Minimum-time control for the test mass release phase of drag-free spacecraft

The gravitational wave is a prediction of the general theory of relativity that can be detected by measuring the relative distance between two test masses (TM) along the geodesics. The TM is fixed by the cage and vent mechanism to avoid collision with the spacecraft cavity during launch. Once the drag-free spacecraft reaches its target orbit, the TM must be released by the grabbing positioning and release mechanism (GPRM) to the cage center of the inertial sensor in a limited amount of time. The capture of the TM by means of the electrostatic suspensions is crucial for the mission. However, the GPRM inevitably generates large release errors and the low electrostatic force makes the TM capture control a difficult task. Moreover, the experimental results of the LISA Pathfinder mission have shown that the TM release velocities were well higher than the ones expected. The previous robust sliding mode controller needs to be optimized. In a research article recently published in Space: Science & Technology, scholars from Tohoku University, Sun Yat-sen University, and DFH Satellite Co. Ltd together propose a minimum-time capture control method for the TM release phase of drag-free spacecraft.

First, the relative dynamics of the TM with drag-free spacecraft, the electrostatic force model and the disturbing force model are described. The relative dynamics of TM with respect to the drag-free spacecraft is establish in the cage reference frame (CRF; O_CX_CY_CZ_C) (as shown in Fig. 1) which centered at the cage center of the inertial sensor, the x axis is parallel to the telescope symmetry axis, the z axis is parallel to that of the LVLH frame, and the y axis also completes the right-hand triad. The dynamics equation for the TM relative motion in the cage frame is based on TH equation. For simplifying the dynamics analysis, it is assumed that the reference orbit of the drag-free spacecraft is circular and total external disturbances d = 0. The Clohessy–Wiltshire (CW) equation is obtained. The inertial sensor with a total of 18 electrodes is generally exploited for the capture control during the TM release phase as shown in Fig. 2, where 12 electrodes are used to generate electrostatic force and the remaining 6 electrodes are used to carry out electrostatic sensing. The control acceleration u by electrostatic force is established on the basis of the generic force on the TM and the TM potential V_TM = 0 assumption. Furthermore, the disturbance of electrostatic coupling force is mainly considered in this paper. The system random noise, voltage noise, and capacitance displacement measurement noise are also considered.

Then, an analytical solution for the optimal-planned trajectory from the initial release point to the center of the inertial sensor is derived and the closed-loop controller is designed to track the optimal release trajectory. To find the analytical solution of the optimal release trajectory, the CW equation is written as the first-order form. The performance index is the terminal time of the TM release, i.e. J = t_f. The Hamiltonian function is derived and analyzed. By analyzing signs of costate variable of the Hamiltonian function, the time-optimal control for the TM capture during release phase can be divided into six segments with constant control. Based on the minimum-time constraint equation and the terminal condition, terminal time t_f can be solved. On the basis of approximate equations sinωt ≈ ωt and cosωt ≈ 1, the TM’s velocity and the corresponding position can be yielded as in Eq. 24 and Eq. 25. Furthermore, considering the disturbing force that acts on the TM, a closed-coop controller for tracking the desired release trajectory should be designed. The TH equation is written as d²r_TM/dt² = f_d. For every segment of the release phase, the analytical solution is obtained. The desired release trajectory is denoted by r_d(t). The tracking error is Δr = r_TM − r_d. A second-order derivative is applied on Δr, and a simple feedback control is employed as f_d = K_pΔr + K_ddΔr/dt + d²r_d/dt². Then, we have d²r/dt² + K_pΔr + K_ddΔr/dt = 0. The real control low V_sj can be solved.

Finally, two numerical simulations are carried out to demonstrate the proposed minimum-time capture control method for the TM release. In simulation, the TM’s mass is 2.45 kg. The position vector of the cage center with respect to the spacecraft is r_c = [0.3/2*√3 m, 0.3/2 m, 0]^T. The gap between the TM and the electrode face is d = 5 mm. The effective area of the electrode is A = 8.1 cm². The maximum controllable voltage is 96 V.

(1) First, the optimal capture control results in two scenarios (Earth-centered and Sun-centered) are compared with that by an hp-adaptive pseudospectral method. Figure 5 describes the TM trajectories from the release point to the center of the inertial sensor in the cage frame under Earth-centered and Sun-centered scenarios, where the blue solid and red dashed trajectories are solved by the proposed analytical method and the hp-adaptive pseudospectral method. The results in Fig. 5 show the good consistency of the two methods. Whereas, the optimal capture time by the analytical method is slightly larger than that by the precise numerical method. However, the computation time of the proposed analytical method is much less than that of the numerical method, which can well satisfy the onboard space-borne real-time control requirements. Therefore, the proposed minimum-time capture control method during the TM release has high accuracy and computing efficiency.

(2) Second, the proposed method is compared with the proportional-integral-derivative (PID) controller in terms of performance such as capture time, stability, and maximum overshoot. A comparison between the proposed optimal control and the PID control is implemented with the same initial values of the Earth-centered case and noises. A Monte Carlo simulation (Fig. 11) is performed to analyze the robustness of the proposed analytical method and validate its feasibility in the TM release phase for the drag-free mission. Also, the TM capture time of two control method is compared. Results show that the capture time by the PID control is about 100 s, whereas the capture time by the proposed method is 45.38 s, less than half of the former. Both methods can successfully release and stabilize the TM to the desired position against the high environmental disturbance and system noises, but the position deviation converges slowly by PID control, with a TM capture time generally needed beyond 100 s to reach the required accuracy. These results give a more detailed view of the goodness and robustness of the control performance of the proposed method.