Robust finite-time control allocation in spacecraft attitude stabilization under actuator misalignment

Abstract

A novel combination of finite time control and control allocation with uncertain configuration matrix due to actuator misalignment is investigated for attitude stabilization of a rigid spacecraft. Finite time controller using nonsingular terminal sliding mode technique is firstly designed as virtual control of control allocator to produce the three axis torques, and can guarantee finite time reachability of given attitude motion of spacecraft in the presence of external disturbances. The convergences of this feedback controller for the resulting closed loop systems are also proven theoretically. Then, under the condition of uncertainty included in the configuration matrix due to actuator misalignment, a robust least squares-based control allocation is employed to deal with the problem of distributing the three axis torques over the available actuators under redundancy, in which the focus of this control allocation is to find the optimal control vector of actuator by minimizing the worst-case residual, under the condition of the uncertainty included in actuator configuration matrix and control constraints like saturation. Simulation results using the orbiting spacecraft model show good performance under external disturbances and even uncertain configuration matrix, which validates the effectiveness and feasibility of the proposed scheme.

Full text

Nonlinear Dyn (2013) 73:53–71
DOI 10.1007/s11071-013-0766-2
ORIGINAL PAPER
Robust finite-time control allocation in spacecraft attitude
stabilization under actuator misalignment
Qinglei Hu ·Bo Li ·Aihua Zhang
Received: 25 June 2012 / Accepted: 8 January 2013 / Published online: 30 January 2013
© Springer Science+Business Media Dordrecht 2013
Abstract A novel combination of finite time con-
trol and control allocation with uncertain configura-
tion matrix due to actuator misalignment is investi-
gated for attitude stabilization of a rigid spacecraft. Fi-
nite time controller using nonsingular terminal sliding
mode technique is firstly designed as virtual control of
control allocator to produce the three axis torques, and
can guarantee finite time reachability of given attitude
motion of spacecraft in the presence of external distur-
bances. The convergences of this feedback controller
for the resulting closed loop systems are also proven
theoretically. Then, under the condition of uncertainty
included in the configuration matrix due to actuator
misalignment, a robust least squares-based control al-
location is employed to deal with the problem of dis-
tributing the three axis torques over the available actu-
ators under redundancy, in which the focus of this con-
trol allocation is to find the optimal control vector of
actuator by minimizing the worst-case residual, under
the condition of the uncertainty included in actuator
configuration matrix and control constraints like satu-
ration. Simulation results using the orbiting spacecraft
model show good performance under external distur-
bances and even uncertain configuration matrix, which
validates the effectiveness and feasibility of the pro-
posed scheme.
Q. Hu ()·B. Li ·A. Zhang
Department of Control Science and Engineering,
Harbin Institute of Technology, Harbin 150001, China
e-mail: huqinglei@hit.edu.cn
Keywords Attitude control ·Finite time ·Control
allocation ·Robust least squares ·Actuator
misalignment
1 Introduction
Attitude control of a spacecraft in space in general is a
process of re-orienting it to a desired attitude or orien-
tation, and plays an important role in achieving space-
craft operational services, such as remote sensing,
communication and variety of space-related research.
Some of these orbiting operations require achieving
these maneuvers within the finite time. However, for
such finite-time and fast rotational maneuvers, the dy-
namics is strongly nonlinear in nature and also af-
fected by various disturbances from the environment
that influence the mission objectives significantly. In
addition, the actuators are not able to provide any
requested joint torque because the available actuator
torque amplitude is limited in actual spacecraft. And
also, the actuator uncertainties due to misalignment
during installation increase the complexity further. All
these in a realistic environment cause a considerable
difficulty in the design of attitude control system for
meeting high precision pointing requirement and de-
sired control performance during the missions, espe-
cially when all these issues are treated simultaneously.
Spacecraft attitude control problem has been stud-
ied extensively in the existing literature. In Refs.
[1,2], the authors show that there exist a simple

54 Q. Hu et al.
linear asymptotically stabilizing proportional-plus-
derivative (PD) control law for the attitude motion of
a rigid body. However, the effects of external distur-
bances are not investigated on the attitude regulation.
To eliminate the offsets arising out of disturbances, the
extended form of PD is presented in Ref. [3] by includ-
ing a nonlinear term such that the global asymptotic
stability is possibly guaranteed. In the sense of optimal
control, an optimal attitude control law for the atti-
tude control problem is presented in Refs. [4,5]. Lya-
punov analysis-based adaptive attitude tracking con-
trol schemes is also presented to compensate for the
unknown rigid body inertia matrix [6–8]. Yet, the im-
plementation of the controller is more complicated due
to the existence of an adaptive estimator. To cope with
both model uncertainties and external disturbances,
variable structure control (VSC) was employed with
different attitude representations to solve the robust at-
titude control problem [9–12]. However, actually, one
problem is that technically speaking, these results con-
sidered in these literatures solve the attitude stabiliza-
tion by implementing the asymptotic stability analysis,
which implies that the system trajectories converge to
the equilibrium with infinite settling time, which is dif-
ficult to implement in practice. For removing this dis-
advantage, finite-time control is an alternative way to
get a fast convergence rate to the origin beside robust
disturbance attenuation. In a related work [13], the au-
thors use the terminal sliding mode control approach
to achieve the attitude finite time convergence that was
found to be effective in simulation studies. However,
the control law contains singular term at crossing zero,
which causes instability of the closed-loop system. A
modified case is presented in a recent paper [14]to
solve this singularity. A similar problem was also con-
sidered by the works [15,16] in the presence of dis-
turbance to converge into a small region of the origin
in finite time.
However, these results have been derived under
the implicit assumption that the actuators are able
to provide any requested joint torque, and also the
torque axis directions and/or input scaling of the actu-
ators (such as gas jets, reaction fly-wheels) are exactly
known. This assumption is rarely satisfied in prac-
tice because of available actuator control power lim-
itation and possible misalignment of the actuators dur-
ing installation. Recognizing these difficulties, several
solutions that take into account actuator constraints
have been extensively studied [17–22]. More specif-
ically, in Refs. [17,18], using VSC technique, the au-
thors formulated robust sliding mode controllers for
global asymptotic stabilization of spacecraft in the
presence of control input saturation and disturbances.
The smooth attitude stabilizing control containing hy-
perbolic tangent functions was also discussed in Ref.
[19]. Ali et al. [20] presented a method to design
a bounded control for spacecraft attitude maneuver
with backstepping control. Ruiter [21] investigated an
adaptive attitude tracking control for a rigid spacecraft
with linearly parameterized disturbances. Recently,
Su and Zheng [22] presented a method to design a
bounded control for spacecraft attitude maneuver with
simple saturated PD structure with disturbance-free
case. For the case of possible actuator misalignment,
in Ref. [23], the authors developed an adaptive con-
trol law to accomplish attitude maneuver in the pres-
ence of relatively small gimbals’ alignment error of
variable speed control moment gyros. In another re-
lated work [24], a nonlinear model reference adaptive
control scheme was presented in the presence of align-
ment errors up to fifteen degrees. An extended Kalman
filter was used to develop methods for on-orbit actua-
tor alignment calibration [25], but no uncertain iner-
tia properties were taken into account. A novel adap-
tive tracking controller was synthesized for Hamilto-
nian systems [26], and that control law is successfully
applied to a spacecraft with both the inertia and the ac-
tuator uncertainties. Another recent paper in Ref. [27]
proposed an adaptive control approach for satellite for-
mation flying, in which backstepping technique is used
to synthesize a controller to handle thrust magnitude
error and misalignment.
Control allocation is able to deal with distributing
the total control demand among the individual actu-
ators while accounting for their constraints under re-
dundant actuators [28]. Loosely speaking, it consists
of using possibly desired control laws that specify only
the total control effort that has to be made to com-
pensate the system, and separately, the one of suit-
ably distributing the desired total control command
over the available actuators, in which the actuator con-
straints like saturation can be taken into account ex-
plicitly. That is to say, if one actuator saturates, and
fails to produce its nominal control effect, another ac-
tuator may be used to make up the difference. This
way, the control capabilities of the actuator suite are
fully exploited before the closed loop performance is
degraded. The general approaches of control alloca-
tion have been deeply investigated in the last decade,

Robust finite-time control allocation in spacecraft attitude stabilization under actuator misalignment 55
and include: daisy chaining [29], linear or nonlinear
programming based on optimization algorithms [30],
direct allocation [31,32], dynamic control allocation
[33,34], etc. Most or partial previous works study
linear control allocation by programming algorithms,
which can be iteratively conducted to minimize the er-
ror between the commands produced by virtual control
law and the moments produced by practical actuator
combinations.
In this paper, an attempt is made to provide a simple
and robust attitude control strategy for spacecraft with
finite time convergence in the presence of external
disturbances, actuator misalignment and even control
constraints. This proposed control law is a novel com-
bination the nonsingular terminal sliding mode with
robust least squares control allocation. For the former
one, it can achieve the desired rotation maneuvering in
finite time and be robust to the external disturbances.
For the latter one, it can deal with the problem of dis-
tributing the total former command into the individual
actuator properly, in which the focus of this control al-
location is to find the optimal control vector of actuator
by minimizing the worst-case residual, under the con-
dition of the uncertainty included in actuator config-
uration matrix and control constraints like saturation.
A key feature of the designed controller ensures both
attitude and velocity convergence in finite time with
simple design procedures and inexpensive online com-
putations such that it is user/designer friendly, which
is of great interest for aerospace industry for real-time
implementation especially when onboard space and
computing power are limited for instance. The ben-
efits of the proposed control method are analytically
authenticated and also validated via simulation study.
The paper is organized as follows. The next section
states spacecraft modeling and control problems. At-
titude control laws are derived in Sect. 3.Nextthe
results of numerical simulations demonstrate various
features of the proposed control law. Finally, the paper
is completed with some concluding comments.
2 Spacecraft modeling and problem formulation
2.1 Spacecraft attitude dynamics
Consider a rigid space system described by the follow-
ing attitude kinematics and dynamics equations [35]:
˙q0
˙q=1
2−qT
q0I+q×ω(1)
J˙ω=−ω×Jω+u(t ) +d(t) (2)
where q0and qare the scalar and vector compo-
nents of the unit quaternion, respectively, with q=
[q1q2q3]T∈R3, satisfying the constraint qTq+q2
0=
1; ω∈R3is the angular velocity of a body-fixed refer-
ence frame of a spacecraft with respective to an inertial
reference frame expressed in the body-fixed reference
frame, Irepresents the identity matrix with proper di-
mensions, and q×(or ω×) denotes a skew-symmetric
matrix, more precisely,
q×=⎡
⎣
0−q3q2
q30−q1
−q2q10⎤
⎦
ω×=⎡
⎣
0−ω3ω2
ω30−ω1
−ω2ω10⎤
⎦
(3)
J∈R3×3is the total inertia matrix of the space-
craft, u(t) =[u1u2u3]T∈R3denotes the combined
control torque produced by the actuators, and d(t) =
[d1d2d3]T∈R3denotes the external disturbance
torque from the environment, which is assumed to be
unknown but bounded.
2.2 Actuator configuration with misalignment
For orbiting spacecraft, loosely speaking, they have
more than three actuators aligned with the spacecraft
body axes. A common configuration with four reac-
tion wheels is shown in Fig. 1, in which three reaction
wheels’ (such as reaction wheels 1, 2, and 3) rotation
axes are orthogonal to the spacecraft ontology shaft
and the fourth one is installed with the equiangular di-
rection with the ontology three axis.
Fig. 1 Configuration of four reaction wheels

56 Q. Hu et al.
Fig. 2 Configuration of four reaction wheels under misalign-
ment
In this sense, employing the configuration of four
reaction wheels shown in Fig. 1, the spacecraft dynam-
ics in Eq. (2) can be written as
J˙ω=−ω×Jω+Dτ (t) +d(t) (4)
where Ddenotes reaction wheel configuration ma-
trix, representing the influence of each wheel on
the angular acceleration of the spacecraft, and τ=
[τ1τ2τ3τ4]Tdenotes the torque produced by the four
reaction wheels. Note that the configuration matrix D
is available for a given spacecraft.
However, in practice, the configuration of actua-
tors will never be perfect, that is to say, whether due
to finite manufacturing tolerances or warping of the
spacecraft structure during launch, some alignment er-
rors can always exist. In this section, actuator align-
ment error is mathematically modeled for the configu-
ration misalignment. Referring to Fig. 2, it is assumed
that the reaction wheel mounted on Xaxis is tilted
over nominal direction with constant angles, α1and
β1; also for other reaction wheels mounted left are
assumed to be tilted over nominal direction with α2,
β2,α3,β3,α4and β4respectively. To this
end, the real reaction wheel torque with misalignment
is expressed as
u=τ1⎡
⎣
cosα1
sin α1cosβ1
sin α1sin β1⎤
⎦+τ2⎡
⎣
sinα2cos β2
cosα2
sinα2sin β2⎤
⎦
+τ3⎡
⎣
sin α3cosβ3
sin α3sin β3
cosα3⎤
⎦
+τ4⎡
⎣
cos(α4+α4)cos(β4+β4)
cos(α4+α4)sin(β4+β4)
sin(α4+α4)⎤
⎦(5)
Generally speaking, the misalignment angle errors
(αi,β
i)are very small in practice, and the follow-
ing relationships are adopted to approximate Eq. (5):
cosαi≈cos βi≈1,
sin αi≈αi,
sin βi≈βi
(6)
Then for the considered actuator configuration, the
configuration matrix Dcan be represented as
D=D0+D (7)
with
D0=⎡
⎣
100cosα4cosβ4
0 1 0 cosα4sin β4
001 sinα4⎤
⎦(8a)
D =⎡
⎣
0α2cosβ2α3cos β3−α4sin α4cos β4−β4cos α4sin β4
α1cosβ10α3sin β3−α4sin α4sin β4+β4cos α4cos β4
α1sin β1α2sin β20α4cos α4⎤
⎦(8b)
where D0denotes the nominal value, and D denotes
the uncertainty of configuration matrix. Accordingly,
the spacecraft dynamics under this uncertainty can be
rewritten as
J˙ω=−ω×Jω+(D0+D)τ (t ) +d(t) (9)

Robust finite-time control allocation in spacecraft attitude stabilization under actuator misalignment 57
For synthesis of control system design, the following
reasonable assumptions are made:
Assumption 1 The external disturbances d(t),in
Eq. (9) include gravitational perturbations, atmo-
spheric drag, etc., and are assumed to be bounded.
Thus, it is reasonable to assume that there always ex-
ists a constant ¯
dsuch that

d(t)
≤¯
d(10)
Assumption 2 Due to physical limitations on the re-
action wheel considered, the control action generated
is limited by the saturation value, i.e. τ(t)∈Ω:= {τ∈
Rm|τi≤τi≤¯τi,i =1,2,3,4}. For simplicity, we as-
sume that actuator output torque has the same con-
straint value (τ ,¯τ),i.e.
τ(t)∈Ω:= τ∈Rm|τ≤τi≤¯τ,i =1,2,3,4(11)
Assumption 3 The uncertainty D due to misalign-
ment is unknown but bounded matrix satisfying
D∞≤ς(12)
for some constant ς.
2.3 Control objective
Consider the spacecraft attitude system given by
Eqs. (1) and (9) with the actuator misalignment under
the Assumptions 1–3. Design a control input τ(t)∈Ω
such that, for all physically realizable initial condi-
tions, the states of the closed-loop system can be stabi-
lized in finite time, which can be expressed as follows:
lim
t→Tq=lim
t→Tω=0 (13)
where Tis the convergent time, which is function of
the initial values of the system states.
3 Control law design
In this section, two steps are split for the controller
design using the principle of control allocation (CA)
[28]: (1) Design a control law specifying which total
control effort to be produced (net torque, etc.), and (2)
Design a control allocator that maps the total control
demand onto individual actuator settings (such as re-
action wheel torques, etc.), as shown in Fig. 3.
From the principle of control allocation, an equiva-
lent representation of Eq. (9) can be written as
J˙ω=−ω×Jω+Buu(t ) +d(t) (14a)
Buu(t) =Bττ(t) (14b)
where u(t) is the virtual control input or called com-
bined control torque produced by the actuators, Buis
the virtual input matrix, and Bτis used to describe dis-
tribution of the physical actuators with Bτ=D+D,
here, representing the influence of each actuator on the
angular acceleration of the spacecraft considered. For
the considered spacecraft, the virtual input matrix Bu
as the identity Bu=I3×3will be defined, for conve-
nience. With such a choice, the virtual input u(t) rep-
resents exactly the total torques produced by the actu-
ators, and the following virtual equivalent plant can be
given:
J˙ω=−ω×Jω+u(t ) +d(t) (15a)
u(t) =(D +D)τ (t ) (15b)
In what follows, we shall develop such an attitude con-
trol law for spacecraft as given in Eqs. (1) and (12)to
achieve the control target given in Sect. 2.3.
3.1 Nonsingular terminal sliding mode-based virtual
finite time feedback controller design
In this section, a nonsingular terminal sliding mode
(NTSM) control algorithm is designed for spacecraft
attitude control to achieve finite time convergence to
the given attitude motion.
Fig. 3 Block diagram for
spacecraft attitude control
with control allocation

58 Q. Hu et al.
3.1.1 Design of sliding surface
For simplicity of expression, the following notions are
introduced and used in the analysis and design of the
NTSM controllers. For any vector x∈Rnand a>0
which is a constant, let
sig(x)a=|x1|asgn(x1) ... |xn|asgn(xn)T
(16a)
|x|a=|x1|a... |xn|aT(16b)
1
dx sig(x)a=a|x1|a−1... a|xn|a−1T(16c)
Under this definition, the following sliding surface is
selected as: [36]
s=sig(˙q)b+βq (17)
where βis a constant, diagonal, positive-definite, con-
trol design matrix chosen by the designer/user, and b
satisfies 1 <b<2.
In view of Eq. (16a), (16b), (16c), the time deriva-
tive of the sliding surface in Eq. (17) yields
˙s=bdiag|˙q|b−1¨q+β˙q
=bdiag|˙q|b−1¨q+1
bβsig(˙q)2−b(18)
Using Eqs. (1) and (15a), (15b), and substituting these
into Eq. (18) yield
˙s=bdiag|˙q|b−1
×1
2q0I+q×J−1−ω×Jω+u+d
−1
4ω2q+1
bβsig(˙q)2−b(19)
3.1.2 NTSM-based virtual feedback controller design
For the derivation of the control law, let us choose a
positive definite Lyapunov function
V=1
2sTs(20)
Then, differentiating Valong the solution of Eq. (15a),
(15b) and using Eqs. (17) and (19)give
˙
V=sTbdiag|˙q|b−11
2q0I+q×J−1
×[−ω×Jω+u+d]
−1
4ω2q+1
bβsig(˙q)2−b(21)
In view of Eq. (21), the control law is defined as
u=ω×Jω−Jq0I+q×−1
×2
bβsig(˙q)2−b−1
2ω2q+ρsgn(s)(22)
where ρis an appropriate constant yet to be deter-
mined.
Then, using the design control law in Eq. (22),
Eq. (21) can be rewritten as
˙
V=1
2sTbdiag|˙q|b−1−ρsgn(s) +d(t)
≤−1
2
2

i=1
b|˙qi|b−1(ρ −¯
d)|si|(23)
Let ρsatisfies ρ> ¯
d, and then Eq. (23) can be further
written as
˙
V≤−1
2
2

i=1
b|˙qi|b−1(ρ −¯
d)|si|<0 (24)
for ˙qi= 0(i =1,2,3), and the condition for Lya-
punov stability is satisfied.
While for the case ˙qi=0, the following analysis
will be further discussed for the stability of the overall
system. Using the designed control law, the following
can be given:
¨q=−1
bβsig(˙q)2−b−1
2ρsgn(s) +1
2¯
d(25)
For ˙qi=0, Eq. (25) can be rewritten as
¨q=−1
2ρsgn(s) +1
2¯
d(26)
It is easy to know that when si>0, ¨q=−1
2ρ+1
2¯
d<
0, and when si<0, ¨q=1
2ρ+1
2¯
d>0, and then we
can conclude that ˙qi=0 is not an attractor. Corre-
spondingly, it can be concluded that the sliding mode
s=0 can be reached at a certain time t∗from any-
where in the phase plane in finite time. Once on the
sliding mode surface, i.e., si=0, one may have
˙qi=−β
1
b
isig(qi)b(27)

Robust finite-time control allocation in spacecraft attitude stabilization under actuator misalignment 59
From this, we can see that ˙qiand qican converge to
the origin along the sliding mode surface at time [14,
36]
ti=t∗+b
b−1β−1
b
iqit∗b−1
b(28)
Therefore, q→0 in finite time, and then ω→0.
Then we have the following conclusion:
Theorem 1 Consider the spacecraft dynamics gov-
erned by Eqs.(1)and (15a), (15b)under the Assump-
tion 1.With the nonsingular terminal sliding surface
defined in Eq.(17)and control law in Eq.(22), the
closed-loop system can be driven onto the sliding sur-
face s=0in finite time.
Proof The proof can be easily obtained from above
analysis and omitted here. 
Remark 1 For the design control law in Eq. (22), there
is no singularity problem when ˙qcrosses zero at a cer-
tain time by implementing the term sig(˙q)2−b, no like
the conventional terminal sliding mode control case.
Remark 2 For the control law in Eq. (22), only three
control parameters β,band ρ, are needed to design,
which means that inexpensive online computations are
required from the point view of a feature of practi-
cal importance in reducing the usage of onboard re-
sources in terms of computing power and memory
size. And also this is user/designer friendly in that is
does not involve a time-consuming design procedure
and demands little redesigning or reprogramming dur-
ing spacecraft operation.
While for the selection of these parameters, ρis
related to the bound of external disturbances. In gen-
eral, this bound is difficult to measure, and therefore
ρis usually chosen large enough to ensure the robust
stability. For βand b, they are related to the conver-
gence time and system performance, and can be se-
lected properly by the requirement of the system per-
formance.
3.2 Robust least squares-based control allocator
design under actuator misalignment
Control allocation (CA) is useful for control of over-
actuated spacecraft systems, and deals with suitably
distributing the desired total control command among
the individual actuators while considering the actua-
tor constraints. Using CA, the actuator selection task
is separated from the regulation task in the control de-
sign. Generally, the CA problem is formulated as an
optimization problem systematically handling: redun-
dant sets of actuators, actuator constraints and mini-
mizing power consumption, etc. There is extensive lit-
erature on CA which discusses different algorithms,
approaches and applications [28–34]. While, for the
case with uncertainty of configuration matrix due to
misalignment, robust least squares-based control allo-
cation is an effective way and well developed in recent
research [37–39]. In what follows, the principle of this
scheme is investigated for the considered plant.
In Refs. [36–38], the considered control allocation
problem can be stated as: Given a configuration ma-
trix D0with its uncertain term D, and virtual con-
trol u(t) designed by Sect. 3.1, the optimal actuator
control vector τcan be described by
τ=arg min
τ≤τ≤¯τmax
D≤ς
(D0+D)τ −u
(29)
subject to D∞≤ς, and τ(t) ∈Ω:= {τ∈Rm|τ≤
τi≤¯τ,i =1,2,3,4}.
For a variable τin the interval of (τ ,¯τ),theworst-
case residual can be given
γ(τ)=max
D∞≤ς
(D0+D)τ −u
(30)
And further the triangle inequality
γ(τ)≤max
D∞≤ςD0τ−v+Dτ 
=D0τ−v+ max
D∞≤ςDτ (31)
Let
D =ς
τετT(32)
where
ε=D0τ−u
D0τ−u,if D0τ=u
I, otherwise (33)
in which Iis a unite vector. In this case, the worst-case
residual in the direction of εcan be rewritten as
γ(τ)=D0τ−u+ςτ(34)

60 Q. Hu et al.
Tabl e 1 Simulation
parameters and initial
condition
Model
parameters
J=[2000.9;0170;0.9015]kg m2,
initial angular velocity ω0=[0;0;0]rads−1,
initial quaternion Q0=[0.9;−0.3;0.26;0.18],
α4=35.26, β4=45, αi=[0.2;0.1;0.2;0.3],βi=[0.2;0.2;0.3;0.2]deg.
NTSM b=1.32, β=0.32, ρ=0.036.
RLSCA ζ=0.4, ¯τ=0.15 N m, τ=−0.15 N m.
PD kpx =6.2, kdx =7.6, kpy =6.0, kdy =6.6, kpz =6.6, kdz =9.6.
CSMC β1=0.6, ρ1=1.8.
and satisfies
D0τ−u+ςτ≤κ(35)
where κis the upper bound of the residual to be min-
imized by finding the optimal τin the interval of
(τ ,¯τ). Thus, this problem can be written as a standard
second-order cone programming problem as
min
τ,μ,κ κ(36)
subject to
D0τ−u≤κ−μ(37a)
ςτ≤μ(37b)
τ≤τi≤¯τ(37c)
with the variables τ,μ, and κ.
Acccordingly, the following statements can be con-
cluded.
Theorem 2 [37–39]The optimal solutionτto the ro-
bust least square control allocation problem is given
by
τ=⎧
⎨
⎩
(ηI +DT
0D0)−1DT
0uif η
=(κ−μ)μς 2
μ2+ς2z>0
D+
0uotherwise
(38)
where η>0, z=τ2+¯τ2,κand μare the opti-
mal solution to the above problem,and D+
0is a right
pseudo-inverse of matrix of matrix D0,with D+
0=
DT
0(D0DT
0)−1.
Proof The proof is omitted here; see Refs. [37–39]for
more details. In the next section, numerical simulation
and comparison are given to verify the success of the
NTSM control law in conjunction with the RLSCA
technique. 
4 Simulation and comparison results
To verify the effectiveness and performance of the pro-
posed attitude stabilization control scheme, numeri-
cal simulations have been carried out using the rigid
spacecraft system given in Eqs. (1) and (9). For all nu-
merical simulation presented in this section, the pa-
rameters are provided in Table 1, and the external dis-
turbances are assumed to be
d(t) =10−3
×⎡
⎣
3cos(10ωdt) +4sin(3ωdt)−10
−1.5sin(2ωdt)+3 cos(5ωdt)+15
3sin(10ωdt)−8sin(4ωdt)+10 ⎤
⎦Nm
(39)
For the purpose of comparison, the widely used PD
controller [1], conventional sliding mode (CSM) con-
trol [12] and also the pseudo-inverse (PI) control al-
location [28] are applied to the problem considered.
More specifically, the simulation results using the fol-
lowing four control schemes are conducted: Case 1:
the PD controller incorporating the PI control alloca-
tor, noted as PD +PI; Case 2: the conventional slid-
ing mode controller (CSM) combined with the PI con-
trol allocator, noted as CSM +PI; Case 3: the pro-
posed nonsingular terminal sliding mode controller
(NTSM) combined with the PI control allocator, noted
as NTSM +PI; Case 4: the proposed NTSM incorpo-
rating the proposed robust least squares control alloca-
tor (RLSCA) noted as NTSM +RLSCA. In addition,
the effect of the actuator misalignment is discussed in
above different cases. While for these design parame-
ters of different control case, they were tuned by trial
and error until the controller shows satisfactory per-
formance considering an admissible thrust magnitude
and convergence of the estimated parameters. All the
computations and plots are performed using the MAT-
LAB/SIMULINK software package.

Robust finite-time control allocation in spacecraft attitude stabilization under actuator misalignment 61
Fig. 4 The time responses of the quaternion q0
4.1 Time responses comparison with/without
misalignment
In this case, firstly, to show the effect of the pro-
posed NTSM incorporating with RLSCA, simulation
was done under the given initial condition. The time
histories of quaternion of spacecraft, angular velocity,
the total required control torque and reaction wheels’
output torques are shown in Figs. 4–17 (solid line)
with/without misalignment. It is noted that an accept-
able desirable orientation response is achieved, and the
spacecraft reached the demanded position with a set-
tling time less than 25 s. Moreover, the time responses
of attitude and velocity are few oscillations, but set-
tle within 25 s even if the external disturbances and
misalignment are considered simultaneously. This il-
lustrates that the designed controller is capable of re-
jecting the disturbances and misalignment while main-
taining the rotational capability of the spacecraft, with
high accuracy less than 0.0003 and 0.0005 (rad/s) for
quaternion and velocity, respectively. In addition, from
Figs. 14,15,16,17 (solid line), it can be easily seen
that the output torque of each reaction wheel is within
the saturation limitation 0.15 N m because the RLSCA
explicitly considers the control saturation constrain.
For the purpose of comparison, the system is also
controlled by using the traditional PD control com-
bined with pseudo-inverse (PI) control allocation. The
same simulation cases are repeated with this controller
and the results of simulation are shown in Fig. 4–17
(dash-dotted line). For this case, it can be observed that
the attitude rotational maneuver can be achieved, but
severe oscillations are excited during maneuvering as
demonstrated in the quaternion and velocity responses
asshowninFig.4–10 (dash-dotted line). Despite the
fact that there still exists some room for improvement
with different design control parameter sets, there is
not much improvement in the attitude and velocity re-
sponses. Moreover, severe saturation phenomenon can
be observed from the time responses of reaction wheel
torques. From the comparison of these results, the per-
formance of the proposed design is better than conven-
tional PD control even if these designs will adapt the
system parameters under the external disturbances.
In addition, comparison using the conventional
sliding mode (CSM) controller designed in Ref. [12]

62 Q. Hu et al.
Fig. 5 The time responses of the quaternion q1
Fig. 6 The time responses of the quaternion q2

Robust finite-time control allocation in spacecraft attitude stabilization under actuator misalignment 63
Fig. 7 The time responses of the quaternion q3
Fig. 8 The time responses of the angular velocity ωx

64 Q. Hu et al.
Fig. 9 The time responses of the angular velocity ωy
Fig. 10 The time responses of the angular velocity ωz

Robust finite-time control allocation in spacecraft attitude stabilization under actuator misalignment 65
Fig. 11 The time responses of actual control torques ux
Fig. 12 The time responses of actual control torques uy

66 Q. Hu et al.
Fig. 13 The time responses of actual control torques uz
Fig. 14 The time responses of actuator torques τ1

Robust finite-time control allocation in spacecraft attitude stabilization under actuator misalignment 67
Fig. 15 The time responses of actuator torques τ2
Fig. 16 The time responses of actuator torques τ3

68 Q. Hu et al.
Fig. 17 The time responses of actuator torques τ4
integrated with PI control allocation is further per-
formed for the system. The same simulation case is
also repeated and the results of simulation are shown
in Fig. 4–17 (dotted line). The CSM controller shows
maneuvering ability, to some degree, because of its
robustness, as one can see in Fig. 4–17 (dotted line)
that the responses of attitude can be improved a lot
with comparison with PD case, but still oscillations
are the results of the control law as compared with the
proposed methods, and also the settling time is longer
than the proposed one, within 35 s. In addition, more
saturation phenomenon can be observed from the time
responses of reaction wheel torque. From these com-
parisons, the performance of the proposed design is
better than the last two even if these designs will adapt
the system parameters under the external disturbances.
In addition, from the point view of practice, the
spacecraft attitude responses using Euler angle
[ψθφ]T(φ,θand ψare, respectively, the roll, pitch,
and yaw angles), are shown as Fig. 18. It is clearly that
results of NTSM +RLSCA has the best performance
among of them with shortest time for achieving the
rotation maneuvers, which further illustrates the effec-
tiveness and feasibility of the proposed method.
4.2 Energy consumption comparison
Fuel or electrical energy saving for spacecraft in or-
bit is of crucial importance to prolong the working
life and achieve the target missions, and therefore the
energy consumption optimal problem should also be
considered for spacecraft attitude control system. To
analyze the system energy consumption, the perfor-
mance function which is derived from the optimal con-
trol allocation problem is defined as
E=1
2T
0τ2dt,
where Tdenotes the simulation time, and T=100 s
is chosen in the simulation. Figure 19 shows the bar
graphic visualizations of the energy consumption per-
formance among the three control cases. From Fig. 19,
it can be seen that the energy consumption using
NTSM +RLSCA developed in this paper is minimal
in the three setting intervals; while, for other case,

Robust finite-time control allocation in spacecraft attitude stabilization under actuator misalignment 69
Fig. 18 The time responses
of the Euler angles
they consume more energy to achieve the desired tar-
gets because there is no energy optimization constraint
in PI, especially, within first 20 s of the simulation
for maneuvering. This illustrates the theoretical results
developed in Sect. 3. In addition, the control perfor-
mance comparison of the these schemes is shown in
Table 2in terms of setting time, steady precision of
error, and the actual torques or energy consumption,
which further shows the acceptable performance using
the proposed control law under the multi-constraints
mentioned previously.
Summarizing all the cases, it is noted that the pro-
posed controllers design method (NTSM +RLSCA)
can significant improve the normal performance than
the, NTSM+PI, CSM +PI or PD+PI method in both
theory and simulations. In addition, extensive simula-
tions were also done using different control parame-
ters, disturbance inputs and even combination of the
reaction wheels. These results show that in the closed-
loop system attitude stabilization control and energy
saving are accomplished in spite of these undesired
effects in the system. Moreover, the flexibility in the
choice of control parameters can be utilized to obtain
desirable performance while meeting the constraints
on the control magnitude and actuator uncertainties.
These control approaches provides the theoretical ba-
Fig. 19 Bar graphic visualizations of the energy consume per-
formance comparisons of four cases
sis for the practical application of the advanced control
theory to spacecraft attitude control system.
5 Conclusion
In this paper finite-time control idea has been incor-
porated into a control allocation framework to solve
the attitude control problem of spacecraft in pres-
ence of uncertain configuration matrix due to mis-

70 Q. Hu et al.
Tabl e 2 Comparison of control performance among the four cases with misalignments
Performance Control
Setting
time of
q(s)
Steady
precision
of q
Setting
time of
ω(s)
Steady
precision
of ω
Steady
precision
of s
Energy consumption
0–20 s 20–40 s 60–100 s
PD +PI 60 5 ×10−370 5 ×10−4None 0.9190 0.3610 0.0936
CSM +PI 35 2 ×10−340 2 ×10−43×10−40.2310 0.0524 0.0132
NTSM +PI 25 3 ×10−430 5 ×10−41×10−40.3870 0.0856 0.0072
NTSM +RLSCA 25 3 ×10−430 5 ×10−41×10−40.2606 0.0502 0.0060
alignment, external disturbances and actuator satura-
tion as well. The proposed finite-time control alloca-
tion scheme, which uses nonsingular terminal sliding
mode technique to design the virtual feedback con-
trol to achieve finite-time convergence to desired at-
titude position under undesired disturbances, and, sep-
arately, employs the robust least squares-based con-
trol allocation scheme to suitably distribute the total
virtual control effort into the active actuators, includ-
ing actuator misalignment and actuator saturation, en-
ables the overall scheme to cope with fast attitude
finite-time convergence with high accuracy. Numeri-
cal implementation of the new controller was also pre-
sented to confirm the advantages and improvements
over existing controllers. The case of uncertain con-
figuration due to actuator misalignment has been con-
sidered here only for a special configuration, and other
forms, such as actuator faults and/or magnitude error,
were not considered. The latter case should be inves-
tigated by control reconfiguration together with actua-
tor hardware redundancy and management, and is the
subject of future research.
Acknowledgements This present work was supported par-
tially by National Natural Science Foundation of China (Project
Nos. 61004072, 61273175), Program for New Century Ex-
cellent Talents in University (NCET-11-0801), Heilongjiang
Province Science Foundation for Youths (QC2012C024) and
the Fundamental Research Funds for the Central Universi-
ties (HIT.NSRIF.2009003, HIT.BRETIII.201212). The authors
would also like to thank the reviewers and the Editor for their
comments and suggestions that helped to improve the paper sig-
nificantly.
References
1. Tsiotras, P.: Further passivity results for the attitude control
problem. IEEE Trans. Autom. Control 43(11), 1597–1600
(1998)
2. Tsiotras, P.: Stabilization and optimality results for the atti-
tude control problem. J. Guid. Control Dyn. 19(4), 772–779
(1996)
3. Subbarao, K.: Nonlinear PID-like controllers for rigid-body
attitude stabilization. J. Astronaut. Sci. 52(1–2), 61–74
(2004)
4. Carrington, C.K., Junkins, J.L.: Optimal nonlinear feed-
back control for spacecraft attitude maneuvers. J. Guid.
Control Dyn. 9(1), 99–107 (1986)
5. Sharma, R., Tewari, A.: Optimal nonlinear tracking of
spacecraft attitude maneuvers. IEEE Trans. Control Syst.
Technol. 12(5), 677–682 (2004)
6. Junkins, J.L., Akella, M.R., Robinett, R.D.: Nonlinear
adaptive control of spacecraft maneuvers. J. Guid. Control
Dyn. 20(6), 1104–1110 (1997)
7. Schaub, H., Akella, M.R., Junkins, J.L.: Adaptive control
of nonlinear attitude motions realizing linear closed loop
dynamics. J. Guid. Control Dyn. 24(1), 95–100 (2001)
8. Ahmed, J., Coppola, V.T., Bernstein, D.: Adaptive asymp-
totic tracking of spacecraft attitude motion with inertia ma-
trix identification. J. Guid. Control Dyn. 21(5), 684–691
(1998)
9. Hu, Q.L., Ma, G.F.: Variable structure control and active
vibration suppression of flexible spacecraft during attitude
maneuver. Aerosp. Sci. Technol. 9(4), 307–317 (2005)
10. Hu, Q.L., Xiao, B.: Fault-tolerant sliding mode attitude
control for flexible spacecraft under loss of actuator effec-
tiveness. Nonlinear Dyn. 64(1–2), 13–23 (2011)
11. Liu, K.F., Xia, Y.Q., Zhu, Z., et al.: Sliding mode attitude
tracking of rigid spacecraft with disturbance. J. Franklin
Inst. 349(2), 413–440 (2012)
12. Chen, Y.P., Lo, S.C.: Sliding-mode controller design
for spacecraft attitude tracking maneuvers. IEEE Trans.
Aerosp. Electron. Syst. 29(4), 1328–1333 (1993)
13. Jin, E.D., Sun, Z.W.: Robust controllers design with finite
time convergence for rigid spacecraft attitude tracking con-
trol. Aerosp. Sci. Technol. 12(4), 324–330 (2008)
14. Li, S.H., Wang, Z., Fei, S.M.: Comments on the paper:
robust controllers design with finite time convergence for
rigid spacecraft attitude tracking control. Aerosp. Sci. Tech-
nol. 15(3), 193–195 (2011)
15. Ding, S.H., Li, S.H.: Stabilization of the attitude of a
rigid spacecraft with external disturbances using finite-time
control techniques. Aerosp. Sci. Technol. 13(4), 256–265
(2009)

Robust finite-time control allocation in spacecraft attitude stabilization under actuator misalignment 71
16. Zhu, Z., Xia, Y.Q., Fu, M.Y.: Attitude stabilization of rigid
spacecraft with finite-time convergence. Int. J. Robust Non-
linear Control 21(6), 686–702 (2011)
17. Boskovic, J.D., Li, S.M., Mehra, R.K.: Robust tracking de-
sign for spacecraft under control input saturation. J. Guid.
Control Dyn. 27(4), 627–633 (2004)
18. Hu, Q.L.: Robust adaptive sliding mode attitude maneuver-
ing and vibration damping of three-axis-stabilized flexible
spacecraft with actuator saturation limits. Nonlinear Dyn.
55(4), 301–321 (2009)
19. Hu, Q.: Sliding mode maneuvering control and active vi-
bration damping of three-axis stabilized flexible spacecraft
with actuator dynamics. Nonlinear Dyn. 52(3), 227–248
(2008)
20. Ali, I., Radice, G., Kim, J.: Backstepping control design
with actuator torque bound for spacecraft attitude maneu-
ver. J. Guid. Control Dyn. 33(1), 254–259 (2010)
21. De, R.A.: Adaptive spacecraft attitude tracking control with
actuator saturation. J. Guid. Control Dyn. 33(5), 1692–
1695 (2010)
22. Su, Y.X., Zheng, C.H.: Globally asymptotic stabilization
of spacecraft with simple saturated proportional-derivative
control. J. Guid. Control Dyn. 34(6), 1932–1935 (2011)
23. Yoon, H., Tsiotras, P.: Adaptive spacecraft attitude tracking
control with actuator uncertainties. In: Proceeding of the
AIAA Guidance, Navigation, and Control Conference and
Exhibit (AIAA-2005-6392), San Francisco, California, 15–
18 Aug. (2005)
24. Scarritt, S.K.: Nonlinear model reference adaptive control
for satellite attitude tracking. In: Proceeding of the AIAA
Guidance, Navigation and Control Conference and Exhibit
(AIAA-2008-7165), Honolulu, Hawaii, 18–21 Aug. (2008)
25. Fosbury, A.M., Nebelecky, C.K.: Spacecraft actuator align-
ment estimation. In: Proceeding of the AIAA Guidance,
Navigation and Control Conference (AIAA-2009-6316),
Chicago, Illinois, 10–13 Aug. (2009)
26. Yoon, H., Agrawal, B.N.: Adaptive control of uncertain
Hamiltonian multi-input multi-output systems: with appli-
cation to spacecraft control. IEEE Trans. Control Syst.
Technol. 17(4), 900–906 (2009)
27. Lima, H.C., Bang, H.: Adaptive control for satellite for-
mation flying under thrust misalignment. Acta Astronaut.
65(1–2), 112–122 (2009)
28. Harkegard, O.: Backstepping and control allocation with
applications to flight control. Ph.D. thesis, Linkoping Univ.,
Linkoping, Sweden (2003)
29. Buffington, J.M., Enns, D.F.: Lyapunov stability analysis of
daisy chain control allocation. J. Guid. Control Dyn. 19(6),
1226–1230 (1996)
30. Durham, W.C.: Constrained control allocation. J. Guid.
Control Dyn. 16(4), 717–725 (1993)
31. Durham, W.C.: Constrained control allocation: three mo-
ment problem. J. Guid. Control Dyn. 17(2), 330–336
(1994)
32. Harkegard, O.: Dynamic control allocation using con-
strained quadratic programming. J. Guid. Control Dyn.
27(6), 1028–1034 (2004)
33. Alwi, H., Edwards, C.: Fault tolerant control using slid-
ing modes with online control allocation. Automatica 44,
1859–1866 (2008)
34. Casavola, A., Garone, E.: Fault tolerant adaptive control al-
location schemes for over-actuated systems. Int. J. Robust
Nonlinear Control 20(17), 1958–1980 (2010)
35. Sidi, M.J.: Spacecraft Dynamics and Control, pp. 88–97.
Cambridge University Press, Cambridge (1997)
36. Feng, Y., Yu, X.H., Man, Z.H.: Nonsingular terminal slid-
ing mode control of rigid manipulator. Automatica 38(12),
2159–2167 (2002)
37. Cui, L., Yang, Y.: Disturbance rejection and robust least-
squares control allocation in flight control system. J. Guid.
Control Dyn. 34(6), 1632–1643 (2011)
38. Ghaoui, L.E., Lebret, H.: Robust solutions to least squares
problems with uncertain data. SIAM J. Matrix Anal. Appl.
18(4), 1035–1064 (1997)
39. Ghaoui, L.E., Oustry, F., Lebret, H.: Robust solutions to un-
certain semi-definite programs. SIAM J. Optim. 9(1), 33–
52 (1998)