COUPLED ORBITAL AND ATTITUDE

CONTROL SIMULATION

Scott E. Lennox

AOE 5984: Advanced Attitude Spacecraft Dynamics and Control

December 12, 2003

INTRODUCTION

In the last few years the space industry has started to change its focus of single large satellite

missions to the use of many smaller satellites flying in formation. Formation flying presents many

interesting and difficult problems that have not been dealt with in the past. One of these topics is

the idea of coupling the spacecraft’s attitude and orbital control systems.

Coupling the attitude and orbital control systems is required for some of these spacecraft because

of the physical constraints that are enforced on the spacecraft as well as the operational constraints

to keep the formation together. An example of a physical constraint is the reduced amount of

space and mass allotted for the orbital propulsion system. It might not be possible for smaller

spacecraft to have complete controllability that a larger spacecraft might have been able to poses.

In many cases there might be one or two thrusters that need to be reoriented in the correct direction

before an orbital maneuver can be executed. Many of these small spacecraft are using low thrust

propulsion systems which require almost continuous thrust. In situations like these, we do not want

the propulsion system firing in the wrong direction and sending the spacecraft in a direction that was

not intended. Firing the thrusters in the wrong direction is a major reason to couple the attitude

and orbital control system. The operations of a formation may also require the spacecraft to have a

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high relative position constraint. This relative position constraint requires a low thrust propulsion

system to be able to change the spacecraft’s position with a high amount of accuracy on the thrust

direction. This high accuracy is coupled with the attitude control system accuracy, thus the need

for a coupling of the attitude and control systems.

The coupling of the attitude and orbital control systems is a relatively new concept and there are

only a few published papers. The first paper published on coupled attitude and orbital control

was written by Wang and Hadaegh in 1996.1 They derive and implement the attitude and orbital

control laws in a simulation of microspacecraft flying in formations. The simulations are concerned

with formation-keeping and relative attitude alignment. The relative attitude alignment of the

microsatellites is derived by finding the relative attitude of each microsatellite in the inertial frame.1

This paper did not discuss if the coupled control system could accomplish formation-maneuvering.

This paper is the first step in coupling the attitude and orbital control systems.

Naasz et al. discussed and performed simulations of an orbital feedback control law and a magnetic

torque coil attitude control system.2 They applied the control laws to the Virginia Tech Ionospheric

Scintillation Measurement Mission, a.k.a. HokieSat, which is part of the Ionospheric Observation

Nanosatellite Formation (ION-F) project.2 The ION-F project will perform formation flying demon-

strations while collecting scientific measurements. The orbital feedback control law is proven to be

globally asymptotically stable in Ref. 3 and Ref. 4. The attitude control system is described in Ref. 5

and Ref. 6. Further research needs to be completed before a coupled attitude and orbital control

system can be used for formation flying missions.

For most space missions, orbital maneuvering is controlled by the ground station. The following

two examples of orbital controllers could be used for autonomous orbital control. Ilgen develops a

Lyapunov-optimal feedback control law for orbital maneuvers.7 This control law uses Gauss’s form of

lagrange’s planetary equations in classical and equinoctial orbital-element forms. Naasz and Hall3,4

develop a nonlinear Lyapunov-based control law with mean-motion control to perform autonomous

2

orbital maneuvers for formation flying. This control law can be used for formation-keeping and

formation-maneuvering.

The dynamics of a spacecraft are nonlinear so it reasonable to assume that a nonlinear controller

would be a more effective solution to control a spacecraft’s attitude motion than a linear controller.

Tsiotras presents eight different nonlinear feedback control laws using Lyapunov functions with

a quadratic and logarithmic terms. These control laws use Euler Parameters, Cayley-Rodrigues

Parameters and Direction Cosines. These controllers are expanded to include Modified Rofrigues

Parameters in Ref. 8. Hall el al. uses a Modified Rodrigues Parameter Lyapunov function to derive

three attitude tracking controllers using thrusters and momentum wheels.8 The three controllers

were proven to be globally asymptotically stable using LaSalle’s Theorem. Schaub et al. uses a

linear closed loop control law to model a spacecraft’s nonlinear dynamics.9 The linear closed loop

control law is found using an open loop nonlinear control law. An adaptive control law is developed

to enforce the closed loop dynamics with large knowledge errors in the moments of inertia and

external disturbances. Xing and Parvez derive a nonlinear Lyapunov controller and a robust sliding

controller for the tracking control problem.10 They convert the tracking control problem into a

regulator problem using relative attitude state equations. Transforming the tracking problem into

a regulator problem simplifies the design procedure of the controllers.10

In this paper we present an attitude estimator (orbital controller), two attitude controllers, and an

orbital estimator (orbit propagator). We then couple these controllers together to preform coupled

attitude and orbital maneuvers simulations.

SYSTEM

We consider the system of two spacecraft, a passive “target” spacecraft and a controllable “ren-

dezvous” spacecraft. The “target” spacecraft is in a parking orbit that is known to the “rendezvous”

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spacecraft. The “rendezvous” spacecraft has one variable thrust thruster and a three-axis momentum

wheel system. We also assume an ideal space environment without disturbance forces or torques.

The coupled attitude and orbital control simulation is comprised of two major functions and two

minor functions. The two major functions are the attitude estimator or orbital controller and the

nonlinear attitude controller. The two minor functions are the bang-bang linear attitude controller

and orbital estimator. We will discuss each of these functions in the following sections.

ATTITUDE ESTIMATOR

The attitude estimator that we use in this study was developed by Naasz and Hall in Ref. 3 and

Ref. 4. Naasz and Hall develop a nonlinear Lyapunov-based control law and a mean motion control

strategy. The derivation of the control law begins with the equations of motion for a point-mass

satellite11

r = − µ

‖r‖3r + ap (1)

where r is the position vector from the mass center of the primary body to the satellite, µ is the

gravitational parameter, and ap is the perturbation accelerations. If we set ap = 0 and write the

equations of motion in terms of orbital elements we have Gauss’ form of the Lagrange’s planetary

equations12

da

dt=

2a2

h

(e sin νur +

p

ruθ

)(2)

de

dt=

1h

(p sin νur + [(p + r) cos ν + re]uθ) (3)

di

dt=

r cos θ

huh (4)

dΩdt

=r sin θ

h sin iuh (5)

dω

dt=

1he

[−p cos νur + (p + r) sin νuθ]−r sin θ cos i

h sin i(6)

dM

dt= n +

b

ahe[(p cos ν − 2re) ur − (p + r) sin νuθ] (7)

4

where a is the semi-major axis, e is the eccentricity, i is the inclination, Ω is the right ascension of

the ascending node, ω is the argument of periapse, M is the mean anomaly, ν is the true anomaly,

θ is the argument of latitude, n is the mean motion, p is the semi-latus rectum, h is the angular

momentum, b is the semi-minor axis, ur, uθ, and uh are the radial, transverse, and the orbit normal

control. These equations can also be written as

œ = f (œ) + G (œ)u (8)

where œ is the vector of orbital elements, [a e i Ω ω M ]T , G (œ) is the input matrix found using

equations (2 - 7), and u is the vector of controls [ur uθ uh]T . The equations of motion of the first

five orbital elements are

η = Gu (9)

where

η =

a− a∗

e− e∗

i− i∗

Ω− Ω∗

ω − ω∗

=

δa

δe

δi

δΩ

δω

(10)

where (·)∗ is the target element and

G =

2a2e sin νh

2a2phr 0

p sin νh

(p+r) cos ν+reh 0

0 0 r cos (ω+ν)h

0 0 r sin (ω+ν)h sin i

−p cos νhe

(p+r) sin νhe − r sin (ω+ν) cos i

h sin i

ur

uθ

uh

(11)

5

Naasz and Hall find a control using a Lyapunov function.4 The proof of globally asymptotic stability

for the control is performed in Ref. 4. The control law is

u = −GT Kη = −

2a2e sin νh

2a2phr 0

p sin νh

(p+r) cos ν+reh 0

0 0 r cos (ω+ν)h

0 0 r sin (ω+ν)h sin i

−p cos νhe

(p+r) sin νhe − r sin (ω+ν) cos i

h sin i

T

Kaδa

Keδe

Kiδi

KΩδΩ

Kωδω

(12)

where Ka, Ke, Ki, KΩ, and Kω are positive gains. The angle errors δΩ and δω are defined between

−π and π. The mean motion control is accomplished by defining a new target semi-major axis, a∗∗,

which forces the mean anomaly error to zero

a∗∗ =(−KnδM +

1a∗3/2

)−2/3

(13)

where Kn is a positive gain and δM is defined between −π and π. We replace δM with δθ in

application, so that the mean motion control properly positions the spacecraft within the orbital

plane.3 The Lyapunov-based control law, equation (12), and the mean motion control, equation

(13), allows for feedback control for spacecraft orbital maneuvers.4

Naasz and Hall develop a gain selection method in Ref. 3 and Ref. 4. The gains for the attitude

estimator are found using

Ka =h2

4a2 (1 + e)21

∆tt(14)

Ke =h2

4p2

1∆tt

(15)

Ki =

[h + eh cos (ω + arcsin e sinω)

p(−1 + e2 sin2 ω

) ]21

∆tt(16)

KΩ =[h sin i (−1 + e sin (ω + arcsin e cos ω))

p (1− e2 cos2 ω)

]2 1∆tt

(17)

Kω =e2h2

p2

(1− e2

4

)1

∆tt(18)

where ∆tt is the length of the thruster firing. Kn is chosen depending on how aggressively we want

to correct the argument of latitude error.4

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ATTITUDE CONTROLLERS

This section is devoted to the derivation of the bang-bang linear attitude controller and the Lyapunov

nonlinear attitude controller that is used by the “rendezvous” spacecraft. To be able to derive these

controllers, we need to define the rotational dynamics and kinematics of the system.

Dynamics

The rotational dynamics of the “rendezvous” spacecraft is defined in Ref. 8. The dynamics of the

system are described as

hB = h×BJ−1 (hB −Aha) + ge (19)

ha = ge (20)

hB = IωB + AIsωs (21)

where hB is the system angular momentum vector, I is the 3 × 3 moment of inertia matrix of the

entire spacecraft, Is is the 3× 3 axial moment of inertia matrix for the momentum wheels, A is the

3×3 matrix containing the axial unit vectors of the momentum wheels, ha is the 3×1 matrix of the

axial angular momenta of the wheels, ge is the 3 × 1 matrix of the external torques applied to the

spacecraft, ga is the 3 × 1 matrix of the internal torques applied to momentum wheels, ωB is the

3× 1 angular velocity matrix of the body frame expressed in the inertial frame, ωs is the 3× 1 axial

angular velocity matrix of the momentum wheels with respect to the body, and J is the positive

definite inertia-like matrix defined as

J = I −AIsAT (22)

We can now define ωB and ha using equations (21) and (22)

ωB = J−1 (hB −Aha) (23)

ha = IsAT ωB + Isωs (24)

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Kinematics

The kinematics of the “rendezvous” spacecraft is also defined by Hall et al. in Ref. 8. We will use

Modified Rodrigues Parameters (MRPs)13 to describe the kinematics of the “rendezvous” spacecraft.

Modified Rodrigues Parameters are defined as

σ = e tan(

Φ4

)(25)

where e is the unit vector along the Euler axis and Φ is the Euler angle.13 Using MRPs, the

differential equations for the kinematics are

σ = G (σ) ω (26)

where

G (σ) =12

(I3 + σ× + σσT − 1 + σT σ

2I3

)(27)

and I3 is the 3×3 identity matrix. Hall et al. show that differential equation for the error kinematics

is

δσ = G (δσ) δω (28)

where δσ is the rotational error between the desired reference frame and the current attitude of the

spacecraft, and δω is the difference in the angular velocities of the desired attitude and the current

attitude.8

Bang-Bang Linear Controller

The bang-bang attitude controller is used to obtain an estimate of the amount of time needed to

complete the attitude maneuver using the nonlinear attitude controller. The bang-bang controller

is derived using Euler’s Law14

h = g (29)

8

where h is the angular momentum about the mass center of the system and g is the net applied

moment about the mass center. We constrain the problem to be a planar problem which leads to

h = g

Imaxθ = gmax (30)

where Imax is the maximum moment of inertia of the system, θ is the angular acceleration about

the moment of inertia axis, and gmax is the maximum applied torque that momentum wheels can

produce on the system. By integrating equation (30) twice we obtain∫ θ

θo

Imaxθdθ =∫ t

to

gmaxdt

Imax

(θ − θo

)= gmax (t− to)∫ θ

θo

Imax

(dθ

dt− θo

)dθ =

∫ t

to

gmax (t− to) dt

Imax (θ − θo)− Imaxθo (t− to) =12gmax (t− to)

2 − gmaxto (t− to) (31)

We define to = 0 and we can rearrange equation (31) into the following form

Imax (θ − θo) =12gmax∆t2 + Imaxθo∆t (32)

We can assume that θo = 0, θ = 0, and θ is defined as a ramping function. At ∆t2 there is a

discontinuity in θ. This discontinuity leads us to just examine the first half of the maneuver, where

Imax

(θo

2− θo

)=

12gmax

(∆t

2

)2

(33)

Solving equation (33) for ∆t produces

∆t = 2

√2Imax

gmax

(θo

2− θo

)(34)

Equation (34) provides an estimate of the time required to complete an attitude maneuver using the

nonlinear controller.

Lyapunov Nonlinear Controller

We use Lyapunov’s method to find a nonlinear attitude controller. Lyapunov stated that “the

solution x ≡ 0 of the system x = f (x), f (0) = 0, is asymptotically stable if there exists a positive-

9

definite function V (x) such that(

∂V∂x

)Tf (x) is negative-definite.”15 Hall et al. present the following

candidate Lyapunov function in Ref. 8

V =12δωT Kδω + 2k1 ln(1 + δσT δσ) (35)

where K = KT , and is a positive-definite matrix, and k1 > 0. This is a positive-definite and

unbounded function in terms of the errors δω and δσ.8 The derivative of V calculation yields

V = δωT Kδω + 4k1δσT δσ

1 + δσT δσ(36)

Using equation (28) we can rewrite V as

V = δωT Kδσ + 4k1δσT G (δσ) δω

1 + δσT δσ(37)

Using equation (27) we can derive the following identity

4k1δσT G(δσ)δω

1 + δσT δσ= k1δωT δσ (38)

Plugging this identity, equation (38), into equation (37) we obtain

V = δωT Kδω + k1δωT δσ (39)

= δωT (Kδω + k1δσ) (40)

Choosing K = J and using equations (19), (20), and (23), we obtain

V = δωT(hB −Aha + k1δσ

)= δωT

(h×BJ−1 (hB −Aha) + ge −Aga + k1δσ

)(41)

We need to pick ge and ga for V to be negative-definite. For this study we use

ge = 0

ga = A−1(h×BJ−1 (hB −Aha) + ge + k1δσ + k2δω

)(42)

When we plug equation (42) into equation (41), we obtain

V = −k2δωT δω (43)

10

with k2 > 0. Using this ga, equation (42), V is negative semi-definite and bounded. To prove that

V is negative-definite we need to use LaSalle’s Theorem.16 Equation (43) also yields

limt→∞

δω = 0 (44)

Since limt→∞ δω = 0, we can conclude that limt→∞ δω = 0. By examining equations (23), and

(44), we can also conclude that limt→∞ hB = 0. This produces

0 = −J−1Aha

0 = −J−1Aga

0 = −J−1k1δσ (45)

From equation (45) we can see that

limt→∞

δσ = 0 (46)

We can now conclude that the dynamic and kinematic errors with the feedback control law,equation

(42), are globally asymptotically stable.

ORBITAL ESTIMATOR

The orbital estimator is used to propagate the orbits of both the “target” spacecraft and the “ren-

dezvous” spacecraft. The orbital estimator uses the f and g expressions in terms of eccentric anomaly

to propagate the spacecraft’s orbit.11 The f and g method requires the input of the initial position

(ro) and velocity (vo) of the spacecraft, and the propagation time (∆t). The f and g equations are11

f = 1− a

ro(1− cos ∆E) (47)

g = ∆t−

√a3

µ(∆E − sin∆E) (48)

f = −√

µa sin∆E

rro(49)

g = 1− a

r(1− cos ∆E) (50)

11

where a is the semi-major axis of the orbit, ∆E is the change in eccentric anomaly, ro is the

magnitude of the initial position vector, and r is the magnitude of the finial position vector. The

final position (r) and velocity (v) vectors are calculated using11

r = fro + gvo (51)

v = fro + gvo (52)

PROGRAM

The coupled orbit and attitude maneuver is simulated using Matlab c©. The overall simulation

architecture is shown in Figure 1. The simulation begins by defining a position and velocity of the

“target” and “rendezvous” spacecraft along with the attitude of the “rendezvous” spacecraft. These

initial conditions are used as the first set of inputs into the control loop. The attitude estimator

uses the position and velocity data to produce an optimal thrust magnitude and direction in the

inertial frame. This thrust direction is converted into a desired attitude of the spacecraft. The

bang-bang attitude controller uses the current attitude of the spacecraft and the desired attitude

to calculate an approximate time to complete the attitude maneuver. This time, t1, is used by the

orbital estimator to determine the position and velocity of spacecraft at t1. The new position and

velocity data are used by the attitude estimator to determine the optimal thrust magnitude and

direction at t1. This thrust direction is used to determine the desired attitude at t1. The Lyapunov

attitude controller uses the attitude of the spacecraft at to and the desired attitude at t1 and runs

until the time, t1 is reached. The current attitude at t1 is compared to the desired attitude at t1

by the attitude check function. This function determines if the current attitude is within a set of

error limits that the user specifies. If the current attitude is not within the error limits, then the

current attitude and the position and velocity of the “target” and “rendezvous” spacecraft are used

as the new inputs of the control loop. If the current attitude is within the error limits, then the

thruster “fires” according to the current attitude. The thruster is considered an ideal thruster, so

12

Figure 1: The overall simulation architecture

the magnitude of the thrust is variable and is determined by the attitude estimator. The thrust is

applied to the “rendezvous” spacecraft in the form of a ∆V over a user specified time interval. The

orbital estimator uses the thrust time along with the current position and velocity of the spacecraft

to determine the position and velocity of the spacecraft at t2. The position and velocity at t2 of

both spacecraft and the attitude of the “rendezvous” spacecraft are used as the new inputs to the

control loop. The user defines the number of orbits to simulate and the control loop keeps track of

the total time of the simulation.

RESULTS

Before we can test the coupled orbital and attitude control simulation we need to test each controller

separately. In the following sections we discuss the attitude estimator, the attitude controller, and

the coupled orbital and attitude control technique simulations.

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Attitude Estimator

To test the orbital estimator, we need to assume that the spacecraft is able to thrust in any direction

that is requested. The initial conditions for the simulation are

a∗ = 6823 (km) e∗ = 0.001 i∗ = 28 Ω∗ = 135 ω∗ = 90 ν = 0

da = 0 de = 0 di = 0 dΩ = 0 dω = 0 dν = −(

26823

)

∆tt = 100 (sec) m = 100 (kg) Kn = 0.05

where (·)∗ are the “target” spacecraft orbital elements, d(·) are the change in orbital elements of the

“rendezvous” spacecraft compared to the “target” spacecraft, ∆tt is the duration of time that the

thruster will fire, m is the mass of the spacecraft, and Kn is the mean motion control gain. Figure 2

shows the position error between the two spacecraft in the orbital frame. The position error between

the two spacecraft becomes zero after approximately 14 orbits. Figure 3 shows the change in semi-

major axis of the “rendezvous” spacecraft throughout the simulation. Figure 4 is the magnitude of

the thrust applied to the “rendezvous” spacecraft. Figure 5 shows the thrust direction in the orbital

frame. The thrust direction changes pretty rapidly over the course of the simulation. These figures

are representative of similar simulations that we performed using different initial conditions. We

conclude that the attitude estimator is effective in formation-keeping and formation-maneuvers.

Attitude Controller

We need to test the nonlinear attitude controller to insure globally asymptotic stability. The follow-

ing initial conditions are used

I =

6 0 0

0 6 0

0 0 10

(kgm2

)Is =

0.5 0 0

0 0.5 0

0 0 0.5

(kgm2

)A =

1 0 0

0 1 0

0 0 1

14

Figure 2: The position error in the orbital frame as seen by the “rendezvous” spacecraft.

Figure 3: The change in semi-major axis of the “rendezvous” spacecraft(a) compared to the “target”

spacecraft(a∗).

15

Figure 4: The magnitude of thrust applied to the “rendezvous” spacecraft.

Figure 5: The thrust direction of the “rendezvous” spacecraft in the orbital frame.

16

Figure 6: The change in the Modified Rodrigues Parameters of the spacecraft.

ωi =

−0.1

0.15

0.05

(

rad

s

)σi =

0

0

0

ω∗ =

0

0

0

(

rad

s

)σ∗ =

0.3

0.2

0.4

k1 = 1 k2 = 2

where I is the moment of inertia matrix of the system, Is is the moment of inertia matrix of the

momentum wheels, A is the axial unit vector matrix of the momentum wheels, ωi is the initial

attitude, σi is the initial angular velocity of the system, σ∗ is the desired attitude, ω∗ is the

desired angular velocity, k1 is the attitude gain, and k2 is the angular velocity gain. Figure 6

shows how the attitude varies throughout the simulation. Figure 7 is the variation of the body

angular velocity. Figure 8 is the applied torque provided by the momentum wheels throughout the

simulation. This simulation is representative of the other attitude simulations. We can conclude

that the nonlinear attitude controller is effective in controlling nonlinear attitude dynamics.

17

Figure 7: The change in the angular velocity of the spacecraft.

Figure 8: The applied torque on the spacecraft.

18

Coupled Attitude and Orbital Control

We have now proven analytically and numerically that the attitude estimator and nonlinear attitude

controller are globally asymptotically stable. We can now couple these controllers. The initial

conditions for the simulation are

a∗ = 6823 (km) e∗ = 0.001 i∗ = 28 Ω∗ = 135 ω∗ = 90 ν = 0

da = 0 de = 0 di = 0 dΩ = 0 dω = 0 dν = −(

26823

)

∆tt = 100 (sec) m = 100 (kg) Kn = 0.05

I =

6 0 0

0 6 0

0 0 10

(kgm2

)Is =

0.5 0 0

0 0.5 0

0 0 0.5

(kgm2

)A =

1 0 0

0 1 0

0 0 1

ωi =

0

0

0

(

rad

s

)σi =

0

0

0

k1 = 1 k2 = 2 T =

−1

0

0

where T is the orientation of the thruster in the body frame of the “rendezvous” spacecraft. Figure

9 shows the relative error between the two spacecraft in the orbital frame. It is very hard to see but

the coupled controller takes a little longer to reach the target spacecraft than the attitude estimator

controller (Figure 2). This is expected, but it is good that the change in time is small (approximately

a half of an orbit). Figure 10 shows the variation of the semi-major axis over the simulation time.

There is not much of a difference between this figure and Figure 3. The thrust magnitude can

be seen in Figure 11. The values for the thrust are low, which is good for propulsion systems on

formation flying spacecraft. Figure 12 shows the attitude error between the desired attitude and

the current attitude that was found after the attitude maneuver was completed and the acceptable

attitude error (5 in all directions for this simulation). The thruster did not fire when the attitude

19

Figure 9: The position error in the orbital frame as seen by the “rendezvous” spacecraft. (coupled

control)

error was greater than the acceptable attitude error. We conclude that the coupled attitude and

orbital control can be used for formation flying missions while not significantly increasing the time

to accomplish orbital maneuvers.

CONCLUSIONS

We have derived and proven asymptotic stability for a Lyapunov-based attitude estimator and a

nonlinear Lyapunov attitude controller using momentum wheels. These controllers were used in

conjunction with a linear attitude controller and an orbit estimator to create a coupled attitude and

orbital control system. Simulations were completed which lead to the conclusion that a coupled con-

trol system will work for formation-keeping and formation-maneuvers for formation flying spacecraft

missions.

20

Figure 10: The change in semi-major axis of the “rendezvous” spacecraft(a) compared to the “target”

spacecraft(a∗). (coupled control)

Figure 11: The magnitude of thrust applied to the “rendezvous” spacecraft. (coupled control)

21

Figure 12: The magnitude of the error between the desired thrust direction and the actual thrust

direction.

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22

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www.aoe.vt.edu/˜chall/courses/aoe4140/, 2003.

[15] P. C. Hughes, Spacecraft Attitude Dynamics. John Wiley & Sons, 1986.

[16] H. K. Khalil, Nonlinear Systems. Macmillan Publishing Company, 1992.

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