Evaluation of Tethered Satellite System Models for Simulating Post-Capture Debris Towing

The increasing amount of space debris poses a significant threat to spacecraft, satellites, and future space missions in Earth’s orbit. This increasing amount of space debris, often composed of defunct satellites or rocket stages, increases the risk of catastrophic collisions, which could trigger a cascading effect known as the Kessler Syndrome. The Kessler Syndrome theorizes that fragments from debris collisions would increase the probability of further collisions to the point of making certain orbits unusable with the formation of a debris belt around Earth. As such, Active Debris Removal (ADR) is needed to prevent the increasing amount of debris from being an overwhelming problem in the near future. Among the proposed ADR methods, tether-net systems are promising in capturing large debris; they require relatively low precision for target capture and can wrap around objects with different geometries and tumbling rates. Tether-net capture scenarios generally involve four phases: rendezvous, capture, detumbling, and removal. The chaser spacecraft approaches the target debris, deploys a net to capture it, reduces its angular velocity to stabilize its motion, and finally removes the debris from orbit, either through atmospheric reentry or relocation to a graveyard orbit. Before deploying the tether-net system in space missions, numerical simulations of its capture dynamics must be performed to evaluate its effectiveness. However, modeling the dynamics of the net when wrapped around the debris involves a large number of degrees of freedom, making high-fidelity net-based simulations very computationally expensive. To reduce the high computational costs, Boonrath et al. [1] performed an optimization study to identify parameters for a tethered satellite system (TSS) with four sub-tethers (rigidly attached to the debris), aiming to approximate high-fidelity debris towing simulations using a full net [1]. The findings of their lower-order model configuration showed satisfactory performance in approximating the dynamics of the high-fidelity system and also significantly reduced the computational cost. This study builds upon the work of Boonrath et al. [1], where the lower-order TSS modeling will be evaluated using (i) new and more attachment point locations of the sub-tethers on the previously-studied debris and (ii) a new target debris geometry. The use of new attachment points (four in total) will demonstrate how changing their locations may affect the overall performance of the lower-order model in matching the dynamics of the high-fidelity model. Additionally, by combining the original four attachment points used in [1] and the four newly proposed in this study (i.e., eight sub-tethers in total), the results will reveal how the use of more sub-tethers will affect the accuracy and validity of the lower-order model. TSS models with four and eight sub-tethers will be employed to simulate the towing of the second stage of the Zenit-2 rocket (previously considered by Boonrath et al. [1]) and the Apollo command and service module (which is representative in terms of shape, mass, and size to many space debris currently in orbit). By using two objects with different geometry, mass, and moment of inertial properties, the dynamic response of the lower-order models can be further evaluated against the high-fidelity model. For all cases, the parameters of the sub-tethers (i.e., length, stiffness, and damping properties) will be optimized to allow the TSS models to best match the high-fidelity simulations. The same optimization cost function used in [1] will be used for this work; the cost function aims to minimize the difference in the relative position of the target with respect to the chaser, the debris angular velocity, and the chaser angular velocity between the lower-order and high-fidelity models. The optimization will be performed using the particle swarm optimization (PSO) algorithm, which is available in MATLAB through the particleswarm function. The results from the lower-order models will be validated both qualitatively and quantitatively against the high-fidelity model. For qualitative analysis, the pose of the systems will be visualized at select instances throughout the simulation and can be directly compared. For quantitative analysis, the dynamic quantities and the absolute error between them will be plotted throughout the duration of the simulation. The corresponding plots will compare the tension of the main-tether, the positions of the chaser and debris, and the angular velocities of the chaser and debris. Together, the qualitative and quantitative results will assess how well the lower-order model resembles the dynamics of the high-fidelity model. Additionally, the computational time required to complete the simulations for both models will be compared to highlight the computational cost efficiency of the lower-ordered model. A comparison will be made between the use of four sub-tethers and eight sub-tethers to show how increasing the number of sub-tethers may better capture the complex dynamic interactions between the debris target and net in the high-fidelity model. [1] Boonrath, A., Singh, T., and Botta, E. M., “Identification of parameters for tethered satellite system to emulate net-captured debris towing,” Acta Astronautica, Vol. 225, 2024, pp. 676–688.