Quaternion-based adaptive attitude control of asteroid-orbiting spacecraft via immersion and invariance

Acta Astronautica 167 (2020) 164–180

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Research paper

Quaternion-based adaptive attitude control of asteroid-orbiting spacecraft via immersion and invariance

T

Keum W. Leea,∗, Sahjendra N. Singhb a b

Dept. of Electronic Engineering, Catholic Kwandong University, Gangneung, Gangwon 25601, Republic of Korea Dept. of Electrical and Computer Engineering, University of Nevada, Las Vegas, NV 89154, USA

ARTICLE INFO

ABSTRACT

Keywords: Adaptive attitude control Asteroid-orbiting spacecraft Adaptive quaternion trajectory regulation Immersion and invariance-based design Noncertainty-equivalence adaptive control Parameter identifier Lyapunov stability Backstepping design

The design of an attitude control system for an asteroid-orbiting satellite via immersion and invariance is the subject of this paper. It is assumed that the asteroid is rotating with a constant rate, and that the inertia parameters of the satellite and the coefficients in the spherical harmonic gravitational potential of the asteroid are not known. The objective is to regulate the quaternion trajectory of the satellite orbiting in an equatorial orbit. Based on the immersion and invariance (I&I) theory, a noncertainty-equivalence adaptive (NCEA) attitude control law is derived. For the design, a backstepping design process involving two steps is used, and filtered signals are constructed to overcome the difficulty in solving certain matrix inequalities of the I&I methodology. The control law includes a stabilizer and an identifier - designed separately. Unlike the classical certaintyequivalence adaptive (CEA) systems, here the estimated parameters include not only the signals obtained from an integral type update law, but also judiciously chosen nonlinear algebraic signals that yield stronger stability properties. By the Lyapunov stability analysis, it is shown that the quaternion trajectories of the disturbance input-free closed-loop system asymptotically converge to the equilibrium point. The control law is effective in regulating the attitude to the equilibrium point with minimal rotation of spacecraft. Also, for the model with disturbance input, uniform ultimate boundedness of system trajectories is established. Simulation results for the attitude control of spacecraft in orbit around asteroid 433 Eros are presented for illustration. These results show that the spacecraft achieves nadir pointing attitude despite uncertainties in the system dynamics.

1. Introduction Missions to asteroids and comets are extremely important for gaining insight into the history of the solar system. The orbital and attitude dynamics of spacecraft, under the influence of the gravitational potential of irregularly-shaped small bodies, are very complex. For a successful mission, it is essential to analyze the orbital motion and attitude perturbations of visiting spacecraft in the gravitational field of asteroids. In the past, the effect of gravitational forces of asteroids on spacecraft orbiting in their vicinity has been studied [1–6]. Researchers have also designed control laws for the orbit control of spacecraft around asteroids based on the optimal control theory [7], Lyapunov stability theory [8,9], variable structure control (VSC) theory [10,11], finite-time control theory [12], and extended state observer (ESO) design methodology [13]. The VSC systems are designed to include switching functions for nullifying the effect of uncertain functions. But the discontinuity in the control law can cause control chattering phenomenon. For the ESO design [13], an observer is constructed to obtain

∗

the estimate of velocity vector of the spacecraft and the lumped uncertain vector function for the controller implementation. Of course, the estimation error can be zero only if extremely high-gain feedback is used. Linear optimal control systems for hovering over a tumbling asteroid have been designed using periodic algebraic Riccati equation as well as Lyapunov-Floquet transformation [14]. An iterative-learning control law for asteroid proximity operations has been proposed in Ref. [15]. Also, adaptive and supertwisting control systems for orbit control around asteroids have been developed [16]. Recently, authors have designed control systems for formation flying on circular and elliptic orbits by exploiting the inherent Hamiltonian structure of the systems [17,18]. Hamiltonian structure-preserving control systems for the circular restricted 3-body problem (CR3BP) have been also designed in Refs. [19–22]. Interestingly, for the sun-Earth/moon-solar sail threebody system, the closed-loop system of Ref. [19] yields stable Lissajous orbits. The Hamiltonian structure-based controller of Ref. [21] - consisting of potential shaping, energy dissipation, and disturbance rejection signals - accomplishes libration point orbit control of the Earth-

Corresponding author. E-mail address: [email protected] (K.W. Lee).

https://doi.org/10.1016/j.actaastro.2019.10.031 Received 10 July 2019; Received in revised form 11 October 2019; Accepted 14 October 2019 Available online 06 November 2019 0094-5765/ © 2019 IAA. Published by Elsevier Ltd. All rights reserved.

Acta Astronautica 167 (2020) 164–180

K.W. Lee and S.N. Singh

moon-spacecraft system. Researchers have also studied the attitude motion of asteroid-orbiting spacecraft [23–27]. It is shown in Refs. [23–25] that nonuniform gravity field of asteroids can cause pitch attitude resonance for certain combinations of orbital and asteroid's angular velocities. A linear control law and an adaptive sliding mode control system for roll, pitch, and yaw angle control have been designed [25,28]. A nonlinear attitude tracking control law for the control of spacecraft with a large captured object (asteroid) has been derived [29]. Authors have analyzed orbitattitude coupled dynamics around small bodies and derived natural sun-synchronous orbit and sun-tracking attitude state [30]. Researchers have also focused on simultaneous position and attitude control of spacecraft in the vicinity of asteroids, and designed control laws for asymptotic and finite time control [31,32], adaptive sliding mode control [33], and robust adaptive control [34]. A Lyapunov-based nonlinear feedback law for orbit and attitude control has been presented in Ref. [35]. Based on a non-canonical Hamiltonian structure of the coupled orbit-attitude dynamics in the proximity of an asteroid, a hovering control law - including potential shaping and energy dissipation signals - has been derived [36]. Recently, Astolfi and Ortega [37] and Astolfi et al. [38] have developed a new noncertainty-equivalence adaptive (NCEA) control system design methodology for the control of uncertain nonlinear systems, based on the notion of immersion and invariance (I&I). The parameter estimates of the I&I-based NCEA systems differ from those of the classical certainty-equivalence adaptive (CEA) systems in an important way. Unlike the certainty-equivalence adaptive (CEA) systems [39], the parameter estimates of the NCEA laws consist of the estimates obtained by an integral type dynamic update law as well as certain judiciously selected state-dependent nonlinear algebraic functions. These algebraic functions of the identifier provide stronger stability properties in the closed-loop system, compared to the classical CEA control systems. But for the design of identifier based on the I&I theory, certain partial differential equations must be solved. This poses obstacle in the design of NCEA controller. Seo and Akella [40] used appropriate filtered signals for synthesis to avoid this obstacle and designed a quaternion-based NCEA attitude control system for a rigid body. The stringent integrability obstacle of the I&I methodology can also be overcome by introducing dynamic scaling parameter in the estimator dynamics [38]. Recently, authors of Ref. [41] have designed a filter-free and a filter-dependent NCEA control system for the attitude control of spacecraft and control of robotic systems by the use of dynamic scaling. For the pose control of a 6-DOF Earth-orbiting spacecraft described by dual quaternion, a NCEA law has been developed [42]. Also, a filterfree NCEA attitude control law has been designed which gives faster tracking error convergence and smaller control signal compared to the filter-dependent control system [43]. An adaptive control system based on the I&I theory have been also designed for the spacecraft formation control [44]. Recently, I&I-based hovering control laws, using spherical harmonics and inertia dyadic gravitational potential models, have been developed [45,46]. Also, a Euler angle-based NCEA attitude control system for the roll, pitch, yaw angle control of a spacecraft orbiting around an asteroid has been designed [47]. But there exists singularity in the attitude kinematics for any choice of set of Euler angles, and the control law cannot accomplish regulation of the attitude angles to the singular point. Of course, one can describe the orientation of the spacecraft using quaternions in order to avoid the singularity problem. It appears from literature that I&I-based attitude control system for spacecraft orbiting around irregularly-shaped asteroids using quaternions has not been developed yet. Thus, in view of the strong stability properties of the NCEA system, it is important to explore the applicability of the immersion and invariance theory for the design of quaternion-based attitude control systems for asteroid-orbiting spacecraft. In this paper, focus is on the development of an adaptive control system for the attitude control of an asteroid-orbiting spacecraft. For

the purpose of design, it is assumed that the inertia parameters of the satellite and the gravitational coefficients appearing in the harmonic expansion of the gravity field of the asteroid are not known, and that the asteroid is rotating with a constant angular rate about a fixed axis. The contribution of this paper is threefold. First, an immersion and invariance (I&I)-based adaptive control law is derived for the attitude control of an asteroid-orbiting spacecraft in the presence of uncertainties. For the purpose of design, quaternion is chosen for the attitude parameterization. The design is completed in two steps of a backstepping design process. It is noted that the attitude dynamics around asteroids are complicated, compared to the rotational dynamics of a rigid body or an Earth-orbiting spacecraft, considered for NCEA law design in Refs. [40–43]. Second, by the Lyapunov analysis, asymptotic regulation of the quaternion trajectories of the disturbance input-free system to the target point is established. Interestingly, it is seen that the closed-loop system's trajectories are eventually confined in an attractive manifold. This special property cannot be achieved by CEA systems. Furthermore, this NCEA control law is effective in regulating the attitude to the equilibrium point with minimal rotation of spacecraft. Also, it is shown that for the model perturbed by external disturbance input, the trajectories of the system are uniformly ultimately bounded. Third, the attitude control of a satellite orbiting around asteroid 433 Eros is considered and numerical results are presented. The simulated responses show that this NCEA law achieves desired quaternion trajectory control of the spacecraft orbiting in circular and elliptic prograde or retrograde orbits, despite large initial attitude errors, uncertainties in the model parameters, and the presence of random disturbance inputs. 2. Spacecraft attitude dynamics and control problem Fig. 1 shows a rigid spacecraft in an equatorial orbit around a rotating asteroid. For describing the orientation of the spacecraft relative to the asteroid, following coordinate frames are defined. An inertial frame is defined by ( XI , YI , ZI ) with its origin at the center of mass of the asteroid. The asteroid is rotating about axis ZI normal to the equatorial plane with a constant rate . An orbital frame and a satellite body-fixed frame, both centered at the center of mass of the spacecraft, are defined by ( X0 , Y0, Z0 ) and ( XB , YB, ZB ) , respectively. The Z0 axis points towards asteroid's center of mass, X0 axis is in the transverse direction in the orbital plane, and Y0 is normal to the orbital plane, according to the right-hand rule (Y0 = Z0 × X0 ). For equatorial orbits, the orbital frame is obtained by a simple rotation of the inertial frame through an angle η, the true anomaly. The axes of the frame ( XB , YB, ZB ) are aligned with the principal axes of the spacecraft. Also, a frame (XI ,

Fig. 1. Asteroid-orbiting spacecraft. 165

Acta Astronautica 167 (2020) 164–180

K.W. Lee and S.N. Singh

YI , ZI ) (not shown in the figure) fixed to the asteroid is defined. Its axis ZI is aligned with ZI . In this study, the orientation of the satellite body-fixed frame ( XB , YB, ZB ) with respect to the orbital frame ( X0 , Y0, Z0 ) is parameterized by the unit quaternion qa = (qT , q4)T R4 , where the vector part of qa is q = (q1, q2 , q3)T R3 and the scalar part is q4 . It satisfies the constrained relationship qa 2 = qT q + q42 = 1. The direction cosine matrix CBO = [c1, c2, c3] R3 × 3 from the orbital frame to the satellite body-fixed frame is given by Ref. [48].

CBO (qa) = (q42 where ci Q (q) is

qT q) I3 × 3 + 2qqT

0 ×

U=

0 q3 q2

q3 0 q1

q2 q1 0

CBO (qa) =

2(q22 + q32 ) 2(q1 q2 + q3 q4)

2(q2 q1

q3 q4) 1

2(q3 q1 + q2 q4)

=[

r1,

r2,

T r 3]

=

2(q1 q3

q2 q4)

2(q12 + q32) 2(q2 q3 + q1 q4)

2(q3 q2

2(q12 + q22)

q1 q4) 1

(2)

0 =

CBO (qa)

0

=

(3)

+ c2

r3

r2

r2

Mg1 =

r1

0

r1

=

5 2

=

a1 (qa ) p

J3) c13 c33 +

5 2

{

2 Jc 5 2 22

+ (J1 + J2

J3) c11 c33

(J2 + J3

(3 + 5 )(J2

J1) c13 c23 +

5 2

{

2 Jc 5 3 32

+ (J2 + J3

J1) c13 c21

(J1

(3 + 5 )(J3

{ 0.4J1 c12

(J1 + J2

J2 ) c23 c33 J3 ) c21 c33 + (J1

J2 + J3 ) c23 c31}] (8)

J1) c13 c31

}

J2 + J3) c11 c23

}

b1 (qa) p

Mg3 = =

µ Rc3

+

(3 + 5 )(J1

µ R c3

(7)

3) + ….]

where Ma is mass of asteroid. Keeping only the most significant gravitational coefficients, the gravity gradient torque components Mgk , (k = 1,2,3) , obtained by Reyhanoglu et al. [35], are

The attitude kinematic differential equations of the satellite relative to the orbital frame can be written in terms of quaternion as

Mg 2 =

1) + 3(r0/ R)2C22cos2 cos(2 )

µ = GMa

(4)

× r CB 0

0

r3

+ 0.5(r0 /R)2C20 (3sin2

where c is the longitude of center of mass of satellite, and R c is radial distance of the center of mass of the satellite from the center of mass of the asteroid. (φ and χ will be used to write the gravitational torque in a compact form.) The asteroid's rotational rate is assumed to be a constant. The gravitational constant μ is

where × r

µ [1 R

= [ 1.5C20 + 9C22cos(2 c )](r0/ R c )2 = 6C22sin(2 c )(r0 / Rc ) 2

The direction cosine matrix CBO satisfies the following differential equation:

CBO = [c1, c2, c3] =

1

0

where μ is the asteroid gravitational parameter, R is the distance of the orbiting particle from the center of primary, r0 is the characteristic length, and δ and λ are the latitude and longitude of the orbiting particle measured in the asteroid-fixed frame. The spherical harmonic coefficients Cj0 , (j = 2,3, ..) and C22, …, characterize the oblateness and equatorial ellipticity of an asteroid, respectively. Here, for simplicity in design, only the most significant coefficients (C20 and C22 ) will be considered. Let the satellite's orbit is within the equatorial plane. Define gravitational coefficient-dependent functions φ and χ as

Let the angular velocity of the satellite relative to the inertial frame, expressed in the body frame, be = [ 1, 2 , 3]T . The angular velocity of the orbital frame with respect to the inertial frame, expressed in the , 0]T , where η is the true anomaly. Then the orbital frame, is 0 = [0, relative angular velocity r of the spacecraft with respect to the orbital coordinate system (X0 , Y0, Z0 ) can be written as [35]. r

1

+ 0.5(r0/ R)3C30sin (5sin2

By expanding, Eq. (1) gives

1

2

The researchers have derived polyhedron, spherical harmonics, and inertia dyadic models to represent gravitational potential of asteroids. For simplicity, in this study, gravitational torque obtained by Reyhanoglu et al. [35] - based on spherical harmonics model of the gravitational potential - is considered. However it is noted that the design methodology of this paper is applicable to attitude dynamical models, derived using other gravity field representations. The harmonic expansion of the gravitational potential U of any arbitrary primary can be expressed as

R3 denotes ith column of CBO and the skew symmetric matrix

Q (q ) =

3

0

3 2

(1)

2q4 Q

=

µ Rc3

(9)

c1 (qa ) p

1

q = 2 [q4 q4 =

r

1 T q 2

where

+ Q r]

p = [J1, J2, J3, C20 J1, C20 J2, C20 J3, C22 J1, C22 J2 , C22 J3]T

(5)

r

is assumed to be an unknown parameter vector, and j1 R9 × 1, j {a, b, c} , are appropriate functions and regressor vectors. (The explicit expressions of these functions are given in the appendix.) T R3 × 9 . Then the gravity gradient torque Define 1 = [ aT1, bT1, cT1] and the rotational dynamics can be written as

The Euler's rotational equations of motion of the spacecraft are given by

J =

×J

+ Mg ( ) + u + d

R9

(6)

where u = [u1, u2 , u3 is the vector of control inputs, J = diag {J1, J2, J3} is the diagonal principal inertia matrix of the satellite, Mg = [Mg1, Mg 2, Mg 3]T is the gravity gradient torque vector, and

]T

Mg = 166

1 (qa ) p

(10)

Acta Astronautica 167 (2020) 164–180

K.W. Lee and S.N. Singh ×J

J =

+

1 (qa ) p

therefore q4 will converge to one. However, if e is zero and <0 , then q4 0 , and therefore q4 will converge to −1. For e (t ) 0 , suppose that k1 is selected sufficiently large so that the following inequalities hold:

(11)

+u+d

The objective in this paper is to design an attitude control law such that in the closed-loop system, the quaternion trajectory asymptotically converges to the target value (q = 0, q4 = ± 1) , and the relative angular velocity r = + c2 converges to zero, despite large uncertainties in the inertia parameters Ji and the harmonic coefficients Ckj . Apparently, then the spacecraft will achieve nadir pointing attitude. It is pointed out that the design approach of this paper is also applicable if higher-order gravitational coefficients are included in the gravitational potential; that will simply increase the dimension of the regressor vectors. It is noted that although here nadir pointing attitude control is considered, the design method can be easily extended for any prescribed smooth time-varying quaternion trajectory tracking control.

k1 q (t ) >

r

J =

V1 (qa) =

+ (q4

J

e

+

=

(0)))2]

=2

2p

e

v

ef

in Eq. (15)

1 1 k1 q4 q + [q4 I3× 3 + Q] 2 2

q4 =

1 k1 q 2

2

1 T q 2

=

=J 1[

= =

¨ef =

¨ ef = e

According to Eq. (19), if

(19) e

is zero and

>0 , then q4

× r c2

v

=

+ c2

v.

(22)

k1 q]

>0

+

1p

× r c2

+ u + J [ ¨c2

k1 q + k2

e

+ k3 (s + ) q] (23)

k3 (s + ) q]

e

×J

× r c2

+ J [ ¨c2

1 [(

+

1

2) p

p + u]

k2

+ u]

k1 q + k2

e

+ k3 (s + ) q]

k2

(24)

k3 (s + ) q

e

(25)

k3 (s + ) q

e

f

(26)

+

ef

+

(27)

e

(28)

uf + u

ef

+J

1[

p + u]

Now, substituting for gives

(18)

e

(21)

where the initial values f (0) , ef (0) , and uf (0) are arbitrary. Note that Eq. (28) will be used only for the purpose of analysis. Differentiating Eq. (27) and substituting Eq. (25) for e gives

Note that if e = 0 , then q will tend to zero. This also implies convergence of q4 to one, because quaternion satisfies qa = 1. Note that for q = 0 , CBO = I3 × 3. Substituting Eq. (16) in Eq. (13), one obtains

q=

+ k3 (s + ) q],

×J

=

uf =

(17)

e

+ Q r]

where is a new regressor matrix. To this end, certain filtered signals are used to avoid limitation of the I&I methodology as follows:

(16)

k1 qT q + qT

e

=J

f

k1 q

r

Then, in view of Eq. (24), the error dynamics Eq. (23) can be written

as

(15)

v)

1 [q 2 4

Define

(14)

2q4

where the gain k1 is a positive constant. Then, substituting gives

V1 =

k1

+ J [ k2

In view of Eq. (15), a stabilizing signal is selected as v

e

(13)

where = sign (q4 (0)) . Inclusion of in the Lyapunov function and sliding variable has been also considered earlier in Refs. [49,50]. Note that q4 = 1 and q4 = 1 represent same orientation. Thus, it is sufficient to regulate qa to [0,0,0,1]T if q4 (0) > 0 , and to [0,0,0, 1]T if q4 (0) < 0 . In this way, the spacecraft will execute minimal rotation to arrive at q4 = 1 or q4 = 1. The derivative of V1 along the solution of Eq. (13) is

V1 = qT (

k1 q =

in Eq. (22), where s is the Laplace variable or a differential operator, and k2 and k3 are positive gains. The gain k2 provides damping in the system, and the parameter k3 will be used to eliminate the effect of signindefinite function qT e , which appears in V1 (Eq. (17)) in step 1. Note that one has (s + ) q = q + q . Then, using Eq. (10) for Mg , Eq. (22) takes a modified from

v]

sign (q4

e (t )

= J + J [ ¨c2 + c2 v] ×J + M + u + J [ ¨ c g 2

e

J [k2

Now, consider a Lyapunov function V1 (qa) given by

[qT q

(20)

Now, similar to Ref. [40], let us add and subtract

is a virtual stabilizing signal. Then using the kinematic where v differential equation, one obtains

+

] > 0, if > 0 ] < 0, if < 0

e

Now, consider the dynamics of e = r Differentiating e , and using Eqs. (6) and (21) gives

R3

1 q = 2 [q4 I3 × 3 + Q][ e 1 q4 = 2 qT ( e + v )

=

v

(12)

v

e

Then also attitude regulation to the equilibrium state with minimal rotation of the spacecraft will be possible. In the next step, u is designed to force e to zero. Step 2: Differentiating v in Eq. (16) and using Eq. (5) gives

In this section, a noncertainty-equivalence adaptive attitude control law based on the I&I methodology [38] for nadir attitude pointing is developed. However here certain filters are introduced as proposed in Ref. [40] to circumvent the stringent requirement related to the solution of certain partial differential equations and matrix inequality of [38]. The design is completed in two steps of a backstepping design process. For the derivation of the control law, first, it is assumed that the disturbance torque d is zero. (Later a simple modification in the control law can be made to achieve robustness in the closed-loop system.) Step 1: In the first step, it is desired to regulate qa to the equilibrium point (q = 0, q4 = 1) or (q = 0, q4 = 1) using a virtual control input. Define a new coordinate as

=

qT e] = q [k1 q qT e] = q [ k1 q +

2

Of course, these two inequalities hold if

3. Noncertainty-equivalence adaptive control system

e

2

min [ k1 q max [ k1 q

ef

+J

1 [(

f

+

k2 ,

e,

f )p

e

and u in Eq. (29) from Eqs. (26)–(28)

+ uf + uf ]

k3 (s + ) q Then Eq. (30) can be written as

0 , and 167

(29)

k3 (s + ) q

k2 (

ef

+

ef )

(30)

Acta Astronautica 167 (2020) 164–180

K.W. Lee and S.N. Singh

d [ dt

1(

J

ef

fp

+ uf ) + k2

ef

+ k3 q] = [

J

ef

1(

fp

+ uf ) + k2

ef

+ k3 q]

k2

k3 q +

ef

(32)

o (t )

=[

ef

(0)

1(

J

f (0) p

+ uf (0)) + k2

ef

(0) + k3 q (0)] e

ef

=J

fp

+ uf )

k2

with

+ (

ef ,

ef

=

1

J

fz

k2

k3 q

ef

ef )

T ef

= V1 + [

ef /2]

Vs = =

k1

k1 q

qT

+

2

e

k2

T ef

+ 2

ef

[ J T ef

+

1

fz 1

[ J

k2

ef

Using Eq. (27), one can write the last term

qT

e

qT [

=

ef

+

e

= qT [ J

1

e

e

fz

k3 q +

ef

1 k1 q 2

Vs =

k1 q

2

(k3 + k2

2

2

+ k2

k2

ef

) qT

ef

+ (0.5k1 1 + 0.5k 2 1)

ef

2

[ k3 q

T ef

+ qT ] J

2

1

2 1 max (J )

(37)

T 1 z f ef J

qT J

1

fz

k2 2 k1 q 2

+

ef

2

+

J

J

1

1

fz

2k1

(46)

k3 q]

ef

(47)

1zT z /2,

(48)

>0

zT

T 1 fz fJ

min (J

1)

2

fz

(49)

ef ,

(50)

z ) = Vs + Ve

1

fz

1)

max (J

fz

the derivative of V can be bounded as 2 min (J

1)

fz

2

k3 q

(51)

Because > 0 is a free parameter, for a given l * > 0 , one can select ν to satisfy the following inequality:

fz

[(0.5k1 1 + 0.5k 2 1)

2 1 max (J )

min (J

1)]

(52)

l*

Thus, using Eq. (52) in (51) gives

2

V

(41)

2k2

k2

Then using Eqs. (43) and (49), and noting that

J

(40)

fz

T f[

(45)

k3 q]

ef

T 1 fz fJ

V (qa ,

Using Young's inequality, one has 2

(see Eq. (26)). Now an update law is selected as

k2

where min (.) [ max (.)] denotes minimum [maximum] eigenvalue of a matrix (.), respectively. Because Ve is a positive definite function of z and Ve 0 , it follows that the equilibrium point z = 0 is globally uniformly stable. Now, for the stability analysis of the closed-loop control system, consider a composite Lyapunov function

Substituting Eq. (39) in (37) and arranging terms gives

V

+

fz

Ve =

(39)

ef ]

f

1

J

(36)

from Eq. (35) in (38) gives

ef

T f[

The selected function Ve (z ) provides a measure of the parameter p . Its derivative along the solution of Eq. (47) takes error z = pˆ + the form

(38)

k2

+

(35)

as

ef ]

Now, substituting for

qT

qT

will be

T f ef

=

Ve (z ) =

k3 q]

k3 q] + qT

fz

ef

For the stability analysis of the identifier, consider a Lyapunov function

Then the derivative of Vs can be written as

qT q

= 0 , then q and

(44)

T f ef

z=

For the stability analysis, consider a Lyapunov function

Vs (qa ,

fz

Substituting Eq. (46) in (45) gives

p . Then sub-

Define the parameter vector error as z = pˆ + stituting the control signal Eq. (34) in (33) gives

f

pˆ =

(34)

f ))

2

k3 q

> 0 . Consider the dynamics of the parameter error p . Its derivative is

z = pˆ +

The parameter vector p is not known. Consider an estimate of p as pˆ + ( ef , f ) R9 , where pˆ is obtained by an integral law, and β is an algebraic function. The inclusion of β provides stronger stability properties. In view of Eq. (33), a control law is selected as

ˆ f (p

fz

ef

where z = pˆ +

3.1. Control law

uf =

T f

=

(33)

k3 q

ef

1

Now, an identifier is designed to obtain an adaptation law for pˆ . The algebraic part of the parameter estimate is selected to be

t

In the following analysis, for simplicity, this exponentially decaying signal will be ignored because asymptotic properties of the closed-loop system do not depend on o [39,40]. Setting o to zero, Eq. (32) gives 1(

J

3.2. Parameter estimation and closed-loop stability

where o (t )

2

1 1 + 2k1 2k2

+

ef

In view of Eq. (43), it follows that if bounded.

The solution of this linear differential equation is

= J 1 ( f p + uf )

2

+ k2

(43)

(31)

ef

2

1 k1 q 2

Vs

1 k1 q 2

2

2

+ k2

ef

2

l*

fz

2

k3 q

(53)

Because V is a positive definite function of (qa , ef , z ) and V 0 , it follows that qa , ef , and z are bounded. Integrating Eq. (53), one finds that q , ef , and f z are square integrable functions. Also, the derivatives of (q, ef , f z ) are bounded. Thus, invoking Barbalat's lemma, it follows that q, ef and f z converge to zero. This also implies convergence of e to zero. As q and e converge to zero, v (see Eq. (16)) and the relative angular velocity vector r tend to zero. Then

2

(42)

By selecting the controller gains and the filter pole such that k3 + k2 = 0 , and using the inequalities Eqs. (41) and (42), Eq. (40) can be written as 168

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Fig. 2. Uncontrolled spacecraft in circular prograde orbit (R c = 50 ): quaternion trajectory.

according to Eq. (3), it follows that ω converges to the angular velocity , 0]T of the orbital frame. Of course, convergence of q vector 0 = [0, to zero implies that q4 asymptotically tends to ± 1, and therefore the spacecraft achieves nadir pointing equilibrium attitude. Based on these arguments, the following theorem can be stated.

systems [39], such z dependent terms do not appear in the Lyapunov derivative. Now for the implementation of the controller, u is determined. In ˆ + ) view of Eq. (28), one has u = uf + uf . Substituting uf = f (p from Eq. (34) in this equation gives

Theorem 1. Consider the closed-loop system including the spacecraft model Eqs. (5) and (11) with d = 0 , control input Eq. (34), and the update law Eq. (46). Then in the closed-loop system, the system trajectories beginning from any given initial condition (qa (0), (0)) , converge to the equilibrium point (q = 0, q4 ± 1, 0) . Thus nadir pointing attitude regulation is achieved by the NCEA law.

u=

ˆ f (p

=

(pˆ + )

ˆ f (p

+ )

ˆ f (p

ˆ f (p

+ )

+ ) (54)

+ )

(Eq. (54) has been obtained by using Eq. (26) for (= Tf ef ) and using Eqs. (26) and (27) gives

= (

Remark 1. It is interesting to note that there exists a manifold

= {(

M = { f z = 0}

T f ef T f

+

+

T f T)

f .)

Differentiating

ef )

+

ef

T f(

ef

+

e )}

(55)

Now substituting Eq. (46) for pˆ and Eq. (55) in (54), it can be easily seen that the control signal u takes the form

which attracts the trajectories of the closed-loop system. Such an attractive manifold does not exist in the CEA systems. This is a special property of NCEA systems. It is also noted that the algebraic ( ef , f ) of the parameter estimate causes nonlinear function l * f z 2 in the appearance of a negative semi-definite function Lyapunov derivative (see Eq. (53)). This term provides stronger stability properties in the closed-loop system. In the traditional CEA

u=

(pˆ + )

f

T f {(k2

)

ef

+ k3 q +

e}

(56)

This completes the I&I-based design. Remark 2. In this section, the control law was designed assuming that the disturbance input d (t ) was zero. Although the NCEA law is 2 in sufficiently robust due to negative semi-definite function fz 169

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K.W. Lee and S.N. Singh

Fig. 3. Uncontrolled spacecraft in circular retrograde orbit (R c = 50 ): quaternion trajectory.

Eq. (53), one can introduce σ-modification in the update law Eq. (46) for strengthening the robustness in the presence of disturbance input d (t ) . The modified adaptation law for implementation is given by

pˆ =

T f ef

T f[

k2

ef

k3 q]

(pˆ + )

circular or elliptical. The radial distance of the satellite's center of mass from the center of mass of the asteroid and the orbital rate are given by

Rc ( ) =

(57)

=

where σ is a small positive real number. (A robustness analysis of the closed-loop system perturbed by d (t ) with or without σ-modification is provided in Appendix B and C, respectively. Also, stability of the closed0 is discussed in Remark 3 in the loop system when d = 0 but appendix.)

a (1 e 2) 1 + ecos ) µ

ph3

(1 + e cos )2

where a is the semi-major axis, e is the eccentricity, and ph = a (1 e 2) is the semilatus rectum. For simplicity, all simulated responses of the closed-loop system have been obtained using the adaptation law Eq. (46) to demonstrate robust performance of the control law without σmodification (i. e., = 0 ).

4. Simulation results In this section, simulation results for the open- and closed-loop system are presented. For the purpose of illustration, it is assumed that the satellite is in an equatorial orbit around asteroid 433 Eros. The principal moments of inertia of the spacecraft are ( J1, J2 , J3) [Kg m2], and its mass is 600 [kg]. The remaining parameters of the model given in Ref. [23] are: r0 = 9.933 [km], C20 = 0.0878, and C22 = 0.0439 . The gravitational parameter of 433 Eros and its rotation rate used for simulation are µ = 4.4650 × 10 4 [km3/s2] and = 3.312 × 10 4 [rad/s], respectively. For simulation, it assumed that the spacecraft's orbit is

4.1. Uncontrolled satellite in prograde and retrograde circular orbit: R c = 50 [km] d = 0 First, the open-loop system (Eqs. (5) and (6)), with the input u and the disturbance input d both set to zero, is simulated. The satellite is t ). The orbit assumed to be in a prograde circular orbit (i.e., c = parameters are assumed to be (a = Rc , e ) =(50 [Km], 0). The initial conditions are assumed to be (qT (0), q4 (0)) = (0.5, 0.5, 0.5, 0.5), and T (0) = (0.0004, 0.0004, 0.0004) [rad/sec]. Fig. 2 shows the quaternion 170

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Fig. 4. Adaptive control of spacecraft in prograde circular orbit (R c = 50) [km].

trajectory of the open-loop system for the satellite in prograde orbit t )). The responses for the spacecraft on retro(longitude c = grade circular orbit (longitude c = + t) ) for the same initial conditions have been also obtained. These are shown in Fig. 3. One observes in Figs. 2 and 3 that the unit quaternion trajectory of the openloop system undergoes bounded oscillations. Apparently, it is essential to suppress these oscillatory responses and force qa to the equilibrium value (q = 0, q4 ± 1) .

k2 = 0.3, k3 = 0.2 , and the filter parameter α is 0.5. The adaptation gain is set as = 5500 . The initial value of parameter estimate is arbitrarily set as pˆ (0) = 09 × 1. This is rather not a good choice, but this initial estimate has been assumed to show the robustness of the control law. The filter initial values are ef (0) = 03 × 1 and f (0) = 03 × 9. These design parameters have been selected by the observation of simulated responses. First, it is assumed that the spacecraft is in a prograde circular orbit (R c = 50 [km]). Selected responses are shown in Fig. 4. One observes smooth convergence of q (t ) to zero in about 20 s. The peak magnitude of torque upm = max [{ ui , i = 1,2,3} is 3.4770 [Nm], and the maximum

4.2. Adaptive control in prograde and retrograde circular orbit: e = 0 , a = 50 [km] d = 0

value u 2m = max u12 + u 22 + u32 is 5.1635 [Nm]. The estimated parameter vector pˆ + = z + p remains bounded, and its norm converges to a constant value. It is interesting to observe that although the parameter estimate pˆ + converges to a nonzero value, the closed-loop system's trajectories indeed converge to the manifold M defined by f z = 0 (see Fig. 4).

Now, the closed-loop system including the attitude dynamics Eqs. (5) and (6) with d = 0 , the control law (Eq. (56)), and the adaptation law Eq. (46) is simulated. The initial conditions are qa (0) = (0.5, 0.5, 0.5, 0.5)T , and (0) = (0.0004, 0.0004, 0.0004)T [rad/ sec], respectively. The selected feedback gains are k1 = 0.1, 171

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Fig. 5. Adaptive control of spacecraft in retrograde circular orbit (R c = 50) [km].

Now, for the same initial conditions, simulation is done for the spacecraft in a retrograde circular orbit (R c = 50 [km]). The feedback gains, which were used for attitude control in the prograde orbit, are retained. Selected responses are shown in Fig. 5. It is seen that attitude regulation is smoothly accomplished. The maximum values are upm = 3.4770 [Nm] and u 2m = 5.1635[Nm]. Once again it is seen that the closed-loop systems trajectories converge to the manifold M .

the NCEA control law. As predicted, the trajectories converge to the manifold M . Simulation for the spacecraft in the circular retrograde orbit has been also done. The responses are somewhat similar to those of Fig. 6. (These responses are not shown here.) 4.4. Minimal rotation adaptive control, spacecraft in prograde circular orbit: negative q4 (0) = 0.5, R c = 30 [km] d = 0

4.3. Adaptive control in prograde circular orbit: e = 0 , R c = 30 [km] d = 0

Now, the effectiveness of the control law for regulation to the equilibrium point with minimal rotation of spacecraft is examined. For this purpose, the initial condition qa (0) = [0.5, 0.5, 0.5, 0.5]T . Note that now the initial value of q4 (0) is negative unlike Case 4.2 and 4.3. The controller gains and remaining initial conditions of Case 4.2 are retained. Simulated responses in Fig. 7 show convergence of qa to the equilibrium point. Interestingly, now q4 (t ) remains negative and converges to −1 with minimal rotation of spacecraft.

Now, simulation is done for the spacecraft orbiting in a prograde circular orbit of radius R c = 30 [km]. The controller parameters designed for the Case 4.2 are retained. Selected responses are shown in Fig. 6. Again, the quaternion trajectory asymptotically converges to the equilibrium point. It is interesting to note that even for R c = 30 [km] (orbit close to the asteroid), the response time remains of the same order (20 s). The maximum values are upm= 3.4765 [Nm] and u 2m = 5.1644 [Nm]. Of course, this has been possible due the use of 172

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Fig. 6. Adaptive control of spacecraft in prograde circular orbit (small R c = 30) [km].

4.5. Adaptive control, spacecraft in prograde elliptic orbit perturbed by random disturbance torque: e = 0.2 , a = 30 [km]

waveforms of the disturbance inputs (d1, d2, d3) are shown in Fig. 8 (f). (Disturbance torque caused by solar radiation pressure can be expected to be of similar order (10 3 [Nm] or even smaller order [30].) The initial conditions and controller used for Case 4.2 are retained. For simplicity, sigma modification is not used ( = 0 ). Fig. 8 shows simulated responses over [0, 50] [sec]. One can observe that quaternion trajectory converges close to (q = 0, q4 = 1) in about 20 [seconds]. Of course, it is possible to reduce the convergence time and terminal tracking error, but then larger control input magnitude is needed. To examine the robustness of the closed-loop system, simulation has been also performed for t [0,50] [sec] using stronger disturbance moments. For this purpose, disturbance inputs of larger magnitudes have been obtained by multiplying the random signals di (t ), i = 1,2,3, shown in Fig. 8 (f), by scaling factor md = 5,10,20,40 , and 100. For md = 100 , the maximum magnitude of d (t ) becomes of the order of 10 1 [Nm], which is very large. However, it will be used here simply for verifying the robustness of the controller. Based on the observation of

Disturbance moments caused by Sun, third body, unmodelled higher-order gravity field, etc, affect the rotational motion of spacecraft. However, solar radiation pressure has dominant effect on attitude motion of asteroid-orbiting satellite. Of course, computation of net environmental disturbance torque, which depends on the physical and geometrical characteristics of spacecraft, is not easy. Therefore performance of the NCEA law is examined in the presence of random disturbance moment vector d (t ) = [d1, d2 , d3]T . For simulation, distinct signals di , (i = 1,2,3) , are generated by passing white noise through a 4 filter transfer function 25 × 10 . The mean value and variance of the s + 0.6s + 1 white noise are 0 and 1, respectively. The satellite is assumed to be in an elliptic orbit. The eccentricity e and the semi-major axis a are 0.2 and 30 [km], respectively. First, simulation is done over a short period for t [0,50] [sec]. The 173

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K.W. Lee and S.N. Singh

Fig. 7. Minimal rotation control of spacecraft in prograde circular orbit (small R c = 30) [km].

responses, the maximum value of q (t ) and minimum value of q4 (t ) in the terminal phase (t [40,50]), (denoted as q max and q4, min ) are collected in Table 1 in Appendix D. Note that at the equilibrium point, one has q = 0 and q4 = 1. Thus, the deviations q max or q4, min from the equilibrium values 0 and 1, respectively, correspond to the largest attitude error in the interval [40,50] [sec]. In Table 1, it is seen that in the terminal phase q4 is almost 1 and q is close to zero. This shows effectiveness of the adaptive law in accurate nadir pointing control, despite action of random moments of varying strengths. The closed-loop system, perturbed by persistent random d (t ), t [0,5000] [sec], was also simulated. The values of q max and q4, min , observed over the terminal phase [3000, 5000] [sec], are summarized in Table 2 in Appendix D for different values of md . It is seen that q max and q4, min remain close to desirable equilibrium values 0 and 1, respectively. Also, one can observe in Table 2 that in the terminal phase, q max increases and q4, min decreases monotonically with md . Thus, as expected, it is seen that stronger d (t ) is causing larger attitude error

after decay of initial transients. Fig. 9 shows a three-dimensional plot of (q1, q2 , q3) in the terminal phase [3000, 5000] [sec] for the choice of strong d (t ) (i. e., md = 100). It can be observed that, despite this large random disturbance torque of the order of 10 1 [Nm], the trajectory q (t ) , beginning from qa (0) = (0.5, 0.5, 0.5, 0.5)T , remains confined in a small set of ultimate boundedness, u = {(q1, q2 , q3) [ 5,5.5]10 3 × [ 6,6]10 3 × [ 6.5, 6]10 3} , surrounding the origin in the terminal phase [3000, 5000] [sec]. (Of course, for practical values of disturbance inputs of order 10 3 [Nm], the attitude error will be relatively small (see Table 2).) Simulation using different initial conditions was also performed. In these cases also, the NCEA law accomplished the regulation of the attitude angles. 5. Conclusion In this paper, based on the immersion and invariance methodology, 174

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K.W. Lee and S.N. Singh

Fig. 8. Adaptive control of spacecraft in prograde elliptical orbit (e = 0.2, a = 30 [km]):random d (t ) .

a noncertainty-equivalence adaptive law for the attitude control of a satellite orbiting in an equatorial circular and elliptic orbits around an asteroid was derived. The attitude of the satellite was parameterized using unit quaternion. The objective was to regulate the quaternion trajectory so as to achieve nadir pointing. For the design, a backstepping design method involving two steps was used. Based on the Lyapunov analysis, a control module and a parameter identifier were designed separately, and convergence of system trajectories of the disturbance input-free system to the equilibrium point was established. For the synthesis, certain filtered signals were introduced. Interestingly, it was found that the trajectories of the closed-loop system converge to an attractive manifold in the state space. This was achieved by the inclusion of an algebraic nonlinear function in the parameter estimate. This is a special property of the NCEA systems. The designed control law is effective in regulating the attitude to the equilibrium point with minimal rotation of spacecraft. A robustness analysis of the closed-loop system in the presence of disturbance input was also done, and uniform ultimate boundedness of trajectories was established. Simulation results were obtained for the attitude control of a satellite orbiting around 433 Eros. Simulation results confirmed that the NCEA law is capable of precise attitude control despite parametric uncertainties in the system. Simulation over a long period in the presence of persistent disturbance moments confirmed convergence of the quaternion trajectories in a small set of ultimate boundedness.

Fig. 9. Quaternions ( perturbed by strong random torque (md = 100) in terminal phase, t [3000,5000] [sec]: (e = 0.2, a = 30 [km]).

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Appendix A. The functions

j1

= P1 C20 + P2 (t ) C22 and For simplicity, one can write φ and χ as C (t ) = 6sin(2 c )(r0/ R c )2 . Then Mg1 can be expressed as Mg1 =

µ

(3 + 5P1 C20 + 5P2 C22 )(J3

Rc3

+ 2.5CC22 { 0.4J1 c12

(J1 + J2

a11 a12

=

3µ Rc3

a14

= =0

a15

=

a16

=

a17

=

a13

J3) c21 c33 + (J1

a18

=

a19

=

a1

=[

a1

(A.1)

J2 + J3) c23 c31}] of the regressor

a1

are

c23 c33

a12 5µ Rc3

P1 c23 c33

a15 5µC 2Rc3 µ Rc3 µ Rc3

(0.4c12 + c21 c33

{ 5P2 c23 c33

c23 c31)

2.5C (c21 c33 + c23 c31)}

{5P2 c23 c33 + 2.5C (c21 c33 + c23 c31)} a12, …, a19]

a11,

The second gravity gradient torque component is

Mg 2 =

µ

(3 + 5P1 C20 + 5P2 C22 )(J1

Rc3

+ (J1 + J2

J3) c11 c33

(J2 + J3

J3) c13 c33 + 2.5CC22

0.4J2 c22

J1) c13 c31}

Based on this representation, one finds that b11

=

3µ Rc3

b13

=0 =

b14

=

b12

c13 c33

b11

5µ

P1 c13 c33 Rc3

b16

=0 =

b17

=

{5P2 c13 c33 Rc3

=

5µC

b18 b19

=

b15

b14

µ

µ Rc3

+ 2.5C (c11 c33 + c13 c31)}

0.4c22 + c11 c33

2Rc3

c13 c31

{ 5P2 c13 c33 + 2.5C ( c11 c33 + c13 c31)}

Thus b1

=[

b11,

b12, …,

b19]

Finally, Mg3 takes the form

Mg 3 =

µ Rc3

+ (J2 + J3

(3 + 5P1 C20 + 5P2 C22 )(J2 J1) c13 c21

(J1

1.5(r0/ R c )2 , P2 (t ) = 9cos(2 c )(r0/R c )2 , and

J2 ) c23 c33

In view of this equation, the elements

=0

= C (t ) C22 , where P1 =

J1) c13 c23 + 2.5CC22

0.4J3 c32

J2 + J3) c11 c23}

Thus, one gets

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K.W. Lee and S.N. Singh 3µ

=

c11

Rc3

c13

= =0

c14

=

c12

c11 5µP1 Rc3

c16

= =0

c17

=

c18

=

c19

=

c15

c13 c23

c13 c23

c14

µ Rc3

{ 5P2 c13 c23

µ

2.5C (c13 c21 + c11 c23)}

5P2 c13 c23 + 2.5C (c13 c21 + c11 c23)

Rc3 5µC

( 0.4c32 + c13 c21

2Rc3

c11 c23)

and c1

=[

c11,

c12, …, c19]

B. Stability analysis of closed-loop system including σ-modification and d (t )

0

For simplicity in Section 3, the external disturbance input d (t ) was ignored. Here stability properties of the closed-loop system including σmodification in the presence of d (t ) is examined. (Because the details of derivation can be completed by following Section 3, only modified relevant equations will be presented.) One begins with the model Eq. (11) perturbed by the disturbance input d (t ) . Step 1 of derivation is still valid. Because d (t ) appears additively with u, Eq. (25) takes the form e

1[

=J

p + u + d]

k2

(A.2)

k3 (s + ) q

e

The differential equations (Eqs. (26)–(28)) to obtain filtered signals are still valid. But filtered error Eq. (33) becomes ef

1(

=J

fp

+ uf + d f )

k2

(A.3)

k3 q

ef

where df (to be used only for analysis) satisfies

df =

(A.4)

df + d

For a practical case, d is bounded. Thus, df is also bounded. The control law uf = remain unchanged, but Eq. (35) becomes ef

=

1(

J

fz

df )

k2

ˆ f (p

+ (

ef ,

f ))

and

=

ef

given in Eqs. (34) and (44) (A.5)

k3 q

ef

T f

Now, the derivative of the Lyapunov function Vs in Eq. (40) including df -dependent terms takes the form

Vs =

k1 q

2

k2

ef

) qT

(k3 + k2

2

T ef

[ k3 q

ef

+ qT ] J

1(

fz

df )

2

(A.6)

Note that the design parameters satisfy k3 + k2 T 1 ef J ( f z

qT J

1(

k2 4

df )

fz

2

2

k1 q 4

df )

+

ef

J

+

1

J

fz

2

k2 1

fz

k1

2

+

+

= 0 . Using Young's inequality, one has. 2

k2 4

k1 q 4

2

J 1df

+

ef

J

+

1d

2

(A.7)

k2 f

2

(A.8)

k1

and using these inequalities in Eq. (A.6) gives

1 k1 q 2

Vs where

1d

1 1 + k1 k2

2

2

+ k2

2

1 1 + k1 k2

+

ef

J

1

fz

2

k3 q

+

1d

2

J 1df

1d

For any bounded disturbance input d (t ) , 1d > 0 certainly exists. Now, the stability analysis for the identifier including the modified update law Eq. (57) is examined. Because d the z becomes

z = pˆ +

(A.9)

> 0 provides a bound on the uncertain function J 1d f according to

T f ef

+

T f[

J

1(

fz

df )

k2

ef

0 , Eq. (45) for the derivative of (A.10)

k3 q]

Using Eq. (57) in (A.10) gives

z=

T 1 fJ ( fz

df )

(A.11)

(z + p)

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Thus, the derivative of Ve (z ) = T 1 fJ ( fz

zT

Ve =

df )

2

( z

2 1)

min (J

=

l2

fz 1 2

2

fz

where l1 > 0 , l2 =

z

+

zT p)

2

+ 2

l1 +

fz

2

1 4l1

+

1 ( 2

J 1df

z

2

p 2) (A.12)

2d

l1, and

1)

min (J

along the solution of Eq. (A. 11) gives

1zT z /2

2d

=

2

J 1df

4l1

+

2

1 2

p

is a constant. Note that one can select l1 such that l2 is positive. Thus, Ve < 0 , if

1.

z > 2 2d ( ) This implies that z will remain bounded. Now, using Eqs. (A.9) and (A.12) gives the derivative of the composite Lyapunov function V = Vs + Ve as 2

1 2

V l2

k1 q

1 2

2

fz

2

+ k2 2

z

+

ef

+

(

1 k1

+

1 k2

2

)

1

J

2

fz

k3 q

+

1d

(A.13)

2d

Define *

and

d

=

(

V

(

k1 + k2 k1 k2

2 1 max (J )

= l2

1d

1 k 2 1

1 k 2 1 *(

+ k3

+ k3 q

*(

2d .

+

2 ef

)

2

2

2 ef 2)

+ z

+ z 2) +

+

2 *

ef

1 k 2 2

q ef

2

1 k 2 2

q

+ 2

Note that one can choose the free parameter ν sufficiently large such that 2

)

>0

1 2

1 2

fz

*

is positive. Then Eq. (A.13) gives

2

z

+

d

2

z

+

d

d

(A.14)

d

Sq = {qa : qa = 1} , and z where = min {(0.5k1 + k3), 0.5k2, 0.5 } . Define x a = [ a set a = {xa : ef d( d( 2 = 1 , one has q 2 1 for all t. Consider a level set E = {(qa , x a) (Sq × R6): V = Ca, Ca > 0} for a proper choice of aq = Sq × a . Noting that qa Ca such that aq E . An upper bound on V = Vs + Ve , subject to ef 2 = d / * , z 2 = d / * , and qa = 1, can be used as a value of Ca . Thus, one 1) ( * ) 1 selects Ca = 2 + 0.5(1 + , where the maximum value of V1 is 2 for (q4 = 0, q = 1) (see Eq. (14)). The level set E = Ca contains aq . d Note that if (qa , xa ) is not in E, then q Sq but x a a . Therefore in view of Eq. (A.14), one has V < 0 over the period x a a , and eventually x a will be confined in a . This establishes uniform ultimate boundedness of system trajectories in the set E. T ef ,

*

zT ]T

R12 ,

*) 1 ,

*) 1 } ,

Remark 3. Now, let us examine the effect of the modified update law Eq. (57) on the stability of the closed-loop system if d (t ) 0 , then the derivative of V obtained in Eq. (A.14) becomes derivation, one can show that if d = 0 but

V

*(

ef

2

+ z 2) +

2

0 . Using the above (A.15)

ds

p . Note that ds is smaller than d . In this case, as expected, the level set E will be smaller. Using the arguments used in derivation where ds = 0.5 of Appendix B, one can establish uniform ultimate boundedness of system trajectories. 2

C. Stability analysis of closed-loop system perturbed by d (t )

0 without σ-modification

To this end, effectiveness of the I&I-based adaptive law, without the use of σ-modification, in the presence of disturbance torque, is examined. For this purpose, by setting = 0 in Eq. (A.12), one obtains Ve as

Ve where

l2

fz

2d

=

2

1 4l1

+

(A.16)

2d

J 1df

2

is a constant. It follows from Eq. (A.16) that Ve is negative whenever . In view of Eqs. (A.9) and (A.16), if

= 0 , V given in

Eq. (A.14) takes a simplified form

V

k2 2

2 ef

*

2 fz

+

d

(A.17)

where d = 1d + 2d . In this case, it follows that ef , and f z will remain uniformly bounded. Of course, if ef is bounded, then ω, and f will be bounded. Thus, if f is not identically zero or z does not lie in the null space of f for all t > 0 , then z will be also bounded. Simulation results show that indeed, z remains bounded if d 0 and = 0 . This establishes stronger stability properties of the identifier compared to traditional adaptive systems without σ-modification despite the presence of d (t ) . It may be pointed out that this robustness analysis provides only conservative bounds on the uncertain functions and the size of region of ultimate boundedness. In fact, simulation results confirm that NCEA law achieves small terminal quaternion trajectory error and the size of the region in which the trajectories are ultimately confined, despite random disturbance moments of various strengths.

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D. Performance of the quaternion-based NCEA attitude control law in the presence of disturbance input Table 1

Robustness of the closed-loop system: random d (t ) of various strengths over a short period t factor md = 1, 5, 10, 20, 40, 100. Factor md

q

1

5

10 20

40 100

max ,

t

[40,50] (sec)

1.15834013× 10

4

3.61792834× 10

4

1.80465245× 10

4

7.24228892× 10 0.00144812 0.00361045

4

q4, min, t

0.99999390 0.99999389 0.99999378

0.99999316 0.99998827

Robustness of the closed-loop system: random d (t ) of various strengths over a long period t md = 1, 5, 10, 20, 40, 100.

1

5

10 20 40 100

q

max ,

t

[3000,5000][sec]

1.02509236× 10

5.11926092× 10 0.00102085 0.00202124 0.00390338 0.00879279

[40,50] (sec)

0.99999389

Table 2

Factor md

[0,50] [sec];

[0,5000] [sec]; factor

q4, min, t

[3000,5000][sec]

0.99999390

4

0.99999387

4

0.99999360 0.99999235 0.99998726 0.99995587

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