A generalized super-twisting algorithm-based adaptive fixed-time controller for spacecraft pose tracking

Kejie Gong; Yong Wang; Yurui Duan; Yafei Mei; Yu Jiang; Da Luo

doi:10.1515/astro-2022-0229

Open Access Published by De Gruyter Open Access January 5, 2024

A generalized super-twisting algorithm-based adaptive fixed-time controller for spacecraft pose tracking

Kejie Gong , Yong Wang , Yurui Duan , Yafei Mei , Yu Jiang and Da Luo

From the journal Open Astronomy

https://doi.org/10.1515/astro-2022-0229

Abstract

A generalized super-twisting second-order sliding mode adaptive fixed-time control law, which is used for spacecraft pose tracking in the presence of internal and external uncertainties, is proposed. Lie group SE(3) (for special Euclidean group), which is the configuration space for rigid body motion, is used for modeling the six-degrees-of-freedom dynamics of spacecraft. A fixed-time sliding mode surface is proposed and applied to design an generalized super-twisting sliding mode control law. A novel dual-layer adaption law for the controller is proposed to the ensure the gains varying rapidly with the disturbance. The adaptive second-order sliding mode controller guarantees a uniform exact convergence for the closed-loop tracking control system with less energy consumption. Numerical simulations are performed to demonstrate the excellent performances of the control law.

Keywords: spacecraft pose tracking; lie group SE(3); generalized super-twisting algorithm; second-order sliding mode; fixed-time control

1 Introduction

As spacecraft proximity and spacecraft formation flying require high-precision and high-efficiency control increasingly, spacecraft pose control is playing an increasingly crucial role in current and future space missions. Coupled attitude and position modeling and control can meet these requirements.

For spacecraft six-degrees-of-freedom (6-DOF) modeling, vector form (Huang and Jia 2017), dual-quaternion (Nixon and Shtessel 2021), and Lie group SE(3) (Lee and Vukovich 2015a, Lee and Vukovich 2015b, Lee and Vukovich 2017, Lee et al. 2017) are used to describe the coupled rotational and translational motion. Euler angle (Xing et al. 2010), MRPs (modified Rodriguez parameters) (Huang and Jia 2017), and quaternion (Guzzetti and Howell 2017) are usually used in a vector form to represent the attitude of a spacecraft. Nevertheless, there exists singularity at some angles in Euler angle and MRPs, and ambiguity in quaternion. Furthermore, the separate describing of position and attitude of vector form makes the controllers complicated and reduces the spacecraft’s computational efficiency (Zhang et al. 2018). Dual-quaternion and Lie Group SE(3) represent the attitude and position motion in compact forms. Lie group SE(3) represents the pose uniquely and globally, which is prior to vector form or dual-quaternion. The exponential coordinate of Lie group SE(3) is an effective way to represent the configuration tracking error. Owing to the advantages of Lie group SE(3), an increasing number of researchers have investigated the coupled spacecraft attitude and orbit control in the framework of SE(3) in recent years (Zhang et al. 2018, Gong et al. 2020, Zhang and Yang 2021, Gong et al. 2022, Mei et al. 2022a, Mei et al. 2022b, Ren et al. 2022). In this research, Lie group SE(3) is used for modeling the coupled spacecraft motion, and its exponential coordinate is used for denoting the pose tracking error.

A spacecraft that moves in outer space is affected by various environmental disturbances and internal uncertainties (inertial parametric uncertainties, actuator failures, and misalignments). Precise maneuver for a spacecraft calls for a control scheme with strong robustness. In many cases, the inertia parameters may not be accurately known, and external disturbances are usually unknown. Disturbance observer (DO) is widely used to observe the disturbance and compensate in the control law to improve the control accuracy further. Some researchers treated the internal and external disturbances as the compound disturbance of the control system and designed a DO for compensation to realize the control performance of a high accuracy (Yan and Wu 2017, Li et al. 2019, Cheng et al. 2022, Pukdeboon 2019, Mei et al. 2022a).

Another way to obtain high accuracy is the high-order sliding mode control (HOSMC). The first-order sliding mode control usually provides worse performance compared with HOSMC (Pukdeboon 2019). A number of HOSMCs are related to the super-twisting algorithm (STA) that usually has a second-order form. The STA (Levant 1993), which was first proposed as a continuous control law, allows to compensate Lipschitz perturbations exactly and ensure finite-time convergence. Since then, the STA has achieved great developments with various forms. For instance, generalized super-twisting algorithms (GSTAs) (Moreno and Osorio 2008, Moreno 2009, Cruz-Zavala et al. 2011) and adaptive STAs (Shtessel et al. 2010, Shtessel et al. 2012, Edwards and Shtessel 2014a, Edwards and Shtessel 2014b, Edwards and Shtessel 2015, Yang et al. 2016, Mei et al. 2022a) have been proposed by researchers.

Moreno (Moreno and Osorio 2008, Moreno 2009) proposed two different forms of GSTAs by adding a linear correction term. The GSTAs are more robust and rapidly convergent than the original one. Conventional super twisting algorithm like in the study by Shtessel et al. (2010) and fast super twisting algorithm like in studies by Moreno and Osorio (2008), Moreno (2009) are finite-time convergent. Super-twisting-based fixed-time convergent sliding mode algorithms are also proposed in the studies by Cruz-Zavala et al. (2011), Guzman and Moreno (2015), Basin et al. (2016), and Li et al. (2017), which can also be called GSTA or super twisting-like algorithm. Cruz-Zavala et al. (2011) proposed a robust uniform convergent GSTA that includes high-degree terms with a convergence time bounded by a certain constant.

Adaptation methodology is proposed to improve the performance of STA (Utkin and Poznyak 2013). Utkin and Poznyak (2013) introduced an equivalent control by a low-pass filter to realize the updating of gains. Shtessel et al. (2010, 2012), and Edwards and Shtessel (2014a,b, 2015) proposed several adaptive STAs based on the original one. Different from standard form-based adaptive STA, Yang et al. (2016) proposed an adaptive fast dual-layer STA by adding an additional linear term. Li et al. (2017) proposed an adaptive fixed-time generalized super-twisting DO with single-layer adaption law based on an equivalent control-based scheme.

STA and its improved versions have been widely used as observers (Dong et al. 2017b, Shi et al. 2021, Li et al. 2017), controllers (Dong et al. 2017a, Sellali et al. 2019, Fei and Feng 2019, Ghasemi et al. 2019, Shi et al. 2021), or differentiators (Salgado et al. 2014) in the automatic control research field. It is worth noting that Ghasemi et al. (2019) first proposed an adaptive super-twisting control of a multi-agent system on the Lie Group SE(3). In the field of spacecraft control, STA is also used to design controllers (Pukdeboon 2012, Lu and Xia 2014, Chen and Geng 2015, Torres et al. 2019, Zhang et al. 2020). Pukdeboon (2012) applied the traditional super-twisting controller and the generalized one for spacecraft formation flying. Lu and Xia (2014) proposed an adaptive gain super-twisting controller for a spacecraft attitude control system, which can provide rapidly, robustness, accuracy, anti-chattering, and anti-wasting energy simultaneously for a closed-loop spacecraft system. Chen and Geng (2015) designed a super-twisting controller for on-orbit servicing to non-cooperative target. Torres et al. (2019) used the super-twisting sliding mode control for a high-fidelity 6-DOF rendezvous mission between a chaser spacecraft and a passive target. Zhang et al. (2020) proposed an adaptive super-twisting controller for orbiting around irregular shape small bodies. Nixon and Shtessel (2021) proposed an adaptive dual-layer continuous super-twisting controller for a satellite formation modeled on dual quaternion. In the field of spacecraft control on Lie group SE(3), STA has also been used in recent years. Mei et al. (2022a) proposed a GSTA controller to achieve a finite-time stabilization in spacecraft 6-DOF control on SE(3).

From the view of stabilization time, finite-time control and fixed-time control has been continuously applied in spacecraft control (Jiang et al. 2016, Khodaverdian and Malekzadeh 2023) or other fields (Abadi 2023). Due to the essence of super-twisting sliding mode control, it usually lead to finite-time or fixed-time stabilization. From the existing articles, super-twisting sliding mode control in spacecraft is getting increasing attention. Naturally, the excellent performance of super-twisting control, as described in Lu and Xia (2014), should be worthy of focusing on. To the author’s knowledge, a generalized super-twisting controller for spacecraft motion with fixed-time convergence has not been applied.

Due to the existence of robust term in the STA, the corresponding controllers can achieve high accuracy and attenuate chattering in the closed-loop system. External disturbance and model uncertainties are considered in most spacecraft control research. Generally, there is no need to design a specific scheme to deal with the compound disturbance in super-twisting control algorithms if the parameters are appropriately tuned, as the robust term can restrain the disturbance to some degree and thus improve the accuracy.

The contributions of this study are as follows: a fixed-time convergent adaptive second-order sliding control scheme is first applied in spacecraft 6-DOF control that is modeled on Lie group SE(3). The scheme guarantees fixed-time convergence and high accuracy of the spacecraft closed-loop control system. A new dual-layer adaption law is first applied for the GSTA. The adaption technology can accurately estimate the compound disturbance of the spacecraft control system and provide good robustness to smooth or non-smooth interferences. The proposed control scheme possesses better performances than the existing inertia parametric estimation-based or upper bound estimation-based ones.

The structure of this article is as follows: Section 2 presents the background and some preliminaries. Section 3 proposed a GSTA with dual-layer adaption law. Section 4 designs a relevant adaptive control law for spacecraft pose tracking. Section 5 presents the numerical simulations of the proposed controllers. The last two sections provide the discussion and conclusion of this work.

2 Background and preliminaries

2.1 Dynamics of rigid spacecraft on SE(3)

This subsection presents the integrated kinematics and dynamics of a spacecraft. As the spacecraft dynamics on SE(3) has been described in detail in other literature, it is presented in brevity. Some reference frames need to be defined. The Earth-Centered-Inertial (ECI) reference frame is used for describing absolute motion of a spacecraft around the earth, which is denoted by { E } = { E x , E y , E z } . The body-fixed frames of the leader and follower spacecraft are denoted by { B 0 } = { B x 0 , B y 0 , B z 0 } and { B } = { B x , B y , B z } , respectively.

The configuration space of a rigid spacecraft motion is SE(3) (for special Euclidean group), which is the semi-direct product space of R 3 and SO ( 3 ) . R 3 and SO ( 3 ) are used for describing the translational and rotational motion, respectively. Thus, the pose of spacecraft in space is described as follows:

(1) g = R p 0 1 × 3 1 ,

where R denotes the rotation matrix from the body-fixed frame { B } to the inertial frame { E } and p is the position vector in the ECI frame.

The unified velocity of a spacecraft is defined as ξ = [ ω T , v T ] T ∈ R 6 , where ω is rotational velocity, and v is translational velocity, both expressed in the body-fixed frame. The compact form of the kinematics is denoted as follows:

(2) g ˙ = g ξ ∨ ,

where ( ⋅ ) ∨ denotes the mapping from R 6 to se (3), which is expressed as follows:

(3) ξ ∨ = ω × v 0 1 × 3 0 ∈ se ( 3 ) ,

where se ( 3 ) is the Lie algebra of SE(3), which is also called the tangent space at identity. The operator [ ⋅ ] × denotes the skew-symmetric transformation of a vector in R 3 .

The adjoint operator of g ∈ SE ( 3 ) is defined as follows:

(4) Ad g = R 0 3 × 3 [ p × R ] R ∈ R 6 × 6 .

Note that Ad g is invertible, and its inverse is denoted as Ad g − 1 . The adjoint and co-adjoint operators of se (3) are defined as follows:

(5) ad ξ = ω × 0 3 × 3 v × ω × ∈ R 6 × 6 .

(6) ad ξ * = ( ad ξ ) T = − ω × − v × 0 3 × 3 − ω × .

The rotational and translational dynamics of a rigid spacecraft are denoted as follows:

(7) J ω ˙ = J [ ω × ] ω + M g ( p , R ) + T c ( p , R , v , ω ) + T d

(8) m v ˙ = m [ v × ] ω + F g ( p , R ) + F c ( p , R , v , ω ) + F d ,

where T c ∈ R 3 and F c ∈ R 3 denote the control torque and control force, respectively. M g ∈ R 3 and F g ∈ R 3 denote the gravity gradient moment and gravity force, respectively. T d ∈ R 3 and F d ∈ R 3 denote external torques and forces acting on the spacecraft, respectively.

The Earth’s oblateness with J 2 is taken into consideration. M g and F g expressed in the spacecraft body-fixed frame are modeled as follows (Junkins and Schaub 2009):

(9) F g = F G + F J 2 ,

(10) M g = 3 μ ∥ p ∥ 5 ( [ b × ] J b ) ,

(11) F G = − m μ ∥ p ∥ 3 b − 3 μ ∥ p ∥ 5 J b + 15 2 μ ( b T J b ) ∥ p ∥ 7 b ,

(12) F J 2 = − 3 μ m J 2 R e 2 2 ∥ p ∥ 5 R T p x 1 − 5 p z 2 ∥ p ∥ 2 p y 1 − 5 p z 2 ∥ p ∥ 2 p z 3 − 5 p z 2 ∥ p ∥ 2 ,

where b = R T p , J = 0.5 I 3 + J . μ = 398600.44 km 3 s − 2 is the gravitational constant of the Earth. p x , p y , and p z are the components of p , J 2 = 0.00108263 , and R e = 6,378 km is the Earth’s equatorial radius.

With the help of co-adjoint operator of se ( 3 ) , the dynamics equation in a compact form can be denoted as follows (Lee et al. 2017):

(13) I ξ ˙ = ad ξ * I ξ + τ g + τ c + τ d ,

where τ g = [ M g T , F g T ] T , τ c = [ T c T , F c T ] T , τ d = [ T d T , F d T ] T , I = diag ( J , m I 3 ) .

2.2 Relative dynamics between two spacecrafts

Assuming that there exists a leader spacecraft whose pose and velocity are denoted as g 0 and ξ 0 . The leader is supposed not to be subjected to any disturbances. Based on previous preliminaries, the relative coupled rotational and translational dynamics on SE(3) are derived as follows. The actual relative pose between the two spacecrafts is denoted as h ∈ SE(3) and the desired relative pose as h d ∈ SE(3) . The desired relative pose is supposed to be a fixed configuration. The actual relative pose is given by h = ( g 0 ) − 1 g , and then the pose tracking error can be expressed as follows (Lee et al. 2017):

(14) h e = h d − 1 h = h d − 1 ( g 0 ) − 1 g .

The configuration tracking error of the spacecraft tracking system expressed by exponential coordinate is denoted as Lee et al. (2017):

(15) η ˜ ∨ = logm ( ( h d ) − 1 h ) ,

where logm is the logarithm map from SE(3) to se ( 3 ) . The exponential coordinate of the pose tracking error can also be expressed as η ˜ = [ Θ ˜ T , β ˜ T ] T , where Θ ˜ ∈ R 3 and β ˜ ∈ R 3 denotes the attitude and position of the pose tracking error, respectively, which can be computed by equations in the study by Bullo and Murray (1995). The mapping from SE(3) to se ( 3 ) is bijective when ∥ Θ ∥ < π .

Taking the time derivative of Eq. (14), and substituting the follower’s kinematics Eq. (2) and the leader’s one, the velocity tracking error can be derived and denoted as follows (Lee et al. 2017):

(16) ξ ˜ = ξ − Ad h − 1 ξ 0 ,

where Ad h − 1 ξ 0 denotes the leader’s velocity expressed in follower’s body-fixed frame.

The error kinematics in exponential coordinate is expressed as follows (Bullo and Murray 1995):

(17) η ˜ ˙ = G ( η ˜ ) ξ ˜ .

The detailed expression of G ( η ) is presented in study by Bullo and Murray (1995) and omitted here for brevity. Note that η ˜ is not uniquely defined when ‖ Θ ‖ is exactly π rad.

Taking the time derivative of Eq. (16), the error acceleration ξ ˜ ˙ can be written as follows (Lee et al. 2017):

(18) ξ ˜ ˙ = ξ ˙ + ad ξ ˜ Ad h − 1 ξ 0 − Ad h − 1 ξ ˙ 0 .

Substituting the follower’s dynamics Eq. (13) into Eq. (18) yields the error dynamics of the spacecraft 6-DOF tracking system.

(19) I ξ ˜ ˙ = ad ξ * I ξ + I ( ad ξ ˜ Ad h − 1 ξ 0 − Ad h − 1 ξ ˙ 0 ) + τ g + τ c + τ d .

Nevertheless, the inertia uncertainties cannot be ignored in practical spacecraft control, especially large uncertainties. In this work, the nominal inertia matrix I , total inertia matrix I 1 , and the uncertainty of the inertia matrix Δ I are considered, whose relationship is expressed as I 1 = I + Δ I . In addition, ( I + Δ I ) − 1 is presented as I − 1 + Δ I ˜ . The error acceleration can be derived from Eq. (19),

(20) ξ ˜ ˙ = H + d ˜ + I − 1 τ g ( I ) + I − 1 τ c ,

where

(21) H = I − 1 ad ξ * I ξ + ad ξ ˜ Ad h − 1 ξ 0 − Ad h − 1 ξ ˙ 0 ,

(22) d ˜ = Δ I ( ad ξ * I 1 ξ + τ g ( I ) + τ c ) + I − 1 ad ξ * Δ I ξ + I 1 − 1 ( τ d + τ g ( I 1 ) − τ g ( I ) ) ,

where d ˜ is the compound disturbance of the dynamics system. Let u = τ c , where u is to be utilized in the following control law design section.

Assumption 1

(Li et al. 2019) Both the parametric uncertainties and the external disturbances are bounded in practice. The actuators are unable to generate infinite velocities (including linear and angular velocities). Thus, the unknown dynamics is bounded. In addition, the compound disturbance d ˜ is assumed to be a continuous Lipschitz signal with a Lipschitz constant Δ , i.e., ‖ d ˜ ‖ ⩽ Δ holds almost everywhere.

Some notations are defined at the beginning. Let ∣ ⋅ ∣ denote the absolute value. ‖ ⋅ ‖ denotes the 2-norm of a vector. A vector x = [ x 1 , x 2 , … , x n ] T and constant γ define the function sig ( x ) γ = [ ∣ x 1 ∣ γ sgn ( x 1 ) , ∣ x 2 ∣ γ sgn ( x 2 ) , … , ∣ x n ∣ γ sgn ( x n ) ] T , where sgn ( ⋅ ) is the signum function.

3 Adaptive uniform exact convergent second-order sliding mode algorithm

3.1 Fixed gain-based uniform robust algorithm

The GSTA form in the study by Cruz-Zavala et al. (2011) is denoted as follows:

(23) z ˙ 1 = − k 1 ϕ 1 ( z 1 ) + z 2 z ˙ 2 = − k 2 ϕ 2 ( z 1 ) + χ ( t ) ,

where χ ( t ) is the uncertain term. ϕ 1 ( x ) and ϕ 2 ( x ) are denoted as follows:

(24) ϕ 1 ( x ) = sig ( x ) 1 2 + μ sig ( x ) 3 2 ϕ 2 ( x ) = 1 2 ( sgn ( x ) + 4 μ sig ( x ) + 3 μ 2 sig ( x ) 2 ) ,

where μ ⩾ 0 . If μ = 0 , Eq. (23) becomes the conventional STA. The high-degree terms, such as sig ( x ) 3 2 and sig ( x ) 2 , provide for uniform convergence in the algorithm. The algorithm can also be called the GSTA. Assuming that the first-order derivative of the uncertain term satisfies the boundary condition that χ ˙ ( t ) ⩽ L , where L > 0 is a positive constant (Cruz-Zavala et al. 2011). If k 1 and k 2 are in the set,

(25) K = ( k 1 , k 2 ) ∈ R 2 ∣ 0 < k 1 ≤ 2 L , k 2 > k 1 2 4 + 4 L 2 k 1 2 ∪ { ( k 1 , k 2 ) ∈ R 2 ∣ k 1 > 2 L , k 2 > 2 L } .

Then the differential system Eq. (23) is uniform exact convergent. In other words, the convergence time is independent of the system’s initial conditions. The fixed-time convergence has been strictly proved in Cruz-Zavala et al. (2011).

3.2 Adaptive gain-based uniform robust algorithm

A fixed-time adaptive algorithm with single-layer adaption law proposed in the study by Li et al. (2017) is presented as follows:

(26) z ˙ 1 = − k 1 ( t ) ϕ 1 ( z 1 ) + z 2 + ϕ 0 ( z 1 , L ) z ˙ 2 = − k 2 ( t ) ϕ 2 ( z 1 ) + χ ( t ) ,

where ϕ 1 ( x ) and ϕ 2 ( x ) are the same as presented in Eq. (24). Note that k 1 ( t ) and k 2 ( t ) are time varying, ϕ 0 ( z 1 , L ) is an adaptive compensation term, and they are defined as follows:

(27) k 1 ( t ) = k 10 L ( t ) , k 2 ( t ) = k 20 L ( t ) ,

(28) ϕ 0 ( z 1 , L ( t ) ) = − L ˙ ( t ) L ( t ) ϕ 1 ϕ 1 ′ = − L ˙ ( t ) L ( t ) 2 + 2 μ ∣ z 1 ∣ 1 + 3 μ ∣ z 1 ∣ z 1 ,

where L ˙ ( t ) denotes the first-order time derivative of L ( t ) , k 10 , and k 20 are positive constants. The adaptive parameter L ( t ) , in theory, varies with the disturbance. If a system with the asymptotically stable property has negative degree homogeneity in 0-limit and positive degree homogeneity in ∞ -limit, it is fixed-time stable (Lopez-Ramirez et al. 2016). Easy to find that the system Eq. (26) has the property of bi-limit homogeneity. Thus, the system mentioned earlier is fixed-time convergent with homogenous theory (Li et al. 2017). It should be noted that the single-layer adaption law in the study by Li et al. (2017) takes similar forms as equations from Eq. (29) to Eq. (32). For the sake of brevity, the detailed proof of the single-layer adaption law is omitted.

3.3 Dual-layer adaption law

Different from Li et al. (2017), a dual-layer adaption law of L ( t ) is proposed in this work, described as follows. The equivalent control u e q ( t ) is the average value of the switching signal k 2 ( t ) ϕ 2 ( z 1 ) must take to maintain sliding mode. It can be closely approximated by a real-time low-pass filtering of the signal k 2 ( t ) ϕ 2 ( z 1 ) . The filter is designed as follows:

(29) τ u ˙ e q = sig ( − k 20 ϕ 2 ( z 1 , L ) − u e q ( t ) ) 9 ∕ 11 + sig ( − k 20 ϕ 2 ( z 1 , L ) − u e q ( t ) ) 7 ∕ 5 ,

where 0 < τ ≪ 1 is a small positive filter constant. By filtering the signal − k 20 ϕ 2 , the estimation of the unknown disturbance χ ( t ) can be obtained as u e q . First, a variable δ ( t ) is designed as follows:

(30) δ ( t ) = L ( t ) − 1 a k 20 ∣ u e q ( t ) ∣ − ε ,

where a > 0 is a positive constant and satisfies that 0 < a k 20 < 1 , ε > 0 is a small positive constant.

(31) L ( t ) = l 0 + l ( t ) ,

where l 0 > 0 is the initial value of L ( t ) and the adaption law of l ( t ) is given as follows:

(32) l ˙ ( t ) = − ρ 1 ( t ) sgn ( δ ( t ) ) − ρ 2 ( t ) sig ( δ ( t ) ) 7 ∕ 5 , ∣ δ ( t ) ∣ > 1 ; − ρ 1 ( t ) sgn ( δ ( t ) ) − ρ 2 ( t ) sgn ( δ ( t ) ) , ∣ δ ( t ) ∣ ⩽ 1 ,

where the time-varying gains ρ 1 ( t ) , ρ 2 ( t ) are defined as follows:

(33) ρ 1 ( t ) = r 10 + r 1 ( t ) , ρ 2 ( t ) = r 20 + r 2 ( t ) ,

where r 10 > 0 and r 20 > 0 are the initial values of ρ 1 ( t ) and ρ 2 ( t ) , respectively. The adaption laws of r 1 ( t ) and r 2 ( t ) are given as follows:

(34) r ˙ 1 ( t ) = γ 1 ∣ δ ( t ) ∣ , r ˙ 2 ( t ) = γ 2 ∣ δ ( t ) ∣ ,

where γ 1 > 0 , γ 2 > 0 are positive constants. It is worth noting that coefficients ρ 1 ( t ) and ρ 2 ( t ) in the aforementioned adaption law keep as constants in the study by Li et al. (2017), and in which, the expression of l ˙ ( t ) only takes the first line of Eq. (32). This modification, different from the previous research, is a breakthrough of existing related adaption law. The following is proof of the dual-layer adaption law.

Theorem 1

Assuming that the disturbance χ ( t ) and its time derivative are bounded, that is, ∣ χ ( t ) ∣ ⩽ a 0 < ∞ and ∣ χ ˙ ( t ) ∣ ⩽ a 1 < ∞ , where a 0 and a 1 are unknown positive scalars (Li et al. 2017). Assuming that the estimation error of the filter Eq. (29) is zero (Li et al. 2017). The dual adaption law represented by Eqs. (29)–(34) can make the inequality L ( t ) > χ ( t ) holds.

Proof

When the estimation error of the filter is zero, δ ( t ) is denoted as follows:

(35) δ ( t ) = L ( t ) − 1 a k 20 ∣ χ ( t ) ∣ − ε .

Taking its time derivative and combining Eqs. (32) and (33), if ∣ δ ( t ) ∣ > 1 ,

(36) δ ( t ) δ ˙ ( t ) = δ ( t ) l ˙ ( t ) − δ ( t ) a k 20 χ ˙ ( t ) sgn ( χ ( t ) ) ⩽ δ ( t ) ( − ρ 1 ( t ) sgn ( δ ( t ) ) − ρ 2 ( t ) sig ( δ ( t ) ) 7 ∕ 5 ) + a 1 a k 20 ∣ δ ( t ) ∣ = − ( r 10 + r 1 ( t ) ) ∣ δ ( t ) ∣ − ( r 20 + r 2 ( t ) ) ∣ δ ( t ) ∣ 12 ∕ 5 + a 1 a k 20 ∣ δ ( t ) ∣ ,

if ∣ δ ( t ) ∣ ⩽ 1 ,

(37) δ ( t ) δ ˙ ( t ) = − ( r 10 + r 1 ( t ) ) ∣ δ ( t ) ∣ − ( r 20 + r 2 ( t ) ) ∣ δ ( t ) ∣ + a 1 a k 20 ∣ δ ( t ) ∣ .

Defining the following variables,

(38) e 1 ( t ) = a 1 a k 20 − r 1 ( t ) , e 2 ( t ) = a 1 a k 20 − r 2 ( t ) .

It can be deduced that the following equalities holds.

(39) e 1 ( t ) e ˙ 1 ( t ) = − γ 1 e 1 ( t ) ∣ δ ( t ) ∣ , e 2 ( t ) e ˙ 2 ( t ) = − γ 2 e 2 ( t ) ∣ δ ( t ) ∣ .

Next, the Lyapunov functions are designed to analyse the convergence of ∣ δ ( t ) ∣ , e 1 ( t ) , and e 2 ( t ) . Two essential functions are defined as follows:

(40) V 1 = 1 2 δ 2 + 1 2 γ 1 e 1 2 ,

(41) V 2 = 1 2 δ 2 + 1 2 γ 2 e 2 2 ,

1) In the interval δ > 1 , the Lyapunov function is defined as follows:

(42) V 0 = V 1 + V 2 = δ 2 + 1 2 γ 1 e 1 2 + 1 2 γ 2 e 2 2 .

The time derivative of the aforementioned function is as follows:

(43) V ˙ 0 = a 1 a k 20 − ( r 10 + r 1 ( t ) ) − ( r 20 + r 2 ( t ) ) ∣ δ ( t ) ∣ 7 ∕ 5 − e 1 ( t ) ∣ δ ( t ) ∣ + a 1 a k 20 − ( r 10 + r 1 ( t ) ) − ( r 20 + r 2 ( t ) ) ∣ δ ( t ) ∣ 7 ∕ 5 − e 2 ( t ) ∣ δ ( t ) ∣ .

According to Eq. (38), the inequality a 1 a k 20 − r 2 ( t ) ∣ δ ( t ) ∣ 7 ∕ 5 − e 2 ( t ) < 0 holds for δ ( t ) > 1 , and the equality a 1 a k 20 − r 1 ( t ) − e 1 ( t ) = 0 holds for all δ ( t ) . Zooming the right-hand side of (43) yields

(44) V ˙ 0 ⩽ − ( ρ 1 ( t ) + r 20 ) ∣ δ ( t ) ∣ − ( ρ 2 ( t ) + r 10 ) ∣ δ ( t ) ∣ 12 ∕ 5 .

Due to the facts that r 1 ( t ) and r 2 ( t ) remain increasing and the initial values r 10 > 0 and r 20 > 0 , ρ 1 ( t ) > 0 and ρ 2 ( t ) > 0 hold. Thus, it can be obtained that V ˙ 0 < 0 during the whole process. The aforementioned inequality demonstrates that in the interval ∣ δ ( t ) ∣ ∈ ( 1 , + ∞ ) , the adaption law could drive V 0 decreasing until the trajectory of ∣ δ ( t ) ∣ move to ∣ δ ( t ) ∣ = 1 . Moreover, ∣ δ ( t ) ∣ 12 ∕ 5 dominate in the interval, this term makes V 1 converge more rapidly. ∣ δ ( t ) ∣ will converge to ∣ δ ( t ) ∣ = 1 in finite time for a definitely known initial value greater than 1.

In addition, for bounded initial values r 10 and r 20 , ρ 1 ( t ) and ρ 2 ( t ) are also bounded during the interval δ ( t ) > 1 although they remain increasing. When ∣ δ ( t ) ∣ = 1 is obtained, V ˙ 0 < 0 still holds, V 0 will continue decreasing.

2) In the interval δ ⩽ 1 , we still choose V 0 as the Lyapunov function. According to Eq. (37), the time derivative of V 0 is expressed as follows:

(45) V ˙ 0 = a 1 a k 20 − ( r 10 + r 1 ( t ) ) − ( r 20 + r 2 ( t ) ) − e 1 ( t ) ∣ δ ( t ) ∣ + a 1 a k 20 − ( r 10 + r 1 ( t ) ) − ( r 20 + r 2 ( t ) ) − e 2 ( t ) ∣ δ ( t ) ∣ .

According to Eq. (38), the equalities a 1 a k 20 − r 1 ( t ) − e 1 ( t ) = 0 and a 1 a k 20 − r 2 ( t ) − e 2 ( t ) = 0 hold for all ∣ δ ∣ ⩽ 1 . Thus,

(46) V ˙ 0 = − ( ρ 1 ( t ) + r 20 ) ∣ δ ( t ) ∣ − ( ρ 2 ( t ) + r 10 ) ∣ δ ( t ) ∣ .

Easy to find that V ˙ 0 < 0 , then V ˙ 0 remains decreasing. Accordingly, the adaption law could drive the trajectories of ∣ δ ( t ) ∣ , e 1 ( t ) , and e 2 ( t ) move to ∣ δ ( t ) ∣ = 0 , e 1 ( t ) = 1 and e 2 ( t ) = 0 , and achieve a finite-time property of converging to the neighbourhood of zero.

Combining (44) and (46), the value of V ˙ 0 will always decrease until ∣ δ ( t ) ∣ decrease to 0. To sum up with the above analysis, it can be obtained that lim t → + ∞ δ ( t ) = 0 . For all t > 0 , there exist e 1 ( t ) ⩽ sup t > 0 e 1 ( t ) < + ∞ and e 2 ( t ) ⩽ sup t > 0 e 2 ( t ) < + ∞ . And according to the powers of ∣ δ ( t ) ∣ in Eqs. (44) and (46), it can be discovered that the dual adaption law is finite-time convergent for a known initial value of δ ( t ) . Thus, there must exist a finite time t 0 such that ∣ δ ( t ) ∣ ⩽ ε ∕ 2 for all time t > t 0 . Even so, the finite-time property of δ ( t ) will not affect the fixed-time performance of the adaptive GSTA Eq. (26). According to Eq. (35),

(47) ∣ δ ( t ) ∣ ⩽ L ( t ) − ∣ χ ( t ) ∣ a k 20 − ε ⩽ ε 2 .

And thus,

(48) L ( t ) − ∣ χ ( t ) ∣ a k 20 − ε ⩾ − ε 2 .

It follows that

(49) L ( t ) ⩾ ε 2 + ∣ u e q ∣ a k 20 ⩾ ∣ χ ( t ) ∣ .

In addition, according to the expression of δ ( t ) , there exist

(50) ∣ δ ( t ) ∣ = L ( t ) − 1 a k 20 ∣ u e q ∣ − ε .

Thus,

(51) L ( t ) ⩽ ∣ δ ( t ) ∣ + 1 a k 20 ∣ u e q ∣ + ε ⩽ ∣ δ ( t ) ∣ + 1 a k 20 ∣ χ ( t ) ∣ + ε .

Since that 0 < a k 20 ⩽ 1 holds and ∣ δ ( t ) ∣ is bounded, the adaptive term L ( t ) is robust to the disturbance χ ( t ) and meanwhile, remains bounded.

4 Second-order fixed-time pose feedback control scheme

4.1 Fixed-time sliding mode surface design

A fixed-time type sliding mode surface, which is proposed by using the velocity and exponential coordinate of pose tracking error, is denoted as follows:

(52) s = ξ ˜ + φ ,

where φ is defined as follows:

(53) φ = c 1 sig ( η ˜ ) 1 + γ + c 2 η ˜ + c 3 sig ( η ˜ ) 1 − γ ,

where 0 < γ < 1 , positive-definite matrix c j is defined as c j = diag ( c j 1 , c j i , … , c j 6 ) , j ∈ { 1 , 2 , 3 } , i ∈ { 1 , 2 , … , 6 } and the inequality 4 c ¯ m 1 c m 3 > c m 2 2 holds, where c ¯ m 1 = 6 − γ ∕ 2 c m 1 . Let c m j denotes the minimum element of c j . During the sliding phase, i.e., s = 0 , there exists

(54) ξ ˜ = − c 1 sig ( η ˜ ) 1 + γ − c 2 η ˜ − c 3 sig ( η ˜ ) 1 − γ .

The stability proof of dynamics system Eq. (54) is presented as follows.

Theorem 2

If the initial state of the pose tracking subsystem Eq. (17) satisfies that ‖ Θ ˜ ‖ ∈ [ 0 , π ) , such that η ˜ in the system Eq. (54) will converge to the equilibrium η ˜ = 0 in fixed-time with an almost globally convergence. The settling time satisfies that

(55) T ⩽ 4 γ 4 c ¯ m 1 c m 3 − c m 2 2 π 2 − arctan c m 3 4 c ¯ m 1 c m 3 − c m 2 2 .

Proof

The quadratic function V = 1 2 η ˜ T η ˜ is chosen as the Lyapunov candidate function. Taking the time derivative yields

(56) V ˙ = η ˜ T G ( η ˜ ) ξ ˜ .

Substituting Eq. (54) into the aforementioned equality

(57) V ˙ = η ˜ T G ( η ˜ ) ( − c 1 sig ( η ˜ ) 1 + γ − c 2 η ˜ − c 3 sig ( η ˜ ) 1 − γ ) ,

(58) ⩽ − ‖ λ min ( G ) ‖ ( c ¯ m 1 ( η ˜ T η ˜ ) 1 + γ ∕ 2 + c m 2 η ˜ T η ˜ + c m 3 ( η ˜ T η ˜ ) 1 − γ ∕ 2 ) ,

(59) = − ‖ λ min ( G ) ‖ ( c ¯ m 1 V 1 + γ ∕ 2 + c m 2 V + c m 3 V 1 − γ ∕ 2 ) ,

where ‖ λ min ( G ) ‖ denotes the eigenvalues’ minimum 2-norm of the matrix G ( η ˜ ) . Noting that G ( η ˜ ) is a positive definite matrix (Zhang et al. 2018). In addition, it can be deduced that G ( η ˜ ) has two eigenvalues of one and two pairs of conjugate complex eigenvalues and the real parts of which are positive. Therefore, V ˙ ⩽ 0 , and it can be concluded that η ˜ = 0 is almost globally convergent except those points that Θ ˜ = π . According to the expand expression of G ( η ˜ ) (Lee et al. 2017), the eigenvalues’ minimum 2-norm of G ( η ˜ ) can be computed, i.e., ‖ λ min ( G ) ‖ = 1 . Thus,

(60) V ˙ ⩽ − ( c ¯ m 1 V 1 + γ ∕ 2 + c m 2 V + c m 3 V 1 − γ ∕ 2 ) .

Deforming the aforementioned inequality yields

(61) d t ⩽ − d V c ¯ m 1 V 1 + γ ∕ 2 + c m 2 V + c m 3 V 1 − γ ∕ 2 ,

(62) = − 2 d V γ ∕ 2 V γ ∕ 2 − 1 ( c ¯ m 1 V 1 + γ ∕ 2 + c m 2 V + c m 3 V 1 − γ ∕ 2 ) ,

(63) = − 2 d V 1 γ γ ∕ 2 ( c ¯ m 1 V γ + c m 2 V γ ∕ 2 + c m 3 ) .

Let ϖ = V γ ∕ 2 , then d ϖ = γ 2 V γ ∕ 2 − 1 d V , and Eq. (63) can be converted to

(64) d t ⩽ − 2 d ϖ γ ( c ¯ m 1 ϖ 2 + c m 2 ϖ + c m 3 ) = − 2 d ϖ γ c ¯ m 1 ϖ + c m 2 2 c ¯ m 1 2 + 4 c ¯ m 1 c m 3 − c m 2 2 2 c m 1 2 ,

where V ( T ) = 0 means at time T , the system has converged to the equilibrium η ˜ = 0 . Then the settling time of the system can be obtained by integrating two sides of Eq. (64) from 0 to T .

(65) T ⩽ 4 γ 4 c ¯ m 1 c m 3 − c m 2 2 × arctan 2 c ¯ m 1 V γ ∕ 2 ( 0 ) + c m 2 4 c ¯ m 1 c m 3 − c m 2 2 − arctan c m 2 4 c ¯ m 1 c m 3 − c m 2 2 .

Due to the fact that arctan 2 c ¯ m 1 V γ ∕ 2 ( 0 ) + c m 2 4 c ¯ m 1 c m 3 − c m 2 2 ∈ 0 , π 2 , the differential system Eq. (54) has a definite upper bound of settling time. Thus, the trajectory of the subsystem Eq. (17) is almost globally stable with a fixed-time property.

4.2 Fixed-time attitude and position feedback control scheme

Taking the derivative of φ yields

(66) φ ˙ = ( ( 1 + γ ) diag ( c 1 ∣ η ˜ ∣ γ ) + c 2 + ( 1 − γ ) diag ( c 3 ∣ η ˜ ∣ − γ ) ) η ˜ ˙ .

Then the time derivative of s can be presented as follows:

(67) s ˙ = ξ ˜ ˙ + φ ˙ = H + I − 1 τ g + I − 1 u + d ˜ + φ ˙ = H 1 + I − 1 u + d ˜ ,

where H 1 is denoted as follows:

(68) H 1 = H + I − 1 τ g + φ ˙ .

An adaptive second-order sliding mode control law is designed in this research, which is denoted as follows:

(69) u ( t ) = u e ( t ) + u s ( t ) ,

where u e ( t ) denotes the equivalent control term, and u s ( t ) denotes the sliding mode control term.

(70) u e ( t ) = − I H 1 , u s ( t ) = I [ u s 1 , u s i , … , u s 6 ] T ,

(71) u s i = − k 10 , i L i ( t ) ϕ 1 i ( s i ) + ϕ 0 i ( s i , L i ( t ) ) − ∫ 0 t k 20 , i L i ( t ) ϕ 2 i ( s i ) d τ ,

where ϕ 1 i ( s i ) and ϕ 2 i ( s i ) are defined the same as Eq. (24).

(72) ϕ 0 i ( s i , L i ) = − L i ˙ ( t ) L i ( t ) 2 + 2 μ ∣ s i ∣ 1 + 3 μ ∣ s i ∣ s i ϕ 1 i ( s i ) = sig ( s i ) 1 2 + μ i sig ( s i ) 3 2 ϕ 2 i ( s i ) = 1 2 sgn ( s i ) + 2 μ i sig ( s i ) + 3 2 μ i 2 sig ( s i ) 2 .

The adaption laws of L i ( t ) are designed as follows,

(73) τ u ˙ e q , i = sig ( − k 20 , i L i ( t ) ϕ 2 i ( s i ) − u e q , i ) 9 11 + sig ( − k 20 , i L i ( t ) ϕ 2 i ( s i ) − u e q , i ) 7 5 L i ( t ) = l 0 i + l i ( t ) l ˙ i ( t ) = − ( r 10 , i + r 1 i ) sgn ( δ i ( t ) ) − ( r 20 , i + r 2 i ) sig ( δ i ( t ) ) 7 5 , ∣ δ ( t ) ∣ > 1 ; − ( r 10 , i + r 1 i ) sgn ( δ i ( t ) ) − ( r 20 , i + r 2 i ) sgn ( δ i ( t ) ) , ∣ δ ( t ) ∣ ⩽ 1 . r ˙ 1 i ( t ) = γ 1 i ∣ δ i ( t ) ∣ , r ˙ 2 i ( t ) = γ 2 i ∣ δ i ( t ) ∣ δ i ( t ) = l 0 i + l i ( t ) − 1 a i k 20 , i ∣ u e q , i ( t ) ∣ − ε i .

Theorem 3

Considering the spacecraft pose tracking error system Eqs. (17) and (20), assuming that d ˜ and its first-order derivative d ˜ ˙ are bounded by some constant. Driven by the adaptive second-order sliding mode control law Eq. (69) with dual-layer adaption law Eq. (73), the sliding mode surface s ( t ) will converge to neighborhood that contains the equilibrium within a fixed time.

Proof

According to the sliding mode surface Eq. (67),

(74) s ˙ = H 1 + I − 1 u + d ˜ .

Plugging the control law Eq. (69) into Eq. (74) yields the differential system s ˙ ( t ) . For convenience, considering the scalar form of s ˙ ( t ) ( i = 1 , 2 , … , 6 )

(75) s ˙ i ( t ) = − k 10 , i L i ( t ) ϕ 1 i ( s i ) + ϕ 0 i ( s i , L i ( t ) ) − ∫ 0 t k 20 , i L i ( t ) ϕ 2 i ( s i ) d τ + d ˜ i .

Let z 1 i = s i , z 2 i = − ∫ 0 t k 20 , i L i ( t ) ϕ 2 i ( s i ) d τ + d ˜ i . The time derivatives of z 1 i , z 2 i are as follows:

(76) z ˙ 1 i = − k 10 , i L i ( t ) ϕ 1 i ( s i ) + ϕ 0 i ( s i , L i ( t ) ) + z 2 i z ˙ 2 i = − k 20 , i L i ( t ) ϕ 2 i ( s i ) + d ˜ ˙ i .

Easily finding that Eq. (76) have the same differential structure as Eq. (26). On account of bounded compound disturbance, according to Theorem 1, L i ( t ) will update with time and holds robustness to d ˜ i , i.e., L i ( t ) > d ˜ i . Furthermore, according to Li et al. (2017), and previous analyses, differential system Eq. (76) has a fixed-time convergence property. Therefore, the adaptive second-order control law Eq. (69) can drive the sliding surface variable s ( t ) to converge to a neighborhood near the equilibrium point.

Different from the DO-based control schemes (Zhang et al. 2018, 2021), the proposed second-order sliding control scheme in this research do not consider any disturbance estimation. While the adaption technology of the second-order controller provides an estimation of the compound disturbance essentially. Thus, the integral term is essentially an estimator and can provide compensation.

The adaptive second-order control law in this article is named “ASOSMC”. To demonstrate the performance of the adaptive control scheme, other two schemes are considered for comparison, of which the gains are fixed. Only the sliding mode part u s ( t ) is different among the three control laws. Thus, only the sliding mode part of each control law is present. The non-adaptive super-twisting control law with invariant gains based on Eq. (23) is designed and named “SOSMC”, the sliding mode control term u s of “SOSMC” is denoted as follows:

(77) u s i = − k 1 i ϕ 1 i ( s i ) − ∫ 0 t k 2 i ϕ 2 i ( s i ) d τ ,

where the functions ϕ 1 i ( s i ) and ϕ 2 i ( s i ) are expressed in Eq. (24). In addition, the invariant gain-based first-order sliding mode control law is proposed for comparison, i.e., the integral part in Eq. (77) is removed, which is named “FOSMC” and denoted as follows:

(78) u s i = − k 1 i ϕ 1 i ( s i ) .

The sliding mode control term u s of the four different controllers are summarized in Table 1. Therefore, combing the equivalent terms with Eqs. (69) and (70) can generate three kinds of control schemes.

Table 1

Control laws and relevant formulas

Scheme	Formulas related to sliding mode term u s
ASOSMC	(69)(70)(71)(72)(73)
SOSMC	(77)(24)
FOSMC	(78)(24)
AFTSMC	(80)

In addition to the three control schemes mentioned above, an adaptive fixed-time terminal sliding mode controller (“AFTSMC”) (Huang and Jia 2017) for spacecraft pose is simulated with the same condition for comparison. In the study by Huang and Jia (2017), a fixed-time sliding mode surface and a fixed-time reaching law are proposed. Moreover, the inertia uncertainties and the upper bound of disturbance are estimated by adaptive estimation technology. We transformed the controller in the study by Huang and Jia (2017) to fit the scenario in this work. Here, we present the critical parts of the controller. The fixed-time sliding mode surface is

(79) s = ξ ˜ ˙ + c 4 sig ( η ˜ ) 1 + γ + c 5 sig ( η ˜ ) 1 − γ .

The form of the controller is given as follows:

(80) u = c 6 s 0.5 + c 7 s 1.5 + Y ( ϑ 0 + Δ ϑ ˆ ) + u a d p ,

where Y ( ϑ + Δ ϑ ) is the equivalent term relevant with inertia parameters, and u a d p is the estimated term to resist disturbances. The details of both two terms are omitted here for brevity. Δ ϑ ˆ is the estimation of the inertia uncertainty.

Remark 1

The coefficient of the sliding mode part of “SOSMC” is selected using the trial-error method. The controller “ASOSMC” is first simulated. Then, the values of k 1 i and k 2 i in “SOSMC” is selected to guarantee the same steady-state accuracy as “ASOSMC”.

Remark 2

For fairness, some parameters of controller “AFTSMC” for comparison should be consistent with that in this work. Note that a non-singular fixed-time sliding mode surface is proposed in Huang and Jia (2017), while we do not consider the non-singular one but one as expressed in Eq. (79). The two powers in the sliding mode surface of “AFTSMC” should be ( 1 + γ ) and ( 1 − γ ) . The two powers in the reaching law should be selected as 1 2 and 3 2 , as presented in Eq. (80). For the value selections of c 4 and c 5 , we follow the rule of c 1 + c 2 + c 3 = c 4 + c 5 .

As is known, the existence of the sign function in Eqs. (72) and (73) will lead to chattering effects. Therefore, the sign function “sgn(x)” is replaced by the saturation function “ sat ( x , δ s ) ,” which is denoted as follows:

(81) sat ( x , δ s ) = x ∕ δ s , ∣ x ∕ δ s ∣ ⩽ 1 ; sgn ( x ) , ∣ x ∕ δ s ∣ > 1 ,

where δ s is a sufficiently small positive constant, and the value is selected as 2.5 × 1 0 − 5 in this study.

5 Results

To demonstrate the effectiveness of the proposed control schemes. Numerical simulations are performed for a rigid spacecraft’s close pose tracking with inertia uncertainties and external disturbances.

5.1 Initial parameters of the simulations

A large attitude maneuver and a close position maneuver are considered in this simulation. The leader spacecraft is assumed to move on a highly elliptical orbit, and the follower moves on the leader’s neighborhood orbit. Orbital elements in Table 2 and the relative states in Table 3 define the initial pose and velocity of the spacecraft tracking system. The leader spacecraft’s initial orientation is assumed to align its body-fixed frame with the RTN (Tangent-Transverse-Normal) coordinate system. Assuming that the leader moves in the gravity field modeled like the follower but is subject to no uncertainties or maneuvers. The desired position of the follower is +5 m at the x -axis in the leader’s body fixed frame, which means that the desired relative position remains stationary. Moreover, the desired orientation of the follower is assumed to keep in synchronization with the leader.

Table 2

Initial orbital elements of the leader

Orbital element	value
Semi-major axis (km)	26628
Eccentricity	0.7417
Inclination ( ∘ )	63.4
Argument of perigee ( ∘ )	210
RAAN ( ∘ )	0
True anomaly ( ∘ )	90

Table 3

Initial motion states

Initial states	Values
Leader’s angular velocity (rad/s)	[ 0 , 0 , 0.0012 ] T
Follower’s angular velocity (rad/s)	[ 0.01 , 8 , ‒ 10 ] T × 1 0 ‒ 4
Relative position (m)	[ 20 , 15 , 15 ] T
Relative attitude (rad)	2 π ∕ 3
Relative principal rotation axis	[ 1 , 2 , 3 ] T
Relative linear velocity (m/s)	[ ‒ 0.015 , ‒ 0.27 , ‒ 0.07 ] T

The inertia parameters of the leader spacecraft are given as follows:

m 0 = 110 kg , J 0 = 21.7 − 0.2 − 0.5 − 0.2 22.3 − 0.3 − 0.5 − 0.3 25.5 kg ⋅ m 2 .

The nominal inertia parameters of the follower spacecraft are follows:

m = 100 kg , J = 23 − 0.4 − 0.5 − 0.2 25 − 0.3 − 0.5 − 0.3 22 kg m − 2 .

The inertia parametric uncertainties of the follower spacecraft are defined as follows:

Δ m = 10 kg ,

Δ J = 0.2 diag ( sin ( 0.1 t ) ) + 0.5 , sin ( 0.2 t ) + 0.5 , sin ( 0.3 t ) + 0.5 kg ⋅ m 2 .

The external disturbances torque and force are described as follows:

(82) T d = 0.02 [ sin ( 0.2 t ) , sin ( 0.2 t + π ∕ 6 ) , sin ( 0.2 t + π ∕ 4 ) ] T F d = 0.1 [ sin ( 0.2 t + π ∕ 9 ) , sin ( 0.2 t + π ∕ 3 ) , cos ( 0.2 t ) ] T .

Moreover, extra abrupt disturbance signals are considered to demonstrate the robustness of the proposed controller resists the non-smooth disturbance. A pulse signal with 0.3 N m − 1 that acts from 120 to 140 s is added on the Z channel of disturbance torque. Another pulse signal with 1N that acts from 140 to 160 s is added on the Z channel of disturbance force.

If the controller (69) is expressed by components form as u ( t ) = [ u 1 , u 2 , u 3 , u 4 , u 5 , u 6 ] . Then, the control torque and control forces in Eqs. (7) and (8) can be expressed as follows:

(83) T c = J [ u 1 , u 2 , u 3 ] T F c = m [ u 4 , u 5 , u 6 ] T .

In practical engineering, the actuators cannot output arbitrary large control torques or forces. In this work, the control inputs of the closed-loop system are assumed to be bounded by some constants, which are ∣ T c i ∣ ⩽ T c m a x = 0.5 N ⋅ m and ∣ F c i ∣ ⩽ F c m a x = 5 N , respectively.

Fourth-order Runge-Kutta method with a step size of 0.02s is used for the numerical simulations, and the simulation time is t f = 200 s . The parameters of the control scheme are presented in Table 4. The parameters of the sliding mode surface corresponding to the first three controllers are γ = 0.2 , c 1 = 0.01 diag ( [ 2 , 2 , 2 , 1 , 1 , 1 ] ) , c 2 = 0.01 diag ( [ 6 , 6 , 6 , 2 , 2 , 2 ] ) , and c 3 = 0.01 diag ( [ 12 , 12 , 12 , 5 , 5 , 5 ] ) . The parameters of the sliding mode surface for “AFTSMC” are γ = 0.2 , c 4 = 0.01 diag ( [ 8 , 8 , 8 , 3 , 3 , 3 ] ) , c 5 = 0.01 diag ( [ 12 , 12 , 12 , 5 , 5 , 5 ] ) .

Table 4

Controllers’ parameters for the simulations

Control scheme	Values
ASOSMC	a i = 2 , μ 0 i = 0.1 , ε i = 0.15 , τ = 0.05 , k 10 , i = 0.4 , k 20 , i = 0.1 , l 0 i = 1 0 ‒ 5 , r 10 , i = r 20 , i = 1 0 ‒ 3 , γ 1 i = γ 2 i = 3 × 1 0 ‒ 3
SOSMC	k 1 i = 0.45 , k 2 i = 0.012 , μ = 0.1
FOSMC	k 1 i = 1 , k 2 i = 0 , μ = 0.1
AFTSMC	c 6 = 5 c 7 = 4 diag ( [ 1 , 1 , 1 , 5 , 5 , 5 ] )

5.2 Performance comparisons of the controllers

In this subsection, the simulation results of the three controllers are presented. To distinctly present the evolutions of rotational and translational motion in the maneuvering process, the Y -axis of the pose and velocities are converted to a logarithmic scale base 10. Figure 1(a) and (b) present the attitude and position tracking error, respectively. Figure 2(a) and (b) show the norm of angular velocity and translational velocity, respectively.

Figure 1

Pose tracking error of controllers. (a) Attitude tracking errors and (b) position tracking errors.

Figure 2

Velocity tracking error of controllers. (a) Rotational velocity errors and (b) translational velocity errors.

As one can see from Figures 1 and 2, except for the action range of the abrupt disturbance, the attitude and position tracking errors of “ASOSMC” and “SOSMC” reach to extremely high accuracies that less than 4 × 1 0 − 7 rad and 1.5 × 1 0 − 6 m, respectively. The steady-state accuracy of “FOSMC” and “AFTSMC” is worse than that of “ASOSMC” and “SOSMC”. One can also find that the convergence rate of “SOSMC” is the fastest.

The responses to the pulse interferences of the four schemes can also be seen in Figures 1 and 2. The peak values of the responses are presented in the last two rows in Table 5. One can see the slightest perturbations of attitude and position errors due to the pulse interferences appear in “ASOSMC”. “SOSMC” possesses similar performances. The attitude and position errors of “ASOSMC” recover to steady-state quicker than other schemes. It is a pity that “AFTSMC” maintains the worst performance among the four schemes in the tracking control, no matter the tracking error or the robustness to pulse interferences, even though the estimation technology is used in the scheme.

Table 5

Control performance comparisons

Performance indices	ASOSMC	SOSMC	FOSMC	AFTSMC
‖ θ ˜ ‖ ( rad )	4 × 1 0 ‒ 7	4 × 1 0 ‒ 7	5 × 1 0 ‒ 6	1 × 1 0 ‒ 5
‖ β ˜ ‖ ( m )	1.5 × 1 0 ‒ 6	1.5 × 1 0 ‒ 6	5 × 1 0 ‒ 6	1 × 1 0 ‒ 5
t s t , A (s)	36.6	34.8	36.6	40.2
t s t , p (s)	114.5	106.5	114.5	116.2
∫ 0 t f ‖ T c ‖ d t ( N ⋅ s )	20.38	22.42	24.12	27.42
∫ 0 t f ‖ F c ‖ d t ( N ⋅ m ⋅ s )	285.65	318.20	292.73	355.71
‖ θ ˜ ‖ peak ( rad )	1.7 × 1 0 ‒ 3	4 × 1 0 ‒ 3	2.2 × 1 0 ‒ 3	1.7 × 1 0 ‒ 2
‖ β ˜ ‖ peak ( m )	0.9 × 1 0 ‒ 3	1.85 × 1 0 ‒ 3	3 × 1 0 ‒ 3	1 × 1 0 ‒ 2

Integration of norms of control torque and force are shown in Figure 3(a) and (b), which are chosen as the energy consumption indicators of the schemes.

Figure 3

Integration of norms of the control signals. (a) Integration of norm of torque and (b) integration of norm of force.

Table 5 presents the detailed control performances. t s t , A and t s t , p in Table 5 denote the convergence time of attitude and position as 1 × 1 0 − 5 rad∕s and 1 × 1 0 − 4 m∕s , respectively. It can be found that in Table 5 that other three control schemes cost more energy than the “ASOSMC”, as also demonstrated in Figure 3. And one can also find that “AFTSMC” consumes the most control energy in the four controllers. As mentioned in Remark 1, to achieve the same steady-state accuracy level as “ASOSMC”, the coefficients of sliding mode surface in “SOSMC” must be tuned relatively large. Larger coefficients may lead to more significant control signals in “SOSMC” than in “ASOSMC”. Thus, “SOSMC” costs more energy than “ASOSMC”. In a word, “ASOSMC” can achieve the same or higher steady-state accuracy compared with other schemes but cost less energy.

The saturation durations of the actuators corresponding to the three control schemes are present in Table 6 and intuitively shown in Figures 4, 5, 6 7. t s ( T i ) or t s ( F i ) denotes the saturation duration of each control channel. Total saturation durations of rotational and translational channels are also computed. As clearly shown in Table 6, the total saturation durations of “ASOSMC” is less than that of the other three schemes.

Table 6

Saturation duration of each control channel

Scheme	t s ( T x )	t s ( T y )	t s ( T z )	t s ( F x )	t s ( F y )	t s ( F z )	∑ t s ( T i )	∑ t s ( F i )
ASOSMC	1.65	2.51	4.24	7.34	7.08	7.83	8.40	22.25
SOSMC	2.14	3.01	7.69	14.69	5.74	11.51	12.84	31.94
FOSMC	2.08	3.88	6.62	8.20	8.04	8.02	12.58	24.26
AFTSMC	1.71	2.28	4.69	10.45	6.26	7.91	8.68	24.62

Figure 4

Control signals of “ASOSMC”. (a) Control torque of “ASOSMC” and (b) control force of “ASOSMC”.

Figure 5

Control signals of “SOSMC”. (a) Control torque “SOSMC” and (b) control force “SOSMC”.

Figure 6

Control signals of “FOSMC”. (a) Control torque “FOSMC” and (b) control force “FOSMC”.

Figure 7

Control signals of “AFTSMC”. (a) Control torque “AFTSMC” and (b) control force “AFTSMC”.

The curves of all components of the adaptive parameter L ( t ) are presented in Figure 8. The value of L i ( t ) monotonously increases in the early stage and reaches a relatively stationary state, which could account for the slower convergence rate of “ASOSMC” compared with “SOSMC”. The coefficients of the sliding mode surface vary with L i , and determine the rapidity and robustness of the controller. Then, we analyse the adaptive term k 20 , i L i ( t ) ϕ 2 , i ( s i ) . The curve of each component is shown in Figure 9. As a matter of fact, k 20 , i L i ( t ) ϕ 2 , i ( s i ) denotes the estimation of the first-order derivative of the compound disturbance, i.e., d ˜ ˙ i . Considering the compound disturbance denoted by Eq. (22). As the pose and velocity tracking errors are extremely close to 0, the external disturbances by Eq. (82) dominate in the steady state. From Figure 9, it can be found that the amplitude and frequency match the first-order derivative of Eq. (82) if it is left divided by inertial matrix. It means the adaption law can estimate the compound disturbance to some extent.

$Figure 8 Adaptive parameter L i ( t ) {L}_{i}\left(t) .$

Figure 8

Adaptive parameter L i ( t ) .

$Figure 9 Robust term k 20 , i L i ( t ) ϕ 2 , i ( s i ) {k}_{20,i}{L}_{i}\left(t){\phi }_{2,i}\left({s}_{i}) .$

Figure 9

Robust term k 20 , i L i ( t ) ϕ 2 , i ( s i ) .

The estimation of inertia uncertainties of “AFTSMC” is shown in Figure 10. The upper bound estimations of disturbance are presented in Figures 11 and 12. Actually, the estimations of these parameters are strongly dependent on the coefficients of the adaption law. Even though the coefficients are properly tuned, the adaption law can still not exactly estimate the unknown parameters.

Figure 10

Estimation of inertia uncertainties.

Figure 11

Disturbance upper bound estimation of attitude channel.

Figure 12

Disturbance upper bound estimation of position channel.

6 Discussion

As mentioned earlier, the existing controller “AFTSMC” cannot guarantees the mass and inertia matrix estimates converge to their true values. This is to be expected as this reference motion is not designed to provide persistence of excitation. Moreover, in “AFTSMC”, the upper bounds of the disturbances are estimated by first-order estimators and applied in adaptive term u a d p (Huang and Jia 2017). Thus, the controller “AFTSMC” may provide inaccurate compensation to resist the disturbance and inertia parametric uncertainties and lead to the worst pose and velocity tracking errors of “AFTSMC”, as shown in Figures 1 and 2. Obviously, the responses of tracking errors to pulse interference in “AFTSMC”, as shown in Figure 1, are larger than other three controllers. These again demonstrate that the first-order estimators in “AFTSMC” can not provide superior estimations. In addition, the upper bound estimations in “AFTSMC” also lead to larger control signals and then, lead to more energy consumption compared with other three controllers. Due to the adaptive scheme, the attitude and position errors of “ASOSMC” converge quicker than “SOSMC” after pulse interference. As is known, the integral term in proportional-integral-derivative control contributes to reduce the dynamic error. This accounts for the reason that the steady-state error in “SOSMC” and “ASOSMC” are smaller than other two controllers.

To summarize the proposed control law in this article. First, the high-order control scheme is strongly robust to the uncertainties, including smooth and non-smooth interferences, and guarantees a very high steady-state control accuracy for attitude and position control. Owing to the integration parts of the second-order sliding mode controller (“SOSMC” and “ASOSMC”), the tracking error of the closed-loop system could be reduced to a very high accuracy than the first-order controller (“FOSMC”) and the controller (“AFTSMC”) in the study by Huang and Jia (2017). Second, the adaptive second-order schemes generate smaller control signals but do not essentially influence the convergence time of the states compared with the non-adaptive second-order one. Owing to adaptive gains, the adaptive second-order control scheme costs less energy to a certain extent without losing accuracy. And thus, to some extent, the adaptive second-order method reduces the saturation durations of the actuators. This may further optimize the transient performance of the closed-loop system. Third, the compound disturbance of the spacecraft closed-loop system can be accurately estimated by the adaptive algorithm, and can be compensated effectively in the integral term. The adaptive control scheme in this article possesses better performances than the inertia parametric estimation-based or upper bound estimation-based ones.

7 Conclusion

The control problem of spacecraft pose tracking in the presence of internal and external disturbances is studied. A fixed-time sliding mode surface is designed to ensure the almost globally fixed-time convergence of the tracking pose error over the space SE ( 3 ) × R 6 . The adaptive second-order control law guarantees a fixed-time uniform exact convergence property of the closed-loop system. Besides, the convergence property of the dual-layer adaption law corresponding to the control scheme is proved. Simulations are performed to demonstrate the proposed control scheme’s excellent performance, as discussed in the previous section.

While, there exists also some limitations of this work. The relative motion measurement errors are assumed zero, and all states are assumed to be available in controller design. The measurement noises of the sensors and partial state feedback are not considered in this work. In future work, the noises of the sensors, velocities-free (with unknown angular velocity and linear velocity) can be considered in the spacecraft 6-DOF control, to make the controller more close to engineering application.

Acknowledgments

We gratefully acknowledge the reviewers for their helpful and constructive suggestions that helped us substantially improve the article.

Funding information: Funded by State Key Laboratory of Geo-Information Engineering (NO. SKLGIE2021-M-1-2) and National Natural Science Foundation of China (No. U21B2050).
Author contributions: K.G.: conceptualization, methodology, software, validation, data curation, writing-original draft, visualization, and project administration; Y.W.: formal analysis, resources, writing-original draft, writing review and editing; Y.D.: software, formal analysis, data curation, writing review and editing; Y.M.: methodology, software, validation,writing review and editing; Y.J.: conceptualization, methodology, supervision, project administration, and funding acquisition; D.L: resources and writing-review and editing.
Conflict of interest: The authors declare no conflict of interest.

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Received: 2023-05-07

Revised: 2023-10-30

Accepted: 2023-10-31

Published Online: 2024-01-05

This work is licensed under the Creative Commons Attribution 4.0 International License.

A generalized super-twisting algorithm-based adaptive fixed-time controller for spacecraft pose tracking

Abstract

1 Introduction

2 Background and preliminaries

2.1 Dynamics of rigid spacecraft on SE(3)

2.2 Relative dynamics between two spacecrafts

Assumption 1

3 Adaptive uniform exact convergent second-order sliding mode algorithm

3.1 Fixed gain-based uniform robust algorithm

3.2 Adaptive gain-based uniform robust algorithm

3.3 Dual-layer adaption law

Theorem 1

Proof

4 Second-order fixed-time pose feedback control scheme

4.1 Fixed-time sliding mode surface design

Theorem 2

Proof

4.2 Fixed-time attitude and position feedback control scheme

Theorem 3

Proof

Remark 1

Remark 2

5 Results

5.1 Initial parameters of the simulations

5.2 Performance comparisons of the controllers

6 Discussion

7 Conclusion

Acknowledgments

References

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