Estimation of Motion and Shape of Asteroid Itokawa

Noboru Muranaka*,Makoto Maruya†, Hiroshi Ohyama†, Masashi Uo†, Hideo Morita‡,Takashi Kubota§, Tatsuaki Hashimoto§, Jun Saito§, and Jun'ichiro Kawaguchi§

For each image obtained during from September 30th to October 26th, 2006 (Total 139 images from 1 to 11Km distance), the attitude and position of the spacecraft HAYABUSA are estimated relative to the asteroid Itokawa. By combining these attitudes with those determined in an inertial coordinates by onboard star tracker, the asteroid spin axis direction in the inertial space is determined. Additionally, under the assumption of the spacecraft free flight motion, the spin axis, i.e. the asteroid center of mass, is located in the asteroid coordinate. Using these results the Itokawa shape model is created.

小惑星Itokawaの運動と形状の推定 概要：2005年9月30日～10月26日に亘って小惑星Itokawaから1～11Km の距離で取得された139枚の画像を用

いて、小惑星に相対的な画像取得位置とカメラ姿勢を推定した。スタートラッカで決定される慣性空間に対

する画像ごとのカメラ姿勢の情報と組み合わせることにより小惑星Itokawaの自転軸の方向を決定した。ま

た、探査機の直線運動の仮定のもとに小惑星の自転軸位置（質量中心）も推定した。本講演では、以上の小

惑星Itokawaの運動の推定結果を報告するとともに、カメラ位置・姿勢にもとづく小惑星形状の推定の結果

も報告する。

I. Introduction This paper describes the process of creating Itokawa shape and surface topography model during Hayabusa's

proximity observation. Before Hayabusa's approach to Itokawa, Ostro model was obtained by radar observation from the Earth4. This observation suggested that Itokawa is not simple ellipse shape, however the first detailed image of Itokawa taken by Hayabusa revealed its asymmetric unique shape. This irregular shape stimulated scientists, and it also challenged the guidance and navigation team to extremely precise control of Hayabusa.

Equipment used for Hayabusa's proximity observation is shown in Fig.1. Main instrument is a camera with a telescope (ONC-T). Its resolution is 1024 pixels by 1024 pixels and FOV is 5.7 degrees by 5.7 degrees. A laser altimeter (Lidar) is used to obtain range data from Hayabusa to Itokawa surface that was used to determine absolute scale of the shape model. Other instruments such as Near infrared spectrometer are omitted from this figure.

Figure 1. Observation of Itokawa by Hayabusa.

The total process of estimating the Itokawa motion and creating its shape and surface topography model is shown in

Fig.2. It can be divided into two processes. First half is to estimate camera motion relative to Itokawa and determining its spin axis, which is described in section two. Latter half of the process is to estimate 3D shape of Itokawa and to analyze the topography, which is described in section three. The inputs of this total process are images from the camera (ONC-T) and Hayabusa attitude data from the onboard star tracker, and range data from the laser altimeter (Lidar) . The outputs are GCP data, spin axis data, shape model data and a gradient map of Itokawa. In Hayabusa project, four global

* CosmoLogic, Ltd. 3-5-7 Kirigaoka, Midoriku, Yokohama 226-0016, Japan † NEC TOSHIBA Space Systems, Ltd., 1-10 Nissincho, Fuchu, Tokyo, 183-8551, Japan ‡ NEC Aerospace Systems, Ltd., 1-10 Nissincho, Fuchu, Tokyo 183-8501, Japan § Institute of Space and Astronautical Science (ISAS), JAXA, 3-1-1 Yoshinodai, Sagamihara, Kanagawa 229, Japan

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mapping teams were involved: guidance and navigation team, AMICA team, LIDAR team, and JPL team3,5. This paper describes the work of the guidance and navigation team.

Figure 2. Total Process of Creating Itokawa Shape and Surface Topography Model.

II. Determining Itokawa Spin Axis and Estimating Relative Camera Motion A. Asteroid Fixed Coordinate Systems

The asteroid fixed coordinate systems (AF) to describe the camera attitude and position and GCP’s position on the asteroid surface are defined at first such that Z-axis is parallel to the JPL announced spin axis, X axis normal to the Z-axis, and Y axis to complete the right handed reference frame. The direction of the Z-axis in J2000EQ reference frame is as follows; Alpha (J2000EQ) = 87.43 degrees Delta (J2000EQ) = -66.57 degrees X-axis is set so that the XZ-plane includes the GCP001 (named “USUDA”) i.e. the reference of the asteroid longitude. The origin of the coordinates is set at the vector sum point of all GCP’s determined through the coarse and fine estimation processing described next. (AF1) After determining the asteroid spin axis in the inertial frame J2000EQ, the Z-axis is redefined in the following direction; Alpha (J2000EQ) = 90.638 degrees Delta (J2000EQ) = -66.170 degrees The X and Y axes are defined in the same manner as for the AF1 coordinates. (AF2)

After the spin axis offset determination in XY plane, the origin of the coordinates above is translated by the amounts of the offset. Assuming the asteroid density is uniform, the center of mass along the Z-axis is calculated and its offset is used to translate in the Z-axis direction. As the GCP001 is located far from the asteroid principal axes, X-axis is re-defined so that XZ plane includes the GCP043 (named “Dark Polder”), which defines the final asteroid-fixed coordinates. (AF3) B. Camera Relative Attitude and Position Estimation (Coarse) Three images which include more than four common GCP’s are treated as the data processing unit (image set). The camera attitude and position of one of image among three is selected as the attitude and position reference, and the attitude and position of the other two images (and also the GCP positions) are estimated with respect to the reference (Fig.1). One of GCP’s within the reference image is selected as the reference GCP, the distance of which from the camera is set to unity. Its true scale is determined by the asteroid shape model and LIDAR data comparison in the step of “Model Scaling” shown in Fig.2. For the GCP’s which exist in more than two images and not used for the above processing, their positions are determined by the stereo matching method, of which results are used in the next fine estimation.

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By processing the image

sets and connecting their results, all image attitudes and positions relative to the reference image are determined (CF0: Camera Fixed Frame at Image#0). At the same time the locations of GCP’s covering the whole asteroid surface are also determined along with the camera attitude and positions. An example of the GCP’s distribution on one asteroid side is shown in Fig.4. The results obtained in the above processing are the

camera attitude and position and GCP’s positions relative to the image#0 position and attitude. They are transferred to the asteroid fixed frame (AF1) by the rotation and translation operation. One of the estimation results of the camera position is shown in Fig.5, which shows circular motion around the asteroid mainly due to the asteroid spinning motion. In the figure the arrow at each point shows the camera attitude.

Camera attitude D1 and position R1 for image#1

Zc

Xc

Yc

GCP

l0lipi

R1

l1,ip1,i R2

D2

D1 Camera attitude D2 and position R2 for image#2

Camera attitude and position for image#0 (reference) (Xc-Yc-Zc): Reference Frame

Figure 3. Definition of Coordinate in the Attitude and Position Estimation.

Figure 4. Example of GCP’s on Asteroid Surface.

Figure 5. Coarse Estimation Results of Camera Positions and Attitude.

C. Camera Relative Attitude and Position Estimation (Fine)

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The camera attitudes and positions and GCP’s positions in the CF0 coordinate system are finely estimated using the observation equations described in this section. The state vector is expressed by

δδ⎡ ⎤

= ⎢ ⎥⎣ ⎦

sx

z (1)

where

⎥⎦

⎤⎢⎣

⎡=

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

=

−−

m

mm

M

i

N

i εr

z

z

z

z

z

s

s

s

sδδ

δ

δ

δ

δ

δ

δ

δ

δ

δ ,,

1

1

1

1

M

M

M

M

(2)

N is total number of the GCP’s and M is total number of images used. δsi is the position error vector for the i-th GCP defined as follows;

(3) iiiiiii ll ssspp δδ +=+= ˆˆˆwhere li, and pi are slant range and unit direction vector of the i-th GCP, and corresponding quantities with hats are the calculated slant range and direction vector. The coarse estimation results are used as initial values at the start of the fine estimation. The GCP#0 is treated as the reference, so the error vector for this GCP is not included in the error vector δs in Equation (2). The error vector δzm is the 6-dimension vector comprised with position error vector δrm (3-dimension) and attitude error vector δεm (3-dimension) of the m-th image. These error vectors are defined as follows;

( ) mmm

mmm

I DεD

rrrˆ

ˆ

×+=

+=

δ

δ (4)

where rm= Rm /l0. The quantities with hats are calculated relative position vector and attitude matrix of m-th image. The error vector δs defined in Equation (2) has dimension of 3(N-1) and each error vector δr and δε has dimension

of 3(M-1) excluding the image#0 position and attitude as the reference image. Then, the state vector x defined in Equation (1), which is composed of the all unknown variables (camera attitude and position and GCP’s positions) has the dimension of 3N+6M-9 (=L). For the number of the images M of 139 and the number of GCP’s N of 43, the state vector x has the dimension of 954.

Unifying the observation equations for all GCP position data on all images, the equation for the state vector x is expressed as follows,

yAx = (5) The least square estimate of Equation (5) for the state vector x is given by the following equation.

(6) ( ) yAAAx TT 1−=

where (ATA)-1:LxL matrix, AT:Lx2(N-1) matrix. As the observation equations are

non-linear, Equation (6) needs six to seven times iterations to obtain the stable solution. Once the solution is obtained, using all image data yTy (residual sum) is calculated for the evaluation of the fitness to all observed GCP positions. The solution is searched around the p0, which provides minimum yTy, and output the result as a final best-fit solution, which takes into account the GCP#0 reading error.

Figure 6. O-C plot of the GCP’s Direction after Fine Estimation.

The O-C plots (GCP Observed direction - Calculated directions) are shown in Fig.6. The O-C’s are of the order of +/- 0.0002 degrees, which are consistent with the GCP centre reading errors on the images.

D. Spin Axis Determination and Attitude and Position Corrections

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The camera attitude and position for each image and GCP’s positions thus obtained through the fine estimation processing are presented in the image#0 reference frame (CF0). These are transferred to the asteroid fixed coordinates frame AF1, of which origin is located at the centre of the GCP’s, or the average of all GCP’s vector.

As the camera inertial attitude by the onboard star tracker is available for each image, the asteroid attitude against the inertial reference frame can be calculated using the following equation (Fig.7).

mT

mm CDA = (7)

where Dm is optically determined camera attitude against the asteroid through the coarse and fine estimations, Cm is the inertial attitude determined by the onboard star tracker. The plots of the Z-axis of the AF1 coordinates are shown in Fig.8 (left). The figure shows the coning motion of the Z-axis around the true spin axis with cone angle of 1.24degrees. Excluding the large errors seen in the direction of 200degrees on the plane (considered as the coupling errors between the attitude and position), the true spin axis is estimated as already described in A of

this section. The estimation error is +/- 0.025 degrees in both right ascension and declination directions. In a statistical sense, the asteroid nutation is not observed, of which angle is estimated less than 0.3 degrees.

Figure7. Transformation Matrix from Inertial Reference to Asteroid Fixed Reference

The temporal slopes of the X or Y axis phase angle provide the asteroid spin period. The X-axis phase difference calculated from the JPL spin period model of 12.13237hours is also plotted in Fig.8 (right). As the time span is short, the meaningful difference was not observed (the time span as long as one year is required to obtain the precise period of order of 0.00001 hours).

Re-defining the asteroid fixed coordinate system so that Z-axis coincides with the newly estimated spin axis and assuming the smooth asteroid rotation around this axis (no nutation), the camera attitudes are corrected.

Figure 8 AF1 Z-Axis Motion in Inertial Space (left) and X-Axis Phase Slope with respect to JPL Model (right).

E. Spin Axis Offset Determination and Camera Attitude and Position Corrections

As the camera position relative to the asteroid suffers the cyclic variation, if the asteroid Z-axis has the offset against the asteroid centre of mass as shown in Fig.9. Using the free motion arc data the Z-axis offset can be estimated.

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Figure 9. Asteroid Center of Mass Offset and Spacecraft Motion.

Appling the least square method to the free arcs, the asteroid center of mass position was obtained in the AF1

reference frame as,

maymax

CM

CM

9.11sin3.6cos

==−==

ϕϕ

The optically determined camera position and spin axis offset model fitting are shown in Fig.10, where the camera motion, expressed in an inertial coordinate and centered at AF1 origin, is comprised with three free arcs. The arcs cannot be connected by the free motion as the RCS firings were carried out between the arcs

The position is plotted in Fig11 along with the mass center estimated from the asteroid polygon model, which is described in the following sections. The distance of 3.4m on the XY plane between the offsets point and center of mass point shows the adequacy of both estimate: offset and center of mass , and also shows the accuracies of order of several meters throughout the asteroid motion determination and shape reconstruction.

Figure 10 Estimates Camera (Spacecraft) Motion and Spin Axis Offset Model Fitting Results.

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Figure11 Estimated Spin Axis Position on AF2 Coordinates and Center of Volume Position.

III. Asteroid Shape Modeling A. Shape from Limb Profiling

A limb profile of an asteroid image provides basic information about the asteroid shape, and it can be easily

extracted by pixel value shresholding. So, the authors adopted this simple, robust method as a first step for modeling the asteroid. In order to improve this method, limbs are discriminated into true limb (or Sun-side limb) and "false limb" (Fig.12). Only true limb provides correct shape information, on the contrary false limb may appear inside the asteroid area in the images and it may disturb shape recognition. This limb discrimination can be performed using the data from the onboard sun sensor. From a limb profile of an asteroid image, one "view volume", which circumscribes the three dimensional extent of an asteroid, can be produced. By intersecting several view volumes, 3D shape of the asteroid gradually appears. With the increase in the number of images used, the shape model approaches to the real shape (Fig.13). Figure 14 shows the shape model made from 258 view volumes (258 images). The limb profile of the model agrees with the contour of Itokawa images, however artificial ridges appear between large boulders. This is an inevitable weak point for limb profiling.

Figure 12. Discrimination of Limbs in Itokawa images.

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Figure 13. Process of Shape Modeling Figure 14. Verification of the Shape Model

by Limb Profiling Generated by Limb Profiling

B. Shape from Stereo Measurement and Merging Models Stereo measurement is another effective method to obtain 3D shape of an object, and it was adopted as a second step for modeling the asteroid. As the asteroid rotates with its spin axis, a sequence of stereo images is captured without intentional maneuver of a spacecraft. Stereo method can measure a part of the asteroid that is commonly visible from two cameras. It means that it is potentially capable of creating about half of asteroid shape at one measurement. This is a strong point for stereo measurements. On the other hand, a weak point is that the accuracy of stereo matching depends on the texture on the asteroid surface. If a surface area has not distinctive texture features, image matching may not perform well at the area and a produced model may contain noise. Figure 15 (above) shows an example of stereo image pair, taken from 7 km away from the asteroid and the angle between the two images is about 15 degrees. Figure 15 (below) shows the result of stereo measurement. It indicates that stereo matching performed successfully. The opposite side of Itokawa was also modeled with another stereo image pair, so we have two partial shape models made by stereo method.

Figure 15. Shape Modeling by Stereo Measurement

Figure 16. Merging Models to Generate Final Model.

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The third step of shape modeling is merging three shape models we have created so far (the model created by limb profiling or "L-model", and the two partial shape models made by stereo method) to generate a final model. In this step, L-model is used as limit of the three dimensional extent of the asteroid. Some noises on the two partial shape model are removed using this limit. On the other hand, detailed shape on two partial models, which cannot be modeled by limb profiling, is transferred to the final model. For example, artificial ridges on L-model are removed on the final model.

The final step of shape modeling is scaling. The scale of models created so far is relative, and it is changed into absolute by calculating the scale of AF3 using Lidar data, which has absolute values. Using the following equation, the scale of AF3 is calculated.

⎟⎠⎞

⎜⎝⎛ −⋅=

→→→

babcscaleac

where a is the position of Hayabusa represented by AF3, b is the AF3 origin, and c is Lidar measurement point (Fig.17).

b. AF3 (Relative scale)

Asteroid Surface

Lidar Measurement Data (Absolute scale)

a. Hayabusa (AF3 relative position)

c. Lidar Measurement Point

Radious (Relative scale)

Figure 17. Scaling using Lidar Data

C. Description of the Final Model

Figure 18 shows wire frame representation of the final shape model. It also indicates GCP positions as small pink sphere (two meter in diameter). Measuring the gap between the shape model and GCP positions is one of techniques to estimate the accuracy of the shape model. The gap of this figure suggests that the shape model has accuracy of several meters. Table 1 shows the dimensions of the final model.

Table. 1 Dimensions of the Final Shape Model. --------------------------------------------------------------- Size

- Axis X: 540m Y: 256m Z: 216m - Bounding Box X: 562m Y: 293m Z: 239m

Volume 1.73x107 m3 Surface Area 4.01x105 m2 ---------------------------------------------------------------

Figure 18. The Shape of Final Model.

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IV. Conclusion The authors described the process of creating Itokawa shape and surface topography model. It was performed successfully and contributed to navigation and guidance of Hayabusa.

Acknowledgments The authors wish to thank Asteroid Multi-band Imaging Camera (AMICA) science team for the service of camera

imaging data.

References 1Maruya , et al., "Modeling and Analyzing Itokawa Topography for Hayabusa Touchdown and Sample Collection", Lunar and Planetary Science Conference, 2006. 2Makoto Maruya, et al., "Estimation of Motion and Shape of Asteroid Based on Image Sequences", 14th International Symposium on Space Flight Dynamics – ISSFD, 1999. 3Hirohide Demura, et al., "Pole and Global Shape of 25143 Itokawa" Science 2 June 2006:Vol. 312. no. 5778, pp. 1347 - 1349. 4Ostro,S.J. et al., "Radar observations of asteroid 25143 Itokawa (1998 SF36)", Meteoritics & Planetary Science 39:407-424,2004.5Robert W.Gaskell, "Landmark Navigation and Target Characterization in a Simulated Itokawa Encounter", AAS/AIAA Astrodynamics Specialist Conference, 2005.

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