I. Introduction
The Hayabusa mission is Japanfs first sample return mission from a Near Earth Object (NEO). The Hayabusa
spacecraft (the original code name is MUSES-C) was launched by M-V rocket on May 9, 2003 and following
successive Earth swing-by, had a rendezvous with asteroid 25143 Itokawa on September 12, 2005.
Usually, the radiometric data, e.g. range, Doppler, ƒ¢DOR are used for orbit determination of the spacecraft in
deep space (Moyer 2000). Furthermore, in the case of the rendezvous mission or fly-by mission, when a spacecraft
moves near the target and can get the optical information, we can use both radiometric and optical data for orbit
determination. In the Hayabsua mission, during the cruise phase (Earth to Itokawa), the 2-way X-band range and
Doppler data were used. On the other hand, during the approach phase and the gate-position phase, and the homeposition
phase, we could obtain the optical data of Itokawa by means of star tracker (STT) or optical navigation
camera (ONC)(Kominato et al. 2006), and at close proximity phase (e.g., navigation test, rehearsal for touchdown,
touchdown), we could obtain the position data calculated from the optical data by means of feature-point matching
techniques (Morita et al. 2006).
II. Gravity Model of Itokawa
In this analysis, I used different gravity models for Itokawa: 1) Point-mass, 2) Spherical Harmonics (Kaula 2000),
and 3) Polyhedron gravity model (Werner and Scheeres, 1997). Each model has its advantages and disadvantages.
For instance, however a point mass model is easy to use, this model cannot consider the shape of asteroid. In the
case of a spherical harmonics gravity model, can consider the shape model, but cannot be used when a spacecraft is
close to the surface of the asteroid. In the case of polyhedron gravity model, however this gravity model provides the
gravity close to the surface of the asteroid, requires a precise 3D shape model of the asteroid.
When the spacecraft is located at the gate-position or home-position, it is sufficient to use the point-mass gravity
model. In the case when the spacecraft is close to the asteroid, the distribution of the asteroidfs mass should be taken
into account for the calculation. It is important to note that the error of the spherical-harmonics gravity model
increases when evaluating the gravity field close to the modelfs radius and is no longer guaranteed to converge in
the circumscribing sphere. Therefore, in order to analyze the orbit of the spacecraft in this region, I adopted the
polyhedron gravity model. In fact, the distance between Hayabusa and Itokawa varies from about 20 km (GP) to 0
km (touchdown), so I used the appropriate gravity model, which satisfies each condition.
III. Effect of Attitude Control
Originally, Hayabusa adopted a zero-momentum method for attitude control by using three reaction wheels
(RW-X, RW-Y, RW-Z for each axis). The RW-X had broken down on 31 July, before the spacecraft arrived at
Itokawa, thus the attitude control mode changed to a dual reaction wheel (DRW) mode. Unfortunately, the RW-Y
also broke down when descending from GP to HP (2 October 2005). Therefore we used the bi-propellant thrusters
for attitude control, which is referred to as the gThruster modeh. However, this attitude control generated small
unintended accelerations, which affected the orbit of the spacecraft and yielded the increase of the number of ascent
ƒ¢V (i.e. increase the fuel consumption).
The evolution of the velocity of Hayabusa (HP-Z component in HP coordinate system) is shown in Fig. 1.
These values are calculated from the Doppler data, which acquired during the visible interval. The DRW mode was
used until 2 October, then the RW-Y broken down after the loss of signal (LOS) of this pass and the RCS mode was
Fig. 1 Velocity of Hayabusa in the Home-position coordinate system
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adopted afterward. The gradient of the velocity (i.e. acceleration) of 3 and 4 October 2005 (just after a breakdown of
the RW-Y, and we did a trouble solving about the attitude control), is obviously larger than that of 2 October 2005.
In order to transmit the HK data and the science instrumentfs data to ground station (we usually used high gain
antenna (HGA) for this purpose), a narrow deadband attitude control was essential. When a deadband was narrow,
the change rate of the velocity of Hayabusa (i.e. the acceleration) was obviously larger than that of when a deadband
was relatively wide. We anticipated that these accelerations were caused by the thrust imbalance between upper side
thrusters and bottom side thrusters.
IV. Thrust Imbalance Model
In order to avoid the contamination of the
asteroid by the carbon, Hayabusa spacecraft adopted
the newly developed RCS (Ohminami et al. 2006). In
fact, this RCS is unlike any other in the world. The
RCS is bi-propellant system (Hydrazine, N2H4
/Nitrogen tetroxide, N2O4) and the nominal thrust is
20 N with a pulse time of 19.53 ms. The
configuration of the RCS is shown in Fig. 2. We
usually use the +Z direction thrusters and -Z direction
thrusters for attitude control. Because of these
configurations, the acceleration caused by the
imbalance of the thrust between +Z direction and -Z
direction, act on the Z-axis direction mainly. In fact,
the magnitude of this acceleration can be as large as
that of solar radiation pressure. Fig. 2 Configuration of Reaction Control System
When we evaluate the orbit of Hayabusa over a relatively long arc, it is necessary to estimate the effect of RCS
attitude control maneuvers precisely. In order to calculate an a priori RCS model, I used house-keeping data about
the history of attitude control actuation (i.e., the number of pulses of each thruster), and the temperature of each
thruster injector.
As a first step, I adopted a simple RCS model as the imbalance of the thrust level between the upper thruster
with thrust
!
F
1 placed at the +Z panel of spacecraft and the lower thruster with thrust
!
F
2 placed at the -Z panel of the
spacecraft (Fig. 2).
!
fRCS = F1 " F2 (1)
Then, we can calculate the net acceleration due to the imbalance between the upper and the lower thrusters as
follows:
!
aRCS =
fRCS " #
M
(2)
where,
!
" is the ratio of the total thrust time per unit time, and
!
M is the mass of the spacecraft. It is important to
mention here that how to calculate the
!
" . We obtain the RCS information from the HK data, in particular the
number of pulses executed in a given interval. Thus, we can accumulate the number of pulses, which are used for
attitude control. Then we can calculate the
!
" (the number of total pulses divided by time of interval). In the case of
adding this acceleration into the dynamics model, the attitude of the spacecraft is also taken into account for the
calculation.
V. Evaluation of the Effect of Attitude Control
Next, let us evaluate the imbalance forces by using the actual data set. This time, I used the data from two
passes, which have been acquired on 9 October 2005 and 15 October 2005. The temperatures of the injectors of the -
Z panel thrusters are +35‹C and +45‹C, respectively. The ratios of the total thrust time per unit of time are
calculated from the RCS house keeping data. In order to evaluate the thrust imbalance parameter
!
fRCS , this parameter
is added to the estimated parameters (position and velocity of Hayabusa at epoch time). The solar radiation pressure
coefficient and the gravity parameter of Itokawa are treated as constant values, which have already been estimated in
the previous analysis. The radiometric data (range and Doppler) and the optical data (ONC) are used for orbit
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determination. In order to verify the effectiveness of this method, I compare the residuals of Doppler measurements
of the following two cases: 1) take into account the
!
a
RCS , and 2) exclude the
!
a
RCS
Fig. 3 shows the residuals of Doppler measurements about 2 cases on 9 October 2005, case1 is red and case2 is
blue. It is clear that, when the accelerations caused by the attitude control
!
a
RCS is taken into account for the
calculation, the residuals of Doppler are smaller than when they are excluded. On the other hand, Fig. 4 shows the
residuals of Doppler measurements about 2 cases on 15 October 2005. It is again clear that, when the
!
a
RCS is taken
into account, the Doppler residuals are smaller than in the excluded case.
Fig. 3 Residuals of Doppler (2005/10/09) Fig. 4 Residuals of Doppler (2005/10/15)
As the result of orbit determination, the imbalance parameters are estimated for the two temperatures (Table 1).
These results are consistent with the actual motion of the spacecraft.
Table 1 Thrust Imbalance Parameter
Data Set Temperature (-Z Panel) Thrust Imbalance
f (N)
9 October 2005 +35 ‹C 0.45 } 0.08
15 October 2005 +45 ‹C -0.57 } 0.06
VI. Orbit Determination of Hayabusa at Close Proximity Phase
During the mission phase (the period that Hayabusa rendezvous with Itokawa), there are five opportunities for
descent to the asteroid surface or near the surface (Kawaguchi 2006). The radiometric and optical data obtained
during these periods provide us the actual gravity environment of Itokawa. In these intervals, we calculate the
position of the spacecraft based on Ground Control Point (GCP i.e. feature points).
A. Orbit Determination with Spherical Harmonics Gravity Model
At first, a spherical harmonics gravity model is used for orbit determination of Hayabusa. The estimated
parameters are: a) the position and velocity of Hayabusa at epoch time, b) gravity parameter of Itokawa, c) solar
radiation pressure coefficients, d) ƒ¢Vfs (i.e. orbital maneuvers), f) thrust imbalance parameter
!
fRCS , and g) gravity
coefficients (4th order and degree). The measurements data are: a) position of Hayabusa relative to Itokawa, which
are calculated based on optical data, and b) Doppler data.
I have not estimated these values simultaneously, but adopted a step-by-step approach. The residuals of
GCPNAV and the Doppler measurements for 2 cases: 1) use the constant gravity coefficients, 2) estimate the gravity
coefficients, are shown in Fig. 5. The designation gConsth refer to the constant gravity coefficients), and case2 refer
to the estimated gravity coefficients. The residuals of both GCPNAV and Doppler get smaller, when the gravity
coefficients are treated as estimated parameter.
The final result for the trajectory of Hayabusa around Itokawa is shown in Fig. 6. The observed data are in red
and the computed data are in blue, respectively. The correspondence can be seems to be very good for this case. The
final part (close to Itokawa) cannot be computed precisely by a spherical harmonics gravity model, because it is too
close to the surface.
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Fig. 5 Residuals of Doppler and GCPNAV measurements (2005/11/12)
B. Orbit Determination with Polyhedron Gravity Model
When the spacecraft moves inside of circumscribing sphere, we cannot calculate the gravity of Itokawa
precisely by using spherical harmonics gravity model. Therefore, I used the polyhedron gravity model (Werner and
Scheeres, 1996) for calculation. This polyhedron gravity model permits us to evaluate the gravity and potential by
taking account of the mass distribution of Itokawa.
In this analysis, I adopted the shape model of Itokawa constructed by R. Gaskell (2006) for the polyhedron
gravity model. The original shape model has 3,145,728 facets and 1,579,014 vertices. In addition, lower resolution
models also exist and I used the model with 49,152 facets and 25,350 vertices. Total volume is 0.117 km3 with
uncertainties of less than one percent. The resolution of the last model is adequate for this analysis.
The estimated parameters are: a) the position and velocity of Hayabusa at epoch time, b) gravity parameter of
Itokawa, c) solar radiation pressure coefficients, d) ƒ¢Vfs (i.e. orbital maneuvers), and f) thrust imbalance parameter
!
fRCS . I used the results of the previous analysis (spherical harmonics) as a priori values (e.g., the position and the
velocity of Hayabusa at epoch, gravity parameter ƒÊ, solar radiation pressure coefficient CR, and ƒ¢Vfs).
Fig. 7 shows the gravity map of Itokawa, in terms of the magnitude of the gravity on a sphere of 0.4 km radious
around Itokawa calculated by using polyhedron gravity model.
VII. Conclusion
However, in the case of the Hayabusa mission, the spacecraft adopted a hovering approach for the scientific
observation of Itokawa and there were not only a number of orbital maneuvers but also many attitude control
actuations, I have successfully estimated the orbit of Hayabusa in the vicinity of asteroid Itokawa, while considering
the small accelerations, such as: gravity of Itokawa at the gate position and home position phase, effect of attitude
control by bi-propellant thrusters, and effect of solar radiation pressure.
In addition, the effect of attitude control by bi-propellant thrusters is evaluated. In order to take into account the
thrust imbalance, an acceleration model for the effect of attitude control is introduced.
I have also attempted to evaluate the actual gravity of Itokawa, the mass, the coefficients of spherical
harmonics, and the polyhedron gravity model.
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Fig. 6 Trajectory of Hayabusa in Itokawa
body-fixed coordinate system (2005/11/12)
Fig. 7 Gravity map of Itokawa
(Polyhedron Gravity Model)