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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

dq=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

2r2δ 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

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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=sTJ˙

ω−˙

ωd+rsgn[δq4]δ˙

ρ+˜

JTΓ˙

˜

J

=sTFJ+u−˜

JTΓ˙

ˆ

J

=sTF˜

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

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7 Copyright c

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