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Research Contract Summary. The research project, sponsored by General Dynamics C4 Systems, entails the design and implementation estimation and control algorithms for a Chaser vehicle in a reference frame relative to a Target vehicle. Algorithms are to be implemented using Simulink, with the intent
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1. Design of Guidance and Control Algorithms for Autonomous Rendezvous and Proximity Operations Jessica Williams
Department of Aerospace Engineering
and Engineering Mechanics
The University of Texas at Austin
Research Group Meeting
November 20, 2007
2. Research Contract Summary The research project, sponsored by General Dynamics C4 Systems, entails the design and implementation estimation and control algorithms for a Chaser vehicle in a reference frame relative to a Target vehicle. Algorithms are to be implemented using Simulink, with the intent to convert models into embedded C code for use on a real-time flight processor. I was supported by this contract through Summer 2007.
Work on the estimation task was performed from 2006-2007 and a summary package was delivered in Summer 2007. Subsequent work has been performed by Jack Goetz.
Work on the control task has been performed starting Fall 2007 and comprises the bulk of my research work.
3. Research ContractStatement of Purpose In February 2007, two main research contract objectives were identified:
The purposes of this effort are to define a suite of algorithms that can provide metric knowledge of a space vehicle (SV) in proximity to a host vehicle (HV) or to a specified orbit condition (SOC), and to provide the orbital maneuvering sequence that will control the relative motion of the SV during proximity operations about a HV or to rendezvous with a SOC.
These algorithms will be incorporated into flight software (FSW) by General Dynamics personnel and integrated into a testbed that will be utilized to demonstrate key performance parameters (KPP) associated with mission scenarios defined by program pursuit goals.
4. Past WorkEstimation Work performed in 2006 2007 focused on the Estimation task. A Kalman filter was designed in Simulink to estimate the absolute state of a vehicle in the IJK frame and the relative state of a vehicle in the RCO (relative) frame.
5. Matlab M-FileDescription t: Current time
t0: Initial simulation time
initstate: Initial state of orbit in ECI frame
sampleRate: Rate at which range data is imported
dPts: number of data points processed
mu: Gravitational constant
Xstar: Nominal reference trajectory (position and velocity)
Xstar0: Initial Nominal reference trajectory (position and velocity)
xhat0: initial estimate of correction to the nominal trajectory
xhat: Estimate of correction to the nominal trajectory
xbar: Correction to the state (xhat) propagated forward in time
K: Kalman gain
Pbar: Error covariance matrix
Po: Initial Error covariance matrix (cov = s2)
P: Error covariance matrix, initialized by P = Po
Htilda: Observation-state mapping matrix
G: Observation-state relationship (model)
Phi: State Transition Matrix F
Y: Observation vector (range measurement)
6. Simulink ModelData Generation Data is generated using CW propagation of the initial conditions, plus adding zero mean Gaussian white noise to the relative state vector. Angles are calculated using quadrant checks.
7. Simulink ModelKalman Filter Logic The relativeKalman model opens to a parameter declaration space and a masked subsystem.
8. Simulink ModelEstimation Algorithm The perform estimation subsystem is shown. Each block contains either a masked subsystem or an Embedded Matlab function to perform estimation algorithm.
9. Simulink ModelResults The evolution of the state correction is displayed. Final parameter values are output to Matlab workspace for filter performance evaluation.
10. EstimationProblems and Solutions The relative estimator I built had several problems/errors... These were corrected by Jack as part of the Fall 2007 research deliverable.
Data Stores in Simulink are not available until after the simulation has stopped, even within the simulation itself. Dont use a data store to save a value you would like to use at the next time step. Reading/writing to the same data store at a simulation time step resulted in an error message... every time.
The estimation algorithms required particular values at the previous time step. Time delay blocks are needed to retrieve this information.
The initial guess for the covariance matrix was way too high... estimated relative position and velocity would eventually. A smaller covariance repaired this.
Noise was added to data using zero-mean white Gaussian noise blocks. It turns out that these values were correlated.
11. Current WorkNavigation and Control Current work has focused on navigation/control aspect of a Chaser vehicle in a relative orbit about a Target vehicle.
12. Current WorkNavigation and Control A Linear Quadratic Regulator (LQR) control algorithm has been designed and implemented to keep a vehicle within a defined keep-in zone for stationkeeping maneuvers. This work has been performed using Simulink. Optimization has not been included in the routine*.
13. Current WorkInertial Equations of Motion The integral equations of motion are derived from Newtons Law of Gravitation, plus the inclusion of any disturbing forces and control forces.
14. Current WorkRelative Equations of Motion The Clohessy-Wiltshre equations are defined in a coordinate frame referenced relative to a vehicle in a circular orbit about a central body (the Earth).
15. Current WorkRelative Equations of Motion (CW) Solving the unperturbed linearized Clohessy-Wiltshre (CW) equations with zero external forces (f = 0) yields the simplified matrix results, where the current state in the relative frame can be determined from the initial state in the relative frame, the angular rate of the Target vehicle, and the time elapsed from the initial state to the current state.
16. Current WorkRelative Equations of Motion (Parameterized) The CW equations can be parameterized as a function of initial conditions only. This is a convenient geometric representation of the Chasers relative orbit about the Target vehicle.
17. Current WorkTargeting As referenced from Irvin1, a targeting routine can be designed to transfer a Target vehicle from one relative orbit to another desired relative orbit about a Target vehicle.
18. Current WorkKeep-In Zone The keep-in zone can be defined at any orientation and location relative to the Target vehicle.
19. Current ResultsTest Simulation: Initial Conditions Initial conditions are defined for the Target vehicle (inertial frame ICs) and for the Chaser vehicle (relative frame ICs)
20. Current ResultsTest Scenario The scenario was propagated over a 5 day interval. The vehicle states were propagated in the inertial frame, with the Chaser inertial state being converted into a relative state at each time step for control algorithm input.
4 cases have been investigated, for a particular choice of initial conditions, control gain, and boundary actuation range:
No perturbations, no control
No perturbations, control
Perturbations, no control
Perturbations, control
21. Current ResultsTest Simulation: Uncontrolled Case With Gravitational and Drag perturbations included, the Chaser vehicle drifts away from the keep-in zone over a 5 day simulation time.
22. Current ResultsTest Simulation: Controlled Case When control input is added, the Chaser vehicle is kept within or within a maximum bound of the keep-in region for the simulation time.
23. Current ResultsExample Simulation
24. Current ResultsExample Simulation
25. Current ResultsExample Simulation
26. Current ResultsExample Simulation
27. Current ResultsOptimization
28. Future WorkHover Orbit Problem
29. Hover OrbitRelation To and Adaptation From Current Work The current LQR routine performs continuous impulsive actuation at the boundary of a defined keep-in zone. To accommodate the impulsive maneuver hover orbit, the hover region shall be defined at an arbitrary location relative to the Target vehicle origin. Impulsive (optimal) maneuvers will be performed at the boundary of the 3-dimensional hover region.
30. Deliverables The following packages were delivered to General Dynamics in support of the research contract.
August 2007
Absolute estimator Simulink algorithm
Relative estimator Simulink algorithm
Documentation
Model Summary Document
Read-me Document
December 2007
Stationkeeping Simulink algorithm
Documentation
Algorithm Description Document
Algorithm Operation Document
Algorithm Test Case Document
31. Questions?