Adaptive Sliding Mode Spacecraft Attitude Control

Abstract

An adaptive sliding mode spacecraft attitude controller is derived in this paper. It has the advantage of not requiring knowledge of the inertia of the spacecraft, and rejecting unexpected external disturbances, with global asymptotic position and velocity tracking. The sliding manifold is designed using optimal control analysis of the quaternion kinematics. The sliding mode control law and the parameter adaptation law are designed using Lyapunov stability. Numerical simulations are performed to demonstrate both the nominal and the robust performance.

Full text

Proceedings of the ASME 2014 Dynamic Systems and Control Conference
DSCC 2014
October 22-24, 2014, San Antonio, TX, USA
DSCC2014-5979
ADAPTIVE SLIDING MODE SPACECRAFT ATTITUDE CONTROL
Yizhou Wang, Xu Chen∗
, Masayoshi Tomizuka
Mechanical Systems Control Laboratory
Department of Mechanical Engineering
University of California
Berkeley, California 94720
Email: {yzhwang, maxchen, tomizuka}@berkeley.edu
ABSTRACT
An adaptive sliding mode spacecraft attitude controller is
derived in this paper. It has the advantage of not requiring
knowledge of the inertia of the spacecraft, and rejecting unex-
pected external disturbances, with global asymptotic position
and velocity tracking. The sliding manifold is designed using op-
timal control analysis of the quaternion kinematics. The sliding
mode control law and the parameter adaptation law are designed
using Lyapunov stability. Numerical simulations are performed
to demonstrate both the nominal and the robust performance.
1 INTRODUCTION
The attitude control problem has attracted much attention as
it involves highly nonlinear characteristics of the governing mo-
tion equations. From the perspective of control, feedback control
laws are sought for the purpose of asymptotic trajectory tracking,
with the ability to reject unexpected external disturbances, and
be insensitive to parameter variations. Previous efforts have
been devoted to developing both open-loop and closed-loop
control strategies. Although the open-loop formulation makes
it easier to incorporate some optimal criterion, the resulting
performance is inevitably sensitive to system uncertainties.
Closed-loop control has been investigated to deal with both
single-axis small angle rotations and three-axis large angle
maneuvers. The latter problem is much more challenging as a
larger region of operations makes the linear approximation of
nonlinear dynamics invalid.
The simplest large-angle maneuver uses quaternion and
∗Xu Chen is now an Assistant Professor in Department of Mechanical Engi-
neering, University of Connecticut.
velocity feedback similar to a proportional derivative con-
troller [1]. Model-based control techniques are also investigated
such as sliding-mode control [2], and adaptive control [3].
Sliding-mode control was investigated for the purpose of robust
attitude tracking for various attitude parameterizations (Ro-
drigues parameters [4,5], Modified Rodrigues parameters [6],
quaternions [7–9]). Adaptive attitude tracking control based
on Lyapunov stability was studied for quaternions [10,11] and
rotation matrices [12].
In this paper, we develop an adaptive sliding-mode attitude
tracking controller. A similar methodology has been applied to
control of robot manipulators by Slotine et. al. [13]. Although
the attitude dynamics cannot be directly transformed to the
form of robot dynamics, the design of sliding manifold and the
construction of a Lyapunov function in the present work achieve
similar performance specifications. Compared with the existing
sliding-mode attitude controller, our approach does not require
any inertial information. The use of a sliding manifold reduces
the design complexity and makes the controller have a simple
form.
Unit quaternion is used to parameterize rotations since it is
the minimal singularity-free rotation representation. Based on
the quaternion kinematic relation, a sliding manifold is chosen
according to the optimality criterion proposed in [14]. The
dynamic equations of motion of the spacecraft actuated by
either thrusters or momentum wheels are considered. Global
asymptotic stability is shown using Lyapunov stability analysis.
The remainder of the paper is organized as follows. In
Sec.2, the quaternion kinematics and the spacecraft dynamics
1 Copyright c
2014 by ASME

FIGURE 1. Definitions of the coordinate frames
are reviewed. In Sec.3, an optimal sliding manifold is presented
along with its optimality proof and stability analysis. Then, a
Lyapunov stability analysis is used to derive an asymptotically
stabilizing sliding control law and a parameter adaptation law.
A robust controller is designed to reject external disturbances.
In Sec.4, numerical simulations are shown to demonstrate the
closed-loop performance of the proposed controller.
2 PRELIMINARIES
2.1 Coordinate frames
We define three coordinate frames of interest. The inertial
frame of reference, in which Newton’s law is satisfied, is
denoted as Fr. We attach three mutually perpendicular axes
to the spacecraft, and call this the body-fixed frame Fb. The
spacecraft is modeled as a rigid body actuated by either thrusters
or momentum wheels in three orthogonal directions. The body
axes are chosen to coincide the directions of actuations. The
desired spacecraft attitude is described by a frame denoted Fd.
The frame definitions are depicted in Figure 1.
2.2 Kinematics
The unit quaternion is used to describe the spacecraft atti-
tude,
q=ρ
q4=ˆesin(θ/2)
cos(θ/2)(1)
where ˆeis a unit vector representing the axis of rotation, θis the
angle of rotation from Frto Fb. It has to satisfy the following
unity norm constraint,
||q||2
2=ρTρ+q2
4=1 (2)
If the angular velocity of the spacecraft with respect to Fr, ex-
pressed in Fb, is denoted as ω, the quaternion kinematic equa-
tion is given by,
˙q=1
2Ξ(q)ω(3)
where
Ξ(q) = q4I3×3+ [ρ×]
−ρT
[a×] =

0−a3a2
a30−a1
−a2a10

,a∈R3
The matrix Ξ(•)obeys the following properties,
ΞT(q)Ξ(q) = I3×3
ΞT(a)a=03×1,∀a∈R4
d
dt ΞT(q)˙q=ΞT(q)¨q
(4)
From these properties, one can show that if the desired attitude
trajectory is specified by qd= [ρT
d,q4d]T, the desired angular ve-
locity ωdmust obey,
ωd=2ΞT(qd)˙qd
˙
ωd=2ΞT(qd)¨qd
(5)
From the quaternion definition, one can see that qand −qrepre-
sent the same physical rotation. Hence compared with algebraic
subtraction, the error calculated from quaternion multiplication
provides a better way because it resolves the sign ambiguity. The
quaternion error and multiplication are defined as,
δq=q⊗q−1
d=Ξ(q−1
d),q−1
dq=δ ρ
δq4=ΞT(qd)q
qTqd
q−1
d=−ρd
qd4(6)
δqrepresents the rotation from Fdto Fb.
2.3 Dynamics
For a spacecraft having its three thrusters aligned with the
body axes, the dynamic equations of motion are given by
J˙
ω=−[ω×]Jω+u(7)
where Jis the positive definite inertial matrix of the spacecraft,
uis the torque generated by the thrusters.
2 Copyright c
2014 by ASME

If the spacecraft has three orthogonal momentum wheels
instead, the equations of motion become,
(J−Jw)˙
ω=−[ω×](Jω+Jwv)−u
Jw(˙
ω+˙v) = u(8)
where Jwis the diagonal inertial matrix of the wheels, vis the
wheel angular velocity, and unow is the torque applied to the
wheel.
Eqn. (3) and one of (7) or (8) complete the dynamic model of
the plant. The control objective is to make q→qdand ω→ωd
as t→∞.
3 METHODOLOGY
We apply sliding mode control for controller synthesis. The
idea of sliding mode control is to allow the transformation of a
controller design problem for a general n-th order system to a
simple stabilization problem with reduced order, i.e., stabilizing
the dynamics associated with the switching function. Then for
the equivalent reduced-order system, intuitive feedback control
strategies can be applied.
Sliding mode control design consists of two steps: (i) de-
sign a stable sliding manifold on which the control objective is
achieved, and (ii) design a reaching law and the corresponding
control input so that the switching function is attracted to 0.
Crassidis et. al. proposed an optimal sliding manifold [14]. The
optimality is evaluated when we only consider the quaternion
kinematic equation and treat ωas the input. The following
functional is minimized,
J∗(q(t),t) = min
ωZ∞
t
1
2r2δ ρ Tδ ρ + (ω−ωd)T(ω−ωd)dτ
(9)
subject to the kinematic constraint,
˙q=1
2Ξ(q)ω
and the endpoint constraint,
δq(∞) = [ 0,0,0,1]T(10)
where r>0 is the weighting factor, qdand ωdsatisfy Eqn.(5).
Without loss of generality, we only consider δq4(t)≥0.
There exists two main approaches to optimal control [15],
via the calculus of variations (the maximum principle) or
Lagrangian L(q,ω,t)1
2r2δ ρ Tδ ρ +1
2(ω−ωd)T(ω−ωd)
Lagrange multipliers λ∈R4
Hamiltonian H(q,λ,ω,t)L+1
2λTΞ(q)ω
State equation ˙q=1
2Ξ(q)ω
Costate equation ˙
λ=−∂H
∂q
Stationary condition ∂H
∂ ω =0
Boundary condition δq(∞) = [ 0,0,0,1]T
TABLE 1. The necessary conditions for optimality from calculus of
variations
dynamic programming (the principle of optimality).
3.1 Calculus of variations
In [16], the necessary conditions for optimality are derived
from calculus of variations. For the functional minimization
problem in Eqn.(9), the necessary conditions are summarized in
Table 1. It can be shown by direct substitution that the following
optimal angular velocity ω∗,
ω∗=ωd−rΞT(qd)q(11)
with λ∗=−2rqd, satisfies all the conditions except the boundary
condition. To prove the satisfaction of the boundary condition,
we use the kinematic equation for δq,
δ˙
ρ=1
2δq4(ω−ωd) + 1
2[δ ρ ×](ω+ωd)
δ˙q4=−1
2(ω−ωd)TΞT(qd)q
(12)
and a Lyapunov function candidate,
V=1
2δ ρ Tδ ρ (13)
The Lie derivative taken with respect to the kinematic relation is,
˙
V=−1
2rδq4δ ρ Tδ ρ ≤0 (14)
The Lyapunov function value will keep decreasing until δ ρ =
03×1and δq4=±1. From Eqn.(12) and the minimizer ω∗,δq4
can converge only to 1 since,
δ˙q4=1
2r(1−δq2
4)≥0,=0 only if δq4=1 (15)
3 Copyright c
2014 by ASME

Therefore, all the necessary conditions are satisfied. The optimal
value J∗(q(t),t)can be derived to be,
J∗(q(t),t) = Z∞
t
r2(1−δq2
4)dτ
=2rZ∞
t
r
2(1−δq2
4)dτ
=2rZ∞
t
δ˙q4τ
=2r(1−δq4(t))
(16)
3.2 Dynamic programming
We can also prove optimality by showing that ω∗satisfies
the following Hamilton-Jacobi-Bellman partial differential equa-
tion,
∂J∗
∂t(q,t) = −H(q,∂J∗(q,t)
∂q,ω∗,t)(17)
Proof : We expand ∂J∗
∂t(q,t)by the chain rule,
LHS =∂J∗(q,t)
∂qd
dqd
dt
=−rqTΞ(qd)ωd
(18)
On the other hand, by substituting ∂J∗
∂q=−2rqdinto the Hamil-
tonian, the right-hand side becomes,
H(q,−2rqd,ω∗,t) = r2δ ρTδ ρ −rqT
dΞ(q)(ωd−rΞT(qd)q)
=−rqTΞ(qd)ωd
(19)
3.3 Optimal sliding surface
For optimal tracking performance, it is natural to select the
following sliding manifold s(q,ω,t) = 03×1,
s(q,ω,t) = (ω−ωd) + rsgn[δq4]ΞT(qd)q=03×1(20)
Note that sgn[δq4(t)] is added for generality. The stability of
this sliding manifold has already been seen from the boundary
condition, i.e. q→qdas t→∞. In view of the sliding condition,
we can further show the velocity tracking,
ω=ωd−rΞT(qd)q
| {z }
δ ρ→0
→ωd,t→∞(21)
Therefore, both of the control objectives are satisfied as long as
qand ωare confined in the sliding manifold.
3.4 Linearity in system parameters
In Eqns (7) and (8), the inertia parameters Ji j ,Jwij , where
i,j=1,2,3, appear linearly. To make this more explicit, we fol-
low [10] to use the following linear operator L:R37→ R3×6act-
ing on any three-dimension vector a= [a1,a2,a3]Tby,
L(a) = 

a1000a3a2
0a20a30a1
0 0 a3a2a10

(22)
For J=JT, it follows easily that,


J11 J12 J13
J21 J22 J23
J31 J32 J33


|{z }
J
a=L(a)








J11
J22
J33
J23
J13
J12








| {z }
J
(23)
We denote the column vector of the inertia parameters as J.
3.5 Sliding mode controller and parameter adaptation
law
After finding a stable sliding manifold, we need to design a
sliding control law and a parameter adaptation law to make the
sliding manifold attractive. We propose the following laws for
the thruster model,
u=−Fˆ
J−Ks
˙
ˆ
J=Γ−1FTs
(24)
where
F=−[ω×]L(ω)−L(˙
ωd) + L(rsgn[δq4]δ˙
ρ)(25)
Γ,Kare constant positive-definite matrices of compatible
dimensions, ˆ
Jis the on-line estimate of the spacecraft inertia.
Proposition: The proposed control law achieves asymp-
totic trajectory tracking.
Proof : Assume that δq4is non-zero for a finite time, we
have the time derivative of the switching function,
˙s=˙
ω−˙
ωd+rsgn[δq4]δ˙
ρ(26)
Define the parameter estimate error to be ˜
J=J−ˆ
J. We assume
that the inertial matrix of the spacecraft Jis constant, then
˙
˜
J=−˙
ˆ
J(27)
4 Copyright c
2014 by ASME

Consider the following Lyapunov function candidate, which is a
positive-definite function of the switching function and the pa-
rameter error,
V=1
2sTJs +1
2
˜
JTΓ˜
J(28)
Its Lie derivative can be written as,
˙
V=sTJ˙
ω−˙
ωd+rsgn[δq4]δ˙
ρ+˜
JTΓ˙
˜
J
=sTFJ+u−˜
JTΓ˙
ˆ
J
=sTF˜
J−Ks−˜
JTΓΓ−1FTs
=−sTKs
(29)
which shows that ˙
Vis negative semi-finite. Hence sand ˆ
Jare
bounded. Invoke Barbalat’s lemma,
¨
V=−2sTK˙s(30)
The boundedness of ˙scan be seen by combing Eqns.(7), (24)
and (26). This implies the uniform continuity of ˙
V, hence we
conclude that ˙
V→0. Equivalently s→0 as t→∞. Furthermore,
to analyze the convergence of parameter estimation, we consider
J˙s+Ks =F˜
J(31)
All the terms except ˙sare uniformly continuous. Thus ˙sis uni-
formly continuous. From Barbalat’s lemma again, ˙s→0. There-
fore,
F˜
J→0 (32)
To enforce the asymptotic parameter estimation, i.e. ˜
J→0, the
following persistent excitation condition must be satisfied,
Zt+T
t
FT(δq,ω,ωd,˙
ωd)F(δq,ω,ωd,˙
ωd)dτ≥εI6×6,∀t≥to
(33)
where T,to,εare some positive scalars.
For the momentum wheel model, the sliding control law
and parameter adaptation laws are,
u=−Fˆ
J−Gˆ
Jw−Ks
˙
ˆ
J=Γ−1FTs
˙
ˆ
Jw=Γ−1GTs
G=−[ω×]L(v) + L(˙
ωd)−L(rsgn[δq4]δ˙
ρ)
(34)
The same stability analysis can be performed by using the fol-
lowing slightly modified Lyapunov function candidate V0,
V0=1
2sTJs +1
2
˜
JTΓ˜
J+1
2
˜
JT
wΓ˜
Jw(35)
where ˜
Jw=Jw−ˆ
Jwrepresents the estimate error of the wheel
inertia.
In summary, we have used the Lyapunov stability theory
to show that the sliding manifold is always attractive under the
proposed sliding control law and the parameter adaptation law.
The control objective is achieved. Note that we do not require
the knowledge of the inertial matrix. Also we note that although
the parameter adaptation law is converging to a constant, that
estimate does not necessarily converge to the true inertia of the
spacecraft. The system should be subject to persistent excitation.
3.6 Robust controller
To take into account unexpected external disturbances in
practice, we slightly modify Eqn. (7) by adding a combined dis-
turbance input dthat can be from air drag, solar pressure, gravity
gradient, magnetic field, spherical harmonics,
J˙
ω=−[ω×]Jω+u+d(36)
Although dis unknown, but its magnitude has known bounds
D∈R3,
|di(t)| ≤ Di,∀t>0,i={1,2,3}(37)
The robust sliding mode controller is given by,
u=−Fˆ
J−Ks −ksgn(s)(38)
where ki=Di+ηifor i=1,2,3 and ηi’s are non-negative con-
stants. With the same Lyapunov function used before, we can
show that the Lie derivative is now,
˙
V≤ −
3
∑
i=1
ηi|si|−sTK s (39)
Again, the state variables are guaranteed to reach the sliding
manifold regardless of unknown disturbances. To avoid control
chattering after reaching the sliding manifold, saturation func-
tions can replace sign functions [2].
4 Numerical Simulations
In this section, we show the proposed controller perfor-
mance through numerical simulations. The proposed controller
5 Copyright c
2014 by ASME

is used to control the attitude of the Microwave Anisotropy Probe
(MAP) spacecraft [6]. We assume that quaternions and angular
velocities are available for full-state feedback. The desired atti-
tude profile is specified in 3-1-3 Euler angles {φ,θ,ψ},
˙
φ=0.001745 rad/sec
θ=0.3927 rad
˙
ψ=0.04859 rad/sec
(40)
Then φ,ψcan be obtained by integration. The desired quaternion
trajectory can be computed by converting Euler angle parameter-
ization to unit quaternions,
qd=




sin(θ
2)cos(φ−ψ
2)
sin(θ
2)sin(φ−ψ
2)
cos(θ
2)sin(φ+ψ
2)
cos(θ
2)cos(φ+ψ
2)





(41)
By numerically differentiating qd, we can compute ˙qd,¨qdand
ωd,˙
ωdby Eqn. (5). We let the actual inertial matrix Jbe,
J=

20 5 1
5 17 3
1 3 15

(42)
Our proposed controller does not require knowledge of the inertia
of the spacecraft, so we use an initial estimate ˆ
J(0)of the inertia
with 30% error,
ˆ
J(0) = 

26 1.6 1.4
1.6 13 1.2
1.4 1.2 8.5

(43)
Furthermore, 90oerror angle is used along [1/√3,1/√3,1/√3]T
as the initial condition,
q(0) = 




1
√3sin(π/4)
1
√3sin(π/4)
1
√3sin(π/4)
cos(π/4)




⊗qd(0)(44)
and the spacecraft is at rest initially ω(0)=[0,0,0]T. The
controller parameters are set to be, r=3,K=10 ·I3×3,Q=I6×6.
4.1 Nominal performance
Without external disturbance, we have shown that the state
will be driven to the sliding manifold using Barbalat’s lemma,
0 5 10 15
0
0.5
1
1.5
2
2.5
||s(t)||
time (sec)
FIGURE 2. Plot of the norm of switching funciton s(t)which con-
verges to 0
0 10 20 30
0.2
0.4
0.6
0.8
1
q1
time (sec) 0 10 20 30
−0.5
0
0.5
1
q2
time (sec)
0 10 20 30
0
0.2
0.4
0.6
0.8
q3
time (sec) 0 10 20 30
0.4
0.6
0.8
1
q4
time (sec)
qd(t)
q(t)
FIGURE 3. Plot of q(t)and qd(t)showing asymptotic quaternion
tracking
and consequently the control objectives are achieved. The con-
vergence of the switching function is shown in Figure 2. The
asymptotic quaternion and velocity tracking performance are
plotted in Figures 3and 4respectively. The controlled thruster
torque is shown in Figure 5.
4.2 Robust performance
If the spacecraft is subject to disturbances, the robust sliding
mode controller in Eqn. (38) should be used. In our simulation,
the following disturbance is used,
d(t) = sin(t),−1,cos(t)T(45)
6 Copyright c
2014 by ASME

0 5 10 15
−0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
ω−ωd(rad/sec)
time (sec)
δω1
δω2
δω3
FIGURE 4. Plot of the error velocity δω =ω−ωdshowing asymp-
totic velocity tracking
0 5 10 15
−20
−15
−10
−5
0
5
time (sec)
u(N·m)
u1
u2
u3
FIGURE 5. Plot of the controlled thruster torque
Therefore, in the controller design Di=1,i={1,2,3}.ηis cho-
sen to be 03×1for simplicity. The robsut quaternion and velocity
tracking are shown in Figures 6and 7. The robust control input
is shown in Figure 8.
5 CONCLUSION
In this paper, an adaptive sliding mode attitude controller is
designed for asymptotic quaternion and velocity tracking, which
assumes no inertial information and can reject unknown external
disturbances. The stability was shown through a Lyapunov anal-
ysis. Both the nominal performance and the robust performance
are demonstrated in numerical simulations.
ACKNOWLEDGMENT
The authors would like to thank King Abdulaziz City of Sci-
ence and Technology for supporting this research.
0 10 20 30
0.2
0.4
0.6
0.8
1
q1
time (sec) 0 10 20 30
−0.5
0
0.5
1
q2
time (sec)
0 10 20 30
0
0.2
0.4
0.6
0.8
q3
time (sec) 0 10 20 30
0.4
0.6
0.8
1
q4
time (sec)
qd(t)
q(t)
FIGURE 6. Plot of q(t)and qd(t)with the presence of disturbances
0 5 10 15
−0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
ω−ωd(rad/sec)
time (sec)
δω1
δω2
δω3
FIGURE 7. Plot of the error velocity δω with the presence of distur-
bances
REFERENCES
[1] Wie, B., and Barba, P. M., 1985. “Quaternion feedback for
spacecraft large angle maneuvers”. Journal of Guidance,
Control, and Dynamics, 8(3), pp. 360–365.
[2] Slotine, J.-J. E., Li, W., et al., 1991. Applied nonlinear
control, Vol. 199. Prentice-Hall Englewood Cliffs, NJ.
[3] Narendra, K. S., and Annaswamy, A. M., 2012. Stable
adaptive systems. Courier Dover Publications.
[4] DWYER, III, T. A., and Sira-Ramirez, H., 1988. “Variable-
structure control of spacecraft attitude maneuvers”. Journal
of Guidance, Control, and Dynamics, 11(3), pp. 262–270.
[5] Chen, Y.-P., and Lo, S.-C., 1993. “Sliding-mode con-
troller design for spacecraft attitude tracking maneuvers”.
Aerospace and Electronic Systems, IEEE Transactions on,
7 Copyright c
2014 by ASME

0 5 10 15
−20
−15
−10
−5
0
5
10
time (sec)
u(N·m)
u1
u2
u3
FIGURE 8. Plot of the robust control input
29(4), pp. 1328–1333.
[6] Crassidis, J. L., and Markley, F. L., 1996. “Sliding mode
control using modified rodrigues parameters”. Journal of
Guidance, Control, and Dynamics, 19(6), pp. 1381–1383.
[7] Vadali, S., 1986. “Variable-structure control of spacecraft
large-angle maneuvers”. Journal of Guidance, Control, and
Dynamics, 9(2), pp. 235–239.
[8] Lo, S.-C., and Chen, Y.-P., 1995. “Smooth sliding-mode
control for spacecraft attitude tracking maneuvers”. Jour-
nal of Guidance, Control, and Dynamics, 18(6), pp. 1345–
1349.
[9] Robinett, R. D., and Parker, G. G., 1996. “Spacecraft eu-
ler parameter tracking of large-angle maneuvers via sliding
mode control”. Journal of guidance, control, and dynamics,
19(3), pp. 702–703.
[10] Ahmed, J., Coppola, V. T., and Bernstein, D. S., 1998.
“Adaptive asymptotic tracking of spacecraft attitude motion
with inertia matrix identification”. Journal of Guidance,
Control, and Dynamics, 21(5), pp. 684–691.
[11] Costic, B., Dawson, D., De Queiroz, M., and Kapila, V.,
2001. “Quaternion-based adaptive attitude tracking con-
troller without velocity measurements”. Journal of Guid-
ance, Control, and Dynamics, 24(6), pp. 1214–1222.
[12] Sanyal, A., Fosbury, A., Chaturvedi, N., and Bernstein, D.,
2009. “Inertia-free spacecraft attitude tracking with distur-
bance rejection and almost global stabilization”. Journal of
Guidance, Control, and Dynamics, 32(4), pp. 1167–1178.
[13] Slotine, J.-J. E., and Li, W., 1987. “On the adaptive con-
trol of robot manipulators”. The International Journal of
Robotics Research, 6(3), pp. 49–59.
[14] Crassidis, J. L., Vadali, S. R., and Markley, F. L., 2000.
“Optimal variable-structure control tracking of spacecraft
maneuvers”. Journal of Guidance, Control, and Dynamics,
23(3), pp. 564–566.
[15] S.S.Sastry, 1996. “Lectures in optimal control and dynam-
ics games”. Notes for the course EECS290A, Advanced
topics in control theory, University of California Berkeley.
[16] Crassidis, J. L., and Junkins, J. L., 2011. Optimal estima-
tion of dynamic systems. CRC press.
8 Copyright c
2014 by ASME