## 1 Introduction  
Traditional mechanisms use articulated joints to connect their rigid body elements and to achieve their motion, while compliant mechanisms obtain their motion through the deformation of the compliant joints [1, 2]. Compliant mechanisms have many advantages over traditional mechanisms, such as no backlash or friction, higher precision, and reliability, reduced number of components, and assembly time [3, 4]. Lamina emergent mechanisms (LEMs) are a kind of compliant mechanisms that are manufactured from sheet goods with motion out of the fabrication plane [5]. Researchers have designed many joints for LEMs. Finding suitable compliant joints [6] can be a key step in designing LEMs. [PERSON] et al. [7] first proposed the design concept of utilizing torsional deflection to obtain out-of-plane motions from planar structures, leading to a new group of compliant joints for LEMs called lamina emergent torsional (LET) joints. The LET joints and their equivalent stiffness models were presented in Ref. [8]. Although LET joints have been used in many LEMs [9, 10] due to their planar configurations and large displacement ranges, they yield considerable parasitic motions when subject to tensile or compression loads. In order to solve this problem, three joints were introduced in [11]: Inverted Lamina Emergent joint (I-LEJ), Tension Lamina Emergent joint (T-LEJ), and Inverted tension Lamina Emergent joint (IT-LEJ). Theydemonstrate higher stiffness in tensile or compression loads conditions, at the cost of limited bending flexibility compared to the LET joint and requiring more material area. [PERSON] et al. presented compound joints [12] and two lattice flexure types, called X-type and V-type [13], which can reduce bending stiffness and be validated through physical prototypes. [PERSON]. [14] proposed the shape factor of compliant arrays (CA) combined with the material and geometrical properties. Moreover, the shape factor is applied to the analysis of two special CA (Unidirectional one-way flexible array and two-way flexible array), which effectively showed the balance between elastic bending and strength. [PERSON] et al. [15] designed a Triple-LET flexure joint to achieve larger deformation, and an application on the slider-crank mechanism was finished. [PERSON]. [16] presented an Outside-Deployed Lamina Emergent Joint OD-LEJ (OD-LEJ) with a smaller equivalent spring constant. OD-LEJ can achieve nearly 170.17\({}^{\circ}\) of rotation without plastic deformation, and it can do better than LET joint in large angular displacement. In terms of characteristic parameters, [PERSON]. [17] designed a double C-type flexure hinge and proposed a method to remove some materials of the flexure hinges in certain rules, which ensured anti-tensile property and improved bending property. The performance of the double C-type flexure hinges was compared through the finite element simulation analysis. [PERSON] et al. [18] introduced a type of LET joint called membrane-enhanced LET joints (M-LET joints). M-LET can maintain large angular displacements and mitigate parasitic motions and low buckling resistance. Several different mechanisms are proposed to describe the M-LET joint better. Recently, some LET joints with Spatial motion have also been developed that provide more possibilities for applying LEMs [19; 20; 21]. With the development of technology, some new analysis methods of compliant mechanisms with Spatial motion have gradually emerged, such as the spatial Beam constraint model (BCM) [22], the Spatial Chained beam-constraint-model (CSBCM) [23], or the Chained Power Series Model [24], etc. In terms of application, [PERSON] et al. [25] used rigid-body replacement to synthesize a LEM Audi A4 cup holder. [PERSON] et al. [26] introduced spherical LEMs and discussed the fundamentals of spherical LEMs. The paper showed different spherical 4R LEMs, and classification was made on spherical 4R LEMs. [PERSON] et al.