Tether Assisted Rendezvous with for Satellites with Small Relative Inclinations

Rendezvous and capture of a payload at the end of a librating or rotating space tether is an exciting concept whose implementation would see revolutionary changes in space utilization. This paper addresses two of the most important control problems affecting the feasibility of this concept. The first problem, how to control the tether motion to enable rendezvous even when there is some small relative inclination between the primary satellite and the target payload, proves to be an extremely complex task. Conventional optimal control solution methods are not suitable for the job; the only successful approach is to combine an homotopy and Legendre pseudospectral method. With this approach, rendezvous with relative inclinations of up to 1.5 may be achieved in as few as 2 orbits. The second problem, how to control the tether dynamics following capture when some residual relative velocity exists between the tether tip and the payload, is solved using a wave-absorbing controller. Using tether offset at the primary satellite, the waveabsorbing controller can damp out a relative impact velocity of 7.5m/s normal to the tether. The effect of these two innovations is a significant advance in the cause of tether assisted rendezvous. INTRODUCTION The concept of using tethers to catch a payload in orbit is now well known. Despite the appeal in terms of its elegance and potential for dramatic reductions in launch costs, this application appears to have recently lost favor. There are two primary reasons for this, one of which is a somewhat conservative political climate that must ultimately change in time. The second and more important cause is a better understanding of the complex dynamics and control challenges faced by this concept. Some of these challenges are addressed in this paper. At the outset, when initial studies into this concept were presented, the promise was fantastic. By making a range of (not unreasonable) assumptions, it was possible to show that capture was achievable and that it could enable transfer from a ballistic trajectory into higher orbits, or even to the Moon or Mars. These wide-ranging studies of the effect of various feasible scenarios set a firm basis for the further development of the concept but did not adequately address control issues related to capture, particularly in light of likely perturbations. Particular problems that remain to be addressed include how to control the tether to rendezvous with a passive payload that does not approach with the desired trajectory (note that an active payload does not necessarily solve this problem as the fuel required to correct the trajectory is likely to be large). The authors have developed a guidance algorithm for inplane capture, but no out-of-plane solution has yet been presented in the literature. The out-of-plane case is a particularly difficult one since even a small difference in the relative inclination of the satellite and payload can result in significant relative velocity in the out-of-plane direction. Also, the development of a robust, effective and reusable capturing mechanism remains an issue. Associated with this is the need to allow adequate time for docking. This was addressed by Stuart and the authors for the inplane case, however, out-of-plane effects have so far been ignored. Finally, there is the need to address post-capture dynamics that may arise from the impact of a payload with a small relative velocity at the tether tip. In this paper, we attempt to address the question of whether rendezvous and capture is still possible if there is some small relative inclination between the primary satellite (with the tether) and the target payload. Various algorithms, including conjugate gradient and simulated annealing were applied to this problem, using only tether tension as an input and assuming some small, initial out-of-plane component. The effectiveness of these algorithms was severely limited by the long guidance interval. This limitation was overcome via the application of a Legendre 1 54th International Astronautical Congress of the International Astronautical Federation, the International Academy of Astronautics, and the International Institute of Space Law 29 September 3 October 2003, Bremen, Germany IAC-03-A.P.09 Copyright © 2003 by the author(s). Published by the American Institute of Aeronautics and Astronautics, Inc., with permission. Released to IAF/IAA/AIAA to publish in all forms. pseudospectral method, which is described in detail in the Section “Guidance Algorithm”. A number of authors have given consideration to methods for damping lateral oscillations in the tether. However, these have not specifically been applied to the problem of damping post-capture tether oscillations. In this paper, a wave absorbing controller is applied to the problem. MATHEMATICAL MODELS Control Model To derive an open loop control law for out-of-plane rendezvous requires a simple model of the tether that expresses the fundamental dynamics of the system. The model selected is illustrated in Fig. 1. It assumes that the tether is a rigid, massless link that may change in length via tension control at the primary satellite. The primary satellite is constrained to follow a circular orbit with radius, R, angular rate ω and true anomaly ν. The generalized coordinates are selected as the orthogonal polar coordinates: r, θ, and φ. The alignment of the tether is given by starting with the tether along the positive x-axis, rotating by θ about the positive z-axis (in-plane) and then by φ about the rotated y-axis (out-of-plane).