Adventures on the Interface of Dynamics and Control John L. Junkins Texas A&M University Theodore von Kármán Lecture

A Roadmap: Mechanics and Control of Dynamical Systems Implementation Details of Available Sensors, Actuators, and Computers Control Laws Feedback Gains, Estimator Gains, Switching Functions... Control Design Algorithms Numerical & Analytical Methods … Robust Algorithms Optimality Principles Min Time, Min Energy, Min Error, Min Sensitivity,... Simulation/Validation Robustness to Model Errors & Disturbances … Will it Work? Mathematical Model ODES, PDES & Hybrid Math Models Model Errors ANALYSIS Now What ? NOVA Spacecraf t Actual Dynamical System

Outline NOVA Attitude Maneuvers. Dynamical insights proved crucial in the first on-orbit application of optimal control theory. “Just how nonlinear is it, anyway?!” Role of coordinate selection in the validity of linearizations in orbit/attitude dynamics. A Novel Approach to Multibody Dynamics. A new method is introduced =>> a thing of beauty! Recent Attitude Estimation & Control Results On the marriage of stability & control theory. Faster, Better, Cheaper Missions =>> Yeah, Right!

NOVA Spacecraft Minimum Time Magnetic Attitude Maneuvers John L. Junkins Connie K. Carrington Charles E. Williams 1981 AAS G&C Conf. Keystone Colorado

Geometry & Coordinates

Reduced Order Magnetic Maneuvers: Optimal Control Formulation 6

NOVA Extremal Field Map

Nonlinear Evolution of Uncertainty in Orbital Mechanics

9 Just How Nonlinear Is It? Consider Role of Coordinate Choice on variation of “the degree of nonlinearity.” But how do you measure =>> “The Degree of Nonlinearity?” Study validity of linear error theory as a function of coordinate choice for orbit and attitude dynamics problems.

Nonlinear Transformation of Surfaces of Constant Probability Density

Elements of Linear Error Theory The first 2 statistical moments of x are: x belongs to the Gaussian density fct.: The x surfaces of constant probability: The above results hold, with some degree of approximation for the case of non-linear transformations of the form y belongs to the Gaussian density fct.: The moments transform as: The y surfaces of constant probability:, in which case,, the locally evaluated Jacobian. These developments also generalize exactly for linear dynamical systems, and approximately for nonlinear dynamical systems. Question: How do we measure “degree of nonlinearity?” y=Ax+by=Ax+b

As Regards Algebraic Systems: How Do We Measure Nonlinearity? Essential Idea is Simple: Monte Carlo Sample Points The ith of N points on the 3-  ellipsoid in X-space. If the transformation F (x) is locally linear then Introduce the Static Measure of Nonlinearity If  s is << 0.1 (say), then F (x) is near - linear over the set of N low probability (3  ) points; if  s is > 0.2, F (x) is highly non linear.

Variation of Dynamic Nonlinearity Index for the Perturbed Orbit Problem Dynamic Nonlinearity Index Variation for Various Coordinate Choices where In particular: LINERARIZATION IN RECTANGULAR COORDINATES LINERARIZATION IN POLAR COORDINATES LINERARIZATION IN ORBIT ELEMENT SPACE

Validity of Constant Probability Surfaces Various Orbital Coordinates

Euler’s Principal Rotation Theorem Consider the orthogonal projections: The principal rotation angle satisfies: where Euler’s Theorem leads to the eigen value problem:

Principal Rotation Coordinates Quaternion (Euler Parameters) Classical Rodrigues Parameters Modified Rodrigues Parameters

Validity of Linear Error Theory for Various Attitude Coordinates

Validity of Constant Probability Surfaces for Various Attitude Coordinates Not able to locate images

19 An Expert Opinion: (by Lord Kelvin) “ Quaternions came from Hamilton after his really good work had been done; and though beautifully ingenious, have been an unmixed evil to those who have touched them in any way…” The Vector is a useless offshoot from quaternions, and has never been of the slightest use to any creature.” Yeah, …, Right! Moral: This demonstrates clearly that even the brightest person can be so ignorant!

Cayley Transformation A general n x n orthogonal matrix C can be parameterized in terms of the (n 2 -n)/2 distinct elements in a unique skew-symmetric matrix Q as: Forward Transformation: Inverse Transformation: Notice the unusual feature that the forward and inverse transformations are identical! Also notice the commutativity … very elegant. We have verified that: leads to the kinematic equations : We (Junkins, Schaub, Tsiotras) have also verified that these equations are the generalization of the Classical Rodrigues Parameters to the case of n x n orthogonal matrices. Analogous results have been established for the Modified Rodrigues Parameters.

Freewing -- do you want this slide?

Motivation: Orthogonal Quasi-Coordinate Formulation for Multibody Dynamics Kinetic energy structure for n dof mechanical system : The equations of motion has the form: At any instant, the n x n the mass matrix M = M T > 0 can be factored as: Except possibly near repeated ’s,  ( t ) and C ( t ) are smooth fcts of time. Near repeated ’s some rows of C ( t ) may lose uniqueness  but  the sub - space in which they lie remains unique. This motivates: Find  ( t ) and C ( t ) by solving kinematic differential equations  Permits us to avoid the numerical inversion of M (x(t)).

An Orthogonal Quasi-Coordinate Formulation for Multibody Dynamics Kinetic energy structure for n dof mechanical system: Where we have introduced the quasi - velocity v In [Junkins and Schaub, 1997], we show that instead of solving We can transform the system eqns to the elegant form : with Well this square root formulation looks pretty, … but is it useful?!

So: How well does it work? Example Application

Freewing Tilt Body Nonlinear Response