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Design of Guidance and Control Algorithms for Autonomous Rendezvous and Proximity Operations

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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Design of Guidance and Control Algorithms for Autonomous Rendezvous and Proximity Operations

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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 Contract Statement 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 Work Estimation 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-File Description 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 Model Data 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 Model Kalman Filter Logic The “relativeKalman” model opens to a parameter declaration space and a masked subsystem.

    8. Simulink Model Estimation 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 Model Results The evolution of the state correction is displayed. Final parameter values are output to Matlab workspace for filter performance evaluation.

    10. Estimation Problems 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. Don’t 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 Work Navigation and Control Current work has focused on navigation/control aspect of a Chaser vehicle in a relative orbit about a Target vehicle.

    12. Current Work Navigation 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 Work Inertial Equations of Motion The integral equations of motion are derived from Newton’s Law of Gravitation, plus the inclusion of any disturbing forces and control forces.

    14. Current Work Relative 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 Work Relative 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 Work Relative 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 Chaser’s relative orbit about the Target vehicle.

    17. Current Work Targeting 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 Work Keep-In Zone The keep-in zone can be defined at any orientation and location relative to the Target vehicle.

    19. Current Results Test Simulation: Initial Conditions Initial conditions are defined for the Target vehicle (inertial frame IC’s) and for the Chaser vehicle (relative frame IC’s)

    20. Current Results Test 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 Results Test 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 Results Test 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 Results Example Simulation

    24. Current Results Example Simulation

    25. Current Results Example Simulation

    26. Current Results Example Simulation

    27. Current Results Optimization

    28. Future Work Hover Orbit Problem

    29. Hover Orbit Relation 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?

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