V.J. Modi

A preferred coordinate system and the associated orientation representation in attitude dynamics

Abstract Librational equations of motion when expressed in terms of local coordinates presented in this paper yield a set of rational differential equations devoid of planes of singularity. For a specified level of accuracy in numerical integration, a rational set requires less CPU time than an equivalent transcendental set. To interpret the results of the integration, the time evolution of orientation is presented as a curve in the three dimensional topological space RP3. The above new local coordinates is an example of a globally defined nonsingular rational parametrization of space of rotations suitable for problems of dynamics involving general rotations.

On the deployment dynamics of tether connected two-body systems☆

Abstract A general formulation of the deployment dynamics of tether connected two-body systems taking into account the three dimensional librational motion, longitudinal and transverse vibrations of the continuous tether, eccentricity of the orbit and aerodynamic drag in a rotating, oblate atmosphere, is presented. Three simple deployment procedures (uniform, exponential and their combination) are considered. In the beginning analytical solutions are obtained for the degenerate case of negligible vibrations in a circular orbit, which help in establishing trends for the more general situation. This is followed by the general dynamics of the system investigated through integration of the linearized autonomous equations of motion numerically. Typical plots are presented which describe the effects of various system parameters on the tether dynamics during deployment of a subsatellite from the Space Shuttle.

Attitude dynamics of three-body tethered systems

Abstract The dynamics of a tether-connected three-body system is investigated. The system is treated as a double-pendulum and the motion in the orbital plane is considered for the two cases of fixed-length tethers and variable-length tethers. For fixed-length tethers, it is noted that there are four possible equilibrium configurations: both tethers aligned along the local vertical; both tethers horizontal; and for certain combinations of parameters, two other configurations where one tether is along the local vertical while the other is inclined to the local vertical. Only the vertical equilibrium configuration is stable. Frequencies of oscillations around the stable configuration and corresponding modes are given in the paper. The dynamic response of the system during deployment of the three-body constellation is obtained. Dynamical behaviour during transportation of a cargo from one end-body to the other is also studied.

On the control of the space shuttle based tethered systems

Abstract Spatial dynamics of the Space Shuttle based tethered satellite system is investigated using a nonlinear model that accounts for the aerodynamic drag in a rotating oblate atmosphere. Results show that the normally unstable retrieval manoeuver can be stabilized satisfactorily using a nonlinear tether tension control strategy which depends on the tether length, its variation with time and pitch rate. Effectiveness of the control is illustrated through an example involving a 100 km tether supporting a proposed satellite for charting the earths magnetic field.

Three-dimensional dynamics and control of tether-connected N-body systems☆

Abstract General three-dimensional motion of tethered N-body systems is considered. The equations of motion derived are valid for large motion, variable length and arbitrary orbit, but the tethers are assumed to be massless and straight. It is shown that the out-of-plane libration frequencies are always related to the in-plane librational frequencies by the relation ( ω o j Ω 2 =( ω i j Ω ) 2 + 1 , j = 1, 2, …, N − 1, where Ω is the orbital frequency. Typical transient response of three-body tethered systems is shown. Possible reel rate laws to control both in-plane and out-of-plane motions are presented.

Transient attitude dynamics of satellites with deploying flexible appendages

Abstract The paper presents a general formulation for librational dynamics of satellites with an arbitrary number, types, and orientation of deploying flexible appendages. The generalized force term is incorporated making the formulation applicable to a wide variety of situations where aerodynamic forces, solar radiation, earths magnetic field, etc. become significant. In particular, the case of a beam-type flexible appendage deploying from a satellite in an arbitrary orbit is considered. The corresponding nonlinear, non-autonomous equations for in-plane and out-of-plane vibrations are derived, allowing for the variation of mass density and flexural rigidity along the length with time dependent deployment velocity and spin rate. Next, the attention is focused on the linearized analysis of the in-plane vibrational equation using the assumed-mode method and its substantiation through numerical integration. Finally, the paper presents results for both steady-state and transient attitude behaviour for a representative gravity gradient configuration for a range of initial conditions and system parameters. Results show the combined effect of flexibility and deployment on the dynamics of the system to be substantial. Disturbance of the appendage can excite large amplitude librations. On the other hand, the converse situation is not necessarily true. Furthermore, Coriolis loading, induced by the extending appendages, can become a limiting factor in arriving at a deployment strategy; an effect not pointed out in the literature.

On vibrations of orbiting tethers

Abstract The paper considers three dimensional transverse and longitudinal oscillations of a tether connecting a subsatellite to the shuttle. Attention is focused on the dynamics during the terminal phase of retrieval of the subsatellite. Nonlinearity in the strain-displacement relation is taken into account since it is important and helpful during this phase. Retrieval schemes that can assist in arresting the growth of vibrations are obtained by simplified analysis and validated through numerical solution of the original equations.

ROBUST ATTITUDE AND VIBRATION CONTROL OF THE SPACE STATION

Abstract Simultaneous attitude and vibration control of a space platform is studied using the LQG/LTR approach with the linearized equations obtained through a novel finite difference procedure. Results show the control strategy to be quite effective in damping a wide variety of disturbances including the Shuttle docking, manipulator maneuvers, solar array tracking and initial displacements imparted to the system as well as its subassemblies.

On the Control of Tethered Satellite Systems

Abstract A mathematical model is proposed for studying the dynamics of the Tethered Satellite System (TSS) consisting of a plate-type space station from which a tether supported subsatellite is deployed or retrieved. The rigid body dynamics of the tether, subsatellite and space station are analyzed accounting for the mass of the tether as well as a three-dimensional offset of its point of attachment. Controllability of the linearized equations is established numerically and a comparative study of three different control strategies conducted. The strategies employ thrusters, tension in the tether line or motion of the offset of the attachement to achieve control of the system subjected to relatively large initial disturbances. Results suggest that, in the stationkeeping mode, the tension control strategy damps a given disturbance in the shortest time, however, at an expense of the energy. On the other hand, the offset control proves to be the most efficient in terms of energy consumption, but now the response to disturbance persists over a longer duration. In addition, the performance of the thruster control, tension control, and offset control strategies, as well as their combinations are analyzed during retrieval of the tether. Results suggest that the thruster-offset hybrid controller is the most effective in damping given disturbances.

Tether-platform coupled control

Abstract From the control point of view, tethered systems pose several challenges, the major one pertaining to the regulation of the unstable system dynamics during the retrieval phase. On the other hand, the system configuration permits design of controllers using length rate, tension and offset schemes, which are not feasible with other satellites. Here “offset” refers to the time dependent variation of the tether attachment point at the platform end. The present paper studies several applications of the offset scheme in controlling the tethered systems. To that end, planar equations of motion of a space platform based Tethered Satellite System (TSS) are derived by the Lagrangian procedure. This is followed by representative results aimed at the offset control of platform pitch, tether attitude and vibration motions. The offset scheme is used for simultaneous control of platform and tether pitch motion. Finally the attention is directed towards simultaneous regulation of the platform pitch and longitudinal tether vibration. The numerical results clearly show considerable promise for the offset control scheme in regulating tether, platform and combined tether-platform dynamics.