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*Paul Cefola, 2016-04-08 23:42 *

Added the Tesseral Linear Combination Model Improvement Task (pjc)

# SOCIS¶

Starting a few years ago, ESA has been holding an annual Summer Of Code In Space (SOCIS). The main SOCIS site for year 2016 is located here: http://sophia.estec.esa.int/socis2016/.

The Orekit project proposes the following projects for students. Note that even after the mentor application date (2016-04-10T23:00:00Z) expires, this list will be updated. So if you are interested you can always come back to see new proposals. Also, if anyone has new project ideas, do not hesitate to discuss them on the development mailing lists.

Here is the current list of ideas:

- Add continuous maneuvers to DSST propagator
- Add partial derivatives matrices to DSST propagator
- Add ground antenna center of phase correction to orbit determination
- Add ground points displacements to orbit determination

## Add continuous maneuvers to DSST propagator¶

The Draper Semi-Analytical Satellite Theory (DSST) is a very high performance propagation model. It performs

propagation by using numerical integration for the mean elements, using very large step sizes to achieve

high speed and by using analytical models for short period terms. It is well suited for a large number of

applications, from operations and station-keeping to long term analysis and end of life estimation.

The Orekit implementation of this model includes the traditional force models (zonal and tesseral terms

for gravity field, radiation pressure, atmospheric drag, Sun and Moon third body attraction). All these

models contribute to both mean elements and short period terms. Maneuvers can also be handled, but currently

only using impulsive maneuvers.

This limitation on maneuvers prevents the use of DSST for low thrust electric propulsion as the duration of

the thrust is long with respect to orbital period.

As low-thrust propulsion can be modeled as an acceleration and as Orekit implementation of DSST can consider

general acceleration models contribution to both mean elements and short periods, it should be easy to

add a low-thrust model to DSST.

Two mentors will follow this project.

## Add partial derivatives matrices to DSST propagator¶

The Draper Semi-Analytical Satellite Theory (DSST) is a very high performance propagation model. It performs

propagation by using numerical integration for the mean elements, using very large step sizes to achieve

high speed and by using analytical models for short period terms. It is well suited for a large number of

applications, from operations and station-keeping to long term analysis and end of life estimation.

The Orekit implementation of this model implements propagation of the state vector at any date, including mean

elements and short period terms. In order to be able to perform orbit determination based on DSST propoagtion

model, the partial derivatives of the state vector are needed. These partial derivatives correspond to

two Jacobians matrices. The first matrix is the transition matrix representing the sensitivity of the

current state vector at computation date with respect to the initial state vector. It basically allows to

deduce how the initial state vector should be changed if we want the current state vector to change in

a desired direction in order to reduce measurement residuals. The second matrix is the Jacobian of the

state vector with respect to some model parameters for specific force models (drag and radiation pressure).

It is used to compute how much these parameters should be changed to also reduce the measurements residuals.

Both matrices are important because in real operational cases, both the orbital parameters and the

non-conservative forces parameters are estimated as neither are perfectly known beforehand.

This project will be based on recently published work that will be made available to student. It is a

difficult project that requires good mathematical knowledge (analysis, calculus, a background in space

flight dynamics will greatly help). The project will be limited to implement the derivatives computation.

Using the computed derivatives in a complete orbit determination process will be done later on in a

second phase, as it will probably not fit withing the time frame. Even considering only the matrices

will be implemented, it is expected to be rather long and requires a student that can work fast.

Two mentors will follow this project.

## Add ground antenna center of phase correction to orbit determination¶

Orbit determination is a fairly recent feature in Orekit (it will be available at SOCIS start time).

This feature allows computing orbital parameters as well as meta-data (propagation models coefficients

like drag, stations coordinates, biases, ...) from a set of observation performed on ground or on

board.

Ground stations measurements really depend on the antenna center of phase which is slightly offset

with respect to the fixed ground station location. The offset changes as the antenna points to

different directions while tracking spacecrafts. The dependency between this offset and the

pointing direction is not taken into account in the initial version of orbit determination.

The aim of the project would be to add this effect in the ground-based measurements. This

mainly involves modeling a few geometrical transforms (rotations, translations) and evaluate

the compensation to be added when evaluating the theoretical measurement used to compute

residuals in the orbit determination process.

## Add ground points displacements to orbit determination¶

Orbit determination is a fairly recent feature in Orekit (it will be available at SOCIS start time).

This feature allows computing orbital parameters as well as meta-data (propagation models coefficients

like drag, stations coordinates, biases, ...) from a set of observation performed on ground or on

board.

As tidals effects slightly change Earth shape in the diurnal and semidiurnal frequencies, ground

station experience small displacements each day, of up to 60cm order of magnitude. These effects

should be taken into account when considering measurements.

The aim of this project would be to add this effect in the ground-based measurements. This

is mainly an implementation of the algorithms described in chapter 7, Displacement of reference

points, from IERS conventions 2010 http://www.iers.org/IERS/EN/Publications/TechnicalNotes/tn36.html?nn=94912.

Note that this is a difficult subject.

## Improve the Tesseral Linear Combination Short-Period Motion Model in the DSST propagator¶

The Draper Semi-Analytical Satellite Theory (DSST) is a very high performance propagation model. It performs propagation by using numerical integration for the mean elements, using very large step sizes to achieve high speed and by using analytical models for short period terms. It is well suited for a large number of applications, from operations and station-keeping to long term analysis and end of life estimation.

The Orekit implementation of this model includes the traditional force models (zonal and tesseral terms in the geopotential, radiation pressure, atmospheric drag, Sun and Moon third body attraction). All of these models contribute to both the mean element and short period motions.

For the tesseral terms, there are three models: (1) the tesseral resonance (included in the mean element rates), (2) the tesseral m-daily contributions to the short-period motion, and (3) the tesseral linear combination terms (also a short-period motion). Both the tesseral resonance and the tesseral linear combination terms require a general form of the Hansen coefficients (an eccentricity function) and derivatives of the Hansen coefficients. The Hansen coefficients and the derivatives are computed via recursions. The recursions are initialized via the Newcomb operators. Modified forms for the Hansen coefficients and the Newcomb operators were developed for the DSST to maintain good performance for high eccentricity cases.

“Shallow resonance” cases occur with orbits that have long period repeat ground tracks. Such long repeat ground tracks are frequent in the orbits designed for earth observation. In these cases, the important tesseral linear combination terms are: (1) low degree and order terms in the geopotential, and (2) high degree and order terms centered around the shallow resonance order.

The current DSST algorithm, to model high order shallow resonant terms together with their side bands, requires computation of many negligible intermediate terms in order to include the high degree and order terms. This impacts the efficiency of the DSST. In this task, we want to develop an architecture for the tesseral linear combination terms that allows separate, selectable, regions for the short-periodic tesseral terms: one region for low degree and order terms and another region(s) for shallow resonance side band terms.

In a subsequent phase, we would like to add a model for the J2/shallow resonance second order coupling terms.

In the current task, we would also like to revisit the partitioning of the Hansen coefficient/Newcomb operator computations between the Newcomb operator database and the DSST orbit propagator. The current architecture is based on the data memory available to orbit propagation application programs in the early 1980s. This effort may involve research into recent work on (1) the Hansen coefficients and (2) rational function approximation.

Note that this is a difficult subject.

Two mentors will follow this project.