The Mobility Analysis of a 3-CCR Parallel Manipulator with Three Screw-Type Terminal Constraints

Abstract

In the traditional three DOFs (degrees of freedom) PMs (parallel manipulators), the terminal constraints are constraint forces and torques. A recent study showed that the terminal constraints of some three DOFs PMs are screw-type constraints. However, determining the mobility of this class of PMs remains challenging. In order to solve the above problems, this paper discusses the mobility of PMs having independent screw-type terminal constraints by introducing the principal screws. Firstly, taking a 3-CCR (C: cylindrical joint, R: revolute joint) PM as an example, the problem of identifying the mobility of PMs having screw-type constraints is proposed. Secondly, combined with the analytical expression of the terminal constraint and theory of quadratic curve decomposition, the mobility determination approach is given. Finally, a numerical example for solving the principal screws and determining the mobility of the 3-CCR PM is provided. Furthermore, the axodes of four poses with three helical DOFs and four poses with three rotational DOFs were plotted. The results show that there is a special phenomenon in which three rotational DOFs and three helical DOFs exist alternately in the workspace of this PM. This paper provides a method for directly identifying the independent helical DOFs in parallel manipulators and for studying the distribution of helical DOFs within the workspace.

1. Introduction

Mobility analysis is one of the most fundamental issues in type synthesis, kinematic, and dynamic studies of parallel manipulators [1]. Mobility analysis includes the number of degrees of freedom and motion property of the end-effector.

The 3-CCR PM with three DOFs has attracted much attention from scholars. Hervé and Karouia [2] analyzed the assembly conditions and singular configurations of non-overconstrained 3-CCR PMs with three spherical DOFs based on the Lie group method. Callegari et al. [3] proposed a 3-RCC PM with three translational DOFs and investigated its dynamic and position/force hybrid control. Chaker et al. [4] investigated the impact of clearance and manufacturing errors on the accuracy of a 3-CCR PM with three spherical DOFs. Rodriguez-Leal et al. [5] analyzed the instantaneous mobility of the 3-RCC PM with three translational DOFs using screw theory. Shen et al. [6] studied the motion characteristics of 3-RCC mechanisms with three spherical DOFs using the position and orientation characteristics method. Ma et al. [7] applied the 3-RCC PM with three translational DOFs to the vehicle seat suspension and effectively reduced its vibrations. The existing research on 3-CCR PMs mainly focused on the PMs with constraint forces and torques, while studies of the 3-CCR PMs having screw-type constraints have not been attempted. Studying the mobility of the 3-CCR PM with three screw-type constraints is beneficial for gaining a deeper understanding of the types of DOFs in PMs under the influence of multiple screw-type constraints. This promotes the application of PMs with multiple independent helical DOFs, such as in minimally invasive surgical robots and biomimetic joints.

During the past 150 years, scholars have made intensive efforts in the mobility analysis of mechanisms. The corresponding studies include screw theory-based methods [8,9,10,11], graph methods [12,13], Lie group-based methods [14,15], linear transformation-based methods [16], position and orientation characteristics (POC) methods [17], geometric algebra-based methods [18,19], and computer-aided methods [20,21,22]. Screw theory-based methods establish a general approach for analyzing the DOFs of mechanisms using the theory of the screw systems and reciprocal relationships [8,9,10,11], which have been widely applied to the mobility analysis of lower-mobility PMs, metamorphic PMs, and hybrid mechanisms. Graph-based methods provide a visual representation of screw systems using numerical tools to map various geometric relationships between constraints and DOFs [12,13]. This method establishes a one-to-one correspondence between the properties of DOFs, motion screw systems, and constraint screw systems through graphical representations. Lie group-based methods employ Euclidean subgroups to describe the fundamental motions of kinematic joints and end-effectors using multiplication and intersection operations between subgroups to calculate the DOFs [14,15]. The linear transformation-based method requires the establishment of a Jacobian matrix that maps the joint velocity space to the end-effector output velocity. The DOFs are determined by the rank of this Jacobian matrix [16]. Other methods, such as geometric algebra and Clifford Algebra, are also used to analyze the DOFs of PMs [18,19]. Furthermore, with the rapid development of computer technology, automated mobility analysis methods based on CAD technology and machine learning-based DOFs prediction methods have also begun to attract attention [20,21,22].

Up to now, these DOFs determination methods have primarily focused on PMs with only forces/torque constraints. It is not easy to obtain DOFs simply through the traditional process for PMs that have screw-type terminal constraints. When using screw theory-based methods and atlas methods for mobility analysis, it is necessary to determine the type of DOFs based on the geometric relationships between constraints. However, when multiple screw-type constraints exist at the end-effector, not only do the geometric relationships between screw-type constraints affect the type of DOF, but the pitch values of screw-type constraints also influence the identification of DOFs (detailed analysis is provided in Section 3.1). Therefore, these two types of methods are not suitable for analyzing the DOFs in PMs with three screw-type constraints.

Currently, determining the principal screws of the motion screw system is a viable approach to analyzing the mobility properties of PMs [23,24,25,26,27,28,29,30,31,32,33,34,35,36,37]. Principal screws are defined as a set of mutually orthogonal and intersecting screws. The number of principal screws equals the order of the screw system. Furthermore, any screw within the screw system can be represented as a linear combination of these principal screws.

The methods for solving principal screws can be divided into two main categories. The first type is based on the geometric properties of principal screws being orthogonal and intersecting with each other. Hunt [23] proposed a method for constructing principal screws from given screws, which is applicable to second and third-order screw systems. Based on this, Hunt [24] further investigated geometric construction methods and classified screw systems of different orders. Tsai et al. [25] studied special bases for second- and third-order screw systems, providing geometric conditions and classification methods. Huang et al. [26,27] utilized the theory of quadric decomposition to identify the principal screws of a third-order screw system and applied this method to analyze the principal screws of the 3-RPS PM.

The second category is based on the eigenvalue and eigenvector analysis, which typically relies on numerical computations. Rico and Duffy [28,29] introduced an algebraic method based on dual vectors to identify principal screws and introduced the concept of DOFs partitioning. Parkin [30] proposed a method for solving principal screws applicable to two- to five-order screw systems. Bandyopadhyay and Ghosal [31,32] proposed a generalized eigenvalue method based on dual-number algebra to determine principal screws, providing examples for solving the principal screws of serial, parallel, and hybrid mechanisms. Salgado et al. [33,34] developed an algorithm for solving principal screws based on the decomposition of input and output spaces. Abdel-Baky and Al-Ghefari [35] proposed a method for solving principal screws using dual quaternions. Chen et al. [36] introduced a general-special decomposition method to determine the standard bases of screw systems. Archer and Hopkins [37] introduced a numerical algorithm based on singular value decomposition for solving principal screws.

However, when using the principal screw solution algorithm to analyze mobility, the solving process is numerical, which prevents the establishment of analytical relationships between the structure of the mechanism and the types of DOFs. This approach is not conducive to subsequent research on how changes in structural parameters affect the types of DOF.

This paper addresses the mobility analysis problem of a 3-CCR PM with three screw-type constraints. By solving for the principal screws of the constraint screw system, the special DOFs phenomena of the 3-CCR PM are investigated. The remainder of this paper is organized as follows: Section 2 provides a detailed description of the differences between the mobility analysis of the 3-CCR PM having screw-type terminal constraints and traditional PMs. Section 3 introduces the analytical expressions of the terminal screw-type constraints and the corresponding principal screws and establishes the DOFs identifying approach for the 3-CCR PM. Section 4 gives the numerical example for the DOFs determination of the 3-CCR PM. Finally, some concluding remarks are given in Section 5.

2. Problem Description

The terminal constraint refers to the constraints exerted on the end-effector of a manipulator. In screw theory, one independent terminal constraint $^r at the end-effector of a mechanism is represented as following [1]:

$${\$}^r=\left\{\begin{array}{cc}\begin{array}{l}[\begin{array}{cc}{s}^T& {(r\times s)}^T\end{array}]\\ [\begin{array}{cc}{\mathsf{0}}^T& s^T\end{array}]\\ [\begin{array}{cc}{s}^T& {(r\times s+hs)}^T\end{array}]\end{array}& \begin{array}{l}\mathrm{constraint}\mathrm{force}\\ \mathrm{constraint}\mathrm{torque}\\ \mathrm{constraint}\mathrm{screw}\end{array}\end{array}\right.$$

(1)

where s represents the direction vector of $^r, r denotes the position vector from the reference point to any point on the axis of $^r, and h is a scalar known as the pitch of the screw.

For the existing lower-mobility PMs, almost all terminal constraints are constraint forces or constraint torques. Recent research has shown that there are some PMs [38] with the terminal constraint is one screw pitch in each limb. For PMs with only constraint forces/torque constraints, the DOFs can be visually distinguished based on the geometric relationship of the terminal constraints in each limb [8]. However, for PMs with screw-type terminal constraints, the same geometric relationships of the terminal constraints may still correspond to multiple different types of DOFs. It is not possible to obtain DOFs simply through the traditional process. The mobility problem of this class of PMs is difficult and has not been solved yet.

This paper takes a 3-CCR PM as an example to discuss the DOFs problem of this class of PMs. The 3-CCR PM is shown in Figure 1. This PM consists of a fixed pyramid with O as its center, a moving platform, and three CCR limbs connecting the two platforms. The moving platform is an equilateral triangle with three vertexes, which are A_i(i = 1, 2, 3). The i-th CCR limb connects the fixed pyramid with the moving platform by two cylindrical joints C_i1 and C_i2, one linkages l_i between C_i1 and C_i2, and there is one revolute joint R at point A_i. The joint C_ij is equivalent to a P joint P_ij, and an R joint R_ij. P_i1(i = 1, 2, 3) intersects at point O. R_i3 are symmetrically distributed on the moving platform and intersects at a point outside the moving platform.

Let P_ij, R_ij(j = 1,2,……) represent the j-th P or R joint in the i-th limbs. Let “||”, “⊥” and “|” represent the parallel, perpendicular, and collinear relations, respectively. The geometric relations among the joints in the 3-CCR can be expressed as follows:

P₁₁⊥P₂₁, P₂₁⊥P₃₁, P₁₁⊥P₃₁, R_i1|P_i1, R_i1⊥R_i2, R_i1⊥l_i, R_i2⊥l_i, R_i2|P_i2, R_i2⊥R_i3

(2)

The terminal constraint in each CCR limb of this PM is a screw constraint [38]. The three constraint screws imposed on the moving platform form a constraint screw system. In reference [39], the screw systems have been divided into 50 categories. For this PM, there are eight types of possible constraint screw systems (as shown in

Table 1. The relationships between the pitch of the principal screws, the DOFs, and the atlas of the screw system.

No	Relations Among hα, hβ, and hγ	DOFs Type	Type of the Three-Order Screw System	Sketch of Motion Screw System [39]	Sketch of Constraint Screw System [39]
1	hα = hβ, =0 > hγ, hα > hβ, =hγ = 0	3R	7th type
2	hα, hβ, hγ are mutually unequal and of opposite signs	3R	8th type
3	hα > hβ = 0 > hγ	3R	9th type
4	hα > hβ > hγ = 0, hα = 0 > hβ > hγ	3R	10th type
5	hα, hβ, hγ are mutually unequal and of opposite signs	3H	11th type
6	hα > hβ = hγ = 0, hα = hβ = 0 > hγ	2R1H	12th type
7	hα = hβ > hγ = 0, hα = 0 > hβ > hγ	1R2H	13th type
8	hα = hβ > hγ > 0, 0 > hα > hβ = hγ	3H	14th type

) corresponding to the 7th to 14th types in reference. The eight constraint screw systems belong to the three-order screw systems. Table 1 shows the relationships between the pitch of the principal screws h_α, h_β, and h_γ of the constraint screw system and the DOFs corresponding to various situations, as well as the corresponding atlas of screw systems. In Table 1, yellow lines indicate helical DOFs, blue lines indicate rotational DOFs, green lines indicate screw-type constraints, and red lines indicate force constraints.

For the determination of the constraint screw system, a principal screw analysis of the constraint screw system is required. The constraint type can be determined based on the pitch values of the principal screws. Furthermore, the DOFs of the PMs can be determined based on the relationship between the type of constraint screw system and the DOF, as shown in Table 1.

The terminal constraints of this PM may correspond to eight different screw systems, and the identification of principal screws is performed at a specific pose. Thus, the DOFs determination of this PM needs to be carried out in different discrete poses in the workspace.

3. Terminal Constraint and DOFs Determination Based on Principal Screws Analysis

3.1. Analytical Expression of the Terminal Constraint

Establish the fixed coordinate system O-XYZ at point O with Z⊥P₃₁, X||P₂₁, Y||P₁₁ are satisfied. Establish the moving coordinate system o-xyz at point o with x||A₃A₁ and y|| oA₂ are satisfied, z axis can be determined by x and y (see Figure 1).

Let R_ij, P_ij denote the unit vectors of the j-th R joint in i-th limbs, r_i denotes the position vector from a reference point to an arbitrary point on the axis of the R joint. Let ${\$}_i^r$ be the terminal constraint of i-th limbs, ${\mathit{s}}_i^r$ and ${\mathit{s}}_{i0}^r$ be the direction and moment components of ${\$}_i^r$, respectively. From the reciprocal relation between the motion and the constraint screw system, it leads to the following:

$$\left[\begin{array}{cc}{R}_{i1}^T& {(r_{i1}\times R_{i1})}^T\\ R_{i2}^T& {(r_{i2}\times R_{i2})}^T\\ R_{i3}^T& {(r_{i3}\times R_{i3})}^T\\ {\mathsf{0}}^T& P_{i1}^T\\ {\mathsf{0}}^T& P_{i2}^T\end{array}\right]\left[\begin{array}{c}{s}_{i0}^r\\ s_i^r\end{array}\right]=0$$

(3)

From Equation (3), ${\mathit{s}}_i^r$ can be solved as follows:

$$s_i^r=P_{i1}\times P_{i2}$$

(4)

In the general case, R_i1, R_i2, and R_i3 are linearly independent. Substituting Equation (4) into Equation (3), ${\mathit{s}}_{i0}^r$ can be solved as follows:

$${\mathit{s}}_{i0}^r={\displaystyle \frac{-[({\mathit{r}}_{i1}\times {\mathit{R}}_{i1})\cdot s_i^r]{\mathit{R}}_{i2}\times {\mathit{R}}_{i3}-[({\mathit{r}}_{i2}\times {\mathit{R}}_{i2})\cdot s_i^r]{\mathit{R}}_{i3}\times {\mathit{R}}_{i1}-[({\mathit{r}}_{i3}\times {\mathit{R}}_{i3})\cdot s_i^r]{\mathit{R}}_{i1}\times {\mathit{R}}_{i2}}{{\mathit{R}}_{i1}\times {\mathit{R}}_{i2}\cdot {\mathit{R}}_{i3}}}$$

(5)

From Equation (2), it can be concluded that some joint vectors of the 3-CCR PM satisfy the following relationship:

$$\begin{array}{l}{\mathit{r}}_{i1}=d_{i1}{P}_{i1}, {\mathit{r}}_{i2}={\mathit{r}}_{i1}+l_i{L}_i, {\mathit{r}}_{i3}={\mathit{r}}_{i1}+l_i{L}_i+d_{i2}{P}_{i2}, {\mathit{R}}_{i2}={\mathit{R}}_{i3}\times {\mathit{R}}_{i1}/\left|{\mathit{R}}_{i3}\times {\mathit{R}}_{i1}\right|,\\ P_{i1}\cdot P_{i2}=0, P_{i2}\cdot {\mathit{R}}_{i3}=0, L_i=P_{i1}\times P_{i2}\end{array}$$

(6)

Substituting Equation (6) into Equation (5), it leads to the following:

$${\mathit{s}}_{i0}^r=d_{i1}{P}_{i1}\times L_i-{\displaystyle \frac{d_{i2}{P}_{i1}\cdot {\mathit{R}}_{i3}}{L_i\cdot {\mathit{R}}_{i3}}}{L}_i$$

(7)

From Equations (4) and (7), it is known that the physical meaning of ${\$}_i^r$ is that the terminal constraint in the i-th limbs is a screw-type constraint passing through the intersection point of the link l_i and the joint P_i1, along the direction of l_i, with a pitch − d_i2P_i1·R_i3/L_i·R_i3. The terminal constraints are illustrated by the green lines in Figure 1.

Equations (4) and (7) demonstrate the analytical relationships between the screw-type terminal constraint and the joint vectors in each limb, laying the foundation for establishing the principal screws of the terminal constraint screw system. According to Equation (7), it can be seen that the expression for the screw-type constraints of this 3-CCR mechanism is very simple. This makes the 3-CCR mechanism suitable as an example for studying the relationship between independent screw-type constraints and DOFs.

3.2. Calculating the Principal Screws of the Constraint Screw System

The constraint screw ${\$}_i^r$ of each limb form a constraint screw system, and any constraint screw in this system can be expressed by the linear combination of ${\$}_i^r$ as follows:

$${\$}^r=\left[\begin{array}{c}{s}^r\\ s_0^r\end{array}\right]=\left[\begin{array}{ccc}{s}_1^r& s_2^r& s_3^r\\ s_{10}^r& s_{20}^r& s_{30}^r\end{array}\right]\left[\begin{array}{c}{k}_1\\ k_2\\ k_3\end{array}\right]$$

(8)

where k₁, k₂, and k₃ are three coefficients.

The pitch of $^r can be expressed as follows:

$$h={\displaystyle \frac{\left[\begin{array}{ccc}{k}_1& k_2& k_3\end{array}\right]{\left[\begin{array}{ccc}{s}_1^r& s_2^r& s_3^r\end{array}\right]}^T\left[\begin{array}{ccc}{s}_{10}^r& s_{20}^r& s_{30}^r\end{array}\right]{\left[\begin{array}{ccc}{k}_1& k_2& k_3\end{array}\right]}^T}{\left[\begin{array}{ccc}{k}_1& k_2& k_3\end{array}\right]{\left[\begin{array}{ccc}{s}_1^r& s_2^r& s_3^r\end{array}\right]}^T\left[\begin{array}{ccc}{s}_1^r& s_2^r& s_3^r\end{array}\right]{\left[\begin{array}{ccc}{k}_1& k_2& k_3\end{array}\right]}^T}}$$

(9)

Suppose k₃ ≠ 0, u = k₁/k₃, w = k₂/k₃, Equation (9) can be written as follows:

$$h={\displaystyle \frac{\left[\begin{array}{ccc}u& w& 1\end{array}\right]{\left[\begin{array}{ccc}{s}_1^r& s_2^r& s_3^r\end{array}\right]}^T\left[\begin{array}{ccc}{s}_{10}^r& s_{20}^r& s_{30}^r\end{array}\right]{\left[\begin{array}{ccc}u& w& 1\end{array}\right]}^T}{\left[\begin{array}{ccc}u& w& 1\end{array}\right]{\left[\begin{array}{ccc}{s}_1^r& s_2^r& s_3^r\end{array}\right]}^T\left[\begin{array}{ccc}{s}_1^r& s_2^r& s_3^r\end{array}\right]{\left[\begin{array}{ccc}u& w& 1\end{array}\right]}^T}}$$

(10)

Expanding Equation (10) leads to the following:

$$a_{11}{u}^2+2a_{12}uw+a_{22}{w}^2+2a_{13}u+2a_{23}w+a_{33}=0$$

(11)

where

$$\begin{array}{l}{a}_{11}=-s_1^r\cdot s_{10}^r+h{s}_1^r\cdot s_1^r, a_{12}={\displaystyle \frac{-s_2^r\cdot s_{10}^r-s_1^r\cdot s_{20}^r+2h{s}_2^r\cdot s_1^r}{2}}, a_{22}=-s_2^r\cdot s_{20}^r+h{s}_2^r\cdot s_2^r,\\ a_{13}={\displaystyle \frac{-s_1^r\cdot s_{30}^r-s_3^r\cdot s_{10}^r+2h{s}_3^r\cdot s_1^r}{2}}, a_{23}={\displaystyle \frac{-s_3^r\cdot s_{20}^r-s_2^r\cdot s_{30}^r+2h{s}_3^r\cdot s_2^r}{2}}, a_{33}=-s_3^r\cdot s_{30}^r+h{s}_3^r\cdot s_3^r\end{array}$$

(12)

The quadratic curve expressed by Equation (11) degenerates into two straight lines when the value of h equals the pitch of the principal screw. According to the theory of quadratic curve decomposition, this occurs if and only if the determinant of the coefficient matrix is zero.

$$\left|\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right|=0, (a_{ij}=a_{ji})$$

(13)

By expanding Equation (13), we can obtain the following:

$$c_1{h}^3+c_2{h}^2+c_{3}h+c_4=0$$

(14)

where

$$\begin{array}{l}\begin{array}{lll}{c}_1& =& 4(s_3^r\cdot s_3^r)(s_2^r\cdot s_1^r)2-16(s_3^r\cdot s_1^r)(s_3^r\cdot s_2^r)(s_2^r\cdot s_1^r)+4(s_2^r\cdot s_2^r)(s_3^r\cdot s_1^r)2-\\ & & 2(s_1^r\cdot s_1^r)[(s_2^r\cdot s_2^r)(s_3^r\cdot s_3^r)-4(s_3^r\cdot s_2^r)2]\end{array}\\ \begin{array}{lll}{c}_2& =& [-4(s_2^r\cdot s_1^r)(s_3^r\cdot s_3^r)+8(s_3^r\cdot s_1^r)(s_3^r\cdot s_2^r)](s_1^r\cdot s_{20}^r+s_2^r\cdot s_{10}^r)+\\ & & \left[8\right(s_2^r\cdot s_1^r)(s_3^r\cdot s_2^r)-4(s_2^r\cdot s_2^r)(s_3^r\cdot s_1^r)](s_1^r\cdot s_{30}^r+s_3^r\cdot s_{10}^r)+\\ & & [-8(s_2^r\cdot s_1^r)(s_3^r\cdot s_2^r)+8(s_2^r\cdot s_1^r)(s_3^r\cdot s_1^r)](s_2^r\cdot s_{30}^r+s_3^r\cdot s_{20}^r)-\\ & & 4(s_3^r\cdot s_{30}^r)(s_2^r\cdot s_1^r)2-4(s_2^r\cdot s_{20}^r)(s_3^r\cdot s_1^r)-8(s_3^r\cdot s_2^r)2(s_1^r\cdot s_{10}^r)+2(s_2^r\cdot s_2^r)(s_3^r\cdot s_3^r)(s_1^r\cdot s_{10}^r)+\\ & & 2(s_1^r\cdot s_1^r)[(s_2^r\cdot s_2^r)(s_3^r\cdot s_{30}^r)+(s_2^r\cdot s_{20}^r)(s_3^r\cdot s_3^r)]\end{array}\\ \begin{array}{lll}{c}_3& =& [(s_1^r\cdot s_{20}^r)2+(s_2^r\cdot s_{10}^r)2](s_3^r\cdot s_3^r)+[(s_1^r\cdot s_{30}^r)2+(s_3^r\cdot s_{10}^r)2\left]\right(s_2^r\cdot s_2^r)+\\ & & 2[(s_2^r\cdot s_{30}^r)2+(s_3^r\cdot s_{20}^r)2](s_1^r\cdot s_1^r)+[-4(s_1^r\cdot s_{30}^r+s_3^r\cdot s_{10}^r)(s_3^r\cdot s_2^r)+4(s_2^r\cdot s_1^r)(s_3^r\cdot s_{30}^r)+\\ & & 2(s_2^r\cdot s_{10}^r)(s_3^r\cdot s_3^r)-4(s_2^r\cdot s_{30}^r+s_3^r\cdot s_{20}^r)(s_3^r\cdot s_1^r)](s_1^r\cdot s_{20}^r)+[-4(s_2^r\cdot s_1^r)(s_2^r\cdot s_{30}^r+s_3^r\cdot s_{20}^r)-\\ & & 4(s_2^r\cdot s_{10}^r)(s_3^r\cdot s_2^r)+2(s_2^r\cdot s_2^r)(s_3^r\cdot s_{10}^r)+4(s_2^r\cdot s_{20}^r)(s_3^r\cdot s_1^r)](s_1^r\cdot s_{30}^r)+[4(s_2^r\cdot s_1^r)(s_3^r\cdot s_{30}^r)-\\ & & 4(s_2^r\cdot s_{30}^r+s_3^r\cdot s_{20}^r)(s_3^r\cdot s_1^r)-4(s_3^r\cdot s_{10}^r)(s_3^r\cdot s_2^r)](s_2^r\cdot s_{10}^r)+[4(s_1^r\cdot s_1^r)(s_3^r\cdot s_{20}^r)+\\ & & 4(s_2^r\cdot s_{20}^r)(s_3^r\cdot s_1^r)](s_3^r\cdot s_{10}^r)-2(s_2^r\cdot s_{20}^r)(s_3^r\cdot s_{30}^r)(s_1^r\cdot s_1^r)+8(s_3^r\cdot s_{20}^r)(s_3^r\cdot s_2^r)(s_1^r\cdot s_{10}^r)+\\ & & [-2(s_2^r\cdot s_2^r)(s_3^r\cdot s_{30}^r)-2(s_2^r\cdot s_{20}^r)(s_3^r\cdot s_3^r)](s_1^r\cdot s_{10}^r)\end{array}\\ \begin{array}{lll}{c}_4& =& -[(s_1^r\cdot s_{20}^r)2+(s_2^r\cdot s_{10}^r)2](s_3^r\cdot s_{30}^r)-[(s_1^r\cdot s_{30}^r)2+(s_3^r\cdot s_{10}^r)2](s_2^r\cdot s_{20}^r)\\ & & -2(s_1^r\cdot s_{10}^r)[(s_2^r\cdot s_{30}^r)2+(s_3^r\cdot s_{20}^r)2]+2(s_2^r\cdot s_{30}^r+s_3^r\cdot s_{20}^r)(s_2^r\cdot s_{10}^r)[(s_1^r\cdot s_{30}^r)+(s_3^r\cdot s_{10}^r)]+\\ & & 2(s_2^r\cdot s_{30}^r+s_3^r\cdot s_{20}^r)(s_1^r\cdot s_{30}^r)+2(s_3^r\cdot s_{10}^r)(s_2^r\cdot s_{30}^r+s_3^r\cdot s_{20}^r)(s_1^r\cdot s_{20}^r)-2(s_2^r\cdot s_{10}^r)(s_3^r\cdot s_{30}^r)\\ & & -2(s_2^r\cdot s_{20}^r)(s_3^r\cdot s_{10}^r)(s_1^r\cdot s_{30}^r)-4(s_1^r\cdot s_{10}^r)(s_2^r\cdot s_{30}^r)(s_3^r\cdot s_{20}^r)+2(s_1^r\cdot s_{10}^r)(s_2^r\cdot s_{20}^r)(s_3^r\cdot s_{30}^r)\end{array}\end{array}$$

where c_i(i = 1,…,4) only depends on ${\mathit{s}}_i^r$ and ${\mathit{s}}_{i0}^r$.

The three solutions h₁, h₂, and h₃ of Equation (14) are the pitch values of the principal screws, which can be expressed as follows:

$$\left[h_1, h_2, h_3\right]=\left\{\begin{array}{l}\left[-{\displaystyle \frac{c_3}{c_2}}, -{\displaystyle \frac{c_3}{c_2}}, -{\displaystyle \frac{c_3}{c_2}}\right], A=0, B=0\\ \left[-{\displaystyle \frac{c_2}{c_1}}+K, -{\displaystyle \frac{K}{2}}, -{\displaystyle \frac{K}{2}}\right], B^2-4AC=0, A\ne 0, B\ne 0\\ \left[{\displaystyle \frac{-c_2-2\sqrt{A}\cos (\theta /3)}{3c_1}}, {\displaystyle \frac{-c_2+\sqrt{A}M}{3c_1}}, {\displaystyle \frac{-c_2+\sqrt{A}N}{3c_1}}\right], B^2-4AC<0\end{array}\right.$$

(15)

where

$$\begin{array}{l}A=c_2^2-3c_1{c}_3, B=c_2{c}_3-9c_1{c}_4, C=c_3^2-3c_2{c}_{4}K={\displaystyle \frac{c_2{c}_3-9c_1{c}_4}{c_2^2-3c_1{c}_3}}, M=\cos {\displaystyle \frac{\theta }{3}}+\sqrt{3}\sin {\displaystyle \frac{\theta }{3}}, \\ N=\cos {\displaystyle \frac{\theta }{3}}-\sqrt{3}\sin {\displaystyle \frac{\theta }{3}}, T={\displaystyle \frac{2A{c}_2-3B{c}_1}{2\sqrt{A^3}}}, \theta =\arccos T\end{array}$$

By substituting h_j = 1, 2, 3 in Equation (15) into Equation (12), the corresponding values a_11,j, a_12,j, a_13,j, a_22,j, a_23,j, and a_33,j can be obtained. Then, by substituting these values into Equation (11), a bivariate quadratic equation in terms of u_j, v_j is derived. The solutions of this equation are shown as the following [40]:

$$u_j={\displaystyle \frac{a_{22,j}{a}_{13,j}-a_{12,j}{a}_{23,j}}{a_{12,j}^2-a_{11,j}{a}_{22,j}}}, v_j=-{\displaystyle \frac{a_{23,j}+a_{12,j}{u}_j}{a_{22,j}}}$$

(16)

By considering u = k₁/k₃, w = k₂/k₃ and by substituting Equation (16) into Equation (8), the three principal screws ${\$}_{pj}^r$(j = 1,2,3) corresponding to the terminal constraint system can be obtained:

$${\$}_{pj}^r=\left[\begin{array}{cc}{(u_j{s}_1^r+v_j{s}_2^r+s_3^r)}^T& {(u_j{s}_{10}^r+v_j{s}_{20}^r+s_{30}^r)}^T\end{array}\right], j=1,2,3$$

(17)

Equation (17) is the analytical expression of ${\$}_{pj}^r$ with respect to ${\$}_i^r$, and Equation (15) is the analytical expression of h_j with respect to ${\$}_i^r$. By substituting Equations (4) and (5) into Equation (15), h_i can obtained, and then by substituting the obtained h_i into Equation (17), the analytical expression of ${\$}_{pj}^r$ can be obtained.

To verify whether the solution of the principal screw is correct, one can check if the elements of ${\$}_{pj}^r$ satisfy the perpendicular and intersecting conditions. The specific verification conditions are listed as follows:

$$\begin{array}{l}{s}_{p1}^r\cdot s_{p2}^r=0, s_{p1}^r\cdot s_{p3}^r=0, s_{p2}^r\cdot s_{p3}^r=0, \\ s_{p1}^r\cdot s_{p20}^r+s_{p2}^r\cdot s_{p10}^r=0, s_{p1}^r\cdot s_{p30}^r+s_{p3}^r\cdot s_{p10}^r=0, s_{p2}^r\cdot s_{p30}^r+s_{p3}^r\cdot s_{p20}^r=0\end{array}$$

(18)

3.3. Mobility Identifying Approach

For the obtained values of h_j, they are categorized based on their numerical magnitudes and denoted as h_α, h_β, and h_γ.

$$h_{\alpha }=\max (h_1,h_2,h_3), h_{\gamma }=\min (h_1,h_2,h_3)$$

(19)

Then, the specific types of DOFs of the PM can be identified based on h_α, h_β and h_γ, which are categorized into 8 cases as shown in Table 1.

When h_α = h_β = 0 > h_γ or h_α > h_β = h_γ = 0, the constraint screw system of the PM is classified as type 7. When h_α, h_β, and h_γ are all distinct, and one pitch has a different sign from the other two, the constraint screw system is classified as type 8. When h_α > h_β = 0 > h_γ, the constraint screw system is classified as type 9. When h_α > h_β > h_γ = 0 or h_α = 0 > h_β > h_γ, the constraint screw system is classified as type 10. When h_α, h_β, and h_γ are all distinct and have the same sign, the constraint screw system is classified as type 11. When h_α > h_β = h_γ = 0 or h_α > h_β = h_γ = 0, the constraint screw system is classified as type 12. When h_α = h_β > h_γ = 0 or h_α = h_β > h_γ = 0, the constraint screw system is classified as type 13. When h_α, h_β, and h_γ have the same sign, and only two of them are equal, the end-effector constraint screw system is classified as type 14.

Each type of third-order constraint system uniquely corresponds to one type of third-order motion system [39]. As shown in Table 1, the green lines represent the screw-type constraints, the red lines represent the constraint forces, the blue lines represent the motion screws, and the yellow lines represent the rotations. The above motion screw systems all contain independent helical DOFs. The axes of these helical DOFs are distributed in the cluster straight generatrices of an infinite number of one-sheeted hyperboloids, with the principal generator and the plane of the circular cross-section of each one-sheeted hyperboloid coincide, respectively.

For the 7th and 8th motion three-systems, except motion screws, there also exists an infinite number of rotational DOFs, which are distributed in the same cluster of straight generatrices of a one-sheeted hyperboloid. For the 9th motion system, there similarly exists an infinite number of rotational DOFs, which form two planar pencils of lines intersecting at two common points, respectively. For the 12th motion system, there exists an infinite number of rotational DOFs, which form a planar pencil intersecting at a common point. For the 10th and 13th motion systems, only one rotational DOFs exists. For the 11th and 14th motion systems, no rotational DOFs exist.

Based on the theory of Grassmann line geometry, the dimension of the independent DOFs of the motion systems is 3. Due to the coexistence of multiple rotational and helical DOFs in the above-mentioned motion screw systems, in order to clarify the corresponding DOFs type for each motion screw system, this paper provides a criterion for the DOFs determination of this class of PMs:

For an n-order motion screw system, the independent rotation and translation DOFs should be determined first. If the independent rotation and translation DOFs is a, then the independent helical DOFs is n-a.

Thus, among these 8 cases, the 7th, 8th, and 9th motion systems have three linearly independent rotational DOFs. The 12th motion system has two linearly independent rotational DOFs and one helical DOF. The 10th and 13th motion systems have one rotational DOFs and two linearly independent helical DOFs. Only the 11th and 14th motion screw systems have three linearly independent pure helical DOFs.

Based on the above analysis, the steps for the mobility analysis of PMs with three screw-type constraints are as follows:

(1)

Express the joint vectors. Establish the expression of joint vectors in terms of the independent pose parameters of the end-effector. This step can be achieved through a kinematic inverse solution.

(2)

Compute workspace and feasible joint vectors. Calculate the workspace of the PM and solve for the expressions of joint vectors under feasible poses within the workspace.

(3)

Solve pitches of principal screws for discrete poses. Substitute the expressions of the joint vectors sequentially into Equations (4), (5), and (16)–(18) to solve for the principal screws and their pitches of the constraint screw system under feasible poses.

(4)

Determine the type of DOFs. Utilize the one-to-one correspondence between the principal screws of the constraint screw system and the types of DOFs (as shown in Table 1) to determine the DOFs at each feasible pose, thereby inferring the full-cycle mobility of the PM.

Figure 2 shows the above steps.

4. Mobility Analysis of 3-CCR PM

This section analyzes the mobility of the 3-CCR mechanism according to the steps for mobility analysis of PMs with three screw-type constraints. The terminal constraint and DOFs properties of the 3-CCR PM may vary in different poses, so this section establishes a position solution model for the 3-CCR PM. Based on this, combined with the terminal constraint and DOFs determination method obtained in the previous section, the terminal constraints and DOFs of this PM at different poses in the workspace are determined.

The position vector A_i of point A_i in {n₀} can be expressed as follows:

$$\begin{array}{l}{A}_i={}_{n_1}{}^{n_0}\mathbf{R}{}^{n_1}A_i+o, {}^{n_1}A_i=e_2\left[\begin{array}{c}{c}_{\theta }\\ s_{\theta }\\ 0\end{array}\right], {}_{n_1}{}^{n_0}\mathbf{R}=\left[\begin{array}{ccc}{x}_l& y_l& z_l\\ x_m& y_m& z_m\\ x_n& y_n& z_n\end{array}\right], o=\left[\begin{array}{c}{X}_o\\ Y_o\\ Z_o\end{array}\right],\\ \theta =\pi /6+(2\pi /3)(i-1), i=1,2,3\end{array}$$

(20)

where A_i and ${}^{n_1}A_i$ are the position vectors of A_i in O-XYZ and o-xyz, respectively. o is the position vector of o in o-xyz. ${}_{n_1}{}^{n_0}\mathbf{R}$ is the rotation matrix, and (x_l, x_m, x_n, y_l, y_m, y_n, z_l, z_m, z_n) are the rotation elements. e₁ denotes the distances from o to A_i.

From Equation (20), ${}^{n_1}A_{i2}$ and A_ij can be expressed as follows:

$$\begin{array}{l}{}^{n_1}A_1={\displaystyle \frac{e_2}{2}}\left[\begin{array}{c}q\\ -1\\ 0\end{array}\right],{}^{n_1}A_2=\left[\begin{array}{c}0\\ e_2\\ 0\end{array}\right],{}^{n_1}A_3={\displaystyle \frac{e_2}{2}}\left[\begin{array}{c}-q\\ -1\\ 0\end{array}\right],q=\sqrt{3},\\ A_{11}={\displaystyle \frac{1}{2}}\left[\begin{array}{c}q{e}_2{x}_l-e_2{y}_l+2X_{o1}\\ q{e}_2{x}_m-e_2{y}_m+2Y_{o1}\\ q{e}_2{x}_n-e_2{y}_n+2Z_{o1}\end{array}\right],A_2=\left[\begin{array}{c}{e}_2{y}_l+X_{o1}\\ e_2{y}_m+Y_{o1}\\ e_2{y}_n+Z_{o1}\end{array}\right],A_3={\displaystyle \frac{1}{2}}\left[\begin{array}{c}-q{e}_2{x}_l-e_2{y}_l+2X_{o1}\\ -q{e}_2{x}_m-e_2{y}_m+2Y_{o1}\\ -q{e}_2{x}_n-e_2{y}_n+2Z_{o1}\end{array}\right]\end{array}$$

(21)

The direction vectors of Pi1 and Ri1 in {n0} can be expressed as follows:

$$P_{11}=R_{11}=\left[\begin{array}{c}0\\ 1\\ 0\end{array}\right], P_{21}=R_{21}=\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right], P_{31}=R_{31}=\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right]$$

(22)

According to Equation (2), the direction vectors of Pi2 and Ri2 in {n0} can be expressed as follows:

$$P_{i2}=R_{i2}={\displaystyle \frac{R_{i3}\times R_{i1}}{\sqrt{(R_{i3}\times R_{i1})\cdot (R_{i3}\times R_{i1})}}}$$

(23)

The direction vector of R_i3 in {n₀} can be expressed as follows:

$$R_{i3}={}_{n_1}{}^{n_0}\mathbf{R}{}^{n_1}R_{i3},{}^{n_1}R_{i3}=\left[\begin{array}{c}\sin \left(\pi /4\right)\cos \left(\theta \right)\\ \sin \left(\pi /4\right)sin\left(\theta \right)\\ \cos \left(\pi /4\right)\end{array}\right]$$

(24)

Equations (21)–(24) are the joint vector expressions with respect to the pose parameters of the moving platform.

From Equation (2), the geometric constraint equations of this PM can be expressed as follows:

$$r_i\cdot P_{i1}=0, r_i\cdot P_{i2}=0, i=1,2,3$$

(25)

Considering Equation (2), Equation (25) can be simplified as follows:

$$A_{i1}\cdot P_{i1}-d_{i1}=0, A_{i1}\cdot P_{i2}-d_{i2}=0$$

(26)

By substituting Equations (21)–(23) into Equation (26), it leads to the following:

$$\begin{array}{l}{d}_{11}={\displaystyle \frac{\sqrt{3}{x}_{l}e}{2}}-{\displaystyle \frac{y_{l}e}{2}}+X_o,d_{21}=y_{l}e+Y_o,d_{31}=-{\displaystyle \frac{\sqrt{3}{x}_{n}e}{2}}-{\displaystyle \frac{y_{n}e}{2}}+Z_o,\\ d_{12}={\displaystyle \frac{\sqrt{3}(Y_o{x}_n-Z_o{x}_m-y_{l}e)+2Y_o(x_l{y}_m-y_l{x}_m)+2Z_o(x_l{y}_n-y_l{x}_n)-Y_o{y}_n+Z_o{y}_m-e{x}_l}{\sqrt{-2\sqrt{3}{x}_l(2y_n{x}_m-2x_n{y}_m-y_l)+x_l^2+3y_l^2+4y_l(y_n{x}_m-x_n{y}_m)+4}}},\\ d_{22}={\displaystyle \frac{X_o(x_l{y}_m-x_m{y}_l)-Z_o(x_m{y}_n-x_n{y}_m)+X_o{y}_n-Z_o{y}_l-e{x}_m}{\sqrt{2y_m(x_l{y}_n-2x_n{y}_l)+x_m^2+1}}},\\ d_{32}={\displaystyle \frac{\sqrt{3}(X_o{x}_m-Y_o{x}_l-y_{n}e)+2X_o(x_l{y}_n-y_l{x}_n)+2Y_o(x_m{y}_n-y_m{x}_n)-X_o{y}_m+Y_o{y}_l-e{x}_n}{\sqrt{4\sqrt{3}[x_l(y_m{x}_n+y_l)-x_m(2y_l{x}_n-y_m)]-x_l^2-x_m^2-3y_l^2-y_m^2+4y_n(y_m{x}_l-x_m{y}_l)+8}}}\end{array}$$

(27)

According to Equation (2), the dimension constraint equation of this PM can be obtained as follows:

$$r_i=(A_{i1}-d_{i2}{P}_{i2})-d_{i1}{P}_{i1}, L=\left|r_i\right|, i=1,2,3$$

(28)

Substituting Equations (24) and (27) into Equation (28) and then simplifying it yields the following:

$$(R_{i3}\times R_{i1})\cdot (R_{i3}\times R_{i1})[A_{i1}\cdot A_{i1}-{(A_{i1}\cdot P_{i1})}^2-L^2]-{[A_{i1}\cdot (R_{i3}\times R_{i1})]}^2=0$$

(29)

Substituting Equations (21)–(23) into Equation (29) leads to the following:

$$\begin{array}{l}4[2\sqrt{3}(-2x_l{x}_m{y}_n+2x_m{x}_n{y}_l-x_m{y}_m)+4y_m(x_l{y}_n-x_n{y}_l)-(x_m^2+3y_m^2-4)]Y_o^2\\ +4\{\sqrt{3}[(4x_n{y}_m+2y_l)x_l-4x_m{x}_n{y}_l+2x_m{y}_m]+x_l^2-4x_l{y}_m{y}_n+3y_l^2+4x_m{y}_l{y}_n+x_m^2+3y_m^2\}{Z}_o^2+\\ 8\{\sqrt{3}[4x_m{y}_m{x}_l+2(-2x_m^2+1)y_l-(y_n{x}_m+x_n{y}_m)]-2(2y_m^2-1)x_l+\\ (4x_m{y}_m{y}_l-x_m{x}_n-4y_m{y}_n)\}{Y}_o{Z}_o+4e\{\sqrt{3}[(-8y_l{y}_n+x_m)x_l^2+(8x_n{y}_l^2+2y_l{y}_m-2x_n)x_l\\ -x_m{y}_l^2+2y_n{y}_l+4x_m]+12y_n{x}_l^2+(-12x_n{y}_l+3y_m)x_l^2+(4y_l^2{y}_n+2x_m{y}_l-14y_n)x_l-4x_n{y}_l^3-\\ 3y_m{y}_l^2+10y_l{x}_n-4y_m\}{Y}_o+4e[(32y_l{y}_m+4x_n)x_l^2+(-32x_m{y}_l^2+8y_l{y}_n+8x_m)x_l-4x_n{y}_l^2-\\ 8y_m{y}_l+16x_n-12y_m{x}_l^3+(12x_m{y}_l+3y_n)x_l^2+(-4y_l^2{y}_m+2x_n{y}_l+14y_m)x_l+4x_m{y}_l^3-3y_n{y}_l^2-\\ 10x_m{y}_l-4y_n]Z_o+4\sqrt{3}{e}^2{x}_l[(3y_n{x}_m-3x_n{y}_m-y_l)x_l^2+y_l^3+3(x_m{y}_n-x_n{y}_m)y_l^2+2y_l-4y_n{x}_m+\\ 4x_n{y}_m]-16L^2+4\sqrt{3}{x}_l[-2L^2{y}_l+4L^2(x_m{y}_n-x_n{y}_m)]+e^2\{-3x_l^4+[-12+2y_l^2+36(-x_m{y}_n+\\ x_n{y}_m)y_l]x_l^2+16-3y_l^4+4(-x_m{y}_n+x_n{y}_m)y_l^3-4y_l^2+16(x_m{y}_n-x_n{y}_m)y_l\}-4L^2{x}_l^2-12L^2{y}_l^2-\\ 16L^2(x_m{y}_n-x_n{y}_m)y_l=0\end{array}$$

(30)

$$\begin{array}{l}\begin{array}{l}(2x_m{y}_l{y}_n-2x_n{y}_l{y}_m-x_l^2+1)X_o^2+2(2x_l{y}_l{y}_m-2x_m{y}_l^2-x_l{x}_n+x_m)X_o{Z}_o+\\ 2[((2x_l{y}_l{y}_n-2x_n{y}_l^2+x_l{x}_m)y_m+2(x_m{y}_n+y_l)]X_o+(2x_l{y}_m{y}_n-2x_m{y}_l{y}_n+x_l^2+x_m^2)Z_o^2+\\ [-4e{x}_l{y}_m^3+4e{x}_m{y}_l{y}_m^2+(2x_m{x}_n+4x_l)e_{ym}+(-2x_m{y}_l+2y_n)e]Z_o+(-2x_l{y}_n+2x_n{y}_l)e^2{y}_m^3+\\ (-x_m^2-1)e^2{y}_m^2+[(2x_l{y}_n-2x_n{y}_l)e^2-2L^2(x_l{y}_n-x_n{y}_l)]y_m+e^2-L^2(x_m^2+1)=0\end{array}\\ \begin{array}{l}4[\sqrt{3}(2x_l{y}_l+4\sqrt{3}(x_n{y}_m-x_m{y}_n)x_l-3y_l^2+4(+x_n{y}_m-x_m{y}_n)y_l-x_l^2+4]X_o^2+\\ 8\{\sqrt{3}[(-4x_{l}xn+x_m)y_l+4y_n{x}_l^2+y_m{x}_l-2y_n]-4x_n{y}_l^2+(4x_l{y}_n-3y_m)y_l-x_l{x}_m+3x_n\}{X}_o{Y}_o+\\ 4\{[(8x_m{x}_n+x_l)y_l^2+(-8x_l{x}_n{y}_m+2x_m{y}_m)y_l+x_l^3+(x_m^2-y_m^2-4)x_l-2y_m{y}_n-6x_m{x}_n]\sqrt{3}e+\end{array}\end{array}$$

(31)

$$\begin{array}{l}e[3y_l^3+4y_n{x}_m{y}_l^2+(-5x_l^2+12x_m{x}_n{x}_l-4y_m{y}_n{x}_l+3y_m^2-3x_m^2-4)y_l-12x_l^2{x}_n{y}_m-2x_m{y}_m{x}_l-\\ 2x_n{y}_m+6y_n{x}_m]\}{X}_o+4[\sqrt{3}(4x_l{x}_m{y}_n-4x_m{x}_n{y}_l+2x_m{y}_m)-4x_n{y}_l{y}_m+4x_l{y}_m{y}_n-3y_m^2-x_m^2+4]Y_o^2\\ +[-4x_m{y}_l^2+(32x_m{x}_n{y}_m+8x_l{y}_m+8y_n)y_l+4x_l^2{x}_m+(-32x_n{y}_m^2+24x_n)x_l+4x_m{y}_m^2-\\ (4(-x_m^2+16))x_m]+4e[4x_n{y}_l^3+(-4x_l{y}_n+3y_m)y_l^2+(12x_m^2{x}_n+4x_n{y}_m^2-2x_l{x}_m-2x_n)y_l-\\ 3y_m{x}_l^2+(-12x_m{x}_n{y}_m-4y_m^2{y}_n-6y_n)x_l+3y_m^3+(-5x_m^2-4)y_m]\sqrt{3}e{Y}_o+\\ (4e^2[(-3x_m{x}_n-x_l)y_l^3+y_m{y}_l^2(3x_l{x}_n-x_m)+(x_l^3-3x_l^2{x}_m{x}_n+x_m^2{x}_l-y_m^2{x}_l+2x_l-3x_m^3{x}_n+\\ 3y_m^2{x}_n{x}_m-6x_n{x}_m)y_l+y_m(3x_l^3{x}_n+x_l^2{x}_m+3x_m^2{x}_n{x}_l+3y_m^2{x}_n{x}_l-2x_n{x}_{ll}+x_m^3-y_m^2{x}_m+2x_m)]+\\ 8L^2[y_l(2x_m{x}_n-x_l)-\sqrt{3}(2x_n{y}_m{x}_l+x_m{y}_m)]+(-3y_l^4-4(x_m{y}_n+x_n{y}_m)y_l^3+\\ [2x_l^2+(-36x_m{x}_n+8y_m{y}_n)x_l-6y_m^2+10]y_l^2+4y_l[9x_l^2{x}_n{y}_m+x_m{y}_m{x}_l-x_n{y}_m^3+\\ (-9x_m^2{x}_n+x_n)y_m-3y_n{x}_m]-3x_l^4-6(x_m^2-3)x_l^2+4y_m(9x_m{x}_n{y}_m+y_m^2{y}_n+3y_n)x_l+(-3y_m^4+\\ 2x_m^2{y}_m^2+5y_m^2-3x_m^4+18x_m^2-6)e^2+4L^2(3y_l^2+4x_m{y}_l{y}_n+x_l^2-4x_l{y}_m{y}_n+3y_m^2+x_m^2-8)=0\end{array}$$

(32)

Using ZYX Euler rotations with α, β, and γ are three Euler angles, the rotation matrix of this PM can be expressed as follows:

$${}_{n_1}{}^{n_0}\mathbf{R}=\left[\begin{array}{ccc}\cos (\beta )\cos (\gamma )& \sin (\alpha )\sin (\beta )\cos (\gamma )-\cos (\alpha )\sin (\gamma )& \cos (\alpha )\sin (\beta )\cos (\gamma )+\sin (\alpha )\sin (\gamma )\\ \cos (\beta )\sin (\gamma )& \sin (\alpha )\sin (\beta )\sin (\gamma )+\cos (\alpha )\cos (\gamma )& \cos (\alpha )\sin (\beta )\sin (\gamma )-\sin (\alpha )\cos (\gamma )\\ -\sin (\beta )& \cos (\beta )\sin (\alpha )& \cos (\beta )\cos (\alpha )\end{array}\right]$$

(33)

Substituting Equation (33) into Equations (30)–(32) yields three constraint equations containing the terminal pose parameters X_o, Y_o, Z_o, α, β, and γ. The position solution of this PM is relatively complex and requires solving nonlinear equations. In this paper, the numerical method is used to solve the position solution, and then the DOFs at different points in the workspace are identified. The specific operation is as follows:

Set X_o, Y_o, Z_o ∈ [10 mm, 100 mm], and set the constraint conditions as d_i1, d_i2 ∈ [10 mm, 100 mm]. For the given X_o, Y_o, Z_o, solve α, β, γ that satisfy Equations (30)–(32) using a numerical method. Substitute the pose parameters X_o, Y_o, Z_o, α, β, γ into Equation (27), calculate d_i1, d_i2, and verify the values according to the constraint conditions. If the given points satisfy the constraint conditions, substitute the pose parameters into Equations (21)–(24), obtain the joint vectors at this pose, and then solve the principal screws and the pitches of the constraint screw system using Equations (4), (5), (15) and (17). Finally, determine the DOFs considering the relationship between the principal screws of the constraint screw system and DOFs types. The points with different DOFs are drawn in the workspace, as shown in Figure 3.

In Figure 3, the blue dots indicate that the 3-CCR PM at this pose has three rotational DOFs corresponding to the 8th screw system, and the magenta dots indicate that the 3-CCR PM at this pose has three helical DOFs corresponding to the 11th screw system. The region in the workspace of the 3-CCR PM that only contains 3H DOFs is shown in Figure 3b. To better display the distribution of points in the workspace, the points closer to the origin are set to have higher transparency. This result indicates that the DOFs property of the 3-CCR PM is variable in the workspace, and there are alternating forms of 3H and 3R DOFs in the workspace.

In order to intuitively demonstrate the independent helical DOFs of the 3-CCR PM, four poses corresponding to different d_i1 that have 3H DOFs are shown in

Table 2. The principal screw information of 3-CCR PM with 3H DOFs in some poses.

No	Principal Screws of the Constraint Screw System			Pitch of the Principal Screws	Axode and Pose	No	Principal Screws of the Constraint Screw System			Pitch of the Principal Screws	Axode and Pose
	${\$}_{\mathit{p}1}^{\mathit{r}}$	${\$}_{\mathit{p}2}^{\mathit{r}}$	${\$}_{\mathit{p}3}^{\mathit{r}}$				${\$}_{\mathit{p}1}^{\mathit{r}}$	${\$}_{\mathit{p}2}^{\mathit{r}}$	${\$}_{\mathit{p}3}^{\mathit{r}}$
1	$\left[\begin{array}{ccc}0.5483& -0.6458& -0.5313\\ 0.6288& -0.1004& 0.7710\\ 0.5513& 0.7569& -0.3511\\ 13.6980& -152.2053& -154.0919\\ 61.5057& -93.4742& 131.8786\\ 60.2177& 202.5824& -31.6441\end{array}\right]$			h1 = 79.39, h2 = 261.01, h3 = 194.66	Figure 4a	2	$\left[\begin{array}{ccc}0.3232& -0.3086& 0.8946\\ 0.5468& 0.8324& 0.0896\\ 0.7723& -0.4602& -0.4378\\ 36.4089& -102.7470& 112.0896\\ 19.6546& 153.2852& 69.1029\\ 78.4009& -25.4459& -95.8437\end{array}\right]$			h1 = 83.07, h2 = 171.02, h3 = 148.43	Figure 4b
3	$\left[\begin{array}{ccc}0.6149& 0.1769& -0.7685\\ 0.6649& 0.4078& 0.6258\\ 0.4241& -0.8958& 0.1331\\ 31.2971& 23.8994& -198.1323\\ 55.3421& 155.5982& 108.0792\\ 73.0361& -190.5953& 69.9514\end{array}\right]$			h1 = 87.01, h2 = 238.41, h3 = 229.22	Figure 4c	4	$\left[\begin{array}{ccc}0.7708& -0.2128& 0.6005\\ 0.2154& 0.9741& 0.0688\\ 0.5996& -0.0763& -0.7967\\ 74.4607& -291.5886& 85.9335\\ 140.4947& 604.4967& 92.5120\\ -42.6582& -49.5566& -291.0721\end{array}\right]$			h1 = 62.08, h2 = 654.68, h3 = 289.85	Figure 4d

. The first position parameters are given as X_o = 30.25 mm, Y_o = 55 mm, and Z_o = 86.5 mm. The active limbs are solved as d₁₁ = 40.91 mm, d₂₁ = 67.98 mm, and d₃₁ = 97.60 mm. The second position parameters are given as X_o = 48.25 mm, Y_o = 46 mm, and Z_o = 41.5 mm. The active limbs are solved as d₁₁ = 60.60 mm, d₂₁ = 57.87 mm, and d₃₁ = 53.73 mm. The third position parameters are given as X_o = 66.24 mm, Y_o = 34.75 mm, and Z_o = 66.25 mm. The active limbs are solved as d₁₁ = 77.07 mm, d₂₁ = 46.41 mm, and d₃₁ = 78.90 mm. The fourth position parameters are given as X_o = 25.75 mm, Y_o = 65.25 mm, and Z_o = 55 mm. The active limbs are solved as d₁₁ = 38.07 mm, d₂₁ = 79.42 mm, and d₃₁ = 65.77 mm.

After solving for the types of DOFs corresponding to feasible poses in the workspace of the 3-CCR mechanism, the author selected four poses with 3R DOFs. The X_o, Y_o, and Z_o values of these poses, along with the calculated α, β, and γ values, are substituted into Equations (22)–(23) to obtain the joint vector values for those poses. Then, the joint vector values were sequentially substituted into Equations (4), (5), and (16)–(18) to calculate the values of ${\$}_{pj}^r$ and h_j shown in Table 2.

The principal screws and the pitch of the constraint screw system in these poses are shown in the third and fourth columns of Table 2. The principal screws of the motion screw system in each pose can be calculated based on screw theory. The pitches of the principal screws in the motion screw system are denoted as $h_1^m$, $h_2^m$, $h_3^m$, where the superscript m indicates that these are the principal screws of the motion screw system. The calculations yield $h_1^m$ = −h₁, $h_2^m$ = −h₂, $h_3^m$ = −h₃. Next, MATLAB (R2022b) was used to plot the principal screws of the motion screw systems, as shown by the blue lines in Figure 4, in which the black labeled numbers represent the pitch of the principal screws of the motion screw system.

The method from reference [26] was utilized to generate the axodes of helical DOFs with specific pitches within the motion screw systems at the four poses. To verify whether there are any rotational DOFs at the selected poses, we specifically plotted the axodes corresponding to a pitch of h^m = 0 (representing rotational DOF) using a yellow surface. Additionally, the axodes corresponding to three different pitches in [min($h_i^m$), max($h_i^m$)] were plotted and represented by blue surfaces, as shown in Figure 4. The red labeled numbers in Figure 4 indicate the pitch corresponding to the DOFs within the axodes.

The results indicate that no yellow surfaces were generated under the four poses listed in Table 2, which implies that no rotational DOFs exist in the four poses. And the helical DOFs with the same pitch are distributed on the same hyperboloid of one sheet. Under the aforementioned four poses, the motion screw system of the 3-CCR mechanism contains only helical DOFs with negative pitches.

For comparison, four other poses of the 3-CCR mechanism with 3R DOFs are chosen, as presented in

Table 3. The principal screw information of 3-CCR PM with 3R DOFs in some poses.

No	Principal Screws of the Constraint Screw System			Pitch of the Principal Screws	Axode and Pose	No	Principal Screws of the Constraint Screw System			Pitch of the Principal Screws	Axode and Pose
	${\$}_{\mathit{p}1}^{\mathit{r}}$	${\$}_{\mathit{p}2}^{\mathit{r}}$	${\$}_{\mathit{p}3}^{\mathit{r}}$				${\$}_{\mathit{p}1}^{\mathit{r}}$	${\$}_{\mathit{p}2}^{\mathit{r}}$	${\$}_{\mathit{p}3}^{\mathit{r}}$
1	$\left[\begin{array}{ccc}0.5630& -0.8238& 0.0669\\ 0.4712& 0.3864& 0.7929\\ 0.6790& 0.4149& -0.6056\\ -48.1367& -80.2520& -28.9863\\ -9.0332& -8.7737& 37.0681\\ -26.7398& 35.7026& -10.2465\end{array}\right]$			h1 = −49.51, h2 = 77.53, h3 = 33.66	Figure 5a	2	$\left[\begin{array}{ccc}0.3026& -0.8358& -0.4582\\ 0.6494& -0.1711& 0.7409\\ 0.6976& 0.5218& -0.4910\\ -14.2520& -82.3201& -78.6830\\ -10.4869& -39.0884& 34.7061\\ -60.8703& 97.9753& -39.2412\end{array}\right]$			h1 = −53.59, h2 = 126.61, h3 = 81.03	Figure 5b
3	$\left[\begin{array}{ccc}0.4674& -0.8669& -0.1733\\ 0.5713& 0.1467& 0.8075\\ 0.6746& 0.4764& -0.5638\\ -13.7403& -27.3765& -32.7556\\ -13.2806& -17.2739& 18.0980\\ -25.9073& 37.1252& -1.9140\end{array}\right]$			h1 = −31.49, h2 = 38.89, h3 = 21.37	Figure 5c	4	$\left[\begin{array}{ccc}0.4877& 0.6744& -0.5543\\ 0.6171& 0.1828& 0.7653\\ 0.6175& -0.7154& -0.3271\\ -12.8834& 6.5771& -37.9300\\ -19.4629& 27.7837& 14.6133\\ -27.6635& -45.1821& 13.2599\end{array}\right]$			h1 = −35.38, h2 = 41.84, h3 = 27.87	Figure 5d

.

The first position parameters are given as X_o = 30.25 mm, Y_o = 43.75 mm, and Z_o = 73 mm. The active limbs are solved as d₁₁ = 29.59 mm, d₂₁ = 48.07 mm, and d₃₁ = 77.31 mm. The second position parameters are given as X_o = 28 mm, Y_o = 84.25 mm, and Z_o = 66.25 mm. The active limbs are solved as d₁₁ = 35.53 mm, d₂₁ = 94.36 mm, and d₃₁ = 70.26 mm. The third position parameters are given as X_o = 30.25 mm, Y_o = 46 mm, and Z_o = 43.75 mm. The active limbs are solved as d₁₁ = 27.33 mm, d₂₁ = 46.30 mm, and d₃₁ = 40.74 mm. The fourth position parameters are given as X_o = 37 mm, Y_o = 50.50 mm, and Z_o = 39.25 mm. The active limbs are solved as d₁₁ = 36.62 mm, d₂₁ = 51.52 mm, and d₃₁ = 37.05 mm.

The principal screws of the motion screw system in each pose with 3R DOFs can be calculated. Then, the principal screws of the motion screw systems were plotted by MATLAB, as shown by the blue lines in Figure 5. The axodes of helical DOFs with specific pitches within the motion screw systems at the four poses were generated, as shown in Figure 5, where the yellow one-sheeted hyperboloid represents the spatial distribution of all rotational DOFs.

The results show that, for each pose listed in Table 3, the rotational DOFs in the motion screw system are distributed on a single one-sheeted hyperboloid, and the helical DOFs with the same pitch are distributed on the same hyperboloid of one sheet.

5. Conclusions

The contribution of this paper lies in proposing an approach for identifying the DOFs of PMs with screw-type terminal constraints and revealing a special phenomenon that the 3R DOFs and 3H DOFs coexist in the 3-CCR PM.

For the 3-CCR PM with three screw-type terminal constraints, based on the reciprocal relationship between the DOFs and terminal constraints, the analytical expression of the terminal constraints of the 3-CCR PM is obtained by solving the symbolic equation system. The results show that the terminal constraint in each limb of the PM is a screw-type constraint. Based on the theory of quadratic curve decomposition, the principal screws and the corresponding pitches of the constraint system were solved, and the DOFs determination approach was given, considering the relations of the motion and constraint systems. An analytical expression for the joint vectors and six-dimensional terminal pose parameters of the 3-CCR PM was established using the vector method. The DOFs distribution situations in the workspace of the 3-CCR PM were plotted, revealing the special phenomenon that three rotation and three helical DOFs coexist in the workspace. The distribution surfaces corresponding to the helical DOFs with specific pitches under three poses were drawn. The results showed that helical DOFs with the same pitches were distributed in the same hyperboloid of one sheet, while the pitches in different hyperboloid of one sheet were different. The study provides an effective approach for determining the DOFs of the PMs with screw-type terminal constraints.

This study helps explain the special phenomenon where PMs with three screw-based constraints exhibit three rotational DOFs. It provides a theoretical basis for developing three rotational DOFs PMs with screw-type constraints, helping to overcome the limitations of traditional three rotational DOFs PMs that can only use the force constraint limbs. This study facilitates the development of parallel biomimetic joints with three helical DOFs. For example, the center of rotation of the human ankle joint changes during actual movement. Using three helical DOFs PMs allows for designing biomimetic joints that mimic these motion characteristics, aiding humanoid robots in achieving more coordinated and natural ankle movements.