Convex Optimization for Spacecraft Attitude Alignment of Laser Link Acquisition Under Uncertainties

Abstract

This paper addresses the critical multiple-uncertainty challenge in laser link acquisition for space gravitational wave detection missions—a key bottleneck where spacecraft attitude alignment for laser link establishment is perturbed by inherent random disturbances in such missions, while also needing to balance ultra-high attitude precision, fuel efficiency, and compliance with engineering constraints. To tackle this, a convex optimization-based attitude control strategy integrating covariance control and free terminal time optimization is proposed. Specifically, a stochastic attitude dynamics model is first established to explicitly incorporate the aforementioned random disturbances. Subsequently, an objective function is designed to simultaneously minimize terminal state error and fuel consumption, with three key constraints (covariance constraints, pointing constraints, and torque saturation constraints) integrated into the convex optimization framework. Furthermore, to resolve non-convex terms in chance constraints, this study employs a hierarchical convexification method that combines Schur’s complementary theorem, second-order cone relaxation, and Taylor expansion techniques. This approach ensures lossless relaxation, renders the optimization problem computationally tractable without sacrificing solution accuracy, and overcomes the shortcomings of traditional convexification methods in handling chance constraints. Finally, numerical simulations demonstrate that the proposed method adheres to engineering constraints while maintaining spacecraft attitude errors below 1 μrad under environmental uncertainties. This study provides a convex optimization solution for laser link acquisition in space gravitational wave detection missions considering uncertainty conditions, and its framework can be extended to the optimal design of other stochastically uncertain systems.

1. Introduction

Gravitational waves are a phenomenon predicted by Einstein‘s general theory of relativity, which can reveal some of the extreme physical phenomena in the universe, such as black hole mergers and neutron star mergers [1,2,3]. Gravitational wave detection technologies are primarily categorized into two types: ground-based and space-based systems.

The former represents one of the most technologically mature approaches to date. It leverages laser interferometers deployed on the ground to measure the minute spacetime perturbations induced by the propagation of gravitational waves. In contrast, the latter is specifically designed to detect low-frequency gravitational waves (spanning 0.1 mHz to 1 Hz) [4,5,6]. Such low-frequency signals typically originate from supermassive celestial objects (e.g., supermassive black hole mergers), which cannot be captured by ground-based detectors [7,8,9,10].

The core principle of gravitational wave detection is to measure the minute changes in the length of the detector arms caused by the spatial and temporal distortions induced by gravitational waves [10,11,12,13]. The primary method employed is laser interferometry, which utilizes the coherence of laser light to achieve ultra-precise distance measurements [2,3]. For example, facilities like LISA use laser interferometers to detect tiny changes in spacetime, which rely on the establishment of high-precision laser links [7,10,14], can be viewed as an attitude planning problem. It is worth noting that gravitational wave detection requires laser links to possess extremely high stability and precision [15]. Even very small spatial and temporal changes (such as spatial and temporal fluctuations caused by gravitational waves) can cause changes in the laser interference fringes, and therefore, the stability of the optical link needs to be precisely controlled [16,17,18].

Currently, various algorithms have been considered for attitude planning, including the Memorizable Smoothed Functional Algorithm (MSFA) [19,20], Safe Experimentation Dynamics Algorithm (SED) [21,22], Simultaneous Perturbation Stochastic Approximation (SPSA) [23,24], and convex optimization [25,26,27,28,29,30,31,32]. While MSFA, SED, and SPSA can meet certain requirements, they have theoretical limitations and practical constraints. In contrast, convex optimization offers efficiency, robustness, and global optimality, supported by a solid theoretical foundation and proven engineering adaptability. In particular, it has gained extensive application in landing attitude planning scenarios [27,28,29,30,31], establishing the basis for high-precision attitude adjustments.

Indeed, the process of laser link establishment typically involves three steps: constellation capture control, relative position capture, and high-precision attitude adjustment. A key challenge is achieving rapid, high-precision pointing during the initial acquisition phase. To ensure spacecraft rapidly and accurately reach target states in gravitational wave detection missions, Zhao [25] proposed a sequential penalized convex relaxation algorithm under a convex optimization framework. This algorithm shortens spacecraft attitude alignment time and reduces the time-varying uncertainty region in subsequent steps. In addition, based on Ref. [25], a multistage sequential penalized convex relaxation method was developed in Ref. [26], which used double-layer programming to handle integer variables, a sampling-based approach to remodel the reacquisition problem into a convex optimization problem with minimum-function constraints.

However, in the laser link acquisition process, spacecraft attitude alignment only accounts for pointing constraints and torque constraints under hard constraint conditions [33,34]. In actual missions, certain constraints may be violated within a small probability range. For example, the no-go zone constraint set to protect optical sensors from damage by strong light sources does not allow the sensor to reach the threshold and be damaged instantaneously [12,26]. Additionally, uncertainties such as solar radiation pressure and actuator noise exist during attitude alignment. Gong [35,36,37,38] argued that chance constraints are more suitable for addressing problems involving such uncertainties. Unlike common hard constraints, the mean and covariance cannot be decoupled in covariance control with chance constraints [38]. Okamoto et al. [37] proposed a convex programming method to solve the optimal covariance control problem for stochastic discrete-time linear systems subject to chance constraints, advancing the problem of predictive controllers’ design for stochastic models with chance constraints [39,40,41].

While research on uncertainty in spacecraft attitude alignment problems remains limited, optimization methodologies for uncertainty-involved problems in random systems are already well-established in other fields [13,37,42,43,44,45]. Zhang proposed the chance-constrained sequential convex programming (CC-SCP) algorithm [35], which designs a smooth and differential approximation function to transform chance constraints into constraints capable of being handled by a convex optimization method and proves the feasibility of the exact convex relaxation. Szmuk et al. [46] proposed a continuous convexification algorithm for the fuel-optimized power landing problem under aerodynamic drag and a new type of nonconvex control constraints including free terminal time. Successive convexification reformulated the problem into a series of iterative second-order cone programming (SOCP) problems [47,48,49,50] to estimate the time of flight. These problems are solved with an interior point method (IPM) solver, and reliable convergence is typically achieved in only a few iterations. Ridderhof [48] modeled the spacecraft dynamics as a stochastic system, solving optimal trajectories and feedback laws under state mean and covariance constraints [18,40], and solving the optimal control problem with robustness to disturbances.

Given the extremely high requirements for attitude accuracy in gravitational wave detection missions [7,8,14], the aforementioned methods for solving stochastic system optimization problems cannot be directly applied to spacecraft attitude alignment. Inspired by Ref. [43], this paper establishes a stochastic attitude dynamics model to address uncertainties in real missions, which employs covariance control by constraining the state covariance matrix to probabilistically satisfy attitude alignment accuracy requirements. Main contributions are as follows:

(1)

Existing spacecraft attitude dynamics models predominantly rely on deterministic frameworks. In contrast, the proposed stochastic attitude dynamics model explicitly incorporates random disturbances prevalent in space gravitational wave detection missions. This approach more fully reflects real-world environmental uncertainties and better approximates the operational conditions of laser link acquisition.

(2)

Building upon existing work that solely considers deterministic hard constraints, this paper’s strategy combines stochastic constraints with covariance control. This integration enables explicit quantification of the impact of random perturbations, ensuring probabilistic satisfaction of key performance indicators. This approach avoids the overly conservative or infeasible outcomes associated with deterministic constraint methods alone.

(3)

Compared with traditional convexification methods for chance constraints, this paper proposes a hierarchical convexification approach that integrates the Schur complement theorem, second-order cone relaxation, and Taylor expansion. This method not only effectively handles the non-convex terms in chance constraints but also ensures the losslessness of relaxation. Consequently, it renders the optimization problem computationally tractable without compromising solution accuracy.

The remainder of this paper is organized as follows. Section 2 provides an overview of the stochastic dynamics of the spacecraft attitude and the closed-loop covariance control, containing primal nonconvex constraints for the gravitational wave detection mission. Section 3 relaxes the above dynamics, nonconvex constraints, and control constraints to convexify the problem. Section 4 demonstrates the spacecraft attitude control results and verifies the effectiveness and accuracy of the method. Section 5 summarizes the paper.

2. Problem Formulation

2.1. Stochastic System Dynamics

For the rigid spacecraft, to avoid the singularity problem and nonlinearity, a quaternion-based spacecraft attitude dynamics equation is used [26]:

$$\begin{array}{l}\dot{\mathit{q}}={\displaystyle \frac{1}{2}}\mathbf{\Omega }\left(\mathit{\omega }\right)\mathit{q}\\ \mathit{J}\dot{\mathit{\omega }}=\left(J\mathit{\omega }\right)\times \mathit{\omega }+\mathit{u}\end{array}$$

(1)

where $\mathit{q}={[q_0,q_1,q_2,q_3]}^{\mathrm{T}}={[q_0,{\underset{\_}{\mathit{q}}}^{\mathrm{T}}]}^{\mathrm{T}}$ is the unit quaternion for spacecraft attitude, ${\underset{\_}{\mathit{q}}}^{\mathrm{T}}$ is the vector part of quaternions, quaternions should satisfy the unit constraint, meaning the L2 norm of a quaternion is 1. $\mathit{\omega }={[{\omega }_1,{\omega }_2,{\omega }_3]}^{\mathrm{T}}$ is the angular velocity, $\mathit{J}=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left(J_1,J_2,J_3\right)$ is the inertia tensor, $\mathit{u}={[u_1,u_2,u_3]}^{\mathrm{T}}$ is the control torque. Furthermore,

$$\mathbf{\Omega }\left(\mathit{\omega }\right)=\left(\begin{array}{cccc}0& -{\omega }_1& -{\omega }_2& -{\omega }_3\\ {\omega }_1& 0& {\omega }_3& -{\omega }_2\\ {\omega }_2& -{\omega }_3& 0& {\omega }_1\\ {\omega }_3& {\omega }_2& -{\omega }_1& 0\end{array}\right)=\left(\begin{array}{cc}0& -{\mathit{\omega }}^{\mathrm{T}}\\ \mathit{\omega }& -\mathit{\omega }\times \end{array}\right)$$

(2)

Define the state vector $\mathit{x}\triangleq [\mathit{q};\mathit{\omega }]$, and the above attitude dynamics can be expressed in the following form:

$$f\left(\mathit{x},\mathit{u},t\right)=\left(\begin{array}{c}{\displaystyle \frac{1}{2}}\left(-{\omega }_1{q}_1-{\omega }_2{q}_2-{\omega }_3{q}_3\right)\\ {\displaystyle \frac{1}{2}}\left({\omega }_1{q}_0+{\omega }_3{q}_2-{\omega }_2{q}_3\right)\\ {\displaystyle \frac{1}{2}}\left({\omega }_2{q}_0-{\omega }_3{q}_1+{\omega }_1{q}_3\right)\\ {\displaystyle \frac{1}{2}}\left({\omega }_3{q}_0+{\omega }_2{q}_1-{\omega }_1{q}_2\right)\\ {\displaystyle \frac{1}{J_1}}\left(-{\omega }_3{J}_2{\omega }_2+{\omega }_2{J}_3{\omega }_3+u_1\right)\\ {\displaystyle \frac{1}{J_2}}\left({\omega }_3{J}_1{\omega }_1-{\omega }_1{J}_3{\omega }_3+u_2\right)\\ {\displaystyle \frac{1}{J_3}}\left(-{\omega }_2{J}_1{\omega }_1+{\omega }_1{J}_2{\omega }_2+u_3\right)\end{array}\right)$$

(3)

Let us consider a nonlinear stochastic system [48]

$$\mathrm{d}x=f\left(\mathit{x},\mathit{u},t\right)\mathrm{d}t+\mathit{G}\left(t\right)\mathrm{d}W$$

(4)

where $\mathit{x}ϵR^{n_x}$ is the state vector, $\mathit{u}ϵR^{n_u}$ is the control vector, t is the time, and $W\left(t\right)ϵR^{n_w}$ is the multidimensional Brownian process. The initial state is modeled as a Gaussian distribution with known mean ${\mathit{\mu }}_0$ and covariance ${\mathit{P}}_0$

$$\mathit{x}\left(t_0\right)={\mathit{x}}_0\sim N\left({\mathit{\mu }}_0,{\mathit{P}}_0\right)$$

(5)

Since quaternions must satisfy a normalization constraint, linearizing Equation (3) requires additional consideration of the quaternion component. While preserving the physical meaning of quaternions, a multiplicative error quaternion in the body frame is employed.

$$\mathsf{\delta }\mathit{q}=\mathit{q}\otimes {\widehat{\mathit{q}}}^{-1}$$

(6)

where δq is error quaternion, $\widehat{\mathit{q}}$ is estimated quaternion. Based on the properties and computations of quaternions, leads to

$$\mathsf{\delta }\dot{\mathit{q}}=-\left[\begin{array}{c}0\\ \left[\widehat{\mathit{\omega }}\times \right]\mathsf{\delta }\underset{\_}{\mathit{q}}\end{array}\right]+{\displaystyle \frac{1}{2}}\left[\begin{array}{c}-\mathsf{\delta }{\omega }_3\\ -\mathsf{\delta }{\omega }_2\\ \mathsf{\delta }{\omega }_1\\ 0\end{array}\right]\otimes \mathsf{\delta }\mathit{q}$$

(7)

The first-order approximation yields the following linearized model:

$$\begin{array}{l}\mathsf{\delta }{\dot{q}}_0=0\\ \mathsf{\delta }\dot{\underset{\_}{\mathit{q}}}=-\left[\widehat{\mathit{\omega }}\times \right]\mathsf{\delta }\underset{\_}{\mathit{q}}+{\displaystyle \frac{1}{2}}\mathsf{\delta }\mathit{\omega }\end{array}$$

(8)

The estimation error of angular velocity is defined as

$$\mathsf{\delta }\mathit{\omega }=-\left(\mathsf{\delta }\mathit{\beta }+\zeta \right)$$

(9)

where including the bias estimation error $\mathsf{\delta }\mathit{\beta }=\mathit{\beta }-\widehat{\mathit{\beta }}$, and ζ is the measurement noise with zero mean white noise processes, $\mathit{\beta }$ is a bias vector.

Therefore, the bias estimation dynamics are governed by

$$\mathsf{\delta }\dot{\mathit{\beta }}=\zeta$$

(10)

Define the state $\mathsf{\Delta }\mathit{x}\triangleq [\mathsf{\delta }\mathit{q};\mathsf{\delta }\mathit{\beta }]$, the equations can be assembled as

$$\Delta \dot{\mathit{x}}=\mathit{A}\Delta \mathit{x}+\mathit{G}\mathit{w}$$

(11)

A linear stochastic system is obtained by linearizing the nonlinear stochastic system

$$\mathrm{d}\mathit{x}=\left(\mathit{A}\left(t\right)\mathit{x}+\mathit{B}\left(t\right)\mathit{u}+\mathit{c}\left(t\right)\right)\mathrm{d}t+\mathit{G}\left(t\right)\mathrm{d}W$$

(12)

The coefficient matrix of the linear stochastic system is specified as

$$\begin{array}{l}\mathit{A}\left(t\right)={\left[\begin{array}{cc}{\mathit{A}}_{11}& {\mathit{A}}_{12}\\ {\mathbf{0}}_{3\times 4}& {\mathbf{0}}_{3\times 3}\end{array}\right]}_{7\times 7},\mathit{B}\left(t\right)={\left[\begin{array}{c}{\mathbf{0}}_{4\times 3}\\ {\mathit{B}}_2\end{array}\right]}_{7\times 3}\\ \mathit{G}\left(t\right)={\left[\begin{array}{cc}{\mathbf{0}}_{4\times 4}& {\mathbf{0}}_{4\times 3}\\ {\mathbf{0}}_{3\times 4}& \mathit{g}{\mathit{I}}_3\end{array}\right]}_{7\times 7}\end{array}$$

(13)

where g is a random disturbance, and the block matrix is

$$\begin{array}{l}{\mathit{A}}_{11}=\left(\begin{array}{cc}0& {\mathbf{0}}_{1\times 3}\\ {\mathbf{0}}_{3\times 1}& -\widehat{\mathit{\omega }}\times \end{array}\right),{\mathit{A}}_{12}={\displaystyle \frac{1}{2}}\left(\begin{array}{c}{\mathbf{I}}_3\\ {\mathbf{0}}_{1\times 3}\end{array}\right)\\ {\mathit{B}}_2=J^{-1}\end{array}$$

(14)

The time variable $tϵ[t_0,t_f]$ is divided into N parts, that is,

$$t_0<t_1<\cdots <t_{\mathrm{N}-2}<t_{\mathrm{N}-1}=t_f$$

(15)

Control torque is kept discrete using the zero-order hold (ZOH)

$$\mathit{u}\left(t\right)=\mathit{u}\left(t_k\right),\forall t\in \left[t_{k-1},t_k\right],k=1,2,\dots ,\mathrm{N}-1$$

(16)

Define ${\mathit{x}}_k\triangleq \mathit{x}\left(t_k\right)$ and ${\mathit{u}}_k\triangleq \mathit{u}\left(t_k\right)$, the discrete-time linear stochastic systems are

$${\mathit{x}}_{k+1}={\mathit{A}}_k{\mathit{x}}_k+{\mathit{B}}_k{\mathit{u}}_k+{\mathit{G}}_k{\mathit{w}}_k$$

(17)

where

$${\mathit{A}}_k=\mathbf{\Phi }\left(t_k,t_{k+1}\right)$$

(18)

$${\mathit{B}}_k={\displaystyle {\int }_{t_k}^{t_{k+1}}\mathbf{\Phi }\left(s,t_{k+1}\right)}\mathit{B}\left(s\right)\mathrm{d}s$$

(19)

And, ${\mathit{G}}_k$ is a given matrix satisfying ${\mathit{G}}_k{\mathit{w}}_k$ is a random Gaussian vector having covariance [43]

$${\mathit{Q}}_k={\displaystyle {\int }_{t_k}^{t_{k+1}}\mathbf{\Phi }\left(s,t_{k+1}\right)}\mathit{G}\left(s\right)\mathit{w}{\mathit{w}}^{\mathrm{T}}{\mathit{G}}^{\mathrm{T}}\left(s\right){\mathbf{\Phi }}^{\mathrm{T}}\left(s,t_{k+1}\right)\mathrm{d}s$$

(20)

with $\mathbf{\Phi }\left(t_k,t_{k+1}\right)$ denoting the state transfer matrix from $t_k$ to $t_{k+1}$.

2.2. Constraints

The attitude pointing problem for gravitational wave detection requires that the attitude maneuvers of each spacecraft arrive at the terminal quaternion from the initial quaternion measured by the inertial sensors and the angular velocity, which is determined by the time-varying orbital state, geometrical constraints, and initial time and state, and that the terminal angular velocity of the spacecraft is given in advance by the on-board system. The initial and terminal constraints of the system are represented by the following equation

$$\mathit{q}\left(t_0\right)={\mathit{q}}_0,\mathit{\omega }\left(t_0\right)={\mathit{\omega }}_0,\mathit{\omega }\left(t_f\right)={\mathit{\omega }}_f$$

(21)

$$\mathit{{\rm Z}}\left(\mathit{q}\left(t_f\right),\mathit{r}\left(t_f\right),t_f\right)=0$$

(22)

In gravitational wave detection missions, the spacecraft terminal state and the terminal time of attitude maneuvers are highly correlated. The terminal time determines the orbital state of the spacecraft and the geometric position of the payload in space, and this relationship defines the target terminal attitude of the spacecraft. The relative position between each of the two spacecraft payloads is derived from the following equation:

$${\mathit{r}}_{\mathrm{pay},i,i+1}={\displaystyle \frac{{\mathit{r}}_{i+1}-{\mathit{r}}_i}{‖{\mathit{r}}_{i+1}-{\mathit{r}}_i‖}},i=1,2,3$$

(23)

$$\begin{array}{l}{\mathit{r}}_{\mathrm{pay},i,i+1,x}=\cos {\theta }_{i,i+1}\cdot \cos {\phi }_{i,i+1}\\ {\mathit{r}}_{\mathrm{pay},i,i+1,y}=\cos {\theta }_{i,i+1}\cdot \sin {\phi }_{i,i+1}\\ {\mathit{r}}_{\mathrm{pay},i,i+1,z}=\sin {\theta }_{i,i+1}\end{array}$$

(24)

where i indicates three spacecraft, ${\theta }_{i,i+1}$ and ${\phi }_{i,i+1}$ denote the pitch and yaw angles of the payload pointing from spacecraft i to spacecraft i + 1, respectively. Also, it is assumed that the payload on each spacecraft has fixed angles $∆{\theta }_{i,i+1}$ and ${∆\phi }_{i,i+1}$ relative to the spacecraft’s body attitude. The quaternion of the spacecraft can be obtained by Equations (24) and (25).

In gravitational wave detection missions, the spacecraft’s terminal attitude is determined by its orbital position, which evolves over time. Since the computational steps in the terminal quaternion constraint are too complex and nonlinear, a polynomial fit to the terminal quaternion with respect to the terminal time point yields

$$\mathit{q}\left(t_f\right)=P_q^n\left(t_f\right)$$

(25)

Considering the quaternion unit constraint:

$$‖\mathit{q}‖=1$$

(26)

Several inductive sensors are fixed to the spacecraft body when the spacecraft performs exploration missions. Bright celestial objects (strong light sources such as the Sun) can damage or saturate the sensor, and to protect the optical component system, it is necessary to avoid pointing in sensitive directions, known as forbidden constraints. Accordingly, some inductive sensors need to point to a specific area to accomplish a task (e.g., laser alignment), and these types of constraints are referred to as mandatory constraints. The mathematical expressions for mandatory and forbidden constraints are as follows

$${\mathit{r}}_{\mathrm{B}}^{\mathrm{T}}\mathit{v}\le \cos \eta$$

(27)

$${\mathit{r}}_{\mathrm{B}}^{\mathrm{T}}\mathit{p}\ge \cos \lambda$$

(28)

where ${\mathit{r}}_{\mathrm{B}}={[r_{{\mathrm{B}}_1},r_{{\mathrm{B}}_2},r_{{\mathrm{B}}_3}]}^{\mathrm{T}}$ denotes the direction vector of the sensor in the body coordinate system, $\mathsf{\nu }$ is the direction vector of the bright celestial object in the inertial coordinate system, $\eta$ and $\lambda$ are the sensor’s point of view, and p is the direction vector of the observation center in the body coordinate system.

Considering the relative motion between the spacecraft and the Sun:

$$\mathit{v}={\mathit{C}}_{\mathrm{B}\mathrm{I}}{\mathit{r}}_{\mathrm{I}}={\mathit{r}}_{\mathrm{I}}-2{\underset{\_}{\mathit{q}}}^{\mathrm{T}}\underset{\_}{\mathit{q}}{\mathit{r}}_{\mathrm{I}}+2\underset{\_}{\mathit{q}}{\underset{\_}{\mathit{q}}}^{\mathrm{T}}{\mathit{r}}_{\mathrm{I}}+2q_0\left({\mathit{r}}_{\mathrm{I}}^{\times }\underset{\_}{\mathit{q}}\right)$$

(29)

where C_BI represents the attitude transfer matrix from inertial coordinate system to body coordinate system; ${\mathit{r}}_{\mathrm{I}}={[r_1,r_2,r_3]}^{\mathrm{T}}$ is the direction vector of the bright celestial object in inertial coordinate system; $\underset{\_}{\mathit{q}}={[q_1,q_2,q_3]}^{\mathrm{T}}$ is the vector part of quaternion; ${\mathit{r}}_{\mathrm{I}}^{\times }$ is cross product matrix of ${\mathit{r}}_{\mathrm{I}}$. Then, expand Equation (29)

$${\mathit{C}}_{\mathrm{BI}}{\mathit{r}}_{\mathrm{I}}=\left(1-2q_1^2-2q_2^2-2q_3^2\right)\left[\begin{array}{c}{r}_1\\ r_2\\ r_3\end{array}\right]+2\left[\begin{array}{ccc}{q}_1^2& q_1{q}_2& q_1{q}_3\\ q_1{q}_2& q_2^2& q_2{q}_3\\ q_1{q}_3& q_2{q}_3& q_3^2\end{array}\right]\left[\begin{array}{c}{r}_1\\ r_2\\ r_3\end{array}\right]+2q_0\left[\begin{array}{c}-r_3{q}_2+r_2{q}_3\\ r_3{q}_1-r_1{q}_3\\ -r_2{q}_1+r_1{q}_2\end{array}\right]$$

(30)

Substitute Equation (30) into Equation (27)

$$\begin{array}{cc}\hfill {\mathit{r}}_{\mathrm{B}}^{\mathrm{T}}\left({\mathit{C}}_{\mathrm{BI}}{\mathit{r}}_{\mathrm{I}}\right)-\cos \eta =& \left(q_0^2+q_1^2-q_2^2-q_3^2\right)r_{{\mathrm{B}}_1}{r}_1+\left(q_0^2-q_1^2+q_2^2-q_3^2\right)r_{{\mathrm{B}}_2}{r}_2+\left(q_0^2-q_1^2-q_2^2+q_3^2\right)r_{{\mathrm{B}}_3}{r}_3\hfill \\ & +\left(2q_0{q}_3+2q_1{q}_2\right)r_{{\mathrm{B}}_1}{r}_2+\left(-2q_0{q}_2+2q_1{q}_3\right)r_{{\mathrm{B}}_1}{r}_3+\left(-2q_0{q}_3+2q_1{q}_2\right)r_{{\mathrm{B}}_2}{r}_1\hfill \\ & +\left(2q_0{q}_1+2q_2{q}_3\right)r_{{\mathrm{B}}_2}{r}_3+\left(2q_0{q}_2+2q_1{q}_3\right)r_{{\mathrm{B}}_3}{r}_1+\left(-2q_0{q}_1+2q_2{q}_3\right)r_{{\mathrm{B}}_3}{r}_2\hfill \\ & -\cos \eta \hfill \end{array}$$

(31)

The above equations Equation (27) can be simplified to a quadratic constraint form

$${\mathit{r}}_{\mathrm{B}}^{\mathrm{T}}\left({\mathit{C}}_{\mathrm{BI}}{\mathit{r}}_{\mathrm{I}}\right)-\cos \eta =\left[\begin{array}{cccc}{q}_0& q_1& q_2& q_3\end{array}\right]{\mathit{Y}}_1\left[\begin{array}{c}{q}_0\\ q_1\\ q_2\\ q_3\end{array}\right]={\mathit{q}}^{\mathrm{T}}{\mathit{Y}}_1\mathit{q}\le 0$$

(32)

where

$${\mathit{Y}}_1=\left(\begin{array}{cc}{\mathit{r}}_I^{\mathrm{T}}{\mathit{r}}_{\mathrm{B}}-\cos \eta & {\left({\mathit{r}}_{\mathrm{B}}\times {\mathit{r}}_{\mathrm{I}}^{\mathrm{T}}\right)}^{\mathrm{T}}\\ {\mathit{r}}_{\mathrm{B}}\times {\mathit{r}}_{\mathrm{I}}^{\mathrm{T}}& {\mathit{r}}_{\mathrm{I}}{\mathit{r}}_{\mathrm{B}}^{\mathrm{T}}+{\mathit{r}}_{\mathrm{B}}{\mathit{r}}_{\mathrm{I}}^{\mathrm{T}}-\left({\mathit{r}}_{\mathrm{I}}^{\mathrm{T}}{\mathit{r}}_{\mathrm{B}}+\cos \eta \right){\mathbf{I}}_3\end{array}\right)$$

(33)

Similarly, Equation (28) can be simplified to

$${\mathit{q}}^{\mathrm{T}}{\mathit{Y}}_2\mathit{q}\ge 0$$

(34)

$${\mathit{Y}}_2=\left(\begin{array}{cc}{\mathit{r}}_{\mathrm{o}}^{\mathrm{T}}{\mathit{r}}_{\mathrm{B}}-\cos \lambda & {\left(r_{\mathrm{B}}\times {\mathit{r}}_{\mathrm{o}}^{\mathrm{T}}\right)}^{\mathrm{T}}\\ {\mathit{r}}_{\mathrm{B}}\times {\mathit{r}}_{\mathrm{o}}^{\mathrm{T}}& {\mathit{r}}_{\mathrm{o}}{r}_{\mathrm{B}}^{\mathrm{T}}+{\mathit{r}}_{\mathrm{B}}{\mathit{r}}_{\mathrm{o}}^{\mathrm{T}}-\left({\mathit{r}}_{\mathrm{o}}^{\mathrm{T}}{\mathit{r}}_{\mathrm{B}}+\cos \lambda \right){\mathbf{I}}_3\end{array}\right)$$

(35)

where $\mathit{p}={\mathit{C}}_{\mathrm{B}\mathrm{I}}{\mathit{r}}_{\mathrm{o}}$, and r_o is the centric vector of observation region in inertial coordinate system.

Since the state of the system is Gaussian distributed, boundary constraints Equation (22) are transformed into two-part constraints on mean and covariance

$${\overline{\mathit{x}}}_0=\overline{\mathit{x}}\left(t_0\right),{\mathit{P}}_0=\mathit{P}\left(t_0\right)$$

(36)

$${\overline{\mathit{x}}}_f=\overline{\mathit{x}}\left(t_f\right),{\mathit{P}}_{\mathrm{N}}\le {\mathit{P}}_f=\mathit{P}\left(t_f\right)$$

(37)

Note that the condition on the covariance P is not an equational constraint, and the symbols > 0 and ≥ 0 indicate that P is a positive definite or positive semidefinite matrix.

Traditional deterministic constraints may be too conservative or unsatisfiable in stochastic systems in which states or parameters (perturbations, noise, etc.) have probability distributions. Chance constraints are a widely used form of constraints in stochastic optimization and robust control that allow constraints to be violated with small probability rather than strictly holding under all conditions. Relaxing the constraints into probabilistic form through chance constraints enhances system flexibility and is very useful when dealing with uncertainty.

The closed-loop control law u of the system is also Gaussian distributed with mean $\mathsf{\upsilon }$ and covariance $KP{K}^{\mathrm{T}}$. Consider the maximum control torque constraint

$$P\left\{‖{\mathit{u}}_k‖\le u_{\max }\right\}\ge p$$

(38)

where p is the desired probability.

Individual chance constraints that satisfy a Gaussian distribution can be equivalently transformed into more conservative forms of constraints [43]

$$‖{\nu }_k‖+\gamma \left(p\right)\sqrt{\rho \left({\mathit{K}}_k{\mathit{P}}_k{\mathit{K}}_k^{\mathrm{T}}\right)}\le u_{\max }$$

(39)

where

$$\gamma \left(p\right)=\left\{\begin{array}{cc}\sqrt{2\log \left({\displaystyle \frac{1}{1-p}}\right)}& \begin{array}{cc}\mathrm{if}& n_u\le 2\end{array}\\ \sqrt{2\log \left({\displaystyle \frac{1}{1-p}}\right)}+\sqrt{n_u}& \begin{array}{cc}\mathrm{if}& n_u>2\end{array}\end{array}\right.$$

(40)

Similarly, forbidden constraints and mandatory constraints should also be forms of chance constraints

$$P\left\{{\mathit{x}}^{\mathrm{T}}\mathit{Y}\mathit{x}<0\right\}\ge p$$

(41)

where $\mathit{Y}=\overline{\mathit{y}}+\Delta$, $\overline{\mathit{y}}$ is the mean value matrix, and $\Delta ~\mathcal{N}\left(0,\Sigma \right)$ satisfies the Gaussian distribution, quadratic form chance constraints can be converted to hard constraints

$${\mathit{x}}^{\mathrm{T}}\overline{\mathit{y}}\mathit{x}+{\mathit{\Gamma }}^{-1}\left(p\right)\sqrt{{\left(\mathrm{vec}(\mathit{x}{\mathit{x}}^{\mathrm{T}})\right)}^{\mathrm{T}}\Sigma \left(\mathrm{vec}(\mathit{x}{\mathit{x}}^{\mathrm{T}})\right)}\le 0$$

(42)

where Γ⁻¹ is the inverse cumulative distribution function of the standard normal distribution.

2.3. Original Problem

In modern control theory, open-loop control causes the system state covariance to increase over time, uncertainty builds up over time independently of the control law, and to satisfy the system stability and the terminal state covariance within a certain range, the state feedback control law is used.

$${\mathit{u}}_k={\mathit{\upsilon }}_k+{\mathit{K}}_k\left({\mathit{x}}_k-{\mathit{\mu }}_k\right)$$

(43)

where ${\mathit{\upsilon }}_k$ is the feedforward gain matrix calculated in advance based on the reference value, and ${\mathit{K}}_k$ is the feedback gain matrix calculated based on the state error. The mean and covariance of a linear stochastic system need to satisfy the constraints

$${\mathit{\mu }}_{k+1}={\mathit{A}}_k{\mathit{\mu }}_k+{\mathit{B}}_k{\mathit{\upsilon }}_k$$

(44)

$${\mathit{P}}_{k+1}=\left({\mathit{A}}_k+{\mathit{B}}_k{\mathit{K}}_k\right){\mathit{P}}_k{\left({\mathit{A}}_k+{\mathit{B}}_k{\mathit{K}}_k\right)}^{\mathrm{T}}+{\mathit{Q}}_k$$

(45)

Specifically, the attitude planning problem discussed in this paper can be expressed as the following Problem P1.

Problem P1

Minimize

$$J_{P_1}={\displaystyle \sum _{k=1}^{\mathrm{N}}‖{\mathit{u}}_k‖}$$

(46)

Subject to

$${\mathit{x}}_k\sim N\left({\mathit{\mu }}_k,{\mathit{P}}_k\right)$$

(47)

$${\mathit{u}}_k\sim N\left({\mathit{\nu }}_k,{\mathit{K}}_k{\mathit{P}}_k{\mathit{K}}_k^{\mathrm{T}}\right)$$

(48)

$${\mathit{\mu }}_{k+1}={\mathit{A}}_k{\mathit{\mu }}_k+{\mathit{B}}_k{\mathit{\upsilon }}_k$$

(49)

$${\mathit{P}}_{k+1}=\left({\mathit{A}}_k+{\mathit{B}}_k{\mathit{K}}_k\right){\mathit{P}}_k{\left({\mathit{A}}_k+{\mathit{B}}_k{\mathit{K}}_k\right)}^{\mathrm{T}}+{\mathit{Q}}_k$$

(50)

$${\overline{\mathit{x}}}_0=\overline{\mathit{x}}\left(t_0\right),{\mathit{P}}_0=\mathit{P}\left(t_0\right)$$

(51)

$$\mathit{q}\left(t_f\right)=P_q^n\left(t_f\right)$$

(52)

$${\overline{\mathit{x}}}_f=\overline{\mathit{x}}\left(t_f\right),{\mathit{P}}_{\mathrm{N}}\le {\mathit{P}}_f=\mathit{P}\left(t_f\right)$$

(53)

$$P\left\{‖{\mathit{u}}_k‖\le u_{\max }\right\}\ge p$$

(54)

$$P\left\{{\mathit{x}}^{\mathrm{T}}\mathit{Y}\mathit{x}<0\right\}\ge p$$

(55)

3. Convexification

In this section, the non-convex part of the original problem is convexified and reformulated. We will introduce the relaxation of the covariance dynamics, the convexification of the non-convex terms of the chance constraints, and the final convex optimization problem.

3.1. Relaxation of Covariance Dynamics

To eliminate the non-convex term due to the multiplication of the two terms in Equation (50), a new variable ${\mathit{S}}_k\in R^{n_u\times n_x}$ is introduced such that it satisfies

$${\mathit{S}}_k={\mathit{K}}_k{\mathit{P}}_k$$

(56)

In this way, ${\mathit{S}}_k$ instead of ${\mathit{K}}_k$ as the optimization variable, the matrix ${\mathit{S}}_k$ is orthogonal, and the covariance constraint Equation (50) can be written as

$${\mathit{P}}_{k+1}={\mathit{A}}_k{\mathit{P}}_k{\mathit{A}}_k^{\mathrm{T}}+{\mathit{A}}_k{\mathit{S}}_k^{\mathrm{T}}{\mathit{B}}_k^{\mathrm{T}}+{\mathit{B}}_k{\mathit{S}}_k{\mathit{A}}_k^{\mathrm{T}}+{\mathit{B}}_k{\mathit{S}}_k{{\mathit{P}}_k}^{-1}{\mathit{S}}_k^{\mathrm{T}}{\mathit{B}}_k^{\mathrm{T}}+{\mathit{Q}}_k$$

(57)

However, the current covariance constraint remains a nonlinear nonconvex equation; consider relaxing its constraints to

$${\mathit{P}}_{k+1}\ge {\mathit{A}}_k{\mathit{P}}_k{\mathit{A}}_k^{\mathrm{T}}+{\mathit{A}}_k{\mathit{S}}_k^{\mathrm{T}}{\mathit{B}}_k^{\mathrm{T}}+{\mathit{B}}_k{\mathit{S}}_k{\mathit{A}}_k^{\mathrm{T}}+{\mathit{B}}_k{\mathit{S}}_k{{\mathit{P}}_k}^{-1}{\mathit{S}}_k^{\mathrm{T}}{\mathit{B}}_k^{\mathrm{T}}+{\mathit{Q}}_k$$

(58)

Schur’s complement theorem is often used to transform nonlinear constraints in optimization problems into linear matrix inequalities (LMIs), which are used to determine the positivity of block matrices. Equation (58) can be recast as a linear matrix inequality

$$\left[\begin{array}{cc}{\mathit{P}}_k& {{\mathit{S}}_k}^{\mathrm{T}}{{\mathit{B}}_k}^{\mathrm{T}}\\ {\mathit{B}}_k{\mathit{S}}_k& {\mathit{H}}_k\end{array}\right]\ge 0$$

(59)

where

$${\mathit{H}}_k={\mathit{P}}_{k+1}-\left({\mathit{A}}_k{\mathit{P}}_k{{\mathit{A}}_k}^{\mathrm{T}}+{\mathit{A}}_k{{\mathit{S}}_k}^{\mathrm{T}}{{\mathit{B}}_k}^{\mathrm{T}}+{\mathit{B}}_k{\mathit{S}}_k{{\mathit{A}}_k}^{\mathrm{T}}+{\mathit{Q}}_k\right)$$

(60)

Equation (59) is a semidefinite cone constraint that can be solved by a suitable convex optimization algorithm. The analysis based on optimal control theory proves that the relaxation method is effective and lossless; the complexity of the optimization is greatly reduced while ensuring that the relaxation problem and the original problem have the same solution.

Add the following penalty term to the objective function

$$J_H={\alpha }_H{\displaystyle \sum _{k=1}^N\mathrm{trace}\left({\mathit{H}}_k\right)}$$

(61)

where α_H is the penalty coefficient for H_k.

3.2. Chance Constrain Convexity Strategy

The paradigms in Equation (39) are non-convex terms and can be replaced by non-negative auxiliary variables

$${\nu }_k^{\mathrm{T}}{\nu }_k\le {\tilde{\nu }}_k^2$$

(62)

By Schur’s complement theorem, Equation (39) can be equated to

$$\left[\begin{array}{cc}{\tilde{\rho }}_k{\mathit{I}}_{n_u}& {\mathit{S}}_k\\ {\mathit{S}}_k^{\mathrm{T}}& {\mathit{P}}_k\end{array}\right]\ge 0$$

(63)

where ${\tilde{\rho }}_k$ are non-negative auxiliary variables.

Equation (39) can be written as a linear inequality constraint

$${\tilde{\nu }}_k+\gamma \left(p\right){\tau }_k\le u_{\max }$$

(64)

where ${\tau }_k>0$ is the arithmetic square root of ${\tilde{\rho }}_k$

$${\tilde{\rho }}_k={\tau }_k^2$$

(65)

To obtain a convexification problem, a first-order Taylor expansion of the quadratic equation constraints Equation (65) is performed around the reference value ${\tau }_k$

$${\tilde{\rho }}_k={\overline{\tau }}_k^2+2{\overline{\tau }}_k\left({\tau }_k-{\overline{\tau }}_k\right)+{\chi }_{\tau }$$

(66)

Add a penalty term to the objective function

$$J_{\tau }={\alpha }_{\tau }{\displaystyle \sum _{k=1}^N\left|{\chi }_{\tau }\right|}$$

(67)

where α_τ is the penalty coefficient for the slack term χ_τ.

Introducing matrix variables $\mathit{X}=\mathit{x}{\mathit{x}}^{\mathrm{T}}$, where $\mathit{X}\sim \mathrm{N}(\mathsf{\Lambda },\mathsf{\Xi })$, the quadratic constraint is equivalent to

$$\overline{y}\Lambda +{\mathit{\Gamma }}^{-1}(p)\sqrt{{\Lambda }^{\mathrm{T}}\Sigma \Lambda }\le 0$$

(68)

Convexify the second term using Schur complement and second-order cone constraint (SOC)

$$‖\sqrt{\Sigma }\Lambda ‖\le {\displaystyle \frac{t}{{\mathit{\Gamma }}^{-1}(p)}},t\ge {\overline{y}}^{\mathrm{T}}\Lambda$$

(69)

$$X\ge 0$$

(70)

Similarly, add a penalty to the objective function

$$J_y={\alpha }_y{\displaystyle \sum _{k=1}^N\left|{\chi }_y\right|}$$

(71)

where α_y is the penalty coefficient for the slack term χ_y.

3.3. Convex Optimization Problem

We formulate the attitude pointing control problem as a convex optimization problem: Efficient solvers such as Convex Optimization Toolbox (CVX) can be used to solve this problem due to its convex structure.

This method enables robust laser link acquisition by leveraging covariance propagation to predict attitude uncertainty and by embedding probabilistic constraints into a tractable convex optimization framework. Unlike deterministic approaches, our formulation explicitly accounts for stochastic effects during the attitude alignment process, thereby improving reliability in uncertain environments.

Problem P1 can be reformulated as a convex problem P2.

Problem P2

Minimize

$$J_{P_2}={\displaystyle \sum _{k=1}^N‖{\nu }_k+\gamma \left(p\right){\tau }_k‖}+J_H+J_y+J_{\tau }$$

(72)

Subject to

$${\mathit{x}}_k\sim N\left({\mathit{\mu }}_k,{\mathit{P}}_k\right)$$

(73)

$${\mathit{u}}_k\sim N\left({\nu }_k,{\mathit{K}}_k{\mathit{P}}_k{\mathit{K}}_k^{\mathrm{T}}\right)$$

(74)

$${\mathit{\mu }}_{k+1}={\mathit{A}}_k{\mathit{\mu }}_k+{\mathit{B}}_k{\mathit{\upsilon }}_k$$

(75)

$$\left[\begin{array}{cc}{\mathit{P}}_k& {\mathit{S}}_k^{\mathrm{T}}{\mathit{B}}_k^{\mathrm{T}}\\ {\mathit{B}}_k{\mathit{S}}_k& {\mathit{H}}_k\end{array}\right]\ge 0$$

(76)

$${\mathit{H}}_k={\mathit{P}}_{k+1}-\left({\mathit{A}}_k{\mathit{P}}_k{\mathit{A}}_k^{\mathrm{T}}+{\mathit{A}}_k{\mathit{S}}_k^{\mathrm{T}}{\mathit{B}}_k^{\mathrm{T}}+{\mathit{B}}_k{\mathit{S}}_k{\mathit{A}}_k^{\mathrm{T}}+{\mathit{Q}}_k\right)$$

(77)

$$J_H={\alpha }_H{\displaystyle \sum _{k=1}^N\mathrm{trace}\left({\mathit{H}}_k\right)}$$

(78)

$${\overline{\mathit{x}}}_0=\overline{\mathit{x}}\left(t_0\right),{\mathit{P}}_0=\mathit{P}\left(t_0\right)$$

(79)

$$\mathit{q}\left(t_f\right)=P_q^n\left(t_f\right)$$

(80)

$${\overline{\mathit{x}}}_f=\overline{\mathit{x}}\left(t_f\right),{\mathit{P}}_N\le {\mathit{P}}_f=\mathit{P}\left(t_f\right)$$

(81)

$${\tilde{\nu }}_k+\gamma \left(p\right){\tau }_k\le u_{\max }$$

(82)

$$\left[\begin{array}{cc}{\tilde{\rho }}_k{I}_{n_u}& S_k\\ S_k^{\mathrm{T}}& P_k\end{array}\right]\ge 0$$

(83)

$${\tilde{\rho }}_k={\overline{\tau }}_k^2+2{\overline{\tau }}_k\left({\tau }_k-{\overline{\tau }}_k\right)+{\chi }_{\tau }$$

(84)

$$J_{\tau }={\alpha }_{\tau }{\displaystyle \sum _{k=1}^N\left|{\chi }_{\tau }\right|}$$

(85)

$$\overline{y}\Lambda +{\mathbf{\Phi }}^{-1}(p)\sqrt{{\Lambda }^{\mathrm{T}}\Sigma \Lambda }\le 0$$

(86)

$${\tilde{y}}_k={\overline{\eta }}_k^2+2{\overline{\eta }}_k\left({\eta }_k-{\overline{\eta }}_k\right)+{\chi }_y$$

(87)

$$J_y={\alpha }_y{\displaystyle \sum _{k=1}^N\left|{\chi }_y\right|}$$

(88)

The entire solution process for P2 can be referred to in the flow chart Figure 1.

4. Simulation Results

In this section, the effectiveness of the proposed attitude planning problem, which considers uncertainties in the laser link establishment process, is simulated and verified based on the LISA mission. The spacecraft orbital parameters refer to existing studies on the LISA mission [26], and the specific data are shown in

Table 1. Parameters for gravity wave detection constellation design.

Item	Symbol	Value
arm-length of constellation	km	2.5 × 106
changing of internal angle	degree	±1
changing rate of internal angle	nrad/s	<5

. The convex optimization problem is solved using the CVX toolbox (version 2.2) and the Mosek Optimization Tools (MOSEK) solver. MATLAB (version R2019b)’s CVX is a powerful tool designed for modeling and solving convex optimization problems. CVX supports the formulation of problems such as LMIs and SOCP. MOSEK is a commercial convex optimization solver (version 10.1) employing the interior point method (IPM) with a convergence accuracy set to 10⁻⁸. Its computational speed meets the real-time control requirements for spacecraft in orbit [43].

In gravitational wave detection missions, to ensure high attitude control accuracy, a combination of coarse and minor adjustments is typically employed for high-accuracy attitude alignment. Correspondingly, the process of establishing the laser links can be divided into three phases. The paper addresses the objectives of the first phase, which relies on differential wave-front sensing (DWS) technology. Due to the physical limitations of DWS, it can only directly measure information in two dimensions, both of which must be perpendicular to the laser propagation direction. In the spacecraft body coordinate system, these correspond to the yaw angle and pitch angle. These two angles directly reflect the alignment state of the laser link and are critical parameters for ensuring the accuracy of laser interferometry. It should be noted that this approach does not negate the overall requirement for three axis stabilization of the spacecraft. Rather, it represents a targeted choice based on the technical characteristics of the laser measurement mode. The roll angle does not need to be measured via DWS technology. Control of the roll angle will be implemented in other phases to ensure the spacecraft maintains overall three axis stability.

Based on the reference orbit of the spacecraft formation, the relative attitude angles between the payloads (optical components) of the spacecraft formation and the corresponding terminal quaternions can be calculated. During the spacecraft attitude alignment process considered in this paper, only the pitch angle and yaw angle need to be considered in relation to changes in Euler angles. Figure 2 shows the time-dependent changes in the relative attitude angles between the payloads of the two spacecraft. By calculating the corresponding quaternions from the relative attitude angles, the time-varying terminal quaternions are then fitted as a polynomial function of time. Figure 3 illustrates the results of fitting the time-varying quaternions using polynomials of different orders. The time-varying terminal quaternion curve with the best fitting performance is selected as the constraint condition for the spacecraft’s terminal quaternions.

The numerical simulation in this paper is based on data measurements from the star tracker (STR) during the establishment of the laser link. The parameters utilized in the algorithm proposed in this paper are presented in

Table 2. The parameters utilized in the algorithm.

Parameters	Unit	Value
P0	[/, /, /, /, (rad/s)2, (rad/s)2, (rad/s)2]	diag(10−9, 10−9, 10−9, 10−9, 10−10, 10−10, 10−10)
Pf	[/, /, /, /, (rad/s)2, (rad/s)2, (rad/s)2]	diag(10−9, 10−9, 10−9, 10−9, 10−10, 10−10, 10−10)
g	Nm/μN	10−6
p		0.95
nu		3
nx		7
αy, αH, ατ		104
Δt	s	0.12
τk	Nm	0.01
ε		10−6

. The duration of attitude alignment affects the initial errors and control difficulty of other sensors in the subsequent gravitational wave detection phase, so it is desirable to achieve the target state more quickly. The initial states of the three spacecraft [26] and their corresponding forced constraints and prohibited constraints in space are shown in

Table 3. Initial states of the spacecraft.

Spacecraft	Constraint Name	Constraint Parameter
all	initial time	0 s
	translational inertial	diag(551.25, 450.83, 450.83) kg∙m2
	forbidden cone 1 axis	[0.5000, −0.8660, 0]T
	forbidden cone 2 axis	[0.2432, 0.9077, −0.3420]T
	forbidden cone 3 axis	[0.4924, 0.0868, −0.8660]T
	forbidden cone 1 half angle	30 deg
	forbidden cone 2 half angle	30 deg
	forbidden cone 3 half angle	30 deg
	maximum value of control	7 × 10−3 Nm
spacecraft 1	initial quaternion	[−0.73224, 0.07897, 0.42632, 0.52521]T
	initial angle velocity	[0.08606, 0.10219, 0.06650] rad/s
	mandatory cone axis	[−0.7660, 0.6428, 0]T
	mandatory cone half angle	60 deg
spacecraft 2	initial quaternion	[−0.73224, 0.07897, 0.42632, 0.52521]T
	initial angle velocity	[0.08606, 0.10219, 0.06650] rad/s
	mandatory cone axis	[−0.7660, 0.6428, 0]T
	mandatory cone half angle	60 deg
spacecraft 3	initial quaternion	[−0.73224, 0.07897, 0.42632, 0.52521]T
	initial angle velocity	[0.08606, 0.10219, 0.06650] rad/s
	mandatory cone axis	[−0.7660, 0.6428, 0]T
	mandatory cone half angle	60 deg

. The simulation assumes the presence of three bright targets as prohibitive constraints, each with a field of view half-angle of 30°. The spacecraft’s mission is to align with the other two spacecraft and obtain attitude information. Figure 4 illustrates the three-dimensional attitude pointing of the three spacecraft under the aforementioned pointing constraint conditions. The simulation results demonstrate that the spacecraft can achieve attitude alignment—i.e., adjust its attitude to the terminal state at the corresponding time—while satisfying the constraints.

During the establishment of the laser link, two primary sources of uncertainty were considered: solar radiation pressure and microthruster noise errors. These are treated as Gaussian-distributed random terms in a linear random system, with mean and covariance values of 1 μN, 0.1 μN, and 0.1 (μN)², 0.01 (μN)²/√Hz, respectively. The maximum torque that the microthruster can provide during attitude adjustment is 7 mN. Attitude alignment between spacecraft is achieved through feedback control. The changes in thruster thrust torque and spacecraft angular velocity during attitude alignment are shown in Figure 5 and Figure 6. When the spacecraft formation attitude alignment is complete, the relative angular velocity between spacecraft should be 0. Figure 7 specifically shows the changes in the error quaternion over time during the attitude alignment process. It can be seen that the spacecraft attitude reaches a steady state within 80 s, achieving alignment. The steady-state spacecraft state meets the alignment accuracy requirement of 10⁻⁶ rad. As shown in the enlarged section of Figure 7, the simulation results meet the attitude error constraint requirement. According to the termination criterion of the convex optimization problem, the optimization process is terminated once the attitude error falls below the preset tolerance value and the optimization objective is achieved. Consequently, no new error calculation results are produced after the solver ceases iteration.

5. Conclusions

This paper addresses the random uncertainties present in the data acquisition phase of space gravitational wave detection laser links by proposing a convex optimization attitude alignment strategy based on covariance control. To address uncertainties in real-world missions, a stochastic attitude dynamics model is first established. Unlike existing deterministic models that fail to reflect actual variability, this stochastic model better aligns with the operational conditions of laser link acquisition. Based on this stochastic model, the study integrates covariance control into a convex optimization framework, designing an objective function that simultaneously minimizes terminal attitude error and fuel consumption. Three key constraints are incorporated: covariance constraints probabilistically guarantee sub-micro-arcsecond laser pointing accuracy; second-order cone relaxation of pointing constraints addresses non-convexity in the field of view; and torque saturation constraints ensure the physical feasibility of microthruster outputs.

To overcome the computational challenges of nonconvex terms common in traditional convexification methods, this study further proposes a hierarchical convexification approach. It integrates the Schur complement theorem, second-order cone relaxation, and Taylor expansion techniques to achieve lossless relaxation of nonconvex constraints while maintaining solution accuracy. Numerical simulation results demonstrate that under the influence of random disturbance uncertainties, the strategy successfully achieves precise pointing of the spacecraft’s attitude from its initial state to a time-varying terminal state influenced by orbital geometric relationships. In summary, this work provides an effective technical solution for high-precision attitude alignment of laser links in space gravitational wave detection.