A fundamental tenet of mechanism design is to minimize the number of actuators required to perform a given operational task. Therefore, the challenge lies in designing such mechanisms to achieve the prescribed motion output with fewer actuators [
1]. Generally, under-actuated mechanisms are more cost-effective than general multimobility mechanisms with the same operational requirements. As a type of mechanism that can operate in an under-actuated state, metamorphic mechanisms were proposed more than two decades ago [
2] and attracted significant interest. Li et al. [
3] presented a method for the structural synthesis of metamorphic mechanisms based on configuration transformations. Liu et al. [
4] determined methods for self-locking by applying geometric constraints to kinematic joints. Focusing on a class of improved link-type shape-shifting modular robot, Liu et al. [
5] studied a transformation technique whose feasibility was verified by experimental results. Wang et al. [
6] proposed a method based on matrix calculations to achieve structural synthesis for the source mechanisms of metamorphic mechanisms. Dai and Wang presented an approach for synthesizing a metamorphic palm and characterized the grasp ability of a novel metamorphic robotic hand [
7]. Lan and Du [
8] introduced a new adjacency matrix that can be used to trace the structural changes, find all the possible configurations, and calculate the degrees of freedom (DOFs) of a metamorphic mechanism. Ding [
9] studied topological and metamorphic principles with the same parts and symmetrical structure. Zhang and Ding et al. [
10] proposed a comprehensive symbolic matrix for characterizing the topology of a metamorphic mechanism involving variations of links and axial orientations of the kinematic joints. Dai et al. [
11] introduced an EU-elementary matrix operation to produce the configuration transformation for modeling topological changes of metamorphic mechanisms. Li et al. [
12] presented an augmented adjacency matrix to describe the topological configuration of metamorphic mechanisms. Zhang et al. [
13] studied the synthesis and configuration design of metamorphic mechanisms based on biological modeling and genetic evolution and analyzed constrained metamorphic techniques and geometric constraint principles [
14]. Lan [
15] studied the structural components and evolution of planar metamorphic mechanisms. Zhang et al. [
16] researched a metamorphic kinematic chair with the capacity to change mobility, with the DOFs changing from 1 to 3. Yang and Deng et al. [
17] introduced a new matrix description with respect to increasing or unchanging links in configuration transformation, and then presented a new method for the structural synthesis of metamorphic mechanisms. Gan et al. [
18,
19] invented a reconfigurable Hooke joint, the rT joint, the direction of whose rotation axis can be altered. They then proposed two types of metamorphic parallel mechanisms assembled with these rT joints. Zhang et al. [
20] presented a synthesis method for metamorphic mechanisms based on constraint variation. Valsamos, Moulianitis, and Aspragathos presented a method for the kinematic synthesis of structure topologies for modular metamorphic serial manipulators [
21]. Yang et al. [
22] proposed a practical synthesis method for metamorphic mechanisms and studied its application to broken strand repair operations on extra-high-voltage (EHV) power transmission lines. Kong et al. [
23,
24] presented structural syntheses of a class of variable-DOF single-loop mechanisms and 3-4R two-DOF parallel mechanisms. Li et al. [
25‐
27] introduced the augmented Assur group used in the structural theory of metamorphic mechanisms, which is similar to the Assur groups in general planar mechanisms. Li and Dai [
28] presented an equivalent resistance gradient model for metamorphic mechanisms. From this model, the connections between the equivalent resistance, constraint characteristics, and architectures of metamorphic joints were established. Tian et al. [
29] presented a configuration synthesis method for metamorphic mechanisms based on functional analyses. Based on a type of lockable spherical joint, Guo et al. [
30] proposed a new 3RRlS metamorphic parallel mechanism and analyzed its four configurations. Wei and Dai [
31] presented a novel approach for synthesizing metamorphic parallel mechanisms with the ability to reconfigure motion between 1R2T and 2R1T, and then the Lie group was introduced to study type synthesis using the transformation configuration space for reconfigurable parallel mechanisms [
32]. Kong, Gosselin, and Angeles et al. [
33‐
35] studied the type synthesis of parallel mechanisms, which provided a meaningful reference for the synthesis study of metamorphic mechanisms. Li et al. [
36,
37] studied the structural synthesis of compliant metamorphic mechanisms based on adjacency matrix operations, which can also be used in the design of new compliant metamorphic mechanisms.
At present, studies on the structural synthesis of metamorphic mechanisms are more focused on metamorphic techniques and the structural synthesis of source mechanisms for metamorphic mechanisms. An effective method to obtain the expected working stages of metamorphic mechanisms is to design different constraint forms and structures of metamorphic joints to provide the required geometric and force constraints. Several questions remain unanswered for a given source mechanism and the constraint force demands of the metamorphic working stages. For example, which constraint forms and structures of metamorphic joints should be selected? How many architectures of metamorphic mechanisms correspond to the different metamorphic joints? Based on the equivalent resistance gradient model of metamorphic mechanisms proposed in Ref. [
28], this study investigates the relationship between kinematic status, resistant forces, and constraint forms and structures of metamorphic joints. A practical structural synthesis method is then presented to determine the different constraint forms and structures of constrained metamorphic joints. Finally, the combinations are assembled to obtain structures for a multi-DOF planar metamorphic mechanism with a single actuator to satisfy the functional and operational requirements for a practical working task. The remainder of this paper is organized as follows: Section
2 introduces the kinematic and constraint status matrix of metamorphic joints based on the metamorphic cyclogram of the metamorphic mechanism. Section
3 reviews the equivalent resistance gradient model originally presented in Ref. [
28] and an equivalent resistance matrix is presented. In Section
4, several typical constrained structures for metamorphic joints are given, and a structural synthesis matrix of the metamorphic mechanism is proposed. A kinematic diagram synthesis of source metamorphic mechanisms based on linkage groups is studied, and a systematic structural design approach of metamorphic mechanisms considering the forms and structures of metamorphic joints is presented in Section
5. Finally, Section
6 provides two examples to verify the feasibility and practicality of the proposed design approach. Finally, a discussion is presented in Section
7 and Section
8 concludes the paper.