Minimum-Time Reorientation of an Asymmetric Rigid BodyMinimum-Time Reorientation of an Asymmetric Rigid Body Andrew Fleming∗ Space and Naval Warfare Systems Command, Washington,

Calhoun: The NPS Institutional Archive

Faculty and Researcher Publications Faculty and Researcher Publications Collection

2008-08-18

Minimum-Time Reorientation of an Asymmetric Rigid Body

Fleming, Andrew

The American Institute of Aeronautics and Astronautics (AIAA)

http://hdl.handle.net/10945/29660

Minimum-Time Reorientation of an Asymmetric RigidBody

Andrew Fleming∗

Space and Naval Warfare Systems Command, Washington, DC, 20003, USA

Pooya Sekhavat†

US Naval Postgraduate School, Monterey, CA, 93943, USA

I. Michael Ross‡

US Naval Postgraduate School, Monterey, CA, 93943, USA

Minimum-time solutions are developed for the rest-to-rest reorientation of an asymmet-ric rigid-body. The optimality of the open-loop solutions are demonstrated by applicationof Pontryagin’s Minimum Principle. Bellman’s theory is used to further demonstrate op-timality while extending open-loop theory to real-time application. The open-loop timeoptimal control is, next, used to construct the closed-loop Carathéodory-π control solutionfor a similar maneuver. Closed-loop results presented for the system with and withoutparameter uncertainties verify the successful implementation of the method in practicalapplications.

I. Introduction

In their ground-breaking work of 1993, Bilimoria and Wie1 closed the door on the eigenaxis maneuveras the minimum-time spacecraft reorientation solution. Previous intuitive, straight-line, shortest-distance-between-two-points thinking gave way to optimal control theory and solutions to such problems ceased toseem obvious. Posing a tri-axisymmetric (cubical or spherical) rigid body with independent torque generationallowed them to decouple the problem dynamics. Their work demonstrated that, barring some specialsituations, the eigenaxis rotation is not the minimum-time solution to the rest-to-rest spacecraft reorientationproblem.

Later, in 1999, Shen and Tsiotras2 addressed the problem of reorientating the symmetry axis of anaxisymmetric rigid body. They utilized a unique kinematic parameterization,3 assumed only two controltorques and used a cascaded numerical method to identify minimum-time solutions. They noted a majordifficulty with their methodology was developing initial guesses for the costates which do not, in general,have intuitive physical interpretation.

Livenh and Wie4 presented an extensive analytical analysis of the asymmetric reorientation problemunder constant body-fixed torques. Additionally, the work of Proulx and Ross5 determined an admissibleswitching structure which was illustrated by the traversal of a unit cube. Using this to limit the searchspace a combination of a genetic algorithm and pseudospectral method was used to obtain the optimalsolution. Additionally, they suggested a method of evaluating the “optimality” of a solution by evaluatingthe Hamiltonian derived from the costates obtained through the Covector Mapping Theorem.6 This methodof evaluating compliance with Pontryagin’s Principle is employed in this work.

The unifying theme of all these works is that the minimum-time reorientation problem presents uniquechallenges that have held the interest of engineers for years. The general case of the minimum-time, inde-pendent torque asymmetric reorientation maneuver had no numerical solution before the work of Flemingin 2004.7 In this paper we will examine the time-optimal reorientation of a rigid asymmetric body un-der the influence of three-independent torques. Open-loop solutions to the problem will be developed and

∗Astronautical Engineer, Space and Naval Warfare Systems Command, Washington, DC and AIAA Member.†Research Scientist, Mechanical and Astronautical Engineering, NPS, and AIAA Member.‡Professor, Mechanical and Astronautical Engineering, NPS, and AIAA Member.

1 of 12

American Institute of Aeronautics and Astronautics

AIAA Guidance, Navigation and Control Conference and Exhibit18 - 21 August 2008, Honolulu, Hawaii

AIAA 2008-7012

Copyright © 2008 by A. Fleming, P. Sekhavat, I.M. Ross. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

Dow

nloa

ded

by N

AV

AL

PO

STG

RA

DU

AT

E S

CH

OO

L o

n Ja

nuar

y 22

, 201

3 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

008-

7012

solved for sample problem. Optimality of the open-loop solution will be demonstrated by the application ofPontryagin’s Minimum Principle as well as Bellman’s principle of optimality. The second half of the paperaddresses the closed-loop optimal control problem using the recently introduced Pseudospectral feedbackcontrol scheme.8 It is shown that the closed-loop optimal control is capable of reorienting the rigid body inthe presence of parameter uncertainties.

Following this introduction, the dynamical model is developed in section II. In section III, the optimalcontrol problem is formulated and the necessary conditions arising from the application of the Pontryagin’sMinimum Principle are developed. In section IV, open-loop numerical results are presented and analyzed.The optimality of the solution is verified via Pontryagin’s Principle and Bellman’s principle of optimality.Section V extends the results to closed-loop implementation of Carathéodory-π control solutions throughreal-time applications. Conclusions are presented in section VI.

II. Dynamical Model

The rigid-body reorientation problem dynamics are commonly represented by Euler’s equation.

Iω̇ + ω × Iω = Mext (1)

When the moment of inertia and angular velocity are expressed in the principal axis frame, Euler’s equationcan be expanded to:9

M1 = Ixω̇1 + (Iz − Iy)ω2ω3

M2 = Iyω̇2 + (Ix − Iz)ω1ω3 (2)M3 = Izω̇3 + (Iy − Ix)ω1ω2

Defining the state vector of the asymmetric rigid-body as:

x =

[qω

]∈ R7 (3)

The state equations of motion in assumed inertial space are then given as follows:

q̇1 =12

[ω1q4 − ω2q3 + ω3q2]

q̇2 =12

[ω1q3 + ω2q4 − ω3q1]

q̇3 =12

[−ω1q2 + ω2q1 + ω3q4]

q̇4 =12

[−ω1q1 − ω2q2 − ω3q3] (4)

ω̇1 =u1

Ix−

(Iz − Iy

Ix

)ω2ω3

ω̇2 =u2

Iy−

(Ix − Iz

Iy

)ω1ω3

ω̇3 =u3

Iz−

(Iy − Ix

Iz

)ω1ω2

It is notable that the quaternion kinematics are non-linear ordinary differential equations and the eule-rian dynamics are coupled nonlinear differential equations. However, this formulation and the difficultiesassociated with obtaining solutions to all but the simplest geometries is well known.10

III. Time-optimal Maneuvers

A. Problem Statement

The optimal control problem is stated as, determine the state control function pair, t → (x,u) ∈ R7 × R3,that will drive the spacecraft from its initial position given by x(t0) = x0 to its final position given by

2 of 12

American Institute of Aeronautics and Astronautics

Dow

nloa

ded

by N

AV

AL

PO

STG

RA

DU

AT

E S

CH

OO

L o

n Ja

nuar

y 22

, 201

3 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

008-

7012

x(tf ) = xf while minimizing the cost function,

J (x (·) ,u (·) , tf ) = tf − t0 (5)

where x(·) and u(·) are in appropriate function spaces that will be clarified shortly. The constraints for theproblem are the dynamics given by equations (4). The control space, U, is given by the box constraints,

U := {u ∈ R3 : ‖u‖∞ ≤ 50 N ·m}(6)

B. Pontryagin’s Necessary Conditions

Application of the Minimum Principle allows us to develop the necessary conditions for the optimal solution.The control Hamiltonian11 for the asymmetric spacecraft is given by,

H (λ,x,u, t) =λq1

2(ω1q4 − ω2q3 + ω3q2) +

λq2

2(ω1q3 + ω2q4 − ω3q1) +

λq3

2(−ω1q2 + ω2q1 + ω3q4) +

λq4

2(−ω1q1 − ω2q2 − ω3q3) +

λω1

(u1

Ix−

(Iz − Iy

Ix

)ω2ω3

)+ λω2

(u2

Iy−

(Ix − Iz

Iy

)ω1ω3

)+

λω3

(u3

Iz−

(Ix − Iy

Iz

)ω1ω2

)where the subscripts on the Lagrange multipliers have been selected to aid in bookkeeping.

The adjoint equations are obtained by differentiating the negative of the Hamiltonian with respect to thestates and are given by:

˙λq1 =12

(λq2ω3 − λq3ω2 + λq4ω1)

˙λq2 =12

(−λq1ω3 + λq3ω1 + λq4ω2)

˙λq3 =12

(λq1ω2 − λq2ω1 + λq4ω3)

˙λq4 =12

(−λq1ω1 − λq2ω2 − λq3ω3) (7)

˙λω1 =12

(−λq1q4 − λq2q3 + λq3q2 + λq4q1) + λω2

(Ix − Iz

Iy

)ω3 + λω3

(Iy − Ix

Iz

)ω2

˙λω2 =12

(λq1q3 − λq2q4 − λq3q1 + λq4q2) + λω1

(Iz − Iy

Ix

)ω3 + λω3

(Iy − Ix

Iz

)ω1

˙λω3 =12

(−λq1q2 + λq2q1 − λq3q4 + λq4q3) + λω1

(Iz − Iy

Ix

)ω2 + λω2

(Ix − Iz

Iy

)ω1

However, since the state variables are specified at both the initial and final conditions, the adjoint variableswill be free or unspecified at both initial and final conditions. Therefore, the adjoint equations and terminaltransversality of the adjoint variables provide no new information which will aid in our solution to theproblem.

Applying the Hamiltonian minimization condition,

(HMC)

{Minimize

uH(λ,x,u)

Subject to u ∈ U

3 of 12

American Institute of Aeronautics and Astronautics

Dow

nloa

ded

by N

AV

AL

PO

STG

RA

DU

AT

E S

CH

OO

L o

n Ja

nuar

y 22

, 201

3 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

008-

7012

we have,

λω1

Ix+ µ1 = 0

λω2

Iy+ µ2 = 0 (8)

λω3

Iz︸︷︷︸Si

+µ3 = 0

where µi i = 1, 2, 3 are the Karush-Kuhn-Tucker (KKT) multipliers7,12 associated with Problem HMC thatsatisfy the complementary conditions,

µi

≤ 0 if ui = −50≥ 0 if ui = 50= 0 if −50 < ui < 50

(9)

Thus Si = −µi serve as switching functions. The reader may note that these switching functions are nodifferent than those of the tri-axisymmetric spacecraft.1 The case when the switching function equals zerofor a non-zero period of time was rigorously examined by Bilimoria and Wie1 and shown not to be timeoptimal for the inertial symmetric case. Additionally, Shen and Tsiotras2 examined the axisymmetric caseand determined that second-order singular arcs and infinite-order singular arcs are possible for certain specificboundary conditions. However, in general, both controls can not be zero.

Thus we are left with a switching function that determines when the optimal control u∗ will switchbetween its extreme values. For this reason the control profile is called bang-bang.13

From the Hamiltonian evolution equation,

dHdt

=∂H

∂t= 0 (10)

where H is the lower Hamiltonian,H(λ,x) := min

u∈UH(λ,x,u) (11)

it is clear that H is a constant over time. Combining this result with the Hamiltonian value condition,

H [tf ] +∂E

∂tf= 0 (12)

where E is the end point lagrangian defined as the end point cost adjoined with the end manifold the finalvalue of the lower Hamiltonian is -1. Thus, any candidate optimal solution must have the property that Hbe a constant with value of -1 over the interval of the maneuver.

C. Solution Method

This optimal control problem is a functionally smooth nonlinear optimal control problem; that is, the func-tions involved in the problem formulation are all smooth (differentiable). In recent years, it has becomepossible to routinely solve smooth optimal control problems. More importantly, extremality of the computedsolutions can be rigorously verified by application of Pontryagin’s Principle; i.e., examining the necessaryconditions. It is worth emphasizing that such verifications of optimality can be performed without solvingthe difficult two-point-boundary value problem. In fact, solutions can be computed quite readily by animplementation of the Covector Mapping Principle.14,15 The covector mapping theorem for the Legendrepseudospectral method is implemented in the software package, DIDO.16

The Legendre Pseudospectral method is based on approximating the unknown functions by weightedinterpolants, where the interpolating points are the Lobatto points of Legendre polynomials (Legendre-Gauss-Lobatto (LGL) points). Although there are a variety of PS methods, we choose the LGL/PS methodsince the problem under consideration is a finite horizon problem with non-homogeneous end points.17 Forcomplete details on the selection of PS methods see reference [17] and the references contained therein.

4 of 12

American Institute of Aeronautics and Astronautics

Dow

nloa

ded

by N

AV

AL

PO

STG

RA

DU

AT

E S

CH

OO

L o

n Ja

nuar

y 22

, 201

3 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

008-

7012

Throughout this paper, all of the computational results were obtained by way of DIDO. DIDO is aminimalists’ approach to solving optimal control problems; only the problem formulation is required in muchthe same way as writing it on a piece of paper with pencil. All the dual variables required for verificationof optimality are automatically generated by DIDO. In Sec. V, the results of these verification tests areillustrated. For an introduction to the Covector Mapping Principle, please see Refs. 14 and 15.

IV. Open-Loop Results and Discussion

A. Open-loop Solution and Analysis

For numerical simulation we have chosen NASA’s X-ray Timing Explorer (XTE) spacecraft shown in Fig. 1.Using the spacecraft moment of inertia parameters provided in Table 1 and the control constraints defined

Figure 1. NASA X-ray Timing Explorer (XTE) Spacecraft

in Eq. (6) we seek the optimal control solution u∗ and associated state trajectories for the minimum-timereorientation maneuver. The maneuver under consideration is a 150 degree roll about the x-body axis. The

Parameter Value UnitsIx 5621 Kg ∗m2

Iy 4547 Kg ∗m2

Iz 2364 Kg ∗m2

Table 1. Data for the axisymmetric model

initial and final conditions of the reorientation are given as:

x = [q1, q2, q3, q4, ω1, ω2, ω3]T

x0 = [0, 0, 0, 1, 0, 0, 0]T (13)

xf =[sin(φ)

2, 0, 0,

cos(φ)2

, 0, 0, 0]T

where, φ is the eigenaxis rotation angle.The candidate control solution obtained is shown in Fig. 2. The candidate solution clearly displays bang-

bang characteristics in all three axes as our intuition might have led us to expect. Before evaluating theoptimality of the candidate solution, its feasibility is independently evaluated. A feasible solution must drivethe spacecraft from its known initial state to the desired end state. The initial conditions and control solutionare used as input to a MATLABr ODE45 propagation subroutine which uses an explicit one-step Runge-Kutta medium order (4th to 5th order) solver18 to verify that the control solution drives the system from

5 of 12

American Institute of Aeronautics and Astronautics

Dow

nloa

ded

by N

AV

AL

PO

STG

RA

DU

AT

E S

CH

OO

L o

n Ja

nuar

y 22

, 201

3 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

008-

7012

Figure 2. Asymmetric Spacecraft Time-optimal Control Solution

the given initial conditions to the desired final conditions. A linear interpolation was used to approximatethe control values between LGL points. Propagation results are shown in Fig. 3 and Fig. 4. The originalsolution obtained is shown in solid lines overlaid with the propagated states shown as ‘+’ marks below.

Figure 3. Asymmetric Spacecraft Quaternion Solution Validation by Propagation

It is easy to see that not only does the dynamic system propagate to the desired end state but that thepseudospectral approximation of the states closely matches the propagated results.

Having determined that the candidate solution presented in Fig. 2 is feasible, we next examine thenecessary conditions for optimality. Recall that equation (8) and the complementarity conditions of equation(9) define the switching structure of the control vector and define a relationship between the costate dynamicsand KKT multipliers. An inspection of the switching functions and their relationship to the control behaviorverifies that the control-constraint pair meet the KKT conditions. Switching functions for each axis areshown, overlaid with the control solution (Fig. 5, Fig. 6, and Fig. 7).

As previously stated, the lower Hamiltonian must be a constant and numerically equal to −1 over theperiod of maneuver. This necessary condition is indeed met with small numeric variations as illustrated ifFig. 8.

Our analysis of the solution indicates that it is a feasible solution to the time-optimal reorientationproblem. Additionally, the solution meets the necessary conditions for optimality derived from Pontryagin’sminimum principle. The optimal time required to complete the maneuver is 28.6 seconds.

6 of 12

American Institute of Aeronautics and Astronautics

Dow

nloa

ded

by N

AV

AL

PO

STG

RA

DU

AT

E S

CH

OO

L o

n Ja

nuar

y 22

, 201

3 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

008-

7012

Figure 4. Asymmetric Spacecraft Angular Rate Solution Validation by Propagation

Figure 5. Asymmetric Spacecraft X-axis Switching Function and Control Solution

Figure 6. Asymmetric Spacecraft Y-axis Switching Function and Control Solution

7 of 12

American Institute of Aeronautics and Astronautics

Dow

nloa

ded

by N

AV

AL

PO

STG

RA

DU

AT

E S

CH

OO

L o

n Ja

nuar

y 22

, 201

3 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

008-

7012

Figure 7. Asymmetric Spacecraft Z-axis Switching Function and Control Solution

Figure 8. Asymmetric Spacecraft Time-optimal Maneuver Solution Hamiltonian

8 of 12

American Institute of Aeronautics and Astronautics

Dow

nloa

ded

by N

AV

AL

PO

STG

RA

DU

AT

E S

CH

OO

L o

n Ja

nuar

y 22

, 201

3 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

008-

7012

B. Bellman’s Principle of Optimality

According to the Bellman’s principle of optimality, given an optimal trajectory from a point A to a pointB, the trajectory to point B from a point C lying on the A-B optimal trajectory is also optimal. Thisprinciple can be used to verify the optimality of the open-loop optimal solutions through recalculation ofthe optimal solution using an intermediate point on the original trajectory as the new initial condition. Forthe original open-loop solution to be optimal, the new partial-maneuver solutions must exactly lie on theoriginal complete maneuver trajectory. Figures 9 overlays 3 open-loop optimal trajectories: the originalopen-loop trajectory derived for the complete maneuver (solid line), a second open-loop trajectory derivedfor the system starting from the state values of the original maneuver at t=10.14 s (dotted lines), and athird open-loop trajectory obtained for the system with initial state values of the original complete maneuverat t=17.96 s (circled markers). The perfect overlay of the second and third plots on the overall maneuvertrajectory demonstrates that the original open-loop solution satisfies the Bellman principle of optimality.The control trajectories corresponding to each maneuver are shown in Fig. 10.

Figure 9. Verifying Bellman’s Principle of Optimality on the States

Figure 10. Verifying Bellman’s Principle of Optimality on the Controls

V. Closed-Loop Results and Discussion

As a result of the recent breakthroughs in pseudospectral (PS) control, feedback optimal control can nowbe achieved by recognizing that closed-loop does not necessarily imply closed-form solutions. Given thatpseudospectral methods can demonstrably generate open-loop optimal solutions in fractions of a second toa few seconds, one premise of this work is to show that the closed-loop optimal feedback control can beobtained by real-time computation of open-loop optimal solutions. The control discontinuities in the open-loop segments (see Fig. 2) are addressed by defining a solution over the sample segment in the standard

9 of 12

American Institute of Aeronautics and Astronautics

Dow

nloa

ded

by N

AV

AL

PO

STG

RA

DU

AT

E S

CH

OO

L o

n Ja

nuar

y 22

, 201

3 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

008-

7012

Carathéodory sense, and then glue the pieces in the same manner as in the π-trajectory. This concept isintroduced as a Carathéodory-π trajectory, i.e., when open-loop controls are generated fast enough, closedloop control can be achieved via generating Carathéodory-π solutions.8

A. Closed-Loop Control for System with Exact Parameters

The clock-based Carathéodory-π feedback control response for the system with known parameters is shownin Fig. 11. It is clear that the open-loop and closed-loop responses are very similar. Note that even whenthe system parameters are exact and there is no exogenous disturbance torque, there are still some sensormeasurement errors in the system that are resembled by the Runge-Kutta state propagation error in thesimulation results. This inevitable source of disturbance is the cause of the minor differences between theopen loop and closed-loop trajectories. The control trajectories corresponding to Fig. 11 are shown in Fig. 12.A key desirable feature inherent in such control algorithm is the fact that it is ”gain-free” and does not requirethe user to select or tune any controller gain; rather, ”designer functions” would be automatically generatedat the fundamental computational level.

Figure 11. Open-Loop vs. Closed-Loop State Trajectories for the System with True Parameters

Figure 12. Open-Loop vs. Closed-Loop Control Trajectories for the System with True Parameters

B. Closed-Loop Control in the Presence of Parameter Uncertainties

Next we assume that the rigid body’s real moments of inertia, Ix, Iy, and Iz are 2% less, 7.8% more, and8.5% more than the known amounts tabulated in Table 1, respectively. Applying the open-loop optimalcontrol on such real system results in a maneuver that is neither feasible nor optimal.

Figure 13 shows the closed-loop system response for the system with real parameters. The figure showsthat closed-loop optimal control scheme counteracts the effects of parameter uncertainties and successfully

10 of 12

American Institute of Aeronautics and Astronautics

Dow

nloa

ded

by N

AV

AL

PO

STG

RA

DU

AT

E S

CH

OO

L o

n Ja

nuar

y 22

, 201

3 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

008-

7012

completes the otherwise infeasible maneuver in about 50 seconds (vice 30 s in ideal case). The correspondingcontrol trajectory is shown in Fig. 14.

Figure 13. Closed-Loop State Trajectories in the Presence of Parameter Uncertainties

Figure 14. Closed-Loop Control Trajectories in the Presence of Parameter Uncertainties

VI. Conclusions and Future Work

In this paper the Legendre Psuedospectral Method was applied to the problem of asymmetric spacecraftreorientation. Both open-loop and closed-loop optimal responses were derived and validated. Using thereusable software package DIDO16 greatly simplifies the computational requirements while still demonstrat-ing spectral convergence to the original Bolza problem.19

Feasibility and optimality of the solution were verified using Pontryagin’s Minimum Principle and Bell-man’s Principle of Optimality. This, combined with engineering judgement leads to the conclusion that themaneuver is the time-optimal solution.

The first control solution shown is an open-loop solution to the optimal control problem. In actualimplementation, the control system must have the capability to compensate for unanticipated disturbancetorques and spacecraft modeling and sensor imperfections. This was achieved by employing the closed-loopCarathéodory-π solution concept for a system with and without parameter uncertainties.

A similar algorithm, based on this concept has been successfully implemented for the reorientation ofspacecraft with magnetic torque rod actuators.20 For this more computationally intense problem, solutionswere obtained at rates approaching 5Hz. This solution rate clearly demonstrates the utility of closed-loop Carathéodory-π solution concept in modern satellite systems. The resulting agility, accompanied byincreased autonomy as the control system plans optimal maneuvers (vice operator planning) will result inincreased mission effectiveness. For future systems, improved control system performance can be translatedinto reduced actuator requirements. The resultant mass reduction represents a significant cost savings again

11 of 12

American Institute of Aeronautics and Astronautics

Dow

nloa

ded

by N

AV

AL

PO

STG

RA

DU

AT

E S

CH

OO

L o

n Ja

nuar

y 22

, 201

3 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

008-

7012

without a reduction in performance.

References1Bilimoria, K.D., and Wie, B., "Time-Optimal Three-Axis Reorientation of Rigid Spacecraft," Journal of Guidance,

Control, and Dynamics, Vol. 16, No. 3, 1993, pp. 446-452.2Shen, H. and Tsiotras, P., "Time-Optimal Control of Axisymmetric Rigid Spacecraft Using Two Controls," Journal of

Guidance, Control, and Dynamics, Vol. 22, No. 5, 1999, pp. 682-694.3Tsiotras, P., and Longuski, J.M., "A New Parameterization of the Attitude Kinematics," The Journal of the Astronautical

Sciences, Vol. 43, No. 3., 1995, pp. 243-263.4Livneh, R. and Wie, B., "New Results for an Asymmetric Rigid Body with Constant Body-Fixed Torques," Journal of

Guidance, Control, and Dynamics, Vol. 20, No. 5, 1997, pp. 873-881.5Proulx, R. and Ross, I. M., "Time-Optimal Reorientation of Asymmetric Rigid Bodies," AAS/AIAA Astrodynamics

Specialist Conference, Quebec City, Canada, 30 July - 2 August, 2001.6Fahroo, F., Ross, I.M., “Costate Estimation by a Legendre Pseudospectral Method,” Journal of Guidance, Control, and

Dynamics, Vol. 24, No. 2, 2001, pp. 270-277.7Fleming, A., "Real-time Optimal Slew Maneuver Design and Control." Astronautical Engineer’s Thesis, US Naval Post-

graduate School, 2004.8Ross, I. M., Sekhavat, P., Fleming, A., and Gong, Q., "Optimal feedback control: foundations, examples, and experimental

results for a new approach," Journal of Guidance, Control, and Dynamics, Vol. 31, No. 2, 2008, pp. 307-321.9Sidi, M.J., Spacecraft Dynamics and Control, Cambridge University Press, New York, NY, 1997.

10Wiesel, W.E., Spaceflight Dynamics, Irwin/McGraw-Hill. Boston, MA, 1997.11Pontryagin, L.S., Boltyanskii, V.G., Gramkrelidze, R.V. and Mishchenko, E.F., The Mathematical Theory of Optimal

Processes, John Wiley and Sons, Inc., New York, NY, 1962.12Bazaraa, M. S., Sherali, H. D., and Shetty, C. M., Nonlinear Programming: Theory and Algorithms, John Wiley and

Sons, Inc., 2006.13Bryson, Arthur, E. and Ho, Y., Applied Optimal Control, Taylor & Francis Publishing, New York, NY, 1975.14Ross, I. M., “A Roadmap for Optimal Control: The Right Way to Commute,” Annals of the New York Academy of

Sciences, Vol. 1065, New York, N.Y., January 2006.15Ross, I. M., “A Historical Introduction to the Covector Mapping Principle,” AAS/AIAA Astrodynamics Specialist Con-

ference, Tahoe, NV, August 8-11, 2005, Paper AAS 05-332.16Ross, I. M., "A Beginner’s Guide to DIDO: A MATLAB Application Package for Solving Optimal Control Problems,"

Elissar Technical Report TR-711,, http://www.elissar.biz, 2007.17Fahroo, F. and Ross, I. M., "On Discrete-time Optimality Conditions for Pseudospectral Methods," Proceedings of the

AIAA/AAS Astrodynamics Specialist Conference, Keystone, CO, August 2006.18Hanselman, D., and Littlefield, B., Mastering MATLABr 6, Prentice Hall, Upper Saddle River, NJ, 2001.19Ross, I. M. and Fahroo, F.,“Convergence of Pseudospectral Discretizations for Optimal Control Problems,” Proceedings

of the 40th IEEE Conference on Decision and Control, December 2001, Orlando, FL.20Sekhavat, P., Fleming, A., and Ross, I.M., "Time-Optimal Nonlinear Feedback Control for the NPSAT1 Spacecraft," Pro-

ceedings of the 2005 IEEE/ASME International Conference on Advanced Intelligent Mechatronics, July 23-28, 2005, Monterey,CA.

12 of 12

American Institute of Aeronautics and Astronautics

Dow

nloa

ded

by N

AV

AL

PO

STG

RA

DU

AT

E S

CH

OO

L o

n Ja

nuar

y 22

, 201

3 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

008-

7012