T. Alan Lovell

Calibration of Linearized Solutions for Satellite Relative Motion

F OR a wide variety of space missions, it is desirable to describe the motion of a “deputy” satellite relative to a “chief” satellite. Cartesian coordinates and orbital-element differences are two methods to describe this motion. Cartesian coordinates offer an advantage in direct, intuitive feel for the relative motion. The equations of motion for each description are nonlinear, but can be linearized for close proximity between the chief and deputy. The linearized equations offer an advantage in simplicity. This Note is focused on two observations. First, the linearized descriptions for Cartesian coordinates and orbital-element differences share an equivalence. Second, this equivalence can be used in a simple manner to calibrate the Cartesian initial conditions such that linearization error is greatly reduced comparedwith initial conditions calculated purely fromkinematics. The first observationwas reported by Sengupta and Vadali [1]. However, this observation appears to have been underappreciated, because the second observation follows directly from the first but has not been previously published. These calibrated initial conditions greatly increase the domain of validity of the linearized approximation and provide far greater accuracy in matching the nonlinear solution over a larger range of separations. The calibration process leverages the lower nonlinearity of the orbital-element differences, which was reported by Junkins et al. [2]. These observations are related to additional work in the literature describing the degree of nonlinearity in various parameterizations of a given system [3–6]. Much of that work, however, focused on calculating an index of nonlinearity directly from properties of the nonlinear representations. This Note is focused on the linearization error in comparing nonlinear and linearized forms of each representation. In the following sections, the nonlinear and linearized descriptions of relative motion in Cartesian coordinates and orbital-element differences are reviewed. Then, the equivalence of the linearized descriptions is shown. Finally, the calibration of the Cartesian initial conditions is described and several numerical examples are shown. In the following, parameters related to the deputy’s orbit about the central body are indicated with a subscript d. For notational compactness, parameters related to the chief’s orbit about the central body are left without subscript.

Virtual-Chief Generalization of Hill–Clohessy–Wiltshire to Elliptic Orbits

T HE assumptions made in deriving the Hill–Clohessy–Wiltshire (HCW) equations for spacecraft relative motion are a chief satellite in a circular orbit, a deputy satellite in close proximity to the chief, and both satellites obeying Keplerian motion [1,2]. Additional descriptions have been derived to relax these assumptions. Specifically, analytical solutions have been developed for relative motion in elliptic orbits [3–5]. Other studies have considered nonlinear relative motion, see, e.g., [6,7]. Additional work has considered relative motion in the presence of non-Keplerian perturbations [8,9]. However, the HCW equations remain a popular starting point to describe relative motion, and this motivates a desire to understand the significance of the assumptions involved. Themain contribution of this Note is pedagogical in understanding the circularity assumption, proximity assumption, and their interrelation. This Note presents an intuitive approach to approximately generalize the HCW equations to chiefs in elliptic orbits. The approach defines a virtual chief as a circularized version of the actual chief. The motion of both the chief and deputy relative to the virtual chief can then be described by the HCWequations without violating the circularity assumption. The approach is equivalent to propagating the relative motion in a frame with constant angular velocity, and then transforming the solution into the chief’s local-vertical localhorizontal (LVLH) frame. This transformation becomes aLyapunov– Floquet transformation of the HCW solution, and it results in significant improvement in accuracy compared to the original HCW solution. Throughout this Note, orbital parameters of the deputy and virtual chief are indicated with subscripts D and VC, respectively, and parameters related to the chief’s orbit are left without subscripts. II. Background

Initial relative-orbit determination using heterogeneous TDOA

This paper presents a solution for the initial orbit determination of a space-based RF transmitter using time-difference-of-arrival (TDOA) measurements obtained from space-based receivers in known orbits. Orbit determination requires six independent TDOA measurements spread over time. The TDOA measurement can be expressed as a quadratic equation for the instantaneous transmitter position. The orbital motion of the transmitter is linearized relative to a reference orbit, which allows the TDOA measurement to be transformed to a quadratic equation for the relative position and velocity components at a chosen initial time. The system of six quadratic equations are solved using homotopy continuation.

RF localization solution using heterogeneous TDOA

This paper presents a solution for the localization of an RF transmitter using time-difference-of-arrival (TDOA) measurements that do not share any common receiver location. Localization requires three independent TDOA measurements, and previous solutions have assumed that the three measurements all share a common receiver location. Relaxing this assumption is here referred to as heterogeneous TDOA. The approach presented formulates the TDOA measurements as a system of three quadratic equations for the transmitter position components, and solves these equations using the Macaulay resultant. The heterogeneous TDOAs allow for improved measurement geometry resulting in more accurate localization. The solution also allows for localization by two moving receivers, such as orbiting satellites, collecting TDOA measurements at three instants in time.

Use of Cartesian-Coordinate Calibration for Satellite Relative-Motion Control

M ANY of the foundational developments in spacecraft navigation and guidance that were established by Dr. Richard H. Battin and colleagues at theMIT Instrumentation Laboratorywere posed in terms of Cartesian components of the position and velocity vectors. This representation provided advantages in direct relationships with the observation data (for use in navigation algorithms) and with maneuver accelerations and velocity changes (for use in guidance algorithms). Another advantage was the easy inclusion of perturbative accelerations into numerical trajectory solutions, famously using the early development of onboard digital computers. Of courseDr. Battin’swork alsomade prodigious use of orbital elements as another motion representation, taking advantage of their convenient encapsulation of Keplerian motion, such as in the solution of Lambert’s problem [1]. An example of this balanced approach is the combination of the Apollo cross-product steering, which compared the current velocity vector to a required impulsive velocity to determine the velocity-to-be-gained to steer the craft during maneuvers, and the Lambert guidance, which used orbital-elementbased solutions to determine the required impulsive velocity [2,3]. In continuation of this philosophy, this paper presents another approach to combine the use of Cartesian and orbital-element representations in the control of satellite relative motion. The motion of a “deputy” satellite relative to a “chief” satellite can be described using either Cartesian coordinates or orbital-element differences. The evolution of both descriptions obeys nonlinear dynamics, and either set of equations of motion can be linearized for close proximity. The transformation between the two descriptions is a nonlinear mapping, and this mapping can also be linearized for close proximity. These transformations have previously been used to define a calibrated set of Cartesian states. Linearized propagation of these calibrated Cartesian states can contain lower linearization error than linearized propagation of the true Cartesian states [4,5]. Control laws for the motion of the deputy relative to the chief are often designed using the linearized dynamic model for Cartesian coordinates. Implementation of these controllers in the presence of the nonlinear dynamics can result in degraded performance, increased fuel consumption, or even instability. However, the fact that the calibrated Cartesian states can provide a better linear approximation of the impending motion suggests their use in control-law implementation. The calibration process is related to the lower nonlinearity of the orbital-element differences, which was reported by Junkins et al. [6], and additional work in the literature describing the degree of nonlinearity in multiple parameterizations of various systems [7–10]. An implication of that work was that controllers should be designed using parameterizations with low nonlinearity. Here, however, it is shown that even a controller designed using a highly nonlinear parameterization can be improved using the calibration process. This paper demonstrates the improvements in performance, reduction of fuel consumption, and extension of the domain of validity of linearized control laws using calibrated Cartesian coordinates. First, the Cartesian coordinate calibration is reviewed. Next, a linear-quadratic control law designed using linearized dynamics is reviewed, and the implementation of this control law in the presence of nonlinear dynamics is illustrated. Finally, single-impulse rendezvous design using linearized dynamics is reviewed, and the implementation of the maneuver in the presence of nonlinear dynamics is illustrated. In the following, parameters related to the deputy’s orbit about the central body are indicated with a subscript d. For notational compactness, parameters related to the chief’s orbit about the central body are left without subscript.

A study of the re-entry orbit discrepancy involving tethered satellites

Abstract This paper describes an investigation of the motion of objects in near-Earth orbit that have a high probability of being identified as re-entering the Earths atmosphere. In the case of two or more satellites tethered together, each objects motion deviates from the traditional Keplerian-like motion of a single untethered body, due to the tension force in the tether. Consequently, classical identification and motion prediction techniques applied to a tethered object may produce results that indicate that the object is on course to re-enter when it actually is not, or vice-versa. In this study the factors that cause tethered bodies to behave differently than expected, and how significant these factors must be to cause a discrepancy regarding re-entry, are determined. A candidate re-entry identification methodology based on this foundation is then formulated.

Angles Only Initial Orbit Determination: Comparison of Relative Dynamics and Inertial Dynamics Approaches with Error Analysis

This paper investigates methods based on relative orbital dynamics to determine the motion of a space object using line-of-sight measurements collected by a space-based observer. The so-called “initial relative orbit determination” methods are typically applied to scenarios involving close proximity between the observer and the space object, i.e. scenarios in which the observer and space object are in very similar orbits. However, these methods are mathematically applicable to larger separation scenarios in which the observer and space object are in very different orbits. Previous work demonstrated an initial relative orbit determination algorithm that incorporates a closed-form relative motion solution with secondorder accuracy. This paper introduces a similar algorithm incorporating a third-order solution and compares its performance to the second-order method over various simulated test cases involving different observer-object scenarios. In particular, the sensitivity of these algorithms to measurement error and measurement sample rate is explored.

On Kalman filtering and observability in nonlinear sequential relative orbit estimation

To study the effects of incorporating higher-order nonlinearities with different measurement types on observability and filter performance in sequential relative orbit estimation, an extended Kalma...

Lyapunov-Floquet control of satellite relative motion in elliptic orbits

This paper proposes a method for continuous-thrust control of satellite formations in elliptic orbits. A previously calculated Lyapunov-Floquet transformation relates the linearized equations of relative motion for elliptic chief orbits to the Hill-Clohessy-Wiltshire equations describing circular chiefs. Using a control law based on Lyapunov-Floquet theory, a time-varying feedback gain is computed that drives a deputy satellite toward rendezvous with an elliptic chief. This control law stabilizes the relative motion across a wide range of chief eccentricities.