VARIATIONAL EQUATIONS FOR ORBIT DETERMINATION BY DIFFERENTIAL CORRECTION

Dennis Milbert 

Updated:  2024-oct-14


Orbit Determination

One may consider an orbit as an object's 3-D position and velocity when modeled by a collection of force models. The principal model is Earth gravitation, supplemented by lunisolar perturbations, solar radiation pressure, atmospheric drag, etc... The force models only model object accelleration, and are expressed as second-order differential equations. The key issue is that a realistic set of force models does not possess simple parametric functional solutions { e.g. y=f(x) } for the position and velocity. For such complex cases, we must turn to numerical integration for numerical solutions to the differential equations. Also, differential equations are often solved, analytically or numerically, as initial value problems; which lead to different trajectories depending upon the choice of initial values.

Differential Correction

Differential correction (DC) is the change of position and velocity in the future from a change in one's initial values of position and velocity. These derivatives are expressed as a time-dependent state transition matrix (STM). More exactly, DC frequently refers to not only the STM derivatives, but also their contribution in a parameter estimation algorithm (e.g. least-squares estimation). Thus, discrete measurements of an orbital object in time may be related to an optimal solution of the initial values, and an associated optimal orbit.

The DC method is also extended to encompass change of position and velocity in the future from a change in the estimates of one or more force model parameters; like gravitational or atmospheric drag variation. These derivatives are expressed as a time-dependent sensitivity matrix, S(t), and also participate in a least-squares algorithm.

In its most general form, differential correction (DC) may be applied, not only to orbit determination, but to any physical process that is formulated with a differential equation model, and which does not possess analytic solutions.

Variational Equations

As mentioned above, both the state transition matrix (STM) and the sensitivity matrix, S(t) are time-dependent. Their evolution in time is also described by differential equations. These differential equations are called variational equations. These equations are auxiliary to the force model differential equations.

The force model differential equations describe evolution of an object's position and velocity. The variational differential equations describe evolution of the derivatives of an object's position and velocity with respect to the force model initial values of position and velocity, and the force model parameters (if any).

Also note that since variational equations are differential equations, they will evolve in time, and will also possess initial values. The initial values of the STM are the identity matrix; and, the initial values of the sensitivity matrix, S(t), are all zero when time, t, is the initial time associated with the initial values, t0.

Fine Print: There is a level of abstraction in variational equations that makes first encounters difficult. I wrote this study in large measure to provide practical examples in variational equation solutions as a part of the least-squares estimation process.

Footnote: Variational equations are not restricted to orbit determination. They arise from any differential equation that is expressed as an initial value problem.

The Question

Over the past 15 years, a source has argued that the sensitivity matrix, S(t), can not be initialized to zero. This was a bold assertion, since it contradicted standard practice for orbit determination by the differential correction (DC) method. Further, it contradicted the DC solution method used in many other non-orbit applications.

Which viewpoint is correct?

The Analysis

I numerically tested a simple problem: 2-D ballistic motion that happened to have a closed form (analytic) solution. (This was inspired by the Uniform Gravity Field Model of Section 1.2 of Tapley, et.al (2004).)

Results showed that the sensitivity matrix, S(t), must be initialized to zero.

I extended the analysis by numerically testing 3-D central force (Keplerian) motion, which also had an exact analytic solution.

Results showed that the sensitivity matrix, S(t), must be initialized to zero.

I shared these results with a valued colleague, Prof. Christopher Jekeli. He had many insightful comments. In the course of our discussions, I algebraically derived the state transition matrix (STM) and the sensitivity matrix, S(t), for both a simple and a forced harmonic oscillator (where the start position is at time, t=0). These derivations showed that S(t=0) must be all zeros. And, no surprise, numerical tests of the oscillator problems showed the sensitivity matrix, S(t), must be initialized to zero (as indicated by the algebra).

In the course of my work, I found that concrete examples of variational equations, particularly as they related to differential correction, were lacking in the literature. This caused me document my work as a guidebook, so that a student can more readily grasp how to use variational equations to solve estimation problems for dynamical systems governed by differential equations.

Prof. Jekeli was so interested in my writeup, he derived a new proof that, in general, the sensitivity matrix, S(t), must be initialized to zero. I provided some useful comments to him, and he insisted that I be the second author of a scientific note he was writing.

After valuable suggestions by a reviewer, we enlarged the note to a paper, to appeal to a wider audience. And, prior to submission, Prof. Jekeli insisted that I be the first author of this revised paper. But please note, the new general proof on S(t) initialization is the brainchild of Prof. Jekeli.

The Study

The 61 page study is entitled: Variational Equations for Orbit Determination by Differential Correction. (1.02MB).

The study explores the sensitivity matrix question both numerically and mathematically. This includes relating an general mathematical derivation of the variational equations, and their initial values, made by other authors.

This study can be considered as a supplement to an advanced course in satellite geodesy/orbit determination. It main virtue is that it details the mathematical setup of both force models and variational equations for a number of physical problems. It demonstrates the process in the least-squares estimation of optimal initial values and supporting parameters.

Readers should be comfortable with differential calculus and differential equations. Readers should also be familiar with least squares and simple matrix algebra. Knowledge of some simple physics will aid in understanding the examples. Knowledge of the analytic solution to Kepler's Problem (as found in an introductory satellite geodesy/orbit determination course) will help with one example.

The Paper

The paper is available at the Journal of Geodesy website: On the initialization of the sensitivity matrix in variational equations.

The paper contains the new general proof, by Prof. Jekeli, that the sensitivity matrix variational equations must be set to zero at the initial time.

References

Tapley, Byron D., Bob E. Schutz, and George H. Born, 2004: Statistical Orbit Determination. Elsevier Academic Press, Burlington, MA, 547 pp.

To Contact Me

My e-mail user name is the first initial of my first name, followed by all the letters of my last name (see above), followed by the digit 'five'. My mail service is 'gmail', and it is a 'dot-com"'. Sorry for not spelling out my e-mail address, but I try to keep the spam-bots from fingering me.

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