# Design and Kinematic Analysis of Deployable Antenna for Bionic Bird Tail Feather
## 1 Introduction  
Space antennas find their primary application in satellites, spacecraft, and other similar vehicles, serving as the essential link for two-way communication between these vehicles and ground stations. Functioning as the \"ears\" or \"loudspeakers\" of these spacecraft [1], space antennas have evolved into indispensable components that significantly impact and shape the performance of entire wireless communication systems and, in many cases, the spacecraft as a whole. The deployable structure of space antennas is a critical component that facilitates antenna deployment and ensures the precision and stability of the antenna surface. The deployable mechanism occupies a substantial portion of the antenna's overall mass, serving as the structural backbone that directly influences the antenna's stiffness and natural frequency. Various deployable antenna mechanisms have been developed, including the ring truss type, unit frame type, folding rib type, winding rib type, cable tension type, ring column type, and flexible self-rebound type, among others [2; 3; 4; 5; 6; 7; 8; 9; 10; 11]. However, the current deployable antenna unfolding methods are mostly fixed and primarily involve circumferential unfolding. Therefore, there is a need for innovative unfolding methods that can expand upon the existing configurations, drawing inspiration from nature's own mechanisms. In the natural world, numerous organisms exhibit fascinating unfolding behaviors that are crucial for their survival and functionality. [PERSON] [12], for instance, meticulously observed the metamorphic unfolding of wing veins in various insect species, which can expand up to forty times larger than their original size. This phenomenon, known as spreading, is achieved by an influx of body fluids into the wing veins. Plants also demonstrate unfolding mechanisms, such as the unrolling of leaves in species like hornbeam andbeech [13], as shown in Figure 1, among other examples. The earwig's wing exhibits the highest \"folding ratio\" in the animal kingdom, expanding its body surface area tenfold, with a central wing joint structure that allows stable folding and unfolding without the need for muscle power. Researchers at ETH Zurich, Switzerland, and Purdue University, USA, have even used 3D printing to create a foldable bionic wing structure, as shown in Figure 2, which closely resembling an earwig's wing and is capable of remaining stable when unfolded and folded with a simple touch [14]. Furthermore, [PERSON] and [PERSON] [15] designed a foldable micro-flapping wing inspired by the hindwing folding action of the rhinoceros beetle during resting and hindwing unfolding during flight, leveraging the beetle's biological characteristics for passive fluttering. In 1994, [PERSON] et al. investigated the mechanical properties of the skin of bat wings from different wing structures, studying the unfolding of bat wings [16], as shown in Figure 3.  
Another striking example of a deployable structure in nature is the flycatcher [17], which, due to challenging environmental conditions, closes its leaf-like structure to capture insects for nutrients. This process takes around 40 days, and the curvature of the leaf reverses before and after closing to form a typical bistable structure, as illustrated in Figure 4. A similar mechanism can also be observed in the vocal organs of cicadas. The number of petals and the curvature factor of the curve have a direct impact on the size of the folded cross-section of film reflectors, as demonstrated in Figure 5[18]. Petal folding also has broader applications, such as in petal satellite antennas, where the transmitting surface consists of sails and steel plates on a rotating substrate, offering advantages like compact storage, lightweight design,  
Figure 1: A typical hornbeam leaf, showing the corrugations. Figure 3: Diagram of ventral view of a bat wing. Figure 2: Foldable bionic wing. In addition, the deployment of space antennas faces problems related to hydrodynamics due to turbulence in the space airflow.