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ORIGINAL RESEARCH article

Front. Robot. AI, 14 February 2023
Sec. Space Robotics

Fixed-time regulation of spacecraft orbit and attitude coordination with optimal actuation allocation using dual quaternion

Lichao SunLichao Sun1Yanpei HuangYanpei Huang2Haolin FeiHaolin Fei3Bo XiaoBo Xiao4Eric M. YeatmanEric M. Yeatman5Allahyar Montazeri
Allahyar Montazeri3*Ziwei WangZiwei Wang3
  • 1School of Education, Communication & Society, King’s College London, London, United Kingdom
  • 2Department of Bioengineering, Imperial College London, London, United Kingdom
  • 3School of Engineering, Lancaster University, Lancaster, United Kingdom
  • 4Department of Computing, Imperial College London, London, United Kingdom
  • 5Department of Electrical and Electronic Engineering, Imperial College London, London, United Kingdom

On-orbit service spacecraft with redundant actuators need to overcome orbital and attitude coupling when performing proximity maneuvers. In addition, transient/steady-state performance is required to fulfill the user-defined requirements. To these ends, this paper introduces a fixed-time tracking regulation and actuation allocation scheme for redundantly actuated spacecraft. The coupling effect of translational and rotational motions is described by dual quaternion. Based on this, we propose a non-singular fast terminal sliding mode controller to guarantee fixed-time tracking performance in the presence of external disturbances and system uncertainties, where the settling time is only dependent on user-defined control parameters rather than initial values. The unwinding problem caused by the redundancy of dual quaternion is handled by a novel attitude error function. Moreover, optimal quadratic programming is incorporated into null space pseudo-inverse control allocation that ensures the actuation smoothness and never violates the maximum output capability of each actuator. Numerical simulations on a spacecraft platform with symmetric thruster configuration demonstrate the validity of the proposed approach.

1 Introduction

Orbit and attitude coordination benefits the motion accuracy and efficiency of spacecraft by controlling orientation and position simultaneously. Several works in non-cooperative target capture (Huang et al., 2006; Zhang et al., 2017) and space teleoperation (Wang et al., 2019a) have demonstrated that these tasks can be performed through separate orientation and position control loops through the corresponding actuators, which ease the stability analysis and control synthesis. However, they might not be directly applied in proximate on-orbit servicing tasks due to the coupling effect between translation and orientation motions. On the one hand, rotation motion driven by reaction flywheels will result in orbital motion and vice versa. On the other hand, space load (e.g. space manipulator) motion will impact the pose of the whole system subject to the conservation of angular momentum (Wang et al., 2021).

How to model the coordinated dynamic that involves translation and orientation motions of spacecraft remains challenging for stability analysis and control design. Dual quaternion (DQ) is an alternative to describe the aforementioned coupling effect (Brodsky and Shoham, 1999). With a DQ-based velocity-free controller (Filipe and Tsiotras, 2013), the relative position and attitude were globally asymptotic stable for rigid body motion. The DQ-based modeling also becomes promising in spacecraft formation (Nixon and Shtessel, 2022) and perception (Srivatsan et al., 2016; Reynolds et al., 2020), whereas lacks the consideration of transient-state performance. As an important metric to evaluate transient-state convergence performance, settling time has gradually transformed from control objective to control parameter, thus allowing for more flexible control structures in terms of finite/fixed-time stability (Bhat and Bernstein, 2000; Zhu et al., 2011; Polyakov et al., 2015; Chen et al., 2022; Wang et al., 2022). In terms of faster convergence performance and robustness against disturbances, recent works in (Wang et al., 2020; 2019b) have witnessed that terminal sliding mode control (TSMC) can provide a route for finite-time stability. However, the settling time relies on accurate initial values, which might be limited to practical control implementation. Since noise signal is inevitable for measurement in practice, the resulting settling time tends to enlarge the estimation conservatism.

In addition, spacecraft are generally equipped with redundant sets of actuators in terms of safety. Control allocation plays an important role in over-actuated systems to distribute the control output among the redundant actuators. The fixed, single-gimbal, and double-gimbal thruster configurations were discussed in (Servidia, 2010). In order to improve the torque capacity during maneuvers by determining the initial wheel angular momentum, maximizing the efficiency of torque distribution for low-capacity reaction wheel assemblies was discussed in (Choi et al., 2008). In (Schaub and Lappas, 2009), an optimal torque distribution strategy was developed for reaction wheels to minimize the instantaneous electrical power requirement. In order to allocate the moments of the three axes to the corresponding control surfaces, robust least-square control allocation for unstructured and structured uncertainties was considered with a combination of H2/H feedback and feedforward control (Cui and Yang, 2011).

Motivated by the above observation, we use DQ to develop a non-singular fixed-time terminal sliding mode control (NFxTSMC) strategy for a 6-degree-of-freedom (DoF) spacecraft with settling time requirement, which can overcome the unwinding problem induced by DQ. Moreover, an optimal null-space based pseudo-inverse (ONSPI) control allocation strategy facilitates alleviating the physical restrictions on actuation characteristics. The main contributions of this paper are presented as follows:

1) The DQ-based control scheme features fixed-time and unwinding-free convergence while handling the coupling between translation and orientation motions. Compared with the previous work (Sun et al., 2022), we have shown the scalability of the proposed control scheme from finite-time to fixed-time stability, where the settling time only relies on the user-defined control parameters rather than initial values.

2) Control allocation strategy in a framework of optimal quadratic programming can address the output constraint of the redundant actuators. Compared with the traditional pseudo-inverse (PI) method, multiple constraints can be incorporated into the cost function that provides superior control allocation performances. Optimal quadratic programming is employed in the null space pseudo inverse control allocation that ensures the actuation smoothness and never violates the maximum output capability of each actuator.

The remainder of this paper is organized as follows. In Section 2, the relative kinematics and dynamics are derived for a class of 6-DoF orbit and attitude coordination spacecraft systems based on DQ. In Section 3, an NFxTSMC is proposed considering external disturbances, system uncertainties, and singularity phenomenon. Furthermore, the control allocation strategy is designed in Section 4. In Section 5, numerical simulations on a platform of spacecraft rendezvous and docking have demonstrated the effectiveness of the proposed control and actuator allocation method, followed by conclusions drawn in Section 6.

2 Preliminaries

2.1 Dual quaternion

In order to describe the translational and rotational motion simultaneously, we introduce DQ (Brodsky and Shoham, 1999): q̂η+ϵξ, where η and ξ are the real and dual part, respectively, and ϵ the dual operator such that ϵ2 = 0, ϵ ≠ 0. Here, η and ξ are both quaternions. In the following, ()̂ stands for the DQ variable. Taking â=a+ϵa and b̂=b+ϵb as an example, the following operators (Wang and Sun, 2012) are used throughout this paper:

â±b̂a±b+ϵa±b,â*a+ϵa,âTaT+ϵaT,â1a1+ϵa1,(1)
â×b̂a×b+ϵa×b+a×b,âb̂=âb̂ab+ϵab,(2)
<â|b̂>ab+ab,â|b̂aTb+aTb,(3)
âb̂abab+ϵab+ab+a×b,(4)
âb̂i.f.f.abANDab.(5)

Given the desired pose q̂d, the DQ error can be expressed as:

q̂e=q̂d*q̂=qe+ϵ12qepe(6)

where qe and pe stand for the quaternion and position errors, respectively. Taking the time derivative of q̂e yields:

q̂̇e=12q̂eω̂e,ω̂eω̂q̂e*ω̂dq̂e,(7)

in which ω̂ and ω̂d are the actual and desired velocity motors. Therefore, taking the time derivative of ω̂e, we can obtain the DQ error dynamics (Wang and Sun, 2012):

M̂ω̂̇e=û+M̂ω̂e×q̂e*ω̂dq̂eq̂e*ω̂̇dq̂eω̂×M̂ω̂(8)

where M̂=mddϵI+ϵJ is the dual inertial matrix, m the mass, J the inertial matrix, I the identity matrix with appropriate dimensions. û is dual force motor such that

û=ûc+ûd=uc+ud+ϵτc+τd,(9)

where uc and τc are the control force and torque to be designed. ud and τd are the external force and torque disturbance.

2.2 Necessary lemmas

Lemma 1 (Bhat and Bernstein, 2000). Consider a class of continuous-time systems

ẋ=fx,t,(10)

where xU is the system state and f the non-linear function. If there exists a continuously differentiable function V:UR+{0} such that V̇(x)+γVα(x)0, with γ, α ∈ (0, 1), then the equilibrium of the system trajectory (10) is globally finite-time stable and the settling time T satisfies TV01αγ(1α), in which V0 is the initial value of V.

Lemma 2 (Polyakov, 2011). The equilibrium of the system trajectory (10) is globally fixed-time stable if there exist positive constants: α, β, p, q, k, with pk < 1 and qk > 1, such that

V̇xαVpx+βVqxk,(11)

and the settling time T follows:

T1αk1pk+1βkqk1.(12)

3 Control design and analysis

Considering the 6-DoF trajectory tracking task, the control objective can be stated as follows: design a DQ-based controller ûc=uc+ϵτc such that the relative error states of a class of spacecraft systems 8) converge within the fixed time, under all time and physically realizable initial conditions. That is, ξ̂e(qe)v+ϵ(qe)v[0,0,0]T+ϵ[0,0,0]T and ω̂e0̂ for tT, where (⋅)v denotes the vector part of quaternion and T is the settling time. In the existence of external disturbances and system uncertainties, the error dynamics can be equivalently expressed as:

ξ̂̇e=Θ̂q̂eω̂e,M̂0ω̂̇e=ûc+M̂0ω̂e×q̂e*ω̂dq̂eq̂e*ω̂̇dq̂eω̂×M̂0ω̂+Φ̂,(13)

where Θ̂(q̂e)=12(η̂eI+ξ̂e×) with η̂̇e=12η̂eTω̂e. (⋅)× represents the cross product operator. M̂0 and ΔM̂ the nominal and uncertain part of M̂ such that M̂=M̂0+ΔM̂ and therefore Φ̂ can be written as:

Φ̂=ΔM̂ω̂̇e+ΔM̂ω̂e×q̂e*ω̂dq̂eq̂e*ω̂̇dq̂eω̂×ΔM̂ω̂+ûd.(14)

In order to stabilize ξ̂e and ω̂e within fixed time, traditional sliding mode (Wang et al., 2018) can be modified as dual form:

Ŝi=Si+ϵSi=ξ̂̇ei+b̂isigξ̂eiα,i=1,2,3,(15)

where Ŝ=[Ŝ1,Ŝ2,Ŝ3]T, sig(ξ̂ei)α=sig(ξei)α+ϵsig(ξei)α, sig()=sgn()α, and sgn(⋅) is the sign function. α ∈ (0, 1). b̂i=bi+ϵbi with bi and bi being positive constants. Based on the terminal sliding mode (15), the resulting TSMC can be then designed as a similar structure in (Dong et al., 2016). However, the implementation of the above algorithm tends to generate excessive control torque since the inclusion of negative exponential terms may lead to singularities.

Assumption 1. The real and dual parts of M̂0 are positive-definite, bounded, and invertible.

Assumption 2. Φ̂ is bounded by an unknown dual constant such that Φ̂d̂Ŝ.

Remark 1. Since the real and dual parts of M̂0 represent the mass and inertial of the rigid-body spacecraft system, Assumption 1 can always hold. Since each term in Φ̂ is subject to the measurement range of physical sensors, there exists an upper limitation for Φ̂ that is bounded, which has been also validated in existing literature (Xiao and Yin, 2016).

Inspired by (Wang et al., 2009), a variant of the non-singular sliding mode in dual form is proposed as follows:

Ŝi=ξ̂̇ei+α̂1iξ̂ei+α̂2iŜai,i=1,2,3,(16)

where α̂1i=α1i+ϵα1i, α̂2i=α2i+ϵα2i α1i, α1i, α2i, and α2i are positive constants. Ŝai is the auxiliary terminal sliding mode:

Sai=sigξeip1,ifS̄i=0orS̄i0,ξeiδr1ξei+r2sigξei2,ifS̄i0,ξei<δ(17)
Sai=sigξeip1,ifS̄i=0orS̄i0,ξeiδr1ξei+r2sigξei2,ifS̄i0,ξei<δ(18)
S̄̂i=S̄i+ϵS̄i=ξ̂̇ei+α̂1iξ̂ei+α̂2isigξ̂eip1,(19)

where r1=(2p1)δp11, r1=(2p1)δp11, r2=(p11)δp12, and r2=(2p1)δp12 δ and δ′ are small positive constants. p1 ∈ (0.5, 1). On this basis, the NFxTSMC is designed as:

ûc=M̂0ω̂e×q̂e*ω̂dq̂eq̂e*ω̂̇dq̂e+ω̂×M̂0ω̂K̂sigŜM̂0Θ̂q̂e1α̂1ξ̂̇e+Θ̂̇q̂eω̂e+α̂2Wξ̂eΘ̂q̂eω̂e+α̂3sigŜp2+α̂4sigŜp3(20)

where α̂3=α3+ϵα3, α̂4=α4+ϵα4, and K̂=K+ϵK α3, α3, α4, α4, K, K′, and p2 are positive constants, with p2 ∈ (0.5, 1) and p3 > 1. W(ξ̂e) is a diagonal matrix whose ith entry in the main diagonal is

Wξei=p1ξeip11,ifS̄i=0orS̄i0,ξeiδr1+2r2ξei,ifS̄i0,ξei<δ(21)
Wξei=p1ξeip11,ifS̄i=0orS̄i0,ξeiδr1+2r2ξei,ifS̄i0,ξei<δ(22)

Theorem 1. Given the bounded external disturbances and system uncertainties (14), if the control law (20) is adopted, then the relative DQ and velocity motors of the DQ-based spacecraft system 8) are guaranteed to converge within fixed time T1 + T2, which will be provided later.

Proof. We will validate the fixed-time performance through the following two steps: i) the system trajectory reaches the sliding mode surface within a fixed time under any initial conditions, and ii) within the fixed time, the system trajectory converges to the equilibrium point alongside the sliding mode surface.

Step i): We adopt the following Lyapunov function candidate: V1=12[Ŝ|M̂0Ŝ]. Taking the time-derivative of V1 and substituting (13) into it yields

V̇1=Ŝ|M̂0ξ̂̈e+M̂0α̂1ξ̂̇e+M̂0α̂2Ŝ̇a=Ŝ|M̂0Θ̂̇q̂eω̂e+M̂0α̂1ξ̂̇e+M̂0α̂2Ŝ̇a+M̂0Θ̂q̂e×M̂01ûc+M̂0ω̂e×q̂e*ω̂dq̂eq̂e*ω̂̇dq̂eω̂×M̂0ω̂+Φ̂.(23)

Substituting the controller (20) into V̇1, we can derive

V̇1Ŝ|M̂0α̂2Ŝ̇aM̂0Θ̂q̂eM̂01K̂sigŜM̂0α̂2Wξ̂eΘ̂q̂eω̂e+α̂3sigŜp2+α̂4sigŜp3+M̂0Θ̂q̂eM̂01d̂ŜŜ|M̂0α̂3sigŜp2M̂0α̂4sigŜp3μ̲α3Sip2+1+α3Sip2+1+α4Sip3+1+α4Sip3+1μ1V1p2+12μ2V1p3+12(24)

where μ̲=min{σmin(J),m}, σmin(⋅) is the minimum eigenvalue, μ1=2p2+12μ̲min{α3,α3}, and μ2=2p3+12μ̲min{α4,α4}. Thus, using the Lemma 2, the state trajectory will reach the sliding mode within fixed time T1, where the settling time can be expressed as:

T12μ11p2+2μ2p31.(25)

It therefore implies that Ŝi=0̂ for i = 1, 2, 3 after T1. It is worth pointing out that the coefficients in the upper bound of T1 are only determined by the user-defined parameters, which are independent of initial conditions. When t > T1, one can obtain

ξ̂̇ei=α̂1iξ̂eiα̂2iŜai.(26)

Step ii): Consider the following Lyapunov function: V2=12<ξ̂e|ξ̂e>. Taking the time derivation of V2, we have

V̇2=i=13α1iξei2+α1iξei2+α2iξeip1+1+α2iξeip1+1μ3V21+p12,(27)

where μ3=min{α1i,α1i,α2i,α2i} for i = 1, 2, 3. According to the Lemma 1, the system trajectory on the sliding mode is guaranteed to converge to equilibrium within finite time T2, namely ξ̂e[0,0,0]T+ϵ[0,0,0]T, ω̂e0̂, and T22V21p12(T1)/μ3(1p1).

Remark 2. The system trajectory will enter the asymptotic sliding mode from the terminal sliding one when the sliding mode variables in (20) approach zero. This mechanism ensures singularity-free performance in the convergence procedure. In terms of the parameter selection rule, the error states will converge within the fixed settling time if larger μ1, μ2, μ3, and smaller p1 and p2 are chosen. K and K′ are suggested to be large enough for robustness against external disturbances and system uncertainties.

Remark 3. Traditional TSMC generates negative exponential terms of state variables and can therefore lead to singularities. In contrast, the proposed controller (20) is non-singular due to the following facts: state variables are not small enough to cause singularity for S̄̂0̂; in terms of S̄̂=0̂, the dual controller can be transformed as:

ûc=M̂0ω̂e×q̂e*ω̂dq̂eq̂e*ω̂̇dq̂e+ω̂×M̂0ω̂K̂sigŜM̂0Θ̂q̂e1α̂1ξ̂̇e+Θ̂̇q̂eω̂e+p1×α̂2sigξ̂e2p11+α̂3sigŜp2+α̂4sigŜp3.(28)

Therefore, the singularity phenomenon will not occur if p1 ∈ (0.5, 1).

The double value of quaternions results in the unwinding problem of attitude slewing, thereby degrading the global stability of the closed-loop system (Zheng et al., 2017). Here, an attitude error function is employed to overcome the unwinding problem as follows:

ϕ=2λ1+λ2qe02expqe02qe02+μ,(29)
er=expqe02qe02+μqe0qe0+2μqe0+μ2qev.(30)

where λ1, λ2, and μ are positive constants. qe0 is the real part of qe. The proposed attitude error vector is obviously continuous and bounded with θ ∈ [−π, π], which guarantees the response rate of the attitude error vector and the continuity of the attitude error function simultaneously. Thus, the anti-unwinding NFxTSMC can be obtained by replacing the original attitude error function and vector by (29)-(30). It can be derived that the anti-unwinding state is updated as ξ̂e*=er+ϵ(erpe).

4 Optimal control torque allocation strategy

To improve reliability and safety, redundant actuators are often equipped with spacecraft systems to provide corresponding forces and torques. Inspired by (Gersh and Peck, 2009), consider the following constraint condition in dual framework

ûct=D̂ûat(31)

where ûa denotes the actuation output, and D̂ the control allocation matrix. Without consideration of the actuator installment faults, the PI control allocation strategy can be ideally presented as follows

ûat=D̂ûct.(32)

where D̂=D̂T(D̂D̂T)1 is the Moore–Penrose inverse of D̂. The linear mapping between ûa(t) and ûc(t) is presented through the PI control allocation. However, the solution given by (32) may not satisfy the practical thruster range with the limitation of the thruster configuration (Tang et al., 2011). Thus, the optimal solution can be improved by employing the null space of the control allocation matrix

ûat=D̂ûct+ζ̂t(33)

where D̂ζ̂(t)=0̂, namely Null(D̂)={ζ̂(t)|D̂ζ̂(t)=0̂}. Thus, the thruster output can be adjusted to the available range with the proper choice of ζ̂(t). Furthermore, ζ̂ can be expressed as: ζ̂(t)=χ̂(t)Γ̂, where χ̂(t)=[χ̂1(t),χ̂2(t),,χ̂n6(t)] is the basic solution of null space, and Γ̂=[Γ̂1,Γ̂2,,Γ̂n6]T is the undetermined coefficient. Considering the smoothness of the actuator outputs (Hu et al., 2014; Li et al., 2015; Bai et al., 2022), the ONSPI control allocation can be described as an optimization problem

minζ̂tJζ̂t=12ζ̂t|L̂ζ̂t+12ûatûat1|Q̂ûatûat1+12ûatûat2|R̂ûatûat2s.t.Ĝ1tζ̂tĜ2t(34)

where Ĝ1(t)=ûmin(t)D̂ûc(t), Ĝ2(t)=ûmax(t)D̂ûc(t), and ûmin(t) and ûmax(t) are the known minimum and maximum outputs of the actuators, respectively. L̂, Q̂, and R̂ are positive and diagonal weighting matrices with appropriate dimensions, respectively. With the Lagrange multipliers φ̂10̂ and φ̂20̂, we can construct the Lagrangian function corresponding to the constrained optimization problem (34)

Lζ̂t,φ̂1,φ̂2=12ζ̂t|L̂ζ̂t+12ûatûat1|×Q̂ûatûat1+12ûatûa×t2|R̂ûatûat2+φ̂1|ζ̂tĜ1t+φ̂2|ζ̂tĜ2t,(35)

leading to the Karush–Kuhn–Tucker (KKT) condition as follows:

Ĝ1tζ̂tĜ2t,φ̂10̂,φ̂20̂,ζ̂tLζ̂t,φ̂1,φ̂2=0̂φ̂1|ζ̂tĜ1t=0,φ̂2|ζ̂tĜ2t=0,(36)

where one of the feasible solutions can be represented as:

ζ̂t=ÂQ̂ζ̂t1+R̂ζ̂t2+B̂(37)

where Â=(L̂+Q̂+R̂)1 and B̂=Q̂D̂ûc(t1)+R̂D̂ûc(t2)(Q̂+R̂)D̂ûc(t).

5 Simulation results

To verify the effectiveness of the proposed NFxTSMC (20), simulations have been carried out using the rigid-body spacecraft system governed by (8). The TSMC (Dong et al., 2016) and non-singular fast terminal sliding mode control (NFTSMC) (Sun et al., 2022) are carried out for comparison, where the identical initial values are selected for different controllers. In particular, the control parameters are chosen for purpose of the same settling time. To eliminate the chattering phenomenon caused by the sign function in (20), the hyperbolic tangent function is used as a substitution.

It is assumed that the spacecraft moves in a circular orbit with a height of 42240 km. The initial relative attitude and position of the spacecraft are chosen as ρe(0) = [−20,−10,−10]Tm, qe(0) = [0.6245,0.5,0.5196,0.3]T, ρed = [0,0,0]Tm, qed = [1,0,0,0]T, ωe(0) = [0,0,0]Trad/s. The external disturbance force and torque are ud = 10–2 × [6 + 3 sin(0.6t),5 + 4 sin(0.9t),4 + sin(0.5t)]TN and τd = 10–5 × [2 + 50 sin(0.8t),3 + 30 sin(0.5t),1 + 70 sin(0.3t)]TNm. The nominal mass and inertia are m0 = 100 kg and J0 = diag{18, 18, 24}kgm2 while the actual ones are m = 95 kg and J = diag{17, 17, 22}kgm2. The control parameters are set as: α = 0.67, p1 = 0.8, p2 = 0.9, p3 = 1.2, δ̂=0.5+ϵ0.0001, K̂=1.2+ϵ1.2, α̂1=α̂2=b̂=0.2+ϵ0.2, α̂3=α̂4=20+ϵ20. L̂=200I10+ϵ200I10, Q̂=10I10+ϵ10I10, R̂=20I10+ϵ20I10, and I10R10×10 is the inertial matrix.

Figure 1; Figure 2 represent the time responses of relative position and attitude errors under the effect of the three controllers. Under the same initial values, the relative position errors driven by the TSMC and NFTSMC converge within 78s and 22s, respectively. In contrast, the proposed NFxTSMC realizes the fastest convergence performance (19s) due to the fact that the convergence rate and accuracy are simultaneously considered in the sliding mode and controller design. In Figure 2, the relative quaternion errors in the proposed controller converge with less overshoot and a higher convergence rate compared with other methods. It is noted that the singularity phenomenon is eliminated in the proposed controller. In order to further validate the fixed-time performance provided by NFxTSMC, we introduce different initial values as follows:

FIGURE 1
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FIGURE 1. Time responses of ρe for different controllers.

FIGURE 2
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FIGURE 2. Time responses of qe for different controllers.

Case1: ρe(0) = [−250,−150,100]Tm, qe(0) = [0.7,0.4,0.3,0.51]T, and ωe(0) = [0,0,0]Trad/s;

Case2: ρe(0) = [2500,−1500,−1000]Tm, qe(0) = [0.3,0.4,0.3,0.81]T, and ωe(0) = [0.2,−0.3,0.5]Trad/s.

The control parameters and objectives in Cases 1 and 2 remain identical to the above settings. It can be seen from Figure 3 that initial values do not impact the settling time (10s) and the corresponding translational convergence performance, while Figure 4 demonstrates the identical settling time of attitude variables under different initial conditions. It, therefore, validates that the proposed NFxTSMC ensures fixed-time stability without the need for exact initial values. Moreover, Figures 14 indicate NFxTSMC can realize robust transient-state performance with less overshoot.

FIGURE 3
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FIGURE 3. Time responses of ρe in Cases 1 and 2.

FIGURE 4
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FIGURE 4. Time responses of qe in Cases 1 and 2.

The symmetric thruster configuration (Sun et al., 2022) (see Figure 5) is employed to test the proposed ONSPI control allocation scheme. Attitude and orbit control corresponding to thrusters with respect to x, y, and z axles are summarized in Table  1.

FIGURE 5
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FIGURE 5. Symmetric thruster configuration (STC).

TABLE 1
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TABLE 1. Thruster configuration corresponding to STC.

Figures 69 show the practical thruster output in the STC, where Ti corresponds to the ith thruster for i = 1, 2, … , 16. As observed, pair-mounted actuators can provide symmetrical thrusts. The feasible solution can be found in the pseudo-inverse method within the thrust limitation (20 N) in the STC, where the negative values can also be offered by the thruster from the other direction. Compared with the conventional PI method, the ONSPI method can satisfy control allocation requirements despite control force limitations. Similarly, it is demonstrated in Figures 69 that the ONSPI approach can generate smooth actuator output and provide closed-looped stability against external disturbances.

FIGURE 6
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FIGURE 6. Thruster forces of #1-#4 in the STC.

FIGURE 7
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FIGURE 7. Thruster forces of #5-#8 in the STC.

FIGURE 8
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FIGURE 8. Thruster forces of #9-#12 in the STC.

FIGURE 9
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FIGURE 9. Thruster forces of #13-#16 in the STC.

6 Conclusion

In this paper, we extend the result in (Sun et al., 2022) from finite-time to fixed-time stability, where the settling time of the spacecraft system is only dependent on user-defined control parameters rather than initial values. The proposed non-singular fixed-time control law provides a more accurate and robust estimation of the settling time compared with finite-time control. Thus, it will benefit the application scenarios with measurement errors and unknown environments. Meanwhile, we demonstrate the scalability of the developed non-singular fixed-time control framework which facilitates alleviating the unwinding problem. Furthermore, the disadvantages of the traditional pseudo-inverse method are eliminated by the optimal quadratic programming, which ensures that all the practical actuator outputs are subject to limitation. Finally, numerical simulations to evaluate the overall performances for non-singularity, fast tracking, high accuracy, uncertainty resistance, and fixed-time stability have verified the effectiveness of the proposed method. The actuator faults and fault-tolerant coordinated controller will be considered in future work.

Data availability statement

The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author.

Author contributions

Conceptualization, ZW and LS; methodology, LS and YH; software, ZW; validation, LS, YH, BX, and HF; formal analysis, YH, BX, and HF; resources, EY; data curation, ZW; writing—original draft preparation, LS; writing—review and editing, LS, YH, ZW, BX, HF, and AM; visualization, ZW and BX; supervision, ZW and EY; project administration, ZW, AM, and EY All authors have read and agreed to the published version of the manuscript.

Funding

This work was supported in part by the United Kingdom. EPSRC under Grant EP/V027379/1.

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

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Keywords: dual quaternion, spacecraft control, fixed-time stability, control torque allocation, optimization

Citation: Sun L, Huang Y, Fei H, Xiao B, Yeatman EM, Montazeri A and Wang Z (2023) Fixed-time regulation of spacecraft orbit and attitude coordination with optimal actuation allocation using dual quaternion. Front. Robot. AI 10:1138115. doi: 10.3389/frobt.2023.1138115

Received: 05 January 2023; Accepted: 27 January 2023;
Published: 14 February 2023.

Edited by:

Xueqian Wang, Tsinghua University, China

Reviewed by:

Nan Ma, Loughborough University, United Kingdom
Xiuming Yao, Beijing Jiaotong University, China

Copyright © 2023 Sun, Huang, Fei, Xiao, Yeatman, Montazeri and Wang. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Allahyar Montazeri, a.montazeri@lancaster.ac.uk

Disclaimer: All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.