P. Palmer

A Unification of Models of Tethered Satellites

In this paper, different conservative models of tethered satellites are related mathematically, and it is established in what limit they may provide useful insight into the underlying dynamics. An infinite dimensional model is linked to a finite dimensional model, the slack-spring model, through a conjecture on the singular perturbation of tether thickness. The slack-spring model is then naturally related to a billiard model in the limit of an inextensible spring. Next, the motion of a dumbbell model, which is lowest in the hierarchy of models, is identified within the motion of the billiard model through a theorem on the existence of invariant curves by exploiting Mosers twist map theorem. Finally, numerical computations provide insight into the dynamics of the billiard model.

Optimal satellite attitude control: a geometric approach

Optimal nonlinear control remains one of the most challenging subjects in control theory despite a long research history. In this paper, we present a geometric optimal control approach, which circumvents the tedious task of numerically solving online the Hamilton Jacobi Bellman (HJB) partial differential equation, which represents the dynamic programming formulation of the nonlinear global optimal control problem. Our approach makes implementation of nonlinear optimal attitude control practically feasible with low computational demand onboard a satellite. Optimal stabilizing state feedbacks are obtained from the construction of a Control Lyapunov function. Based on a phase space analysis, two natural dual optimal control objectives are considered to illustrate the application of this approach to satellite attitude control: Minimizing the norm of the control torque subject to a constraint on the convergence rate of a Lyapunov function, then maximizing the convergence rate of a Lyapunov function subject to a constraint on the control torque. Both approaches provide ease of implementation and achieve robust optimal trade-offs between attitude control rapidity and torque expenditure, without computational issues.

Exact steering law for pyramid-type four control moment gyro systems

An exact approach for gimbal steering based on generalised-inverse for a cluster of Control Moment Gyros (CMG) is presented. Feedback gains are calculated from analytical solutions of simplified model for desired closed-loop attitude dynamics of a satellite and corresponding angular momentum response of CMGs. It is highly desirable to be able to use full angular momentum workspace of CMG cluster for rapid slew manoeuvres. However, the troublesome internal elliptic singularities restrict the angular momentum workspace in most of the pseudo-inverse-based steering logics. Therefore, we propose a Generalised Inverse Steering Logic (GISL) different from Moore-Penrose inverse but exact unlike variants of Singularity Robust laws. The proposed method gives exact control while avoiding internal elliptic singularities and using full momentum capability of the CMG cluster. The important features of the proposed steering law are demonstrated by performing simulations.

An Implementation of the Logarithmic Hamiltonian Method for Artificial Satellite Orbit Determination

We discuss the use of a recently discovered exact two-body leapfrog for accurate symplectic integration of perturbed two-body motion and for the computation of the state-transition matrix. We pay special attention to artificial satellite orbit determination and describe in detail the evaluation of the perturbing acceleration. Inclusion of air drag and other non-canonical forces are also discussed. The main advantage of this new formulation is conceptual simplicity, for easy programming and high accuracy for orbits with large eccentricity. The method has been evaluated in real artificial satellite orbit determinations.

The Persistence of a Slow Manifold with Bifurcation

This paper considers the persistence of a slow manifold with bifurcation in a slow-fast two degree of freedom Hamiltonian system. In particular, we consider a system with a supercritical pitchfork bifurcation in the fast space which is unfolded by the slow coordinate. The model system is motivated by tethered satellites. It is shown that an almost full measure subset of a neighborhood of the slow manifolds normally elliptic branches persists in an adiabatic sense. We prove this using averaging and a blow-up near the bifurcation.

Dynamic Systems Approach to the Lander Descent Problem

T HE gravity turn landing scheme was originally developed for the 1966–1968 Lunar Surveyor landing mission [1]. Since then it has been applied in many landing missions such as the Viking Lander and Mars Polar Lander [2]. The descent method has the advantage of being near fuel optimal while guaranteeing a vertical landing [3,4]. However, themethod requires a control system that can apply the thrust antiparallel to the instantaneous velocity vector for the entire descent. This Note focuses on the dynamic aspects of the landing scheme. The immediate problem here is to solve for the control that takes the lander from a circular, or near-circular, initial parking orbit to the surface of the planet with zero final relative velocity and zero flightpath angle. In the past, this problem has been studied primarily through various assumptions. The studies can generally be divided into two groups corresponding to two different assumptions: 1) descent from low altitude and 2) aflat planet. In [3,5] both assumptions are considered and compared for the case of a constant thrustto-weight ratio. Using the first assumption, the authors argue for the validity of approximating the full nonlinear gravity with a constant gravity while maintaining, or at least an approximation of, the Coriolis force. This way, one of the equations decouple from the remaining two equations, which can then be solved by quadrature. Using the second assumption (see also [4,6]), both the gravity gradient and the Coriolis forces are neglected. This truncation can be solved analytically. In particular, the authors in [6] obtain analytical solutions with an inclusion of a quadratic air drag. It is shown that the effect of the drag is to increase the effective thrust–weight ratio so that the descent is completed with a lower thrust compared with the vacuum case. In this Note, the problem is considered without any assumptions on the gravity and the Coriolis forces. The approach undertaken here is also more geometric and includes an initial qualitative analysis. Furthermore, a closed-form solution is derived for the case of a controller that depends upon the flight-path angle. Finally, a numerical analysis is provided for the full problem with constant thrust to mass ratio. Through an appropriate choice of scaling the numerical solution of the required control is presented through a single planar plot, regardless of the parameters of the problem. Model

Attitude Determination through Image Registration Model and Test-case for Novel Attitude System in Low Earth Orbit

[Abstract] The mass and cost requirements of high-accuracy satellite pointing systems often inhibit the potential application of smaller and more affordable satellites. There is therefore an increasing need to develop high accuracy attitude systems that do not breach small satellite mass and cost constraints. This paper proposes a novel method of attitude determination using imagery from two canted, Earth pointing, push-broom sensors. The effects of attitude on inter-imager shifts are modelled, with model inversions proving the techniques potential, given an adequate registration scheme, for determining attitude over 3-axes. Simulated results are presented alongside real data from the Surrey Satellite Technology Ltd (SSTL) Disaster Monitoring Constellation (DMC), confirming the viability of using Earth observational cameras to measure attitude, rotation rates or onboard vibrations to a high degree of accuracy. Since the technique is capable of working with conventional onboard imaging sensors, the implementation costs and additional payload mass of such a system are deemed negligible. I. Introduction OR Earth observation the quality of imagery is dependant upon the stability of the platform and accuracy of attitude telemetry. Small rotations of the satellite will result in large displacements for imager ground projections, while changes in rate of rotation will stretch or shorten pixels of pushbroom scanners. High accuracy attitude systems however, incur mass and cost constraints that limit the scope of imaging applications for small or more inexpensive satellites. A need therefore exists for low mass low cost attitude systems capable of obtaining high accuracy attitude telemetry, especially during the period of image capture, onboard small satellites. The Surrey Space Centre, at the University of Surrey, has been studying the effects of attitude perturbations on pushbroom-imagery using the Surrey Satellite Technology Ltd (SSTL) Disaster Monitoring Constellation (DMC) 1 . This paper details the development of such research and proposes a novel method of attitude determination using imagery from two canted, Earth pointing, push-broom sensors viewing the same area of the Earth surface. Through registration of Earth features and analysis of perspective or timing based distortions, attitude or rotation rates can be determined over 3-axes. The technique is applicable to existing satellites with an appropriately angled sensor pair, such as onboard the DMC multi-spectral imager. Although much work has been done on image-based spacecraft pointing it is often related to rendezvous monitoring and flyby missions 2,3 . The potential for onboard calibration of satellite imagery in Low Earth Orbit (LEO) has been recognised by the European Space Agency (ESA) however. In order to reduce the effects of rate induced distortions within imagery, ESA have proposed SmartScan 4 , a novel imaging system that is designed to use additional imaging sensors to detect and compensate for motion about the focal plane of the imagers. Although currently untested in space, such studies help to highlight the potential of using Earth-pointing cameras to determine attitude. This paper models the effect of attitude and rates on registration shifts, as measured between images from angularly displaced pushbroom imagery. A model inversion methodology is presented allowing extraction of attitude from imagery. Simulated results are presented, proving the potential of the technique, given a suitable sub-pixel level registration scheme, to separate attitude over 3 axes. Attitude estimates extracted from DMC imagery, captured during yaw and pitch rate manoeuvres, is compared to onboard telemetry, confirming the

Onboard Semianalytic Approach to Collision-Free Formation Reconfiguration

Allowing satellite formations the flexibility to perform path planning operations onboard each spacecraft can significantly reduce the ground operations burden and increase the responsiveness of the formation to reconfiguration events. This paper develops a semianalytic approach to rapid onboard, fuel-minimized, collision-free path generation for satellites in a formation. A sequential optimization approach to collision avoidance is examined. Through hardware testing and comparison with other approaches, this analytic method shows viability for onboard, fuel-minimized, collision-free path generation.

Analysis of fidelities of linearized orbital models using least squares

Satellites orbiting in Low Earth Orbit (LEO) are accelerated by Earth gravity and dominant orbital perturbations due to Earth oblateness and atmospheric drag. The equations of motion describing such a motion are highly nonlinear in nature. Linearized orbital models only approximate these nonlinear dynamics. The difference between a linearized model and full nonlinear dynamical equations of motion is termed as process noise. To determine the accuracy of these approximate models, we need to compare these with numerical integration of the full non-linear dynamical equations. Linearized solution propagations are characterized by a set of initial conditions which determine orbital evolution. The question arises on how to choose the initial conditions of the analytical approximation appropriate to a given choice of initial conditions for the numerically propagated orbit such that the process noise is minimized. An algorithm is developed, based upon the statistical method of nonlinear least squares to compare linearized orbital models for relative and absolute satellite dynamical motion with numerically propagated orbits to evaluate their accuracy. Due to recent interest in formation flying missions a comparison of accuracies of linearized relative orbital models i.e., Hill-Clohessy-Wiltshire (HCW) equations [1], J2 Modified Hills equations by Schweighart-Sedwick (SS) [2], and for absolute orbital models described by analytical equations for Keplers problem [3] and Epicycle Model by Hashida and Palmer [4] have been carried out.1 2

Path planning for fuel-optimal collision-free formation flying trajectories

Rapid collision identification and avoidance is a necessary capability for distributed implementation of path planning tools on formation flying spacecraft. This paper presents an approach for rapid collision identification, and fuel optimal trajectory shaping around collisions, while in natural motion and along reconfiguration trajectories. Collision identification and trajectory shaping is also explored with respect to multiple maneuvering craft. The Hill-Clohessy-Wiltshire relative motion dynamics are utilized. A semi-analytic approach is used which exploits a full decoupling of the equations of relative motion into phase-space. Algorithm implementation as part of on-orbit control toolbox allows for near real-time fuel-minimized collision-free maneuvers planning. 1 2