Kyle T. Alfriend

State Transition Matrix of Relative Motion for the Perturbed Noncircular Reference Orbit

A precise analytic solution that includes the effects of the reference orbit eccentricity and differential perturbations is needed for the relative motion of formation-flying satellites. As a result of the spherical Earthand circular reference orbit assumptions, Hills equations, which have often been used for describing relative motion, are insufficient for the long-term prediction of the relative motion. A new approach, called the geometric method, is developed to obtain the state transition matrix for the relative motion that includes the effects caused by the reference orbit eccentricity and the differential gravitational perturbations. The geometric method uses the relationship between the relative states and differential orbital elements to obtain the state transition matrix instead of directly solving the complex relative motion differential equations. The state transition matrices are derived for both mean and osculating elements with the primary gravitational perturbation that results from the equatorial bulge term J 2 . Although the results are based on the J 2 effects, the approach can be extended easily to include other perturbing forces.

J2 Invariant Relative Orbits for Spacecraft Formations

An analytic method is presented to establish J2 invariant relative orbits. Working with mean orbit elements, the secular drift of the longitude of the ascending node and the sum of the argument of perigee and mean anomaly are set equal between two neighboring orbits. By having both orbits drift at equal angular rates on the average, they will not separate over time due to the J2 influence. Two first order conditions are established between the differences in momenta elements (semi-major axis, eccentricity and inclination angle) that guarantee that the drift rates of two neighboring orbits are equal on the average. Differences in the longitude of the ascending node, argument of perigee and initial mean anomaly can be set at will, as long as they are setup in mean element space. For near polar orbits, enforcing both momenta element constraints may result in impractically large relative orbits. It this case it is shown that dropping the equal ascending node rate requirement still avoids considerable relative orbit drift and provides substantial fuel savings.

Impulsive Feedback Control to Establish Specific Mean Orbit Elements of Spacecraft Formations

An impulsive feedback control is developed to establish specie c relative orbits for spacecraft formation e ying. The relative orbit tracking errors are expressed in terms of mean orbit elements. The feedback control, based on Gauss’ s variational equations of motion, allows specie c orbit elements or orbit element sets to be controlled with minimal impact on the remaining osculating orbit elements. This is advantageous when J2-invariant orbits are to be controlled, where only the argument of perigee and mean anomaly will drift apart at equal and opposite rates. The advantage of this impulsive feedback control, compared to optimal control solutions, is that it can operate with little computational effort and in a near-optimal manner, whilerequiring only a marginal penalty in fuel cost. When applied to the spacecraft formation e ying problem, this control could also be used to perform general orbit corrections. Formulas are developed providing accurate estimates of the sensitivities of the mean semimajor axis and mean eccentricity with respect to the osculating inclination angle. With these sensitivities, the tracking error in semimajor axis, eccentricity, and inclination angle can be canceled within one orbit.

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.

Probability of Collision Error Analysis

The decision for the International Space Station (ISS) to maneuver to avoid a potential collision with another space object will be based on the probability of collision, PC. The calculation of PC requires the covariance of both objects at conjunction. It is well known that the covariance computed by US Space Command is optimistic (too small), especially at altitudes where atmospheric drag is the dominant perturbation, because its computation assumes there are no dynamic model errors. In this paper the effect of errors in the covariance on PC and the sensitivity of PC to the encounter geometry are investigated.

Probability of Collision Between Space Objects

The International Space Station is being designed to perform debris avoidance maneuvers based on certain criteria developed from the probability of collision Pc. Existing methods to determine the Pc are based on the dee nitionofacollision/conjunctionplanethatcontainsallofthepositionuncertaintyassociatedwiththeproblem.In thispaperwedevelopa directand more naturalway of obtaining probability ofcollision and presentanalternative but equivalent dee nition for Pc that leads to the same results obtained earlier. Because debris avoidance is crucial for every orbiting asset in low Earth orbit, a study of this nature helps to establish the equivalence of different methods for risk assessment and evaluation. HE International Space Station (ISS) shall continuously face the threat of collision with orbiting debris. Hence, there needs to be a comprehensive methodology that can assess the risks posed byindividualdebrisencountersand suggestmaneuverswhenneces- sary. Such a study shall not only benee t the ISS, but also any future orbital asset placed in low Earth orbits. Thus, although we refer to the ISS in the rest of our paper, the analysis presented here holds true for any other orbiting asset of size and orbit comparable with that of the upcoming ISS. Space shuttle (SS) maneuvers are commanded to avoid poten- tial collisions with cataloged space objects (maintained by the U.S. Space Command ) whenever the estimated conjunction with an ob- jectfalls within a box centered on the estimated SSposition. The di- mensionsofthisconjunctionboxare §5 kmin thein-track direction and §2kmintheradialandout-of-planedirections.Thedimensions of such a conjunction box are probably based on prior estimates of position error covariances. The determination was made that this simple criterion, or any other known deterministic criterion 1,2 when applied to the ISS, would result in too many maneuvers. 3 In addi- tion, unnecessary maneuvers waste fuel and hamper the micrograv- ity experiments onboard the ISS. Although the size of the conjunc- tion box could be decreased to decrease the manuever rate, such a step clearly increases the risk to unacceptable levels. Therefore, the ISS needs a more rigorous probability-based approach for collision avoidance. 4

Hybrid Cartesian and Orbit Element Feedback Law for Formation Flying Spacecraft

A spacecraft formation e ying control strategy is discussed where the desired orbit is prescribed in terms of specie c orbit element differences and the relative orbit is measured in terms of Cartesian coordinates of the rotating reference frame centered on the chief satellite. A direct method to map orbit element differences to their corresponding local Cartesian coordinates is presented. A numerical study illustrates theaccuracy with which this linearized transformation performs this coordinate transformation. A hybrid continuous feedback control law is thendevelopedthathasthedesired relativeorbitgeometryexplicitlygivenin termsoforbitelementdifferencesand the actual orbit given in terms of local Cartesian coordinates. A numerical simulation illustrates the performance and limitations of such feedback control laws. Using the linearized mapping between the relative orbit coordinates causes only a small performance penalty. However, it is advantageous to work in mean element space when determining the relative orbit tracking error.

Evaluation and Comparison of Relative Motion Theories

A modeling error index is introduced for evaluating and comparing the accuracy of various theories of the relative motion of satellites to determine the effect of modeling errors on the various theories. The derived index does not require linearization of the equations of motion, and so nonlinear theories can also be evaluated. The index can be thought of as proportional to the percentage error; consequently, the smaller the index, the more accurate the theory. The results show that not including the reference orbit eccentricity and differential gravitational perturbations has a major effect on the accuracy of the theory, and the nonlinear effects are much smaller except for very large relative motion orbits. The two key parameters in the evaluation are the eccentricity of the reference orbit and the relative motion orbit size. The theories compared are Hill’s equations, a small eccentricity state transition matrix, a non-J2 state transition matrix, the Gim–Alfriend state transition matrix, the unit sphere approach, and the Yan–Alfriend nonlinear method. The numerical results show the sequence of the index from high to low should be Hill’s equation, non-J2, small eccentricity, Gim–Alfriend state transition matrix index, with the unit sphere approach and the Yan–Alfriend nonlinear method having the lowest index and equivalent performance.

Adaptive Sigma Point Filtering for State and Parameter Estimation

This article presents practical adaptive nonlinear filters for recursive estimation of the state and parameter of nonlinear systems with unknown noise statistics. The adaptive nonlinear filters combine adaptive estimation techniques for system noise statistics with the nonlinear filters that include the unscented Kalman filter and divided difference filter. The purpose of the integrated filters is to not only compensate for the nonlinearity effects neglected from linearization by utilizing nonlinear filters, but also to take into account the system modeling errors by adaptively estimating the noise statistics and unknown parameters. Simulation results indicate that the advantages of the adaptive filters make these attractive alternatives to the standard nonlinear filters for the state and unknown parameter estimation in the orbit determination.

Magnetic Attitude Control System for Dual-Spin Satellites

A closed-loop control law is developed for a dual-spin satellite control system which utilizes the interaction of the geomagnetic field with the satellite dipole parallel to the spin axis. The control law consists of the linear combination of the pitch axis component of the rate of change of the geomagnetic field and the product of the roll angle and roll axis component of the geomagnetic field. Application of the method of multiple time scales yields approximate solutions for the feedback gains in terms of the system parameters. Approximate solutions are also obtained for the response of the system to disturbance torques. A comparison of the approximate solutions and numerical solutions obtained by numerical integration of the exact equations of motion is then given.