In this paper, a new synthesis method of 2R3T (R denotes rotation and T denotes translation) overconstrained and non-overconstrained parallel mechanisms (PMs) with three branched chains based on the displacement sub-manifold method is presented. Firstly, the displacement sub-manifolds of mechanisms were determined based on 2R3T motions. Subsequently, the displacement sub-manifolds of the branched chains were derived using the displacement sub-manifold theory, and their corresponding motion diagrams were provided. Additionally, a comprehensive analysis of non-overconstrained 2R3T PMs with a single-constraint branched chain was conducted, and the type synthesis of overconstrained 2R3T PMs with two or three identical constraints was also performed, accompanied by the presentation of partial mechanism diagrams. Finally, the number of DOF (degrees of freedom) of the mechanism was calculated using the modified Kutzbach–Grübler equation for a new PMs,and the screw theory was used to verify the kinematic characteristics, proving this new method's correctness.

The parallel mechanisms (PMs) have the advantages of high rigidity, high bearing capacity, and small cumulative error and are widely used in aerospace, machining, medical rehabilitation, bionics, and other fields (Li et al., 2021; Zhou et al., 2022; L. Wang et al., 2022; Niu et al., 2023). According to the constraints of branched chains in relation to the moving platform, PMs can be divided into overconstrained and non-overconstrained PMs. The overconstrained PMs have two or more identical constraints, such as redundant constraints that can improve the stiffness of the mechanism, which has attracted extensive research by scholars on overconstrained PMs in recent years. Chen et al. (2021) proposed a new overconstrained PM without parasitic motion, clarified the reasons for the non-parasitic motion characteristics of the mechanism, and studied its kinematics and dynamics. Ibrahim et al. (2015) designed a 4-DOF, endoscopic, dexterous PM for minimally invasive surgery, deduced the mechanism's forward and inverse kinematic solutions, and studied the singularity using the screw theory (Sun et al., 2018). As an example, a 2-DOF overconstrained PM was applied to an assembly line, and the kinematic calibration problem of an overconstrained PM was studied. Du et al. (2022) established the geometric error model of the 2UPR–RPU (where U, P, R, C, and S represent universal, prismatic, revolute, cylindrical, and spherical joints, respectively) overconstrained PM and performed a sensitivity analysis on the error source.

Although the overconstrained PMs have excellent stiffness and load-carrying capacity, they are often required to meet special geometric conditions and are very sensitive to part-manufacturing and assembly errors. If these conditions cannot be met, the mechanism will not only be challenged to ensure motion accuracy but will also be unable to maintain the designed motion characteristics. Therefore, it is equally important to study non-overconstrained PMs. Huang et al. (2011) proposed a systematic synthesis method of symmetric non-overconstrained 3-DOF translational PMs using the screw theory. Ye et al. (2022) presented a type synthesis method of 4-DOF non-overconstrained parallel mechanisms (PMs) with symmetrical structures using screw theory. Ye and Hu (2021) proposed a novel 3-DOF RPU+UPU+SPU PM, and its complete kinematics and stiffness are studied. Kuo et al. (2014) proposed a non-overconstrained 3-DOF parallel-positioning mechanism. The inverse and forward kinematic solutions of the mechanism are provided.

Type synthesis is the essential factor that determines the function and performance of mechanical equipment, and it is also the first step in the exploration of the development of new processing equipment and is worthy of in-depth study. Type synthesis seeks the specific topology structure of the mechanism under the constraint of the number and characteristics of the desired DOF, and its core is to describe mechanism motion patterns. Therefore, based on the different description methods, the existing type synthesis methods can be divided into two categories: instantaneous motion-based methods and finite motion-based methods. Among them, the instantaneous motion-based methods include the constrained screw synthesis method (S. Wang et al., 2022; X. Li et al., 2022), the map method (Lu and Ye, 2017), and the differential geometry synthesis method (Meng et al., 2007; Li et al., 2011). These methods can only describe the motion of mechanisms in instantaneous states and cannot express the motion characteristics of mechanisms in continuous motion processes. The finite motion-based methods include the displacement sub-group and/or sub-manifold synthesis method (Hervé, 1999), the GF (generalized function) synthesis method (Zhang et al., 2018), the linear transformation method (Gogu, 2009), the POC (position and orientation characteristic) set method (Jin and Yang, 2004), and the finite screw method (Yang et al., 2016). These methods can describe the continuous motion of mechanisms, avoiding the synthesis result being an instantaneous mechanism, and do not require checks regarding full-cycle mobility.

In 1978, Hervé (1978) introduced Lie group theory into mechanism analysis, laying the theoretical foundation for using Lie group theory to analyze the DOF of mechanisms. In 2004, Li et al. (2004) systematically expounded the method's general theory and process. Based on this theory, Li et al. (2017) studied the equivalent mechanism of 3-DOF RPR branched-chain motion. L. Li et al. (2022) synthesized a 3-DOF parallel mechanical branch with double branches. Wei and Dai (2019) studied a reconfigurable parallel mechanism and the configuration transformation problem. Note that, among the numerous studies on type synthesis, there are relatively few works that are focused on 2R3T PMs. Most of the existing 2R3T PMs are composed of a single constrained branched chain and multiple unconstrained branched chains, and there are many joints in 2R3T PMs, such as the 2UPS+UPU PM (Rong et al., 2018), the 5PSS+UPU PM (Li et al., 2019), and the 4UCU+UCR PM (Luo et al., 2021). The joint is the weak part of the mechanism, which is the main reason for the deformation and clearance of the mechanism; thus, a mechanism with a small number of joints theoretically has better accuracy and stiffness. Therefore, an effective way to reduce the number of joints and to improve the accuracy and stiffness of the mechanism is to replace unconstrained branch chains with constrained branched chains and make the mechanisms become overconstrained PMs. Unfortunately, there are few reports on the configuration of overconstrained 2R3T PMs.

This paper studies various combinations of constrained branched chains and proposes a type synthesis method for non-overconstrained and overconstrained 2R3T PMs. The rest of this article is organized as follows. In Sect. 2, based on the displacement sub-manifold method, the synthesis steps of 2R3T PMs are proposed, as are branched-chain bonds

In 1978, Hervé (1978) enumerated all 12 kinds of displacement sub-groups, as shown in Table 1. It also can be readily proven that the displacement sub-manifold

Displacement sub-groups.

Synthesis flow chart of a 2R3T PM based on the displacement sub-manifold method.

According to the representation method of the displacement sub-manifold, the displacement sub-manifold of rigid-body motion with a 2R3T property is

Motion diagram of

The 3-D displacement sub-group G(

Generators of

From Table 2, we can get the displacement sub-manifolds

Based on the displacement sub-manifold method, the

Motion diagram of

From the above analysis, we can see that there are two types of

For conciseness, this paper defines the UPS, URS, PRPS, and PRRS branched chains as 6

Category 6–6–5

By combining a single constrained branch chain 5

According to the characteristic of the constrained force and/or torque of the branches of the PMs with limited DOF, Ye and Hu (2021) proposed a rule for judging the constrained force and/or torque:

In each leg, the constrained forces should be perpendicular to all P joints and coplanar with all R joints.

The constrained torques should be perpendicular to all R joints in each leg.

Let R

This section selects the unconstrained branched chain 63 and the same constrained branched chains 5

Category 5

Category 5

In Sect. 3.1, four kinds of unconstrained branched chains are listed so, in this section, there are 7

According to the constrained force and/or torque judgment rules (a) and (b), in Fig. 5, there exists a constrained torque

The multi-constraint branched-chain

In Sect. 3.2, the kinematic bond of branched chains 1 and 2 is

Motion diagram of 2-

In Sect. 3.2, it is obtained that the

Category 5

According to the constrained force and/or torque judgment rules (a) and (b), in Fig. 7, a constrained torque

According to Sect. 2, the 5-D displacement sub-group

Category G

Category 6

Combined with the previous content, a total of 868 new types are synthesized in this paper, of which there are 378 PMs of

In order to verify the correctness of the type synthesis method, this section adopts the modified Kutzbach–Grübler equation to calculate the number of DOF and uses the screw theory to verify the property of DOF. Taking 2

CAD model of 2

The physical prototype of the 2

Motion tests of physical prototypes.

The number of DOF of the 2

In practical engineering applications, it is vital to consider the feasibility of fabrication and assembly and the requirements of the task's degrees of freedom. The PMs with two constraint branched chains shown in Fig. 5 have three advantages:

The PMs have fewer joints (16 kinematic joints), which reduces errors due to joint gaps.

Fewer kinematic joints reduce the manufacturing costs and assembly difficulties of the PMs, which is more convenient for manufacturing and practical application.

The double-drive PMs contain two double-drive branch chains (i.e., there are two drivers on one branch chain), and this type of PM reduces the number of branch chains while maintaining the DOF, resulting in a high degree of dexterity.

Based on the 2

In order to obtain 5-DOF PMs with fewer joints, a three-branch 2R3T PM type synthesis method is proposed, and the validity of the method is verified with a new 2R3T PM with three branched chains, as an example.

The displacement sub-groups

The PMs synthesized in this paper can be categorized into three types: non-overconstrained PMs, overconstrained PMs with two constrained branched chains, and overconstrained PMs with three constrained branched chains. The analysis shows that the constraints of the three types of PMs are the same. Specifically, the direction of constrained torque is mutually perpendicular to all revolute joints in the constrained branched chains.

A class of three branched-chain 2R3T PMs with the fewest kinematic joints (15 kinematic joints) is synthesized based on the type synthesis method proposed in this paper. The number and property of the DOF of the typical PM are verified using the screw theory and the modified Kutzbach–Grübler equation.

Future research will focus on the stiffness analysis, dynamics analysis, and control algorithms of the 2

The data that support the findings of this study are available from the corresponding author upon reasonable request.

YR and XZ conceptualized the study, wrote the original draft of the paper, and reviewed and edited the paper. TD and HW assisted with the theory. All the authors read and approved the final paper.

The contact author has declared that none of the authors has any competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.

We would like to thank the Hebei Natural Science Foundation (grant no. E2021203018) for the financial support.

This research has been supported by the Natural Science Foundation of Hebei Province (grant no. E2021203018).

This paper was edited by Haiyang Li and reviewed by three anonymous referees.