International Journal of Aerospace Engineering

Volume 2015 (2015), Article ID 475742, 6 pages

http://dx.doi.org/10.1155/2015/475742

## Approximate State Transition Matrix and Secular Orbit Model

Flight Dynamics Group, ISRO Satellite Centre, Bangalore 560 017, India

Received 21 September 2014; Revised 24 February 2015; Accepted 24 February 2015

Academic Editor: Christopher J. Damaren

Copyright © 2015 M. P. Ramachandran. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The state transition matrix (STM) is a part of the onboard orbit determination system. It is used to control the satellite’s orbital motion to a predefined reference orbit. Firstly in this paper a simple orbit model that captures the secular behavior of the orbital motion in the presence of all perturbation forces is derived. Next, an approximate STM to match the secular effects in the orbit due to oblate earth effect and later in the presence of all perturbation forces is derived. Numerical experiments are provided for illustration.

#### 1. Introduction

Autonomous orbit control in satellites is possible with the present onboard technological advancements. The Global Positioning System receiver solution gives the satellite position measurement in Cartesian frame. State propagating equations along with the measurement equations in the linear filter then estimate the orbit. State transition matrix (STM) is used in the state update equations. A reference orbit model is available onboard. Using the receiver orbit solution the absolute orbit control system then ensures the satellite motion to this reference orbit model in the earth centered fixed Cartesian reference frame. This control enables the satellite to achieve the required orientation too. In orbit determination system, STM of two-body dynamics as suggested in [1] is usually used. Yet it will be always desirable to match the complete dynamics especially to improve the accuracy and scalability of the navigation system [2].

The orbital motion of the satellite is made up of secular or mean motion along with short and long periodic motions [3, page 571]. When we include the complete dynamics as reference orbit we have to use continuous control. This requires more fuel. Continuous maneuver can also disturb the payload functioning. On the other hand mean motion (without periodic motions) as a reference orbit is more suited for orbit control by impulse thrusting. This is adopted in formation flying [4, Chap 10] besides that the mean motion is used to derive the initial conditions. It is noted that orbit control is usually executed as a function of time [5] instead of true anomaly. In the control system, the state measurements in Cartesian frame are usually updated in time space. Subsequently the STM derived here then updates these states.

STM henceforth shall mean absolute STM unless mentioned. It may be noted that Vallado [3, page 748] has discussed the STM for two-body orbital motion. In [6] a STM including the oblate earth effects using equinoctial mean elements and then applying interpolation is obtained. The present note brings out a STM that is in Cartesian frame as an alternative to [6] and considers only secular effects. We note in the literature that the STM that is in Cartesian frame is derived in [7] and it includes secular and periodic effects. Here the periodic effects are neglected. Further the STM derived here is extendable to accommodate secular along track effects in the presence of all perturbations. This is simpler than the expansion based method of deriving the STM as in [8].

It is important to note that secular forces due to oblate effect are considered in relative motion as in formation flying which is based on geometric approach [9]. The absolute transition matrix derived here can further be used to derive relative transition matrix as in [10]. This work is beyond the scope of this paper.

#### 2. Secular Acceleration

Consider the equation of motionwhere denotes the second derivative with respect to time of , the position vector in the inertial frame. The disturbing potential [11] iswhere , is the instantaneous declination, is the radius of the earth, is the gravitational constant, and is the magnitude of the position vector, . The vector represents other perturbation forces due to the inhomogeneous mass distribution of the earth, third body forces due to sun and moon besides solar radiation pressure and atmosphere drag forces. The potential is axisymmetric about the -axis and is independent of azimuth angle . The Lagrange’s planetary equation of motion is invoked and the following relations are deduced. The Keplerian elements are averaged over an orbit. The first-order secular motion that neglects periodic effects is described bywhere are, respectively, the semimajor axis, eccentricity, and inclination are invariant over the duration of interest:And are argument of perigee, the longitude of ascending node, and mean anomaly, respectively. The equation of the centre enables getting the true anomaly :In (3a) and (3b) we note thatEquation (4) is used when the eccentricity is not large. The longitude of the ascending node varies linearly with incremental time, . Orbit models in the satellite for control purposes need to have the cross-track motion that is predominant due to . The argument of perigee along with the true anomaly gives the argument of latitude which isHere, the subscript 2 has been added to denote the model. When all forces are included and solution of (1) is obtained, the instantaneous argument of latitude is denoted by . Here, in this paper, a proposal is made to add a polynomial function to the argument of latitude , in (6) over every orbit. This is a mean variation of the differential argument of latitude and could be, say, a quadratic or cubic power of time and is denoted as [3, page 570, 652]. Here, in this note, is a least squares fit over one orbital period and it accommodates the secular difference. This is defined asThe residue between and is periodic, which is incidentally not required for control. This correction enables (3a) and (3b) along with (7) to match the secular effects when all perturbation is present to a reasonable accuracy specially along track. Next a STM that matches the orbit model in (3a) and (3b) and later the secular effect in the presence of all forces is derived.

The following equations are used to transform from the orbital frame to the Cartesian frame :where unit vectors , respectively, are in the radial, tangential, that is, along the direction of motion (along-track) and normal to the orbital plane (see Figure 1).