Context
Detecting abrupt changes consists in identifying the instants when the statistical
behavior of a system changes significantly. This problem is crucial in a
wide range of engineering applications, where faults, environmental changes, or
operational mode switches may occur. In a real time setting, detecting such
changes must be performed on-line, i.e. whenever a sensor provides information
on the system. Different approaches have been proposed for this purpose over
the past 50 years. However, in many problems, systems can be described by
variables submitted to some geometrical constraints that classical approaches
fail to take into account. For instance, in the context of autonomous navigation,
a dynamic vehicle is equipped with accelerometers and gyrometers that,
after some pre-processing, yield the position and the attitude of the vehicle, i.e.
its orientation with respect to a reference frame. It takes the form of either a
rotation matrix or a quaternion, both of which take values on specific spaces
called Lie groups [8]. A variety of algorithms has recently been introduced to
infer such parameters on-line while preserving their properties. Most of them
are variants of the Kalman filter [1][2]. Nevertheless, in the case of highly maneuvering
vehicles or due to external constraints, they may experience value
jumps that should be detected, not only to ensure seamless navigation, but also
to obtain valuable information on the mobile and its environment.
Targeted application
In this thesis, we focus on satellite attitude control which consists in maintaining
the orientation of a satellite in space [4]. This problem is of paramount importance,
for instance in communication systems to ensure reliable communication
with Earth: antenna misalignment can indeed seriously impair data transmission.
Also, in remote sensing, Earth imaging sensors embedded on satellites
must be oriented toward an observation area. Generally, this orientation is
measured by a star sensor [3], often hybridized with an Inertial Measurement
Unit (IMU). In situations where a satellite experiences some failures, its attitude
can suddenly change and it is therefore essential to detect these variations.
Objectives of the thesis
This thesis aims to develop new methods for on-line change detection when the
data and/or the parameters of interest lie on matrix Lie groups. The latter have
the specificity of being equipped with both a Riemannian and a group structure
[7]. They make it possible to leverage mathematical tools (especially from the
fields of statistics and optimization). Practical examples are the set of rotation
matrices and the set of orthonormal matrices.
In this thesis, we focus on systems whose behavior can be captured through
assumed dynamic models. Their unknown states are then sequentially inferred
from the observations by a filter, usually a Kalman or particle filter [6]. Two
classes of approaches will be investigated for change detection:
- multiple-model based methods,
- statistical-test based methods.
a) In the first case, several candidate evolution models are considered and the
main challenge lies in identifying the instants when a transition occurs from one
dynamic model to another. Among state-of-the art methods, we can cite Jump
State Markov Model (JSMM) or Interacting Multiple Model (IMM) which are
formulated in a Bayesian filtering framework [9, 5, 7]. Although these methods
are well established in the literature, they are dedicated to Euclidean variables
and the objective of this thesis will be to generalize them within a Lie group
framework. b) In the second setting, the outputs of the filtering algorithm are
monitored to flag any deviations in their statistical behavior. Usually, residuals
defined as the difference between the actual observations and their predictions
are leveraged. Classical detection tests designed for that purpose are dedicated
to probability distributions defined on vector-spaces. Again, the objective will
be to adapt them to Lie-group defined quantities.
The algorithms developed will be theorized and then numerically validated
on our satellite attitude control problem: in the case of maneuvers, a dynamic
change on the orientation or/and the position of the satellite can be indeed modeled
on the Lie groups SO(3) or SE(3). By simulating synthetic observations
on SO(3) provided by a star sensor, the proposed framework will be tested. If
possible, the approaches will also be applied to real star sensor data.
General informations
Skills: statistical signal processing, recursive estimation, Kalman filtering
theory. Knowledge about information geometry will be an added-value.
- Programming tools: MATLAB and /or PYTHON.
- Required curriculum: MSc. or Eng. diploma in applied mathematics,
computer sciences or statistics. - Starting period: October 2026.
- Applications (CV, motivation letter, grades, recommendation letter) and
informal inquiries are to be e-mailed to Samy Labsir (Associate Professor,
IPSA/TéSA, Toulouse, Researcher associate, ISAE-SUPAERO, Toulouse), samy.labsir@ipsa.fr and Audrey Giremus , audrey.giremus@u-bordeaux.fr, (Full Professor, University of Bordeaux). - Location: TéSA laboratory, Toulouse, France and IMS Laboratory, Talence,
France. - Salary: 1500 euros net per month.
References
[1] Kalman R. E. A new approach to linear filtering and prediction problems.
Transactions of the ASME – Journla of Basic Enigneering, 82:35–45, 1960.
[2] Kalman R. E. and Bucy R. S. New results in linear filtering and prediction
theory. Transactions of the ASME – Journla of Basic Enigneering, 83:95–
107, 1961.
[3] Michael J. Lichter. Star tracker accuracy improvement and optimization
for attitude measurement. Nasa technical report, Air Force Institute of
Technology & NASA Glenn Research Center, 2020.
[4] F. Landis Markley and John L. Crassidis. Fundamentals of Spacecraft Attitude
Determination and Control. Springer, New York, NY, 2014.
[5] Blom H. A. P. and Bar-Shalom Y. The interacting multiple model algorithm
for systems with markovian switching coefficients. IEEE Transactions on
Automatic Control, 33(8):780–783, 1988.
[6] Arulampalam M. S., Maskell S., Gordon N., and Clapp T. A tutorial on
particle filters for online nonlinera/non-Gaussian Bayesian tracking. IEEE
Transactions on Signal Processing, 50(2):174–188, 2002.
[7] Dingler S. State estimation with the interacting multiple model (IMM)
method, 2022.
[8] Karlheinz Spindler. Optimal control on Lie groups with applications to
attitude control. Mathematics of Control, Signals, and Systems, 11(3):197–
219, 1998.
[9] Bar-Shalom Y., Rong Li X., and Kirubarajan T. Estimation with Applications
to Tracking and Navigation: Theory Algorithms and Software. John
Wiley and Sons, 2001.
