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

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

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

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

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

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

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

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

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

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

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. … 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. “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 …”

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

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

Dynamic Model u Spacecraft is a point-mass >Only position dynamics u 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 u Navigation strategy motivated by marching bands >Designated band leaders provide basic reference path >Band members navigate by tracking certain leaders

Dynamic Model u Schematic representation of the MSFF system R(t) : Position vector of leader from Earth center  (t) : Position vector of follower relative to leader

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

Dynamic Model u Dynamics of leader spacecraft u Dynamics of follower spacecraft M : Earth mass m l, m f : Spacecraft masses F dl (t), F df (t): Disturbance force vectors (3x1) u l (t), u f (t): Control input vectors (3x1)

Dynamic Model u Nonlinear, relative position dynamics F d (t) : Composite disturbance force

Dynamic Model u Dynamics are given w.r.t. inertial coordinate frame u Spacecraft masses vary slowly in time due to fuel consumption and payload variations  m l and m f are constant parameters u Disturbance forces result from solar radiation, aerodynamics, and magnetic field; hence, vary slowly in time  F d is a constant vector

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

Nonlinear Control u 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 u Control system will need to download a new linear controller u Control design based on nonlinear model >Same controller valid for formation reconfiguration and formation- keeping Extrapolates “valid” operating range

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

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

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

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

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

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

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

Nonlinear Control (Stability Analysis) u 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-like function ( V(t)  0 ) l Examine function’s time variation  Stable  Unstable

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

Simulation Results u System parameters u Leader spacecraft in natural orbit around the Earth  Radius: x 10 7 m  Angular velocity:  = x rad/s (orbit period = 24 h)  No control required ( u l = 0 )

Simulation Results (Unnatural Trajectory) u Initial position and velocity of follower relative to leader u Desired relative trajectory  Follower is commanded to move around leader in a circular orbit of radius 100 m with angular velocity 4   Parameter estimates initialized to 50% of actual parameter values 

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

Simulation Results (Unnatural Trajectory) Position Tracking Errors Parameter Estimates

Simulation Results (Unnatural Trajectory) Control Forces  Maximum magnitude = 0.02 N

Simulation Results (Natural Trajectory) u Follower commanded to move in natural, elliptical orbit around the Earth with orbit period = 24 h >Typical of formation-keeping  Elliptical orbit for  d (t)  obtained by integrating  Relative dynamics with u l = u f = 0 and F d = 0 >Proper initial conditions must be selected  Parameter estimates initialized to zero 

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

Simulation Results (Natural Trajectory) Control Forces  Maximum magnitude = 4 x N

Fuel Consumption Issue u 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 u 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

Fuel Consumption Issue u Formation-keeping >On/off thrusters may suffice >When “off”, orbital mechanics maintain natural orbit u 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

Fuel Consumption Issue u Simple, ad-hoc solution to reduce fuel consumption  Let q d (t) = [x d (t), y d (t), z d (t)] T be a desired spacecraft trajectory  Define a ball centered at {x d (t), y d (t), z d (t)} with radius 

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

Fuel Consumption Issue On/Off Type Control Algorithm Goal: Control spacecraft position such that it never leaves the ball q s (t) = [x s (t), y s (t), z s (t)] T : spacecraft position 1. If  2. If  Control off Control on

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

Fuel Consumption Issue u 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 u Case 2 (spacecraft outside ball) may occur during initialization of formation reconfiguration u Trade-off between tracking performance and fuel consumption >Asymptotic tracking vs. bounded tracking with less fuel

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

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