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Observability of Light Curve Inversion for Shape and Feature Determination Exemplified by a Case Analysis

The Journal of the Astronautical Sciences volume 69, pages 537–569 (2022)Cite this article

Abstract

As the resident space object population continues to grow, Space Situational Awareness becomes most important for reducing the risk of collision among these objects. Obtaining object characteristic information, such as shape or reflectivity properties among other aspects, is essential for precise orbit propagation and object identification. Measurements of object brightness over time, or so-called light curve measurements, have a rich history of use for characterizing astronomical objects. If light curve measurements do not sufficiently capture the geometry of a system, the resulting shape and characteristic estimates from light curve inversion are not guaranteed to be accurate. Previous methods for increasing the likelihood of sufficient sampling involve acquisition of unfeasibly large amounts of light curve data, which binds valuable sensor resources to focus on one object for long periods of time. In this paper, observability is defined for the shape inversion problem from light curve measurements with a diffuse reflection model. This opens the horizon for efficient and effective object characterization on a routine basis within a sensor network, thus making dedicated, several night-long observations of one object for characterization obsolete. A realistic orbit and attitude motion are implemented to determine whether supplied light curve measurements of an Atlas V upper stage are sufficient for light curve inversion.

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Acknowledgements

This work was funded through the Department of Defense Science, Mathematics And Research for Transformation (SMART) Scholarship-for-Service Program. The views expressed in this article are those of the authors and do not necessarily reflect the official policy or position of the Air Force, the Department of Defense, or the U.S. Government. The authors thank the Astronomical Institute of the University of Bern (AIUB) Switzerland for providing some of the light curve measurements used in this paper.

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  1. Air Force Research Laboratory, Kirtland AFB, NM, 87113, USA

    Alex M. Friedman

  2. School of Aeronautics and Astronautics, Purdue University, West Lafayette, IN, 47906, USA

    Carolin Frueh

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Appendix A: EGI Observability of a LEO Object

Appendix A: EGI Observability of a LEO Object

The following simulations consider a LEO object, with orbital elements and epoch defined in Table 6. Cartesian position and velocity are propagated with two-body motion. Two attitude profiles are applied: one is a simple, single-axis rotation about the body z-axis and one is a more complex, three-axis rotation. The rotation rates for each of the two attitude profiles are given in Table 7. An example of the three-axis rotation motion is shown in Fig. 23. The color gradients depict the motion of the body-fixed axes over time. The body z-axis is tilted 30∘ from the inertial z-axis, and the rotation rate about the body z-axis is defined in Table 7.

Table 6 LEO orbit definition
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Table 7 LEO attitude profiles
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Fig. 23
figure 23

Body-fixed axes over time for the three-axis rotation case

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The analysis of the LEO object focuses on the impact of the time between measurements and the EGI tessellation number on the observability of the EGI minimization. Therefore, observability analysis is performed with a range of times between measurements and EGI tessellation numbers, given in Table 8.

Table 8 Measurement spacing and EGI tessellation numbers analyzed for the LEO object
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A.1 Single-Axis Rotation Systems

The analysis of the EGI minimization observability begins with a simulation of a LEO object with a simple, single-axis rotation. The rank of the observability Gramian is calculated for a range of measurement spacing cases as defined in Table 8. For each case, measurements are spaced equally for a total analysis time of two hours. Figure 24 depicts the LEO simulation observability results with each line representing a different measurement spacing. This analysis is performed with a tessellation number of seven, which is equal to 294 surface normal directions on the EGI. The black dashed line in Fig. 24 represents the number of EGI surface normal directions.

For this system to be observable, the rank of the observability Gramian must be equal to the number of EGI surface normal directions. All of the measurement spacing cases in Fig. 24 do not reach full rank due to insufficient sampling of the EGI as the observer-object-Sun geometry and attitude change. Figure 25 depicts the three smallest measurement spacing cases with a shorter analysis time. When the time between measurements is one second, there is a plateau in the observability Gramian rank curve from approximately 0.05 to 0.14 hours from the analysis epoch. During this plateau, more measurements are acquired, but the system geometry has not change sufficiently to increase the rank of the observability Gramian.

Fig. 24
figure 24

Rank of the observability Gramian versus time for the LEO object with single-axis rotation and 294 surface normal directions

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Fig. 25
figure 25

Rank of the observability Gramian versus time for the LEO object with shorter analysis time showing more detail

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The observability Gramian rank is compared to the number of measurements in Fig. 26 for selecting efficient measurement spacing. As with the observability Gramian rank versus time, Fig. 27 shows how redundant measurements exist for the 1 second and 2.5 second measurement spacing cases. The 1 second measurement spacing case has approximately 300 redundant measurements which do not increase the rank of the observability Gramian. Measurement spacing cases with longer times between each measurement result in fewer measurements to reach the maximum observability Gramian rank for this system.

Fig. 26
figure 26

Rank of the observability Gramian versus number of measurements for the LEO object with single-axis rotation and 294 surface normal directions

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Fig. 27
figure 27

Rank of the observability Gramian versus number of measurements for the LEO object with single-axis rotation and 294 surface normal directions, zoomed in to show detail

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The rank deficiency of the observability Gramian is also demonstrated with the 3D representation of the linear independence of the reflection matrix columns over time in Fig. 28. The colorbar represents the measurement indices in the analysis, and in this case, also the analysis time because the one second measurement spacing case is depicted. The EGI is defined by a tessellation number of seven, resulting in 294 surface normal directions. Note that the EGI is viewed in the body frame viewed down the observer vector towards the object. The visualization appears to show how the object becomes visible to the observer, but this is not the case. The color change of the facets over time is an indication of the linear independence of the reflection matrix columns.

Fig. 28
figure 28

3D visualization of the linearly independent columns in the reflection matrix for the LEO object with single-axis rotation, 294 surface normal directions, and one second between measurements, where the colorbar represents measurement indices

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From the 3D visualization, the period of redundant measurements for increasing the rank, which is equivalent to the plateau in Fig. 25, is clearly shown. The bottom row of Fig. 28 contains a gap of color between blue and yellow which is another way the measurement redundancy is shown with the one second measurement spacing case. The specific regions of insufficient sampling of the system geometry are apparent in the 3D visualization. A gray region at the bottom of the EGI remains after all of the measurements have been accumulated, indicating that the columns of the reflection matrix associated with those specific surface normal directions have not been sampled sufficiently.

A.2 Three-Axis Rotation Systems

The observability of the EGI minimization is performed for the same LEO object with a more complex attitude profile. When the LEO object has an attitude profile defined by three-axis rotation, it is possible to sufficiently sample the system geometry for EGI minimization, as shown by the rank of the observability Gramian over time in Fig. 29. Similar to the single-axis rotation results, the 1 second and 2.5 second cases have measurements which are redundant in terms of increasing the observability Gramian matrix rank.

Figure 30 shows the rank of the observability Gramian versus the number of measurements. The 5, 10, and 20 second measurement spacing cases are all on top of one another and contain no redundant measurements, thus indicating that these measurement spacing cases require the theoretical minimum number of measurements for achieving full rank. The impact of the EGI tessellation number for the LEO object with three-axis rotation is comparable to the single-axis rotation results [10].

Fig. 29
figure 29

Rank of the observability Gramian versus time for the LEO object with three-axis rotation and 294 surface normal directions

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Fig. 30
figure 30

Rank of the observability Gramian versus number of measurements for the LEO object with three-axis rotation and 294 surface normal directions

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The 3D visualizations of the LEO object with three-axis rotation in Fig. 31 are similar to the single-axis rotation EGI visualizations. However, the three-axis rotation visualizations do not contain a gray region at the bottom of the EGI at the end of the analysis. The similarity is a result of the 32.8∘ inclination of the LEO object. The 30∘ axis of rotation for the three-axis attitude profile appears as a nearly flat spin in Fig. 31 because of the inclination, but the slight difference between the axis of rotation and the inclination results in the bottom of the EGI becoming visible and illuminated to the observer. While the orbit for the LEO object remains the same compared to the single-axis rotation cases, the change in the attitude profiles from simple, single-axis rotation to more complex, three-axis rotation results in a system which is sufficiently sampled for EGI minimization.

Fig. 31
figure 31

3D visualization of the linearly independent columns in the reflection matrix for the LEO object with three-axis rotation, 294 surface normal directions, and 1 second between measurements, where the colorbar represents measurement indices

Full size image

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Friedman, A.M., Frueh, C. Observability of Light Curve Inversion for Shape and Feature Determination Exemplified by a Case Analysis. J Astronaut Sci 69, 537–569 (2022). https://doi.org/10.1007/s40295-021-00293-w

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Keywords

  • Observability
  • Space situational awareness
  • Light curves
  • Sensor tasking