# Integrated 6-DOF Orbit-Attitude Dynamical Modeling and Control Using Geometric Mechanics

Authors: Ling Jiang; Yue Wang; Shijie Xu
Journal: International Journal of Aerospace Engineering
 (2017)

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## Abstract

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


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## Body

## 1. Introduction

The integrated 6-DOF orbit-attitude dynamical modeling and control of spacecraft have shown great importance in various space missions, such as formation flying [1–7], proximity operations [8–14], and proximity operations about minor celestial bodies [15–17].Necessity of the integrated approach for spacecraft’s 6-DOF motion lies in two aspects. Firstly, the orbit and attitude motions are actually kinematically coupled in spacecraft relative motions [1, 7] and are dynamically coupled due to effects of 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-mass ratio (HAMR) objects caused by the solar radiation pressure (SRP) [22, 23]. Secondly, and more importantly, proximity operations and some formation flying missions, such as proximity rendezvous, docking, landing on and sampling an asteroid, interferometric observation, and coordinated pointing, usually require the relative position and attitude of spacecraft to follow the desired trajectory simultaneously.Unlike the traditional approach that treats orbit and attitude motions separately, such as Zhang et al. [24, 25], the integrated approach models and controls the orbit and attitude motions in a unified framework. Therefore, integrated orbit-attitude dynamical modeling and control will capture the system’s dynamics better and will lead the 6-DOF motion in the phase space more accurately and effectively. That is, the integrated approach yields better performances than the separate one in terms of accuracy, efficiency, and agility.There are two main challenges in the integrated dynamical modeling and control. The first one is to find an adequate unified mathematical representation for the 6-DOF orbit-attitude motion, which must be amenable for the application of various nonlinear control theories. The unified representation of the 6-DOF motion is also required to be applicable to a wide range of spacecraft maneuvers.The second challenge is that the 6-DOF orbit-attitude motion is highly nonlinear and is subjected to external disturbances and parameter uncertainties. Consequently, adequate control theories are needed in the controller design, such as the sliding mode control [9], adaptive control [12, 13], adaptive sliding mode control [3, 17], adaptive terminal sliding mode control [2, 14], finite-time control [5, 16, 26], and state-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 rigid body and the configuration space is the Lie groupSE(3), the special Euclidean group, which is the set of position and attitude of a rigid body moving in a three-dimensional Euclidean space. The dual quaternion has been widely used as a unified representation of the 6-DOF rigid body motion, but the constraints between its elements need to be dealt with carefully [10, 11]. Sanyal et al. [8, 27] have used the matrix form of the Lie group SE(3) in the dynamical modeling of rigid body motion and achieved controllers with almost global convergence, which is the best that can be achieved for a system evolving on a noncontractible state space, like the 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 controller design.Recently, exponential coordinates of Lie groupSE(3) have been used as a unified representation of the 6-DOF orbit-attitude (rigid body) motion in spacecraft formation flying and asteroid hovering [6, 9, 14–17]. The exponential coordinates are in vector form with six elements, and thus, unlike the dual quaternion, have no constraint between elements. Besides, compared with the matrix form of SE(3), the set of exponential coordinates with a vector form is more convenient for applications of nonlinear control theories. In the mentioned studies, the exponential coordinates are used to describe the 6-DOF kinematics of the spacecraft, and the control scheme is designed to reduce errors of configuration and velocities from almost any given initial state, except those that differ in orientation by a π radian rotation from the desired states where exponential coordinates for the attitude are not uniquely defined [6]. Because the set of such initial states is an embedded lower-dimensional subspace of the state space, the tracking control scheme designed in terms of exponential coordinates can achieve almost global convergence, which makes it applicable in practice. Exponential coordinates of SE(3) used by Lee et al. [6, 15, 16] have provided a good unified representation for the 6-DOF motion with several advantages: having a vector form, having no constraint between elements, being convenient for controller design, and achieving almost global convergence in the controller.In the present paper, based on the results by Bullo and Murray [28], we will improve the dynamical modeling method for the 6-DOF rigid body motion suggested by Lee et al. [6, 15, 16]. Lee et al. used exponential coordinates of SE(3) only in the kinematics of spacecraft, that is, only gave the relation between the first-order time derivative of exponential coordinates and the (angular) velocity of spacecraft. We will use exponential coordinates of SE(3) on the dynamical level further, that is, give the relation between the second-order time derivative of exponential coordinates and the (angular) acceleration of spacecraft. By using this relation on the dynamical level, the system can be written as a second-order system with a compact form. Then, the set of exponential coordinates becomes more convenient for controller design and a broader range of control theories, such as higher-order sliding modes, can be applied.Another improvement suggested in the current work is to introduce a desired trajectory for the 6-DOF motion, that is, the guidance onSE(3). The controller will lead the system to follow the desired trajectory until reaching the desired final state. In the earlier studies by Lee et al. [6, 15, 16], Lee [9], and Lee and Vukovich [14, 17], the controller aimed at the desired final state directly. Therefore, the true trajectory to the final state could not be specified. Compared with Lee’s approach, our approach has two advantages: firstly, it is applicable to diverse spacecraft proximity operations requiring a specific 6-DOF orbit-attitude trajectory, for example, approaching along an obstacle-free trajectory; secondly, the initial tracking error is much smaller and the control saturation in the beginning can be avoided.Based on the second-order system in terms of exponential coordinates ofSE(3), a simple asymptotic tracking control law with almost global convergence is designed for a demonstration purpose. Finally, both the integrated dynamical modeling and control law are applied to body-fixed orbit-attitude hovering over an asteroid.


## 2. Integrated Modeling of 6-DOF Relative Dynamics

As shown in Figure1, the 6-DOF orbit-attitude (rigid body) motion of a spacecraft in proximity of the reference body is considered. The reference body can be another spacecraft in spacecraft formation flying and proximity operations or the asteroid in minor celestial body proximity operations. The inertial frame is denoted by SIF={e1,e2,e3} with OIF as its origin. The body-fixed principal-axis frames of the reference body and spacecraft are denoted by SRB={u,v,w} and SSC={i,j,k} with ORB and OSC as their origins, respectively.Figure 1
The spacecraft moving in proximity of the reference body.


### 2.1. Dynamics of Spacecraft

The attitude of the spacecraft with respect to the inertial frameSIF is described by matrix ASC:(1)ASC=αSC,βSC,γSCT∈SO3,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 is also 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 of the spacecraft with respect to the origin OIF expressed in the inertial frame SIF is denoted by rSC.The configuration space of 6-DOF orbit-attitude motion of the spacecraft is the Lie group:(2)QSC=SE3,known as the special Euclidean group of three-dimensional space with elements (ASC,rSC), which is the semidirect product of SO(3) and R3, that is, SO(3)⋉R3. The velocity phase space of 6-DOF motion of the spacecraft is the tangent bundle TSE(3), which can be represented by SE(3)×se(3) through the left translation, where se(3) is the Lie algebra of Lie group SE(3).The configuration of spacecraft on the special Euclidean groupSE(3) can be represented by the following 4×4 matrix:(3)gSC=ASCrSC01∈SE3.The velocities of spacecraft, which are contained within Lie algebrase(3), can be represented by the 6×1 vector:(4)ξSC=ΩSCVSC∈R6,where ΩSC and VSC are the spacecraft’s angular velocity and velocity with respect to the inertial frame, respectively, both expressed in its body-fixed frame SSC.Then, the 6-DOF orbit-attitude kinematics of the spacecraft can be given by [15](5)g˙SC=gSCξSC∨,where the map ·∨:R6→se(3) is the Lie algebra isomorphism defined by(6)ξSC∨=ΩSC×VSC00∈se3.The map·×:R3→so(3) is the Lie algebra isomorphism, which is also the cross-product operator for 3×1 vectors, defined by(7)w×=wxwywz×=0-wzwywz0-wx-wywx0∈so3,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] (8)ISCξ˙SC=adξSC∗ISCξSC+τSCn+τSCc,where ISC is the combined matrix of the mass and inertia of the spacecraft given by(9)ISC=ISC00mSCI3×3∈R6×6,ISC and mSC are the inertia tensor and mass of the spacecraft, respectively, and I3×3 is the 3×3 identity matrix. The adjoint operator adξSC and coadjoint operator adξSC∗ on the Lie algebra se(3) can be given as matrixes:(10)adξSC=ΩSC×0VSC×ΩSC×∈R6×6,adξSC∗=adξSCT=-ΩSC×-VSC×0-ΩSC×∈R6×6.The environmental torqueTSCn and force FSCn acting on the spacecraft, both expressed in the body-fixed frame SSC, are written together in vector τSCn:(11)τSCn=TSCnFSCn∈R6.Keep in mind that, in minor celestial body proximity operations, the environmental torque and forceτSCn include the gravitational torque and force by the minor body, while, in spacecraft formation flying and proximity operations, the gravitational torque and force between two spacecraft will be neglected.The control torqueTSCc and force FSCc acting on the spacecraft, both expressed in the body-fixed frame SSC, are written together in vector τSCc:(12)τSCc=TSCcFSCc∈R6.


### 2.2. Dynamics of Reference Body

The 6-DOF orbit-attitude dynamics of the reference body is described similarly as that of the spacecraft. The attitude of reference body with respect to the inertial frameSIF is described by matrix ARB: (13)ARB=αRB,βRB,γRBT∈SO3,where αRB, βRB, and γRB are coordinates of unit vectors e1, e2, and e3 expressed in the body-fixed frame of reference body SRB, respectively. The position vector of the reference body with respect to the origin OIF expressed in the inertial frame SIF is denoted by rRB. The configuration space of reference body, QRB, is also Lie group SE(3) with elements (ARB,rRB). The configuration of the reference body is denoted by(14)gRB=ARBrRB01∈SE3.The velocities of the reference body are represented by(15)ξRB=ΩRBVRB∈R6,where ΩRB and VRB are angular velocity and velocity of the reference body with 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 reference body is given by(16)g˙RB=gRBξRB∨,where(17)ξRB∨=ΩRB×VRB00∈se3.The 6-DOF orbit-attitude dynamics of the reference body can be given by(18)IRBξ˙RB=adξRB∗IRBξRB+τRB,where IRB is the combined matrix of the mass and inertia of the reference body(19)IRB=IRB00mRBI3×3∈R6×6,IRB and mRB are the inertia tensor and mass of the reference body, respectively, and(20)adξRB∗=-ΩRB×-VRB×0-ΩRB×∈R6×6.Resultant torqueTRB and force FRB acting on the reference body, both expressed in the body-fixed frame SRB, are written together in vector τRB:(21)τRB=TRBFRB∈R6.The torque and forceτRB will include control torque and force acting on the reference body, if they exist, in spacecraft formation flying and proximity operations.


### 2.3. Relative Dynamics of Spacecraft

In space missions, what really matters is the relative dynamics of spacecraft with respect to the reference body, rather than the absolute dynamics of spacecraft in the inertial space. The 6-DOF relative dynamics of spacecraft, which will be written in a similar form with the absolute dynamics, can be obtained based on the results in Sections2.1 and 2.2.The relative configuration of spacecraft with respect to the reference body, which is also on the special Euclidean groupSE(3), is denoted by(22)gR=gRB-1gSC=ARrR01∈SE3,where matrix AR is the relative attitude of spacecraft with respect to the reference body, which is also the coordinate transformation matrix from the body-fixed frame of spacecraft SSC to the body-fixed frame of reference body SRB, and vector rR is the relative position of spacecraft with respect to the reference body ORB, expressed in the body-fixed frame of reference body SRB.The relative attitude matrixAR can be written as(23)AR=αR,βR,γRT,where αR, βR, and γR are coordinates of unit vectors u, v, and w of the reference body’s body-fixed frame SRB expressed in the spacecraft’s body-fixed frame SSC, respectively.According to (3) and (14), the relative configuration of spacecraft can be obtained as(24)gR=gRB-1gSC=ARB-1-ARB-1rRB01ASCrSC01=ARB-1ASCARB-1rSC-rRB01.Then, the 6-DOF relative orbit-attitude kinematics of the spacecraft with respect to the reference body can be given by(25)g˙R=gRB-1g˙SC+ddtgRB-1gSC=gRB-1g˙SC-gRB-1g˙RBgRB-1gSC=gRB-1gSCξSC∨-gRB-1gRBξRB∨gRB-1gSC=gRξSC∨-ξRB∨gR=gRξSC∨-gR-1ξRB∨gR.The relative kinematics of spacecraft (25) can be written in the same form as the absolute kinematics (5) and (16):(26)g˙R=gRξR∨,where the 6×1 vector ξR is the relative velocity of spacecraft:(27)ξR=ΩRVR∈R6.Vectors ΩR and VR are relative angular velocity and relative velocity of spacecraft with respect to the reference body, respectively, both expressed in the body-fixed frame of spacecraft SSC.According to (25) and (26), we can obtain the relative velocity of spacecraft ξR∨ as follows:(28)ξR∨=ξSC∨-gR-1ξRB∨gR,which can be written in a compact form(29)ξR=ξSC-AdgR-1ξRB.The operator AdgR-1, the adjoint action of gR-1∈SE(3) on ξRB∨∈se(3), satisfies(30)AdgR-1ξRB∨=gR-1ξRB∨gR,and the matrix form of operator AdgR-1 can be given by (31)AdgR-1=AR-10-AR-1rR×AR-1AR-1∈R6×6.As for the 6-DOF relative orbit-attitude dynamics of the spacecraft with respect to the reference body, the time derivative of relative velocityξR, that is, the relative acceleration of the spacecraft, can be obtained according to (29):(32)ξ˙R=ξ˙SC-ddtAdgR-1ξRB.According to Lee et al. [15],(33)ddtAdgR-1ξRB=-adξRAdgR-1ξRB+AdgR-1ξ˙RB,where the adjoint operator adξR on the Lie algebra se(3) is defined in (10).Therefore, the relative acceleration of the spacecraftξ˙R can be written as(34)ξ˙R=ξ˙SC+adξRAdgR-1ξRB-AdgR-1ξ˙RB.According to (8) and (34), the 6-DOF relative dynamics of the spacecraft with respect to the reference body can be given by(35)ISCξ˙R=ISCξ˙SC+ISCadξRAdgR-1ξRB-AdgR-1ξ˙RB=adξSC∗ISCξSC+τSCn+τSCc+ISCadξRAdgR-1ξRB-AdgR-1ξ˙RB.Equations (26) and (35) form complete equations for the 6-DOF relative motion.


## 2.1. Dynamics of Spacecraft

The attitude of the spacecraft with respect to the inertial frameSIF is described by matrix ASC:(1)ASC=αSC,βSC,γSCT∈SO3,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 is also 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 of the spacecraft with respect to the origin OIF expressed in the inertial frame SIF is denoted by rSC.The configuration space of 6-DOF orbit-attitude motion of the spacecraft is the Lie group:(2)QSC=SE3,known as the special Euclidean group of three-dimensional space with elements (ASC,rSC), which is the semidirect product of SO(3) and R3, that is, SO(3)⋉R3. The velocity phase space of 6-DOF motion of the spacecraft is the tangent bundle TSE(3), which can be represented by SE(3)×se(3) through the left translation, where se(3) is the Lie algebra of Lie group SE(3).The configuration of spacecraft on the special Euclidean groupSE(3) can be represented by the following 4×4 matrix:(3)gSC=ASCrSC01∈SE3.The velocities of spacecraft, which are contained within Lie algebrase(3), can be represented by the 6×1 vector:(4)ξSC=ΩSCVSC∈R6,where ΩSC and VSC are the spacecraft’s angular velocity and velocity with respect to the inertial frame, respectively, both expressed in its body-fixed frame SSC.Then, the 6-DOF orbit-attitude kinematics of the spacecraft can be given by [15](5)g˙SC=gSCξSC∨,where the map ·∨:R6→se(3) is the Lie algebra isomorphism defined by(6)ξSC∨=ΩSC×VSC00∈se3.The map·×:R3→so(3) is the Lie algebra isomorphism, which is also the cross-product operator for 3×1 vectors, defined by(7)w×=wxwywz×=0-wzwywz0-wx-wywx0∈so3,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] (8)ISCξ˙SC=adξSC∗ISCξSC+τSCn+τSCc,where ISC is the combined matrix of the mass and inertia of the spacecraft given by(9)ISC=ISC00mSCI3×3∈R6×6,ISC and mSC are the inertia tensor and mass of the spacecraft, respectively, and I3×3 is the 3×3 identity matrix. The adjoint operator adξSC and coadjoint operator adξSC∗ on the Lie algebra se(3) can be given as matrixes:(10)adξSC=ΩSC×0VSC×ΩSC×∈R6×6,adξSC∗=adξSCT=-ΩSC×-VSC×0-ΩSC×∈R6×6.The environmental torqueTSCn and force FSCn acting on the spacecraft, both expressed in the body-fixed frame SSC, are written together in vector τSCn:(11)τSCn=TSCnFSCn∈R6.Keep in mind that, in minor celestial body proximity operations, the environmental torque and forceτSCn include the gravitational torque and force by the minor body, while, in spacecraft formation flying and proximity operations, the gravitational torque and force between two spacecraft will be neglected.The control torqueTSCc and force FSCc acting on the spacecraft, both expressed in the body-fixed frame SSC, are written together in vector τSCc:(12)τSCc=TSCcFSCc∈R6.


## 2.2. Dynamics of Reference Body

The 6-DOF orbit-attitude dynamics of the reference body is described similarly as that of the spacecraft. The attitude of reference body with respect to the inertial frameSIF is described by matrix ARB: (13)ARB=αRB,βRB,γRBT∈SO3,where αRB, βRB, and γRB are coordinates of unit vectors e1, e2, and e3 expressed in the body-fixed frame of reference body SRB, respectively. The position vector of the reference body with respect to the origin OIF expressed in the inertial frame SIF is denoted by rRB. The configuration space of reference body, QRB, is also Lie group SE(3) with elements (ARB,rRB). The configuration of the reference body is denoted by(14)gRB=ARBrRB01∈SE3.The velocities of the reference body are represented by(15)ξRB=ΩRBVRB∈R6,where ΩRB and VRB are angular velocity and velocity of the reference body with 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 reference body is given by(16)g˙RB=gRBξRB∨,where(17)ξRB∨=ΩRB×VRB00∈se3.The 6-DOF orbit-attitude dynamics of the reference body can be given by(18)IRBξ˙RB=adξRB∗IRBξRB+τRB,where IRB is the combined matrix of the mass and inertia of the reference body(19)IRB=IRB00mRBI3×3∈R6×6,IRB and mRB are the inertia tensor and mass of the reference body, respectively, and(20)adξRB∗=-ΩRB×-VRB×0-ΩRB×∈R6×6.Resultant torqueTRB and force FRB acting on the reference body, both expressed in the body-fixed frame SRB, are written together in vector τRB:(21)τRB=TRBFRB∈R6.The torque and forceτRB will include control torque and force acting on the reference body, if they exist, in spacecraft formation flying and proximity operations.


## 2.3. Relative Dynamics of Spacecraft

In space missions, what really matters is the relative dynamics of spacecraft with respect to the reference body, rather than the absolute dynamics of spacecraft in the inertial space. The 6-DOF relative dynamics of spacecraft, which will be written in a similar form with the absolute dynamics, can be obtained based on the results in Sections2.1 and 2.2.The relative configuration of spacecraft with respect to the reference body, which is also on the special Euclidean groupSE(3), is denoted by(22)gR=gRB-1gSC=ARrR01∈SE3,where matrix AR is the relative attitude of spacecraft with respect to the reference body, which is also the coordinate transformation matrix from the body-fixed frame of spacecraft SSC to the body-fixed frame of reference body SRB, and vector rR is the relative position of spacecraft with respect to the reference body ORB, expressed in the body-fixed frame of reference body SRB.The relative attitude matrixAR can be written as(23)AR=αR,βR,γRT,where αR, βR, and γR are coordinates of unit vectors u, v, and w of the reference body’s body-fixed frame SRB expressed in the spacecraft’s body-fixed frame SSC, respectively.According to (3) and (14), the relative configuration of spacecraft can be obtained as(24)gR=gRB-1gSC=ARB-1-ARB-1rRB01ASCrSC01=ARB-1ASCARB-1rSC-rRB01.Then, the 6-DOF relative orbit-attitude kinematics of the spacecraft with respect to the reference body can be given by(25)g˙R=gRB-1g˙SC+ddtgRB-1gSC=gRB-1g˙SC-gRB-1g˙RBgRB-1gSC=gRB-1gSCξSC∨-gRB-1gRBξRB∨gRB-1gSC=gRξSC∨-ξRB∨gR=gRξSC∨-gR-1ξRB∨gR.The relative kinematics of spacecraft (25) can be written in the same form as the absolute kinematics (5) and (16):(26)g˙R=gRξR∨,where the 6×1 vector ξR is the relative velocity of spacecraft:(27)ξR=ΩRVR∈R6.Vectors ΩR and VR are relative angular velocity and relative velocity of spacecraft with respect to the reference body, respectively, both expressed in the body-fixed frame of spacecraft SSC.According to (25) and (26), we can obtain the relative velocity of spacecraft ξR∨ as follows:(28)ξR∨=ξSC∨-gR-1ξRB∨gR,which can be written in a compact form(29)ξR=ξSC-AdgR-1ξRB.The operator AdgR-1, the adjoint action of gR-1∈SE(3) on ξRB∨∈se(3), satisfies(30)AdgR-1ξRB∨=gR-1ξRB∨gR,and the matrix form of operator AdgR-1 can be given by (31)AdgR-1=AR-10-AR-1rR×AR-1AR-1∈R6×6.As for the 6-DOF relative orbit-attitude dynamics of the spacecraft with respect to the reference body, the time derivative of relative velocityξR, that is, the relative acceleration of the spacecraft, can be obtained according to (29):(32)ξ˙R=ξ˙SC-ddtAdgR-1ξRB.According to Lee et al. [15],(33)ddtAdgR-1ξRB=-adξRAdgR-1ξRB+AdgR-1ξ˙RB,where the adjoint operator adξR on the Lie algebra se(3) is defined in (10).Therefore, the relative acceleration of the spacecraftξ˙R can be written as(34)ξ˙R=ξ˙SC+adξRAdgR-1ξRB-AdgR-1ξ˙RB.According to (8) and (34), the 6-DOF relative dynamics of the spacecraft with respect to the reference body can be given by(35)ISCξ˙R=ISCξ˙SC+ISCadξRAdgR-1ξRB-AdgR-1ξ˙RB=adξSC∗ISCξSC+τSCn+τSCc+ISCadξRAdgR-1ξRB-AdgR-1ξ˙RB.Equations (26) and (35) form complete equations for the 6-DOF relative motion.


## 3. Tracking Error Modeling and Control Law Design

In formation flying and proximity operations, the 6-DOF relative orbit-attitude motion of the spacecraft with respect to the reference body is usually required to track the desired trajectory. The goal of the controller is to reduce the tracking error. Here, introducing a desired trajectory for the 6-DOF orbit-attitude motion, that is, guidance onSE(3), is an improvement compared with the earlier works by Lee et al. [6, 15, 16] and Lee and Vukovich [14, 17], where the controllers aimed at the desired final state directly. In our approach with a reference trajectory, the true orbit-attitude trajectory to the final state can be specified, and the control saturation caused by the large initial tracking error can be avoided.


### 3.1. Desired Relative Trajectory and Tracking Error

The desired trajectory is denoted bygdt,ξdt,ξ˙dt, where(36)g˙d=gdξd∨.The goal of the orbit-attitude motion controller is gRt,ξRt→gdt,ξdt.The tracking error in configuration space of the relative motion, which is also on the special Euclidean groupSE(3), is defined by(37)ge=gd-1gR=Aere01∈SE3.By using the same method as in Section2.3, the kinematics of tracking error of the spacecraft’s relative motion is given by(38)g˙e=geξe∨,where the 6×1 vector ξe is the tracking error in the relative velocity:(39)ξe=ΩeVe=ξR-Adge-1ξd.The time derivative of velocity tracking errorξe can be obtained:(40)ξ˙e=ξ˙R+adξeAdge-1ξd-Adge-1ξ˙d.According to (35), the dynamics of the 6-DOF orbit-attitude tracking error can be given by(41)ISCξ˙e=adξSC∗ISCξSC+τSCn+τSCc+ISCadξeAdge-1ξd-Adge-1ξ˙d+adξRAdgR-1ξRB-AdgR-1ξ˙RB,where the motion of reference body gRBt,ξRBt,ξ˙RBt and the desired relative motion gdt,ξdt,ξ˙dt are known.By using the following relations,(42)gR=gdge,gSC=gRBgR,ξR=ξe+Adge-1ξd,ξSC=ξR+AdgR-1ξRB,(38) and (41) can form complete equations for the tracking error ge,ξe of the 6-DOF relative motion. The goal of the controller is to achieve ge→I4×4 and ξe→0, where I4×4 is the 4×4 identity matrix.


### 3.2. Tracking Error in Exponential Coordinates of SE(3)

By using exponential coordinates ofSE(3), the tracking error in configuration space of the orbit-attitude motion can be denoted by(43)ηe∨=logge=logAere01∈se3,where the vector of exponential coordinates ηe is denoted by(44)ηe=Θebe∈R6,and the corresponding element of the Lie algebra se(3) is denoted by(45)ηe∨=Θe×be00∈se3.The specific expressions of the logarithm maplog:SE(3)→se(3) are given by(46)Θe×=ϕ2sin⁡ϕAe-AeT,be=A-1Θere,where ϕ satisfies cos⁡ϕ=1/2trAe-1, ϕ<π, and(47)A-1Θe=I3×3-12Θe×+1ϕ2-1+cos⁡ϕ2ϕsin⁡ϕΘe×Θe×.Actually, Θe=ϕn is the principal rotation vector of attitude matrix Ae; ϕ=Θe and unit vector n are the principal rotation angle and axis, respectively. The logarithm map log:SE(3)→se(3) is bijective when the principal rotation angle of Ae is less than a π radian, that is, ϕ<π, but it is not uniquely defined when ϕ is exactly a π radian [15].The kinematics of tracking error of the spacecraft’s relative motion (38) can be rewritten in terms of exponential coordinates ηe:(48)η˙e=Gηeξe.The kinematical matrixGηe is given by [28](49)Gηe=I6×6+12adηe+Aϕadηe2+Bϕadηe4,where I6×6 is the 6×6 identity matrix:(50)adηe=Θe×0be×Θe×,(51)Aϕ=2ϕ21-αϕ+12ϕ2αϕ-βϕ,(52)Bϕ=1ϕ41-αϕ+12ϕ4αϕ-βϕ,(53)αϕ=ϕ2cotϕ2,(54)βϕ=ϕ/2sin⁡ϕ/22.Equations (48) and (41) can form complete equations of motion for the tracking error of spacecraft’s 6-DOF relative motion in terms of exponential coordinates ηe,ξe. The goal of the controller is to achieve ηe→0 and ξe→0.


### 3.3. Dynamics of Tracking Error as a Second-Order System

In equations of motion (48) and (41), exponential coordinates of SE(3) are used only on the kinematical level. That is, (48) has only given the relation between the first-order time derivative of exponential coordinates and the velocity. In the following, we will extend exponential coordinates to the dynamical level, by giving the relation between the second-order time derivative of exponential coordinates and the acceleration. This extension is an important improvement compared with earlier works by Lee et al. [6, 15, 16].By taking time derivative of both sides of the kinematics of tracking error (48), we can have the second-order derivative of exponential coordinates:(55)η¨e=Gηeξ˙e+G˙ηeξe,where, according to (49), G˙ηe can be obtained as(56)G˙ηe=12ad˙ηe+A˙ϕadηe2+Aϕad˙ηeadηe+adηead˙ηe+B˙ϕadηe4+Bϕad˙ηeadηe3+adηead˙ηeadηe2+adηe2ad˙ηeadηe+adηe3ad˙ηe.According to (50), ad˙ηe in (56) is given by(57)ad˙ηe=Θ˙e×0b˙e×Θ˙e×,where Θ˙e and b˙e are given by (48). By using (51) and (52), A˙ϕ and B˙ϕ in (56) can be derived as (58)A˙ϕ=dAdϕdϕdt=1ϕ332αϕ+32βϕ+αϕβϕ-4Θe·Θ˙eϕ,B˙ϕ=dBdϕdϕdt=1ϕ532αϕ+32βϕ+αϕβϕ-4Θe·Θ˙eϕ.Then, (55) and (41) can form a second-order equation of tracking error of the 6-DOF relative motion in terms of exponential coordinates ηe,ξe:(59)η¨e=Gηeξ˙e+G˙ηeξe,where ξ˙e, Gηe, and G˙ηe are given by (41), (49), and (56), respectively.


### 3.4. Integrated Tracking Control Law with Almost Global Convergence

Based on the dynamics of tracking error of the 6-DOF relative motion, appropriate control theories can be applied, and integrated tracking control laws can be designed. To show the application potential of our integrated 6-DOF dynamical modeling (59) in terms of exponential coordinates, a simple integrated tracking control law will be designed for a demonstration purpose.Since the exponential coordinates are not uniquely defined when the rotation angle is exactly aπ radian, the controllers can work with almost all initial tracking errors, except those with a rotation angle of a π radian. Because the set of such initial states is an embedded lower-dimensional subspace of the state space, the tracking control law designed in terms of exponential coordinates can achieve almost global convergence and then is applicable to space missions and operations in practice.By using a simple feedback scheme, the continuous tracking control lawτSCc for the integrated orbit-attitude motion of spacecraft can be designed as(60)τSCc=-adξSC∗ISCξSC-τSCn-ISCG-1ηeG˙ηeξe+Kdη˙e+Kpηe-ISCadξeAdge-1ξd-Adge-1ξ˙d+adξRAdgR-1ξRB-AdgR-1ξ˙RB,where the control gains Kd and Kp can be tuned to achieve desired stability and dynamic performance of the closed-loop system.The dynamics of closed-loop system describing tracking error of the 6-DOF relative motion is obtained by substituting the control law (60) into the second-order equation of tracking error (59):(61)η¨e+Kdη˙e+Kpηe=0,which is a homogeneous linear differential equation with constant coefficients. By choosing appropriate control gains Kd and Kp in the control law (60), the tracking error in configuration space ηe and its derivative η˙e will both converge to 0 asymptotically, and, according to (48), the tracking error in velocity ξe will also converge to 0 asymptotically.Therefore, with the feedback control law (60), the spacecraft’s 6-DOF relative motion will converge to the desired trajectory asymptotically from almost all initial tracking errors, except those with a rotation angle of a π radian. Thus, the control law (60) is an integrated asymptotic tracking control law with almost global convergence.


## 3.1. Desired Relative Trajectory and Tracking Error

The desired trajectory is denoted bygdt,ξdt,ξ˙dt, where(36)g˙d=gdξd∨.The goal of the orbit-attitude motion controller is gRt,ξRt→gdt,ξdt.The tracking error in configuration space of the relative motion, which is also on the special Euclidean groupSE(3), is defined by(37)ge=gd-1gR=Aere01∈SE3.By using the same method as in Section2.3, the kinematics of tracking error of the spacecraft’s relative motion is given by(38)g˙e=geξe∨,where the 6×1 vector ξe is the tracking error in the relative velocity:(39)ξe=ΩeVe=ξR-Adge-1ξd.The time derivative of velocity tracking errorξe can be obtained:(40)ξ˙e=ξ˙R+adξeAdge-1ξd-Adge-1ξ˙d.According to (35), the dynamics of the 6-DOF orbit-attitude tracking error can be given by(41)ISCξ˙e=adξSC∗ISCξSC+τSCn+τSCc+ISCadξeAdge-1ξd-Adge-1ξ˙d+adξRAdgR-1ξRB-AdgR-1ξ˙RB,where the motion of reference body gRBt,ξRBt,ξ˙RBt and the desired relative motion gdt,ξdt,ξ˙dt are known.By using the following relations,(42)gR=gdge,gSC=gRBgR,ξR=ξe+Adge-1ξd,ξSC=ξR+AdgR-1ξRB,(38) and (41) can form complete equations for the tracking error ge,ξe of the 6-DOF relative motion. The goal of the controller is to achieve ge→I4×4 and ξe→0, where I4×4 is the 4×4 identity matrix.


## 3.2. Tracking Error in Exponential Coordinates of SE(3)

By using exponential coordinates ofSE(3), the tracking error in configuration space of the orbit-attitude motion can be denoted by(43)ηe∨=logge=logAere01∈se3,where the vector of exponential coordinates ηe is denoted by(44)ηe=Θebe∈R6,and the corresponding element of the Lie algebra se(3) is denoted by(45)ηe∨=Θe×be00∈se3.The specific expressions of the logarithm maplog:SE(3)→se(3) are given by(46)Θe×=ϕ2sin⁡ϕAe-AeT,be=A-1Θere,where ϕ satisfies cos⁡ϕ=1/2trAe-1, ϕ<π, and(47)A-1Θe=I3×3-12Θe×+1ϕ2-1+cos⁡ϕ2ϕsin⁡ϕΘe×Θe×.Actually, Θe=ϕn is the principal rotation vector of attitude matrix Ae; ϕ=Θe and unit vector n are the principal rotation angle and axis, respectively. The logarithm map log:SE(3)→se(3) is bijective when the principal rotation angle of Ae is less than a π radian, that is, ϕ<π, but it is not uniquely defined when ϕ is exactly a π radian [15].The kinematics of tracking error of the spacecraft’s relative motion (38) can be rewritten in terms of exponential coordinates ηe:(48)η˙e=Gηeξe.The kinematical matrixGηe is given by [28](49)Gηe=I6×6+12adηe+Aϕadηe2+Bϕadηe4,where I6×6 is the 6×6 identity matrix:(50)adηe=Θe×0be×Θe×,(51)Aϕ=2ϕ21-αϕ+12ϕ2αϕ-βϕ,(52)Bϕ=1ϕ41-αϕ+12ϕ4αϕ-βϕ,(53)αϕ=ϕ2cotϕ2,(54)βϕ=ϕ/2sin⁡ϕ/22.Equations (48) and (41) can form complete equations of motion for the tracking error of spacecraft’s 6-DOF relative motion in terms of exponential coordinates ηe,ξe. The goal of the controller is to achieve ηe→0 and ξe→0.


## 3.3. Dynamics of Tracking Error as a Second-Order System

In equations of motion (48) and (41), exponential coordinates of SE(3) are used only on the kinematical level. That is, (48) has only given the relation between the first-order time derivative of exponential coordinates and the velocity. In the following, we will extend exponential coordinates to the dynamical level, by giving the relation between the second-order time derivative of exponential coordinates and the acceleration. This extension is an important improvement compared with earlier works by Lee et al. [6, 15, 16].By taking time derivative of both sides of the kinematics of tracking error (48), we can have the second-order derivative of exponential coordinates:(55)η¨e=Gηeξ˙e+G˙ηeξe,where, according to (49), G˙ηe can be obtained as(56)G˙ηe=12ad˙ηe+A˙ϕadηe2+Aϕad˙ηeadηe+adηead˙ηe+B˙ϕadηe4+Bϕad˙ηeadηe3+adηead˙ηeadηe2+adηe2ad˙ηeadηe+adηe3ad˙ηe.According to (50), ad˙ηe in (56) is given by(57)ad˙ηe=Θ˙e×0b˙e×Θ˙e×,where Θ˙e and b˙e are given by (48). By using (51) and (52), A˙ϕ and B˙ϕ in (56) can be derived as (58)A˙ϕ=dAdϕdϕdt=1ϕ332αϕ+32βϕ+αϕβϕ-4Θe·Θ˙eϕ,B˙ϕ=dBdϕdϕdt=1ϕ532αϕ+32βϕ+αϕβϕ-4Θe·Θ˙eϕ.Then, (55) and (41) can form a second-order equation of tracking error of the 6-DOF relative motion in terms of exponential coordinates ηe,ξe:(59)η¨e=Gηeξ˙e+G˙ηeξe,where ξ˙e, Gηe, and G˙ηe are given by (41), (49), and (56), respectively.


## 3.4. Integrated Tracking Control Law with Almost Global Convergence

Based on the dynamics of tracking error of the 6-DOF relative motion, appropriate control theories can be applied, and integrated tracking control laws can be designed. To show the application potential of our integrated 6-DOF dynamical modeling (59) in terms of exponential coordinates, a simple integrated tracking control law will be designed for a demonstration purpose.Since the exponential coordinates are not uniquely defined when the rotation angle is exactly aπ radian, the controllers can work with almost all initial tracking errors, except those with a rotation angle of a π radian. Because the set of such initial states is an embedded lower-dimensional subspace of the state space, the tracking control law designed in terms of exponential coordinates can achieve almost global convergence and then is applicable to space missions and operations in practice.By using a simple feedback scheme, the continuous tracking control lawτSCc for the integrated orbit-attitude motion of spacecraft can be designed as(60)τSCc=-adξSC∗ISCξSC-τSCn-ISCG-1ηeG˙ηeξe+Kdη˙e+Kpηe-ISCadξeAdge-1ξd-Adge-1ξ˙d+adξRAdgR-1ξRB-AdgR-1ξ˙RB,where the control gains Kd and Kp can be tuned to achieve desired stability and dynamic performance of the closed-loop system.The dynamics of closed-loop system describing tracking error of the 6-DOF relative motion is obtained by substituting the control law (60) into the second-order equation of tracking error (59):(61)η¨e+Kdη˙e+Kpηe=0,which is a homogeneous linear differential equation with constant coefficients. By choosing appropriate control gains Kd and Kp in the control law (60), the tracking error in configuration space ηe and its derivative η˙e will both converge to 0 asymptotically, and, according to (48), the tracking error in velocity ξe will also converge to 0 asymptotically.Therefore, with the feedback control law (60), the spacecraft’s 6-DOF relative motion will converge to the desired trajectory asymptotically from almost all initial tracking errors, except those with a rotation angle of a π radian. Thus, the control law (60) is an integrated asymptotic tracking control law with almost global convergence.


## 4. Body-Fixed Orbit-Attitude Hovering over an Asteroid

In this section, we will apply integrated dynamical modeling and tracking control law for the spacecraft’s 6-DOF motion obtained above to the body-fixed orbit-attitude hovering over an asteroid.The body-fixed orbit-attitude hovering means that both the position and attitude of the spacecraft are kept to be stationary 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 noncanonical Hamiltonian system within the framework of gravitationally coupled orbit-attitude dynamics, in which the spacecraft was considered as a rigid body. Wang and Xu [29] have proposed a noncanonical Hamiltonian structure-based feedback control law. Here, we will use the asymptotic tracking control law designed in Section 3.4 to achieve the orbit-attitude hovering and make comparisons with the controller in Wang and Xu [29].


### 4.1. System Description

It is assumed that the center of mass of the asteroid, that is, the reference body in earlier parts of the paper, is stationary in the inertial space, and the asteroid is rotating with a constant angular velocityωT around its maximum-moment principal axis, which is assumed to be w axis without loss of generality. That is to say, the external torque and force acting on the asteroid, τRB, are zeros. Thus, the 6-DOF dynamics of the reference body can be simplified as(62)gRB0=I4×4,ξRB≡0,0,ωT,0,0,0T,ξ˙RB=0.To make comparisons with the controller in Wang and Xu [29], a similar mission scenario is chosen in the following. The gravity field of the asteroid is approximated by a second degree and order-gravity field with harmonics C20 and C22. The parameters of the asteroid are chosen to be the same as in Wang and Xu [29]:(63)μ=94m3/s2,C20=-0.1,C22=0.04,ae=400m,ωT=2.9089×10-4s-1,where μ=GmRB, G is the gravitational constant, and ae is the mean equatorial radius of the asteroid.Since the spacecraft is in the close proximity of the asteroid 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 are considered in the environmental torque and forceτSCn. That is to say, τSCn depends only on the relative configuration of spacecraft with respect to the asteroid:(64)τSCn=τSCngR.The explicit formulation ofτSCngR is given by [19](65)TSCn=3μRR5RR×ISCRR,FSCn=-μmSCRR2R-R+3μ2RR45R-RTISCR-R-trISC+τ0mSC1-5γR·R-R2-10τ2mSCαR·R-R2-βR·R-R2R-R-2ISCR-R+2τ0mSCγR·R-RγR+4τ2mSCαR·R-RαR-βR·R-RβR,where RR=AR-1rR is the relative position of the spacecraft with respect to the asteroid expressed in the body-fixed frame of spacecraft SSC, R-R is the unit vector along RR, τ0=ae2C20, and τ2=ae2C22. In the environmental force FSCn, the perturbation caused by the gravitational orbit-attitude coupling of the spacecraft has been included [30].The parameters of spacecraft are chosen as(66)mSC=1×103kg,ISC=200010001.6×103kg·m2.The hovering position-attitudegRH, that is, the relative configuration of spacecraft at hovering, is chosen to be the same as in Wang and Xu [29]:(67)gRH=ARHrRH01∈SE3,where(68)ARH=αRH,βRH,γRHT,rRH=ARHRRH,αRH=0.9659,0,-0.2588T,βRH=0.067,0.9659,0.25T,γRH=0.25,-0.2588,0.933T,RRH=5000.9798,0,0.2Tm.In the simulation, the desired trajectorygdt,ξdt,ξ˙dt is simply chosen as the desired hovering position-attitude:(69)gdt≡gRH,ξdt≡0,ξ˙dt≡0.A better reference trajectory connecting the initial position-attitude and the hovering position-attitude, which is easier to track, can be designed in the future through studies on the guidance on SE(3), which is not the main concern of this paper.The initial relative configuration of spacecraft with respect to the asteroidgR0 is chosen as(70)gR0=AR0rR001∈SE3,where(71)AR0=ARHL,L=cos⁡π9-sin⁡π90sin⁡π9cos⁡π90001cos⁡-π200sin⁡-π20010-sin⁡-π200cos⁡-π201000cos⁡π18-sin⁡π180sin⁡π18cos⁡π18,rR0=rRH+200,100,250Tm.The initial relative velocity of spacecraft with respect to the asteroidξR0 is chosen as(72)ξR0=ΩR0VR0=-0.06s-1,0.05s-1,0.09s-1,-1m/s,2m/s,1.5m/sT.In simulations, absolute motions of the spacecraft and asteroid in the inertial space will be simulated separately. The relative motion of spacecraft and its tracking error that appear in the control law will be calculated by using absolute motions and the desired trajectory. With this approach, not only the control law but also the dynamical modeling method can be verified.To initiate the simulation of spacecraft’s absolute motion, the initial configuration of spacecraft in the inertial space can be calculated by(73)gSC0=gRB0gR0,and the initial velocity of spacecraft in the inertial space can be calculated by(74)ξSC0=ξR0+AdgR0-1ξRB0.The control gainsKd and Kp in the control law (60) are chosen as(75)Kd=0.04I6×6,Kp=4×10-4I6×6.The maximum control torque and force that can be provided by the spacecraft’s control system are set to be 10 N·m and 10 N for each axis, respectively.


### 4.2. Numerical Simulation Results

The exponential coordinates of the position-attitude tracking errorηe=ΘeT,beTT and their norms are shown in Figures 2 and 3, respectively, and the tracking errors of the spacecraft’s angular velocity and velocity ξe=ΩeT,VeTT are shown in Figures 4 and 5, respectively. It can be seen that the motion of spacecraft converges to the hovering position-attitude and keeps motionless relative to the asteroid after about 900 seconds. The attitude motion 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, while the orbital motion has an error of about 10 m and 0.15 m/s within 600 seconds and an error of about 0.05 m and 1 × 10−3 m/s within 1000 seconds. The attitude motion has a faster converge than the orbital motion.Figure 2
Exponential coordinates of attitude tracking error and its norm.Figure 3
Exponential coordinates of position tracking error and its norm.Figure 4
Tracking error of the angular velocity of spacecraft.Figure 5
Tracking error of the velocity of spacecraft.The trajectory of spacecraft in the body-fixed frame of asteroid in Figure6 shows that it does not approach the hovering position directly due to the initial velocity error. In the figure, the hovering position is denoted by the star (∗), the initial position of spacecraft is denoted by the circle (○), and the final position of spacecraft is denoted by the pentagram (☆). Notice that the star (∗) is overlapped with the pentagram (☆) and cannot be distinguished.Figure 6
Trajectory of the spacecraft in the asteroid body-fixed frame.The components of control torque and forceTSCc and FSCc, both expressed in the spacecraft body-fixed frame SSC, are shown in Figures 7 and 8, respectively. The control torque is within the spacecraft’s control capacity during the whole trajectory, whereas the thrust saturation of the control force occurs on all three axes before 600 seconds. Although thrust saturation occurs for the control force, the controller is still effective.Figure 7
Components of control torque on the three axes of the spacecraft.Figure 8
Components of control force on the three axes of the spacecraft.As the spacecraft is approaching to the hovering configuration, the control torque and force are converging to values that are needed to balance the gravitational and centrifugal torque and force at hovering position-attitude. Since the gravity gradient torqueTSCn is quite small, the control torque TSCc converges to nearly zero.Compared with the hovering controller in Wang and Xu [29], the controller proposed in this paper has not utilized the Hamiltonian structure of the system. On one hand, the controller in this paper (60) is much more complicated than that in Wang and Xu [29], which is consisted of two potential shapings and one energy dissipation. However, on the other hand, the closed-loop system in this paper is a linear system and converges much faster than that in Wang and Xu [29], which is a noncanonical Hamiltonian system with energy dissipation, oscillating around the hovering position-attitude for a long duration before the final convergence.


## 4.1. System Description

It is assumed that the center of mass of the asteroid, that is, the reference body in earlier parts of the paper, is stationary in the inertial space, and the asteroid is rotating with a constant angular velocityωT around its maximum-moment principal axis, which is assumed to be w axis without loss of generality. That is to say, the external torque and force acting on the asteroid, τRB, are zeros. Thus, the 6-DOF dynamics of the reference body can be simplified as(62)gRB0=I4×4,ξRB≡0,0,ωT,0,0,0T,ξ˙RB=0.To make comparisons with the controller in Wang and Xu [29], a similar mission scenario is chosen in the following. The gravity field of the asteroid is approximated by a second degree and order-gravity field with harmonics C20 and C22. The parameters of the asteroid are chosen to be the same as in Wang and Xu [29]:(63)μ=94m3/s2,C20=-0.1,C22=0.04,ae=400m,ωT=2.9089×10-4s-1,where μ=GmRB, G is the gravitational constant, and ae is the mean equatorial radius of the asteroid.Since the spacecraft is in the close proximity of the asteroid 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 are considered in the environmental torque and forceτSCn. That is to say, τSCn depends only on the relative configuration of spacecraft with respect to the asteroid:(64)τSCn=τSCngR.The explicit formulation ofτSCngR is given by [19](65)TSCn=3μRR5RR×ISCRR,FSCn=-μmSCRR2R-R+3μ2RR45R-RTISCR-R-trISC+τ0mSC1-5γR·R-R2-10τ2mSCαR·R-R2-βR·R-R2R-R-2ISCR-R+2τ0mSCγR·R-RγR+4τ2mSCαR·R-RαR-βR·R-RβR,where RR=AR-1rR is the relative position of the spacecraft with respect to the asteroid expressed in the body-fixed frame of spacecraft SSC, R-R is the unit vector along RR, τ0=ae2C20, and τ2=ae2C22. In the environmental force FSCn, the perturbation caused by the gravitational orbit-attitude coupling of the spacecraft has been included [30].The parameters of spacecraft are chosen as(66)mSC=1×103kg,ISC=200010001.6×103kg·m2.The hovering position-attitudegRH, that is, the relative configuration of spacecraft at hovering, is chosen to be the same as in Wang and Xu [29]:(67)gRH=ARHrRH01∈SE3,where(68)ARH=αRH,βRH,γRHT,rRH=ARHRRH,αRH=0.9659,0,-0.2588T,βRH=0.067,0.9659,0.25T,γRH=0.25,-0.2588,0.933T,RRH=5000.9798,0,0.2Tm.In the simulation, the desired trajectorygdt,ξdt,ξ˙dt is simply chosen as the desired hovering position-attitude:(69)gdt≡gRH,ξdt≡0,ξ˙dt≡0.A better reference trajectory connecting the initial position-attitude and the hovering position-attitude, which is easier to track, can be designed in the future through studies on the guidance on SE(3), which is not the main concern of this paper.The initial relative configuration of spacecraft with respect to the asteroidgR0 is chosen as(70)gR0=AR0rR001∈SE3,where(71)AR0=ARHL,L=cos⁡π9-sin⁡π90sin⁡π9cos⁡π90001cos⁡-π200sin⁡-π20010-sin⁡-π200cos⁡-π201000cos⁡π18-sin⁡π180sin⁡π18cos⁡π18,rR0=rRH+200,100,250Tm.The initial relative velocity of spacecraft with respect to the asteroidξR0 is chosen as(72)ξR0=ΩR0VR0=-0.06s-1,0.05s-1,0.09s-1,-1m/s,2m/s,1.5m/sT.In simulations, absolute motions of the spacecraft and asteroid in the inertial space will be simulated separately. The relative motion of spacecraft and its tracking error that appear in the control law will be calculated by using absolute motions and the desired trajectory. With this approach, not only the control law but also the dynamical modeling method can be verified.To initiate the simulation of spacecraft’s absolute motion, the initial configuration of spacecraft in the inertial space can be calculated by(73)gSC0=gRB0gR0,and the initial velocity of spacecraft in the inertial space can be calculated by(74)ξSC0=ξR0+AdgR0-1ξRB0.The control gainsKd and Kp in the control law (60) are chosen as(75)Kd=0.04I6×6,Kp=4×10-4I6×6.The maximum control torque and force that can be provided by the spacecraft’s control system are set to be 10 N·m and 10 N for each axis, respectively.


## 4.2. Numerical Simulation Results

The exponential coordinates of the position-attitude tracking errorηe=ΘeT,beTT and their norms are shown in Figures 2 and 3, respectively, and the tracking errors of the spacecraft’s angular velocity and velocity ξe=ΩeT,VeTT are shown in Figures 4 and 5, respectively. It can be seen that the motion of spacecraft converges to the hovering position-attitude and keeps motionless relative to the asteroid after about 900 seconds. The attitude motion 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, while the orbital motion has an error of about 10 m and 0.15 m/s within 600 seconds and an error of about 0.05 m and 1 × 10−3 m/s within 1000 seconds. The attitude motion has a faster converge than the orbital motion.Figure 2
Exponential coordinates of attitude tracking error and its norm.Figure 3
Exponential coordinates of position tracking error and its norm.Figure 4
Tracking error of the angular velocity of spacecraft.Figure 5
Tracking error of the velocity of spacecraft.The trajectory of spacecraft in the body-fixed frame of asteroid in Figure6 shows that it does not approach the hovering position directly due to the initial velocity error. In the figure, the hovering position is denoted by the star (∗), the initial position of spacecraft is denoted by the circle (○), and the final position of spacecraft is denoted by the pentagram (☆). Notice that the star (∗) is overlapped with the pentagram (☆) and cannot be distinguished.Figure 6
Trajectory of the spacecraft in the asteroid body-fixed frame.The components of control torque and forceTSCc and FSCc, both expressed in the spacecraft body-fixed frame SSC, are shown in Figures 7 and 8, respectively. The control torque is within the spacecraft’s control capacity during the whole trajectory, whereas the thrust saturation of the control force occurs on all three axes before 600 seconds. Although thrust saturation occurs for the control force, the controller is still effective.Figure 7
Components of control torque on the three axes of the spacecraft.Figure 8
Components of control force on the three axes of the spacecraft.As the spacecraft is approaching to the hovering configuration, the control torque and force are converging to values that are needed to balance the gravitational and centrifugal torque and force at hovering position-attitude. Since the gravity gradient torqueTSCn is quite small, the control torque TSCc converges to nearly zero.Compared with the hovering controller in Wang and Xu [29], the controller proposed in this paper has not utilized the Hamiltonian structure of the system. On one hand, the controller in this paper (60) is much more complicated than that in Wang and Xu [29], which is consisted of two potential shapings and one energy dissipation. However, on the other hand, the closed-loop system in this paper is a linear system and converges much faster than that in Wang and Xu [29], which is a noncanonical Hamiltonian system with energy dissipation, oscillating around the hovering position-attitude for a long duration before the final convergence.


## 5. Conclusions

The integrated 6-DOF orbit-attitude dynamical modeling and controller design for the relative motion of spacecraft with respect to a reference body have been studied in the framework of geometric mechanics in the present paper. The configuration and velocity of the relative motion have been represented by the Lie groupSE(3) and also its exponential coordinates. By using exponential coordinates of SE(3) on the dynamical level, the dynamics of tracking error have been formulated as a second-order system with a quite compact form. A simple asymptotic tracking control law with almost global convergence was designed for a demonstration purpose. Both the integrated dynamical modeling and control law were verified in an asteroid hovering mission scenario. To verify the dynamical modeling method, individual motions of the spacecraft and asteroid in the inertial space have been simulated, rather than the equations of motion describing the relative dynamics.



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*Source: 4034328-2017-05-30.xml*