Shock and Vibration

Volume 2018, Article ID 3129834, 16 pages

https://doi.org/10.1155/2018/3129834

## Attitude Maneuvering and Vibration Reducing Control of Flexible Spacecraft Subject to Actuator Saturation and Misalignment

^{1}National Key Laboratory of Science and Technology on Space Intelligent Control, Beijing Institute of Control Engineering, Beijing, China^{2}Department of Automation, School of Information Science and Technology, Tsinghua University, Beijing, China^{3}Department of Strategic Missile and Underwater Weapon, Naval Submarine Academy, Qingdao, China

Correspondence should be addressed to Tao Zhang; nc.ude.auhgnist@gnahzoat

Received 18 October 2017; Revised 17 June 2018; Accepted 5 July 2018; Published 5 September 2018

Academic Editor: Jussi Sopanen

Copyright © 2018 Jiawei Tao 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

A robust adaptive constrained control scheme is proposed for flexible spacecraft attitude maneuver and vibration suppression, in which multiple constraints are simultaneously considered, such as uncertain inertia parameters, external disturbance, unmeasured elastic vibration, actuator saturation, and even actuator misalignment. More specifically, a novel path planning scheme based on quintic polynomial transition is firstly developed to realize smooth acceleration variate and therefore decrease the vibration of flexible appendages. Secondly, an elastic modal estimator is employed to estimate the unmeasured variables, such as the modal position and velocity. Thirdly, an adaptive updating technique is used to spare the extra knowledge of system parameters and the bound of the external disturbance. In addition, an auxiliary design system is constructed to address the actuator saturation problem, and a compensation term is synthesized and integrated into the controller to handle the actuator misalignment. Finally, overall system stabilization is proved within the framework of Lyapunov theory, and numerical simulation results are presented to illustrate the effectiveness of the proposed scheme.

#### 1. Introduction

Flexible spacecraft with large flexible structures is usually expected to achieve high pointing and fast attitude maneuvering in future space missions. However, the attitude maneuvering operation will introduce certain levels of vibration to flexible appendages due to the rigid-flexible coupling effect, which will deteriorate its pointing performance. For flexible spacecrafts, the governing differential equations for attitude kinematics and dynamics are strongly nonlinear in nature. The attitude maneuvering problem is further complicated by the uncertainty of spacecraft inertia parameters due to onboard payload motion and fuel consumption. Furthermore, it is also affected by various external disturbances that influence the mission objectives significantly. Additionally, the actuator misalignment during installation and actuator saturation increases the complexity and difficulty further.

All these factors in a realistic environment cause a considerable difficulty in the design of attitude control system for meeting high-precision pointing requirement and desired control performance during the attitude maneuver process, especially when all these issues are treated simultaneously.

Over the last few decades, considerable works have been found for vibration suppression and attitude control system design. A nonlinear state feedback attitude control law [1] in combination with path planning is proposed. Specifically, the planned attitude maneuver path is smooth, and thus, appendages’ vibration excited by attitude maneuver can be attenuated greatly. Specially, variable structure control (VSC) is known as an efficient way to deal with system uncertainty and external disturbance and has been applied to attitude control problem of flexible spacecrafts in [2, 3].

However, these design methods require the information on the bounds on the uncertainties or disturbances for the computation of control gains. Recently, to overcome the drawbacks of each method, a combination of these techniques with adaptation mechanism to tune the controller gains are also studied for flexible spacecrafts with parameter uncertainties and disturbances [4–6]. A new adaptive system [4] for rotational maneuver and vibration suppression of an orbiting spacecraft with flexible appendages has been designed. Then adaptive output regulation of the closed-loop system is accomplished in spite of parameter uncertainties and disturbances. A new variable structure control approach [5] has been proposed for attitude control and vibration suppression of flexible spacecrafts during attitude maneuvering, and the adaptive version of the proposed controller is achieved through releasing the limitation of knowing the bounds of the uncertainties and perturbations in advance.

Relevant drawback of these control strategies is either the extra necessity to measure the modal variables or to treat the rigid-flexible coupling effect as an additional disturbance acting on a rigid structure. With regard to the latter situation, an extended state observer [7, 8] is designed to estimate and thus to attenuate the total disturbance in finite time, including an external disturbance torque and the coupling effect. As a result, the prior knowledge of the total disturbance is not required.

Unfortunately, in some cases, the availability of the measured modal variables is an unrealistic hypothesis due to the impracticability of using appropriate sensors or the economical requisite. For removing this disadvantage, reconstructing the unmeasured modal position and velocity by means of appropriate dynamics is an alternative way. A class of nonlinear controllers incorporating modal state estimator [9–13] has been derived for spacecraft with flexible appendages. It does not ask for measures of the modal variables, but only uses the parameters describing the attitude and the spacecraft angular velocity. The controller derived then uses estimates of the modal variables and its rate to avoid direct measurement.

However, a typical feature in all of the mentioned attitude-control schemes and methods is that the control device is assumed to be able to produce big enough control torque without taking actuator saturation into account. Extensive results pertaining to spacecraft attitude control systems containing actuator saturation nonlinearities have been presented in [14–22]. A robust variable structure controller in [14, 15] has been skillfully designed to control the spacecraft attitude under input saturation. However, these control schemes lose the generality to nonlinear flexible spacecraft system. A modified adaptive backstepping attitude controller [17, 18] is developed considering external disturbances and input saturation. But the way to include the effects of the flexible dynamics in the lumped disturbance for the rigid dynamics deprives the controller of a direct compensation of the dynamic terms caused by the flexibility.

A typical feature in the previous approaches is that they do not consider actuator misalignments. However, whether due to finite manufacturing tolerances or warping of the spacecraft structure during launch, some actuator alignment error will definitely exist in practice. This problem may cause mission performance to degrade and thus pose significant risk to the successful operation of the spacecraft. Therefore, it is desirable to design a control scheme to handle the actuator misalignments. Unfortunately, there has been insufficient research on attitude control in the presence of actuator misalignments. An adaptive control law [23] is developed to accomplish attitude maneuver in the presence of relatively small gimbals’ alignment error of variable speed control moment gyros.

More specially, an adaptive control approach [24] was proposed for satellite formation flying. The backstepping technique is used to synthesize the controller, and the thrust misalignment is successfully handled. In another related work, a nonlinear model reference adaptive control scheme [25] is tested in the presence of misalignment errors up to fifteen degrees. Although an extended Kalman filter is used in another approach to develop methods for on-orbit actuator alignment calibration, uncertain inertia properties have not been taken into account.

With a view to handle the above challenges and potential problems simultaneously, a new constrained robust adaptive control scheme is proposed in this paper. The main contributions are summarized as follows:(1)A novel path planning scheme based on quantic polynomial transition is applied to smooth acceleration variate and therefore decrease the vibration of flexible appendages.(2)This paper investigates the feasibility of attitude maneuver and vibration suppression in the presence of uncertain inertia parameters, external disturbances, unmeasured elastic vibrations, actuator saturation, and even actuator misalignment simultaneously, which has not been previously examined. As compared with the controller in [16] which only considers attitude stabilization, that is, rest-to-rest maneuver of a flexible spacecraft, the control scheme in the present paper can be applied to handle attitude tracking control problem. The controller in [6] investigates the attitude tracking problem for flexible spacecrafts, while it does not take the actuator saturation and actuator misalignment into consideration explicitly.(3)The proposed controller is designed without requiring prior knowledge of the measurement of vibration variables, of the parameter uncertainties, and the upper bound of external disturbance. Robust control terms are synthesized to compensate all uncertainties including parametric uncertainties, unmeasured elastic vibration, external disturbances, actuator saturation, and actuator misalignment.

The rest of this paper is organized as follows: Section 2 states flexible spacecraft modelling with actuator misalignment and control problem formulation. A constrained robust adaptive-control scheme is proposed in Section 3. Numerical simulation results are presented in Section 4 to demonstrate the effectiveness and superiority of the proposed control scheme. Finally, conclusion is given in Section 5.

#### 2. Flexible Spacecraft Modelling and Problem Formulation

##### 2.1. Kinematics and Dynamics Equation

This section briefly reviews the quaternion mathematical description of the attitude motion of a flexible spacecraft.

The attitude kinematic equation of the spacecraft can be written in terms of unit quaternion for global representation without singularities as follows [26]:where represents the attitude quaternion, and is an identity matrix of the dimension specified by its subscript.

The dynamic model of a flexible spacecraft is governed by the following differential equations:where is the spacecraft angular velocity in the body-fixed frame, is the skew-symmetric matrix of , is the modal coordinate vector of the flexible appendages, the matrix denotes the total inertia of spacecraft moment, is the coupling matrix between the flexible appendages and the rigid spacecraft, is the control torque, is the external disturbance torque vector, and and denote the damping and stiffness matrices, respectively, and they are defined aswhere is the number of elastic modes considered, is the natural frequency, and is the corresponding damping ratio.

For simplicity of the development, let us first introduce the following variable representing the total angular velocity expressed in modal variables, so the dynamics of the flexible spacecraft from (2) to (3) can be further expressed aswhere , with as the contribution of the flexible parts to the total inertia matrix.

##### 2.2. Attitude Tracking Model

Let be the unit quaternion representing the desired attitude, be the desired angular velocity.

Then, the quaternion error and angular velocity error are given bywhere is the inverse of the desired attitude unit quaternion, the symbol is the operator for quaternion multiplication; with , we havewhere is the transformation matrix from the desired coordinate frame to the body coordinate frame with

From (1) to (10), tracking error dynamic equation can be obtained as follows:where and are given by

##### 2.3. Actuator Uncertainty Model

In practical aerospace engineering, redundant actuators are fixed to improve the reliability of the attitude control system. The considered spacecraft is controlled by using four reaction wheels. The configuration structure of four actuators in [27, 28] is adopted. Three reaction wheels are fixed orthogonally aligned with the axes of the body-fixed frame. The fourth, redundant, actuator is mounted skewed at and .

With this configuration, the total control torque can be calculated aswhere denotes the nominal torque generated by the four reaction wheels, and is the reaction wheel configuration matrix, representing the influence of each wheel on the angular acceleration of the spacecraft.

However, in practice, the knowledge of orthogonal configuration of actuator will never be perfect. Due to finite-manufacturing tolerances or warping of the spacecraft structure during launch, actuator misalignments may exist. The reaction wheel mounted on the *X*-axis is offset from the nominal direction by constant angles, and . The reaction wheels mounted on *Y*-axis and *Z*-axis are assumed to be tilted away from their nominal directions by , , , and , while the redundant reaction wheel is tilted from its nominal direction by and .

Then, the real control torque acting on spacecrafts with misalignment is expressed as

The definition of the misalignment angles suggests that are small angles. Hence, the following relationships are used to approximate (16):

Then for the considered actuator configuration, the configuration matrix can be represented aswhere denotes the nominal configuration matrix, and denotes the actuator misalignment. They can be written as

In view of (15) and (18), tracking error dynamics in (12) can be rewritten as

To facilitate control system design, the following assumptions and lemmas are presented and will be used in the subsequent developments.

*Assumption 1. *The components of external disturbance in (21) are assumed to be bounded by a set of unknown bounded constants, that is

*Assumption 2. *Due to physical limitations on the reaction wheels considered, the maximum output torque of each actuator output torque has the same limit value , that is [29]where and represent the known saturation levels of reaction wheels.

*Assumption 3. *The uncertainty due to misalignment is an unknown but bounded matrix satisfyingwhere is a known positive constant.

*Remark 1. *According to (20), every element of actuator misalignment matrix is a function of misalignment angle errors which are small in practice, then the Frobenius norm of is bounded by a known quantity . It is easy to find a feasible in practice.

Lemma 1. *For arbitrary positive constant and variable , the following inequality holds [30, 31]with and .*

##### 2.4. Control Problem Formulation

Given any initial attitude and angular velocity, the control objective can be stated as considering the flexible spacecraft attitude system described by (1)–(3), design a torque command such that the following goals are met in the presence of uncertain inertia parameters, external disturbance, unmeasured elastic vibration, actuator saturation (23), and even actuator misalignment:(1)The attitude orientation and angular velocity tracking errors are driven to zero, or a small set containing the origin.(2)The vibration induced by the maneuver rotation should be attenuated as soon as possible in the presence of parametric uncertainties, external disturbance, actuator saturation, and even actuator misalignment.

#### 3. Robust Adaptive Constrained Controller Design

The proposed attitude control scheme for the flexible spacecraft is shown in Figure 1, in which the controller is composed of an auxiliary design system to compensate the effect of actuator saturation and modal estimator to estimate unmeasured modal position and velocity, as well as path planning, which will be given in the following subsections.