Louis S. Breger

Safe Trajectories for Autonomous Rendezvous of Spacecraft

ight experiments suggest that safety considerations will play an important role in the success of future missions. This paper presents a method for online generation of safe, fuel-optimized rendezvous trajectories that guarantee collision avoidance for a large class of anomalous system behaviors. We examine the cost of imposing safety as a problem constraint and of additional constraints that guarantee innite horizon passive collision avoidance while enabling future docking retries. Tradeos between passive and active approaches to safety are examined. A convex formulation of the collision avoidance algorithm is introduced and shown to provide much faster solutions with only a small additional fuel expense. Numerous examples using both rotating and nonrotating targets are presented to demonstrate the overall benets of incorporating these safety constraints when compared to nominal trajectory design techniques.

Distributed coordination and control of formation flying spacecraft

Formation flying is a key technology for many planned space missions that will use multiple spacecraft to perform distributed observations. This paper extends pro vious work on the design of a highly distributed formation flying control system that uses linear programming to determine minimum fuel trajectories for the spacecraft to remain within some specified tolerance of their “desired points”. The primary contribution of this paper is that it presents a direct procedure for calculating the fleet reference point (called the virtual center) that can be used to determine the desired states for each vehicle in the fleet. The calculation of this virtual center is based on measurements available from the relative navigation sensing system (carrier-phase differential GPS) developed for this application. The selection of the reference point includes a weighting on fuel use across the fleet, which facilitates increased coordination and cooperation within the decentralized control system. Full nonlinear simulations are presented to demonstrate the reduction in fuel use that can be obtained with this improved cooperation.

Gauss's Variational Equation-Based Dynamics and Control for Formation Flying Spacecraft

Formation flying is an enabling technology for many future space missions, and this paper presents several modeling and control extensions that would enhance the efficiency of many of these missions. In particular, a new linear time-varying form of the equations of relative motion is developed from Gauss’s variational equations. These new equations of motion are further extended to account for the effects of J2, and the linearizing assumptions are shown to be consistent with typical formation flying scenarios. It is then shown how these models can be used to initialize general formation configurations and can be embedded in an online, optimization-based, model predictive controller. A convex linear approach for initializing fuel-optimized partially J2 invariant orbits is developed and compared with analytic approaches. All control methods are validated using a commercial numerical propagator. Thesimulationresultsillustratethatformation flyingusingthismodelpredictivecontrollerwithJ2-modifiedGauss’s variational equations requires fuel use that is comparable to using unmodified Gauss’s variational equations in simulations that do not include the J2 effects. Nomenclature a = semimajor axis b = semiminor axis e = eccentricity h = angular momentum i = inclination M = mean motion n = orbit frequency p = semilatus rectum r = magnitude of radius vector � = argument of latitude � = right ascension of the ascending node ! = argument of perigee

Nonlinearity in Sensor Fusion: Divergence Issues in EKF, modified truncated SOF, and UKF

Relative navigation is a challenging technological component of many planned NASA and ESA missions. It typically uses recursive filters to fuse measurements (e.g., range and angle) from sensors with contrasting accuracies to estimate the vehicle state vectors in real time. The tendency of Extended Kalman filter to diverge under these conditions is well documented in the literature. As such, we have investigated the application of the modified truncated Second-Order Filter (mtSOF) and the Unscented Kalman filter (UKF) to those mission scenarios using numerical simulations of a representative experimental configuration: estimation of a static position in space using distance and angle measurements. These simulation results showed that the mtSOF and UKF may also converge to an incorrect state estimate. A detailed study establishes the divergence process of the mtSOF and UKF, and designs new strategies that improve the accuracy of these filters.

Model predictive control of spacecraft formations with sensing noise

This paper extends previous analysis on the impact of sensing noise upon the performance of model predictive control of formation flying spacecraft. We present a method of predicting the performance of the closed-loop system in the presence of sensing noise and demonstrate its effectiveness for the spacecraft relative motion problem. Its performance predictions are verified through simulation and used to analyze the effects of formation flying mission parameter trades without recourse to extensive numerical simulation. Ideal values for several model predictive control parameters are identified.

8 – Cooperative Spacecraft Formation Flying: Model Predictive Control with Open- and Closed-Loop Robustness

This chapter discusses the cooperative spacecraft formation flying. Formation flying of multiple spacecraft is an enabling technology for many future space science missions including enhanced stellar optical interferometers and virtual platforms for Earth observations. Controlling a formation will require several considerations beyond those of a single spacecraft. Key among these is the increased emphasis on fuel savings for a fleet of vehicles because the spacecraft must typically be kept in an accurate formation for periods on the order of hours or days, and the performance of the formation should degrade gracefully as one or more of the spacecraft runs out of fuel. This chapter presents a model predictive controller that is particularly well-suited to formation flying spacecraft because it explicitly minimizes fuel use, exploits the well-known orbital dynamics environment, and naturally incorporates constraints.

J2-Modified GVE-Based MPC for Formation Flying Spacecraft

Formation flying is an enabling technology for many future space missions. This paper presents an MPC controller that uses dynamics based on a modified version of Gauss’ Variational Equations which incorporates osculating J2 effects. A linear parameter-varying version of existing dynamics is developed, creating a highly accurate model that can easily be embedded in the MPC controller design. The linearization assumptions are shown to be consistent with typical formation flying scenarios. The controller is demonstrated on an MMS-like formation flying mission in a highly elliptical orbit using a commercial orbit propagator with realistic disturbances (including J2). These simulations show that formation flying using MPC with J2-modified GVEs requires fuel expenditures comparable to using unmodified GVEs in simulations with no J2 effects.

Analytical Performance Prediction for Robust Constrained Model Predictive Control

This paper presents a new analysis tool for predicting the closed-loop performance of a robust constrained model predictive control (MPC) scheme. Currently, performance is typically evaluated by numerical simulation, leading to an extensive computation when investigating the effect of controller parameters, such as the horizon length, the cost weightings and the constraint settings. The analytic method, in this paper, avoids this computational burden, thus enabling a rapid study of the trades between the design parameters and the performance. Previous work developed an MPC formulation employing constraint tightening to achieve robust feasibility and constraint satisfaction despite the action of an unknown but bounded disturbance. This paper shows that the expected performance of that controller can be predicted using a combination of the gains of two linear systems, the optimal control for the unconstrained system, and a candidate policy used in performing the constraint tightening. The method also accounts for the possible mismatch between the predicted level of disturbance and the actual level encountered. The analytic results are compared with simulation results for several examples and are shown to provide accurate predictions of performance and its variation with the system parameters.

Nonlinear Models of Relative Dynamics

This chapter presents commonly used nonlinear models for relative perturbed and unperturbed motion. The chapter presents a variety of models for simulating relative motion in perturbed or unperturbed elliptic orbits. In the absence of perturbations, bounded relative motion between any two spacecrafts in elliptic Keplerian orbits can be straightforwardly found from the energy matching condition. This chapter suggests a rather simple method for generating bounded relative motion between any two spacecrafts for unperturbed Keplerian motion. The underlying methodology relies on the concept of orbital-period commensurability. The chapter presents a single-impulse formation-keeping method by using the inherent freedom of the energy matching condition. The chapter provides an insight into the resulting formation-keeping maneuver by utilizing the classical orbital elements. The chapter elaborates the energy matching condition, and equations of relative motion under the influence of J2. A simple framework is designed for both initialization and initialization-error correction for spacecraft formation flying. This framework yields bounded relative motion between any two spacecraft flying on arbitrary elliptic orbits. The chapter also introduces a set of useful transformations for converting the relative motion states between the L and I frames.

Partial J2 Invariance for Spacecraft Formations

This paper presents a method for finding spacecraft formation initial conditions (ICs) that minimize the drift resulting from J2 disturbances and also minimize the fuel required to attain those ICs. A third goal that the formation remain in a particular geometry can also be considered. The approach uses linear optimization, but is valid for highly eccentric and widely spaced orbits. Optimization allows for the intelligent selection of degrees of freedom in already existing invariance conditions, as well as the minimization of different types of drift. Results are compared to J2-invariance conditions in the literature and the method is shown to find relative orbits with slightly lower levels of drift that require significantly less ∆V to obtain.