Michael Tillerson

Relative Dynamics and Control of Spacecraft Formations in Eccentric Orbits

Formation eying is a key technology for both deep-space and orbital applications that involve multiple spacecraft. Many future space applications will beneet from using formation e ying technologies to perform distributed observations (e.g., synthetic apertures for Earth mapping interferometry) and to provide improved coverage for communication and surveillance. Previous research has focused on designing passive apertures for these formation e ying missions assuming a circular reference orbit. Those design approaches are extended and a complete initialization procedure for a large e eet of vehicles with an eccentric reference orbit is presented. The main result is derived from the homogenous solutions of the linearized relative equations of motion for the spacecraft. These solutions are used to end the necessary conditions on the initial states that produce T-periodic solutions that have the vehicles returning to the initial relative states at the end of each orbit, that is, v(t0)=v(t0+T). This periodicity condition and the resulting initialization procedure are originally given (in compact form) at the reference orbit perigee, butthis is alsogeneralized to enable initialization atanypoint around thereference orbit. In particular, an algorithm is given that minimizes the fuel cost associated with initializingthe vehicle states (primarily the in-track and radial relative velocities) to values that are consistent with periodic relative motion. These algorithms extend and generalize previously published solutions for passive aperture forming with circular orbits. The periodicity condition and the homogenous solutions can also be used to estimate relative motion errors and the approximate fuel cost associated with neglecting the eccentricity in the reference orbit. The nonlinear simulations presented clearlyshowthatignoringthereferenceorbiteccentricitygeneratesanerrorthatiscomparabletothedisturbances caused by differential gravity accelerations.

COORDINATION AND CONTROL OF MULTIPLE UAVs

This paper addresses the problems of autonomous task allocation and trajectory planning for a fleet of UAVs. Two methods are compared for solving the optimization that combines task assignment, subjected to UAV capability constraints, and path planning, subjected to dynamics, avoidance and timing constraints. Both sub-problems are non-convex and the two are strongly-coupled. The first method expresses the entire problem as a single mixed-integer linear program (MILP) that can be solved using available software. This method is guaranteed to find the globally-optimal solution to the problem, but is computationally intensive. The second method employs an approximation for rapid computation of the cost of many different trajectories. This enables the assignment and trajectory problems to be decoupled and partially distributed, offering much faster computation. The paper presents several examples to compare the performance and computational results from these two algorithms.

Cooperative path planning for multiple UAVs in dynamic and uncertain environments

This paper addresses the problem of cooperative path planning for a fleet of unmanned aerial vehicles (UAVs). The paths are optimized to account for uncertainty/adversaries in the environment by modeling the probability of UAV loss. The approach extends prior work by coupling the failure probabilities for each UAV to the selected missions for all other UAVs. In order to maximize the expected mission score, this stochastic formulation designs coordination plans that optimally exploit the coupling effects of cooperation between UAVs to improve survival probabilities. This allocation is shown to recover real-world air operations planning strategies, and to provide significant improvements over approaches that do not correctly account for UAV attrition. The algorithm is implemented in an approximate decomposition approach that uses straight-line paths to estimate the time-of-flight and risk for each mission. The task allocation for the UAVs is then posed as a mixed-integer linear program that can be solved using CPLEX.

Multi-Task Allocation and Path Planning for Cooperating UAVs

This paper presents results on the guidance and control of fleets of cooperating Unmanned Aerial Vehicles (UAVs). A key challenge for these systems is to develop an overall control system architecture that can perform optimal coordination of the fleet, evaluate the overall fleet performance in real-time, and quickly reconfigure to account for changes in the environment or the fleet. The optimal fleet coordination problem includes team composition and goal assignment, resource allocation, and trajectory optimization. These are complicated optimization problems for scenarios with many vehicles, obstacles, and targets. Furthermore, these problems are strongly coupled, and optimal coordination plans cannot be achieved if this coupling is ignored. This paper presents an approach to the combined resource allocation and trajectory optimization aspects of the fleet coordination problem which calculates and communicates the key information that couples the two. Also, this approach permits some steps to be distributed between parallel processing platforms for faster solution. This algorithm estimates the cost of various trajectory options using the distributed platforms and then solves a centralized assignment problem to minimize the mission completion time. The detailed trajectory planning for this assignment can then be distributed back to the platforms. During execution, the coordination and control system reacts to changes in the fleet or the environment. The overall approach is demonstrated on several example scenarios to show multi-task allocation and cooperative path planning.

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.

Formation Flying Control in Eccentric Orbits

This paper extends recent work on the control of a formation of spacecraft orbiting about an eccentric reference orbit. The approach uses the equations for periodic relative motion that were previously developed from Lawden’s original work. This paper presents fuel/time-optimal algorithms for the low-level station-keeping of one satellite with respect to another satellite in the presence of disturbances. The station-keeping algorithm is optimized by posing it as a linear programming problem. The primary extension of this paper is to present the solution to the linear programming problem using the time-varying linearized dynamics that occur for an eccentric reference orbit. Numerous nonlinear simulations were performed to demonstrate the effectiveness of this overall control approach. The results indicate that, even in the presence of differential J2 disturbances, our formation flying control approach is very effective, requiring a ∆V =5 —15 mm/s/orbit, depending on the scenario. The simulations also show that Lawden’s equations are necessary for determining the desired state for periodic relative motion, but Hill’s equations are sufficient for the linear programming control problem. This result is important because using the time-invariant Hill’s equations significantly reduces the computational effort required to formulate the linear program. 1

Advanced guidance algorithms for spacecraft formation-keeping

This paper presents advanced formation-keeping guidance algorithms that use linear programming (LP) to determine fuel-optimal control inputs and state trajectories. The overall formation-keeping problem is analyzed in terms of two key issues: (i) what dynamics model should be used to specify the desired state to maintain a passive aperture; and (ii) what dynamics model should be used in the LP to represent the motion about this state. Several linearized models of the relative dynamics are considered in this analysis, including Hills equations for circular orbits, modified linear dynamics that partially account for the J/sub 2/ effects, and Lawdens equations for eccentric orbits. A controller is developed for formation-keeping using each of these models. A modified LP formulation is presented to include robustness to sensor noise while ensuring a feasible solution. The guidance algorithms are implemented in numerous very detailed nonlinear simulations that demonstrate effective control in the presence of all expected disturbances and sensor noises. The average fuel cost for the formation-keeping maneuvers over a two week simulation is on the order of 4 mm/s per orbit.

Analysis of the impact of sensor noise on formation flying control

This paper analyzes the impact of sensing noise on formation flying control algorithms developed for distributed spacecraft systems. The key issue is that the sensing errors cause uncertainty in the initial conditions of the trajectory planning process which is at the core of the fuel-optimization algorithm. To account for these sensing errors, modifications are presented to the station-keeping optimization algorithms that have been developed using linear programming. The approach robustified the design of the control inputs to velocity errors, but this is achieved at the expense of using much shorter design horizons. The modified control approach is demonstrated using a realistic nonlinear simulation environment. The results from these simulations confirm that noise in the relative velocity measurements will play a crucial role in the fleet performance and/or the fuel cost.

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.

NEW FORMATION FLYING TESTBED FOR ANALYZING DISTRIBUTED ESTIMATION AND CONTROL ARCHITECTURES

Formation °ying spacecraft has been identied as an enabling technology for many future NASA and DoD space missions. However, this is still, as yet, an unproven technology. Thus, to minimize the risk associated with using this new technology on future missions, testbeds are required that enable a com- prehensive simulation and validation. This paper presents an innovative hardware-in-the-loop testbed that can be used to analyze estimation and control architectures for formation °ying spacecraft. The testbed consists of multiple computers that each em- ulate a spacecraft in the °eet. These computers are restricted to communicate via serial cables to emu- late the actual inter-spacecraft communications ex- pected on-orbit. All estimation and control algo- rithms are implemented in MATLAB, which greatly enhances its °exibility/recongurability and provides an excellent environment for rapidly comparing nu- merous control and estimation algorithms and ar- chitectures. A multi-tasking/multi-thread software environment is simulated by simultaneously running several instances of MATLAB on each computer. The paper presents the initial simulation results us- ing a highly distributed estimation, coordination, and control architecture for a °eet of 4 spacecraft.