integrated 6-dof orbit-attitude dynamical modeling and...

Research ArticleIntegrated 6-DOF Orbit-Attitude Dynamical Modeling andControl Using Geometric Mechanics

Ling Jiang, YueWang, and Shijie Xu

School of Astronautics, Beihang University, Beijing 100191, China

Correspondence should be addressed to Yue Wang; [email protected]

Received 6 March 2017; Accepted 8 May 2017; Published 30 May 2017

Academic Editor: Enrico C. Lorenzini

Copyright © 2017 Ling Jiang et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The integrated 6-DOF orbit-attitude dynamical modeling and control have shown great importance in various missions, forexample, formation flying and proximity operations. The integrated approach yields better performances than the separate one interms of accuracy, efficiency, and agility. One challenge in the integrated approach is to find a unified representation for the 6-DOFmotion with configuration space SE(3). Recently, exponential coordinates of SE(3) have been used in dynamics and control of the6-DOFmotion, however, only on the kinematical level. In this paper, we will improve the current method by adopting exponentialcoordinates on the dynamical level, by giving the relation between the second-order derivative of exponential coordinates andspacecraft’s accelerations. In this way, the 6-DOF motion in terms of exponential coordinates can be written as a second-ordersystemwith a quite compact form, to which a broader range of control theories, such as higher-order slidingmodes, can be applied.For a demonstration purpose, a simple asymptotic tracking control law with almost global convergence is designed. Finally, theintegrated modeling and control are applied to the body-fixed hovering over an asteroid and verified by a simulation, in whichabsolute motions of the spacecraft and asteroid are simulated separately.

1. Introduction

The integrated 6-DOF orbit-attitude dynamical modelingand control of spacecraft have shown great importancein various space missions, such as formation flying [1–7],proximity operations [8–14], and proximity operations aboutminor celestial bodies [15–17].

Necessity of the integrated approach for spacecraft’s 6-DOFmotion lies in two aspects. Firstly, the orbit and attitudemotions are actually kinematically coupled in spacecraft rela-tive motions [1, 7] and are dynamically coupled due to effectsof external forces and torques, such as the gravitational orbit-attitude coupling in close proximity of minor celestial bodies[18–21] and the orbit-attitude coupling of high area-to-massratio (HAMR) objects caused by the solar radiation pressure(SRP) [22, 23]. Secondly, and more importantly, proximityoperations and some formation flying missions, such asproximity rendezvous, docking, landing on and samplingan asteroid, interferometric observation, and coordinatedpointing, usually require the relative position and attitude ofspacecraft to follow the desired trajectory simultaneously.

Unlike the traditional approach that treats orbit andattitude motions separately, such as Zhang et al. [24, 25], theintegrated approach models and controls the orbit and atti-tude motions in a unified framework. Therefore, integratedorbit-attitude dynamical modeling and control will capturethe system’s dynamics better and will lead the 6-DOFmotionin the phase space more accurately and effectively. That is,the integrated approach yields better performances than theseparate one in terms of accuracy, efficiency, and agility.

There are two main challenges in the integrated dynami-cal modeling and control. The first one is to find an adequateunified mathematical representation for the 6-DOF orbit-attitude motion, which must be amenable for the applicationof various nonlinear control theories.The unified representa-tion of the 6-DOF motion is also required to be applicable toa wide range of spacecraft maneuvers.

The second challenge is that the 6-DOF orbit-attitudemotion is highly nonlinear and is subjected to external distur-bances and parameter uncertainties. Consequently, adequatecontrol theories are needed in the controller design, suchas the sliding mode control [9], adaptive control [12, 13],

HindawiInternational Journal of Aerospace EngineeringVolume 2017, Article ID 4034328, 13 pageshttps://doi.org/10.1155/2017/4034328

https://doi.org/10.1155/2017/4034328

2 International Journal of Aerospace Engineering

adaptive sliding mode control [3, 17], adaptive terminalslidingmode control [2, 14], finite-time control [5, 16, 26], andstate-dependent Riccati equation (SDRE) method [4, 7].

As for the unified mathematical representation of the 6-DOF motion, the spacecraft can be considered as a rigidbody and the configuration space is the Lie group SE(3),the special Euclidean group, which is the set of positionand attitude of a rigid body moving in a three-dimensionalEuclidean space. The dual quaternion has been widely usedas a unified representation of the 6-DOF rigid body motion,but the constraints between its elements need to be dealtwith carefully [10, 11]. Sanyal et al. [8, 27] have used thematrix formof the Lie group SE(3) in the dynamicalmodelingof rigid body motion and achieved controllers with almostglobal convergence, which is the best that can be achievedfor a system evolving on a noncontractible state space, likethe 6-DOF rigid body motion, with continuous feedback[6]. However, due to the complexity of matrix calculations,the matrix form of SE(3) is not convenient for controllerdesign.

Recently, exponential coordinates of Lie group SE(3)have been used as a unified representation of the 6-DOForbit-attitude (rigid body) motion in spacecraft formationflying and asteroid hovering [6, 9, 14–17]. The exponentialcoordinates are in vector form with six elements, and thus,unlike the dual quaternion, have no constraint betweenelements. Besides, compared with the matrix form of SE(3),the set of exponential coordinates with a vector form is moreconvenient for applications of nonlinear control theories. Inthe mentioned studies, the exponential coordinates are usedto describe the 6-DOF kinematics of the spacecraft, and thecontrol scheme is designed to reduce errors of configurationand velocities from almost any given initial state, except thosethat differ in orientation by a 𝜋 radian rotation from thedesired states where exponential coordinates for the attitudeare not uniquely defined [6]. Because the set of such initialstates is an embedded lower-dimensional subspace of thestate space, the tracking control scheme designed in termsof exponential coordinates can achieve almost global con-vergence, which makes it applicable in practice. Exponentialcoordinates of SE(3) used by Lee et al. [6, 15, 16] have provideda goodunified representation for the 6-DOFmotionwith sev-eral advantages: having a vector form, having no constraintbetween elements, being convenient for controller design,and achieving almost global convergence in the controller.

In the present paper, based on the results by Bulloand Murray [28], we will improve the dynamical modelingmethod for the 6-DOF rigid bodymotion suggested by Lee etal. [6, 15, 16]. Lee et al. used exponential coordinates of SE(3)only in the kinematics of spacecraft, that is, only gave therelation between the first-order time derivative of exponentialcoordinates and the (angular) velocity of spacecraft. We willuse exponential coordinates of SE(3) on the dynamical levelfurther, that is, give the relation between the second-ordertime derivative of exponential coordinates and the (angular)acceleration of spacecraft. By using this relation on thedynamical level, the system can be written as a second-ordersystem with a compact form. Then, the set of exponentialcoordinates becomes more convenient for controller design

jk

irRw

u

v

r２＂

r３＃

O）＆

e1

e2

e3

O３＃

O２＂

Reference body

Inertial frame

Spacecra�

Figure 1:The spacecraft moving in proximity of the reference body.

and a broader range of control theories, such as higher-ordersliding modes, can be applied.

Another improvement suggested in the current work is tointroduce a desired trajectory for the 6-DOF motion, that is,the guidance on SE(3). The controller will lead the system tofollow the desired trajectory until reaching the desired finalstate. In the earlier studies by Lee et al. [6, 15, 16], Lee [9], andLee and Vukovich [14, 17], the controller aimed at the desiredfinal state directly. Therefore, the true trajectory to the finalstate could not be specified. Compared with Lee’s approach,our approach has two advantages: firstly, it is applicable todiverse spacecraft proximity operations requiring a specific6-DOF orbit-attitude trajectory, for example, approachingalong an obstacle-free trajectory; secondly, the initial trackingerror is much smaller and the control saturation in thebeginning can be avoided.

Based on the second-order system in terms of exponentialcoordinates of SE(3), a simple asymptotic tracking controllaw with almost global convergence is designed for a demon-stration purpose. Finally, both the integrated dynamicalmodeling and control law are applied to body-fixed orbit-attitude hovering over an asteroid.

2. Integrated Modeling of 6-DOFRelative Dynamics

As shown in Figure 1, the 6-DOF orbit-attitude (rigid body)motion of a spacecraft in proximity of the reference body isconsidered. The reference body can be another spacecraft inspacecraft formation flying and proximity operations or theasteroid in minor celestial body proximity operations. Theinertial frame is denoted by 𝑆IF = {e1, e2, e3} with 𝑂IF as itsorigin. The body-fixed principal-axis frames of the referencebody and spacecraft are denoted by 𝑆RB = {u, v,w} and 𝑆SC ={i, j, k} with 𝑂RB and 𝑂SC as their origins, respectively.

2.1. Dynamics of Spacecraft. The attitude of the spacecraftwith respect to the inertial frame 𝑆IF is described by matrixASC:

ASC = [𝛼SC,𝛽SC, 𝛾SC]𝑇 ∈ SO (3) , (1)

International Journal of Aerospace Engineering 3

where 𝛼SC, 𝛽SC, and 𝛾SC are coordinates of unit vectors e1,e2, and e3 of the inertial frame 𝑆IF expressed in the body-fixed frame of spacecraft 𝑆SC, respectively. Matrix ASC isalso the coordinate transformation matrix from the body-fixed frame 𝑆SC to the inertial frame 𝑆IF. SO(3) is the three-dimensional special orthogonal group.The position vector ofthe spacecraft with respect to the origin 𝑂IF expressed in theinertial frame 𝑆IF is denoted by rSC.

The configuration space of 6-DOF orbit-attitude motionof the spacecraft is the Lie group:

𝑄SC = SE (3) , (2)

known as the special Euclidean group of three-dimensionalspace with elements (ASC, rSC), which is the semidirectproduct of SO(3) and R3, that is, SO(3) ⋉ R3. The velocityphase space of 6-DOFmotion of the spacecraft is the tangentbundle TSE(3), which can be represented by SE(3) × se(3)through the left translation, where se(3) is the Lie algebra ofLie group SE(3).

The configuration of spacecraft on the special Euclideangroup SE(3) can be represented by the following 4×4matrix:

gSC = [ASC rSC0 1 ] ∈ SE (3) . (3)

The velocities of spacecraft, which are contained withinLie algebra se(3), can be represented by the 6 × 1 vector:

𝜉SC = [ΩSCVSC

] ∈ R6, (4)

where ΩSC and VSC are the spacecraft’s angular velocity andvelocity with respect to the inertial frame, respectively, bothexpressed in its body-fixed frame 𝑆SC.

Then, the 6-DOF orbit-attitude kinematics of the space-craft can be given by [15]

gSC = gSC (𝜉SC)∨ , (5)

where the map (⋅)∨ : R6 → se(3) is the Lie algebraisomorphism defined by

(𝜉SC)∨ = [(ΩSC)× VSC

0 0 ] ∈ se (3) . (6)

The map (⋅)× : R3 → so(3) is the Lie algebraisomorphism, which is also the cross-product operator for3 × 1 vectors, defined by

(w)× = [[[𝑤𝑥𝑤𝑦𝑤𝑧

]]]

×

= [[[

0 −𝑤𝑧 𝑤𝑦𝑤𝑧 0 −𝑤𝑥−𝑤𝑦 𝑤𝑥 0

]]]

∈ so (3) , (7)

where so(3) is the Lie algebra of Lie group SO(3).The 6-DOF orbit-attitude dynamics of the spacecraft can

be given by [15]

ISC��SC = ad∗𝜉SC (ISC𝜉SC) + 𝜏𝑛SC + 𝜏𝑐SC, (8)

where ISC is the combined matrix of the mass and inertia ofthe spacecraft given by

ISC = [ISC 00 𝑚SCI3×3

] ∈ R6×6, (9)

ISC and𝑚SC are the inertia tensor and mass of the spacecraft,respectively, and I3×3 is the 3 × 3 identity matrix. The adjointoperator ad𝜉SC and coadjoint operator ad

∗𝜉SC

on the Lie algebrase(3) can be given as matrixes:

ad𝜉SC = [(ΩSC)× 0

(VSC)× (ΩSC)×] ∈ R6×6,

ad∗𝜉SC = (ad𝜉SC)𝑇 = [− (ΩSC)× − (VSC)×0 − (ΩSC)×] ∈ R

6×6.(10)

The environmental torqueT𝑛SC and forceF𝑛SC acting on the

spacecraft, both expressed in the body-fixed frame 𝑆SC, arewritten together in vector 𝜏𝑛SC:

𝜏𝑛SC = [T𝑛SCF𝑛SC

] ∈ R6. (11)

Keep in mind that, in minor celestial body proximityoperations, the environmental torque and force 𝜏𝑛SC includethe gravitational torque and force by the minor body, while,in spacecraft formation flying and proximity operations, thegravitational torque and force between two spacecraft will beneglected.

The control torque T𝑐SC and force F𝑐SC acting on thespacecraft, both expressed in the body-fixed frame 𝑆SC, arewritten together in vector 𝜏𝑐SC:

𝜏𝑐SC = [T𝑐SCF𝑐SC

] ∈ R6. (12)

2.2. Dynamics of Reference Body. The 6-DOF orbit-attitudedynamics of the reference body is described similarly as thatof the spacecraft. The attitude of reference body with respectto the inertial frame 𝑆IF is described by matrix ARB:

ARB = [𝛼RB,𝛽RB, 𝛾RB]𝑇 ∈ SO (3) , (13)

where 𝛼RB, 𝛽RB, and 𝛾RB are coordinates of unit vectors e1, e2,and e3 expressed in the body-fixed frame of reference body𝑆RB, respectively. The position vector of the reference bodywith respect to the origin 𝑂IF expressed in the inertial frame𝑆IF is denoted by rRB. The configuration space of referencebody, 𝑄RB, is also Lie group SE(3) with elements (ARB, rRB).The configuration of the reference body is denoted by

gRB = [ARB rRB0 1 ] ∈ SE (3) . (14)

The velocities of the reference body are represented by

𝜉RB = [ΩRBVRB

] ∈ R6, (15)


where ΩRB and VRB are angular velocity and velocity of thereference bodywith respect to the inertial frame, respectively,both expressed in the reference body’s body-fixed frame 𝑆RB.

The 6-DOF orbit-attitude kinematics of the referencebody is given by

gRB = gRB (𝜉RB)∨ , (16)

where

(𝜉RB)∨ = [(ΩRB)× VRB

0 0 ] ∈ se (3) . (17)

The 6-DOF orbit-attitude dynamics of the reference bodycan be given by

IRB��RB = ad∗𝜉RB (IRB𝜉RB) + 𝜏RB, (18)

where IRB is the combined matrix of the mass and inertia ofthe reference body

IRB = [IRB 00 𝑚RBI3×3

] ∈ R6×6, (19)

IRB and 𝑚RB are the inertia tensor and mass of the referencebody, respectively, and

ad∗𝜉RB = [− (ΩRB)× − (VRB)×0 − (ΩRB)×] ∈ R

6×6. (20)

Resultant torque TRB and force FRB acting on the refer-ence body, both expressed in the body-fixed frame 𝑆RB, arewritten together in vector 𝜏RB:

𝜏RB = [TRB

FRB] ∈ R

6. (21)

The torque and force 𝜏RB will include control torque andforce acting on the reference body, if they exist, in spacecraftformation flying and proximity operations.

2.3. Relative Dynamics of Spacecraft. In space missions, whatreally matters is the relative dynamics of spacecraft withrespect to the reference body, rather than the absolutedynamics of spacecraft in the inertial space. The 6-DOFrelative dynamics of spacecraft, which will be written in asimilar form with the absolute dynamics, can be obtainedbased on the results in Sections 2.1 and 2.2.

The relative configuration of spacecraftwith respect to thereference body, which is also on the special Euclidean groupSE(3), is denoted by

g𝑅 = (gRB)−1 gSC = [A𝑅 r𝑅0 1 ] ∈ SE (3) , (22)

where matrix A𝑅 is the relative attitude of spacecraft withrespect to the reference body, which is also the coordinatetransformation matrix from the body-fixed frame of space-craft 𝑆SC to the body-fixed frame of reference body 𝑆RB, and

vector r𝑅 is the relative position of spacecraft with respect tothe reference body𝑂RB, expressed in the body-fixed frame ofreference body 𝑆RB.

The relative attitude matrix A𝑅 can be written as

A𝑅 = [𝛼𝑅,𝛽𝑅, 𝛾𝑅]𝑇 , (23)

where 𝛼𝑅, 𝛽𝑅, and 𝛾𝑅 are coordinates of unit vectors u, v, andw of the reference body’s body-fixed frame 𝑆RB expressed inthe spacecraft’s body-fixed frame 𝑆SC, respectively.

According to (3) and (14), the relative configuration ofspacecraft can be obtained as

g𝑅 = (gRB)−1 gSC = [A−1RB −A−1RBrRB0 1 ][ASC rSC

0 1 ]

= [A−1RBASC A−1RB (rSC − rRB)0 1 ] .

(24)

Then, the 6-DOF relative orbit-attitude kinematics of thespacecraft with respect to the reference body can be given by

g𝑅 = (gRB)−1 gSC + 𝑑𝑑𝑡 [(gRB)−1] gSC= (gRB)−1 gSC − (gRB)−1 gRB (gRB)−1 gSC= (gRB)−1 gSC (𝜉SC)∨

− (gRB)−1 gRB (𝜉RB)∨ (gRB)−1 gSC= g𝑅 (𝜉SC)∨ − (𝜉RB)∨ g𝑅= g𝑅 [(𝜉SC)∨ − (g𝑅)−1 (𝜉RB)∨ g𝑅] .

(25)

The relative kinematics of spacecraft (25) can be writtenin the same form as the absolute kinematics (5) and (16):

g𝑅 = g𝑅 (𝜉𝑅)∨ , (26)

where the 6 × 1 vector 𝜉𝑅 is the relative velocity of spacecraft:𝜉𝑅 = [Ω𝑅

V𝑅] ∈ R

6. (27)

Vectors Ω𝑅 and V𝑅 are relative angular velocity and relativevelocity of spacecraft with respect to the reference body,respectively, both expressed in the body-fixed frame ofspacecraft 𝑆SC.

According to (25) and (26), we can obtain the relativevelocity of spacecraft (𝜉𝑅)∨ as follows:

(𝜉𝑅)∨ = (𝜉SC)∨ − (g𝑅)−1 (𝜉RB)∨ g𝑅, (28)

which can be written in a compact form

𝜉𝑅 = 𝜉SC − Ad(g𝑅)−1𝜉RB. (29)

The operator Ad(g𝑅)−1 , the adjoint action of (g𝑅)−1 ∈ SE(3) on(𝜉RB)∨ ∈ se(3), satisfies(Ad(g𝑅)−1𝜉RB)∨ = (g𝑅)−1 (𝜉RB)∨ g𝑅, (30)


and the matrix form of operator Ad(g𝑅)−1 can be given by

Ad(g𝑅)−1 = [[

A−1𝑅 0

(−A−1𝑅 r𝑅)× A−1𝑅 A−1𝑅]]

∈ R6×6. (31)

As for the 6-DOF relative orbit-attitude dynamics of thespacecraft with respect to the reference body, the time deriva-tive of relative velocity 𝜉𝑅, that is, the relative acceleration ofthe spacecraft, can be obtained according to (29):

��𝑅 = ��SC − 𝑑𝑑𝑡 [Ad(g𝑅)−1𝜉RB] . (32)

According to Lee et al. [15],

𝑑𝑑𝑡 [Ad(g𝑅)−1𝜉RB] = −ad𝜉𝑅Ad(g𝑅)−1𝜉RB + Ad(g𝑅)−1 ��RB, (33)

where the adjoint operator ad𝜉𝑅 on the Lie algebra se(3) isdefined in (10).

Therefore, the relative acceleration of the spacecraft ��𝑅can be written as

��𝑅 = ��SC + ad𝜉𝑅Ad(g𝑅)−1𝜉RB − Ad(g𝑅)−1 ��RB. (34)

According to (8) and (34), the 6-DOF relative dynamicsof the spacecraft with respect to the reference body can begiven by

ISC��𝑅 = ISC��SC + ISC [ad𝜉𝑅Ad(g𝑅)−1𝜉RB − Ad(g𝑅)−1 ��RB]= ad∗𝜉SC ISC𝜉SC + 𝜏𝑛SC + 𝜏𝑐SC

+ ISC {ad𝜉𝑅Ad(g𝑅)−1𝜉RB − Ad(g𝑅)−1 ��RB} .(35)

Equations (26) and (35) form complete equations for the 6-DOF relative motion.

3. Tracking Error Modeling andControl Law Design

In formation flying and proximity operations, the 6-DOFrelative orbit-attitude motion of the spacecraft with respectto the reference body is usually required to track the desiredtrajectory. The goal of the controller is to reduce the trackingerror. Here, introducing a desired trajectory for the 6-DOForbit-attitude motion, that is, guidance on SE(3), is animprovement compared with the earlier works by Lee et al.[6, 15, 16] and Lee andVukovich [14, 17], where the controllersaimed at the desired final state directly. In our approach witha reference trajectory, the true orbit-attitude trajectory to thefinal state can be specified, and the control saturation causedby the large initial tracking error can be avoided.

3.1. Desired Relative Trajectory and Tracking Error. Thedesired trajectory is denoted by (g𝑑(𝑡), 𝜉𝑑(𝑡), ��𝑑(𝑡)), where

g𝑑 = g𝑑 (𝜉𝑑)∨ . (36)

The goal of the orbit-attitude motion controller is (g𝑅(𝑡),𝜉𝑅(𝑡)) → (g𝑑(𝑡), 𝜉𝑑(𝑡)).

The tracking error in configuration space of the relativemotion, which is also on the special Euclidean group SE(3),is defined by

g𝑒 = (g𝑑)−1 g𝑅 = [A𝑒 r𝑒0 1] ∈ SE (3) . (37)

By using the same method as in Section 2.3, the kinemat-ics of tracking error of the spacecraft’s relativemotion is givenby

g𝑒 = g𝑒 (𝜉𝑒)∨ , (38)

where the 6 × 1 vector 𝜉𝑒 is the tracking error in the relativevelocity:

𝜉𝑒 = [Ω𝑒V𝑒

] = 𝜉𝑅 − Ad(g𝑒)−1𝜉𝑑. (39)

The time derivative of velocity tracking error 𝜉𝑒 can beobtained:

��𝑒 = ��𝑅 + ad𝜉𝑒Ad(g𝑒)−1𝜉𝑑 − Ad(g𝑒)−1 ��𝑑. (40)

According to (35), the dynamics of the 6-DOF orbit-attitude tracking error can be given by

ISC��𝑒 = ad∗𝜉SC ISC𝜉SC + 𝜏𝑛SC + 𝜏𝑐SC + ISC {ad𝜉𝑒Ad(g𝑒)−1𝜉𝑑− Ad(g𝑒)−1 ��𝑑 + ad𝜉𝑅Ad(g𝑅)−1𝜉RB − Ad(g𝑅)−1 ��RB} ,

(41)

where the motion of reference body (gRB(𝑡), 𝜉RB(𝑡), ��RB(𝑡))and the desired relative motion (g𝑑(𝑡), 𝜉𝑑(𝑡), ��𝑑(𝑡)) areknown.

By using the following relations,

g𝑅 = g𝑑g𝑒,gSC = gRBg𝑅,𝜉𝑅 = 𝜉𝑒 + Ad(g𝑒)−1𝜉𝑑,𝜉SC = 𝜉𝑅 + Ad(g𝑅)−1𝜉RB,

(42)

(38) and (41) can form complete equations for the trackingerror (g𝑒, 𝜉𝑒) of the 6-DOF relative motion. The goal of thecontroller is to achieve g𝑒 → I4×4 and 𝜉𝑒 → 0, where I4×4 isthe 4 × 4 identity matrix.

3.2. Tracking Error in Exponential Coordinates of SE(3). Byusing exponential coordinates of SE(3), the tracking errorin configuration space of the orbit-attitude motion can bedenoted by

(𝜂𝑒)∨ = log (g𝑒) = log([A𝑒 r𝑒0 1]) ∈ se (3) , (43)

where the vector of exponential coordinates 𝜂𝑒 is denoted by

𝜂𝑒 = [Θ𝑒b𝑒

] ∈ R6, (44)


and the corresponding element of the Lie algebra se(3) isdenoted by

(𝜂𝑒)∨ = [(Θ𝑒)× b𝑒0 0 ] ∈ se (3) . (45)

The specific expressions of the logarithm map log :SE(3) → se(3) are given by

(Θ𝑒)× = 𝜙2 sin𝜙 [A𝑒 − A𝑇𝑒 ] ,b𝑒 = A−1 (Θ𝑒) r𝑒,

(46)

where 𝜙 satisfies cos𝜙 = (1/2)[tr(A𝑒) − 1], 𝜙 < 𝜋, andA−1 (Θ𝑒) = I3×3 − 12 (Θ𝑒)×

+ ( 1𝜙2 − 1 + cos𝜙2𝜙 sin𝜙 ) (Θ𝑒)× (Θ𝑒)× .(47)

Actually, Θ𝑒 = 𝜙n is the principal rotation vector of attitudematrix A𝑒; 𝜙 = |Θ𝑒| and unit vector n are the principalrotation angle and axis, respectively. The logarithmmap log :SE(3) → se(3) is bijectivewhen the principal rotation angle ofA𝑒 is less than a 𝜋 radian, that is, 𝜙 < 𝜋, but it is not uniquelydefined when 𝜙 is exactly a 𝜋 radian [15].

The kinematics of tracking error of the spacecraft’s rel-ative motion (38) can be rewritten in terms of exponentialcoordinates 𝜂𝑒:

��𝑒 = G (𝜂𝑒) 𝜉𝑒. (48)

The kinematical matrix G(𝜂𝑒) is given by [28]

G (𝜂𝑒) = I6×6 + 12ad𝜂𝑒 + 𝐴 (𝜙) (ad𝜂𝑒

)2+ 𝐵 (𝜙) (ad𝜂

𝑒

)4 ,(49)

where I6×6 is the 6×6 identity matrix:

ad𝜂𝑒

= [(Θ𝑒)× 0

(b𝑒)× (Θ𝑒)×] , (50)

𝐴 (𝜙) = 2𝜙2 [1 − 𝛼 (𝜙)] + 12𝜙2 [𝛼 (𝜙) − 𝛽 (𝜙)] , (51)

𝐵 (𝜙) = 1𝜙4 [1 − 𝛼 (𝜙)] + 12𝜙4 [𝛼 (𝜙) − 𝛽 (𝜙)] , (52)

𝛼 (𝜙) = 𝜙2 cot(𝜙2) , (53)

𝛽 (𝜙) = [ (𝜙/2)sin (𝜙/2)]

2 . (54)

Equations (48) and (41) can form complete equations ofmotion for the tracking error of spacecraft’s 6-DOF relativemotion in terms of exponential coordinates (𝜂𝑒, 𝜉𝑒). The goalof the controller is to achieve 𝜂𝑒 → 0 and 𝜉𝑒 → 0.

3.3. Dynamics of Tracking Error as a Second-Order System. Inequations ofmotion (48) and (41), exponential coordinates ofSE(3) are used only on the kinematical level. That is, (48) hasonly given the relation between the first-order time derivativeof exponential coordinates and the velocity. In the following,we will extend exponential coordinates to the dynamicallevel, by giving the relation between the second-order timederivative of exponential coordinates and the acceleration.This extension is an important improvement compared withearlier works by Lee et al. [6, 15, 16].

By taking time derivative of both sides of the kinematicsof tracking error (48), we can have the second-order deriva-tive of exponential coordinates:

��𝑒 = G (𝜂𝑒) ��𝑒 + G (𝜂𝑒) 𝜉𝑒, (55)

where, according to (49), G(𝜂𝑒) can be obtained as

G (𝜂𝑒) = 12ad𝜂𝑒 + �� (𝜙) (ad𝜂𝑒

)2 + 𝐴 (𝜙) (ad𝜂𝑒

ad𝜂𝑒

+ ad𝜂𝑒

ad𝜂𝑒

) + �� (𝜙) (ad𝜂𝑒

)4 + 𝐵 (𝜙) [ad𝜂𝑒

(ad𝜂𝑒

)3+ ad𝜂

𝑒

ad𝜂𝑒

(ad𝜂𝑒

)2 + (ad𝜂𝑒

)2 ad𝜂𝑒

ad𝜂𝑒

+ (ad𝜂𝑒

)3 ad𝜂𝑒

] .

(56)

According to (50), ad𝜂𝑒

in (56) is given by

ad𝜂𝑒

= [[(Θ𝑒)× 0

(b𝑒)× (Θ𝑒)×]]

, (57)

where Θ𝑒 and b𝑒 are given by (48). By using (51) and (52),��(𝜙) and ��(𝜙) in (56) can be derived as

�� (𝜙) = 𝑑𝐴𝑑𝜙 𝑑𝜙𝑑𝑡= 1𝜙3 [32𝛼 (𝜙) + 32𝛽 (𝜙) + 𝛼 (𝜙) 𝛽 (𝜙) − 4] Θ𝑒 ⋅ Θ𝑒𝜙 ,

�� (𝜙) = 𝑑𝐵𝑑𝜙 𝑑𝜙𝑑𝑡= 1𝜙5 [32𝛼 (𝜙) + 32𝛽 (𝜙) + 𝛼 (𝜙) 𝛽 (𝜙) − 4] Θ𝑒 ⋅ Θ𝑒𝜙 .

(58)

Then, (55) and (41) can form a second-order equationof tracking error of the 6-DOF relative motion in terms ofexponential coordinates (𝜂𝑒, 𝜉𝑒):

��𝑒 = G (𝜂𝑒) ��𝑒 + G (𝜂𝑒) 𝜉𝑒, (59)

where ��𝑒, G(𝜂𝑒), and G(𝜂𝑒) are given by (41), (49), and (56),respectively.

3.4. Integrated Tracking Control Law with Almost GlobalConvergence. Based on the dynamics of tracking error of


the 6-DOF relative motion, appropriate control theories canbe applied, and integrated tracking control laws can bedesigned. To show the application potential of our integrated6-DOF dynamical modeling (59) in terms of exponentialcoordinates, a simple integrated tracking control law will bedesigned for a demonstration purpose.

Since the exponential coordinates are not uniquelydefined when the rotation angle is exactly a 𝜋 radian, thecontrollers can work with almost all initial tracking errors,except those with a rotation angle of a 𝜋 radian. Because theset of such initial states is an embedded lower-dimensionalsubspace of the state space, the tracking control law designedin terms of exponential coordinates can achieve almost globalconvergence and then is applicable to space missions andoperations in practice.

By using a simple feedback scheme, the continuous track-ing control law 𝜏𝑐SC for the integrated orbit-attitudemotion ofspacecraft can be designed as

𝜏𝑐SC = −ad∗𝜉SC ISC𝜉SC − 𝜏𝑛SC − ISCG−1 (𝜂𝑒) [G (𝜂𝑒) 𝜉𝑒

+ K𝑑��𝑒 + K𝑝𝜂𝑒] − ISC {ad𝜉𝑒Ad(g𝑒)−1𝜉𝑑 − Ad(g𝑒)−1 ��𝑑

+ ad𝜉𝑅Ad(g𝑅)−1𝜉RB − Ad(g𝑅)−1 ��RB} ,(60)

where the control gains K𝑑 and K𝑝 can be tuned to achievedesired stability and dynamic performance of the closed-loopsystem.

The dynamics of closed-loop system describing trackingerror of the 6-DOF relativemotion is obtained by substitutingthe control law (60) into the second-order equation oftracking error (59):

��𝑒 + K𝑑��𝑒 + K𝑝𝜂𝑒 = 0, (61)

which is a homogeneous linear differential equation withconstant coefficients. By choosing appropriate control gainsK𝑑 and K𝑝 in the control law (60), the tracking error inconfiguration space 𝜂𝑒 and its derivative ��𝑒 will both convergeto 0 asymptotically, and, according to (48), the tracking errorin velocity 𝜉𝑒 will also converge to 0 asymptotically.

Therefore, with the feedback control law (60), the space-craft’s 6-DOF relative motion will converge to the desiredtrajectory asymptotically from almost all initial trackingerrors, except those with a rotation angle of a 𝜋 radian. Thus,the control law (60) is an integrated asymptotic trackingcontrol law with almost global convergence.

4. Body-Fixed Orbit-Attitude Hoveringover an Asteroid

In this section, we will apply integrated dynamical modelingand tracking control law for the spacecraft’s 6-DOF motionobtained above to the body-fixed orbit-attitude hovering overan asteroid.

The body-fixed orbit-attitude hovering means that boththe position and attitude of the spacecraft are kept to bestationary in the asteroid body-fixed frame [29]. The orbit-attitude hovering was modeled also in an integrated manner

by Wang and Xu [29] by using the theory of noncanonicalHamiltonian system within the framework of gravitationallycoupled orbit-attitude dynamics, in which the spacecraft wasconsidered as a rigid body.Wang andXu [29] have proposed anoncanonical Hamiltonian structure-based feedback controllaw. Here, we will use the asymptotic tracking control lawdesigned in Section 3.4 to achieve the orbit-attitude hoveringand make comparisons with the controller in Wang and Xu[29].

4.1. System Description. It is assumed that the center of massof the asteroid, that is, the reference body in earlier parts ofthe paper, is stationary in the inertial space, and the asteroidis rotating with a constant angular velocity 𝜔𝑇 around itsmaximum-moment principal axis, which is assumed to bew axis without loss of generality. That is to say, the externaltorque and force acting on the asteroid, 𝜏RB, are zeros. Thus,the 6-DOF dynamics of the reference body can be simplifiedas

gRB0 = I4×4,𝜉RB ≡ [0, 0, 𝜔𝑇, 0, 0, 0]𝑇 ,��RB = 0.

(62)

To make comparisons with the controller in Wang andXu [29], a similar mission scenario is chosen in the following.The gravity field of the asteroid is approximated by a seconddegree and order-gravity field with harmonics 𝐶20 and 𝐶22.The parameters of the asteroid are chosen to be the same asin Wang and Xu [29]:

𝜇 = 94m3/s2,𝐶20 = −0.1,𝐶22 = 0.04,𝑎𝑒 = 400m,𝜔𝑇 = 2.9089 × 10−4 s−1,

(63)

where 𝜇 = 𝐺𝑚RB,𝐺 is the gravitational constant, and 𝑎𝑒 is themean equatorial radius of the asteroid.

Since the spacecraft is in the close proximity of theasteroid and is assumed to have a low area-to-mass ratio,perturbations of solar gravity and SRP are negligible. Thus,only the gravitational torque and force by asteroid areconsidered in the environmental torque and force 𝜏𝑛SC. Thatis to say, 𝜏𝑛SC depends only on the relative configuration ofspacecraft with respect to the asteroid:

𝜏𝑛SC = 𝜏𝑛SC (g𝑅) . (64)

The explicit formulation of 𝜏𝑛SC(g𝑅) is given by [19]

T𝑛SC = 3𝜇𝑅5𝑅R𝑅 × ISCR𝑅,F𝑛SC = −𝜇𝑚SC𝑅2𝑅 R𝑅 + 3𝜇2𝑅4𝑅 {[5R

𝑇

𝑅ISCR𝑅 − tr (ISC)


+ 𝜏0𝑚SC (1 − 5 (𝛾𝑅 ⋅ R𝑅)2)− 10𝜏2𝑚SC ((𝛼𝑅 ⋅ R𝑅)2 − (𝛽𝑅 ⋅ R𝑅)2)]R𝑅− 2ISCR𝑅 + 2𝜏0𝑚SC (𝛾𝑅 ⋅ R𝑅) 𝛾𝑅+ 4𝜏2𝑚SC ((𝛼𝑅 ⋅ R𝑅)𝛼𝑅 − (𝛽𝑅 ⋅ R𝑅)𝛽𝑅)} ,

(65)

where R𝑅 = A−1𝑅 r𝑅 is the relative position of the spacecraftwith respect to the asteroid expressed in the body-fixed frameof spacecraft 𝑆SC, R𝑅 is the unit vector along R𝑅, 𝜏0 =𝑎2𝑒𝐶20, and 𝜏2 = 𝑎2𝑒𝐶22. In the environmental force F𝑛SC,the perturbation caused by the gravitational orbit-attitudecoupling of the spacecraft has been included [30].

The parameters of spacecraft are chosen as

𝑚SC = 1 × 103 kg,

ISC = [[[2 0 00 1 00 0 1.6

]]]

× 103 kg⋅m2. (66)

The hovering position-attitude gRH, that is, the relativeconfiguration of spacecraft at hovering, is chosen to be thesame as in Wang and Xu [29]:

gRH = [ARH rRH0 1 ] ∈ SE (3) , (67)

where

ARH = [𝛼RH,𝛽RH, 𝛾RH]𝑇 ,rRH = ARHRRH,𝛼RH = [0.9659, 0, −0.2588]𝑇 ,𝛽RH = [0.067, 0.9659, 0.25]𝑇 ,𝛾RH = [0.25, −0.2588, 0.933]𝑇 ,RRH = 500 [0.9798, 0, 0.2]𝑇m.

(68)

In the simulation, the desired trajectory (g𝑑(𝑡), 𝜉𝑑(𝑡),��𝑑(𝑡)) is simply chosen as the desired hovering position-attitude:

g𝑑 (𝑡) ≡ gRH,𝜉𝑑 (𝑡) ≡ 0,��𝑑 (𝑡) ≡ 0.

(69)

A better reference trajectory connecting the initial position-attitude and the hovering position-attitude, which is easierto track, can be designed in the future through studies onthe guidance on SE(3), which is not the main concern of thispaper.

The initial relative configuration of spacecraft withrespect to the asteroid g𝑅0 is chosen as

g𝑅0 = [A𝑅0 r𝑅00 1 ] ∈ SE (3) , (70)

where

A𝑅0 = ARHL,

L = [[[[[[

cos(𝜋9 ) − sin(𝜋9 ) 0sin(𝜋9 ) cos(𝜋9 ) 0

0 0 1

]]]]]]

[[[[[[

cos(− 𝜋20) 0 sin(− 𝜋20)0 1 0− sin(− 𝜋20) 0 cos(− 𝜋20)

]]]]]]

[[[[[[

1 0 00 cos( 𝜋18) − sin( 𝜋18)0 sin( 𝜋18) cos( 𝜋18)

]]]]]]

,

r𝑅0 = rRH + [200, 100, 250]𝑇m.

(71)

The initial relative velocity of spacecraft with respect tothe asteroid 𝜉𝑅0 is chosen as

𝜉𝑅0 = [Ω𝑅0V𝑅0

] = [−0.06 s−1, 0.05 s−1, 0.09 s−1,

− 1m/s, 2m/s, 1.5m/s]𝑇 .(72)

In simulations, absolute motions of the spacecraft andasteroid in the inertial space will be simulated separately.The

relativemotion of spacecraft and its tracking error that appearin the control lawwill be calculated by using absolutemotionsand the desired trajectory. With this approach, not only thecontrol law but also the dynamical modeling method can beverified.

To initiate the simulation of spacecraft’s absolute motion,the initial configuration of spacecraft in the inertial space canbe calculated by

gSC0 = gRB0g𝑅0, (73)


0 100 200 300 400 500 600 700 800 900 1000−1.5−1

−0.50

0.51

1.52

2.5

600 650 700 750 800 850 900 950 1000−3−2−1

0123456

Time (second)

Time (second)

its n

orm

(rad

)At

titud

e tra

ckin

g er

ror a

nd

×10−4

its n

orm

(rad

)At

titud

e tra

ckin

g er

ror a

nd

xe

ye

ze

Figure 2: Exponential coordinates of attitude tracking error and itsnorm.

and the initial velocity of spacecraft in the inertial space canbe calculated by

𝜉SC0 = 𝜉𝑅0 + Ad(g𝑅0)−1𝜉RB0. (74)

The control gains K𝑑 and K𝑝 in the control law (60) arechosen as

K𝑑 = 0.04I6×6,K𝑝 = 4 × 10−4I6×6. (75)

The maximum control torque and force that can beprovided by the spacecraft’s control system are set to be10N⋅m and 10N for each axis, respectively.

4.2. Numerical Simulation Results. The exponential coordi-nates of the position-attitude tracking error 𝜂𝑒 = [Θ𝑇𝑒 , b𝑇𝑒 ]𝑇and their norms are shown in Figures 2 and 3, respectively,and the tracking errors of the spacecraft’s angular velocityand velocity 𝜉𝑒 = [Ω𝑇𝑒 ,V𝑇𝑒 ]𝑇 are shown in Figures 4 and 5,respectively. It can be seen that the motion of spacecraft con-verges to the hovering position-attitude and keepsmotionlessrelative to the asteroid after about 900 seconds. The attitudemotion of spacecraft has an error of about 0.026 degrees and 5× 10−4 degree/s within 600 seconds and an error of about 1.5×10−5 degrees and 3 × 10−7 degree/s within 1000 seconds, whilethe orbital motion has an error of about 10m and 0.15m/s

0 100 200 300 400 500 600 700 800 900 1000

600 650 700 750 800 850 900 950 1000Time (second)

Time (second)

its n

orm

(m)

Posit

ion

trac

king

erro

r and

its n

orm

(m)

Posit

ion

trac

king

erro

r and

−15

−10

−5

0

5

10

15

−300−200−100

0100200300400500

be

bxe

bye

bze

Figure 3: Exponential coordinates of position tracking error and itsnorm.

0 100 200 300 400 500 600 700 800 900 1000

600 650 700 750 800 850 900 950 1000Time (second)

Time (second)

velo

city

(deg

/s)

Trac

king

erro

r of a

ngul

arve

loci

ty (d

eg/s

)Tr

acki

ng er

ror o

f ang

ular

×10−4

−4−3−2−1

0123456

−4−3−2−1

01234

xe

ye

ze

Figure 4: Tracking error of the angular velocity of spacecraft.


0 100 200 300 400 500 600 700 800 900 1000

600 650 700 750 800 850 900 950 1000Time (second)

Time (second)

Trac

king

erro

r of v

eloci

ty (m

/s)

Trac

king

erro

r of v

eloci

ty (m

/s)

−2−1.5−1

−0.50

0.51

1.52

2.5

−0.15

−0.1

−0.05

0

0.05

0.1

Vxe

Vye

Vze

Figure 5: Tracking error of the velocity of spacecraft.

within 600 seconds and an error of about 0.05m and 1 ×10−3m/swithin 1000 seconds.The attitudemotion has a fasterconverge than the orbital motion.

The trajectory of spacecraft in the body-fixed frame ofasteroid in Figure 6 shows that it does not approach thehovering position directly due to the initial velocity error. Inthe figure, the hovering position is denoted by the star (∗), theinitial position of spacecraft is denoted by the circle (I), andthe final position of spacecraft is denoted by the pentagram(f). Notice that the star (∗) is overlapped with the pentagram(f) and cannot be distinguished.

The components of control torque and force T𝑐SC and F𝑐SC,both expressed in the spacecraft body-fixed frame 𝑆SC, areshown in Figures 7 and 8, respectively. The control torqueis within the spacecraft’s control capacity during the wholetrajectory, whereas the thrust saturation of the control forceoccurs on all three axes before 600 seconds. Although thrustsaturation occurs for the control force, the controller is stilleffective.

As the spacecraft is approaching to the hovering configu-ration, the control torque and force are converging to valuesthat are needed to balance the gravitational and centrifugaltorque and force at hovering position-attitude. Since thegravity gradient torque T𝑛SC is quite small, the control torqueT𝑐SC converges to nearly zero.

Compared with the hovering controller in Wang and Xu[29], the controller proposed in this paper has not utilizedthe Hamiltonian structure of the system. On one hand, the

controller in this paper (60) is much more complicated thanthat inWang and Xu [29], which is consisted of two potentialshapings and one energy dissipation. However, on the otherhand, the closed-loop system in this paper is a linear systemand converges much faster than that in Wang and Xu [29],which is a noncanonical Hamiltonian system with energydissipation, oscillating around the hovering position-attitudefor a long duration before the final convergence.

5. Conclusions

The integrated 6-DOF orbit-attitude dynamical modelingand controller design for the relative motion of spacecraftwith respect to a reference body have been studied in theframework of geometric mechanics in the present paper. Theconfiguration and velocity of the relative motion have beenrepresented by the Lie group SE(3) and also its exponentialcoordinates. By using exponential coordinates of SE(3) on thedynamical level, the dynamics of tracking error have beenformulated as a second-order system with a quite compactform. A simple asymptotic tracking control law with almostglobal convergence was designed for a demonstration pur-pose. Both the integrated dynamical modeling and controllawwere verified in an asteroid hoveringmission scenario. Toverify the dynamical modeling method, individual motionsof the spacecraft and asteroid in the inertial space have beensimulated, rather than the equations ofmotion describing therelative dynamics.


−500

0500

−500

0

500

−600

−400

−200

0

200

400

600

Trajectory in asteroid frame

v-direction (m)

w-di

rect

ion

(m)

u-direction (m)−600 −400 −200 0 200 400 600

−600

−400

−200

0

200

400

600


v-di

rect

ion

(m)

u-direction (m)

−600 −400 −200 0 200 400 600

−600

−400

−200

0

200

400

600


u-direction (m)

w-di

rect

ion

(m)

−600 −400 −200 0 200 400 600

−600

−400

−200

0

200

400

600


v-direction (m)

w-di

rect

ion

(m)

Figure 6: Trajectory of the spacecraft in the asteroid body-fixed frame.

0 100 200 300 400 500 600 700 800 900 1000−5

0

5

10

Time (second)

Con

trol t

orqu

e (N·m

)

T c,x３＃

T c,y３＃

T c,z３＃

Figure 7: Components of control torque on the three axes of the spacecraft.


0 100 200 300 400 500 600 700 800 900 1000

−10

−5

0

5

10

Time (second)

Con

trol f

orce

(N)

Fc,x３＃

Fc,y３＃

Fc,z３＃

Figure 8: Components of control force on the three axes of thespacecraft.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work has been supported by the National NaturalScience Foundation of China under Grants 11432001 and11602009 and the Fundamental Research Funds for theCentral Universities.

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RotatingMachinery


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Journal of

Volume 201

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VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



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Advances inOptoElectronics

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Volume 2014

The Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014

SensorsJournal of


Modelling & Simulation in EngineeringHindawi Publishing Corporation http://www.hindawi.com Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks


integrated 6-dof orbit-attitude dynamical modeling and...

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