Finite-Time Control forSpacecraft Body-FixedHovering Over an Asteroid

DAERO LEEAMIT K. SANYALNew Mexico State UniversityLas Cruces, NM, USA

ERIC A. BUTCHERUniversity of ArizonaTucson, AZ, USA

DANIEL J. SCHEERESUniversity of Colorado at BoulderBoulder, CO, USA

A finite-time control scheme for autonomous body-fixed hoveringof a rigid spacecraft over a tumbling asteroid is presented. Therelative configuration between the spacecraft and asteroid isdescribed in terms of exponential coordinates on the Lie group SE(3),which is the configuration space for the spacecraft. With a Lyapunovstability analysis, the finite-time convergence of the proposed controlscheme for the closed-loop system is proved. Numerical simulationsvalidate the performance of the proposed control scheme.

Manuscript received March 12, 2014; revised June 4, 2014; released forpublication July 18, 2014.

DOI. No. 10.1109/TAES.2014.140197.

Refereeing of this contribution was handled by J. Li.

The authors acknowledge funding for this research through NSF grantCMMI 1131643 and NASA grant NNX11AQ35A.

Authors’ addresses: D. Lee, A. Sanyal, New Mexico State University,Aerospace Engineering, Las Cruces, NM 88003, E-mail:(daerolee@nmsu.edu).

0018-9251/15/$26.00 C© 2015 IEEE

I. INTRODUCTION

Controlled hovering motion of spacecraft over anasteroid is essential for performing scientific explorationsof small bodies such as asteroids and comets. Controlledhovering near an asteroid can be obtained by feedbackcontrol that compensates for gravitational and rotationalaccelerations and fixes the position [1, 2] and orientationof the spacecraft either in an asteroid body-fixedcoordinate frame or in a heliocentric orbital coordinateframe. In general, there are two approaches in controlledhovering motion at an asteroid: near-inertial hovering andbody-fixed hovering [2–5]. In near-inertial hovering, thespacecraft is stationed at a fixed or desired positionrelative to the asteroid in the sun–asteroid frame, implyingthat the asteroid rotates beneath the spacecraft. Inbody-fixed hovering, the spacecraft stays at a fixed pose(position and orientation) relative to the rotating asteroid.The Hayabusa mission to Itokawa [6, 7] implemented amixture of these two approaches, maintaining anorbit-fixed location far from the asteroid and transitioninginto a body-fixed frame during its descent to the asteroidsurface [2]. Body-fixed hovering is essential for thespacecraft to sample a small body surface by controllingits motion in the asteroid body-fixed frame. Hoveringtrajectories can be implemented for many rotation periods(hours) of the asteroid with a modest control thrust, sincethe gravitational attraction is relatively weak at theasteroid. In general, the asteroid rotation period is on theorder of hours to days at most [2]. This work considers therelative orientation (attitude) of the spacecraft in additionto the relative position, so that body-fixed imagers on thespacecraft can obtain the desired science data from abody-fixed hovering mission. Previous studies onspacecraft hovering over an asteroid have primarily dealtwith achieving the desired relative position of apoint-mass spacecraft with respect to the asteroid, withoutconsidering the relative attitude [1, 3–5, 8–10]. On theother hand, [11] presents a body-fixed hovering schemefor a rigid-body spacecraft hovering over an asteroid withasymptotic convergence to the desired relative pose. Thisimplies that the control objective can be achieved ininfinite time. The hovering maneuver is obtained for auniformly rotating asteroid whose rotation rate is constantin the asteroid body-fixed frame [2, 12, 13]. Thus, thespacecraft can execute coupled translational and rotationalmaneuvers in three-dimensional Euclidean spacewhile tracking a desired trajectory in the Lie group ofrigid-body translations and rotations, SE(3) [14–21]. Theasymptotic tracking control scheme for this body-fixedhovering scheme was adapted from the asymptotictracking control scheme for spacecraft formationflying [11].

Most of the existing asymptotic control techniquesensure that the closed-loop system dynamics of acontrolled system are Lipschitz continuous, which impliesuniqueness of system solutions in forward and backwardtimes [22]. Hence, convergence to an equilibrium state is

506 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 51, NO. 1 JANUARY 2015

achieved over an infinite time interval. In order to achieveconvergence in finite time, the closed-loop systemdynamics needs to be non-Lipschitzian, giving rise tononuniqueness of solutions in backward time. Uniquenessof solutions in forward time, however, can be preserved inthe case of finite-time convergence. Alternatively,discontinuous finite-time stabilizing feedback controllershave also been developed in the literature [23–25]. Inpractical implementations, discontinuous feedbackcontrollers can lead to chattering behavior due to plantuncertainties or measurement imperfections. Suchcontrollers may also excite unmodeled high-frequencydynamics when used, for instance, to control lightlydamped structures [22, 26]. However, a continuousfinite-time control provides an ideal solution to resolvethis problem [27]. It also requires that the closed-loopsystem be stable in the sense of Lyapunov and the systemstates be able to converge to the equilibrium in a finitetime [28].

This study was motivated by the need for a continuousfinite-time stable control scheme to achieve body-fixedhovering over a tumbling asteroid in six degrees offreedom, guaranteeing finite-time convergence withoutany discontinuity. In this paper, a continuous finite-timetracking control scheme is proposed in the framework ofgeometric mechanics. This single finite-time controlscheme deals with the coupled translational androtational motion of spacecraft in configuration, withouthaving to design separate controllers for orbital andattitude motion when the degrees of freedom are coupledat the natural dynamics level as well as due to controlthrust generated by body-fixed actuators. When allactuators (thrusters and attitude actuators) are fixed to thespacecraft body (which is almost always the case), thecontrol scheme for orbital motion needs knowledge of theattitude. Also, because the gravity gradient torque dependson the location in orbit for noncircular orbits, the attitudecontrol depends on the orbital position. Since thegravitational field of the asteroid is the primary anddominant effect term of the disturbances on the spacecraftover an asteroid [29], the gravity force and torques areincluded as the known external disturbances in thespacecraft dynamics.

A continuous finite-time control scheme in theframework of geometric mechanics is proposed forbody-fixed spacecraft hovering over an asteroid,performing the coupled translational and rotationalmaneuvers simultaneously and achieving tracking in finitetime [30]. Geometric mechanics provide a powerful andefficient framework that is currently used to analyze thecoupled translational (position) and rotational (attitude)motion of spacecraft and to design control schemes withlarge domains of convergence. The asteroid’s trajectory isassumed to be obtained from the known dynamics modelof a rigid body in a central gravity field. Then, thebody-fixed spacecraft hovering is performed withreference to the asteroid. The desired trajectory for thespacecraft is obtained by having a constant desired relative

configuration between the spacecraft trajectory and theasteroid trajectory. The relative configuration (pose) of thespacecraft with respect to the asteroid is represented byexponential coordinates on SE(3). The exponentialcoordinates are obtained from the inverse of theexponential mapping from the Lie algebra se(3) to the Liegroup SE(3). An important advantage of exponentialcoordinates versus these other coordinates is that they donot have singularities in their kinematics and they can treatlarge maneuvers with continuous state feedback. Thesecoordinates also do not need a redundant parameterizationlike the four-parameter unit quaternion description ofrigid-body attitude. The kinematics of the exponentialcoordinates are used to propagate the exponentialcoordinate and generate the reference trajectory.Note that se(3), which is the tangent space at the identityelement of SE(3), is isomorphic as a vector space to R

6.The finite-time control scheme is designed to reduce therelative configuration and relative velocities in finite timeautonomously from almost any given initial state, exceptthose that differ in orientation by a rotation of π radiansfrom the desired states at the initial time. Note that theexponential coordinates for the relative attitude(orientation) are not uniquely defined in this case[18–20]. Since the set of such initial states is anembedded lower dimensional subspace of the statespace, this tracking control scheme is therefore almostglobal in its convergence over the state space [14] infinite time.

In comparison to previous work, the contribution ofthis study is summarized as follows. Continuousfinite-time stable control for autonomous body-fixedhovering of a rigid spacecraft over an asteroid is proposedin the framework of geometric mechanics without havingto design separate controllers for the position and attitudemaneuvers. This finite-time control can perform coupledtranslational and rotational maneuvers over an asteroid infinite time. The spacecraft is considered as a rigid bodywhose motion is described by both position and attitudeinstead of a point mass. Then the states are updated by thefeedback control law such that the relative configurationand velocities can converge to zero without using explicitreference states or the help of explicit guidance, unlike inconventional control schemes [31–34]. Thus, coupledtranslational and rotational maneuvers are performedautonomously from any given configuration and velocityconditions to achieve the desired relative configurationand velocities in finite time.

II. RIGID-BODY DYNAMICS MODELS

The dynamics of the asteroid and spacecraft are nowdescribed. The asteroid is modeled as a rigid body thatorbits around the sun. The asteroid is assumed to move inunconstrained motion in this environment. The spacecraftis modeled as a rigid body that approaches the asteroid.The desired trajectory for the spacecraft is obtained byspecifying its relative pose (position and orientation) withrespect to the asteroid.

LEE ET AL.: FINITE-TIME CONTROL FOR SPACECRAFT BODY-FIXED HOVERING OVER AN ASTEROID 507

A. Dynamics Model and State Trajectory of the Asteroid

The asteroid is modeled as a rigid body that orbitsaround the sun in a central gravity field. The configurationspace of the asteroid is the special Euclidean group SE(3),which is the set of all translational and rotational motionsof a rigid body. SE(3) is also a Lie group and can beexpressed as the semidirect product SE (3) = R × SE (3),where R

3 is the three-dimensional real Euclidean space ofpositions of the center of mass of the body and SO(3) isthe Lie group of orientations of the rigid body. Asuperscript (·)0 is used to specify the asteroid states andparameters. The asteroid attitude is represented by therotation matrix R0 ∈ SO(3), which transforms a vector in aasteroid body-fixed frame to the inertial frame. Theasteroid’s position is expressed by the inertial positionvector b0 ∈ R

3 from the origin of the inertial frame (i.e.the sun) to the center of mass of the asteroid. Translationaland angular velocities of the asteroid are expressed by thevectors ν0 ∈ R

3 and �0 ∈ R3, respectively, as represented

in its body-fixed frame.The kinematics of the asteroid are expressed as

b0 = R0ν0,R0 = R0(�0)×, (1)

where the operator (·)× : R3 → so(3) is the cross product

operator defined by

ν× =

⎡⎢⎣

v1

v2

v3

⎤⎥⎦

×

=

⎡⎢⎣

0 −v3 v2

v3 0 −v1

−v2 v1 0

⎤⎥⎦ .

Here, so(3) is the Lie algebra of SO(3), which isrepresented as the linear space of 3 × 3 skew-symmetricmatrices. The Lie algebra of SE(3), denoted as se(3), is asix-dimensional vector space that is tangent to SE(3) at theidentity element. The algebra se(3) is a semidirect productof R

3 and se(3) and is isomorphic to the vector space R6.

Let the asteroid’s mass be chosen as m0 and its inertiamatrix be J0 in its body frame. The dynamics of theasteroid are given by

m0v0 = m0(ν0)×�0 + F0G(b0, R0), (2)

J 0�0 = J 0(�0)�0 + M0

G(b0, R0), (3)

where F0G,M0

G ∈ R3 denote, respectively, the gravity

force [35] and gravity gradient torque on the asteroid dueto the sun, as given by

F0G = −

(m0μS

‖b0‖3

)p0 − 3

(μS

‖b0‖5

)J 0 p0

+15

2

(μS(( p0)TJ 0 p0)

‖b0‖7

)p0, (4)

M0G = 3

(μS

‖b0‖5

)(( p0)×J 0 p0), (5)

where p0 = (R0)Tb0, J 0 = (1/2) trace(J 0)I + J 0, μS isthe gravitational parameter of the sun, and I is a 3 × 3

identity matrix. The state space of the asteroid isTSE(3) � SE(3) × se(3) and its motion states are denotedby (b0, R0, ν0, �0). Here se(3) denotes the Lie algebra(tangent space at the identity) of the Lie group SE(3), andse(3) is isomorphic to R

6 as a vector space. Theconfiguration of the asteroid on SE(3) can also berepresented by the following 4 × 4 matrix:

g0 =[

R0 b0

01×3 1

]∈ SE(3), (6)

where 0m × n denotes an m × n null matrix. We also denotethe vector of body velocities of the asteroid by

ξ 0 =[

�0

ν0

]∈ R

6. (7)

Thereafter, the kinematics of the asteroid in (1) can beexpressed as follows:

g0 =g0(ξ 0)∨, where(ξ 0)∨ =[�0 ν0

01×3 0

]∈ se(3). (8)

We express the mass and inertia properties assigned tothe asteroid by a 6 × 6 matrix, and the vector of gravityforces and torques by a 6 × 1 vector, as follows:

ϕ0g =

[M0

g

F0g

]∈ R

6, I0 =

[J 0 03×3

03×3 m0I

]∈ R

6×6. (9)

We represent the adjoint operator and coadjointrepresentations where the adjoint action of SE(3) on se(3)is defined in the same manner as that between a Lie groupand its corresponding Lie algebra. The adjoint actions of g∈ SE(3) on X ∈ se(3) are given by

Adg =[R 03×3

(b)×R R

]∈ R

6×6, s.t.AdgX| = (gXg−1)|,

(10)

where (·)| : se(3) → R6 is the inverse of the vector-space

isomorphism (·)∨ : R6 → se(3). The adjoint

representation of se(3) is expressed in matrix form as

adξ 0 =[

(�0)× 03×3

(ν0)× (�0)×

]∈ R

6×6. (11)

The coadjoint operator is described on the dual of theLie algebra, which can be identified with R

6 via thevector-space isomorphism (·)|. Therefore, the coadjointrepresentation can be expressed in matrix form as

ad∗ξ 0 = (adξ 0 )T =

[−(�0)× −(ν0)×

03×3 −(�0)×

]. (12)

Using the coadjoint operator, we can compactlyexpress the dynamics model of a spacecraft or other objectmodeled as a rigid body. The dynamics of the asteroid in

508 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 51, NO. 1 JANUARY 2015

(2) and (3) are expressed in this compact form as

I0ξ

0 = ad�ξ 0I

0ξ 0 + ϕ0G. (13)

The kinematics in (8), the dynamics in (13), and theknown initial states (g0(t0), ξ 0(t0)) at time t0 can be used togenerate the state trajectory of the asteroid for time t ≥ t0.

B. Spacecraft Dynamics

The configuration of the spacecraft is given by theposition vector from the origin of the heliocentric inertialframe to the center of mass of the spacecraft (denoted byb ∈ R

3), and the attitude is given by the rotation matrixfrom a body-fixed coordinate frame to the heliocentricinertial frame (denoted R ∈ SO(3)). The kinematics for thespacecraft take the same form as the kinematics for theasteroid, and are given by

g = g(ξ )∨, where (ξ )∨ =[

(�)× ν

01×3 0

]∈ se(3) and

g =[

R b

01×3 1

]∈ SE(3). (14)

Hence, g ∈ SE(3) is the configuration, ν ∈ R3 is the

translational velocity, and � ∈ R3 is the angular velocity

of the spacecraft, both vectors being expressed in thespacecraft’s body frame. The state space for eachspacecraft’s motion is SE(3) × se(3). Letφc : SE × se(3) → R

3 denote the feedback control forceon the spacecraft, and let τ c : SE(3) × se(3) → R

3 denotethe feedback control torque on the spacecraft. Let thespacecraft’s mass be chosen as m and its inertia matrix beJ in its body-fixed frame. The dynamics equations ofmotion for the spacecraft are therefore given as follows:

mv = m(ν)×� + FGS(b, R) + FGa

(r, R)

+ φc(b, R, ν, �), (15)

J � = J (�)� + MGS(b, R) + MGa

(r, R)

+ τ c(b, R, ν, �), (16)

where FGS, MGS

∈ R3 denote the gravity force and

gravity gradient torque on the spacecraft due to the sun,respectively; FGa

, MGa∈ R

3 denote the gravity force andgravity gradient torque on the spacecraft due to theasteroid, respectively; r = [x y z]T is the relative positionvector of an arbitrary point on the spacecraft in theasteroid body-fixed frame; rc = R0T(b − b0) = [xc yc zc]T

is the relative position vector of the center of mass of thespacecraft in the asteroid body-fixed frame; andR = RTR0 is the rotation matrix that transforms from theasteroid body-fixed frame to the spacecraft body-fixedframe. The coordinate frames and geometric relationbetween the asteroid and spacecraft are illustrated inFig. 1. The unit vectors of the asteroid and spacecraftbody-fixed frames are given by (b0

x, b0y, b0

z) and (bx, by,bz), respectively. And (ox, oy, oz) are unit vectors for the

Fig. 1. Coordinate frames and geometric relation.

spacecraft orbital frame or local vertical and localhorizontal (LVLH) frame, while (er, eδ , eλ) are unitvectors associated with the spherical coordinate system r,δ, λ as shown in Fig. 1. The gravity force and gradienttorque on the spacecraft due to the sun have the same formas the gravity force and gradient torque on the asteroid,which are given by (4) and (5). Note that unlike thedynamics models of terrestrial unmanned vehicles (as in[14]) or spacecraft in (nearly) circular orbits (as in [36]),for spacecraft moving in general orbits or trajectories thegravity forces and torques vary with the location (inertialposition vector b) of the spacecraft. The gravity force onthe spacecraft by the asteroid FGa

is described using thesecond-degree and second-order spherical harmonicgravity field in the asteroid body-fixed frame. Thegravitational force potential of second degree and order inthe asteroid body-fixed frame [2, 13] is

U2 =− μa

‖r‖3

[C20

(1 − 3

2cos2 δ

)+ 3C22 cos2 δ cos 2λ

],

(17)

where δ is the spacecraft’s latitude, λ is the longitude ofthe orbiting particle measured in the asteroid body-fixedframe, μa is the gravitational parameter of the asteroid,‖r‖ =

√(x2 + y2 + z2), and C20 and C22 are the

second-degree and second-order gravity coefficients. Thegravity potential of second degree and order in the asteroidbody-fixed frame in Cartesian coordinates [13] is

U2 =−μaC20(x2 + y2−2z2)

2‖r‖5+ 3μaC22(x2−y2)

‖r‖5. (18)

The effective gravity potential function of the asteroidin Cartesian coordinates is

U = μa

‖r‖ + U2 = μa

‖r‖ − μaC20(x2 + y2 − 2z2)

2‖r‖5

+3μaC22(x2 − y2)

‖r‖5. (19)

LEE ET AL.: FINITE-TIME CONTROL FOR SPACECRAFT BODY-FIXED HOVERING OVER AN ASTEROID 509

The first-order partial derivatives of the effective gravitypotential U in (19) about x, y, and z [13] are

∂U

∂x= − μax

‖r‖3− μaC20x

‖r‖5+ 5μaC20x(x2 + y2 − 2z2)

2‖r‖7

+6μaC22x

‖r‖5− 15μaC22x(x2 − y2)

‖r‖7(20)

∂U

∂y= − μay

‖r‖3− μaC20y

‖r‖5+ 5μaC20y(x2 + y2 − 2z2)

2‖r‖7

−6μaC22y

‖r‖5− 15μaC22y(x2 − y2)

‖r‖7(21)

∂U

∂z= − μaz

‖r‖3+ 2μaC20z

‖r‖5+ 5μaC20z(x2 + y2 − 2z2)

2‖r‖7

−15μaC22z(x2 − y2)

‖r‖7. (22)

The first-order partial derivative of the effective gravitypotential U in (19) about the relative position vector r is

∂U

∂ r=

[∂U

∂x

∂U

∂y

∂U

∂z

]T

. (23)

The gravity force due to the effective gravity potential Uof the asteroid in (19) in the asteroid body-fixed frame is

F 0Ga

=∫

∂U

∂ rdm. (24)

Using the rotation matrix R = RTR0 that transforms to thespacecraft body-fixed frame from the asteroid body-fixedframe, the gravity force due to the gravity potential in (19)in the spacecraft body-fixed frame is expressed by

FGa= RTR0

∫∂U

∂ rdm = RF 0

Ga. (25)

The integration of the first columns in (20)–(22) as thevector form denoted by F

p

Gais the primary gravity force

term:

Fp

Ga= R

∫ −μa

‖r‖3rdm = −−mμa

‖rc‖3Rrc − 3

(μa

‖rc‖5

)Jpc

+15

2

(μa( pT

c J pc)

‖rc‖7

)pc, (26)

where pc = Rrc = [xc yc zc]T, which is the relativeposition vector of the center of mass of the spacecraft inthe spacecraft body-fixed frame, and J = (

1/

2)

trace(J )I + J . The gravity force due to the gravity potential in(19) in the spacecraft body-fixed frame can then bemodeled by

FGa= Fp

Ga+ RF2

Ga= −−mμa

‖rc‖3Rrc − 3

(μa

‖rc‖5

)J pc

+15

2

(μa( pT

c J pc)

‖rc‖7

)pc + RF2

Ga, (27)

where

F2Ga

=m

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

−μaC20xc

‖rc‖5+ 5μaC20xc(x2

c + y2c − 2z2

c)

2‖rc‖7

+6μaC22xc

‖rc‖5− 15μaC22xc(x2

c − y2c )

‖rc‖7

−μaC20yc

‖rc‖5+ 5μaC20yc(x2

c + y2c − 2z2

c)

2‖rc‖7

−6μaC22yc

‖rc‖5− 15μaC22yc(x2

c − y2c )

‖rc‖7

2μaC20zc

‖rc‖5+ 5μaC20zc(x2

c + y2c − 2z2

c)

2‖rc‖7

−15μaC22zc(x2c − y2

c )

‖rc‖7

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

.

The gravitational force acting on a particle of mass dmat a distance r from the asteroid center of mass isdescribed by

d FGa= (

d Fp

Ga+ d F2s

Ga

)dm. (28)

The gravity gradient torque due to the primary gravityforce is already derived in (5). Thus, the gravity gradienttorque due to primary gravity force of the asteroid on thespacecraft is given by

Mp

Ga= 3

(μa

‖rc‖5

)(( pc)×J pc). (29)

The gravitational force acting on a particle of mass dm at adistance r from the asteroid center of mass, havinglongitude λ and latitude δ, due to the second degree andorder of gravitational potential is described by [29]:

d F2sGa

=[∂U2

∂rer + 1

r cos δ

∂U2

∂λeλ + 1

2

∂U2

∂δeδ

]dm,

(30)

where U2 is given in (17). The gravity gradient torque onthe spacecraft due to U2 gravitational potential can bedetermined from

M2Ga

=∫

ρ × d F2sGa

. (31)

Evaluation of this gravity gradient torque involvesexpanding the various powers of ‖rc + ρ‖ using thebinomial theorem and neglecting terms involving third-and higher order powers of ‖ρ‖/‖rc‖. Let

M2Ga

=[M2

GaxM2

GayM2

Gaz

]Tdenote the gravity gradient

torque in the spacecraft body-fixed frame (bx, by, bz), andlet the inertia matrix for the spacecraft be given as J =diag[J1 J2 J3]. The orientation of the spacecraft body-fixedframe with respect to the LVLH frame is described by thefollowing rotation matrix:

RL = RTLR (32)

510 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 51, NO. 1 JANUARY 2015

RL =

⎡⎢⎣

cos �s − sin �s 0

sin �s cos �s 0

0 0 1

⎤⎥⎦

⎡⎢⎣

1 0 0

0 cos is − sin is

0 sin is cos is

⎤⎥⎦

×

⎡⎢⎣

cos us − sin us 0

sin us cos us 0

0 0 1

⎤⎥⎦ ,

where RL is the rotation matrix from the LVLH frame ofthe spacecraft to the inertial frame, �s is the rightascension of the ascending node, is is the inclination angle,and us is the argument of the latitude for the spacecraft,which can be computed from the spacecraft orbit [31]. Thesequence of rotations for the rotation matrix RL in (32) isyaw (ψ) around the bx axis, followed by pitch (θ) aroundthe by axis, followed by roll (φ) around the bz axis. Thegravity gradient torque components M2

Gain the spacecraft

body-fixed frame (bx, by, bz) can be written as followsafter some algebra [29]:

M2Gax

= μa

‖rc‖3

[5A(J3 − J2) cos(ψ) cos2 θ sin ψ

+ 5B

(− 2

5J1 cos ψ sin φ + (J1 − J2

+ J3) sin ψ cos2 θ cos φ

)](33)

M2Gay

= μa

‖rc‖3

[5A(J3 − J1) cos(ψ) cos θ sin θ

+ 5B

(− 2

5J2(sin ψ sin θ sin φ − cos ψ cos φ)

+(J2−J1+J3)(sin ψ sin θ sin φ+sin2 θ cos ψ cos φ)

+ (J2 − J3 + J1) cos2 θ cos ψ cos φ

)](34)

M2Gaz

= μa

‖rc‖3

[5A(J1 − J2) sin(ψ) cos θ sin θ

+ 5B

(− 2

5J3(sin ψ cos φ − cos ψ sin θ sin φ)

+ (J2−J1+J3)(cos ψ sin ψ sin φ−sin2 θ sin ψ cos φ)

− (J1 − J2 + J3) cos2 θ sin ψ cos φ

)], (35)

where A and B are defined as

A =[−3

2C20 + 9C22 cos(2λc)

](re

rc

)2

B = [6C22 sin(2λc)]

(re

rc

)2

,

with λc being the longitude of the center of mass of thespacecraft measured in the asteroid boy-fixed frame. Theeffective gravity gradient torque on the spacecraft by theasteroid is then the sum of the primary gravity gradient

moment Mp

Ga=

[M

p

GaxM

p

GayM

p

Gaz

]Tand

M2Ga

=[M2

GaxM2

GayM2

Gaz

]T:

MGa=

[(M

p

Gax+ M2

Gax

) (M

p

Gay+ M2

Gay

)

×(M

p

Gaz+ M2

Gaz

)]T, (36)

where re is the maximum radius of the asteroid. Thedynamics in (15) and (16) can then be expressed in thecompact form

Iξ = ad�ξ Iξ + ϕGS

+ ϕGa+ ϕc, (37)

with

ϕGS=

[MGS

FGS

]∈ R

6, ϕGa=

[MGa

FGa

]∈ R

6,

ϕc =[

τ c

φc

]∈ R

6, I =[

J 03×3

03×3 mI

]∈ R

6×6,

where ϕGS∈ R

6 is the vector of known gravity inputs(torque and force) on the spacecraft due to the sun,ϕGa

∈ R6 is the vector of known gravity inputs on the

spacecraft due to the asteroid, and ϕc ∈ R6 is the vector of

control inputs (torque and force) on the spacecraft.

III. FINITE-TIME TRACKING OF SPACECRAFTHOVERING MANEUVERS

A. Setting Up the Relative Configuration

Let the desired relative configuration of the spacecraftbe given by (hf ∈ SE(3)n), where hf provides theappropriate asteroid–spacecraft separation and relativeorientation. Let (g, ξ ) ∈ SE(3) × R

6 denote the states(configuration and velocities) of the spacecraft. Given theasteroid trajectory generated by (8) and (13), the desiredstates of the spacecraft are

g0d = g0(hf ) and ξ 0d = Ad(hf )−1ξ 0 for t ≥ t0, (38)

and the configuration error between the spacecraft and theasteroid is

h = (g0)−1gk for t ≥ t0. (39)

The exponential coordinate vector η for theconfiguration error of the spacecraft is expressed as

η =[

�

β

]∈ R

6, (40)

where � ∈ R3 and β ∈ R

3 are the exponential coordinatevectors for the attitude tracking error (principal rotationvector) and the position tracking error, respectively. Theconfiguration error of the spacecraft is then expressed inthe exponential coordinates using the logarithm map:

(η)∨ = logm((hf )−1h) = logm((g0d )−1g), (41)

where logm : SE(3) → se(3) is the logarithm map(inverse of the exponential map). This makes η the

LEE ET AL.: FINITE-TIME CONTROL FOR SPACECRAFT BODY-FIXED HOVERING OVER AN ASTEROID 511

exponential coordinate vector giving the relativeconfiguration between the desired configuration and theactual configuration of the spacecraft. Fig. 2 illustrates thedesired relative configuration hf and the body-fixedhovering maneuver of a spacecraft over a uniformlyrotating asteroid. The notations b0

x, b0y, b0

z and bx, by, bz

represent the unit vectors in the asteroid body-fixed frameand the spacecraft body-fixed frame, respectively.

REMARK 1 The logarithm map logm:SE(3) → se(3) isbijective when the principal angle of rotationcorresponding to R(�) has a magnitude less than π

radians, i.e., ‖�‖< π . It is not uniquely defined when‖�‖ is exactly π radians.

The relative velocities between the asteroid andspacecraft are obtained by taking a time derivative of bothsides of (39) and substituting in (8) and (14), which yields

ξ = ξ − Ad(h)−1ξ 0 = ξ − Ad(h0d )−1ξ 0d (42)

where h0d = (hf )−1h = expm((η)∨) and ξ is the relativevelocity of the asteroid with respect to the spacecraft in itsbody coordinate frame. Let us denote the quantities h0 =h(t0), η0 = η(t0), ξ 0 = ξ (t0), and ξ 0 = ξ (t0) at initial timet0, and assume that all these quantities are known; these areknown if the initial state of the spacecraft is known. Thekinematics of exponential coordinates given by [37] are

˙η = G(η)ξ . (43)

Given η = [�TβT]T ∈ R6, the function G(η) can be

expressed as a block-triangular matrix, as follows [38]:

G(η) =[

A(�) 03×3

T (�, β) A(�)

], (44)

where

A(�) = I + 1

2(�)× +

(1

θ2− 1 + cos θ

2θ sin θ

)(�×)2

S(�) = I + 1 − cos θ

θ2(�)× + θ − sin θ

θ3(�×)2

T (�, β) = 1

2(S(�)β)× A(�) +

(1

θ2−1 + cos θ

2θ sin θ

)(�βT

+ (�Tβ)A(�))− (1+cos θ)(θ− sin θ)

2θ sin2 θS(�)β�T

+(

(1 + cos θ)(θ + sin θ)

2θ3 sin2 θ− 2

θ4

)�T�T,

with θ = ‖�‖ being the norm of �, which is the vector ofthe first three components of the exponential coordinatevector η ∈ R

6; therefore, θ corresponds to the principalrotation angle. The exponential coordinate vector of �

and its time derivatives are obtained from Rodrigues’sformula, which is a well-known formula for the rotationmatrix in terms of the exponential coordinates on SO(3):

R(�) = I + sin(θ)

θ(�)× + 1 − cos(θ)

θ2(�×)2. (45)

Fig. 2. Body-fixed spacecraft hovering over asteroid.

The following result is used to prove the main result on thetracking control scheme in the next subsection.

LEMMA 1 The matrix function G(·) : R6 → R

6×6 definedby (44) satisfies the relation [39, 40]

G(η)η = η. (46)

The vector of relative velocities can be expressed as

ξ = ξ −

ξ , where

ξ = Ad(h)−1ξ 0. (47)

The expression for the relative accelerations, i.e., the timederivative of ξ , can be obtained from the time derivative ofξ in (47) as derived in [39, 40], and is given by

.

ξ = ξ + adξ Ad(h)−1ξ 0 − Ad(h)−1

.

ξ 0 . (48)

Substituting the dynamics equation (37) into (48), weobtain the following equation for the time evolution of therelative accelerations:

I

.

ξ = ad�ξ Iξ + ϕGS

+ ϕGa+ ϕc

+ I(adξ Ad(h)−1ξ 0 − Ad(h)−1

.

ξ 0). (49)

B. Trajectory Tracking Errors

The desired state trajectory of the spacecraft requiredto maintain the formation is given by the desiredconfiguration g0 and its time derivative, which gives thedesired velocities ξ 0 = Ad(hf )−1ξ 0 as in (38). Let us denotethe desired state trajectory for time t ≥ t0 by the desiredposition vector in the inertial frame b0(t), the desiredattitude R0(t), and the desired translational and angularvelocity in the body frame, ν0(t) and �0(t). The kinematicsin SE(3) still holds for the desired states. Next, as in [14],we define the trajectory tracking errors of the spacecraft as

a(t) = b(t) − b0(t), position tracking error in theinertial frame;

x(t) = (R0)T(t)ak(t), position tracking error in theasteroid’s body-fixed frame;

Q(t) = (R0)T(t)R(t), attitude tracking error in therotation matrix (orientation);

512 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 51, NO. 1 JANUARY 2015

υ(t) = ν(t) − QT(t)(ν0 + (�0(t))× x(t)), translationalvelocity tracking error; and

�(t) = �(t) − QT(t)�0(t), angular velocity trackingerror.

C. Finite-Time Control Scheme for Body-FixedSpacecraft Hovering

A finite-time control scheme for body-fixed spacecrafthovering over an asteroid in the framework of geometricmechanics is presented such that the spacecraft follows itsdesired trajectory, which maintains a constant relativepose with respect to the asteroid in finite-time. The controlscheme for the spacecraft in the relative configurationachieves the tracking of its desired state trajectory in finitetime, based on its relative motion with respect to theasteroid, as described earlier.

THEOREM 1 Consider the finite-time control scheme forϕc given by

ϕc = −ϕGS− ϕGa

− ad∗ξ Iξ − I

(adξ Adh−1ξ 0 − Adh−1 ξ

0

+KH (η)G(η)ξ) − γ I�

(�TI�)1− 1q

,

� = ξ + Kη

(ηTη)1− 1q

,

(50)

where

H (η) = 1

(ηT η)1− 1q

{I − 2

(1 − 1

q

)ηηT

ηT η

},

K =[

k1I 03×3

03×3 k2I

];

K =[

k1I 03×3

03×3 k2I

]

is a positive definite matrix; k1, k2, and γ are positivescalar values; and q ∈ (1, 2) is a rational number(preferably a ratio of odd integers, to avoid sign matcheswhen taking powers using a computer code). The feedbacksystem given by (43) and (49) and the control scheme in(50) globally stabilizes (η, ξ ) = (0, 0) ∈ R

6 × R6 and

therefore tracks the trajectory (g, ξ ) = (g0, ξ 0) given by(38) in finite time. Moreover, the domain of attraction ofthis trajectory is almost global over the state spaceSE(3) × R

6.The following lemma is used to prove the finite-time

stability and convergence.

LEMMA 2 Suppose there exists a smooth function χ . Ifthere exist c > 0 and 0 < ε < 1 such that χ is positivedefinite and χ + cχε is negative semidefinite, then χ canconverge to zero in finite time. Meanwhile, theconvergence time Tc, which is the time interval to reachχ = 0, satisfies

Tc ≤ χ1−ε0

c(1 − ε), (51)

where χ0 denotes the initial value of χ [27, 28].

PROOF For the closed-loop feedback control given by(50), we propose the following Lyapunov candidatefunction for the kth spacecraft:

V (ξ , η) = 1

2�T

I�. (52)

Note that V (ξ , η) ≥ 0 and V (ξ , η) = 0 if and only if� = 0. Thus, V (ξ , η) is positive definite on R

6 with itsminimum at the desired equilibrium. We evaluate the timederivative of V (ξ , η) along the trajectories of theclosed-loop system. The time derivative of V (ξ , η) is

V (ξ , h, ξ 0, η) = �TI

.

� = �TI

( .

ξ + d

dtK

η

(ηT η)1− 1q

)

= {�T (I.

ξ +I(KH (η).

η}. (53)

Substituting (43) and (49) into (53) yields

V = �T (ad�ξ Iξ + ϕGS

+ ϕGa+ ϕc + I(adξ Ad(h)−1ξ 0

− Ad(h)−1 ξ0 + KH (η)G(η)ξ )). (54)

The time derivative of V along the trajectories of thefeedback system is obtained by substituting the controlscheme (50) into (54):

V = −γ�T

I�

(�TI�))1− 1q

= −γ (�TI�)

1q = −γ

(2

1

2�T

I�

) 1q

= −21q γ V

1q ≤ −2

1q cV

1q , (55)

where c ≥ γ and V = 0 if and only if � = 0. Therefore, thecondition for Lyapunov stability is satisfied and the slidingsurface � is achieved in finite time according to Lemma 2.The corresponding convergence time tc is given by

tc ≤ V1− 1

q

0

21q c

(1 − 1

q

) , (56)

where V0 denotes the initial value of V. Next, thedynamics of the feedback system—in terms of theexponential coordinates describing the motion relative tothe virtual target—are obtained by substituting the controlscheme (50) into (49). These equations of relative motionfor the feedback system are

.

η = G(η)ξ (57)

.

ξ = −KH (η)G(η)ξ − γ �

(�T I�)1− 1q

. (58)

When � = 0, we have

ξ = − K η

(ηT η)1− 1q

. (59)

LEE ET AL.: FINITE-TIME CONTROL FOR SPACECRAFT BODY-FIXED HOVERING OVER AN ASTEROID 513

Substituting (59) into the time derivatives of exponentialcoordinates in (57), we can express it in terms ofexponential coordinates η:

.

η = G(η)

(− K η

(ηT η)1− 1q

). (60)

Substituting (40) and (44) in terms of � and β into (60)and using Lemma 1, we can express the time derivatives ofthe attitude and position vector tracking errors inexponential coordinates, when � converges to zero, as thefollowing:

.

� = − k1

(ηT η)1− 1q

� (61)

.

β = − k1

(ηT η)1− 1q

β − (k2 − k1)

(ηT η)1− 1q

A(�)β. (62)

From (61), we see that � converges exponentially to thezero vector. From (44), we see that when� → 0, A(�) → I . Thus, the time derivative of β

approaches.

β = − k2

(ηT η)1− 1q

β. (63)

Therefore, we see that once � converges to zeroexponentially, we also have β converging to zeroexponentially. Therefore � → 0 implies that η → 0 andξ → 0. Given that η is defined for all relativeconfigurations except those that have a relative orientationwith a principal rotation angle of π radians, this controlscheme can converge to the desired trajectory from almostall initial states except those that have a relative orientationwith a principal angle of π radians. Thus, the control isfinite-time convergent, and its domain of convergence isalmost global on the state space SE(3) × R

6.

REMARK 2 Equations (57) and (58) can be implementedwith a standard integration scheme like Euler’s method orthe Runge-Kutta methods, since η and ξ are vectors in R

6.Thereafter, the relative configuration is obtained from

h = hf expm((η)∨

)(64)

and the absolute configuration of the kth spacecraft isgiven by

g = g0h = g0hf expm((η)∨

). (65)

However, the asymptotic tracking control scheme [11],which was adapted from [40] and developed for thebody-fixed spacecraft hovering over an asteroid in theframework of geometric mechanics, is presented inTheorem 2 to compare with the performance of thefinite-time control scheme.

THEOREM 2 The asymptotic tracking control scheme ϕca

for spacecraft hovering around the asteroid is given by

ϕca= −φGa

− φGS− ad�

ξ Iξ − I(adξ Adh−1ξ 0 − Adh−1 ξ

0

+ KG(η)ξ) − P� (66a)

� = ξ + K η (66b)

where

P =[

p1I 03×3

03×3 p2I

]

and

K =[

k1I 03×3

03×3 k2I

]

are positive definite control gain matrices. The domain ofattraction of this trajectory is almost global over the statespace SE(3) × R

6.

IV. NUMERICAL SIMULATION RESULTS

A spacecraft mission scenario is considered todemonstrate the capability of the proposed finite-timecontrol scheme in Theorem 1. Assume that a tumblingasteroid is in orbit in the heliocentric frame. As ourtumbling asteroid, we choose 4179 Toutatis. Theasymptotic control scheme in Theorem 2 is alsoimplemented, to compare with the performance of thefinite-time control scheme in Theorem 1. The spacecraft isrequired to achieve body-fixed hovering over the tumblingasteroid using both control schemes, to carry out ascientific exploration (illustrated in Fig. 2). Then, thesimulation results are shown together to compare witheach other. The mission scenario aims to make thespacecraft approach 4179 Toutatis in orbit and maintainthe desired relative position in the asteroid body-fixedframe, and meanwhile to make the attitude of thespacecraft almost stationary to the asteroid. For thismission scenario, the trajectory of asteroid 4179 Toutatisis assumed to be known through the spacecraft onboardnavigation. The desired relative position in the asteroidbody-fixed frame is selected as [0 0 10]T km. The desiredattitude is the varying body-fixed attitude of asteroid 4179Toutatis. The spacecraft is required to achieve the desiredrelative configuration and velocities in the asteroidbody-fixed frame from the initial relative position[5 5 15]T km in the asteroid body-fixed frame and thenmaintain them for the remainder of total flight time (4 h).The numerical values of k1, k2, q, and γ are chosen as0.01, 0.01, 24/21, and 0.04, respectively, for the finite-timecontrol scheme. On the other hand, the values of k1, k2, p1,and p2 are all identically chosen as 0.0733 for theasymptotic control scheme. The numerical values used forthe gravitational parameter of the sun μS and thegravitational parameter of asteroid 4179 Toutatis μa are1.3274 × 1011 km3/s2 and 3.3369 × 10−6 km3/s2,respectively. The mass of the asteroid is 5 × 103 kg [41].The numerical values used for the second-degree andsecond-order spherical harmonic field coefficients C20 and

514 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 51, NO. 1 JANUARY 2015

TABLE IInitial Relative Position, Translational Velocity, Relative Attitude, andAngular Velocity of the Spacecraft in the Asteroid Body-Fixed Frame

Initial Conditions Values

Relative position (km) [5 5 15]T

Relative velocity (km/s) [−0.0063 −0.0025 −0.0036]T

Relative attitude (◦) 120.0Relative angular velocity (radians/s) 10−3 × [−0.06 0.26 0.05]T

TABLE IIParameters of the Asteroid 4179 Toutatis and the Spacecraft

Type Mass (kg) Moment of Inertia (kg·m2)

Asteroid 4179Toutatis 5 × 1013 1013 × diag[2.3425 5.9650 6.5025]T

Spacecraft 607 diag[48.5 47.6 51.0]T

C22 are −0.4160 km2 and 0.1811 km2, respectively[2, 41]. The total simulation time is 4 h and the selectedtime step is 0.1 s. Table I shows initial conditions ofrelative position, relative translational velocity, relativeattitude in terms of a principal rotation angle, and relativeangular velocity of the spacecraft with respect to asteroid4179 Toutatis in the asteroid body-fixed frame. Table IIshows the parameters of the asteroid 4179 Toutatis and thespacecraft, respectively.

Fig. 4a shows the body-fixed hovering trajectories over4179 Toutatis in the asteroid body-fixed frame using thefeedback control schemes in (50) and (66a), respectively.The desired relative position between the asteroid and thespacecraft, [0 0 10]T km in the asteroid body-fixed frame,is almost obtained using the finite-time control, whereas itis not fully obtained using the asymptotic tracking control.The approach trajectories in Fig. 3 are created by thefeedback control schemes in (50) and (66a) to reduce therelative configuration between the asteroid and thespacecraft without explicit reference trajectories. Fig. 4bshows a rigid-body trajectory where the spacecraftbody-fixed frames are also plotted along the approachtrajectory, using the finite-time control only in (50), in theasteroid body-fixed frame with respect to the desiredrelative configuration. The rigid-body trajectory in Fig. 4bdemonstrates that the coupled translational and attitudemaneuver is simultaneously performed to achieve thedesired relative configuration.

Fig. 4 presents the norms of the position andtranslational velocity tracking errors. The norm of theposition tracking error ‖a‖ in Fig. 5a using the finite-timecontrol scheme (dotted line) shows that it takes less than1 h to converge to less than 0.1 m norm of the positiontracking error from the starting point, which is 5

√3 km

away from the desired point. The spacecraft continues toapproach the desired relative position and maintains it,leading to a station-keeping maneuver. On the other hand,the norm of the position tracking error ‖a‖ in Fig. 5a usingthe asymptotic control scheme (solid line) shows that itdoes not converge to less than 2000 m even after 4 h. The

norm of the translational velocity tracking errors ‖υ‖ inFig. 6b also shows that they are consistent with the normof the position tracking errors. The norm of thetranslational velocity tracking error ‖υ‖ in Fig. 6b by thefinite-time control scheme (dotted line) converges to lessthan 0.01 m/s within almost 2 h and maintains until thefinal time. On the other hand, the norm of the translationalvelocity tracking errors ‖υ‖ in Fig. 5b by the asymptoticcontrol scheme (solid line) shows that it does not convergeto less than 0.1 m/s even after 4 h. The norms of theposition and translational velocity tracking errors obtainedwith the finite-time control scheme converge to zero,unlike the norms of the position and translational velocitytracking errors obtained with the asymptotic trackingcontrol scheme. Thus, Fig. 4 obviously verifies that theposition and translational velocity tracking errorsconverge to the desired relative position and velocity fasterwhen the finite-time control is used.

Fig. 5 presents the norms of the attitude and angularvelocity tracking errors. The norms of attitude trackingerrors

∥∥�∥∥ using the finite-time control scheme (dotted

line) and the asymptotic tracking control scheme (solidline) in Fig. 6a take about 1 h to converge to 0.1◦ from120◦. The asymptotic tracking control shows fasterconvergence to zero in the norm of attitude tracking error∥∥�

∥∥ than the finite-time control until about 30 min.However, the norms of attitude tracking errors converge tozero faster after 30 min when the finite-time control isused. The norms of angular velocity tracking errors

∥∥�∥∥

in Fig. 6b are also consistent with the norms of the attitudetracking errors. The norms of angular velocity trackingerrors

∥∥�∥∥ using both control schemes take about 1 h to

converge to less than 4 × 10−5 radians/s. However, thenorms of angular velocity tracking errors converge to zerofaster after 1 h when the finite-time control is used. Thus,Fig. 5 verifies that the finite-time control achieves fasterattitude synchronization with the attitude and angularvelocity of the asteroid than the asymptotic trackingcontrol.

Fig. 6 presents the norms of the control forces andtorques. The norm of the control forces in Fig. 7a obviouslyshows that initially large control forces are requiredby both control schemes. However, the norm of the controlforce using the finite-time control is less than the norm ofthe control force using the asymptotic tracking control. Thenorm of the control force using the asymptotic trackingcontrol scheme increases initially about 100 N, which mayresult in the thrust saturation and decrease to 0.1 N after1 h. On the other hand, the norm of the control force usingthe finite-time control scheme is initially less than 50 N anddecreases to 0.36 N after 1 h. Then relatively larger controlforces are continuously used to maintain the desiredrelative position when the finite-time control schemeis used. Thus, the finite-time control scheme producesmore stable transient control force response to performtranslational maneuver than the asymptotic tracking controlscheme, while avoiding thrust saturation. The norms of thecontrol torques in Fig. 7b using both control schemes also

LEE ET AL.: FINITE-TIME CONTROL FOR SPACECRAFT BODY-FIXED HOVERING OVER AN ASTEROID 515

Fig. 3. Body-fixed hovering trajectories over asteroid 4179 Toutatis. (a) Body-fixed hovering trajectory over asteroid. (b) Body-fixed rigid-bodyhovering trajectory.

Fig. 4. Norms of position and translational velocity tracking errors. (a) Norms of position tracking errors. (b) Norms of translational velocitytracking errors.

Fig. 5. Norms of attitude and angular velocity tracking errors. (a) Norms of attitude tracking errors. (b) Norms of angular velocity tracking errors.

516 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 51, NO. 1 JANUARY 2015

Fig. 6. Norms of control forces and torques. (a) Norms of control forces. (b) Norms of control torques.

Fig. 7. Time integrals of norms of control forces and torques.

show that relatively large control torques are initially usedfor about 180 s to synchronize the body-fixed frame of thespacecraft with the body-fixed frame of the asteroid. Thenthe norms of the control torques in Fig. 7b using bothcontrol schemes show that after 180 s relatively very smalltorques are required for the spacecraft to maintain theattitude synchronization with the body-fixed frame of theasteroid. However, the finite-time control scheme requiressmaller transient control response to obtain the attitudesynchronization than the asymptotic tracking controlscheme.

Finally, Fig. 7 presents the time integrals of the normsof control forces and control torques implemented usingboth control schemes. The total control forces and torquesused by the asymptotic tracking control for 4 h are 1.1519× 104 N·s and 0.287 N·m·s, respectively. The total controlforces and torques used by the finite-time control for 4 hare 9.2254 × 103 N·s and 0.2036 N·m·s, respectively. It isobvious that the asymptotic tracking control uses morecontrol forces and toques than the finite-time control. In

other words, the finite-time control can save more controluses than the asymptotic tracking control. This is due tothe fact that initially large transient responses occur whenthe asymptotic tracking control is used. The time integralsof the norm of the control force (solid lines),

∫ t

0

∥∥φ1c

∥∥dt

and∫ t

0

∥∥τ 1c

∥∥dt , continuously increase almost until 1 h, toperform approach and attitude alignment maneuverssimultaneously. On the other hand, relatively smallercontrol forces and torques are used to maintain the desiredrelative configuration after 1 h. Hence, they are almostconstant for the time after the desired relativeconfiguration is obtained.

V. CONCLUSION

Autonomous body-fixed hovering of a spacecraft nearan asteroid is addressed using a continuous finite-timecontrol scheme for rigid-body motion. The maincontribution of this work is a continuous finite-timetracking control scheme that can perform coupledtranslational and rotational maneuvers over a tumblingasteroid and converge in finite time to a desired asteroidbody-fixed position and orientation. The dynamics modeland the control scheme are obtained in the framework ofgeometric mechanics. The finite-time control scheme usesfeedback of the state trajectories (positions, orientations,and velocities) of the asteroid that is being tracked by thespacecraft. The control law is expressed in terms of theexponential coordinates on the Lie group of rigid-bodymotions that give the relative pose of the spacecraft withrespect to the asteroid, as well as the relative velocities.Within the Lyapunov framework, the continuousfinite-time control is proved to guarantee almost globalfinite-time convergence to the desired relativeconfiguration in the nonlinear state space of rigidspacecraft dynamics. A six-degrees-of-freedom simulationwas utilized to demonstrate the tracking performance ofthis finite-time control for body-fixed hovering over atumbling asteroid. As a tumbling asteroid, 4179 Toutatiswas selected for this simulation. The finite-time control

LEE ET AL.: FINITE-TIME CONTROL FOR SPACECRAFT BODY-FIXED HOVERING OVER AN ASTEROID 517

was able to drive the spacecraft to the desired relativeconfiguration and velocities without using explicitreference states in finite time, so that body-fixed hoveringcan be achieved autonomously. An asymptotic trackingcontrol scheme was also implemented to compare with theperformance of the finite-time control scheme. Thefinite-time control showed superior performance to theasymptotic tracking control in terms of faster trackingconvergence, more stable transient response, and lowertransient control effort.

REFERENCES

[1] Sawai, S., Scheeres, D. J., and Broschart, S. B.Control of hovering spacecraft using altimetry.Journal of Guidance, Control, and Dynamics, 25, 4 (2002),786–795.

[2] Scheeres, D. J. Controlled Hovering Motion at an Asteroid. InOrbital Motion in Strongly Perturbed Environments. Berlin:Springer, 2012, ch. 11, 243–254.

[3] Broschart, S. B., and Scheeres, D. J.Boundedness of spacecraft hovering under dead-band controlin time-invariant systems.Journal of Guidance, Control, and Dynamics, 30, 2 (2007),601–610.

[4] Broschart, S. B., and Scheeres, D. J.Control of hovering spacecraft near small bodies: Applicationto Asteroid 25143 Itokawa.Journal of Guidance, Control, and Dynamics, 28, 2 (2005),343–354.

[5] Scheeres, D. J.Stability of hovering orbits around small bodies.Advances in the Astronautical Sciences, 102 (1999),855–875.

[6] Kawaguchi, J., Fujiwara, A., and Uesugi, T.Hayabusa—Its technology and science accomplishmentsummary and Hayabusa-2.Acta Astronautica, 62, 10–11 (2008), 639–647.

[7] Yoshimitsu, T., Kawaguchi, J., Hashimoto, T., Kubota, T., Uo,M., Morita, H., and Shirakawa, K.Hayabusa—Final autonomous descent and landing based ontarget marker tracking.Acta Astronautica, 65, 5–6 (2009), 657–665.

[8] Kubota, T., Hashimoto, T., Uo, M., Maruya, M., and Baba, K.Maneuver strategy for station keeping and global mappingaround an asteroid.Advances in the Astronautical Sciences, 108 (2001),769–780.

[9] Gaudet, B., and Furfaro, R.Robust spacecraft hovering near small bodies in environmentswith unknown dynamics using reinforcement learning.In AIAA/AAS Astrodynamics Specialist Conference,Minneapolis, MN, Aug. 2012.

[10] Berry, K., Sutter, B., May, A., Williams, K., Barbee, B. W.,Beckman, M., and Williams, B.OSIRIS-REx touch-and-go (TAG) mission design andanalysis.In 36th Annual AAS Guidance and Control Conference,Breckenridge, CO, Feb. 2013.

[11] Lee, D., Sanyal, A. K., Butcher, E. A., Williams, K., andScheeres, D. J.Spacecraft hovering control for body-fixed hovering over auniformly rotating asteroid using geometric mechanics.In AAS/AIAA Astrodynamics Specialist Conference, HiltonHead, SC, Aug. 2013.

[12] Scheeres, D. J., Williams, B. G., and Miller, J. K.Evaluation of the dynamic environment of an asteroid:Applications to 433 Eros.

Journal of Guidance, Control, and Dynamics, 23, 3 (2000),466–475.

[13] Hu, W.-D., and Scheeres, D. J.Periodic orbits in rotating second degree and order gravityfields.Chinese Journal of Astronomy and Astrophysics, 8, 1 (2008),108–118.

[14] Sanyal, A., Nordkvist, N., and Chyba, M.An almost global tracking control scheme for maneuverableautonomous vehicles and its discretization.IEEE Transactions on Automatic Control, 56, 2 (Feb. 2011),457–462.

[15] Nordkvist, N., and Sanyal, A. K.A Lie group variational integrator for rigid body motion inSE(3) with applications to underwater vehicle dynamics.In 49th IEEE Conference on Decision and Control, Atlanta,GA, Dec. 2010, 5414–5419.

[16] Sanyal, A., Holguin, L., and Viswanathan, S. P.Guidance and control for spacecraft autonomous chasing andclose proximity maneuvers.In 7th IFAC Symposium on Robust Control Design, Aalborg,Denmark, June 2012, 7, 753–758.DOI. No. 10.3182/20120620-3-DK-2025.00068.

[17] Lee, T., Leok, M., and McClamroch, N. H.Dynamics of connected rigid bodies in a perfect fluid.In American Control Conference, St. Louis, MO, June 2009,408–413.

[18] Marsden, J. E., and Ratiu, T. S. Introduction to Mechanics andSymmetry (2nd ed.). New York: Springer, 1999, 261–308.

[19] Bloch, A. M., Crouch, P., Baillieul, J., and Marsden, J.Mathematical Preliminaries. In Nonholonomic Mechanics andControl. New York: Springer, 2003, ch. 2, 97–104.

[20] Bullo, F., and Lewis, A. D.Chapter title.In Geometric Control of Mechanical Systems. New York:Springer, 2005, ch. 5, 247–298.

[21] Varadarajan, V. S.Chapter title.In Lie Groups, Lie Algebras, and Their Representations (2nded.). New York: Springer, 1984, ch. 2, 84–92.

[22] Bhat, S. P., and Bernstein, D. S.Continuous finite-time stabilization of the translational androtational double integrators.IEEE Transactions on Automatic Control, 43, 5 (May 1998),678–682.

[23] Ryan, E. P.Finite-time stabilization of uncertain nonlinear planar systems.Dynamics and Control, 1, 1 (1991), 83–94.

[24] Park, K.-B., and Tsuji, T.Terminal sliding mode control of second-order nonlinearuncertain systems.International Journal of Robust and Nonlinear Control, 9, 11(Sept. 1999), 769–780.

[25] Wang, J., and Sun, Z.6-DOF robust adaptive terminal sliding mode control forspacecraft formation flying.Acta Astronautica, 73 (2012), 76–87.

[26] Tadikonda, S. S. K., and Baruh, H.Gibbs phenomenon in structural control.AIAA Journal, 29, 9 (1991), 1488–1497.

[27] Wu, S.-N., Sun, X.-Y., Sun, Z.-W., and Chen, C.-C.Robust sliding mode control for spacecraft globalfast-tracking manoeuvre.Journal of Astronomy and Space Sciences, 225, 7 (2012),749–760.

[28] Bhat, S. P., and Bernstein, D. S.Finite-time stability of continuous autonomous systems.SIAM Journal on Control and Optimization, 38, 3 (2000),751–766.

518 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 51, NO. 1 JANUARY 2015

[29] Liang, C., and Li, Y.Attitude analysis and robust adaptive backstepping slidingmode control of spacecrafts orbiting irregular asteroids.Mathematical Problems in Engineering, 2014 (2014), 1–15.

[30] Lee, D., Sanyal, A. K., Butcher, E. A., and Scheeres, D. J.Finite-time control for body-fixed hovering of a rigidspacecraft over an asteroid.Presented at the 24th AAS/AIAA Space Flight MechanicsMeeting, Santa Fe, NM, Jan. 2014; AAS Paper14–221.

[31] Lee, D., Cochran, J. E., Jr., and No, T. S.Robust position and attitude control for spacecraft formationflying.Journal of Aerospace Engineering, 25, 3 (2012), 436–447.

[32] Xin, M., and Pan, H.Integrated nonlinear optimal control of spacecraft in proximityoperations.International Journal of Control, 83, 2 (2010), 347–363.

[33] Zhang, F., and Duan, G.-R.Integrated relative position and attitude control of spacecraftin proximity operation missions.International Journal of Automation and Computing, 9, 4(2012), 342–351.

[34] Park, H.-E., Park, S.-Y., Park, C., and Kim, S.-W.Development of integrated orbit and attitudesoftware-in-the-loop simulator for satellite formation flying.Journal of Astronomy and Space Sciences, 30, 1 (2013),1–10.

[35] Schaub, H., and Junkins, J. L. Analytical Mechanics of SpaceSystems (2nd ed.). Reston, VA: American Institute ofAeronautics and Astronautics, 2009, 188–192.

[36] Sanyal, A. K., Lee, T., Leok, M., and McClamroch, N. H.Global optimal attitude estimation using uncertaintyellipsoids.Systems and Control Letters, 57, 3 (2008), 236–245.

[37] Bullo, F., and Murray, R. M.Proportional derivative (PD) control on the Euclidean group.In Proceedings of 3rd European Control Conference, 1995,1091–1097.

[38] Bras, S., Izadi, M., Silvestre, C., Sanyal, A., and Oliveira, P.Nonlinear observer for 3D rigid body motion.In IEEE 52nd Annual Conference on Decision and Control,Dec. 2013, 2588–2593.

[39] Lee, D., Sanyal, A. K., Butcher, E. A., and Scheeres, D. J.Almost global asymptotic tracking control for spacecraftbody-fixed hovering over an asteroid.Aerospace Science and Technology, 38 (2014), 105–115.

[40] Lee, D., Sanyal, A. K., and Butcher, E. A.Asymptotic tracking control for spacecraft formation flyingwith decentralized collision avoidance.Journal of Guidance, Control, and DynamicsIn press.

[41] Scheeres, D. J., Ostro, S. J., Hudson, R. S., DeJong, E. M., andSuzuki, S.Dynamics of orbits close to Asteroid 4179 Toutatis.Icarus, 132, 1 (1998), 53–79.

Daero Lee is a postdoctoral research associate and instructor in the department ofmechanical and aerospace engineering at New Mexico State University. He receivedB.A. and M.S. degrees in aerospace engineering from Konkuk University, Seoul, Koreain 1999 and 2001. He also received an M.S. degree in aerospace engineering fromAuburn University in 2006. He received a Ph.D. degree in aerospace engineering fromthe Missouri University of Science and Technology in 2009. From 2010 to 2011, he wasa postdoctoral research associate in the wind-energy power-grid adaptation technologycenter at Chonbuk National University and the department of aerospace engineering atKAIST in Korea. His research interests are dynamics, control, and navigation forasteroid explorations, spacecraft formation flying, and rendezvous and docking;geometric/algebraic methods applied to nonlinear systems; optimal control andestimation; and spacecraft trajectory optimization.

Amit K. Sanyal received a B.Tech. degree in aerospace engineering from the IndianInstitute of Technology, Kanpur, in 1999. He received an M.S. degree in aerospaceengineering from Texas A&M University in 2001. He received an M.S. degree inmathematics and a Ph.D. degree in aerospace engineering from the University ofMichigan in 2004. From 2004 to 2006, he was a postdoctoral research associate in themechanical and aerospace engineering department at Arizona State University. From2007 to 2010, he was an assistant professor in mechanical engineering at the Universityof Hawaii. He is currently an assistant professor in mechanical and aerospaceengineering at New Mexico State University. His research interests are geometricmechanics, geometric control, discrete variational mechanics for numerical integrationof mechanical systems, optimal control and estimation, geometric/algebraic methodsapplied to nonlinear systems, spacecraft guidance and control, and control of unmannedvehicles. He is a member of the IEEE Control Systems Society technical committees onAerospace Control and Nonlinear Control, and a past member of the AIAA Guidance,Navigation, and Control Technical Committee.

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Eric A. Butcher is an associate professor in the aerospace and mechanical engineeringdepartment at the University of Arizona. He obtained M.S. and Ph.D. degrees inmechanical engineering from Auburn University and another M.S. degree in aerospaceengineering from the University of Colorado at Boulder. His research interests includeastrodynamics; spacecraft guidance, navigation, and control; nonlinear dynamics andvibrations; and stability, control, and estimation in time-periodic, time-delayed,stochastic, and fractional derivative systems. He has coauthored over 150 refereedjournal and conference papers and is an associate editor for the International Journal ofDynamics and Control.

Daniel J. Scheeres is the A. Richard Seebass Endowed Chair Professor in the departmentof aerospace engineering sciences at the University of Colorado Boulder and a memberof the Colorado Center for Astrodynamics Research. Prior to this, he held facultypositions in aerospace engineering at the University of Michigan (1999–2008) and IowaState University (1997–1999), and was a member of the technical staff in the navigationsystems section at the California Institute of Technology’s Jet Propulsion Laboratory(1992–1997). He was awarded Ph.D. (1992), M.S.E. (1988), and B.S.E (1987) degrees inaerospace engineering from the University of Michigan, and holds a B.S. degree in lettersand engineering from Calvin College (1985). He has authored or coauthored over 200papers, notes, and chapters in peer-reviewed journals and over 250 conference papers.His research interests include space situational awareness; the dynamics, control, andnavigation of spacecraft trajectories; the design of space missions; optimal control;planetary science; celestial mechanics; and dynamical astronomy. He is a fellow of theAmerican Institute of Aeronautics and Astronautics and a fellow of the AmericanAstronautical Society, and serves on the AIAA Astrodynamics Technical Committee.

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