Configuration Synthesis and Lightweight Networking of Deployable Mechanism Based on a Novel Pyramid Module

As one of the cutting-edge technologies in the international aerospace field, space-deployable structures have become an advanced research topic in space science and technology. Over the years, the critical technologies of space deployable structures have made breakthroughs in deployable mechanisms, cable membrane form finding, dynamic analysis, reliable environmental adaptability analysis, and verification [1]. With the development of communication, remote sensing, and navigation satellites, the research on deployable structure technology with large space antennas as the primary research object has attracted much attention. Researchers have paid significant interest to signal harvesters’ equipment [2, 3]. Space deployable antenna mechanism is the critical equipment to support large-scale space antenna. With the characteristics of occupying a small space with a folding pose in the storage and launch stage and then gradually expanding when entering orbit, the antenna has high gain and high efficiency. Its development trend is large-scale, lightweight, and high deployable rate [4‐7]. As an essential type of space deployable antenna, the truss deployable antenna mechanism has been successfully applied in the aerospace field with the advantages of good structural stability and high deployable rate. Typical applications in China are the Beidou-3 satellite [8] and the HJ-1C satellite [9].

The deployable characteristic of the deployable mechanism is the premise of realizing a large unfolding rate. High enough stiffness and stability are the guarantees of supporting the antenna reflector, and the effective combination of large aperture and lightweight is the goal of mechanism design. The deployable mechanism is different from the general series/parallel mechanisms. It is a spatial multi-closed-loop mechanism composed of multiple basic units. Therefore, the basic unit configuration and deployable method of the deployable mechanism that meets the above characteristics still need to be continuously explored. For example, Phocas et al. [10] studied a reconfigurable and deployable linkage mechanism applied to the construction field. Sun et al. [11] studied a ring mesh deployable antenna mechanism based on H-shaped deployable units. Chen et al. [12] demonstrated the deployable mechanism design and networking method based on the classical linkage Bennett. Hofmann et al. [13] conducted design research on the positioner of the deployable spacecraft mechanism. In addition, many scholars have also focused on the design and research of spatial deployable units. Meng et al. [14] proposed a new basic unit with good composability and deployability, and thus constructed a double-layer ring deployable antenna mechanism and a variety of truss array deployable mechanisms. Yang et al. [15] studied a new type of deployable mechanism loaded with composite belt springs, which promoted the application of this type of mechanism in aerospace. Chen et al. [16] proposed a high-stiffness modular planar deployable antenna mechanism. Guo et al. [17] designed a deployable support mechanism for a space planar antenna. Lyu et al. [18] developed an isosceles trapezoidal prism deployable unit, and proposed a construction method using this unit to approximate a smooth plane curve or cylinder. Further, Wang et al. [19] proposed a design method of surface deployable mechanism, which takes the idea that the crease circumference is distributed around the central hub of the regular polygon as the design idea. Takamatsu et al. [20] proposed the concept configuration and construction mode of a deployable truss for a large antenna. Tian et al. [21] innovatively designed a networking method for a rib module and its large-size modular deployable antenna mechanism. Wang et al. [22] proposed a joint transformation configuration synthesis method, and innovatively designed a petal-type spatial deployable mechanism. Huang et al. [23, 24] developed a petal-type deployable mechanism and its improved mechanism. Both mechanisms have good packaging ratios and deployable performance, but the improved mechanism has a simple structure and good mechanical properties. What’s more, Wu et al. [25] studied the nonlinear response of a flexible deployable linkage using the finite particle method. Zhang et al. [26] proposed a new type of hybrid antenna-supporting mechanism and analyzed its motion characteristics. In terms of the research on truss mechanisms, Xu et al. [27] proposed a space multi-loop mechanism configuration synthesis method with constraint chains, synthesized a variety of tetrahedral deployable units suitable for parabolic truss deployable antenna mechanism, and designed a large aperture antenna networking mode. Based on this, Guo et al. [28‐30] proposed a modular tetrahedral truss deployable antenna mechanism and studied its mechanical properties, structural design, and comprehensive performance. The above literature offers various deployable unit mechanisms with excellent performance, and some deployable antenna mechanisms can be networked and applied to aerospace, but there are problems such as insufficient deployable rate and unsatisfactory scalability. Consequently, exploring more new types of deployable mechanisms and networking methods has intense academic research significance.

To obtain a high-performance deployable antenna mechanism, the optimization design of a deployable antenna mechanism is the effective way, especially the optimization design aiming at structural lightweight. Kim et al. [31] proposed a lightweight reconfigurable deformable origami block mechanism, which can be used as a driver to realize fast, reversible, and stable movement. Ze et al. [32] proposed a reversible wireless deployable microrobot. Sofla et al. [33] studied a single degree of freedom(DOF) articulated tetrahedral truss mechanism, and optimized a deformable articulated truss structure through two different tetrahedral arrangement modes. Dai et al. [34] used a genetic algorithm and gradient-based optimizer to optimize the overall structure and component size of a new double-ring deployable antenna mechanism. Further, Dai et al. [35] optimized the link size of the double-ring deployable antenna mechanism under the target of high stiffness and lightweight based on the constraints of dynamic characteristics. Wang et al. [36] studied a spatial cylindrical deployable structure with a large unfolding ratio and few DOFs inspired by Kirigami. They proposed an optimal design method of cylindrical deployable structure parameters that is most suitable for the target surface. Xu et al. [37] proposed a configuration optimization method of a parallel mechanism with few kinematic joints, two rotations, and one shift, based on the limit constraint screw, and constructed a five-axis hybrid robot with the most miniature kinematic joints. Mei et al. [38] applied the lightweight design concept to the parallel mechanism and proposed a lightweight and highly flexible five-axis mobile processing robot. The above literature has proposed various mechanism optimization design methods to achieve mechanism lightweight. Still, there are few reports on the effective lightweight design methods of truss deployable antenna mechanisms.

Deployable mechanisms are mostly complex multi-closed-loop mechanisms, which are problematic in configuration synthesis, motion analysis, lightweight design, and experimental research. It is significant to explore the spatial multi-loop coupling deployable mechanism with a high deployable rate, strong scalability, and light structure. This paper focuses on the design and analysis of a deployable mechanism based on a pyramid unit, which is extensible and lightweight. In the second section, a new type of pyramid deployable unit mechanism and its configuration synthesis method are proposed. In the third section, the motion characteristics of the pyramid deployable unit are analyzed. In the fourth section, the structure design and motion simulation analysis of the deployable mechanism based on the pyramid unit is carried out, and the principal prototypes are developed to complete the experiment.

Based on the configuration synthesis method with adding constraint chain of spatial closed-loop mechanism, the synthesis process of pyramid deployable unit mechanism includes the determination of the folding principle, the design of constraint chains, the addition of virtual constraint chains, and the construction of mechanisms. The configuration synthesis flow chart is shown in Figure 1.

In the 4RRR-4RR unit, the plane of the four revolute axes connected by the four webs and the top node, the three revolute auxiliary axes in the same synchronizing link are parallel to each other. The revolute axes in all synchronizing links are perpendicular to the connecting lines of the nodes at both ends of the synchronizing link, as shown in Figure 12. Establish a reference coordinate system o-xyz at the center of the top node P, the z-axis is vertical to the end face of the top node P, the x-axis is along the center line of the bottom node A and C and points from A to C, and the y-axis is determined according to the right-hand rule.

The 4RRR-4RR pyramid unit can be decomposed into a three-closed-loop mechanism and a RRR (S₁₈, S₁₉, and S₂₀) kinematic chain, in which the three-closed-loop mechanism is composed of three 7R single closed-loop mechanisms sharing two webs, as shown in Figure 13.

Due to the symmetry of the structure, the RRR kinematic chain is a redundant constraint chain; that, it does not affect the DOF of the pyramid unit. Therefore, the DOF of the 4RRR-4RR pyramid unit is the same as that of the three-closed-loop mechanism. Next, the DOF of the three-closed-loop mechanism is analyzed. The structural diagram of the three-closed-loop mechanism and its topology diagram are shown in Figures 14 and 15, respectively.

The three-closed-loop mechanism can be regarded as a parallel mechanism, and the fixed platform (node B) and the moving platform (node A) are connected by two constraint chains. One is the RRR constraint chain (S₂, S₁, and S₇), and the other is RR (S₄, S₃), including a two-closed-loop mechanism and a series constraint chain. The two-closed-loop mechanism can also be regarded as a parallel mechanism. The fixed platform (node B) and the moving platform (node P) are connected by two constraint chains; one is the RR constraint chain (S₅, S₆), and the other is RRR (S₈, S₉, and S₁₀), including a single closed-loop mechanism (S₁₁, S₁₂, S₁₇, S₁₆, S₁₅, S₁₄, S₁₃) and a series constraint chain. The single closed-loop mechanism can also be regarded as a parallel mechanism, and the fixed platform (node C) and the moving platform (node P) are connected by two constraint chains. One is the RR constraint chain (S₁₁, S₁₂), and the other is the RRRRR constraint chain (S₁₃, S₁₄, S₁₅, S₁₆, S₁₇). Firstly, the DOF of the single closed-loop mechanism is analyzed, and a reference coordinate system o_C-x_Cy_Cz_C is established at the center of node C. The x_C-axis is along the DC line, the z_C-axis is perpendicular to the plane of ABCD, and the y_C-axis is determined by the right-hand rule. The kinematic screws of the constraint chains RR and RRRRR in the reference coordinate system are expressed as:where \(\left( {\begin{array}{*{20}c} {a_{i} } & {b_{i} } & {c_{i} } \\ \end{array} } \right)\) refers to the direction vector of S_i, and \(\left( {\begin{array}{*{20}c} {x_{i} } & {y_{i} } & {z_{i} } \\ \end{array} } \right)\) refers to the position vector of the center of S_i in the reference coordinate system.

Calculate the reciprocal screw of Eqs. (4) and (5), respectively, and the constraint screw applied to P can be expressed as:where \(\user2{\$ }_{{{\text{r}}1}}^{P}\) represents the constraint couple perpendicular to the revolute joint in the constraint chain (S₁₃, S₁₄, S₁₅, S₁₆, S₁₇); \(\user2{\$ }_{{{\text{r}}2}}^{P}\) represents the constraint couple along the z-axis; \(\user2{\$ }_{{{\text{r}}3}}^{P}\) represents the constraint couple parallel to the xy plane; \(\user2{\$ }_{{{\text{r}}4}}^{P}\) represents the constraint force parallel to the axis of the revolute joint at both ends of the link E₃ through the origin of the coordinate system, and \(\user2{\$ }_{{{\text{r}}5}}^{P}\) represents the constraint force along the axis of the link E₃ through the origin of the coordinate system.

Taking the union of Eqs. (6) and (7) and solving its reciprocal screw, the kinematic screw of P relative to C can be expressed as:

Therefore, the single closed-loop mechanism (S₁₁, S₁₂, S₁₇, S₁₆, S₁₅, S₁₄, S₁₃) can be replaced by the generalized translational joint represented by \(\user2{\$ }_{{{\text{m}}P}}^{C}\). In the closed-loop mechanism (S₅, S₆, S₈, S₉, S₁₀, \(\user2{\$ }_{{{\text{m}}P}}^{C}\)), node B is the fixed platform, node P is the moving platform, and a reference coordinate system is established at the center of the fixed platform. The kinematic screws of the constraint chains RR (S₅, S₆) and RRRP (S₈, S₉, S₁₀, \(\user2{\$ }_{{{\text{m}}P}}^{C}\)) are represented in the reference coordinate system, respectively:

Calculate the reciprocal screw of Eqs. (9) and (10) respectively, and the constraint screw applied to P can be expressed as:

Taking the union of Eqs. (11) and (12) and solving its reciprocal screw, the kinematic screw of P relative to B can be expressed as:

Therefore, the single closed-loop mechanism (S₅, S₆, S₈, S₉, S₁₀, \(\user2{\$ }_{{{\text{m}}P}}^{C}\)) can be replaced by the generalized translational joint represented by \(\user2{\$ }_{{{\text{m}}P}}^{C}\). In the closed-loop mechanism (S₄, S₃, S₂, S₁, S₇, \(\user2{\$ }_{{{\text{m}}P}}^{C}\)), with node B as the fixed platform, node A as the moving platform, the kinematic screw of constraint chains RRR (S₂, S₁, S₇) and RRP (S₃, S₄, \(\user2{\$ }_{{{\text{m}}P}}^{C}\)) are represented respectively in the reference coordinate system:

According to Eqs. (14) and (15), the constraint screw applied to node A can be obtained as:where \(A = x_{4} b_{3} - y_{4} a_{3} - x_{3} b_{3} + y_{3} a_{3} - {{b_{3} z_{4} C} \mathord{\left/ {\vphantom {{b_{3} z_{4} C} {b_{5} }}} \right. \kern-0pt} {b_{5} }}\),\(B = C - \left( {b_{5} a_{5} \left( {x_{4} b_{3} - y_{4} a_{3} - x_{3} b_{3} + y_{3} a_{3} } \right) - a_{5} b_{3} z_{4} C} \right){{} \mathord{\left/ {\vphantom {{} {\left( {a_{3} z_{4} b_{5} - a_{5} b_{3} z_{4} } \right)}}} \right. \kern-0pt} {\left( {a_{3} z_{4} b_{5} - a_{5} b_{3} z_{4} } \right)}}\), \(C = {{\left( {x_{5} b_{5} - a_{5} y_{5} } \right)} \mathord{\left/ {\vphantom {{\left( {x_{5} b_{5} - a_{5} y_{5} } \right)} {z_{5} }}} \right. \kern-0pt} {z_{5} }}\).

Taking the union of Eqs. (16) and (17) and solving its reciprocal screw, the kinematic screw of A relative to B can be expressed as:

This screw represents movement along the x_C-axis. Similarly, it can be concluded that D also has a DOF of movement along the DC relative to C. In the single closed-loop mechanism (S₅, S₆, S₈, S₉, S₁₀, \(\user2{\$ }_{{{\text{m}}P}}^{C}\)), taking B as the fixed platform and C as the moving platform, using the screw theory, it can be found that C has only one DOF of movement along the BC relative to B. Therefore, considering the removed RRR (S₁₈, S₁₉, S₂₀), the pyramid mechanism can be equivalent to the mechanism shown in Figure 16. The DOF of the mechanism is one.

To sum up, the 4RRR-4RR pyramid unit mechanism is a single DOF overconstrained mechanism, and only one actuation is needed to realize the synchronous folding movement of the four nodes on the bottom with the same pose.