Mathematical Problems in Engineering

Volume 2015, Article ID 947843, 11 pages

http://dx.doi.org/10.1155/2015/947843

## Robust Linear Quadratic Regulator via Sliding Mode Guidance for Spacecraft Orbiting a Tumbling Asteroid

^{1}Department of Control Science and Engineering, Jilin University, Changchun, Jilin 130022, China^{2}Department of Control Engineering, Changchun University of Technology, Changchun, Jilin 130022, China^{3}State Key Laboratory of Management and Control for Complex Systems, Institute of Automation, Chinese Academy of Sciences, Beijing 100190, China

Received 30 June 2015; Revised 1 September 2015; Accepted 17 September 2015

Academic Editor: Eric Florentin

Copyright © 2015 Peng Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Aiming to ensure the stability of the spacecraft with multiuncertainties and mitigate the threat of the initial actuator saturation, a Robust Linear Quadratic Regulator (RLQR) via sliding mode guidance (SMG) for orbiting a tumbling asteroid is proposed in this paper. The orbital motion of the spacecraft near a tumbling asteroid is modelled in the body-fixed frame considering the sun-relative effects, and the orbiting control problem is formulated as a stabilization of a nonlinear time-varying system. RLQR based on the adaptive feedback linearization is proposed to stabilize the spacecraft orbiting with the uncertainties of the asteroid’s rotation and gravitational field. In order to avoid the initial actuator saturation, SMG is applied to generate the transition process trajectory of the closed-loop system. The effectiveness of the proposed control scheme is verified by the simulations of orbiting the asteroid Toutatis 4179.

#### 1. Introduction

The exploration of asteroids has become a hot topic of interests. Equilibria and periodic orbits near asteroids are usually highly unstable due to the irregular shapes and the complex rotation of the asteroids. Hence, the control of a spacecraft in closed proximity of an asteroid is among the most challenging technical problems in the exploration of asteroids [1].

Many researches have paid great attention to the orbital control of the spacecrafts near asteroids. Sawai et al. [2] presented a control based on the one-dimensional altimetry measurements to stabilize a hovering spacecraft. Broschart and Scheeres [3] investigated the stability of realistic hovering control in the body-fixed and the inertial reference frames, respectively. Then, Broschart and Scheeres [4] proposed the sufficient conditions for a dead-band controller to bound spacecraft hovering motion in time-invariant Lagrangian dynamical systems. Wie [5] presented the dynamic modeling and control analysis of multiple gravity tractors in halo orbits. Furfaro et al. [6] investigated a novel closed-loop autonomous guidance law based on multiple sliding surfaces for the soft landing of the spacecraft on the designated point on the asteroid. Liu et al. [7] presented orbital control law for the spacecraft, which consists of PD controller and a nonsingular terminal sliding mode controller, to track the soft landing trajectory. Guelman [8] investigated a simple three-dimensional guidance law for the orbit transfer to a quasi-circular orbit about a rotating small celestial body using continuous thrust. In these previous contributions, asteroids are assumed to be a pure rotation configuration about the axis with its maximum inertia. However, the rotation of asteroids is very complex in practice and lots of them are time-varying [9]. The asteroid with nonprincipal axis rotational motion is so-called “tumbling asteroid” [10]. Nazari et al. [11] investigated the observer based body-frame hovering control over a tumbling asteroid, which is based on the time-varying LQR or the Lyapunov-Floquet transformation and time-invariant LQR, respectively. However, the rotation of the asteroid needs to be modelled in his control, which will cost a huge amount of telescope observing time to be determined [12], and the effect of the gravitational fields uncertainty is not analyzed. Hence, designing a robust control law for the spacecraft under the multiuncertainties of the asteroid’s gravitational field and rotation is urgent.

The initial actuator saturation problem also needs to be considered. The output of the controller may have a high peak at the start, whose magnitude depends on the size of the initial error. As the continuous adjustable thrust is small, the control output may rise over the limitation of the thrust and threaten the stability of the system [13]. Aiming to avoid this problem, arranging the transition process is a good idea to acquire an acceptable controlled quantity. In many researches of the active disturbance rejection controllers, Tracking Differentiator (TD) is applied to arrange the transition process to avoid much larger control outputs [14–17]. However, the stability of the TD is difficult to be approved and the magnitude of the control output via TD is hardly adjustable. The sliding mode control is an important improvement of the control theory [18]. Based on it, the sliding mode guidance (SMG) is confirmed due to its good performance in the orbital control of the spacecraft near asteroids [19]. We apply the SMG to arrange the transition process of the controller. By doing this, the max magnitude of the control output becomes adjustable and the threat of the initial actuator saturation problem is reduced.

In this paper, we proposed RLQR via the SMG for stabilizing the orbit of the spacecraft around a tumbling asteroid, which does not need to model the rotation of the asteroid. Firstly, the spacecraft orbital motion near a tumbling asteroid is modeled as a restricted three-body problem. RLQR based on the adaptive feedback linearization is proposed. The feedback linearization consists of a feedforward control, which is based on the spheric harmonic coefficient model of the asteroids gravitational field, and an adaptive compensator to ensure the robust stability against model uncertainties. Aiming to mitigate the threat of the initial actuator saturation, the SMG is applied to arrange the transition process of the RLQR. Simulations of orbiting the Toutatis 4769 are performed to verify the effectiveness of the proposed controller. The results of applying the RLQR with and without the SMG are compared to show the advantages.

The rest of the paper is organized in the following form. In Section 2, the orbital dynamic of spacecraft orbital motion is modeled and the problem formulation for control is proposed. In Section 3, the RLQR based on the adaptive feedback linearization is proposed. The SMG is applied to arrange the transition process of the proposed RLQR. In Section 4, simulations of orbiting the Toutatis 4769 are performed. Conclusions are drawn in Section 5.

#### 2. Problem Formulation

For the relative orbital motion of a spacecraft near an asteroid, the dynamic model includes the nonspherical gravity field of the asteroid and the solar radiation pressure (SRP) [1]. In most previous researches, the orbital dynamic of the spacecraft was formulated into two regimes: the gravity dominated regime, in which the effects on the spacecraft from the sun were ignored, and the solar dominated regime, in which the gravity of the asteroid was out of consideration. Few researches considered both of the effects from the sun and the asteroid [20]. In this section, the orbital dynamic of the spacecraft near a tumbling asteroid is modeled as a restricted three-body problem with the joint perturbations.

##### 2.1. Orbital Dynamic of the Spacecraft near Tumbling Asteroids

Before the modeling, two relative frames need to be defined. As shown in Figure 1, the inertial frame centered at the mass center of the asteroid. The -axis is parallel to the sun-line. The -axis is coinciding with the direction of the velocity vector of the asteroid. The -, -, and -axis compose the right-handed coordinate system. The body-fixed frame fixes on asteroid with the origin coinciding with the mass center of the asteroid. The , , and are coinciding with the axis of the asteroids maximum, minimum, and intermediate moment of inertia, respectively.