When a rigid body is constrained by several mechanical connections providing n non-redundant constraints, n DOF of the body are removed, correspondingly, while 6–n DOF will remain. In this regard, every constraint wrench in the n system is reciprocal to the twist (6–n) system.

It is clear that the above result complies with Maxwell's principles of constraint. This may be expressed as follows: 1M=6-n, where M is the number of DOF.

In order to visualize the relationship between freedoms and constraints in a mechanical system, Blanding (1999) introduced both constraint lines and freedom lines, as shown in Fig. 1. At the same time, he addressed the rule of complementary patterns that states that every freedom line intersects all constraint lines, as shown in Fig. 2.

Based on the visual constraint-based design method proposed by Blanding, Hopkins and Culpepper extended this method and further proposed a FACT approach (Hopkins and Culpepper, 2011) to achieve a visual type synthesis of compliant mechanisms. For this purpose, they denoted a collection of commonly used freedom and constraint screw sets as freedom spaces (FSs) and constraint spaces (CSs), respectively, and established luxuriant patterns representing the unique mapping between FSs and their complementary CSs.

In this paper, parasitic displacement is used to characterize the precision of compliant mechanisms (Zhao et al., 2012). The compliant mechanism with a large parasitic displacement is a low-precision compliant mechanism. The compliant mechanism with a small parasitic displacement is a high-precision compliant mechanism. Compared with the asymmetric compliant mechanism, a symmetrical compliant mechanism has a smaller parasitic displacement. Thus, the symmetrical compliant mechanism has higher precision.

The schematic diagram and a pseudo-rigid-body model of a symmetric CM are shown in Fig. 3. The schematic diagram and a pseudo-rigid-body model of an asymmetric CM are shown in Fig. 4. The reference (Pucheta and Cardona, 2010) can explain the parasitic displacement equation of the flexible beam. Through a superposition operation, the parasitic displacement Δdy of CM is expressed as follows: 2Δdy=dy1+dy2, where dyi (i=1,2) is the parasitic displacement of a flexible beam.

As shown in Fig. 3b, a symmetric mechanism has two flexible beams, and their parasitic displacements are equal and opposite. Thus, the parasitic displacement relationship for symmetric arrangement of two flexible beams is expressed as follows: 3dy1=-dy2. Similarly, for the asymmetric mechanism, the parasitic displacements of two flexible beams are not equal, and they have a different direction, as shown in Fig. 4b. Thus, the parasitic displacement relationship for a symmetric arrangement of two flexible beams is expressed as follows: 4dy1≠-dy2. The above results show that the parasitic displacements of the symmetric mechanism can be offset. However, the parasitic displacements of the asymmetric mechanism cannot be offset. As a result, compared with asymmetric CMs, symmetric CMs can achieve higher precision. Thus, the arrangement of the mechanism determines the precision of the mechanism. In order to obtain high-precision CPMs, a symmetric arrangement for the compliant motion branch will be ensured.

To ensure that the synthesized mechanism can meet application requirements including motion, precision, and stiffness, a criterion for the type synthesis of the CPM with suitable constrained branch is presented. The following two points should be emphasized for the proposed type synthesis approach.

a.

To ensure that the CPM with a suitable constrained branch can achieve the desired motion, the compliant branch need to meet the following conditions: 5M1=M,6M2≥6, where M1 is the number of DOF of the compliant suitable constrained branch, and M2 is the number of DOF of the compliant active branch. For a 2R1T space posture adjustment mechanism as the example, M is equal to three. In order to achieve the desired motion, M1 and M2 are equal to three and six, respectively. At the same time, in order to reduce errors and simplify the kinematics and dynamics model of the mechanism, the compliant branch needs to be as short as possible.

First, the number of the compliant active branch and compliant suitable constrained branch is determined, respectively. Then, the appropriate arrangement form is obtained according to the principle of symmetry. The key point for the mechanism is the symmetrical arrangement. The parasitic displacement of the mechanism will be lower, and the high-precision configuration will be obtained by symmetrical arrangement.

For a 2R1T space posture adjustment mechanism as the example, the requirement of high stiffness is obtained. In order to ensure that the mechanism can resist shock and vibration in the rocket during launch, the CPMs with a suitable constrained branch are designed. Compared with the usual CPMs, the CPMs with a suitable constrained branch have higher stiffness. When M is equal to three, in order to achieve the high precision, the ns and n1 are equal to four and one, respectively. As the same time, in order to reduce the occupied space of the mechanism, the CPMs with a suitable constrained branch need to have as compact a structure and as small a size as possible.

According to the criterion for the type synthesis of the CPM with a suitable constrained branch, a new synthesis approach is proposed for CPMs with a suitable constrained branch based on the PRBM and FACT methods. According to the given DOF, the number of DOF of the compliant suitable constrained branch, M1, is determined, and the number of DOF of the compliant active branch, M2, is determined. A compliant branch with a different motion is synthesized based on the PRBM and FACT approaches, respectively. According to the criterion for the type synthesis of the CPM with a suitable constrained branch, the number of compliant active and compliant suitable constrained branches are obtained, respectively. According the principle of symmetry, the new configuration of the symmetrical arrangement mechanism is proposed. The degrees of freedom and motion characteristics of the mechanism are verified by the finite element simulation method. The process for the type synthesis of the CPM with a suitable constrained branch is described in Fig. 5.

In order to synthesize the 2R1T compliant parallel mechanism, we need to synthesize the compliant active branch based on the proposed synthesis approach. Thus, the motion chain with the 6 DOF need to be synthesized. Besides, according to Sect. 3.1 of the requirements, the short motion chain is selected for the synthesis. Thus, based on the existing 6 DOF serial mechanisms (SMs) in the reference (Lang et al., 2019), these serial mechanisms with only three kinematic pairs are selected as the original branch. This obtains a series of 6 DOF compliant active branches by the rigid-body-replacement method, and some of the branches are illustrated in Table 1. For PUS as the example, the P denotes the flexible hinge with connected to the fixed platform, the S denotes the flexible hinge with connected to the moving platform, and the P denotes the flexible hinge as the actuated hinge.

According to the high-stiffness requirement in Sect. 3.1, the CPMs with compliant suitable constrained branches are synthesized to improve the stiffness of the mechanisms. In order to synthesize the 2R1T compliant suitable constrained branch, the DOF of the compliant suitable constrained branch need to be determined based on the proposed type synthesis approach. Thus, a 3 DOF compliant suitable constrained branch needs to be synthesized. The total steps are performed for the synthesis process of the 2R1T compliant suitable constrained branch.

Step 1. Specify the desired freedom space (FS) according to the specifications of a synthesized CM and specify the reciprocal constraint space (CS) by the dual rule. For the 2R1T mechanisms, the freedom space and constraint space are shown in Fig. 6.

Step 2. Determine all possible reciprocal subspaces representing the constraints from step 1 and generate a complete fundamental building block (FBB) library by decomposing all reciprocal constraint subspaces. For the constraints of the 2R1T motion compliant branch, the FBBs and the simple set symbol are illustrated in Fig. 7.

Step 3. Obtain all possible subspaces equivalent to the FS based on the set operation for FBBs. Note that, in this paper, the union and intersection of the sets are designated by ∪ and ∩, respectively. For the 2R1T motion compliant branch, the set operation is expressed as follows: 9LN,u=UN1,n∪RN2,u=FN1,u1,n∪RN2,u2=RN1,u1∪RN2,u1∪RN2,u2=RN1,u1∪RN2,u2∪Pn=RN1,u1∪Pn∪RN2,u2=UN1,n∪Pn.

Step 4. Establish the mapping relationship between the FBB and the flexure unit, and select the appropriate flexure unit instead of the FBB. For example, R(N,u) represents a rotational degree of freedom (Yu et al., 2010; Lobontiu, 2002). So, a right circular notched flexible unit (θx) can be represented by R(N,u). P(n) represents a translational degree of freedom (Yang et al., 2021). So, a translational flexible unit (δz) can be represented by P(n). L(N,n) represents a plane constraint with 3 DOF. The flexible straight beam unit (θx, θy, and δz) can be represented by L(N,n) (Jia et al., 2015). Some flexible units are shown in Table 2.

Step 5. Obtain the compliant suitable constrained branch via the serial layout. Yu et al. (2010) proposed that the DOF of the flexible straight beam is changed when the dimension parameters of the flexible straight beam are changed (Jia et al., 2015). In this paper, the dimension parameters of the flexible straight beam are selected. The narrow flexible straight beam unit has 3 DOF and can achieve two rotational motions and one translational motion, respectively. Thus, the beam can realize a 2R1T motion. For the flexible straight beam, part of the compliant suitable constrained branches are shown in Fig. 8a–b. For the circular notched flexure unit, part of the compliant suitable constrained branches are shown in Fig. 8c.

In Fig. 8, three kinds of compliant suitable constrained branches are shown. Among them, each branch has 3 DOF. As shown in Fig. 8a, the L1 compliant suitable constrained branch only has a flexible straight beam. The flexible beam provides three constrained forces, including two parallel constrained forces. The two parallel constrained forces are equivalent to one constrained force and one constrained moment. Thus, this branch can achieve 2R1T motion. Similarly, as shown in Fig. 8b, the L2 compliant suitable constrained branch has two flexible straight beams, and those two flexible straight beams provide two constrained forces and one constrained moment. Thus, the L2 compliant suitable constrained branch can achieve 2R1T motion. As shown in Fig. 8c, the PU compliant suitable constrained beam has a flexible prismatic hinge and a flexible universal joint. The branch can achieve 2R1T motion.

Through steps 1–5, a series of 2R1T compliant suitable constrained branches are synthesized, and some of them are illustrated in Table 3. For L1 as an example, L denotes the flexible straight beam, and 1 denotes the number of the flexible straight beam.

Based on the new type synthesis approach, the number of compliant suitable constrained branches and compliant active branches, respectively, need to been determined. After the synthesis of compliant branches is completed, one compliant suitable constrained branch and two compliant active branches are selected by Eqs. (7) and (8). According to the high-precision requirement in Sect. 3.1, the symmetrical arrangement is selected.

Thus, in order to obtain a 2R1T motion CPM with high precision, this section proposes two special symmetrical arrangements. The first symmetrical arrangement can be called the axis symmetrical arrangement. It is considered as two separated components, where three identical compliant active branches are distributed at 120∘ around the edge of two platforms, and a compliant suitable constrained branch connects to the center of two platforms. Its structure diagram is shown in Fig. 9a. The second symmetrical arrangement can be called the plane symmetrical arrangement. As shown in Fig. 9b, its structure diagram shows that three identical compliant active branches and one compliant suitable constrained branch are distributed at 90∘ around the edge of two platforms.

As a result, based on the synthesized compliant active branches and compliant suitable constrained branches, the arrangements are determined. The axis symmetrical (Zhang et al., 2011) and the plane symmetrical arrangements (Pakzad et al., 2019) can achieve high precision when they move in a principal motion direction. As shown in Table 4, a series of the high-precision and high-stiffness CPMs with a suitable constrained branch are proposed through this symmetrical arrangement, and they are illustrated in Fig. 10.

As shown in Table 4, the new 3 PUS/PU(1) compliant parallel mechanism is proposed. For the 3 PUS/PU(1) compliant parallel mechanism as the example, PUS denotes the PUS compliant active branch, PU denotes the PU compliant suitable constrained branch, and (1) denotes the axis symmetrical arrangement.

In Fig. 10, four kinds of CPMs are demonstrated. Among them, each mechanism has four compliant branches, and the 6 DOF compliant active branches are the PUS compliant branch. As shown in Fig. 10a, 3 PUS/L(1) CPM is an axis symmetrical CPM. The suitable constrained branch has only a narrow flexible straight beam and is applies two constraining forces along the x axis and the y axis and one constraining moment around the z axis on the moving platform. Thus, the new 3 PUS/L1(1) CPM can achieve 2R1T motion. As shown in Fig. 10b, 3 PUS/L2(1) CPM is an axis symmetrical CPM. The suitable constrained branch has two narrow flexible straight beams. The two beams ensure the co-planar plane and apply two constraining forces along the x axis and the y axis and one constraining moment around the z axis on the moving platform. Thus, the new 3 PUS/L2(1) CPM can achieve 2R1T motion. As shown in Fig. 10c, 3 PUS/PU(2) CPM is an plane symmetrical CPM. The suitable constrained branch has a flexible prismatic hinge and a flexible universal joint. The branch applies two constraining forces along the x axis and the y axis and one constraining moment around the z axis on the moving platform. Similarly, as shown in Fig. 10d, 3 PUS/PU(1) CPM has a suitable constrained branch. The branch applies two constraining forces along the x axis and the y axis and one constraining moment around the z axis on the moving platform.

In order to verify the correctness of the proposed synthesis approach, the DOF of the synthesized mechanism is analyzed. In this section, taking the 3 PUS/PU(1) compliant parallel mechanism as the example, a finite element simulation (Zheng et al., 2015; M. Wang et al., 2019) is carried out to validate the DOF and the motion characteristic of 3 PUS/PU(1) compliant parallel mechanism, as shown in Fig. 11.

For the finite element model (FEM) of the 3 PUS/PU(1) compliant parallel mechanism, let the circumradius of the moving platform, rb, be 150 mm, the circumradius of the fixed platform, ra, be 200 mm, and the intersection angle of the compliant active branch with the fixed platform, θ, be 60∘. The material is titanium alloy, with a low density and high strength. The material properties are identified, where Young's modulus E is 96 GPa, Poisson's ratio μ is 0.36, and the density ρ is 4620 kg m-3.

For the 3 PUS/PU(1) compliant parallel mechanism, the von Mises resultant deformations will be obtained based upon different inputs of the three actuators. One actuator is turned on, two actuators are turned on, and three actuators are turned on, respectively, and the results are shown in Fig. 12.

The deformation results of the 3 PUS/PU(1) compliant parallel mechanism are illustrated in Fig. 12. It can be clearly seen that the moving platform rotates around the x axis, as illustrated in Fig. 12a, the moving platform rotates around the y axis, as illustrated in Fig. 12b, and the moving platform translations along the z axis, as illustrated in Fig. 12c. Thus, the 3 PUS/PU(1) compliant parallel mechanism can achieve 2R1T motion. The results demonstrate the correctness and feasibility of the proposed synthesized approach for CPM with a suitable constrained branch.