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Nonlinear Model Predictive Detumbling of Small Satellites with
a Single-axis Magnetorquer
Kota Kondo ∗
Kyushu University, Fukuoka-shi, Fukuoka, 819-0395, Japan
Ilya Kolmanovsky †
University of Michigan, Ann Arbor, Michigan 48109
Yasuhiro Yoshimura‡
Kyushu University, Fukuoka-shi, Fukuoka, 819-0395, Japan
Mai Bando§
Kyushu University, Fukuoka-shi, Fukuoka, 819-0395, Japan
Shuji Nagasaki¶
Kyushu University, Fukuoka-shi, Fukuoka, 819-0395, Japan
Toshiya Hanada‖
Kyushu University, Fukuoka-shi, Fukuoka, 819-0395, Japan
Nomenclature
𝑩,𝑩0= Earth’s magnetic field vector in body and orbital frame, respectively
𝐻= Hamiltonian
𝑖= inclination
𝐽cost = cost function
𝑱= moment of inertia
𝒎= magnetic dipole moment
𝑚max = maximum control input of magnetic torquer
𝑚𝑥= magnetic moment of the single magnetic torquer along the 𝑥-axis
𝑁= discretized step number on prediction horizon
𝑸, 𝑅1, 𝑅2= weight matrices
𝑸t= terminal cost
𝑟= distance between the satellite and the center of the Earth
Related work, "Model Predictive Approach for Detumbling an Underactuated Satellite," presented as AIAA 2020-1433 at AIAA Scitech 2020
Forum, Orlando, FL, 6-10 January 2020
∗Student, Department of Mechanical and Aerospace Engineering; kongon1004@gmail.com, AIAA Student Member
†Professor, Department of Aerospace Engineering; ilya@umich.edu, AIAA Associate Fellow.
‡Assistant Professor, Department of Aeronautics and Astronautics.
§Associate Professor, Department of Aeronautics and Astronautics.
¶Assistant Professor, Department of Aeronautics and Astronautics.
‖Professor, Department of Aeronautics and Astronautics.
𝑻= control torque vector
𝑇𝑠= prediction horizon
𝑼= optimal input matrix
𝑉= Lyapunov function
𝑣= dummy input
𝜆= Lagrange multiplier
𝝎= angular velocity vector
𝜔𝑒= argument of perigee
𝜇= Lagrange multiplier for equality constraint
𝜃= true anomaly
Subscripts
𝑖=𝑖-th time step on prediction horizon
∗= conditions on prediction horizon
I. Introduction
Various
actuators are used in spacecraft to achieve attitude stabilization, including thrusters, momentum wheels,
and control moment gyros. [
1
–
3
]. Small satellites, however, have stringent size, weight, and cost constraints,
which makes many actuator choices prohibitive. Consequently, magnetic torquers have commonly been applied to
spacecraft to attenuate angular rates [
4
,
5
]. Approaches for dealing with under-actuation due to magnetic control torques
dependency on the magnetic field and required high magnetic flux densities have been previously considered in [6, 7].
Generally speaking, control of a satellite that becomes under-actuated as a result of on-board failures has been a
recurrent theme in the literature, see e.g., [
8
,
9
] and references therein. Methods for controlling spacecraft with fewer
actuators than degrees of freedom are increasingly in demand due to the increased number of small satellite launches
[10].
Magnetic torquers have been extensively investigated for momentum management of spacecraft with momentum
wheels [
11
] and for nutation damping of spin satellites [
12
], momentum-biased [
13
], and dual-spin satellites [
14
].
Nonetheless, severely under-actuated small spacecraft that carry only a single-axis magnetic torquer have not been
previously treated.
This note considers the detumbling of a small spacecraft using only a single-axis magnetic torquer. Even with a
three-axis magnetic torquer, the spacecraft is under-actuated, while, in the case of only a single axis magnetic torquer,
the problem is considerably more demanding. Our note examines the feasibility of spacecraft attitude control with a
single-axis magnetic torquer and possible control methods that can be used.
2
Our specific contributions are as follows. We demonstrate, through analysis and simulations, that the conventional
B-dot algorithm for spacecraft detumbling fails with a single-axis magnetic torquer. Also, there has not been any
previous analysis of a satellite’s controllability and stabilizability with a single magnetic actuator. We discuss these
properties; this discussion motivates consideration of more advanced control approaches such as Nonlinear Model
Predictive Control (NMPC) [
15
,
16
]. Closed-loop simulation results with NMPC are reported, which illustrate the
potential of NMPC to perform spacecraft detumbling with a single-axis magnetic torquer. These developments, which
show the improved capability to detumble the spacecraft with NMPC as compared to the classical B-dot law in the case
of a single magnetic torquer, contribute to advancements in small satellite technology.
II. Spacecraft Rotational Dynamics
The body-fixed frame of a rigid spacecraft is assumed to be located at the center of mass and to be aligned with the
principal axes of inertia. The evolution of body frame components of spacecraft angular velocity vector is described by
the classical Euler’s equations [17]:
𝑱¤
𝝎+𝝎×𝑱𝝎 =𝑻(1)
A single-axis magnetic torquer interacts with the Earth’s local magnetic field and generates control torque according
to [18]:
𝑻=𝒎×𝑩(2)
Given a single-axis magnetic torquer with the coil along the
𝑥
-axis, the control torque with the single-axis
magnetorquer is written as
𝑻=[
0
,−𝐵𝑧𝑚𝑥, 𝐵𝑦𝑚𝑥]𝑇
. Thus, no torque is generated about the
𝑥
-axis, which makes
angular rate stabilization challenging. By aggregating the above equations, we obtain
¤𝜔𝑥
¤𝜔𝑦
¤𝜔𝑧
=
1
𝐽𝑥
(𝐽𝑦−𝐽𝑧)𝜔𝑦𝜔𝑧
1
𝐽𝑦
(𝐽𝑧−𝐽𝑥)𝜔𝑧𝜔𝑥−𝐵𝑧𝑚𝑥
𝐽𝑦
1
𝐽𝑧
(𝐽𝑥−𝐽𝑦)𝜔𝑥𝜔𝑦+𝐵𝑦𝑚𝑥
𝐽𝑧
(3)
where 𝑚𝑥is the control input.
III. Detumbling Control Law
A. B-dot Algorithm
The conventional approach to detumbling small satellites with magnetic torquers exploits the B-dot algorithm [
19
].
The B-dot algorithm’s principle is to add damping through control moments, which leads to a reduction in spacecraft
angular velocities. This section analyzes closed-loop stability with the B-dot law in the case of a single-axis magnetic
3
torquer. Considering the Euler’s equations for spacecraft dynamics given is Eq.
(1)
, define a Lyapunov function [
19
] as
𝑉(𝝎)=1
2𝝎𝑇𝑱𝝎 (4)
This Lyapunov function is positive everywhere except when
𝝎=
0( i.e., the equilibrium point). With the use of
Eq. (1), its time derivative along trajectories of the system is found as
¤
𝑉(𝝎)=𝝎𝑇𝑱¤
𝝎(5)
=𝝎𝑇(−𝝎×𝑱𝝎 +𝑻)(6)
=𝝎𝑇𝑻(7)
The conventional B-dot feedback law in [
20
,
21
] generates each axis magnetic dipole moment as follows. For
instance, for the 𝑥-axis,
𝑚𝑥=−𝑚max
¤
𝐵𝑥
|| ¤
𝐵𝑥|| (8)
where 𝑚max is the maximum magnitude of the magnetic dipole moment.
The time derivative of Earth’s magnetic field vector 𝑩with respect to an inertial frame is given by
𝐼¤
𝑩=¤
𝑩+𝝎×𝑩(9)
where the left superscript
𝐼
on
𝑩
indicates ”with respect to an inertial frame.” Assuming sufficiently large angular
velocity,
𝝎
, so that
𝐼¤
𝑩
is small enough in magnitude compared to
¤
𝑩
, where the latter is the derivative of Earth’s
magnetic field vector with respect to a body fixed frame, Eq. (9) can be approximated as [4, 21].
0≈¤
𝑩+𝝎×𝑩(10)
⇒¤
𝑩≈ −𝝎×𝑩(11)
Following [
22
], which treated the case of three single-axis magnetic torquers, suppose we proceed with closed-loop
stabilizability analysis by computing the time derivative of
𝑉
along closed-loop system trajectories. In the case of a
4
single-axis magnetic actuation, we obtain
¤
𝑉(𝝎)=−(𝝎×𝑩)𝑇𝒎(12)
=−𝑚max
|| ¤
𝑩|| (𝝎×𝑩)𝑇[𝜔𝑦𝐵𝑧−𝜔𝑧𝐵𝑦,0,0]𝑇(13)
=−𝑚max
|| ¤
𝑩|| (𝜔𝑦𝐵𝑧−𝜔𝑧𝐵𝑦)2(14)
It is clear that, although
¤
𝑉(𝝎) ≤
0, the expression for
¤
𝑉
does not depend on
𝜔𝑥
. Furthermore, since
𝑉(𝑡)=𝑉(𝝎(𝑡))
is a non-increasing function of
𝑡
,
𝝎
is bounded. Moreover
¥
𝑉
is a continuous function of
𝑩
and
𝝎
, which are bounded.
Hence,
¤
𝑉(𝑡)
is uniformly continuous in time [
23
]. Therefore, by Barbalat’s lemma, we conclude
lim𝑡→∞ ¤
𝑉(𝑡)=
0,
which indicates that in the limit as
𝑡→ ∞
either
𝜔𝑦𝐵𝑧=𝜔𝑧𝐵𝑦=
0or
𝜔𝑦=𝜔𝑧=
0. In the either case, since
𝜔𝑥
can be
arbitrary, B-dot algorithm appears to be incapable of detumbling a satellite with a single-axis magnetic actuation. This
is confirmed by our subsequent numerical simulations.
B. Controllability Analysis
This section discusses the controllability properties of the satellite angular velocity dynamics with the single-axis
magnetic actuation. Note that both three and two-axis magnetically actuated satellites have been shown to be controllable
[
24
,
25
]. The single-axis magnetic actuation case is more challenging since there is only a single control input; this case
has not previously been addressed in the literature.
For the case of single-axis magnetic actuation, local weak controllability in the sense of [
26
,
27
] can be demonstrated.
The weak local controllability is necessary for local controllability. It implies that the set of reachable states at a given
time from a given state starting at another given time instant contains an open neighborhood of the state space. Clearly,
weak local controllability is necessary but not sufficient for local controllability.
Note that equations of motion (3) can be written as
¤
𝝎=𝑓0(𝝎, 𝑡) + 𝑓1(𝝎, 𝑡 )𝑚𝑥(15)
=
1
𝐽𝑥
(𝐽𝑦−𝐽𝑧)𝜔𝑦𝜔𝑧
1
𝐽𝑦
(𝐽𝑧−𝐽𝑥)𝜔𝑧𝜔𝑥
1
𝐽𝑧
(𝐽𝑥−𝐽𝑦)𝜔𝑥𝜔𝑦
+
0
−𝐵𝑧
𝐽𝑦
𝐵𝑦
𝐽𝑧
𝑚𝑥.(16)
The system is time-varying as
𝐵𝑦
and
𝐵𝑧
depend on the spacecraft position in orbit and hence on time. The necessary
and sufficient conditions for local weak controllability of a time-varying nonlinear system can be obtained by extending
the state vector with the time,
𝑡
, as an extra state with the dynamics
¤
𝑡=
1, and analyzing the resulting autonomous
system. This leads to conditions such as Theorem 4 in [28] which we adopt here.
5
Define the operators h𝜉 , 𝜂iand [𝜉, 𝜂]for two time-varying vector fields 𝜉an 𝜂as
h𝜉, 𝜂i ≡ [𝜉 , 𝜂] − 𝜕 𝜉
𝜕𝑡 ,(17)
[𝜉, 𝜂]=𝜕𝜂
𝜕𝜔 𝜉−𝜕𝜉
𝜕𝜔 𝜂. (18)
Note that
[𝜉, 𝜂]
is the conventional Lie Bracket. The controllability distribution
4
can now be defined for our
time-varying nonlinear system based on Algorithm 1.
Algorithm 1: Controllability distribution for time-variant nonlinear systems
Set 40=span{𝑓1}and 𝑘=0;
while dim(4𝑘⊕ h4𝑘, 𝑓0i ⊕ [4𝑘, 𝑓1] ) >dim(4𝑘)do
Set 4𝑘+1=4𝑘⊕ h4𝑘, 𝑓0i ⊕ [4𝑘, 𝑓1];
𝑘=𝑘+1;
end
⊕sums up the span of two vector fields.
The necessary and sufficient conditions for local weak controllability in [
28
] lead to the following result: Algorithm 1
converges in at most 2steps for the system (15) and
42
is nonsingular and has rank 3at a given
𝝎0
and
𝑡0
if and only if
the system is locally weakly controllable from (𝝎0,𝑡0).
By examining the form of
42
(see calculations in the Appendix), we observe that
𝐽𝑦≠𝐽𝑧
(unequal moments of
inertia about the two principal axes which are orthogonal to the axis along which the magnetic actuator is aligned) is a
necessary condition for weak local controllability (and hence for local controllability). This is also apparent from (15)
as the angular velocity component about 𝑥-axis becomes decoupled from the rest of the dynamics if 𝐽𝑦=𝐽𝑧.
Furthermore, the numerical evaluation of
42
along the orbits used in our NMPC simulations with IGRF model (36)
for 𝐵𝑦and 𝐵𝑧confirms that the rank of 42is equal to 3and hence the system is locally weakly controllable.
We note that we cannot establish a stronger property of (small-time) local controllability of the satellite with
a single-axis magnetic actuation with the above analysis. While it appears to hold in our numerical simulations,
such property is much harder to demonstrate and is left to future research. Nevertheless, conditions for local weak
controllability are necessary for local controllability and hence are useful.
C. Stabilizability Analysis
Existing approaches [
17
,
24
,
29
,
30
] to stabilizing the spacecraft with magnetic actuation typically rely on the
application of the theory of averaging [
31
]. Following this route in the case of single-axis magnetic actuation encounters
technical difficulties as we now illustrate.
6
Consider the equations of motion given by Eq. (15) and suppose a control law of the form,
𝑚𝑥=𝜖2¯𝑚𝑥(𝝎
𝜖, 𝑡)(19)
has been specified, where 𝜖 > 0is a small parameter. Let
𝒉=𝝎
𝜖=ℎ𝑥ℎ𝑦ℎ𝑧T
,
then
¤
ℎ𝑥=𝜖1
𝐽𝑥
(𝐽𝑦−𝐽𝑧)ℎ𝑦ℎ𝑧
¤
ℎ𝑦=𝜖1
𝐽𝑦
(𝐽𝑧−𝐽𝑥)ℎ𝑧ℎ𝑥−𝜖
𝐽𝑦
𝐵𝑧(𝑡)¯𝑚𝑥(𝒉, 𝑡)(20)
¤
ℎ𝑧=𝜖1
𝐽𝑧
(𝐽𝑥−𝐽𝑦)ℎ𝑥ℎ𝑦+𝜖
𝐽𝑧
𝐵𝑦(𝑡)¯𝑚𝑥(𝒉, 𝑡)
is in the form to which the theory of averaging [31] can be applied. Letting
𝑢𝑦(𝒉)=−𝜖
𝐽𝑦
lim
𝑇→∞
1
𝑇𝑇
0
𝐵𝑧(𝑡)¯𝑚𝑥(𝒉, 𝑡)𝑑𝑡
𝑢𝑧(𝒉)=𝜖
𝐽𝑧
lim
𝑇→∞
1
𝑇𝑇
0
𝐵𝑦(𝑡)¯𝑚𝑥(𝒉, 𝑡)𝑑𝑡
and assuming that the limits exist, the averaged version of Eq. (20) has the form,
¤
¯
ℎ𝑥=𝜖1
𝐽𝑥
(𝐽𝑦−𝐽𝑧)¯
ℎ𝑦¯
ℎ𝑧
¤
¯
ℎ𝑦=𝜖1
𝐽𝑦
(𝐽𝑧−𝐽𝑥)¯
ℎ𝑧¯
ℎ𝑥+𝑢𝑦(¯
𝒉)(21)
¤
¯
ℎ𝑧=𝜖1
𝐽𝑧
(𝐽𝑥−𝐽𝑦)¯
ℎ𝑥¯
ℎ𝑦+𝑢𝑧(¯
𝒉)
By the straightforward application of Brockett’s necessary condition [
32
], the averaged system given by Eq.
(21)
is
not smoothly or even continuously stabilizable, i.e., a stabilizing time-invariant control law given by
𝑢𝑦(¯
𝒉)
,
𝑢𝑧(¯
𝒉)
, if
exists, would have to be discontinuous. This conclusion is consistent with the interpretation of the averaged system
dynamics as similar to a rigid body controlled by two control torques; such a system is known to be not smoothly or even
continuously stabilizable by time-invariant feedback laws. Consequently,
¯𝑚𝑥(𝝎
𝜖, 𝑡)
would have to be discontinuous as a
function of
𝝎
. Unfortunately, the theory of averaging [
31
,
33
] assumes smoothness (twice continuous differentiability)
in [31] of the right hand side of ordinary differential equations being averaged.
7
Hence, there is a complication in using the classical averaging theory to develop conventional control laws for the
case of single-axis magnetic actuation. On the other hand, NMPC, if suitably formulated, is able to stabilize systems
that are not smoothly or even continuously stabilizable, including underactuated spacecraft [34].
D. NMPC Formulation
This section formulates an NMPC approach to detumbling the satellite with the nonlinear dynamics represented by
Eq.(3) based on the following receding horizon optimal control problem,
minimize 𝐽cost =1
2𝝎T(𝑡+𝑇𝑠)𝑸t𝝎(𝑡+𝑇𝑠) + 𝑡+𝑇𝑠
𝑡{1
2(𝝎T(𝜏)𝑸𝝎 (𝜏) + 𝑅1𝑚2
𝑥(𝜏)) − 𝑅2𝑣(𝜏) }𝑑𝜏
subject to 𝑱¤
𝝎+𝝎×𝑱𝝎 =𝑻
𝑻=𝒎×𝑩,where 𝒎=[𝑚𝑥,0,0]T
𝑚2
𝑥+𝑣2−𝑚2
max =0
(22)
where
𝑡
is the current time instant,
𝑇𝑠
is the prediction horizon,
𝑸
,
𝑸t
,
𝑅1
, and
𝑅2
are positive-definite weight matrices,
and
𝑸t
is terminal cost. The auxiliary input,
𝑣
, is introduced following [
35
] to enforce the control constraints by recasting
them as equality constraints in Eq.
(22)
. The negative sign preceding
𝑅2
in the cost function being minimized promotes
keeping
𝑣
positive and control constraints strictly satisfied. This receding horizon optimal control problem is chosen as
it is synergistic with the continuation/generalized minimal residual method (C/GMRES) method [
36
]. Reference [
37
]
provides a comparison of different strategies to handle inequality constraints in such a setting.
E. NMPC with C/GMRES Algorithm
The C/GMRES method [
36
] is applied to design NMPC based on Eq.
(22)
. As C/GMRES method has small
computational footprint, its use is advantageous in small satellites with limited computational and electric power.
Following C/GMRES method, the problem is first discretized as follows:
𝝎∗
𝑖+1(𝑡)=𝝎∗
𝑖(𝑡) + 𝑓(𝝎∗
𝑖(𝑡),𝒖∗
𝑖(𝑡))Δ𝜏(23)
𝝎∗
0(𝑡)=𝝎(𝑡)(24)
𝐶(𝝎∗
𝑖(𝑡),𝒖∗
𝑖(𝑡)) =0(25)
𝐽cost =𝜓(𝝎∗
𝑁(𝑡)) +
𝑁−1
𝑖=0
𝐿(𝝎∗
𝑖(𝑡),𝒖∗
𝑖(𝑡))Δ𝜏(26)
where
𝑓(𝝎,𝒖)
is the right hand side of equations of motion in Eq.
(22)
,
𝐶(𝝎,𝒖)
is the equality constraint in Eq.
(22)
,
𝐿(𝝎,𝒖)=1
2(𝝎𝑇𝑸𝝎 +𝑅1𝑚2
𝑥) − 𝑅2𝑣,
and
Δ𝜏=𝑇𝑠/𝑁
. Setting the initial state of the discretized problem to the current
8
angular velocity vector as
𝝎∗
0(𝑡)=𝝎(𝑡)
, a sequence of control inputs
{𝒖∗
𝑖(𝑡)} 𝑁−1
𝑖=0
is found at each time instant
𝑡
; then
the control given to the system is based on the first element of this sequence and is defined as 𝒖(𝑡)=𝒖∗
0(𝑡).
The solution of the discretized problem is based on introducing the Hamiltonian, 𝐻, as
𝐻(𝝎,𝝀,𝒖,𝝁)=𝐿(𝝎,𝒖) + 𝝀𝑇𝑓(𝝎,𝒖) + 𝝁𝑇𝐶(𝝎,𝒖)(27)
where
𝝀
is the vector of co-states and
𝝁
is the Lagrange multiplier associated with the equality constraint. The
first-order necessary conditions for optimality dictate [
38
] that
{𝒖∗
𝑖(𝑡)} 𝑁−1
𝑖=0
,
{𝝁∗
𝑖(𝑡)} 𝑁−1
𝑖=0
,
{𝝀∗
𝑖(𝑡)} 𝑁−1
𝑖=0
, satisfy the
following conditions:
𝐻𝒖(𝝎∗
𝑖(𝑡),𝝀∗
𝑖+1(𝑡),𝒖∗
𝑖(𝑡),𝝁∗
𝑖(𝑡)) =0(28)
𝝀∗
𝑖(𝑡)=𝝀∗
𝑖+1(𝑡) + 𝐻𝑇
𝝎(𝝎∗
𝑖(𝑡),𝝀∗
𝑖+1(𝑡),𝒖∗
𝑖(𝑡),𝝁∗
𝑖(𝑡))Δ𝜏(29)
𝝀∗
𝑁(𝑡)=𝜓𝑇
𝝎(𝝎∗
𝑵(𝑡)) (30)
To determine
{𝒖∗
𝑖(𝑡)} 𝑁−1
𝑖=0
and
{𝝁∗
𝑖(𝑡)} 𝑁−1
𝑖=0
, which satisfy Eqs.(23–25) and (28–30), we define a vector of the inputs
and multipliers in Eq. (31) as
𝑼(𝑡)=[𝑚𝑥∗
0(𝑡), 𝑣∗
0(𝑡), 𝜇∗
0(𝑡), . . . , 𝑚 𝑥∗
𝑁−1(𝑡), 𝑣∗
𝑁−1(𝑡), 𝜇∗
𝑁−1(𝑡)]𝑇(31)
This vector has to satisfy the equation,
𝑭(𝑼(𝑡),𝝎(𝑡), 𝑡) ≡
(𝜕𝐻
𝜕𝑢 )𝑇(𝝎∗
0(𝑡), 𝑢∗
0(𝑡), 𝜆∗
1(𝑡), 𝑡)
𝑚2
𝑥0+𝑣2
0−𝑚2
max =0
.
.
.
(𝜕𝐻
𝜕𝑢 )𝑇(𝝎∗
𝑵−1(𝑡), 𝑢∗
𝑁−1(𝑡), 𝜆∗
𝑁(𝑡), 𝑡 +𝑇)
𝑚2
𝑥 𝑁 −1+𝑣2
𝑁−1−𝑚2
max
=0,(32)
where 𝐻=1
2(𝝎𝑇𝑸𝝎 +𝑅1𝑚2
𝑥) − 𝑅2𝑣+𝜆𝑥{1
𝐽𝑥
(𝐽𝑦−𝐽𝑧)𝜔𝑦𝜔𝑧} + 𝜆𝑦{1
𝐽𝑦
(𝐽𝑧−𝐽𝑥)𝜔𝑧𝜔𝑥−𝐵𝑧𝑚𝑥}
+𝜆𝑧{1
𝐽𝑧
(𝐽𝑥−𝐽𝑦)𝜔𝑥𝜔𝑦+𝐵𝑦𝑚𝑥} + 𝜇{𝑚2
𝑥+𝑣2−𝑚2
max}
(33)
In C/GMRES [36, 39], Eq. (32), which has to haold at each time instant, 𝑡, is replaced by a stabilized version,
𝑑
𝑑𝑡 𝑭(𝑼(𝑡),𝝎(𝑡), 𝑡 )=−𝜁𝑭(𝑼(𝑡),𝝎(𝑡), 𝑡) (𝜁 > 0)(34)
9
and then by
𝜕𝑭
𝜕𝑼
¤
𝑼(𝑡)=−𝜁𝑭−𝜕𝑭
𝜕𝑼
¤
𝝎(𝑡) − 𝜕𝑭
𝜕𝑡 (35)
where arguments are omitted.
Finally, ¤
𝑈can be determined from Eq. (35) with C/GMRES resulting in a form of a predictor-corrector strategy.
IV. Simulation Results
This section presents simulation results of both the B-dot algorithm and NMPC for a spacecraft in a sun-synchronous
orbit with orbital elements given in Table 1.
A. Orbital Elements
Table 1 Six elements of Aeolus (sun synchronous orbit)
Semi-major axis 6691.6 [km]
Eccentricity 0.00046440
Inclination 96.700[deg]
Right Ascension of Ascending Node 100.90 [deg]
Argument of perigee 119.70 [deg]
Mean anomaly 240.49 [deg]
B. Earth’s Magnetic Field Model
We employ International Geomagnetic Reference Field (IGRF) [
40
] as an Earth’s magnetic model in our simulations.
However, to reduce the onboard computational complexity, NMPC uses a simpler magnetic dipole model described in
Eq. (36) [41].
𝐵0𝑥
𝐵0𝑦
𝐵0𝑧
=𝐷𝑚
3
2sin 𝑖sin 2𝜂
−3
2sin 𝑖cos 2𝜂−1
3
−cos 𝑖
(36)
where
𝜂=𝜃+𝜔𝑒
,
𝐷𝑚=−𝑀𝑒
𝑟3
,
𝑀𝑒=
8
.
1
×
10
25
[gauss
·
cm
3
],
𝑟
is a distance between the satellite and the center of the
Earth,
𝜃
is true anomaly, and
𝜔𝑒
is argument of perigee. Figure 1 shows that there is a slight discrepancy between the
two models on the sun-synchronous orbit, which, as we will see from the simulation results, will not preclude NMPC
controller from achieving detumbling.
10
Fig. 1 Magnetic field in dipole and IGRF on the sun-synchronous orbit.
C. Comparison of B-dot and NMPC on Asymmetric Satellite
This section demonstrates NMPC’s advantages over the B-dot algorithm, using a general satellite model whose
moments of inertia are given in Table 2. The maximum magnetic moment is set to 1.0 [A ·m2].
Table 2 Moment of inertia of the asymmetric satellite
Moments of inertia 𝐽𝑥𝐽𝑦𝐽𝑧
Value [kg ·m2] 0.020 0.030 0.040
1. The B-dot algorithm
As shown in Eq.8, the B-dot law finds control inputs depending the Earth’s magnetic field. Following the common
practice, to avoid the B-dot law generating unnecessary control inputs, its implementation is based on
𝑚𝑥=
0,if ¤
𝐵𝑥<10−7
−𝑚max
¤
𝐵𝑥
| | ¤
𝐵𝑥| | ,otherwise
(37)
11
2. NMPC
We here demonstrate NMPC’s capability of detumbling the spacecraft. Simulations are conducted with the NMPC
parameters listed in Table 3, which were determined by trial and error.
Table 3 NMPC properties for the asymmetric satellite
𝑇𝑠𝑄 𝑄t𝑅1𝑅2𝑁Δ𝜏
10 [sec] diag([104,102,5×10]) diag([104,102,5×10]) 10−110−110 1.0 [sec]
where
𝑇𝑠
is prediction horizon,
𝑄
,
𝑄t𝑅1
, and
𝑅2
are weight matrices,
𝑁
is discretized step number along prediction
horizon, and Δ𝜏=𝑇𝑠/𝑁.
The initial conditions for the four study cases for which simulation results are reported below are chosen randomly
with angular velocity components between -3.0 and 3.0 [deg/s]. The initial angular velocities and the results are all
given in Table 4. These four simulations are representatives of a larger number of simulation case studies that we have
performed.
Table 4 Initial conditions and results
𝜔𝑥[deg/s] 𝜔𝑦[deg/s] 𝜔𝑧[deg/s] B-dot detumbled? NMPC detumbled?
Case 1 2.429286 2.878490 -0.366780 No Yes
Case 2 -1.576299 -0.246907 2.778531 No Yes
Case 3 0.047150 -2.486905 -1.425107 Yes Yes
Case 4 -0.626909 -0.795380 2.927892 No No
Figure 2 to 5 present the simulation results from the four case studies. When the magnitude of all the angular
velocities are less than 0.10 [deg/s], the simulations are set to be terminated in both B-dot and NMPC simulations. The
simulations are run for the maximum time of 150 [min].
12
Figure 3 also reports an example where the B-dot law does not achieve all axes detumbling, but the NMPC algorithm
does. Note that in the beginning of the trajectory, NMPC increases
𝜔𝑧
to be able to reduce
𝜔𝑥
, which is a challenging
variable to control.
Fig. 4 Case 3: time history of angular velocities.
Figure 4 is a case where both the B-dot method and the NMPC approach achieve detumbling. This requires about
150 [min] for the B-dot algorithm, while NMPC is able to achieve this within a much shorter time period.
Finally, Fig. 5 showcases a plot where neither the B-dot nor NMPC are able to detumble the spacecraft within the
allocated time of 150 [min]. In the case of the B-dot algorithm,
𝜔𝑥
persists at a nonzero value. NMPC, on the other
hand, is able to gradually attenuate all angular velocity components but is not able to fully detumble the spacecraft
within the given time.
14
Fig. 5 Case 4: time history of angular velocities.
Fig. 6 Control input comparison.
Figs. 6 shows the time histories of control inputs in all the four cases. As can be seen, the NMPC controller requires
much smaller control inputs. This can be advantageous in terms of reduced energy consumption and reduced electrical
15
disturbance.
V. Conclusions
The satellite’s detumbling with only on a single-axis magnetic actuator is a challenging problem, in particular,
requiring a different approach to stabilization than for three-axis and two-axis magnetic actuation systems. The necessary
conditions for the controllability of spacecraft angular velocities involve: (1) spacecraft not having equal moments of
inertia about the two principal axes that are orthogonal to the axis along which the magnetic actuator is aligned and (2)
satisfying the specific rank controllability conditions derived in the note. The latter have been shown to hold numerically
for the spacecraft and the orbit considered in the simulations. The classical B-dot law appears to be incapable of
eliminating spacecraft rotational motion in the simulations, which has also been predicted from the theoretical analysis.
The Nonlinear Model Predictive Control (NMPC) strategy based on the continuation/generalized minimal residual
(C/GMRES) method has been shown to achieve detumbling within the allocated time through simulations in most
cases. The possibility of detumbling spacecraft with only the single-axis magnetic coil opens the possibility for small
spacecraft missions with stringent cost and packaging constraints.
Appendix
Below are the controllability distributions in Algorithm 1:
40=span{𝑓1}(38)
=span{[0,−𝐵𝑧
𝐽𝑦
,𝐵𝑦
𝐽𝑧
]𝑇}(39)
41=40⊕ h40, 𝑓0i ⊕ [40, 𝑓1](40)
=span{𝑓1} ⊕
1
𝐽𝑥𝐽𝑧
𝐵𝑦𝜔𝑦(𝐽𝑦−𝐽𝑧) − 1
𝐽𝑥𝐽𝑦
𝐵𝑧𝜔𝑧(𝐽𝑦−𝐽𝑧)
1
𝐽𝑦
𝜕𝐵𝑧
𝜕𝑡 −1
𝐽𝑦𝐽𝑧
𝐵𝑦𝜔𝑥(𝐽𝑥−𝐽𝑧)
−1
𝐽𝑧
𝜕𝐵𝑦
𝜕𝑡 −1
𝐽𝑦𝐽𝑧
𝐵𝑧𝜔𝑥(𝐽𝑥−𝐽𝑦)
(41)
=span{𝑓1, 𝑔1}(42)
16
where 𝑔1is the second term in Eq. (41).
42=41⊕ h41, 𝑓0i ⊕ [41, 𝑓1](43)
=span{𝑓1, 𝑔1} ⊕ span{h 𝑓1, 𝑓0i,h𝑔1, 𝑓0i} ⊕ span{[ 𝑓1, 𝑓1],[𝑔1, 𝑓1]} (44)
=span{𝑓1, 𝑔1} ⊕ span{𝑔1, 𝑔2} ⊕ span{0, 𝑔3}(45)
=span{𝑓1, 𝑔1, 𝑔2, 𝑔3}(46)
where 𝑔2=h𝑔1, 𝑓0iand 𝑔3=[𝑔1, 𝑓1]. The explicit calculations of 𝑔2and 𝑔3give:
𝑔2=
−2
𝐽𝑥𝐽𝑦𝐽𝑧
(𝐽𝑦−𝐽𝑧)(𝐽𝑦
𝜕𝐵𝑦
𝜕𝑡 𝜔𝑦−𝐽𝑧𝜕𝐵𝑧
𝜕𝑡 𝜔𝑧)
−1
𝐽𝑥𝐽2
𝑦𝐽𝑧
(𝐽𝑦(𝐵𝑧𝐽2
𝑥𝜔2
𝑥−2𝜕𝐵𝑦
𝜕𝑡 𝐽2
𝑥𝜔𝑥−𝐵𝑧𝐽𝑥𝐽𝑧𝜔2
𝑥+2𝜕𝐵𝑦
𝜕𝑡 𝐽𝑥𝐽𝑧𝜔𝑥−𝐵𝑧𝐽𝑥𝐽𝑧𝜔2
𝑧
+𝜕2𝐵𝑧
𝜕𝑡2𝐽𝑥𝐽𝑧+𝐵𝑧𝐽2
𝑧𝜔2
𝑧) − 𝐵𝑧𝐽3
𝑥𝜔2
𝑥−𝐵𝑧𝐽3
𝑧𝜔2
𝑧+𝐵𝑧𝐽2
𝑥𝐽𝑧𝜔2
𝑥+𝐵𝑧𝐽𝑥𝐽2
𝑧𝜔2
𝑧)
1
𝐽𝑥𝐽𝑦𝐽2
𝑧
(𝐽𝑧(𝐵𝑦𝐽2
𝑥𝜔2
𝑥+2𝜕𝐵𝑧
𝜕𝑡 𝐽2
𝑥𝜔𝑥−𝐵𝑦𝐽𝑥𝐽𝑦𝜔2
𝑥−2𝜕𝐵𝑧
𝜕𝑡 𝐽𝑥𝐽𝑦𝜔𝑥−𝐵𝑦𝐽𝑥𝐽𝑦𝜔2
𝑦
+𝜕2𝐵𝑦
𝜕𝑡2𝐽𝑥𝐽𝑦+𝐵𝑦𝐽2
𝑦𝜔2
𝑦) − 𝐵𝑦𝐽3
𝑥𝜔2
𝑥−𝐵𝑦𝐽3
𝑦𝜔2
𝑦+𝐵𝑦𝐽2
𝑥𝐽𝑦𝜔2
𝑥+𝐵𝑦𝐽𝑥𝐽2
𝑦𝜔2
𝑦)
(47)
(48)
𝑔3=
2
𝐽𝑥𝐽𝑦𝐽𝑧
𝐵𝑦𝐵𝑧(𝐽𝑦−𝐽𝑧)
0
0
(49)
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