## 1 Introduction  
In the wake of rapid development of information technology (IT), artificial intelligence (AI), and unmanned equipment, as well as their common application in the military field, the traditional combat style, in which each platform uses its own sensors and weapon systems to detect, track, and strike targets individually, can no longer meet the needs of digital warfare. As an emerging combat style, cooperative operations can organically link various geographically dispersed sensors, command and control systems, and weapons systems into a cross-platform information network. In this manner, it is possible to connect and share battlefield information, obtain real-time situational awareness of the battlefield, and improve the integrated combat effectiveness. Cooperative combat has received increasing academic attention in recent years. However, current research on cooperative operations mainly focuses on multi-missile cooperative guidance [1; 2; 3], formation control [4; 5; 6], formation design [7; 8; 9], path planning [10; 11; 12], and multitarget assignment [13; 14; 15]. Research on air-ground cooperative operations is currently lacking.  
Unmanned aerial vehicles (UAVs) and unmanned ground vehicles (UGVs) are the two most representative objects in a cluster system. Although UAVs can quickly reconnoiter a wide area, they are limited by endurance and flight altitude, which prevents them from carrying out their given tasks in special environments. For UGVs, they are able to approach targets at a close range and have a high endurance, but are slower and have a limited field of view. The advantages and disadvantages of UAVs and UGVs are summarized in Table 1. For a UAV-UGV cooperative formation, they can make full use of the advantages of spatial distribution, parallel execution of tasks, functional distribution, and fault tolerance, so that they can complement each other to compensate for the shortcomings of a single type of object and effectively improve the outcome of cooperative operations. For example, in the reconnaissance of complex terrain conditions, combat reconnaissance vehicles are often unable to detect effectively owing to obstacle blockages in the surrounding environment, and may even suffer communication losses between multiple vehicles. Introducing multiple UAVs to allow for formation control to achieve companion-flight cooperative reconnaissance can provide a large range of environmental information for the overall reconnaissance decisions of ground vehicles. Additionally, UAVs can be used as a communication relay in the case of ground communication obstacles to realize the complementary advantages and cooperation of the cluster system [16,17].  
Based on this demand for engineering applications, many scholars have conducted research on the air-ground cooperative control between UAVs and UGVs. As an emerging research field, its theoretical knowledge and technical aspects involve the intersection of several disciplines and technical fields, such as embedded systems, aerodynamics, and control theory, which includes path tracking, formation clustering, automatic landing, and other research fields characterized by many research contents and difficulties [18,19]. This highlights that many problems need to be solved and explored in this area. In particular, the heterogeneity of UAVs and UGVs, for example, having different working fields, different physical models, and different technical indicators [20, 21, 22], poses new challenges to cooperative control between UAVs and UGVs [23,24].  
Hence, this study investigates the air-ground cooperative time-varying formation-keeping control problem of a heterogeneous cluster system composed of a UGV and a UAV in the scenario of air-ground cooperative combat, which enables the UAV to track the motion trajectory of the UGV.