Constrained Slews for Single-Axis Pointing

Abstract

T HERE is a significant interest in the determination of maneuver strategies for underactuated satellites, where the availability of only two torque components prevents the spacecraft from performing arbitrary slews. The problem is representative of those situations in which an actuator failure [1,2] or inherent physical constraints (e.g., use of magnetic torquers only as attitude effectors that deliver a command torque along directions perpendicular to the externalmagneticfield [3,4]) result in constraints onto the admissible rotations and full three-axis control is not attainable. In these cases, the available torque is constrained on a plane, perpendicular to the direction b about which it is not possible to rotate the spacecraft, where b is prescribed in either the body-fixed frame (for actuator failure) or in the fixed target frame (as for the direction of the magnetic field). In the present note it is shown that, regardless of the direction of b, there always exists an eigenaxis rotation [5] that provides for the exact pointing of a given body-fixed axis toward a prescribed target direction, identified by the unit vectors
and , respectively. The direction of the admissible rotation eigenaxis and the amplitude of the resulting rotation are analytically determined for any possible combination of the unit vectors
, , and b. In [6], a similar problem is dealt with, and its solution proposed as a means for planning feasible maneuvers for underactuated spacecraft, where the overall angular displacement from a specified target frame is minimized by rotating the body frame about a nonnominal Euler axis ĝ. The analytical solution of Giulietti and Tortora [6] provides a reorientation strategy in which the final misalignment error grows with the angle between e and ĝ, where e is the nominal eigenaxis. Thus, the optimal nonnominal eigenaxis ĝ lies along the projection of e onto the plane of the admissible rotation axes, perpendicular to the direction of b, where the angle between e and the admissible ĝ is minimum [6]. The resulting final pointing error grows linearly with the desired nominal rotation about e, when is small, reaching a limit value cos 1 ĝ e for a desired rotation . As a consequence, the misalignment error may attain rather large values when the angle between ĝ and e and the required reorientation angle are both large. The actual limits for the maximum tolerable attitude error will strongly depend on accuracy requirements for the considered application, but the exact acquisition of a prescribed attitude may not be mandatory, especially in the presence of failures, when a deterioration of system performance is expected and may be tolerated. Conversely, accurate pointing of a single direction
, such as the boresight of a sensor payload or a directional antenna, may be vital for the mission or, at least, sufficient for a minimal level of operations in those critical situations where arbitrary attitude maneuvers cannot be accomplished. As an example, orbit maneuvers require pointing of the thruster nozzle with an accuracy within a fraction of a degree from the desired v direction, whereas the roll angle of the spacecraft around the thrust vector line has no effect on the outcome of the thrust pulse. By application of an approach similar to that discussed in [6], based on the identification of a suitable eigenaxis ĝ and corresponding rotation
, this paper demonstrates that it is possible to analytically define a rotation which exactly aims a body-fixed axis toward a prescribed (yet arbitrary) target direction. The approach is initially derived by choosing an ad hoc body reference frame FB, where the first axis of FB is parallel to the unit vector
, which must be pointed along the target direction . This choice makes the derivation of the pointing strategy simpler from the mathematical standpoint. It will then be generalized for arbitrary sensor and target directions by use of a suitable coordinate transformation matrix. The derivation of the eigenaxis rotation that allows for the desired alignmentwill be discussed in the next section. Some examples of the resulting rotation strategies will then be given in the Results section, where a comparison with the minimum misalignment provided by the solution proposed in [6] is also presented. The Conclusions section ends the note.