Immersion and Invariance Adaptive Control for Spacecraft Pose Tracking via Dual Quaternions

(is paper addresses the simultaneous attitude and position tracking of a target spacecraft in the presence of general unknown bounded disturbances in the framework of dual quaternions, which provides a concise and integrated description of the coupled rotational and translational motions. By virtue of the newly introduced dual direction cosine matrix, the dimension of the dual quaternion-based relative motion dynamics written in vector/matrix form can be lowered to six. Treating the disturbances as unknown parameters, a modular adaptive pose tracking control scheme composed of two separately designed parts is then derived. One part is the adaptive disturbance estimator designed based on the immersion and invariance theory. Driven by the disturbance estimation errors, it can realize exponential convergence of the estimations and has the nice “parameter lock” property, which can hardly be expected in the conventional certainty equivalent adaptive controllers. (e other part is a proportional-derivative-like pose tracking controller where the estimated disturbances are directly used. (e closed-loop stability of the relative motion system under different kinds of disturbances is proven by Lyapunov stability analysis. Simulations and comparisons with two previous dual quaternion-based controllers demonstrate the novel features and performance improvements of the proposed control scheme.


Introduction
Simultaneously tracking the desired attitude and position (pose) with respect to (w.r.t) a target spacecraft with high precision is a crucial and indispensable technology for a broad range of space proximity missions, such as on-orbit service, rendezvous and docking, and so on [1][2][3][4]. is problem is intractable since it necessitates the six-degree-of-freedom (six-DOF) control of spacecraft relative motion, where exists strong kinematic and dynamic couplings between the relative rotation and translation.
Among various mathematical tools for describing the coupled rotation and translation, such as homogeneous transformation matrix [5][6][7], quaternion or modified Rodrigues parameters plus position vector [8][9][10][11][12], and so on, dual quaternion has shown to be well suitable to represent the six-DOF motion because it not only is a compact, global nonsingular, and computationally efficient parameterization for the rigid body pose but also captures complex couplings between rotation and translation in a natural and implicit way [13][14][15][16][17]. Dual quaternions have already been successfully applied to kinematic analysis in various research fields, such as robot calibration [18], navigation [19], pose estimation [20,21], and inverse kinematics [22].
By virtue of the dual inertial operator introduced by Brodsky and Shoham [23], Wang et al. [24] first derived relative dynamics between two spacecraft using dual quaternions. Replacing the dual inertial operator by an invertible block diagonal matrix, Filipe and Tsiotras [25] rewrote the dual quaternion-based dynamic equation in a more convenient 8-dimensional (8D) matrix/vector form. Based on these two types of governing equations given in [24] or [25], lots of control laws have been proposed to deal with the pose tracking problem under various situations [26][27][28][29][30][31][32][33][34][35], among which the unknown disturbance, as an inevitable factor in practice, has been intensively studied in recent years. To attenuate small amplitudes of disturbances, proportional-derivative-(PD-) like controllers were derived in [26][27][28]. However, this robustness against disturbances is at the expense of high proportional gains, which is undesirable. Several sliding mode controllers (SMCs) were proposed in [29][30][31][32] to reject general bounded disturbances. However, there is often a trade-off between the chattering suppression and control precision. In [33], the internal model-based control was specially utilized to compensate for sinusoidal and constant disturbances, while the frequencies of all disturbances must be exactly known. Regarding disturbances as unknown parameters, adaptive control laws based on certainty equivalence (CE) principle were designed in [27,28]. Since the parameter adaption laws are driven by the tracking errors of the system states, dynamic behaviours of the parameter estimation are unexpected. Besides, even if the corresponding true values are reached, the estimations still keep updating unless the state tracking errors are zero.
us, the CE-based adaptive control often leads to degradation of the closed-loop performance [36,37].
As a novel noncertainty equivalent (NCE) method for adaptive control of nonlinear systems with uncertain parameters [38][39][40][41], immersion and invariance (I&I) adaptive control has great potential to overcome many limitations resulting from the CE-based methods. It has been effectively utilized in the attitude [36,[42][43][44][45] and/or orbit control [46][47][48][49] with unknown inertial or mass parameters and has shown significant improvements in both closed-loop performance and adjustability of the estimation convergence process. However, little attention has been paid to its application in tackling unknown disturbances in the six-DOF motion control.
Motivated by the above observations, this paper investigates the I&I adaptive pose tracking control of spacecraft subject to general bounded unknown disturbances in the framework of dual quaternions. Similar to [19], the matrix/ vector form of dual number algebraic operations is adopted. However, by introducing a dual direction cosine matrix and a 6D antiblock dual inertial matrix, the swap operations induced by the 8D diagonal dual inertial matrix in [25] can be avoided, and dimensions of the resultant system equations can be lowered to six. On this basis, an estimation law is designed based on I&I adaptive control theory to compensate for the general unknown disturbances, which enables the overall adaptive control law to take a modular form.
at is, the dynamics of parameter estimations can be adjusted independently of the applied control laws and can realize "parameter lock," which are impossible with the CE-based adaptive control. e overall control scheme is completed by combining the estimated disturbances with a PD-like pose tracking controller. Simulation results verify the effectiveness and benefits of the proposed adaptive control strategy when compared with a CEbased adaptive controller and a SMC. e rest of the paper is organized as follows. Section 2 introduces necessary preliminaries of quaternions and dual quaternions. On this basis, the dual quaternion-based model of relative motion between two spacecraft written in 6D vector/ matrix form is derived in Section 3. In Section 4, firstly the I&I-based adaptive disturbance estimation law is derived and then the complete adaptive pose tracking controller is designed along with Lyapunov stability proof. Simulation and comparison results are presented in Section 5 to demonstrate the features and effectiveness of the proposed adaptive control scheme. Section 6 concludes this paper.

Quaternions.
A quaternion is defined as q � q 0 q T T ∈ H, where q 0 ∈ R and q ∈ R 3 are its scalar and vector parts. Vectors can be viewed as vector quaternions which are quaternions with zero scalar parts and vice versa. q * � [q 0 − q T ] T is the conjugate of q. 1 H � 1 0 T T and 0 H � 0 0 T T are unit and zero quaternions where 0 � 0 0 0 T . A unit quaternion satisfies qq * � 1 H . For any two quaternions a � a 0 a T T and b � b 0 b T T , some necessary operations are given as follows: vector part: vec(a) � a. (1) If the attitude of frame F X w.r.t. F Y is represented by the unit quaternion q XY , then the coordinate transform of a vector r is given as r X � q * XY r Y q XY , where r � 0 r T T is the quaternion form of r and is used to match the dimensions of the quaternion and vector. e 3D form coordinate transform can be given as where C XY is the direction cosine matrix (DCM); T(·, ·): H × H ⟶ R 3×3 is defined as

Dual Numbers and Dual Vectors.
A dual number is defined as a � a r + ϵa d ∈ R, where a r , a d ∈ R are the real and dual parts. ϵ is the dual unit and satisfies ε 2 � 0, ε ≠ 0. When real and dual parts are both vectors, a dual number is extended to a dual vector with a form a � a r + εa d ∈ R 3 where a r , a d ∈ R 3 . e zero dual vector is denoted by 0 � 0 + ε0. Since R 3 is isomorphic to R 6 [21], a dual vector can be equivalently denoted as a vector form a � [a T r a T d ] T . For any a, b ∈ R 3 , some important operations are defined as follows: Swap: a s � a d + εa r , Corresponding product: a ⊙ a � a r a r + εa d a d , 2 Complexity Based on the above definitions, the following properties can be obtained: a × a � 0.

Dual Quaternions.
A dual quaternion is defined as q � q r + εq d ∈ H, with q r , q d ∈ H. Dual quaternions composed of vector quaternions can be viewed as dual vectors and vice versa. H is isomorphic to R 8 , and thus q can also be denoted as q � q T r q T d T . q * � q * r + εq * d is the conjugate q. 1 � 1 + ε0 is the unit dual quaternion. Unit dual quaternions satisfy qq * � 1. Apart from the cross product, operations given in (4) are still valid for dual quaternions a and b. Additionally, there are vector part : vec(a) � a � a r + εa d . (6) Besides, there are the following properties: Supposing the unit quaternion q XY represents the pose of F X w.r.t. F Y , then the coordinate transform of a dual vector r is given by r X � q * XY r Y q XY , which is written in a 8D form to match the dimensions of the dual quaternion and dual vector. By introducing the dual direction cosine matrix (DDCM) C XY , which is a counterpart of C XY in dual form, the 6D form coordinate transform can be given as where

Spacecraft Dynamics Based on Dual Quaternions
e following frames are needed: F I is the Earth centred inertial frame; F B and F T denote the body fixed frame of the chaser and target spacecraft, respectively, and their origins locate at the center of mass (c.m.) of the corresponding spacecraft and axes point along the principal axes of inertias; F O is the orbit frame, whose origin is at c.m. of the spacecraft, x O axis points along the radius direction, and z O is normal to the orbit plane. Hereafter, the superscript of a quantity denotes the frame in which it is expressed.
If the attitude and displacement of F B w.r.t. F I are represented by q BI and vector t BI , then the pose of F B w.r.t. F I can be described as a unit dual quaternion as follows: where t B BI � 0 (t B BI ) T T . Taking derivative of (9), the kinematic equation of a rigid body in terms of dual quaternion is obtained as where ω B BI is the dual quaternion form of dual velocity BI is the linear velocity. By replacing F B with F T , kinematic equation of the target spacecraft can be described in the same way.
To describe the six-DOF dynamics of the spacecraft in a 6D vector/matrix form, instead of using the 8 × 8 diagonal dual inertial matrix given in [25], the novel 6 × 6 antidiagonal dual inertial matrix is defined as where m B and J B are the mass and inertial matrix of the chaser spacecraft. By virtue of (11), the dynamic equation of the spacecraft can be given as where f B � f B + ετ B is the total external dual force exerted on the spacecraft and f B and τ B are the corresponding force and torque. e relative pose between F B and F T is calculated by Differentiating (13), the relative kinematics can be obtained as where ω B BT � ω B BI − q * BT ω T TI q BT is the relative dual velocity in dual quaternion form. By using the DDCM defined in (8), the 6D form of relative dual velocity can be calculated by Substituting (15) into (12), the dual quaternion-based relative dynamics written in 6D form can be obtained as (16) Besides, (16) can be further expressed as where S � 0 3×3 S 1 S 2 0 3×3 and S 1 and S 2 are skew-symmetric matrices which are given by Complexity Remark 1. Compared to the 8D dynamic equations given in [25], by virtue of the novel dual inertial matrix defined in (11) and the DDCM defined in (8), the dynamic equations given in (13) and (16) avoid swap operation on the dual velocity induced by the diagonal dual inertial matrix given in [25], and their dimensions are lowered to six. For a spacecraft that orbits the Earth, f B can be expressed as ∇g , and f B J 2 are gravitational force, gravity gradient torque, and oblateness perturbation force written in dual form and can be calculated by f where μ � 398600 g and a B J2 are given as where R e � 6378.137 kmis the Earth's mean equatorial radius; is the time-varying dual disturbance force due to atmosphere drag, solar radiation, third bodies, and so on. It is assumed that dual disturbance and its first derivative are bounded. Note that a dual number (vector or quaternion) is bounded if and only if its real and dual parts are both bounded.

I&I Adaptive Control Law Design
e pose tracking control of spacecraft subject to unknown disturbances is considered in this paper. Designed based on the I&I adaptive control theory, the overall control strategy takes a modular structure which includes a disturbance estimator and a pose tracking control law, designed separately.

Disturbance
where ξ � ξ f + ϵξ τ is the dynamic part of the estimation and β(M B ω B BI ) is a continuous vector function to be specified. e disturbance estimation error is defined as Combined with (12), the time derivative of (23) can be obtained aŝ Based on the observation of (24), the update law of ξ can be selected as Substituting (25) into (24) yieldŝ To stabilize the subsystem of estimation error, the function β can be chosen as with λ > 0. Substituting (27) into (26), we get . Its time derivative along (28) can be obtained as Since _ f B d is bounded, the ultimate bound of disturbance estimation error can be made arbitrarily small by choosing large enough λ.When the disturbances are constant, there is It can be readily obtained from (31) that f B d is exponentially convergent. In summary, the disturbance estimator designed based on the I&I adaptive method is given as 4 Complexity Remark 2. According the I&I adaptive control theory, the designs of disturbance estimation and pose tracking law are separate. us, the proposed disturbance estimator (32) can be combined with different control laws.

Pose Tracking Control Law.
Combined with the disturbance estimation given by (32), the PD-like adaptive pose tracking control law is designed as where Remark 3. q BT is the vector part of relative attitude quaternion q BT , and t B BT is the relative position vector; therefore, p B BT can also reflect the pose tracking error. e relationships between p B BT , q B BT , and ω B BT are given in the following lemmas.
Proof. According to the definition of circle product in (4), Lemma 1 can be easily verified, which is omitted here. □

Lemma 2. For any
Proof. Applying properties given in (4) and (7), there is (a⊙ When the disturbances are time-varying, the first result of this paper is presented in the following theorem.

Theorem 1.
Consider the closed-loop system composed of relative motion models (14) and (16), disturbance estimator (32), and control law (33). e tracking error states p B BT and ω B BT converge to arbitrarily small neighbourhood of the origin by choosing large enough λ.
Proof. Consider the following candidate Lyapunov function: where k � k 1 + c ⊙ k 2 , c � c ω + ε2c v , and c ω and c v are small positive constants. To show the positive definiteness of V, with Lemma 1, it can be obtained that where c ω � ‖ω B TI ‖ is the upper bound of the target's angular velocity that is unknown, BT ‖] T , and

Complexity
It can be easily verified that the positive definiteness of A can be guaranteed by small enough c ω and c v . Applying Lemma 2, the time derivative of V can be given as where . Based on the definition of S in (18) and Substituting (39) into (38) yields where Similarly, it can be verified that the positives definiteness of A ′ can be ensured by sufficiently small c ω and c v . Denote the minimum eigenvalue of A ′ as λ min (A ′ );, then, there is Based on (41) and (30), it can be obtained that It can be seen from (42) that when c ω and c v have been decided, the ultimate bound of x is entirely determined by the ultimate bound of f d B . erefore, the ultimate bound of x can be made smaller by properly increasing the value of λ. □ Remark 4. To guarantee the positive definiteness of A in (37) and A ′ in (40), the ranges of c ω and c v can be calculated as 6 Complexity Note that in the proposed I&I adaptive control scheme, since c ω and c v are only used in the stability analysis rather than in the control law, they are not related to the control effect. From (43), it can be seen that the main constraints are the upper bounds and as long as c ω and c v are close enough 0, the positive definiteness of A and A ′ can always be guaranteed. erefore, the exact upper bounds of c ω and c v are actually not needed, which is also an advantage of the I&I adaptive control scheme.
When the disturbances are constant, the second result of this paper is described in the following theorem. (14) and (16), disturbance estimator (32), and control law (33). en, the tracking error states p B BT ⟶ 0, ω B BT ⟶ 0 as t ⟶ ∞ for all initial conditions. Proof. Consider the following candidate Lyapunov function:

Theorem 2. Consider the closed-loop system composed of relative motion models
e time derivative of V 2 can be calculated as where Young's inequality has been applied, k 2 � min k ω , k v , and χ ∈ (0, k 2 ). Integrating both sides of When the disturbances are constant, according to (31), ‖f B d ‖ 2 is exponentially convergent; therefore, the right hand of (46) is integrable on [0, ∞]. As a result, V 2 (t) is bounded on [0, ∞]. Hence, the system states q BT and ω B BT are also bounded.
On the other hand, when f B d � 0, for same candidate Lyapunov function

Simulation Results
e effectiveness and features of the proposed I&I adaptive pose tracking control scheme composed of (32) and (33) are verified by numerical simulations and comparisons with two existing dual quaternion-based controllers. In the case of constant disturbances, the proposed control law is compared with a CEbased adaptive controller designed in [28]. In the case of timevarying disturbances, it is compared with a SMC given in [32]. e objective is to track the pose of a target spacecraft, which is orbiting in a highly eccentric Molniya orbit, and the orbit elements are given in Table 1. e target body fixed frame is assumed to be aligned with its orbit frame. e mass and inertial matrix of the chaser spacecraft are given as e gains for the pose tracking controller (33) are selected as k q � 1, k r � 2.8, k ω � 6, and k v � 20.
In order to evaluate the control effort, the energy consumption is defined as where θ � 1(m) + ε1 is used to unify the units of the energy to N 2 m 2 s.

Simulations in the Case of Constant
Disturbance. e adaptive disturbance estimation law designed based on certainty equivalence principle is given as [28] where α � α f + εα τ , α f and α τ are positive adaptive gains, and c � c ω + ε2c v is the same as that defined in (35). e CE-Complexity based adaptive controller for comparison is composed of pose tracking law (33) and the disturbance estimation law (48). As stated in Remark 4, in the proposed I&I adaptive control scheme, c is only used for stability analysis; therefore, the positive definiteness of A and A ′ can always be ensured by small enough c ω and c v without knowing their exactly upper bounds. However, in the CE-based control law, c is also involved in the adaption law as shown in (48), whose value will also influence the estimation effect. us, for the CE-based control law, the upper bounds of c ω and c v have to be specified, which means the value of c ω needs to be known in advance.
According to the target orbit elements, there is c ω � 0.0004. en, based on (43), the ranges of c ω and c v can be calculated as 0 < c ω < 0.172 and 0 < c v < 0.295. e parameters for the CE controller are set as c � 0.17 + ε0. 29 and α � 0.28 + ε0. 4.
Note that to guarantee the positive definiteness of A in (37) and A ′ in (40), c ω and c v have to be small enough. In the proposed I&I adaptive control scheme, c is only used for stability analysis; therefore, the positive definiteness of A and A ′ can always be ensured by small enough c ω and c v without knowing the exactly value of c ω . However, in the CE-based control law, c is also involved in the adaption law as shown in (48), whose value will also influence the estimation effect.
us, for the CE-based control law, the value ranges of c ω and c v have to specified, which needs the value of c ω in advance.
According to the target orbit elements, there is c ω � 0.0004. en, based on (43) Figure 1 shows the global result of the tracking trajectory. Figures 2-7 present the time responses of relative attitude and position, relative angular velocity and linear velocity, and control force and torque under the two adaptive controllers which are composed of the same pose tracking control law and different disturbance estimation laws. Although both controllers achieve the tracking control objective, the proposed controller has smoother transient responses, shorter settling time, and higher tracking precision. After 120 s, the errors of relative attitude and position of the proposed controller are smaller than 5 × 10 − 7 and 1.5 × 10 − 4 m and those of the CE-based controller are 1.5 × 10 − 4 and 4 × 10 − 3 m. e energy costs of the proposed and the CE-based controllers are 29.6 N 2 m 2 s and 34.1 N 2 m 2 s, and the former is 86.8 % of the latter. Such significant performance improvements without increasing the control efforts are due to the fact that I&I-based adaption law can estimate the disturbances quickly, smoothly, and accurately. As shown in Figures 8 and 9, disturbance estimation errors under the I&I-based adaption law converge to zero at about 30 s in exponential form while those under the CE-based adaption law converge after 100 s with large overshoots and oscillatory transient responses, which lead to the deterioration of the relative motion states and more energy consumption.
By comparison of the I&I-based and CE-based adaption laws (32) and (48), it can be observed that the I&I-based one is driven by the estimation errors rather than the system states like the CE-based one.
is brings the I&I-based adaptive estimator another nice property, that is, the socalled "parameter lock," which means when the true values are reached, the parameter estimations automatically stop updating and get locked on their corresponding true values.
is feature of the I&I adaption law is clearly shown in Figure 10 where the initial disturbance estimations of both estimators are set as the true values. It can be seen that the CE-based adaption law still drives the estimation away from the true value while the I&I one can keep the estimation staying at the true values.
With this modification, the tracking errors can only converge to a neighbourhood of the origin.
Note that to effectively reject the disturbance, k σ has to satisfy 0.05 sin 0.02t + π 6 + 0.03 0.06 sin 0.03t + π 2 + 0.02 − 0.04 sin(0.05t + π) − 0.05 e upper bounds of disturbances are set as D f � 1 and D τ � 0.1. e other parameters for the SMC are set as k a � 0.3 + ε0.4k k � 3 + ε8k a � 0.1 + ε1, and Δ � 0.003. Δ is set as the minimum value which can eliminate the chattering. And the control gains of the SMC are tuned such that its settling time is approximately identical with that of the proposed controller. e time histories of relative motion states under two controllers are presented in Figures 11 and 12. e steady state errors of relative attitude and position and relative angular and linear velocities of proposed controller are around 1 × 10 − 4 , 5 × 10 − 4 m, 5 × 10 − 6 rad/s, and 4 × 10 − 5 m/s and those of the SMC are around 4 × 10 − 3 , 0.02 m, 3 × 10 − 4 rad/s, and 1 × 10 − 3 m/s, which are much larger. It can be seen that in the presence of quite large time-varying disturbances, the SMC has to sacrifice the control precision for chattering alleviation while the proposed I&I adaptive controller can still achieve relatively high control precision. e control forces and torques under the two controllers are shown in Figures 13 and 14. e proposed controller generates smaller and smoother control forces and torques than the SMC. Energy consumptions of the proposed controller and SMC for a total simulation time of 400 s are 26.4 N 2 m 2 s and 36.6 N 2 m 2 s, respectively. e energy consumption of the proposed controller is 72.1% of that of the SMC.
Besides, Figure 15 compares the converge process of I&I-based disturbance estimation errors with different λ and verifies that the final estimation errors for the timevarying disturbances can be made smaller by choosing a larger λ.

Conclusion
e six-degree-of-freedom relative motion control between two rigid spacecraft in the presence of general unknown bounded disturbances is investigated based on dual quaternion description. e newly introduced dual direction cosine matrix enables the integrated relative dynamics to be written in a six-dimensional vector/matrix form. Adopting an adaptive perspective, a noncertainty equivalent disturbance estimator is designed based on the immersion and invariance (I&I) adaptive control theory. For constant disturbances and time-varying disturbances with bounded first derivatives, the estimation errors exponentially converge to the origin and adjustable small neighbourhood around the origin at the prescribed rate, respectively. Since the I&Ibased adaption law is driven by the disturbance estimation errors, convergence of the estimation is independent of the applied control law and can realize "parameter lock" when the true values are reached. Combining the disturbance estimation with a proportional-derivative-like pose tracking control law yields the complete I&I adaptive pose tracking controller. Simulation results demonstrate the effectiveness of the proposed adaptive control scheme in tracking the target pose with high precision under general unknown bounded disturbances. Comparisons with the existing certainty equivalence-based adaptive controller and sliding mode controller illustrate the novel features of the proposed controller and show that owing to the quick and accurate estimation of the disturbances, the proposed controller can generate smoother transient performance and higher control precision with less energy cost.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.