1.Introduction

With rapid development of aerospace technology,new space missions,like space rendezvous and docking(RVD),satellite formation flying(SFF),on-orbit maintenance and so on,have proposed higher requirements for control accuracy of relative position and attitude.To complete these tasks,an accurate model of 6 degree-of-freedom(6-DOF)relative motion must be established.For simplicity,traditionally,position and attitude are separately modeled and controlled,without considering their coupling.However,to realize high accurate control of relative position and attitude,their coupling must be deeply explored.

Recently,there are mainly two ways in research of position and attitude coupling.One is‘independent-coupled’modeling.First separately model the relative position and attitude motion.Then the coupling terms are artificially introduced and finally the coupled dynamics is built.In[1–10],the coupling is mainly reflected in the impact of attitude error on position.However,this modeling method can hardly include all coupling terms.What’s worse,most of them do not have clear physical meaning or formation causes.

The other way is building an integrated model using dual quaternion.The Charles theory shows that the general motion of any space rigid body can be realized by the rotation around one axis and the translation along this axis.This combination of rotation and translation is called screw motion,which is often used to describe the general motion of the space rigid body[11].To describe the screw motion,dual quaternion has proved to be the most efficient and compact method with the least computation[12].Dual quaternion has already been successfully applied in space robot[13,14],mechanism kinematics[15,16],inertial navigation[17,18]and so on.

However,in close on-orbit service,there are not much application.Due to the novelty and special operation rules of dual quaternion,using it to build the integrated dynamics model of relative motion is very difficult.In[19–25],the integrated models of relative motion based on dual quaternion either are too complicated to understand or have no clear modeling process.Until now,there is no unified model of relative motion based on dual quaternion,which also hinders the consequent deeper research.

2.Mathematical foundation

2.1 Quaternion

Quaternion is the extension of the complex number in four dimensional(4D)real space R4,which is defined as

If the scalar part qsis 0,the quaternion is called vector quaternion,which is equal to a vector.If the vector part qvis 0,the quaternion is called scalar quaternion,which is equal to a scalar.Thus vector and scalar quaternion can inter convert with the normal vector and scalar according to specific computation requirements.

Based on the Euler finite rotation theory,supposing coordinate A rotates angle Φ around axis e and becomes coordinate B,then this rotating coordinate transform can be described by quaternion which is defined by Euler parameters,as

As(2)shows,when used to describe rotation, the quaternion’s four parameters are not independent,which satisfy the following constraint:

Quaternion which satisfies(3)is called unit quaternion.

纳入和排除标准[2] ：①所有患者神经功能损害体征持续时间超过1 h，NIHSS评分≥4分，颅脑CT检查发现患者双侧基底节区多发性腔隙性脑梗死；②排除3个月内有卒中史和重大颅脑外伤史、可疑蛛网膜下腔出血、颅内出血既往史、颅内肿瘤、及型出血倾向、活动性内出血、血压异常升高（收缩压超过180 mmHg或舒张压超过100 mmHg）、不符合溶栓治疗适应征的患者。

The operation rules of quaternion used in this paper are given as follows,where q,qaand qbare arbitrary quaternions.

Conjunction

（4）查询模块。该模块主要完成用户对的搜索内容的认别及对认别结果的反馈。分别制定以“内容”为主的查询规则（关键句查询）和以“主题”为主的查询规则（关键字、词查询），设定查询控制方式并与信息采集模块、索引模块的信息抓取和索引方式对应，通过对关键字、词、句进行精确解读，建立与索引文件的联系和信息比较，便于用户完成筛选和获取所需信息。

如今，李淑荣负责的川东北市场，不仅让胜利测井在国内最大规模的页岩气市场—中石化涪陵页岩气勘探测井领域拥有较高的市场份额，而且在贵州、云南等页岩气、煤层气市场占据越来越大的市场空间，成为胜利测井保效创效新的经济增长点。

In matrix form,quaternion multiplication can be expressed as

For three arbitrary quaternions p,q,r,their multiplication satisfies

Suppose vector r is represented asin coordinate A and asin coordinate B.Then there is

If the corresponding direction cosine matrix isCba,then there is

In the coordinate conversion of 3D vector or vector quaternion,using(9)to replace(7),which can make the dimension of this computation decrease from four to three.

2.2 Dual number and dual vector

In 1873,Clifford proposed the concept of dual number.Dual number is defined as an ordered real number pair

As(20)and(25)show,no matter defined by either form,quaternion describing screw motion is always a unit quaternion.

In 1901,Study extended the concept of the dual number to space geometry to express the relationship between two lines,which is represented by dual angle

where the real part θ and the dual part d respectively represent the angle and distance between two lines,as Fig.1 shows.

岭南建筑是岭南文化的现象和表征，也是岭南文化的典型载体。“余地三弓红雨足，荫天一角绿云深”，余荫山房外封闭、内开放的空间布局与疏密有致的植物景观配置高度凝练和浓缩了岭南建筑的审美特征，充分表现出岭南建筑“求真而传神，求实而写意”的艺术风格[16]，体现了岭南人崇尚自然真趣的审美趣味。受到岭南地形和湿热多雨气候的影响，余荫山房以庭院为中心来进行空间围合，即可巧妙地解决炎热潮湿的岭南地区的通风、防晒等问题，又可争取园内景观最大化，在庭院中巧妙布置山、石、水池[17]，点缀建筑小品，并与环绕的建筑一起共同形成“满院绿荫人迹少，半窗红日鸟声多”宁静而优美的环境。

Fig.1 Dual angle

According to dual quaternion algebra,the dual function can be expanded into Taylor series.As for dual anglethere are

A dual vector is composed by three dual numbers,whose set is denoted by.The dualvector is of ten called motor and written as

Operation rules of the dual number and motor are similar to those of the real number and vector in linear algebra.Just beware ε2=0.

Definition 1 If aTb=0 andthe dual vector is called unit motor,which is denoted by

The geometric interpretation of the unit motor is showed in Fig.2.It represents space position of line N relative to origin O.The real part a0is a unit vector parallel to N and the dual part describes line N’s moment relative to origin O,where p is the position vector of the arbitrary point on line N.

Fig.2 Unit motor

2.3 Dual quaternion

Dual quaternion can be defined as a dual number whose parameters are all quaternion

Also it can be defined as a quaternion whose parameters are all dual number

The dual quaternion set is denoted by Hd.In computation,it is often expressed in form of an 8D vector.

The operation rules of dual quaternion are similar to those of quaternion,dual numbers and dual vectors.Just beware ε2=0.For simplicity,only some important rules are given as follows.

Multiplication If is called unit dual quaternion,which satisfies

Similar to quaternion,multiplication of dual quaternion can be expressed in matrix form as

As for arbitrary three dual quaternions,their multiplication satisfies

As Fig.3 shows,describe the screw motion of the space rigid body by the motion of the coordinate.Coordinate A firstly rotates Φ around screw axis,then shifts d alongand finally becomes coordinate B.

Fig.3 Coordinates transform in dual quaternion

According to geometric meaning,this screw motion can be expressed by dual quaternion which is defined as

whereare the dual angle and the screw axis.

Compared with(2),we can see that dual quaternion and quaternion have the similar form when defined by geometric meaning.And there is

Dual quaternion which describes screw motion can also be interpreted as:coordinate A firstly rotates q and then shifts rb.

When defined by(21),the real part q and dual part q′satisfy

The real part of(22)is 0.Thus according to the rule of quaternion multiplication,there is

Notice that q is a quaternion describing rotation,so there is

Equations(23)and(24)are two restraints of dual quaternion which is used to describe screw motion.They make the 8D space composed by dual quaternion decrease to a 6D space,and there is

where a and b are respectively the real and dual parts of.ε is the dual operator,which satisfies ε2=0 andThe dual number set is denoted by

If the dual vector is expressed asin coordinate A and asin coordinate B,then there is

Besides,as for dual quaternion defined by(21),its logarithm is given as

Noting that the logarithm maps dual quaternion to a 6D vector.

遵循直观性原则的目的是培养学生对课本知识的感性认知，使学生能更透彻的理解课本内容。例如在讲解“多普勒效应”时，教师可以播放一段汽车鸣笛的视频，之后要求学生思考，如果一辆汽车边鸣笛边从你旁边经过，你是否能靠听觉判断出这辆汽车是靠近你还是远离你，你的判断依据是什么？之后可通过多媒体向学生演示人们听到汽车鸣笛时的声调变化。

By comparison of sections 2.1 and 2.3,it can be seen that,when describing motion,dual quaternion has many parallels with quaternion.In fact,according to the Kotelnikov transference principle,dual quaternion completely inherits properties of quaternion as shown in Table 1.

平遥县2014年汾河灌区渠系配套项目实施区域主要位于县城西北部，涉及5个乡镇8个行政村，包括四个片区。一是香乐乡北薛靳、青落、陶屯、云家庄村4村；二是宁固镇净化、魏乐2村；三是古陶镇新庄村；四是洪善镇新营村。

Table 1 Comparison of quaternion and dual quaternion

Item Quaternion Dual quaternion Degree of freedom 3 6 Coordinate transform︿ab=︿q∗°︿aa°︿q Charles theory Geometric meaning rb=q∗°ra°q Euler theory» cos(Φ/2)esin(Φ/2)– "cos(︿Φ/2)︿Lsin(︿Φ/2)i

3.Coordinates

As shown in Fig.4,to establish the model of relative motion between the chaser and the target,some necessary coordinates are defined.

Fig.4 Coordinates

Earth centered inertial(ECI):denoted as frame I.It is a nonaccelerating reference frame with its origin in the center of the earth,xipointing to the vernal equinox,and zi pointing to the north pole.yiis given by the right-hand rule.

Orbital coordinate:denoted as frame O.Its origin is at the centroid of spacecraft.zopoints from the origin to the earth center.xois vertical to zoin the orbital plane.yo is parallel to the negative normal direction of the orbital plane.

从表15知，p=0.000<0.01，回归模型非常显著.但是从表16中看到，系数b1，b2，b3，b4相应的显著性概率均大于0.05，所以没有一个变量在模型中是重要变量，因此需要对变量进行筛选，采用逐步回归法重新建立回归模型.方法是在前面四步的基础上，增加：

Notice that the dot product returns a scalar,the cross product returns a 3D vector and the logarithm maps quaternion to 3D space.

Suppose the position vector and attitude quaternion of frame B are r and q respectively.Then the spacecraft’s spatial position can be described by dual quaternion as

4.Motion model based on dual quaternion

4.1 Integrated kinematics

In the following sections,attitude,position,angular velocity and velocity all mean frame B’s motion relative to frame I.

where riand rbare respectively expressions of r in frame I and frameB.Notice that Xarepresents expression of X in frame A.

Take the derivative of(28).Then bring in the quaternion kinematic equation and use the cross product rule,we can get

Equation(29)is kinematic equation described by dual quaternion.Similarly,another form can be deduced as

Now we get the full description of integrated kinematic equation by dual quaternion as

Body fixed coordinate:denoted as frame B.It is centered at the centroid of spacecraft.Its three axes are parallel to three principal axes of inertia.When the spacecraft is at the state of the earth oriented attitude,frame B coincides with frame O.The target’s body fixed coordinate is denoted as frame T and the chaser’s is denoted as frame C.

4.2 Time derivative of vector

Suppose coordinate A rotates at

relative to inertial space.u is a changing vector.

Then the time derivative of u in inertial space can be written as

where dau/dt represents the time changerate of u relative to the vector base namely the relative derivative of u in coordinate A.

In coordinate A,the component-wise form of(35)is

where(du/dt)ais the component vector of du/dt expressed in coordinate A,dua/dt is the derivative of the component vector of u expressed in A,namely the relative derivative of u in coordinate A.is coordinate A’s angular velocity expressed in A.

Extend this property to two changing coordinatesA and B,we can get

4.3 Integrated dynamics

If at one moment,a rigid body’s angle velocity is ω,the velocity of arbitrary point A on it is vA.Then its velocity motor can be described as

where ω is a sliding vector,which is not related to reference point and rAis the position vector of A,so vAis a location vector which will change with the referencepoint.

YANG Yang, JU Zhi-guo, YUAN Ming-yuan, LI Rong-xian, ZHANG Hui-qun, NING Zhong-ping, FANG Ming, LI Xin-ming

Suppose at one moment,the force and the torque exerted on point A are fAand MA.Apparently,the force is a sliding vector and the torque is a location vector.Thus the force motor can be expressed as

The rigid body’s momentum motor is

As shown in(40),linear momentum HLis a sliding vector while v is a location vector,the angular momentum is a location vector while the angular velocityω is a sliding vector.We can see that when mass and moment of inertial are involved,the real part and dual part of velocity motor and momentum motor are exchanged.Thus we introduce operator d/dε,which has an opposite effect of ε,shown as

(1)原煤直接作为电煤销售受阻，严重制约矿井生产。由于那罗寨煤矿煤层赋存条件较差，原煤灰分在44.5%左右，硫分达到2.7%上下，难以符合电煤质量要求。但若原煤全级入选，精煤产率低、矸石泥化严重、煤泥水系统生产压力大，效益低。

Then we introduce dual mass as

传统理论认为法律趋同有两种方式：一种是被动地接受其他国家的法律，又称移植，对应上文法律的强行趋同；另一种是主动地接受其他国家的法律，又称继受，对应上文法律的自发趋同。该理论同时主张，不论是移植还是继受，都是经济落后国家对经济发达国家法律制度的趋同。并且，不同法律体系的移植和继受都从法律制度的改变开始，该制度可以脱离一国的法律文化而产生，但要使制度有稳定性和适应性则需要由该法律制度所表征的法律文化与本土原先的法律文化消解冲突并相互融合，最终才能在新产生的法律文化上彰显制度的生命力。[注]参见米健：《当今与未来世界法律体系》，第14-19页。

葛根是苏伯维尔的君药，而葛根素则是葛根的有效成分，研究表明其具有解热〔4〕，镇痛〔5〕，抗菌、抗感染〔6〕，降血压〔7〕，降血糖、血脂〔8〕，抗氧化，抗肿瘤，解酒〔9〕等作用，与苏伯维尔的功能主治一致，所以本实验选择葛根素作为苏伯维尔水提工艺的含量测定指标。为考察提取情况，首先对葛根素含量进行考察，其次对浸膏得率进行考察，试验分析得出：葛根素含量比浸膏得率更好地反映药材的提取情况，故设计葛根素含量与浸膏得率的权重系数为8：2。为确保水提工艺的合理性，本实验进行了验证试验，结果表明，正交试验优选出的水提工艺合理可行。

where m is the mass of the rigid body,J is the moment of inertia relative to the centroid.The inverse of is defined as

Now,the spacecraft’s momentum motor relative to centroid can be rewritten as

where f and M are the net force and torque acted on the centroid of the spacecraft.

学生的个体差异大，给生物课堂教学增加了难度。在具体的生物教学过程中，学生的个性化差异比较突出，学生生物基础比较差，学习能力也有着不同，这些都给生物课堂教学带来了很大的阻力，教师教学过程中很难兼顾到每个学生。这是当前高中生物课堂教学面临的一个重要的难题。

According to(36),rewritte(45)in frame B as follows:

whereare respectively the gravity, gravity gradient torque, control force and control torque exerted on the spacecraft.

Equations(31)and(46)are the full integrated model of a single spacecraft’s position and attitude coupled motion.

秦铁崖叹服。他拿出风云八虎致江云飞的战书，展示给两人看：“秦某专为此战而来。江云飞来不了，我来也一样。”

5.Integrated modeling based on relative error

Error of relative motion is defined as the space position of frame C relative to frame T.And the model based on relative error is established in frame C.

5.1 Relative kinematics

Suppose the target and chaser are in circular orbit,their spatial positions areThen their difference can be defined by multiplication as

where qtand qcare the target and chaser’s attitude quaternions.The spatial position error between two spacecrafts is

where are rotational quaternion errors, is the translational quaternion error.We can see from(48)that the coupling between posit on and attitude has been integrated in the spatial position error described by dual quaternion.

In later sections,according to section 2.1,for coordinate transform of the 3D vector,replace Qctwith Cct,which is calculated by(9).

Differentiating(48)and bringing in(31),we can get

This is the kinematic equation based on the error of relative motion.

5.2 Relative dynamics

Use the velocity motor error and target’s velocity motor to express the chaser’s velocity as

Equations(50)and(55)are kinematic and dynamic equations based on relative error.

Given the convenience of designing a controller,analyze the coupling effect,and decompose(50)and(55)into real and dual parts.Firstly,give the following definitions:

Based on these definitions,decomposition results of(50)can be directly given as

The decomposition of(55)is the difficulty and focus of the whole modeling process.Given the clarity and simplicity of derivation,the right four terms of(55)are decomposed separately as follows.

The first term:

Taking(58),(59),(61),(62)together and bringing in the definitions in(56),the real and dual parts of(59)can be written as

where rtand rcare the orbit radii of the target and the chaser,respectively.Given the distance between them can be neglected in comparison with the orbit radius,so we deem thatWe can see that attitude error parametersare involved in many terms in(64),which describes the translational motion.We can also see from(64)that the coupling between attitude and position is intricate.However,dual quaternion can describe the coupled relative motion in an integrated form,which is simple,compact and accurate as shown in(50)and(55),so we do not have to artificially add many complicated coupling terms.

5.3 Comparison with traditional coupled modeling method

In[3],relative position and attitude are separately modeled by different methods.And the coupling is divided in to coupled control inputs and coupled control instruction.Then relative position and attitude are artificially coupled via transformation between different coordinates. In[26],considering more coupling terms,the coupled model is much more complex than that in[3].

Based on the above analysis,this traditional modeling method has the following disadvantages.First,during the modeling process,detailed information of the target is needed,such as the target’s mass,moment of inertia,attitude quaternion and so on.As to non-cooperative targets, in practice, it is almost impossible to get so much information about the target,especially mass and moment of inertia.Second,to add the coupling terms,there are much coordinates transformations between ECI,body fixed frame and Hill frame,which are not only complicated but difficult to obtain accurate transformation relationships.Third,there is no unified and accurate model to describe the coupled relative motion.Because all coupling terms are added artificially,when using different modeling methods or considering different coupling factors,the final coupled models are all different,such as the model in[3],[4]and[26].

Using dual quaternion,these defects can be conquered at utmost.First,the coupled model can be built without much detailed information of the target like mass and moment of inertia.Second,coordinates transformation is only between two body fixed frames,which is easier and more feasible.Third, there is a unified and compact model to describe the coupled relative motion.

5.4 Coupled control

Dynamics described by(55)is strong coupled and nonlinear.Given the reliability and practicability,we adopt the generalized proportion-derivative(PD)control law.

5.4.1 Design of control law

The design object of the controller is to stabilize the close loop system described by(50)and(55).Namely0+ε0 and.For convenience,rewrite(55)as

Definition 2 Supposingaand b are vectors of the same dimensions,define a new multiplication ‘.∗’as

After decomposing(67)into real and dual parts,bring them into(63)and(64)respectively,then we can get

where Φeand e are Euler parameters transferred from qe.Based on the classic control theory,parameters of the generalized PD control law can be chosen as follows:

whereare respectively angular frequency and damping of rotation and translation.

5.4.2 Simulation analysis

According to the former hypothesis,the velocity and angular velocity of the target can be described in frame T as

Other initial parameters are given in Table 2.represents the desired relative position.In order to be intuitive,initial errors of attitude angles and initial relative position are given instead of initial quaternions.The initial dual quaternion can be easily calculated based on the relation between attitude angle and quaternion,and by(48).

Table 2 Simulation parameters

Parameter Value mc/kg 200 Jc/(kg·m2) diag(300,200,250)rt/km 6 900■φx φy φz i/(°)■i■-5.73 -2.86 6.86 rce0/m -120 -60 50 iT ωce0/(rad/s)■0.008 -0.005 0.006 iT ˙rce0/(m/s)■0.255 0.385 0.075 iT rcew/m■-20 0 0 iT

Simulation results are shown in Fig.5–Fig.10.These quantities are all described in frame C.In Fig.5,ψx,ψy,ψzrepresent the angle attitude errors in three axes.In Fig.6,rex,rey,rez represent the relative positions in three axes.

Fig.5 Attitude angle error curve

Fig.6 Relative position curve

Fig.7 Angular velocity error curve

Fig.8 Velocity error curve

Fig.9 Control torque curve

Fig.10 Control force curve

In Fig.7,wex,wey,wez represent the angular velocity errors in three axes.In Fig.8,vex,vey,vez represent the velocity errors in three axes.In Fig.9,Tx,Ty,Tz represent the control torques in three axes.In Fig.10,Fx,Fy,Fx represent the control torques in three axes.

To show the control precision clearer, the orders of magnitude of different controlled variables are given in Table3.And we can see that the control accuracy at 300 s has already been very high.

Table 3 Control results at time of 300 s

Controlled variable Order of magnitude Attitude angle/(˚) 10-5 Relative position/m 10-2 Angular velocity/(rad/s) 10-6 Velocity/(m/s) 10-3

6.Conclusions

To handle the coupling between position and attitude during the proximity relative motion of two spacecrafts,this paper uses dual quaternion to establish an integrated model of coupled relative posit on and attitude motion.As to the coupled relative motion,it has a unified description which is more compact,simpler and needs less target information than that using the traditional method.Firstly,introduce the foundation of dual quaternion systematically.Based on this,the detailed and clear modeling process are given,which provides an important reference for future researches.Then the 6D model is decomposed into two parts,which shows the couplings clearly and the decomposing form model can greatly facilitate the simulation model building in Matlab.Finally,a generalized PD control law is designed based on the integrated model established by dual quaternion.Simulation results show that the control law is effective and of high accuracy.

References

[1]SEGAL S,GURFIL P.Effect of kinematic rotation-translation coupling on relative spacecraft translational dynamics.Journal of Guidance Control&Dynamics,2012,32(3):1045–1050.

[2]XIN M,PAN H.Nonlinear optimal control of spacecraft approaching a tumbling target.Aerospace Science and Technology,2011,15(2):79–89.

[3]LU W,GENG Y H.Coupled control of relative position and attitude for on-orbit servicing spacecraft with respect to target.Acta Aeronautica et Astronautica Sinica,2011,32(5):857–865.(in Chinese)

[4]ZHANG Q Z,JIN Y Q,KANG Z Y,et al.Coupled control of relative position and attitude for servicing spacecraft approaching the target in close proximity.Systems Engineering and Electronics,2015,37(1):141–147.(in Chinese)

[5]SUN S H,JIA Y M.Attitude and orbit control of spacecrafts for motion reconstruction of flying around and approaching the tumbling target.CAAI Transaction on Intelligent Systems,2016,11(6):818–826.

[6]STANSBERY D T,CLOUTIER J R.Position and attitudecontrol of a spacecraft using the state-dependent Riccati equation technique.Proc.of the American Control Conference,2000:1867–1871.

[7]XING Y,CAO X,ZHANG S,et al.Relative position and attitude estimation for satellite formation with coupled translational and rotational dynamics.Acta Astronautica,2010,67(3):455–467.

[8]YANS U,KANGLI.Coupled relative attitude-orbit dynamics and control of anear satellites.Aerospace Control&Application,2014,40(4):20–25.(in Chinese)

[9]CHEN W,JING W,XIA X.Autonomous robust orbital and attitude coupling control of final proximity in rendezvous.Proc.of the IEEE International Symposium on Systems and Control in Aerospace and Astronautics,2009:1–6.

[10]ZHENG Z,ZHOU H,CHEN B,et al.Sliding mode control for satellite proximity operations with coupled attitude and orbit dynamics.Proc.of the IEEE International Conference on Intelligent Control and Information Processing,2011:824–829.

[11]SELIG J M.Geometric fundamentals of robotics.New York:Springer,2005.

[12]FISCHER I.Dual-number methods in kinematics,statics and dynamics.Boca Raton,USA:CRC Press,1998.

[13]FIGUEREDO L F C,ADORNO B V,ISHIHARA J Y,et al.Robust kinematic control of manipulator robots using dual quaternion representation.Proc.of the IEEE International Conference on Robotics and Automation,2013:1949–1955.

[14]GAO Y,LIU X P.Robot base frame calibration method based on dual quaternion.Journal of Mechanical&Electrical Engineering,2017,34(3):310–314.

[15]GAN D,LIAO Q,WEI S,et al.Dual quaternion based inverse kinematics of the general spatial 7R mechanism.Journal of Mechanical Engineering Science,2008,222:1593–1598.(in Chinese)

[16]NI Z S,LIAO Q Z,WU X X.General 6R robot inverse solution algorithm based on a quaternion matrix and a Groebner base.Journal of Tsinghua University(Science and Technology),2013,53(5):683–687.

[17]BRANETS V N,SHMYGLEVSKY I P.Introduction to strapdown inertial navigation systems.Moscow:Nanka,1992.

[18]ZHAO Y L,QIAO B,JIN Y Q,et al.An integrated relative navigation algorithm for spacecraft based on binocular vision and inertial measurement.Aerospace Control,2016,34(4):47–52.(in Chinese)

[19]FILIPE N,TSIOTRAS P.Adaptive position and attitudetracking controller for satellite proximity operations using dual quaternions.Journal of Guidance Control&Dynamics,2014,38(4):566–577.

[20]ZHU Z X,MA J J,FAN R S.Synchronization control of relative motion for spacecraft with screw theory based description.Acta Aeronautica et Astronautica Sinica,2016,37(9):2788–2798.

[21]WANG J Y,LIANG H Z,SUN Z W,et al.Relative motion coupled control based on dual quaternion.Aerospace Science&Technology,2013,25(1):102–113.

[22]WANG Q B,TANG S,MA K,et al.Relative position and attitude estimation of spacecrafts based on dual quaternion for rendezvous and docking.Acta Astronautica,2013,91(10):237–244.

[23]YANG Y D,JING W X,ZHANG Z.A new method for orbit and attitude coupling control problem of flexible spacecraft based on dual quaternions.Journal of Astronautics,2016,37(8):946–956.(in Chinese)

[24]GUI H,VUKOVICH G.Dual-quaternion based adaptive motion tracking of spacecraft with reduced control effort.Nonlinear Dynamics,2015,83(1/2):1–18.

[25]WANG X,YU C,LIN Z.A dual quaternion solution to attitude and position control for rigid-body coordination.Proc.of the IEEE International Conference on Bioinformatics and Biomedicine,2013:38–42.

[26]CHEN B L,GENG Y H.Relative motion coupled dynamic modeling between two docking ports.Systems Engineering and Electronics,2014,36(4):714–720.(in Chinese)