Recursive dynamic modeling for attitude control of fully flexible spacecraft with central body and appendages

Abstract

A spacecraft composed of a central body and flexible appendages is a typical rigid-flexible coupling system. Current modeling methods generally regard the central body as rigid. As the central body becomes larger and more flexible, the elastic deformation of the central body may not be ignored. In this study, considering the flexibility effect of the central body and based on the recursive kinematics, a complete coupled dynamic model of the orbit, attitude, and elastic deformation of a fully flexible spacecraft is established. Compared with traditional methods, the model has smaller generalized coordinate dimension and can more clearly reveal the kinematic relationship between the appendages and the central body. By comparing the simulation results of typical working conditions with the ADAMS software, the correctness of the built model is verified. The PD controller for the attitude of the spacecraft is used, and three models of different rigid/flexible settings are compared in detail. The dynamic response of the system in uncontrolled and controlled states are discussed. Numerical simulation results show that the flexibility of the central body poses a certain influence on the attitude response of the spacecraft, the flexible vibration of the appendages, and the attitude control accuracy.

Introduction

With the gradual implementation of major aerospace projects, the requirements for the functions and performance of the spacecraft payload have continued to increase, and the structure of the spacecraft is continuously developing toward the direction of large scale and complexity. Spacecraft usually adopt low-rigidity and lightweight flexible structures to reduce the total mass. The coupled dynamic characteristics of flexible structure vibration should be considered when performing the dynamic analysis and control system design of flexible spacecraft.

The modeling method for the spacecraft with “center-appendage” configuration has been relatively mature after years of development. Early spacecraft mostly used body-mounted solar arrays and small antennas, and the moment of inertia and mass of the flexible attachments accounted for a remarkably small proportion of the entire spacecraft. Rigid body assumptions were generally used to model the spacecraft.^1–5

With the application of large-scale solar panels and antennas on spacecrafts, the influences of flexibility on characteristics and attitude of the whole system cannot be ignored. In many practical engineering applications, a considerable amount of losses has been attributed to the influence of elastic deformation. For example, the image quality of the “Landsat 4” observer was deteriorated due to the interference of the flexible solar panel drive system and the attitude controller on the rotating parts. ⁶ The periodic attitude disturbances of the satellite body emerged after the launch of FY-3A due to the coupling vibration of the solar panels. ⁷ Moreover, the motion forms of the space structure are also complex and diverse when they are in large scales, such as large maneuvering,^8–10 space movement of large manipulator,^11–13 rendezvous and docking,^14–16 and large-scale appendage deployment and locking.^17–19 For example, the flexible vibration of the solar panels poses a strong coupling effect on the attitude of the spacecraft body and the joint motion of the robotic arm for large or substantially large spacecraft platforms with enormous flexible solar panels, such as GEO communication satellites. ²⁰ Therefore, these problems cannot ignore the influence of structural deformation or vibration characteristics. Using the traditional rigid multibody dynamics method to solve the above problems is difficult. The method of rigid-flexible coupling dynamics should be used for research,^21–23 in which the hybrid-coordinate method is usually used and the natural modes of the partial vibrations of elastic elements are employed to describe the deformations.

When deriving equations of motion of multibody systems, an important consideration is to determine what analytical method to use to arrive at the equations. A variety of methods are used in these dynamic modeling approaches, ²⁴ including Lagrangian equations, ²⁵ Hamilton principle, ²⁶ Newton-Euler equations, ²⁷ the virtual work principle, ²⁸ Kane method, ² and Gibbs-Appell formulation. ²⁹ Each has its own advantages and disadvantages. For example, the Lagrangian approach has become a widely used method in the aerospace engineering due to its simple form; however, complex algebraic operations are required to convert from generalized coordinates to the desired angular velocity vector form, leading to great workload when seeking derivative and partial derivative. ²⁴

The recursive modeling method^30–34 programmatically sets up dynamic equations for chain, tree, and closed-loop flexible multibody systems in the forward kinematics recursion process, expressed in relative and modal coordinates. It has the advantages of traditional absolute coordinate methods and relative joint coordinate methods and overcomes their shortcomings. The form of the results is suitable for simulating variable structure multibody systems. Due to the independence of the recursive sets on each chain, this method has easily developed into a parallel computing method.

In most of the previous studies, the flexibility of appendages, such as robotic arms or solar panels, is currently mainly considered in the modeling of spacecrafts, and the center hub is generally regarded as a rigid body. However, few modeling studies are conducted systematically and comprehensively on the fully flexible spacecraft with a flexible central body. The dynamic characteristics of the central body appear as low natural frequencies and dense modes when the size of the central body becomes increasingly large, or the central body comprises flexible cabins and trusses with low rigidity. If the central body is still considered a rigid body, then it will not be able to characterize the overall flexibility parameters of the system accurately, and inaccurate structural dynamics parameters will lead to deviations in the frequency detuning design of the overall control system of the spacecraft.

In this work, a large spacecraft composed of a central body and four solar panels is studied, considering the flexibility effect of the central body and solar panels. The recursive dynamic equations of the system are derived, and numerical simulations of several typical situations are carried out to reveal the influence of the flexibility of the central body on the dynamic response of the system. The paper is organized as follows: in section “Recursive kinematics of the system” and section “Equations of motion,” the complete dynamic model of the system is derived on the basis of recursive kinematics and the principle of velocity variation. The corresponding simplified dynamic model and the expression form of the coupling coefficients are introduced. In section “Model verification,” the simulation of typical working conditions with initial deformation of the central body reveals that the presented model is in good agreement with the results of the ADAMS software, which verifies the accuracy of the proposed model. In section “Numerical simulation,” the reciprocating vibration of the solar panels after deployment is used as the disturbance input of the system, and the system dynamic responses using the three models called “rigid hub + rigid panel” (RHRP), “rigid hub + flexible panel” (RHFP), and “flexible hub + flexible panel” (FHFP) are compared in detail, and differences under uncontrolled and controlled attitudes are comprehensively discussed. Numerical simulation results show that the flexibility of the central body poses a certain influence on the dynamic response and attitude control accuracy of the spacecraft. Finally, the main conclusions are drawn in section “Conclusion.”

Recursive kinematics of the system

Figure 1 is a schematic of a combined spacecraft system with solar panels. Solar panels B₂, B₃, B₄, and B₅ are connected with the central body B₁. The central body and the solar panels are treated with flexible bodies. The inconsistent grid colors of the central body are used to distinguish between cabin and connecting sections. The red and gray grids represent the connecting sections, while the green grid represents the cabin section. C and C₁ are respectively the centroids of the spacecraft system and the central body when not deformed. C _i (i = 2, …, 5) is the centroid of the corresponding solar panel when not deformed. Floating reference frame C-xyz is established at the centroid of the spacecraft system C, while C _i -x_iy_iz_i is established at the centroid of C _i in each component B_i. O-x₀y₀z₀ is the inertial frame.

Figure 1.

Configuration and coordinate system definition of fully flexible spacecraft.

The flexible body B _i (i = 1,…,5) is discretized through the lumped mass finite element method. B _i contains l _i nodes after discretization. Considering the kth node, the node mass is $m_i^k$ , and the displacement vector relative to the centroid C _i is ${\mathit{\rho }}_i^k$ . The node k is at the position vector ${\mathit{\rho }}_{i0}^k$ when it is not deformed, and the node deformation vector is recorded as ${\mathit{u}}_i^k$ , as shown in Figure 2. The position vector of the system centroid C is denoted as $\mathit{r}$ . The position vectors of the component centroid C _i and the node k are recorded as ${\mathit{r}}_i$ and ${\mathit{r}}_i^k$ , respectively. P _i indicates the hinge point of the connection appendage B _i with central body B₁.

Figure 2.

Recursive kinematics between solar panels and central body.

The deformation of the node k is described by modal coordinates based on the hypothesis of small deformation,

$${\mathit{u}}_i^k={\mathit{\Phi }}_i^k{\mathit{a}}_i$$

(1)

where ${\mathit{\Phi }}_i^k$ and ${\mathit{a}}_i$ are respectively the modal vector matrix of node k and the modal coordinate matrix of B _i . Assuming that the s order modes are retained,

$${\mathit{\Phi }}_i^k=\left(\begin{array}{ccc}{\mathit{\phi }}_{i1}^k\hfill & \cdots \hfill & {\mathit{\phi }}_{\mathrm{is}}^k\hfill \end{array}\right)\in {\mathit{R}}^{3\times s}$$

(2)

$${\mathit{a}}_i={\left(\begin{array}{ccc}{a}_{i1}\hfill & \cdots \hfill & a_{\mathrm{is}}\hfill \end{array}\right)}^T\in {\mathit{R}}^{s\times 1}$$

(3)

The position vector of any point on the central body ${\mathit{r}}_1^k$ can be expressed as

$${\mathit{r}}_1^k={\mathit{r}}_1+{\mathit{\rho }}_1^k=\mathit{r}-{\mathit{\rho }}_1^c+{\mathit{\rho }}_1^k$$

(4)

The position vector of any point on the flexible appendage ${\mathit{r}}_i^k$ can be expressed as

$$\begin{array}{c}\multicolumn{1}{c}{{\mathit{r}}_i^k={\mathit{r}}_1+{\mathit{\rho }}_1^{P_i}-{\mathit{\rho }}_i^{P_i}+{\mathit{\rho }}_i^k=\mathit{r}-{\mathit{\rho }}_1^c+{\mathit{\rho }}_1^{P_i}-{\mathit{\rho }}_i^{P_i}}\\ \multicolumn{1}{c}{+{\mathit{\rho }}_i^k(i=2,\dots ,5)}\end{array}$$

(5)

The relative displacement vector ${\mathit{\rho }}_i^k$ can be further expressed as

$${\mathit{\rho }}_i^k={\mathit{\rho }}_{i0}^k+{\mathit{u}}_i^k={\mathit{\rho }}_{i0}^k+{\mathit{\Phi }}_i^k\mathit{a}$$

(6)

The absolute velocity of node k can be obtained by differentiating the vector ${\mathit{r}}_i^k$ considering time

$${\stackrel{·}{\mathit{r}}}_1^k=\stackrel{·}{\mathit{r}}-(-{\tilde{\rho }}_1^c+{\tilde{\rho }}_1^k)\mathit{\omega }+(-{\mathit{\Phi }}_1^c+{\mathit{\Phi }}_1^k){\stackrel{·}{\mathit{a}}}_1$$

(7)

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{\mathit{r}}}_i^k=\stackrel{·}{\mathit{r}}-(-{\tilde{\mathit{\rho }}}_1^c+{\tilde{\mathit{\rho }}}_1^{P_i}-{\tilde{\mathit{\rho }}}_i^{P_i}+{\tilde{\mathit{\rho }}}_i^k)\mathit{\omega }}\\ \multicolumn{1}{c}{+(-{\mathit{\Phi }}_1^c+{\mathit{\Phi }}_1^{P_i}){\stackrel{·}{\mathit{a}}}_1+({\mathit{\Phi }}_i^k-{\mathit{\Phi }}_i^{P_i}){\stackrel{·}{\mathit{a}}}_i(i=2,\dots ,5)}\end{array}$$

(8)

where $\tilde{\rho }$ represents the square coordinate matrix of the array $\mathit{\rho }$ , and its form is as follows:

$$\tilde{\rho }=\left[\begin{array}{ccc}0\hfill & -{\rho }_3\hfill & {\rho }_2\hfill \\ \multicolumn{1}{c}{{\rho }_3}& 0\hfill & -{\rho }_1\hfill \\ \multicolumn{1}{c}{-{\rho }_2}& {\rho }_1\hfill & 0\hfill \end{array}\right]$$

(9)

The acceleration expression of node k can be obtained by differentiating the vector from equations (7) and (8):

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{··}{\mathit{r}}}_1^k=\stackrel{··}{\mathit{r}}-(-{\tilde{\mathit{\rho }}}_1^c+{\tilde{\mathit{\rho }}}_1^k)\stackrel{·}{\mathit{\omega }}+(-{\mathit{\Phi }}_1^c+{\mathit{\Phi }}_1^k){\stackrel{··}{\mathit{a}}}_1}\\ \multicolumn{1}{c}{+\tilde{\omega }\tilde{\omega }(-{\mathit{\rho }}_1^c+{\mathit{\rho }}_1^k)+2\tilde{\omega }(-{\mathit{\Phi }}_1^c+{\mathit{\Phi }}_1^k){\stackrel{·}{\mathit{a}}}_1}\end{array}$$

(10)

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{··}{\mathit{r}}}_i^k=\stackrel{··}{\mathit{r}}-(-{\tilde{\mathit{\rho }}}_1^c+{\tilde{\mathit{\rho }}}_1^{P_i}-{\tilde{\mathit{\rho }}}_i^{P_i}+{\tilde{\mathit{\rho }}}_i^k)\stackrel{·}{\mathit{\omega }}}\\ \multicolumn{1}{c}{+(-{\mathit{\Phi }}_1^c+{\mathit{\Phi }}_1^{P_i}){\stackrel{··}{\mathit{a}}}_1+(-{\mathit{\Phi }}_i^{P_i}+{\mathit{\Phi }}_i^k){\stackrel{··}{\mathit{a}}}_i}\\ \multicolumn{1}{c}{+\tilde{\mathit{\omega }}\tilde{\mathit{\omega }}(-{\tilde{\mathit{\rho }}}_1^c+{\tilde{\mathit{\rho }}}_1^{P_i}-{\tilde{\mathit{\rho }}}_i^{P_i}+{\tilde{\mathit{\rho }}}_i^k)}\\ \multicolumn{1}{c}{+2\tilde{\omega }[(-{\mathit{\Phi }}_1^c+{\mathit{\Phi }}_1^{P_i}){\stackrel{·}{\mathit{a}}}_1+(-{\mathit{\Phi }}_i^{P_i}+{\mathit{\Phi }}_i^k){\stackrel{·}{\mathit{a}}}_i]}\\ \multicolumn{1}{c}{(i=2,\dots ,5)}\end{array}$$

(11)

Equation (7) is rewritten into matrix form

$${\stackrel{·}{\mathit{r}}}_1^k={\mathit{B}}_1^k{\mathit{v}}_1$$

(12)

where

$${\mathit{v}}_1={\left(\begin{array}{ccc}{\stackrel{·}{\mathit{r}}}^{\mathrm{T}}\hfill & {\mathit{\omega }}^{\mathrm{T}}\hfill & {\stackrel{·}{\mathit{a}}}_1\hfill \end{array}\right)}^{\mathrm{T}}$$

(13)

$${\mathit{B}}_1^k=\left[\begin{array}{ccc}{\mathit{I}}_3\hfill & ({\tilde{\rho }}_1^c-{\tilde{\rho }}_1^k)\hfill & ({\mathit{\Phi }}_1^k-{\mathit{\Phi }}_1^c)\hfill \end{array}\right]$$

(14)

The recursive form of node k velocity can be obtained from equation (8),

$${\stackrel{·}{\mathit{r}}}_i^k={\mathit{T}}_{i1}^k{\mathit{v}}_1+{\mathit{U}}_i^k{\stackrel{·}{\mathit{a}}}_i(i=2,\dots ,5)$$

(15)

where

$${\mathit{T}}_{i1}^k=\left[\begin{array}{ccc}{\mathit{I}}_3\hfill & ({\tilde{\mathit{\rho }}}_1^c-{\tilde{\mathit{\rho }}}_1^{P_i}+{\tilde{\mathit{\rho }}}_i^{P_i}-{\tilde{\mathit{\rho }}}_i^k)\hfill & (-{\mathit{\Phi }}_1^c+{\mathit{\Phi }}_1^{P_i})\hfill \end{array}\right]$$

(16)

$${\mathit{U}}_i^k=({\mathit{\Phi }}_i^k-{\mathit{\Phi }}_i^{P_i})$$

(17)

Similarly, equation (10) is rewritten into matrix form

$${\stackrel{··}{\mathit{r}}}_1^k={\mathit{B}}_1^k{\stackrel{·}{\mathit{v}}}_1+{\mathit{w}}_1^k$$

(18)

where

$${\mathit{w}}_1^k=\tilde{\mathit{\omega }}\tilde{\mathit{\omega }}(-{\mathit{\rho }}_1^c+{\mathit{\rho }}_1^k)+2\tilde{\mathit{\omega }}(-{\mathit{\Phi }}_1^c+{\mathit{\Phi }}_1^k){\stackrel{·}{\mathit{a}}}_1$$

(19)

The recursive form of node k acceleration can be obtained from equation (11):

$${\stackrel{··}{\mathit{r}}}_i^k={\mathit{T}}_{i1}^k{\stackrel{·}{\mathit{v}}}_1+{\mathit{U}}_i^k{\stackrel{··}{\mathit{a}}}_i+{\mathit{w}}_i^k(i=2,\dots ,5)$$

(20)

where

$$\begin{array}{c}\multicolumn{1}{c}{{\mathit{w}}_i^k=\tilde{\mathit{\omega }}\tilde{\mathit{\omega }}(-{\tilde{\mathit{\rho }}}_1^c+{\tilde{\mathit{\rho }}}_1^{P_i}-{\tilde{\mathit{\rho }}}_i^{P_i}+{\tilde{\mathit{\rho }}}_i^k)}\\ \multicolumn{1}{c}{+2\tilde{\mathit{\omega }}[(-{\mathit{\Phi }}_1^c+{\mathit{\Phi }}_1^{P_i}){\stackrel{·}{\mathit{a}}}_1+(-{\mathit{\Phi }}_i^{P_i}+{\mathit{\Phi }}_i^k){\stackrel{·}{\mathit{a}}}_i]}\end{array}$$

(21)

Equation (20) directly reveals the recursive kinematic relationship between the central body and the appendages at the acceleration level.

Equations of motion

According to the principle of velocity variation, the dynamic equation of the spacecraft system is

$$\sum _{i=1}^5\sum _{k=1}^l[\delta {\stackrel{·}{\mathit{r}}}_i^{k\mathrm{T}}(-m_i^k{\stackrel{··}{\mathit{r}}}_i^k+{\mathit{F}}_i^k)-\delta {\stackrel{·}{\mathit{\epsilon }}}_i^{k\mathrm{T}}{\mathit{\sigma }}_i^k]=0$$

(22)

where ${\mathit{F}}_i^k$ is the external force acting on the node k, and ${\mathit{\epsilon }}_i^k$ and ${\mathit{\sigma }}_i^k$ are the strain and stress of the node k, respectively. According to the structural dynamics, the total virtual power of the internal force of the body B _i can be expressed as follows

$$\sum _{k=1}^{l_i}\delta {\stackrel{·}{\mathit{\epsilon }}}_i^{k\mathrm{T}}{\mathit{\sigma }}_i^k=\delta {\stackrel{·}{\mathit{a}}}_i^{\mathrm{T}}({\mathit{C}}_{\mathrm{ai}}{\stackrel{·}{\mathit{a}}}_i+{\mathit{K}}_{\mathrm{ai}}{\mathit{a}}_i)$$

(23)

where C _a and K _a are the modal damping and modal stiffness matrices of the body B _i , respectively.

Equation (22) is recognized to obtain

$$\mathit{\delta }{\mathit{v}}^{\mathrm{T}}(-\mathit{M}\stackrel{·}{\mathit{v}}-\mathit{\varsigma }+{\mathit{f}}^o-{\mathit{f}}^u)=0$$

(24)

where

$$\mathit{v}={\left[\begin{array}{ccccc}{\mathit{v}}_1^{\mathrm{T}}\hfill & {\stackrel{·}{\mathit{a}}}_2^{\mathrm{T}}\hfill & {\stackrel{·}{\mathit{a}}}_3^{\mathrm{T}}\hfill & {\stackrel{·}{\mathit{a}}}_4^{\mathrm{T}}\hfill & {\stackrel{·}{\mathit{a}}}_5^{\mathrm{T}}\hfill \end{array}\right]}^{\mathrm{T}}$$

(25)

For this system, $\mathit{\delta }\mathit{v}$ is an independent variation. Thus, equation (24) can be written as

$$\mathit{M}\stackrel{·}{\mathit{v}}=-\mathit{\varsigma }+{\mathit{f}}^o-{\mathit{f}}^u$$

(26)

The generalized mass matrix is symmetric:

$$\mathit{M}=\left[\begin{array}{ccccc}{\mathit{M}}_{11}\hfill & {\mathit{M}}_{12}\hfill & {\mathit{M}}_{13}\hfill & {\mathit{M}}_{14}\hfill & {\mathit{M}}_{15}\hfill \\ \multicolumn{1}{c}{{\mathit{M}}_{21}}& {\mathit{M}}_{22}\hfill & {\mathit{M}}_{23}\hfill & {\mathit{M}}_{24}\hfill & {\mathit{M}}_{25}\hfill \\ \multicolumn{1}{c}{{\mathit{M}}_{31}}& {\mathit{M}}_{32}\hfill & {\mathit{M}}_{33}\hfill & {\mathit{M}}_{34}\hfill & {\mathit{M}}_{35}\hfill \\ \multicolumn{1}{c}{{\mathit{M}}_{41}}& {\mathit{M}}_{42}\hfill & {\mathit{M}}_{43}\hfill & {\mathit{M}}_{44}\hfill & {\mathit{M}}_{45}\hfill \\ \multicolumn{1}{c}{{\mathit{M}}_{51}}& {\mathit{M}}_{52}\hfill & {\mathit{M}}_{53}\hfill & {\mathit{M}}_{54}\hfill & {\mathit{M}}_{55}\hfill \end{array}\right]$$

(27)

where

$${\mathit{M}}_{11}=\sum _{k=1}^{l_1}{m}_1^k{\mathit{B}}_1^{k\mathrm{T}}{\mathit{B}}_1^k+\sum _{i=2}^5\sum _{k=1}^{l_i}{m}_i^k{\mathit{T}}_i^{k\mathrm{T}}{\mathit{T}}_i^k,$$

$${\mathit{M}}_{12}=\sum _{k=1}^{l_2}{m}_2^k{\mathit{T}}_{21}^{k\mathrm{T}}{\mathit{U}}_2^k,$$

$${\mathit{M}}_{13}=\sum _{k=1}^{l_3}{m}_3^k{\mathit{T}}_{31}^{k\mathrm{T}}{\mathit{U}}_3^k,$$

$${\mathit{M}}_{14}=\sum _{k=1}^{l_4}{m}_4^k{\mathit{T}}_{41}^{k\mathrm{T}}{\mathit{U}}_4^k,$$

$${\mathit{M}}_{15}=\sum _{k=1}^{l_5}{m}_3^k{\mathit{T}}_{51}^{k\mathrm{T}}{\mathit{U}}_5^k,$$

$${\mathit{M}}_{22}=\sum _{k=1}^{l_2}{m}_2^k{\mathit{U}}_2^{k\mathrm{T}}{\mathit{U}}_2^k,$$

$${\mathit{M}}_{23}=0,$$

$${\mathit{M}}_{24}=0,$$

$${\mathit{M}}_{25}=0,$$

$${\mathit{M}}_{33}=\sum _{k=1}^{l_3}{m}_3^k{\mathit{U}}_3^{k\mathrm{T}}{\mathit{U}}_3^k,$$

$${\mathit{M}}_{34}=0,$$

$${\mathit{M}}_{35}=0,$$

$${\mathit{M}}_{44}=\sum _{k=1}^{l_4}{m}_4^k{\mathit{U}}_4^{k\mathrm{T}}{\mathit{U}}_4^k,$$

$${\mathit{M}}_{45}=0,$$

$${\mathit{M}}_{55}=\sum _{k=1}^{l_5}{m}_{55}^k{\mathit{U}}_5^{k\mathrm{T}}{\mathit{U}}_5^k.$$

The generalized inertial force array is as follows:

$$\mathit{\varsigma }={\left[\begin{array}{ccccc}{\mathit{\varsigma }}_1^{\mathrm{T}}\hfill & {\mathit{\varsigma }}_2^{\mathrm{T}}\hfill & {\mathit{\varsigma }}_3^{\mathrm{T}}\hfill & {\mathit{\varsigma }}_4^{\mathrm{T}}\hfill & {\mathit{\varsigma }}_5^{\mathrm{T}}\hfill \end{array}\right]}^{\mathrm{T}}$$

(28)

where

$${\mathit{\varsigma }}_1=\sum _{k=1}^{l_1}{m}_1^k{\mathit{B}}_1^{k\mathrm{T}}{\mathit{w}}_1^k+\sum _{i=2}^5\sum _{k=1}^{l_i}{m}_i^k{\mathit{T}}_{i1}^{k\mathrm{T}}{\mathit{w}}_i^k$$

$${\mathit{\varsigma }}_2=\sum _{k=1}^{l_2}{m}_2^k{\mathit{U}}_2^{k\mathrm{T}}{\mathit{w}}_2^k$$

$${\mathit{\varsigma }}_3=\sum _{k=1}^{l_3}{m}_3^k{\mathit{U}}_3^{k\mathrm{T}}{\mathit{w}}_3^k$$

$${\mathit{\varsigma }}_4=\sum _{k=1}^{l_4}{m}_4^k{\mathit{U}}_4^{k\mathrm{T}}{\mathit{w}}_4^k$$

$${\mathit{\varsigma }}_5=\sum _{k=1}^{l_5}{m}_5^k{\mathit{U}}_5^{k\mathrm{T}}{\mathit{w}}_5^k$$

The generalized external force array is

$${\mathit{f}}^o={\left[\begin{array}{ccccc}{\mathit{f}}_1^{o\mathrm{T}}\hfill & {{\mathit{f}}_2^o}^{\mathrm{T}}\hfill & {{\mathit{f}}_3^o}^{\mathrm{T}}\hfill & {{\mathit{f}}_4^o}^{\mathrm{T}}\hfill & {{\mathit{f}}_5^o}^{\mathrm{T}}\hfill \end{array}\right]}^{\mathrm{T}}$$

(29)

where

$${\mathit{f}}_1^o=\sum _{k=1}^{l_1}{\mathit{B}}_1^{k\mathrm{T}}{\mathit{F}}_1^k+\sum _{i=2}^5\sum _{k=1}^{l_i}{\mathit{T}}_{i1}^{k\mathrm{T}}{\mathit{F}}_i^k$$

$${\mathit{f}}_2^o=\sum _{k=1}^{l_2}{\mathit{U}}_2^{k\mathrm{T}}{\mathit{F}}_2^k$$

$${\mathit{f}}_3^o=\sum _{k=1}^{l_3}{\mathit{U}}_3^{k\mathrm{T}}{\mathit{F}}_3^k$$

$${\mathit{f}}_4^o=\sum _{k=1}^{l_4}{\mathit{U}}_4^{k\mathrm{T}}{\mathit{F}}_4^k$$

$${\mathit{f}}_5^o=\sum _{k=1}^{l_5}{\mathit{U}}_5^{k\mathrm{T}}{\mathit{F}}_5^k$$

The generalized internal force array is

$${{\mathit{f}}^u}^{\mathrm{T}}={\left[\begin{array}{ccccc}{\mathit{f}}_1^{u\mathrm{T}}\hfill & {{\mathit{f}}_2^u}^{\mathrm{T}}\hfill & {{\mathit{f}}_3^u}^{\mathrm{T}}\hfill & {{\mathit{f}}_4^u}^{\mathrm{T}}\hfill & {{\mathit{f}}_5^u}^{\mathrm{T}}\hfill \end{array}\right]}^{\mathrm{T}}$$

(30)

where

$${\mathit{f}}_1^u={\left[\begin{array}{ccc}{0}^{\mathrm{T}}\hfill & 0^{\mathrm{T}}\hfill & {({\mathit{C}}_{a1}{\stackrel{·}{\mathit{a}}}_1+{\mathit{K}}_{a1}{\mathit{a}}_1)}^{\mathrm{T}}\hfill \end{array}\right]}^{\mathrm{T}}$$

$${\mathit{f}}_2^u=({\mathit{C}}_{a2}{\stackrel{·}{\mathit{a}}}_2+{\mathit{K}}_{a3}{\mathit{a}}_2{)}^{\mathrm{T}}$$

$${\mathit{f}}_3^u=({\mathit{C}}_{a3}{\stackrel{·}{\mathit{a}}}_3+{\mathit{K}}_{a3}{\mathit{a}}_3{)}^{\mathrm{T}}$$

$${\mathit{f}}_4^u=({\mathit{C}}_{a4}{\stackrel{·}{\mathit{a}}}_4+{\mathit{K}}_{a4}{\mathit{a}}_4{)}^{\mathrm{T}}$$

$${\mathit{f}}_5^u=({\mathit{C}}_{a5}{\stackrel{·}{\mathit{a}}}_5+{\mathit{K}}_{a5}{\mathit{a}}_5{)}^{\mathrm{T}}$$

The above modeling method based on recursive kinematics also considers the flexibility of the central body and the four solar panels to derive a complete dynamic model of a fully flexible spacecraft and retain the high-order terms of the large-scale rigid body motion and elastic vibration. The coupling response of the two can be calculated accurately. However, spacecraft attitude control requires high real-time performance, and equation (26) requires a large amount of calculation. Equation (26) is simplified in this study based on the following assumptions.

• (a) The elastic deformation of the central body and the flexible appendages are small deformations;

• (b) The centroid of the system is not markedly changed, and the translation and rotation of the system around the centroid can be decoupled;

• (c) The attitude angular velocity is small.

Except for the gyroscopic moment term, ignoring the second-order and above high-order terms of $\mathit{\omega }$ , ${\mathit{a}}_i$ , ${\stackrel{·}{\mathit{a}}}_i$ (i = 1, …, 5), equation (26) can be simplified into the following form:

$$\begin{array}{c}\multicolumn{1}{c}{m\stackrel{··}{\mathit{r}}+{\mathit{F}}_{\mathit{ta}1}{\stackrel{··}{\mathit{a}}}_1+\sum _{\mathit{i}=\mathit{2}}^{\mathit{5}}{\mathit{F}}_{\mathit{tai}}{\stackrel{··}{\mathit{a}}}_{\mathit{i}}=\mathit{F}}\\ \multicolumn{1}{c}{\mathit{J}\stackrel{·}{\mathit{\omega }}+\tilde{\omega }J\mathit{\omega }+{\mathit{F}}_{\mathit{ra}1}{\stackrel{··}{\mathit{a}}}_1+\sum _{\mathit{i}=\mathit{2}}^{\mathit{5}}{\mathit{F}}_{\mathit{rai}}{\stackrel{··}{\mathit{a}}}_{\mathit{i}}=\mathit{T}}\\ \multicolumn{1}{c}{{\mathit{m}}_{11}{\stackrel{··}{\mathit{a}}}_1+{\mathit{F}}_{\mathrm{ta}1}^{\mathrm{T}}\stackrel{··}{\mathit{r}}+{\mathit{F}}_{\mathrm{ra}1}^{\mathrm{T}}\stackrel{·}{\omega }+\sum _{i=2}^5{\mathit{F}}_{{\mathit{a}}_1{\mathit{a}}_i}{\stackrel{··}{\mathit{a}}}_{\mathit{i}}+{\mathit{K}}_{a1}{\mathit{a}}_1}\\ \multicolumn{1}{c}{+{\mathit{C}}_{a1}{\stackrel{·}{\mathit{a}}}_1=\sum _{k=1}^{l_1}{\mathit{Q}}_2^{\mathrm{T}}{\mathit{F}}_1^k}\\ \multicolumn{1}{c}{{\mathit{m}}_{22}{\stackrel{··}{\mathit{a}}}_2+{\mathit{F}}_{\mathrm{ta}2}^{\mathrm{T}}\stackrel{··}{\mathit{r}}+{\mathit{F}}_{\mathrm{ra}2}^{\mathrm{T}}\stackrel{·}{\omega }+{\mathit{F}}_{a_1{a}_2}^{\mathrm{T}}{\stackrel{··}{\mathit{a}}}_1+{\mathit{K}}_{a2}{\mathit{a}}_2+{\mathit{C}}_{a2}{\stackrel{·}{\mathit{a}}}_2=0}\\ \multicolumn{1}{c}{{\mathit{m}}_{33}{\stackrel{··}{\mathit{a}}}_3+{\mathit{F}}_{\mathrm{ta}3}^{\mathrm{T}}\stackrel{··}{\mathit{r}}+{\mathit{F}}_{\mathrm{ra}3}^{\mathrm{T}}\stackrel{·}{\omega }+{\mathit{F}}_{a_1{a}_3}^{\mathrm{T}}{\stackrel{··}{\mathit{a}}}_1+{\mathit{K}}_{a3}{\mathit{a}}_3+{\mathit{C}}_{a3}{\stackrel{·}{\mathit{a}}}_3=0}\\ \multicolumn{1}{c}{{\mathit{m}}_{44}{\stackrel{··}{\mathit{a}}}_4+{\mathit{F}}_{\mathrm{ta}4}^{\mathrm{T}}\stackrel{··}{\mathit{r}}+{\mathit{F}}_{\mathrm{ra}4}^{\mathrm{T}}\stackrel{·}{\omega }+{\mathit{F}}_{a_1{a}_4}^{\mathrm{T}}{\stackrel{··}{\mathit{a}}}_1+{\mathit{K}}_{a4}{\mathit{a}}_4+{\mathit{C}}_{a4}{\stackrel{·}{\mathit{a}}}_4=0}\\ \multicolumn{1}{c}{{\mathit{m}}_{55}{\stackrel{··}{\mathit{a}}}_2+{\mathit{F}}_{\mathrm{ta}5}^{\mathrm{T}}\stackrel{··}{\mathit{r}}+{\mathit{F}}_{\mathrm{ra}5}^{\mathrm{T}}\stackrel{·}{\omega }+{\mathit{F}}_{a_1{a}_5}^{\mathrm{T}}{\stackrel{··}{\mathit{a}}}_1+{\mathit{K}}_{a5}{\mathit{a}}_5+{\mathit{C}}_{a5}{\stackrel{·}{\mathit{a}}}_5=0}\end{array}$$

(31)

where $\mathit{F}$ is the resultant external force received by the system, and $\mathit{T}$ is the resultant external moment received by the system.

Equation (31) is the spacecraft dynamics equation form of “flexible hub + flexible panel” (FHFP). The expressions of the coefficients in the formula are defined as follows.

Total mass of the spacecraft:

$$m=\sum _{i=1}^5{m}_i$$

(32)

Inertia of the spacecraft relative to the centroid of the system:

$$\mathit{J}=\sum _{k=1}^{l_1}{m}_1^k{\mathit{Q}}_1^{\mathrm{T}}{\mathit{Q}}_1+\sum _{i=2}^5\sum _{k=1}^{l_i}{m}_{\mathrm{ik}}{\mathit{Q}}_{i3}^{\mathrm{T}}{\mathit{Q}}_{i3}$$

(33)

Modal mass matrix of each part:

$${\mathit{m}}_{11}=\sum _{k=1}^{l_1}{m}_1^k{\mathit{Q}}_2^{\mathrm{T}}{\mathit{Q}}_2+\sum _{i=2}^5\sum _{k=1}^{l_i}{m}_i^k{\mathit{Q}}_{\mathit{i}4}^{\mathrm{T}}{\mathit{Q}}_{i4}$$

(34)

$${\mathit{m}}_{\mathrm{ii}}=\sum _{k=1}^{l_i}{m}_i^k{\mathit{Q}}_{\iota 5}^{\mathrm{T}}{\mathit{Q}}_{i5}(i=2,\dots ,5)$$

(35)

The flexible coupling coefficient between the vibration of the central body B₁ and the translation of the spacecraft:

$${\mathit{F}}_{\mathrm{ta}1}=\sum _{k=1}^{l_1}{m}_1^k{\mathit{Q}}_2+\sum _{i=2}^5\sum _{k=1}^{l_k}{m}_i^k{\mathit{Q}}_{i4}$$

(36)

The flexible coupling coefficient between the vibration of the panel B _i (i = 2, …, 5) and the translation of the spacecraft:

$${\mathit{F}}_{\mathrm{tai}}=\sum _{k=1}^{l_k}{m}_i^k{\mathit{Q}}_{i5}$$

(37)

The flexible coupling coefficient between the vibration of the central body B₁ and the rotation of the spacecraft:

$${\mathit{F}}_{\mathrm{ra}1}=\sum _{k=1}^{l_1}{m}_1^k{\mathit{Q}}_1^{\mathrm{T}}{\mathit{Q}}_2+\sum _{i=2}^5\sum _{k=1}^{l_k}{m}_i^k{\mathit{Q}}_{i3}^{\mathrm{T}}{\mathit{Q}}_{i4}$$

(38)

The flexible coupling coefficient between the vibration of the panel B _i (i = 2, …, 5) and the rotation of the spacecraft:

$${\mathit{F}}_{\mathrm{rai}}=\sum _{k=1}^{l_k}{m}_i^k{\mathit{Q}}_{i3}^{\mathrm{T}}{\mathit{Q}}_{i5}$$

(39)

The flexible coupling coefficient between the vibration of the central body B₁ and the vibration of the panel B _i (i = 2, …, 5):

$${\mathit{F}}_{a_1{a}_i}=\sum _{k=1}^{l_k}{m}_{\mathrm{ik}}{\mathit{Q}}_{i4}^{\mathrm{T}}{\mathit{Q}}_{i5}$$

(40)

where

$${\mathit{Q}}_1={\tilde{\rho }}_{10}^c-{\tilde{\rho }}_{10}^k$$

$${\mathit{Q}}_2=-({\mathit{\Phi }}_1^c-{\mathit{\Phi }}_1^k)$$

$${\mathit{Q}}_{i3}=({\tilde{\rho }}_{10}^c-{\tilde{\rho }}_{10}^{P_i})+({\tilde{\rho }}_{i0}^{P_i}-{\tilde{\rho }}_{i0}^k)$$

$${\mathit{Q}}_{i4}=-({\mathit{\Phi }}_1^c-{\mathit{\Phi }}_1^{P_i})$$

$${\mathit{Q}}_{i5}=-({\mathit{\Phi }}_i^{P_i}-{\mathit{\Phi }}_i^k)$$

If the flexibility of the central body is not considered, then let ${\stackrel{··}{\mathit{a}}}_1=0$ , ${\stackrel{·}{\mathit{a}}}_1=0$ , and ${\mathit{a}}_1=0$ , and equation (31) degenerates into the spacecraft dynamics equation of “rigid hub + flexible panel” (RHFP):

$$\begin{array}{c}\multicolumn{1}{c}{m\stackrel{··}{\mathit{r}}+\sum _{\mathit{i}=\mathit{2}}^{\mathit{5}}{\mathit{F}}_{\mathit{tai}}{\stackrel{··}{\mathit{a}}}_{\mathit{i}}=\mathit{F}}\\ \multicolumn{1}{c}{\mathit{J}\stackrel{·}{\mathit{\omega }}+\tilde{\omega }J\mathit{\omega }+\sum _{\mathit{i}=\mathit{2}}^{\mathit{5}}{\mathit{F}}_{\mathit{rai}}{\stackrel{··}{\mathit{a}}}_{\mathit{i}}=\mathit{T}}\\ \multicolumn{1}{c}{{\mathit{m}}_{22}{\stackrel{··}{\mathit{a}}}_2+{\mathit{F}}_{\mathrm{ta}2}^{\mathrm{T}}\stackrel{··}{\mathit{r}}+{\mathit{F}}_{\mathrm{ra}2}^{\mathrm{T}}\stackrel{·}{\omega }+{\mathit{K}}_{a2}{\mathit{a}}_2+{\mathit{C}}_{a1}{\stackrel{·}{\mathit{a}}}_2=0}\\ \multicolumn{1}{c}{{\mathit{m}}_{33}{\stackrel{··}{\mathit{a}}}_3+{\mathit{F}}_{\mathrm{ta}3}^{\mathrm{T}}\stackrel{··}{\mathit{r}}+{\mathit{F}}_{\mathrm{ra}3}^{\mathrm{T}}\stackrel{·}{\omega }+{\mathit{K}}_{a3}{\mathit{a}}_3+{\mathit{C}}_{a3}{\stackrel{·}{\mathit{a}}}_3=0}\\ \multicolumn{1}{c}{{\mathit{m}}_{44}{\stackrel{··}{\mathit{a}}}_4+{\mathit{F}}_{\mathrm{ta}4}^{\mathrm{T}}\stackrel{··}{\mathit{r}}+{\mathit{F}}_{\mathrm{ra}4}^{\mathrm{T}}\stackrel{·}{\omega }+{\mathit{K}}_{a4}{\mathit{a}}_4+{\mathit{C}}_{a4}{\stackrel{·}{\mathit{a}}}_4=0}\\ \multicolumn{1}{c}{{\mathit{m}}_{55}{\stackrel{··}{\mathit{a}}}_5+{\mathit{F}}_{\mathrm{ta}5}^{\mathrm{T}}\stackrel{··}{\mathit{r}}+{\mathit{F}}_{\mathrm{ra}5}^{\mathrm{T}}\stackrel{·}{\omega }+{\mathit{K}}_{a5}{\mathit{a}}_5+{\mathit{C}}_{a5}{\stackrel{·}{\mathit{a}}}_5=0}\end{array}$$

(41)

Furthermore, if the flexibility of the appendage is not considered, then let ${\stackrel{··}{\mathit{a}}}_i=0$ , ${\stackrel{·}{\mathit{a}}}_i=0$ , and ${\mathit{a}}_i=0$ (i = 2, …, 5), and equation (41) is degenerated into the spacecraft dynamics equation of “rigid hub + rigid panel” (RHRP):

$$\begin{array}{c}\multicolumn{1}{c}{m\stackrel{··}{\mathit{r}}=\mathit{F}}\\ \multicolumn{1}{c}{\mathit{J}\stackrel{·}{\mathit{\omega }}+\tilde{\omega }J\omega =\mathit{T}}\end{array}$$

(42)

Model verification

The geometry and mass inertia parameters of each part of the spacecraft are given in Table 1. The central body and four solar panels are discretized by finite element method, and modal analysis is performed. The central body and the panels retain the first 10 modes to participate in the calculation. The modal and frequency information are shown in Table 2. The table indicates that the central body also exhibits the characteristics of low frequency and dense modes, and the frequencies of some modes overlap with the frequency range of the flexible solar panels.

Table 1.

Physical parameters of the spacecraft system.

Object type	Length (m)	Width (m)	Thickness (m)	Radius (m)	Mass (kg)	Inertia
						Jxx (kg m2)	Jyy (kg m2)	Jzz (kg m2)
Hub	16.8	–	–	2.5	37,057.5	8.72e4	1.31e6	1.31e6
Panel	7.27	2	0.115	–	59.9	19.0	305.9	286.9

Table 2.

Finite element discretization, modal, and frequency information.

Object information	Hub	Panel
Nodes	2736	837
Modes	10	10
Frequency (Hz)	1.18, 1.18, 3.35,7.51, 10.81, 10.81, 12.48,12.48, 15.96, 33.03	0.18, 0.98, 1.02,1.14, 2.88,3.31, 5.14, 5.79,6.39, 7.15

The simulation results of the FHFP model based on equation (31) are compared with the commercial software ADAMS to verify the accuracy of the proposed model. The working conditions are set up as follows. Initial angular velocity is set as ${\omega }_x=-0.5\mathrm{deg}/\mathrm{s}$ , ${\omega }_y=-0.5\mathrm{deg}/\mathrm{s}$ , and ${\omega }_z=-0.5\mathrm{deg}/\mathrm{s}$ under the inertial frame. The square waveform of moment is applied in the inertial frame x-, y-, and z-directions, in which the peaks are 100, 1000, and 1000 Nm, respectively. In each second, 0–0.4 s moment is the peak value, where 0.4–1 s is 0. The applied moment time is 10 s, and the simulation time is 20 s. The law of moment driven in the x-direction is shown in Figure 3.

Figure 3.

Moment driving law in the x-direction.

The initial value of the central body modal coordinate ${\mathit{a}}_1(1:3)=10$ is set to compare the simulation results when the central body undergoes flexible deformation; that is, the initial deformation to the first three modes is applied. The first three modes correspond to bending in the y-direction, bending in the z-direction, and twisting in the x-direction, as shown in Figure 4. These modes ensure the presence of initial deformations in the three directions, and the maximum deformation is $u_{1max}=max({\mathit{\Phi }}_1{\mathit{a}}_1)=0.184\mathrm{m}$ . In the Adams software, the initial values of modal coordinates can be set through the setting of the Modal Neutral File.

Figure 4.

Schematic of the first three vibration modes of the central body: (a) first vibration mode: bending in the y-direction, (b) second vibration mode: bending in the z-direction, and (c) third vibration mode: twisting in the x-direction.

Figure 5 shows the simulation results of the angular velocity under the inertial frame. The FHFP model proposed in this research is in good agreement with the ADAMS simulation results. The slight difference in high-frequency vibration is due to the use of a fully coupled model in ADAMS, while the FHFP model discards the high-order coupling terms of angular velocity and deformation. The FHFP model will be used in the following research to study the influence of the flexibility of the central body on the dynamic response of the system.

Figure 5.

Angular velocity of spacecraft in all directions under inertial frame: (a) angular velocity in x direction, (b) angular velocity in y direction, and (c) angular velocity in z direction.

Numerical simulation

The structure will be disturbed during spacecraft operation in orbit, such as large attitude maneuvers, rendezvous and docking, solar panel deployment, and orientation to the sun, and the operation accuracy is affected. Previous studies only focused on the flexibility characteristics of appendages. The flexibility of the central body and appendages will be simultaneously considered in this research based on the FHFP model, which will be compared with the RHFP and RHRP model.

Take the deployment of a spacecraft’s solar panel as an example. Flexural vibration will occur for a period after the solar panel is deployed in place, which will disturb the attitude of the spacecraft. This scenario will be taken as a typical working condition in the current study. Assuming that the solar panel B₂ undergoes initial deformation, and the initial value of the modal coordinates ${\mathit{a}}_2(1:10)=1$ is given to the solar panel B₂. Thus, the solar panel will produce complex deformation with multimode superposition. The maximum deformation is $u_{2max}=max({\mathit{\Phi }}_2{\mathit{a}}_2)=0.79\mathrm{m}$ , as shown in Figure 6. The initial angular velocity of the spacecraft is set as follows: ${\omega }_x=-1\mathrm{deg}/\mathrm{s}$ , ${\omega }_y=-1\mathrm{deg}/\mathrm{s}$ , and ${\omega }_z=-1\mathrm{deg}/\mathrm{s}$ .

Figure 6.

Schematic of the initial configuration of a single solar panel deformation.

Uncontrolled central body

The condition that the central body is free is considered; that is, the central body is unaffected by the controlled torque. Then, the dynamic responses calculated by the three models of RHRP, RHFP, and FHFP are compared.

Figure 7 shows the results of the attitude response of the spacecraft under the inertial frame. Figure 7(a), (c), and (e) are the results of the angular velocities in three directions. The result is an ideal one without disturbance because the deformation characteristics of the solar panel cannot be considered in the RHRP model. The results of RHFP and FHFP reveal that the deformation of the solar panel introduces a strong disturbance to the attitude angular velocity of the spacecraft. The partially enlarged view shows that the amplitude of the attitude disturbance in the FHFP model is slightly larger than that in the RHFP model during the initial period. This finding is due to the changes in flexibility parameters of the spacecraft system and the reduction in overall structural stiffness after considering the flexibility of the central body. High-frequency disturbances are gradually attenuated under the action of structural damping. The vibration in the FHFP model is attenuated faster than that in the RHFP model. The vibration amplitude is already smaller than that in the RHFP after approximately 8 s. This result is due to the increase in damping characteristics of the central body caused by FHFP, which accelerates the overall energy dissipation of the system. Figure 7(b), (d), and (f) show the Euler angular displacement in the order of 3-2-1. The angular displacements calculated by the three models are similar with only slight deviations.

Figure 7.

Attitude response of the spacecraft in an uncontrolled state: (a) angular velocity in x direction, (b) angular displacement in x direction, (c) angular velocity in y direction, (d) angular displacement in y direction, (e) angular velocity in z direction, and (f) angular displacement in z direction.

The flexible vibration of the central body and each solar panel is further investigated. Figure 6 shows that the center point of one end of the central body B₁ is N₁, and the outer endpoint of each solar panel B _i is N _i . The deformation results of each point on the floating coordinate system of the corresponding object are presented below.

The deformation of point N₁ can only be calculated by the FHFP model as shown in Figure 8 because the RHRP and RHFP models ignore the flexibility of the central body. Deformation in the $z_1$ -direction is the largest, with an amplitude of 2.7 mm. The flexible vibration amplitude of the central body is small because the mass and inertia of the central body is substantially larger than that of a single solar panel.

Figure 8.

Deformation of point N₁ on the central body.

Figure 9 shows the flexural response curve of point N₂–N₅ on the solar panel. The coupling force term caused by the flexibility of the central body for solar panel B₂ is not observed due to its large initial deformation, and the calculation results of RHFP and FHFP are close. The calculation results of the two models for B₃ and B₄ are different. On the one hand, the FHFP result reveals the presence of numerous high-frequency responses. On the other hand, the deflection amplitude calculated by FHFP is remarkably larger than that by RHFP. Interestingly, for B₅, which is symmetric around the origin with B₂, the deformation curves of the end point N₅ calculated by the two models are almost the same. The flexibility of the central body exerts a remarkable impact on the deformation of B₃ and B₄, while that on B₂ and B₅ exerts a small impact. This phenomenon may be due to the high-degree coupling of certain vibration modes of the central body with the deformation of B₃ and B₄.

Figure 9.

Flexibility response of typical positions of each component: (a) panel B₂, (b) panel B₃, (c) panel B₄, and (d) panel B₅.

Overall, the flexibility characteristics of the central body will affect the overall flexibility parameters, such as stiffness and damping characteristics, despite only undergoing slight flexible vibrations, thus affecting the attitude characteristics of the spacecraft. Meanwhile, the vibration of the central body is coupled with that of the appendages, which affects the vibration characteristics of the appendages.

Controlled central body

Introducing the attitude controller of the spacecraft is necessary when the spacecraft is performing high-precision missions in orbit. The current study aims to discuss the dynamic response characteristics of the system under controlled conditions. Therefore, a simple linear PD control method, which controls the attitude by applying torque on the three degrees of freedom of rotation of the spacecraft, is adopted in this study.

Taking the attitude information of the spacecraft as the control feedback, the control force array can be written as:

$$\mathit{\tau }(t)=k_p\mathit{Jy}(t)+k_d\mathit{J}\stackrel{·}{\mathit{y}}(t),$$

(43)

where $\mathit{y}(t)$ and $\stackrel{·}{\mathit{y}}(t)$ are the drift amount and velocity of the spacecraft attitude, respectively. $\mathit{J}$ is the inertia matrix of the spacecraft. $k_p$ and $k_d$ are the control parameters of the PD controller. $k_p=10$ , $k_d=2$ are taken in this study.

Figure 10 shows the attitude angular velocity and angular displacement of the spacecraft under controlled state. The result reveals that the attitude angular velocity and the attitude angle quickly reach near zero, indicating that the attitude of the spacecraft has been effectively controlled. Specifically, the attitude control accuracy in the y- and z-directions is better than that in the x-direction. This finding is due to the initial disturbance (the deformation of the solar panel) applied in this case, which mainly acts on the x-axis rotation direction. The simulation results of the three models are also compared. The RHRP model ignores the flexibility of all components and cannot reflect the high-frequency jitter of the attitude deviation. If the RHRP model is used in the design of the controller, then satisfying the high-precision control requirements will be difficult. The attitude responses of the RHFP and FHFP models are close, and the amplitude of the attitude angular velocity vibration of the FHFP is slightly larger than that of the RHFP model, which is consistent with the result of the uncontrolled state in the previous section. From the angular displacement perspective, the attitude angle amplitude calculated by RHFP and FHFP is quite different in the x-direction with an initially large disturbance. This phenomenon indicates that considering the flexibility of the central body will affect the accuracy of attitude control.

Figure 10.

Attitude response of the spacecraft in a controlled state: (a) angular velocity in x direction, (b) angular displacement in x direction, (c) angular velocity in y direction, (d) angular displacement in y direction, (e) angular velocity in z direction, and (f) angular displacement in z direction.

The disturbance to the entire system is still small in this case because the mass of the single solar panel is substantially smaller than that of the central body. The above analysis indicates that the influence of the flexibility of the central body on the accuracy of attitude control will not be negligible when the scale of the flexible appendages becomes large or the external disturbance to the central body becomes severe.

Conclusion

The central body is generally treated as a rigid body in the current spacecraft attitude and orbital dynamics. However, a large central body and low stiffness present the characteristics of low frequency and dense modes; thus, the elastic deformation will not be negligible. The influence of the flexibility of the appendages and the central body of the spacecraft on the attitude dynamic response of the spacecraft is considered in this study.

First, the recursive kinematics relationship between the solar panel and the central body at the position, speed, and acceleration level is given in accordance with the system topology, and the complete dynamic model of the system is derived on the basis of the principle of velocity variation. Compared with the Cartesian coordinate method, this model has no constraint equation between the appendages and the central body, which reduces the generalized coordinate dimension and is suitable for the needs of attitude control. Compared with the Lagrangian equation form most commonly used in the spacecraft field, the kinematic relationship of the acceleration level can be observed. The corresponding simplified dynamic model and the expression form of the coupling coefficients are introduced on the basis of the small angular velocity and deformation and the ignored high-order terms of angular velocity and deformation. Subsequently, the simulation of typical working conditions with initial deformation of the central body reveals that the FHFP model is in good agreement with the results of the ADAMS software, which verifies the accuracy of the proposed model. Finally, the reciprocating vibration of the solar panels after deployment is used as the disturbance input of the system, and the system dynamic response of the three models of RHRP, RHFP, and FHFP under uncontrolled and controlled attitudes is comprehensively discussed. Numerical simulation results show that the flexibility of the central body poses a certain influence on the dynamic response and attitude control accuracy of the spacecraft.

The main conclusions in this paper are as follows. (1) Under the basic assumptions of small deformation and angular velocity, the calculation results of the simplified model proposed in this paper and the complex model considering all coupling terms are in good agreement, which verifies the effectiveness of the proposed model in this paper. (2) The flexibility of the central body can affect the attitude response of the spacecraft, the flexible vibration of the appendages, and the attitude control accuracy in the presence of a low fundamental frequency of the central body and the existence of a disturbance input. The flexibility effect of the central body should be considered during attitude control. The influence of the central body flexibility on the system under different mass, stiffness, and disturbance characteristics will be further explored in the subsequent research.