Aerospace and Electronic Systems Magazine April 2017 - 30
Supervised Learning Algorithms for Spacecraft Attitude Determination and Control System Health Monitoring
Kernel function (non-linear cases).
ration is possible. Kernel functions also called "kernel trick" are
defined as follows:
k xi , x j xi · x j
Recall (2); now the optimization problem is modified as follows:
i j · yi · y j k xi , x j
2 i, j
but subjected to C ≥ αi ≥ 0 and ∑iαi · yi = 0.
Now the inner product φ(xi) · φ(xj) can be computed by kernel
functions without going through the map of φ(x) explicitly. α is
obtained from (4) (using a QP solver), and C is a regularization
parameter that penalizes the error during the optimization. C represents the tradeoff between error and margin. It is chosen by the
user; larger C means a higher penalty to errors. Figure 2 shows
nonlinear kernel function mapping philosophy. Among the large
choice of kernel functions , the two kernel nonlinear functions
1. Polynomial Kernel:
k xi , x j
xiT · x j 1 ,
where n is the degree of inner product kernel.
2. Gaussian Radial Basis Function (RBF):
k xi , x j
exp xiT x j
where σ = −1 / (2γ2) and γ is the kernel width parameter.
A more comprehensive explanation with some mathematical
concepts may be found in .
EXPERIMENTAL SETTINGS AND DATA
Telemetry contains massive data acquired from on-orbit sensory
measurements that indicate spacecraft SOH. Anomaly detection
algorithms/techniques described in the previous section were applied to the ADCS of remote sensing spacecraft owned by National
Authority for Remote Sensing and Space Sciences (NARSS) of
Egypt. In our design work, there are more than 1150 different types
of telemetry channels to be analyzed by ground operations. These
represent values measured by sensors and equipment, so they are
classified as: analogue/numerical values for variables (temperature, angular velocity, orientation angles, etc.), discrete or binary
(devices on/off), and categorical (instruments status, operations
modes) values of parameters/variables acquired from spacecraft in
The main purpose of ADCS is to orientate the main structure
of the spacecraft at the desired angle(s) within required accuracy.
Attitude of spacecraft can be represented in different ways with a
set of variables such as Euler angles, angular velocity, and quaternions, etc. ADCS composition contains angular velocity meter
(AVM) blocks, which are mounted in such a way that three AVMs
measuring axes are collinear to body-fixed axes, and the fourth
one is a backup. The system also had a magnetometer (MM), magnetorquers (MT), reaction wheels (RW), and star sensor. Figure 3
shows the space system structure. In our work we used only the
analog telemetry channels that indicate the measurements of interest, namely the telemetry channels related to the aforementioned
equipments and devices, since they are the most important for detecting anomalies in the ADCS subsystem. Table 1 introduces the
list of telemetry frames that were analyzed.
PLS-DA ALGORITHM ARCHITECTURE
The PLS-DA algorithm is mainly applied as a supervised learning
classification tool for spacecraft anomaly detection. The algorithm
coded in MATLAB and its strategy is to manage both nominal and
faulty operation status acquired via the ADCS telemetry data to
provide fault detection alarm for the system. The methodology for
classification operates in two phases. The first is the training phase,
which learns the classifier using the available labeled training data.
The second is the testing phase, which classifies a test instance as
nominal or an outlier using the classifier. The framework of the
algorithm is enumerated in the following steps.
1. Collect the ADCS telemetry from sensory orbit of the normal
operation as the training data.
2. Autoscaling (standardization) the training data to the unit
variance and mean centering (variance = 1 and mean = 0) to
unify the influence of the features before building the model.
A data set is composed of n rows and m columns. The n rows
are called observations here and the m columns are called
variables. Geometrically we can represent the observations as
points in a multidimensional space where the variables define
the axes . The lengths of the axes are determined by the
scaling of the variables. The results of PLS-DA modeling are
scale dependent. Selecting the scaling of the variables is therefore an important step in fitting projection models such as PLSDA. If one has no prior information about the importance of
the variables, autoscaling all variables to unit variance (UV) is
recommended. This is equivalent to making all of the variable
axes have the same length, which gives all the variables equal
importance. These measures allow the model to decide the affect of each variable regarding their real influence. If one has
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