Method and implementation of the inertial frame bundle block adjustment

129 Pages·2017·3.35 MB·English

by  Steffi Burmeister

Determining Rotational Elements of Planetary
Bodies
Method and Implementation of an
Inertial Frame Bundle Block Adjustment
vorgelegt von
Dipl.-Math.
Steffi Burmeister
geb. in Schwerin
von der Fakultät VI – Planen Bauen Umwelt
der Technischen Universität Berlin
zur Erlangung des akademischen Grades
Doktorin der Naturwissenschaften
– Dr.-rer.-nat. –
genehmigte Dissertation
Promotionsausschuss:
Vorsitzender: Prof. Dr. Frank Neitzel
Gutachter: Prof. Dr. Jürgen Oberst
Gutachter: Prof. Dr. Jürgen Kusche
Tag der wissenschaftlichen Aussprache: 27. Januar 2017
Berlin 2017

Acknowledgement
I’d like to thank Professor J. Oberst for the opportunity to write a PhD thesis
and do research on a fascinating and inspiring topic, for the gained insight in
the field of planetary geodesy and the knowledge that I was able to gather
during the past four years of work under his supervision.
I am also grateful to my colleagues Dr. K. Willner, A. Pasewaldt and to F.
Preusker for various discussions and especially for providing the data sets of
Phobos and Vesta used for this work.
Finally I thank Simon Fear for his clear opinion about tables, and my family
and friends for their curiosity, encouragements and kindness.

Abstract
The rotational elements of a planetary body state its orientation in space with
respect to the International Celestial Reference Frame (ICRF). They are used
for computing control point networks (CPNs) and for creating e.g. maps or
terrain models of the body. This work describes the inertial frame bundle
block adjustment. It is a method to determine rotational elements of planetary
bodies in a direct and analytical way from image point measurements. Simul-
taneously with the rotational parameters, a corresponding CPN is computed
and the external orientation data of the camera are improved. In contrary to
the classical photogrammetric bundle block adjustment, here the interdepen-
dence of the body’s rotational model and the camera’s external orientation is
resolved. The method was tested on simulated data and applied to two real
science cases. The amplitude of forced libration, which contributes to the ori-
entation of the prime meridian, was determined for the Martian moon Phobos.
Using combined images of the missions Viking and Mars Express, an ampli-
tude of 1:13(cid:14) (cid:6)0:13 was computed together with a CPN of 680 points and an
uncertainty of ? 55 m. The amplitude is in agreement with previous results,
derived by indirect methods (e.g. Willner et al., 2010; Oberst et al., 2014).
Secondly, the pole axis orientation of Vesta was confirmed using images of the
current Dawn mission. The computed CPN of 82806 points is compared with
the body-fixed solution of Preusker et al. (2012), showing that both networks
are plausible within the average intersection error of 10 m. The software which
was implemented is focusing on the application to large data sets. It could be
achieved that the costs of memory and running time depend only on the num-
ber of images. In the example of Vesta with 5440 images, 164 hours (out of
168) are saved with respect to a reference adjustment software. The method
is suitable to study and refine rotational models of bodies with solid surfaces.
adjustment, bundle block adjustment ICRF, forced libration amplitude,
Phobos, Vesta, parallel inversion
Zusammenfassung
Die Rotationselemente eines planetaren Körpers geben seine Orientierung im
Raum in Bezug auf den internationalen Himmelsreferenzrahmen (engl. ICRF)
an. Sie sind notwendig für die Berechnung von Kontrollpunktnetzen (KPN)
und für die Erstellung von z.B. Karten oder Geländemodellen des Körpers.
Diese Arbeit beschreibt den Bündelblockausgleich im inertialen Referenzrah-
men. Das ist eine Methode, um Rotationselemente planetarer Körper direkt
und analytisch anhand photogrammetrischer Messungen zu bestimmen. Si-
multan wird ein KPN berechnet und die externen Orientierungen der Kamera
werden verbessert. Im Gegensatz zum klassischen körperfesten Bündelblock-
ausgleich ist hier die bestehende Abhängigkeit zwischen dem Rotationsmo-
dell des Körpers und der Kameraorientierung aufgelöst. Die Methode wurde
anhand einer Simulation getestet und auf zwei reale Datensätze angewandt.
Die Amplitude der erzwungenen Libration, die sich auf die Orientierung des
Hauptmeridians auswirkt, wurde für den Marsmond Phobos bestimmt. Mittels
Bilddaten aus den Missionen Viking und Mars Express ist eine Amplitude von
1;13(cid:14) ((cid:6)0;13) berechnet worden, das zugehörige KPN mit 680 Punkten hat
eine Unsicherheit von ? 55 m. Die Amplitude stimmt mit den Ergebnissen
von Willner et al. (2010) und Oberst et al. (2014), die durch indirekte Me-
thoden erhalten wurden, überein. Die Polachsenorientierung von Vesta wurde
mittels Bilddaten aus der aktuellen Dawn Mission bestätigt. Das berechnete
KPN mit 82806 Punkten wurde verglichen mit dem Ergebnis von Preusker et
al. (2012) aus einem körperfesten Bündelblockausgleich; beide Lösungen sind
innerhalb des mittleren Schnittfehlers von 10 m plausibel. Die Software wurde
mit Schwerpunkt auf große Datensätze implementiert. Dabei konnte erreicht
werden, dass die Kosten für Speicher und Laufzeit nur von der Anzahl der Bil-
der abhängig sind. Beim Vesta-Beispiel mit 5440 Bildern werden im Vergleich
zu einer Referenz-Software 164 von 168 Stunden eingespart. Die Methode ist
geeignet, um Rotationsmodelle von planetaren Körpern mit fester Oberfläche
zu studieren und zu verbessern.
Ausgleichsrechnung,BündelblockausgleichICRF,Librationsamplitude,Pho-
bos, Vesta, parallele Inversion
Contents
1 Introduction 1
1.1 Motivation and content . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Reference frames . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.1 International Celestial Reference Frame (ICRF) . . . . . 5
1.2.2 Hipparcos Celestial Reference Frame (HCRF) . . . . . . 7
1.2.3 Spacecraft and camera reference frame . . . . . . . . . . 7
1.2.4 Body-fixed reference frame . . . . . . . . . . . . . . . . . 9
1.3 Rotational elements . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.4 Control point network analysis . . . . . . . . . . . . . . . . . . . 13
1.5 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.6 Missions and data . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2 Definitions and Foundations 23
2.1 Mathematical foundations and notation . . . . . . . . . . . . . . 23
2.2 Conversion between reference frames . . . . . . . . . . . . . . . 25
2.3 Body-fixed bundle block adjustment . . . . . . . . . . . . . . . . 28
2.3.1 The functional model . . . . . . . . . . . . . . . . . . . . 30
2.3.2 The stochastic model . . . . . . . . . . . . . . . . . . . . 34
3 Inertial Frame Bundle Block Adjustment 37
3.1 The extended functional model . . . . . . . . . . . . . . . . . . 37
3.2 Derivatives of rotational elements . . . . . . . . . . . . . . . . . 39
3.3 Derivatives of classical parameters . . . . . . . . . . . . . . . . . 42
3.4 Test case - a simulation of Phobos . . . . . . . . . . . . . . . . . 43
3.5 Numerical stability . . . . . . . . . . . . . . . . . . . . . . . . . 48
i
4 Application to Large Data Sets 51
4.1 Sparse matrix compression . . . . . . . . . . . . . . . . . . . . . 53
4.2 Optimisation of running time . . . . . . . . . . . . . . . . . . . 60
4.2.1 The splitting technique . . . . . . . . . . . . . . . . . . . 60
4.2.2 The inverse decomposition method . . . . . . . . . . . . 61
4.2.3 Sufficiency of the inverse decomposition . . . . . . . . . . 63
4.2.4 An iterative inverse decomposition algorithm . . . . . . . 66
4.2.5 Parallel computation . . . . . . . . . . . . . . . . . . . . 71
4.3 Implementation of the decomposition algorithm in OpenCL . . . 73
4.3.1 Memory design . . . . . . . . . . . . . . . . . . . . . . . 74
4.3.2 Tile size . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.3.3 Work-groups and work-items . . . . . . . . . . . . . . . . 75
4.3.4 The parallel decomposition algorithm . . . . . . . . . . . 76
5 Application to the science cases Phobos and Vesta 81
5.1 Phobos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.1.1 Forced libration amplitude . . . . . . . . . . . . . . . . . 83
5.1.2 Control point network for Phobos . . . . . . . . . . . . . 83
5.2 Vesta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.2.1 Pole axis orientation . . . . . . . . . . . . . . . . . . . . 87
5.2.2 Comparison with previous CPN solution . . . . . . . . . 88
5.2.3 Surface spherical harmonics analysis . . . . . . . . . . . 90
6 Conclusions and Outlook 97
6.1 Summary and conclusions . . . . . . . . . . . . . . . . . . . . . 97
6.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
Bibliography 101
A Appendix 105
A.1 Proof of the split technique . . . . . . . . . . . . . . . . . . . . 105
A.2 Backward operation in split mode . . . . . . . . . . . . . . . . . 106
A.3 Phobos tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
A.4 Centre of figure . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
B Code examples for Compressed Sparse Matrices 115
Chapter 1
Introduction
1.1 Motivation and content
Since the final decades of the 20th century, spacecraft have been launched to
studyobjectsinthesolarsystem. Inmanycasesthespacecraftcarriesanimag-
ing system to gather optical information about the target. For this thesis the
focus is set on planetary bodies with solid surfaces - that means planets, dwarf
planets, moons, asteroids or comets which are suitable for a photogrammetric
survey. In the field of planetary geodesy images are used to study properties
of planetary bodies, e.g. topography, shape and rotation. Major objectives
of these studies are the definition of reference systems, the determination of
rotational parameters as well as mapping and modelling of planetary surfaces.
This includes in particular the computation of control point networks (CPNs)
which are a first-order realisation of the body’s reference system and define
the body-fixed reference frame by three-dimensional coordinates. While a body
moves in space, it changes its orientation e.g. by self-rotation, the precession
of the pole axis or librations. These changes in orientation are described by
the rotational model of the body in terms of functions of time, formulated with
respect to an inertial reference system. These functions indicate the direction
of the pole axis as well as the orientation of the body’s prime meridian. Any
error in the rotational model contributes to a misalignment between optical
measurements and the corresponding ground coordinates. Hence, in order to
create stable maps of high quality, a precise knowledge of the rotational pa-
rameters is desired.
1
2
In the past century spin poles, rates and sense of rotation of the large asteroids
were mostly determined by earth-based measurements of lightcurves and an
analysis which included a theoretical shape model. Magnusson (1986) gives a
good overview about the applied methods and the literature of that era. He
consolidated some of the common methods and proposed a procedure which
is here referred to as (parameter) range scan method. There, a series of anal-
yses with trial values for the parameters to be determined is performed, and
the data are statistically evaluated such that a best-fit yields the final result.
This approach, gradually improved e.g. by De Angelis (1993), was widely
used. With the available image data of the Hubble Space Telescope Camera
and the spacecraft missions, the trial values are typically evaluated with re-
spect to a computed CPN or the related improvements of image observations
(Thomas et al., 1995). A special effect, that has been verified for a few moons
in the solar system, is the forced libration of satellites. This small variation
in the mean self rotation rate depends on the satellite’s moments of inertia
and the gravitational torque that a primary body exerts on its satellite. The
amplitude of the forced libration contributes to the orientation of the prime
meridian and is therefore an important rotational parameter. An angular error
in the model about the prime rotation axis will lead to a positional error of
the control points; this error is proportional to the size of the body. Further-
more, since the magnitude of this libration is related to the magnitude of the
gravitational torque, conclusions about the internal structure of a body can
be drawn (Murray and Dermott, 1999). By applying the range scan method
with a bundle block adjustment or similar type of CPN analysis, amplitudes of
forced libration have been obtained for the Martian moon Phobos (Duxbury
and Callahan, 1989; Willner et al., 2010, a.o.) as well as for the Saturnian
moons Janus, Epimetheus (Tiscareno et al., 2009) and Mimas (Tajeddine et
al., 2013). An alternative method has been introduced by Oberst et al. (2014)
who used a rotation model of Phobos without longitudinal libration and eval-
uated the spacecraft position residuals over the anomalistic period1 after a
bundle block adjustment. With respect to Phobos also a theoretical approach
has been used, where the equations of motion are solved by numerical integra-
tion and the magnitude of forced libration could be determined as one of the
1pericenter to pericenter