## 1 Introduction  
In the aerospace sector, there is an increasing demand for a deployment mechanism that can be used in areas such as satellite communications, space science and deep space exploration [1; 2; 3; 4]. The spiral tube and actuator for controlled extension and retraction (STACER) mechanism is widely used in the aerospace industry due to its advantages, such as a large exhibition ratio, strong thermal symmetry along the ring, and a simply equipped structure. Over 650 STACER units have been used in the past 30 years [5; 6; 7; 8]. As shown in Figure 1, it is a layered elastic circular tube structure which is spirally deformed with a thin metal strip at a fixed inclination angle. In the deploying process through its own elastic potential energy, the STACER achieves a rigid support character due to the tight contact of the strip layers [9]. A typical deployment mechanism with a large exhibition ratio includes STACER, a tape spring and the storable tubular extendible member (STEM) [10; 11; 12; 13]. The current research on such a mechanism mainly focuses on the optimization of configuration parameters, the analysis of mechanical characteristics during the deploying and gathering processes, and the influence of environmental factors on its stability [14; 15; 16; 17]. According to the configuration characteristics of the tape spring, three kinds of gathering forms are proposed. [PERSON] [18] studied the deploying process of one-dimensional and two-dimensional gathering forms and found that the tape spring would form an elastic gathering region with zero transverse curvature and constant longitudinal curvature. According to this morphological feature, the relationship between the bending moment and the rotation angle of the bending region is obtained. [PERSON] [19] studied the influences of the shape parameters, such as the curvature, thickness, and wrap angle of the tape spring, on the peak bending moment, steady bending moment, and strain during the bending process in order to optimize these parameters of the tape spring according to the influence tendency. [PERSON] and [PERSON] [20] pointed out that the standard ratio of diameter to wall thickness when designing a STEM is limited by the yield strain of the material. The properties of the STEM cross-section (centroid, shear center, moment of inertia, etc.) are affected by the envelope angle of the section. During the configuration of an optimization design, the mechanical properties of different configurations during the deployment process are quite different. The differences may determine whether the deployment mechanism can be deployed successfully. [PERSON] [21] established a nonlinear explicit dynamics finite element model for studying the process of STEM expansion. The effects of different curling methods and curl tightness on the unfolding process were studied. The positive curl and tight curl were beneficial to the deployment process of STEM. [PERSON] [22; 23] found that the displacement of the tape spring and the thrust curve presents significant hysteresis characteristics after the coil spring is gathered in a winding manner through both experimental and simulation methods. It also determines whether the entire mechanism can be successfully deployed. The impact arm designed by [PERSON] [5] based on STACER mechanism proves that the thrust force is constant during the deployment of STACER. It indicates that the STACER mechanism is essentially a constant force spring. It is also one of the criteria for judging whether a STACER mechanism is qualified for employment. [PERSON] [9] proposed to establish a multi-body flexible dynamics simulation analysis model of STACER, and fully studied the deformation process, load state, and energy variation law during its deployment. Through theoretical analysis, a simplified formula for calculating the thrust force was proposed, which provides a theoretical support for the design and application of the deployment mechanism.