Motion Branch Transformation of 3(Rc)PU Parallel Mechanism with Reconfigurable Joint and Kinematic Performance Index

Abstract

A novel 3(Rc)PU parallel mechanism based on the reconfigurable circular groove cross joint (Rc joint) is proposed. The motion branch transformation relationship and constraint performance evaluation method for the mechanism are studied. According to constrained screw analysis, the (Rc)PU limb provides a constraint couple to the moving platform in one motion mode and no constraint to the platform in another motion mode. According to the two configurations of the limb, the 3(Rc)PU mechanism has four different motion branches, namely 3T, 1R3T, 2R3T, and 3R3T. Based on the input selection principle, the input selection scheme of the 3(Rc)PU mechanism is determined, and it is concluded that, to achieve stable motion, at least two input drive pairs need to be applied on each limb. A unified kinematic model with four motion branches is established, and the geometric constraint equations of the 3(Rc)PU parallel mechanism are derived. After selecting the input actuated joints, the local minimization transmission index (LMTI) of the reconfigurable mechanism under different motion branches is established to evaluate the motion/force transmission performance of the reconfigurable mechanism under different motion branches. The conclusion shows that the LMTI values satisfy the corresponding constraint conditions, and the 3(Rc)PU parallel mechanism has good motion/force transmission performance.

1. Introduction

The multi-configuration changes in reconfigurable mechanisms enable them to transform their own motion manifold by switching between different configurations so as to adapt to different working conditions, different tasks, and different functions with greater flexibility and useability [1,2]. They have an extremely high application value for realizing the engineering application goal of “one machine for multiple purposes or one machine for multiple configurations” [3,4].

Reconfigurable mechanisms include metamorphic mechanisms and motion bifurcation mechanisms; the difference between them is mainly whether the topological structure of the mechanism has changed. There are six basic evolutionary rules of the metamorphic mechanism: (1) the mechanism’s number of limbs changes [5]; (2) the relative position of each limb and platform of the mechanism changes [6]; (3) the parameters (number, relative position, and geometric parameters) in each limb of the mechanism change [7]; (4) the parameters of the kinematic pairs (type, quantity, direction, and relative position) in each limb of the mechanism change [8]; (5) the moving platform is transformed into a non-rigid series open-loop mechanism or a closed-loop mechanism with an integrated end-effector [9]; (6) the mechanism switches between the reference and moving plane [10]. The basic evolution rule of motion of bifurcation mechanisms is to change the instantaneous freedom of the mechanism by switching different motion modes through a singular configuration. Metamorphic parallel mechanisms (PMs for short) [11] have adaptive and reconfigurable characteristics. Among them, a kind of metamorphic mechanism changes from the parameters of kinematic pairs in the limb to the principle of switching motion modes [12], such as reconfigurable joints. The representative reconfigurable joints include vA joints [13,14], which switch between two different motion modes by changing the angle between adjacent joint axes; rT joints [15,16,17,18,19,20], in which the link motion can be changed from rotation in pitch and yaw directions to rotation in pitch and roll directions; and rR joints [21] for one-dimensional rotation in different directions and in which lockable joints [22,23,24,25] of freedom can be changed by locking kinematic pairs.

A conventional parallel robot is characterized by connecting two platforms using several series limbs. Parallel robots with multiple closed loops have the advantages of stable operation, high stiffness, and large load capacity. Different from traditional rigid parallel robots, the reconfigurable platform can be connected according to a series of open-loop linkage mechanisms in series or a closed-loop linkage mechanism integrated with the end actuator. By changing the shape of the non-rigid moving platform and kinematic pairs, the end actuator can be equipped with multiple motion modes [26].

For conventional rigid PMs, the number of driving parts should be equal to the degrees of freedom of the whole mechanism, and they should be evenly distributed on the kinematic pairs of each limb. For reconfigurable mechanisms, due to their unique topology, a multi-redundantly actuated system is often adopted to enable the switch between multiple motion modes; that is, the driver is greater than or equal to the instantaneous degrees of freedom of the mechanism [27], so as to realize the redundant actuation of the mechanism. By adding redundantly actuated limbs or joints, redundantly actuated PMs can achieve the optimal distribution of robot driving force, reduce the demand on the mechanical strength of the robot and the load capacity of the driving unit, effectively avoid or eliminate the singularity, reduce the influence of the PM’s joint clearance on the robot motion accuracy, improve the robot motion accuracy, and increase the operability and reliability of the mechanism [28,29].

Multi-redundantly actuated PMs have become a research hotspot in the field of parallel mechanisms. At present, research on these PMs mostly focuses on conventional rigid PMs. WANG [30] proposed a 6PUS+SP mechanism to achieve the required mandibular motion; CHENG [31] proposed a 6RSS+SP mechanism to model the masseter, temporalis, and pterygoid; and ZHAO [32] proposed an 8PSS mechanism to improve the dynamic performance.

BOUDREAU [33] added a 3RRR driving joint to make it a mechanism with a redundancy of three. SHAHIDI [34] proposed an n-PRPaR mechanism with multiple redundancies by transforming the Delta mechanism.

Due to the characteristics of various configurations of metamorphic mechanisms, their own topology is changed in the deformation process. In the process of motion characteristic analysis, input selection is generally required. The selection of a mechanism’s input actuated joints should adhere to the following principle: the input actuated joints of the metamorphic mechanism should be greater than or equal to the degree of freedom if the input actuated joints of the mechanism are to achieve stable drive. The number of degrees of freedom will change under different motion modes, so that the mechanism will have to actuate redundancy in one or more motion modes.

This paper presents a novel 3(Rc)PU PM based on an Rc joint and investigates its kinematics and performance. Induced by configuration changes of the Rc joint, the 3(Rc)PU PM can switch between 3T, 1R3T, 2R3T, and 3R3T motion modes; when the 3(Rc)PU PM mechanism is converted to 3T, 1R3T, 2R3T motion mode, it belongs to the redundant drive parallel mechanism, and the multi-redundant drive performance analysis of the mechanism can realize the optimal driving force allocation of the reconfigurable mechanism. It has promising potential to be used for constructing a reconfigurable production center to provide various solutions for product production. This paper is organized as follows. A novel reconfigurable kinematic pair (Rc joint) is proposed in Section 2. The Rc joint is connected to P and U pairs to form a reconfigurable kinematic chain (Rc)PU limb, and then a 3(Rc)PU PM with four kinds of motion modes is constructed from three (Rc)PU limbs. Based on screw theory, the screw system of the mechanism under the general motion configuration is established, and the freedom of the reconfigurable limb under different motion configurations and the motion characteristics of the PM under different motion modes are analyzed in Section 3. The forward and inverse kinematics of all motion modes of 3(Rc)PU PMs are analyzed and solved according to the closed-loop equation in Section 4, and the multi-redundantly actuated performance in different motion modes of the novel 3(Rc)PU PM is analyzed in Section 5. Conclusions are drawn in Section 6.

2. Constraints of the Reconfigurable (Rc)PU Limb

The reconfigurable (Rc)PU limb is formed by the reconfigurable circular groove cross joint (Rc joint), a P pair (prismatic pair), and a U pair (universal joint). The limb switches between different motion modes based on the evolutionary rule (4) of the metamorphic mechanism [8]. As shown in Figure 1, the Rc joint has three axes of rotation at the same central point: the outer axis (Axis 1), the inner axis (Axis 2), and the side axis (Axis 3). The motion characteristics of the Rc joint are changed by altering the angle and configuration between the outer axis and the inner axis. The outer axis (Axis 1) is a fixed axis; its direction is unchanged, and the inner ends of the outer axis are connected to a circular groove through a revolute pair. The inner axis (Axis 2) is placed inside the circular groove, where it can rotate, and then the angle between the inner axis and the outer axis is changed; the position of the inner axis in the circular groove can be determined with the positioning hole.

The angle change of the outer axis (Axis 1) and the inner axis (Axis 2) leads to different types of kinematic pairs of Rc joints. For the convenience of analysis, special configurations with two collinear axes and two perpendicular axes are selected as the two types of Rc joints. Figure 2a shows the configuration of the (Rc)PU limb when the two axes are collinear; in this case, the (Rc)PU limb is equivalent to the motion characteristics of the UPU limb and is called the (Rc)₁PU limb. Figure 2b shows the configuration of the (Rc)PU limb when the two axes are perpendicular; in this case, the (Rc)PU limb is equivalent to the motion characteristics of the UPS limb and is called the (Rc)₂PU limb.

In the evolution of mechanisms, constraints are the decisive factor. Taking the (Rc)PU limb moving to a certain angle as the initial state of the limb, the geometric constraints of two different isomorphic states in the initial state of the limb are analyzed. As shown in Figure 2a, a local coordinate system A-x₁y₁z₁ is established at the center A of the circular groove of the Rc joint. The outer axis of the Rc joint is in the direction of the x₁ axis, the perpendicular direction of the joint support base is in the direction of the y₁ axis, the vertical direction of the joint support side wall is in the direction of the z₁ axis, and the distance between the center of the Rc joint circular groove and the U pair is a (constant). Then, the kinematic screw system of the (Rc)₁PU limb can be expressed as follows:

$${}^1\mathbb{S}=\{\begin{array}{l}{\mathit{S}}_{11}={\left(1, 0, 0; 0, 0, 0\right)}^{\mathrm{T}}\\ {\mathit{S}}_{12}={\left(0, \mathrm{c}{\alpha }_1, \mathrm{s}{\alpha }_1; 0, 0, 0\right)}^{\mathrm{T}}\\ {\mathit{S}}_{13}={\left(0, 0, 0;-\mathrm{s}{\alpha }_2, \mathrm{c}\alpha \mathrm{c}{\alpha }_2,-\mathrm{s}\alpha \mathrm{c}{\alpha }_2\right)}^{\mathrm{T}}\\ {\mathit{S}}_{14}={\left(1, 0, 0; 0,-a\mathrm{s}{\alpha }_1\mathrm{c}{\alpha }_2,-a\mathrm{c}{\alpha }_1\mathrm{c}{\alpha }_2\right)}^{\mathrm{T}}\\ {\mathit{S}}_{15}={\left(0, \mathrm{c}{\alpha }_1, \mathrm{s}{\alpha }_1; 2a\mathrm{s}{\alpha }_1\mathrm{c}{\alpha }_1\mathrm{c}{\alpha }_2, a\mathrm{s}{\alpha }_1\mathrm{s}{\alpha }_2,-a\mathrm{c}{\alpha }_1\mathrm{s}{\alpha }_2\right)}^{\mathrm{T}}\end{array}$$

(1)

where c is cos and s is sin;

α₁ is the angle between the axis of screw S₁₃ and plane A-x₁y₁;

α₂ is the angle between the axis of screw S₁₃ and plane A-y₁z₁.

The kinematic screw of the (Rc)₁PU limb can form a fifth-order screw system. From the reciprocity between the kinematic screw system and the constrained screw system, the constrained screw of the (Rc)₁PU limb is

$${}^1\mathbb{S}^r={\left(0, 0, 0; 0, \cos {\alpha }_1,-\sin {\alpha }_1\right)}^{\mathrm{T}}$$

(2)

It can be seen from the constrained screw system of the (Rc)₁PU limb that the limb provides a constraint couple acting on the moving platform, and the constraint couple restricts the rotational motion of the U pair center of the (Rc)₁PU limb in the direction of the prismatic pair axis. The (Rc)₁PU limb has five degrees of freedom.

Figure 2b is the local coordinate system and screw distribution diagram of the (Rc)₂PU limb. The (Rc)₂PU screw’s system of motion can be expressed as follows:

$${}^2\mathbb{S}=\{\begin{array}{l}{\mathit{S}}_{21}={\left(1, 0, 0; 0, 0, 0\right)}^{\mathrm{T}}\\ {\mathit{S}}_{22}={\left(0, \mathrm{s}{\alpha }_1,-\mathrm{c}{\alpha }_1;0, 0, 0\right)}^{\mathrm{T}}\\ {\mathit{S}}_{23}={\left(0, \mathrm{c}{\alpha }_1, \mathrm{s}{\alpha }_1; 0, 0, 0\right)}^{\mathrm{T}}\\ {\mathit{S}}_{24}={\left(0, 0, 0;-\mathrm{s}{\alpha }_2,\mathrm{c}{\alpha }_1\mathrm{c}{\alpha }_2,-\mathrm{s}{\alpha }_1\mathrm{c}{\alpha }_2\right)}^{\mathrm{T}}\\ {\mathit{S}}_{25}={\left(\mathrm{c}{\alpha }_2, \mathrm{s}{\alpha }_2, 0; a\mathrm{s}{\alpha }_1\mathrm{s}{\alpha }_2\mathrm{c}{\alpha }_2,-a\mathrm{s}{\alpha }_1{\mathrm{c}}^2{\alpha }_2,-a{\mathrm{s}}^2{\alpha }_2-a\mathrm{c}{\alpha }_1{\mathrm{c}}^2{\alpha }_2\right)}^{\mathrm{T}}\\ {\mathit{S}}_{26}={\left(-\mathrm{s}{\alpha }_2, \mathrm{c}{\alpha }_1\mathrm{c}{\alpha }_2,-\mathrm{s}{\alpha }_1\mathrm{c}{\alpha }_2; 0, 0, 0\right)}^{\mathrm{T}}\end{array}$$

(3)

The (Rc)₂PU limb’s screw system is a six-order screw system, and the cardinality of the spanning multiset is always greater than or equal to the order of the corresponding screw system. In this case, the screw system of motion has no repeating elements, so $\mathrm{card}\langle \mathbb{S}\rangle =6$, the constraint screw system of reciprocity with it is empty, and the limb does not exert any constraint effect on the moving platform connected to it.

3. Motion Branch Transformation of 3(Rc)PU PM

3.1. Moving Platform Constraints of 3(Rc)PU PM

Figure 3a shows the structural diagram of the 3(Rc)PU PM. Three (Rc)PU limbs are connected to the two platforms, and the connection points of the three limbs and the moving platform are B₁, B₂, and B₃. The connection points to the static platform are A₁, A₂, and A₃. The distributions of the kinematic pairs on the static platform and the moving platform are shown in Figure 3b.

The center of the circumscribed circle of ΔA₁A₂A₃ is O_n, and its radius is R. A fixed coordinate system O_n-XYZ of the 3(Rc)PU PM is connected to point O_n, with the X axis on plane A₁A₂A₃ and parallel to the outer axis of the Rc joint at point A₁. The Z axis is on plane A₁A₂A₃ and is parallel to the outer axis of the Rc joint at point A₂, and the direction of the Y axis conforms to the right-hand rule. The center of the circumscribed circle of ΔB₁B₂B₃ is O_u and its radius is r; the moving coordinate system O_u-xyz of the 3(Rc)PU PM is established at point O_u, with the x, y, and z axes in the same direction as the X, Y, and Z axes, respectively, in the fixed coordinate system O_n-XYZ.

Using d = [x, y, z]^T to represent the shift operator of the position vector ^up to the position vector ⁿp, and using the rotation matrix ${}_n{}^u\mathit{R}$ to represent the shift operator of the position vector ^up to the position vector ⁿp, let the matrix ${}_n{}^u\mathit{R}$ be

$${}_n{}^u\mathit{R}=\left[\begin{array}{ccc}{r}_{11}& r_{12}& r_{13}\\ r_{21}& r_{22}& r_{23}\\ r_{31}& r_{32}& r_{33}\end{array}\right]$$

(4)

In vector space, the description of a point between different coordinate systems can be regarded as the result of an affine mapping of rotation and displacement. The position vector ^up of any point in the moving coordinate system O_u-xyz and the position vector ⁿp of that point in the fixed coordinate system O_n-XYZ can be described using the following equation:

$${}^n\mathit{p}={}_n{}^u\mathit{R}{}^u\mathit{p}+\mathit{d}$$

(5)

As shown in Figure 3, the position vectors of points B₁, B₂, and B₃ in the moving coordinate system O_u-xyz are as follows:

$${}^u\mathit{p}_{B1}={\left(0,0,r\right)}^{\mathrm{T}}, {}^u\mathit{p}_{B2}={\left(\sqrt{3}r/2,0,-r/2\right)}^{\mathrm{T}}, {}^u\mathit{p}_{B3}={\left(-\sqrt{3}r/2,0,-r/2\right)}^{\mathrm{T}}.$$

(6)

Then, the position vectors of points B₁, B₂, and B₃ in the fixed coordinate system O_n-XYZ can be obtained from Equation (5) as follows:

$$\{\begin{array}{l}{}^n\mathit{p}_{B1}={\left[r{r}_{13}+x, r{r}_{23}+y, r{r}_{33}+z \right]}^{\mathrm{T}}\\ {}^n\mathit{p}_{B2}={\left[\frac{\sqrt{3}}{2}r{r}_{11}-\frac{1}{2}r{r}_{13}+x, \frac{\sqrt{3}}{2}r{r}_{21}-\frac{1}{2}r{r}_{23}+y, \frac{\sqrt{3}}{2}r{r}_{31}-\frac{1}{2}r{r}_{33}+z\right]}^{\mathrm{T}}\\ {}^n\mathit{p}_{B3}={\left[-\frac{\sqrt{3}}{2}r{r}_{11}-\frac{1}{2}r{r}_{13}+x,-\frac{\sqrt{3}}{2}r{r}_{21}-\frac{1}{2}r{r}_{23}+y,-\frac{\sqrt{3}}{2}r{r}_{31}-\frac{1}{2}r{r}_{33}+z\right]}^{\mathrm{T}}\end{array}$$

(7)

The (Rc)PU limb has two different isomorphisms wherein the (Rc)₁PU limb provides a constraint couple to the moving platform connected to it, and the direction of the constraint couple passes through the secondary center B of the ball and coincides with screw S₁₃. Therefore, the three binding screws acting on the moving platform are

$${\mathit{W}}_i=\left(\begin{array}{l}{\mathit{f}}_i\\ {\mathit{m}}_i\end{array}\right)=f\left(\begin{array}{l}{\mathit{s}}_{Bi}\\ {\mathit{s}}_{0Bi}\end{array}\right)=f\left(\begin{array}{l}{\mathit{s}}_{Bi}\\ {}^n\mathit{p}_{Bi}\times {\mathit{s}}_{Bi}+h{\mathit{s}}_{Bi}\end{array}\right) \mathit{i}=1,2,3$$

(8)

In Equation (8), f_i represents the pure force, m_i represents the moment of force, f represents the magnitude of the force, s_Bi represents the attitude vector of the line vector where the constraint force is located, s_0Bi represents the line distance of the attitude vector s_Bi to the origin, ⁿp_Bi represents the position vector of point B_i in the fixed coordinate system O_n-XYZ, h represents the spinning moment, etc. From the constrained screw of the (Rc)₁PU limb, h = 0 can be obtained. The secondary part of the constraint moment of force is the position vector of reference point B_i in the fixed coordinate system. Therefore, the attitude vector and line distance of these three constraint force screws are

$${\mathit{s}}_{B1}={\mathit{s}}_{B2}={\mathit{s}}_{B3}={\left[0,0,0\right]}^{\mathrm{T}}$$

(9)

$${\mathit{s}}_{0B1}={}^n\mathit{p}_{B1},{\mathit{s}}_{0B2}={}^n\mathit{p}_{B2},{\mathit{s}}_{0B3}={}^n\mathit{p}_{B3}$$

(10)

3.2. Bifurcated Motion Mode Identification of 3(Rc)PU PM

According to the two kinematic configurations of the (Rc)PU limb, it can be concluded through linear independence analysis of the reconfigurable circular groove cross joint that the 3(Rc)PU parallel mechanism has four different motion branches under the initial configuration, namely 3T motion branches (3(Rc)₁PU configuration), the 1R3T motion branch (2(Rc)₁PU-(Rc)₂PU configuration), the 2R3T motion branch ((Rc)₁PU-2(Rc)₂PU configuration), and the 3R3T motion branch (3(Rc)₂PU configuration).

Figure 4 shows four configurations of the 3(Rc)PU parallel mechanism. As shown in Figure 4a, when the three limbs of the 3(Rc)PU PM are all (Rc)₁PU limbs, then the moving platform will be acted on by three constraint force vectors. The constraint screw system composed of these three constraint force vectors is as follows:

$$\mathit{W}={\mathit{W}}_1\cup {\mathit{W}}_2\cup {\mathit{W}}_3=\left(\begin{array}{l}{\mathit{s}}_{B1}\\ {\mathit{s}}_{0B1}\end{array}\right)\cup \left(\begin{array}{l}{\mathit{s}}_{B2}\\ {\mathit{s}}_{0B2}\end{array}\right)\cup \left(\begin{array}{l}{\mathit{s}}_{B3}\\ {\mathit{s}}_{0B3}\end{array}\right)$$

(11)

Due to the reciprocity between the constraint screw and the motion screw, the screw system in 3T motion mode of the 3(Rc)PU PM can be obtained as shown:

$${\mathit{S}}_1={\left(0, 0, 0; 1, 0, 0\right)}^T,{\mathit{S}}_2={\left(0, 0, 0; 0, 1, 0\right)}^T,{\mathit{S}}_3={\left(0, 0, 0; 0, 0, 1\right)}^T$$

(12)

The kinematic description of the moving platform in 3T motion mode can be obtained, as shown in Figure 4. At this time, the moving platform has three translation degrees of freedom along the X axis, Y axis, and Z axis.

As shown in Figure 4b, when the limb structure of the 3(Rc)PU PM is in the 2(Rc)₁PU-(Rc)₂PU configuration, the moving platform will be subject to the action of two constraint vectors. The constrained screw system composed of these two constraint vectors is as follows:

$$\mathit{W}={\mathit{W}}_1\cup {\mathit{W}}_2|{\mathit{W}}_1\cup {\mathit{W}}_3|{\mathit{W}}_2\cup {\mathit{W}}_3=\left(\begin{array}{l}{\mathit{s}}_{B1}\\ {\mathit{s}}_{0B1}\end{array}\right)\cup \left(\begin{array}{l}{\mathit{s}}_{B2}\\ {\mathit{s}}_{0B2}\end{array}\right)|\left(\begin{array}{l}{\mathit{s}}_{B1}\\ {\mathit{s}}_{0B1}\end{array}\right)\cup \left(\begin{array}{l}{\mathit{s}}_{B3}\\ {\mathit{s}}_{0B3}\end{array}\right)|\left(\begin{array}{l}{\mathit{s}}_{B2}\\ {\mathit{s}}_{0B2}\end{array}\right)\cup \left(\begin{array}{l}{\mathit{s}}_{B3}\\ {\mathit{s}}_{0B3}\end{array}\right)$$

(13)

In Equation (13), the symbol “∪” represents a union, and the symbol “|”represents the logical operator “or”. The screw system of the 3(Rc)PU PM in the 2(Rc)₁PU-(Rc)₂PU configuration is obtained:

$$\begin{array}{l}{\mathit{S}}_1={\left(0, 0, 0; 1, 0, 0\right)}^T,{\mathit{S}}_2={\left(0, 0, 0; 0, 1, 0\right)}^T,{\mathit{S}}_3={\left(0, 0, 0; 0, 0, 1\right)}^T,\\ {\mathit{S}}_4={\left(\frac{a_1}{\sqrt{a_1^2+b_1^2+1}}, \frac{b_1}{\sqrt{a_1^2+b_1^2+1}}, \frac{1}{\sqrt{a_1^2+b_1^2+1}}; 0, 0, 0\right)}^T\end{array}$$

(14)

where

$$\begin{array}{l}{a}_1=\frac{3r_{31}y-r_{33}y-3r_{21}z+r_{23}z-3r{r}_{21}{r}_{33}+3r{r}_{23}{r}_{31}}{3r_{21}x-r_{23}x-3r_{11}y+r_{13}y-3r{r}_{11}{r}_{23}+3r{r}_{13}{r}_{21}},\\ b_1=\frac{-\left(3r_{31}x-r_{33}x-3r_{11}z+r_{13}z-3r{r}_{11}{r}_{33}+3r{r}_{13}{r}_{31}\right)}{3r_{21}x-r_{23}x-3r_{11}y+r_{13}y-3r{r}_{11}{r}_{23}+3r{r}_{13}{r}_{21}}\end{array}$$

The kinematic description of the moving platform of the reconfigurable PM in 1R3T motion mode is shown as 1R3T. At this time, the moving platform has three translation degrees of freedom along the X axis, Y axis, and Z axis and one rotation degree of freedom.

When the limb structure is (Rc)₁PU-2(Rc)₂PU, the moving platform will be subject to a constraint force vector:

$$\mathit{W}={\mathit{W}}_1|{\mathit{W}}_2|{\mathit{W}}_3=\left(\begin{array}{l}{\mathit{s}}_{B1}\\ {\mathit{s}}_{0B1}\end{array}\right)|\left(\begin{array}{l}{\mathit{s}}_{B2}\\ {\mathit{s}}_{0B2}\end{array}\right)|\left(\begin{array}{l}{\mathit{s}}_{B3}\\ {\mathit{s}}_{0B3}\end{array}\right)$$

(15)

The motion screw system of the 3(Rc)PU PM in the 2(Rc)₁PU-(Rc)₂PU structures is

$$\begin{array}{l}{\mathit{S}}_1={\left(0, 0, 0; 1, 0, 0\right)}^T,{\mathit{S}}_2={\left(0, 0, 0; 0, 1, 0\right)}^T,{\mathit{S}}_3={\left(0, 0, 0; 0, 0, 1\right)}^T,\\ {\mathit{S}}_4={\left(\frac{a_2}{\sqrt{a_2^2+1}}, \frac{1}{\sqrt{a_2^2+1}}, 0; 0, 0, 0\right)}^T,{\mathit{S}}_5={\left(\frac{b_2}{\sqrt{b_2^2+1}}, 0, \frac{1}{\sqrt{b_2^2+1}}; 0, 0, 0\right)}^T\end{array}$$

(16)

where

$$\begin{array}{l}{a}_2=-(y+r{r}_{23})/(x+r{r}_{13})\\ b_2=-(y+r{r}_{23})/(x+r{r}_{13})\end{array}$$

The kinematic description of the reconfigurable PM’s moving platform in the 2R3T motion branch is shown in Figure 4c. At this time, the moving platform has three translation degrees of freedom along the X axis, Y axis, and Z axis and two rotation degrees of freedom.

When the three limbs of the 3(Rc)PU PM are all (Rc)₂PU limbs, that is, the limb structure is the 3(Rc)₂PU configuration, the unconstrained action is applied to the moving platform, and the degree of freedom of the 3(Rc)PU PM in this motion branch is six. The kinematic description of the reconfigurable PM’s moving platform is shown in Figure 4d. At this time, the moving platform has three translation degrees of freedom along the X, Y, and Z axes and three rotation degrees of freedom.

The various mobility configurations of the 3(Rc)PU PM increase its adaptability in tasks with multi-stages. Figure 5 shows a concept of a reconfigurable production center by installing the 3(Rc)PU PM into the production platform to produce products of different specifications to meet the individual needs of the customer. When the product is relatively complex, the reconfigurable production center can be configured as 5-DOF or 6-DOF to perform the task; when the product is relatively simple, in order to avoid consuming a lot of energy, the reconfigurable production center can be configured as 3-DOF to complete the corresponding production actions. The reconfigurable production center would provide different solutions for the productive process of products.

4. Kinematic Model

4.1. Inverse Kinematics of 3(Rc)PU PM

The inverse kinematics of the mechanism are obtained from the position parameter {X, Y, Z, α, β, γ} of the moving platform. X, Y, and Z are the displacement of reference point O_u of the moving platform along the coordinate axis in the fixed coordinate system, and α, β, and γ are the angles of the moving platform around the coordinate axis in the fixed coordinate system. On the (Rc)₁PU limb, one of the outer axes and prismatic pairs can be used as the drive of the reconfigurable PM on this motion mode. The (Rc)₂PU limb does not provide constraints on the moving platform, so each of the three limbs must provide at least two constraints to completely limit the reconfigurable PM’s full degree of freedom. Based on the input selection principle, the angles rotated by the outer axis and the displacement of prismatic pair are selected as the input parameters for inverse kinematic analysis of the (Rc)PU limb, which are

{θ₁, θ₂, θ₃, l₁, l₂, l₃}.

where l₁, l₂, and l₃ are respectively the length of line vectors A₁B₁, A₂B₂, and A₃B₃, that is, the length of the prismatic pair where the linear motor is located. θ₁, θ₂, and θ₃ are respectively the angle between the projection of line vectors A₁B₁, A₂B₂, and A₃B₃ on the plane A₁O_nB₁, A₂O_nB₂, and A₃O_nB₃ and the XO_nZ plane. The angle at which the outer axis is rotated is the difference between the values of θ₁, θ₂, and θ₃ at the initial position and the values at the target position.

The rotation matrix R represented by the Euler angle is

$$R=R(X,\gamma )R(Y,\beta )R(Z,\alpha )=\left[\begin{array}{ccc}\mathrm{c}\gamma \mathrm{c}\beta & \mathrm{c}\gamma \mathrm{s}\beta \mathrm{s}\alpha -\mathrm{s}\gamma \mathrm{c}\alpha & \mathrm{c}\gamma \mathrm{s}\beta \mathrm{c}\alpha +\mathrm{s}\gamma \mathrm{s}\alpha \\ \mathrm{s}\gamma \mathrm{c}\beta & \mathrm{s}\gamma \mathrm{s}\beta \mathrm{s}\alpha +\mathrm{c}\gamma \mathrm{c}\alpha & \mathrm{s}\gamma \mathrm{s}\beta \mathrm{c}\alpha -\mathrm{c}\gamma \mathrm{s}\alpha \\ -\mathrm{s}\beta & \mathrm{c}\beta \mathrm{s}\alpha & \mathrm{c}\beta \mathrm{c}\alpha \end{array}\right]$$

(17)

Taking the closed-loop motion chain formed by each limb with the static platform and the moving platform as a loop, the closed-loop vector equation of this loop can be obtained as follows:

$$A_i{B}_i+O_n{A}_i-O_n{B}_i=0, (i=1, 2, 3)$$

(18)

It should be noted that A_iB_i, O_nA_i, and O_nB_i are all vectors in the fixed coordinate system O_n-XYZ.

In the fixed coordinate system O_n-XYZ, the position vector of the center A_i of the Rc limb can be expressed as

$${\mathit{O}}_n{\mathit{A}}_1={\left[0, 0, R\right]}^{\mathrm{T}}, {\mathit{O}}_n{\mathit{A}}_2={\left[\sqrt{3}R/2, 0,-R/2\right]}^{\mathrm{T}}, {\mathit{O}}_n{\mathit{A}}_3={\left[-\sqrt{3}R/2, 0,-R/2\right]}^{\mathrm{T}}.$$

(19)

The position vector of the center B_i of each limb’s universal joint in the moving coordinate system O_u-xyz can be expressed as

$${\mathit{O}}_u{\mathit{B}}_1={\left(0, 0, r_1\right)}^{\mathrm{T}},{\mathit{O}}_u{\mathit{B}}_2={\left(\sqrt{3}{r}_1/2, 0,-r_1/2\right)}^{\mathrm{T}},{\mathit{O}}_u{\mathit{B}}_3={\left(-\sqrt{3}{r}_1/2, 0,-r_1/2\right)}^{\mathrm{T}}$$

The position vector of the origin O_u of the moving coordinate system O_u-xyz in the fixed coordinate system O_n-XYZ is

$${\mathit{O}}_n{\mathit{O}}_u={\left(X,Y,Z\right)}^{\mathrm{T}}$$

The position vector O_uB_i (i = 1, 2, 3) of the center B_i of each limb’s universal joint in the moving coordinate system O_u-xyz can be converted to the description O_nB_i (i = 1, 2, 3) in the fixed coordinate system O_n-XYZ:

$${\mathit{O}}_n{\mathit{B}}_1={\left[m_1, n_1, d_1\right]}^{\mathrm{T}},{\mathit{O}}_n{\mathit{B}}_2={\left[m_2, n_2, d_2\right]}^{\mathrm{T}},{\mathit{O}}_n{\mathit{B}}_3={\left[m_3, n_3, d_3\right]}^{\mathrm{T}}.$$

(20)

where

$$\begin{array}{l}{m}_1=\left[(c\gamma \mathrm{s}\beta \mathrm{c}\alpha +\mathrm{s}\gamma \mathrm{s}\alpha )r+X\right],n_1=\left[(\mathrm{s}\gamma \mathrm{s}\beta \mathrm{c}\alpha -\mathrm{c}\gamma \mathrm{s}\alpha )r+Y\right],d_1=\left[\mathrm{c}\beta \mathrm{c}\alpha r-R+Z\right],\\ m_2=\left[\left(\frac{\sqrt{3}}{2}\mathrm{c}\gamma \mathrm{c}\beta -\frac{1}{2}\mathrm{c}\gamma \mathrm{s}\beta \mathrm{c}\alpha -\frac{1}{2}\mathrm{s}\gamma \mathrm{s}\alpha \right)r-\frac{\sqrt{3}R}{2}+X\right],n_2=\left[\left(\frac{\sqrt{3}}{2}\mathrm{s}\gamma \mathrm{c}\beta -\frac{1}{2}\mathrm{s}\gamma \mathrm{s}\beta \mathrm{c}\alpha +\frac{1}{2}\mathrm{c}\gamma \mathrm{s}\alpha \right)r+Y\right],\\ d_2=\left[\left(-\frac{\sqrt{3}}{2}\mathrm{s}\beta -\frac{1}{2}\mathrm{c}\beta \mathrm{c}\alpha \right)r+\frac{R}{2}+Z\right],m_3=\left[\left(-\frac{\sqrt{3}}{2}\mathrm{c}\gamma \mathrm{c}\beta -\frac{1}{2}\mathrm{c}\gamma \mathrm{s}\beta \mathrm{c}\alpha -\frac{1}{2}\mathrm{s}\gamma \mathrm{s}\alpha \right)r+\frac{\sqrt{3}R}{2}+X\right],\\ n_3=\left[\left(-\frac{\sqrt{3}}{2}\mathrm{s}\gamma \mathrm{c}\beta -\frac{1}{2}\mathrm{s}\gamma \mathrm{s}\beta \mathrm{c}\alpha +\frac{1}{2}\mathrm{c}\gamma \mathrm{s}\alpha \right)r+Y\right],d_3=\left[\left(\frac{\sqrt{3}}{2}\mathrm{s}\beta -\frac{1}{2}\mathrm{c}\beta \mathrm{c}\alpha \right)r+\frac{R}{2}+Z\right].\end{array}$$

The inverse kinematic solution of the 3(Rc)PU PM can be obtained from Equation (21):

$$l_1=\left|A_1{B}_1\right|=\sqrt{m_1^2+n_1^2+d_1^2},l_2=\left|A_2{B}_2\right|=\sqrt{m_2^2+n_2^2+d_2^2},l_3=\left|A_3{B}_3\right|=\sqrt{m_3^2+n_3^2+d_3^2}.$$

(21)

When all limbs are (Rc)₁PU limbs, θ₁ = θ₂ = θ₃ = 0. When all limbs are (Rc)₂PU limbs,

$$\{\begin{array}{c}{\theta }_1=\arccos \frac{\left|A_1{B}_1\cdot (0,0,1)\right|}{\left|A_1{B}_1\right|}=\arccos \frac{d_1}{\sqrt{m_1^2+n_1^2+d_1^2}},\\ {\theta }_2=\arccos \frac{\left|A_2{B}_2\cdot (1,0,0)\right|}{\left|A_2{B}_2\right|}=\arccos \frac{m_2}{\sqrt{m_2^2+n_2^2+d_2^2}},\\ {\theta }_3=\arccos \frac{\left|A_3{B}_3\cdot (1,0,0)\right|}{\left|A_3{B}_3\right|}=\arccos \frac{m_3}{\sqrt{m_3^2+n_3^2+d_3^2}}.\end{array}$$

(22)

The input parameters of each motion branch of the 3(Rc)PU parallel mechanism are shown in

Table 1. The inverse solution input parameters in various motion modes.

Serial No.	Motion Mode	Inverse Solution Input Parameter Composition
1	3T	{l1, l2, l3}
2	1R3T	{θ2, l1, l2, l3}
3	2R3T	{θ2, θ3, l1, l2, l3}
4	3R3T	{θ1, θ2, θ3, l1, l2, l3}

.

4.2. Forward Kinematics of 3(Rc)PU PM

The position parameters {X, Y, Z, α, β, γ} of the moving platform are obtained from the input parameters {θ₁, θ₂, θ₃, l₁, l₂, l₃} of the actuated joint, which is the forward kinematics analysis of the 3(Rc)PU PM. The geometric parameter constraint equation based on the moving platform in the fixed coordinate system O_n-XYZ is established. According to the cosine theorem, the geometric constraint equation on the moving platform is

$$\{\begin{array}{l}{\left(X_{O_n{B}_1}-X_{O_n{B}_2}\right)}^2+{\left(Y_{O_n{B}_1}-Y_{O_n{B}_2}\right)}^2+{\left(Y_{O_n{B}_1}-Y_{O_n{B}_2}\right)}^2={\left(\sqrt{3}r\right)}^2\\ {\left(X_{O_n{B}_2}-X_{O_n{B}_3}\right)}^2+{\left(Y_{O_n{B}_2}-Y_{O_n{B}_3}\right)}^2+{\left(Y_{O_n{B}_2}-Y_{O_n{B}_3}\right)}^2={\left(\sqrt{3}r\right)}^2\\ {\left(X_{O_n{B}_3}-X_{O_n{B}_1}\right)}^2+{\left(Y_{O_n{B}_3}-Y_{O_n{B}_1}\right)}^2+{\left(Y_{O_n{B}_3}-Y_{O_n{B}_1}\right)}^2={\left(\sqrt{3}r\right)}^2\end{array}$$

(23)

By substituting the vector Equation (20) in the fixed coordinate system O_n-XYZ into Equation (23) and substituting the position parameters in the equation with cosθ_i = (1 − t_i²)/(1+ t_i²), sinθ_i = 2t_i/(1 + t_i²), a system of equations can be obtained:

$$\{\begin{array}{l}{f}_1(1, t_1^2, t_2^2, t_1{t}_2, t_1^2{t}_2^2)=0\\ f_2(1, t_2^2, t_3^3, t_2{t}_3, t_2^2{t}_3^2)=0\\ f_3(1, t_1^2, t_3^2, t_1{t}_3, t_1^2{t}_3^2)=0\end{array}$$

(24)

where f_i(·) is a linear equation for t_i, whose coefficients depend only on known structure parameters. The Sylvester junction elimination method was used to eliminate some parameters, and the solutions of all equation parameters (t₁, t₂, t₃) were obtained. The results were substituted into the geometric constraint equation θ_i = 2arctan(t_i), and the angles between each limb and the static platform plane were obtained; these angles can also be regarded as the output variables of the PM. The forward kinematic solution of the 3(Rc)PU PM is as follows:

$$\{\begin{array}{l}\mathit{w}=({\mathit{O}}_n{\mathit{B}}_2-{\mathit{O}}_n{\mathit{B}}_1)\times ({\mathit{O}}_n{\mathit{B}}_3-{\mathit{O}}_n{\mathit{B}}_1)/\Vert ({\mathit{O}}_n{\mathit{B}}_2-{\mathit{O}}_n{\mathit{B}}_1)\times ({\mathit{O}}_n{\mathit{B}}_3-{\mathit{O}}_n{\mathit{B}}_1)\Vert \\ \mathit{u}=({\mathit{O}}_n{\mathit{B}}_2-{\mathit{O}}_n{\mathit{B}}_3)/\Vert ({\mathit{O}}_n{\mathit{B}}_2-{\mathit{O}}_n{\mathit{B}}_3)\Vert \\ \mathit{v}=(\mathit{w}\times \mathit{u})\\ \mathit{R}=(\mathit{u},\mathit{v},\mathit{w})\\ \mathit{P}={\mathit{O}}_n{\mathit{B}}_1+r\mathit{v}\end{array}$$

(25)

where R is the attitude matrix of the moving platform (when the mechanism is in 3T motion mode, the rotation matrix is zero matrix), and P is the position vector of the center of the moving platform shape in the fixed coordinate system O_n-XYZ.

5. Kinematic Performance Analysis

The essential function of a parallel mechanism is to output motion and resist external load. The motion/force transmission and constraint characteristics reflect the essential characteristics of parallel mechanisms. The 3(Rc)PU parallel mechanism has four kinds of motion branches, and it is necessary to study the motion/force transmission performance of the parallel mechanism in different motion modes.

By selecting the P pair on each limb and the outer axis on the Rc joint as the driving joint, the input actuation screw of the mechanism can be obtained by solving the inverse kinematics of the PM. The input actuation screw system of the mechanism can be obtained using Equations (21) and (22):

$$\mathit{T}=\{\begin{array}{l}{\mathit{S}}_{A1}={\left(0, 0, 0;{ m}_1, n_1, d_1\right)}^T\\ {\mathit{S}}_{A2}={\left(0, 0, 0;{ m}_2, n_2, d_2\right)}^T\\ {\mathit{S}}_{A3}={\left(0, 0, 0; m_3, n_3, d_3\right)}^T\\ {\mathit{S}}_{A4}={\left(1, 0, 0; 0, R, 0\right)}^T\\ {\mathit{S}}_{A5}={\left(0, 0, 1; 0,-\sqrt{3}R/2, 0\right)}^T\\ {\mathit{S}}_{A6}={\left(0, 0, 1; 0, \sqrt{3}R/2, 0\right)}^T\end{array}$$

(26)

Since the transmission wrench screw exerted by each limb on the moving platform must pass through the joint center point of each limb, the U joint center of the (Rc)PU limb is selected as the characteristic point of the transmission wrench screw of each limb, and the screw system of the mechanism’s transmission force on the moving platform can be obtained as

$$\mathit{T}=\{\begin{array}{l}{\mathit{S}}_{T1}={\left(m_1,n_1,d_1;-{\mathrm{Rn}}_1,R{m}_1,0\right)}^T\\ {\mathit{S}}_{T2}={\left(m_2,n_2,d_2;R{n}_2/2,-\sqrt{3}R{d}_2/2-R{m}_2/2,\sqrt{3}R{n}_2/2\right)}^T\\ {\mathit{S}}_{T3}={\left(m_3,n_3,d_3;R{n}_3/2,\sqrt{3}R{d}_3/2-R{m}_3/2,-\sqrt{3}R{n}_3/2\right)}^T\\ {\mathit{S}}_{T4}={\left(0,-d_1,n_1;n_1^2{l}_1+d_1^2{l}_1+d_{1}R,-m_1{n}_1{l}_1,-m_1{d}_1{l}_1\right)}^T\\ {\mathit{S}}_{T5}={\left(-n_2,m_2,0;-m_2{d}_2{l}_2+R{m}_2/2,-n_2{d}_2{l}_2+R{n}_2/2,m_2^2{l}_2+\sqrt{3}R{m}_2/2+n_2^2{l}_2\right)}^T\\ {\mathit{S}}_{T6}={\left(n_3,-m_3,0; m_3{d}_3{l}_3-R{m}_3/2,n_3{d}_3{l}_3-R{n}_3/2,-m_3^2{l}_3+\sqrt{3}R{m}_3/2-n_3^2{l}_3\right)}^T\end{array}$$

(27)

Equation (27) is the set of transmission wrench screws obtained by locking the actuated joints on each limb. Among them, S_T1, S_T2, and S_T3 are the transmission wrench screws corresponding to the prismatic input actuated joints on limbs I, II, and III, and S_T4, S_T5, and S_T6 are the transmission wrench screws corresponding to the input actuated joints on the outer axis of limbs I, II, and III.

The input transmission performance index of the 3(Rc)PU PM can be obtained from Equations (26) and (27):

$${\psi }_i=\frac{\left|S_{Ai}\circ S_{Ti}\right|}{{\left|S_{Ai}\circ S_{Ti}\right|}_{\max }}, (i=1, 2, \cdots , n)$$

(28)

In Equation (28), the value of n is related to the motion mode of the 3(Rc)PU parallel mechanism and is the mechanism’s number of input actuated joints in each motion mode. When the parallel mechanism is in 3T motion mode, the relation between the output twist screw and the constrained screw corresponding to limb I is as follows:

$$\{\begin{array}{l}{S}_{O1}\circ S_{Ti}=0,(i=1, 2)\\ S_{O1}\circ W_j=0,(j=1, 2, 3)\end{array}$$

(29)

The output twist screw corresponding to limb I of the parallel mechanism in 3T motion mode is

$$S_{O1}=\left(0, 0, 0;-n_2{d}_1+n_1{d}_2, m_2{d}_1-m_1{d}_2, m_1{n}_2-m_2{n}_1\right)$$

(30)

Similarly, the other two output twist screws under the 3T motion branch and the output twist screw under other motion modes can be obtained. The output transmission performance index η_1g of the parallel mechanism can be obtained:

$${\eta }_{1i}=\frac{\left|S_{Ti}\circ S_{Oi}\right|}{{\left|S_{Ti}\circ S_{Oi}\right|}_{\max }},(i=1, 2, \cdots , n).$$

(31)

Combinations of the mechanism’s input twist screw, transmission wrench screw, and output twist screw under different motion branches are shown in

Table 2. Screw combinations of mechanisms under different motion branches.

Motion Mode	Input Twist Screw	Transmission Wrench Screw	Output Twist Screw
3T	SA1, SA2, SA3	ST1, ST2, ST3	SO1, SO2, SO3
1R3T	SA1, SA2, SA3, SA4	ST1, ST2, ST3, ST4	SO1, SO2, SO3, SO4
2R3T	SA1, SA2, SA3, SA4, SA5	ST1, ST2, ST3, ST4, ST5	SO1, SO2, SO3, SO4, SO5
3R3T	SA1, SA2, SA3, SA4, SA5, SA6	ST1, ST2, ST3, ST4, ST5, ST6	SO1, SO2, SO3, SO4, SO5, SO6

.

The output transmission performance of the 3(Rc)PU PM under different motion modes can be obtained as

$${\eta }_k=\frac{1}{N}{\displaystyle \sum _{g=1}^N{\eta }_{kg}} , (k=1, 2, \cdots , n).$$

(32)

The local minimization transmission index (LMTI for short) can be used to evaluate the motion/force transmission performance of the redundantly actuated PM. According to Equations (28) and (32), the LMTI can be expressed as follows:

$$\gamma =\min \left\{{\psi }_1, {\psi }_2, \cdots , {\psi }_n, {\eta }_k\right\},(k=1, 2, \cdots , n).$$

(33)

where ${\psi }_1, {\psi }_2, \cdots , {\psi }_n$ are the input transmission performance indices of the 3(Rc)PU PM.

With the structural parameters set as R = 360 mm and r = 220 mm, the motion/force transmission performance of the 3(Rc)PU parallel mechanism under different positions was analyzed. When the parallel mechanism is in the 3T motion branch, let α = 0, β = 0, and γ = 0, and take the center of the moving platform in the fixed coordinate system Y direction Y = 400 mm and Y = 600 mm as an example to calculate the motion/force transmission performance of the (Rc)PU PM in the 3T motion mode, as shown in Figure 6.

As can be seen from Figure 6, when the mechanism is in the 3T motion branch, the LMTI value gradually increases with the increase in the Y value. In the XO_nZ plane, the LMTI value is the largest when the 3(Rc)PU PM is in the area near (X, Z)= (0, −65), and the motion/force transmission efficiency of the mechanism is the highest. Similarly, when the parallel mechanism is in the 1R3T motion branch, let α = 0, β = 0, and γ = 30°, and take the center of the moving platform in the fixed coordinate system Y direction Y = 400 mm and Y = 600 mm as an example to calculate the motion/force transmission performance of the parallel mechanism in the 1R3T motion branch. The performance distribution curve is shown in Figure 7. The motion/force transmission performance distribution curves are shown in Figure 8 and Figure 9 for when the parallel mechanism is in the 2R3T moving branch and the 3R3T motion branch, respectively.

It can be seen from Figure 6, Figure 7, Figure 8 and Figure 9 that the LMTI value of the 3(Rc)PU PM gradually increases with the Y value in the accessible working space of the mechanism. At the same time, there is no aggregation of the LMTI distribution isolines in each motion branch, and the LMTI values are all greater than 0.5, indicating that the 3(Rc)PU PM has good motion/force transmission performance.

6. Conclusions

A reconfigurable circular groove cross joint (called an Rc joint) is designed, which has kinematic characteristics similar to the spherical pair and universal joint and can freely switch configurations between the two kinematic pairs. By changing the configuration of the (Rc)PU limb, the bifurcation motion identification of the 3(Rc)PU PM is carried out. It shows that the reconfigurable PM has four different motion branches, namely 3T, 1R3T, 2R3T, and 3R3T motion branches. These branches can be switched freely.

The unified kinematics model of the 3(Rc)PU parallel mechanism with multiple motion branches is established, and the forward and inverse kinematics are calculated. After determining the input drive pairs of the 3(Rc)PU PM, the transmission screw system of the mechanism is derived, and the local minimization transmission index (LMTI) of the mechanism under the 3T, 1R3T, 2R3T, and 3R3T motion branches is established to evaluate the motion/force transmission performance of the mechanism. The LMTI value satisfies the corresponding constraint conditions, indicating that the 3(Rc)PU PM has good motion/force transmission performance.