Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

55, 3, pp. 1055-1066, Warsaw 2017
DOI: 10.15632/jtam-pl.55.3.1055

ULTIMATE STATE BOUNDEDNESS OF UNDERACTUATED SPACECRAFT

SUBJECT TO AN UNMATCHED DISTURBANCE

Rouzbeh Moradi, Alireza Alikhani

Aerospace Research Institute (Ministry of Science, Research and Technology), Tehran, Iran

e-mail: roozbeh moradi aerospace@yahoo.com; aalikhani@ari.ac.ir (corresponding author)

Mohsen Fathi Jegarkandi

Department of Aerospace Engineering, Sharif University of Technology, Tehran, Iran

e-mail: fathi@sharif.edu

Ultimate state boundedness for underactuated spacecraft subject to large non-matched di-
sturbances is attained. First, non-smooth time-invariant state feedback control laws that
make the origin asymptotically stable are obtained. Then, the controller is extended to
make the closed-loop system globally uniformly ultimately bounded under the following
conditions: 1) the disturbances acting on the directly actuated states are known and 2) the
disturbance acting on the unactuated state is bounded and its profile need not be known.
Finally, numerical simulations are presented to verify the analytical results. A large step di-
sturbance is considered, and it is shown that the proposed controller makes the closed-loop
system globally uniformly ultimately bounded. The proposedmethod is rather general and
can be extended to other systems.

Keywords:underactuated spacecraft stabilization,non-matcheddisturbances, globaluniform
ultimate boundedness

1. Introduction

A mechanical system is underactuated when the number of independent control inputs is less
than the number of degrees of freedom to be controlled. Considering the stabilization of such
systems, an extensive amountof studies hasbeenpublished in the literature.Choukchou-Braham
et al. (2013), Olfati-Saber (2001), Aneke (2003), Spong (1998), Fantoni and Lozano (2002) and
Liu and Yu (2013) are just a few examples.

The linearized model of a majority of underactuated systems, especially in the absence of
gravitational terms, is not controllable near equilibriumpoints (Choukchou-Braham et al., 2013).
This leads to the well-known fact that most underactuated systems do not satisfy Brockett’s
necessary condition for smooth feedback stabilization (Choukchou-Braham et al., 2013). In order
to deal with this problem, non-smooth feedbacks have been proposed to stabilize underactuated
mechanical systems (Reyhanoglu et al., 2000).

In addition to the smoothness of feedback control laws, underactuation leads to another
important challenge: attenuation of disturbances. The severity of this problem increases when
the disturbances are non-matched, i.e. span{P} /∈ span{B}, whereP andB are the disturbance
and control matrices, respectively (Astolfi and Rapaport, 1998).

Reducing the effects of disturbances on the stabilization of underactuated spacecraft has
been considered by several papers. Astolfi and Rapaport (1998) considered robust stabilization
of the angular velocity of a rigid body subject to external disturbances usingL2-gain analysis.
Several propositions were proved, and the robust stabilization problem was solved in a region
having a hole. Floquet et al. (2000) used higher order sliding mode control (variable structure



Ultimate state boundedness of underactuated spacecraft... 1057

control) to make the origin asymptotically stable for underactuated spacecraft within a finite
time. However, it was assumed that no disturbance was exerted on the unactuated axis, i.e.
they satisfiedmatching conditions. Karami and Sassani (2000) used the backstepping technique
to asymptotically stabilize the angular velocity and attitude of underactuated spacecraft. The
considered step disturbance had a very low magnitude. Zhang et al. (2008) considered the spin
stabilization problem of underactuated spacecraft subject to sinusoidal disturbances. All of the
disturbances were considered to be sinusoidal. Wang et al. (2003) considered stabilization of
the angular velocity and attitude of underactuated spacecraft under sinusoidal disturbance.
Although three exogenous disturbances were considered to be exerted on the spacecraft, the
controller design was based onmatched type disturbances.
In this paper, a new method is proposed to find non-smooth time-invariant state feedback

control laws for underactuated spacecraft. Then, the controller is extended to make the closed-
loop system globally uniformly ultimately bounded (GUUB). The proposedmethod is based on
a combination of feedback linearization and Lyapunov stability theory. To avoid singularity of
control inputs near the equilibrium points, a thin boundary layer is defined. It is assumed that
outside the boundary layer, the extended controller is applied to the system. However, inside
the boundary layer, the terms leading to singularity are canceled.
The present paper has two contributions: First, the proposed method is rather general and

can be extended to other systems and second, the large step disturbance is considered to verify
the controller performance.Most of the previous works have considered sinusoidal disturbances,
which are not as severe as the step disturbance for the underactuated spacecraft.
The rest of this paper consists of the following sections: Section 2 provides non-smooth

time-invariant state feedback control of underactuated spacecraft. In Section 3, the controller is
extended to make the closed-loop systemGUUB subject to non-matched disturbances. Finally,
Section 4 presents numerical simulations and discussions to verify the analytical results.

2. Time-invariant non-smooth state feedback control of underactuated spacecraft

In the principal coordinate system, the rigid spacecraft angular velocity equations are described
by the following expressions (Sidi, 2000)

ṗ=α1qr+u1+dp dp =
Tdp
Jx

q̇=α2pr+u2+dq dq =
Tdq
Jy

ṙ=α3pq+u3+dr dr =
Tdr
Jz

(2.1)

where [p,q,r] are angular velocities of the spacecraft, [u1,u2,u3] are normalized control inputs
and [Tdp,Tdq,Tdr] are external disturbances. α1, α2 and α3 are fractions of moments of inertia
and are assumed to be constant. Their values are obtained from the following set of equations

α1 =
Jy −Jz
Jx

α2 =
Jz −Jx
Jy

α3 =
Jx−Jy
Jz

(2.2)

where [Jx,Jy,Jz] are principal moments of inertia of the rigid body along the principal body
axis. The relation between control moments and inputs are given by the following equations

u1 =
Mx
Jx

u2 =
My
Jy

u3 =
Mz
Jz

(2.3)

[Mx,My,Mz] are three control moments acting on the spacecraft, and are assumed to be pro-
duced by thrusters.



1058 R.Moradi et al.

Without loss of generality, it is assumed that the third state (r) is unactuated (u3 =0). On
the other hand, it is assumed that αi 6=0 ∀i=1,2,3.
For now, the disturbances are not considered in the controller design. Therefore, Eqs. (2.1)

will be simplified to Eqs. (2.4)

ṗ=α1qr+u1 q̇=α2pr+u2 ṙ=α3pq (2.4)

In order to obtain a virtual control input for r, q is bisected into two parts

q= aq1+ bq2 a,b∈R (2.5)

a and b are constant real numbers that are presented to show that any linear combination of q1
and q2 will lead to the same results. This fact will be confirmed shortly.
Inserting q into the second and third rows of Eqs. (2.4) leads to

ṗ=α1qr+u1 aq̇1 =α2pr+u2− bq̇2 ṙ=α3p(aq1+ bq2) (2.6)

The following definition is introduced

q̇2 =w (2.7)

w is a scalar variable that is used to stabilize the unactuated state (r). The goal is now to
determine q2 that forces r to approach the origin. This q2 is denoted as q2,des.
In order for r to be exponentially stabilized, the following relation must hold

ṙ=−krr (2.8)

Considering the last row of Eqs. (2.6) and (2.8), the following equation is obtained

−krr=α3p(aq1+ bq2,des) (2.9)

Solving for q2,des results in

q2,des =
1

b
φ−
a

b
q1 (2.10)

where φ=−krr/(α3p).
It can be easily shown that

lim
t→∞
q= lim

t→∞
φ (2.11)

Therefore, it is possible to tune the controller parameters tomake sure that the steady-state
value of q becomes zero.
Now, the following linear combination of q2 and q2,des is introduced to transform Eqs. (2.6)

into a virtually fully actuated form

z= cq2+dq2,des c,d∈R (2.12)

c andd are constant real numbers, and their values have direct influence on the stability andper-
formance of the closed-loop system. The validity and importance of this statement will become
clear at the end of this Section. As stated previously, q2,des is the virtual control input.

Note 1: If c=−d and z=0, q2 will be equal to q2,des, which is the ideal case. This is equivalent
to saying that the stabilization of z is equivalent to the stabilization of r. According to
this point, the equation for ṙ will be replaced by ż.



Ultimate state boundedness of underactuated spacecraft... 1059

Differentiating Eq. (2.12) with respect to time together with Eq. (2.7) results in

ż= cq̇2+dq̇2,des = cw+dq̇2,des (2.13)

Replacing the third equation of Eqs. (2.6) with Eq. (2.13) leads to

ṗ=α1qr+u1 q̇1 =
1

a
(α2pr+u2− bq̇2) ż= cw+dq̇2,des (2.14)

The time derivative of q2,des can be obtained by partial differentiation of Eq. (2.10)

q̇2,des =
1

b

∂φ

∂p
ṗ−
a

b
q̇1 (2.15)

It is assumed that (∂φ/∂r)/ṙ≈ 0 in comparison to the other terms. The reason for this assump-
tion is to rewrite ż in terms of the original variables.

Using Eqs. (2.5), (2.7), (2.15) and performing somemathematical operations, alongwith the
third row of Eqs. (2.4), ż will be simplified to

ż=(c+d)w+
d

b

∂φ

∂p
ṗ−
d

b
q̇ (2.16)

Therefore, the entire set of Eqs. (2.14) in a virtually fully actuated formwill be given as

ṗ=α1qr+u1 q̇1 =
1

a
(α2pr+u2− bw) ż=(c+d)w+

d

b

∂φ

∂p
ṗ−
d

b
q̇ (2.17)

According to the third row, in order for z to be stabilizable, c should not be equal to −d.
However, this is in contradiction with the previously made conclusion (Note 1). In order to
alleviate this problem, it will be assumed that c≈−d.

Using feedback linearization and expecting exponential convergence from z i.e. ż = −kzz,
w is obtained as follows

w=
1

c+d

(

−kzz−
d

b

∂φ

∂p
ṗ+
d

b
q̇
)

(2.18)

Insertingw in the second equation of Eqs. (2.17), u2 is obtained as follows

u2 = aq̇1−α2pr+
b

c+d

(

−kzz−
d

b

kpkr(−r)

α3p
−
d

b
kqq
)

(2.19)

UsingEq. (2.12) and considering the fact that limt→∞q2,des =0, the above equation is simplified
to

u2 = aq̇1−α2pr+
b

c+d

(

−kzcq2−
d

b

kpkr(−r)

α3p
−
d

b
kqq
)

(2.20)

The reason for assuming limt→∞q2,des =0 can be inferred from Eq. (2.10). φ is a function that
can converge to zero through tuning the controller parameters (kp < kr or equivalently, the
convergence rate of p less than the convergence rate of r). On the other hand, according to the
dynamics imposed on q1 (q̇1 = −kq1q1), this variable will also converge to zero. Considering
these facts, u2 is

u2 =−akq1q1−α2pr−
b

c+d
kzcq2−

d

c+d

kpkr(−r)

α3p
−
d

c+d
kqq (2.21)



1060 R.Moradi et al.

Assuming kq1 = [c/(c+d)]kz = [c/(c+d)]kq and according to Eq. (2.5), the final equation for
u1 and u2 is

u1 =−kpp−α1qr u2 =−kqq−α2pr+
d

c+d

kpkrr

α3p
(2.22)

Since the procedure used to obtain this controller is based on linear state bisection (Eq. (2.5)),
this controller is called LSB.
In accordance with Eq. (2.22), u2 consists of two parts

u21 =−kqq−α2pr u22 =
d

c+d

kpkrr

α3p
(2.23)

u21 and u22 are used to stabilize q and r, respectively. The form of u2 is consistent with the one
proposed by Reyhanoglu (1996).
According to Eqs. (2.23), u2 does not depend on a and b. According to Eq. (2.1), the com-

ponents of dp and dq can be easily counteracted by the direct control input vectors, u1 and u2.
Therefore, the component of disturbances on the unactuated state (dr) plays the key role in the
controller design.
Non-smooth time-invariant state feedback control laws (Eq. (2.22)) make the origin asymp-

totically stable for the disturbance-free system (Eqs. (2.4)). In the next Section, this controller
is extended to provide GUUB, in the presence of non-matched disturbances.

3. Underactuated spacecraft angular velocity ultimate boundedness in presence

of non-matched disturbances

Definition 1 (Astolfi and Rapaport (1998)): The disturbances are non-matched when
span{P} /∈ span{B}, where P and B are the disturbance and control matrices, respecti-
vely.

Assumption 1: dp and dq can be unbounded, but should be known.

Assumption 2: dr should be bounded, and its maximum value should be known.

In order to extend the controller and to ensure that the states are GUUB, the following
procedure is proposed:

1) Since u2 has been selected to stabilize r, u1 is used to provide GUUB.

2) A candidate Lyapunov function (CLF) is proposed.

3) The derivative of this CLF along the trajectories of the closed-loop system is evaluated.

4) u1 is used to ensure GUUB of the states for the perturbed closed-loop system.

Consider Eqs. (2.4) with disturbances

ṗ=u1+α1qr+dp q̇=u2+α2pr+dq ṙ=α3pq+dr (3.1)

It has been shown in Section 2 that u1 and u2 (Eq. (2.22)) make the origin asymptotically
stable for Eqs. (3.1) without the presence of disturbances.
After eliminating dp and dq using direct control inputs, the closed-loop systemwill be

ṗ=u′1 q̇=−kqq+
d

c+d

kpkrr

α3p
ṙ=α3pq+dr (3.2)

Now, the goal is to find u′1 that makes the closed-loop system GUUB.



Ultimate state boundedness of underactuated spacecraft... 1061

In order to solve this problem, the following CLF is proposed

V =
1

2

[

p q r
]







kp 0 0
0 kq 0
0 0 kr













p
q
r






=
1

2
(kpp

2+kqq
2+krr

2) (3.3)

The derivative of V along the trajectories (Eqs. (3.2)) is given by

V̇ =
∂V

∂x
ẋ= kppu

′

1−k
2
qq
2+

d

c+d

kpkqkr
α3

rq

p
+α3krpqr+krrdr

¬ kppu
′

1−k
2
qq
2+

d

c+d

kpkqkr
α3

rq

p
+α3krpqr+krrMd

(3.4)

whereMd is the maximum absolute value of dr. If u
′

1 is selected as

u′1 =
1

kpp

(

−
d

c+d

kpkqkr
α3

rq

p
−α3krpqr−k

2
pp
2−k2rr

2
)

p 6=0 (3.5)

V̇ will result in the following equation

V̇ ¬−k2pp
2−k2qq

2−k2rr
2+krrMd =−w2(x)+krrMd (3.6)

Since

−w2(x)¬−min(k
2
p,k
2
q,k
2
r)‖x‖

2
2 (3.7)

and at the same time r¬‖x‖2, V̇ will satisfy the following inequality

V̇ ¬−min(k2p,k
2
q,k
2
r)‖x‖

2
2+krMd‖x‖2 (3.8)

Therefore, if the following inequality holds

Md ¬
min(k2p,k

2
q,k
2
r)

2kr
‖x‖2 (3.9)

equivalently

‖x‖2 ­
2kr

min(k2p,k
2
q,k
2
r)
Md (3.10)

V̇ will satisfy the following relation

V̇ ¬−
1

2
min(k2p,k

2
q,k
2
r)‖x‖

2
2 =−w1(x) (3.11)

wherew1(x) is apositive definite function.Equations (3.3) and (3.11) confirmthat the conditions
of theorem 4.18 (Khalil, 2001) are satisfied. This means that the states of Eqs. (3.2) become
GUUB for the state-feedback control law given by Eq. (3.5).
An important parameter is introduced: thp0 or the thicknesses of the boundary layer. This

parameter is selected such that the control inputs never reach the singular point and. at the
same time, the states reach the vicinity of equilibrium points with good quality.

Note 2: In order tomake the control input (Eq. (3.5)) smoother, especially near the equilibrium
point, the terms that contain p in their denominator are neglected. As will be shown in
the simulation Section, this simplification leads tomore implementable control inputs and
still provides GUUB of the states.



1062 R.Moradi et al.

Depending on the magnitude of p and the value chosen for thp0, two different cases occur
during simulations:
— for |p|>thp0

u1 =
−α3krqr

kp
−kpp−α1qr−dp

u2 =−kqq+
d

c+d

kpkr
α3

r

p
−α2pr−dq

(3.12)

— for |p| ¬ thp0

u1 =
−α3krqr

kp
−kpp−α1qr−dp

u2 =−kqq−α2pr−dq

(3.13)

The above controllers are extended forms of LSB. Therefore, they will be called ELSB. In
order to verify the analytical results, several simulations are carried out, and the results are
presented in the next Section.

4. Simulations

The system and controller parameters are presented in Table 1. c and d are selected as 1
and−0.94 to −0.90, respectively.

Table 1. System and controller parameters

Initial conditions Boundary layer Controller Moments of inertia
[deg/s] thickness [deg/s] coefficients [kgm2]

p0 =8 kp =0.05 Jx =449.5

q0 =−6 thp0 =0.1 kq =0.1 Jy =264.6

r0 =7 kr =0.1 Jz =312.5

Three scenarios are considered for simulation. These scenarios are presented in Table 2.

Table 2. Simulation scenarios

Scenario Controller Disturbance

First scenario LSB step

Second scenario ELSB step

Third scenario LSB sinusoidal

Acomparison of thefirst and second scenarios analyzes the capabilities of theLSBandELSB
in dealing with a large step disturbance. A comparison of the first and third scenarios shows
severity of the step disturbance compared to the sinusoidal disturbance.

First scenario

The component of disturbances exerted on the unactuated axis is considered as the following
step function

dr =
1

Jz
(4.1)

Therefore, the external disturbance (Tdr) exerted on the unactuated axis is 1Nm.



Ultimate state boundedness of underactuated spacecraft... 1063

As stated inAssumption 1, thedisturbances exerted on the actuated states arenot important
as long as they are known. Therefore, they are not considered here.

The LSB response is illustrated in Figs. 1 and 2.

As illustrated in Fig. 1, LSB is not able to provide GUUB for the states. The reason can
be explained by Eq. (2.22). The controller tries to make the origin an asymptotically stable
equilibrium for the closed-loop system.However, due to the presence of a large step disturbance
on the unactuated axis, r increases in an unacceptable way. This important example shows the
adverse effects of not considering large non-matcheddisturbances in the controller design.Due to
the non-smooth nature of the control inputs, a jump in the control moment is observed (Fig. 2).

Note 3: In order to reduce the adverse effects of sudden changes in the controlmoments and to
make them more implementable, thickness of the boundary layer should increase, at the
expense of less response quality.

Second scenario

In order to provide GUUB, the ELSB is used. The response and control moments of this
controller are illustrated in Figs. 3 and 4:

According to Fig. 3, the ELSB makes the closed-loop system GUUB. As stated in Note 2,
the simplifications have led to implementable control moments. At the same time, the states are
bounded.

Third scenario

It is assumed that the following sinusoidal disturbance is exerted on the unactuated axis

dr =
1

Jz
sin
(2π

50
t
)

(4.2)

In comparison to Eq. (4.1), the amplitudes of the disturbances are the same.

The results of the simulation are shown in Figs. 5 and 6.

A comparison of Fig. 1 and Fig. 5 shows that the closed-loop system is bounded for the
sinusoidal disturbance. However, for the step disturbance, the response of the unactuated axis
becomes unbounded.

Finally, it can be concluded that the LSB makes the origin asymptotically stable for the
disturbance-free closed-loop system. Therefore, in the absence of non-matched disturbances, the
origin is asymptotically stable.However, this controllerwill not provide satisfactory performance
when large non-matched disturbances are considered. On the other hand, the ELSB attenuates
the effects of non-matched disturbances andmakes the closed-loop systemGUUB.

5. Conclusion

A mechanical system is underactuated when the number of independent control inputs is less
than the number of degrees of freedom to be controlled. The presence of uncontrollable modes
in their linearized models prevents them from being smooth state feedback stabilizable. The
problems increase when disturbances, especially of the non-matched type, enter into the pro-
blem. In this paper, non-smooth time-invariant state feedback control laws have been obtained
that made the origin asymptotically stable for underactuated spacecraft. Then, these control
laws have been extended to make the closed-loop system GUUB. Simulation results have been
presented to verify the analytical solutions.



1064 R.Moradi et al.

Fig. 1. Response of the LSB (first scenario)

Fig. 2. Control moments



Ultimate state boundedness of underactuated spacecraft... 1065

Fig. 3. Response of the ELSB (second scenario)

Fig. 4. Control moments



1066 R.Moradi et al.

Fig. 5. Response of the LSB (third scenario)

Fig. 6. Control moments



Ultimate state boundedness of underactuated spacecraft... 1067

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Manuscript received April 26, 2016; accepted for print April 10, 2017