+-----------------------------------+-----------------------------------+
| ![](media/image1.png)             | # AAS 03-026                      |
+-----------------------------------+-----------------------------------+

Deep Impact Flyby Spacecraft Instrument Pointing in the Presence of an
Inactive Impactor Spacecraft

Daniel G. Kubitschek, Nick Mastrodemos, and Stephen P. Synnott

Jet Propulsion Laboratory

California Institute of Technology

**26th ANNUAL AAS GUIDANCE AND CONTROL CONFERENCE**

February 5-9, 2003 Sponsored by

Breckenridge, Colorado Rocky Mountain Section

![](media/image2.wmf){width="2.0in" height="1.2770833333333333in"}

AAS Publications Office, P.O. Box 28130 - San Diego, California 92198

**AAS 03-026**

DEEP IMPACT FLYBY SPACECRAFT INSTRUMENT POINTING IN THE PRESENCE OF AN
INACTIVE IMPACTOR SPACECRAFT

**Daniel G. Kubitschek, Nick Mastrodemos, and Stephen P. Synnott**

**Jet Propulsion Laboratory**

######### **California Institute of Technology**

> The engineering goal of the Deep Impact mission is to impact comet
> Tempel 1 on July 4, 2005, with a 350 kg active Impactor spacecraft
> (s/c). The relative velocity will be just over 10 km/s. The impact is
> expected to excavate a crater of approximately 20 m deep and 100 m
> wide. The science objective is that of exposing the interior material
> and understanding the properties of the nucleus. In order to achieve
> the engineering goal and science objective, Deep Impact will use the
> autonomous optical navigation (AutoNav) software system to guide the
> Impactor s/c to Tempel 1 intercept near the center of brightness (CB),
> while a second s/c, the Flyby s/c, uses identical software to
> determine its comet-relative trajectory providing the attitude
> determination and control system (ADCS) with the relative position
> information necessary to point the High Resolution Instrument (HRI)
> and Medium Resolution Instrument (MRI) at the expected impact site
> during encounter.
>
> If the Impactor s/c is determined to be functioning improperly prior
> to release, the issue of predicting the impact location to correctly
> point the instruments at key science epochs (TOI: Time of Impact; and
> TOFI: Time of Final Imaging), becomes important and therefore must be
> studied. This relies, fundamentally, on the ability to determine the
> trajectory of the Impactor s/c relative to the Flyby s/c by treating
> the Impactor s/c as an optical beacon, relative to which the Flyby
> s/c's trajectory is estimated using images of the Impactor s/c.
> Simulation results show that for an inactive Impactor s/c, the
> pointing error is improved from 519 μrad (1 μrad = 10^-6^ radians) to
> 72.3 μrad (3σ) at TOI, and from 3.96 mrad (1 mrad = 10^-3^ radians) to
> 441 μrad (3σ) at TOFI. When compared to the baseline CB
> targeting/tracking approach, results show the pointing error
> contribution due to knowledge of the impact location changes from 60
> μrad to 72.3 μrad (3σ) at TOI, and from 495 μrad to 441 μrad (3σ) at
> TOFI. This paper deals only with the pointing error contribution due
> to errors in predicting the impact location and describes the
> acquisition of optical data of the Impactor s/c and associated errors
> using the Flyby instruments; the expected uncertainty in predicting
> the impact location and the resulting pointing errors; and the
> algorithm for autonomously computing a pointing correction during
> encounter.

**INTRODUCTION**

The engineering goal of the Deep Impact mission is to impact comet
Tempel 1 on July 4, 2005, with a 350 kg active Impactor spacecraft
(s/c). The impact velocity will be just over 10 km/s. The impact is
expected to excavate a crater of approximately 20 m deep and 100 m wide.
A second spacecraft, the Flyby s/c, is responsible for delivering the
Impactor s/c and will perform a slowing maneuver (deflection maneuver),
following Impactor release, to observe the impact event, ejecta plume
expansion, and crater formation, which will take place over a period of
approximately 800 seconds. Figure 1 shows the flight system
configuration with the Impactor s/c stowed in the lower portion of the
Flyby structure. Figure 2 shows the encounter geometry for the Deep
Impact mission. The science objective is that of exposing the interior
material and understanding the properties of the nucleus.

![](media/image4.wmf){width="2.1347222222222224in" height="2.20625in"}

**Figure 1** Deep Impact flight system configuration showing the
instrument platform, High Gain Antenna, ITS boresight and solar array^2^
![](media/image1.wmf)

**Figure 2** Tempel 1 encounter geometry for the Deep Impact mission

Deep Impact will use the on-board autonomous optical navigation
(AutoNav) software system^1^ to guide the Impactor s/c to Tempel 1
intercept near the center of brightness (CB). The Flyby s/c uses
identical software to determine its comet-relative trajectory in order
to provide the attitude determination and control system (ADCS) with the
relative position information necessary to point the High Resolution
Instrument (HRI) and Medium Resolution Instrument (MRI) at the nucleus
CB during encounter. The Impactor s/c and the Flyby s/c operate in an
independent fashion with the Flyby s/c pointing the MRI/HRI instruments
at the impact site in an indirect way by assuming that the Impactor s/c
will impact at or near the nucleus CB. Figures 3 and 4 show simulated
images of the comet nucleus using the Impactor ITS camera and the Flyby
MRI/HRI cameras along their respective trajectories.

**ITS at E-2hrs ITS at E-5 min**

![](media/image2.jpeg){width="2.407638888888889in"
height="2.2090277777777776in"}
![](media/image3.jpeg){width="2.3826388888888888in"
height="2.2194444444444446in"}

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![](media/image5.jpeg){width="2.415277777777778in"
height="2.2090277777777776in"}

**ITS at E-1 min ITS at E-30 sec**

**Figure 3** Simulated ITS images of the comet nucleus (based on
Halley-Stooke data) during encounter where E- designates time to impact

**MRI at E-2 hrs MRI at E-0 sec MRI at E+800 sec**

![](media/image6.jpeg){width="1.8111111111111111in"
height="1.7895833333333333in"}
![](media/image7.jpeg){width="1.8020833333333333in"
height="1.7916666666666667in"}
![](media/image8.jpeg){width="1.707638888888889in"
height="1.7916666666666667in"}

![](media/image9.jpeg){width="1.8111111111111111in"
height="1.7895833333333333in"}
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height="1.7916666666666667in"}
![](media/image11.jpeg){width="1.707638888888889in"
height="1.7909722222222222in"}

> **HRI at E-2 hrs HRI at E-0 sec HRI at E+800 sec**

**Figure 4** Simulated MRI and HRI images of the comet nucleus (based on
Halley-Stooke data) during encounter where E- designates time to impact.
Note the apparent rotation of the nucleus, seen in the MRI images, as
the Flyby s/c passes underneath

Two key science epochs drive system performance during the Tempel 1
encounter: Time of impact (TOI), and Time of final crater imaging
(TOFI). To obtain the highest possible temporal resolution imaging at
the TOI, the HRI will be operated in a 128x128 pixel sub-frame mode (see
figure 5) which allows images to be taken more rapidly than if they were
full-frame exposures. The HRI instrument has a 1008x1008 active pixel
charged-couple device (CCD) detector with a pixel scale of 2 μrad/pixel
(1 μrad = 10^-6^ radians), giving it a 2 mrad (1 mrad = 10^-3^ radians)
field-of-view (FOV). Therefore, overall pointing error must not exceed
128 μrad at TOI in order to capture the impact event in the HRI 128x128
pixel sub-frame. At the time of final imaging, the Flyby s/c will be at
a range of 700 -- 1000 km from the surface of the nucleus. At this
range, the HRI FOV covers only a small portion of the nucleus (mean
nucleus radius is estimated to be 2.6 km with an approximate axial ratio
of no larger than 2:1). Pointing at both of these science epochs
requires good knowledge of the impact site relative to the observed CB.
Analysis shows that for an active Impactor maneuvering to intercept the
CB, pointing at TOI, to capture the impact site in the 128x128 pixel
subframe, is achievable and less than 100 μrad (3σ) of which 60 μrad
(3σ) are due to uncertainties in knowledge of the actual impact site
location. Studies have shown that the probability of capturing a
high-resolution image of the fully developed crater is 97% for the
expected HRI instrument performance^3^.

Due to the short lifetime requirement (7 days) of the non-redundant
Impactor s/c, the inactive Impactor failure scenario has warranted
attention. There are a number of considerations that must be addressed
under this contingency, broadly divided into two aspects: 1) achieving
impact in an illuminated area with an inactive Impactor s/c; and 2)
autonomously predicting the impact location of the inactive Impactor s/c
to compute and apply a pointing correction relative to the CB in an
effort to minimize Flyby pointing performance degradation at the key
science epochs. The first problem is addressed by fine-tuning the
trajectory of the flight system to intercept the nucleus of Tempel 1
prior to the release of the Impactor s/c. If it is determined that the
Impactor s/c is not in an operational condition prior to release, then
the maneuver to fine-tune the trajectory will be delayed to allow for
additional optical navigation data collection and the release of the
Impactor will be postponed to E-12 hrs to increase the probability of an
illuminated impact. Here we discuss the method for solving the second
problem. I

If we consider only the contribution of errors in knowledge of the
impact site location, the baseline targeting and tracking approach
provides TOI and TOFI pointing performance as shown in table 1. In
addition, table 1 shows that for an inactive Impactor s/c our ability to
point the Flyby s/c instruments is substantially degraded due to
uncertainties in knowledge of where the impact will occur relative to
the nucleus center of brightness as observed from the Flyby s/c during
the last 2 hrs of the encounter. The results shown in table 1 for a
maneuvering Impactor s/c are only part of a larger pointing error budget
and must be combined, in a root-sum-square (RSS) sense with errors such
as knowledge of the Flyby s/c's position relative to the observed CB,
motion of the CB due to nucleus rotation, ADCS alignment errors, ADCS
alignment drifts, and ADCS control errors to arrive at the total
pointing error at TOI and TOFI.

**Table 1**

Expected Flyby s/c pointing errors due to impact site location
uncertainties at TOI and TOFI for the baseline targeting/tracking
approach and for the current inactive

Impactor s/c scenario

+---------------------+----------------------+------------------------+
| ####### Approach    | ####### TO           | ####### T              |
|                     | I Pointing 3σ Error  | OFI Pointing 3σ Error  |
|                     |                      |                        |
|                     | **(μrad)**           | **(μrad)**             |
+---------------------+----------------------+------------------------+
| ####### Baseline    | 60                   | 495                    |
|                     |                      |                        |
| ####### (Ma         |                      |                        |
| neuvering Impactor) |                      |                        |
+---------------------+----------------------+------------------------+
| ######              | 519                  | 3960                   |
| # Inactive Impactor |                      |                        |
+---------------------+----------------------+------------------------+

![](media/image12.jpeg){width="5.118897637795276in"
height="4.351968503937008in"}

**Figure 5** MRI image at TOI showing the HRI FOV (blue), the 512x512
pixel HRI subframe (green) and the 128x128 pixel HRI subframe (red)
within which the impact site must reside at TOI

**IMPACT SITE PREDICTION FOR AN INACTIVE IMPACTOR**

The basic problem with an inactive Impactor s/c is that it cannot
maneuver itself to impact at a location that is expected, independently,
by Flyby s/c. Although the Impactor s/c has a high likelihood of
impacting somewhere on the surface (studies show \~ 95% probability of
delivering an inactive Impactor on an impact trajectory^3^), pointing
the narrow FOV HRI instrument with the Flyby s/c is degraded: the Flyby
s/c will point at the CB which may be as much as 4.5 km (3σ) from the
actual impact site; an approach that may only capture crater images with
the wider FOV MRI instrument (7 m resolution) instead of capturing
images with the desired spatial resolution (3.4 m) obtained using the
HRI instrument.

The basic idea of impact site prediction relies on the fact that
following release, the inactive Impactor's trajectory remains
unperturbed until impact. The Flyby s/c will initiate a deflection
maneuver which is designed to control the Flyby miss-distance to 500 ±
50 km and slow the Flyby spacecraft to provide 800 ± 20 seconds of
science imaging from the time of impact to the time of shield mode entry
prior to passage through the inner coma dust environment (shield mode
occurs approximately 50 sec before the Flyby s/c reaches it's closest
approach point). If the Impactor s/c is healthy, then the deflection
maneuver will nominally take place at E-23:48 hrs (12 min after Impactor
release), where E- designates time of impact, and will be \~ 102 m/s in
magnitude. This maneuver results in execution errors that map to a 32 km
B-plane position error (3σ) at encounter and an 8 sec (3σ)
time-of-flight (TOF) error.

The orthogonal triad that represents the orientation of the B-plane
coordinate system relative to the International Celestial Reference
Frame (ICRF), as seen in figure 6, is determined as follows:

$$\hat{s} = \frac{{\overline{V}}_{\infty}}{\left| {\overline{V}}_{\infty} \right|}$$

Here, ${\overline{V}}_{\infty}$ is the comet-relative velocity in the
inertial frame of reference. The unit vector,$\hat{t}$, is derived by
rotating $\hat{s}$ by and angle of -90° about the ICRF$\hat{z}$axis,
projecting the result onto the equatorial plane, and normalizing (The
unit vector $\hat{t}$ is always in the ICRF *x-y* plane).

$$\begin{matrix}
{\hat{t}}^{'} & & \begin{bmatrix}
0 & 1 & 0 \\
 - 1 & 0 & 0 \\
0 & 0 & 1
\end{bmatrix}\ \hat{s} = \begin{pmatrix}
s_{y} \\
 - s_{x} \\
s_{z}
\end{pmatrix} \\
 & & \\
\hat{t} & & \frac{1}{\sqrt{s_{y}^{2} + s_{x}^{2}}}\begin{pmatrix}
s_{y} \\
 - s_{x} \\
0
\end{pmatrix}
\end{matrix}$$

And finally,$\hat{r}$is the cross product of $\hat{s}$ and $\hat{t}$.

$\hat{r} = \hat{s} \times \hat{t}$.

The transformation from ICRF to B-plane coordinates is given by:

$R_{\text{ICRF} \rightarrow \text{Bplane}} = \begin{bmatrix}
\hat{t} & \hat{r} & \hat{s}
\end{bmatrix}^{T} = \begin{bmatrix}
t_{x} & t_{y} & t_{z} \\
r_{x} & r_{y} & r_{z} \\
s_{x} & s_{y} & s_{z}
\end{bmatrix}$.

![](media/image13.wmf)

**Figure 6** Definition and orientation of the B-plane relative to the
International Celestial Reference Frame (ICRF)

If, following the deflection maneuver, the trajectory of the Flyby s/c
can be estimated relative to that of the Impactor, then the Flyby's
trajectory can be integrated and its position evaluated relative to the
Impactor s/c at TOI. The key is to treat the Impactor as an optical
beacon, relative to which the Flyby s/c's trajectory may be estimated
using optical images of the Impactor. The expectation is that these
images will provide very accurate determination of the Flyby's
trajectory crosstrack to the line-of-sight relative to the Impactor.
Alongtrack information will come primarily from radiometric tracking of
the Flyby (range and Doppler measurements). This post-deflection radio
tracking is part of the baseline approach and serves to evaluate the
need for a contingent deflection trim maneuver, at E-12 hrs, as well as
to reduce the TOF error introduced by the deflection burn. A summary of
the post-deflection Flyby navigation plan is as follows:

1.  Continue radio tracking of the Flyby s/c for 6 hrs after end of
    deflection burn

2.  Image the Impactor from the Flyby, beginning 35 minutes after
    release

3.  Estimate the Flyby's heliocentric trajectory with a radio-only
    solution

4.  Estimate the Flyby's Impactor-relative trajectory with an optical
    solution using initial conditions obtained from the radio-only
    solution

**OBSERVING THE IMPACTOR SPACECRAFT**

Following separation, the Impactor remains on a trajectory, which is
essentially the incoming asymptote relative to comet Tempel 1. The Flyby
s/c slows by \~ 101 m/s with an \~ 5.7 m/s velocity component
perpendicular to the incoming asymptote, giving a constant inertial view
angle of the Impactor from the Flyby s/c of \~ 3° with respect to the
Impactor's comet-relative trajectory. This angle does not change until
impact. Images of the Impactor and background stars will be acquired,
beginning 35 minutes after release, and processed on the ground. The
accuracy of the orbit determination based on these optical data depends,
among other things, on the Signal-to-Noise Ratio (SNR) of the background
stars and the observability of the Impactor s/c using the Flyby
instruments.

**Impactor Spacecraft Surface Properties**

The Impactor has an irregular shape and a mix of materials with varying
reflective properties, e.g. metal, paint, thermal blanket etc. The ITS
anti-boresight side of the Impactor s/c bus, the one that for most of
the time will be facing the Flyby s/c, has a hexagonal shape (figure 7).
The ITS boresight direction has a hemispherical shape made up of copper
plates, with six panels connecting the two sides. An accurate
determination of the optical signal strength obtained from the Impactor
at a given orientation would require knowledge of the reflective
properties, exact shape modeling and attitude history. The Impactor,
during the time it is imaged from the Flyby, may be turning to point the
ITS off-nucleus to obtain calibration images, or in the case that it is
inactive it will be tumbling at an unknown rate and direction. The
overall phase darkening of the Impactor at different attitudes will vary
in a complicated way, requiring a modeling process beyond the scope of a
contingency treatment. The approach is not to compute the exact signal
from the Impactor, but rather to establish whether reasonable, as well
as conservative assumptions, result in a sufficient SNR.

![](media/image14.png){width="2.6013888888888888in"
height="2.1118055555555557in"}

**Figure 7** Impactor s/c flight system configuration^4^

Our simplified assumption is an irregular shape with an average diameter
of 0.8 m and albedo of \~ 0.3 based on drawings and data from Ball
Aerospace Technologies Corporation (BATC). Assuming that we view mostly
a flat Impactor surface and that the single particles on the surface
scatter isotropically, we may adopt a Lommel-Seeliger reflection law
with an average incidence angle, i, equal to the phase angle ≈ 65°; and
an average emission angle, e, equal to the view angle from the Flyby s/c
≈ 3°. The integrated intensity at that phase angle compared to intensity
at a phase of 0° is: cos(i)/(cos(i) + cos(e)) = 0.29 corresponding to a
phase darkening of ≈ 1.3 magnitudes or a linear phase coefficient of
0.02 mag/degree. By parameterizing the phase darkening of the Impactor
with a phase coefficient, we can consider a variety of more conservative
phase functions: 0.04 mag/degree, a value typical for comets and
asteroids, or 0.06 mag/degree corresponding to some of the most
phase-darkened objects, such as comet Encke^5^.

**Impactor Spacecraft Optical Signal**

Figures 8 and 9 show the expected SNR in the peak pixel for a 1 second
and 3 second exposure, as a function of time for 3 different phase
functions for the HRI and MRI, respectively. The SNR is computed with
the current data for each instrument and with the rather conservative
assumption of 4 DN (≈ 120 e-) of read noise. As shown in figure 8, HRI
imaging guarantees a high SNR for many hours after separation. For the
MRI, however, the SNR is significantly lower. To maintain a value of SNR
≥ 7, which will typically guarantee a center-finding accuracy of \~ 0.1
pixels (1σ), we may need to consider longer exposures. Figure 9 shows
that a 3 second exposure can provide a SNR ≥ 7 for \~ 3 hours of
imaging. In general, the choice of camera (MRI, HRI) depends on many
factors, which include:

1.  Sufficient signal from at least two background stars. The FOV of the
    MRI (10 mrad) is 5 times larger than that of the HRI (2 mrad) FOV,
    but the HRI can detect stars fainter by at least 4 magnitudes
    compared to the MRI. Depending on the approach trajectory,
    background star availability may favor one instrument over the
    other.

2.  Ability to acquire the Impactor in a narrow HRI FOV. Because of
    errors in the deflection maneuver, which map to errors in the Flyby
    s/c position at time of Impactor acquisition, the Impactor will not
    necessarily be at the predicted location during imaging.

These considerations will be examined in detail after the approach
asymptote of the flight system becomes known (i.e. after launch).
Another issue, which is common to both instruments, is that of obtaining
star-relative Impactor images in the presence of an extended diffuse
coma, which is foreground to the stars and background to the Impactor.
Based on the Deep Impact Science Team predictions for the coma
brightness, a worst-case scenario whereby all of the observed outgassing
is the result of a narrow jet, gives the peak pixel brightness due to
coma of up to 80,000 e-/s in the pixels adjacent to the nucleus. For a
late release at E-12 hrs, imaging of the Impactor could extend to as
late as E-9 hrs. At that time the Impactor will be projected, relative
to the Flyby instruments, in a direction where the background coma is \~
15,000 km from the nucleus. At that distance the coma is quite faint.
Assuming the coma brightness decreases as r^-1^ with distance from the
nucleus, the coma background will be \~ 3 - 3.5 DN for an exposure of 1
-- 3 sec (i.e. comparable to the CCD system noise). This will result in
a small decrease of the Impactor's optical signal relative to the
background, and therefore, has no affect on center-finding accuracy.
Figure 10 shows a simulated image of the Impactor s/c as seen in the HRI
instrument at a range of approximately 180 km and with an exposure
duration sufficient to bring the signal up near full-well (16,383 DN).

![](media/image15.png){width="4.186805555555556in" height="3.40625in"}

**Figure 8** SNR of the Impactor s/c as observed with the HRI

![](media/image16.png){width="4.332638888888889in"
height="3.3333333333333335in"}

**Figure 9** SNR of the Impactor s/c as observed with the MRI

![](media/image17.jpeg){width="5.4125in" height="3.6694444444444443in"}

**Figure 10** Simulated subframe image of the Impactor s/c in the HRI at
a range of 179.9 km and phase of \~ 65°; blue dot represents location of
CB

###### SIMULATIONS AND MONTE CARLO BASED PERFORMANCE ANALYSIS

Given that sufficient optical signal from the Impactor s/c is expected,
as shown in the previous section, a suite of MATLAB algorithms have been
developed to:

1.  Simulate images of the Impactor s/c using either the MRI or HRI
    instrument

2.  Integrate the Impactor and Flyby s/c truth trajectories

3.  Perform image processing and orbit determination^6^

4.  Evaluate the Flyby position error (true position -- estimated
    position) at the key science epochs (TOI and TOFI)

5.  Compute the pointing error at TOI and TOFI

6.  Perform a Monte Carlo analysis to assess the pointing error due to
    the uncertainty in the impact site location at the key science
    epochs

The random parameters in the Monte Carlo simulations are the Flyby
trajectory deflection maneuver errors, and the center-finding errors
described previously (0.1 -- 0.2 pixels; 1 -- 2 μrad for the MRI). The
epoch time is taken to be Release plus 35 minutes (start OpNav imaging
arc) which results in a range of \~ 180 km from the Flyby s/c to the
Impactor s/c. The deflection burn is modeled as an impulsive maneuver
occurring at release plus 5 minutes, with maneuver execution errors
randomly sampled in the alongtrack direction (0.50 m/s, 1σ), relative to
comet Tempel 1, and crosstrack directions (0.12 m/s, 1σ). These errors
represent the expected flight system deflection maneuver execution
capability and are used to integrate the Flyby s/c truth trajectory.

The beginning of Impactor image acquisition occurs at Release plus 35
minutes and ends at Release plus 1 hour and 35 minutes to give a batch
orbit determination (OD) arc length of 1 hour. Images are sampled at a
15 min interval providing 5 observations that are used in the estimation
process. At the beginning of image acquisition, differences between the
Flyby s/c's true position and the Flyby s/c's expected position result
in pointing errors on the order of 3.8 mrad (3σ). Here, the position
error (3σ) results in a pointing error that is approximately 2 times
larger than the HRI FOV. Since immediate detection of the Impactor is
not guaranteed with the HRI, it will be preferable to use the MRI
throughout. In 500 simulation runs, the uncertainty in the initial
position of the Impactor never exceeded the MRI FOV. The maximum
pointing error was found to be 6.5 mrad at initial image acquisition.
The maximum dispersion from the MRI boresight was found to be 650.5
pixels or approximate 6.5 mrads in 500 simulation runs. Figure 11 shows
the dispersion of locations of the Impactor in the MRI FOV at initial
acquisition. The HRI FOV is superimposed (blue) and shows that the
associated Flyby s/c maneuver execution errors require use of the MRI
instrument for Impactor tracking to be assured of capturing the Impactor
observations.

![](media/image18.jpeg){width="4.7875in" height="4.072916666666667in"}

**Figure 11** Dispersion of Impactor locations in the MRI FOV at initial
image acquisition (Release + 35 min) for 500 simulation runs; the HRI
FOV is superimposed in blue

After computing the orbit solution, the estimated Flyby position is
evaluated at TOI (Release + 24 hrs). The angular difference between the
unit direction vectors from the true Flyby s/c position and estimated
Flyby position, relative to the Impactor s/c gives the pointing error at
TOI.

$$\vartheta_{\text{TOI}_{\text{Error}}} = \text{arccos}\left( {\hat{u}}_{\text{True}} \cdot {\hat{u}}_{\text{Est}} \right)$$

where

$${\hat{u}}_{\text{True}} = \frac{\overline{X}\left( t_{\text{TOI}} \right)_{\text{Flyby} \rightarrow \text{Impactor}_{\text{True}}}}{\left| \overline{X}\left( t_{\text{TOI}} \right)_{\text{Flyby} \rightarrow \text{Impactor}_{\text{True}}} \right|},\begin{matrix}
 & 
\end{matrix}{\hat{u}}_{\text{Est}} = \frac{\overline{X}\left( t_{\text{TOI}} \right)_{\text{Flyby} \rightarrow \text{Impactor}_{\text{Est}}}}{\left| \overline{X}\left( t_{\text{TOI}} \right)_{\text{Flyby} \rightarrow \text{Impactor}_{\text{Est}}} \right|}$$

Here,$\overline{X}\left( t_{\text{TOI}} \right)_{\text{Flyby} \rightarrow \text{Impactor}_{\text{True}}}$
and
$\overline{X}\left( t_{\text{TOI}} \right)_{\text{Flyby} \rightarrow \text{Impactor}_{\text{Est}}}$
are the true and estimated position of the Flyby s/c, relative to the
Impactor s/c at TOI, respectively.

At TOFI (TOI + 750 sec) the pointing error is computed differently,
since there is a discontinuity in the Impactor's trajectory at that
time. Following impact, the B-plane error vector, $\Delta\overline{x}$,
remains constant and represents that contribution to the pointing error,
which is due to the uncertainty in our knowledge of where the impact
site is located. Here we compute the B-plane error vector by first
transforming the vector difference between the Flyby s/c's estimated
position and the Flyby s/c's true position relative to the Impactor s/c
at TOI into the B-plane reference frame.

$$\Delta\overline{r} = \left\lbrack R_{\text{ICRF} \rightarrow \text{Bplane}} \right\rbrack\left( \overline{X}\left( t_{\text{TOI}} \right)_{\text{Flyby} \rightarrow \text{Impactor\_True}} - \overline{X}\left( t_{\text{TOI}} \right)_{\text{Flyby} \rightarrow \text{Impactor\_Est}} \right)$$

The error vector is then the projection of $\Delta\overline{r}$ onto the
B-plane to give the direction. The projection is scaled by the range and
pointing error at TOI to give the magnitude.

$$\Delta\overline{B} = \frac{\vartheta_{\text{TOI}_{\text{err}}}\left| \overline{X}\left( t_{\text{TOI}} \right) \right|}{\sqrt{\Delta r(1)^{2} + \Delta r(2)^{2}}}\begin{pmatrix}
\Delta r(1) \\
\Delta r(2) \\
0
\end{pmatrix}$$

Transforming the projection into the ICRF frame gives the inertial
correction vector that is used to evaluate the pointing error at TOFI.

$$\Delta\overline{x} = \lbrack R\rbrack_{\text{ICRF} \rightarrow \text{Bplane}}^{T}\Delta\overline{B}$$

Using the error vector and the true Flyby s/c range to the comet nucleus
at TOFI, the pointing error is computed from the law of cosines as

$$\vartheta_{\text{TOFI}_{\text{Error}}} = \text{arccos}\left( \frac{R_{\text{Flyby} \rightarrow \text{Comet}}^{2} + R_{\text{Flyby} \rightarrow \text{Impact\_Sit}e}^{2} - \left| \Delta\overline{x} \right|^{2}}{2R_{\text{Flyby} \rightarrow \text{Comet}}R_{\text{Flyby} \rightarrow \text{Impact\_Sit}e}} \right)$$

where

$$R_{\text{Flyby} \rightarrow \text{Comet}} = \left| \overline{X}\left( t_{\text{TOFI}} \right)_{\text{Flyby} \rightarrow \text{Comet}_{\text{True}}} \right|$$

and

$$R_{\text{Flyby} \rightarrow \text{Impact\_Site}} = \left| \overline{X}\left( t_{\text{TOFI}} \right)_{\text{Flyby} \rightarrow \text{Comet}_{\text{True}}} + \Delta\overline{x} \right|$$

Table 2 shows the error contribution of the uncertainty in the impact
site location to the overall pointing error at TOI and TOFI. The OD
arclength is 1 hr with observations taken every 15 min to give a total
of 5 optical navigation (OpNav) observations. The following section will
describe the autonomous algorithm that is necessary to point the MRI and
HRI instruments at the key science epochs using the estimated Flyby
s/c's trajectory.

**Table 2**

Monte Carlo results for Flyby s/c pointing error contribution due to
impact site location uncertainty at TOI and TOFI based on 500 simulation
runs

+------------------+-----------------+---------------+----------------+
| ####### K        | ####### Min     | ####### Max   | ######         |
| ey Science Epoch |                 |               | # 3\*RMS Error |
|                  | **(μrad)**      | **(μrad)**    |                |
|                  |                 |               | **(μrad)**     |
+------------------+-----------------+---------------+----------------+
| ####             | 3.47            | 41.2          | 72.3           |
| ### TOI Pointing |                 |               |                |
+------------------+-----------------+---------------+----------------+
| #####            | 21.3            | 328           | 441            |
| ## TOFI Pointing |                 |               |                |
|                  |                 |               |                |
| (TCA-100 sec)    |                 |               |                |
+------------------+-----------------+---------------+----------------+

###### POINTING ERROR SENSITIVITY TO OPNAV ORBIT DETERMINATION 

For this approach, pointing errors depend directly upon the accuracy of
the orbit determination process using optical images of the Impactor
s/c. To fully understand the problem and to assess the best approach, we
consider the sensitivity to OD arc length, center-finding errors, and
range from the Flyby s/c to the Impactor s/c during the OpNav imaging
arc.

To assess the sensitivity to the OD arc length, 3 different OD batch
lengths were considered. Each batch has the same starting epoch
corresponding to release (R) + 35 min and the same imaging interval of 1
image per 15 min. Table 3 shows the results for 1 hr, 2 hr, and 3 hr OD
batch solutions.

**Table 3**

Sensitivity to OD Arc Length

+----------------------+-----------------------+----------------------+
| **OD Arc Length**    | **RMS**               | **RMS**              |
|                      |                       |                      |
| **Number of          | **TOI Pointing Error  | **TOFI Pointing      |
| Observations**       | (μrad)**              | Error (μrad)**       |
+----------------------+-----------------------+----------------------+
| **1 hr arc ⇒ 5       | 24.1                  | 147                  |
| OpNavs**             |                       |                      |
|                      |                       |                      |
| **500 sim runs**     |                       |                      |
+----------------------+-----------------------+----------------------+
| **2 hr arc ⇒ 9       | 45.8                  | 297                  |
| OpNavs**             |                       |                      |
|                      |                       |                      |
| **100 sim runs**     |                       |                      |
+----------------------+-----------------------+----------------------+
| **3 hr arc ⇒ 13      | 37.5                  | 231                  |
| OpNavs**             |                       |                      |
|                      |                       |                      |
| **100 sim runs**     |                       |                      |
+----------------------+-----------------------+----------------------+

This analysis assumes center-finding errors of 0.1-0.2 pixels, which
represents a random angular measurement error of 1-2 μrad (1σ) in the
MRI instrument. Using the HRI, this error would be 0.2-0.4 μrad. Here,
we considered a single 1 hr OD batch arc length starting at R + 35
minutes with 1 observation every 15 min. Table 4 shows the results for 3
different center-finding uncertainties: 2 μrad (0.2 MRI pixels), 10 μrad
(1 MRI pixel), and 20 μrad (2 MRI pixels).

**Table 4**

Sensitivity to center-finding errors

+----------------------+-----------------------+----------------------+
| **Center-finding     | **RMS**               | **RMS**              |
| Error**              |                       |                      |
|                      | **TOI Pointing Error  | **TOFI Pointing      |
| **(1σ)**             | (μrad)**              | Error (μrad)**       |
+----------------------+-----------------------+----------------------+
| **2 μrad**           | 24.1                  | 147                  |
|                      |                       |                      |
| **(0.2 MRI pixels)** |                       |                      |
|                      |                       |                      |
| **500 sim runs**     |                       |                      |
+----------------------+-----------------------+----------------------+
| **10 μrad**          | 47.0                  | 289                  |
|                      |                       |                      |
| **(1 MRI pixel)**    |                       |                      |
|                      |                       |                      |
| **100 sim runs**     |                       |                      |
+----------------------+-----------------------+----------------------+
| **20 μrad**          | 88.2                  | 547                  |
|                      |                       |                      |
| **(2 MRI pixels)**   |                       |                      |
|                      |                       |                      |
| **100 sim runs**     |                       |                      |
+----------------------+-----------------------+----------------------+

Finally, the sensitivity to the Impactor range during OpNav imaging was
considered. Three different 1 hr OD arcs were considered: the first
batch started at R + 35 min and ended 1 hr later; the second batch
started at R + 1 hr 35 min and ended 1 hr later (R + 2 hr 35 min); and
the third batch started at R + 2 hr 35 min and ended at 3 hr 35 min.
Table 5 shows the RMS pointing errors at TOI and TOFI for each of the 3
OD arcs. The degradation with range is not unexpected since the OD
process relies on angular measurements whose spatial error increases as
the range increases.

**Table 5**

Sensitivity to Impactor range during OpNav imaging

+-----------------+---------------+------------------+----------------+
| **OD Arc        | **OD Arc      | **RMS**          | **RMS**        |
| Start**         | End**         |                  |                |
|                 |               | **TOI Pointing   | **TOFI         |
|                 |               | Error (μrad)**   | Pointing Error |
|                 |               |                  | (μrad)**       |
+-----------------+---------------+------------------+----------------+
| **R + 35 min**  | ####### R     | 24.1             | 147            |
|                 | + 1 hr 35 min |                  |                |
| **500 sim       |               |                  |                |
| runs**          |               |                  |                |
+-----------------+---------------+------------------+----------------+
| **R + 1 hr 35   | ####### R     | 28.7             | 176            |
| min**           | + 2 hr 35 min |                  |                |
|                 |               |                  |                |
| **100 sim       |               |                  |                |
| runs**          |               |                  |                |
+-----------------+---------------+------------------+----------------+
| **R + 2 hr 35   | ####### R     | 29.0             | 178            |
| min**           | + 3 hr 35 min |                  |                |
|                 |               |                  |                |
| **100 sim       |               |                  |                |
| runs**          |               |                  |                |
+-----------------+---------------+------------------+----------------+

###### THE AUTONOMOUS POINTING CORRECTION ALGORITHM

Until now we have discussed the process of ground-based OpNav to
determined the location of the Impactor s/c relative to the Flyby s/c
and the uncertainty in the location of the impact site, which results in
pointing errors at the key science epochs. The next step is to develop
an algorithm for computing a pointing correction on-board the Flyby s/c
and in real time so that the ADCS software can adjust the MRI/HRI
pointing during encounter.

Once the Flyby s/c's inertial, heliocentric trajectory has been
determined based on optical observations of the Impactor s/c, the
Chebyshev coefficients that represent the Flyby s/c and the Chebyshev
coefficients that represent the Impactor s/c trajectory as a function of
time throughout the entire encounter period, are uploaded to the AutoNav
flight software on the Flyby s/c. As shown in figure 2, AutoNav acquires
and processes images of comet Tempel 1 starting 2 hrs prior to the
expected TOI. Every minute, AutoNav updates its position and velocity
based on observations of the CB of comet Tempel 1 and generates an
ephemeris in the form of Chebyshev coefficients. These coefficients are
passed to the ADCS software, which evaluates the position of the Flyby
s/c and the position of Tempel 1 (also from a set of Chebyshev
coefficients) and computes the desired pointing direction to center the
comet CB in the HRI FOV. With every AutoNav OD update, the difference
between the Flyby s/c's ground-based inertial position (based on optical
observations of the Impactor s/c) and AutoNav's updated estimate of the
Flyby s/c's inertial position (based on optical observations of
Tempel 1) is computed and the resulting vector is applied directly to
correct the Impactor's heliocentric position as shown in figure 12.

![](media/image2.wmf)

**Figure 12** Graphical depiction of the inertial pointing correction,
∆*P*, which is computed by AutoNav. All state vectors correspond to the
same AutoNav OD update epoch time (i.e. 5 min prior to estimated TOI);
B-plane is normal to view

The difference between the Flyby s/c AutoNav solution (based on images
of the comet) relative to the inertial frame and the Flyby s/c GroundNav
solution (based on images of the Impactor s/c) relative to the inertial
frame is computed to be

$$\Delta\overline{X}(t_{\text{OD}})_{\text{Flyby}} = \overline{X}(t_{\text{OD}})_{\text{Flyby},\text{AutoNav}} - \overline{X}(t_{\text{OD}})_{\text{Flyby},\text{GroundNav}}$$

This correction is applied directly to the Impactor s/c's inertial state
obtained by evaluating the Impactor s/c ephemeris, which resides
on-board the Flyby s/c in the form of Chebyshev coefficients, at
$t_{\text{OD}}$

$$\overline{X}(t_{\text{OD}})_{\text{Impactor\_update}} = \overline{X}(t_{\text{OD}})_{\text{Impactor, GroundNav}} + \Delta\overline{X}(t_{\text{OD}})_{\text{Flyby}}$$

The updated Impactor s/c state is then mapped into B-plane coordinates,
where $B \cdot r,B \cdot t$ give the 2-D pointing correction (the impact
location), relative to the observed CB, in the B-plane frame. Since the
orientation of the B-plane is known relative to the ICRF, the 2-D
vector, $\Delta P$, is transformed into an inertial 3-D pointing
correction via the transpose of
$R_{\text{ICRF} \rightarrow \text{Bplane}}$ and passed to ADCS for
application.

$${\overline{X}}_{\text{Impactor\_update}} = \begin{pmatrix}
B \cdot r \\
B \cdot t \\
\text{TOF}
\end{pmatrix} \Rightarrow \Delta P = \begin{pmatrix}
{\Delta P}_{x} \\
{\Delta P}_{y} \\
{\Delta P}_{z}
\end{pmatrix}_{\text{ICRF}}$$

The process of computing the pointing correction occurs with every OD
update until TOI after which time the applied correction will remain
constant.

**SUMMARY**

Predicting the impact location to point the Flyby s/c instruments in the
presence of an inactive Impactor s/c can be achieved by treating the
Impactor s/c as an optical beacon, estimating the Flyby s/c's inertial
trajectory using images of the Impactor, uploading the estimated
trajectory to the Flyby s/c's AutoNav software and autonomously
computing a pointing correction for ADCS. This correction addresses only
that component of the overall pointing error at key science epochs (TOI
and TOFI), which is due to the uncertainty in the impact site location
for an inactive Impactor s/c. Simulations show that uncertainties in the
Flyby's position relative to the Impactor at the time of initial image
acquisition require use of the MRI instrument for Impactor optical
navigation purposes and acquisition of OpNav images as soon as possible
following release of the Impactor s/c provide the best measurements.
Results also show that for an inactive Impactor s/c, the pointing error
due to impact site location uncertainty is driven back down to a level
comparable with that of the baseline approach of a maneuvering Impactor
s/c that will target the nucleus center of brightness. By treating the
Impactor as an optical beacon pointing errors improve from 519 μrad to
72.3 μrad (3σ) at TOI, and from 3.96 mrad to 441 μrad (3σ) at TOFI; this
is the difference between a mission with seriously degraded science
return and a mission with flight system performance that equals the
current baseline approach, provided that the inactive Impactor impacts
in an illuminated area. The maneuvering Impactor design was selected to
first, improve the probability of impacting in an illuminated area and
second, impacting in a location, which is viewable from the Flyby s/c.

**AKNOWLEDGEMENTS**

The authors would like to acknowledge Bob Gaskell, NASA-Jet Propulsion
Laboratory, for his work on modeling cometary nuclei for simulation
analysis and for providing us with a sphere model of the Impactor s/c
and Robert Werner, NASA-Jet Propulsion Laboratory for his comments and
suggestions. The research described in this paper was carried out at the
Jet Propulsion Laboratory, California Institute of Technology, under
contract with the National Aeronautics and Space Administration.

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