With the booming development of satellite applications, the giant constellations of low Earth orbit (LEO) satellites have introduced challenges for the data relay service. The multi-beam satellite not only offers concurrent access to a large number of objects, but can also meet the high data requirements toward specific coverage of the LEO constellation. However, the multi-beam satellite often faces the mismatch problem of spot allocation and data requirements, which can cause an overload traffic jam or a waste of resources. An optimization algorithm on spot beam allocation is necessary to automatically place the spot centers with appropriate beam widths in line with the density of the traffic demands and to realize the uniformity of the beam occupation. Compared with the conventional K-means algorithm, two adjustable parameters α and β are introduced: one for tuning the ratio of two components making up the distance matrix, and the other for setting the obligatory minimum number of objects per beam. In this paper, the whole process of the proposed method is demonstrated, including the establishment of the low-orbit satellite constellation model, the extraction of the distribution features, and the implementation and evaluation of the modified K-means algorithm. The results prove the validity of the proposed algorithm. A larger value of β with a relative smaller value of α tends to obtain the uniformity of beam occupation; the minimum standard deviation of objects per beam is achieved when α is 0.2 and β is 0.8. This demonstrates that the uniformity of objects per beam can be realized by adjusting the parameters of the distance determination matrix and the obligatory minimal number of objects in each beam. The impact of parameter range on the results is also analyzed.

For the rigid impact and flexible impact in space operation tasks, impact dynamic models between two objects are established in this paper, laying the model foundation for controlling or suppressing the impact. For the capture task between a grapple shaft and a rigid body, the impact dynamic model is established based on the Zhiying–Qishao model. Moreover, by introducing a friction factor into the original impact model, an improved dynamic model between two rigid bodies is proposed. For the capture task with flexible impact, an impact dynamic model between the grapple shaft and a flexible wire rope is established based on the dynamic model of the flexible wire rope. The ground experiments and simulations are carried out with two objects on an air flow table. The experiment results validate the impact dynamic model proposed in this paper.