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
1
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”
3
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
6
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)
7
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.
13
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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[16] H. K. Khalil, Nonlinear Systems. Macmillan Publishing Company, 1992.
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