adventures on the interface of dynamics and...

Copyright ©1997, American Institute of Aeronautics and Astronautics, Inc.

AIAA Meeting Papers on Disc, January 1997A9715102, AIAA Paper 97-0002

Adventures on the interface of dynamics and control

John L. JunkinsTexas A & M Univ., College Station

AIAA, Aerospace Sciences Meeting & Exhibit, 35th, Reno, NV, Jan. 6-9, 1997

Controllers for dynamical systems are best designed, on the average, by analysts whose expertise encompasses both modelingof the class of systems under discussion and the methodology for designing controllers. Good decisions on issues such asselection of coordinates, modelling approximations, selection/design and location of sensors and actuators, and definition ofmeaningful performance measures, require expertise beyond algebraic/computational methods for control design. Thiscommon sense viewpoint underlies a significant trend toward a unified development of analysis methodology for mechanicsand control, especially as regards spacecraft dynamics, stability, estimation, and control. This paper overviews recentdevelopments and discusses several 'adventures' which illustrate mechanics-based insights useful in modern applications. Thediscussion spans orbital dynamics, attitude dynamics and control, and nonlinear multibody dynamics and control, andconsiders both conceptual developments and results from several successful missions. (Author)

AIAA-97-0002

ADVENTURES ON THE INTERFACEOF DYNAMICS AND CONTROL*

John L. JunkinstTexas A&M University, College Station, TX 77843-3141

Controllers for dynamical systems are best designed, on the average, by an-alysts whose expertise encompasses both modeling of the class of systemsunder discussion and methodology for designing controllers. Good decisionson issues such as selection of coordinates, modelling approximations, selec-tion/design and location of sensors and actuators, and definition of meaning-ful performance measures require expertise beyond algebraic/computationalmethods for control design. This common sense viewpoint underlies a signifi-cant trend toward a unified development of analysis methodology for mechan-ics and control, especially as regards spacecraft dynamics, stability, estima-tion, and control. This paper overviews recent developments and discussesseveral "adventures" which illustrate mechanics-based insights useful in mod-ern applications. The discussion spans orbital dynamics, attitude dynamicsand control, and nonlinear multibody dynamics and control, and considersboth conceptual developments and results from several successful missions.

I. Introduction

DYNAMICS AND CONTROL analysis and as-sociated design methods have matured rapidly

over the past three decades. In spite of signifi-cant advances, however, developing reliable and well-optimized controllers for aerospace vehicles remainsan iterative and labor intensive art form. This is be-cause the increasing size, structural flexibility, anddynamical complexity, coupled with competing de-mands for greater precision and autonomy continueto outstrip our ability to adequately model these sys-tems. A second set of difficulties complicate prac-tical applications: The traditional discipline spe-cializations result in few engineers having sufficientbreadth and depth of expertise to efficiently re-solve problems involving coupling between mechan-ics, controls, electro-mechanics, electro-optics, andcomputer sub-systems.

It is evident that crucial issues lie on the inter-face of traditional disciplines. These are difficult toresolve efficiently because of the implicitly requiredcoupled effort of a group of engineers. Respondingto these challenges, engineers with increased breadthand depth have emerged over the past two decades.Researchers working on the mechanics and controlsinterface have published hundreds of papers (see the

'Copyright ©1996 by John L. Junkins. Published by theAmerican Institute of Aeronautics and Astronautics, Inc.with permission. This paper documents the January 1997Theodore van Karman Lecture presented at the AerospaceSciences Meeting in Reno, NV.

^ George Eppright Chair Professor, Aerospace EngineeringDepartment, Fellow AIAA.

survey by Hyland et al.1) and several texts2 7; thesedocument the multidisciplinary arena: mechanicsand control of dynamical systems.

In analysis of nonlinear mechanical systems, aset of fundamental modeling issues surround thechoice of coordinates used to describe the systemdynamics. These decisions impact design and syn-thesis of controllers, because a given physical sys-tem may be described rigorously by sets of regular,near-linear differential equations, or equivalently, inthe Lagrangian generalized coordinates sense, byhighly nonlinear equations containing singularities,with the "degree of nonlinearity" depending stronglyupon the coordinates selected. Much of the early his-tory of dynamics addressed the quest for judiciouscoordinates under natural forces like gravity (con-sider the field of celestial mechanics).

Coordinate Transformation IssuesClassical coordinate transformation methodology,

e.g., Hamilton's Canonical Transformation Theory,and Hamilton-Jacobi Theory^ focus on finding ig-norable coordinates and analytical integrals of themotion, rather than finding coordinates governed bythe most nearly linear differential equations, overa state space domain of interest. This methodol-ogy was developed for open loop initial and/or two-point boundary value problems considering (typi-cally) conservative systems. These remarks are mosteasily understood when one considers the classicalLagrangian viewpoint wherein 'coordinates' definethe location of all mass in the system (e.g., thesystem's geometric configuration). Another level of

1American Institute of Aeronautics and Astronautics

transformation generality was established by Hamil-ton; his phase space coordinates are essentially thecanonical version of modern state variables. In thisview of state space analysis, coordinates define bothposition (geometric configuration) and velocity (ormomentum), and coordinate transformation impliesa contact transformation operation where one set ofposition/velocity state space coordinates are nonlin-early mapped into another.8

Feedback Linearization considers an additional levelof generality, simultaneously transforming position,velocity and control variables. The idea is to 'ab-sorb' all nonlinear terms in a given differential equa-tion model into judicious nonlinear state transforma-tions and feedback laws. Important early work in-cludes Krener,9 Hermann and Krener,10 Brockett,11

Jakubczyk and Respondek,12 Hunt and Su,13 Junk-ins et al,14 Hunt, et al15 and Isidori, et al.16 Excel-lent introductory treatments are given by Isidori,17

Nijmeijer and van der Schaft,18 and Slotine andLi.19 For certain problems we can establish asymp-totic stability, disturbance decoupling, noninteract-ing control and output regulation. These transfor-mations are generally exact (i.e., not Taylor's series).In some cases, the transformations can be found byinspection by simply specifying part of the controlsto cancel all nonlinear terms appearing in the equa-tions of motion, the remaining part of the controlcan be an additive linear state feedback, resultingin a linear closed loop system. =J> The linear con-trol is mapped back through the inverse nonlineartransformation to generate actuator commands.

Feedback linearizing transformations can be ob-tained as the solution of a set of coupled partialdifferential equations. Assuming a solution can beobtained, the major practical problem is that ex-act cancellation on the nonlinearities requires per-fect knowledge of the system dynamics. Cancella-tion is not exact in applications and the resultingfeedback linearized input-output behavior may be apoor approximation of the actual system's behav-ior. In the worst cases, instability may result. Oneapproach to enhance robustness is to apply adap-tive techniques to achieve asymptotic behavior inthe presence of model uncertainties, e.g. Nam andArapostathis,20 Sastry and Isidori,21 Taylor et al.22

Marino and Tomei23 and Kanellakopoulos et al.24

Estimation-based adaptive schemes have been pro-posed by Pomet and Praly,25 Compion and Bastu,28

Han et al27 and Teel et al.28

An interesting application of feedback lineariza-tion is attitude control and momentum managementof a spacecraft in low-earth orbit, e.g., consider theinternational space station (ISS) project. The dif-

ficulty lies in keeping the spacecraft in an earth-pointing attitude using momentum exchange actu-ators for control while simultaneously desaturatingthe same actuators working against gravity gradi-ent torques. The problem is amenable to feedbacklinearization, c.f. Sheen and Bishop29'30 and Dziel-ski et al.31 Other related works include Singh andlyer32 and Singh and Bossart.33 An easily appreci-ated source of uncertainty in the ISS attitude controland momentum management problem lies in poorknowledge of the inertia matrix. Conceptually it ispossible to predict the inertia matrix as a functionof the locations and motions of the ISS components.However, rather than attempting to model to highprecision the 30 year evolution of the ISS, it seemsmore sensible to design a robust controller that guar-antees stablility for a wide range of mass properties.Sheen and Bishop30 applied direct adaptive meth-ods; the ISS attitude and control moment gyro mo-mentum were controlled with large uncertainties inthe inertias and attitude errors outside the linearrange. The indirect adaptive approach embodies adistinct parameter identification module. Paynterand Bishop34 show Kalman filter parameter identifi-cation with feedback linearization control comparesfavorably with direct adaptive control.

Overview of Spacecraft Attitude ManeuversAttitude maneuvers represent an interesting class

of nonlinear spacecraft control problems where the-ory, computation, ground experiment, and on-orbitimplementation have converged over the past twodecades. It is an area wherein spacecraft dynam-ics plays a particularity important role in obtainingthe necessary insights to achieve useful results fromapplication of stability and control theory. Theseproblems inherently suffer from the triple curse ofnonlinearity, high dimensionality, and model errors.Furthermore, application of optimal control is diffi-cult because often no single performance measurecan capture the mission objectives, For example,minimum time, minimum energy, minimum sensi-tivity to errors, and minimum vibration are inher-ently competitive performance measures. Importantrecent literature on this class of problems are theworks by Wie, Junkins, Schaub, Mukherjee, Robi-nett, et al.3'4'6'35'44 In Section IV of this paper weconsider recent developments and mission results.

The first on-orbit implementation of a Pontryag-in's Principle derived optimal control was the 1981minimum time magnetic attitude maneuvers for theNOVA family of navigation spacecraft6'35 depictedin Figs. 1 and 2. The NOVA vehicle spins at ~ 5 rpmand an electro-magnet aligned with the spin axisinteracts with the earth's magnetic field to gener-

American Institute of Aeronautics and Astronautics

Fig. 1 NOVA Navy Navigation Satellite

Fig. 2 Orientation of the NOVA Spin Axis

ate a torque transverse to the spin axis and precessthe spacecraft slowly. The polarity of the electro-magnet is the control variable and the problem iseasily stated: Given the spacecraft spin axis point-ing direction (a0, S0) = initial right ascension anddeclination at time t0, determine the polarity his-tory — 1 < p(t) < 1 required to achieve a minimumtime reorientation of the spin axis to a desired tar-get state (CXT^ST)- A direct assault on this problemwith Pontryagin's Principle6 leads to a 12th ordernonlinear two-point boundary value problem that isnot compatible with near-real time implementation.However, it has been shown35 that the use of a pas-sive damper reduces nutation quickly compared toa slow precession of the angular momentum vector.This observation permits a two time-scale approxi-

240 * 360a 240 * 360

Fig. 3 Extremal Field Map of Minimum Time Maneuvers

mation and results in a dramatic order reduction toa 4th order system. The optimum control is gener-ated as a non-linear bang-bang controller with mul-tiple polarity (p(t) — il) switches determined asthe sign of a nonlinear switching function. Solvingthe two point point boundary value problem is re-duced to finding a single unknown 0 < 70 < 360°that generates the entire family of extremal trajec-tories. We define a special inertial frame (a*, 5*)defined so that the initial state is on the equatorand the target state is the origin, this simplifies thesolution of the two point boundary problem. In factit is reduced to a "video game" interface where thegraphical solution is obtained in several "screens"as illustrated in Fig. 3. Note that the solution isobtained by sweeping J0 at some interval t\j0 be-tween chosen limits limits, and graphically observ-ing the 70-region where the trajectory passes nearestthe origin. It is easy to isolate the origin to withina fraction of a degree within three or four "screens";this is the procedure implemented successfully forthe NOVA missions. In Fig. 3, it is evident thatthere are two local solutions, this is because onecan travel south from San Francisco to Los Ange-les, but one can also traverse the great circle passingover the North Pole, if persistent! The cusps arepolarity reversal points where the direction of pre-cession reverses. The maneuver shown for 70 ~ 30°required 19 switches and over 7 hours to re-orientthe pointing direction from (a0 = 45.2°,<50 = 35.1")to (aT = -42.7°,ST = -31.4°).

This elegant application of optimal control the-ory was formulated just a few months prior to thelaunch of the first NOVA spacecraft in May 1981and represents an excellent example of the synergybetween dynamics and controls. It replaced a non-linear programming approach (iterating a nonlinearattitude simulation searching on 10 to 30 unknown


switch times) that did not converge reliably andwas not compatible with real-time mission support.The above described approach replaces this nonlin-ear programming approach to switch time optimiza-tion with the bounded one dimensional search. Forthis mission, given attitude determination informa-tion at the beginning of a ten minute pass of thesatellite over the Laurel, Maryland Johns HopkinsApplied Physics Laboratory's mission control cen-ter, this algorithm could be executed in about fiveminutes of real time and up-loaded to the spacecraftcomputer while it was still above the horizon, for im-plementation over the next few hours. In the actualmission, the first (> 7 hours) maneuver was followedby a trim maneuver of several minutes to null finalerrors of ~ 1°, these small residual errors were pri-marily due to the approximate 4th order sphericalharmonic model used for the geomagnetic field.

Overview of this PaperWith the above introduction, we turn to some

more recent developments and present some resultsof relatively broad significance. While the issuesdiscussed are presented in the context of spacecraftdynamics, estimation, and control, they are devel-oped in a way that reveals broadly useful concepts.Most of the developments address aspects of howto transform or approximate the system dynamicsto advance toward solution of important problems.Coordinate choice is one major common issue, weaddress some aspects of this subject first.

It is obvious that an infinity of coordinates aregenerally possible; seeking the most nearly linear de-scription of a given system's dynamics establishes anattractive philosophical point of view for coordinatetransformations and/or choosing between availablecoordinate sets. This suggests that before resortingto feedback linearization and/or adaptive controllerdesign for a given system, we should first seek thebest generalized coordinates, vis-a-vis smallness ofnonlinearities and in view of the application underconsideration. A useful starting point is to developa broadly applicable measure of the degree of non-linearity, and compute this measure for various co-ordinates to help inform these decisions. Of course,coordinate selection and transformation are but oneof many issues which must be addressed in dynamicsand control analysis, and while it plays a role in eachof the "adventures" discussed herein, we will discussother important mechanics-based insights for con-troller design in the context of particular problems.

In Sec. II we introduce error propagation conceptsand apply them to approximation of the nonlinearerror analysis in orbital and attitude dynamics. Sev-eral coordinate choices and their relative merits are

studied, vis-a-vis the validity of linear error theory.The methodolology is validated through use of theMonte Carlo Method.

In Sec. Ill, we present a recently introducedorthogonal quasi-coordinate formulation for multi-body dynamics. We show how to generate, bysolving kinematic-like differential equations, the in-stantaneous spectral decomposition of a configurationvariable mass matrix typical of those arising in largedeformations of robot manipulators. Through thisformulation, it is possible to by-pass inversion ofthe configuration-variable mass matrix; and definequasi-coordinates for which the mass matrix is anidentity matrix. Solving these differential equationsleads to a robust simulation of nonlinear multibodysystem dynamics. This formulation is easy to par-allelize and therefore, is an attractive approach forhigh dimensional systems.

In Sec. IV, we consider some recent research ondesign of globally stabilizing feedback control lawsfor large angle spacecraft re-orientation maneuvers.These results utilize Lyapunov stability theory insome interesting ways. To compliment results fromanalytical and computational studies, we consideron-orbit results from the recent Clementine lunarmission.45

II. Nonlinear Systems Error Propagation

Figure 4 is a qualitative sketch showing how a sur-face of constant probability evolves during the orbitof a near-earth spacecraft, where the motion initi-ates with a known Gaussian distribution of errorsin inital position and velocity. It is known46 thatthese surfaces of constant probability are typicallywell approximated over small time intervals by ellip-soids, however over longer times, the region of un-certainty becomes banana-shaped, and eventually,wraps around the earth in a (qualitative) toroidaltube of uncertainty. We show that the shape of confi-dence surface approximations and therefore their va-lidity is strongly dependent on the coordinate choice.We also note the conceptual equivalence of propagat-ing initial condition uncertainty of a single object tothe distribution dynamics (or "cloud dynamics") ofparticles (idealized as identical) following an explo-sion. Thus the evolution of uncertainty is an approx-imate dual of the planetary ring formation problem.

Error Propagation in Orbital MechanicsFigure 5 shows the distribution of 200 Monte

Carlo positions, projected into the nominal orbitplane, at quarter period intervals of one completenominal orbit, where T is time measured in fractionsof the nominal un-perturbed orbital period. Also


Gaussianapproximation of99% confidenceellipsoid l—-V

actual 99%confidencesurface

99% confidenceellipsoid(approximatelyGaussian)

Fig. 4 Evolution of Uncertainty in Orbital Mechanics

3-0 Surfaces Computed Based Upon Linearization of theDifferential Equation* for: ---------- RECTANGULAR COORDINATES

- - - - - - - - - - - - - POLAR COORDINATES———————— ORBIT ELEMENTS

8060

8040

=•28020>,

8000

= 0.25

-8000

-8010

=—8020

••-8030

-8040

-3320 -3300 -3280 -3260x(km)

_____T = (V75

150

10050i.

1. 0-50

-100

-160

= 0.50

400

200

I 0

-200

-1005 -1000x(km)

T = l . f

-3400 -3200x (km)

IS 6670 6675 6680 6685x(km)

Fig. 5 Analytical Estimation of 3-cr Confidence Surfaces

shown are three analytical predictions of the 3-<rconfidence boundaries. Three analytical confidencepredictions use methods described below with threecoordinate choices for linearization of the dynamics:(1) Rectangular Coordinates, (2) Polar Coordinates,and (3) Classical Orbital Elements (variation of pa-rameters method). In this study, only initial condi-tion uncertainty is considered.

The dynamical model46 includes perturbations byearth oblatcness (J2) and atmospheric drag. It isobvious by inspection of Fig. 5 at r = 1 that the con-fidence surfaces predicted by linear error theory inpolar coordinates and orbital elements are superiorto the linear error theory in rectangular coordinates.

For the numerical simulations underlying Fig. 5,the following parameters for the nominal orbit areassumed. At injection, the nominal perigee altitudeis 300 km, the orbit eccentricity is 0.2, the inclina-tion of nominal orbital plane is 10", the longitude of

=> =>

Fig. 6 Confidence Surfaces Mapped Through NonlinearCoordinate Transformations

ascending node is 45°, and the argument of perigeeis 20°. The mass of each sample Monte-Carlo parti-cle is taken to be 0.5 kg with a radius of 0.17 m. Theballistic drag coefficient corresponding to the orbit-ing particle is taken equal to 1.2. The uncertaintyof the initial conditions, for the purposes of generat-ing this example were assumed as un-correlated zeromean Gaussian errors with the initial position errorstandard deviation of 2 km and the initial velocityerror standard deviation of 0.002 km/s, in all threecomponents (equatorial rectangular coordinates).

To see how linear error theory can be used toapproximate non-ellipsoidal confidence boundaries,shown in Fig, 5, note that an ellipsoid (using linearerror theory), when mapped through a nonlinear co-ordinate transformation is not an ellipsoid! This ideais captured qualitatively in Fig. 6. Finding a coordi-nate transformation which enhances the the validityof linear error theory is of great consequence, be-cause it may allow us to by-pass the expense andassociated convergence difficulties of a Monte Carloprocess.Judicious Use of Linear Error Theory Approxima-tions for Nonlinear Dynamical Systems

Consider a linear algebraic model of the form

y = Ax + b (1)

where y 6 7^mxl and x 6 ft"*1 are random vectors,whereas b e 7lmxl and A e 7£mxn are known con-stant matrices. Question: If we know the first twostatistical moments of x

(2)

what are the corresponding statistical moments ofy? The answer is well known,47 the mean and co-variance moments of y are:

y = Ax. + b, (3)

Furthermore, if x = x+<5x belongs to a multidimen-sional Gaussian probability density distribution

(4)


then y belongs to another Gaussian distribution ob-tained fromEq. (4) by replacing (z, x, n) by (y,y, m)(denoted by x -4 y). Observe that the surfaces ofconstant probability density are the two hyperellip-soids

* x =r~ •y, (5)

and that r = 3 defines the 3-cr error surface. Notethe integral of the density function (Eq. (4)) overthe interior of the volume bounded by Eq. (5) is theprobability that 6x is contained within the ellipsoidof Eq. (5). For an n — 6 dimensional 3-cr ellipsoid,the probability that SxTP~x

lSx < 32 is ~ 0.95.The above results apply with some degree of ap-

proximation for nonlinear transformations in lieu ofEq. (1) where A is interpreted as a locally evalu-ated Jacobian: A — |̂ , and 6y = A<5x. The valid-ity of the linearity assumption is of course problem-dependent, and that is precisely the focus of the cur-rent discussion.

Consider now a nonlinear dynamical system mod-eled exactly by two alternative systems of nth orderdifferential equations of the form

x = f (t, x), x(i0) = xc; y = g(t, y), y(t0) = y0

(6)where x(i) and y(i) denote state vectors at timet related by the forward/inverse contact coordinatetransformations:

= G(x); x = F(y) (7)

Consider also the small departure motion <5x(t) =x(i) — x(t) and <Jy(<) = y(t) — y(t) near referencetrajectories x(<) and y(<). The S() motions are ap-proximately governed by the linear equations

<9fl«xL (8)

5x(t) = t0); Sy(t) = *y(t,t0)6y(te)(9)

where the state transition matrices <bx — $x(t,t0)and $5, = $y(t,t0) satisfy

*-[!],••' *•-[£],'• <10)with identity matrices as initial conditions.

Now consider the case of uncertain initial condi-tions on Eqs. (6). Suppose that initial position errorsare represented as additive Gaussian errors rfx0 witha known covariance matrix PXaXo- Suppose the er-rors are sufficiently small that the corresponding er-ror covariance matrix of <5y0 can be determined from

linearizing Eq. (7) so we can approximate PyoV<1 asthe static covariance propagation

p _•r ~

dG (ii)The dynamic propagation of the covariance matri-

ces along the nonlinear trajectories can be computedfor the expectations Pxx(t) — E{5x(t)5x(t)T} andPVy(t) = E{Sy(t)Sy(t) } using linear error theoryvia the similarity transformations

.[*,(Mo)] (12)

The linear error theory estimated 3-<r confidenceellipsoids can be computed from the above covari-ance estimates by substituting directly into Eq. (5).Alternatively, we can transform either of the ellip-soids, at any instant, through the nonlinear coor-dinate tranformations of Eq. (7) and obtain a non-ellipsoidal surface. If linear error theory is more validin the y coordinates than the x coordinates =>• weexpect that the nonlinearly transformed confidencesurfaces to be more valid than the correspondingquadratic surface obtained from linear error theoryapplied directly to the differential equations for <5x,e.g., the ellipsoid of Eq. (5), using Pxx fromEq. (12).

In the context of nonlinear statistics, how do wemeasure the severity of nonlinearity? We do this bymapping through the nonlinear transformation a setof non-random points on the 3-a confidence surfaceand record the largest departure in a measure of non-linearity. For example, for a 6-dimensional space, wecould select the 2n — 12 points x, = x + &X; at the•f ends of the semi axes of the 3-cr hyperellipsoid.More generally, the idea is to select a finite num-ber of extreme points that collectively constitute aset of worst cases. Linearity means that the Jaco-bian would be everywhere constant and this extremesample should provide a conservative basis for eval-uating linearity errors. We introduce measures ofstatic and dynamic nonlinearity. The static indexis for nonlinear coordinate transformations, whereasthe dynamic index is for nonlinear differential equa-tions. For the static transformation of Eq. (7), wedefine the following static nonlinearity index:

vs ~ sup (13)

Solving the nonlinear differential Eqs. (6), from thefinite set (e.g., 2n) of initial conditions x,-(i0) =x(*o) + S\i(t0) on the 3-a initial error ellipsoid, wedenote the state transition matrix evaluated at timet along the ith trajectory Xi(t) as $,-(£, t0] = ffr^j,

is evaluated along x(t). We define


LINEARIZATION INRECTANGULARCOORDINATES

-...„....„..._..IN POLARCOORDINATES

LINEARIZATIONIN ORBITELEMENT SPACE

O.S 1 1.2 1.4

Non-dimensional lima, T

Fig. 7 Dynamic Nonlinearity Indices: Two OrbitVariation for Three Coordinate Choices

the dynamic nonlinearity index v(t, to) to measurethe relative variation of 3>;(i, tc) as follows

v(t,t0) = sup- (14)

Here, we abbreviate $z(t, t0) and <&y(t, t0) as$(i, i0), with the understanding that there is anequation of the form of Eq. (14) for each coordinatechoice.

In the above context, the sup (•) operator extractsthe maximum value of (•) over the finite set of 2npoints sampled at time t (on the trajectories Xj(t)which ensue from the initial conditions from the Inextrema selected from the surface of the initial 3-<rhyperellipsoid), and || • ||2 denotes the Frobeniousnorm of a matrix.Nonlinearity Indices: Numerical Results

Using the static indices, we first verify that thesmall initial covariance matrices can be reliablymapped between coordinate sets using the similaritytransformation in the form of Eq. (11), because thelargest value of vs for transforming between Rectan-gular Coordinates, Polar Coordinates, and Orbit El-ements is initially found to be ~ 10~3. We note thatthe subsequent growth of the error volume results inmuch larger static indices, however, and more im-portantly, the time history of the three dynamic in-dices defined in Eq. (14) corresponding to linearizingthe dynamics about the nominal orbit for the threecoordinate choices, show large variations as seen inFig. 7. It is evident that i/(t, t0) varies by an or-der of magnitude and this can be used to anticipate

that covariance approximations made in rectangu-lar coordinates are less valid than polar coordinates,which in turn are less valid than covariance approx-imations using the linearized orbit element differen-tial equations. In fact, over two full orbits, a max-imum of ~ 2% variation in $(t, t0) and an averageof only ~ 0.4% suggests that linear error theory inorbit element space approximates well the actual er-ror dynamics. Observe the qualitatively appealingcharacteristic of all three i>'s in Fig. 7 =$• the mo-tion is significantly more nonlinear near perigee pas-sage where gravitational and drag nonlinearities area maximum on the nominal orbit. When the con-fidence ellipsoids from the orbit element space arenonlinearly mapped into rectangular coordinates, wesee that they indeed are in excellent agreement withthe 200 Monte Carlo trajectories (Fig. 5). See Junk-ins et al46 for the three dynamic equations in theform of Eqs. (6), coordinate transformations in theform of Eqs. (7) and complete discussion of this ap-plication.Error Propagation in Rotational Kinematics

Substantial historical effort has led to a wide fam-ily of coordinate sets for representing angular mo-tion.48'3 For a given time history of angular velocityw(t) = [wi(i),w2(t)> W3(*)]T! the orientation historyis described exactly by many alternative sets of kine-matic differential equations. Frequently, the Eulerparameters (quaternions), the Euler angles, the Ro-drigues parameters and the modified Rodngues pa-rameters are used to parameterize the attitude kine-matics.

Euler Parameters. To make the discussion spe-cific, consider the direction cosine matrix whichprojects orthogonal body unit vectors {b} ontospace fixed orthogonal unit vectors {n} in the sense{b} — [C]{n|. Let $ denote the principal rotationangle and e be a unit vector along the axis of prin-cipal rotation (Euler's theorem).3

The Euler parameters are defined as

/30 = cos($/2), fa = li sin($/2), i = 1, 2,3(15)

This redundant four dimensional parameterizationsatisfies the constraint (3$ + f)\ + /?| + /3| — I.Euler 3-2-1 Angles. The transformation fromthe Euler angles Q(t] = [<j>(t),e(t},4>(t)\T tothe Euler parameters, with the abbreviations:(s = sin, c = cos, t = tan), is defined by


and the kinematic differential equations associatedwith the Buler angles take the form

0 =

The initial conditions on Eq. (18) are(0(t0),e(t0),t/)(t0)) = (00A,</>o). It can be recog-nized that this parameterization becomes singularat 8(t) = ±7r/2. For the present discussion, theprescribed angular velocities u(t) are chosen suchthat this singular condition is never reached. ThusEq. (18) is assumed to remain non-singular at alltimes.

Departures motion 8Q(t) ~ 0(i) — Q(t) in theEuler angles from a nominal trajectory 0(t) satis-fies linearized differential equations^ derivable by lin-earizing Eq. (18) (50 (t0) = 00 - 00):

1 '0i = Al o

oSQ

(18)Rodrigues Parameters. The Rodrigues parameterseliminate the constraint associated with the Eulerparameters, by introducing the ratio of the Eulerparameters as new coordinates

f = l,2,3 (19)

Letting q(f) = [qi(t),q-i(t),q*(i)}T € U3, the kine-matic equations associated with the Rodrigues pa-rameters take the form

q = (20)

where / denotes the 3 x 3 identity matrix and thesymbol [•] denotes the 3 x 3 skew-symmetric matrix

0-qa

-q-tgi0

(21)

The vector q of the Rodrigues parameters is relatedto the principal eigenaxis e and rotation angle $ as

q = e tan($/2) (22)

As it is evident from Eq. (23), the Rodrigues param-eters cannot be used to describe principal rotations

of more than IT. To provide a simplest non-linearmotion that does not encounter singularities for anyof the paramcterizations, we consider a prescribedangular velocity vector u(t) such that |$| < TT issatisfied at all times. Prom Eq. (21), the linearizeddeparture motion <5q(i) = q(tf) — q(t) from a nominalset of Rodrigues parameters q(t) is governed by

(23)

Modified, Rodrigues Parameters. If instead of usingEq. (20), one eliminates the Euler parameter con-straint by the transformation

ffi = /?,-/(! + ft,) = &itan($/4), i = 1,2,3 (24)

the result is the following set of differential equationsfor the modified Rodrigues parameter vector a(t) =[ffi(t),ff2(t),a3(t)]T eft3

a = 1 [(1 - erTa)J + 2[a] + 2<r<rT] u, a(t0) = a0

(25)Both the classical, Eq. (20), and the modified Ro-drigues parameters, Eq. (25), can be derived fromthe Euler parameters via stereographic projection.49

The modified Rodrigues parameters, although veryclosely related to the Rodrigues parameters, aremuch superior to them as regards singularity avoid-ance, since they aren't limited to rotations of |$| <IT. A direct transformation from the modified Ro-drigues parameters to the Rodrigues parameterscould be expressed as

(26)

For |$| < 7T/2, we have a « |q « j$e, with theapproximation becoming exact as |$| — > 0 and be-coming poor for |$| » Tr/2. Moreover, for a givenprincipal rotation angle $, tan($/4) « $/4 is clearlya much better approximation than tan($/2) as $/2,so we anticipate a kinematics to linearize withsmaller errors than q kinematics. Thus in appli-cations where linearization errors are important, wemight anticipate it to be advantageous to use themodified Rodrigues parameters rather than the clas-sical Rodrigues parameters. The modified Rodriguesparameters have been generalized further in a recentpaper by Tsiotras.50

Linearized equations (for large w, small Sa =rr — 7f) governing departures in the modified Ro-drigues parameters from a nominal motion Tf(t) canbe derived from Eq. (26) as

(27)


Numerical Study of Attitude Error PropagationFor the purpose of illustration, we consider a pre-

scribed angular velocity time history. We choose thenominal Euler angle motion

(j)(t) = siri 3£ cos 5£9(t) = 0.4?r sin 5*

- 0.5 cos 5t {0.1 +sin 3<}3

(28)

It can be readily recognized, for this motion, thatQ(t] never takes the values ±?r/2 and thus the cho-sen nominal Euler angles do not encounter singular-ities in Eqs. (17). The nominal anglular rates 6(t]can be obtained by straight-forward differentiationof Eq. (28) with respect to time, and from Eq. (17)we can obtain the corresponding angular velocities

sin 9 sin ip cos rf> 0cos6 _ 0_ 1

— sin 6 cos tp sin ij} 0(29)

The history of the body components of uj(t) for time0 < t < 10 are shown in Fig. 8.

time, t

Fig. 8 Angular Velocity History in Body Axes

The Euler principal rotation angle is obtainedfrom the direction cosine matrix from cos$ =(trace((7) — l)/2. For the choice of the angular ve-locities we made above, |$| < n and hence neitherthe Rodrigues parameters or the modified Rodriguesparameters encounter a singularity.

In Fig. 9, we present the time histories of the nom-inal Euler angles defined in Eq. (28) and the corre-sponding nominal Rodrigues parameters and nomi-nal modified Rodrigues parameters. To expose theeffect of various attitude coordinate choices on thevalidity of Linear Error Propagation, 200 Monte-Carlo initial errors were simulated using a Gaussian

1.0

fin ft-1.01.5

ew o,-1.6.1.0

V (0 0.-1.0

•0.40.8

5,0)0-0.80.2

2 4 6time, t

Nominal Roddy!** P»r«nrt«f«

4 6 8 10time, /

Nominal ModHltd RodrlguM Paramturo0.2 r

8,(0 0-0.20.4

6,0) o-0.40.3

«,(!) 0-0.3

Fig. 9 Nominal Motion: Euler Angles 0(t), Rodrigues Pa-rameters q(£), Modified Rodrigues Parameters a(t)

Fig. 10 3-<r Confidence Surfaces Snap Shot at t = 0.0s

random number generator with an initial error stan-dard deviation of 2° in the Euler angles. Initial er-rors in the three angles are assumed to be statis-tically independent. Before discussing the dynamicerror propagation results, we consider the validity ofmapping initial uncertainty prescribed in the Eulerangles to the Rodrigues parameters and the modi-fied Rodrigues parameters, so we can establish con-sistent initial uncertainties in all three coordinatesets. The static nonlinearity index i/3, at time t = 0was computed from Eq. (13) for the various coor-dinate transformations; the largest value of v was« 10~3. It is therefore evident that the static indexof nonlinearity is "small" for all the static transfor-mations over the given volumes of uncertainty =>we conclude that Gaussian statistics prescribed inthe original coordinates (Euler angles) are mappedwith good approximation to Gaussian statistics inthe Rodrigues parameters and modified Rodriguesparameters.


Tig. 11 3-<r Confidence Surfaces Snap Shot at t = 0.5s

Mont̂ Carto Polnla

Eutor (3-2-1 ]AngBS

> nodr£Llea Paramotan

• Mo. RodriguBs Pamm«tara

Fig. 12 3-0- Confidence Surfaces Snap Shot at t = 9.5s

During simulations, uncertainty is propagated us-ing Eq. (12), for all the three coordinate sets. Three"snap shots" of the 3-<r constant probability surfacesat t = Os, 0.5s and 9.5s from Linear Error Theoryare shown in Figs. 10, 11 and 12. The two Rodriguesellipsoidal 3-<r surfaces are mapped into correspond-ing non-ellipsoidal surfaces in the Euler angle space.All three 3-cr surfaces of constant probability are dis-played, along with the 200 Monte-Carlo simulationpoints; for ease of visualization, projections of the3-cr surfaces and the Monte-Carlo integrated pointsinto the (S<f>, 58), (5<j), <5i/>), (69,54>) planes are shown.For very early evolutions of the nonlinear probabilitydistribution of errors, the linear error theory uncer-tainty propagation is valid for all three coordinatesets.

Figures 11 and 12 indicate that the longer termprediction of the 3-<r surface from propagation in theRodrigues parameters and the modified Rodriguesparameters captures the true nature of the actualnonlinear/non-Gaussian error distribution in Eulerangle space, to a much better degree when comparedwith direct linear error theory propagation using thelinearized Euler angle differential equations. The

Fig. 13 Dynamic Nonlinearity Index Variation

performance of Linear Error Theory in different co-ordinates can be anticipated without the expense ofa Monte Carlo process by computing the dynamicnonlinearity index v(t, t0) for the three coordinatesystems as a function of time (Fig. 13). The com-putations of v(t, t0) are based on 12 initial condi-tions chosen from the initial three-sigma error ellip-soid. The linearization-based predictions using Ro-drigues parameters and modified Rodrigues param-eters compared favorably with Monte-Carlo calcu-lations (errors less than 1%), whereas linearizationapproximations in the Euler angles were very erraticwith errors ranging from ~ 20 to ~ 100%.

Figures 6 and 13 provide clear evidence that notall linearizations are created equal. For both or-bital mechanics and attitude dynamics => evaluatingthe linearity indices allows confident decisions be-tween competing coordinate choices vis a vis appli-cability of Linear Error Theory. These results havesignificant implications in applictions of estimationand control theory. For example, the linearity andGaussian assumptions underlying Kalman filter de-sign strongly favor the modified Rodrigues parame-ters over any set of Euler angles.

III. Quasi-coordinates for Multibody SystemsConsider the class of nonlinear dynamical systems

whose kinetic energy (T) has the structure

T = ixT[M(x)]x + T! (t, x, x) + T0(t, x) (30)

where x(i) £ 7£" is a configuration vector of inde-pendent generalized coordinates, M(x) = MT(x) e•j^nxn ig a positive definite configuration-variablemass matrix, Ti (t, x, x) depends linearly on x, andT0(t, x) does not contain x. For this class of sys-


terns, being acted on by control forces u(t) € 7£m,the differential equations can be brought to the form

= diag(Si), T = di

Af(x)x=F(t,x,i,u) (31)

We assume that the mass matrix M = MT > 0 isan analytic function of x(t), F(t, x,x, u) € 72." is acontinuous function of all arguments. It is evidentthat any discontinuity in the arbitrary control inputsu(() can instantaneously affect x(t) but not x(t) orx(i). Therefore M = M(x, x) is also continuous, butM — M(x, x, x) may be discontinuous, dependingon u(t). For any instant t, we observe that M canbe decomposed into it's spectral factors C, and A as

M = CTA.C <=> A = CMCT (32)

where A = diag(\i) and CT £ -Jlnxn is the or-thogonal matrix of eigenvectors. Obviously, sinceM — M(x(i)), A and C are implicitly time vary-ing functions of x(t). Except possibly near re-peated AJ'S, we anticipate that both A and C willbe smooth functions of x(t). It is evident that ifwe can generate C'(t), and A(i) by solving differen-tial equations governing their time evolution, thendue to the orthogonality of C (CTC = CCT = /),we can avoid the necessity of numerically invertingthe large configuration-variable M(x(t)), becauseM-1 = CT\~1C. Since M = MT > 0, the eigen-values A; will always be positive. Because A* > 0we can always define s; = ^f\l and the correspond-ing matrix S = diag(si). This allows M to be fac-tored as M = CTSTSC = WTW, where W = SCis the matrix square root of M. Therefore keep-ing track of C and S is equivalent to a square rootalgorithm with most of the advantages.51 The algo-rithm presented by Oshman,51 however, is not ro-bust if there are repeated crossings of the eigenval-ues of M(x(i)), we discuss below a way to solve thisproblem. Note that the inverse of M is alternativelywritten M~l = CTS~2C.

In a recent study,52 we found it attractive to in-troduce a quasi-coordinate velocity transformation

v(t) = SCx(t) => X(t) :

satisfying the dynamical equation

), (33)

(34)

To make the above differential equation meaning-ful we require means for continuously computing theeigenfactors S, C. Recent papers8'51"59 have estab-lished the insight needed to write the following dif-ferential equations for C and S

where

for

(36)

(37)

(38)

For the case of near-repeated eigenvalues, it is nec-essary to introduce a stabilization method to inte-grate Eqs. (33)-(35). This can be accomplished byusing differential Jacobi rotations to establish (C, S)correctors for integration steps near repeated eigen-values.52

We note that the bulk of the computational effortis the matrix product (\f*ij\ = CMCT), but sincein reality, each nij is an independently computableinner product of rows of C with M=> this process isinherently parallelizable in contrast to, for example,solution for x in Eq. (31). This property enablesthis approach to be easily mapped onto massivelyparallel machines to be employed on a large classof high-dimensioned nonlinear multi-body dynam-ics and large dynamic deployment/re-configurationproblems.

For the analogous problem of spacecraft attitudekinematics, C € 7£3x3 is the orthogonal direction co-sine matrix and the distinct elements of fly are thebody components of angular velocity. So this lineof thinking lifts C(t) £ 7£3x3 rotational kinemat-ics to analogous kinematic equations which describethe evolution of higher dimensioned orthogonal ma-trices C(t) e 72."xn which locally factor the sym-metric positive definite mass matrix. We note that

Nonlinear McdumiMn

C = -SIC, and 5 = (35)

Fig. 14 A Conservative Nonlinear Multibody System

near-repeated eigenvalues indicate symmetry condi-tions in which the principal axes of M and there-fore the corresponding rows of C are not generallyunique. However, we observe52 that C, C, S, S mustgenerate M and M. => The M requirement usuallymakes the eigenvectors (C) unique (unless \j —I A,and Xj —t \i occur simultaneously, i.e., not merely


eigenvalue crossing, but the more rare event of eigen-value osculation). Numerical difficulties will occur insome Aj —» At- situations, however, and a stabiliza-tion scheme is needed to robustly solve Eqs. (35).Invoking quadradicly convergent differential Jacobirotations52 near repeated eigenvalues modifies theapproximation for C and 5, as needed, to smoothlytrack both M(t) and M(t). As a practical matter,the occasional lack of uniqueness of C is not trouble-some, since the algorithm validity relies on the con-vergence to any instantaneously diagonalizing trans-formation.

Discussion of Figures:Fig. (a) The angular responsehistory is characterized byfrequent elbow snap—throughswheretheOi = 62. Thescsnap —through events are characterized

by high 'jerk' large - regions.

Angular Free Response

eangularaccelerationis very nonlinear due to the'highjerk regions' near every elbow snapthrough, one would anticipate asmaller step sizes in these regionsto maintain integration accuracy.

Flg.(c) The quasi - coordinateacceleration history is smootherthan the 8(0 history, esp. =*near elbow snap - throughs,anticipate larger step sizes fora given accuracy. => ^"'effect*.

Anyilar Acceleration Hiilcny

Quasi ~coordl(UteAccelentloni>|andi>i

e integration was done witha fixed step size, note the max errors- snap - throughs. 3 methods, solve:Method 1: v = S~*CF(&) + (SMelhod2:8=CT.r2CF(9)

2.0.10'"

1.5.10"1

fe 1.0-10 "

5.0-10"

0.0

Energy Errors for 1M 10 f of Integration: ——— nwttiodl 1

~*Jt7^ W«-̂ *»«™ n"

j(....................l..Jilfc- ———— D- ———— **t>

lO"*

10-*10'10

101"10"tff"

.. . -IM.S.No™ »f FKU7.ft<f>t*lfer.-l*'.'——— mtttiod 1 :-— -m«hod2

...... ..":r:.!n.̂ »(!?....L—.-.̂ ...... .....;..j22r

=:;;;:;Fig.(«) Thelongtermstabiliryofintegration vs step size evident, noteespecially square root methods, =>Method 1 is always superiorioMethod 3 (conventional algorithm).Ultintateprtcision is vastly superior.

Fig. 15 Nonlinear Response

To provide some insight to the the potential sig-nificance of the above developments, we consider anexample.60 Shown in Fig. 14 is a simple multibodyproblem, the mass matrix of this 2dof system de-pends on the cosine of the elbow angle, we show inFig. 15 the free nonlinear response of this system.

We consider the nonlinear free vibration prob-

lem starting from the initial conditions: #i(0) =0,6»2(0) = 60°,0,-(0) = 0. For simplicity, the un-stretched spring length is taken as zero. Since theinitial configuration is not an equilibrium position,the system will undergo nonlinear free vibration witha known exact energy integral; errors in this ex-act integral can be used as a rigorous measure ofintegration stability. The system parameters, con-straint conditions, and initial conditions were variedto study effects of nonliinear snap-through motionsand near-repeated mass matrix eigenvalues. For thecase of constrained motion (right figure of Fig. 14),the mass matrix eigenvalues crossed once on eachoscillation.

The focus of this example is on stability of thealgorithm, and even this simple case shows clear ad-vantages. The numerical solution was computed bya standard 4th order Runge-Kutta algorithm, andfor especially simple discussion, we did not makeuse of the usual variable step size control one wouldemploy for problems with a nonlinear accelerationhistory. Referring to Fig. 15, note the remark-able enhancement of the numerical solution achievedthrough the use of the above quasi-coordinate for-mulation (Method 1), in comparison to direct inte-gration (Method 3) of angular accelerations by in-verting Eq. (31). Method 2 is a hybrid in which the(C(t), S(t)} solution is used to algebraicly generatethe inverse of M(x), but angular acceleration is in-tegrated instead of the quasi-coordinate vector. Thequasi-coordinate accelerations are much smootherthan the angular accelerations, especially near el-bow snap-throughs, and the resulting integration isthree orders of magnitude more precise than Method3 and almost one order of magnitude more precisethan Method 2 for any given step size. Note furtherthat the ultimate precision is improved dramaticallyover the traditional Method 3, again by three or-ders of magnitude in this low dof example, and thisapproach has shown exceptional stability on all ex-amples studied to date.

IV. Feedback Control of Attitude Maneuvers

During the past two decades, there has been an ex-citing series of dynamics/controls adventures whichaddress various aspects of How to maneuver a space-craft from orientation state A to orientation stateB. Prior to the 1980s, most attitude control lawswere designed based upon linearized dynamics and,uncoupled one-loop-at-a-time yaw-pitch-roll controldesign strategies. Because spacecraft were mostlyrigid and maneuvers were usually slow, these ideal-izations were sufficient to solve practical problemsand led to sufficiently robust control laws that could


be implemented in available computers. The ad-vent of large flexible structures and requirementsfor rapid maneuvers and vibration arrest have posednew challenges which have been partially met withan impressive set of recent theoretical, computa-tional and hardware developments. This family ofproblems is complicated by several considerations:(i) nonlinearity, (ii) structural flexibility, (iii) evolu-tionary variable geometry structures (iv) model un-certainty, (v) high dimensionality and (vi) practicallimitations on real-time computing, sensor/actuatorlimitations, and difficulty in defining optimality. Be-cause of the need for autonomy and reliability in theface of these difficulties, these problems have servedas a very important technology driver in dynamicsand control. As is evident below, mechanics-derivedinsights have been crucial in determining the mostappropriate ways to harness control theory and ef-fect impressive/realizable solutions.

To capture the spirit of recent analytical devel-opments, consider the attitude dynamics of a rigidspacecraft described by the principal axis form ofEuler's rotational equations of motion

(39)

along with the kinematical equations for a chosenset of coordinates. We could choose the Euler an-gles, however, this would lead to transcendental dif-ferential equations containing a singularity. A betterselection for large motion control design is any one ofthe sets associated with Euler's Principal RotationTheorem (e.g., the Euler parameters, the Rodriguesparameters, or the modified Rodrigues parameters).We select the latter set, so the 6th order system isdescribed by Eqs. (39), and Eqs. (25).

The most important step in applying a Lyapunovapproach in control design is the selection of the Lya-punov function to measure errors from the targetstate. This issue is treated in several texts and re-cent papers.3'40'61'82 For the purposes of illustration,we adopt the following Lyapunov function:

V = -wT[diag(Ii}]u + 2k0ln[l + <rTa]

„ (40)

= kinetic energy +2k0ln 1 + tan I —

We note that the Lyapunov function V is posi-tive definite and vanishes only at the target state(the origin). The idea is to find the structure

of a globally asymptotically stabilizing control lawUi(wi, (jj%, (jJz, <TI, o"2) ""3) by requiring w,- to cause V tobe strictly negative along the closed loop trajectoriesof the system. In this case, differentiating Eq. (40)leads directly to the modified "power" equation

V = (ui + k0 a,-) = WT (u + k0 a) (41)

and requiring V to be non-positive means that theparenthetic term (u,- + k0 or,-) have the opposite signof w;, so the simplest stabilizing control law obtainedby setting u + k0 a = —Gu> has the linear feedbackform:

u = -k0 a -Gui (42)where the constant gains kc > 0 and G — GT > 0can be selected by some optimization process sub-ject to the positivity requirement. Remarkably, thissimple linear feedback law stabilizes the general non-linear dynamics. The derivative of V is:

V = -uTGu (43)

Since V does not contain a, it is clear that V is onlynegative semi-definite and we cannot immediatelyclaim asymptotic stability. However, a useful theo-rem3'42 says that if the higher derivatives of V areevaluated on the set Z where V = 0, and the firstnon-zero V time derivative is odd and negative defi-nite on Z, then we have asymptotic stability in spiteof the apparent possibility that V may vanish. Phys-ically, this proves that the motion does not remain atpossible apogee states where V kisses zero, becausethese states are not equilibrium points. In this case,the only equilibrium state of the closed loop systemis the origin and it can be verified that the first non-zero derivative of V on {Z ; ui — 0, a ^ 0} is thethird derivative and it is negative definite on Z, so weindeed have asymptotic stability for all states within±2n of the target state. It is easy to verify that co-ordinate selection is crucial in the above process offinding globally stabilizing control laws. We found asimple linear feedback law gives stability in the large,choosing any set of Euler angles would be an illumi-nating exercise that would not lead to these elegantresults. Over and above the proof of stability, thereare other implicit advantages. In particular, sincethe modified Rodrigues are very near-linear threeparameter description of motion, the domain of va-lidity of the optimal control gains (if we linearizeabout the target state and apply optimal linear con-trol theory3) is much larger than, for example, anychoice of Euler angles. The insight to write the mea-sure of attitude error as ln(l + aTa) in Eq. (40) isdue to Tsiotras.62


More generally, analogous recent developmentshave been carried out for distributed parameter andtime varying systems, and validated through ana-lytical, numerical, and laboratory experimental re-search, and these have been augmented with con-cepts from adaptive control theory to make thisapproach robust in the presence of model errorsand disturbances.3'38"44'52'61-83 In particular, the re-sults by Schaub and Junkins63 provide key insightsinto the construction of attitude penalty functionsfor stability and control analysis, to eliminate theusual dependence of the definition of attitude er-ror on the choice of attitude coordinates. We in-troduced63 a universal attitude error measure J =(3 — trace(C))/4, where C is the direction cosinematrix projecting the current body axes onto theirtarget orientation. This positive error measure canbe expressed in any coordinate set and, for example,as a function of the principal rotation $ and e issimply J = sin2($/2). Thus a 180° degree rotationfrom the target orientation about any axis gives themaximum value of J — 1. This function is graphedin Fig. 16.

naxis") from the initial target state, the angular ac-celeration history of the reference maneuver beingspecified as bang-bang, bang-off-bang, and torque-shaped variations of these strategies. The trackingfeedback laws are designed using Lyapunov meth-ods to ensure stability and robustness with respectto model errors and disturbances.

489 NThraster. .. 5.3 & 22.7 N Thrusters

...,..••••-—" Star Trackers

High Gain Parabolic Antenna

Fig. 17 Clementine Spacecraft

180

QUATERNIONS

Fig. 16 Universal Attitude Penalty Function

The marriage of Lyapunov stability theory withcontroller design methods has resulted in a unifiedapproach that applies to nonlinear, distributed pa-rameter, and time varying control law design. Thedevelopments consider not simply regulation in thevicinity of a fixed target equilibrium state, but alsotracking of nonlinear reference maneuvers and modelreference adaptive control. The reference maneu-vers, in practical applications, are typically designedby rotating about the principal rotation axis ("eige-

ii,[rpm]1000

Time [sec]

WHEEL SPEEDS

20) 300Time [sec]

Fig. 18 Clementine On-Orbit Maneuver Results

To illustrate the maturity of this approach, con-sider the recent Clementine mission. The Clemen-tine spacecraft, shown in Fig. 17, was launched Jan-uary 25, 1994. This was the first post-Apollo re-turn to the moon and was primarily a mapping mis-


sion with a requirement for numerous large angle,three dimensional pointing maneuvers to point themapping camera to track specific targets of inter-est on the lunar surface. Attitude control was ac-complished by torquing three orthogonal reactionwheels. Attitude estimation made use of two solidstate star cameras and real-time, on-board star iden-tification utilizing an approach that represents a re-finement of the concepts introduced by Junkins etal.,64 this approach was also utilized in the recentlylaunched Near Earth Asteroid Rendezvous (NEAR)mission.65

The attitude control law for Clementine was de-signed by Creamer45'66 using methodology similar toapproaches of Wie,39 and Junkins et al.3>38'40'52 Aunique feature of Creamer's approach is the incor-poration of integral feedback to null motor friction-induced steady state errors. The reference maneu-vers designed for Clementine were eigenaxis rota-tions of the ncar-minimum-fuel bang-off-bang type.Inverse dynamics was used to solve for a refer-ence feed-forward torque profile to drive the reactionwheels. Based upon real time state estimation, per-turbation feedback control proportional to the er-ror quaternion (Euler parameter) estimates, angu-lar velocity tracking error estimates and the inte-gral of the error quaternion estimate was superim-posed on the feed-forward reference torque history.The stability characteristics were analytically veri-fied by Lyapunov analysis and simulation studies.The on-orbit implementation was remarkably suc-cessful, with over 1000 large angle maneuvers per-formed without significant anomalies other than oc-casional maximum wheel speed saturations. A typ-ical comparison of simulated versus actual on-orbitresults is shown in Fig. 18. As is evident, the wheelspeeds and quaternions are almost graphically iden-tical to the oil-orbit maneuver results.

The Clementine and NEAR missions clearlydemonstrate that by virtue of advances in concep-tual understanding, hardware design, and renewal ofan aggressive "can-do philosophy," we have entered anew lower cost, higher performance, and shorter de-velopment age for small exploratory space missions.The success of these missions indicate that Faster,Better, and Cheaper are not just salesmanship buzzwords, they describe the culmination of impressiveadvances on several fronts. This on-going evolutionpromises even greater impact in the future.

V. Concluding Remarks

This paper provides an overview of recent progressin spacecraft dynamics, estimation and control. Thediscussion synthesizes a broad set of topics and in-

cludes analytical and computational research as wellas several on-orbit implementations. The presenta-tion emphasizes unified developments in methodol-ogy for analytical dynamics, stability, control, andestimation. The primary conclusion that can bedrawn from the several "adventures" discussed isthat the field of spacecraft dynamics and control ismaturing rapidly => many of the most interestingchallenges arise on the traditional disciplinary inter-faces.

In preparing this paper, the author has been fre-quently reminded of the multidimensional joy he hasexperienced over the past three decades while work-ing in this rapidly changing field, often time-sharingbasic research with active involvement in develop-ment efforts and missions. Let us never un-learn thelessons of our most dramatic decade of progress (theApollo program) - theoretical research in dynamicsand control methodology and advanced flight imple-mentations not only can comfortably co-exist, theybelong to the same set.

VI. Acknowledgements

This work has benefitted from collaboration withmany colleagues. I am delighted to mention thefollowing individuals who have most directly con-tributed to this work: M.R. Akella, K.T. Al-friend, R.H. Bishop, J.E. Cochran, N.G. Creamer,J.N. Juang, A.J. Kurdila, Y. Kim, V.J. Modi,H.S. Morton, R. Mukherjee, Z.H. Rahman,R.D. Robinett III, H. Schaub, M.D. Shuster,T.E. Strikwerda, P. Tsiotras, J.D. Turner, andS.R. Vadali. My sponsorship by the following or-ganizations has been vital: the University of Vir-ginia, Virginia Polytechnic Institute, Texas A&MUniversity, NASA, Naval Surface Weapons Center,Sandia National Laboratory, Air Force Office of Sci-entific Research, Phillips Laboratory, United SpaceAlliance, Lockheed Martin Corporation, the TexasAdvanced Technology Program, and the Texas En-gineering Experiment Station. Finally, I have en-joyed working closely with over fifty graduate stu-dents whose years of inspired scholarship underwrotemany of the results discussed herein.

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46. Junkins, J. L., Akclla, M. R., and Alfriend, K. T., Non-Gaussian Error Propagation in Orbital Mechanics, toappear in Journal of the Astronautical Sciences, 1997.

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48. Shuster, M., "A Survey of Attitude Representations,"Journal of Ike Astronautical Sciences, Vol. 41, No 4,Oct-Dec 1993, pp. 439-518.

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50. Tsiotras, P., Junkins, J. L., and Schaub, H., "HigherOrder Cayley Transforms with Applications to AttitudeRepresentations," AAS Astrodynamics Specialist Con-ference, San Diego, CA, July 1996. Paper No. AAS 96-3628.

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53. Schaub, H., Tsiotras, P., and Junkins, J. L., "Princi-pal Rotation Representations of Proper NxN Orthog-onal Matrices," International Journal of EngineeringScience, Vol. 33, No. 15, Elsevier Science Ltd., GreatBritain, 1995, pp. 2277-2295.

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61. Junkins, J. L., Akella, M. R., and Robinett, R. D., "Non-linear Adaptive Control of Spacecraft Near-Minimum-time Maneuvers," AAS Space Flight Mechanics Confer-ence, Huntsville, AL, Feb. 1997.

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63. Schaub, H., Robinett, R. D., and Junkins, J. L., "NewAttitude Penalty Functions For Spacecraft Optimal Con-trol Problems," AIAA Guidance, Navigation and Con-trol Conference, San Diego, CA, July 29-31 1996. PaperNo. AIAA 96-3792.

64. Junkins, J. L., White, C. C., and Turner, J. D., "StarPattern Recognition for Real Time Attitude Determina-tion," J. of the Astron. Sciences, Vol. 25, No. 3, Sep.1977, pp. 251-270.

65. Strikwerda, T., "Near Earth Asteroid Rendezvous Guid-ance and Control System," SAE Aerospace Control andGuidance Systems Meeting, Nashville, Tennessee, Octo-ber 9-11 1996.

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