Finite-time command-filtered approximation-free attitude tracking control of rigid spacecraft

In this paper, a finite-time command-filtered approximation-free attitude tracking control strategy is proposed for rigid spacecraft. A novel finite-time prescribed performance function is first constructed to ensure that the attitude tracking errors converge to the predefined region in finite time. Then, a finite-time error compensation mechanism is constructed and incorporated into the backstepping control design, such that the differentiation of virtual control signals in recursive steps can be avoided to overcome the singularity issue. Compared with most of approximation-based attitude control methods, less computational burden and lower complexity are guaranteed by the proposed approximation-free control scheme due to the avoidance of using any function approximations. Simulations are given to illustrate the efficiency of the proposed method.


Introduction
The attitude control of a rigid spacecraft is a significant and practical problem in various applications, such as formation flying, satellite surveillance, and earth observation [1]. A number of relevant works have been investigated for spacecraft attitude stabilization or tracking problems, including adaptive control [2], sliding mode control [3], robust control [4], event-triggered control [5], output feedback control [6], etc. All of the aforementioned methods are concerned with asymptotic stabilization or tracking of the spacecraft attitude, meaning that the attitudes will track the desired trajectories as time approaches the infinity.
Different from the aforementioned schemes, the finite-time control can guarantee the tracking error convergence within a finite time and has been widely implemented on spacecraft attitude control systems and other nonlinear systems (see [7][8][9][10][11][12][13], and references therein). In [11], a fast nonsingular terminal sliding mode control scheme was presented for the uncertain spacecraft, such that the convergence of attitude tracking errors could be achieved within finite time.
In [12], an adaptive finite-time control with adding a power integrator was developed to guarantee that tracking errors could converge to the desired regions in finite time. In [13], an adaptive finite-time control law was presented by combining an integral disturbance observer and a terminal sliding mode surface, and the chattering problem could be eliminated.
In most of the above-mentioned existing schemes, it was assumed that the models of rigid spacecraft were exactly or partially known. However, for practical spacecraft applications, system uncertainties and constraints have important influences on the safe operation and system performance. The existence of these factors makes it challenging to design an autonomous attitude controller, which is required to ensure a fast convergence rate of the tracking errors and satisfactory flight performance of the controlled spacecraft. To handle the uncertainties existing in the system dynamics, some function approximators including neural networks (NNs) and fuzzy logic systems (FLSs) are usually involved in the adaptive controllers design (see [14][15][16][17][18][19][20][21][22], and references therein). In [20], a Chebyshev neural network-based finite-time control scheme was developed by constructing a terminal sliding mode surface, such that the uniform ultimate boundedness (UUB) of attitude tracking errors was guaranteed. In [21], a fast nonsingular terminal sliding mode control law was designed for rigid spacecraft, and the lumped system uncertainty including unknown inertia matrix and thrusters faults was approximated by the FLSs. Nevertheless, those approximation-based control approaches in literature [14][15][16][17][18][19][20][21][22] may impose demanding computational burden due to their complicated structures in the implementation.
In addition, to achieve guaranteed transient tracking response, a constructive technique with a prescribed performance function (PPF) was presented in [23], where the transient tracking error can be rigorously maintained within a prescribed boundary. With the satisfactory characteristics, the PPF-based control has been applied to various nonlinear systems [24][25][26][27][28]. However, PPFs in the aforementioned literature can only constrain the error convergence in exponential form, meaning that the system states can be maintained to the predefined region within infinite settling time.
Recently, several finite-time prescribed performance functions (FTPPFs) have been presented to guarantee that the system states can be maintained to a predefined region in finite time [29][30][31], but the approximation tools including FLSs and NNs may lead to the increase of computational burden. In [32], an approximationfree control (AFC) was developed for nonlinear systems without using any function approximations (i.e., NNs and FLSs), and a proportional-like controller was designed with a simplified structure. Owing to this attractive property, the AFC method is easier to implement in various applications [33][34][35][36][37][38]. In [36], an adaptive error-constrained finite-time controller was presented for Lagrangian systems, where a piecewise function should be required in the controller design to handle the singularity problem, which would lead to a complex stability analysis. In [37], an approximationfree control scheme was proposed for rigid spacecraft, and the state variables can asymptotically converge to a small region around the origin. In [38], a modelfree prescribed performance control approach was presented for flexible spacecraft to ensure the uniform ultimate boundedness (UUB) of attitude tracking errors, and the invertibility of model matrix should be always guaranteed. In [39], a finite-time approximation-free control scheme was proposed for the attitude tracking of quadrotor unmanned aerial vehicles, in which the PPF was in an exponential form, such that only the asymptotic convergence of tracking errors to the predefined region can be guaranteed.
Motivated by the above discussions, this paper proposes a novel finite-time approximation-free control (FTAFC) scheme for the attitude tracking of rigid spacecraft. The main contributions are summarized as follows: (1) A novel finite-time prescribed performance function (FTPPF) is presented, such that the attitude tracking errors can be constrained to converge into the predefined region within finite time.
(2) A finite-time approximation-free control (FTAFC) scheme is proposed by constructing a finite-time error compensation mechanism, and the singularity problem is directly avoided in the control design without using any piecewise functions. (3) Compared with the approximation-based control approaches in literature [14][15][16][17][18][19][20][21][22], the unknown spacecraft dynamics can be accommodated effectively without employing any function approximations, and thus the proposed controller has less computational burden.
The remainder of the paper is structured as follows. Section 2 introduces the problem formulation and some preliminaries. The controller design and stability analysis are given in Sect. 3. Comparative simulations are provided in Sect. 4, followed by some conclusions given in Sect. 5.

Attitude dynamics and kinematics of rigid spacecraft
The attitude of a rigid spacecraft can be described by Modified Rodrigues Parameters (MRPs), which is formulated as [40] with σ = [σ 1 , σ 2 , σ 3 ] T ∈ R 3 being the spacecraft attitude, (·) ∈ R and n = [n x , n y , n z ] T ∈ R 3 being the Euler angle and Euler axis, respectively. The MRPs vector σ can be mapped to its shadow counterpart σ s with the following relationship [41] σ s = 1 σ T σ σ (2) and σ is guaranteed to be bounded within a unit sphere through switching the MRPs to σ s in (2) as σ T σ > 1, such that the global singularity-free rotation representation is ensured. In terms of MRPs (1), the spacecraft attitude kinematic and dynamic equations are given bẏ where G(σ ) ∈ R 3×3 is a Jacobian matrix defined by , ω 2 , ω 3 ] T ∈R 3 represents the angular velocity; J ∈ R 3×3 denotes a symmetric, positive and bounded inertia matrix; u = [u 1 , u 2 , u 3 ] T ∈R 3 is the control torque, and d = [d 1 , d 2 , d 3 ] T ∈R 3 represents unknown but bounded disturbances.

Relative attitude error dynamics and kinematics
Define σ d = [σ d1 , σ d2 , σ d3 ] T ∈ R 3 as the MRPs of the desired attitude, and ω d = [ω d1 , ω d2 , ω d3 ] T ∈ R 3 as a time-varying and continuous desired angular velocity. In this paper, it is assumed that ω d and its first-order differentiationω d are both continuous and bounded.
The attitude tracking error and angular velocity error represented by MRPs are defined as e = [e 1 , e 2 , e 3 ] T ∈ R 3 and ω e = [ω e1 , ω e2 , ω e3 ] T ∈ R 3 , respectively, which can be formulated as [41] where 2 is the corresponding direction cosine matrix.
From (3)-(6), the dynamics and kinematics of the relative attitude error are formulated aṡ e = G(e)ω e (7) Then, Eq. (8) can be rewritten aṡ The time derivative ofė along (7) yields e =Ġ(e)ω e + G(e)ω e Define x 1 = e, x 2 =ė, and then Eq. (10) can be rewritten as The Jacobin matrix G(x 1 ) has the following properties [42]: Remark 1 There are some different attitude representations to describe the attitude of a spacecraft, including unit quaternion and MRPs, etc. [43]. Although the unit quaternion can be employed for global representation of the attitude of a spacecraft without singularities, there is a norm constraint imposed on the four parameters. Compared with the unit quaternion, the MRPs-based attitude description is effective for eigenaxis rotations up to 360 • and more computationally efficient in the real-time attitude calculation. Consequently, the MRPs are used to describe spacecraft attitude system (3) and (4) in this paper.

Preliminaries
Consider the initial value problem [44] asζ with h: + × Ω ζ → n being a continuous function vector, and Ω ζ ∈ n being a nonempty open set.
Definition 1 [44]: The solution of initial value problem (15) is maximal if it has no proper right extension that is also a solution of (15).
Proposition 1 [44]: When the conditions of Lemma 1 hold, for a maximal solution

Finite-time command filter
To construct a finite-time error compensation mechanism, the following command filter is employed in the controller design [45] x c = υ where α is the input of the finite-time command filter, and x c and υ are the outputs; β 1 and β 2 are positive constants. As proved in [45], by properly choosing β 1 and β 2 , the following equalities hold within a finite time where α 0 is the ideal input without noise (i.e., α 0 = α).
In practice, the measured input is usually affected by noise such that α = α 0 holds, and then the following result is obtained.

Finite-time approximation-free control
In this section, a finite-time approximation-free control (FTAFC) is developed for system (11) to maintain the transient tracking performance as well as steady-state tracking performance, where the finite-time stability analysis is also provided.

Finite-time prescribed performance function
To maintain the attitude tracking errors of the spacecraft system within a predefined time and boundary, an essential definition is defined first.

Definition 2
If a smooth function ρ(t) satisfies the following conditions: being an arbitrarily small positive constant; and ρ(t) = ρ T 0 for ∀t ≥ T 0 with T 0 being the settling time.
Then, this function is called finite-time prescribed performance function (FTPPF).
According to Definition 2, a novel FTPPF ρ(t) in this paper is constructed as where ρ 0 = ρ(0) is the initial value of FTPPF. From (19), it can be deduced that lim which means that the FTPPF ρ(t) is a smooth function.

Remark 2
The PPF in literature [33][34][35][36][37][38] is designed in form of , and a is a positive constant. It is worth noting that (20) is an exponential form, which means that the settling time of PPF is infinite and the convergence rate of PPF is slower.
The corresponding comparison of different prescribed performance functions is shown in Fig. 1. It can be seen from Fig. 1 that the settling time of FTPPF is fixed (i.e., T 0 =0.5s), while the settling time of PPF is infinite. Then, the prescribed bound of the attitude tracking error e i with i = 1, 2, 3 is retained by the following constrained condition: where δ and δ are positive constants denoting the upper and lower bounds of overshoot, respectively. To retain predefined response (21), a strictly monotonically increasing and smooth function S(ζ i ) is chosen to satisfy the following conditions: where Then, the control problem with error boundary condition (21) is transformed into an equivalent "unconstrained" control problem of the transformed error ζ i (t). According to (22), the transformation function S(ζ i ) can be chosen as and thus the inverse function of (23) is given by Remark 3 In contrast to the PPFs in [33][34][35][36][37][38], the advantage of proposed FTPPF (19) is that it can converge to the predefined region within a finite time, and the settling time can be specified by the designer, which is more efficient for practical applications.

Controller design
To ensure the spacecraft finite-time transient tracking performance as well as steady-state tracking performance, improved error transformation function (24) is incorporated into the control design. The detailed design steps are given as follows.
Define the intermediate error z 2 as where z 2 = [z 21 , z 22 , z 23 ] T , and x c is the output of finite-time command filter, which is designed aṡ where β 1 and β 2 are positive constants, x ci and υ i , respectively, denote the finite-time command filter's outputs for estimating the virtual control law α i and its derivativeα i to be designed below.

Remark 4
It is well known that function approximators including FLSs and RBFNNs are widely employed to approximate the nonlinear dynamic in the controlled systems, but there are not general guidelines for the parameter tuning of FLSs and RBFNN weights. Besides, the design of the adaptation laws also takes a long time to ensure the convergence, leading to potentially sluggish transient responses [46]. From (32) and (42), it is seen that the proposed control structure has a simple cascade proportional-like structure without using any function approximations, such that the computational costs in the control implementation can be reduced.
Remark 5 Different from the existing classical AFC schemes [32][33][34][35][36][37][38], in which the error convergence is guaranteed when the time goes to infinity, controller (42) can ensure the finite-time convergence of the attitude tracking errors without triggering any potential singularity issues. In addition, the design of error compensation mechanism (34) is to compensate for the influence of the filtering error x ci − α i , i = 1, 2, 3 caused by finite-time command filter (16).

Stability analysis
To analyze the finite-time stability of the closed-loop control system, some essential lemmas are first provided as follows.

Lemma 3 [11]: Given a systemẋ = f (x), if there exist a continuous function V (x) and scalars
The settling time is bounded by Lemma 4 [41]: For all positive numbers x i , i = 1, 2, ..., n and there exists a constant 0 < p < 1, such that the following inequality holds The following two theorems are proved to show the main results of this paper. (34), the signals ξ 1 can converge to origin within the finite time.

Theorem 1 Consider error compensation mechanisms
Proof Choose a Lyapunov function candidate as Substituting (34) into (49) and according to the Lemma 2 result iṅ where 1 = 2k 1 g 1m is a positive constant, 2 = √ 2(l 1 − )g m1 > 0 holds when choosing appropriate constant l 1 to satisfy l 1 − > 0. According to Lemma 3, it is concluded that ξ 1i can converge to origin with the following finite settling time This completes the proof.
Proof The whole proof process contains four steps. In Step 1-Step 3, all signals in the closed-loop system are proved to be bounded, and the proof of the finite-time convergence of the attitude tracking error e i is given in Step 4.
Step 2: Select the Lyapunov function as Differentiating (58) yieldṡ Substituting (43) into (59) yieldṡ where Following the similar analysis given in Step 1 and using Lemma 1 and Lemma 2, it is obtained that the signals ζ 2i , ρ 2 ,ρ 2 , x ci andẋ ci are bounded for ∀t ∈ [0, τ max ). Moreover, the external disturbance d is bounded, and Φ(x 1 , x 2 , ω d ,ω d ) is a continuous function of x 1 , x 2 , ω d andω d , which is bounded for ∀t ∈ [0, τ max ). Then, from the extreme value theorem in [32], there is a positive constantF 2 > 0, such that the following inequality holds for ∀t ∈ [0, τ max ) Substituting (61) into (60) yieldṡ , which implies thatV 2 is negative for the case s 2i >F 2 k 2 . Consequently, it can be concluded that ε 2i will converge to a compact set Thus, ε 2i is bounded, and the actual control law u in (42) is also bounded for ∀t ∈ [0, τ max ).
Step 3: In this step, the case of τ max = +∞ will be discussed. From (57) and (63), it is obtained that 6 is nonempty and compact. According to the definition of Ω ζ , it is clear thatΩ ζ ⊂ Ω ζ holds. Thus, if we assume τ max < ∞ and consider the fact Ω ζ ⊂ Ω ζ , Proposition 1 indicates that there exists a time t 1 ∈ [0, τ max ) such that ζ(t 1 ) / ∈Ω ζ , which creates an obvious contradiction. Therefore, the opposite of the assumption is true, i.e., τ max = ∞ holds. Then, it can be concluded that all signals in the closed-loop systems are bounded, and the attitude tracking error e i is maintained within the predefined boundary for ∀t ≥ 0.
Step 4: Construct the following Lyapunov function From (56) and (62), the derivative of V 4 is obtained aṡ Introducing two auxiliary termsF where k 1 =k 11 +k 12 , k 2 =k 21 +k 22 , μ 1 = min{2g m1 k 11 , 2g m2 k 21 }, μ 2 = 2 1+γ 2 · min{τ 1 g m1 , τ 2 g m2 }, and ϕ = in finite time, where 0 < θ < 1. It means that s 1i and ε 2i will converge to a sufficiently small region around zero within the finite time T 2 given by From (64)-(68), it is obtained that s 1i and ε 2i can both converge into a small region within a finite time. According to (33) and (50), the constrained errors ε 1i can also converge into a sufficiently small region around zero within a finite time. According to the definition x 2 = z 2 + x c and ω e = G(x 1 ) −1 x 2 , it can be obtained that x 2 is bounded, which further implies that angular velocity error ω e is bounded. Consequently, the attitude tracking error e i is maintained within prescribed region (21) and converge into the sufficiently small region around origin in finite time T 3 ≤ T 1 + T 2 . This completes the proof.
Remark 6 From (67), it can be seen that the error convergence region is determined by the choice of some control parameters, such as k 1 , k 2 , τ 1 and τ 2 , which can be specified by the designer in advance. Generally, in order to guarantee the attitude tracking error e i , i = 1, 2, 3 converges into a sufficiently small region, the larger values of k 1 , k 2 , τ 1 and τ 2 are selected in (32) and (42), respectively. However, too large values of k 1 , k 2 , τ 1 and τ 2 may result in a high control gain. Consequently, the selections of k 1 , k 2 , τ 1 and τ 2 should be considered comprehensively between the error convergence region and control gain.

Simulation results
In this section, two different control schemes are provided, including the proposed FTAFC (M1) in this paper, and standard AFC (M2) in [34]. Detailed control configurations and corresponding discussions are given as follows.
(3) The M2 scheme In the M2 scheme [34], the virtual controller and actual controller are designed  Figure 2 shows the comparison of the attitude tracking error e 1 . From Fig. 2, it is seen that the proposed M1 scheme can guarantee a better tracking performance than M2, that is a faster tracking speed and smaller tracking errors can be achieved with M1 scheme. Moreover, M1 and M2 strictly retain the attitude tracking errors within region (21) specified by the FTPPF and PPF, respectively. However, the settling time of FTPPF in M1 is fixed, while the settling time of PPF in M2 is infinite, and the improved transient convergence over the standard AFC can be achieved by the suggested finite-time compensation scheme in the FTAFC. Similar results can be found in Fig. 3, which describes the comparison of the attitude tracking error e 2 . The comparison of the attitude tracking error e 3 is shown in Fig. 4, and again one can find that the proposed M1 scheme can guarantee a better tracking performance with relatively smaller steady-state tracking error than M2 scheme. The angular velocity error ω e that a better steady-state attitude tracking performance is achieved with the proposed M1 scheme in this paper.

Conclusion
This paper has proposed a finite-time command-filtered approximation-free attitude tracking control scheme for rigid spacecraft without using any function approximations. A novel finite-time PPF is first presented to ensure that the attitude tracking errors converge into the predefined region within finite time. To overcome the singularity caused by differentiating the virtual control signals, a finite-time error compensation mechanism is Intermediate error z 2 of the M1 scheme developed and incorporated into the recursive control design. However, the state values of attitude and angular velocity should be available to design the finite-time controller which may limit its practical applications. Consequently, the finite-time approximation-free attitude tracking control without velocity measurements is still a challenging work and deserves to be further investigated in future work.