate the velocity in the same clockwise direction. Once transients
due to initial velocity errors wear off, the combination of these two
effects will cause the spacecraft to hit the same boundary on suc-
cessive occasions until the nonlinear effects of gravity and changes
in dead-band orientation eventually turn it around. This successive
bouncing motion is linearly unstable and leads to large oscilla-
tions from the nominal hovering point. Examination of numerical
simulations shows that this bouncing movement always takes over
in the steady state for hovering above the leading edge of an ellipsoid
subject to the GDTS w/OL controller.
Figure 4b illustrates the dynamics above the trailing edge. Above
the bodys trailing edge, the relative orientation of n
db
and v
c
is op-
posite, such that at the minimum altitude boundary, a component of
the control thrust is applied in the positive t
db
direction and at the
maximum altitude boundary, thrust is applied in the negative t
db
di-
rection. As opposed to above the leading edge, the thrust component
along t
db
in this conguration causes the velocity vector reected off
the dead-band boundary to be rotated counterclockwise from where
it would have been if there were no transverse thrust component. In
this case, the clockwise rotation of the velocity vector caused by the
Coriolis force will counter the effect of the control thrust, encour-
aging the spacecraft to move toward the other dead-band boundary.
Again, in the steady state, this motion always takes over above an
ellipsoids trailing edge, causing a chattering effect, where the
spacecraft never hits the same boundary on successive occasions.
This chattering has a stabilizing effect that keeps the spacecraft close
to the initial hovering point.
The analysis here is done in the equatorial plane for the sake of
simplicity, but this effect occurs with weakening strength all the way
to the bodys XZ and Y Z planes. In these planes, thrust does not
directly induce motion that causes disruptive Coriolis accelerations.
This is why our numerical results in the XZ and Y Z planes agree
well with the analytical predictions, in whose derivation Coriolis
forces essentially do not exist. The strength of this Coriolis effect
above either the leading or trailing edge is determined by , , and
the angle between n
db
and v
c
. If we dene this angle as , the less
negative is (measured from n
db
and positive for orientations such
that v
T
c
t
db
>0), the weaker the destabilizing effect of control thrust
will be above the leading edge, whereas the more positive is, the
stronger the focusing effect will be above the trailing edge. If we use
a symmetric, universal method of choosing control direction, such
as gravitational direction or initial acceleration, we can postulate
that some optimal control law exists that will adequately focus the
leading edge dynamics, while not weakening the focusing effect off
the trailing edge too much. This optimal control direction would be
a function of small-body parameters as well as the hovering dura-
tion. Ideally, a spacecraft would use a method of choosing control
direction above the leading and trailing edges such that the angle
is always positive. We have been able to show through numeri-
cal simulation that shifting the control direction above the leading
edge so that is positive does indeed produce the desired stabiliz-
ing effect. It should be noted that should never approach 90 deg
because there would then be no thrust in the direction normal to the
dead-band to maintain the spacecrafts altitude.
Numerical Simulations of Hovering with Initial Acceleration
Thrusting and Normal Sensing Control
We have also performed numerical simulations using another im-
plementation of the body-xed hovering solution that we call initial
acceleration thrusting and normal sensing (IATNS) hovering con-
but we eliminate the open-loop thrust that cancels the spacecrafts
nominal accelerations. For the IATNS controller, we also change
the direction of the dead-band control thrust to be aligned with the
nominal acceleration vector to more directly counter the natural mo-
tion of the spacecraft. The direction of altitude measurement is also
changed from along v
3
to measure altitude in the direction normal
to the surface at the hovering point. This is the most robust direction
in which to measure altitude because the component of altitude that
changes with the surface topology is minimized for small deviations
in spacecraft position.
The IATNS controller offers advantages over the GDTS w/OL
controller. The elimination of the open-loop thrust will result in fuel
savings because with an open-loop thrust, any inaccuracies in its ap-
plication will result in a constant push toward one side of the dead
band that must be negated with thrust from the closed-loop con-
troller, effectively causing the control to work against itself. Also,
removing the open loop causes the spacecrafts motion to proceed
very closely along the initial acceleration direction from the initial
hovering point. This lessens the deviation from the initial hovering
point due to transient effects caused by initial velocity errors because
the spacecraft will more quickly move into steady-state motion in
the dead band. Sensing altitude in the direction normal to the surface
at the initial hovering point keeps the dead-band orientation vector
n
db
closer to the control direction v
c
, which our analysis suggests
should improve performance above the bodys leading edge. This
controller may also offer advantages in a surface-descent scenario,
as there will exist some angle between the thruster plume and the
surface normal. This angle could help avoid contamination of the
surface regolith by the thrusters in a sample collecting mission and
allow descent imaging to be less obscured by the thruster plume.
The thrust vector for the IATNS controller is
T =
_
_
_
T
m
a
0
, if h
0
h >
T
m
a
0
, if h
0
h <
0, otherwise (15)
where
a
0
=
_
V
x

0
+
2
x
0
,
V
y

0
+
2
y
0
,
V
z

0
_
T
_
_
_
_
_
_
V
x

0
+
2
x
0
,
V
y

0
+
2
y
0
,
V
z

0
_
T
_
_
_
_
(16)
With some thought, we expect that the IATNS controller will fail
catastrophically near the resonance radius of the body. As the space-
craft hovering point approaches the resonance radius, the initial ac-
celeration vector a
0
will turn 180 deg as the nominal acceleration
transitions from pointing toward the body to away. Therefore, this
348 BROSCHART AND SCHEERES
a)
b)
Fig. 4 Dead-band dynamics above an ellipsoid under GDTS w/OL control: a) leading and b) trailing edges.
controller will be limited to use inside the bodys resonance radius
or near the rotation axis.
Using the same ellipsoidal small-body and system parameters
used in our simulations for the GDTS w/OL controller, we have
simulated spacecraft trajectories subject to the IATNS controller in
the XY, XZ, and Y Z planes. Again, we ran 10 different simulations
with random velocity errors at each initial hovering point.
We found the IATNS body-xed hovering controller to work very
well at altitudes inside the resonance radius. As expected, the con-
troller quickly becomes less stable as the initial position nears the
area where the initial acceleration rotates away from the body. In all
three planes, initial positions inside this threshold were found to be
stable. For comparison, Fig. 5 shows the results of simulations in
the XY plane after 50,000 s. In the XY plane, the analytical stability
conditions dene the threshold where the initial acceleration vector
turns away from the body. The bold line in the gure outlines this
region. As we hoped, the IATNS controller reduces the deviations
associated with the Coriolis forces discussed in the preceding sec-
tion above the bodys leading edge compared to the GDTS w/OL
controller. By reducing the control thrust component along the dead
band, the instability related to Coriolis forces above the leading
edge is weakened and develops more slowly. In this 50,000-s gure,
a slight degradation of stability is just beginning to appear above the
leading edge. In the data from the 20,000-s runs, this degradation of
stability cannot be seen. Similarly, the stabilizing effect above the
trailing edge should be slightly weakened by the realignment of the
BROSCHART AND SCHEERES 349
Fig. 5 Stability results for IATNS body-xed hovering controller. Angular deviation 50,000 s (XY plane).
vector n
db
for this controller. However, the angle is still negative
and therefore a stabilizing effect remains, as demonstrated in our
numerical results.
Stability of Inertial Hovering
We now look at the equations of motion for inertial hovering in
solution [Eq. (5)], we obtain the perturbation equations
x 2 y
2
x =

2
V
x
2

0
x +

2
V
xy

0
y +

2
V
xz

0
z
(17)
y +2 x
2
y =

2
V
xy

0
x +

2
V
y
2

0
y +

2
V
yz

0
z
(18)
z =

2
V
xz

0
x +

2
V
yz

0
y +

2
V
z
2

0
z (19)
This is a periodic, time-varying linear system. We can analyze
the stability of this system using Floquet theory, which suggests
that the state transition matrix is of the form = P(t ) exp(Mt ),
where P(t ) is a periodic matrix and M is a constant matrix. We can
evaluate the stability of the system by examining the eigenvalues
of M. This is done by evaluating the state transition matrix after
one period of motion, that is, when P is the identity matrix. We can
numericallycalculate the state transitionmatrixusingour simulation
package. Further, we note that this is a Hamiltonian system, and
thus, the eigenvalues of M, or Floquet multipliers, must come in
complex conjugate and inverse pairs. This means that for stability,
the Floquet multipliers of the systemmust all lie on the unit circle in
the complex plane. Multipliers with magnitude different from unity
imply instability.
Without performing any calculations, we expect that the system
will have at least two unstable multipliers corresponding to motion
in roughly the radial direction. If there are some perturbations in
this direction, the gravitational attraction the spacecraft feels will
either increase or decrease in the same way that the nominal thrust
will be inaccurate, resulting in an unchecked acceleration.
We can validate this insight by looking at the simplied case of
inertial hovering over a rotating point mass. Because this potential
eld is spherically symmetric, it will remain constant throughout
the maneuver. The equations of motion in the inertial frame for this
system are
x =
V
x
+ T
x
=

r
3
x + T
x
(20)
y =
V
y
+ T
y
=

r
3
y + T
y
(21)
z =
V
z
+ T
z
=

r
3
z + T
z
(22)
where r =(x
2
+y
2
+z
2
)
1
2
is the radial distance fromthe point mass.
If we linearize about a nominal position [x
0
, y
0
, z
0
]
T
and thrust
vector,
T
0
=
_

V
x

0
,
V
y

0
,
V
z

0
_
T
we obtain the following perturbation equations:
_
_
_

z
_

_
=
_
_
_
_
_
_
_
_
_
3x
2
r
5

1
r
3
3xy
r
5
3xz
r
5
3xy
r
5
3y
2
r
5

1
r
3
3yz
r
5
3xz
r
5
3yz
r
5
3z
2
r
5

1
r
3
_

0
_
_
x
y
z
_
_
(23)
Because the gravitational eld is spherically symmetric, we can
assume, without loss of generality, that the initial position is of the
350 BROSCHART AND SCHEERES
Fig. 6 Regions of instability for inertial hovering above a 15 7 6 ellipsoid for various rotation rates, radius vs latitude, T
Eros
=5.27 h.
form [x
0
, 0, 0]
T
. These equations then reduce to
_
_
_

z
_

_
=
_
_
_
_
_
_
_
_
_
2
x
3
0
0 0
0
1
x
3
0
0
0 0
1
x
3
0
_

_
_
_
x
y
z
_
_
(24)
These are three decoupled, second-order differential equations.
The eigenvector along the radial direction is unstable with eigenval-
ues

(2/x
3
0
) and the two eigenvectors transverse to the radial
direction have stable eigenvalues equal to i

(/x
3
0
).
Here, a stabilizing controller is necessary for the radial direc-
tion, as predicted, and the transverse directions are always stable.
The stability of the transverse eigenvalues will change in the more
general case, when the potential distribution varies as the body
rotates. For this discussion of stability, we will ignore the inher-
ently unstable eigenvalues, assuming existence of a stabilizing con-
troller in the radial direction (similar to the tight control assumed in
Ref. 5).
For the ellipsoidal shape case, we have numerically integrated the
perturbation equations [Eqs. (1719)] for one period of motion and
determined the eigenvalues of the state transistion matrix. Integra-
tions were again performed using Matlab with a relative tolerance
of 10
8
(m) and an absolute tolerance of 10
11
(m). We found that
inertial-frame hovering, subject to a stabilizing controller in the ra-
dial direction, is stable at most radial distances and inclinations. The
exception to this is a region near the resonance radius that extends
around the body, forming a nearly spherical shell of instability.
The cross-hatched regions in Fig. 6 show the shape of this unsta-
ble region, which is axially symmetric about z for different rotation
rates around an ellipsoidal body measuring 15 7 6 km with a
density of 2.3 g/cm
3
. The resonance radius corresponding to each
rotation rate is shown as a vertical dashed line.
It can be seen that the region of instability moves with the chang-
ing resonance radius [Eq. (11)], a function of rotation rate. As lat-
itude increases from the ellipsoids equator to its pole, the radius
of unstable hovering becomes smaller and the strength of the in-
stability is marginally decreased. The radial distance covered by
the instability also decreases as the rotation rate decreases. Figure 7
shows the magnitude of the four transverse eigenvalues (correspond-
ing to the two transverse directions in position space) vs the radius
in the equatorial plane for selected rotation rates. Again, the cor-
relation between the primary instability and the resonance radius
is clearly shown. This gure also shows that the magnitude of the
instability increases for faster rotation rates. We note that the mag-
nitude of the instabilities that exist near the resonance radii is small
(generally <2). Beyond the instability at the bodys resonance ra-
dius, we have found hovering to be stable in all cases up to the
point where our assumptions break down and third-body effects of
the sun become an issue. The qualitative properties of this ellip-
soidal example are reective of the other ellipsoidal shapes we have
tested.
One may expect there to be instabilities near other harmonics of
the rotation rate. Our data have revealed that relatively weak in-
stabilities may exist inside the resonance radius at these secondary
resonances. However, as the ratio of hovering radius to resonance
radius decreases, the condition number of the state transition matrix
that gives us our stability result increases exponentially. The sec-
ondary harmonic instabilities we have found for the ellipsoidal body
case lie too far inside the resonance radius to be considered numer-
ically accurate. Therefore, the stability of hovering well inside the
resonance radius remains unknown. We have found no secondary
harmonics outside of the resonance radius, where the result is nu-
merically tractable.
Numerical Support
The analytical result described above is supported very well by
numerical simulation. For these simulations, we used an ellipsoidal
shape model with dead-band control on distance from the plane
passing through the origin dened by the vectors z and r
0
z in the
inertial frame. The control direction was dened as being constant
and in the direction of the initial hovering point.
Because the control direction is constant and the dead-band con-
trol connes the motion of the spacecraft to a thin region between
two planes, we can imagine that, assuming the thrusters have ad-
equate authority, all inertial hovering trajectories will at least have
BROSCHART AND SCHEERES 351
Fig. 7 Magnitude of Floquet multipliers for inertial hovering above a 15 7 6 ellipsoid for various rotation rates. Latitude =0

.
bounded time responses. That is, as the spacecraft moves far from
the initial point, the gravitational attraction will increasingly pull
the spacecraft back toward center. This phenomenon is not apparent
in the linearized analysis but is ultimately true in the full nonlinear
case. However, oscillations of this magnitude generally would not
be acceptable during a mission to a small body. For the purposes
of our analysis, we will dene instability not as unbounded motion
but as growth in the periodic oscillation of the spacecraft about the
nominal point. This view is consistent with the linearized stability
used in the analytical analysis.
We performed simulations of inertial hovering trajectories over
a range of radial distances along the equatorial plane. The primary
feature of the previous results, instability near the resonance ra-
dius, was clearly shown for trajectories with a target radius near
the resonance radius. Numerical simulation also conrmed that the
linear analytical stability analysis for hovering outside the reso-
nance radius produced accurate results; that is, inertial hovering is
cillations in the position of the spacecraft in these regions were
seen to be bounded for relatively long simulation times (35 days).
In these simulations, we also found that in regions very near the
body, oscillations from the nominal point were quite large in mag-
nitude, though bounded in the long term. In general, the bound
on the spacecrafts oscillations about the nominal point for sta-
our analytical work did not yield insight to the stability of iner-
tial hovering very near the body. Our numerical results have shown
inertial hovering to be nonlinearly stable very near the body, but
with large (on the order of the small-body size) oscillations in
position.
Hovering Above Asteroid Itokawa
On 9 May 2003, JAXA successfully launched the MUSES-C
spacecraft, renamed Hayabusa after launch, toward (25143)
Itokawa.
10
In this mission, the spacecraft will rendezvous with the
asteroid, spend some time in inertial hovering, and then descend to
the surface to collect regolith samples to be returned to Earth. Be-
cause of the applicability of our work to this mission, we nowpresent
results for a spacecraft operating near Itokawa subject to our control
strategies. We have simulated motion above the 6098-vertex poly-
hedral shape model of Itokawa developed by Ostro et al.,
11
using
their estimates of density (2.5 g/cm
3
) and rotation period (12.132 h).
Body-Fixed Hovering
To verify that the qualitative results found for the ellipsoidal case
apply to more realistic small-body shapes, we have performed sim-
ulations of body-xed hovering above Itokawa subject to the two
controllers discussed earlier with a of 5 m. Our approach was to
use initial positions at a range of altitudes along various radial lines
extending from the center of the body to capture the qualitative sta-
bility properties found in the ellipsoidal case. Again, we quantify
stability by average maximum angular deviation over 20 runs with
different initial velocity errors.
Recall that in the ellipsoidal case, the GDTS w/OL controller
[Eq. (12)] agreed well with the analytical stability criteria in the
XZ and Y Z planes. In the equatorial plane, there were unstable re-
gions found above the bodys leading edge that satised the stability
criteria and the stable region was extended above the trailing edge
for shorter duration hovering. Our analysis has suggested that these
leading- andtrailing-edge effects are present at all latitudes, weaken-
ing as the hovering point moves away fromthe equator. These results
are supported by numerical simulations of GDTS w/OL hovering
above Itokawa.
For the radial line extending at 30

latitude and 45

longitude
(measured from the positive x axis), we found the GDTS w/OL
controller to behave in line with the stability criteria. There is a
clearly visible increase in the average angular deviation at both
20,000 and 50,000 s when the initial radius moves beyond the region
satisfying the stability criteria. As the initial radius approaches the
limit of the analytically stable region, a slow increase in average
deviation is found, which we attribute to the weakened, out-of-plane,
Figure 8 shows angular deviation versus radius for GDTS w/OL
hovering at 5

latitude and 45

## longitude, that is, above the leading

edge near the equatorial plane. The radii satisfying the analytical sta-
bility criteria are indicated with squares at the bottomof the plot. We
can see that the average deviation starts to increase somewhat inside
the region satisfying the sufciency criteria for the 20,000-s runs.
For the 50,000-s runs, the effect is more dramatic; destabilization
occurs well inside the limit of analytical stability and causes severe
degradation in performance. These ndings support the results of
the ellipsoidal analysis.
Simulations for the radial line at 0

latitude and 30

longi-
tude, that is, off the trailing edge, show a region of stability extend-
ing well beyond the threshold suggested by the analytical stability
criteria after 20,000 s. In the 50,000-s case, the angular deviation
352 BROSCHART AND SCHEERES
Fig. 8 Stability of body-xed hovering above Asteroid (25143) Itokawa as a function of radius, GDTS w/OL controller (5

latitude, 45

longitude).
Fig. 9 Stability of body-xed hovering above Asteroid (25143) Itokawa as a function of radius, IATNS controller (0

latitude, 30

longitude).
begins to increase just inside the limit of the region of predicted
stability. Beyond the analytical stability threshold, deviations are
noticably smaller than in the unstable regions of the other two
cases.
We also implemented the IATNS controller [Eq. (15)] above
Itokawa. Again, we found the results to be in agreement with our
ndings in the ellipsoidal case. That is, the IATNS controller is con-
sistently stable for 20,000- and 50,000-s simulations in all directions
in a region near the body extending almost to the analytical stability
threshold. Beyond this point, the controller becomes very unstable.
Figure 9 shows a typical example (0

latitude and 30

longitude).
These results for hovering above Itokawa have conrmed the
qualitative aspects of our ndings for body-xed hovering in the
ellipsoidal case subject to both the GDTS w/OL and IATNS con-
trollers. These results suggest that the qualitative properties of the
stability regions produced by these controllers can be extended to
hovering over realistic asteroid shapes.
Inertial-Frame Hovering
Alarge portion of the Hayabusa mission will be dedicated to iner-
tial hovering, during which instrumentation onboard the spacecraft
BROSCHART AND SCHEERES 353
Fig. 10 Magnitude of Floquet multipliers for inertial hovering above Asteroid (25143) Itokawa, 15

latitude.
Table 1 Asteroid (25143) Itokawa and Hayabusa
mission parameters
a
Parameter Value
Orbital inclination, deg 1.728
Longitude of ascending node,

70.921
Orbital eccentricity 0.280
Orbital period 1.521 Julian yr
Periapsis passage 2001 May 4.601
Argument of periapsis, deg 161.029
Orbital semimajor axis, AU 1.323
Rotation period, h (Ref. 11) 12.132
Rotation pole, deg (Ref. 12) 355, 84
Hayabusa time of arrival (s/c) 61505
Hayabusa time of departure (s/c) 11205
a
Courtesy of Horizons Website,
13
Japan Aerospace Explo-
ration Agency.
will be used to map the asteroids surface. For this phase of the
mission, the Hayabusa mission planners currently intend to imple-
ment a controller consisting of three orthogonally oriented position
dead-band controllers to maintain spacecraft position.
6
This type of
control effectively forms a box in position space that contains the
spacecrafts trajectory.
Our previous analysis suggests that the spacecraft can maintain
position in inertial space with only a single-dimensional controller
at altitudes outside the bodys resonance radius. If this is the case, it
should result in improved fuel efciency and operational simplicity
for this part of the Hayabusa mission. The current plan is for the
Hayabusa spacecraft to hover at an altitude of approximately 20 km
over Itokawa.
7
Our analysis predicts that inertial hovering at this
altitude is stable.
For our results to be as accurate as possible, it was necessary
to compute the appropriate latitude at which the spacecraft would
be hovering. The spacecraft will be operating in the vicinity of the
Earthasteroid line for purposes of communication. Given the orbit
of Itokawa and the appropriate mission data (Table 1) (Refs. 1113),
we calculate that the Earthasteroid line will move between 14.8

## latitude upon the spacecrafts arrival and 16.8

at departure. Our
analysis is performed at 15

latitude.
Figure 10 shows the analytical stability results obtained by exam-
ining the eigenvalues of the state transition matrix after one period
of inertial hovering at 15

## latitude over Itokawa for a range of radial

distances. We can see that the primary instability exists in the region
near the bodys resonance radius of 579 m. Again, we can note that
the magnitude of this instability is reasonably small. Interestingly,
we see two secondary instabilities near radius 450-m. These results
are numerically valid (as opposed to the secondary instabilities seen
in the ellipsoidal case). Without a more general result, it is difcult
to say whether these results validate the idea of secondary harmonic
instabilities suggested in the ellipsoidal case or if they exist only
because of a specic parameter of this models shape. Beyond the
resonance radius, all inertial hovering is stable. We can conclude
that the Hayabusa mission plan to inertially hover at an altitude of
20 km is safely within the region of stable inertial hovering subject
to one-dimensional control when solar gravitation and solar radia-
tion pressure are ignored. The effect of these perturbations will be
determined in future work.
Conclusions
Analytical stability results for hovering in the body-xed frame
were compared to numerical simulation data with realistic hovering
control. We found that without the assumption of innitely tight
dead-band control used in the development of the analytical sta-
bility conditions, Coriolis accelerations destabilize hovering above
the leading edges of ellipsoidal small bodies. The effect of this
destabilization depends on the angle between the direction of thrust
application and the dead-band orientation. Numerical results for a
second realistic controller that uses this knowledge to improve hov-
ering performance near the small body are presented.
Analytical and numerical analyses of hovering in the inertial
frame have been performed. We found that inertial hovering is stable
at all radii outside the region near the resonance radius up to the point
where our two-body assumption breaks down. A shell of unstable
inertial hovering positions surrounds the small body at a distance
near the bodys resonance radius. This instability is strongest in the
equatorial plane and strengthens with increasing rotation rates. In
most cases, however, the strength of this instability remains mild and
it would likely be possible for a spacecraft to move safely across the
region.
354 BROSCHART AND SCHEERES
The concepts of body-xed and inertial-frame hovering have
been applied to Asteroid (25143) Itokawa, the target of the recently
launched Hayabusa mission. Our qualitative results for hovering
control above ellipsoidal bodies were found to transfer well to the
more general polyhedral shape of Itokawa, suggesting that the con-
trol methodologies presented here may be applied to hovering above
real small bodies.
Acknowledgments
The research described in this paper was sponsored by the Inter-
planetary Network Technology Programby a grant to the University
of Michigan fromthe Jet Propulsion Laboratory, California Institute
of Technology, which is under contract with NASA. S. Broschart
was also supported by the Graduate Student Researchers Program
at the Jet Propulsion Laboratory. The authors thank Shujiro Sawai
of the Japan Aerospace Exploration Agency for providing details
on the Hayabusa mission.
References
1
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