Research Background

1. Research Background Nonlinear Control of Electro-Mechanical Systems Experimental Validation Design Analysis Robot Manipulators Magnetic Bearings Flexible Mechanical Systems (PDE) Aerospace Systems - Constrained robots - Electrically driven robots - Flexible joint robots - Formation Flying - Attitude Control - VTOL

2. Nonlinear Control of Multiple Spacecraft Flying in Formation Dr. Marcio S. de Queiroz

3. Outline • Multiple Spacecraft Formation Flying (MSFF) Concept • Dynamic Model • Nonlinear Control • Simulation Results • Fuel Consumption Issue • Ongoing and Future Research

4. MSFF Concept • Distribute the functionality of a large spacecraft among an array of highly-coordinated, autonomous micro-spacecraft (“Virtual spacecraft”) Virtual spacecraft Large, specialized spacecraft

5. MSFF Concept • Virtual spacecraft • Mission hardware function spread across micro-spacecraft • Coordination via software • Analogous to network of PCs vs. mainframe

6. MSFF Concept • Why? • Micro-spacecraft are less expensive • Mass production • Low weight/volume for launch • Increases the baseline of scientific instruments • Widens coverage area of satellites • Reduces ground support • Micro-spacecraft are autonomous • MEMS are an enabling technology • Micro-instruments, micro-propulsion

7. MSFF Concept • Why? (cont.) • Flexible architecture • Robustness, redundancy, and reconfigurability • Minimizes effects of failure • Multi-mission capability Reduces mission cost and increases performance

8. MSFF Concept • Current related applications • Spacecraft rendezvous • Satellite recovery and servicing • Potential future applications • Surveillance • Earth surface mapping • Space-based communication system • Interferometer

9. MSFF Concept • Facts • MSFF idea was first proposed in 1984 • Has not yet been flight-tested • NASA’s New Millennium Interferometer (NMI) • Formation of 3 spacecraft for long baseline optical stellar interferometry • NASA’s Earth Orbiter-1 (EO-1) • Formation of 2 spacecraft with the Landsat 7 satellite for stereo imaging

10. MSFF Concept • More facts • AFOSR’s TechSat 21 • Several application missions to demonstrate MSFF paradigm • Micro-satellite dimensions: 2-7 meters, weight: < 100 kg • AFOSR/DARPA University Nanosatellite Program • NMSU/ASU/UC 3 Corner SAT

11. … next month … the Air Force launches a fleet of tiny … satellites made of miniature components - diminutive machines that could … work together in groups to replace or supplement larger spacecraft. Researchers are exploring methods to … use midget spacecraft - some weighing less than a pound and hardly larger than a pack of cards - that could be used alone to perform simple tasks or flown in formations to execute more complex ones. “We’re talking about fully integrated satellites that could be mass produced cheaply by the hundreds and sent into space to perform a of variety tasks.” If one or several of the machines in a formation fails, others in the group could redistribute themselves and the continue performing the same task ... Peter Panetta of NASA’s Goddard Space Flight, agrees, saying there is a growing interest in increasingly smaller … spacecraft. “This isn’t just a fad. A lot of people see this as the future …”

12. MSFF Concept • Guidance and control challenges • Reliable onboard sensing to determine relative position/attitude • Global positioning system (GPS) • High-level control • Fleet path planning, navigation strategy • Fault-tolerance schemes • Centralized vs. decentralized control • Low-level control • Accurate control of the relative position/attitude(NMI mission: order of a centimeter; EO-1 mission: order of 10-20 meters) • Should be fuel-efficient High-level control Low-level control Sensing

13. MSFF Concept • Two-phase, low-level control operation • Formation reconfiguration • Spacecraft are commanded to their respective positions and orientations in the formation • Requires propulsion • Formation-keeping • Once in formation, spacecraft move in their respective natural orbits about the Earth • Maintained by orbital mechanics and propulsion

14. Dynamic Model • Spacecraft is a point-mass • Only position dynamics • MSFF fleet composed of a leader-follower pair • Leader provides a reference motion trajectory • Follower navigates in neighborhood of leader according to a desired, relative trajectory • Navigation strategy motivated by marching bands • Designated band leaders provide basic reference path • Band members navigate by tracking certain leaders

15. Dynamic Model • Schematic representation of the MSFF system R(t): Position vector of leader from Earth center r(t): Position vector of follower relative to leader

16. Dynamic Model • Newton’s law of gravitation Two bodies attract each other with a force acting along the line that joins them G:Universal gravitational constant

17. Dynamic Model • Dynamics of leader spacecraft • Dynamics of follower spacecraft M:Earth massml, mf: Spacecraft masses Fdl (t), Fdf (t): Disturbance force vectors (3x1) ul(t), uf (t): Control input vectors (3x1)

18. Dynamic Model • Nonlinear, relative position dynamics Fd(t): Composite disturbance force

19. Dynamic Model • Dynamics are given w.r.t. inertial coordinate frame • Spacecraft masses vary slowly in time due to fuel consumption and payload variations • mland mf are constant parameters • Disturbance forces result from solar radiation, aerodynamics, and magnetic field; hence, vary slowly in time • Fdis a constant vector

20. Nonlinear Control • Common practice: • Linearize relative position dynamics • Hill’s or Clohessy-Wiltshire equations • Design standard, linear controllers • Assumptions • for all time • Leader in circular orbit around the Earth • Reasonable approach for formation-keeping

21. Nonlinear Control • Problems with linearized dynamics • Initial position of follower relative to leader may be large • During formation reconfiguration maneuvers, leader will not be in circular orbit • Control system will need to download a new linear controller • Control design based on nonlinear model • Same controller valid for formation reconfiguration andformation-keeping Extrapolates “valid” operating range

22. Nonlinear Control • Significant contributions can be made to advance MSFF technology by exploiting nonlinear control • Several issues tailored for nonlinear control • Dynamic model is nonlinear • Higher performance under broader operating conditions • Trajectory tracking problem • Reconfiguration maneuvers, collision avoidance, minimize fuel • Uncertainties in system model • Mass, inertia, disturbance, drag • Expensive sensor technology (GPS) may limit state info • Actuator saturation • Physical limit or need to minimize fuel

23. Nonlinear Control • Goal: Design a new class of MSFF controllers that addresses these issues • Theoretical tools: Lyapunov-based control design • Easily handle nonlinearities • Flexible • Tracking or setpoint problems • Adaptive or robust controllers for uncertainties • Output feedback controllers for lack of full-state feedback • Bounded controllers for actuator saturation • Guaranteed stability properties • Implementation tools: Low-cost and computational power of microprocessors

24. Nonlinear Control (Design) • Adaptive tracking control objective • Given the nonlinear MSFF dynamics and a desired position trajectory of follower w.r.t. leader, rd(t). Design uf(t) such that • Assumption: Spacecraft masses and disturbance forces are not known precisely

25. Nonlinear Control (Design) • Property: Dynamics can be parameterized Known matrix: Unknown, constant parameter vector:

26. Nonlinear Control (Design) • Quantify control objective • Position tracking error: • Control objective is then • Parameter estimation error: • is a dynamic, parameter estimate • Filtered tracking error: • L > 0 is a constant, diagonal, control gain matrix • Allows 2nd-order dynamic equation to be written as a 1st-order • If r(t) 0 then e(t) 0

27. Nonlinear Control (Design) • Write dynamics in terms of r(t) • Substitute for using dynamic equation, and apply parameterization property

28. Nonlinear Control (Design) • Adaptive control law (standard, “robot” adaptive controller) • K > 0 is constant, diagonal, control gain matrix • G > 0 is constant, diagonal, adaptation gain matrix • Closed-loop system dynamics Linear feedback stabilizing term Attempts to “cancel” Helps “cancel”

29. Nonlinear Control(Stability Analysis) • Lyapunov Stability Analysis • If system’s total energy is continuously dissipating system eventually goes to an equilibrium point • Determination of system’s stability properties • Construct a scalar, energy-likefunction (V(t) 0) • Examine function’s time variation  Stable  Unstable

30. Nonlinear Control (Stability Analysis) • Define the non-negative function • Differentiate V along closed-loop dynamics • Apply Barbalat’s lemma Position tracking error is asymptotically stable

31. Simulation Results • System parameters • Leader spacecraft in natural orbit around the Earth • Radius: 4.224x107m • Angular velocity: w= 7.272 x 10-5 rad/s (orbit period = 24 h) • No control required (ul = 0)

32. Simulation Results (Unnatural Trajectory) • Initial position and velocity of follower relative to leader • Desired relative trajectory • Follower is commanded to move around leader in a circular orbit of radius 100m with angular velocity 4w • Parameter estimates initialized to 50% of actual parameter values( )

33. Simulation Results (Unnatural Trajectory) Relative Trajectory (‘*’ denotes leader spacecraft)

34. Simulation Results (Unnatural Trajectory) Position Tracking Errors Parameter Estimates

35. Simulation Results (Unnatural Trajectory) Control Forces • Maximum magnitude =0.02 N

36. Simulation Results (Natural Trajectory) • Follower commanded to move in natural, elliptical orbit around the Earth with orbit period = 24 h • Typical of formation-keeping • Elliptical orbit for rd(t)obtained by integrating • Relative dynamics with ul = uf = 0 and Fd = 0 • Proper initial conditions must be selected • Parameter estimates initialized to zero( )

37. Simulation Results (Natural Trajectory) Position Tracking Errors Parameter Estimates • Disturbance estimates converge to actual values

38. Simulation Results (Natural Trajectory) Control Forces • Maximum magnitude =4 x 10-5 N

39. Fuel Consumption Issue • Continuous thruster • Ideal scenario • Control amplitude can be continuously modulated but maximum amplitude is limited • Nonlinear saturation control results apparently can be applied with guaranteed closed-loop stability • On/off type thruster • Currently, a more realistic scenario • Control amplitude can be modulated only for certain periods of time • Not clear how to rigorously address closed-loop stability under a pulse-type, nonlinear control law

40. Fuel Consumption Issue • Formation-keeping • On/off thrusters may suffice • When “off”, orbital mechanics maintain natural orbit • Formation reconfiguration • Demanding maneuvers will require significant control effort • When on/off thrusters are used, obvious trade-off between performance and fuel consumption • Reconfiguration may last for only short periods of time

41. Fuel Consumption Issue • Simple, ad-hoc solution to reduce fuel consumption • Let qd(t) = [xd(t), yd(t), zd(t)]Tbe a desired spacecraft trajectory • Define a ball centered at{xd(t), yd(t), zd(t)}with radiuse

42. Fuel Consumption Issue Gradient at a point q* = {x*, y*, z*}on the ball surface:

43. Fuel Consumption Issue On/Off Type Control Algorithm Goal:Control spacecraft position such that it never leaves the ball qs(t) = [xs(t), ys(t), zs(t)]T: spacecraft position 1. If 2. If Control off Control on

44. Fuel Consumption Issue On/Off Type Control Algorithm (cont.) 3. If • If  • Else  Control off Control on

45. Fuel Consumption Issue • Control on means: • Control is set to the designed nonlinear control • Left on for some finite time interval T • Algorithm is resumed only after T has expired • Case 2 (spacecraft outside ball) may occur during initialization of formation reconfiguration • Trade-off between tracking performance and fuel consumption • Asymptotic tracking vs. bounded tracking with less fuel

46. Ongoing and Future Research • Account for spacecraft attitude dynamics • MSFF position/attitude tracking controller • 4-parameter kinematic representation (quaternions) • Account for higher-order gravitational perturbations (J2 effect) and atmospheric drag • Output feedback controller • Only GPS position measurements • No GPS “estimation” architecture for velocity • Formation control of autonomous vehicles • Aircraft, ships, underwater vehicles, mobile robots

47. Ongoing and Future Research • Testbed for preliminary experiments • 3 DOF • DC motor-propeller pairs provide actuation • Optical encoders sense the 3 angular positions