H Dong

Q Hu

M R Akella
email: makella@mail.utexas.edu

Abstract-Based on the dual-quaternion description, a smooth six-degree-of-freedom observer is proposed to estimate the incorporating linear and angular velocity, called the dual angular velocity, for a rigid body. To establish the observer, some important properties of dual vectors and dual quaternions are established, additionally, the kinematics of dual transformation matrices is deduced, and the transition relationship between dual quaternions and dual transformation matrices is subsequently analyzed. An important feature of the observer is that all estimated states are ensured to be C ∞ continuous, and estimation errors are shown to exhibit asymptotic convergence. Furthermore, to achieve tracking control objectives, the proposed observer is combined with an independently designed proportional-derivativelike feedback control law (using full-state feedback), and a special Lyapunov "strictification" process is employed to ensure a separation property between the observer and the controller, which further guarantees almost global asymptotic stability of the closed-loop dynamics. Numerical simulation results for a prototypical spacecraft pose tracking mission application are presented to illustrate the effectiveness and robustness of the proposed method.

Index Terms-Dual quaternion, Observer, Lyapunov strictification, 6-DOF control, Tracking control.

I. INTRODUCTION

F OR various motion control problems of practical impor- tance, such as on-orbit missions of spacecrafts (including monitoring, surveillance, refueling, on-orbit assembly), a follower spacecraft is often required to track both the timevarying relative positions and the reference attitude trajectories accurately and synchronously with respect to a leader spacecraft (i.e., six-degree-of-freedom (6-DOF) tracking or pose tracking). Beard et al. [START_REF] Beard | A coordination architecture for spacecraft formation control[END_REF] proposed a coordination architecture for spacecraft formation tracking control problems, in which the leader-following strategy and the virtual-structure approach are introduced. A 6-DOF synchronization scheme of spacecraft formations for deep-space mission applications was presented in Ref. [START_REF] Krogstad | 6-DOF mutual synchronization of formation flying spacecraft[END_REF]. Kristiansen et al. [START_REF] Kristiansen | Spacecraft coordination control in 6DOF: integrator backstepping vs passivity-based control[END_REF] introduced backstepping and passivity control methods for 6-DOF tracking problems, and Lv et al. [START_REF] Lv | 6-DOF synchronized control for spacecraft formation flying with input constraint and parameter uncertainties[END_REF] addressed the input constraint problem under the similar background.

Six-degree-of freedom spacecraft controllers based on the dual-number/dual-quaternion description have recently drawn significant attention in the literature. Qualitatively speaking, dual quaternions are the extension of the traditional Euler quaternions, but different in the sense that they can be used to describe not only the rotational motion of a rigid body, but also the translational motion synchronously. Compared to other 6-DOF description methods, dual quaternions have clear physical meanings and compact forms, and the intrinsic couplings of the relative translational motion with the rotational one are taken into account automatically in the dual-quaternion-based model. Another appealing property of dual quaternions is that the algebraic similarity between dual quaternions and quaternions could help designers a lot in the design process of control methods. Wang et al. [START_REF] Wang | 6-DOF robust adaptive terminal sliding mode control for spacecraft formation flying[END_REF], [START_REF] Wang | Finite-time control for spacecraft formation with dual-number-based description[END_REF] provided detailed discussions of the dual-quaternion-based modeling process for the integrated 6-DOF kinematics and dynamics of rigid bodies, and they also proposed several finite-time control laws by using sliding mode methods. Based on the dual quaternion formulation, Filipe and Tsiotras [START_REF] Filipe | Adaptive position and attitude-tracking controller for satellite proximity operations using dual quaternions[END_REF] presented a robust control method with additional mass and inertia identification mechanisms. Refs. [START_REF] Dong | Dual-quaternion based fault-tolerant control for spacecraft formation flying with finite-time convergence[END_REF] and [START_REF] Dong | Dual-quaternion-based fault-tolerant control for spacecraft tracking with finite-time convergence[END_REF] further introduced several fault-tolerant control schemes for 6-DOF tracking operations of spacecrafts in the presence of actuator faults.

From a practical standpoint, due to the slow sampling rates and inherent noise characteristics of sensors, and also possible strict constraints on the cost and space (volume) available for sensors installation in case of cube/nano-satellites, reliable linear and angular velocity measurements may not always be feasible/available. For attitude tracking control problems, when angular velocities are unavailable, the passivity property of the rigid-body attitude dynamics allows control objectives to still be accomplished using only output (dynamic) feedback [START_REF] Lizarralde | Attitude control without angular velocity measurement: A passivity approach[END_REF], [START_REF] Tsiotras | Further passivity results for the attitude control problem[END_REF], [START_REF] Akella | Rigid body attitude tracking without angular velocity feedback[END_REF]. Another widely-studied method to solve this problem is designing observers to estimate angular velocities and establishing proper observer-controller architectures to achieve tracking control goals [START_REF] Seo | Separation property for the rigid-body attitude tracking control problem[END_REF], [START_REF] Song | Quaternion observer-based model-independent attitude tracking control of spacecraft[END_REF], [START_REF] Xiao | Attitude stabilization of spacecrafts under actuator saturation and partial loss of control effectiveness[END_REF]. To our best knowledge, the first nonlinear angular velocity observer for rigid bodies under the quaternion kinematics description was proposed by Salcudean [START_REF] Salcudean | A globally convergent angular velocity observer for rigid body motion[END_REF]. Zou derived distributed [START_REF] Zou | Distributed attitude synchronization and tracking control for multiple rigid bodies[END_REF] and finite-time stable [START_REF]Finite-time output feedback attitude tracking control for rigid spacecraft[END_REF] observers for attitude tracking control problems. But because the fact that separation properties are usually difficult to be established for nonlinear systems, stability analyses of closed-loop systems with observer-controller architectures always involved in nontrivial theoretical complexities. Nicosia et al. [START_REF] Nicosia | Nonlinear observer and output feedback attitude control of spacecraft[END_REF] presented a nonlinear observer and guaranteed asymptotically bounded stability for the closed-loop system.

Recently, based on a switching logic, the separation property of the attitude tracking system with a nonlinear observer and an independently designed proportional-derivative-like (PDlike) controller was synthesized in Ref. [START_REF] Chunodkar | Switching angular velocity observer for rigid-body attitude stabilization and tracking control[END_REF], and the closedloop system was rendered to be almost global stable. Ref. [START_REF] Akella | Partial lyapunov strictification: smooth angular velocity observers for attitude tracking control[END_REF] further extended this result and guaranteed C ∞ continuity for all estimated states by the way of circumventing the need for switching within the angular velocity observer structure. However, when concerning the controller development and observer design for 6-DOF tracking control problems without both linear and angular velocities measurements, relevant studies are relatively limited. Ref. [START_REF] Filipe | Pose tracking without linear-and angular-velocity feedback using dual quaternions[END_REF] extended the attitudeonly result given in [START_REF] Akella | Rigid body attitude tracking without angular velocity feedback[END_REF], and proposed a velocity-free outputfeedback controller under the dual quaternion formulation. Based on the vectrix formalism, a 6-DOF output feedback control law was introduced in Ref. [START_REF] Wong | Output feedback control for spacecraft formation flying with coupled translation and attitude dynamics[END_REF], and the closed-loop dynamics was guaranteed to be locally asymptotically stable. But these results are all given upon the passivity control strategy, so can't provide real-time linear/angular velocities estimations.

In this paper, significantly building upon former results given in Refs. [START_REF] Chunodkar | Switching angular velocity observer for rigid-body attitude stabilization and tracking control[END_REF] and [START_REF] Akella | Partial lyapunov strictification: smooth angular velocity observers for attitude tracking control[END_REF], a novel 6-DOF observer under the dual-quaternion-based description is proposed to estimate the incorporating linear and angular velocity (i.e., the dual angular velocity) of a rigid body. The special structure of the observer ensures C ∞ continuity of all estimated states, and guarantees the global asymptotic convergence of estimation errors irrespective of prescribed control inputs. To establish the observer, some important mathematical properties of dual vectors, dual quaternions and dual transformation matrices are presented and proved, the kinematics of dual transformation matrices and the transition relationship between dual quaternions and dual transformation matrices are subsequently deduced. These results provide important building blocks for the establishment of all the main results in the paper.

Furthermore, the proposed observer is shown to satisfy a "separation" property when combined with an independently designed PD-like controller. This is achieved by utilizing a partial Lyapunov "strictification" strategy [START_REF] Malisoff | Constructions of strict Lyapunov functions[END_REF], [START_REF] Mazenc | A simplified design for strict lyapunov functions under matrosov conditions[END_REF], [START_REF] Mazenc | Lyapunov functions for time-varying systems satisfying generalized conditions of matrosov theorem[END_REF], in which a nonstrict Lyapunov function could be transformed into a strict one whose derivative contains additional nonpositive terms of system states. Specifically, a strictificationlike analysis is carried out for the controller part Lyapunovlike function to guarantee the boundedness of all tracking state errors. Subsequently, this result is employed to conduct a further strictification process to obtain a partially strict Lyapunov-like function for observer. The word "partially" is used to emphasize that, for a dual quaternion, only the vector component of it is contained in the "strictified" Lyapunov-like function's time derivative. Finally, by employing a composite function, consisting of both the new partially strict observer Lyapunov-like function and the controller one, the combined observer-controller scheme is proved to render almost global 1asymptotic convergence of all tracking errors, and the separation property is guaranteed accordingly.

It is noteworthy that, Refs. [START_REF] Chunodkar | Switching angular velocity observer for rigid-body attitude stabilization and tracking control[END_REF] and [START_REF] Akella | Partial lyapunov strictification: smooth angular velocity observers for attitude tracking control[END_REF] (in which only 3-DOF attitude tracking problems are considered), especially the strictification process in Ref. [START_REF] Akella | Partial lyapunov strictification: smooth angular velocity observers for attitude tracking control[END_REF], highly rely on the boundedness property of traditional quaternions ( q = 1), but dual quaternions don't inherit this property due to containing position information. This fact and also the intrinsic couplings between the orientational motion and the translational motion increase the difficulties and complexities for both the observer design and the subsequent controller development. From these points of view, the strictification process and the observercontroller construction presented in this paper are more general. To the best knowledge of the authors, this is the first time a dual angular velocity observer is designed, and also the first time a separation property is established for the rigid-body dynamics under the dual-quaternion formulation.

The remainder of the paper is organized as follows. In Section II, commonly-used definitions and operations of dual numbers and dual quaternions are reviewed, and some important properties of them are introduced and proved, the kinematics of dual transformation matrices and the transit relationship between dual quaternions and dual transformation matrices are also derived, and then the dual-quaternion-based relative kinematics and dynamics of rigid bodies are introduced. Based on these mathematical preliminaries, a novel dual angular velocity observer is proposed in Section III, and the convergence of estimation errors is analyzed. In Section IV, the observer is combined with a PD-like controller, by a special strictification process, the proof of the separation property and the stability analysis for the closed-loop system are presented. Then, numerical simulations for a prototypical pose tracking task of spacecrafts are presented in Section V to illustrate the effectiveness of the proposed method. Finally the paper is completed with some conclusions in Section VI.

Throughout the paper, R and R are employed to denote the sets of real numbers and dual numbers, respectively. Superscript • implies the corresponding quantity belongs to the dual number set. Right subscripts • r and • d denote the real part and the dual part of a dual number, respectively. Right superscript • x means the corresponding vector is expressed in a frame X . H and Ĥ are the sets of quaternions and dual quaternions, respectively. Ĥs denotes the scalar-part set of dual quaternions, while Ĥv is the vector-part set. Note that Ĥs ⊂ R and Ĥv ⊂ R3 . The notation • refers to the Euclidean norm, and • T refers to the transpose of vectors and matrices.

II. MATHEMATICAL PRELIMINARIES

A. Dual Numbers and Dual Vectors

The concept of dual numbers was first introduced by Clifford [START_REF] Clifford | A preliminary sketch of biquaternions[END_REF], then named and perfected by Study [START_REF] Study | Von den bewegungen und umlegungen[END_REF]. The definition of a dual number is given as follows.

â = a r + εa d (1)
where â ∈ R is a dual number, a r , a d ∈ R are called the real part and the dual part of â, respectively. ε is referred to a special dual unit with rules:

ε 2 = 0 but ε = 0 (2)
Throughout the paper, â1 ≥ â2 implies both the real part and the dual part of â1 are larger or equal to â2 , where â1 , â2 ∈ R.

Dual vectors are a class of dual numbers whose real and dual parts are vectors:

â = a r + εa d (3)
where a r , a d ∈ R n are the real part and dual part of â, respectively. The zero dual vector 0n ∈ Rn is defined as 0n = 0 n + ε0 n , where 0 n denotes the n-dimensional zero vector. Some basic operations and properties of dual numbers and dual vectors are introduced as follows.

λâ = λa r + ελa d (4) âT = a T r + εa T d (5) ââ = a r a r + ε(a r a d + a d a r ) (6)
âs = a d + εa r , (â s ) s = â (7) â = a r + ε a d , (8)
â1 ± â2 = a r1 ± a r2 + ε(a d1 ± a d2 ) (9)
â1 • â2 = âT 1 â2 = a 1r • a 2r + ε(a 1r • a 2d + a 1d • a 2r ) (10) Ââ = A r a r + ε(A r a d + A d a r ) (11)
â1 • â2 = a T 1r a 2r + a T 1d a 2d , âs 1 • âs 2 = â1 • â2 (12)
where λ ∈ R, â1 , â2 ∈ Rn , and  = A r + εA d ∈ Rm×n is called a dual matrix. When â1 , â2 ∈ R3 , one further has

â1 × â2 = -â 2 × â1 = a 1r × a 2r + ε(a 1r × a 2d + a 1d × a 2r ) (13)
Another important property of dual vectors is that the dual cross product can also be written as the multiplication of a skew-symmetric matrix and a vector: â × b = Ŝ(â) b, where Ŝ(â) = -ŜT (â) = S(a r ) + εS(a d ) is called the dual skewsymmetric matrix of the dual vector â, and satisfies

Ŝ(â) = 0 -â 3 â2 â3 0 -â 1 -â 2 â1 0 = 0 -a 3r a 2r a 3r 0 -a 1r -a 2r a 1r 0 + ε 0 -a 3d a 2d a 3d 0 -a 1d -a 2d a 1d 0 (14)
and here âi = a ir + εa id , i = 1, 2, 3 are entries of â, with â = [â 1 , â2 , â3 ] T . Some other important properties of dual vectors are presented as follows.

Lemma

1: â × ( b × ĉ) + b × (ĉ × â) + ĉ × (â × b) = 03 . Lemma 2: â × ( b × ĉ) = b(â • ĉ) -ĉ(â • b). Lemma 3: (â × b) • â = 03 , â • (â × b) = 03 . Lemma 4: âs • (â × b) = 03 . Proof of Lemmas 1-4: See Appendix A. Lemma 5 [22]: â • ( b × ĉ) = bs • (ĉ × âs ) = ĉs • (â s × b).

B. Quaternion and Dual Quaternion

The unit quaternion is the most commonly-used method to describe the relative attitude motion between two reference frames. The definition of it is q = [η, ξ T ] T , where η ∈ R and ξ = [ξ 1 , ξ 2 , ξ 3 ] T ∈ R 3 are called the scalar part and the vector part of q, respectively, and satisfies η 2 + ξ T ξ = 1. The multiplication of two quaternions

q 1 = [η 1 , ξ T 1 ] T and q 2 = [η 2 , ξ T 2 ]
T is defined as:

q 1 ⊗ q 2 = [η 1 η 2 -ξ T 1 ξ 2 , (η 1 ξ 2 + η 2 ξ 1 + ξ 1 × ξ 2 ) T ] T (15)
Based on the definition of quaternions, the coordinates of any real vector a ∈ R 3 in a frame Y can be calculated from the coordinates of another frame X of that same vector [START_REF] Schaub | Analytical mechanics of space systems[END_REF] [0, (a y ) T ] T = q * yx ⊗ [0, (a x ) T ] T ⊗ q yx [START_REF] Salcudean | A globally convergent angular velocity observer for rigid body motion[END_REF] In which q yx is the quaternion of the frame Y with respect to the frame X , q * yx = [η yx , -ξ T yx ] T is the conjugate quaternion of q yx . Eq. ( 16) can also be represented by

a y = C(q yx )a x (17)
and here

C(q yx ) = I 3×3 -2η yx S(ξ yx ) + 2S(ξ yx )S(ξ yx ) (18)
is called the transformation matrix from X to Y. The definition of dual quaternions are based on both quaternions and dual vectors, as a generic example, the dual quaternion of Y with respect to X is [START_REF] Wang | Finite-time control for spacecraft formation with dual-number-based description[END_REF], [START_REF] Wu | Strapdown inertial navigation system algorithms based on dual quaternions[END_REF]

qyx = q yx + ε 1 2 q yx ⊗ [0, (r y yx ) T ] T (19)
where r y yx is the position vector from the origin of frame X to the origin of frame Y (and expressed in frame Y). qyx can also be written as the combination of a dual scalar and a dual vector: qyx = [η yx , ξT yx ] T , where ηyx ∈ Ĥs and ξyx ∈ Ĥv are called the scalar part and the vector part of qyx , respectively.

Dual quaternions follow the operations of dual vectors, and some other special operations and properties used in this paper are given as follows.

vec( q) = ξ (20)
[ 0, vec( q) T ] T • q = vec( q) • vec( q) (21)

q1 ⊗ q2 = [η 1 η2 -ξT 1 ξ2 , (η 1 ξ2 + η2 ξ1 + ξ1 × ξ2 ) T ] T (22) q * = [η, -ξT ] T , q * ⊗ q = qI (23) q1 • ( q2 ⊗ q3 ) = qs 3 • [ q * 2 ⊗ ( qs 1 )] (24)
In which q1 , q2 , q3 ∈ Ĥ, qI = [1, 0, 0, 0] T + ε[0, 0, 0, 0] T . Furthermore, the kinematics of the dual quaternion qyx satisfies

qyx = 1 2 qyx ⊗ [ 0, ( ωy yx ) T ] T = 1 2 Ê( qyx ) ωy yx ( 25
)
where function

Ê( qyx ) = -ξT yx ηyx I 3×3 + Ŝ( ξyx ) (26)
and ωy yx = ω y yx +ε(v y yx +ω y yx ×r y yx ) is called the dual angular velocity, and here ω y yx and v y yx are the angular velocity and linear velocity of Y with respect to X , respectively.

Remark 1: It is noteworthy that v y yx = ṙy yx , which is the time derivative of r yx with respect to the frame Y and then expressed in the frame Y. As a counterpart, v x yx = ṙx yx is the time derivative of r yx with respect to the frame X and then expressed in the frame X . So actually the rigorous expressions of them should be (y) v y yx and (x) v x yx , respectively, where the left-superscripts denote in which frames these derivatives are obtained. But because throughout the paper, we never define a linear velocity vector in the form (x) v y (which implies this derivative is got with respect to a frame X but then expressed in a frame Y), so v y . = (y) v y and v x . = (x) v x are employed for ease of notation, where v could be any linear velocity vector. Furthermore, for this special notation, the relationship between v y and v x doesn't follow the one given in [START_REF] Zou | Distributed attitude synchronization and tracking control for multiple rigid bodies[END_REF], instead, it should be deduced from the original relationship r y = C(q yx )r x , by taking time derivative for both sides, one can get v y = -ω y yx × r y + C(q yx )v x .

C. Dual-Quaternion-based Transformation

Dual quaternions can be used to describe 6-DOF transformation. For any 3-dimensional dual vector âx = a x r + εa x d ∈ R3 (which is initially expressed in the frame X ), the following equation holds [START_REF] Filipe | Nonlinear pose control and estimation for space proximity operations: An approach based on dual quaternions[END_REF],

vec( q * yx ⊗ [ 0, (â x ) T ] T ⊗ qyx ) = a y r + εa y d + ε(a y r × r y yx ) (27) where a y r = C(q yx )a x r , a y d = C(q yx )a x d .
Eq. ( 27) indicates that the transformation rule upon dual quaternions is a little different with the property given in [START_REF] Salcudean | A globally convergent angular velocity observer for rigid body motion[END_REF]. Specifically, an additional cross term shows up, which stems from the unique structure of dual quaternions.

Based on [START_REF] Bhat | A topological obstruction to continuous global stabilization of rotational motion and the unwinding phenomenon[END_REF], the definition and properties of dual transformation matrices are given in the following proposition.

Proposition 1: The dual quaternion transformation rule given in ( 27) can be expressed by

vec( q * yx ⊗ [ 0, (â x ) T ] T ⊗ qyx ) = Ĉ( qyx )â x (28)
and here

Ĉ( qyx ) = I 3×3 -2η yx Ŝ( ξyx ) + 2 Ŝ( ξyx ) Ŝ( ξyx ) (29)
is called the dual transformation matrix of the frame Y with respect to the frame X , which satisfies the following properties:

1) Ĉ( q * yx ) = ĈT ( qyx ); 2) ĈT ( qyx ) Ĉ( qyx ) = Ĉ( qyx ) ĈT ( qyx ) = I 3×3 + ε0 3×3 ; 3) For any dual vectors â1 , â2 ∈ R3 , Ĉ( qyx )(â x 1 × âx 2 ) = [ Ĉ( qyx )â x 1 ] × [ Ĉ( qyx )â x 2 ] 4) d[ Ĉ( qyx )]/dt = -Ŝ( ωy yx ) Ĉ( qyx ). Proof : See Appendix B.
Remark 2: In Refs. [START_REF] Condurache | Onboard exact solution to the full-body relative orbital motion problem[END_REF] and [START_REF]Dual tensors based solutions for rigid body motion parameterization[END_REF], Condurache and Burlacu introduced a special dual orthogonal tensor-based construction and showed the similar results as given in Proposition 1. By comparison, in this paper, these important facts are proved from a different but straightforward way, in which the relationship between dual quaternions and dual transformation matrices is emphasized, and then it is utilized to deduct the properties of dual transformation matrices. Furthermore, one can also use the kinematics of dual transformation matrices given in Proposition 1 to deduce the kinematics of dual quaternions shown in [START_REF] Mazenc | A simplified design for strict lyapunov functions under matrosov conditions[END_REF], the fundamental reason is that dual quaternions and dual transformation matrices are two equivalent methods to describe the 6-DOF motion between two arbitrary frames, so they can be converted to each other.

D. Dual-Quaternion based Relative Kinematics and Dynamics

In this paper, the 6-DOF relative pose tracking control problem of a leader-follower spacecraft formation is considered. As shown in Fig. 1, three reference frames are employed: the inertial frame, the body-fixed frame of the leader and the bodyfixed frame of the follower, denoted by

I = {X I , Y I , Z I }, L = {X L , Y L , Z L } and B = {X B , Y B , Z B }, respectively.
Within this context, compared with the "absolute" positions and velocities with respect to the inertia frame I, the relative positions and velocities between the leader and the follower are much smaller, and to implement precisely 6-DOF tracking control, the follower should have the ability to measure sufficient relative motion information with respect to the leader. Furthermore, instead of directly tracking the body-fixed frame of the leader (which is actually impractical, because superimposing the frame B onto the frame L will lead to collision), to extend the design flexibility of control objectives, a frame T (named as the target frame) is defined to describe the desired relative pose of the follower with respect to the leader (in Fig. 1, the frame T is blurred to indicate it is a virtual and user-designed frame). 19) and ( 25), the dual quaternion of the frame B with respect to the frame L is

O I Z I X I Y I Z L X L Y L Z B Y B X B r bl Z T Y T X T
qbl . = [η bl , ξT bl ] T = q bl + ε 1 2 q bl ⊗ r b bl ( 30
)
where q bl is the quaternion of B with respect to L, and r b bl ∈ R 3 is the relative position vector from the origin of L to the origin of B. Then the 6-DOF relative kinematics and dynamics can be presented as [START_REF] Wang | 6-DOF robust adaptive terminal sliding mode control for spacecraft formation flying[END_REF], [START_REF] Wang | Finite-time control for spacecraft formation with dual-number-based description[END_REF], [START_REF] Filipe | Adaptive position and attitude-tracking controller for satellite proximity operations using dual quaternions[END_REF],

qbl = 1 2 Ê( qbl ) ωb bl (31) Mb ωb bl = -Ŝ( ωb bi )( Mb ωb bi ) + Mb [ Ŝ( ωb bl ) ωb li -Ĉ( qbl ) ωl li ] + ûb (32)
where ωb bi and ωl li are the dual angular velocities of B and L with respect to I, respectively. A reasonable assumption is that both ωl li and ωl li are bounded and available. ωb bl = ωb biωb li , and here ωb li = Ĉ( qbl ) ωl li . Notice by [START_REF] Bhat | A topological obstruction to continuous global stabilization of rotational motion and the unwinding phenomenon[END_REF] and remark 1, because additional cross terms show up, ωb li doesn't transform every component of ωl li onto the frame B. But these cross terms actually play an important role to establish ωb bl , and it can be proved that ωb bl fully obey the definition of general dual angular velocities:

ωb bl = ω b bl + ε(v b bl + ω b bl × r b bl ) (33)
where ω b bl and v b bl are the relative angular velocity and the relative linear velocity of B with respect to L, respectively. Furthermore, ûb = f b + ετ b is the total dual input applied to the rigid body, and here f b ∈ R 3 and τ b ∈ R 3 are the total force and the total torque applied to the rigid body, respectively. Mb = m b I 3×3 d dε + εJ b is called the dual mass matrix, with m b ∈ R and J b ∈ R 3×3 are the mass and the inertia (expressed in the frame B) of the rigid body, respectively.

Similarly, for the frame B and the target frame T ,

qbt = 1 2 Ê( qbt ) ωb bt ( 34
)
Mb ωb bt = -Ŝ( ωb bi )( Mb ωb bi ) + Mb [ Ŝ( ωb bt ) ωb ti -Ĉ( qbt ) ωt ti ] + ûb (35)
Notice that since frame T is designed based upon L, ωtl is available. And in [START_REF]Dual tensors based solutions for rigid body motion parameterization[END_REF],

ωb ti = Ĉ( qbt ) ωt ti , ωt ti = ωt tl + Ĉ( qtl ) ωl li , ωb bt = ωb bi -ωb ti = ωb bl -ωb tl ,
and here ωb tl = Ĉ( qbt ) ωt tl . The control objective is to superimpose the frame B onto the frame T .

E. Proportional-Derivative-Like Controller

Similar with the traditional attitude tracking control problem, Ref. [START_REF] Filipe | Pose tracking without linear-and angular-velocity feedback using dual quaternions[END_REF] shows that one can also use a PD-like controller to achieve the 6-DOF tracking goal for the system in [START_REF] Condurache | Onboard exact solution to the full-body relative orbital motion problem[END_REF][START_REF]Dual tensors based solutions for rigid body motion parameterization[END_REF], as summarized in Proposition 2.

Proposition 2 [START_REF] Filipe | Pose tracking without linear-and angular-velocity feedback using dual quaternions[END_REF]: Consider the system given in [START_REF] Condurache | Onboard exact solution to the full-body relative orbital motion problem[END_REF][START_REF]Dual tensors based solutions for rigid body motion parameterization[END_REF], design the total dual input applied to the rigid body as

ûb = -k p vec( q * bt ⊗ ( qbt -qI ) s ) -k d ( ωb bt ) s + Mb ( Ĉ( qbt ) ωt ti ) + Ŝ( ωb ti )( Mb ωb ti ) (36)
with k p , k d > 0. It can be guaranteed for any qbt (0) ∈ Ĥ, when t → ∞, ξbt (t) → 03 , and ωb bt (t) → 03 . Remark 3: As discussed in Sec. II.D, ωb bt can be obtained by either an "absolute" way ωb bi -ωb ti or a "relative" way ωb bl -ωb tl . As aforementioned, for the pose tracking control problem of spacecrafts, the relative velocities between spacecrafts has a much smaller order of magnitude than their absolute velocities with respect to the inertia frame. So it is not reasonable to use the "absolute" way to calculate ωb bt , because it will introduce big measurement errors and lead to poor control precisions. Instead, the follower should directly measure the relative pose and linear/angular velocities with respect to the leader. And when sensors for relative linear/angular velocity measurements are unavailable, observers could be established to estimate ωb bl instead of ωb bi . To this end, a smooth observer is designed in the following section.

qe = q * ol ⊗ qbl = q e + ε 1 2 q e ⊗ r b e ( 37
)
ωb e = ωb bl -ωb ol = ωb bi -ωb oi ( 38
)
where q e = q * ol ⊗ q bl and r b e = r b bl -r b ol are the estimation errors of the relative quaternion and position, respectively, ωb ol = Ĉ( qe ) ωo ol is defined for ease of notation, and ωb oi = ωb ol + ωb li . Then, one of the main results of this paper, a smooth observer which guarantees the asymptotic convergence of estimation errors, is summarized in the following theorem.

Theorem 1: Consider the following smooth observer construction, qol = 1 2 Ê( qol )[ ωo ol + λ ĈT ( qe )vec( q * e ⊗ ( qe -qI ) s ) s ] (39)

ωo ol = ĈT ( qe ) M -1 b • [γvec( q * e ⊗ ( qe -qI ) s ) -Ŝ( ωb oi )( Mb ωb oi ) -... λ Mb ( Ŝ(vec( q * e ⊗ ( qe -qI ) s ) s ) ωb ol ) + ... Mb ( Ĉ( qbl ) ωl li ) + Mb ( Ŝ( ωb ol ) ωl li ) + ûb ] (40)
where λ, γ > 0 and

M -1 b = J -1 b d dε + 1 m b I 3×3 ε is the inverse of Mb .
Then for any qol (0) ∈ Ĥ, and ωo ol (0) ∈ R3 , it can be guaranteed ξe (t) → 03 and ωb e (t) → 03 , when t → ∞. Proof : By the definition of qe , one has,

Ĉ( qe ) = Ĉ( qbl ) ĈT ( qol ) (41)
Substituting (39) into the time derivative of (41) yields,

d( Ĉ( qe )) dt = -Ŝ( ωb bl ) Ĉ( qe ) + Ĉ( qe ) Ŝ[ ωo ol + λ ĈT ( qe )vec( q * e ⊗ ( qe -qI ) s ) s ] (42)
Recall the Property 3 given in Proposition 1, for any dual vector â ∈ R3 , one has

Ĉ( qe ) Ŝ[ ωo ol + λ ĈT ( qe )vec( q * e ⊗ ( qe -qI ) s ) s ]â o = Ŝ[ Ĉ( qe )( ωo ol + λ ĈT ( qe )vec( q * e ⊗ ( qe -qI ) s ) s )] • Ĉ( qe )â o (43)
Due to the arbitrariness of âo , one can further get,

Ĉ( qe ) Ŝ[ ωo ol + λ ĈT ( qe )vec( q * e ⊗ ( qe -qI ) s ) s ] = Ŝ[ Ĉ( qe )( ωo ol + λ ĈT ( qe )vec( q * e ⊗ ( qe -qI ) s ) s )] • Ĉ( qe ) (44)
Substitute (44) into (42) and consider (38), one has:

d( Ĉ( qe )) dt = -Ŝ[ ωb e -λvec( q * e ⊗ ( qe -qI ) s ) s ] Ĉ( qe ) (45
) As discussed in Remark 2, (45) can be converted to a equivalent dual-quaternion kinematics:

qe = 1 2 Ê( qe )[ ωb e -λvec( q * e ⊗ ( qe -qI ) s ) s ] = 1 2 qe ⊗ [ ωb e -λvec( q * e ⊗ ( qe -qI ) s ) s ] (46)
Based on these analyses, consider the following Lyapunovlike function candidate:

V o = γ( qe -qI ) • ( qe -qI ) + 1 2 ( ωb e ) s • ( Mb ωb e ) (47)
By the definition of "•" product, it is easy to show V o ( qe = qI , ωb e = 03 ) = 0, and V o > 0 for any qe = qI and ωb e = 03 , so V o is a valid Lyapunov-like candidate. The time differential of V o is,

Vo = 2γ( qe -qI ) • qe + ( ωb e ) s • ( Mb ωb e ) (48)
By [START_REF] Wu | Strapdown inertial navigation system algorithms based on dual quaternions[END_REF]

Considering Lemma 4 and also the property given in [START_REF] Malisoff | Constructions of strict Lyapunov functions[END_REF], then substituting (49) into (48) yields,

Vo = -γλvec( q * e ⊗ ( qe -qI ) s ) • vec( q * e ⊗ ( qe -qI ) s ) + ( ωb e ) s • [-Ŝ( ωb oi )( Mb ωb e ) + Mb ( Ŝ( ωb e ) ωb oi )] (50) Furthermore, -Ŝ( ωb oi )( Mb ωb e ) + Mb ( Ŝ( ωb e ) ωb oi ) = -m b ω b oir × ω b ed -ε[m b ω b oid × ω b ed + ω b oir × (J b ω b er )] + εJ b (ω b er × ω b oir ) + m b (ω b er × ω b oid + ω b ed × ω b oir ) (51
( ωb e ) s • [-Ŝ( ωb oi )( Mb ωb e ) + Mb ( Ŝ( ωb e ) ωb oi )] = m b ω b ed • (ω b er × ω b oid ) -m b ω b er • (ω b oid × ω b ed ) -ω b er • (ω b oir × (J b ω b er )) + ω b er • (J b (ω b er × ω b oir )) = -ω b er • (S(ω b oir )J b ω b er ) -ω b er • (J b S(ω b oir )ω b er ) (52)
Notice S(ω b oir )J b + J b S(ω b oir ) is a skew-symmetric matrix, so that: Recall (46), the vector part of qe can be written as:

-ω b er • (S(ω b oir )J b ω b er ) -ω b er • (J b S(ω b oir )ω b er ) = -ω b er • [(S(ω b oir )J b + J b S(ω b oir ))ω b er ] = 0 (53) Furthermore, Vo = -γλvec( q * e ⊗ ( qe -qI ) s ) • vec( q * e ⊗ ( qe -qI ) s ) = -γλ( 1
ξe = - λ 2 ξe + δ (55)
where

δ = λ 2 ξe - λ 2 (η e I 3×3 + Ŝ( ξe ))vec( q * e ⊗ ( qe -qI ) s ) s + 1 2 (η e I 3×3 + Ŝ( ξe )) ωb e
Eq. (54) shows that δ is a bounded signal consisting of the bounded states ξe and ωb e . Thus (55) is actually an asymptotically stable linear filter with bounded input. Since lim t→∞ ξe (t) = 03 , so lim t→∞ δ(t) = 03 , which results in lim t→∞ ηe ωb e (t) = 03 , and lim t→∞ ηe (t) = ±1 + ε0, so finally we have lim t→∞ ωb e (t) = 03 , and the proof is complete.

IV. OBSERVER-BASED CONTROLLER DESIGN AND LYAPUNOV-LIKE FUNCTION STRICTIFICATION

In this section, based on the proposed observer, an independently designed PD-like controller is introduced to achieve 6-DOF relative tracking objectives, and then the Lyapunov strictification strategy is employed to prove the separation property of the observer-controller construction, and guarantee the almost global asymptotic convergence of tracking errors.

A. Necessity Analysis

The definition of strict Lyapunov function is given firstly. Definition [START_REF] Malisoff | Constructions of strict Lyapunov functions[END_REF]: For a nonautonomous system

ẋ = f (t, x) (56)
A Lyapunov function V (t, x) is a strict Lyapunov function for system (56), if there exists a positive definite function α(•) such that

∂V (t, x) ∂t + ∂V (t, x) ∂x f (t, x) ≤ -α( x ) (57)
By this definition, if one could find a strict Lyapunov function for system (56), then the asymptotic convergence result of x is directly guaranteed.

As mentioned in Sec. II. E, when the relative dual angular velocity ωb bl is available, ωb bt = ωb bl -ωb tl is available, and one can use a PD-like controller given in Proposition 2 to stabilize the closed-loop system. When ωb bl can't be measured, a direct idea is using the estimation value ωb ol to replace it, then the new controller is,

ûb = -k p vec( q * bt ⊗ ( qbt -qI ) s ) -k d ( ωb ol -ωb tl ) s + Mb ( Ĉ( qbt ) ωt ti ) + Ŝ( ωb ti )( Mb ωb ti ) (58)
To analysis the stability of the closed-loop system, consider the following Lyapunov-like function candidate,

V c = k p ( qbt -qI ) • ( qbt -qI ) + 1 2 ( ωb bt ) s • ( Mb ωb bt ) (59)
Recall Lemma 4 and the property given in ( 24), then substitute the dynamics [START_REF]Dual tensors based solutions for rigid body motion parameterization[END_REF] and the control law (58) into the time differential of V c , one has

Vc = -k d ( ωb bt ) • ( ωb ol -ωb tl ) -( ωb bt ) s • [-Ŝ( ωb ti )( Mb ωb bt ) + Mb ( Ŝ( ωb bt ) ωb ti )] (60)
Similar with the analysis given in (51) to (53), it can be proved that

( ωb bt ) s • [-Ŝ( ωb ti )( Mb ωb bt ) + Mb ( Ŝ( ωb bt ) ωb ti )] = 0 (
It can be seen that the cross term k d ωb bt • ωb e is a big obstacle for stability analysis. Furthermore, consider a composite Lyapunov-like function candidate V oc = νV o + V c , where ν ∈ R is a positive constant. By (54) and (63), the time differential of V oc is,

Voc = -νγλς e • ςe -k d ωb bt • ωb bt + k d ωb bt • ωb e ( 64
)
where ςe = vec( q * e ⊗ ( qe -qI ) s ) = 1 2 r b e + εξ e is defined for ease of notation. Eq. (64) shows that the cross term k d ωb bt • ωb e still can't be canceled or dominated because there is no negative term in ωb e , which presents itself as a serious obstacle for further stability analysis.

The strictification strategy will be carried out to solve this problem in next subsection. The objective of strictification is judiciously modifying the Lyapunov-like function candidate V o by introducing a negative term in ωb e in the right part of (64), so one can dominate the cross term k d ωb bt • ωb e . But to achieve this goal, we need to prove the boundedness of ωb bt firstly. What's more, if we can also introduce a negative term of qbt into the time differential of V c , then (64) could directly show the result that both the estimation errors and the tracking errors will converge to zero asymptotically, and consequently the separation property can be established.

B. Boundedness Analysis for the Closed-Loop System

To analyze the boundedness of system states under the controller given in (58), introduce a cross term N c ,

N c = ςbt • ( Mb ωb bt ) (65)
where

ςbt = vec( q * bt ⊗ ( qbt -qI ) s ) = 1 2 r b bt + εξ bt (66)
The differential of N c with respect to time is,

Ṅc =[ 1 2 v b bt + ε(η bd ω b bt + S(ξ bt )ω b bt )]• [m b (v b bt + ω b bt × r b bt ) + εJ b ω b bt ] + ςbt • ( Mb ωb bt ) (67
) Substituting ûb given in (58) into (67) yields,

Ṅc = m b 2 (v b bt + ω b bt × r b bt ) T (v b bt + ω b bt × r b bt ) - m b 2 (ω b bt × r b bt ) T (v b bt + ω b bt × r b bt ) + (η bt ω b bt ) T (J b ω b bt ) + (S(ξ bt )ω b bt ) T (J b ω b bt ) -k p ςbt • ςbt + ςbt • [-Ŝ( ωb bt )( Mb ωb bt ) -Ŝ( ωb ti )( Mb ωb bt ) -Ŝ( ωb bt )( Mb ωb ti ) + Mb ( Ŝ( ωb bt ) ωb ti ) -k d ( ωb bt ) s + k d ( ωb e ) s ] (68)
For further analysis, consider the dual part (attitude part) of controller (58),

τ b = -k p ξ bt -k d ω b bt + k d ω b e + J b C(q bt ) ωb ti + S(ω b ti )(J b ω b ti ) (69) Since ω b
e is guaranteed to be bounded, by straightforward Lyapunov-based analysis, one can readily prove ω b bt is bounded under the control law given in (69) (refer [START_REF] Akella | Partial lyapunov strictification: smooth angular velocity observers for attitude tracking control[END_REF] for details), which means there exists a positive constant c 1 , such that ω b bt ≤ c 1 . Moreover, because ωl li is bounded, and ωt tl is user-designed (which should also be bounded), then ωt ti = ωt tl + ωt li is bounded, and there exists positive constants c 2 and c 3 , such that ω t ti ≤ c 2 and v t ti + ω t ti × r t ti ≤ c 3 . Then recall [START_REF] Bhat | A topological obstruction to continuous global stabilization of rotational motion and the unwinding phenomenon[END_REF], one has

ωb ti ≤ c 2 + ε(c 3 + c 2 r b bt ).
Based on these facts and notice η bt , ξ bt ≤ 1, by applying the Cauchy-Schwarz inequality, we have,

ςbt • [-Ŝ( ωb bt )( Mb ωb bt )] ≤ 1 2 m b c 1 r b bt v b bt + ω b bt × r b bt + J bmax ω b bt 2 (70) ςbt • [-Ŝ( ωb ti )( Mb ωb bt )] ≤ 3 2 m b c 2 r b bt v b bt + ω b bt × r b bt + J bmax c 2 ξ bt ω b bt + m b c 3 ξ bt v b bt + ω b bt × r b bt (71) ςbt • [-Ŝ( ωb bt )( Mb ωb ti ) + Mb ( Ŝ( ωb bt ) ωb ti )] ≤ 3 2 m b c 2 r b bt v b bt + ω b bt × r b bt + 2J bmax c 2 ξ bt ω b bt + m b c 3 ξ bt v b bt + ω b bt × r b bt (72)
and here J bmax is the maximum eigenvalue of J b . So

Ṅc ≤ -k p ςbt • ςbt -k d ςbt • ( ωb e ) s + l 1 r b bt v b bt + ω b bt × r b bt + l 2 ω b bt 2 + l 3 ξ bt v b bt + ω b bt × r b bt + l 4 ξ bt ω b bt + l 5 (v b bt + ω b bt × r b bt ) 2 (73)
where

l 1 = m b c 1 + 3m b c 2 + k d /2, l 2 = 3J bmax , l 3 = 2m b c 3 , l 4 = 3J bmax c 2 + k d , l 5 = m b /2.
Proposition 3: Consider the 6-DOF tracking model given in [START_REF] Condurache | Onboard exact solution to the full-body relative orbital motion problem[END_REF][START_REF]Dual tensors based solutions for rigid body motion parameterization[END_REF], and the control input designed as (58), then system states qbt , ωb bt , ωb bi and ωb oi are uniformly bounded. Proof : Employ the following augmented Lyapunov-like function candidate,

V c1 =ρV c + N c = ρk p ( qbt -qI ) • ( qbt -qI ) + ρ 2 ( ωb bt ) s • ( Mb ωb bt ) + ςbt • ( Mb ωb bt ) (74)
where ρ ∈ R is a constant. By the definition of qbt , ςbt , and ωb bt , and then applying the Binet-Cauchy identity of cross product along with the Cauchy-Schwarz inequality, one can get,

V c1 ≥ρk p ((η bt -1) 2 + ξ bt 2 + 1 4 r bt 2 ) + ρm b 2 v b bt + ω b bt × r b bt 2 + ρJ bmin 2 ω b bt 2 - m b 4 r b bt 2 - m b 4 v b bt + ω b bt × r b bt 2 - J bmax 2 ξ bt 2 - J bmax 2 ω b bt 2 ≥ρk p (η bt -1) 2 + (ρk p - J bmax 2 ) ξ bt 2 + ( ρk p 4 - m b 4 ) r b bt 2 + ( ρJ bmin 2 - J bmax 2 ) ω b bt 2 + ( ρm b 2 - m b 4 ) v b bt + ω b bt × r b bt 2 (75) Choosing ρ ≥ max{ J bmax k p , 2m b k p , 2J bmax J bmin , 1}
one has,

V c1 ≥ρk p (η bt -1) 2 + J bmax 2 ξ bt 2 + m b 4 r b bt 2 + J bmax 2 ω b bt 2 + m b 4 v b bt + ω b bt × r b bt 2 (76)
Eq. (76) guarantees V c1 ≥ 0, and V c1 = 0 only when qbt = qI and ωb bt = 03 , so V c1 is a valid Lyapunov function. Substituting (63) and (73) into the time differential of V c1 , one can get,

Vc1 ≤ -κ 1 ( ξ bt 2 + 1 4 r b bt 2 ) - ρk d 2 ( ω b bt 2 + v b bt + ω b bt × r b bt 2 ) + κ 2 ωb e • ωb e + l 1 r b bt v b bt + ω b bt × r b bt + l 2 ω b bt 2 + l 3 ξ bt v b bt + ω b bt × r b bt + l 4 ξ bt ω b bt + l 5 v b bt + ω b bt × r b bt 2 (77)
where

κ 1 = k p -k d /2 and κ 2 = k d /(2 ) + ρk d /2
, and here is a positive constant. To guarantee κ 1 , κ 2 > 0, choose = k p /k d . Utilize the Cauchy-Schwarz inequality again and complete squares, we have

Vc1 ≤ - κ 1 2 ξ bt 2 - κ 1 8 r b bt 2 -( ρk d 2 - 4l 2 1 κ 1 - l 2 3 κ 1 -l 5 ) v b bt + ω b bt × r b bt 2 -( ρk d 2 - l 2 4 κ 1 -l 2 ) ω b bt 2 + κ 2 ωb e • ωb e -( 4l 2 1 κ 1 v b bt + ω b bt × r b bt 2 -... l 1 r b bt v b bt + ω b bt × r b bt + κ 1 16 r b bt 2 ) -( l 2 3 κ 1 v b bt + ω b bt × r b bt 2 -... l 3 ξ bt v b bt + ω b bt × r b bt + κ 1 4 ξ bt 2 ) -( l 2 4 κ 1 ω b bt 2 -l 4 ξ bt ω b bt + κ 1 4 ξ bt 2 ) ≤ - κ 1 2 ξ bt 2 - κ 1 8 r b bt 2 -( ρk d 2 - 4l 2 1 κ 1 - l 2 3 κ 1 -l 5 ) v b bt + ω b bt × r b bt 2 -( ρk d 2 - l 2 4 κ 1 -l 2 ) ω b bt 2 + κ 2 ωb e • ωb e (78)
If we further modify the value of ρ to

ρ = max{ J max k p , 2m b k p , 2J bmax J bmin , 1, 16l 2 1 κ 1 k d + 4l 2 3 κ 1 k d + 4l 5 k d , ... 4l 2 4 κ 1 k d + 4l 2 k d }
we can further get,

Vc1 ≤ - κ 1 2 ξ bt 2 - κ 1 8 r b bt 2 - ρk d 4 ω b bt 2 - ρk d 4 v b bt + ω b bt × r b bt 2 + κ 2 ωb e • ωb e (79)
An upper bound of V c1 can be obtained from (74),

V c1 ≤ρk p + J bmax 2 + ρk p ξ bt 2 + 5ρk p + m b 4 r b bt 2 + (2ρ + 1)m b 4 v b bt + ω b bt × r b bt 2 + ρJ bmin + J bmax 2 ω b bt 2 (80)
Then since it has been proved in Theorem 1 that ωb e is bounded, there exists positive constant p, such that ωb e • ωb e ≤ p. And by (79) and (80), we can get,

Vc1 ≤ -χV c1 + σ (81)
where

χ = min{ κ 1 2ρk p , κ 1 2(5ρk p + m b ) , ρk d 2(ρJ bmin + J bmax ) , ... ρk d (2ρ + 1)m b }, σ = χ(ρk p + J bmax 2 ) + κ 2 p
So that V c1 is uniformly bounded, which guarantees qbt and ωb bt are uniformly bounded, since ωt ti is bounded, so ωb bi is bounded. Furthermore, because ωb e is bounded, ωb oi is also bounded. The proof is complete.

C. Toward a Strict Observer Lyapunov-Like Function

To conduct the strictification process, consider a new cross term,

N o = -2(η e ξe ) • ωb e (82)
Notice Ŝ( ξe ) ξe = 03 and ηe ηe + ξT e ξe = 1 + ε0, the time differential of N o can be written as: where Proposition 4: For the observer construction given in (39) and (40), the following augmented function V o1 ,

Ṅo = -ωb e •
α 1 = b 7r + 2J bmax J bmin b 4r (b 5r + b 6r ) + 4b 4d b 5d + 2b 4r b 5r , α 2 = b 7d + 2m b J bmin b 4r (b 5d + b 6d ) + 2b 4d b 6r + 2b 4d b 5r , α 3 = 2b 4r b 5d , α 4 = 1 2 b 7r + b 4r (b 5r + b 6r ), β 1 = 2γ J bmin b 4r , β 2 =
V o1 = µV o + N o ( 97
)
is a partially strict Lyapunov-like function for the estimation error dynamics. Where µ > 0 is a positive constant, satisfying,

µ = max{ 2 γ , 4 J bmin , 8 m b , 4(2α 2 3 + 2α 2 4 + β 2 + 2β 3 ) γλ , ... 2α 2 1 + 2α 2 2 + 2β 1 + β 2 γλ }
(98) Proof : By the definition of µ, one has Utilize the Cauchy-Schwarz inequality and complete squares, one has,

V o1 ≥(µγ -1)( q e -q I 2 + 1 4 q e ⊗ r b e 2 ) + ( J bmin 2 µ -1) ω B er 2 + ( m b 2 µ -2) ω B ed 2 ≥0 (
Vo1 ≤ - µγλ 8 ( r b e 2 + 1 2 ξ e 2 ) - 1 2 ( ω b er 2 + ω b ed 2 ) -( µγλ 8 -α 2 3 -α 2 4 - 1 2 β 2 -β 3 ) r b e 2 -( µγλ 2 -α 2 1 -α 2 2 -β 1 - 1 2 β 2 ) ξ e 2 -(α 2 1 ξ e 2 -α 1 ξ e ω b er + 1 4 ω b er ) -(α 2 2 ξ e 2 -α 2 ξ e ω b ed + 1 4 ω b ed ) -(α 2 3 r b e 2 -α 3 r b e ω b er + 1 4 ω b er ) -(α 2 4 r b e 2 -α 4 r b e ω b ed + 1 4 ω b ed ) ≤ - µγλ 8 ( r b e 2 + 4 ξ e 2 ) - 1 2 ( ω b er 2 + ω b ed 2 ) ≤ - µγλ 2 ςe • ςe - 1 2 ωb e • ωb e (
101) So V o1 is now a partially strict Lyapunov-like function, the asymptotic convergence of ξe and ωb e can be directly guaranteed by (101), through the use of the corollary of Barbalat's lemma. The proof is complete.

V. PROOF OF THE SEPARATION PROPERTY

After the strictification process, another important contribution of this paper is organized in Theorem 2.

Theorem 2: Consider the dual-quaternion based relative tracking dynamics in [START_REF] Condurache | Onboard exact solution to the full-body relative orbital motion problem[END_REF][START_REF]Dual tensors based solutions for rigid body motion parameterization[END_REF], with the observer proposed in (39-40), and the total control input designed in (58), then the tracking and estimation errors satisfy lim t→∞ [ ξe (t), ωb e (t), ξbt (t), ωb bt (t)] = 03 . Proof : Define the following Lyapunov-like function candidate, which is the combination of an observer part and a controller part,

V oc1 = νV o1 observer part + V c1 controller part (102)
where ν = 4κ 2 is a positive constant.

According to (79) and ( 101), the time derivative of V oc1 satisfies

Voc1 ≤ - νµγλ 2 ςe • ςe - ν 2 ωb e • ωb e - κ 1 2 ξ bt 2 - κ 1 8 r b bt 2 - ρk d 4 ω b bt 2 - ρk d 4 v b bt + ω b bt × r b bt 2 + κ 2 ωb e • ωb e ≤ - νµγλ 2 ςe • ςe - ν 4 ωb e • ωb e - κ 1 2 ςbt • ςbt - ρk d 4 ωb bt • ωb bt (103) Because V oc1 ≥ 0 and Voc1 ≤ 0, so V oc1 ∈ L ∞ ,
The proof is completed. Remark 4: It's noteworthy that, except the controller and observer parameters k p , k d , λ and γ, all the other introduced parameters (like ν, ρ, χ, , and so on) are employed purely for analysis. Which means these parameters put no restriction on the controller or the observer, and won't influence the performance of the closed-loop system.

Remark 5: The logic procedure in this section is crucial and rigorously ordered. Firstly, Theorem 1 guarantees the boundedness of the estimation error ωb e . And then, as shown in Proposition 3, this result is combined with a strictification-like process, and enables ωb bi and ωb oi to be uniformly bounded. Subsequently, in Proposition 4, the boundedness of ωb bi and ωb oi are utilized to help us get a partially strict observer Lyapunov-like function, and finally the separation property is proved in Theorem 2.

VI. NUMERICAL SIMULATIONS

In this section, based on the observer in given (39-40) and the controller presented in (58), typical simulation examples are illustrated and analyzed. The performance of the proposed observer-controller construction is compared with the PDlike full-state-feedback controller and also the passivity-based output-feedback method introduced in Ref. [START_REF] Filipe | Pose tracking without linear-and angular-velocity feedback using dual quaternions[END_REF]. And then, the effectiveness of the proposed approach is further demonstrated by Monte Carlo simulations.

To optimize the comparison results, a similar simulation scenario with Ref. [START_REF] Filipe | Pose tracking without linear-and angular-velocity feedback using dual quaternions[END_REF] is employed, in which the follower spacecraft is required to track an elliptical motion around the leader while the X B -axis of it precisely pointing to the leader. In the simulation, the inertia frame I is chosen as the Earth-centered inertial (ECI) frame, and the leader spacecraft is running on a Molniya orbit, the orbital elements of the leader are given in Table I. The body-fixed frame of the leader (the frame L) is always coincide with it's Local-Vertical-Local Horizontal (LVLH) frame (the definition of the LVLH frame is: xaxis is along the radius vector of the leader, z-axis is along the angular momentum vector, and y-axis completes the right-handed system). The target frame T is designed as: q tl (0) = [0, 0, 0, 1] T , r l tl = [a e , 0, 0] T m, ω l tl = [0, 0, a e b e ω 0 /(a 2 e cos 2 (ω 0 t) + b 2 e sin 2 (ω 0 t))] T rad/s, v l tl = [-a e ω 0 sin(ω 0 t), b e ω 0 cos(ω 0 t), 0] T m/s, a e = 15m, b e = 25m. And ω 0 denotes the (unperturbed) mean orbital angular velocity of the leader, so ω 0 = µ/a 3 rad/s, where µ = 398600.4405 kg 3 /m 2 is the geocentric gravitational constant, and here a is the semimajor axis of the leader's orbit. Under this design, the target frame T conducts an elliptical motion around the leader, while it x-axis always pointing to the leader.

As mentioned in the paper, the control objective is to superimpose the body-fixed frame of the follower onto the target frame. The nominal mass and the inertia of the follower are assumed to be [START_REF] Dong | Dual-quaternion-based fault-tolerant control for spacecraft tracking with finite-time convergence[END_REF]

m = 400 kg, J b = 55 1.5 -3 1.5 65 -0.5 -3 -0.5 58 kg • m 2
To test the robustness of the proposed method, the gravity gradient torque τ gg and the J2 perturbation force f J2 are employed in the simulation, given by Ref. [START_REF] Filipe | Adaptive position and attitude-tracking controller for satellite proximity operations using dual quaternions[END_REF],

τ gg = 3µ r b bi × (J b r b bi ) r b bi 5 f J2 = m b a b J2 , a i J2 = - 3µJ 2 R 2 e 2 r i bi 4      (1 -5( z i bi r i bi ) 2 )
x i bi

r i bi (1 -5( z i bi r i bi ) 2 ) y i bi r i bi (3 -5( z i bi r i bi ) 2 ) z i bi r i bi     
where x i bi , y i bi , z i bi are the corresponding components of r i bi , J 2 = 0.0010826267, and R e = 6378.137 km is the mean equatorial radius of the Earth.

Furthermore, during the proximity process, the mass ejected by the actuators will introduce changes to the follower's nominal mass and inertia, and lead to parameter uncertainties, this issue is also considered in the simulation. Assume the actuators of the follower are thrusters, and they are able to generate variable thrust amplitudes (for example, the coldgas thrusters [START_REF] Stone | Fast variable-amplitude cold gas thruster[END_REF]). The follower has a total of 12 thrusters to implement 6-DOF control, and the configuration of them is given in Fig. 2. The specific impulse of thrusters is assumed to be [START_REF] Stone | Fast variable-amplitude cold gas thruster[END_REF]: SI = 9.80665×70 N•s/kg, and the maximum output of thrusters is 30N. So the mass change induced by every thruster is ∆m j = ( t 0 |F j (t)|dt)/SI, where F j (•) is the output of the corresponding thruster j, and j = 1, 2, ..., 12. A reasonable assumption is that every ∆m j can be regarded as a mass point removed from the position of the corresponding thruster, and then we can further calculate the change of inertia.

The initial conditions are set to: q bl (0) = [0, 0.8, 0.6, 0] T , r l bl (0) = [10, -20, -15] T m, v l bl (0) = [1, 0.5, 0.4] T m/s, ω l bl (0) = [0.05, -0.1, 0.2] T rad/s, qol (0) = qbl (0) and ωo ol (0) = [0, 0, 0, 0, 0, 0, 0, 0] T rad/s. Parameters of the proposed method are k p = 1, k d = 20, λ = 0.5 and γ = 10. For the controller given in Ref. [START_REF] Filipe | Pose tracking without linear-and angular-velocity feedback using dual quaternions[END_REF], the linear time-invariant subsystem is set to A = I 8×8 , B = 5I 8×8 and C = k d I 8×8 . Follow the aforementioned conditions, this subsection demonstrates the simulation results when there is no measurement noises.

Figure 3 shows the estimation errors of angular and linear velocities, it can be seen that the designed observer has a good performance, all estimation errors can converge within 100s, with smooth trajectories and high precisions. The tracking errors of relative positions and quaternions are shown in Fig. 4 and Fig. 5, respectively. And Figure 6 further demonstrates the tracking errors (norms) of linear and angular velocities. These results show that all the three methods can achieve control objectives. For the two velocity-free methods, one can see that the observer-based method introduced in this paper has faster convergence processes and higher precisions with respect to the passivity-based output feedback controller given in Ref. [START_REF] Filipe | Pose tracking without linear-and angular-velocity feedback using dual quaternions[END_REF]. The reason why the controller in Ref. [START_REF] Filipe | Pose tracking without linear-and angular-velocity feedback using dual quaternions[END_REF] needs a longer convergence time maybe become this method is based on the output of a special linear time-invariant system which can be regarded as a low-pass filter, this structure could result in extended convergence processes. Compared with the full-state feedback case, the proposed method in this paper leads to small fluctuations due to the indispensable procedure to eliminate estimation errors. The changes of masses and principle inertias are illustrated in Fig. 7, compared to the full-state feedback controller, the extra mass consumption of the proposed method also comes from the estimation process. The steady-state errors of this simulation case are summarized in Table II. Overall, the performance of the proposed observercontroller method is just slight worse than the nominal fullstate-feedback PD-like controller. It is shown that under the proposed method, the follower could track the predetermined elliptical trajectory around the leader, while it's X B -axis always pointing to the leader. To sum up, the method proposed in this paper successfully completes the estimation and tracking task with reasonable convergence time and good precisions. In this subsection, to further illustrate the robustness of the proposed observer-controller construction, measurement noises of relative quaternions and positions are considered. Given the definition of quaternions [START_REF] Schaub | Analytical mechanics of space systems[END_REF]:

ξ bl = sin( θ bl 2 )e bl , η bl = cos( θ bl 2 ) (106)
where θ bl and e bl are the eigenangle and the eigenaxis associated with q bl , respectively. Measurement noises are generated through the random perturbations of e bl within a spherical cone of a prescribed cone half-angle and uniform distribution centered around the true eigenaxis. The cone half-angle is specified as 0.05 deg. For relative position, zero-mean additive white Gaussian noises with standard deviation of 0.01 m are considered as process noises. After adding these noises to the model, the simulation case given in Sec. VI. A is repeated. The time responses of ξ bt and r bt are given in Figs. 9 and 10, respectively, and semi-logarithmic scales are employed in these two figures to clearly show the performance differences among different methods. One can see that the overall performance of all the three control methods suffer from the presence of measurement noises, and tracking errors can only converge to small residual sets around the origin. The proposed method has an almost same steady-state performance with the full-state PD-like controller, which means it is not adversely affected any worse than PD-like controller under noisy measurements. The estimation errors of the observer are shown in Fig. 11, though the steady estimate errors are also influenced by noisy measurements, but the proposed observer can still render the estimation errors rapidly converge to a small bounded region.

C. Monte Carlo Simulations

Finally, Monte Carlo simulations are employed to further the performance and robustness of the proposed method under external disturbances and parameter uncertainties with variable initial conditions and control parameters.

Table III presents the randomized parameters and initial conditions, and also their random ranges. Notice that the follower can only know the nominal mass and inertia values, and can't get any information about external disturbances. The process of a single simulation is that: First, by Monte Carlo techniques, randomly generate a set of new simulation parameters and initial conditions; then, under the proposed method, the simulation case given in Sec. VI. A is repeated with those new generated random values. Finally, the steady tracking errors and estimation errors are recorded. Under all these conditions, 1000 Monte Carlo runs are conducted. The distributions of tracking errors and estimation errors are given in Figs. 12-14, which shows that the overwhelming majority of simulations have good performance, even in the presence of unknown external disturbances and parameter uncertainties,

J b , kg•m 2 55 ± 10, 1.5 ± 1, -3 ± 1 1.5 ± 1, 65 ± 5, -0.5 ± 0.5 -3 ± 1, -0.5 ± 0.5, 50 ± 6 kp, - [0.25, 2] k d , - [5, 25] λ, - [0.2, 2] γ, - [2, 15] q bl (0), - H r l bl (0), m [-200, -200, -200] T to [200, 200, 200] T ω l bl (0), rad/s [-0.2, -0.2, -0.2] T to [0.2, 0.2, 0.2] T r l bl (0), m/s [-2, -2, -2] T to [2, 2, 2] T
while the tracking and estimation errors of the worst cases are less than 2 × 10 -2 m ( r b bt ), 4 × 10 -4 ( ξ bt ), 4 × 10 -4 m/s ( v b bt ), 3 × 10 -4 rad/s ( ω b bt ), 3 × 10 -4 m/s ( v b e ), and 2 × 10 -4 rad/s ( ω b e ). By further analyzing the simulation conditions of these worst cases, it is found out that their relatively worse performance are caused by the improper observer and controller parameters. For example, for the point on the top of Fig. 12 (the corresponding simulation has the worst position tracking error), the k p of this simulation case is 0.2723, which is too small to guarantee diminutive tracking errors. Simulation results suggest that k p ∈ [0.5, 2], k d ∈ [5k p , 20k p ], λ ∈ [0.2, 1] and γ ∈ [5λ, 20λ] would be reasonable choices, while the final decisions should also base on the initial conditions of the corresponding simulation cases. || (m/s) Fig. 14: Distributions of estimation errors matrices is subsequently analyzed. Based on these mathematical foundations, a smooth observer is proposed to estimate the dual relative angular velocity, and the estimation errors are guaranteed to be global asymptotic convergence irrespective of control inputs. Subsequently, the observer is combined with a separately designed PD-like controller. A partial Lyapunov strictification process is employed to transform nonstrict Lyapunov functions into strict ones whose derivatives contain additional non-positive terms, then the separation property is proved and the almost global asymptotic convergence of the closed-loop system is guaranteed accordingly. Numerical simulations for a prototypical 6-DOF spacecraft pose tracking mission are employed, which shows that the proposed observer-controller construction has an good performance even in the presence of external disturbances, parameter uncertainties and noisy measurements.

By iteratively using the above property, one can obtain:

â × ( b × ĉ) = b(â • ĉ) -ĉ(â • b) (115)
The Lemma 2 is proved. By Eqs. ( 10) and ( 13), and consider the properties of the cross product of real vectors, we have: The Lemma 3 is proved. By ( 7), ( 12) and ( 13 Finally, to analyze the kinematics of the dual transformation matrix, consider a fixed dual vector nx in frame X , for ease of notation, using Ĉyx denote Ĉ( qyx ), and then we have,

d
The proof is complete.

Fig. 1 :

 1 Fig. 1: Reference frames

  ) where ω b oir and ω b oid are the real part and dual part of ωb oi , respectively, while ω b er and ω b ed are the real part and dual part of ωb e , respectively. Then

  Because V o (t) ≥ 0 and Vo (t) ≤ 0, so V o ∈ L ∞ , by the definition of V o , we have qe , ωb e ∈ L ∞ , which implies ξ e , r b e , v b e , ω b e ∈ L ∞ . Eq. (54) also shows that ∞ 0 Vo (σ)dσ exists, and ξ e , r b e ∈ L 2 . Furthermore, by (46), one has ξe , ṙb e ∈ L ∞ . Actually to the corollary of Barbalat's lemma, one can guarantee lim t→∞ ξ e (t) = 0 3 and lim t→∞ r b e (t) = 0 3 , and accordingly, lim t→∞ ξe (t) = 0 3 .

(b 7r ξ e ω b er + 1 2 b 7r r b e ω b ed + b 7d ξ e ω bbmin b 4r ξ e 2 + γ m b b 4d ξ e r b e + γ 2m b b 4r r b e 2 + α 3 r b e ω b er + α 4 r b e ω b ed + β 1 ξ e 2 + β 2 ξ e r b e + β 3 r b e 2 (

 222 b7 (ς e ) s ) • ωb e = (ς e ) s ) • [ ˆ 2 ( ωb e ( ˆ 1 b5 ))] = 2m b J bmin b 4r b 5d ξ e ω b ed + 2J bmax J bmin b 4r b 5r ξ e ω b er + 2b 4d b 5d ξ e ω b er + b 4r b 5d r b e (ς e ) s ) • [ ˆ 2 ( b6 ( ˆ 1 ωb e ))] = 2m b J bmin b 4r b 6d ξ e ω b ed + 2J bmax J bmin b 4r b 6r ξ e ω b er + 2b 4d b 6r ξ e ω b ed + b 4r b 6r r b e (ς e ) s ) • [ b6 ωb e ] = 2b 4r b 6r ξ e ω b er + 2b 4d b 6r ξ e ω b ed + 2b 4d b 6d ξ e ω b er + b 4r b 6r r b e (ς e ) s ) • [γ ˆ 2 ςe ] = 2γ J (95) So Ṅo ≤ -ωb e • ωb e + α 1 ξ e ω b er + α 2 ξ e ω b ed

γ m b b 4d and β 3

 3 = γ 2m b b 4r . The function of cross term N o is shown in the following proposition.

r b e 2 + ξ e 2 ) -ω b er 2 -ω b ed 2 ++ α 4 r b e ω b ed + β 1 ξ e 2 + β 2 ξ e r b e + β 3 r b e 2 (

 222222 99) So V o1 is a valid Lyapunov-like function candidate. According to (54) and (96), the differential of V o1 with respect to time satisfies, Vo1 = µ Vo + Ṅo ≤ -µγλ( 1 4 α 1 ξ e ω b er + α 2 ξ e ω b ed + α 3 r b e ω b er

Fig. 2 :

 2 Fig. 2: Configuration of thrusters

Fig. 3 :Fig. 4 :Fig. 5 :Fig. 6 :Fig. 7 :

 34567 Fig. 3: Estimation errors of dual angular velocites

Fig. 8 :

 8 Fig. 8: 3D illustration under the proposed method

Fig. 9 :

 9 Fig. 9: Time response of r b bt under noisy measurements

Fig. 10 :Fig. 11 :

 1011 Fig. 10: Time response of ξ bt under noisy measurements

Fig. 12 :Fig. 13 :

 1213 Fig. 12: Error distributions of relative dual quaternions

For

  â, b, ĉ ∈ R3 , from Eq. (13), one has:â × ( b × ĉ) = a r × (b r × c r ) + ε[a r × (b r × c d ) + ... a r × (b d × c r ) + a d × (b r × c r )] (107) Similarly: b × (ĉ × â) =b r × (c r × a r ) + ε[b r × (c r × a d ) + ... b r × (c d × a r ) + b d × (c r × a r )] (108) and ĉ × (â × b) =c r × (a r × b r ) + ε[c r × (a r × b d ) + ... c r × (a d × b r ) + c d × (a r × b r )](109) Consider the Jacobi identity of real 3-dimensional vectors:a * × (b × c • ) + b × (c • × a * ) + c • × (a * × b ) = 0 3 (110)where * , , • denote r or d, so by iteratively using this identity, finally one can get:â × ( b × ĉ) + b × (ĉ × â) + ĉ × (â × b) = 03 (111)The Lemma 1 is proved.By Eqs. (6) and (10), one has:b(â • ĉ) =b r (a r • c r ) + ε[b r (a r • c d ) + ... b d (a r • c r ) + b r (a d • c r )](112)andĉ(â• b) = c r (a r •b r )+ε[c d (a r •b r )+c r (a r •b d )+c r (a d •b r )](113) The real 3-dimensional vectors follows the property: a * × (b × c • ) = b (a * • c • ) -c • (a * • b )

  (â × b) • â =(a r × b r ) • a r + ε[(a r × b r ) • a d + ... (a r × b d ) • a r + (a d × b r ) • a r )] =0 + ε[(a r × b r ) • a d + (a d × b r ) • a r ] = 03 (116) Furthermore: â • (â × b) = (â × b) • â = 03(117)

  ), âs • (â × b) =a T d (a r × b r ) + a T r (a d × b r ) + ε0 3 . (10),[START_REF] Seo | Separation property for the rigid-body attitude tracking control problem[END_REF] and[START_REF] Filipe | Pose tracking without linear-and angular-velocity feedback using dual quaternions[END_REF], and also Lemma 3, one can get,q * yx ⊗ [ 0, (â x ) T ] T ⊗ qyx = [η yx , -ξT yx ] T ⊗ [ 0, (â x ) T ] T ⊗ [η yx , ξT yx ] T = [ 0, (( ξT yx âx ) ξyxξyx × ( ξyx × âx ) + ηyx ηyx âx -... 2η yx ( ξyx × âx ) + 2 ξyx × ( ξyx × âx )) T ] T(119) By Lemma 2, and rewrite (119) to the 3-dimensional form, one hasvec( q * yx ⊗ [ 0, (â x ) T ] T ⊗ qyx ) = [(η yx ηyx + ξT yx ξyx )I 3×3 -2η yx Ŝ( ξyx ) + 2 Ŝ( ξyx ) Ŝ( ξyx )]â x (120) Since η yx η yx + ξ T yx ξ yx = 1,and by (19), it can be proved that ηyx ηyx + ξT yx ξyx = qT yx qyx = 1 + ε0(121)Substituting (121) into (120) yieldsq * yx ⊗ âx ⊗ qyx = Ĉ( qyx )â x (122)and the definition of Ĉ( qyx ) isĈ( qyx ) = I 3×3 -2η yx Ŝ( ξyx ) + 2 Ŝ( ξyx ) Ŝ( ξyx ) (123)Properties 1 and 2 in Proposition 1 can be readily proved. By (123), we haveĈ( q * yx ) = I 3×3 + 2η yx Ŝ( ξyx ) + 2 Ŝ( ξyx ) Ŝ( ξyx ) = I T 3×3 -2η yx ŜT ( ξyx ) + 2 ŜT ( ξyx ) ŜT ( ξyx )= ĈT ( qyx ) (124) Eq. (124) shows that, the converse transformation, from Y to X , can be described by ĈT ( qyx ), and so ĈT ( qyx ) Ĉ( qyx ) = Ĉ( qyx ) ĈT ( qyx ) = I 3×3 + ε0 3×3 (125) Furthermore, by[START_REF] Bhat | A topological obstruction to continuous global stabilization of rotational motion and the unwinding phenomenon[END_REF],Ĉ( qyx )(â x 1 × âx 2 ) -[ Ĉ( qyx )â x 1 ] × [ Ĉ( qyx )â x 2 ] = 0 3 + ε(a y 1r × a y 2r ) ×r y yx -ε(a y 1r × r y yx ) × a y 2r -ε(a y 1r ) × (a y 2r × y yx ) = 03 (126) which guarantees the Property 3.

  Partial Lyapunov Strictification: Dual Quaternion based Observer for 6-DOF Tracking Control Hongyang Dong, Qinglei Hu, Member, IEEE, Maruthi R. Akella, Senior Member, IEEE, and Frédéric Mazenc

  III. DUAL ANGULAR VELOCITY OBSERVER DEVELOPMENT Consider an observer-related reference frame O, in which the estimates for ωb bl are generated. Let qol = [η ol , ξT ol ] T ∈ Ĥ and ωo ol ∈ R3 denote the estimated values of qbl and ωb bl , respectively. Furthermore, define qe = [η e , ξT e ] T and ωb e as the estimation error states:

  Since ξ e ≤ 1, so ξe ≤ (ς e ) s . And because ωb e , ξe , ηe , ςe , ωb bi and ωb oi are all proven to be bounded, so there exists positive constant dual numbers bj = b jr + εb jd , j = 1, 2, ..., 6, satisfy ξe ≤ b1 , ωb

	J bmin m b , where J bmax and J bmin are the maximum and minimum d dε + ε 1 eigenvalues of J b , respectively. Then according to (49), a upper bound of ωB e can be obtained,
	ωb e = M -1 b [-Ŝ( ωb e )( Mb	ωb bi ) -Ŝ( ωb oi )( Mb	ωb e ) + ...
	Mb ( Ŝ( ωb e ) ωb oi ) -γ ςe ]
	≤ ˆ 2 ( ωb e ( ˆ 1 b5 )) + ˆ 2 ( b6 ( ˆ 1	ωb e ))
	+ b6	ωb e + γ ˆ 2 ςe
				(89)
	So by (84)-(89), one can get:
	Ṅo ≤ -ωb e • ωb e + ( b7 (ς e ) s ) • ωb e
	+ (2 b4 (ς e ) s ) • [ ˆ 2 ( b5 ˆ 1	ωb e ) + ˆ 2 ( ωb e ( ˆ 1 b6 ))
	+ b6	ωb e + γ ˆ 2 ςe ]
				(90)
	where b7 = 2 b1 b2 + λ b2

ωb e + (( ξT e ξe ) ωb e ) • ωb e + [( ξT e ( ωb e -λ(ς e ) s )) ξe ] • ωb e + (λη e ηe (ς e ) s ) • ωb e -(η e Ŝ( ξe ) ωb e ) • ωb e -2(η e ξe ) • ωb e (83) e ≤ b2 , (ς e ) s ≤ b3 , ηe ≤ b4 , ωb bi < b5 and ωb oi < b6 . Then the following inequations can be established: (( ξT e ξe ) ωb e ) • ωb e ≤ ( b1 b2 (ς e ) s ) • ωb e (84) [( ξT e ( ωb e -λ(ς e ) s )) ξe ] • ωb e ≤ ( b1 b2 (ς e ) s ) • ωb e + (λ b2 1 (ς e ) s ) • ωb e (85) (λη e ηe (ς e ) s ) • ωb e ≤ (λ b2 4 (ς e ) s ) • ωb e (86) -(η e Ŝ( ξe ) ωb e ) • ωb e ≤ ( b2 b4 (ς e ) s ) • ωb e (87) -2(η e ξe ) • ωb e ≤ (2 b4 (ς e ) s ) • ωb e (88) Further define ˆ 1 = m b d dε + εJ bmax and ˆ 2 = 1 1 + λ b2 4 + b2 b4 . Using ω b er and ω b ed denote the real part and dual part of ωb e , respectively, then we can further obtain,

  . Furthermore, recall[START_REF] Schaub | Analytical mechanics of space systems[END_REF],[START_REF] Wu | Strapdown inertial navigation system algorithms based on dual quaternions[END_REF], (46) and (49), we have ςe , ωb e , ςbt , ωb bt ∈ L ∞ . By the corollary of Barbalat's Lemma, these results guarantee

	lim t→∞	[ς e (t), ωb e (t), ςbt (t), ωb bt (t)] = 03	(104)
	And it can be readily proved that ςe (t) = 03 and ςbt (t) = 03
	guarantee ξe (t) = 03 and ξbt (t) = 03 , respectively. So finally
	we get
	lim t→∞	[ ξe (t), ωb e (t), ξbt (t), ωb bt (t)] = 03

and by the definition of V oc1 , we have ςe , ωb e , ςbt , ωb bt ∈ L ∞ . Additionally, since ∞ 0 Voc1 (σ)dσ exists, ςe , ωb e , ςbt , ωb bt ∈ L 2

TABLE I :

 I Orbital Elements of the Leader Spacecraft

	Orbital Elements	Quantity
	Semimajor axis, km	26553.937
	Eccentricity, -	0.729677
	Inclination, deg	63.4
	Argument of perigee, deg	-90
	RAAN, deg	0
	True anomaly, deg	180

TABLE II :

 II Steady Tracking and Estimation Errors

		PD-like	Observer-based	Output feedback
	ξ bt , -	2.5 × 10 -6	5.0 × 10 -6	1.2 × 10 -3
	r b bt , m	1.1 × 10 -4	1.8 × 10 -4	9 × 10 -3
	ω b bt , rad/s	1.2 × 10 -7	5.2 × 10 -6	3.6 × 10 -4
	v b bt , m/s	4.7 × 10 -6	1.2 × 10 -5	1.2 × 10 -4
	ω b e , rad/s	-	1.1 × 10 -7	-
	v b e , m/s	-	1.3 × 10 -6	-

TABLE III :

 III Randomized Parameters and Initial Values

	Parameters	Ranges
	m b , kg	[350, 450]

Due to topological obstructions (noncontractible) of the configuration space of attitude motion SO(3), it is impossible for any continuous statefeedback controller to render global asymptotic stability[START_REF] Bhat | A topological obstruction to continuous global stabilization of rotational motion and the unwinding phenomenon[END_REF],[START_REF] Wen | The attitude control problem[END_REF], so the notion "almost global" is adopted here to imply the stability of attitude motion over an open and dense set in SO(3).


O Spitz

J Wu

M Carras

C W Wong

F Grillot

Controlling the Likelihood of Extreme Pulses in a Quantum Cascade Laser with Optical Feedback and Bias Perturbation

Keywords: OCIS codes: (140, 5965) Semiconductor lasers, quantum cascade; (140, 3070) Infrared and far-infrared lasers

We experimentally generate controllable extreme pulses in a mid-infrared quantum cascade laser with external optical injection and square wave perturbation.

Introduction

The term rogue waves describes random isolated events with amplitudes well above that of neighboring ones, and which occur more often than expected from the distribution of lower amplitude events [START_REF] Onorato | Rogue waves and their generating mechanisms in different physical contexts[END_REF]. Optical rogue waves were first demonstrated experimentally twelve years ago in the context of super-continuum generation in optical fibers [START_REF] Solli | Optical rogue waves[END_REF] and have since been observed in a wide variety of configurations such as semiconductor lasers [START_REF] Bonatto | Deterministic optical rogue waves[END_REF]. In order to detect rogue waves from experimental data, the probability density function (PDF) of an appropriate attribute of the data needs to be analyzed, and crest or wave heights are commonly used. Once extracted from the data, heights are compared to a threshold value above which events can be considered as extreme. This criterion is often defined in terms of the standard deviation of the whole time series frame : any event whose height is higher than the mean value (µ) plus height times the standard deviation (σ) is considered as an extreme event (μ ± 8σ) [START_REF] Bonatto | Deterministic optical rogue waves[END_REF]. We apply this criterion to a mid-infrared quantum cascade laser (QCL) under conventional optical feedback (COF) and asymmetric square low modulation to experimentally report the first extreme pulses in such lasers. We further unveil a method to produce controllable extreme pulses which is highly relevant for sensing applications [START_REF] Jin | Generation of extreme pulses on demand in semiconductor lasers with optical injection[END_REF]. Midinfrared QCLs are intersubband semiconductor lasers and their versatility make them candidates of choice for a wide range of free-space applications such as high-speed communications [START_REF] Pang | Gigabit free-space multi-level signal transmission with a mid-infrared QCL operating at room temperature[END_REF] or light detection and ranging (LIDAR) [START_REF] Lin | Chaotic LIDAR[END_REF].

Device description and experimental apparatus

The QCL under study is a distributed feedback (DFB) laser which needs to be cooled down at cryogenic temperatures in order to be pumped with a continuous bias. At the temperature of boiling nitrogen, the QCL has a current threshold of 331 mA and emits up to 50 mW in optical power when pumped at 950 mA. It emits single mode at 5.45 µm as shown in the optical spectrum of Fig. 1a) retrieved with a Fourier transform infrared spectrometer. COF is applied with a gold plated mirror which defines the external cavity. This cavity is 27 cm long and a midinfrared beam-splitter is placed between the QCL and the mirror. The wave reflected by the beam-splitter hits a high-bandwidth detector (Vigo PEM Mercury-Cadmium-Telluride) working at room temperature. This detector is linked to a low-noise amplifier (RF BAY, Inc LNA-545) with a 500 MHz bandwidth. The electric signal exiting the amplifier is analyzed using both a real-time oscilloscope at one giga sample per second (Atten ADS112CAL) and a b) a)

RF spectrum analyzer (Agilent Technologies CXA N9000A), the latter being used to optimize the alignment of the COF. The QCL is powered with a low-noise source (Wavelength Electronics QCL2000 LAB) and the continuous bias delivered by the source can be modulated with an external signal from a waveform generator (Rigol DG1022Z).

Results and Discussion

The QCL is pumped with a continuous bias at 600 mA and a square wave modulation with a period of 10 µs and a low amplitude of 4 mA is added. This square wave is asymmetric and the upper part of the square lasts for only 2 µs. Spikes with a large amplitude occur in the time trace every 10 µs, as shown in Fig. 2, except for some few cases where the amplitude of spikes is low and actually corresponds to the low amplitude of the square modulation. The dashed blue line in Fig. 2 corresponds to the aforementioned criterion (μ + 8σ) and shows that the spikes with large amplitude are extreme pulses because the maximum of these spikes are above the dashed blue line. It is relevant to notice that, if either the duty cycle of the square signal or the amplitude of the square signal is modified, extreme pulses are still visible in the time traces but it becomes tougher to precisely control their likelihood. Further analysis is required to thoroughly determine the optimal parameters to control extreme pulses. Replacing the square modulation by a sine one gives the same conclusion. If the amplitude of the modulation is too large, the spikes can be controlled but the criterion for extreme events is no longer met as shown in Fig. 3. We thus reported the first extreme pulses in a QCL as well as a method to control the likelihood of such events through a periodic perturbation.

FIG. 1

 1 FIG.1 a) optical spectrum characteristics of the free-running DFB QCL operating at 77 K under a continuous bias of 340 mA; b) experimental apparatus split between the analysis path above and the external optical feedback path below.

FIG. 2 a

 2 FIG. 2 a) Experimental time traces of the QCL under COF and a square wave modulation with a period of 10 µs and an amplitude of 4 mA. The dashed blue line represents the threshold criterion for extreme events; b) close up on an extreme event and the low-amplitude square modulation, the grey arrow represents the amplitude discrepancy between the extreme pulse and the square wave modulation.

This work is supported by the French Defense Agency (DGA), the French ANR program under grant ANR-17-ASMA-0006, the Office of Naval Research and the National Science Foundation. b) a)


O Spitz

A Herdt

M Carras

W Elsäßer

F Grillot

Square Wave Emission in a Mid-infrared Quantum Cascade Oscillator Under Rotated Polarization

Keywords:

Quantum cascade lasers, which are known to only emit a transverse-magnetic wave under free-running operation, can output a square wave with transverse-electric emission under polarization-rotated feedback.

Introduction

The technique of feeding back a rotated polarization to a semiconductor laser, known as cross-polarization reinjection, has proved its ability to rotate the polarization of the light emitted by the semiconductor laser [START_REF] Marconi | Robust square-wave polarization switching in vertical-cavity surface-emitting lasers[END_REF]. Numerical and experimental works carried out on standard interband laser diodes showed that these lasers do not only display a chaotic behavior, which is usual with conventional optical feedback, but can also emit a square wave in both the TE and the TM modes when the feedback ratio is large enough [START_REF] Marconi | Robust square-wave polarization switching in vertical-cavity surface-emitting lasers[END_REF]. The particularity of this square wave is that it is simultaneously present in the TE and TM mode with a phase shift and that the upper part of the square wave can possibly exhibit ultra-fast oscillations beyond the relaxation frequency of the laser diode [START_REF] Uy | Sustained oscillations accompanying polarization switching in laser dynamics[END_REF]. We experimentally apply the cross-polarization reinjection technique to a mid-infrared quantum cascade laser (QCL) and unveil a similar behavior except that the square wave does not display oscillations in the investigated bandwidth between 0.1 and 1000 MHz. QCLs are versatile mid-infrared optical sources used in many industrial applications [START_REF] Capasso | High-Performance Midinfrared Quantum Cascade Lasers[END_REF] and they are renowned to emit a TM wave because of the symmetry of their gain medium and their material anisotropy. Here we demonstrate the first TE emission in a QCL with cross-polarization reinjection.

Device description and experimental setup

The QCL under study is a distributed feedback laser emitting single mode at ~5.7 µm when pumped with a continuous bias of 800 mA at 249 K, as shown on Fig. 1(a). This configuration induces a high internal temperature and this is the reason why the QCL is episide-down mounted on AlN substrate as presented on Fig. 1(b). has a measured absorption of 25% at the laser wavelength. The light that is not absorbed is reflected for 99 % and transmitted for 1%, resulting in a maximum feedback ratio of 35%.

Results and Discussion

Fig. 2(a) and 2(b) show the time trace and RF spectrum analysis when conventional optical feedback is applied into the QCL pumped with a continuous bias. The time trace shows a chaotic pattern called low frequency fluctuations [START_REF] Jumpertz | Chaotic light at mid-infrared wavelength[END_REF], where the RF spectrum is quite broad. This non-linear phenomenon can only be seen in the TM wave since the TE wave does not exist in that case. When rotating the feedback polarization with the aforementioned setup, both the TM and the TE wave display a square wave pattern for high feedback ratios (above 25%). Fig. 3(a) and 3(b) show the characteristics of the TE square wave when recorded with the 50 MHz bandwidth detector. The oscillations on top of the squares are only an artefact of the limited bandwidth and were not spotted with the 1 GHz detector, thus these oscillations are not related to the relaxation oscillations, which have never been spotted in QCLs. However, the low-frequency cutoff of this detector, around 10 MHz, does not allow retrieving the shape of the square wave. We therefore experimentally reported the first TE emission in a QCL with crosspolarization reinjection. The TE emission is characterized by a self-pulsating pattern which could be used to generate random numbers and also to investigate the relaxation frequency of QCLs at several dozens of GHz through wavelength conversion, from mid-infrared wavelength to near-infrared domain.

FIG. 1

 1 FIG.1 a) optical spectrum retrieved with a Fourier transform infrared spectrometer (Bruker Vertex 80V) of the free-running QCL operating at 249 K and 340 mA (9 mA above the current threshold); b) schematic of the emission facet of a QCL which is episide-down mounted with mounting base (1), substrate (2), cladding (3), active region (4), dielectric region (5) and gold metallization (6); c) experimental setup which splits the beam to an analysis path above and an external optical feedback path below.The experimental setup, as presented on Fig.1(c), is made of an analysis path with a Mercury-Cadmium-Telluride detector with a bandwidth of 50 MHz (KMPV50-0.5-J2) or 1000 MHz (Kolmar KV104-0.1E/10 with KA700 preamplifier) for mid-infrared detection. The detector is linked to a real time spectrum analyzer (RSA) with a maximum real time bandwidth of 110 MHz (Tektronix RSA6114A) and a 40 GS/s oscilloscope (Tektronix TDS6154C) for real time analysis and acquirements. The external optical feedback path is set with a gold plated mirror and a quarter-wave plate (QWP) made for a wavelength of 4.5 µm. If the QWP is placed at 45°, the TM wave emitted by the QCL travels back and forth inside the external cavity and turns to a TE wave before being fed back inside the QCL. The external cavity has a total length of around 35 cm. The non-polarizing beam splitter (NPBS)

FIG. 2 :FIG. 3 :

 23 FIG. 2 : Experimental time traces (a) and RF spectrum (b) of the TM wave emitted by the QCL under study when conventional optical feedback is applied. In such case, the QCL does not emit a TE wave

Acknowledgments : This work is supported by the French Defense Agency (DGA) and the French ANR under grant ANR-17-ASMA-0006