Type synthesis of 3-DOF spherical hybrid mechanisms with fixed centers of rotation

Abstract

To improve the load-bearing capacity and the rotation range about the normal of the moving platform in a spherical parallel mechanism (SPM), a type synthesis method for spherical hybrid mechanisms (SHMs) with fixed centers of rotation is proposed by coupling a serial transmission chain with the central passive limb chain of the SPM near its center area. Based on the analysis of the spherical mechanism configuration’s research status with a fixed rotation center, a method for systematically synthesizing SHMs is given, and four types of limb chains are developed. The possible limb constraint systems provided by each type of limb chain are analyzed via screw theory, and the type synthesis of each type of limb chain is carried out. Screening rules are proposed to obtain the preferred limb chains from the limb chain configuration results. By selecting and permuting preferred limb chains based on the mechanism’s center position of rotation and the constraint type, a series of SHMs that rotate unrestricted about the normal of the moving platform are produced. Selecting the RBR-2RRR SHM as an example, its workspace, singularity, dexterity, and stiffness are analyzed to verify the effectiveness of the research. This work enriches the configuration types of SHMs and provides theoretical support for the design and applications of SHMs in engineering.

Keywords

Spherical hybrid mechanism spherical parallel mechanism type synthesis screw theory ball cage joint

Introduction

The 3-DOF spherical mechanism is used to attain three-dimensional rotation and has wide applications in fields such as space pointing mechanisms, bionic joints, and medical rehabilitation. The parallel and hybrid configuration schemes have advantages such as high precision, high stiffness, and high dynamic motion compared with the serial configuration and have always been a research hotspot that attracts much attention from academia and industry.

Many scholars have conducted in-depth and extensive research on the configuration of the spherical parallel mechanism (SPM). Gosselin first proposed a camera positioning mechanism named Agile Eye¹ (the topological structure is a 3-RRR overconstrained SPM). Karouia and Hervé² obtained three classes of non-overconstrained SPMs, 3-P_lRR, 3-RP_lR, and 3-RRP_l, by replacing one R pair in the RRR limb chain with a P_l pair (where P_l denotes a planar pair or planar sub-chain). Di Gregorio³ and Karouia and Hervé⁴ obtained four types of non-overconstrained SPMs, 3-RUU, 3-RCC, 3-CRC, and 3-CCR, by adding an inactive rotation pair and inactive prismatic pair to the RRR limb chain. Fang and Tsai⁵ carried out the type synthesis of 3-limb chain symmetrical non-overconstrained SPMs based on screw theory and obtained typical mechanisms, including 3-RRS, 3-RSR, 3-URU, 3-URC, 3-UPC, and 3-CRC. Kong and Gosselin⁶ synthesized symmetrical SPMs with and without inactive kinematic pairs in the limb chain based on screw theory. Karouia and Hervé⁷ synthesized nonsymmetrical non-overconstrained SPMs based on group theory.

By changing one of the revolute pairs in the RRR limb chain to a circular arc-shaped prismatic pair P_c, in which the virtual axis is coaxial with the replaced R pair, a new mechanism configuration can be obtained. By varying the different numbers and positions of substitutions, three major classes (RRP_c, RP_cP_c, P_cP_cP_c) and a total of seven basic limb kinematic chains can be obtained. By analogy, P_cRS, P_cSR, RP_cS and other limb kinematic chains can be obtained. Specifically, Sui et al.⁸ synthesized three types of SPMs: 3-RRP_c, 3-P_cRR, and 3-P_cRP_c. Moreover, Zhao et al.⁹ proposed a 2RP_cR-RRP_c SPM, Zarkandi^10,11 proposed two 3-P_cRR SPMs, and Zhao et al.¹² proposed a 3-P_cP_cR SPM.

In addition, scholars have proposed using the spherical pair as the main bearing chain of the SPM and carrying out type synthesis with the purpose of improving its rigidity. The configurations visible in the literature include 3-PUS/S,¹³ 3-PSU/S,¹⁴ 3-PSS/S,^15,16 3-UPS/S,¹⁷ 3-SPS/S,^18,19 and 3-UCU/S.²⁰

The abovementioned SPMs have one important characteristic in common: they cannot rotate unrestricted about the normal direction of the moving platform, which limits the SPM’s use in applications such as active ball joints or positioning platforms for machining tools (such as milling or drilling). To give the mechanism a rotation capability of more than a full cycle, two solutions can be implemented based on different configuration principles:

Adopting a coaxial input spherical parallel configuration scheme

The initial revolute pair in the 3-RRR, 3-RSR, and 3-RRS mechanism limb chains is arranged with a coaxial input or changed to a circular arc-shaped prismatic pair. When an appropriate input drive is used, the mechanism’s moving platform can rotate about its normal without being limited, but this scheme has higher requirements for drive coordination control. The mechanisms seen in the literature are 3-RRR,²¹ 3-RSS,²² 3-P_cRR,²³ 3-P_cSR,²⁴ 3-P_cRS,²⁵ 3-P_cUS/S,²⁶ and 3-P_cSS/S.²⁷

Adopting a spherical hybrid configuration scheme

Xu et al.²⁸ proposed a 3-UPS/S+R spherical hybrid mechanism (SHM). The terminal rotation motion is driven by a lateral transmission chain. In addition, some scholars have proposed central transmission SHMs. Specifically, Zhang, Hess Coelho, and Du proposed SHMs with topological structures of 2RP_c-2RP_cS-RUR,²⁹ 3RRR-RUR,³⁰ and UPR-UCU-URU,³¹ respectively, for use in underwater vector propulsion devices. Bajaj and Dollar³² proposed a PRU-PSS-RUR SHM for wrist joints. Wu and Zou³³ proposed three types of SHMs with topological structures of 2RRR-RPS-RUR, 2UPU-RPS-RUR, and 2CUU-RPS-RUR for active ball joints or machine tool machining tool heads. However, at present, U-pairs are mostly used in transmission chains, which exhibit periodic fluctuations³⁴ that can cause vibration and wear during motion transmission.

SHMs inherit the characteristics of compact structure, high rigidity, and high precision from parallel mechanisms while possessing complete cycle rotation capabilities that general parallel mechanisms do not have. At present, there is still potential for further exploration of the configuration types of SHMs. Exploring new types of SHMs is very significant for expanding the application range of spherical mechanisms in engineering fields.

In this paper, a type synthesis method is proposed for SHMs with fixed rotation centers by coupling a serial transmission chain with the central passive limb chain of the SPM near the center area of the SPM. Based on the analysis of the spherical mechanism configuration’s research status with a fixed rotation center, a method for systematically synthesizing SHMs is given, and four types of limb chains are developed. Based on screw theory, the possible limb constraint systems provided by each type of limb chain are analyzed, and the type synthesis of each type of limb chain is carried out. Screening rules are proposed to obtain the preferred limb chains from the limb chain configuration results. Depending on the difference in the mechanism’s center position of rotation and the constraint type, a series of SHMs that rotate unrestricted about the normal direction of the moving platform are produced by selecting the corresponding preferred limb chain for permutations and combinations.

Constraint-synthesis method based on screw theory and synthesis ideas for SHMs

Constraint-synthesis method based on screw theory

In screw theory,³⁵ any screw can be represented as $$ = (S; S_{0})$ , with Plücker coordinates $(L, M, N; P, Q, R)$ . When $LP + MQ + NR = 0$ , these coordinates can represent the kinematic screw of a revolute pair or the constraint force applied to a rigid body. When L, M, and N are all 0 and P, Q, and R are not all 0, these coordinates can represent the kinematic screw of a prismatic pair or the constraint force couple applied to a rigid body.

If two screws $$ = (S; S_{0})$ and $$^{r} = (S^{r}; S_{0}^{r})$ satisfy the following expression, then $$$ and $$^{r}$ are reciprocal screws:

\begin{matrix} $ ° $^{r} = {SS}_{0}^{r} + S^{r} S_{0} = L P_{r} + M Q_{r} + N R_{r} + P L_{r} \\ + Q M_{r} + R N_{r} = 0 \end{matrix}

(1)

In this case, $$$ and $$^{r}$ can represent a rigid body’s kinematic screw and constraint screw, respectively. If the reciprocal product is 0, the constraint screw applied to the rigid body does not work on the kinematic screw, which indicates that $$^{r}$ does not restrict the movement of $$$ .

All possible motion forms of the end component in space are linear combinations of all the kinematic screws $$_{i}$ in the series chain, that is,

ω_{j} $_{j} = ω_{1} $_{1} + ω_{2} $_{2} + \dots + ω_{m} $_{m}

(2)

The linear space composed of all kinematic screws $$_{i}$ is called the limb screw system $\hat{$}$ of the series chain. The maximum linearly independent number n of all elements in $\hat{$}$ is called the screw system’s rank, denoted as rank( $\hat{$}$ ) = n. Any n linearly independent screws $$_{1}$ , $$_{2}$ , …, $$_{n}$ in $\hat{$}$ can be used as a basis of $\hat{$}$ , and any element in $\hat{$}$ can be obtained by linearly combining the basis screws. Therefore, after obtaining a basis of $\hat{$}$ , other equivalent kinematic chains can be constructed by linearly combining them.

If there is a set of screws $$_{i}^{r}$ that satisfies $$_{i}^{r} ° $_{j} = 0$ for any $$_{j} \in \hat{$}$ , then the linear space composed of all screws $$_{i}^{r}$ is called the limb constraint system ${\hat{$}}^{r} .$ If the parallel mechanism has k limb chains, the mechanism constraint system ${\hat{$}}_{m}^{r}$ is the union of the limb constraint systems ${\hat{$}}_{i}^{r}$ , that is,

{\hat{$}}_{m}^{r} = \cup_{i = 1}^{k} {\hat{$}}_{i}^{r}, {\hat{$}}_{i}^{r} \subseteq {\hat{$}}_{m}^{r}

(3)

The type synthesis method based on screw theory³⁵ is as follows: First, according to the kinematic screw system of the target mechanism, the mechanism constraint system can be obtained, and the number of limb chains and each limb constraint system can be determined. Second, according to the limb constraint system, the limb screw system can be obtained and used for the type synthesis of the limb chain. Finally, the obtained limb chains can be assembled according to certain geometric requirements to obtain the final mechanism configuration and verify the non-instantaneity of the mechanism.

Synthesis ideas for SHMs

As shown in Figure 1, the SHM can be seen as an integrated coupling between a central transmission limb chain in a series configuration and an SPM (with 2 or 3 DOFs) with a central ball pair passive limb chain by sharing the same rotational center.

Figure 1.

Composition schematic of an SHM.

Among them, the central transmission limb chain has two swing DOFs and one full-cycle rotation DOF. It exists both as a series mechanism and as a passive limb chain of a parallel mechanism. The advantages of this coupling design are as follows: (1) the drive that controls the rotation of the end can be arranged close to the fixed platform to reduce the end load and improve dynamic performance; (2) the central transmission limb chain can be used as the central passive limb chain of the SPM for external force unloading and improving the linear displacement stiffness of the mechanism; and (3) the design makes full use of the blank area near the center of the SPM to improve the space utilization rate of the mechanism.

When the parallel part mechanism has 2 DOFs, the central S-pair passive limb chain does not constrain the rotation motion of the mechanism. Therefore, there must be a constraint limb chain that constrains the rotation of the moving platform about its normal. As shown in Figure 1, the constraint limb chains that make the four points Aoca always coplanar (first type coplanar constraint) are defined as type I limb chains, and the constraint limb chains that make the four points BAab always coplanar (second type coplanar constraint) are defined as type II limb chains. Type I and type II limb chains can eliminate the rotation DOF of the moving platform about its normal through their respective coplanar constraints. In addition, if the moving platform has a definite motion, it must also have a non-constraint limb chain. The limb chains that do not constrain the rotation of the moving platform about its normal are defined as type III limb chains. Type III limb chains may not provide any constraint but may also provide an overconstraint that is the same or partially the same as that of the central S-pair passive limb chain. When the parallel part mechanism has 3 DOFs, there are at least three type III limb chains in the mechanism, and there are no type I or type II limb chains.

The synthesis idea is as follows: First, the SHM can be deconstructed into a series mechanism and an SPM with a central S-pair passive limb chain. Second, in the serial part, the central transmission limb chain’s configuration can be synthesized according to its DOFs and task requirements. Third, in the parallel mechanism part, all possible limb constraint systems provided by type I, type II, and type III limb chains can be analyzed according to the constraint task of the limb chain, the corresponding limb screw systems can then be solved, and type synthesis for various types of limb chains can be carried out. Finally, the obtained central transmission limb chain and type I, type II, and type III limb chains can be reasonably configured to obtain an SHM configuration that meets the requirements.

Synthesis and preferred selection of limb chains

Symbol description

To more clearly describe the synthesized limb chain types, the following rules are established: The symbol main body represents the type of kinematic pair, where R, P, C, U, and S represent a revolute pair, prismatic pair, cylindrical pair, universal joint, and ball joint, respectively. The upper left or upper right symbols represent the direction of the R-pair centerline or P-pair. The lower left or lower right symbol indicates that the kinematic pair axis must pass through this point. Moreover, i, j, and k represent arbitrary directions of the spatial axes. $({}^{i}R {}^{j}R)_{N}$ and $({}^{i}R {}^{j}R {}^{k}R)_{N}$ represent the 2R and 3R spherical sub-chains, respectively. $\overset{⌢}{PR}$ indicates that the P-pair direction is parallel or coaxial with the R-pair axis.

For example, ${}^{x}{R_{N}}$ represents a revolute pair with an axis parallel to the x-axis and passing through point N. ${}^{yz}{C_{N}}$ represents a cylindrical pair with an axis parallel to the yoz plane and passing through point N. ${}_{N}^{yz}{\overset{⌢}{PU}}_{M}^{x}$ indicates that the direction of the prismatic pair is parallel or coaxial with the first rotating axis of the U-pair. The first rotating axis of the U-pair is parallel to the yoz plane and passes through point N, and the second rotating axis is parallel to the x-axis and passes through point M.

The focus of limb chain synthesis in this article is on basic kinematic pairs and commonly used multi-DOF kinematic pairs. The helical pair H and the circular arc-shaped prismatic pair P_c are not studied in this paper.

Synthesis of central transmission limb chains

Unequal speed central transmission limb chain

The central transmission limb chain needs to meet two conditions: Condition 1 is to constrain the moving platform of the parallel mechanism to generate spherical motion about point o; and Condition 2 is to transmit a rotational motion through the fixed platform and the moving platform of the parallel mechanism to the end of the hybrid mechanism.

According to Condition 1, the moving platform can be connected to the fixed platform through an S-pair. According to Condition 2, for the central limb chain to transmit rotational motion, the S-pair can be changed to a U-pair, and the R-pair can be set at the connection between the central limb chain and the fixed platform and the moving platform of the parallel mechanism to obtain the ${}^{z}{RU} {}^{z}R$ limb chain, as shown in Figure 2(a).

Figure 2.

Central transmission limb chain: (a) ${}^{z}{RU} {}^{z}R$ and (b) ${}^{z}{RB} {}^{z}R$ .

Constant velocity central transmission limb chain

By replacing the U-joint in the ${}^{z}{RU} {}^{z}R$ limb chain with a constant velocity joint, a constant velocity central transmission limb chain can be obtained. There are many types of constant velocity joints, among which the ball cage constant velocity joint³⁶ has the advantage of being able to rotate around the center, high rigidity, and high strength. It is the preferred alternative to the U-joint. At present, there is no recognized abbreviation for the ball cage constant velocity joint. In this article, it is referred to as the B-pair. A schematic diagram of its mechanism can be obtained by modifying the schematic diagram of the S-pair. In summary, the constant-velocity central transmission limb chain is ${}^{z}{RB} {}^{z}R$ , as shown in Figure 2(b).

Synthesis of type I limb chains

Figure 3 is a partial view of a 2-DOF SPM with a central S-pair passive limb chain. The fixed and moving platforms are fixed to the fixed coordinate system $o - xyz$ and the moving coordinate system $o - x_{1} y_{1} z_{1}$ , respectively, and their initial postures are the same. $o_{2}$ is the starting point of the type I limb chain. N is a point on the y-axis with coordinates $(0, - y_{0}, 0)$ . $$_{11}^{r}$ to $$_{15}^{r}$ are the structural constraints provided by the type I limb chain to the moving platform. By making the four points o₂, o, c, and a constant coplanar (the first type of coplanar constraint), the type I limb chains eliminate the rotational DOF of the SPM around the $z_{1}$ axis. Moreover, the type I limb chain may provide six types of constraints.

Figure 3.

Possible constraint types provided by the type I limb chain.

Case 1: A constraint force couple $$_{11}^{r}$

In this instance, the limb chain provides a constraint force couple $$_{11}^{r}$ , initially parallel to the z-axis. When the moving platform moves, it is always parallel to the plane $yo z_{1}$ and perpendicular to the y-axis.

In the limb coordinate system with $o_{2}$ as the origin, according to equation (3), the following expression can be obtained:

{\hat{$}}_{1 - 1}^{r} = {$_{11}^{r}} = (0 0 0; 0 0 1)

(4)

The reciprocal screw for ${\hat{$}}_{1 - 1}^{r}$ is calculated according to equation (1), and the limb screw system is expressed as follows:

{\hat{$}}_{1 - 1} = {\begin{matrix} $_{11} = (1 0 0; 0 0 0) \\ $_{12} = (0 1 0; 0 0 0) \\ $_{13} = (0 0 0; 1 0 0) \\ $_{14} = (0 0 0; 0 1 0) \\ $_{15} = (0 0 0; 0 0 1) \end{matrix}

(5)

According to equation (2), by linearly combining the basic screws in ${\hat{$}}_{1 - 1}$ , different forms of limb chains can be obtained. At the initial posture, the specific limb chain configuration is presented in Table 1. Some limb chains in Table 1 need to satisfy the following additional conditions: ① At least one prismatic pair must have a motion component along the x-axis. ② At least one prismatic pair must have a motion component along the y-axis. Meanwhile, the kinematic pairs in the limb chain should be linearly independent.

Table 1.

Configuration of the type I limb chains in Case 1.

Type	Notes	Basic limb chains	Limb chains with multi-DOF kinematic pairs
2R3P	–	${{}^{y}R {}^{i}P {}^{j}P {}^{k}P} {}^{x}R$	${{}^{y}{U^{x}} {}^{i}P {}^{j}P {}^{k}P}$	${}^{y}C {{}^{x}R {}^{i}P {}^{j}P}$	${}^{i}P {}^{y}C {{}^{x}R {}^{j}P}$
		${{}^{y}R {}^{i}P {}^{j}P} {}^{x}R {}^{k}P$	${}^{i}P {}^{j}P {}^{y}C {}^{x}R$	${{}^{y}R {}^{i}P {}^{j}P} {}^{x}C$	${{}^{y}R {}^{i}P} {}^{x}C {}^{j}P$
		${{}^{y}R {}^{i}P} {}^{x}R {}^{j}P {}^{k}P$	${}^{y}R {}^{x}C {}^{i}P {}^{j}P$	${}^{y}C {}^{x}C {}^{i}P$	${{}^{y}C {}^{i}P} {}^{x}C$
		${}^{y}R {}^{x}R {}^{i}P {}^{j}P {}^{k}P$
3R2P	①	${}^{y}R {{}^{i}P {}^{j}P {}^{x}R {}^{x}R}$	${{}^{i}P {}^{y}C} {}^{x}R {}^{x}R$	${}^{y}C {}^{x}R {{}^{i}P {}^{x}R}$	${}^{i}P {}^{y}R {{}^{x}C {}^{x}R}$
		${}^{i}P {}^{y}R {{}^{j}P {}^{x}R {}^{x}R}$	${}^{y}R {{}^{i}P {}^{x}C {}^{x}R}$	${}^{i}P {}^{j}P {}^{y}{U^{x}} {}^{x}R$	${}^{i}P {}^{y}{U^{x}} {{}^{j}P {}^{x}R}$
		${}^{i}P {}^{j}P {}^{y}R {}^{x}R {}^{x}R$	${}^{y}{U^{x}} {{}^{i}P {}^{j}P {}^{x}R}$	${}^{y}C {{}^{x}C {}^{x}R}$	${}^{i}P {}^{y}{U^{x}} {}^{x}C$
			${}^{y}{U^{x}} {{}^{i}P {}^{x}C}$
	②	${}^{y}R {}^{y}R {}^{x}R {}^{i}P {}^{j}P$	${}^{y}R {{}^{y}{U^{x}} {}^{i}P {}^{j}P}$	${}^{y}C {{}^{y}{U^{x}} {}^{i}P}$	${}^{i}P {}^{y}C {}^{y}{U^{x}}$
		${{}^{y}R {}^{y}R {}^{i}P} {}^{x}R {}^{j}P$	${}^{i}P {}^{y}R {{}^{y}{U^{x}} {}^{j}P}$	${{}^{y}R {}^{y}C} {}^{x}C$	${{}^{y}R {}^{i}P {}^{y}R} {}^{x}C$
		${{}^{y}R {}^{y}R {}^{i}P {}^{j}P} {}^{x}R$	${}^{y}R {}^{y}R {}^{x}C {}^{i}P$	${{}^{y}R {}^{y}C} {}^{x}R {}^{i}P$	${{}^{y}R {}^{i}P {}^{y}C} {}^{x}R$
4R1P	①	${}^{i}P {}^{y}R {}^{x}R {}^{x}R {}^{x}R$	${}^{y}R {^{x} C {}^{x}R {}^{x}R}$	${}^{y}{U^{x}} {{}^{x}C {}^{x}R}$	${}^{i}P {}^{y}{U^{x}} {}^{x}R {}^{x}R$
		${}^{y}R {{}^{i}P {}^{x}R {}^{x}R {}^{x}R}$	${}^{y}{U^{x}} {{}^{i}P {}^{x}R {}^{x}R}$
	②	${{}^{y}R {}^{y}R {}^{i}P} {}^{y}R {}^{x}R$	${{}^{y}C {}^{y}R {}^{y}R} {}^{x}R$	${{}^{i}P {}^{y}R {}^{y}R} {}^{y}{U^{x}}$	${}^{y}R {}^{y}R {}^{y}{U^{x}} {}^{i}P$
		${}^{y}R {}^{y}R {}^{y}R {{}^{x}R {}^{i}P}$	${{}^{y}C {}^{y}R} {}^{y}{U^{x}}$
	–	${}^{y}R {}^{y}R {{}^{i}P {}^{x}R {}^{x}R}$	${}^{y}R {}^{y}R {{}^{x}C {}^{x}R}$	${{}^{y}R {}^{y}C} {}^{x}R {}^{x}R$	${{}^{i}P {}^{y}R} {}^{y}{U^{x}} {}^{x}R$
		${{}^{i}P {}^{y}R} {}^{y}R {}^{x}R {}^{x}R$	${}^{y}R {}^{y}{U^{x}} {{}^{i}P {}^{x}R}$	${}^{y}R {}^{y}{U^{x}} {}^{x}C$	${}^{y}C {}^{y}{U^{x}} {}^{x}R$
5R	–	${}^{y}R {}^{y}R [{}^{y}R / {}^{x}R] {}^{x}R {}^{x}R$	${}^{y}R {}^{y}{U^{x}} {}^{x}R {}^{x}R$	${}^{y}R {}^{y}R {}^{y}{U^{x}} {}^{x}R$

To save space, two symbols, {} and [], are used in this article to represent limb chains. The first symbol, {}, represents any possible arrangement of the elements in {}, while [] indicates that only one element in [] is taken. For example, $A {B [C / D]}$ represents four permutations: ABC, ACB, ABD, and ADB.

Case 2: A constraint force $$_{12}^{r}$

In this instance, the limb chain provides a constraint force $$_{12}^{r}$ , which always intersects perpendicularly with the y-axis at point N. At the initial moment, it is parallel to the x-axis. When the moving platform moves, it always passes through point N and is perpendicular to the plane $yo z_{1}$ . It works together with the S-pair in the central limb chain to constrain the rotational DOF of the moving platform about the z₁-axis.

In the limb coordinate system with N as the origin, the following expression can be obtained:

{\hat{$}}_{1 - 2}^{r} = {$_{12}^{r}} = (1 0 0; 0 0 0)

(6)

According to equation (1), the limb screw system ${\hat{$}}_{1 - 2}$ that is reciprocal with ${\hat{$}}_{1 - 2}^{r}$ can be obtained:

{\hat{$}}_{1 - 2} = {$_{11}, $_{12}, $_{14}, $_{15}, $_{16}}

(7)

where $$_{16} = (0 0 1; 0 0 0)$ .

By linearly combining ${\hat{$}}_{1 - 2}$ , the limb chain configuration is obtained and presented in Table 2. Among them, point N in the limb chain cannot coincide with point o, which is the mechanism’s rotation center.

Table 2.

Configuration of the type I limb chains in Case 2.

Type	Basic limb Chains	Limb chains with multi-DOF kinematic pairs
3R2P	$({}^{i}R {}^{j}R {}^{k}R)_{N} {}^{yz}P {}^{yz}P$	$S_{N} {}^{yz}P {}^{yz}P$	${{{}^{i}R}_{N} {}_{N}^{j}{U_{N}^{k}}} {}^{yz}P {}^{yz}P$	${}_{N}^{i}{U_{N}^{j}} [^{yz} {\overset{⌢}{PR}}_{N} /^{yz} C_{N}]^{yz} P$
	${(^{i} R {}^{j}R)}_{N} {}^{yz}{\overset{⌢}{P R}}_{N} {}^{yz}P$	${}^{y}{\overset{⌢}{RP}}_{N} {{}^{i}R}_{N} {}^{yz}{C_{N}}$	${}^{y}P {^{i} R_{N} {}_{N}^{j}{U_{N}^{k}}} {}^{yz}P$	${}^{y}{P_{N}^{i}} U_{N}^{j} [{}^{yz}{C_{N}} /^{yz} {\overset{⌢}{PR}}_{N}]$
	${}^{y}P ({}^{i}R {}^{j}R {}^{k}R)_{N} {}^{yz}P$	${}^{y}P S_{N} {}^{yz}P$	${}^{y}P ({}^{i}R {}^{j}R)_{N} {}^{yz}{C_{N}}$	${{}^{y}C}_{N} [({}^{i}R {}^{j}R)_{N} /_{N}^{i} U_{N}^{j}] {}^{yz}P$
	${}^{y}P ({{}^{i}R}_{N} {}^{j}R)_{N} {}^{yz}{\overset{⌢}{PR}}_{N}$	${}^{y}{\overset{⌢}{RP}}_{N} {}_{N}^{i}{U_{N}^{j}} {}^{yz}P$	$({}^{i}R {}^{j}R)_{N} {}^{yz}{C_{N}} {}^{yz}P$	${}^{y}{C_{N}} {}^{i}{R_{N}} [^{yz} C_{N} /^{yz} {\overset{⌢}{PR}}_{N}]$
	${}^{y}{\overset{⌢}{RP}}_{N} ({}^{i}R {}^{j}R)_{N} {}^{yz}P$
	${}^{y}{\overset{⌢}{RP}}_{N} {{}^{i}R}_{N} {}^{yz}{\overset{⌢}{PR}}_{N}$
	$({}^{i}R {}^{j}R)_{N} {{}^{yz}P {}^{x}R {}^{yz}P}$	${}_{N}^{i}{U_{N}^{j}} {{}^{yz}P {}^{x}R {}^{yz}P}$	${{}^{i}R}_{N}_{N}^{yz} U^{x} {}^{yz}P {}^{yz}P$	${{}^{i}R}_{N} {}^{yz}{C_{N}} {{}^{yz}P {}^{x}R}$
	${{}^{i}R}_{N} {}^{yz}{\overset{⌢}{PR}}_{N} {{}^{yz}P {}^{x}R}$	${{}^{i}R}_{N}_{N}^{yz} {\overset{⌢}{PU}}^{x} {}^{yz}P$	${{}^{y}R}_{N} {{}^{yz}P {{}^{yz}P}_{N}^{yz} U^{x}}$	${{}^{y}R}_{N} {}^{yz}P {}^{yz}{C_{N}} {}^{x}R$
	${{}^{y}R}_{N} {}^{yz}P {{}^{yz}P}^{yz} R_{N} {}^{x}R$	${{}^{y}C}_{N} {{}^{yz}P}^{yz} R_{N} {}^{x}R$	${}^{y}{C_{N}} {}^{yz}{R_{N}} {^{x} R {}^{yz}P}$	${{}^{y}C}_{N} {}^{yz}{C_{N}} {}^{x}R$
	${{}^{y}R}_{N} {{}^{yz}P}^{yz} R_{N} {{}^{x}R {}^{yz}P}$	${{}^{y}C}_{N} {{{}^{yz}P}_{N}^{yz} U^{x}}$	${}^{y}P {}_{N}^{i}{U_{N}^{j}} {{}^{yz}P {}^{x}R}$	${}^{y}P {{}^{i}R}_{N} {}^{yz}{C_{N}} {}^{x}R$
	${{}^{y}R}_{N} {{}^{yz}R}_{N} {{}^{x}R {}^{yz}P {}^{yz}P}$	${}^{y}P {{}^{i}R}_{N} {}_{N}^{yz}{U^{x}} {}^{yz}P$	${}^{y}P {{}^{i}R}_{N} {}_{N}^{yz}{\overset{⌢}{PU}}^{x}$	${}^{y}P {{}^{y}R}_{N} {^{yz} P {}_{N}^{yz}{U^{x}}}$
	${}^{y}P ({}^{i}R {}^{j}R)_{N} {{}^{yz}P {}^{x}R}$
	${}^{y}P {{}^{y}R}_{N} {{}^{yz}P}^{yz} R_{N} {}^{x}R$
4R1P	$({}^{i}R {}^{j}R {}^{k}R)_{N} {{}^{yz}P {}^{x}R}$	$({}^{i}R {}^{j}R)_{N} {}^{yz}{C_{N}} {}^{x}R$	$({}^{i}R {}^{j}R)_{N} {}_{N}^{k}{U^{x}} {}^{yz}P$	${{{}^{i}R}_{N} {}_{N}^{j}{U_{N}^{k}}} {{}^{yz}P {}^{x}R}$
	${}^{y}P (^{i} R {}^{j}R {}^{k}R)_{N} {}^{x}R$	${}_{N}^{i}{U_{N}^{j}} {}^{yz}{C_{N}} {}^{x}R$	${}^{y}P {{{}^{i}R}_{N} {}_{N}^{j}{U_{N}^{k}}} {}^{x}R$	${}^{y}P {[(}^{i} R {}^{j}R)_{N} /_{N}^{i} U_{N}^{j}] {}_{N}^{k}{U^{x}}$
	${}^{y}{\overset{⌢}{RP}}_{N} {(^{i} R {}^{j}R)}_{N} {}^{x}R$	$S_{N} {{}^{yz}P {}^{x}R}$	${}^{y}P S_{N} {}^{x}R$	$[{}^{y}{\overset{⌢}{RP}}_{N} / {{}^{y}C}_{N}] {{}^{i}R}_{N} {}_{N}^{yz}{U^{x}}$
	$({}^{i}R {}^{j}R)_{N} {}^{yz}{\overset{⌢}{PR}}_{N} {}^{x}R$	${(^{i} R {}^{j}R)}_{N} {}_{N}^{yz}{\overset{⌢}{PU}}^{x}$	${}^{y}{\overset{⌢}{RP}}_{N} {}_{N}^{i}{U_{N}^{j}} {}^{x}R$	${{}^{y}C}_{N} [({}^{i}R {}^{j}R)_{N} /_{N}^{i} U_{N}^{j}] {}^{x}R$
		${}_{N}^{i}{U_{N}^{j}} {}^{yz}{\overset{⌢}{PR}}_{N} {}^{x}R$	${}_{N}^{i}{U_{N}^{j}} {}_{N}^{yz}{\overset{⌢}{PU}}^{x}$	${}_{N}^{i}{U_{N}^{j}} {}_{N}^{k}{U^{x}} {}^{yz}P$
	$({}^{i}R {}^{j}R)_{N} {{}^{x}R {}^{yz}P {}^{x}R}$	${}_{N}^{i}{U_{N}^{j}} {{}^{x}R {}^{yz}P {}^{x}R}$	${{}^{i}R}_{N} {}_{N}^{yz}{U^{x}} {{}^{yz}P {}^{x}R}$	${{}^{y}R}_{N} {{}^{yz}P}_{N}^{yz} U^{x} {}^{x}R$
	${}^{y}P ({}^{i}R {}^{j}R)_{N} {}^{x}R {}^{x}R$	${}^{y}P {}_{N}^{i}{U_{N}^{j}} {}^{x}R {}^{x}R$	${}^{y}P {{}^{i}R}_{N} {}_{N}^{yz}{U^{x}} {}^{x}R$	${{}^{y}C}_{N} {}_{N}^{yz}{U^{x}} {}^{x}R$
	${{}^{y}R}_{N} {{}^{yz}P}^{yz} R_{N} {}^{x}R {}^{x}R$	${{}^{y}C}_{N} {{}^{i}R}_{N} {}^{x}R {}^{x}R$	${{}^{i}R}_{N} {}^{yz}{C_{N}} {}^{x}R {}^{x}R$	${}^{i}{R_{N}} {}_{N}^{yz}{\overset{⌢}{PU}}^{x} {}^{x}R$
	${{}^{y}R}_{N} {}^{y}P {{}^{i}R}_{N} {}^{x}R {}^{x}R$
	${}^{i}{R_{N}} {}^{yz}{\overset{⌢}{PR}}_{N} {}^{x}R {}^{x}R$
5R	$({}^{i}R {}^{j}R {}^{k}R)_{N} {}^{x}R {}^{x}R$	$S_{N} {}^{x}R {}^{x}R$	${{{}^{i}R}_{N} {}_{N}^{j}{U_{N}^{k}}} {}^{x}R {}^{x}R$	$[({}^{i}R {}^{j}R)_{N} /_{N}^{i} U_{N}^{j}] {}_{N}^{k}{U^{x}} {}^{x}R$
	$({}^{i}R {}^{j}R)_{N} {}^{x}R {}^{x}R {}^{x}R$	${}_{N}^{i}{U_{N}^{j}} {}^{x}R {}^{x}R {}^{x}R$	${{}^{i}R}_{N} {}_{N}^{yz}{U^{x}} {}^{x}R {}^{x}R$

Case 3: A constraint force couple $$_{11}^{r}$ or a constraint force $$_{12}^{r}$ , depending on the instantaneous spatial position of the limb chain

Although the spatial orientations of the constraints in the above two situations change with the motion of the moving platform, the nature of the constraints does not change. In this section, the following situation is discussed: the constraint properties provided by the limb chain switch between the constraint force $$_{12}^{r}$ and constraint couple $$_{11}^{r}$ as the limb chain moves. Although the constraint properties provided by this type of limb chain have changed, the constraint effect still eliminates the rotational DOF of the moving platform around the z₁-axis.

To provide a clearer explanation of the process of switching constraint properties, the ${}_{N}^{y}{U^{x}} {}^{yz}P {{}^{x}U}_{N}^{yz}$ and ${}_{N}^{y}{U^{x}} {}^{x}R {{}^{x}U}_{N}^{yz}$ limb chains are used as examples in Figure 4. In these two limb chains, the spatial orientation of the first rotation axis (rotation axis a) of the starting U-pair remains unchanged, while the spatial orientation of the second rotation axis (rotation axis b) of the end U-pair changes with the limb chain end posture. In general, the extension of Axis b intersects with Axis a, and the limb chain provides a constraint force $$_{12}^{r}$ at this point. When Axis b is parallel to Axis a, the constraint provided by the limb chain switches from a constraint force $$_{12}^{r}$ to a constraint couple $$_{11}^{r}$ . When the direction of Axis b crosses the parallel position, the extensions of Axis b and Axis a intersect again, and the constraint provided by the limb chain changes from a constraint couple $$_{11}^{r}$ to a constraint force $$_{12}^{r}$ again. The specific limb chain configurations with the above properties are shown in Table 3. Among them, point N in the limb chain cannot coincide with point o, which is the mechanism’s rotation center.

Figure 4.

Transformation of the constraint force and constraint couple.

Table 3.

Configuration of the type I limb chains in Case 3.

Type	Basic limb chains	Limb chains with multi-DOF kinematic pairs
3R2P	${{}^{y}R}_{N} {}^{x}R {{}^{yz}P {{}^{yz}P}^{yz} R_{N}}$	${}_{N}^{y}{U^{x}} {{}^{yz}P {{}^{yz}P}^{yz} R_{N}}$	${{}^{y}R}_{N} {{{}^{x}U}_{N}^{yz} {}^{yz}P {}^{yz}P}$	${{}^{y}R}_{N} {}^{x}R {{}^{yz}P {}^{yz}{C_{N}}}$
	${{}^{y}R}_{N} {}^{yz}P {}^{yz}P {}^{x}R {{}^{yz}R}_{N}$	${{}^{y}R}_{N} {}^{yz}P {}^{x}R {}^{yz}{C_{N}}$	${}_{N}^{y}{U^{x}} {{}^{yz}P {}^{yz}{C_{N}}}$	${{}^{y}C}_{N} {}^{x}R {{{}^{yz}R}_{N} {}^{yz}P}$
	${{}^{y}R}_{N} {}^{yz}P {}^{x}R {{{}^{yz}P}^{yz} R_{N}}$	${{}^{y}C}_{N} {}^{yz}P {}^{x}R {{}^{yz}R}_{N}$	${{}^{y}C}_{N} {}^{x}R {}^{yz}{C_{N}}$	${{}^{y}C}_{N} {{}^{yz}P {{}^{x}U}_{N}^{yz}}$
	${}^{y}P {{}^{y}R}_{N} {}^{yz}P {}^{x}R {{}^{yz}R}_{N}$	${}^{y}P {}_{N}^{y}{U^{x}} {{{}^{yz}P}^{yz} R_{N}}$	${}^{y}P {}_{N}^{y}{U^{x}} {}^{yz}{C_{N}}$	${}^{y}P {{}^{y}R}_{N} {{}^{yz}P {{}^{x}U}_{N}^{yz}}$
	${}^{y}P {{}^{y}R}_{N} {}^{x}R {{{}^{yz}R}_{N} {}^{yz}P}$	${}^{y}P {{}^{y}R}_{N} {{}^{yz}{C_{N}} {}^{x}R}$	${}^{y}P {{}^{y}R}_{N} {}^{x}R {}^{yz}{C_{N}}$
4R1P	${{}^{y}R}_{N} {}^{yz}P {}^{x}R {{{}^{yz}R}_{N} {}^{x}R}$	${{}^{y}R}_{N} {}^{x}R {{}^{yz}{C_{N}} {}^{x}R}$	${}_{N}^{y}{U^{x}} {{}^{yz}{C_{N}} {}^{x}R}$	${}_{N}^{y}{U^{x}} {{{}^{yz}R}_{N} {}^{yz}P {}^{x}R}$
	${{}^{y}R}_{N} {}^{x}R {{{}^{yz}R}_{N} {}^{yz}P {}^{x}R}$	${{}^{y}R}_{N} {{}^{x}R {}^{yz}P {{}^{x}U}_{N}^{yz}}$	${{}^{y}R}_{N} {}^{yz}P {}^{x}R {}_{N}^{yz}{U^{x}}$	${{}^{y}R}_{N} {}^{x}R {{{}^{yz}P}_{N}^{yz} U^{x}}$
	${}^{y}P {{}^{y}R}_{N} {}^{x}R {{{}^{yz}R}_{N} {}^{x}R}$	${}_{N}^{y}{U^{x}} {{}^{yz}P {{}^{x}U}_{N}^{yz}}$	${}_{N}^{y}{U^{x}} {{{}^{yz}P}_{N}^{yz} U^{x}}$	${{}^{y}C}_{N} {}^{x}R {{}^{x}R {{}^{yz}R}_{N}}$
		${{}^{y}P {{}^{y}R}_{N}} {{}^{x}R {{}^{x}U}_{N}^{yz}}$	${{}^{y}C}_{N} {{}^{x}R {{}^{x}U}_{N}^{yz}}$	${{}^{y}P {{}^{y}R}_{N}} {}^{x}R {}_{N}^{yz}{U^{x}}$
		${{}^{y}C}_{N} {}^{x}R {}_{N}^{yz}{U^{x}}$	${}^{y}P {}_{N}^{y}{U^{x}} {{}^{x}R {{}^{yz}R}_{N}}$	${}^{y}P {}_{N}^{y}{U^{x}} [{}_{N}^{yz}{U^{x}} / {{}^{x}U}_{N}^{yz}]$
5R	${{}^{y}R}_{N} {}^{x}R {{}^{x}R {}^{x}R {{}^{yz}R}_{N}}$	${}_{N}^{y}{U^{x}} {{{}^{yz}R}_{N} {}^{x}R {}^{x}R}$	${{}^{y}R}_{N} {{}^{x}R {}^{x}R {{}^{x}U}_{N}^{yz}}$	${{}^{y}R}_{N} {}^{x}R {{}^{x}R {}_{N}^{yz}{U^{x}}}$
		${}_{N}^{y}{U^{x}} {{}^{x}R {}_{N}^{yz}{U^{x}}}$	${}_{N}^{y}{U^{x}} {{}^{x}R {{}^{x}U}_{N}^{yz}}$

Case 4: Both a constraint force couple $$_{11}^{r}$ and a constraint force $$_{12}^{r}$

In this instance, in the limb coordinate system with N as the origin, the following expression can be obtained:

{\hat{$}}_{1 - 4}^{r} = {$_{11}^{r}, $_{12}^{r}}

(8)

After calculating the reciprocal screw for ${\hat{$}}_{1 - 4}^{r}$ , the limb screw system can be obtained as follows:

{\hat{$}}_{1 - 4} = {$_{11}, $_{12}, $_{14}, $_{15}}

(9)

Similarly, by linearly combining ${\hat{$}}_{1 - 4}$ , the limb chain configuration can be obtained and presented in Table 4. Among them, point N in the limb chain cannot coincide with point o, which is the mechanism’s rotation center.

Table 4.

The type I limb chains in Case 4.

Type	Basic limb chains	Limb chains with multi-DOF kinematic pairs
2R2P	${{}^{y}P {{}^{y}R}_{N}}^{x} R_{N} {}^{yz}P$	${{}^{y}C}_{N} {{}^{x}R}_{N} {}^{yz}P$
	${{}^{y}R}_{N} {{}^{x}R}_{N} {}^{yz}P {}^{yz}P$	${}_{N}^{y}{U_{N}^{x}} {}^{yz}P {}^{yz}P$
	${}^{y}P {{}^{y}R}_{N} {{}^{yz}P {}^{x}R}$	${{}^{y}C}_{N} {}^{yz}P {}^{x}R$
	${{}^{y}R}_{N} {{}^{yz}P {}^{yz}P {}^{x}R}$	${{}^{y}C}_{N} {}^{x}R {}^{yz}P$
3R1P	${{}^{y}P {{}^{y}R}_{N}} {{}^{x}R}_{N} {}^{x}R$	${}^{y}P {}_{N}^{y}{U_{N}^{x}} {}^{x}R$
	${{}^{y}R}_{N} {{}^{x}R}_{N} {}^{yz}P {}^{x}R$	${{}^{y}C}_{N} {{}^{x}R}_{N} {}^{x}R$
	${{}^{y}R}_{N} {{}^{x}R}_{N} {}^{x}R {}^{yz}P$	${}_{N}^{y}{U_{N}^{x}} {{}^{yz}P {}^{x}R}$
	${{}^{y}P {{}^{y}R}_{N}} {}^{x}R {}^{x}R$	${{}^{y}C}_{N} {}^{x}R {}^{x}R$
	${{}^{y}R}_{N} {{}^{x}R}_{N} {{}^{yz}P {}^{x}R}$
4R	${{}^{y}R}_{N} {{}^{x}R}_{N} {}^{x}R {}^{x}R$	${}_{N}^{y}{U_{N}^{x}} {}^{x}R {}^{x}R$
	${{}^{y}R}_{N} {}^{x}R {}^{x}R {}^{x}R$

Case 5: Three orthogonal constraint forces $$_{13}^{r}$ , $$_{14}^{r}$ , and $$_{15}^{r}$ passing through point o and a constraint force couple $$_{11}^{r}$

In this instance, the limb screw system of ${\hat{$}}_{1 - 5}$ can be easily obtained as follows:

{\hat{$}}_{1 - 5} = {$_{11}, $_{12}}

(10)

The limb chain configuration obtained is $({}^{xy}R {}^{xy}R)_{O}$ .

Case 6: A constraint force couple $$_{11}^{r}$ and any one or two constraint forces from $$_{13}^{r}$ , $$_{14}^{r}$ , and $$_{15}^{r}$

By obtaining the limb screw system and taking linear combinations, the following basic limb kinematic chains can be obtained:

\begin{matrix} {{{}^{x}R}_{o} {{}^{y}R}_{o} {}^{x}P {}^{y}P}, {{{}^{x}R}_{o} {{}^{y}R}_{o} {}^{x}P {}^{z}P}, {{{}^{x}R}_{o} {{}^{y}R}_{o} {}^{y}P {}^{z}P}, \\ {{{}^{x}R}_{o} {{}^{y}R}_{o} {}^{x}P}, {{{}^{x}R}_{o} {{}^{y}R}_{o} {}^{y}P}, {{{}^{x}R}_{o} {{}^{y}R}_{o} {}^{z}P} . \end{matrix}

Due to the presence of the inactive prismatic pair (which does not move when the moving platform rotates), the above limb chains are not adopted. By further combining the above basic limb kinematic chains, the effective limb chain configurations ${{}^{xy}{C_{O}} {}^{xy}{R_{O}}}$ and ${}^{xy}{C_{O}} {}^{xy}{C_{O}}$ can be obtained.

Synthesis of type II limb chains

As shown in Figure 1, Type II limb chains ensure that points A, B, b, and a are coplanar (second type coplanar constraint), eliminating the rotational DOF of the moving platform about its normal. Figure 5 shows the front view of the plane ABba. AB and ab are line segments on the fixed platform and the moving platform, respectively. It is assumed that $o' - x_{2} y_{2} z_{2}$ is the local coordinate system in the plane ABba, where $z_{2}$ is perpendicular to the plane ABba, $y_{2}$ is parallel to the edge AB, and $x_{2}$ is perpendicular to the edge AB.

Figure 5.

Kinematic constraints on ab.

To ensure that ab and AB remain coplanar, the type II limb chain needs to provide a constraint force $$_{21}^{r}$ parallel to the axis $z_{2}$ and two constraint couples $$_{22}^{r}$ and $$_{23}^{r}$ parallel to the axes $x_{2}$ and $y_{2}$ , respectively. Furthermore, the kinematic screw system can be obtained as follows:

{\hat{$}}_{2} = {\begin{matrix} $_{21} = (0 0 1; 0 0 0) \\ $_{22} = (0 0 0; 1 0 0) \\ $_{23} = (0 0 0; 0 1 0) \end{matrix}

(11)

Linearly combining the basic screws in ${\hat{$}}_{2}$ can obtain three types of basic limb kinematic chains, ${P_{1} P_{1} R_{1}}$ , ${R_{1} P_{1} R_{1}}$ , and R₁R₁R₁, where P₁ is in the plane ABba and R₁ is perpendicular to the plane ABba. At the same time, when ab moves in space, the angles between the plane ABba and the fixed platform and the moving platform change. Therefore, the plane ABba should be linked to the fixed platform and the moving platform with rotating pairs R₂ and R₃, respectively. The axes of R₂ and R₃ are parallel to AB and ab, respectively. The configuration of type II limb chains is shown in Table 5. In the table, ${{}^{1}U}^{3}$ indicates that the first rotation axis line of the U-pair is perpendicular to the plane ABba and that the second rotation axis line is parallel to ab. ${{}^{2}U}^{1}$ is defined analogously.

Table 5.

Configuration of type II limb chains.

Type	Basic limb chains	Limb chains with multi-DOF kinematic pairs
3R2P	$R_{2} {P_{1} P_{1} R_{1}} R_{3}$	$R_{2} P_{1} P_{1} {{}^{1}U}^{3}$	${{}^{2}U}^{1} P_{1} P_{1} R_{3}$	$C_{2} P_{1} [{{}^{1}U}^{3} / R_{1} R_{3}]$
		$C_{2} R_{1} P_{1} R_{3}$	$R_{2} P_{1} R_{1} C_{3}$	$[{{}^{2}U}^{1} / R_{2} R_{1}] P_{1} C_{3}$
4R1P	$R_{2} {R_{1} P_{1} R_{1}} R_{3}$	$R_{2} {P_{1} R_{1}} {{}^{1}U}^{3}$	${{}^{2}U}^{1} {P_{1} R_{1}} R_{3}$	${{}^{2}U}^{1} P_{1} {{}^{1}U}^{3}$
		$C_{2} R_{1} R_{1} R_{3}$	$C_{2} R_{1} {{}^{1}U}^{3}$	$[{{}^{2}U}^{1} / R_{2} R_{1}] R_{1} C_{3}$
5R	$R_{2} R_{1} R_{1} R_{1} R_{3}$	$R_{2} R_{1} R_{1} {{}^{1}U}^{3}$	${{}^{2}U}^{1} R_{1} R_{1} R_{3}$	${{}^{2}U}^{1} R_{1} {{}^{1}U}^{3}$

Synthesis of type III limb chains

Type III limb chains do not affect the rotational DOF of the moving platform, so they either only provide overconstraints that are the same or partially the same as those provided by central passive limb chain or do not provide any constraints at all. As shown in Figure 6, $$_{31}^{r}$ to $$_{33}^{r}$ are the structural constraints provided by the Type III limb chain to the moving platform, with the following four combinations:

Figure 6.

Possible constraint types provided by the type III limb chain.

Case 1: A constraint force $$_{31}^{r}$ that always passes through point o

In this instance, the limb chain provides a constraint force $$_{31}^{r}$ that always passes through point o. The synthesis method of this type of limb chain is the same as that of Case 2 for the type I limb chains. Similarly, three types of limb chain configurations can be obtained: 3R2P, 4R1P, and 5R. Furthermore, each type of limb chain can be divided into two subclasses according to the different number of rotation axes passing through point o. These two subclasses are discussed below:

Three revolute pair axes passing through point o

Linearly combining the basic screws can obtain limb chain configurations such as $({}^{i}R {}^{j}R {}^{k}R)_{O} {}^{y}P {}^{z}P$ and ${(^{i} R^{j} R)}_{O} {}^{yz}{\overset{⌢}{PR}}_{O} {}^{yz}P$ . Because the 3R spherical sub-chain meets all the motion needs of the moving platform, the prismatic pairs in the limb chain are inactive joints. Therefore, the above limb chains are not adopted. By further combining the basic limb kinematic chains, feasible limb chain configurations ${{{}^{i}C}_{O} {{}^{j}R}_{O} {{}^{k}C}_{O}}$ , $S ({}^{i}R {}^{j}R)_{O}$ , ${{}^{i}R}_{O} S {{}^{j}R}_{O}$ , and $({}^{i}R {}^{j}R)_{O} S$ can be obtained.

Two revolute pair axes passing through point o

In this case, the axes of the revolute pairs in the limb chain intersect at point o or are parallel to the constraint force $$_{31}^{r}$ , and the prismatic pair is always perpendicular to the constraint force $$_{31}^{r}$ . The limb chain configuration obtained is shown in Table 6. To reduce the article’s length, the symbol ↔ is used in Table 6 to indicate that the reverse order of this configuration is also feasible. For example, $\underset{\leftrightarrow}{{}_{O}^{xz}{U^{y}} {}^{xz}P {}^{xz}P {{}^{xyz}R}_{O}}$ indicates that both ${}_{O}^{xz}{U^{y}} {}^{xz}P {}^{xz}P {{}^{xyz}R}_{O}$ and ${{}^{xyz}R}_{O} {}^{xz}P {{}^{xz}P}^{y} U_{O}^{xz}$ can be used as limb chain configurations. The reverse order limb chain is not only a reversal of the forward order of kinematic pairs but also a reversal of the connection relationship between kinematic pairs. For example, ${}_{O}^{xz}{U^{y}}$ in forward order should be represented as ${{}^{y}U}_{O}^{xz}$ in reverse order.

Table 6.

Configuration of the type III limb chains in Case 1.

Type	Basic limb chains	Limb chains with multi-DOF kinematic pairs
3R2P	${{}^{i}R}_{O} {{}^{xz}P {}^{y}R {}^{xz}P} {{}^{j}R}_{O}$	$\underset{\leftrightarrow}{{{}^{xz}C}_{O} {^{xz} P {}^{y}R} {{}^{i}R}_{O}}$	$\underset{\leftrightarrow}{{{}^{i}R}_{O} {{}^{xz}C}_{O} {^{xz} P {}^{y}R}}$	$\underset{\leftrightarrow}{{{}^{xz}C}_{O} {}^{y}R {^{xz} {\overset{⌢}{PR}}_{O}}}$
	$\underset{\leftrightarrow}{{{}^{i}R}_{O} {{}^{j}R}_{O} {^{xz} P {}^{y}R {}^{xz}P}}$	$\underset{\leftrightarrow}{{}_{O}^{xz}{U^{y}}^{xz} P^{xz} P {{}^{i}R}_{O}}$	$\underset{\leftrightarrow}{{{}^{i}R}_{O} {}_{O}^{xz}{U^{y}}^{xz} P^{xz} P}$	$\underset{\leftrightarrow}{{{}^{i}R}_{O} {}_{O}^{xz}{\overset{⌢}{PU}}^{y} {}^{xz}P}$
	${^{xz} {\overset{⌢}{PR}}_{O}} {}^{y}R {{}^{xz}{\overset{⌢}{PR}}_{O}}$	$\underset{\leftrightarrow}{{}_{O}^{xz}{U^{y}} {}^{xz}{P^{xz}} {\overset{⌢}{PR}}_{O}}$	$\underset{\leftrightarrow}{{{}^{i}R}_{O} {}^{xz}{P^{y}} {\overset{⌢}{UP}}_{O}^{xz}}$	$\underset{\leftrightarrow}{{}_{O}^{xz}{\overset{⌢}{PU}}^{y} {}^{xz}{\overset{⌢}{RP}}_{O}}$
	$\underset{\leftrightarrow}{{{}^{i}R}_{O} {}^{xz}{\overset{⌢}{PR}}_{O} {^{xz} P {}^{y}R}}$	${{}^{xz}C}_{O} {}^{y}R {{}^{xz}C}_{O}$	$\underset{\leftrightarrow}{{{}^{xz}C}_{O} {}^{xz}{P^{y}} U_{O}^{xz}}$
4R1P	$\underset{\leftrightarrow}{i R_{O} {^{xz} P {}^{y}R} {}^{xz}{\overset{⌢}{PR}}_{O}}$	$\underset{\leftrightarrow}{{{}^{i}R}_{O} {}_{O}^{xz}{U^{y}} {^{xz} P {}^{y}R}}$	$\underset{\leftrightarrow}{{}_{O}^{xz}{U^{y}} {^{xz} P {}^{y}R} {{}^{i}R}_{O}}$	$\underset{\leftrightarrow}{{}_{O}^{xz}{\overset{⌢}{PU}}^{y} {}^{y}R {{}^{i}R}_{O}}$
	$\underset{\leftrightarrow}{{{}^{i}R}_{O} {{}^{j}R}_{O} {^{y} R {}^{xz}P {}^{y}R}}$	$\underset{\leftrightarrow}{{}^{xz}{\overset{⌢}{PR}}_{O} {{}^{y}R}^{y} U_{O}^{xz}}$	$\underset{\leftrightarrow}{{}^{y}{R^{y}} {\overset{⌢}{UP}}_{O}^{xz} {}^{i}{R_{O}}}$	$\underset{\leftrightarrow}{{}^{y}R {}^{y}R {{}^{xz}C}_{O} {{}^{i}R}_{O}}$
	${{}^{i}R}_{O} {{}^{y}R {}^{xz}P {}^{y}R} {{}^{j}R}_{O}$	$\underset{\leftrightarrow}{{{}^{i}R}_{O} {}^{y}R {}^{y}R {{}^{xz}C}_{O}}$	$\underset{\leftrightarrow}{{}_{O}^{xz}{U^{y}}^{y} {\overset{⌢}{UP}}_{O}^{xz}}$	$\underset{\leftrightarrow}{{}_{O}^{xz}{U^{y}} {}^{y}R {{}^{xz}C}_{O}}$
	$\underset{\leftrightarrow}{{}^{y}R {}^{y}R {}^{xz}{\overset{⌢}{PR}}_{O} {{}^{i}R}_{O}}$	${}_{O}^{xz}{U^{y}} {}^{xz}P {{}^{y}U}_{O}^{xz}$
	$\underset{\leftrightarrow}{{{}^{i}R}_{O} {}^{y}R {}^{y}R {}^{xz}{\overset{⌢}{PR}}_{O}}$
5R	$\underset{\leftrightarrow}{{{}^{i}R}_{O} {{}^{j}R}_{O} {}^{y}R {}^{y}R {}^{y}R}$	$\underset{\leftrightarrow}{{{}^{i}R}_{O} {}_{O}^{xz}{U^{y}} {}^{y}R {}^{y}R}$	$\underset{\leftrightarrow}{{}_{O}^{xz}{U^{y}} {}^{y}R {}^{y}R {{}^{i}R}_{O}}$	${}_{O}^{xz}{U^{y}} {{}^{y}R}^{y} U_{O}^{xz}$
	${{}^{i}R}_{O} {}^{y}R {}^{y}R {}^{y}R {{}^{j}R}_{O}$

Case 2: Two constraint forces $$_{31}^{r}$ and $$_{32}^{r}$ that always pass through point o

In this instance, the limb chain provides two constraint forces passing through point o. The limb screw system can be easily obtained as follows:

{\hat{$}}_{3 - 2} = {\begin{matrix} $_{31} = (1 0 0; 0 0 0) \\ $_{32} = (0 1 0; 0 0 0) \\ $_{33} = (0 0 1; 0 0 0) \\ $_{34} = (0 0 0; 0 0 1) \end{matrix}

(12)

Linearly combining the basic screws in ${\hat{$}}_{3 - 2}$ can obtain limb chain configurations such as $({}^{i}R {}^{j}R {}^{k}R)_{O} {}^{z}P$ and ${{}^{i}R}_{O} {{}^{j}R}_{O} {}^{y}R {{}^{k}R}^{k} R_{O}$ . Additionally, the above limb chains are not adopted since they contain inactive joints. By further combining the basic limb kinematic chains, a feasible limb chain configuration ${{{}^{i}C}_{O} {{}^{j}R}_{O} {}^{k}{R_{O}}}$ can be obtained.

Case 3: Three constraint forces $$_{31}^{r}$ , $$_{32}^{r}$ , and $$_{33}^{r}$ that always pass through point o

In this instance, the limb chain provides three constraint forces $$_{31}^{r}$ , $$_{32}^{r}$ , and $$_{33}^{r}$ that always pass through point o. The limb chain obtained is a spherical sub-chain $({}^{i}R {}^{j}R {}^{k}R)_{O}$ .

Case 4: No constraints are provided

In this instance, the limb chain does not provide any constraints on the moving platform, that is, it is an unconstrained limb chain. The widely used unconstrained limb chains are ${URS}$ , ${UPS}$ , ${UCU}$ , and ${SPS}$ .

Limb chain screening principle

In the above text, we used screw theory to synthesize the limb chain configurations corresponding to different constraint conditions. In this section, we formulate some optimization rules to screen out limb chain configurations with simple compositions and reasonable layouts. The details are provided in the following discussion:

(1) Two bar-three pair or three bar-four pair limb chains containing multi-DOF kinematic pairs (C, U, S) should be preferred. Limb chains composed of basic kinematic pairs often have disadvantages such as many components, complex forms, and self-interference. Replacing basic kinematic pairs with multi-DOF kinematic pairs can reduce the total number of components in a limb chain and simplify its complexity.

(2) The limb chain should contain only one driving joint and should be as close to the fixed platform as possible. An excessive number of driving devices matched with driving pairs will increase the mechanism’s inertia constant and reduce its dynamic characteristics. In addition, placing the drive joint close to the fixed platform can reduce the limb chain’s center of gravity height, thereby improving the mechanism’s dynamic performance. In the selection of driving joints, the revolute pair R can be used as a driving pair or passive pair. The U-pair and S-pair are passive pairs. The P-pairs are mostly used as driving pairs. When used as a passive pair, it often needs to be equipped with a guiding mechanism to avoid motion jamming.

(3) The limb chain should minimize space occupation and have a simple layout. Limb chains composed of the same kinematic pairs with different arrangements can have completely different occupation effects during movement. For example, ${}^{y}P {}^{x}R {}^{x}R$ and ${}^{x}R {}^{yz}P {}^{x}R$ have the same constituent elements, but when performing the same movement, ${}^{y}P {}^{x}R {}^{x}R$ requires a larger space in the y-direction than the latter.

Based on the above screening conditions, the preferred limb chains are shown in Table 7, where joints with underlines ¯ are driving joints.

Table 7.

Preferred limb chains.

Type	Constraint	Configuration of limb chains
Central limb	${\hat{$}}_{3 - 3}^{r}$	${}^{z}{\underline{R}} U {}^{z}R$	${}^{z}{\underline{R}} B {}^{z}R$
Type I	${\hat{$}}_{1 - 2}^{r}$	${}_{N}^{y}{U_{N}^{x}}^{yz} {\underline{C}}_{N} {}^{x}R$	$[_{N}^{x} U_{N}^{yz} /_{N}^{yz} U_{N}^{x}] {{}^{yz}R}_{N} {}^{yz}{\underline{P}} {}^{x}R$	${{}^{y}R}_{N} [_{N}^{x} U_{N}^{yz} /_{N}^{yz} U_{N}^{x}] {}^{yz}{\underline{P}} {}^{x}R$
		$S_{N} [{}^{yz}{\underline{P}} / {}^{x}{\underline{R}}] {}^{x}R$	${{}^{y}R}_{N} [_{N}^{x} U_{N}^{yz} /_{N}^{yz} U_{N}^{x}] {}^{x}{\underline{R}} {}^{x}R$	$[_{N}^{x} U_{N}^{yz} /_{N}^{yz} U_{N}^{x}] {{}^{yz}R}_{N} {}^{x}{\underline{R}} {}^{x}R$
	${\hat{$}}_{1 - 3}^{r}$	${}_{N}^{y}{U^{x}} {}^{yz}{\underline{P}} {{}^{x}U}_{N}^{yz}$	${}_{N}^{y}{U^{x}} {}^{x}{\underline{R}} {{}^{x}U}_{N}^{yz}$
	${\hat{$}}_{1 - 4}^{r}$	${}_{N}^{y}{U_{N}^{x}} {}^{yz}{\underline{P}} {}^{x}R$	${}_{N}^{y}{U_{N}^{x}} {}^{x}{\underline{R}} {}^{x}R$
	${\hat{$}}_{1 - 5}^{r}$	$({}^{y}{\underline{R}} {}^{x}R)_{O}$
Type II	${\hat{$}}_{2}^{r}$	$U \underline{P} U$	$U \underline{R} U$
Type III	${\hat{$}}_{3 - 1}^{r}$	${}_{O}^{xz}{U^{y}} {}^{xz}{\underline{P}} {{}^{y}U}_{O}^{xz}$	${}_{O}^{xz}{U^{y}} {}^{y}{\underline{R}} {{}^{y}U}_{O}^{xz}$	${}^{i}{\underline{R}}_{O} {S {{}^{j}R}_{O}}$
	${\hat{$}}_{3 - 3}^{r}$	$(^{i} \underline{R} {}^{j}R {}^{k}R)_{O}$
	–	$U [\underline{R} / \underline{P}] S$	$S [\underline{R} / \underline{P}] U$	$U \underline{C} U$

Type synthesis of SHMs

Based on the aforementioned analysis, a series of SHM configurations can be obtained by assembling the limb chains presented in Table 7 under certain conditions. According to the different positions of the rotation center in the mechanism, SHMs can be classified into two types: SHMs with a rotation center close to the fixed platform and those with a rotation center close to the moving platform.

Selection and spatial distribution of limb chains

Collating the obtained types reveals that the spatial distribution of SHM limb chains can be classified into three categories, as shown in Figure 7.

Figure 7.

The spatial distribution of outer limb chains: (a) type I and II limb chains, (b) type II and III limb chains, and (c) only type III limb chains.

The limb chain composition types corresponding to each spatial layout are as follows:

(1) The outer limb chains are type I and type III limb chains (Figure 7(a)), in which type I limb chains make Aoca four-point constant coplanar and constrain the redundant rotational DOF of the SPM moving platform, and type III limb chains provide another driving input needed for stable motion of the mechanism. The assembly conditions between these chains are as follows:① The origin points $o_{2}$ or N of the Type I limb chain coordinate system are set on the y-axis. Point O in the limb chain coincides with the rotation center of the SHM, and each kinematic pair is arranged in the direction specified by its superscript and subscript. ② The configuration of type III limb chains is obtained in the xoz plane. When constructing the mechanism, it can be arranged in the xoz plane first, and then the entire limb chain can be rotated around the z-axis to the desired plane. Point O in the limb chain coincides with the rotation center of the SHM. ③ The recommended distribution angle τ between the two chains is 90°, as this configuration provides good support for the SPM moving platform. Additionally, the mechanism layout can be determined through specific optimization objective functions for the mechanism’s dimensional synthesis.

(2) The outer chains consist of type II and type III limb chains (Figure 7(b)). The type II limb chain ensures that the four points ABba remain coplanar, thereby eliminating redundant rotational DOF in the SPM moving platform. The type III limb chain provides an additional drive input necessary for stable motion of the mechanism. The assembly conditions between these limb chains are as follows:

④ The axes of the starting and ending revolute pairs of the type II limb chain are arranged on a set of parallel straight lines AB and ab (AB and ab are located on the fixed and moving platforms of SPM, respectively), and each kinematic pair is arranged in the direction specified by its superscript and subscript. Only unconstrained limb chains in the type III limb chain can be arranged within the plane ABba, while the remaining type III limb chains are still arranged in the direction specified by their superscript and subscript. ⑤ The spatial layout of the limb chain is generally shown in Figure 7(b), and the specific structural parameters need to be determined through dimensional synthesis of ad, cd, AD, oD, and oc.

(3) All three outer limb chains consist of a type III limb chain (Figure 7(c)), which forms the requested SHM by coupling with the center limb chain. The distribution of the limb chains is as follows: ⑥ evenly distributed circumferentially.

In addition, in all three cases mentioned above, the following conditions are met: ⑦ Both ends of the central limb chain pass through the fixed and moving platforms of the SPM and are connected to them through a revolute pair. The rotation center of the U-pair or B-pair coincides with the rotation center of the SHM.

Type synthesis of the rotation center close to the fixed platform

When the rotation center is arranged close to the fixed platform, the mechanism can have a larger swing radius and swing line displacement space. By arbitrarily selecting central limb chains (CLCs), type I, type II, and type III limb chains from Table 7 and arranging and combining them according to the limb chain composition and assembly conditions shown in Table 8, a series of SHMs with rotation centers close to the fixed platform can be obtained. The typical configurations and their examples are shown in Table 8 and Figures 8 to 10.

Table 8.

Limb chain composition, assembly conditions, and typical configurations.

Limb composition	Assembly conditions	Typical configurations
CLC + I + III	①,②,③,⑦	R[U/B]R-[SRR/URR]-[URS/SRU], R[U/B]R-RR-[RRR/RSR/RRS]
		R[U/B]R-[UCR/SPR/UPR]-[UCU/UPS/SPU]
CLC + II + III	②,④,⑤,⑦	R[U/B]R-UPU-[UCU/UPS/SPU], R[U/B]R-URU-[URS/SRU]
CLC + 2II	④,⑤,⑦	R[U/B]R-2UPU, R[U/B]R-2URU
CLC + 3III	②,⑥,⑦	R[U/B]R-3[UCU/UPS/SPU/RRR/RSR/RRS/URS/SRU]

Figure 8.

Examples of the SHMs containing type I limb chains: (a) RUR-SRR-SRU, (b) RUR-URR-URS, (c) RBR-RR-RRR, (d) RBR-UCR-UCU, (e) RBR-SPR-UPS, and (f) RBR-UPR-UPS.

Figure 9.

Examples of the SHMs containing type II limb chains: (a) RBR-UPU-UPS, (b) RUR-URU-URS, and (c) RBR-2UPU.

Figure 10.

Examples of the SHMs containing type III limb chains only: (a) RBR-3RSR, (b) RBR-3RRR, and (c) RBR-3UCU.

Type synthesis of the rotation center close to the moving platform

By arranging the rotation center close to the moving platform, new configurations can be obtained. At the same time, the rotation radius decreases, and the mechanism theoretically has better dynamic characteristics. The synthesis process is the same as before and involves first selecting limb chains based on the screening principle and then arranging and combining them to obtain the target configuration.

Preferred limb chain configurations

Based on the synthesized limb chains and combined with the established screening principle, the preferred limb chains can be obtained when the rotation center is close to the moving platform, as shown in Table 9.

Table 9.

Preferred limb chains when the rotation center is close to the moving platform.

Type	Constraint	Configuration of limb chains
CLC	${\hat{$}}_{3 - 3}^{r}$	${}^{z}{\underline{R}} U {}^{z}R$	${}^{z}{\underline{R}} B {}^{z}R$
Type I	${\hat{$}}_{1 - 2}^{r}$	${}^{x}R {}^{yz}{\underline{P}} S_{N}$	${}^{x}R {}^{yz}{\underline{P}} [_{N}^{x} U_{N}^{yz} /_{N}^{yz} U_{N}^{x}] {{}^{y}R}_{N}$	${}^{x}R {}^{yz}{\underline{P}} {{}^{yz}R}_{N} [_{N}^{x} U_{N}^{yz} /_{N}^{yz} U_{N}^{x}]$
		$[^{z} \underline{P} / {}^{x}{\underline{R}}] {}^{x}R S_{N}$	${}^{z}{\underline{P}} {}^{x}R [_{N}^{x} U_{N}^{yz} /_{N}^{yz} U_{N}^{x}] {{}^{y}R}_{N}$	${}^{z}{\underline{P}} {}^{x}R {{}^{yz}R}_{N} [_{N}^{x} U_{N}^{yz} /_{N}^{yz} U_{N}^{x}]$
		${{}^{x}R}^{yz} {\underline{C}}_{N} {}_{N}^{x}{U_{N}^{y}}$	${}^{x}{\underline{R}} {}^{x}R [_{N}^{x} U_{N}^{yz} /_{N}^{yz} U_{N}^{x}] {{}^{y}R}_{N}$	${}^{x}{\underline{R}} {}^{x}R {{}^{yz}R}_{N} [_{N}^{x} U_{N}^{yz} /_{N}^{yz} U_{N}^{x}]$
	${\hat{$}}_{1 - 3}^{r}$	${}^{z}{\underline{C}}_{N} {{}^{x}R}^{x} U_{N}^{y}$	${}^{z}{\underline{P}} {}_{N}^{yz}{U^{x}}^{x} U_{N}^{y}$	${}_{N}^{yz}{U^{x}} [{}^{yz}{\underline{P}} / {}^{x}{\underline{R}}]^{x} U_{N}^{y}$
	${\hat{$}}_{1 - 4}^{r}$	${}^{x}R {}^{yz}{\underline{P}} {}_{N}^{x}{U_{N}^{y}}$	${}^{z}{\underline{P}} {}^{x}R {}_{N}^{x}{U_{N}^{y}}$	${}^{x}{\underline{R}} {}^{x}R {}_{N}^{x}{U_{N}^{y}}$
	${\hat{$}}_{1 - 5}^{r}$	$({}^{x}{\underline{R}}^{y} {R)}_{O}$
Type II	${\hat{$}}_{2}^{r}$	$U \underline{P} U$	$U \underline{R} U$
Type III	${\hat{$}}_{3 - 1}^{r}$	${}^{i}{\underline{R}}_{O} {S {{}^{j}R}_{O}}$	${}_{O}^{xz}{U^{y}} {}^{y}{\underline{R}} {{}^{y}U}_{O}^{xz}$	${}_{O}^{xz}{U^{y}}^{xz} \underline{P} {{}^{y}U}_{O}^{xz}$
	${\hat{$}}_{3 - 3}^{r}$	$(^{i} \underline{R} {{}^{j}R}^{k} {R)}_{O}$
	–	$S \underline{P} U$	${\underline{R} U} S$	${U \underline{P}} S$
		${S \underline{R}} U$	${U \underline{C}} U$	$\underline{P} S S$

Type synthesis

By arbitrarily selecting central limb chains (CLCs), type I, type II, and type III limb chains from Table 9 and arranging and combining them according to the limb chain composition and assembly conditions shown in Table 10, a series of SHMs with rotation centers close to the moving platform can be obtained. The typical configurations and their examples are shown in Table 10 and Figures 11 to 13.

Table 10.

Limb composition, assembly conditions, and typical configurations.

Limb composition	Assembly condition	Typical configurations
CLC + I + III	①,②,③,⑦	R[U/B]R-[RRU/RRS]-[RSU/RUS/URS/SRU]R[U/B]R-RR-[RRR-RSR-RRS]R[U/B]R-[RPS/RCU/RPU]-[SPU/UPS/UCU]R[U/B]R-[PRS/PRU/CRU]-[PUS/CUU/PSS]
CLC + II + III	②,④,⑤,⑦	R[U/B]R-UPU-[UCU/UPS/SPU], R[U/B]R-URU-[URS/SRU]
CLC + 2II	④,⑤,⑦	R[U/B]R-2UPU, R[U/B]R-2URU
CLC + 3III	②,⑥,⑦	R[U/B]R-3[SPU/UPS/UCU], R[U/B]R-3[PUS/CUU/PSS],R[U/B]R-3[RRR/RRS/RSR], R[U/B]R-3[SRU/URS/RUS/RSU]

Figure 11.

Examples of the SHMs containing type I limb chains: (a) RBR-RCU-UCU, (b) RBR-RPU-SPU, (c) RUR-PRS-PUS, (d) RBR-PRU-PUS, (e) RBR-CRU-CUU, and (f) RBR-RRU-RSU.

Figure 12.

Examples of the SHMs containing type II limb chains: (a) RBR-UPU-UPS, (b) RUR-URU-URS, and (c) RBR-2UPU.

Figure 13.

Examples of the SHMs containing type III limb chains only: (a) RBR-3RUS, (b) RUR-3PUS, and (c) RBR-3CUU.

Characteristics of various configurations

(1) From the perspective of the central limb chain: The rotational speed transmitted by the RUR limb chain is not uniform, and periodic fluctuations occur,³⁴ making it difficult to accurately control the angular displacement of its rotation during spatial motion. The configuration using such a limb chain can be used for situations with low transmission torque (due to the lower stiffness of the U-pair) and low-precision control of the output angular displacement and angular velocity. The ball cage joint has the advantages of a high transmission torque, strong tensile and compressive strength, constant speed transmission, and easy control, making the RBR limb chain an excellent transmission and unloading limb chain. Such limb chains can be used in scenarios with high transmission torque and high-precision control of the output angular displacement and angular velocity. (2) From the perspective of the position of the rotation center: When the rotation center is close to the fixed platform, the mechanism has a large rotation radius and a large linear displacement space. When the rotation center is close to the moving platform, the rotation radius of the mechanism decreases, and the dynamic performance is enhanced. (3) From the perspective of constraint type: The configuration obtained based on the first type of coplanar constraint exhibits partial kinematic decoupling because the rotation axis of the first type of coplanar constraint is the y-axis and the rotation around the y-axis is only controlled by the type III limb chain. The configuration obtained based on the second type of coplanar constraint shows decoupling only for the motion of the central limb chain. However, when the mechanism is only composed of a central transmission limb chain and two type II limb chains, the mechanism can be subject to more overconstraints that restrict the rotation of the moving platform with the minimum number of limb chains, thus yielding greater structural stiffness. The configuration with only type III limb chains uses a uniform distribution of the same limb chains, and its performance is characterized by central symmetry in spatial distribution. The recommended configurations from different perspectives are shown in Table 11. Users can select appropriate configurations from the corresponding classification based on the characteristics of the application scenario.

Table 11.

Recommended configurations from various perspectives.

Outer limbs	Large rotation radius	Example	Small rotation radius	Example
Type I and III	R[U/B]R-UPR-UPS	Figure 8(f)	R[U/B]R-PRS-PUS	Figure 11(c)
Only Type II	R[U/B]R-2UPU	Figure 9(c)	R[U/B]R-2UPU	Figure 12(c)
Only type III	R[U/B]R-3RSR	Figure 10(a)	R[U/B]R-3PUS	Figure 13(b)

An example of the obtained SHMs

Figure 14 shows a RBR-2RRR SHM that can be used for vector propulsion devices or human biomimetic joints. The coordinate systems o-XYZ and o-xyz are established with the rotation center of the mechanism as the origin. The initial attitudes of the two coordinate systems coincide and are fixedly connected to the fixed platform and the moving platform. The Z-axis is perpendicular to the fixed platform upwards, and the X-axis is located in the plane where the Z-axis and rotation axis $R_{1}^{1}$ are located. $R_{1}^{i}$ , $R_{2}^{i}$ , and $R_{3}$ are the unit direction vectors of each revolute pair, where the superscript i denotes the i-th limb chain, where i = 1, 2. The structural parameters between the revolute pairs are shown in Figure 14. In the initial state, the plane $R_{1}^{i} {oR}_{2}^{i}$ is coplanar with the Z-axis, and $R_{2}^{i}$ points upwards. $θ_{i}$ is the input angle of each limb chain.

Figure 14.

Structural parameters of RBR-2RRR.

Inverse kinematics and simulation verification

The global coordinates of $R_{2}^{i}$ and $R_{3}$ in o-XYZ are expressed as follows:

R_{2}^{i} = Rot (z, n_{i}) Rot (y, π - φ_{1}) Rot (z, θ_{i}) Rot (y, - φ_{2}) [0 0 1]^{T}

(13)

R_{3} = Rot (z, α) Rot (y, β) [0 0 1]^{T}

(14)

where $Rot (y, θ_{i})$ and $Rot (z, θ_{i})$ are the rotation transformation matrices around the y- and z-axes, respectively. $θ_{i}$ is the input angle of the mechanism. $n_{i}$ is the angle between the $R_{1}^{i} oZ$ plane and the $XoZ$ plane, where $n_{1} = 0 °$ , $n_{2} = 120 °$ . $α$ , $β$ , and $γ$ are the ZYZ Euler angle parameters of o-xyz relative to o-XYZ.

From the geometrical relationship between the revolute pairs $R_{2}^{i}$ and $R_{3}$ , the following expression can be obtained:

R_{2}^{i} \cdot (R_{3})^{T} = \cos φ_{3}

(15)

Substituting equations (13) and (14) into equation (15) yields the following expression:

θ_{i} = 2 \arctan (\frac{a \pm \sqrt{(a^{2} + b^{2} - c^{2})}}{b + c}) i = 1, 2

(16)

where

$a = - s (α - n_{i}) s β s φ_{2}$

$b = s φ_{2} (c (α - n_{i}) s β c φ_{1} + c β s φ_{1})$

$c = c φ_{2} (c β c φ_{1} - c (α - n_{i}) s β s φ_{1}) + c φ_{3}$

In the central limb chain, the output shaft and input shaft do not rotate relative to each other. Therefore, the output shaft needs to rotate $- α$ around the z-axis during the last attitude transformation to offset the effect of rotating $α$ around the z-axis first when using the ZYZ Euler angle transformation. Therefore, the following expression can be obtained:

θ_{3} = α + γ

(17)

Next, the correctness of the inverse kinematics solution is verified through virtual prototype simulation in ADAMS. The parameters of the output axis attitude angle are set to $α = π t / 5$ , $β = π t / 36$ , $γ = - 7 π t / 60$ , and the simulation time is 10 s. The simulation curve of the input angle $θ_{i}$ is shown in Figure 15(b). Meanwhile, the theoretical curve of the input angle $θ_{i}$ can be plotted in MATLAB using equations (16) and (17), as shown in Figure 15(a). The numerical values and variation trends of the theoretical curve and simulation curve are the same, which verifies the correctness of the inverse kinematics solution and the feasibility of the mechanism’s motion.

Figure 15.

Input angle $θ_{i}$ : (a) theoretical curves in MATLAB and (b) simulation curves in ADAMS.

Singularity and workspace analysis

Limb singularity and platform singularity³⁷ occur in the RBR-2RRR mechanism. The condition for limb singularity to occur is that the revolute pairs $R_{1}^{i}$ , $R_{2}^{i}$ , and $R_{3}$ on the same limb chain are coplanar, as shown in Figure 16(a) and (b). The condition for parallel singularity to occur is that the revolute pairs $R_{2}^{1}$ , $R_{3}$ , and $R_{2}^{2}$ are coplanar, as shown in Figure 16(c) and (d).

Figure 16.

Singularity and workspace: (a) the first type of limb singularity, (b) the second type of limb singularity, (c) the first type of platform singularity, (d) the second type of platform singularity, and (e) the singular configuration and workspace.

If we want the workspace to avoid limb singularity and platform singularity, the structural parameters of the RBR-2RRR mechanism need to meet the following conditions:

{\begin{matrix} φ_{1} + φ_{2} - φ_{3} < π - β \\ φ_{1} + φ_{2} + φ_{3} > π + β \end{matrix}

(18)

{\begin{matrix} | R_{2}^{1} R_{2}^{2} | \geq \sin φ_{3} \sqrt{2 - 2 \cos δ} \\ | R_{2}^{1} R_{2}^{2} | \leq \sin φ_{3} \sqrt{2 + 2 \cos δ} \end{matrix}

(19)

where $δ$ is the minimum angle between the planes $R_{2}^{1} o R_{3}$ and $R_{2}^{2} o R_{3}$ , and it is set to avoid platform singularity.

The target workspace is set to $β$ =50°, and the structural parameters of the RBR-2RRR mechanism are set to $φ_{1}$ =80°, $φ_{2}$ = $φ_{3}$ , and $δ$ =3°. By combining equations (18) and (19), we can determine that $φ_{2}$ >83.8°. Without considering other optimization indicators, $φ_{2}$ = 85° is utilized. The distribution of the workspace and singular configuration are implemented in a polar coordinate system $(α, β)$ , as shown in Figure 16(e). The black and red solid lines are the limb singularity configuration and platform singularity configuration of the mechanism, respectively. The green area is the workspace of the RBR-2RRR, and its three-dimensional solid is a spherical cone with a cone angle of 100°. The rotation range of the mechanism around the moving platform’s normal direction in the entire workspace is ±180°. Compared with the agile eye’s ±70° swing angle and ±30° spin angle attitude space,¹ although the swing space of RBR-2RRR is somewhat reduced, its ability to rotate around the normal direction of the moving platform is significantly improved.

Dexterity analysis

By taking the derivative of equations (15) and (17), the following expression can be obtained:

[\begin{matrix} A_{11} & A_{12} & 0 \\ A_{21} & A_{22} & 0 \\ 1 & 0 & 1 \end{matrix}] [\begin{matrix} \overset{\cdot}{α} \\ \overset{\cdot}{β} \\ \overset{\cdot}{γ} \end{matrix}] = [\begin{matrix} B_{11} \\ B_{22} \\ 1 \end{matrix}] [\begin{matrix} {\overset{\cdot}{θ}}_{1} \\ {\overset{\cdot}{θ}}_{2} \\ {\overset{\cdot}{θ}}_{3} \end{matrix}] or A {\overset{\cdot}{θ}}_{α β γ} = B \overset{\cdot}{θ}

(20)

where

$A_{i 1} = s β (s (α - n_{i}) (c φ_{1} c θ_{i} s φ_{2} + s φ_{1} c φ_{2}) + c (α - n_{i}) s θ_{i} s φ_{2})$ , i=1, 2

$A_{i 2} = c β (s (α - n_{i}) s θ_{i} s φ_{2} - c (α - n_{i}) (c φ_{1} c θ_{i} s φ_{2} + s φ_{1} c φ_{2})) + s β (s φ_{1} c θ_{i} s φ_{2} - c φ_{1} c φ_{2})$ , i=1, 2

$B_{ii} = - s φ_{2} (s β (c (α - n_{i}) c φ_{1} s θ_{i} + s (α - n_{i}) c θ_{i}) + c β s φ_{1} s θ_{i})$ , i=1, 2

To this end, the kinematic Jacobian matrix of the mechanism is expressed as follows:

J = Φ A^{- 1} B, Φ = [\begin{matrix} 0 & - s α & c α s β \\ 0 & c α & s α s β \\ 1 & 0 & c β \end{matrix}]

(21)

where $Φ$ is the transformation matrix between the ZYZ Euler angle rates and angular velocities.

The condition numbers of the Jacobi matrix,³⁸ the kinematic conditioning index,³⁸ and the global conditioning index³⁹ of the mechanism are respectively represented as follows:

k (J) = ‖ J ‖ ‖ J^{- 1} ‖ (J is a nonsingular square matrix)

(22)

D = k (J)_{min}^{- 1} \times 100 %

(23)

η_{J} = \int_{W} \frac{1}{k_{J}} dW / \int_{W} dW

(24)

$k (J)$ is calculated within $α \in [0, 360]$ , $β \in [0, 50]$ , and its distribution within the reachable workspace is obtained, as shown in Figure 17. Among them, $k (J)_{min}$ is 1.4834, obtained at $α$ =132° and $β$ =32.04°. Further calculation shows that the dexterity indicators D and $η_{J}$ of the mechanism are 67.41% and 0.5334, respectively.

Figure 17.

The distribution of $k (J)$ in the reachable workspace.

Stiffness comparison

The deformation response of the RBR-2RRR mechanism and the 3-RRR mechanism to the same load is compared in this section utilizing the ANSYS static analysis module. The comparison demonstrates that the proposed type synthesis method can enhance the stiffness of the spherical mechanism.

In the modeling stage, the structural parameters of the RBR-2RRR mechanism are consistent with those in section “Singularity and workspace analysis.” The 3-RRR mechanism adopts the second set of structural parameters optimized for stiffness in Table 1 of the literature.⁴⁰ Specifically, the angle between adjacent revolute pairs is 90°, and the angle between the revolute pairs’ axes on the moving platform and the fixed platform and the normal of the platform is 60°. The width and thickness of the limb chains and the rotation shaft radius of the revolute pair are the same in the two mechanisms.

The settings in the ANSYS static analysis module are as follows: the materials for all components are structural steel, except for the outer race, inner race, steel ball, and cage in the ball cage joint, which are 42CrMo, 42CrMo, GCr15, and 20CrMnTi, respectively. The starting revolute pair of each limb chain is bound to the fixed platform, and the contact between the other revolute pairs is set to no separation. The contact between the inner and outer surfaces of the cage is set to be frictionless, and the contact between the steel ball and the inner and outer raceways of the ball cage joint is set to be friction contact with a friction coefficient of 0.1. The unit size is 2 mm, and automatic mesh generation is performed. Finally, a perpendicular load of 2000 N is applied to the moving platforms of both mechanisms.

As shown in Figure 18(a), the maximum deformation of the RBR-2RRR mechanism is 0.03 mm, which occurs in the plane where the moving platform bears the load. The maximum stress value is 638.11 MPa, located on the steel ball inside the ball cage joint of the central transmission limb chain (Figure 18(b)). The maximum deformation of the 3-RRR mechanism is 0.446 mm, which occurs in the plane where the moving platform bears the load (Figure 18(c)). The maximum stress value is 224.91 MPa, located on the shaft of the middle rotating pair (Figure 18(d)). Meanwhile, by comparing Figure 18(b) and (d), it can be seen that under vertical load, compared to the 3-RRR mechanism, the force transmission of the RBR-2RRR mechanism mainly occurs on the central transmission limb chain, and the load force on the outer limb chain of the mechanism is very small, which realizes the unloading of the external force.

Figure 18.

Deformation and stress results: (a) total deformation of RBR-2RRR SHM, (b) equivalent stress of RBR-2RRR SHM, (c) total deformation of 3-RRR SPM, and (d) equivalent stress of 3-RRR SPM.

In summary, in the case that the width and thickness of the connecting rod are approximately the same and the radius of the rotation shaft is the same, the deformation of the RBR-2RRR SHM is much smaller than that of the 3-RRR SPM after applying the same load along the normal direction of the moving platform to both mechanisms, which indicates that the type synthesis method proposed in this paper can improve the stiffness of the spherical mechanism.

Conclusion

In this article, a type synthesis method for SHMs with fixed rotation centers is proposed by coupling a serial transmission limb chain with the central passive limb chain of the SPM near the center area of the SPM, which improves the load-bearing capacity and the rotation range about the normal of the moving platform of the SPM. Based on the analysis of the spherical mechanism configuration’s research status with a fixed rotation on the center, a method for systematically synthesizing SHMs is given, and four types of limb chains are developed. Based on screw theory, the possible limb constraint systems provided by each type of limb chain are analyzed, and the type synthesis of each type of limb chain is carried out. Screening rules are proposed to obtain the preferred limb chains from the limb chain configuration results. Depending on the difference in the mechanism’s center of rotation and the constraint type, a series of SHMs with different fixed centers of rotation, unrestricted normal rotation around the moving platform, and high load-bearing capacity were obtained by selecting the preferred limb chains from the corresponding types of limb chains via permutation and combination. In this work, the configuration types of SHMs have been improved, providing theoretical support for their design and engineering applications.

Meanwhile, it should be pointed out that the overall workspace of the SHMs obtained are mainly determined by the central transmission limb chain. Due to the limitation of the deflection angle of the central transmission limb chain, the mechanism has a slightly smaller swing space (generally, it can reach ±50°, and with a special ball cage joint,⁴¹ it can reach ±60°). However, within the allowable swing range, the configuration obtained in this study can still be used as a supplement to the existing configuration and has considerable development potential.

Footnotes

Appendix

Notation

B	Ball cage constant velocity joint
C	Cylindrical pair
H	Screw pair
P and P_c	Prismatic pair and circular arc-shaped prismatic pair
R	Revolute pair
S	Spherical pair
U	Universal joint
$$$ and $$^{r}$	Kinematic screw and constraint screw
$\hat{$}$ and ${\hat{$}}^{r}$	Screw system and constraint system

Handling Editor: Chenhui Liang

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China [grant number 52075466].

ORCID iDs

Shuyang Shi

Kai Li

Zheng Ma

Yulin Zhou

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Type synthesis of 3-DOF spherical hybrid mechanisms with fixed centers of rotation

Abstract

Keywords

Introduction

Adopting a coaxial input spherical parallel configuration scheme

Adopting a spherical hybrid configuration scheme

Constraint-synthesis method based on screw theory and synthesis ideas for SHMs

Constraint-synthesis method based on screw theory

Synthesis ideas for SHMs

Synthesis and preferred selection of limb chains

Symbol description

Synthesis of central transmission limb chains

Unequal speed central transmission limb chain

Constant velocity central transmission limb chain

Synthesis of type I limb chains

Case 1: A constraint force couple $ 11 r

Case 2: A constraint force $ 12 r

Case 3: A constraint force couple $ 11 r or a constraint force $ 12 r , depending on the instantaneous spatial position of the limb chain

Case 4: Both a constraint force couple $ 11 r and a constraint force $ 12 r

Case 5: Three orthogonal constraint forces $ 13 r , $ 14 r , and $ 15 r passing through point o and a constraint force couple $ 11 r

Case 6: A constraint force couple $ 11 r and any one or two constraint forces from $ 13 r , $ 14 r , and $ 15 r

Synthesis of type II limb chains

Synthesis of type III limb chains

Case 1: A constraint force $ 31 r that always passes through point o

Three revolute pair axes passing through point o

Two revolute pair axes passing through point o

Case 2: Two constraint forces $ 31 r and $ 32 r that always pass through point o

Case 3: Three constraint forces $ 31 r , $ 32 r , and $ 33 r that always pass through point o

Case 4: No constraints are provided

Limb chain screening principle

Type synthesis of SHMs

Selection and spatial distribution of limb chains

Type synthesis of the rotation center close to the fixed platform

Type synthesis of the rotation center close to the moving platform

Preferred limb chain configurations

Type synthesis

Characteristics of various configurations

An example of the obtained SHMs

Inverse kinematics and simulation verification

Singularity and workspace analysis

Dexterity analysis

Stiffness comparison

Conclusion

Footnotes

Appendix

Declaration of conflicting interests

Funding

ORCID iDs

References

Case 1: A constraint force couple $$_{11}^{r}$

Case 2: A constraint force $$_{12}^{r}$

Case 3: A constraint force couple $$_{11}^{r}$ or a constraint force $$_{12}^{r}$ , depending on the instantaneous spatial position of the limb chain

Case 4: Both a constraint force couple $$_{11}^{r}$ and a constraint force $$_{12}^{r}$

Case 5: Three orthogonal constraint forces $$_{13}^{r}$ , $$_{14}^{r}$ , and $$_{15}^{r}$ passing through point o and a constraint force couple $$_{11}^{r}$

Case 6: A constraint force couple $$_{11}^{r}$ and any one or two constraint forces from $$_{13}^{r}$ , $$_{14}^{r}$ , and $$_{15}^{r}$

Case 1: A constraint force $$_{31}^{r}$ that always passes through point o

Case 2: Two constraint forces $$_{31}^{r}$ and $$_{32}^{r}$ that always pass through point o

Case 3: Three constraint forces $$_{31}^{r}$ , $$_{32}^{r}$ , and $$_{33}^{r}$ that always pass through point o