Srinivas R. Vadali

Formation establishment and reconfiguration using impulsive control

We analyze spacecraft formation establishment and reconfiguration problems for two-body orbits. The desired formations are characterized by nonsingular orbital-elemental differences. An analytical, two-impulse solution is proposed for achieving the desired orbital-elemental differences. Gausss variational equations are used to compute the corresponding impulse magnitudes analytically and the resulting solutions can be easily implemented using onboard computational resources. It is also shown that the cost obtained from the analytical solution differs by less than 1 % from that obtained by numerical optimization.

Optimal Variable-Structure Control Tracking of Spacecraft Maneuvers

An optimal control approach using variable-structure (sliding-mode) tracking for large angle spacecraft maneuvers is presented. The approach expands upon a previously derived regulation result using a quaternion parameterization for the kinematic equations of motion. This parameterization is used since it is free of singularities. The main contribution of this paper is the utilization of a simple term in the control law that produces a maneuver to the reference attitude trajectory in the shortest distance. Also, a multiplicative error quaternion between the desired and actual attitude is used to derive the control law. Sliding-mode switching surfaces are derived using an optimal-control analysis. Control laws are given using either external torque commands or reaction wheel commands. Global asymptotic stability is shown for both cases using a Lyapunov analysis. Simulation results are shown which use the new control strategy to stabilize the motion of the Microwave Anisotropy Probe spacecraft.

Relative Motion and the Geometry of Formations in Keplerian Elliptic Orbits with Arbitrary Eccentricity

The well-known Hill-Clohessy-Wiltshire equations that are used for the design of formation flight relative orbits are based on a circular reference orbit Classical solutions such as the projected circular or general circular relative orbit are no longer valid in the presence of eccentricity. This paper studies the effects of eccentricity on relative motion for Keplerian orbits. A new linear condition for bounded motion in relative position coordinates is derived that is valid for arbitrary eccentricities and epoch of the reference orbit. It is shown that the solutions to the Tschauner-Hempel equations that are used for rendezvous in elliptic orbits are directly related to the description of relative motion using small orbital element differences. A meaningful geometric parameterization for relative motion near a Keplerian elliptic orbit of arbitrary eccentricity is also developed. The eccentricity-induced effects are studied and exploited to obtain desired shapes of the relative orbit. Equations relating these parameters to initial conditions and differential classical and nonsingular elements are also derived. This parameterization is very useful for the analysis of more complicated models, such as the nonlinear relative motion problem.

Periodic relative motion near a keplerian elliptic orbit with nonlinear differential gravity

This paper presents a perturbation approach for determining relative motion initial conditions for periodic motion in the vicinity of a Keplerian elliptic orbit of arbitrary eccentricity, as well as an analytical solution for the relative orbit that accounts for quadratic nonlinearities in the differential gravitational acceleration. The analytical solution is obtained in the phase space of the rotating coordinate system, centered at the reference satellite, and is developed in terms of a small parameter relating relative orbit size, and semimajor axis and eccentricity of the reference orbit. The results derived are applicable for arbitrary epoch of the reference satellite. Relative orbits generated using the methodology of this paper remain bounded over much longer periods in comparison to the results obtained using other approximations found in the literature, because the semimajor axes of the satellites are shown to be matched to the second order in the small parameter. The derived expressions thus serve as excellent guesses for initiating a numerical procedure for matching the semimajor axes of the two satellites. Several examples support the claims in this paper.

Optimal Finite-Time Feedback Controllers for Nonlinear Systems with Terminal Constraints

This paper presents a dynamic-programming approach for determining finite-time optimal feedback controllers for nonlinear systems with nonlinear terminal constraints. The method utilizes a polynomial series expansion of the cost-to-go function with time-dependent gains that operate on the state variables and constraint Lagrange multipliers. These gains are computed from backward integration of differential equations with terminal boundary conditions, derived from the constraint specifications. The differential equations for the gains are independent of the states. The Lagrange multipliers at any particular time are evaluated from the knowledge of the current state and the gain values. Several numerical examples are considered to demonstrate the applications of this methodology. The accuracy of the method is ascertained by comparing the results with those obtained by using open-loop solutions to the respective problems. Finally, results of the application of the developed methodology to a spacecraft detumbling problem are presented.

Averaged Relative Motion and Applications to Formation Flight Near Perturbed Orbits

DOI: 10.2514/1.30620 This paper presents expressions for describing averaged relative motion between two satellites in neighboring orbits around an oblate planet. The theory assumes small relative distances between the satellites, but is uniformly valid for all elliptic orbits as well as the special case of a circular reference orbit by the use of nonsingular orbital elements. These expressions are useful when the short-periodic variations in relative position and velocity are of limited interest and, instead, the time-averaged behavior of the states is sought. The averaged expressions also provide insight into the effects of an oblate planet on bounded relative motion. For example, a bias term due to oblateness effects, hitherto unreported, has been identified in the radial position, which can be accounted for in the reference trajectory. Application of these expressions is shown in the derivation of an analytical filter that removes short-periodic variations in relative states without the use of tuned numerical filters, one for each frequency of interest,whicharenormallyusedfordisturbanceaccommodationincontrolsystemdesign.Theuseofthisanalytical filter is demonstrated for formation keeping on a prescribed relative trajectory.

Optimal Design of Satellite Formation Relative Motion Orbits Using Least-Squares Methods

Formation flying relative orbit design can be achieved by determining six initial conditions in the local-vertical- local-horizontal frame or, equivalently, six differential orbital elements. In this paper, two novel approaches are proposed to design perturbed satellite formation relative motion orbits using least-squares techniques. First, it is shown that the initial conditions required to approximate a desired formation geometry can be analytically solved for by using the Gim―Alfriend state transition matrix in conjunction with a linear least-squares approach. An improvement to this method is obtained by using the Gaussian least-squares differential correction approach and numerical integration of the equations of motion of the two satellites. Numerical results are presented to demonstrate the applications of the two approaches.

Near-Optimal Feedback Rendezvous in Elliptic Orbits Accounting for Nonlinear Differential Gravity

DOI: 10.2514/1.26650 This paper presents a novel approach to the design of near-optimal feedback control laws, for minimum-fuel rendezvous between satellites in elliptic orbits of arbitrary eccentricity. The rendezvous problem for the nonlinear differential gravity model is solved by the application of neighboring optimal feedback control methodology used in conjunction with a nominal trajectory, obtained by solving the related minimum-fuel feedback control problem for the linear Tschauner–Hempel equations, analytically. This novel closed-form solution is used to determine the best values of the final true anomaly by examining its effect on the cost to go for rendezvous. The neighboring feedback controllawaccountingfornonlineardifferentialgravityisobtainedbyusingageneralizedsweepmethod,validwhen the reference solution does not satisfy the first-order necessary conditions for optimality, exactly. Several numerical examples are analyzed to demonstrate the efficacy of the method.

Formation Flying Satellites: Control by an Astrodynamicist

Satellites flying in formation is a concept being pursued by the Air Force and NASA. Potential periodic formation orbits have been identified using Hills (or Clohessy Wiltshire) equations. Unfortunately the gravitational perturbations destroy the periodicity of the orbits and control will be required to maintain the desired orbits. Since fuel will be one of the major factors limiting the system lifetime it is imperative that fuel consumption be minimized. To maximize lifetime we not only need to find those orbits which require minimum fuel we also need for each satellite to have equal fuel consumption and this average amount needs to be minimized. Thus, control of the system has to be addressed, not just control of each satellite. In this paper control of the individual satellites as well as the constellation is addressed from an astrodynamics perspective.

An inverse-free technique for attitude control of spacecraft using CMGs

Abstract This paper presents a technique for CMG-based attitude control of spacecraft that does not require calculating the pseudo-inverse of the CMG Jacobian matrix. In order to obtain a steering law, the transpose of the Jacobian matrix is used rather than its pseudo-inverse. The transpose-based method alleviates some of the problems inherent in the existing steering laws that use the pseudo-inverse, such as high gimbal rates near singularities. The proposed method provides viable rates for the CMGs even when the gimbals pass through a singular configuration. The validity of the steering law is demonstrated using Liapunovs stability theorem, and it is shown that the method provides superior results when compared with conventional steering laws that use the pseudo-inverse of the CMG Jacobian Matrix. In addition, the redundant feature of the 4-gimbal CMG system is exploited to force the gimbals to attain certain specified angles while the spacecrafts attitude is being controlled. This is done by adding appropriate null motion to the CMGs. The specified angles for the gimbals can be arbitrary as long as their net momentum is consistent with the requirements of the spacecraft.