International Journal of Aerospace Engineering

Volume 2018, Article ID 2825681, 18 pages

https://doi.org/10.1155/2018/2825681

## Consideration of Three-Dimensional Attitude and Position Control for a Free-Floating Rigid Body Using Three Thrusters

^{1}Department of Mechanical and Aerospace Engineering, Tottori, Tottori University, 4-101 Koyama-cho, Tottri 680-8552, Japan^{2}Department of Aeronautics and Astronautics, Fukuoka, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka 819-0395, Japan

Correspondence should be addressed to Takashi Matsuno; pj.ca.u-irottot.hcem@ustam

Received 22 March 2018; Accepted 28 August 2018; Published 29 October 2018

Academic Editor: Paolo Gasbarri

Copyright © 2018 Takashi Matsuno 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

The study examines the control algorithm of a three-dimensional attitude and position of a free-floating rigid body with three thruster forces in which the force directions are fixed with respect to the body. This study provides a theory to develop a control method of an underactuated satellite with the minimum thruster number. In the procedure, three switching controllers are used in conjunction with motion planning in the final angular-rate deceleration phase to individually control the six state variables to the target values. The switching controllers have a hierarchical structure by using invariant manifolds as switching surfaces. The state variables in higher class manifolds that include lower class ones are adjusted by repeatedly adding intentional disturbances while the lower class state variables are returned to the original values by using lower class invariant manifolds. This study describes methods to define the invariant manifolds and also the intentional disturbance for achieving the forementioned control strategy. Finally, the motion planning in the angular-rate deceleration phase from a remained single-axis rotation finalizes the six state values of the body to the target values. Numerical simulations verify the proposed method.

#### 1. Introduction

What is the minimum number of thrusters required to control both the attitude and position of a rigid satellite? What strategies can be applied and what kind of motion is controlled for a satellite in which there are few remaining thrusters? These questions motivated us to pursue the current investigation because nonlinear underactuated control makes these satellites fault-tolerant with respect to thruster malfunctions and enables continuity of the respective missions.

Most satellites are equipped with a sufficient number of thrusters to control the attitude and position in three dimensions because of the following reasons. In typical thrusters, the force directions are fixed to a satellite body and the magnitudes are restricted as positive. This implies that another thruster is required to generate a force in the opposite direction. Consequently, more than twelve thrusters are required for a satellite to guarantee the six degrees of freedom in motion and to ensure the robustness of the system in simple control logic. However, the number of thrusters can be reduced by applying nonlinear control methods. Hence, combined attitude and position control based on the nonlinear motion of a satellite can achieve a significant reduction in the number of actuators.

A few underactuated systems in which the degrees of freedom of motion exceed the number of actuators can be potentially completely controlled by utilizing their nonholonomic constraints [1, 2]. A free-floating satellite is a type of an underactuated system, and its attitude and position dynamics are represented as nonintegrable second-order differential equations that act as nonholonomic constraints [3]. Several intensive studies focused on the attitude motion control of a satellite by utilizing nonholonomic constraints. Several extant studies indicate that two independent torques successfully control a three-dimensional satellite attitude. Tsiotras and Longuski [4] proposed new attitude parameters termed as “-parameters,” and then Tsiotras and Doumtchenko [5] developed a discontinuous control law that utilized the parameters. In another study, Morin and Samson [6] designed a smooth time-varying feedback control. Recent studies shifted to more practical ones including a controller for the underactuated CMG satellite system [7], implementation of the kinematic planning scheme [8], and an optimal process for state stabilization [9]. The strategies indicate that two bidirectional control torques (i.e., both positive and negative directional torques are generated) with independent magnitude can control three attitude parameters of a satellite. Therefore, the problem of satellite attitude control is transformed into the number of thrusters that are required to generate independent torques with respect to the two principal axes. Sidi [10] presented a pioneering study related to this problem that numerically demonstrated that four thrusters produce independent torques with respect to three principal axes. In [11], we analytically confirmed the forementioned results and indicated that three thrusters that satisfy a configuration produce bidirectional torques with respect to two independent axes. This implies that when the nonholonomic dynamics of a satellite are considered, three thrusters control the three-dimensional attitude of a satellite. Hence, Matsuno et al. [11] and Yoshimura et al. [12] demonstrated an attitude control strategy and developed feedback controllers although a zero or unidirectional (only positive or negative directional) torque was generated under the thruster configuration around the third axis owing to the coupling effect with the other two axes.

The present study focuses on the simultaneous attitude and position control of an underactuated satellite. In two-dimensional motion, at least two thrusters are required to accelerate and decelerate the rotation of a satellite. Essentially, the same type of controllers in [13–15] is applied to obtain a controller with two thrusters for the in-plane attitude and position motion of a free-floating satellite. The study numerically verified that a switching controller that utilizes an invariant manifold achieves a two-dimensional attitude and position control by using positive albeit variable magnitudes of thruster forces. The study was extended in [16] for two on/off types of thrusters in which the input magnitudes are fixed.

In contrast to the two-dimensional cases, only a few studies discuss three-dimensional attitude and position control for underactuated satellites. With respect to a satellite with four thrusters, the previous study [17] described a design procedure of motion planning for the simultaneous control of the attitude and position. However, in the case with three thrusters, the three-dimensional attitude and position control is quite challenging because of the following reason. Three thruster configurations that guarantee bidirectional control torques for two principal axes generate dependent torque with respect to the third principal axis of a satellite. Specifically, the attitude control proposed in the present requires a series of complex nonlinear processes. Furthermore, thruster forces are unidirectional, and thus, the controllability of the system is uncertain. It should be noted that although the theorem of Sussman [18] or Goodwine and Burdick [19] provides sufficient conditions for the controllability of nonholonomic systems, they cannot be applied in this study owing to the nonnegative restriction of the thruster forces.

Given the above background, the present study develops a switching controller by using three invariant manifolds and provides a motion planning scheme in the final phase. This paper deals with the problem to control a free-floating rigid body in three-dimensional space using three thruster forces (hereafter, we refer to this type of rigid body model as “FFRB”). In this simplified problem, the proposed method achieves both attitude and position control by using only three thrusters in which the directions are fixed with respect to the body. The effectiveness of the method is numerically certified in ideal conditions (constant mass and nonlimited positive thruster force). This study is the first to demonstrate that the three-dimensional attitude and position of a rigid body can be controlled with three fixed thrusters, although the following problems remain in the future. The proposed controller includes a few heuristic processes; rigorous certification of the control strategy is required, and it sometimes results in an extremely long controlling time to the target states. We believe that the concept of this controller explained in this study contributes to significantly decreasing number of thrusters in practical underactuated satellites.

The remainder of this paper is organized as follows. In Section 2, the formulations of the kinematic and dynamic equations of a FFRB are discussed. The procedure consists of a combination of the three hierarchical feedback steps and motion planning. Furthermore, the effects of the design parameters in each step are discussed. In Section 4, numerical simulations are conducted to verify the proposed controller, and a summary of the study is presented in Section 5.

#### 2. Three-Dimensional Model of a FFRB (Free-Floating Rigid Body with Three Force Inputs)

##### 2.1. Thruster Configuration

In order to simplify the discussion in Section 3, this subsection defines the three-thruster configuration for controlling a FFRB. It should be noted that the main purpose of this study is to demonstrate that a range of three-thruster configurations controls both the position and the attitude of a FFRB to the target values.

It is assumed that a rigid body has the three thrusters that satisfy the following conditions (i)–(iii) for its torques. (i)Bidirectional control torques are generated about principal axes I and II of the FFRB.

This condition implies that the attitude motion in three dimensions is controlled by applying nonlinear control methods utilizing nonholonomic constraint of a FFRB unless the inertial moments of principal axes I and II are equal. (It is widely known that the attitude motion around the principal axis III is uncontrollable when the inertial moments are equal.) (ii)With respect to the principal axis III, a unidirectional control torque is generated whenever control torques are applied around the principal axes I and II.

This condition simplifies the discussion in Section 3. Furthermore, with respect to the translational motion control, the following condition is assumed. (iii)In the thruster configuration, the induced force along the principal axis III is bidirectional.

First, it is proven below that a range of three-thruster configurations satisfy the conditions (i) and (ii). We consider a thruster placed at and oriented to with respect to the body frame (see Figure 1(a)), and it generates a torque moment with respect to the mass center. Three thrusters form a tetrahedron with their moments, and a range of three-thruster configurations have a projection of the tetrahedron on the I–II plane including the mass center inside the body as shown in Figure 1(b). Subsequently, the combination of the three forces in positive or zero values generates their resultant torque within the extension of the tetrahedron. Thus, this type of three-thruster configurations satisfies the conditions (i) and (ii). It is noted that a vertex of the tetrahedron is the mass center of the FFRB, and therefore, it is impossible to generate bidirectional control torques for all three principal axes with three thrusters as shown in [11, 12].