Adaptive integral sliding mode control for spacecraft attitude tracking with actuator uncertainty

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Abstract

The attitude tracking of a rigid spacecraft with actuator uncertainties, such as actuator faults, alignment errors, and saturation constraints is examined. In addition, the unknown external disturbances and spacecraft inertia are also taken into consideration. A novel integral-terminal-sliding mode (ITSM), which is singularity free compared to traditional TSM, is designed such that the attitude tracking error converges to zero in finite time on the ITSM. An adaptive technique is then utilized to develop an adaptive ITSM controller (AITSMC), which achieves finite-time attitude tracking in the presence of some or all of the above uncertainties. Numerical examples are presented to demonstrate the effectiveness of the proposed method.

Introduction

Spacecraft attitude tracking control is a key enabling technology for many spacecraft missions and has received considerable attention in recent years. Many advanced control methods have been utilized to deal with the nonlinear attitude dynamics and achieve attitude tracking even in the presence of internal system uncertainties and external disturbances. Among the approaches applied to the problem, sliding mode control (SMC) is effective in handling various modeling uncertainties [1], [2] and can flexibly interface with other control methods, such as adaptive control [3], [4], [5], [6], disturbance observers [6], fuzzy control [7], etc., to attain better control performance. In [3], [4], [5], [6], [7], some or all of uncertain spacecraft inertia and unknown disturbance torques as well as actuator saturation were taken into account. These control laws are all based on a linear sliding mode (LSM), on which the system states slide to the desired equilibrium in infinite time.

Finite-time stability, implying finite-time convergence of the system trajectories, can yield faster convergence and better disturbance rejection than asymptotic or exponential stability [8]. In order to achieve finite-time convergence, terminal sliding mode control (TSMC), which utilizes nonsmooth functions such as fractional power functions as the sliding surface, was proposed [9]. TSMC-based attitude tracking laws were then derived in [10], [11], showing significant increase in robustness compared to LSM-based control laws. TSMC, however, can encounter a singularity problem because of the differentiation of a non-smooth function. To overcome the singularity problem, the nonsingular TSMC (NTSMC) was proposed in [12] and then applied to spacecraft attitude control [13]. In addition, faster TSMC/NTSMC (FTSMC/FNTSMC) was also proposed to enhance the convergence speed of traditional TSMC and NTSMC when the system states are far from equilibrium [14]. The LSMC and TSMC together with its variants are all first order in nature and suffer from the chattering problem when applied with a discontinuous function. In [15], higher-order sliding mode control laws were designed to achieve finite-time attitude tracking as well as overcome the chattering problem. All the above sliding mode methods have a reaching phase when the system states are initially outside the sliding surface. In contrast, the integral-SMC (ISMC) can be designed such that the system states start exactly from sliding surface for all initial conditions, thus eliminating the reaching phase [16], [17]. As a result, robustness is assured from outset.

None of the above studies, however, considered unexpected actuator faults, which are possible and even likely for operational spacecraft. Given the difficult space environment and high launch/operation cost of spacecraft, it is not feasible to repair or replace faulty actuators for most spacecraft. Therefore, fault-tolerant control (FTC), which is mainly based on software design rather than hardware replacement, is an appealing approach to deal with actuator failure and increase system reliability. An active FTC scheme for satellite attitude control was designed in [18], which requires a fault-detection and isolation (FDI) mechanism, while passive FTC methods were proposed in [19], [20], which require no FDI mechanism. Note that the upper bound of disturbance torques was required in [20]. In [22], an ISMC for attitude stabilization was designed with an adaptive algorithm to compensate for unknown external disturbances and actuator faults. The ISMC in [21], however, assumes known spacecraft inertias and does not consider actuator saturation. In contrast, the indirect adaptive FTC designed in [22] can accommodate unknown disturbances and actuator saturation simultaneously. The preceding fault-tolerant attitude control schemes can only yield asymptotic convergence of the attitude errors while the TSMC and NTSMC methods were used in [23] and [24], respectively, to achieve finite-time fault-tolerant attitude control. The quaternion-based NTSMC designed in [24], however, has a restriction that the scalar part of the quaternion cannot be zero.

Meanwhile, actuator alignment errors are quite common in practical applications due to limits on hardware fabrication and installation. The deviation of attitude actuators, such as thrusters, reaction wheels (RWs) or control moment gyros (CMGs), from their nominal positions can introduce undesired output torque errors. Even if the magnitude of an output torque can be precisely controlled, its direction can differ from the desired direction due to actuator misalignment. This problem must be dealt with for the effective and high precision attitude control. By taking RW misalignment into account, fault-tolerant attitude control is approached via LSMC in [25] and TSMC in [26], [27], respectively. Specifically, an optimization algorithm was used in [27] to allocate the command torque to redundant RWs. The TSMC-based control laws designed in [26], [27], however, encounter a singularity problem, as mentioned above. In [28], [29], the misalignment of CMGs were considered and adaptive attitude tracking laws were proposed. In contrast to the rotation-only control considered in the previous studies, simultaneous attitude and position tracking under thruster misalignment was addressed in [30].

This paper investigates the attitude tracking of a rigid spacecraft actuated by RWs using the ISMC method. Apart from the unknown spacecraft inertias and external disturbances, various actuator uncertainties, such as the sudden faults, alignment errors, and saturation constraints, are taken into account. A novel integral-terminal-sliding mode (ITSM) is proposed based on modified Rodrigues parameters (MRPs) and then developed with adaptive techniques into an adaptive ITSM controller (AITSMC) to handle the preceding system uncertainties. The main contribution of this paper is twofold:

1) It is shown how to combine the adding a power integrator (API) method and ISM techniques to derive an ITSM for finite-time attitude control. As a key step in the design, the system dynamics on the ITSM is deliberately constructed to be equivalent to the closed-loop dynamics produced by the API-based controller in [31]. As a result, the ITSM not only accomplishes finite convergence time but also avoids the singularity problem, in contrast to the conventional TSM. Moreover, the equivalent control on the ITSM is the same as the API-based controller.

2) The AITSMC is continuous and thus free from chattering and more importantly involves an adaptive algorithm to dynamically estimate and compensate for the total system uncertainty. If the actuator alignment error is restricted to a certain region, it is proven that the AITSMC can drive system states into a small neighborhood of the designed sliding mode in finite time. Therefore, the attitude tracking error is stabilized approximately along the ITSM to a neighborhood about zero, which can be made arbitrarily small. In addition, the AITSMC is efficient in computation since it only includes one adaptive gain and the adaptive algorithm has a simple form.

This paper proceeds as follows. Section 2 establishes the models of the spacecraft attitude dynamics, and actuator faults and misalignment. In Section 3, a novel ITSM is designed and then an AITSMC is developed by means of an adaptive technique. Section 4 demonstrates the application and effectiveness of the proposed methods via numerical simulations. Conclusions are summarized in Section 5.

Section snippets

Definitions and lemmas

Definition 2.1

(Finite-time stability) [8] : Consider the system ẋ=f(x), xn, f(0)=0, where f(x) is continuous on n. Denote by U a neighborhood of x=0. Then, the origin is said to be finite-time stable if the origin is 1) Lyapunov stable in U and 2) finite-time convergent in U. The finite-time convergence means the existence of a function T:U[0,+) such that, for any initial condition x0U, the solution of the system denoted by x(t,x0) satisfies x(t,x0)U for t[0,T(x0)) and limtT(x0)x(t,x0)=0. If U=n,

Controller design

The objective in this section is to design a command torque τc such that the tracking error (σe(t),ωe(t)) converges to a small neighborhood about zero in finite time, even in the presence of some or all of the above uncertainties. To this end, an ITSM ensuring finite-time convergence is designed first. With this ITSM, a tracking law is derived by using adaptive techniques to deal with all modeling uncertainties.

Numerical examples

In this section, simulation results are presented to illustrate the performance of the proposed control method. The spacecraft inertia matrix is given byJ=[10121200.520.515]kg·m2.

The external disturbance is assumed to be d(t)=[sin(0.05t),0.5sin(0.05t),cos(0.05t)]T×102 Nm. The objective is to track a time-varying trajectory given by σd(t)=12tan(13π80)[cos(ω0t),sin(ω0t),1]T, where ω0=0.008 rad/s. The torque limit of each RW is set as τmax=0.2 N m. The initial conditions are σ(0)=[0.4,0.2,0.3]T

Conclusions

An adaptive integral-terminal-sliding mode controller (AITSMC) was proposed for spacecraft attitude tracking. Unknown spacecraft inertias and external disturbances as well as various actuator uncertainties, such as actuator faults, misalignment and saturation constraints, were taken into account. No a priori information about these uncertainties is required for the AITSMC due to its adaptive feature. It was shown that the spacecraft attitude tracking error can be stabilized by the AITSMC to a

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