Adaptive Integral-Sliding-Mode Control for Spacecraft Single-Axis Pointing Using Two Torques

This paper deals with the attitude control problem of a nearly axis-symmetric spacecraft actuated by two torques perpendicular to its symmetry axis. As a result, the spacecraft symmetry axis is unactuated and the rotation about it is uncontrollable. Our objective aims to stabilize the symmetry axis to an arbitrary inertial direction irrespective of the spinning motion about it. First, a non-smooth controller is derived via homogeneous techniques to align the symmetry axis in finite time when there are no uncertainties. To compensate for perturbations induced by uncertain inertias, unknown external disturbances and actuator faults, an adaptive integral sliding mode controller is developed by combining adaptive control and integral sliding modes with the non-smooth controller. The resultant adaptive controller can stabilize the system states into a small neighborhood around the sliding mode. Consequently, the performance of the non-smooth controller can be approximately recovered, even in the presence of uncertainties, which ensures significant robustness and high control accuracy. Numerical examples are presented to verify the effectiveness and advantages of the proposed methods.


I. INTRODUCTION
The attitude motion of a rigid spacecraft has 3 degrees-offreedom and it becomes underactuated when the available control torques are fewer than three. The attitude control of underactuated spacecraft has attracted increasing attention of researchers because it can provide significant tolerance to partial actuator failures, as occurred in the Dawn spacecraft and the Kepler telescope [1], [2], and greatly improve the reliability of the attitude control system.
Many studies have considered the attitude control of spacecraft underactuated by internal torques, such as reaction wheels and control moment gyros [3]. In this case, the angular momentum conservation constraint is a critical challenge and various approaches have been proposed to handle it and meanwhile achieve single-axis pointing [4], [5], three-axis attitude stabilization [6]- [11] or tracking control [12].
In contrast, the spacecraft underactuated by external torques, such as reaction thrusters, does not have the The associate editor coordinating the review of this manuscript and approving it for publication was Youqing Wang . momentum conservation problem and its attitude control is generally addressed from two perspectives. The first seeks discontinous or time-varying control laws to achieve threeaxis attitude stabilization [13]- [19] when the system is fully controllable, for example, when only two external control torques are available and the spacecraft is asymmetric, or when the spacecraft is axis-symmetric and the torques are not normal to the symmetry axis [20], [21]. The second perspective aims to stabilize a reduced yet fully controllable subsystem of the original uncontrollable system. For example, if the spacecraft is axis-symmetric and the two torques are perpendicular to the symmetry axis, the rotation about the symmetry axis is not controllable. In addition, it is common that the uncontrollable symmetry axis has nonzero angular velocity for on-orbit spacecraft. To deal with this scenario, spin-axis stabilization was proposed to stabilize the symmetry axis of the spacecraft to an arbitrary target inertial direction irrespective of the spinning motion about the symmetry axis [22]- [24]. Such type of attitude control can have practical applications when the spacecraft symmetry axis is a normal vector of the solar panel, or the line-of-sight of an antenna, a telescope or a camera, etc. For these cases, the main concern is to align the line-of-sight of the spacecraft along the target direction while the rotation motion about it is irrelevant.
Tsiotras and Longuski [22], Tsiotras [23] first derived proportional-derivative feedback laws asymptotically stabilizing the spinning axis of an axis-symmetric spacecraft with two torques. In spite of their remarkable simplicity, these algorithms have poor robustness against inertia uncertainties and external disturbances. For better robustness, a spinstabilization law was developed in [25] by means of the internal modal principle to reject sinusoidal disturbances with known frequencies. A computational H ∞ feedback law was designed in [26] to point the spacecraft symmetry axis while suppressing the influence of unknown disturbances. Neither of the designs in [25], [26] considered the inertia uncertainties. A more recent work by [27] presented an adaptive sliding mode controller for spin-stabilization with significantly improved robustness to system uncertainties. The studies in [14], [22], [23], [26], [27] all utilized the (w, z)-coordinate to describe the spacecraft attitude. The resultant attitude kinematics have the appealing feature that the rotation of the symmetry axis is decoupled from the rotation about it, and therefore significantly facilitates spin-stabilization law design for axis-symmetric spacecraft.
None of spin-stabilization laws derived in [14], [22], [23], [25]- [27] consider the influence of uncertain inertia, unknown disturbances, and actuator faults all together. In addition, they are all asymptotically convergent. Compared to asymptotic controllers, finite-time control methods ensure convergence in finite time and hence faster convergence speed and better robustness to uncertainties. Typical finitetime control design methods include homogeneous theory, adding a power integrator techniques, terminal sliding modes [28], and higher-order sliding modes [29], [30]. Various finite-time attitude control laws were designed for fully actuated spacecraft [31]- [34]. Recently, finite-time angular velocity observers were designed in [18], [19] and then used to obtain velocity-free three-axis attitude stabilization laws with two torques. The combined observer-controller, however, ensures local asymptotic stability instead of finite-time stability. Therefore, the finite-time attitude control of spacecraft with two torques still needs further research.
This paper investigates the spin-axis stabilization of an axis-symmetric spacecraft controlled by two external torques normal to the symmetry axis. The symmetry axis is also the spinning axis that is required to point to a desired inertial direction. A Hölder-continuous controller with a nonlinear proportional-derivative structure is designed by means of the homogeneous theory to stabilize the spinning axis to the target direction in finite time, when there is no system uncertainties. It is then developed into an adaptive controller with significant robustness against uncertain inertia, unknown external disturbances, and actuator faults by combining integral sliding mode (ISM) methods and adaptive techniques. The main contributions of the paper are twofold.
1) The ISM surface is designed such that the sliding motion evolves exactly as the uncertainty-free closedloop system under the Hölder-continuous controller.
In other words, on the sliding mode the spinning axis of the spacecraft will converge to the target direction in finite time. As a result, the parameter tuning for the proposed ISM is greatly simplified because it is similar to that for the Hölder-continuous controller and has a clear physical meaning. 2) The proposed adaptive ISM controller can actively estimate and compensate for system uncertainties such as uncertain inertia, unknown external disturbances and actuator faults in real time. As a result, the performance of the Hölder-continuous controller can be approximately recovered even in the presence of unknown perturbations, and the spacecraft spinning axis can be pointed to the target direction with high precision. Numerical examples are presented to demonstrate the effectiveness and advantages of the proposed methods compared to the existing spin-stablization controllers derived in [25], [27]. This paper proceeds as follows: Section II presents some useful lemmas and the attitude equations of a nearly axissymmetric spacecraft actuated by two torques. In section III, a Hölder-continuous controller is designed when there are no uncertainties, and then developed into an adaptive ISM controller, to stabilize the spacecraft symmetry axis to any inertial direction. Numerical examples are given in Section IV and conclusions are summarized in Section V.

A. FINITE-TIME STABILITY
Consider the systeṁ where f : x ∈ V → [f 1 (x), · · · , f n (x)] T ∈ R n is continuous on an open neighborhood V of the origin. System (1) is finitetime stable if it is Lyapunov stable and every solution x(t, x 0 ) starting from any initial condition x 0 ∈ V converges to the origin in a finite time T (x 0 ) ≥ 0.
Definition 1: Consider system (1) and denote by U a neighborhood of x = 0 n . The origin of system (1) is said to be finite-time stable if it is 1) Lyapunov stable in U and VOLUME 8, 2020 2) finite-time convergent in U , that is, for any initial condition x 0 ∈ U there exists a time T (x 0 ) ≥ 0 such that the solution satisfies lim t→T (x 0 ) x(x 0 , t) = 0 n . If U = R n , then the origin is globally finite-time stable.
Given λ > 0 and a weight vector r = [r 1 , · · · , r n ] T ∈ R n , define a dilation operator r λ : R n → R n as r λ x = [λ r 1 x 1 , · · · , λ r n x n ] T , where x = [x 1 , · · · , x n ] T ∈ R n . Then, system (1), or equivalently the vector field f (x) is said to be homogeneous of degree l ∈ R with respect to the dilation r for any i ∈ I n , x ∈ V , and λ > 0, and l satisfies l > − min{r 1 , · · · , r n }. The following result has been proven in [35].
Lemma 1: Consider the following continuous systeṁ where f (0 n ) =f (0 n ) = 0 n . Assume that the unperturbed systemẋ = f (x) is homogeneous of degree l < 0 with respect to a dilation r λ and is asymptotically stable. Then, the origin of system (1) System (2) is globally finite-time stable if it is globally asymptotically stable and locally finite-time stable.  T ∈ R 3 be the direction cosines ofb 3 in F I , the attitude of F B relative to F I can be parameterized by means of the (w, z)-coordinates with [36]

Consider a rigid spacecraft attached with a body frame F
and z ∈ R being a rotation angle aboutî 1 . The attitude kinematics in terms of (w, z)-coordinates and the Euler dynamics of the spacecraft are given by [36] is the spacecraft angular velocity expressed in F B , J ∈ R 3×3 is the spacecraft inertia matrix, τ ∈ R 3 is the control torque, and d ∈ R 3 denotes the external disturbance torque. E(t) = diag{e 1 (t), e 2 (t), e 3 (t)} is the health indicator matrix of the actuators with 0 ≤ e i (t) ≤ 1, i = 1, 2, 3. In particular, e i (t) = 1 means that the ith actuator is completely health and can produce control torques as commanded. 0 < e i (t) < 1 means that the ith actuator encounters the fault of fading control torques while e i (t) = 0 corresponds to the case that the ith actuator completely fails. As shown in Fig. 1, we assume in the following that the spacecraft is nearly axis-symmetric and only two torques perpendicular to the symmetry axis are available for control, i.e., 0 < e 1 (t), e 2 (t) ≤ 1 and e 3 (t) = 0 for all t ≥ 0. Without loss of generality, let the body axes {b 1 ,b 2 ,b 3 } align with the principal axes of the spacecraft, andb 3 along the unactuated symmetry axis. In other words, The control objective is to point the symmetry axisb 3 to a target direction fixed in the inertial space irrespective of the spinning motion aboutb 3 . Note that the target direction can be an arbitrary direction in the inertial space. Note that the choice of the inertial frame is very flexible. We can always choose an inertial frame such thatî 3 coincides with the target direction. Therefore, the objective can, without loss of generality, reduce to stabilizing the symmetry axisb 3 toî 3 .
Note that w 1 and w 2 now describe the direction of the symmetric axisb 3 in the inertial space andb 3 is aligned witĥ i 3 if and only if w 1 = w 2 = 0. The equations governing the evolution ofb 3 can be derived as [27] w = S( 3 )w + F(w) 12 (5) is a skew-symmetric matrix given by and F(w) is a symmetric matrix given by Since the spacecraft is nearly axis-symmetric aboutb 3 , it follows that J 1 = J 2 and 3 (t) The angular velocity along the other two axes will not affect the symmetry axis. In other words, the angular velocity about the symmetry axis remains constant if the disturbance torque alongb 3 is always zero, and is time-varying otherwise.
The reduced system given in (5) and (6) completely describes the evolution of the symmetry axisb 3 in 3-dimensional space. It has two degrees-of-freedom and two control inputs, and, therefore, is a fully actuated system. Assumption 1: The spacecraft is nearly axis-symmetric and its inertia is constant.
Assumptions 1-3 are reasonable for realistic spacecraft and also adopted in [27]. Assumption 4 means that the perturbations induced by actuator faults and inertia uncertainties cannot be arbitrarily large in order to attain effective attitude control. The remaining control authority of the actuators needs to be able to suppress the perturbing effect, which is also reasonable.

III. CONTROLLER DESIGN
In this section, we aim to design control laws that can point the unactuated spinning axisb 3 to an arbitrary inertial direction even in the presence of the above uncertainties. This objective is attained by stabilizing the reduced system given in (5) and (6) to the origin or a small neighborhood of the origin. Note that the global stability or global convergence discussed in the following refers to the behavior of the reduced states, i.e., (w, 12 ), instead of the 6-dimensional states of the full attitude motion system given in (4), i.e., (w, z, ).
Proof: Since the global finite-time stability is a combination of global asymptotic stability and local finite-time stability. We intend to first show that the closed-loop system is globally asymptotically stable and then locally finite-time stable.
1) Global Asymptotic Stability: Choose a Lyapunov function candidate as follows: where V 1 = 0 if and only if 12 = 0 2 and w = 0 2 . The time derivative of V 1 is computed aṡ Note that the following equalities hold: Hence, substituting (5) and (9) into (11) yieldṡ which implies that the closed-loop system is Lyapunov stable, and lim t→∞ V 1 (t) exists and is finite. Hence, w(t) and 12 (t) are uniformly bounded. It then follows from (6) thaṫ 12 (t) is uniformly bounded and thus 12 (t) is uniformly continuous. Recall the basic mathematical fact that a continuous function in a bounded space is uniformly continuous and the composition of two uniform continuous functions remains uniform continuous. It is then derived thatV 1 (t) is a uniform continuous function of t and Barablat's lemma [37] can be used to conclude that lim t→∞V1 (t) = 0 and thus lim t→∞˙ 12 (t) = 0 2 . Similar analyses can be used to show that˙ 12 (t) is uniformly continuous as well. Once again, Barablat's lemma is applied to obtain lim t→∞ w(t) = 0 2 . Therefore, the closed-loop system is globally asymptotically stable.

B. ADAPTIVE ISM CONTROLLER WITH UNCERTAINTIES
Next, consider the case that the external disturbance is nonzero and Assumptions 2 and 3 hold. In addition, denote by J * = diag{J * 1 , J * 2 , J * 3 } the known nominal values of the spacecraft inertia. Accordingly, the nominal values of J 12 and J a are given by Inspired by the structural characteristic of the closed-loop dynamics given in (9), an integral sliding mode surface is designed as (14) The above sliding mode relies on the nominal inertia of the spacecraft and the true inertia is not required. Since s(0) = 0 for all initial conditions, the trajectory always starts from the sliding mode. This, however, does not mean that the closedloop trajectory always stays on the sliding mode s = 0 ever since the beginning time. The perturbations due to system uncertainties and measurement noise can drive the system trajectory away from the sliding mode. Later, adaptation techniques will be used to dynamically adjust the compensation for uncertainties according to the deviation from the sliding mode.
Once the system states are confined on the sliding mode since a certain moment, i.e., s(t) ≡ 0 2 , we have which is equivalent to − k 2 sat α 2 ( 12 ). (15) Evidently, the system dynamics on the sliding mode have a form similar to (9). The only difference is that the true inertia parameters are now replaced with their known nominal values. As a result, the stability results of Theorem 1 also hold for the system dynamics on the sliding mode. Thus, (14) gives a non-singular terminal sliding mode, as stated in the following lemma.
Differentiating (14) and substituting τ * 12 with (7) leads tȯ s = J * 12˙ 12 − J * a S( 3 ) 12 − τ * 12 . Employing (6) and letting Since the inertias J 12 , J 12 , J a , and J a are all constants as stated in Assumption 1, it follows that η > 0 is a constant. Due to the presence of uncertainties, η is unknown. An adaptively tuned variableη is introduced as an estimate of η in order to compensate for the lumped unknown perturbations. A novel adaptive controller based on the ISM given in (14) is then designed as where k 3 > 0 and where > 0 is a constant. The adaptive variableη is updated by the following equatioṅ where ρ 1 , ρ 2 > 0. The control torque given in (18) consists of three parts. The nominal part τ * 12 denotes the torques required on the sliding mode and the linear part −k 3 s provides torques pulling the system trajectory to the sliding mode. In particular, the adaptive part −η(1+ 12 )s a aims to compensate for the perturbations. When s is stabilized to the vicinity of zero, the performance of controller (7) is approximately recovered. The above observation enables us to obtain some rules on parameter tuning. Increasing k 1 and k 2 enlarges the spring force and damping effect of the sliding mode dynamics, respectively. Increasing k 3 amplifies the stabilizing effect to the ISM surface and meanwhile the magnitude of required control torques. ρ 1 and ρ 2 determine the sensitivity of the updating law on state errors. Increasing ρ 2 enables to adjust the parameter estimateη quickly when state errors are large. In contrast, increasing ρ 1 enforces the damping in the updating law and can avoid excessive growth of the parameter estimate. With the boundary layer given in (19), the control torque is continuous and the chattering effect can be mitigated by properly selecting the parameter . Generally, decreasing the values of can reduce the ultimate state errors but increase the risk of inducing chattering effect in control torques.
The proposed adaptive ISM controller incorporates a finite-time convergent sliding mode dynamics and an adaptive algorithm to estimate and compensate for the perturbations. Hence, it has better robustness against uncertain inertia, unknown disturbances, and actuator faults than the existing methods derived in [22], [23], [25]- [27]. Note that these methods did not take actuator faults into consideration. In addition, the proposed controller has a clear physical meaning and thus its gain tuning is greatly simplified.
The stability of the entire closed-loop system is summarized in the following theorem.
Substitutingη with (20) and (25) into (23) and invoking In addition, it is direct to verify that Consequently, (26) further leads tȯ Recalling the expression of s a given in (19), it can be seen that χη(1+ 12 ) s (1− s a ) = 0 whenη(1+ 12 ) s > . Whenη(1 + 12 ) s ≤ , the following inequality can be verified: Letting we can then deduce from (27) thaṫ where γ 1 = min{2χk 3 , ρ 1 } > 0 according to Assumption 4. It can then be concluded by means of Theorem 4.18 in [37] that the states of the closed-loop system are all uniformly bounded and ultimately stabilized to the following region Noting that s ≤ √ 2V 2 , we can finally deduce that the sliding function is stabilized to the region given by (21). The conclusion in Theorem 2 is now obtained.

IV. NUMERICAL EXAMPLES
In this section, numerical simulations are performed to the demonstrate the efficacy of the proposed methods. The first example compares the performance of the Hölder-continuous controller given by (7) and the adaptive controller given by (18)- (20). More precisely, the Hölder-continuous controller is simulated in the absence of uncertainties while the adaptive ISM controller is simulated in the presence of multiple uncertainties. The second example compares the performance of the proposed adaptive ISM controller with three existing spin-axis stabilization controllers, namely, the optimal controller derived in [23], the internal-model-based controller derived in [25] and the adaptive sliding mode controller designed in [27].
Consider a nearly axis-symmetric spacecraft withb 3 -axis being the symmetry axis. The symmetry axis is unactuated and there are two control torques alongb 1 andb 2 , which are perpendicular tob 3 -axis. The torque saturation limit is 2 N·m. The true inertia matrix of the spacecraft is while the nominal inertia used for control is assumed to be J * = diag{32, 32, 65} kg · m 2 . Additionally, the external disturbance acting on the spacecraft is given by  1 t), respectively. The desired inertial direction is [0, 0, 1] T . The influence of measurement noise is considered in the simulations. The measured attitude and angular velocity used for control law implementation are assumed to be w m = w + n w and m = + n , where n w and n are white Gaussian noise processes subject to the distributions N (0 2 , (1 × 10 −4 ) 2 I 2 ) and N (0 2 , (1 × 10 −6 rad/s) 2 I 2 ), respectively. with no uncertainties, i.e., J = J * = diag{32, 32, 65} kg · m 2 , d = 0 3 , and E 12 (t) = I 2 . Following this, the adaptive ISM controller is simulated by considering the inertia uncertainties, external disturbances, and actuator faults specified previously. The control gains for controller (7) are set to k 1 = 4, k 2 = 12, α 1 = 0.6 and α 2 = 0.75. For the adaptive controller, we set k 3 = 0.01, ρ 1 = 0.002, ρ 2 = 0.05, = 0.01η(0) = 0.01 and maintain k 1 , k 2 , α 1 and α 2 the same as controller (7). Fig. 2 compares the resultant responses of w coordinate, angular velocity , and command torque τ 12 by controller (7) and the adaptive ISM controller. When there are no uncertainties, controller (7) yields closed-loop dynamics given by (9), which is equivalent to the ideal sliding dynamics on the ISM s = 0 2 , i.e., (15). Therefore, the first simulation and its results given in Figs. 2a, 2c, and 2e correspond to the ideal performance of the adaptive ISM controller when s = 0 2 is maintained precisely from the beginning to the end of the simulation. In this case, the symmetry axis is stabilized to the desired direction in about 40 seconds. Figs. 2b, 2d, and 2f plot the actual responses of w, , and τ 12 of the adaptive ISM controller under multiple system uncertainties. Fig. 3 plots the time histories of the sliding variable s and adaptive parameterη. Despite the influence of multiple system uncertainties, the actual performance of w and has very minor difference with the ideal performance. The reason can be found from Fig. 3. The initial angular velocity and system uncertainties bring the attitude trajectory away from the sliding mode s = 0 2 . This triggers the adaptive algorithm to increaseη so that the attitude trajectory is pulled back to the vicinity of s = 0 2 after 15 seconds. In order to reject the influence of system uncertainties, the resultant command torque is nonzero and time-varying in the steady phase (Fig. 2f) as opposed to the ideal case (Fig. 2e), which is zero in the steady phase. Therefore, the adaptive ISM controller can quickly and efficiently compensate for the system uncertainties and approximately recover the ideal performance specified by the ISM dynamics.
The simulation results are given in Figs. 4-6, which compare the time histories of w , 12 , and τ 12 by the four methods. The proposed method and the methods in [23], [27] attain a convergence time of about 40 seconds for w and 12 . The method in [25] yields the longest convergence time. We can also see that the proposed method achieves much smoother transient responses than the other three methods. Table 1 summarizes and compares the steady-state accuracy attained by the four control methods. The residual errors  of w and 12 for the proposed method are 1.08 × 10 −5 and 2.31 × 10 −5 rad/s, respectively, which are far smaller than those for the other three methods. Note that the methods in [23], [25] exhibit poor robustness because they rely on precise inertia matrix, fully healthy actuators, and satisfactory disturbance compensation. The sliding mode controller in [27] utilizes adaptive techniques to reject disturbances but the sliding mode motion ensures merely asymptotic convergence. In summary, the proposed method has better robustness against inertia uncertainties, external disturbances, and actuator faults. This advantage is due to the finite-time convergent sliding mode dynamics and the adaptive mechanism for perturbation compensation. As a result, the proposed adaptive ISM controller ensures better transient responses and higher steady-state accuracy.

V. CONCLUSION
Two controllers were proposed to stabilize the spinning axis of an axis-symmetric spacecraft to any desired inertial direction using two torques normal to the symmetry axis. The first is a non-smooth yet continuous controller with a nonlinear proportional-derivative structure, which was developed via the homogeneous theory without any uncertainties. It was further utilized to construct an integral sliding mode surface, based on which a novel adaptive controller was developed to deal with uncertain inertia, unknown disturbances and actuator faults. The adaptive controller can approximately recover the performance of the non-smooth controller to achieve high pointing accuracy of the spacecraft spinning axis, even in the presence of multiple uncertainties. The effectiveness of the proposed methods was demonstrated by numerical examples.