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

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
Simulation parameters are defined in the Matlab
workspace.

• 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.

Chaser
Target
7
Simulink ModelKalman Filter Logic

• The relativeKalman model opens to a parameter
  declaration space and a masked subsystem.

A digital clock, which updates at the sampleRate
declared in the workspace, is compared to a ramp
input. If the two are the same (meaning a
measurement update has occurred), the algorithm
logic is true and the estimation process
occurs.
The estimation algorithm is in a masked
subsystem. Inputs to the estimation algorithm are
range and angle data at a given time, as well as
the sample rate of the system.
Parameters declared in workspace are initialized
to be Data Stores in Simulink model. The use of
Data Stores for model parameters is still under
review updating the Data Store Write block after
the Data Store Read block has been read is
causing problems.
This simulation will be run from a call in the
Matlab M-file. Running the simulation
independently does not initialize the model
parameters, and an error message will be
displayed.
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.

(5) Measurement Update
(1) Parameter Routing
(6) View evolution of state updates
(3) Time Update
(4) Kalman Gain Calculation
(2) State Transition Matrix Calculation
(7) Parameter Updates and Workspace Variables
9
Simulink ModelResults

• The evolution of the state correction is
  displayed. Final parameter values are output to
  Matlab workspace for filter performance
  evaluation.

The nominal relative state can also be plotted
against the true relative state in the Matlab
workspace following completion of the simulation.

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.

Youll see why later...
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.

Unperturbed motion can be propagated in the IJK
frame
State Vector contains position and velocity of
vehicle in IJK frame
If vehicle is perturbed, the acceleration of the
body can be propagated in the IJK frame using
modified terms
Force is per unit mass
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).

Equations of motion are obtained by solving the
relative expressions  
Procedure entails binomial expansion, eliminating
non first-order terms to obtain the familiar
result  
Image courtesy Vallado
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.

Function of initial conditions!
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.

Choosing a 0 creates a closed relative orbit!
Relative coordinates propagate over time as
functions of these parameters
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.

A control input of R 1012 reaches a nearly
steady-state error after being propagated for
10000 seconds
A control input of R 108 reaches a nearly
steady-state error after being propagated for
3000 seconds.
1Irvin, D., A Study of Linear vs. Nonlinear
Control Techniques for the Reconfiguration of
Satellite Formations, MS Thesis, Air Force
Institute of Technology, Wright Patterson Air
Force Base, Ohio, March 2001.
18
Current WorkKeep-In Zone

• The keep-in zone can be defined at any
  orientation and location relative to the Target
  vehicle.

For the hover orbit, the keep-in zone will be an
ellipsoid defined at an arbitrary orientation
relative to the Target vehicle.
For the stationkeeping problem, the keep-in zone
will be a sphere centered at the Target vehicle
(origin of relative frame).
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)

Target Inertial state (Classical Orbital
Elements) k 6678
semimajor axis km e 0
eccentricity inc 10pi()/180
inclination rad AP 0pi()/180
argument of perigee rad LAN
0pi()/180 longitude of ascending
node rad v 10pi()/180 true
anomaly rad w sqrt(mu/k3)
mean motion 1/s
Target relative state (Parameterized) rho
1.9 a 0 b 0 theta
45pi()/180 m 1 n 0
The relative orbit parameters have been
particularly chosen to create a relative orbit
that is closed about the Target vehicle when
propagated using the linearized equations of
relative motion. A Lambert targeting routine
could be used to reach this closed relative orbit
in a mission operation simulation.
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.

Chaser vehicle drifts away from Target vehicle
over simulation period. Along-track drift is due
to atmospheric drag and is the primary disturbing
force.
Radial ?
Along-track ?
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.

Chaser vehicle stays within proximity to Target
vehicle for entire simulation period
Radial ?
Along-track ?
23
Current ResultsExample Simulation
In the uncontrolled case, the distance from the
Target vehicle to the Chaser vehicle grows
exponentially over the simulation time, due
primarily to drag effects. When controlled, the
vehicle stays (nearly) within the prescribed
keep-in zone.
Uncontrolled Chaser
Controlled Chaser
24
Current ResultsExample Simulation
In the uncontrolled case, the Target relative
state is seen to drift away from a desired
(targeted) state in the relative frame that would
keep the Chaser in proximity to the Target. When
controlled, the Target relative state does not
match exactly, but the difference stays bounded.
Uncontrolled Chaser
Controlled Chaser
25
Current ResultsExample Simulation
In the uncontrolled case, the distance between
the vehicles in the IJK frame begins to grow
exponentially. When controlled, this distance
stays bounded.
Uncontrolled Chaser
Controlled Chaser
26
Current ResultsExample Simulation
In the uncontrolled case, the error between the
desired relative state and the actual relative
state begins to increase exponentially,
especially in the along-track direction as the
Chaser semimajor axis is brought down by drag.
The error stays bounded in the controlled case.
Uncontrolled Chaser
Controlled Chaser
27
Current ResultsOptimization
The LQR may keep the Chaser vehicle within the
keep-in zone for the simulation time, but it may
not be optimal or realistic.
250 m/s ?V over a 5 day simulation! No
mission would implement this
Simulation parameters can be varied to create a
more realistic stationkeeping strategy. Ideally
this will be optimized for a minimum ?V cost
function. For now, more suitable parameters can
be chosen.
28
Future WorkHover Orbit Problem

• Future work stems from the Hover Orbit problem
  that has been investigated by Irvin and Cobb1.
  Optimal impulsive burns were calculated and
  implemented in a 2-dimensional relative frame
  using CW dynamics. Impulsive maneuvers are
  performed only at the hover bubble boundary.
• This past body of work can be enhanced by
• Implementing maneuvers in 3-D space.
• Performing impulsive burns at optimal locations
  on a minimum or maximum hover ellipse region.
• Adding perturbations to the simulation model.

1Irvin, D. and Cobb, R. Multiple Leg
Fuel-Optimal Trajectories For Hovering
Satellites, 17th AAS/AIAA Space Flight Mechanics
Meeting Sedona, Arizona, 2007.
Image courtesy Dr. Glenn Lightsey
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?