Attitude-orbit coordinated control of spacecraft formation for heliocentric gravitational-wave detection with large-scale communication delays

Einstein’s general theory of relativity reveals that spacetime can be warped by mass and energy. As a conjecture in this context, the evolution of space-time curvature affected by gravitation will propagate through the vast universe like ripples. There remains a wealth of mysteries about astrophysics that have not been explored yet, and these “ripples” called gravitational waves may revolutionize the investigation for this. Space-based gravitational-wave observations, which broadly adopt heliocentric, geocentric, or libration-point architectures, can extend the scope to celestial bodies that generate valuable millihertz frequency waves and escape the Earth-bound noise. The heliocentric gravitational-wave detection relying on laser interferometry faces dual challenges: 1) the attitude errors couple into the interferometric arm-length measurements, potentially corrupting the picometer-level displacement sensitivity, making the attitude accuracy exceptionally rigorous, while 2) the ultra-long interferometric arms (and thus million-kilometer-scale intersatellite communication distances) introduce significant communication delays. As a key to space-based gravitational-wave observations, the past few years have witnessed the development of attitude-orbit control technique, and numerous control methodologies have sprung up for various space missions. However, current communication delay models and their corresponding compensation methods prove inadequate for heliocentric gravitational-wave detection scenarios. Additionally, in line with adaptive SMC, there are few investigations on attitude-orbit coordinated control for heliocentric gravitational-wave detection spacecraft formation. In a research article recently published in Space: Science & Technology, a team from Central South University proposed an advanced communication delay model and an attitude-orbit coordinated controller for heliocentric gravitational-wave detection spacecraft formation, addressing the attitude-orbit coordinated control issue.

Firstly, an attitude-orbit coupled relative motion model of the formation spacecraft and a comprehensive model of communication delays caused by the large detection arm length and hardware resource constraints are established. The architecture of heliocentric gravitational-wave detection spacecraft formation is shown in Fig. 1, in which the serial number SCi is the ith spacecraft, i ∈ {0, 1, ···, N}, typically N = 3. Communication behaviors among different spacecraft are characterized by a graph . , , and  are the node set, edge set and weighted adjacency matrix, respectively, in which the node indices pertain to a finite index set  and  for . The directed edge , which means that there is an available information path from spacecraft j to i, otherwise a_ij = 0. Moreover, it assumes that the self-loop is unallowable, i.e.,  for . Considering the restricted three-body problem, the relative motions are described in the local-vertical local-horizontal (LVLH) frame F_L. For , define ρ_i ≜ [x_i, y_i, z_i]^T ∈ ℝ³, r₀ ≜ [x₀, y₀, z₀]^T ∈ ℝ³, ρ_i ≜ r_i − r₀. Let ρ_i,d ∈ ℝ³ the desired relative position vector. With the relative position error η_i,1 ≜ ρ_i − ρ_i,d and the relative velocity error η_i,2 ≜ , the position error system of SCi can be expressed in F_L as

(1)

where u_i,o ∈ ℝ³ and u_i,do ∈ ℝ³ are the control accelerations and external disturbances of SCi, respectively. For , the attitude dynamics is described in the body-fixed coordinate frame F_bi, where C_i,e ∈ ℝ^3×3 and q_i ∈ ℝ⁴ are the rotation matrix and the unit attitude quaternion depicting the rotation from the inertial frame F_I to F_bi, respectively. Define q_i,d ∈ ℝ⁴, ω_i,d ∈ ℝ³, q_i,e ≜ [q_i,e,v^T, q_i,e,4]^T, and ω_i,e ∈ ℝ³ are the desired values of attitude, the desired angular velocity, the relative attitude error, and the angular velocity error of SCi, respectively. Setting η_i,3 ≜ q_i,e,v and η_i,4 ≜ , the attitude error system can be rewritten as

(9)

Combining with δ_i,1 ≜ [η_i,1^T, η_i,3^T]^T and δ_i,2 ≜ [η_i,2^T, η_i,4^T]^T, the relative translational and rotational motion dynamics are formulated as

(12)

In the heliocentric gravitational-wave detection mission scenario shown in Fig. 2, the large-scale communication delays of approximately 8.333 s are observed for the spacecraft formation tracking the desired trajectories. This paper clarifies a more flexible modeling method of the communication delays ��_ij(t) acting on the accessible information channels between SCi and SCj, , in which signals ��_ij(t) satisfy

(16)

(17)

where E_ij(t), C_ij(t) stand for the delivery distance and size of the data packet, respectively. υ_a is the speed of light in a vacuum. b_ij(t), p_ij(t), and σ are the network bandwidth, transmit power, and variance of white Gaussian noise in the channels, respectively.

Then, to realize the fast convergence of the formation attitude-orbit errors, a reliable and robust integration design of attitude-orbit coordinated control protocol is elaborated by delay-dependent Lyapunov function. Based on the relative translational and rotational motion dynamics [Eq. (12)] for SCi, the terminal sliding manifold is reconstructed as

(18)

where p, q, g, and h are positive odd scalars, 1 < p/q < g/h < 2. α_i > 0 and β_i > 0 are predetermined diagonal matrices. It is proved via the Lyapunov function in this paper that considering the sliding manifold [Eq. (18)], the error system [Eq. (12)] possesses finite-time stability when s_i = 0. To compel the followers’ outputs ρ_i and q_i to track the virtual leader SC0 in the prespecified shape ρ_i,d and their desired attitude q_i,d, a robust adaptive attitude-orbit coordinated control protocol [Eq. (23)] is proposed.

u_i,c ≜ u_i,e + u_i,s + u_i,r + u_i,v (23)

where

(24)

(25)

(26)

The adaptive update law [Eq. (27)] is provided as

(27)

where  is the estimation of ��_i. The diagonal matrices K_i,1 > 0, K_i,2 > 0, K_i,3 > 0, and scalars k_i,4 > 0, k_i,v > 0, k_i,d > 0, 0 < γ < 1 are to be determined. Note that u_i,c is developed to achieve reliable formation flying during heliocentric gravitational-wave detection. Therein, u_i,e is the equivalent control law that stabilizes the error vectors δ_i,1 and δ_i,2 after s_i = 0; u_i,s guarantes the finite-time reachability of the sliding manifold [Eq. (23)] to the area near the equilibrium; u_i,r is the coordinated control part that optimizes individual dynamics with a more balanced cost and higher efficiency; u_i,v is introduced as the auxiliary control input for delay-effect compensation. Besides, it is proved in this paper that with the attitude-orbit coordinated control protocol [Eq. (23)] and update law [Eq. (27)], considering the existence of large-scale communication delays, parameter uncertainties, and external disturbances in the relative position and attitude error system [Eq. (12)], if there exist scalars ρ_ij ≥ 0, σ_ij < 1 for  such that the following inequalities hold

(28)

(29)

where ρ_ij = ρ_ji when i ≠ j, ρ_ii = 0. Then, all the closed-loop signals that involved the error system [Eq. (12)] can converge to a tiny neighborhood around the equilibrium in finite time. The system schematic diagram of SCi is summarized in Fig. 3.

Finally, numerical simulations are constructed with the typical heliocentric gravitational-wave detection mission scenario, verifying the validity and superiority of the proposed method. In the simulation, the uncertainty of the inertia matrix of SCi, the execution errors, and the secondary gravitational effects of Venus and Jupiter are considered. Set σ² = 10⁻¹⁰, C_ij ∈ [300, 500] kbits, b_ij^mean = 1 GHz, p_ij^max = 1.4 pW. Results show that due to the reasonable assumption C_ij ∈ [300, 500] kbits, Eq. (16) preserves ��_ij(t) boundedness while additionally capturing communication stochasticity beyond fixed delays. Meanwhile, the rapid convergence of all error signals is gotten by t = 300 s. Furthermore, the controller [Eq. (23)] is compared with 3 other controllers: (a) the attitude-orbit coordinated controller for TianQin geocentric gravitational-wave detection in Ref. [25]; (b) the exponential attitude controller for heliocentric gravitational-wave detection in Ref. [26]; and (c) the attitude-orbit cooperative controller for synthetic aperture radar satellite formations in Ref. [48]. The simulations obtained from 3 different cases are averaged and thus displayed in Table 5 and Fig. 4 and Fig. 5. Results in Table 5 reflects the formation tracking level of SCi at t = 1000 s. In Fig. 4, CPI₁ and CPI₂ characterize the formation tracking level and synchronization among individuals, respectively. The rapid convergence of CPI₁ and CPI₂ shows better transient control performance in formation tracking. The energy cost index (ECI) is gathered in Fig. 5. On the whole, our method can track the desired trajectories with less energy cost but better control performance.