Mathematical Problems in Engineering

Volume 2018 (2018), Article ID 3420649, 12 pages

https://doi.org/10.1155/2018/3420649

## A Two-Dimensional Solar Tracking Stationary Guidance Method Based on Feature-Based Time Series

^{1}Shanghai Institute of Microsystem and Information Technology, Chinese Academy of Sciences, Shanghai 200050, China^{2}University of Chinese Academy of Sciences, Beijing 100049, China^{3}Innovative Academy for Microsatellites, Chinese Academy of Sciences, Shanghai 201203, China^{4}School of Information Science and Technology, ShanghaiTech University, Shanghai 201210, China

Correspondence should be addressed to Keke Zhang; moc.mot@89orezkkz

Received 19 December 2017; Accepted 15 March 2018; Published 24 April 2018

Academic Editor: David Bigaud

Copyright © 2018 Keke 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

The amount of satellite energy acquired has a direct impact on operational capacities of the satellite. As for practical high functional density microsatellites, solar tracking guidance design of solar panels plays an extremely important role. Targeted at stationary tracking problems incurred in a new system that utilizes panels mounted in the two-dimensional turntable to acquire energies to the greatest extent, a two-dimensional solar tracking stationary guidance method based on feature-based time series was proposed under the constraint of limited satellite attitude coupling control capability. By analyzing solar vector variation characteristics within an orbit period and solar vector changes within the whole life cycle, such a method could be adopted to establish a two-dimensional solar tracking guidance model based on the feature-based time series to realize automatic switching of feature-based time series and stationary guidance under the circumstance of different angles and the maximum angular velocity control, which was applicable to near-earth orbits of all orbital inclination. It was employed to design a two-dimensional solar tracking stationary guidance system, and a mathematical simulation for guidance performance was carried out in diverse conditions under the background of in-orbit application. The simulation results show that the solar tracking accuracy of two-dimensional stationary guidance reaches 10^{∘} and below under the integrated constraints, which meet engineering application requirements.

#### 1. Introduction

As microsatellites are featured with being light, small, smart, and cheap, they not only have become the research hotspot of satellite technology development and application, but are growing towards a direction of high functional density and practical applications. In order to adapt to multitype tasks and diversified applications, the requirement for higher payload power appropriate for high functional density microsatellites has been increasingly intense for the consideration of higher cost performance, rapid deployment, and extensive applicability. However, energy maximization must be taken into account with an aim to make microsatellites applied in high-power loads. Specific to application characteristics of microsatellites cluster or network, the scheme of energy maximization should be universal. Especially for the typical nonsolar synchronous inclined orbits such as Globalstar system, the solar panels adopting a solar tracking revolving mechanism is the only approach to solving energy acquisition issues provided that the platform itself remains an earth-oriented demand.

In order to maximize the output power improving the efficiency, many methods and devices of sun tracking and maximum power point tracking were proposed, which belong to the special issue of solar energy application in ground photovoltaic systems. There are some fuzzy techniques that were proposed in photovoltaic systems [1–3], such as a method based on simultaneous use of two fuzzy controllers in order to maximize the generated output power of a solar panel. The sun tracking is performed by changing the solar panel orientation in horizontal and vertical directions by two DC motors properly designed. Several solutions for two axis solar tracking systems based on solar maps or based on tetrahedron geometry were studied, which can predict the exact apparent position of the sun or the strongest intensity of visible light [4–6]. Dual-axis solar tracking design utilizes a four-quadrant light dependent resistor (LDR) sensor or adaptive solar sensor and so on. The control of moving the mechanical structure uses a low power microcontroller or a traditional PLC [7–10].

For satellite, the maximum energy acquisition purpose is the same that is achieved generally by solar array drive assembly (SADA). As the microsatellites pursue a high performance-price ratio strictly limiting development costs, it is impossible to select expensive SADA with slip ring. The light-small two-dimensional turntable could be used to carry the solar panels to realize solar tracking. For the convenience of open-loop control, stepping motor was utilized as a drive element. Nevertheless, the stepping motor still has some shortcomings such as poor dynamic behavior, high pulse step overshoot, and great rate ripple, and it is a nonnegligible interference source as far as satellites with a high requirement for attitude stabilization, such as earth observation satellites. Now, satellite rotating mechanism has been extensively investigated at home and abroad, covering mechanism models, ground-based validation, controller design, and attitude control coupling [11–14].

Solving the problem related to rotating mechanism motion stability of solar panels is a key to guaranteeing high pointing accuracy and high attitude stabilization of satellites. Factors that affect rotating mechanism motion stability can be elaborated from the following two aspects. Firstly, cogging torque of the stepping motor leads to instability of control, which can be resolved by means of compensation, and research on this issue has been relatively mature. SPOT satellite of France adopts a great subdivision sine/cosine driving strategy and an accelerometer to measure perturbed moment and implement compensation [15]. Engineers of NASA also introduce an input molding technique [16]. Additionally, an adaptive uniaxial driving strategy based on current compensation has been explored [17]. Secondly, due to instability of input angular velocity guiding turntable motion, the accurate angular velocity in the case of two-dimensional solar tracking becomes unstable. However, such an issue is usually neglected under the circumstance of unlimited capacity so that relevant investigations can be rarely seen as well.

A solar tracking stationary guidance method based on feature-based time series was proposed under the background of microsatellite solar panels loaded with two-dimensional turntable. Through phase division for time within an orbit period, the target velocity was divided into several segments of steady speeds, so as to solve guidance input instability problems and substantially reduce variable speed control. The high-stability and small-disturbance control of the two-dimensional turntable can be fulfilled.

#### 2. Problem Description and Modeling

Concerning a microsatellite that adopts orbital inclination of 55^{∘}, angle (the included angle between solar vector and orbital plane) varies between +78^{∘} and . Simultaneously, the corresponding tasks request the satellite to remain earth-oriented.

Specific to the microsatellite characteristics, when angle is close to , the included angle formed by solar vector and normal of the solar panels should be greater than if the conventional one-dimensional SADA has been utilized. In this case, solar panels are deemed to acquire no energy. Considering that the satellite runs in a 55^{∘} inclined orbit, such a phenomenon exists within a period long enough to incur unbalanced satellite energy supply making the entire satellite unable to run normally. Resultantly, two-dimensional rotation must be conducted to realize solar tracking orientation of the panels under the circumstance that earth-oriented status of the satellite is maintained, which is able to maximize energy acquired. As the two-dimensional turntable is employed, 360-degree continuous rotation cannot be completed; thus an appropriate two-dimensional solar tracking stationary guidance strategy should be formed to guarantee that the satellite can acquire enough energies for operation in a condition of different angles.

##### 2.1. Basic Assumption

The two-dimensional turntable mechanism makes use of a stepping motor. Without loss of generality, friction moment, fluctuating moment, cogging torque, and mechanism dynamics were taken into account, while mutual induction winding, high-order harmonic torque, and external disturbance torque of the motor were ignored. Based on a cogging moment compensation method, velocity stability of the turntable could be improved. In addition, it has been assumed that the angular velocity of the turntable can be matched with the guidance input.

##### 2.2. Constraint Condition

###### 2.2.1. Two-Dimensional Rotation Angle Limitation Constraint

With regard to the microsatellite, area of its solar panels unfolded is very large so that it is able to occlude payload or sensor field of view of its planes and . The maximum rotation angle of solar panels defined according to structural configuration and field of view requirements for satellites can be denoted by . In this case, principal axis of the solar panels (normal direction of the panel) is only able to rotate within a circular cone of semi-cone angle that adopts the satellites radius vector (i.e., the earth’s core points at the satellite) as its axis. Furthermore, the value of was defined to be 90^{∘} dependent on physical design outcomes.

###### 2.2.2. Attitude Control Capacity Constraint

Due to the influence of the capacity possessed by an actuator selected for the microsatellite, the maximum angular velocity and the maximum angular acceleration of the turntable should be no more than 0.2^{∘}/s and 0.01^{∘}/, respectively, for the purpose of ensuring that disturbance to the platform during turntable running satisfies the earth-oriented stabilization control requirement.

###### 2.2.3. Pointing Error Constraint

Within the range of limited maximum rotation angle denoted as , angle tracking error of solar tracking should be no more than 14^{∘} to guarantee that the amount of energy acquired can be maximized.

##### 2.3. Model Input

refers to a normalized solar vector in an orbital coordinate system.

Not only can solar vector be acquired by an on-board sun sensor or obtained through calculations based on sun ephemeris, but its expression in an orbital coordinate system is obtained in line with an attitude transformation relation.

stands for on-board cumulative seconds relative to on-board starting time.

##### 2.4. Model Output

In conformity with input and constraint conditions, feature-based time series and angle/angular velocity series required by turntable control can be autonomously calculated to form target angles and angular velocities controlling the turntable in diverse time points. These series consist of pitching direction control time series , azimuth direction control time series , pitching angular velocity series , and azimuth angular velocity series . The corresponding output model can be expressed in

#### 3. Guidance Law Design

Time series is a set of variables ordered chronically and the solar tracking angular relation can be described by multiple time series in one orbit. Therefore, characteristic time points should be selected to form time series for guidance implementation. Then, guidance requirement of the specified accuracy is fulfilled through density control over the series.

Design purpose of the two-dimensional solar tracking stationary guidance law is to guarantee maximization of energies acquired by the solar panels under constraints.

##### 3.1. Optimal Pointing Guidance Law in Angle Limitation Constraint

Under constraint described in Section 2, that is, in a condition of two-dimensional rotation angle limitation, conical surface formed by the maximum limited angle constituted by the principal axis of solar panels intersects with that formed by rotation of the sun around orbital plane normal of the satellite. Optimal pointing of the solar panels principal axis is the combination of two segments of arcs, as shown in Figure 1. In general cases, two conical surfaces mentioned above directly intersect with each other or intersect in a translation manner among the conical surfaces, as points 1–4 presented in Figure 1.