During the last 13.5 year motion cycle of the interplanetary research project, there were necessary flight path modifications of the Cassini spacecraft. In the order of signal travel time (approximatively 80 minutes) on the Earth-Cassini long sized channel, complex event detection of orbital modifications requires special investigation and analysis of the collected large trajectory dataset. This paper presents a sophisticated, in-depth learning approach for detecting Cassini spacecraft's trajectory modifications in postprocessing mode. The model uses neural networks with Long Short-Term Memory (LSTM) to extract useful data and learn the time series' inner data pattern, together with the penetrability of the LSTM layers distinguish dependencies between the long- and short-term phases.
A complex event’s significance is related to processing several events, followed
by paying attention to discriminate distinct occurrences within a time series of
events [
The trajectory represented contains just the last 13.5 Earth years including approaching phase, SOI event and orbiting around the Saturn of the Cassini spacecraft. This duration is approximately half period time of the Saturn trajectory around the Sun. The star, circle, and square characters mark the Sun, the first sample, and the last sample, respectively. After the SOI, the Cassini trajectory is a dynamic curve compound by an ellipse-based helicoid around Saturn, orbiting around the Sun in its own ellipse. It should be mentioned that the first amount of samples are before SOI, consequently the time interval between SOI and end of project is just 13.3 years. With the established method, we aim to detect events in post processing mode relevant to Cassini’s trajectory modifications. To this end, we put forward our model that uses the capacity of Long-Short Term Memory neural networks to produce useful data and learn the internal data structure of the trajectory time series and leverage the memory dependence LSTM potential.
In section 2 we provide a brief literature review of several related studies within
the field. Section 3 describes extraction of sampling and trajectory
characteristics of the Cassini database (Imaging Science Subsystem (ISS), Saturn EDR Data
Sets (Volume 1 – Volume 116) [
Cassini’s project’s trajectory was separated into three classes of activities
occurring in phases: i) launching and journey to Saturn, ii) approaching and arrival
at Saturn, and iii) science phase. Controlling the trajectory required processing
a variety of status information including certain step length, velocity, etc. The
approaching and arrival process provided complete project trajectory information.
Basic role of trajectory maps in the science phase is to position the spacecraft in a
specific location related to Saturn, which has been meticulously planned and has
suficient entrance conditions consistent with spacecraft velocity and path angle.
Paper [
Paper [
There were two pieces of the spacecraft: Huygens probe and Cassini orbiter.
Cassini-Huygens (C-H) arrived at Saturn in 2004, sending useful data back to
Earth. Huygens moved through Titan’s atmosphere, Saturn’s biggest moon, plunged
down by parachute to the furthest point so far, landed on his surface. Huygens took
samples, testing them, and submitting the findings to Cassini, who then returned
these signals to Earth. Remote sensing devices gathered data from vast distances
remotely. The acquired image data was provided by NASA’s Imaging Science
Subsystem (ISS). The ISS comprises of two detached wide-angle video cameras and a
narrow-angle camera. The dataset of ISS picture volumes comprises an immense
number of images and their corresponding labels. The dataset is freely accessible
by reference [
We evaluated the 116 volumes of the data collection from the above NASA
source. The staring study had time stamp 02:07:06 on February 6, 2004, and end
stamps on September 14, 2017, all clarified in UTC (Coordinated Universal Time).
The C-H project’s five key objectives (i.e., Saturn study, Moon Titan, Saturn
rings, ice satellites, and magnetosphere) involved trajectory maneuvers. Reference
[
With no oficial details on the Solstice mission’s performed trajectory modifications, our forecast is 68.4 percent (average of the previous two) of the scheduled maneuvers, providing 141 trajectory modifications. 4.
Our classifier’s key purpose is to specify complex events from sensory produced data; we expanded the sensory data index to detect temporal semantics for complex event detection. Potential dynamic data set occurrences are where observation sequences shift.
We consider severe trajectory adjustment as the Cassini orbiter velocity vector shifts more than a threshold metric. Since velocity is a measure vector, extreme trajectory occurrences mean satisfying either of the following two conditions: extreme alteration of the velocity direction or of the acceleration vector’s magnitude. 4.1. Modification per Time of the Velocity Vector Direction The speed of the angle modification
Δ between consecutive velocity vectors and +1 is given by the following formula: Δ = Δ cos ︁(
+1· ︁) +1 − ‖ ‖·‖ +1‖ , = 1, 2, . . . , − 1 (4.1) where and +1 are two consecutive velocity vectors of the orbiter, ‖ ‖ is the magnitude of the vector, Δ is the time interval between two consecutive samplings and = 1, 2, . . . ,
− 1. The number of vectors is the total number of samples in the
= 407, 303 − 13, 326 = 393, 977. Value around 0 of the Δ/
Δ means a very small modification of the direction per unit of time. Such cases were at the beginning of the project (see values before 2005 on the Figure 2a).
This amount of samples analysed by us belongs to the last 13.3 years of the C-H project. Starting with the SOI event, the angle of the consecutive samplings of the velocity direction modified in a higher range per unit of time. The spacecraft trajectory has been updated many times, but no specific knowledge on these incidents is accessible to the public. NASA’s open 116-volume archive includes samples of high time dispersion.
Values of the Δ/ Δ in the scale of over 1 rad/sec were sampled in case of relatively short delay time between consecutive samplings. The distribution of the Δ/
Δ is exponential conform to the right hand side histogram of Figure 2. 4.2. Modification of the Acceleration Vector Magnitude Trajectory adjustment happens as the magnitude of velocity vector v ( , , ) varies during successive sampling by a higher value than the threshold ℎ . The (a) Angle modification vs. date.
(b) Histogram of angle modification. estimation of the adjustment magnitude per unit of time of the velocity vector between two consecutive samples (being the moving average acceleration) is centered on the velocity components defined in the database as follows:
= + + = ‖Δ ‖ = ‖ +1 − ‖ , = 1, 2, . . . , Δ +1 − − 1.
The magnitude of velocity modification can be derived using the following relation:
‖ +1 − ‖2 = (Δ , )2 + (Δ , )2 + (Δ , )2 where Δ , , Δ , , Δ , are the modification of the orthogonal velocity components in the sampling interval , and + 1. The acceleration magnitude of the Cassini can be seen on the left-hand side of Figure 3. It can be observed that the distribution of the acceleration magnitude is power function conform to the right-hand side histogram of Figure 3.
The histogram denotes a structure in which the amplitude of acceleration events calculated in a specified magnitude duration is spread through possible magnitude values. Each level within the generated histogram expresses the acceleration rate among the acceleration span. A linear function, resulting in the histogram’s power function dependency, will approximate the log-log scale histogram. 4.3. Complex Event Detection of the Cassini Orbiter Trajectory Our classifier’s key purpose is to specify complex events from sensory produced data; we expanded the sensory data index to detect temporal semantics for complex (4.2) (4.3) (4.4) (a) Acceleration magnitude vs. date.
(b) Histogram of acceleration magnitude. event detection. Potential dynamic data set occurrences are where observation sequences shift. Let’s have trajectory adjustment indexes where set unique events: (4.5) (4.6) (4.7) (4.8) (4.9) = {1 < <
Δ − 1 | Δ
≥ ℎ } = {1 < <
− 1 | ≥ ℎ } = ∪ = ∩ where and are sample indexes of the analyzed NASA database for which the velocity direction modifications or the acceleration magnitude are greater than the corresponding threshold values. Set is used to sense the individual efect simultaneously of the two conditions mentioned in subsections 3.1 and 3.2. If the set ’s cardinality is low, then conditions (4.5) and (4.6) are not strongly dependent and both of them help to detect complex events on the trajectory. The resulting set contains all the sampling indexes detected by the proposed complex event detector.
= 1, 2, . . . ,
After the SOI phase in 2004, the spacecraft performed many trajectory adjustments in compliance with the Earth commands sent by the supervisor squad. Cardinality and of the sets and , respectively, give the number of extreme events called Cassini trajectory modifications dependent on circumstances (4.7) or (4.8) over the last 13.3 years of the project studied. Obviously, number of trajectory variations should be less than the amount of observations listed in Table 2.
Working points are represented with red bubble markers and are placed in the extreme modification coordinates of the gradient of the surfaces. Based on Table 1. the amount of conducted maneuvers is considered 323. By leveraging the dependence of severe values on the threshold values ℎ and ℎ within our model, we may evaluate the working point in three-dimensional space. Figure 4 sets out this dependency as a surface plot. The working point on both surfaces is placed on cordinates with extreme modification of the surfaces. To fulfill the total number of trajectory manoeuvres the values of the thresholds are ( ℎ , ℎ ) = (2.85/, 33.84/ 2). For these threshold values, the cardinality of the set and were found to be ( , ) = (323, 1). It was found that approximately two times more extreme events appear in set than in set . These two sets are not disjunctive because in several samples, both of the conditions (4.5) and (4.6) fulfill. The union of these two sets gives precisely 323 extreme trajectory adjustment cases. In the continuation, we demonstrate the approach for detecting these trajectory maneuvers with recurrent neural network.
(a) Cardinality of set vs. thresholds.
(b) Cardinality of set vs. thresholds.
4.4. Deep Learning Method of Detecting Trajectory
Modifications
The work presented in [
XTrain YTrain XValidate YValidate XTest YTest Index 1 1 (N-1)/2+1 (N-1)/2+1 3*(N-1)/4+1 3*(N-1)/4+1
(N-1)/2 (N-1)/2 3*(N-1)/4 3*(N-1)/4 N-1 N-1 Rows 7 7 7 7 7 7
(N-1)/2 (N-1)/2 (N-1)/4 (N-1)/4 (N-1)/4 (N-1)/4 The input to the RNN system is a sequence of sample ID, ∈ {1, . . . , − 1 = 393,976}, sampling intervals Δ = +1− , modification of the position coordinates (Δ , Δ , Δ ) and modification of the velocity components (Δ , , Δ , , Δ , ) among the last 13.3 years of the studied time interval. The input of the neural network is a 7 type matrix conform to the formula below:
= [ 1, 2, . . . , −1], where column vectors have the following elements:
= [Δ , Δ , Δ , Δ , Δ , , Δ , , Δ , ] .
−1 samples is divided into the subsets of objects conform to Table 3.
We used half of the samples for learning and the remaining half of samples is divided equally for validation and testing. Based on the formula (4.11) each matrix has 7 rows and
columns.
The human control team from Earth sent adjustment orders on the trajectory due to the mission’s numerous science and celestial goals.
The RNN method conducts a binary trajectory sample classification. The sequences of the sampled multidimensional time series will be detected depending on the trajectory’s extreme events. The orbiter’s automatic adjustments have been produced to hold the orbiter on the complicated helicoid described in Section 3.
It is common to stack LSTM layers for better modeling capacity, particularly when a large amount of training data is available based on its ability to manipulate previous computation knowledge. In principle, RNNs should accommodate arbitrary long-term dependencies in the input series to prevent the gradient issue where long-term dependencies occur. Our each neural network machine performance is a binary feature that demonstrates how the Cassini orbiter trajectory is highly changed. Each neural network has formation algorithm style ADAM; the threshold gradient method is L2Norm. Table 4 ofers additional criteria for the used set of neural networks.
The number of classes for the tested neural networks is two to detect the trajectory’s complex events. As the CED conditions (see relations 4.5, 4.6) are fulfilled for any trajectory samples, that sample is classified True, otherwise is classified False.
Training and validation of trajectory research data sets are conducted in the last 13.3 years of the Cassini project using a set of twelve separate recurrent neural networks. Half of them are LSTMs, while the remainders are BiLSTMs. The changing input parameters (mini-batch scale, number of hidden units, number of classes on layer three and layer four) and measured performance metrics (subsequent learning time and the precision of these networks’ identification) are shown in Table 5. It should note that for each RNN type we got detection accuracy over 99 %. Big diference is detected between the learning time of LSTM and BiLSTM network types. It is obvious that having both direction of the signal propagation the BiLSTM requires extra processing relative to the LSTM structure. 654.2 749.1 625.9 748.8 623.2 748.0 1089.8 1172.0 1093.4 1174.7 1092.4 1175.8
99.86 99.78 98.39 99.98 99.92 99.06 98.76 99.95 99.11 99.19 99.36 98.20 RNN
LSTM LSTM LSTM LSTM LSTM LSTM BiLSTM BiLSTM BiLSTM BiLSTM BiLSTM BiLSTM
A popular challenge in machine learning is research and building algorithms that can learn from and render data predictions. These algorithms operate by rendering data-driven forecasts or decisions by creating a theoretical framework from input data. The accuracy levels provided in the deep learning-based model’s spacecraft maneuvering behavior are acquired within the training dataset. The results support the view that deep models are important in the detection of trajectory modifications detection of sensory data.
Figure 6 (left) and Figure 6 (right) visualize the deficit during the learning process. It can be found that L2’s number of secret units afects learning loss. The fewer the number of hidden units are, the greater the loss are. For 10 hidden units, the learning loss doubles. The related activity has LSTM and BiLSTM networks, but the last one is learned on the L2 layer for 100 hidden units.
The ideal learning rate is bound to be exposed to the lack of the results’ learned behavior, which depends on the dataset and architecture used. The loss value shows how efectively or adequately the model performs during each learning iteration. Figure 6a reveals that RNN4 LSTM ofers the best loss relative to other versions. We can see that RNN8 BiLSTM has the strongest loss on the figure’s right hand relative to other versions. The eficiency of identification is the ratio between the number of identified trajectory corrections and the number of real trajectory corrections. In this context we name it accuracy metric like in popular statistical software tool environments.
Generally defined model accuracy after measuring the parameters of the this model is explained in percentage. It is called the model accuracy metric that compares the model forecast with the correct data. Figure 7 represent dependency of the accuracy on the learning time for each of the analysed RNN. All LSTM networks need lower learning time than for BiLSTM networks.
The model accuracy is identified after calculating the parameters of the used model with a percentage style; this is clear in the figure above; as an illustration, RNN4 is hitting an accuracy that exceeds the value of 99.9% with a minimum learning time with MiniBatch size = 5000, 100 hidden units and is able to learn the behaviour with 99.98% accuracy of the trajectory in less than 20.5 minutes on a desktop computer with 64 GB RAM and 12 CPU cores.
(a) Minibatch learning loss (LSTM).
(b) Minibatch learning loss (BiLSTM).
C-H spacecraft trajectory modifications detection method was conducted using LSTM/BiLSTM networks; as far as we know, this study is the first to detect the events of spacecraft trajectory modifications. By the provided results, we show that our test study clearly identifies that LSTM models with the chosen parameters have a reasonable option for specifying Cassini’s trajectory modifications. Their usage and extra stacked layers produce a noticeable boost in increasing detection process eficiency. It is worth mentioning that the models used can be comfortably generalized to cover a large scientific field relevant to estimation. With more specific information, the provided models present a robust processing step in employing the inner features and time-series representation via LSTM time dependencies for precise detection. The proposed detection model was capable of identifying 99.98% of the trajectory modifications of the Cassini orbiter.