The spin-stabilized spacecraft is a convenient platform for the measurement of magnetic and electric fields, plasma properties, and other physical phenomena best observed over a wide baseline. Many spinning spacecraft have radial booms that are constructed from very thin and flexible multi-stranded wire. These radial booms can be quite long, with examples ranging from 25 to 250 m, and are kept under tension by centrifugal force. This design allows the science instrumentation to span a distance that is large in relation to the size of the spacecraft body. However, micrometeoroid or debris impacts and deployment problems are inherent risks. As the contribution to the mass moment of inertia increases with the square of distance, radial boom anomalies can significantly alter the dynamics of the spacecraft despite having low mass. For torques to be balanced at equilibrium, flexible members tend toward orientations that are perpendicular to and radiate outward from the spin axis. Likewise, the spin axis is dependent on the mass distribution of the spacecraft, and therefore on the orientation of each member. This paper presents models for the center of mass (CM) and inertia tensor that account for this mutual dependency of spin axis and radial boom orientation.
Radial boom anomalies are not unknown in the history of spin-stabilized spacecraft. In 1996, a wire boom on the Fast Auroral SnapshoT (FAST) satellite failed to deploy shortly after launch, causing a large coning angle for the duration of the mission. (Coning is defined to be the angle between the nominal body spin axis and the spacecraft major principal axis of inertia [MPA].) Despite this problem, the mission went on to become a success; complete electric field measurements were still possible using the remaining three radial booms and two axial booms.1The IMAGE spacecraft, launched in March 2000, had multiple radial boom breaks: −X antenna damage in October 2000, +Y antenna damage in September 2001, and further +Y antenna damage in September 2004.2 The more recent ARTEMIS mission consists of two of the five spacecraft from the THEMIS constellation that were moved to lunar orbits starting in 2010. In October 2010, ARTEMIS P1 (formerly THEMIS B) lost part of one wire boom, including the spherical probe attached to its end.3 One additional relevant example of a probable micrometeoroid impact is given by the DODGE spacecraft, which was not a spinner. This was a Department of Defense satellite, launched in July 1967. This mission was designed to test gravity-gradient stabilization and included ten retractable booms. In November 1967, DODGE abruptly changed from a gravity-gradient stable attitude to a tumbling mode. This event occurred during an intense Leonid meteor shower, so impact was the probable cause.4 Thanks to the efforts of the respective engineering teams, all these spacecraft were able to carry out their missions and now serve as useful case studies for current and future missions.
The models and techniques presented here will be included in the Attitude Ground System (AGS) for the Magnetospheric Multiscale (MMS) mission.5 The AGS provides flight dynamics attitude ground support services in the control center at the NASA Goddard Space Flight Center (GSFC). Increasing the accuracy of CM and inertia tensor values benefits three major aspects of MMS mission support: the predictions of spin axis precession (needed for ground contact planning), the propagation step in the attitude Kalman filter6 (used to generate definitive attitudes for the science teams), and the CM and inertia products uplinked to the MMS spacecraft (used for onboard attitude and orbit control). The benefits from the new models will be most significant in the event of a boom anomaly, but they will also provide some marginal improvement even for nominal flight operations activities with unbroken booms.
One fundamental assumption made in developing these models is that nutation and boom vibrations are fully damped. This assumption implies three things: the axis of rotation passes through the CM, the MPA is coincident with the axis of rotation, and all wire booms are oriented radially to the axis of rotation. If any of these conditions were not true, there would be unbalanced torques and/or forces driving internal motion of the spacecraft components. Whenever there is internal motion, friction will damp it out in time. For the MMS spacecraft, it is known that boom vibrations dampen to half-amplitude in roughly 12 hours.7 After a disturbance such as a break in one of the booms, the rigid central body of the spacecraft is likely to settle into a new orientation to balance the torques and forces. With this rebalancing, the MPA will almost never coincide exactly with the spacecraft nominal spin axis (for most spinners, the body Z-axis). Therefore, the spacecraft body will be tilted relative to the spin plane.
The main goal of this paper is to develop a method that considers the “fraction remaining” for each boom and solves for the CM and inertia tensor while respecting the mutual dependence of spin axis and boom directions. (The fraction remaining for each radial boom is referred to below as the boom fraction, 𝑓𝑏.) To begin, a spin vector is assumed and its direction is varied via an iterative method until it is coincident with the MPA of the inertia tensor resulting from that geometry. For each spin vector evaluated, the orientations of the booms, and in turn the inertia tensor of the spacecraft, depend upon it. Consequently, each time a spin vector is evaluated, solutions for the CM and boom directions consistent with that spin vector must be found. A separate iterative method is implemented to solve for these parameters. That is, for each step of the “outer iteration,” which solves for consistent spin axis direction and MPA, a nested “inner iteration” solves for the boom geometry. Once the spin axis direction converges to be consistent with the resultant MPA, and all other parameters are consistent, this outer iterative solution will yield a mapping between boom fraction and MPA direction. It is possible to determine the location where a boom was severed using this mapping, if the MPA is measurable on orbit. As an example, the MMS spacecraft includes star cameras that allow for accurate attitude estimation. By taking appropriate spin-averages, the MPA can be estimated.5 The observed change in MPA due to a boom break can be compared to the prediction from the inertia tensor model given here to determine the approximate location where the break occurred.
The organization of the paper is as follows. After this introduction, there is a description of the spacecraft model, which gives an overview of the coordinate frame and the parts of the spacecraft hardware relevant to describing the CM and inertia tensor models. Next, there is a derivation of the inner iteration method for solving for the CM and the radial boom directions. An error analysis of this method shows that conditions are easily met to guarantee convergence. The next two sections present the inertia tensor model and the outer iteration method, which solves for consistent values for the spin axis and the MPA. A numerical acceleration method for the outer iteration is used to reduce the number of iterations required for convergence. Those sections are followed by an analysis showing how to estimate the location where a boom was severed and the uncertainty in that estimate. The paper ends with conclusions and a brief discussion of future work.
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